Divisão De Serviços Técnicos

UNIVERSIDADE FEDERAL DO RIO GRANDE DO NORTE CENTRO DE CIÊNCIAS EXATAS E DA TERRA INSTITUTO DE QUÍMICA PROGRAMA DE PÓS-GRADUAÇÃO EM QUÍMICA

Novos protocolos teóricos utilizados nos estudos das ligações hidrogênio-hidrogênio, na estabilidade do tetraedrano, seus derivados e das reações de diels-alder e na quantificação da afinidade de fármacos

______Norberto de Kássio Vieira Monteiro Tese de Doutorado Natal/RN, outubro de 2015

Norberto de Kássio Vieira Monteiro

NOVOS PROTOCOLOS TEÓRICOS UTILIZADOS NOS ESTUDOS DAS LIGAÇÕES HIDROGÊNIO-HIDROGÊNIO, NA ESTABILIDADE DO TETRAEDRANO, SEUS DERIVADOS E DAS REAÇÕES DE DIELS-ALDER E NA QUANTIFICAÇÃO DA AFINIDADE DE FÁRMACOS.

Orientador: Caio Lima Firme

NATAL/RN

2015

Divisão de Serviços Técnicos

Catalogação da Publicação na Fonte. UFRN Biblioteca Setorial do Instituto de Química

Monteiro, Norberto de Kássio Vieira.

Orientador: Caio Lima Firme.

Tese (Doutorado em Química) - Universidade Federal do Rio Grande do Norte. Centro de Ciências Exatas e da Terra. Programa de Pós-Graduação em Química.

1. Diels-Alder – Tese. 2. Ligação H-H – Tese. 3. Tetraedrano – Tese. 4. CYP17 – Tese I. Firme, Caio Lima. II. Universidade Federal do Rio Grande do Norte. III. Título.

RN/UFRN/BSE- Instituto de Química CDU 544.142.4:615 (043.3)

AGRADECIMENTOS

Às minhas três mães: Marilú, Vitória e Dona Francisca. Sem elas, nem sei o que seria de mim.

Ao meu amor, Richele, a qual tenho o privilégio de chamar de esposa, pela ajuda e por me aguentar nos momentos de ansiedade e estresse nos anos que me dediquei ao doutorado.

Ao meu gordinho! Filhão, você foi a melhor coisa que já aconteceu na minha vida! Papai te ama!

À minha querida afilhada Shara Maria, a qual considero como uma filha.

Aos meus padrinhos que sempre estiveram comigo durante toda a minha vida.

Aos alunos de iniciação científica: Cidinha, Luís, Franklin, Fabrício, Bruno, Elizeu e Valéria.

Aos alunos e amigos da pós-graduação: Gutto, Caio Sena, João Batista, Acássio, Roberta, William, Tamires e Francinaldo.

Aos meus amigos: Elton, Isabelle, Amisson, Clesiane, Aninha e, em especial, Sérgio, que além de amigo também é meu padrinho de casamento.

Aos meus sogros, seu Laércio e Dona Rita, que me tratam como um filho.

À minha cunhada Raynara por me dar caronas à UFRN quando eu estava sem moto.

Muito especialmente, agradeço ao meu orientador Prof. Caio de Lima Firme, pela disponibilidade, paciência, dedicação e profissionalismo e, acima de tudo, por ter acreditado em mim desde o início.

Aos professores Jones de Andrade, Gorette Cavalcante e Luiz Seixas pelas longas e produtivas conversas.

Aos professores Carlos Alberto Martinez e Sibele Berenice pela competência em dirigir o Programa de pós-graduação em Química.

Aos professores componentes da banca de qualificação, Dr. Euzébio Guimarães, Dr. Davi Serradella e Dr. Caio de Lima Firme, pelas correções e aperfeiçoamento desta tese.

Aos professores componentes da banca de defesa de tese, Dr. Euzébio Guimarães, Dr. Davi Serradella, Dr. Cláudio Costa dos Santos, Dra. Thereza A. Soares e Dr. Caio de Lima Firme.

“... com o passar dos tempos, você mudou. Deixou de ser você. Deixou as pessoas apontarem pra sua cara e dizerem que você não é bom. E quando as coisas ficaram difíceis, você começou a procurar alguém para culpar... como uma grande sombra. O mundo não é um mar de rosas. É um lugar ruim e asqueroso... e não importa o quão durão você é... ele te deixará de joelhos e te manterá assim se permitir. Nem você, nem eu, nem ninguém baterá tão forte quanto à vida. Mas isso não se trata de quão forte você pode bater. Se trata de quão forte pode ser atingido e continuar seguindo em frente. Quanto você pode apanhar e continuar seguindo em frente. É assim que a vitória é conquistada!”

Rocky Balboa

“Empty your mind, be formless, shapeless — like water. If you put water in a cup, it

becomes the cup; You put water into a bottle it becomes the bottle; You put it in a teapot

it becomes the teapot. Water can flow or it

can crash. Be water, my friend”.

Lee Jun Fan (Bruce Lee).

RESUMO

Esta tese foi realizada em quatro capítulos, todos a nível teórico, enfocados principalmente na densidade eletrônica. No primeiro capítulo, nos descrevemos a aplicação de um minicurso para estudo das reações de Diels-Alder na Universidade Federal do Rio Grande do Norte. Utilizando ferramentas de química computacional, os estudantes puderam construir um determinado conhecimento e puderam associar importantes aspectos da físico-química e da química orgânica. No segundo capítulo, estudamos um novo tipo de interação química envolvendo átomos de hidrogênio de carga neutra, denominada de ligação hidrogênio-hidrogênio (H-H). Nesse estudo realizado com alcanos complexados, fornece novas e importantes informações acerca das suas estabilidades envolvendo esse tipo de interação. Mostramos que a ligação H-H desempenha um papel secundário na estabilidade de alcanos ramificados em comparação aos seus isômeros menos ramificados ou lineares. No terceiro capítulo, estudamos a estrutura eletrônica e a estabilidade do tetraedrano, de tetraedranos substituídos e seus parentes de silício e germânio. Avaliamos o efeito do substituinte na gaiola de carbono dos derivados do tetraedrano e os resultados indicam que fortes grupos retiradores de elétrons (GRE) causam pequena instabilidade na gaiola carbônica, em contrapartida, fracos GRE resulta em grande instabilidade. Mostramos que na aromaticidade-σ, GRE e grupos doadores de elétrons (GDE) resultam em um decréscimo e em aumento, respectivamente, dos índices de aromaticidade NICS e D3BIA. Em adição, outro fator pode ser utilizado para explicar a estabilidade do tetra- tert-butiltetraedrano, assim como a ligação H-H. GVB e ADMP também foram utilizados para avaliar o efeito de substituintes na gaiola do tetraedrano. No quarto capítulo, nós realizamos uma investigação teórica do efeito inibitório do fármaco abiraterona (ABE), utilizado no tratamento do câncer de próstata, frente à enzima CYP17, comparando as energias de interação e densidade eletrônica desse fármaco com o substrato natural, a pregnenolona (PREG). Dinâmica molecular e docking foram utilizados para obter os complexos CYP17-ABE e CYP17-PREG. Á partir da dinâmica molecular, foi obtido que a ABE possui uma maior tendência de difusão água  sítio de interação da enzima CYP17, quando com a mesma tendência da PREG. Com o método ONIOM (B3LYP:AMBER), encontramos que a energia de interação da ABE é 21,38 kcal mol-1 mais estável em comparação à PREG. Os resultados obtidos através da

QTAIM indicam que essa estabilidade se dá devido a uma maior densidade eletrônica das interações entre a ABE e a CYP17.

Palavras-chave: Diels-Alder; Ligação H-H; Tetraedrano; CYP17.

ABSTRACT

This thesis was performed in four chapters, at the theoretical level, focused mainly on electronic density. In the first chapter, we have applied an undergraduate minicourse of Diels-Alder reaction in Federal University of Rio Grande do Norte. By using computational chemistry tools students could build the knowledge by themselves and they could associate important aspects of physical-chemistry with Organic Chemistry. In the second chapter, we studied a new type of chemical bond between a pair of identical or similar hydrogen atoms that are close to electrical neutrality, known as hydrogen-hydrogen (H-H) bond. In this study performed with complexed alkanes, provides new and important information about their stability involving this type of interaction. We show that the H-H bond playing a secondary role in the stability of branched alkanes in comparison with linear or less branched isomers. In the third chapter, we study the electronic structure and the stability of tetrahedrane, substituted tetrahedranes and silicon and germanium parents, it was evaluated the substituent effect on the carbon cage in the tetrahedrane derivatives and the results indicate that stronger electron withdrawing groups (EWG) makes the tetrahedrane cage slightly unstable while slight EWG causes a greater instability in the tetrahedrane cage. We showed that the sigma aromaticity EWG and electron donating groups (EDG) results in decrease and increase, respectively, of NICS and D3BIA aromaticity indices. In addition, another factor can be utilized to explain the stability of tetra-tert-butyltetrahedrane as well as H- H bond. GVB and ADMP were also used to explain the stability effect of the substituents bonded to the carbon of the tetrahedrane cage. In the fourth chapter, we performed a theoretical investigation of the inhibitory effect of the drug abiraterone (ABE), used in the prostate cancer treatment as CYP17 inhibitor, comparing the interaction energies and electron density of the ABE with the natural substrate, pregnenolone (PREG). Molecular dynamics and docking were used to obtain the CYP1ABE and CYP17-PREG complexes. From molecular dynamics was obtained that the ABE has higher diffusion trend water  CYP17 binding site compared to the

PREG. With the ONIOM (B3LYP:AMBER) method, we find that the interaction electronic energy of ABE is 21.38 kcal mol-1 more stable than PREG. The results obtained by QTAIM indicate that such stability is due a higher electronic density of interactions between ABE and CYP17.

Keywords: Diels-Alder; H-H bond; Tetrahedrane; CYP17.

LISTA DE FIGURAS – REFERÊNCIAL TEÓRICO

Figura 1. Algoritmo mostrado o procedimento autoconsistente para resolução das equações Kohn-Sham...... 30

Figura 2. Gráfico molecular do Ferroceno...... 32

Figura 3. Mapa de contorno da distribuição de carga do cloreto de sódio e o gradiente do campo vetorial ∇ρ...... 33

Figura 4. Orbital Hartree-Fock da molécula H2 para (A) R = 0,741 Å e (B) R = 6,35 Å...... 37

Figura 5. Energias da molécula H2 obtidas à partir dos métodos Hartree-Fock, MO, VB e GVB, comparadas com a energia exata não-relativística...... 37

Figura 6. Representação do método ONIOM de um sistema proteico dividido em duas camadas. Na camada baixa (low layer), os átomos são representados por estruturas do tipo arame (wireframe) e na camada alta (high layer) os átomos são representados por estruturas do tipo bola e bastão (ball-and-stick)...... 38

LISTA DE FIGURAS

Artigo 1 - Teaching thermodynamic, geometric and electronic aspects of Diels-Alder cycloadditions by using computational chemistry – an undergraduate experiment.

Figure 1. (A) Plot of enthalpy change, ∆H, versus Gibbs free energy change, ∆G, of Diels-Alder reactions (1)-(7). (B) Plot of ln K versus ∆H of Diels-Alder reactions (1)- (7)………………………………………………………………………………………48

Figure 2. HOMO and LUMO of buta-1,3-diene and ethene (A); buta-1,3-dien-2-amine and nitroethene (B); and 2-nitrobuta-1,3-diene and ethenamine (C)………………...…50

Figure 3. Optimized geometries of corresponding reactants and products from Diels- Alder reactions 4 (A) and 7 (B). Selected bond lengths, in Angstroms, are depicted………………………………………………………………………….…...…51

LISTA DE FIGURAS

Artigo 2 - Hydrogen-hydrogen bonds in highly branched alkanes and in alkane complexes: a DFT, ab initio, QTAIM and ELF study

Figure 1. Virial graphs of C8H18 isomers. (A) Octane, (B) 3-methylheptane, (C) 3,3 dimethylhexane, (D) 2,2,3-trimethylpentane and (E) 2,2,3,3-tetramethylbutane. The geometry optimization of the molecules were performed at ωB97xd/6-311++G(d,p) level of theory…………………………………………………………………………..76

Figure 2. Randomized, non-optimized and optimized geometries of methane-methane, ethane-ethane, propone-propane and butane-butane complexes depicting their corresponding selected CH--HC dihedral angles, in degrees. The geometry optimization of the molecules were performed at G4/CCSD(T)/6-311++G(d,p) level of theory. The dashed lines represent the selected dihedral angles………………………………….…78

Figure 3. Randomized non-optimized initial geometry and optimized geometry of methane-methane, ethane-ethane, propone-propane, butane-butane, pentane-pentane and hexane-hexane complexes depicting their corresponding selected H-H interatomic distances, in angstroms. The geometry optimization of the molecules were performed at ωB97XD/6-311++G(d,p) level of theory. The dashed lines represent the selected interatomic distances………………………………………………………………...…80

Figure 4. Virial graphs of Alkane complexes optimized at G4/CCSD(T)/6-311++G(d,p) level of theory. Methane-methane (Met-Met), ethane-ethane (Eth-Eth), propane-propane (Pro-Pro) and butane-butane (But-But) along with the charge density, ρ(r), and Laplacian of the electron density, ∇2 ρ(r) at H-H bond critical, in au. The others complexes were not optimized due to lack of computational resources……………….81

Figure 5. Virial graphs of Alkane complexes optimized at ωB97XD/6-311++G(d,p) level of theory. Methane-methane (Met-Met), ethane-ethane (Eth-Eth), propane-propane (Pro-Pro), butane-butane (But-But), pentane-pentane (Pen-Pen) and hexane-hexane (Hex-Hex) complexes along with the charge density, ρ(r), and Laplacian of the electron density, ∇2 ρ(r) at H-H bond critical, in au…………………………………………..…82

Figure 6. Plot of number of H-H bond paths versus ΔG (complex-2*isolated), in kcal/mol, of selected alkanes at ωB97XD/6-311++G(d,p) level of theory…………….83

Figure 7. Plot of number of H-H bond paths, from ωB97XD/6-311++G(d,p) wave function, versus boiling point, in °C, of selected alkanes…………………………...….84

Figure 8. Plot of energy, in hartrees, of a hydrogen atom belonging to H-H bond (EH(H-

H)) in ethane-ethane complex versus the energy (E) in ethane-ethane complex of selected steps in the optimization steps……………………………………………………...…..85

Figure 9. Plot of energy, in hartrees of a hydrogen atom not belonging to H-H bond (EH) in ethane-ethane complex versus the energy (E) in ethane-ethane complex of selected steps in the optimization steps………………………………………………...86

Figure S1. Optimized geometries of alkane molecules. The alkane molecules were optimized at (a) B3LYP/6-311++G(d,p), (b) M062X/6-311++G(d,p), (c) G4 method, (d) G4/CCSD/6-311++G(d,p) (T) and (e) wB97XD/6-311++G(d,p). (1) Methane, (2) ethane, (3) propane, (4) butane, (5) pentane and (6) hexane………………………..….95

Figure S2. Virial graphs of Alkane complexes optimized at B3LYP/6-311G++(d,p) level of theory. Methane-methane (Met-Met), ethane-ethane (Eth-Eth), propane-propane (Pro-Pro), butane-butane (But-But), pentane-pentane (Pen-Pen) and hexane-hexane (Hex-Hex) complexes………………………………………………………………..…96

Figure S3. Virial graphs of Alkane complexes optimized at M062X/6-311G++(d,p) level of theory. Methane-methane (Met-Met), ethane-ethane (Eth-Eth), propane-propane (Pro-Pro), butane-butane (But-But), pentane-pentane (Pen-Pen) and hexane-hexane (Hex-Hex) complexes………………………………………………………………..…97

Figure S4. Virial graphs of Alkane complexes optimized at G4 method. Methane- methane (Met-Met), ethane-ethane (Eth-Eth), propane-propane (Pro-Pro), butane-butane (But-But), pentane-pentane (Pen-Pen) and hexane-hexane (Hex-Hex) complexes……98

Figure S5. Plot of energy (E), in Hartree, of ethane-ethane complex versus of optimization steps………………………………………………………………………99

LISTA DE FIGURAS

Artigo 3 - Stability and Electronic Structure of Substituted Tetrahedranes, Silicon and Germanium Parents – a DFT, ADMP, QTAIM and GVB study.

Figure 1: Optimized geometries of the substituted tetrahedranes (A-F), silicon and germanium cages (G-I) with corresponding C1-C4, C1-C5, C1-Si, C1-N, Si-Si, Si-H, Ge-Ge, Ge-H and Ge-C bond lengths (in Angstroms); values of selected angles (in degree)………………………………………………………………………………107

Figure 2: Virial graphs of substituted tetrahedranes (A-F), silicon and germanium cages (G-I) with the electronic density, ρ(r), Laplacian of the electron density, ∇2 ρ(r), local energy density, H(r) and relative kinetic energy density, G(r)/ρ at cage-substituent bond critical, in au. Values of electronic density, ρ(r), Laplacian of the electron density, ∇2 ρ(r), local energy density, H(r), relative kinetic energy density, G(r)/ρ, delocalization index, DI, bond order, n, and ellipticity, ε, from carbon, silicon or germanium atoms in the tetrahedrane (or parent) cage for 3a – 3f, 4a, 5a and 5b………………………….108

Figure 3: Gibbs free energy difference, in kcal mol-1, from diazocompound (as a reference) to cyclobutadiene and tetrahedrane structures, along with nitrogen, with respect to the syntheses of 3a (A), 3b (B), 3f (C) and 3e (D)…………...……………109

Figure 4: Doubly occupied generalized valence bond (GVB) orbitals of lone pairs of oxygen of nitro (-NO2) functional group substituent (LP1, LP1’, LP2 and LP2’) (A and

B) and singly occupied GVB orbitals of C-C sigma bond [VB(C-C)1, VB(C-C)2 and VB(C-C)’] (B)………………………………………………………………………...113

Figure 5: (A) Plot of NICS versus D3BIA for 3a-3e, 4a, 4b and 5a; (B) Plot of NICS versus D3BIA for 3a-3f, 4a, 4b and 5a……………………………………………….115

Figure 6: Singly occupied generalized valence bond (GVB) orbitals of C-C sigma bond

[VB(C-C)1, VB(C-C)2 and VB(C-C)’] of 3a (A), 3b (B), 3f (C) and cubane (D)…...117

Figure 7: Plot of total energy, in Hartree, versus the time of trajectory, in femtoseconds, of tetrahedrane (3a) using ADMP method. Some structures are depicted at the corresponding selected points of the trajectory…………………………………….…118

Figure 8: Plot of total energy, in Hartree, versus the time of trajectory, in femtoseconds, of 3c using ADMP method. The structure at the end of trajectory is also depicted…..119

Figure 9. Plot of total energy, in Hartree, versus the time of trajectory, in femtoseconds, of 3e using ADMP method. A couple of structures are depicted at the selected corresponding points of the trajectory………………………………………………...119

LISTA DE FIGURAS

Artigo 4 - Docking, Molecular Dynamics, ONIOM, QTAIM: theoretical investigation of inhibitory effect of Abiraterone on CYP17.

Figure 1. Inside the cell, testosterone is converted to DHT and binds to AR. AR-DHT complex undergoes phosphorylation and is translocated to the nucleus in the form of dimer. Inside the nucleus, AR-dimer binds to DNA coregulators leading to transcription activation……………………………………………………………………………...125

Figure 2. Chemical structure of ABE……………………………..………………….126

Figure 3. The protonation states of histidine at pH 7.0. HID (A) HIE (B) and HIP (C)……………………………………………………………………………………..128

Figure 4. Schematic representations of pockets of the CYP17-ABE (A) and CYP17- PREG (B) complexes …………………………………………………………………133

Figure 5. Plots of RMSD (in nm) versus time of simulation (in ns) for CYP17-ABE complex of each protonation state of His 373. HISD (A), HISE (B) and HISH (C)………………………………………………………………………………..……135

Figure 6. Plots of RMSD (A) and χ (B) of the His373 of each protonation state. The data represent the average of five 10-ns snapshots for each protonation state expressed as mean standard deviation. It was considered a p value less than 0.05………..…….135

Figure 7. The binding poses of PREG in the active site of the enzyme CYP17….….136

Figure 8. Plot of RMSD (in nm) versus time of simulation (in ns) for CYP17-PREG complex……………………………………………………………………………….136

Figure 9. ABE, heme group and CYS442 of CYP17-ABE complex (A) and PREG, heme group and CYS442 of CYP17-PREG complex (B)…………………………….137

Figure 10. Optimized structures of the QM (highlighted) and MM (gray lines) layers of the CYP17-ABE (A) and CYP17-PREG (B) complexes and CYP17 enzyme (C) at the ONIOM (B3LYP:AMBER) level……………………………………………………..139

Figure 11. Virial graphs of binding pockets of CYP17-ABE (A) and CYP17-PREG (B) complexes…………………………………………………………………………….139

Figure S1. Optimized geometries of Fe(NH3)2(NH2)2 at B3LYP/6-311G(d,p) level in the triplet and quintet states …………………………………………………………157

Figure S2. Fitting structures of CYP17-ABE (A) and CYP17-PREG (B) complexes before (yellow) and after (red) molecular dynamics simulation. The picture was rendered using VMD……………………………………………………….…………158

Figure S3. B-factors for the CYP17-ABE (A) and CYP17-PREG (B) complexes. The picture was rendered using VMD, “blue” indicates high, “white” indicates intermediate and “red” low values. High temperature factors imply exibility and low ones rigidity……………………………………………………………………………...…158

LISTA DE TABELAS – REFERÊNCIAL TEÓRICO

Tabela 1. Acrônimos, sinais dos autovalores e denominações dos pontos críticos...... 32

LISTA DE TABELAS

Artigo 1 - Teaching thermodynamic, geometric and electronic aspects of Diels-Alder cycloadditions by using computational chemistry – an undergraduate experiment.

Table 1. Values of enthalpy, H, Gibbs free energy, G, in kcal mol-1, and entropy, S, in cal mol-1 K-1 of all studied molecules…………………………………………………..48

Table 2. Gibbs free energy change, ∆G, Enthalpy change, ∆H, entropy change and temperature product, in kcal mol-1 K-1, T∆S, entropy change, in kcal mol-1, ∆S, and equilibrium constant logarithm, ln K, of Diels-Alder reactions (1)–(7)………………..49

Table 3. Energy values of HOMO (EHOMO) and LUMO (ELUMO) and energy gap between HOMO and LUMO orbitals, |EHOMO-ELUMO|, in a.u., of the reagents of the Diels-Alder reactions (1) – (7)……………………………………………….…………50

Table 4. Bond lengths of reagents (r1, r2, r3 and r4) and products (r’1, r’2, r’3 and r’4), in Angstroms, of the studied Diels-Alder reactions…………………………….…………52

LISTA DE TABELAS

Artigo 2 - Hydrogen-hydrogen bonds in highly branched alkanes and in alkane complexes: a DFT, ab initio, QTAIM and ELF study

Table 1. Amount of branching, amount of H-H bonds and heats of combustion (ΔHcomb) in branched alkanes…………………………………………………………………….75

Table 2. Number of H-H bonds, boiling point, in °C, Gibbs energy of alkane complexes

(Gcomplex), two-fold Gibbs energy of the corresponding isolated alkane (2*Gisolated), and complex stability (G(complex-2*isolated)) of the alkane complexes………………………..82

Table 3. ELF value, η(r), electron density, ρ(r), and Laplacian of the electron density, ∇2 ρ(r) at bond critical point of a selected H-H bond in alkane complexes…………....87

LISTA DE TABELAS

Artigo 3 - Stability and Electronic Structure of Substituted Tetrahedranes, Silicon and Germanium Parents – a DFT, ADMP, QTAIM and GVB study

Table 1. Values of Gibbs free energy difference and enthalpy difference for the corresponding tetrahedranes 3a – 3f, 4a, 5a and 5b from diazocompound to cyclobutadiene (and derivatives), ΔG1 and ΔH1, from diazocompound to tetrahedrane

(and derivatives), ΔG2 and ΔH2, and from cyclobutadiene (and derivatives) to tetrahedrane (and derivatives), ΔGf and ΔHf, in kcal mol-1, respectively. Values of delocalization index, DI, bond order, n, and ellipticity, ε, from carbon, silicon or germanium atoms in the tetrahedrane (or parent) cage, bond path angle, in degree, from bonds in the tetrahedrane (or parent) cage for 3a – 3f, 4a, 5a and 5b……………...…110

Table 2. Values of AIM atomic charge, q(Ω), atomic energy, E(Ω), atomic dipole moment, M(Ω), atomic volume, V(Ω) and localization index, LI of carbon, silicon and germanium atoms in tetrahedrane, silicon and germanium derivatives, respectively...111

Table 4. Overlap matrix of selected GVB orbitals of molecule 3f of two oxygen lone pairs (LP1, LP2, LP1’ and LP2’) and of two C-C sigma bonds [VB(C-C)1, VB(C-C)’ and VB(C-C)2]………………………………………………………………………...113

Table 3. Average DI, DI, of the carbon or silicon or germanium atoms in the tetrahedrane (or parent) cage, the corresponding DIUs, the degree of degeneracy (δ) involving the atoms of the tetrahedrane (or parent) cage, the charge density of the cage critical points [ρ(3,+3)], D3BIA and NICS values for 3a, 3b, 3c, 3d, 3e, 3f, 4a, 5a and 5b……………………………………………………………………………………...115

Table 5. Overlap matrix of selected GVB orbitals, VB(C-C)’, VB (C-C)1 and VB (C-

C)2, of 3a, 3b, 3f and cubane. ……………………………………………………...….117

LISTA DE TABELAS

Artigo 4 - Docking, Molecular Dynamics, ONIOM, QTAIM: theoretical investigation of inhibitory effect of Abiraterone on CYP17.

Table 1. Binding energies of CYP17 with PREG…………….…………...……….…135

Table 2. Values of Lennard-Jones potential (VLJ) and Coulomb potential (VC), in kcal mol-1, between CYP17 (p)/water (w) and ligands ABE and PREG…………………..138

Table 3. Values of electronic energy of CYP17 enzyme, ABE, PREG, CYP17-ABE and CYP17-PREG complexes……………………………………………………………..138

Table 4. Electronic density (ρ), in au., of the critical points between abiraterone (or pregnenolone) and CYP17 active site……………………………………………..….139

TABLE S1. Electronic energy, in a.u., bond length Fe-NH2 and bond length Fe-NH3, in angstroms, of triplet and quintet states of the Fe(NH3)2(NH2)2 with different functionals using 6-311G(d,p) basis set……………………………………………………..…….157

LISTA DE ABREVIATURAS, ACRÔNIMOS E SIGLAS EMPREGADAS.

ρ Densidade eletrônica

훁ퟐ흆 Laplaciano da densidade eletrônica

ΔG Variação da energia livre de Gibbs

AMBER Assisted Model Building with Energy Refinement (Campo de força ou pacote de programas de simulação )

BCP Bond Critical Point (Ponto Crítico de Ligação)

DFT Density Functional Theory (Teoria Funcional da Densidade)

GVB Generalized Valence Bond (Ligação de Valência Generalizada)

HF Hartree-Fock

Hb Total Energy Density (Densidade de Energia Total)

QM/MM Mecânica Quântica/Mecânica Molecular

QTAIM Quantum Theory of Atoms in Molecules (Teoria Quântica de Átomos em Moléculas)

RESP Restrained Electrostatic Potential

ABE Abiraterona

PREG Pregnenolona

SUMÁRIO

1. INTRODUÇÃO GERAL ...... 24

2. REFERENCIAL TEÓRICO ...... 25

2.1. TEORIA FUNCIONAL DA DENSIDADE (Density functional theory – DFT) ...... 25 2.2. TEORIA QUÂNTICA DE ÁTOMOS EM MOLÉCULAS (Quantum Theory of Atoms in Molecules - QTAIM) ...... 30 2.3. FUNÇÃO DE LOCALIZAÇÃO ELETRÔNICA (Electron Localization Function - ELF)...... 34 2.4. MÉTODO HARTREE-FOCK (HF) E TEORIA DA LIGAÇÃO DE VALÊNCIA GENERALIZADA (Generalized Valence Bond – GVB)...... 36 2.5. CÁLCULOS DE QM/MM ...... 38 2.6. SIMULAÇÃO MOLECULAR ...... 39 2.7. DOCAGEM MOLECULAR ...... 40 3. CAPÍTULO 1 – Artigo publicado no periódico World Journal of Chemical Education...... 41

4. CAPÍTULO 2 – Artigo publicado no periódico The Journal of Physical Chemistry A...... 70

5. CAPÍTULO 3 – Artigo publicado no periódico New Journal of Chemistry...... 100

6. CAPÍTULO 4 – Formatação de artigo a ser submetido para o periódico The Journal of Physical Chemistry B...... 124

7. REFERÊNCIAS GERAIS ...... 159

1. INTRODUÇÃO GERAL

A química é a ciência que lida com a construção, transformação e propriedades das moléculas. A química teórica é uma subárea da química onde os métodos matemáticos são combinados com as leis fundamentais da física com o objetivo de se estudar processos de relevância química. (Clark, 1985)

A química computacional é uma disciplina que se estende muito além dos limites tradicionais que separam a química, a física, a bioquímica e a ciência da computação, permitindo o estudo aprofundado de átomos, moléculas e macromoléculas através da ferramenta computacional. A química computacional pode atuar como ferramenta de apoio na análise e interpretação de dados experimentais, através de informações que muitas vezes não são possíveis de serem obtidas diretamente de experimentos (Morgon et al., 1995; Morgon et al., 1997; Morgon e Riveros, 1998; Morgon et al., 2000), ou na previsão de propriedades diversas. Além disso, a química computacional pode ser aplicada nas mais diversas áreas da química, assim como: Físico-química, química orgânica, química inorgânica, química analítica, bioquímica, além de ser utilizada como ferramenta de ensino.

Na química orgânica, por exemplo, existem muitas explicações consolidadas acerca da estabilidade de alcanos e sua relação o ponto de fusão e ebulição. A química computacional pode nos fornecer novas e importantes informações acerca do modo de interação dessas moléculas e relacioná-las com as suas propriedades macroscópicas.

A química computacional também pode ser aplicada no desenvolvimento de moléculas de alta energia de interesse militar. Moléculas orgânicas de alta tensão em formato de gaiola podem ser usadas como potenciais explosivos em resposta a estímulos externos. (Katz e Acton, 1973; Zhou et al., 2004)

O estudo teórico na área de bioquímica possui uma gama de aplicações como a análise conformacional de grandes sistemas moleculares de importância biológica; o estudo da interação enzima-substrato e principalmente no planejamento racional de novos fármacos. Ainda existe um novo cenário surgindo com o aparecimento de uma recente estratégia de cálculos híbridos que envolvem a mecânica quântica e a mecânica molecular, conhecida como QM/MM (Quantum Mechanics/Molecular Mechanics). Assim, por exemplo, existe a possibilidade de descrição da quebra e formação de ligações químicas em reações que envolvam grandes sistemas através da mecânica quântica, onde é sabido que a mecânica molecular é falha na descrição das propriedades eletrônicas.

Portanto, para essa tese, foram realizados quatro estudos teóricos das áreas descritas acima, utilizando métodos quânticos, de mecânica molecular e híbridos objetivando um estudo mais aprofundado, assim como fornecer novas perspectivas em cada tema abordado.

2. REFERENCIAL TEÓRICO

2.1. TEORIA FUNCIONAL DA DENSIDADE (Density functional theory – DFT)

A teoria funcional da densidade (DFT) é um dos métodos quanto-mecânicos mais populares e bem- sucedidos da atualidade. (Capelle, 2006) Essa metodologia é rotineiramente aplicada para cálculos de energia de ligação de moléculas na química e para cálculos de estrutura de bandas de sólidos em física. Por outro lado, as primeiras aplicações em campos considerados mais distantes da mecânica quântica, assim como a biologia (Devipriya e Kumaradhas, 2012; Sicolo et al., 2014; Bigler et al., 2015; Jitonnom et al., 2015) e a mineralogia (Deng et al.; Liu et al., 2013; Mikhlin et al., 2014; Kahlenberg et al., 2015) estão começando a aparecer.

Para se ter uma ideia do que é a teoria funcional da densidade, se faz necessário algum conhecimento prévio da mecânica quântica. Na mecânica quântica, toda e qualquer informação relevante a um sistema está contida na sua função de onda, Ψ. Os graus de liberdade nucleares aparecem somente na forma de energia potencial, ν(r), atuando nos elétrons devido à aproximação de Born-Oppenheimer.(Woolley e Sutcliffe, 1977) Devido a isso, a função de onda depende apenas das coordenadas eletrônicas. Essa função de onda não-relativística é calculada a partir da equação de Schrödinger para um elétron se movendo em um potencial ν(r):

ħ2∇2 [− + 휈(풓)] 훹(풓) = 휀훹(풓) (1) 2푚 onde ∇2 é o operador Laplaciano definido como a soma dos operadores diferenciais em coordenadas cartesianas:

휕2 휕2 휕2 ∇2= + + (2) 휕푥2 휕푦2 휕푧2 para um sistema multieletrônico, a equação de Schrödinger se torna:

푁 ħ2∇2 [∑ (− + 휈(풓)) + ∑ 푈(풓 , 풓 )] 훹(풓 , 풓 , … , 풓 ) = 퐸훹(풓 , 풓 , … , 풓 ) (3) 2푚 푖 푗 1 2 푁 1 2 푁 푖 푖<푗

onde N é o número de elétrons e U(ri,rj) é a interação elétron-elétron. Para um sistema Coulômbico, temos:

푞2 푈̂ = ∑ 푈(풓 , 풓 ) = ∑ (4) 푖 푗 |풓 − 풓 | 푖<푗 푖<푗 푖 푗 e o operador energia cinética é dado por: 26

ħ2 푇̂ = − ∑ ∇2 (5) 2푚 푖 푖 e o potencial de interação elétron-núcleo é dado por:

푄푞 푉̂ = ∑ 휈(풓푖) = ∑ (6) |풓푖 − 푹| 푖 푖

Onde Q é a carga nuclear (Pople, 1999) e R é a posição do núcleo.

A função de onda Ψ em si, não é uma observável, pois não é mensurável. Entretanto, o quadrado do valor absoluto da função de onda possui uma interpretação física. (Koch e Holthausen, 2001) Essa interpretação só é possível porque o produto de um número complexo pelo seu complexo conjugado é real e não negativo, onde:

2 |훹(푥⃗1,푥⃗2, … 푥⃗푁,)| 푑푥⃗1푑푥⃗2 … 푑푥⃗푁 (7) representa a probabilidade que os elétrons 1, 2, ..., N são encontrados simultaneamente em volumes

푑푥⃗1푑푥⃗2 … 푑푥⃗푁. Desde que esses elétrons sejam indistinguíveis, não deve haver troca de coordenadas de quaisquer dois elétrons (i e j) se os mesmos são comutados, ou seja:

2 2 |훹(푥⃗1,푥⃗2, … , 푥⃗푖, 푥⃗푗, … , 푥⃗푁,)| = |훹(푥⃗1,푥⃗2, … , 푥⃗푗, 푥⃗푖, … , 푥⃗푁,)| (8)

Isso garante que as funções de onda possuam sinais contrários resultando em funções antissimétricas aplicadas à férmions com spins semi-inteiros. Elétrons são férmions com spin = ½ e Ψ deve ser, portanto, antissimétrica com respeito a permutação das coordenadas espaciais e spin de quaisquer dois elétrons:

훹(푥⃗1,푥⃗2, … , 푥⃗푖, 푥⃗푗, … , 푥⃗푁,) = −훹(푥⃗1,푥⃗2, … , 푥⃗푖, 푥⃗푗, … , 푥⃗푁,) (9) esse princípio da antissimetria, representa uma generalização quanto-mecânica denominada de princípio da exclusão de Pauli (onde dois elétrons não podem ocupar o mesmo estado). Uma consequência lógica da interpretação probabilística da função de onda, é que integrando a eq. (7) em todo o intervalo sobre todas as variáveis, o resultado é igual a um. Em outras palavras, a probabilidade de encontrar os N elétrons em qualquer lugar do espaço corresponde a uma unidade,

2 ∫ … ∫|훹(푥⃗1,푥⃗2, … 푥⃗푁,)| 푑푥⃗1푑푥⃗2 … 푑푥⃗푁 = 1 (10) a função de onda que satisfaz a eq. (10) é dita ser normalizada.

A interpretação probabilística da eq. (10) da função de onda leva diretamente à densidade eletrônica, ρ(r), que é definida como a integral múltipla sobre todas as coordenadas spin e sobre todas as variáveis espaciais de todos os elétrons: 27

2 휌(풓⃗⃗) = 푁 ∫ … ∫|훹(푥⃗1,푥⃗2, … 푥⃗푁,)| 푑푥⃗1푑푥⃗2 … 푑푥⃗푁 (11)

휌(풓⃗⃗) determina a probabilidade de encontrar qualquer um dos N elétrons dentro de um volume 푑(풓⃗⃗⃗⃗ퟏ⃗). 휌(풓⃗⃗) é a densidade de probabilidade, comumente denominada de densidade eletrônica.

Diferentemente de uma função de onda, a densidade eletrônica é uma observável e pode ser mensurada experimentalmente, como por exemplo, através da difração de raios X. (Averkiev et al., 2014) A teoria funcional da densidade é um método baseado na análise do sistema molecular através da densidade eletrônica. O primeiro trabalho com a densidade eletrônica foi realizado no início do século XX com o objetivo de explicar as propriedades térmicas e elétricas dos metais. (Chudnovsky, 2007) Mas foi somente em 1964, através do trabalho de Hohenberg e Kohn (Hohenberg e Kohn, 1964) que mostra que a densidade eletrônica, de fato, determina o operador Hamiltoniano e, assim, todas as propriedades do sistema. Ainda assim, o método precisava buscar os funcionais adequados, os chamados funcionais exatos, necessários para serem incorporados na expressão da energia total eletrônica. Somente com os trabalhos de Kohn-Sham (Kohn e Sham, 1965) melhorando o tratamento matemático das equações de densidade e do funcional, que a teoria funcional da densidade avançou como ferramenta de cálculo teórico para estudo das propriedades moleculares com aplicações computacionais. (Kohn et al., 1996)

O primeiro teorema de Hohenberg-Kohn afirma que a densidade eletrônica determina o potencial externo [ν(r)] devido aos núcleos, assim como também determina o número total de elétrons, N, via sua normalização ∫ 휌(풓)푑푟 = 푁. N e ν(r) determinam o operador Hamiltoniano molecular (Hop) [vide eq. (3)] que, por sua vez, determina a energia do sistema através da equação de Schrödinger 퐻표푝훹 = 퐸훹. A densidade eletrônica determina a energia do sistema e todas as outras propriedades eletrônicas do sistema no estado fundamental. (Geerlings et al., 2003) O esquema 1 mostra a relação existente entre as variáveis descritas acima.

Esquema 1:

Hop E

consequentemente, E é um funcional de ρ: 28

퐸 = 퐸[휌] (12)

O segundo teorema de HK estabelece que o funcional FHK é universal, ou seja, o mesmo é usado para todos os problemas de estrutura eletrônica (Burke, 2007), que é dado por:

퐹퐻퐾 = 푇[휌] + 푉푒푒[휌] (13) onde T[ρ] é o funcional de energia de energia cinética eletrônica e Vee[ρ] é o funcional de interação elétron- elétron. Esse funcional é um componente da expressão do funcional da energia (Geerlings et al., 2003):

퐸[휌] = ∫ 휌(풓)휈(풓)푑풓 + 퐹퐻퐾 (14)

dessa forma, o segundo teorema de HK estabelece um princípio variacional onde 퐸[휌] ≥ 퐸0.

Para contornar o problema dos teoremas de HK, Kohn e Sham propuseram uma abordagem para a aproximação dos funcionais de energia cinética e potencial elétron-elétron. (Kohn e Sham, 1965) Eles introduziram um sistema fictício de N elétrons não-interagentes para serem descritos através de um único determinante em N orbitais ϕi.

Kohn e Sham reescreveram a equação (14) e explicitaram a repulsão eletrônica de Coulomb e definiram uma nova função universal G[ρ]:

1 휌(푟1)휌(푟2) 퐸휈[휌] = 퐺[휌] + ∫ ∫ 푑푟1푑푟2 + ∫ 휌(푟)휈(푟)푑푟 (15) 2 |푟1 − 푟2| em que

퐺[휌] = 푇푠[휌] + 퐸푥푐[휌] (16) onde 푇푠[휌] é o funcional energia cinética de um sistema de elétrons que não interagem e 퐸푥푐[휌] inclui o termo de interação elétron-elétron não-clássica (troca e correlação) e a parte residual da energia cinética,

푇[휌] − 푇푠[휌], em que 푇[휌] é a energia cinética exata para o sistema de elétrons que interagem.

O Hamiltoniano dos elétrons que não interagem com o potencial efetivo, 휈푒푓, é dado por:

1 퐻퐾푆 = − ∇2 + 휈 (17) 2 푒푓

A obtenção da função de onda Kohn-Sham, ΨKS, do estado fundamental do sistema de elétrons que não interagem, descrito pelo Hamiltoniano da equação (17), é dada pelo produto anti-simetrizado de N orbitais de um elétron, ϕi(r), através do determinante de Slater (Morgon e Coutinho, 2007): 29

퐾푆 퐾푆 퐾푆 휙1 (푟1) 휙2 (푟1) … 휙푁 (푟1) 1 휙퐾푆(푟 ) 휙퐾푆(푟 ) … 휙퐾푆(푟 ) 훹퐾푆 = | 1 2 2 2 푁 2 | (18) √푁! ⋮ ⋮ ⋱ ⋮ 퐾푆 퐾푆 퐾푆 휙1 (푟푁) 휙2 (푟푁) … 휙푁 (푟푁)

퐾푆 Por sua vez, os orbitais Kohn-Sham, 휙푖 , são obtidos a partir da equação de Schrödinger para um elétron:

1 (− ∇2 + 휈 ) 휙퐾푆 = 휀 휙퐾푆 (19) 2 푒푓 푖 푖 푖

A relação existente entre o sistema real e o sistema que não interage é dada por:

푁 퐾푆 2 휌푠(푟) = ∑ 2|휙푖 (푟)| = 휌0(푟) (20) 푖 onde ρ0(r) é a densidade eletrônica no estado fundamental. A energia cinética é dada por:

푁 1 푇 [휌] = − ∑〈휙퐾푆|∇2|휙퐾푆〉 (21) 푠 2 푖 푖 푖 e o potencial efetivo é dado por:

휌(푟1) 휈푒푓 = 휈(푟) + 푑푟1 + 휈푥푐(푟) (22) |푟 − 푟1| em que

훿퐸 [휌] 휈 (푟) = 푥푐 (23) 푥푐 훿휌(푟)

A equação (23) é o funcional proveniente da energia de correlação e troca [equação (15)] em relação à densidade.

Das equações acima, percebe-se que o potencial efetivo é dependente da densidade, ρ(r), a qual 퐾푆 depende dos orbitais Kohn-Sham, 휙푖 , que deverão ser encontrados. Para isso, utiliza-se um procedimento autoconsistente que é mostrado na Figura 1.

Primeiramente é construída a densidade inicial, ρ0, para se calcular o potencial efetivo, 휈푒푓, para se obter o Hamiltoniano. Assim, resolve-se a equação de Kohn-Sham para se obter um conjunto de 퐾푆 autofunções, 휙푖 , que irão compor o determinante de Slater. Em seguida é obtida a função de onda Kohn- KS Sham, Ψ , do qual a nova densidade eletrônica, ρ1, pode ser calculada. A densidade recém-calculada, ρn é comparada com a anterior ρn-1. Se ρn ≠ ρn-1 o ciclo é reiniciado até que a densidade não se altere de um ciclo 30 para outro, ou seja, ρn = ρn-1. Quando essa condição for atingida, significa dizer que essa densidade de carga minimiza a energia e a convergência foi alcançada.

Cálculo do potencial efetivo

ρ0

Resolução da equação de KS

Determinante de Slater

Obtenção de ΨKS

Cálculo da nova densidade

Não Sim ρn é a densidade procurada

Figura 1. Algoritmo mostrado o procedimento autoconsistente para resolução das equações Kohn-Sham. Fonte: Autor, 2015.

2.2.TEORIA QUÂNTICA DE ÁTOMOS EM MOLÉCULAS (Quantum Theory of Atoms in Molecules - QTAIM)

A teoria quântica de átomos em moléculas (QTAIM) é um modelo quântico considerado eficiente no estudo da ligação química (Cortés-Guzmán e Bader, 2005) e caracterização das interações intra e intermoleculares. (Grabowski et al., 2004; Oliveira et al., 2009; Monteiro e Firme, 2014).

A teoria quântica de átomos em moléculas (Bader, 1990), desenvolvida pelo Professor Richard F. W. Bader e seus colaboradores, as propriedades observáveis dos átomos constituintes de um sistema molecular estão contidas na sua densidade eletrônica, ρ. As trajetórias da densidade eletrônica são obtidas à partir do vetor gradiente da densidade eletrônica (∇ρ) (Popelier, 2000), que é dada pela primeira derivada da 31 densidade eletrônica sobre todas as coordenadas (Equação 24) e o conjunto de trajetória desse gradiente formam as bacias atômicas (Ω).

푑휌 푑휌 푑휌 ∇휌 = 풊 + 풋 + 풌 (24) 푑푥 푑푦 푑푧

Um ponto crítico (PC) na densidade eletrônica é um ponto no espaço em que cada derivada do ∇ρ é zero (∇ρ = 0). Para se distinguir um ponto crítico de mínimo local, máximo local ou um ponto crítico de sela, são consideradas as derivadas segundas da densidade eletrônica. Existem nove derivadas segundas de ρ(r) que podem ser dispostas em uma matriz Hessiana, mostrada na equação 25 (Bader, 1990):

휕2휌 휕2휌 휕2휌 2 휕푥 휕푥휕푦 휕푥휕푧 휕2휌 휕2휌 휕2휌

∇∇휌 = 2 (25) 휕푦휕푥 휕푦 휕푦휕푧

휕2휌 휕2휌 휕2휌 2 (휕푧휕푥 휕푧휕푦 휕푧 )

A matriz Hessiana pode ser diagonalizada através da rotação do sistema de coordenadas r (x, y, z)  r (x’, y’, z’). Os novos eixos coordenados são chamados de eixos principais da curvatura porque a magnitude das três derivadas da segunda de ρ, calculadas com respeito a esses eixos, são maximizados. A rotação do sistema de coordenadas é acompanhada de uma transformação unitária, r’ = rU, onde U é uma matriz unitária construída à partir de uma conjunto de três equações de autovalores (훁훁흆)풖푖 = 휆푖풖푖 (com 푖 = 1, 2, -1 3) em que 풖푖 é o inésimo autovetor em U. A equação matricial U (훁훁흆)U = Λ transforma a Hessiana na sua forma diagonal, que pode ser escrita como:

휕2휌 0 0 휕푥′2 휆 0 0 휕2휌 1 훬 = 0 0 = ( 0 휆 0 ) (26) 휕푦′2 2 0 0 휆3 휕2휌 0 0 ( 휕푧′2) em que λ1, λ2 e λ3 são as curvaturas da densidade em relação aos três eixos principais x’, y’ e z’.

A soma dos autovalores da matriz Hessiana é conhecida como o Laplaciano da densidade eletrônica (∇2ρ), que é dado pela equação 27:

2 2 2 2 휕 휌 휕 휌 휕 휌 ∇ 휌 = + + = 휆1 + 휆2 + 휆3 (27) 휕푥2 휕푦2 휕푧2

Um ponto crítico pode ser classificado de acordo com o seu ranking, denotado por ω, que é o número de autovalores não-zero da ρ(r) de um ponto crítico e pela assinatura, denotada por σ, que é a soma algébrica dos sinais dos autovalores. Assim, um ponto crítico é caracterizado pelo conjunto de valores (ω,σ), como pode ser exemplificado na Tabela 1. (Firme, 2007) Existem quatro tipo de pontos críticos estáveis, possuindo três autovalores não-zero. O primeiro deles é o ponto crítico (3,-3), denominado de atrator nuclear (Nuclear Attractor – NA), que possui todos os autovalores negativos, sendo a sua densidade eletrônica ρ(r) um máximo local; o segundo ponto crítico é o (3,-1), denominado de ponto crítico de ligação (Bond Critical

Point - BCP), que possui dois autovalores negativos (λ1 e λ2) e um positivo (λ3) com ρ(r) sendo um máximo no plano definido pelos autovetores correspondentes e um mínimo ao longo dos três eixos que é perpendicular a esse plano; o terceiro ponto crítico (3,+1) é o ponto crítico do anel (Ring Critical Point -

RCP) que possui dois autovalores positivos (λ2 e λ3) e um negativo (λ1) com ρ(r) sendo um mínimo e por último o ponto crítico de gaiola (Cage Critical Point – CCP) (3,+3) que possui todos os autovalores positivos, sendo a ρ(r) um mínimo local. (Matta et al., 2007) Todos os pontos críticos estão representados na Figura 2.

Tabela 1. Acrônimos, sinais dos autovalores e denominações dos pontos críticos.

Nome Acrônimo λ1 λ2 λ3 (ω,σ) Atrator Nuclear NA - - - (3,-3)

Ponto Crítico da Ligação BCP - - + (3,-1)

Ponto Crítico do Anel RCP - + + (3,+1)

Ponto Crítico da Gaiola CCP + + + (3,+3)

(3,-1)

(3,+1) (3,+3) (3,-3)

Figura 2. Gráfico molecular do Ferroceno. Fonte: (Firme et al., 2010). 33

Um ponto crítico (3,-3) age como um atrator do campo vetorial do ∇ρ, ou seja, existe uma vizinhança aberta do atrator que é invariante ao fluxo de ∇ρ tal que qualquer caminho do gradiente originado nessa vizinhança aberta termina no atrator. Existem também caminhos de gradiente que terminam ou se originam nos pontos críticos (3,-1). Os caminhos de gradiente que se originam nos pontos críticos (3,-1) definem os caminhos de ligação (Figura 2). Os dois caminhos de gradiente definem uma linha através da densidade eletrônica ligando os núcleos vizinhos ao longo do qual ρ(r) é máximo em relação a qualquer linha vizinha. (Bader et al., 1979). Essa linha é encontrada entre cada par de núcleos cujas bacias atômicas compartilham uma superfície interatômica comum e é chamada de interação atômica. A existência do ponto crítico (3,-1) e a sua linha associada de interação atômica indicam que a densidade eletrônica é acumulada entre os núcleos que estão ligados. Esse acúmulo de carga é a condição necessária quando as forças de Feyanman que atuam nos núcleos e elétrons estão em equilíbrio.(Bader e Fang, 2005) Sendo assim, a presença da linha de interação atômica satisfaz as condições necessárias para que os átomos estejam ligados. Essa linha de interação é denominada de caminho de ligação e o ponto crítico (3,-1) é chamado de ponto crítico de ligação. Mais a frente voltaremos a falar do caminho de ligação.

A densidade eletrônica máxima nas posições dos núcleos resulta em uma topologia originada através das trajetórias da densidade eletrônica são obtidas à partir do vetor gradiente ∇ρ. Essa topologia é particionada do espaço molecular em regiões mononucleares, Ω, denominadas de bacias atômicas. (Matta et al., 2007) A superfície de ligação entre duas bacias atômicas é denominada de superfície de fluxo zero (Figura 3), onde o produto escalar do vetor ∇ρ e o vetor normal [n(r)] se anulam (Equação 28).

Superfície de fluxo zero

Figura 3. Mapa de contorno da distribuição de carga do cloreto de sódio e o gradiente do campo vetorial ∇ρ. Fonte: (Bader, 1990).

∇휌. 푛(푟) = 0 (28) 34

Existe um ponto na superfície de fluxo zero onde os vetores gradiente se anulam (∇ρ = 0), pois esses vetores apontam para os núcleos atômicos e, consequentemente, ocorre formação do ponto crítico de ligação (3,-1).

A presença de uma superfície interatômica de fluxo zero entre dois átomos ligados é sempre acompanhada por uma linha de densidade local máxima, denominada de caminho de ligação. O caminho de ligação é um indicador universal de todos os tipos de ligação química: interações fracas, fortes, camada fechada e aberta.(Bader, 1998)

O conjunto de caminhos de ligação que conectam os núcleos dos átomos ligados em uma geometria de equilíbrio, assim como os seus pontos críticos de ligação, é denominado de gráfico molecular. (Wiberg et al., 1987) O gráfico molecular é o resultado direto das propriedades topológicas principais da distribuição de cargas de um sistema em que o máximo local, os pontos críticos (3,-3), ocorrem nas posições dos núcleos e os pontos críticos (3,-1) que conectam certos pares de núcleos em uma molécula (Figura 2). (Bader, 1990) Por outro lado, existe outro tipo de gráfico, que é considerado um “espelho” do gráfico molecular. Esse tipo de gráfico é definido por um conjunto de linhas com densidade máxima de energia potencial negativa. Em outras palavras, existe uma única linha com densidade máxima de energia potencial negativa conectando os mesmos núcleos de um caminho de ligação. (Keith et al., 1996) Essa linha de “máxima estabilidade” no espaço real é denominada de “caminho virial”. O conjunto de caminhos viriais associados aos pontos críticos constituem o gráfico virial.

2.3. FUNÇÃO DE LOCALIZAÇÃO ELETRÔNICA (Electron Localization Function - ELF)

A função de localização eletrônica (Electron Localization Function – ELF) descrita primeiramente por Becke e Edgecombe (Becke e Edgecombe, 1990) como uma função local que demonstra o quanto o princípio da exclusão de Pauli é eficiente em um dado ponto do espaço molecular. Ou seja, o ELF fornece uma descrição quantitativa do princípio da exclusão de Pauli. O ELF é definido como:

1 ( ) ( ) 휂 푟 = 2 29 , 1 + (퐷⁄퐷ℎ)

em que,

푁 1 1 |∇휌|2 퐷 = ∑|∇ѱ |2 − (30) 2 푖 8 휌 푖=1

e 35

푁 3 퐷 = (3휋2)2/3휌5/3 (31), 휌 = ∑|ѱ |2 (32), ℎ 10 푖 푖=1

onde as somas são todos os (spin) orbitais ѱi(r) monocupados.(Savin, 2005) O valor do ELF é confinado ao intervalo [1,0], onde a tendência para 1 significa que os spins paralelos são amplamente improváveis, e onde há, portanto, uma alta probabilidade de spins opostos. A tendência para zero indica uma alta probabilidade de regiões com os mesmos pares de spin. A função de localização eletrônica fornece um rigoroso suporte para a análise da função de onda e da estrutura eletrônica em moléculas e cristais.(Noury et al., 1999) A topologia da ELF tem sido amplamente empregada no estudo da ligação química.(Noury et al., 1998; Chevreau et al., 2001; Chesnut, 2002; Chevreau, 2004; Rosdahl et al., 2005) O particionamento topológico do campo gradiente da ELF (Häussermann et al., 1994; Silvi e Savin, 1994) fornece bacias de atratores correspondentes às ligação químicas e aos pares de elétrons isolados. Em uma molécula podemos encontrar dois tipos de bacias. A primeira como sendo as bacias de caroço que estão em torno dos núcleos com número atômico Z > 2 denominado de C(A), onde A é o símbolo atômico do elemento. O outro tipo de bacia é denominado de bacia de valência. As bacias de valência são caracterizadas pelo número de camadas de valência que a bacia possui, em outras palavras, pelo número de bacias de caroço que os elementos compartilham nas suas fronteiras. Esse número é chamado de ordem sináptica. Assim, existem bacias monosinápticas, disinápticas, trisinápticas e assim por diante, especificadas como V(A), V(A, X), V(A, X, Y) respectivamente. Bacias monosinápticas correspondem aos pares de elétrons isolados do modelo de Lewis. Bacias polisinápticas correspondem aos pares de elétrons compartilhados do modelo de Lewis.

A função local ELF, η(r), fornece a informação química e ∇η(r) fornece dois tipos de pontos no espaço real: pontos críticos onde (r)=0 e para os demais pontos, (r)≠0. Os pontos críticos são caracterizados pelos seus índices que são os números de autovalores positivos da matriz Hessiana de η(r). A topologia da ELF já foi aplicada no estudo da ligação de hidrogênio.(Fuster e Silvi, 2000; Alikhani e Silvi,

2003; Alikhani et al., 2005; Soler et al., 2005) No complexo FH•••N2 o valor da ELF entre as bacias V(F,H) e V(N) é 0,047. Enquanto para o complexo FH•••NH3 o valor de (r) é 0.226. Esses valores são atribuídos às ligações fracas e fortes respectivamente. Nesse trabalho, a ELF foi utilizada como um método complementar para avaliar a ligação hidrogênio-hidrogênio entre os complexos de alcanos estudados.

2.4. MÉTODO HARTREE-FOCK (HF) E TEORIA DA LIGAÇÃO DE VALÊNCIA GENERALIZADA (Generalized Valence Bond – GVB)

A descrição da estrutura eletrônica de átomos e moléculas em termos de orbitais eletrônicos fornece os fundamentos básicos para teorias mais complexas.(Petersson, 1974) A energia obtida à partir de cálculos baseados em orbitais possuem geralmente um erro de cerca de 0,015 a.u. (10 kcal mol-1) para cada elétron. Embora esses erros compreendam uma pequena fração em relação à energia total, eles são bem significativos para os padrões químicos. Faz-se, portanto, necessário uma descrição mais adequada da estrutura eletrônica dos orbitais para se obter dados mais confiáveis.

Uma aproximação comum das funções de onda das moléculas é o método Hartree-Fock,(Hartree, 1947) em que a função de onda é dada como um produto antissimetrizado (determinante de Slater) das funções espaciais e spin. O produto antissimétrico assegura que o princípio da exclusão de Pauli esteja satisfeito.

Assim, para a molécula de H2, a função de onda Hartree-Fock é dada por:

Â[휙(1)훼(1)휙(2)훽(2)] = 휙(1)휙(2)[훼(1)훽(2) − 훽(1)훼(2)] (33) onde Â é o antissimetrizador (operador determinante), 휙 é o orbital espacial duplamente ocupado otimizado, e α e β são as funções spin. Para o H2, o orbital otimizado 휙 é gerado(Goddard e Ladner, 1971) como mostrado na Figura 4A e pode ser expendido como:

휙 = 휒푙 + 휒푟 (34) onde 휒푙 é a função localizada à esquerda do próton e 휒푟 é a função localizada à direta deste. A expansão da parte espacial da função de onda total é dada por:

휙(1)휙(2) = (휒푟휒푟 + 휒푙휒푙) + (휒푟휒푙 + 휒푙휒푟) (35) a equação (35) se aplica a todas as distâncias internucleares R, enquanto que para grandes distâncias, a função de onda toma a forma:

(휒푟휒푙 + 휒푙휒푟) (36) o que significa que existe probabilidade zero de ambos os elétrons estarem simultaneamente próximos no mesmo núcleo. A função de onda Hartree-Fock se comporta incorretamente para altos valores de R, como mostrado na Figura 5, que compara o método Hartree-Fock com a energia exata. 37

A B

Figura 4. Orbital Hartree-Fock da molécula H2 para (A) R = 0,741 Å e (B) R = 6,35 Å. Fonte: (Goddard e Ladner, 1971).

Assim, o método Hartree-Fock fornece uma descrição pobre com relação à quebra e a formação de uma ligação que é intolerável em estudos de reações químicas, pois esse método força ambos os elétrons a ocuparem um único orbital, enquanto, para grandes separações, teremos dois orbitais monocupados. Uma alternativa à esse problema, é o uso da função de onda da ligação de valência (VB) que já foi mostrada na equação (36), o que leva a uma função de onda apropriada para R = ∞. Entretanto, essa função de onde funciona incorretamente para pequenos valores de R (Figura 5).

Figura 5. Energias da molécula H2 obtidas à partir dos métodos Hartree-Fock, MO, VB e GVB, comparadas com a energia exata não-relativística. Fonte: [(Goddard et al., 1973)].

Assim, nenhum dos métodos (Hartree-Fock e VB), leva a uma descrição satisfatória para todos os valores de R.(Goddard et al., 1973) Com a finalidade de contornar esses problemas, é usado o formalismo da função de onda VB, de modo que a mesma se torna:

퐺푉퐵 훹 = (휙푙휙푟 + 휙푟휙푙) (37)

38 essa função de onda é apropriada para altos valores de R, quando resolvida para orbitais utilizando o campo auto-consistente à cada R tal como no método Hartree-Fock. Isso combina as qualidades tanto do método Hartree-Fock quanto do método VB e leva a uma função de onda que se comporta apropriadamente quando os átomos estão separados assim como para pequenos valores de R. Essa abordagem é chamada de ligação de valência generalizada (generalized valence bond – GVB)(Goddard, 1967; Ladner e Goddard, 1969; Goddard e Ladner, 1971; Hunt et al., 1972) e difere do método Hartree-Fock pois agora existem dois orbitais, um para cada elétron, ao invés de um par de elétrons por orbital.

2.5. CÁLCULOS DE QM/MM

Métodos híbridos combinam duas técnicas computacionais em um cálculo e permitem o estudo de grandes sistemas químicos com uma precisão maior que cálculos de mecânica molecular. (Hall et al., 2008; Senn e Thiel, 2009; Wang et al., 2012) O método QM/MM é uma técnica hibrida que combina um método quanto-mecânico (QM) com um método de mecânica molecular (MM).(Warshel e Levitt, 1976; Singh e Kollman, 1986; Field et al., 1990; Maseras e Morokuma, 1995) O ONIOM(Dapprich et al., 1999a) (N-layer Integrated molecular Orbital molecular Mechanics) é um dos métodos de QM/MM mais empregados e permite a combinação de diferentes métodos quanto-mecânicos assim como métodos clássicos de mecânica molecular. No geral, a região do sistema onde o processo químico (interação química ou reação química) ocorre é tratado com um método de maior acurácia (mecânica quântica ou semi-empírico), e essa região é denominada de camada alta (high layer). O restante do sistema é tratado com MM (camada baixa – low layer) (Figura 6). Muitos estudos com o método ONIOM são focados em sistemas biológicos, assim como em proteínas(Lin e O’malley, 2008; Alzate-Morales et al., 2009; Kitisripanya et al., 2011; Ugarte, 2014) onde o sítio ativo é tratado na camada alta, usando frequentemente a teoria funcional da densidade (Density Functional Theory – DFT) e o restante da proteína é tratado como camada baixa, usando frequentemente o campo de força AMBER.

O ONIOM obtém a energia do sistema simulado através da combinação das energias computadas pelos diferentes métodos teóricos,(Dapprich et al., 1999b) em que a camada baixa engloba todo o sistema molecular incluindo a parte pertencente a camada alta. A equação abaixo apresenta, de modo simplificado, as considerações realizadas em um sistema de dupla camada calculado com o método ONIOM (Equação 38):

퐸푂푁퐼푂푀 = 퐸푙표푤(푅) − 퐸ℎ푖푔ℎ(푆푀) − 퐸푙표푤(푆푀) (38)

Onde a energia final do sistema, EONIOM, corresponde a energia calculada da região com o método de maior precisão, Ehigh(SM), e da energia dessa mesma região calculada com um método de menor precisão, Elow(SM), subtraídas da energia do restante do sistema calculado com o método de menor precisão, Elow(R). O termo “SM” refere-se a expressão small, ou seja, do sistema pequeno tratado com o método QM. O termo “R” vem da palavra real e refere-se a todo o sistema.

2.6. SIMULAÇÃO MOLECULAR

A simulação por dinâmica molecular (DM) é uma das técnicas mais versáteis no estudo de macromoléculas biológicas.(Delano, 2002; Alonso et al., 2006) A DM pode ser caracterizada como um método para gerar as trajetórias de um sistema de N partículas por integração numérica direta das equações de movimento Newtonianas(Burkert e Allinger, 1982; Höltje e Folkers, 1996), ou seja, os átomos são tratados como uma coleção de partículas unidas por forças harmônicas ou elásticas. O conjunto completo dos potenciais de interação entre as partículas é referido como “campo de força”. (Brooks et al., 1990; Van Gunsteren e Berendsen, 1990) O campo de força empírico é conhecido como uma função energia potencial e permite que a energia potencial total do sistema, V(r), seja calculada partir da estrutura tridimensional do sistema. V(r) é descrito como a soma de vários termos de energia, incluindo os termos para os átomos ligados (comprimentos e ângulos de ligação, ângulos diedros) e não-ligados (interações de van der Waals e de Coulomb) e assume a forma geral:

푉(푟) = 푉푙푖푔푎푑표 + 푉푛ã표−푙푖푔푎푑표 (39)

1 푉푙푖푔푎푑표 = ∑ 퐾 (푟 − 푟 )2 + ∑ 퐾 (휃 − 휃 )2 + ∑ 푉 [1 − (−1)푛cos (푛휑 + 훾푛) (40) 푟 0 휃 0 2 푛 푙푖푔푎çõ푒푠 â푛푔푢푙표푠 푑푖푒푑푟표

휎 12 휎 6 푞 푞 푛ã표−푙푖푔푎푑표 푖푗 푖푗 푖 푗 푉 = ∑ 4휀푖푗[( ) − ( ) ] + (41) 푟푖푗 푟푖푗 4휋휀0푟푖푗 푖,푗

Vários campos de força são comumente usados em simulações de dinâmica molecular de sistemas biológicos, incluindo o AMBER(Wang et al., 2004), CHARMM(Brooks et al., 1983) e GROMOS.(Christen et al., 2005) Estes diferem principalmente no modo de parametrização, mas geralmente fornecem resultados semelhantes.

2.7. DOCAGEM MOLECULAR

É amplamente aceito que a atividade farmacológica é obtida através da interação de uma molécula (ligante) em um sítio (pocket) de outra molécula (receptor), comumente uma proteína.(Vieth et al., 1998) O processor computacional capaz de predizer a orientação geométrica de um determinado ligante candidato à fármaco dentro do sítio ativo de uma enzima é denominado de docking molecular.(Cavasotto e Orry, 2007) A ideia geral das técnicas de docking é de gerar um leque de conformações do ligante e ordená-las através de um score com base em suas estabilidades com o receptor.(Alonso et al., 2006)

3. CAPÍTULO 1 – Artigo publicado no periódico World Journal of Chemical Education.

Teaching thermodynamic, geometric and electronic aspects of Diels-Alder cycloadditions by using computational chemistry – an undergraduate experiment.

Norberto K. V. Monteiro* and Caio L. Firme

Institute of Chemistry, Federal University of Rio Grande do Norte, Natal-RN, Brazil, CEP 59078-970.

*Corresponding author. Tel: +55-84-3215-3828; e-mail: [email protected]

ABSTRACT

We have applied an undergraduate minicourse of Diels-Alder reaction in Federal University of Rio Grande do Norte. By using computational chemistry tools (Gaussian, Gaussview and Chemcraft) students could build the knowledge by themselves and they could associate important aspects of physical-chemistry with Organic Chemistry. They have performed a very precise G4 method for the quantum calculations in this 15-hour minicourse. Students were taught the basics of Diels-Alder reaction and the influence of frontier orbital theory on kinetics. They learned how to use the quantum chemistry tools and after all calculations they organized and analyzed the results. Afterwards, by means of the PhD student and teacher guidance, students discussed the results and eventually answered the objective and subjective questionnaires to evaluate the minicourse and their learning. We realized that students could understand the influence of substituent effect on the electronic (and kinetic indirectly), geometric and thermodynamic aspects of Diels- Alder reactions. Their results indicated that Diels-Alder reactions are exergonic and exothermic and that although there is an important entropic contribution, there is a linear relation between Gibbs free energy and enthalpy. They learned that electron withdrawing groups in dienophile and electron donating groups in diene favor these reactions kinetically, and on the other hand, the opposite disfavor these reactions. Considering thermodynamically controlled Diels-Alder reactions, the EDG decrease the equilibrium constant, in relation to reference reaction, unlike the EWG. We believe that this minicourse gave an important contribution for Chemistry education at undergraduate level.

GRAPHIC FOR MANUSCRIPT

HOMOdiene ≡

LUMOdienophile

KEYWORDS: Diels-Alder reaction; Diene; Dienophile; Computational Chemistry; G4; Frontier Orbital Theory

INTRODUCTION

Pericyclic reactions[1,2] are one of three major types of reactions, together with polar and radical reactions. Pericyclic reactions can be divided into four classes, cycloadditions, electrocyclic reactions, sigma-tropic rearrangements and the less common group transfer reactions. All these categories share a common feature that is a concerted mechanism involving a cyclic transition state with a concerted movement of electrons. Concerted periclyclic cycloaddition involves the reaction of two unsaturated molecules with resulting in the formation of a cyclic product with two new σ-bonds.[3] Examples of cycloaddition reactions include the 1,3-dipolar cycloaddition (also referred as Huisgen cycloaddition),[4] nitrone-olefin cycloaddition[5] and cycloadditions like the Diels-Alder reaction which is the most frequently encountered and most studied of all pericyclic reactions (Scheme 1). The number of π-electrons involved in the components characterizes the cycloaddition reactions, and for the three abovementioned cases this would be [2+3], [3+2] and [4+2] respectively. 43

In 1950 the Nobel Prize of chemistry was granted to Otto Diels and Kurt Alder by describing an important reaction that involves the formation of C-C bond in a six membered ring with high stereochemical control, named Diels-Alder reaction.[6] Since 1928, the Diels-Alder reaction has been used in the synthesis of new organic compounds, providing a reasonable way for forming a 6-membered systems.[7–13]

The simplest Diels-Alder reaction involves the reagents butadiene and ethane (dienophile), but it needed 200°C under high pressure although it has low yield[14] (about 18%) (Scheme 1A). Butadiene reacts more easily with acrolein[15], (Scheme 1B). A methyl substituent on the diene, on C1 or C2, the reaction rate increases, e.g. trans-piperylene reacting with acrolein[16,17] at 130°C (Scheme 1C). These last two reactions achieved a yield Close to 80%.

Scheme 1

The standard state thermodynamic functions for Diels-Alder reactions are represented by enthalpy change, Gibbs free energy change and entropy. The exothermic nature of these reactions are the result of converting two weaker π-bonds into two stronger σ-bonds.[18] Because it is an addition reaction it has negative entropy change but Diels-Alder reactions are exergonic due to the high negative entalphy change. 44

The reactivity of Diels-Alder reactions can be reasoned by molecular orbital (MO) theory[19] in which the HOMO of diene donates electrons to the LUMO of the dienophile by overlapping these orbitals. The addition of an EDG to the diene results in a better electron donor (and a better nucleophile) by increasing its HOMO energy. Likewise, a EWG-substituted dienophile generates a better electron acceptor (and a better electrophile) by decreasing its LUMO energy.[20] As a consequence, the frontier molecular orbital theory accounts for the dependence of the Diels-Alder reaction rate on the substituent effect.[21] In the transition state of the Diels-Alder reactions, the HOMO of diene interacts with LUMO of dienophile. The Scheme 2A shows the energy diagram of butadiene and ethene. The Scheme 2B shows the energy diagram of butadiene and a substituted dienophile with an EWG whose energy gap is lower than that in Scheme 2A. Even lower energy gap is reached when an EDG is attached to the C1 or C2 of diene (Scheme 2C). It is well-known that the lower the energy gap between HOMO and LUMO from diene and dienophile, respectively, the faster the corresponding reaction rate.

Scheme 2

LUMO

LUMO LUMO LUMO LUMO

LUMO HOMO HOMO HOMO HOMO HOMO HOMO

(A) (B) (C)

One great challenge for chemistry education research is to improve students' understanding of physical chemical properties of chemical reactions. One very important tool for chemistry education practice is computational chemistry which improves the understanding of microscopic properties and/or provides macroscopic properties more easily. Several successful chemistry education papers focused on using computational chemistry as the main tool for the study of organic reactions. Hessley[22] used the molecular modeling to study reaction mechanisms for the formation of alkyl halides and alcohols in order to prevent the students from memorizing these reactions. Other works used computational chemistry for obtaining the Hammett plots;[23] for studying the controversial Wittig reaction mechanism;[24] for analyzing the origin of solvent effects on the rates of organic reactions;[25] and for investigating the kinetic and thermodynamic aspects of endo vs exo selectivity in the Diels-Alder reactions.[26] However, except for ref. 26 none of these studies have associated the aspects of physical-chemistry of organic reactions of these reactions. 45

In this work, we used the computational chemistry tool in an undergraduate minicourse in order to provide chemistry undergraduate students from Federal University of Rio Grande do Norte better understanding of the influence of substituent groups on the kinetic and thermodynamic properties of Diels- Alder reactions. All results from this minicourse were obtained by the students. Due to relatively short time of the minicourse, each student or pair of students calculated just one reaction out of seven selected reactions and afterwards all results were collected and analyzed as it is presented in this work.

COMPUTATIONAL DETAILS

Seven Diels-Alder reactions were previously selected and the geometries of all studied species were optimized by using standard techniques.[27] Frequency calculations were performed to analyze vibrational modes of the optimized geometry in order to determine whether the resulting geometries are true minima or transition states. The geometry optimization, frequency calculations and the generation of the HOMO and LUMO molecular orbitals of all studied species were calculated by G4 method[28] from Gaussian 09 package.[29] Every molecule´s initial geometry was done with GaussView 5[30] and final geometry was taken from Chemcraft.[31] The G4-based computational method has been successfully used in the study of pericyclic reactions before.[32]

DESCRIPTION OF THE COMPUTATIONAL CHEMISTRY MINICOURSE

We offered a 15-hour minicourse (divided into 3 hours per day) in July, 2015. Ten students participated in the minicourse and according to pre-minicourse questionnaire (see Supporting Information) all participants were undergraduate chemistry students in different grades from Institute of Chemistry of Federal University of Rio Grande do Norte. All students have completed Organic Chemistry 1 which is the main prerequisite for this minicourse. About 50% of students did not have any experience with computational chemistry. This minicourse was part of a teaching project in the Institute of Chemistry in which the first week at the beginning of the every semester is devoted to a series of minicourses where all chemistry students must attend and can choose the minicourse(s) that is (are) most convenient. This minicourse is taught once a year at the beginning of the first or second semester. This project can be part of a regular Organic Chemistry discipline which includes Diels-Alder reaction or it can be similarly given as minicourse. All data shown in this paper were provided by the students from calculations performed in the minicourse. Some time-consuming calculations have lasted nearly one day.

In the first class, the students received a pre-minicourse questionnaire with short-answer questions (Supporting information) in order to obtain student's academic background prior to the minicourse; After 46 pre-minicourse questionnaire, the students were taught the theoretical content about the Diels-Alder reactions; the thermodynamic and kinetic properties for analyzing the Diels-Alder reactions; and they were presented Gaussian 09, GaussView 5 and Chemcraft software. In the second class, the students were grouped (four pairs of students) and each group or individual student performed the optimization and frequency calculations of the reagents and products of one selected Diels-Alder reaction shown in Scheme 2 (The G4 calculations occurred during the second class and in the interval between the second and third classes). In the third class the students analyzed all thermodynamic data in order to obtain student-generated data in Tables 1-2 and the plots in Figures 1 and 3. In the first half of fourth class the students generated the HOMO and LUMO molecular orbitals in order to obtain similar MO representations as shown in Figure 2 and to get their energy values and corresponding energy gaps as it is shown in Table 3. In the second half of fourth class the students analyzed the bond lengths of all reagents and products of the Diels-Alder reactions in order to obtain data in the Table 4 and Figure 4. In the fifth class, all the results were discussed as presented in sections 5 to 8 and the students answered the post-minicourse questionnaire (objective and subjective questions) and the evaluation which are shown in Supporting Information. Eventually, a statistics (in Supporting Information) were generated with the rates for answers given by the students for objective questions and student’s evaluation. Three filled post-minicourse subjective questionnaire forms are shown in Supporting Information.

RESULTS

The Scheme 3 shows all studied Diels-Alder reactions where R1 and R2 represent the substituents of the diene and dienophile respectively and the products cyclohexene derivatives formed which resulted in seven

Diels-Alder reactions. We used two substituents: one electron donating group (EDG) (-NH2) and one electron withdrawing group (EWG) (-NO2). 47

Scheme 3

1. R1 = R2 = H;

2. R1 = NH2 and R2 = H

3. R1 = H and R2 = NO2

4. R1 = NH2 and R2 = NO2

5. R1 = NO2 and R2 = H

6. R1 = H and R2 = NH2

7. R1 = NO2 and R2 = NH2

In Table 1 it is depicted the absolute values of enthalpy (H), Gibbs free energy (G) and entropy (S) of all reagents (dienes and dienophiles) and products (cyclohexene and cyclohexene derivatives) and the corresponding values of enthalpy change (∆H), Gibbs free energy change (∆G), temperature and entropy change product (T∆S) and logarithm of equilibrium constant (ln K) for Diels-Alder reactions (1)-(7) are shown in Table 2. In Figure 1A it is shown the linear relation involving ∆H and ∆G for Diels-Alder reactions (1)-(7) whereas in Figure 1B there is the linear relation of ln K and ∆H for Diels-Alder reactions (1)–(7).

Figure 2 depicts the HOMO of buta-1,3-diene, buta-1,3-dien-2-amine and 2-nitrobuta-1,3-diene and LUMO of ethene, nitroethene and etheneamine (Figure 2A, 2B and 2C respectively). The energy values of HOMO and LUMO of the reagents of the Diels-Alder reactions (1)–(7), along with the corresponding energy gap values, are shown in Table 3.

-1 ΔH (kcal mol ) ln K -52.5 -50 -47.5 -45 -42.5 -40 -37.5 -35 35.0 40.0 45.0 50.0 55.0 60.0 65.0 70.0 -20 -30 -22

-24 Δ -35 G (kcal (kcal G mol

R² = 0.9839 ) 1 -26 - R² = 0.9839 -28 -40

-30 -

-32 1 -45

) H (kcal mol (kcal H

-34 Δ -36 -50 -38 -40 -55 A B

Figure 1. (A) Plot of enthalpy change, ∆H, versus Gibbs free energy change, ∆G, of Diels-Alder reactions (1)-(7). (B) Plot of ln K versus ∆H of Diels-Alder reactions (1)-(7).

Table 1. Values of enthalpy, H, Gibbs free energy, G, in kcal mol-1, and entropy, S, in cal mol-1 K-1 of all studied molecules.

Molecule G/kcal mol-1 H/kcal mol-1 S/cal mol-1 K-1

Ethene -49286.36 -49270.75 52.31 Ethenamine -84010.42 -83991.90 62.02 nitroethene -177588.33 -177567.34 70.25 Buta-1,3-diene -97825.36 -97806.25 63.98 Buta-1,3-dien-1-amine -132549.78 -132527.46 74.63 1-nitrobuta-1,3-diene -226121.56 -226098.50 77.14 Cyclohexene -147141.70 -147119.59 73.92 Cyclohex-2-en-1-amine -181859.99 -181835.93 80.38 Cyclohex-3-en-1-amine -181861.41 -181837.44 80.09 4-nitrocyclohexene -275447.53 -275420.66 89.78 3-nitrocyclohexene -275446.12 -275419.62 88.53 6-nitrocyclohex-2-en-1-amine -310164.56 -310136.10 95.04 2-nitrocyclohex-3-en-1-amine -310164.17 -310135.86 94.52 49

Reaction ∆G/kcal mol-1 ∆H/kcal mol-1 T∆S/kcal mol-1 ∆S/kcal mol-1 K-1 ln K

(1) Buta-1,3-diene + Ethene  Cyclohexene -29.98 -42.59 -12.63 -0.0424 50.61 (2) Buta-1,3-dien-1-amine + Ethene  Cyclohex-2- -23.85 -37.72 -13.88 -0.0466 40.26 en-1-amine (3) Buta-1,3-diene + Nitroethene  4- -33.84 -47.07 -13.25 -0.0445 57.12 nitrocyclohexene (4) Buta-1,3-dien-1-amine + Nitroethene  6- -26.45 -41.29 -14.86 -0.0498 44.65 nitrocyclohex-2-en-1-amine (5) 1-nitrobuta-1,3-diene + Ethene  3- -38.21 -50.37 -12.20 -0.0409 64.49 nitrocyclohexene (6) Buta-1,3-diene + Ethenamine  Cyclohex-3-en- -25.63 -39.29 -13.69 -0.0459 43.26 1-amine (7) 1-nitrobuta-1,3-diene + Ethenamine  2- -32.19 -45.46 -13.31 -0.0446 54.34 nitrocyclohex-3-en-1-amine 50

A B C

Figure 2. HOMO and LUMO of buta-1,3-diene and ethene (A); buta-1,3-dien-2-amine and nitroethene (B); and 2-nitrobuta-1,3-diene and ethenamine (C).

Table 3. Energy values of HOMO (EHOMO) and LUMO (ELUMO) and energy gap between HOMO and

LUMO orbitals, |EHOMO-ELUMO|, in a.u., of the reagents of the Diels-Alder reactions (1) – (7).

Reaction EHOMO/a.u. ELUMO/a.u. |EHOMO-ELUMO|/a.u.

(1) -0.2293 0.0173 0.2466 (2) -0.1867 0.0173 0.2040 (3) -0.2293 -0.0923 0.1370

(4) -0.1867 -0.0923 0.0944

(5) -0.2522 0.0173 0.2695 (6) -0.2293 0.0509 0.2801 (7) -0.2522 0.0509 0.3031

Figure 3 shows the C-C bond length of the diene and C-C bond length of the dienophile of reactions

(4) (Figure 3A) and (7) (Figure 3B) in which these bond lengths are represented in the Scheme 4 (r1, r2, r3, r4, r’1, r’2, r’3 and r’4). The values of C-C bond length for Diels-Alder reactions (1)–(7) are listed in Table 4.

Scheme 4

Figure 3. Optimized geometries of corresponding reactants and products from Diels-Alder reactions 4 (A) and 7 (B). Selected bond lengths, in Angstroms, are depicted.

Table 4. Bond lengths of reagents (r1, r2, r3 and r4) and products (r’1, r’2, r’3 and r’4), in Angstroms, of the studied Diels-Alder reactions.

Reaction r1/Å r2/Å r3/Å r4/Å r’1/Å r’2/Å r’3/Å r’4/Å

(1) 1.336 1.468 1.336 1.327 1.507 1.333 1.507 1.534 (2) 1.345 1.458 1.339 1.327 1.510 1.332 1.506 1.532 (3) 1.336 1.468 1.336 1.323 1.508 1.332 1.506 1.532 (4) 1.345 1.458 1.339 1.323 1.509 1.334 1.504 1.524 (5) 1.330 1.465 1.335 1.327 1.506 1.333 1.504 1.533 (6) 1.336 1.468 1.336 1.335 1.507 1.332 1.507 1.532 (7) 1.330 1.465 1.335 1.335 1.506 1.333 1.504 1.540

THERMODYNAMIC DATA AND EQUILIBRIUM CONSTANT

Enthalpy change values from Table 2 indicate that all reactions are exothermic because two weaker  bonds in reactants are converted to two stronger  bonds in the product. Cycloaddition reactions have negative entropy change owing to the smaller translational and rotational contributions of entropy in cyclohexene adducts compared to those values from reactants (diene and dienophile). The negative entropy does not contribute to the spontaneity of Diels-Alder reacdrptions. However, according to Gibbs free energy change values in Table 2 these reactions are exergonic due to low entropy change values in modulus (see Table 2) and high enthalpy change values in modulus and one can see that the modulus of TS is nearly three times smaller than the modulus of H, in average. Experimental data show that Diels-Alder reactions are exothermic and exergonic, e.g., the reaction of buta-1,3-diene and ethene to form cyclohexene[33] which has an H and G of -40.0 and -27.0 kcal mol-1 respectively; the reaction of cyclopentadiene with itself to form dicyclopentadiene (H and G of -18.4 and -8.2 kcal mol-1 respectively);[34] and the reaction of cyclopentadiene and maleic anhydride (H = -24.8 kcal mol-1).[35] Although the reactions are not the same as those students have calculated, their thermodynamic data is within the same range.

The tendency for a reaction to reach equilibrium at the standard conditions is driven by the standard Gibbs free energy, ΔG°, which is related to the equilibrium constant[36] according to Eq. 1.

∆퐺° = −푅푇푙푛퐾 (1)

There is no relationship between the kinetics and thermodynamics for most of chemical reactions, except for some electron transfer and some radical reactions.[37–39] As previously described, an EDG attached to the diene and a dienophile substituted with a EWG result in an increase in the rate of a Diels- 53

Alder reaction. However, these effects are not observed for the thermodynamic properties as shown in Table 2. The EDG (amino group) thermodynamically disfavors the Diels-Alder reactions (2), (4) and (6) resulting in less exothermic and exergonic reactions and hence lower ln K values compared to the reaction (1). On the other hand, the reactions with EWG (nitro group) substituent [reactions (3), (5) and (7)] have more negative values of ∆G and ∆H and higher values of ln K which thermodynamically favor these reactions.

The plot from Figure 1A has a very good correlation coefficient indicating that the trend in enthalpy change for formation of cyclohexene and derivatives is nearly the same as that from corresponding Gibbs free energy change, which means that entropy change for these cycloaddition reactions does not alter significantly both trends, though TS values are not negligible (see Table 2).

Since G for all reactions is negative, there is a direct relation between modulus of G and equilibrium constant, K, according to expressions in Eq. 2.

∆퐺 (−∆퐺) ∆퐺 − − 퐾 = 푒 푅푇 ∴ ∆퐺 < 0 ∴ 퐾 = 푒 푅푇 ∴ 퐾 = 푒푅푇 (2)

Then, for exergonic Diels-Alder reactions, there is a direct relation between K and |ΔG°|, i.e., the smaller ΔG° (in modulus) the smaller K. Moreover, there is direct relation between K and reaction yield.

The very good correlation coefficient of the Figure 1B indicates that ΔH can be associated with the K as well. Thus, we can associate K with stability factors instead of exergonicity. Then, there is a direct relation between modulus of ΔH (in modulus) and K.

The data in Table 2 indicate that reaction (5) has the highest K value and hence the highest negative values of ΔG and ΔH. On the other hand, the reactions (2) and (6) have the lowest K values and the lowest negative values of ΔG and ΔH. The direct proportionality between K and |ΔG| and between K and |ΔH| can be observed in the decreasing order of K, |ΔG| and |ΔH|: (5) > (3) > (7) > (1) > (4) > (6) > (2).

YIELD AND KINETIC PROPERTIES

The efficiency of a chemical reaction is usually related to reaction yield. In a thermodynamically controlled reaction, the reaction yield depends on the equilibrium constant and not on the reaction rate. In a kinetically controlled reaction, yield depends on the reaction rate and the reaction time. In Diels-Alder reactions, according to frontier molecular orbital theory, the electron-rich diene acts as a nucleophile. The most important orbital in the diene is the Highest Occupied Molecular Orbital (HOMO). For the formation of chemical bond, this orbital interacts with the Lowest Unoccupied Molecular Orbital (LUMO) of the dienophile, which acts as an electrophile. As most Diels-Alder reactions are kinetically controlled, the 54 smaller the HOMO-LUMO energy difference, the greater the charge transfer, the faster the reaction and the higher its yield.2 The energy diagram (Figure 2A) shows the interaction between the HOMO of the diene and the LUMO of the dienophile. The energies of HOMO and LUMO depend on the nature of the substituents in the diene and in the dienophile, in which an EWG decreases the LUMO energy of the dienophile (Figure 2B) bringing it closer to the HOMO energy of the diene. An EDG-substituted diene (Figure 2C) further decreases the energy gap between HOMO and LUMO. Thus, EWG-substituted dienophiles and EDG-substituted dienes result in faster reactions and higher yields in comparison with unsubstituted dienophiles and dienes.

The Table 3 shows the energy values of HOMO (EHOMO) and LUMO (ELUMO) of the reagents of the

Diels-Alder reactions (1)-(7), and the energy gap between HOMO and LUMO (|EHOMO-ELUMO|). The reference reaction (1) has 0.2466 a.u. energy gap. The reactions with EDG in diene [reaction (2)] or EWG in dienophile [reaction (3)] have lower energy gaps as expected and additive effect of both EDG in diene and EWG in dienophile [reaction (4)] imparts further decrease in energy gap. From the analysis of energy gaps in Table 3 and assuming that frontier molecular orbital theory can indicate relatively faster or slower reactions, we could predict lower energy barriers for reactions (2), (3) and (4) which would provide higher yields for these reactions in comparison with reaction (1). On the other hand, reactions (5), (6) and (7) have the highest |HOMO-LUMO| values (0.2695, 0.2801 and 0.3031 a.u. respectively) due to inverse electron demand, i.e. EWG-substituted dienes and/or EDG-substituted dienophiles. Since these reactions have the highest energy gap, they are the slowest reactions. Therefore, the decreasing order of |HOMO-LUMO| is: (7) > (6) > (5) > (1) > (2) > (3) > (4).

Figure 2 shows the HOMO of buta-1,3-diene, buta-1,3-dien-2-amine and 2-nitrobuta-1,3-diene and LUMO of ethene, nitroethene and etheneamine (Figures 2A, 2B and 2C respectively). According to Figures 2B (HOMO of buta-1,3-dien-2-amine and LUMO of nitroethene) and 2C (HOMO of 2-nitrobuta-1,3-diene and LUMO of etheneamine) some changes occur in the size and shape of their corresponding HOMO and LUMO in comparison with those orbitals of their parent molecules depicted in Figure 2A (HOMO of buta-

1,3-diene and LUMO of ethene). When an EDG (-NH3) is attached to the diene, its HOMO includes the nitrogen lone pair participation resulting in a better nucleophile while LUMO in nitroethene includes oxygen lone pair participation and –NO2 vicinal carbon orbital distortion towards nitrogen atom (Figure 2B). On the other hand, HOMO of 2-nitrobuta-1,3-diene (Figure 2C) has no nitrogen lone pair participation and 2- nitrobuta-1,3-diene is probably a weaker nucleophile than buta-1,3-dien-2-amine.

GEOMETRIC CHANGES

Scheme 4 represents a general Diels-Alder reaction in which selected bond lengths of reactants and product are assigned (r1, r2, r3 for dienes, r4 for dienophiles, and r’1, r’2, r’3, r’4 for cyclohexenes). Figure 3 depicts the optimized geometries of reactants and product of reactions (4) (Figure 3A) and (7) (Figure 3B). 55

Surprisingly, data from Table 4 indicate that amino group increases the double bond length of reactants more than nitro group. Moreover, data from Table 4 also confirm that double bonds r1, r3 and r4 in reactants become single bonds in products while the single bond r2 becomes double bond in products.

Except for r’4 in reactions (4) and (7), no other selected bond length of all cyclohexene derivatives is significantly affected by substituent group. Then, the variations of enthalpy change in the reactions (1) to (7) are probably accounted for by the reactants than the product. Figure 3 also shows that cyclohexane derivatives lost the carbon chain planarity from their precursors due to the lost of  bonds in the product.

CONCLUSIONS

The computational chemistry allowed to teach the thermodynamic, electronic (indirectly kinetic) and geometric aspects of Diels-Alder reactions. All of these reactions were exergonic and exothermic. Entropy change has an important contribution on the Gibbs free energy change, however it did not affect the linear relation between the latter and enthalpy change. Then, the thermodynamics of these reactions can be reasoned by the influence of electronic effects on the stability of reactants and product. However, from the analysis of geometrical parameters, stability is more affected by reactants than by products because the bond lengths of reactants vary more significantly in comparison with reference compound due to the substituent effect than those in the corresponding product.

Students could also understand that substituent effects on thermodynamics and kinetics (indirectly from frontier orbital theory) are distinguished. The EDG in diene makes Diels-Alder reaction less favorable thermodynamically, whereas EWG substituent in diene results in more favorable reaction in thermodynamics. On the other hand, EDG in diene and EWG in dienophile favor the Diels-Alder reaction kinetically.

From the statistics of post-minicourse questionnaire and students evaluation (Supporting information) of these minicourse we realized that students could better understand thermodynamic and kinetic aspects of a reaction from the use of computational chemistry. This minicourse also contributed for the Organic Chemistry knowledge by giving students practical and simple tools to develop the knowledge by themselves, in which these tools can also be employed in the teaching of Diels-Alder reactions in the classroom. This minicourse also contributed for the association between physical-chemistry knowledge (quantum chemistry, thermodynamics and kinetics, indirectly) with organic chemistry by means of Diels- Alder reaction.

ASSOCIATED CONTENT

Supporting Information 56

Pre-minicourse, Post-minicourse questionnaires (objective and subjective questions) and students evaluation. This material is available via the Internet at http://www.sciepub.com/

ACKNOWLEDGMENTS

Authors thank Fundação de Apoio à Pesquisa do Estado do Rio Grande do Norte (FAPERN), Coordenação de Ensino de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) and Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) for financial support.

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Supporting Information

for

Teaching aspects of organic chemistry reaction from Diels-Alder cycloadditions by using computational chemistry – an undergraduation experiment.

Norberto K. V. Monteiro* and Caio L. Firme

Institute of Chemistry, Federal University of Rio Grande do Norte, Natal-RN, Brazil, CEP 59078-970.

*Corresponding author. Tel: +55-84-3215-3828; e-mail: [email protected]

PRE-MINICOURSE QUESTIONNAIRE

Teaching thermodynamic, geometric and electronic aspects of Diels-Alder cycloadditions by using computational chemistry – an undergraduate experiment.

FEDERAL UNIVERSITY OF RIO GRANDE DO NORTE

CHEMISTRY INSTITUTE

STUDENT´S NAME:______DATE:______

1) What course are you majoring in? ______

2) What is your grade? ______

3) Which organic chemistry disciplines have you done before? ______60

4) Do you have any previous experience in computational chemistry? ______

POST-MINICOURSE QUESTIONNAIRE – OBJECTIVE QUESTIONS.

1. Did this minicourse help you to understand Diels-Alder reaction? (a) Very much; (b) Considerably;

(c)Moderately; (d) Hardly ever

2. Did this minicourse help you to understand the frontier orbital theory? (a) Very much; (b)

Considerably; (c)Moderately; (d) Hardly ever

3. Did this minicourse help you to understand the relation between frontier orbital theory, kinetic

approach and yield? (a) Very much; (b) Considerably; (c)Moderately; (d) Hardly ever

4. Did this minicourse help you to understand the relation between thermodynamic approach and yield?

(a) Very much; (b) Considerably; (c)Moderately; (d) Hardly ever

5. How do you evaluate the importance of computational chemistry for understanding chemical

reactions? (a) Very important; (b) Important; (c) Moderately important; (d) Not important

STUDENTS EVALUATION

Grades (1-minimum to 5-maximum) that correspond to students opinion about features of the Diels-Alder minicourse. Features Grade (% of students)

1 2 3 4 5

I had the prerequisite knowledge to complete

this minicourse? Depth of content Backgrounds need 61

Visually attractive Multidisciplinary

Easy to be performed

The objectives of this minicourse were clear?

The computational tool really helped in

understanding?

POST-MINICOURSE QUESTIONNAIRE – SUBJECTIVE QUESTIONS.

Teaching thermodynamic, geometric and electronic aspects of Diels-Alder cycloadditions by using computational chemistry – an undergraduate experiment.

FEDERAL UNIVERSITY OF RIO GRANDE DO NORTE CHEMISTRY INSTITUTE

STUDENT´S NAME:______DATE:______

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4. CAPÍTULO 2 – Artigo publicado no periódico The Journal of Physical Chemistry A.

Monteiro, N. K. V.; Firme, C. L. The Journal of Physical Chemistry A 2014, 118, 1730.

Hydrogen-hydrogen bonds in highly branched alkanes and in alkane complexes: a DFT, ab initio, QTAIM and ELF study

Norberto K. V. Monteiro and Caio L. Firme*

Institute of Chemistry, Federal University of Rio Grande do Norte, Natal-RN, Brazil, CEP 59078-970.

*E-mail: [email protected]

ABSTRACT

The hydrogen-hydrogen (H-H) bond or hydrogen-hydrogen bonding is formed by the interaction between a pair of identical or similar hydrogen atoms that are close to electrical neutrality and it yields a stabilizing contribution to the overall molecular energy. This work provides new, important information regarding hydrogen-hydrogen bonds. We report that stability of alkane complexes and boiling point of alkanes are directly related to H-H bond, which means that intermolecular interactions between alkane chains are directional H-H bond, not nondirectional induced dipole-induced dipole. Moreover, we show the existence of intramolecular H-H bonds in highly branched alkanes playing a secondary role on their increased stabilities in comparison with linear or less branched isomers. These results were accomplished by different approaches: density functional theory (DFT), ab initio, quantum theory of atoms in molecules (QTAIM) and electron localization function (ELF).

Keywords: hydrogen-hydrogen bond; H-H bond, hydrogen-hydrogen bonding; alkane; QTAIM; ELF.

INTRODUCTION

Bader defined bond path as an atomic interaction line (AIL) which is mirrored by its corresponding virial path, where the potential energy density is maximally negative1,2. Bader3,4 has stated that bond paths 71 are indicative of bonded interactions, which encompass all kinds of chemical interactions. Bond paths or virial paths can be used to characterize four types of bonded interactions involving hydrogen atom: dihydrogen bond, hydrogen bond, hydride bond and hydrogen-hydrogen bond.

The dihydrogen bond is a bonded interaction consisting of electrostatic interaction between two hydrogen atoms of opposite charges and its formation is restricted to systems that possess a hydridic hydrogen wherein transition metals and boron atom are typical elements that can interact with this hydridic hydrogen5. On the other hand, the hydrogen bond involves only one hydrogen atom and it can be represented by X-H•••Y interaction, where X and Y are electronegative atoms; X-H is named as a proton- donating bond, and Y is an acceptor center.6

An unconventional type of hydrogen bond is named hydride bond, also known as “inverse hydrogen bond”. In the hydride bond, a negatively charged hydrogen atom will be electron donor for non-hydrogen acceptors. The hydride bond consists of a hydride placed between two electrophilic centers.7,8

Recently,9 a new kind of interaction involving hydrogen atoms has been discovered by means of the quantum theory of atoms in molecule (QTAIM): the hydrogen-hydrogen (H-H) bond or hydrogen-hydrogen bonding, wherein a bond path links a pair of identical or similar hydrogen atoms that are close to electrical neutrality, opposing the dihydrogen bonding with positively and negatively charged hydrogens.5,9 It is demonstrated that these interactions are found to link hydrogen atoms bonded to both unsaturated and saturated carbon atoms in hydrocarbon molecules and contribute to molecular stabilization, where each H-H bond makes a stabilizing contribution of up to 10 kcal/mol to the energy of the molecule.9 These interactions are found between the ortho-hydrogen atoms in planar biphenyl, between the hydrogen atoms bonded to the C1-C4 carbon atoms in phenanthrene and other angular polybenzenoids, and between the methyl hydrogen atoms in the cyclobutadiene, tetrahedrane and indacene molecules corseted with tertiary-tetra-butyl groups.9

Some experimental10–14 and computational15–18 studies exemplify interactions between two hydrogen atoms, although not differentiating H-H bond from dihydrogen bond. Nonetheless, the interactions cited in these works occur between similar or identical hydrogen atoms close to electrical neutrality. Some studies used the term “non-bonding repulsive interaction” to describe the H-H bond.

The H-H bond in planar biphenyl was questioned.19 However. no other studied molecular system with hydrogen-hydrogen bond was questioned to the date. Bader20 argued that there is no repulsion in rotamers such as ethane or biphenyl, only decrease of nuclear attraction. In reply21 to Bader´s statements, it was used an arbitrary set of MO´s (which is obviously not univocal22). Nonetheless, QTAIM is dependent upon the electron density which, in turn, arises from a variational calculation and it represents “the distribution that minimizes the energy for any geometry and thus the presence of bond paths represent spatial accumulation of density that lower the energy of the system”, as stated by Bader.20 Then, bond path 72 is an universal indicator of bonded interactions.3 Accordingly, QTAIM has become a method that has already been successfully applied to understand conventional hydrogen bonds,23,24 C-H•••O bonds25 and dihydrogen bonds.26,27

In this work, we show that there are very good linear relations between H-H bond in alkane complexes and their stabilities and between H-H bond in alkane complexes and their boiling points. We also demonstrate that there are H-H bonds in highly branched alkanes and we show that the H-H bonds play a secondary role in the stability of highly branched alkanes. All results in this work were obtained by means of different approaches: the topology of the virial field function based on QTAIM; the critical points of electron localization function (ELF); ab initio and density functional theory.

COMPUTATIONAL DETAILS AND THEORETICAL METHODS

The geometries of the studied species were optimized by using standard techniques.28 Vibrational modes of the optimized geometry were calculated in order to determine whether the resulting geometries are true minima or transition states. Calculations were performed at ωB97XD29, G430, G4/CCSD(T),31 M062X32 and B3LYP33,34 methods with the 6-311++G(d,p) basis set by using Gaussian 09 package,35 including the electron density which was further used for QTAIM calculations. The 6-311++G(d,p) basis set was applied because the inclusion of the diffuse components in the basis set is desired to describe properly the hydrogen- hydrogen bond36. Among the density functionals accounting for dispersion, ωB97XD37,38 contains empirical dispersion terms and also long-range corrections, exhibiting a van der Waals correction. All topological information were calculated by means of AIM2000 software39. The algorithm of AIM2000 for searching critical points is based on Newton-Raphson method which relies heavily on the chosen starting point.40 Every available alternative for increasing calculation accuracy and choosing different starting points was done. Integrations of the atomic basins were calculated in natural coordinates in default options of integration. All integrations yielded 10-3 to 10-4 order of magnitude for the Laplacian of the calculated atomic basin. Atomic energies were calculated from the atomic virial approach. All calculated bond paths are mirrored by their corresponding virial paths (where maximum negative potential energy exists between two atoms). According to atomic virial approach, the bond paths mirrored by virial paths are indicative of bonded interaction. The evaluation of the electron localization function (ELF) values, (r), and topological information of bond critical point (BCP) have been carried out with the TopMoD program.41,42 For each alkane complex, the QTAIM coordinates of bond critical point of a selected H-H bond were used to locate the corresponding ELF BCP. All ELF BCP ´s were confirmed as an index 1 critical point since their 24 corresponding [r(QTAIM BCP coordinates)] is 0.0000. Fuster and Grabowski obtained a 0.9935 coefficient of determination involving ELF H--- BCP distance and QTAIM H--- BCP distance, which reinforces our approach for finding ELF bond critical point of H-H bond from QTAIM coordinates. 73

QTAIM AND VAN DER WAALS INTERACTIONS

The QTAIM is an extension of quantum mechanics to an open system which is made up of atomic basins in a molecule and it can be developed by deriving Heisenberg´s equations of motion from Schrödinger´s equation wherein any atomic property in a molecule can be calculated.43

The quantum theory of atoms in molecules was developed by Bader to study the electronic structure and especially the chemical bond by trespassing the limits of orbital theories when approaching to the physics underlying atoms in a molecule1,2. This quantum model is considered innovative in the study of chemical bonding and characterization of intra- or intermolecular interactions. 44,45 According to the QTAIM quantum-mechanic concepts,46,47 the calculation of gradient of charge density, ∇ρ, of the studied molecular system is performed through the electron density's Hessian matrix in order to find critical points of the charge density.48 The critical points of the charge density are obtained where ∇ρ = 0. The bond critical point is a saddle point on an electron density curvature being a minimum in the direction of the atomic interaction line and a maximum in the two directions perpendicular to it.49–51 The BCP’s give information about the character of different chemical bonds or chemical interactions, for example, metal-ligand interactions, 49,52,53 covalent bonds1,2,54, different kinds of hydrogen bonds,24,55,56 including dihydrogen bonds26,57,58 and H-H bond.36,59 The Laplacian of the charge density (∇2ρ) is obtained from the sum of the three eigenvalues of the 1,2 2 2 Hessian matrix (λ1, λ2 and λ3) of the charge density . The negative (∇ ρ < 0) or positive (∇ ρ > 0) sign of the Laplacian represents concentration and depletion of the charge density, respectively60.

The H-H bonds exhibit characteristics of closed-shell interaction since they have: low value of -2 2 charge density (ρb) (< 6 x 10 au); the ratio │λ1│ / λ3 < 1; ∇ ρ>0; and the positive value for the total energy density Hb that is close to zero. The H-H bond presents BCP topological values similar to those from weak hydrogen bonds61–63.

All properties of matter are somewhat related to its corresponding density of electronic charge ρ(r), where one can associate to its topology delineating atoms and the bonding between them.9 In molecules, the nuclei of bonded atoms are linked by a line along which the electron density is a maximum with respect to any neighboring gradient path and it is termed bond path.9 In truth, according to Bader, a bond path has to be mirrored by its corresponding virial path, otherwise it cannot be regarded as a bond path. In a virial path the potential energy density is maximally stabilizing.64 When the molecular system is also in a stable electrostatic equilibrium, it ensures both a necessary and a sufficient condition for bonding.65 Bader also defined bond path as a measured consequence of the action of the density operator along the trajectories that originate in bond critical point and end in both interacting nuclei.66 The bond path is based on theorems of quantum mechanics that govern the interactions between atoms. Only two forces operate in a molecular system: Feynman and Ehrenfest, which act on the nuclei and electrons, respectively. They both are united in the virial theorem,67 which relates these forces to the energy of the molecule.68 This is the essential physics 74 of all bonded interactions that when combined with the properties of the bond critical point, provides all the features of a chemical bond.

Many studies involving critical points and bond paths in van der Waals and weak interactions were performed.69–71 The van der Waals interactions were identified as directional bonded interactions because of the existence of bond paths in solid chloride and N4S4 which are important to understand the structure of these crystals. Hydrogen-hydrogen bond is also a directional interaction, different from induced dipole- induced dipole interaction which is not directional. Wolstenholme and Cameron72 showed that crystal structures of tetraphenylphosphonium squarate, bianthrone, and bis (benzophenone) azine contains a variety δ+ +δ of C-H ••• H-C interactions, as well as a variety of C-H•••O and C-H•••Cπ interactions. This study established similarities and differences between these weak interactions, leading to a better understanding of H-H bonds. Slater has shown in his paper73 that there are no fundamental differences between van der Waals and covalent bonding, because they have the same quantum mechanism.74 According to Feynman,75 the origin of chemical bond is derived from electron density between nuclei, at the same time the requirement for occurrence of bond path.

ELF

The electron localization function introduced by Becke and Edgecombe76 is measured from the Fermi hole curvature in terms of the local excess kinetic energy density due to Pauli repulsion. It has been shown that ELF is confined to the [1,0] interval, where the tendency to 1 means that parallel spins are highly improbable, otherwise opposite spin pairs are highly probable; while the tendency to zero indicates a high probability of same spin pairs. This interpretation gives ELF function a deep physical meaning. The electron localization function provides a rigorous basis for the analysis of the wave function and electronic structure in molecules and crystals.77 The ELF topology has been widely applied for study of chemical bonding.78–82 The topological partitioning of ELF gradient field83,84 provides two types of basins of attractors which can be thought as corresponding to atomic cores, bonds and lone pairs: core basins surrounding nuclei with atomic number Z > 2 and labeled by C(A) where A is the atomic symbol of the element, and valence basins which are characterized by its synaptic order85 which is the number of the atomic valence shells in which they participate. Therefore, there are monosynaptyc, disynaptyc, trisynaptyc basins etc, labeled V(A), V(A, X), V(A, X, Y), etc, respectively. Monosynaptyc basins correspond to the lone pairs of the Lewis model. Disynaptyc and trisynaptyc basins correspond to two-center and three center bonds respectively.

The ELF local function, (r), provides the chemical information and (r) provides two types of points in real space: wandering points where (r)≠0 and critical points where (r)=0. The critical points are characterized by their index which is the number of positive eigenvalues from hessian matrix of (r). 86–89 The topology of the ELF has been applied in the study of hydrogen bond. In FH•••N2 complex, the value 75 of ELF at the index 1 critical point between the V(F,H) and V(N) basins is 0.047, while for FH•••NH3 the corresponding (r) is 0.226. These values are attributed to weak and strong hydrogen bonds, respectively. In this work, the ELF was used as a complementary method for evaluating hydrogen-hydrogen bond for alkane complexes from methane-methane until hexane-hexane.

RESULTS AND DISCUSSION

We analyzed the H-H bonds as intramolecular interactions in some C8H18 isomers and their contribution to molecular stabilization. Afterwards, we analyzed the H-H bonds as intermolecular interactions in the case alkane-alkane complexes and its relation with complex stability and boiling point.

Octane isomers: We investigated the possible relation between the amount of H-H bonds and branched alkane stabilization based on heat of combustion (ΔHcomb) of the corresponding alkanes. The isomer with the lowest ΔHcomb is the most stable. It is well known that branched alkanes are more stable than 90 their linear isomers. Among some C8H18 isomers (Table 1), the highly branched isomer, 2,2,3,3- tetramethylbutane, is the most stable and the linear isomer octane is the least stable. The heats of combustion were calculated at ωB97xd/6-311++G(d,p) level, since this functional has signiﬁcantly lower absolute errors when considering the noncovalent attractions and empirical dispersion terms.91

Table 1. Amount of branching, amount of H-H bonds and heats of combustion (ΔHcomb) in branched alkanes.

Octane isomers* Branching H-H bonds ΔHcomb(kcal/mol) ΔΔHcomb(kcal/mol) Octane 0 0 -1647.6 2.6 3-methyl-heptane 1 0 -1647.3 2.3 3,3-dimethyl-hexane 2 0 -1646.9 1.9 2,2,3-trimethyl-pentane 3 2 -1645.8 0.8 2,2,3,3-tetramethyl-butane 4 3 -1645.0 0 *Optimized at ωB97XD/6-311++(d,p) level of theory.

Figure 1 shows the virial graphs of some C8H18 isomers (octane, 3-methylheptane, 3-3- dimethylhexane, 2-2-3-trimethylpentane and 2-2-3-3-tetramethylbutane). The averaged value of charge density of the bond critical point, ρb, from the H-H bonds, is 0.0098 au. which is typical for weak hydrogen bonds and higher than that for van der Waals interactions.61

(A) (B)

2 (E)

On going from octane, 3-methyl-heptane to 3,3-dimethyl-hexane, there is a unitary increase of methyl branching accompanying an increasing branching stabilization on C8H18 system (Table 1). However, no H-H bond was found in this set of alkanes. Then, other factors contribute for 3-methyl-heptane and 3,3- dimethyl-hexane stabilizations relative to octane, which is not the focus of this work. In literature, one may find different explanations for the alkane branching stabilization. Gronert’s work92 suggests that germinal steric interactions in linear alkane are relieved on branching. Opposing Gronert´s hypothesis, Wodrich and Schleyer93 attributed branching stabilization to 1,3-alkyl-alkyl stabilizing interaction (so called 77 protobranching). In a more recent work, Kennitz and collaborators94 attribute alkane branching stabilization to a greater electron delocalization in branched alkanes relative to linear alkanes.

Nonetheless, when comparing 3,3-dimethyl-hexane, 2,2,3-trimethyl-pentane and 2,2,3,3-tetramethylbutane, there is an increasing trend of intramolecular H-H bond and branching stabilization (Table 1). It was already been shown that these interactions contribute to energy stabilization in hydrocarbons.9,36 The

2,2,3,3-tetramethylbutane is the most stable branched C8H18 isomer and it has three H-H bonds, while the second most stable C8H18 isomer, 2,2,3-trimethylpentane, has two intermolecular H-H bonds. Moreover, in the series 3-methyl-heptane to 3,3-dimethyl-hexane, the average stabilization, ΔΔHcomb, is 0.35 kcal/mol, while for 3,3-dimethyl-hexane to 2,2,3,3-etramethyl-butane series, the average stabilization is 0.95 kcal/mol, which implies the influence of an additional stabilization factor when leaving the first series and going to the latter.

Additional evidence on stabilization due to H-H bond of highly branched C8H18 isomer was given by comparing hydrogen atomic energy belonging or not to H-H bond. The atomic energy of hydrogen atoms belonging to H-H bond (H1 in Figure 1E) are in average 0.80 kcal/mol lower when compared to the hydrogen atoms not belonging to H-H bond (H2 in Figure 1E). Then, for highly branched alkanes, H-H bond can be regarded as secondary stabilization factor.

One possible reason for the non-existence of H-H bond in octane, 3-methylheptane and 3,3- dimethylhexane can be ascribed to the lack of necessary alignment involving C-H---H-C bonds of vicinal or germinal methyl groups. Further details on this geometrical requirement will be in the following subsection.

Alkane complexes: In this section we evaluated the H-H bonds in linear bimolecular alkane complexes: methane-methane, ethane-ethane, propane-propane, butane-butane, pentane-pentane and hexane- hexane. These alkane complexes were obtained from two different optimization steps: firstly, the optimization of each single alkane molecule (the geometry of each single molecule is in Supporting Information Figure S1); secondly, the optimization of the whole bimolecular complex built from an initial geometry where each single optimized molecule was arbitrarily positioned. The optimizations were done in different levels of theory: ωB97XD/6-311++G(d,p), G4/CCSD(T)/6-311++G(d,p), B3LYP/6-311++G(d,p) (Figure S2, Supporting Information), and M062X/6-311++G(d,p) (Figure S3, Supporting Information). Only optimized structures of alkane complexes obtained from G4/CCSD(T)/6-311++G(d,p) and ωB97XD/6- 311++G(d,p) are depicted in Figures 2 and 3, respectively. We emphasize that the single molecules in Figures 2 and 3 comes from optimized geometries, but both Figures contain randomized non-optimized initial geometry (to the left) and optimized geometry (to the right) for the complex where individual molecules are optimized as above-mentioned. Both (columns) are shown for comparison reasons which depict differences of some geometrical parameters after optimization of the whole complex affording better understanding of H-H bond. 78

In Figure 2, it is depicted the CH---HC dihedral angle for methane-methane, ethane-ethane, propane-propane, butane-butane from G4/CCSD(T)/6-311++G(d,p) level of theory. In this case, geometry optimization proceeded through G4 method. The CCSD(T)/6-311++G(d,p) was used for self-consistent field and density matrix calculations from the G4-optimized geometry (Figure S4, Supporting Information). Due to infrastructure limitations, it was not possible to calculate pentane-pentane and hexane-hexane complexes from G4/CCSD(T)/6-311++G(d,p) level of theory. From Figure 2, one can see that all CH---HC dihedral angle decreases from non-optimized to optimized structures. Except for ethane-ethane complex, this decrease is reasonably considerable: in methane-methane (from 17.53 to 0.34o), in propane-propane (from 30.51 to 8.02o), and in butane-butane (from 65.18 to 18.56o), which tends to point both hydrogen atoms in nearly the same direction. This is a clear indication that hydrogen atoms tend to form an intramolecular interaction.

RANDOMIZED, NON-OPTIMIZED GEOMETRY OPTIMIZED GEOMETRY

Figure 2. Randomized, non-optimized and optimized geometries of methane-methane, ethane-ethane, propone-propane and butane-butane complexes depicting their corresponding selected CH--HC dihedral angles, in degrees. The geometry optimization of the molecules were performed at G4/CCSD(T)/6- 311++G(d,p) level of theory. The dashed lines represent the selected dihedral angles.

In Figure 3, it is depicted selected H-H interatomic distances for methane-methane, ethane-ethane, propane-propane, butane-butane, pentane-pentane and hexane-hexane from ωB97XD/6-311++G(d,p) level of theory where all molecular pairs are closer after optimization. Except from methane-methane complex, the optimized structures of the alkane complexes shown in Figure 3 are notoriously different from those in Figure 2. This is a consequence in the usage of different methods for optimization calculation. The optimization with ωB97XD functional tends to impose a conformation which increases the number of intramolecular H-H bonds in the alkane complex. This is confirmed when comparing the virial graphs of G4 and ωB97XD optimized structures in Figures 4 and 5, respectively.

RANDOMIZED, NON-OPTIMIZED GEOMETRY OPTIMIZED GEOMETRY

Figure 4 shows the virial graphs of the optimized alkane complexes using G4/CCSD(T)/6- 311++G(d,p) level of theory. There are one H-H bond in methane-methane complex and two H-H bonds in 2 other alkane complexes. The values of charge density of BCP (ρb) and the Laplacian of ρb (∇ 휌) indicate that all H-H bond paths from alkane complexes exhibit the characteristics of closed-shell interactions.

Figure 4. Virial graphs of Alkane complexes optimized at G4/CCSD(T)/6-311++G(d,p) level of theory. Methane-methane (Met-Met), ethane-ethane (Eth-Eth), propane-propane (Pro-Pro) and butane-butane (But- But) along with the charge density, ρ(r), and Laplacian of the electron density, ∇2 ρ(r) at H-H bond critical, in au. The others complexes were not optimized due to lack of computational resources.

Figure 5 shows the virial graphs of the optimized alkane complexes using ωB97XD/6-311++G(d,p) level of theory. There is an increasing number of H-H bond as the number of carbon atoms increase in the alkane complex. The number of H-H bond increase by a unit from methane-methane to butane-butane. From butane-butane to hexane-hexane, the increase is two units of H-H bond. In ethane-ethane complex there are also two C-H intramolecular bonds whose virial paths are curved, meaning less attractive and stable -6 -3 interactions, with a low value of ρb (about 8.5 × 10 at 6.7 × 10 ua range values). Accordingly, Table 2 shows the number of H-H bonds in the alkane complexes calculated at ωB97XD/6-311++G(d,p) level of theory, along with the experimental values of the boiling points of the corresponding alkanes and the so- called complex stability, G(complex-2*isolated), derived from Gibbs energy of complex and two-fold Gibbs energy of the corresponding isolated alkane, whose equation is: G= Gcomplex - 2Gisolated. 82

Table 2. Number of H-H bonds, boiling point, in °C, Gibbs energy of alkane complexes (Gcomplex), two-fold

Gibbs energy of the corresponding isolated alkane (2*Gisolated), and complex stability (G(complex-2*isolated)) of the alkane complexes.

Complexes* Number of Boiling Point Gcomplex 2*Gisolated G(complex- H-H bonds (°C) (kcal/mol) (kcal/mol) 2*isolated) (kcal/mol) Met-Met 1 -162 -50848.0 -50847.8 -0.2

Eth-Eth 2 -89 -100184.2 -100182.2 -2.0

Pro-Pro 3 -42 -149522.3 -149520.2 -2.1

But-But 4 0 -198860.6 -198858.2 -2.3

Pen-Pen 6 36 -248201.1 -248196.5 -4.6

Hex-Hex 8 69 -297540.0 -297534.5 -5.4

*Optimized at ωB97XD/6-311++(d,p) level of theory.

0.0050 0.0051 0.0000085 ρ = 0.0032 0.0043 Pro-Pro 0.0034 2 0.0000060 EthEth-Eth-Eth Pro-Pro MetMet--MetMet ρ = −0.0023

But-But PenPen--PenPen HexHex-Hex-Hex But-But 0.0057 0.0045

0.0059 0.0067 0.0045 0.0049

Figure 5. Virial graphs of Alkane complexes optimized at ωB97XD/6-311++G(d,p) level of theory. Methane-methane (Met-Met), ethane-ethane (Eth-Eth), propane-propane (Pro-Pro), butane-butane (But- 83

But), pentane-pentane (Pen-Pen) and hexane-hexane (Hex-Hex) complexes along with the charge density, ρ(r), and Laplacian of the electron density, ∇2 ρ(r) at H-H bond critical, in au.

We can see that as the number of H-H bonds increase from methane-methane to hexane-hexane, the

G(complex-2*isolated) increases as well. This linear relation is plotted in Figure 6, with a reasonably good coefficient of determination. In another words, the hydrogen-hydrogen bonds influence on the stability of alkane complexes.

6.40

) R² = 0.9428 5.60

mol 4.80

kcal/ 4.00

( ) 3.20

isolated 2.40

2* - 1.60

(complex 0.80 G

 0.00 0 2 4 6 8 10 H-H bond paths

Figure 6. Plot of number of H-H bond paths versus ΔG (complex-2*isolated), in kcal/mol, of selected alkanes at ωB97XD/6-311++G(d,p) level of theory.

From Table 2 there is another linear relation: between alkanes’ boiling point and number of H-H bonds, which is plotted in Figure 7, whose coefficient of determination is 0.9052. The boiling point is directly dependent upon the magnitude of the intermolecular interactions which, in turn, is directly related to the complex stability. Each H-H bond in these complexes contribute to decrease on complex electronic energy (Ecomplex) compared to the sum of electronic energy of the two corresponding isolated alkanes

(Eisolated), leading to an increased complex stability and boiling point. As a consequence, as the number of H- 84

H bonds increase, the complex stability increases, which leads to an expected increase in the alkane boiling point. These trends are confirmed with the reasonably good coefficient of determination in Figure 7.

150150 R² = 0.9052 100 100 R² = 0.9052 Met-met 50

C) 50

Eth-Eth 0 0 0 2 4 6 8 10 Pro-Pro -50 0 1 2 3 4 5 6 7 8 -50 • But-But

Boiling point ( point Boiling -100

-100 Pen-Pen Boiling point ( Boilingpoint -150 -150 Hex-Hex -200 -200 H-H bond paths Number of H-H bond paths Figure 7. Plot of number of H-H bond paths, from ωB97XD/6-311++G(d,p) wave function, versus boiling point, in °C, of selected alkanes.

It is well known the correlation between boiling/melting point in linear alkanes with increasing number of carbon atoms.90,95 The intermolecular attractive forces acting in neutral species are known as van der Waals forces based on non-directional induced dipole-induced dipole interactions. However, the results from this work strongly indicate that the intramolecular forces in alkane complexes are consequence of directional attractive H-H bonds.

In order to demonstrate the stabilizing effect of H-H bond, the atomic energies of a hydrogen atoms belonging and not to H-H bond were obtained from atomic virial theorem of QTAIM. Figures 8 and 9 show the energy of a hydrogen atom belonging (EH(H-H)) and not belonging (EH) to H-H bond, respectively, versus energy of the ethane-ethane complex in some selected optimization steps, which includes the first and the last steps of optimization plus five other points (steps 1, 22, 27, 33, 43, 49 and 52). Figure S5, in Supporting Information, shows the plot of energy, E, in hartrees, versus optimization steps of ethane-ethane 85 complex. The decrease of EH(H-H) of hydrogen atom belonging to H-H bond versus E (Figure 8) reveals the stabilizing effect of this type of interaction in the ethane-ethane complex. On the other hand, there is no decrease in overall EH of a hydrogen atom not belonging to H-H bond in ethane-ethane complex (Figure 9). Then, the decreasing trend in alkane complex energy is accompanied with the decreasing trend of hydrogen atoms from H-H bonds, while no overall change in hydrogen energy not belonging to H-H bond occurs when alkane complex reaches the minimum in the potential energy surface.

E (hartrees)

-159.654 -159.653 -159.652 -159.651 -159.650 -159.649

-0.6200 E

-0.6205 H(H

H) (

-0.6210 hartrees)

-0.6215

-0.6220

-0.6225

Figure 8. Plot of energy, in hartrees, of a hydrogen atom belonging to H-H bond (EH(H-H)) in ethane-ethane complex versus the energy (E) in ethane-ethane complex of selected steps in the optimization steps. 86

E (hartrees) -159.654 -159.653 -159.652 -159.651 -159.650 -159.649 -0.6204

-0.6206

-0.6208 E

-0.6210 H ( -0.6212 hartrees)

-0.6214

-0.6216

-0.6218

-0.6220

-0.6222

-0.6224

A complementary ELF analysis of H-H bond in alkane complexes (from methane-methane until hexane-hexane) was carried out. The coordinates of QTAIM bond critical point from a selected H-H bond in each alkane complex was used for locating the corresponding ELF BCP. The calculation of gradient of ELF function at QTAIM BCP coordinates confirmed the ELF BCP for all alkane complexes. These BCP’s are index 1 critical points, (r), whose values are 0.010, 0.015, 0.021, 0.017, 0.026 and 0.025 for the methane- methane, ethane-ethane, propane-propane, butane-butane, pentane-pentane and hexane-hexane complexes, respectively (Table 3). These (r) values are nearly half the (r) values for weak hydrogen bond.89 Furthermore, the electron density and Laplacian of electron density values at these points have topological features of closed-shell interactions. Then, ELF is capable to identify H-H bond as well as QTAIM, indicating that each H-H bond in alkane complexes is weaker than a weak hydrogen bond.

Table 3. ELF value, η(r), electron density, ρ(r), and Laplacian of the electron density, ∇2 ρ(r) at bond critical point of a selected H-H bond in alkane complexes.

Complex η(r) ρ(r) 훁2 ρ(r) Methane-methane 0.010 0.0032 0.0093 Ethane-ethane 0.015 0.0053 0.0180 Propane-propane 0.021 0.0051 0.0137 Butane-butane 0.017 0.0050 0.0158 Pentane-pentane 0.026 0.0067 0.0193 Hexane-hexane 0.025 0.0067 0.0197

CONCLUSIONS

This work gives new insights and understanding about intramolecular and intermolecular interactions in alkanes.

There are H-H bonds in highly branched C8H18 isomers, but not in linear or less branched alkanes. Then, the hydrogen-hydrogen bond plays a secondary role on the stability of branched alkanes which partly accounts for the increased stability of highly branched alkanes when compared with linear or less branched isomers.

We found very good linear relations between H-H bond in complex alkanes and their stabilities and between H-H bond in alkanes and their boiling points. For all studied molecular systems, the hydrogen- hydrogen bonds perform a stabilizing interaction. The hydrogen-hydrogen bond plays an important role on the boiling point of linear alkanes. Intermolecular interactions in alkane complexes are related to H-H bond, not to non-directional induced dipole-induced dipole.

The electron localization function can also be used for identification of hydrogen-hydrogen bond. The ELF topological approach enables a clear identification of hydrogen-hydrogen bonds in alkane complexes whose (r) are smaller than that from a weak hydrogen bond.

ACKNOWLEDGMENTS

Authors thank, Fundação de Apoio à Pesquisa do Estado do Rio Grande do Norte (FAPERN), Coordenação de Ensino de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) and Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) for financial support. 88

ASSOCIATED CONTENT

Supporting Information

Optimized geometries of alkane molecules; Virial graphs of Alkane complexes optimized at B3LYP/6-311++G(d,p), M062X/6-311++G(d,p) levels and G4 method; plot of energy optimization of ethane-ethane complex. This material is available free of charge via the Internet at http://pubs.acs.org.

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TABLE OF CONTENTS

150150 R² = 0.9052 100 100 R² = 0.9052 Met-met 50

C) 50

Eth-Eth 0 0 0 2 4 6 8 10 Pro-Pro -50 0 1 2 3 4 5 6 7 8 -50 • But-But

Boiling point ( point Boiling -100

-100 Pen-Pen Boiling point ( Boilingpoint -150 -150 Hex-Hex -200 -200 H-H bond paths Number of H-H bond paths

Supporting Information

Hydrogen-hydrogen bonds in highly branched alkanes and in alkane complexes: a DFT, ab initio, QTAIM and ELF study

Norberto K. V. Monteiro1, and Caio L. Firme1*

6 5 4 3 2 1 b) a)

MetMet-Met-Met EthEth--EthEth ProPro--Pro

ButBut-ButBut PenPen-Pen-Pen Hex-Hex-Hex

Figure S2. Virial graphs of Alkane complexes optimized at B3LYP/6-311G++(d,p) level of theory. Methane-methane (Met-Met), ethane-ethane (Eth-Eth), propane-propane (Pro-Pro), butane-butane (But- But), pentane-pentane (Pen-Pen) and hexane-hexane (Hex-Hex) complexes.

EthEth--EthEth Pro-Pro MetMet-Met-Met Pro-Pro

ButBut-But-But PenPen--PenPen HexHex--HexHex

Figure S3. Virial graphs of Alkane complexes optimized at M062X/6-311G++(d,p) level of theory. Methane-methane (Met-Met), ethane-ethane (Eth-Eth), propane-propane (Pro-Pro), butane-butane (But- But), pentane-pentane (Pen-Pen) and hexane-hexane (Hex-Hex) complexes.

Eth-Eth ProPro--Pro MetMet-Met-Met Eth-Eth

But-But PenPen-Pen-Pen But-But HexHex-Hex-Hex

Figure S4. Virial graphs of Alkane complexes optimized at G4 method. Methane-methane (Met-Met), ethane-ethane (Eth-Eth), propane-propane (Pro-Pro), butane-butane (But-But), pentane-pentane (Pen-Pen) and hexane-hexane (Hex-Hex) complexes.

Step N

0 10 20 30 40 50 60 70 -159.6500

-159.6505

-159.6510

-159.6515

-159.6520 E (Hartree) -159.6525

-159.6530

-159.6535

-159.6540

Figure S5. Plot of energy (E), in Hartree, of ethane-ethane complex versus of optimization steps.

100

5. CAPÍTULO 3 – Artigo publicado no periódico New Journal of Chemistry.

Monteiro, N. K. V.; de Oliveira, J. F.; Firme, C. L. New Journal of Chemistry 2014, 38, 5892.

Stability and Electronic Structure of Substituted Tetrahedranes, Silicon and Germanium Parents – a DFT, ADMP, QTAIM and GVB study

Norberto K. V. Monteiro, José F. de Oliveira and Caio L. Firme*

Institute of Chemistry, Federal University of Rio Grande do Norte, Natal-RN, Brazil, CEP 59078-970.

*Corresponding author. Tel: +55-84-3215-3828; e-mail: [email protected]

Graphical Abstract

REF = 0

G= -15.5 kcal.mol-1

-1 ΔGf = 20.5 kcal.mol

G = -35.9 kcal.mol-1

Abstract

The electronic structure and the stability of tetrahedrane, substituted tetrahedranes and silicon and germanium parents have been studied at ωB97XD/6-311++G(d,p) level of theory. The quantum theory of atoms in molecules (QTAIM) was used to evaluate the substituent effect on the carbon cage in the tetrahedrane derivatives. The results indicate that electron withdrawing groups (EWG) have two different behaviors, ie., stronger EWG makes the tetrahedrane cage slightly unstable while slight EWG causes a greater instability in the tetrahedrane cage. On the other hand, the sigma electron donating groups, σ-EDG, stabilizes the tetrahedrane cage and π-EDG leads to tetrahedrane disruption. NICS and D3BIA indices were used to evaluate the sigma aromaticity of the studied molecules, where EWGs and EDGs results in decrease and increase, respectively, of both aromaticity indices, showing that sigma aromaticity plays an important role in the stability of tetrahedrane derivatives. Moreover, for tetra-tert-butyltetrahedrane there is another stability factor: hydrogen-hydrogen bonds which imparts a high stabilization in this cage. Generalized 101 valence bond (GVB) was also used to explain the stability effect of the substituents directly bonded to the carbon of the tetrahedrane cage. Moreover, the ADMP simulations are in accordance with our thermodynamic results indicating the unstable and stable cages under dynamic simulation.

Keywords: Tetrahedrane; QTAIM; NICS; D3BIA; GVB; ADMP

1. Introduction

The development of new high energy molecules has aroused great interest of theoreticians and experimentalists, with applications in the military industry and others high-tech fields.1 Cage strained organic molecules can be used as potential explosives in response to various external stimuli such heat, shock or impact.1–4 The highly-strained cage compounds have raised interest due to their high density and high energy. As a consequence, they can also be used as energetic materials, such as hexanitrohexaazaisowurtzitane 1,5 and octanitrocubane.6 Similarly, tetrahedrane is also a high-energy molecule. Despite many attempts to isolate the parent tetrahedron (unsubstituted – C4H4) were not successful,7 Maier et al8 in 1978 were able to synthesize the first tetrahedrane derivative, the tetra-tert- butyltetrahedrane.

Tetrahedrane is one of the most strained organic molecule, with highly symmetrical structure and unusual bonding being known as a Platonic solid. It has a strain energy of 586 kJ.mol-1 according to Wiberg et al 9 which makes this compound thermodynamically unstable. As a consequence, their synthesis and isolation tends to be extremely difficult. Conversely, tetrahedrane has σ-aromaticity10 because it has a large negative NICS (Nucleus Independent Chemical Shift ) value, which can somehow contribute for decreasing the high instability of tetrahedrane cage.

Tetrahedrane and its derivatives can be obtained from the corresponding cyclobutadiene or derivatives. The anti-aromatic and highly-reactive cyclobutadiene has only been isolated at low temperature in inert matrices11 and immobilized more recently at room temperature in hemicarcerand.12 On the other hand, structural parameters of substituted cyclobutadienes have been reported showing pronounced bond alternations, 13,14 although only slight CC bond length alternations (1.464 and 1.482 Å), under room temperature, of tetrakis (tert-butyl) cyclobutadiene 15 have been reported. In fact, measurements at -150 °C showed noticeably larger bond alternation (1.441 and 1.526 Å) for tetrakis (tert-butyl) cyclobutadiene as well.

Tetra-tert-butyltetrahedrane can be formed from tetra-tert-butylcyclobutadiene by photolysis using n- octane as solvent while the reverse reaction takes place thermally with an activation barrier of 26 kcal mol-1 16 . Similarly, an ab initio study of the interconversion of the C4H4 system predicts an activation energy of about 30 kcal mol-1.17 Maier and co-workers succeeded in synthesizing tetrakis (tert-butyl)tetrahedrane by 102 photochemical isomerization of the corresponding cyclobutadiene.8 The reason for its stability is attributed to the voluminous tBu-substituents that avoid the tetrahedrane skeleton from ring-opening due to the van der Waals strain among them, the so-called “corset effect”. However, this effect is lost if one of the tert-butyl substituents is replaced with a smaller group. Indeed, phenyl- and methyl-substituted tetrahedrane derivatives were not detected, even under matrix isolation conditions.18

Recently, it was synthesized the tetrakis (trimethylsilyl) tetrahedrane ((Me3Si)4THD) by 19,20 photochemical isomerization of the corresponding cyclobutadiene ((Me3Si)4CBD), where (Me3Si)4THD has enhanced thermal stability due the four σ-donating trimethylsilyl groups that electronically stabilize the highly strained tetrahedrane skeleton.21,22 Indeed, it was also reported the synthesis of perfluoroaryltetrahedranes with extended σ-π conjugation between the strained tetrahedrane core and the aromatic ring.23 In addition, the insertion of a heteroatom (N, O, S, Ge, Si or a halogen atom) into the tetrahedrane core would be interesting because the tetrahedrane σ framework has the potential to interact with the nonbonding orbitals on the heteroatom, according to molecular orbital theory. However, such heteroatom-substituted tetrahedrane derivatives have remained elusive because of the synthetic difficulty in preparing such molecules.24

In this work tetrahedrane and its derivatives where studied from different tools such as theory of density functional (DFT), quantum theory of atoms in molecules (QTAIM), atom-centered density matrix propagation (ADMP) and generalized valence bond (GVB) in order to investigate their electronic structures, thermochemical and dynamical stabilities that eventually lead to the substituents that enhance the stability of tetrahedrane cage. The results from QTAIM, GVB and the aromaticity indices provided new and important information about the main structural and electronic effects associated with the tetrahedrane cage instability and the reasons why some substituents increase its stability. Moreover, this paper also provides information about the stability of cages containing germanium and silicon parents of tetrahedrane and tetrakis(trifluoromethyl) tetrahedrane.

2. Computational details

The geometries of the studied species were optimized by using standard techniques.25 Frequency calculations were performed to analyze vibrational modes of the optimized geometry in order to determine whether the resulting geometries are true minima or transition states. Calculations were performed at ωB97XD/6-311++G(d,p)26 level of theory by using Gaussian 09 package,27 including the electronic density which was further used for QTAIM calculations. ωB97XD functional incorporates an empirical dispersion term (D) that improves treatment of non-covalent complexes, it uses 100% Hartree Fock (HF) exchange for long-range interactions and it has an adjustable parameter (X) to include short-range exact exchange.28,29 103

Mohan and co-workers30 have shown that ωB97XD functional exhibited better performance in the description of bonded and nonbonded interactions in comparison with other DFT methods. All topological information 31,32 were calculated by means of AIM2000 software.33 The algorithm of AIM2000 for searching critical points is based on Newton-Raphson method which relies heavily on the chosen starting point.34 Integrations of the atomic basins were calculated in natural coordinates with default options of integration.

The valence bond package VB200035, version 2.5, was used for all generalized valence bond, GVB, calculations. The GVB singly occupied orbitals were calculated from VB/6-31G level theory and they were generated by means of Molekel visualization program.36 The package VB2000 generates nonorthogonal singly occupied orbitals from a general implementation of group function theory (GFT)37 and modern VB methods based on high efficiency of the algebrant algorithm.35,38

Nucleus Independent Chemical Shift (NICS)39 and Density, Degeneracy, Delocalization-Based Index Aromaticity (D3BIA)40 were applied to determine the aromaticity in tetrahedrane and its derivatives. D3BIA is based on density of (homo)aromatic ring, degeneracy and uniformity of the electron density among the atoms of the aromatic ring. We will use the same D3BIA formula for homoaromatic species41 where the electron density is associated with the charge density of the ring critical point (for 3c-2e bonding systems, for example) or of the cage critical point (for tridimensional 4c-2e bonding system). The degeneracy (훿) is associated with the atomic energy of the constituent atoms of the aromatic ring of the molecular system. Another important electronic feature in D3BIA is the uniformity of the electronic density calculated from the delocalization index of atoms of the aromatic ring. Then, D3BIA for homoaromatic especies is defined as:

퐷3퐵퐼퐴 = 휌(3, +3) ∙ 퐷퐼푈 ∙ 훿 (1)

Where ρ is the ring density factor of the cage critical point (3,+3), for cage structures, and 훿 is the degree of degeneracy where its maximum value (훿=1) is given when the difference of energy between the atoms of the cage is less than 0.009 au. The formula of 훿 is given in Eq. (2).

푛 훿 = (2) 푁

Where n is the number of atoms whose ∆퐸(Ω) ≤ 0.009푎푢. and N is the total number of atoms in the homoaromatic circuit. The minimum value for n is 1, where there is no atomic pair whose ∆퐸(Ω) ≤ 0.009푎푢.

The DIU is the delocalization index uniformity among bridged atoms given by Eq. (3):

100휎 퐷퐼푈 = 100 − (3) 퐷퐼 104

Where σ is mean deviation and 퐷퐼 is the average of delocalization index between C-C, Si-Si or Ge- Ge bonds of the cage. The DI gives the amount of electron(s) between any atomic pair. 31

The NICS values were calculated with the B3LYP/6-311++G(d,p) through the gauge-independent atomic orbital (GIAO) method42 implemented in Gaussian 09. The magnetic shielding tensor was calculated for ghost atoms located at the geometric center of the cage.

The ab initio molecular dynamics simulations involves quantum chemistry calculations aiming to obtain the potential energy and nuclear forces.43–46 The simulations performed in this work involved the atom-centered density matrix propagation (ADMP)47–50 in order to study the stability of tetrahedrane and selected substituted tetrahedranes. The ADMP calculations were performed at B3LYP/6-31G(d) level of theory in Gaussian 09. A time step of 0.1 fs was used for the ADMP trajectories. The Nosé-Hoover thermostat51,52 was employed to maintain a constant temperature at 298.15K. The default fictitious electron mass was 0.1 amu. A maximum number of 50 steps were used in each trajectory.

3. Results and discussion

Scheme 1 shows the reaction route (diazocompounds 1a-1i  cyclobutadienes 2a-2i  tetrahedranes 3a-3i) to synthesize tetrahedrane (and derivatives) from its precursors diazocompound and cyclobutadiene (and derivatives).

Scheme 1

1 2 3 R1=R2=R3=R4 = -H (1a) R1=R2=R3=R4 = -H (2a) R1=R2=R3=R4 = -H (3a) = -CH (3b) = -CH3 (1b) = -CH3 (2b) 3 = -C(H C) (3c) = -C(H3C) 3 (1c) = -C(H3C) 3 (2c) 3 3 = -CF (3d) = -CF3 (1d) = -CF3 (2d) 3 = -SiF (3e) = -SiF3 (1e) = -SiF3 (2e) 3 1 2 3 4 1 2 3 4 R1=R2=R3=H and R4 = -NO (3f) R =R =R =H and R = -NO2 (1f) R =R =R =H and R = -NO2 (2f) 2 = -NH (3g) = -NH2 (1g) = -NH2 (2g) 2 = -OH (1h) = -OH (2h) = -OH (3h) = -NO (3i) = -NO2 (1i) = -NO2 (2i) 2

Different substituted tetrahedranes were studied: tricyclo[1.1.0.02,4]butane, 3a, 1,2,3,4- tetramethyltricyclo[1.1.0.02,4]butane, 3b, 1,2,3,4-tetrakis(trifluormethyl)tricyclo[1.1.0.02,4]butane, 3c, 1,2,3,4-tetrakis(trifluorosilyl)tricyclo[1.1.0.02,4]butane, 3d, tetra-tert-butyltricyclo[1.1.0.02,4]butane, 3e, 1- 105 nitro tricyclo[1.1.0.02,4]butane, 3f, 1,2,3,4-tetraamino tricyclo[1.1.0.02,4]butane, 3g, 1,2,3,4-tetrahidroxy 2,4 2,4 tricyclo[1.1.0.0 ]butane, 3h, and 1,2,3,4-tetranitro tricyclo[1.1.0.0 ]butane, 3i. In addition, Ge4 and Si4 parents of tetrahedrane, 4a-4b, and tetrakis(trifluoromethyl) tetrahedrane, 5a-5b, were also studied (Scheme 2).

Scheme 2

4 5 R1=R2=R3=R4 = -H (4a) R1=R2=R3=R4 = -H (5a)

= -CF3 (4b) = -CF3 (5b)

In scheme 3 shows the structures of tetraamino-substituted tetrahedrane, 3g, tetrahydroxi-substituted tetrahedrane, 3h, tetranitro-substituted tetrahedrane, 3i, and tetratrifluoromethyl-substituted silicon parent of tetrahedrane, 4b, that were optimized and yielded the corresponding open structures 3g’, 3h’, 3i’ and 4b’, which prevented any structural study with these molecules.

Scheme 3

3g 3h 3i 4b

3g’ 3h’ 3i’ 4b’

106

The Scheme 3 indicates that 3g, 3h, 3i and 4b do not have stable tetrahedrane cage. However, no unique pattern can be established to account for this fact. The –NH2 and –OH groups are electron withdrawing groups (EWG) by inductive effect and electron donating groups (EDG) by resonance effect. On the other hand, the –NO2 and –CF3 do not have dual behavior and they are EWGs. Moreover, for tetrahedrane cage four –CF3 do not disrupt the tetrahedrane cage while four –NO2 groups do so. Conversely, in Si4 tetrahedrane parent, -CF3 disrupts the corresponding cage.

The equilibrium geometries and selected bond lengths, in Angstroms, of tetrahedrane and derivatives are depicted in Figure 1. The corresponding virial graphs are shown in Figure 2, along with several topological data of selected bond critical points. Electronic density, ρ(r), Laplacian of the electron density, ∇2 ρ(r), local energy density, H(r) and relative kinetic energy density, G(r)/ρ values are shown for critical points cage-substituent and electronic density, ρ(r), Laplacian of the electron density, ∇2 ρ(r), local energy density, H(r), relative kinetic energy density, G(r)/ρ, delocalization index, DI, bond order, n, and ellipticity are shown for critical points of carbon, silicon or germanium atoms in the tetrahedrane (or parent) cage. From Figure 1, we can see that the alkyl and silyl groups impart the shortening of C-C bonds from the cage while the –CF3 and –NO2 groups lengthen the corresponding C-C bonds. As a consequence, based only on geometrical information, we can assume that alkyl and silyl groups behave as EWGs. These EWGs and EDGs do not disrupt the tetrahedrane cage.

We have analyzed the topology of the electron density of all species using the quantum theory of atoms in molecules (QTAIM).31,53,54 According to the topological analysis of QTAIM, when ρ of the critical point is relatively high (×10-1 au.) and ∇2ρ < 0, the chemical interaction is defined as shared shell and it is applied to covalent bond. However, other parameters are required to describe the nature of the chemical interaction such as the local energy density, H(r), which is the sum of local kinetic energy density, G(r), and local potential energy density, V(r), respectively, and the ratio G(r)/ ρ. Cremer and Kraka55 suggested that H(r) < 0 and G(r)/ρ < 1 are indicative of shared shell (or covalent) interaction. In Figure 2 the values of ρ, ∇2ρ, H(r) and G(r)/ρ of C-C, Si-Si and Ge-Ge bonds of the tetrahedrane cage (or parent) indicate that all C- C, Si-Si and Ge-Ge bond paths are shared shell interactions for 3a – 3f, 4a, 5a and 5b. The ellipticity (ε) values in the tetrahedranes indicate no cylindric symmetry on the single C-C bond, while for Si-Si and Ge- Ge cages the ε is closer to zero indicating symmetry around the distribution of electronic density of these bonds. 107

8 4 2 6 3 7

(3a) (3b) (3c) A B C

(3f) (3d) (3e) F D E

(4a) (5a) (5b) G H I

Figure 1: Optimized geometries of the substituted tetrahedranes (A-F), silicon and germanium cages

(G-I) with corresponding C1-C4, C1-C5, C1-Si, C1-N, Si-Si, Si-H, Ge-Ge, Ge-H and Ge-C bond lengths (in Angstroms); values of selected angles (in degree). 108

ρ = 0.2539 ρ = 0.2531 ρ = 0.2801 2ρ = 0.1515 2ρ = 0.1494 2ρ = 0.2442 H(r) = -0.217 H(r) = -0.213 H(r) = -0.280 G(r)/ρ = 0.256 ρ = 0.2191 G(r)/ρ = 0.129 G(r)/ρ = 0.252 2ρ = -0.017 48 ρ = 0.2198 H(r) = -0.168 ρ = 0.2230 2ρ = -0.017 G(r)/ρ = 0.842 2 47 ρ = -0.015 H(r) = -0.168 DI = 0.948 H(r) = -0.174 G(r)/ρ = 0.845 n = 0.94 G(r)/ρ = 0.847 DI = 0.980 ε = 0.092 (3a) DI = 1.025 (3b) A n = 0.97 n = 1.02 ε = 0.089 B (3c) ε = 0.105 C

ρ = 0.2748 ρ = 0.1567 2ρ = 0.1835 2ρ = 0.0748 ρ = 0.2316 ρ = 0.2996 H(r) = -0.244 H(r) = -0.115 2ρ = 0.0007 2ρ = 0.1685 G(r)/ρ = 0.222 G(r)/ρ = 0.259 H(r) = -0.190 H(r) = -0.277 G(r)/ρ = 0.819 G(r)/ρ = 0.362 ρ = 0.2241 ρ = 0.2202 DI = 1.012 2ρ = -0.017 2ρ = -0.016 n = 1.00 H(r) = -0.175 H(r) = -0.169 ε = 0.099 G(r)/ρ = 0.857 G(r)/ρ = 0.838 DI = 0.953 DI = 1.016 (3e) (3f) (3d) E F n = 0.94 D n = 1.01 ε = 0.109 ε = 0.213

ρ = 0.1469 ρ = 0.1379 2ρ = 0.0683 ρ = 0.1339 ρ = 0.0695 2ρ = 0.0853 2 2 H(r) = -0.062 ρ = 0.0290 ρ = -0.0035 H(r) = -0.110 G(r)/ρ = 0.595 H(r) = -0.072 H(r) = -0.026 G(r)/ρ = 0.181 G(r)/ρ = 0.325 G(r)/ρ = 0.420 DI = 0.912 n = 1.18 ρ = 0.0783 ρ = 0.0699 2 ε = 0.043 2ρ = 0.0145 ρ = -0.0034 H(r) = -0.036 H(r) = -0.026 G(r)/ρ = 0.280 G(r)/ρ = 0.420 DI = 0.918 DI = 0.951 (4a) n = 1.25 (5a) n = 1.60 G H (5b) ε = 0.039 ε = 0.063 I

The virial graph of tetra-tert-butyltetrahedrane, 3c, in Figure 2C indicates a different reason for its stability rather than the so-called “corset effect” which is a steric repulsion between bulk substituents preventing the breaking of the cage.8 There are 23 intramolecular hydrogen-hydrogen (H-H) bonds between tert-butyl groups according to the virial graph of 3c where the average charge density of the corresponding

critical point, ρH-H, is 0.0041 au. The augmented cooperative effect of these small interactions contributes for 109 the stabilization of the molecular system.56 An important study based on QTAIM, ELF and ab initio calculations demonstrated the stability effect of the H-H bonds in alkane complexes and highly branched alkanes. 57 As to the derivative 3c, according to the values of the hydrogen atomic energy belonging or not to H-H bond, the hydrogens atoms belonging to H-H bond (e.g., H48 in Figure 2C) are in average 2.65 kcal mol-1 lower in energy than those not belonging to H-H bond (e.g., H47 in Figure 1C). This result shows the stability effect of H-H bond in 3c.

Figure 3 shows the Gibbs free energy difference along the coordination reaction from diazo compound (as a reference) to cyclobutadiene and to tetrahedrane structures (plus nitrogen), as depicted in Scheme 1.

-1 ΔG2= -12.3 kcal.mol -1 ΔG2= -15.5 kcal.mol ΔG = -35.9 kcal.mol-1 1 ΔG1= -44.7 kcal/mol

-1 ΔG = 32.3 kcal.mol-1 ΔGf = 20.5 kcal.mol f

(A) (B)

ΔG = -9.7 kcal.mol-1 -1 2 ΔG1= -30.6 kcal.mol -1 ΔG2= -33.5 kcal.mol -1 ΔG1= -32.3 kcal.mol

-1 ΔGf = 22.5 kcal.mol

-1 ΔGf = -2.9 kcal.mol (C) (D)

Data of G1 in Figure 3 indicate that the diazocompounds 1a, 1b, 1f and 1e are higher in Gibbs free energy than the corresponding cyclobutadiene (or cyclobutadiene derivative), 2a, 2b, 2f and 2e, respectively.

Their corresponding formation enthalpy (H1) follows the same trend (Table 1). In fact, all cyclobutadiene derivatives (2a – 2f), plus nitrogen molecule, are more stable than the corresponding diazocompound (Table 1). The same trend is shown for predecessors of silicon tetrahedrane parent 4a, but nothing can be said about 110 the predecessors of germanium tetraedrane parents 5a and 5b because their corresponding diazocompounds were not formed. Likewise, the tetrahedrane and all tetrahedrane derivatives, 3a – 3f, are lower in energy

(both enthalpy and Gibbs free energy) than the corresponding diazocompound, as indicated by G2 and H2, except for H2 related to 3f. The silicon tetrahedrane parent 4a exhibits the same behavior.

Table 1 depicts the values of Gibbs free energy difference and enthalpy difference from diazocompound to cyclobutadiene (and derivatives), G1 and H1, from diazocompound to tetrahedrane

(and derivatives), G2 and H2, and from cyclobutadiene (and derivatives) to tetrahedrane (and derivatives), -1 Gf and Hf, in kcal mol , respectively. Table 1 also shows the delocalization index, DI, bond order, n, and ellipticity, ε, of carbon-carbon, silicon-silicon or germanium-germanium bond in the tetrahedrane (or parent) cage, besides the bond path angle, in degree, in the tetrahedrane (or parent) cage for 3a – 3f, 4a, 5a and 5b. The values of n were obtained from the linear relation between formal bond order (n) and delocalization index,58 which yields very good correlations for carbon-carbon and germanium-germanium bonds, but presents moderate correlation for Si-Si bond. The ellipticity, ε, of a bond critical point indicate whether the corresponding bond has elliptical symmetry (when ε=0, for single or triple bonds) or not.31

Table 1: Values of Gibbs free energy difference and enthalpy difference for the corresponding tetrahedranes 3a – 3f, 4a, 5a and 5b from diazocompound to cyclobutadiene (and derivatives), G1 and H1, from diazocompound to tetrahedrane (and derivatives), G2 and H2, and from cyclobutadiene (and -1 derivatives) to tetrahedrane (and derivatives), ΔGf and ΔHf, in kcal mol , respectively. Values of delocalization index, DI, bond order, n, and ellipticity, ε, from carbon, silicon or germanium atoms in the tetrahedrane (or parent) cage, bond path angle, in degree, from bonds in the tetrahedrane (or parent) cage for 3a – 3f, 4a, 5a and 5b.

Molecule ΔG1 / ΔG2 / ΔH1 / ΔH2 / ΔGf / ΔHf / DI / e n  Bond path kcal mol-1 kcal mol-1 kcal mol-1 kcal mol-1 kcal mol-1 kcal mol-1 angle/o

3a -35.92 -15.45 -27.43 -6.24 20.47 21.19 1.025 1.02 0.105 76.9

3b -44.65 -12.30 -33.77 -2.31 32.35 31.46 0.980 0.97 0.089 79.1

3c -12.26 -27.72 -2.90 -16.48 -15.46 -13.58 0.948 0.94 0.092 78.8

3d -37.51 -15.81 -27.04 -5.50 21.70 21.54 0.953 0.94 0.109 78.6 111

3e -30.59 -33.48 -20.97 -22.71 -2.89 -1.74 1.016 1.01 0.213 72.2

3f -32.28 -9.72 -20.76 3.45 22.55 24.21 1.012(b) 1.00(b) 0.099(b) 80.2(c)

4a -10.27 -8.92 -4.88 -1.11 1.35 3.77 0.918 1.60 0.039 80.6

5a -(a) -(a) -(a) -(a) 7.24 10.60 0.951 1.25 0.063 74.0 5b -(a) -(a) -(a) -(a) -3.23 11.24 0.912 1.18 0.043 75.3 (a) The diazocompound was not obtained. (b) Values from carbon(from cage)-nitrogen(from nitro group). (c) Average value

Table 2 shows the topological values (AIM atomic charge, virial atomic energy, atomic volume and localization index) of the carbon, silicon and germanium atomic basins, , of the tetrahedrane (or parent) cage for 3a – 3f, 4a, 5a and 5b. These values are obtained from the integration of the corresponding atomic basin.

Table 2: Values of AIM atomic charge, q(Ω), atomic energy, E(Ω), atomic dipole moment, M(Ω), atomic volume, V(Ω) and localization index, LI of carbon, silicon and germanium atoms in tetrahedrane, silicon and germanium derivatives, respectively.

Molecule q(Ω) E(Ω) M(Ω) V(Ω) LI 3a -0.121 -38.0936 0.435 6.09 4.059 3b -0.109 -38.1522 0.292 6.10 3.968 3c -0.089 -38.1732 0.300 6.09 3.934 3d 0.037 -38.0509 0.202 6.10 3.844 3e -0.697 -38.2994 1.48 6.69 4.732 3f 0.235(a) -37.9671(a) 0.305(a) 5.76(a) 3.691(a) 4a 0.655 -289.3407 0.889 13.30 11.44 5a 0.283 -2074.3391 0.076 31.66 29.74 5b 0.387 -2074.3590 0.097 31.57 29.66 (a) Values of carbon atom attached to the nitro group.

112

Despite the antiaromatic nature of cyclobutadiene or its derivatives and the σ-aromaticity10 of tetrahedrane cage, in most cases, tetrahedrane and its derivatives are higher in energy (both enthalpy and

Gibbs free energy) than the corresponding cyclobutadiene as it is indicated by Gƒ and Hƒ. The fact that the σ-aromaticity10 of tetrahedrane (or derivatives) does not impart smaller energy in comparison to the corresponding cyclobutadiene (which is antiaromatic) can be attributed to the highly strained cage structure of tetrahedrane and derivatives, except for 3c and 3e which are more stable than 2c and 2e. As for 3c, QTAIM analysis indicates that its relative stability is ascribed to a large amount of stabilizing hydrogen- hydrogen bonds, as aforementioned, while the relative stability of 3e is possibly reasoned by the great charge density donator effect of the –SiF3 group, according to the more negative values of q(C) and E(C) plus higher values of V(C) and LI(C) compared to those from 3a. Nonetheless, the relative stability imparted by hydrogen-hydrogen bond in 3c is nearly 12 kcal mol-1 higher than that bestowed by the donation effect of

–SiF3 group in 3e.

Taking values of any topological property of tetrahedrane 3a as a reference, the DI values of C-C bonds in the tetrahedrane cage of 3e indicates that trifluorosilyl group partially removes the charge density of each C-C bond (Table 1 and Figure 2) where ρ = 0.2801 u.a. and 0.2539 u.a. for 3a and 3e respectively. However, from Table 2, the AIM atomic charge of carbon atom in 3e is 6 times more negatively charged than that from 3a, which is corroborated by a greater localization index and greater atomic volume of carbon atom in 3e. In addition, the virial atomic energy of each carbon atom in 3e is 0.2058 au. smaller than that from 3a. Then, –SiF3 pronouncedly donates charge density to the carbon atoms of the tetrahedrane cage in 3e, although removes partially the charge density in the C-C bonding region. There is also a greater atomic dipole moment in the carbon atoms of 3e in comparison with those from all other studied tetrahedrane derivatives or parents, which can be partially related to a larger amount of charge density within carbon atomic basins in 3e, besides the influence of the electropositive Si(F3) atom bonded to the carbon atom.

Then, –SiF3 in 3e increases the charge density of C atom attached to it but removes the charge density of the adjacent C-C bond.

Conversely, the methyl and t-butyl groups, in 3b and 3c, respectively, are probably EWGs according to carbon atomic charge and localization index, where these values are slightly less negative, than those from 3a. Moreover, methyl and t-butyl groups also partially remove charge density from C-C region bond of tetrahedrane cage, as indicated by the corresponding charge density of BCP (see Figure 2B and 2C), the DI, bond order and ellipticity (see Table 1) in comparison with those values from 3a.

The nitro group in 3f has a more pronounced electron withdrawing effect than alkyl groups as can be easily observed in the positive atomic charge, low LI and higher atomic energy of the carbon atom attached to the nitro group in 3f with respect to those values for the carbon atoms in 3a (Table 2). Nonetheless, 3b is more unstable than 3f, but 3f is slightly more unstable than 3a. Then, one nitro group 113 stabilizes the tetrahedrane cage in comparison with four methyl groups, but four hydrogen substituents (in 3a) still generate a more stable structure than one nitro and three hydrogen substituents (in 3f).

Figure 4 shows the doubly occupied GVB oxygen lone pair of nitro group, LP1 and LP2, and singly occupied C-C sigma orbitals, VB(C-C)’, VB (C-C)1 and VB (C-C)2, of 3f. Table 4 shows the overlap integrals between the GVB orbitals of Figure 4. From GVB orbitals of 3f (Figure 4), one of the oxygen lone pair, named LP2, is directed towards the cage. The average overlap integral between LP2 and C-C bonds in the cage (LP2-CC1) are three-fold greater than the average overlap integral between LP1 (which is not directed towards the cage) and C-C bonds in the cage (LP1-CC1) as indicated in Table 4. Then, we can infer that the influence of both oxygen LP2 lone pairs from four nitro groups may cause a great instability and may lead to the disruption of tetrahedrane cage in 3i yielding 3i’.

VB(C-C)’ VB(C-C)1 LP2

LP1

LP1’ VB(C-C)2

LP2’

(A) (B)

Figure 4: Doubly occupied generalized valence bond (GVB) orbitals of lone pairs of oxygen of nitro (-NO2) functional group substituent (LP1, LP1’, LP2 and LP2’) (A and B) and singly occupied GVB orbitals of C-C sigma bond [VB(C-C)1, VB(C-C)2 and VB(C-C)’] (B).

Table 4: Overlap matrix of selected GVB orbitals of molecule 3f of two oxygen lone pairs (LP1, LP2, LP1’ and LP2’) and of two C-C sigma bonds [VB(C-C)1, VB(C-C)’ and VB(C-C)2].

LP1 LP1’ LP2 LP2’

VB(C-C)1 0.007 0.019 0.054 0.045

VB(C-C)2 0.005 0.005 0.005 0.006 VB(C-C)’ 0.008 0.006 0.034 0.009 114

When comparing topological data between 3d and 3f in Table 2, we observe that the trifluoromethyl group in 3d has a less electron withdrawing effect than the nitro group in 3f, since carbon atomic charge is less positive in 3d, carbon atomic energy is more stable in 3d, and LI is greater in 3d. In addition, the –CF3 group also decreases the charge density in the C-C bonding region as noted from its DI and bond order (Table 1). In terms of molecular energy, the stability of 3d is quite similar to that from 3a and it is slightly higher than that from 3f.

To sum up the energetic influence of the studied substituents in tetrahedrane cage we can note that

EWGs such as –CF3 do not affect the relative stability of the tetrahedrane cage, while with a stronger EWG

(e.g., –NO2) the tetrahedrane cage becomes slightly more unstable. On the other hand, slight EWG such as –

CH3 imparts a higher instability in tetrahedrane cage. Conversely, a -EDG such as –SiF3 makes the tetrahedrane cage more stable, while a -EDG (e.g. –NH2 and –OH) disrupts the tetrahedrane cage. These trends can be observed in the decreasing order of Gƒ (3c > 3e > 3a > 3d > 3f > 3b) and Hƒ (3c > 3e > 3a > 3d > 3f > 3b), without the silicon and germanium parents.

The integration of the atomic basins gives the value of the bond path angle of the electron density function. The bond path angle involving the atoms in the cage of all tetrahedrane derivatives and Si/Ge parents is higher than the corresponding geometric bond angle (Table 1). The increase of the bond angle in cyclopropane ring (which is part of the tetrahedrane cage) decreases its ring strain (or cage strain) which is 59 so-called banana bond. However, there is no linear relation between bond path angle and Gƒ for the studied series.

The tetrahedrane cage with silicon and germanium atoms, 4a and 5a, were also obtained, as well as trifluoromethyl tetrasubstituted germanium cage, 5b. However, the trifluoromethyl tetrasubstituted silicon parent disrupted during optimization procedure (Scheme 3). Formation enthalpy, Hƒ, indicates that they have intermediate stability between the most stable (3c and 3e) and least stable (3a, 3b, 3d and 3f) substituted tetrahedranes. Nonetheless, this stability order does not match the aromaticity order, according to NICS and D3BIA, where the set 4a, 5a and 5b are the least aromatic.

Table 3 shows the topological values (퐷퐼̅̅̅ , DIU, δ and ρ(3,+3)) associated with the D3BIA formula for homoaromatic and cage structures, along with NICS values in the geometrical center of the tetrahedrane (or parent) cage for 3a, 3b, 3c, 3d, 3e, 3f, 4a, 5a and 5b. The aromaticity of the substituted tetrahedranes, germanium and silicon parents were investigated with two aromaticity indices: NICS and D3BIA. Hereafter we mention the increase or decrease of NICS with respect to the aromaticity where aromaticity increases with higher negative values of NICS and decreases with lower negative values (or higher positive values) of NICS. 115

Figure 5 shows the linear relations between NICS and D3BIA where Figure 5A lacks only one molecule, 3f, and Figure 5B has all the studied molecules (3a-3f, 4a, 4b and 5a).

NICS X D3BIA NICS X D3BIA 20 20 D3BIA D3BIA

15 15

10 10 R² = 0.9142 R² = 0.7664 5 5

0 0 -60 -40 -20 0 20 40 -60 -40 -20 0 20 40 NICS / ppm NICS /ppm (B) (A)

Figure 5: (A) Plot of NICS versus D3BIA for 3a-3e, 4a, 4b and 5a; (B) Plot of NICS versus D3BIA for 3a- 3f, 4a, 4b and 5a.

Figure 5A shows a very good linear relation between NICS and D3BIA where 3f was excluded because NICS in 3f is higher (more negative) than it is supposed to be since 3c is 10.81 kcal mol-1 more stable than 3f but they have the same NICS value. Probably, this is the influence of lone pairs of oxygen atoms from nitro group that interact with the tetrahedrane cage of 3f (Figure 4 and Table 4) which enhances its NICS value.

Table 3: Average DI, 퐷퐼̅̅̅, of the carbon or silicon or germanium atoms in the tetrahedrane (or parent) cage, the corresponding DIUs, the degree of degeneracy (δ) involving the atoms of the tetrahedrane (or parent) cage, the charge density of the cage critical points [ρ(3,+3)], D3BIA and NICS values for 3a, 3b, 3c, 3d, 3e, 3f, 4a, 5a and 5b.

Molecule 푫푰̅̅̅̅ DIU δ ρ(3,+3)/au D3BIA NICS / ppm 3a 1.025 99.97 1.0 0.182 18.24 -48.53 3b 0.979 99.89 1.0 0.176 17.61 -45.31 3c 0.954 99.99 1.0 0.177 17.68 -44.95 3d 0.961 99.98 1.0 0.182 18.21 -47.45 3e 1.029 99.97 1.0 0.187 18.67 -54.54 3f 0.997 96.77 0.5 0.181 8.77 -45.30 4a 0.933 99.95 1.0 0.046 4.57 -1.03 5a 0.951 99.99 1.0 0.040 4.00 1.85 5b 0.911 99.93 1.0 0.039 3.88 32.33

116

The alkyl groups (in 3b and 3c) decrease both NICS and D3BIA in comparison with 3a. Likewise, the trifluoromethyl and nitro groups in 3d and 3f, respectively, also decrease both aromaticity indices with respect to 3a. On the other hand, the trifluorosilyl group increases both NICS and D3BIA in 3e. Then, we may assume that EWGs decrease the aromaticity of the tetrahedrane cage while the EDG increases its aromaticity. The decreasing order of formation enthalpy and NICS for 3a-3f is nearly the same (except for 3c where hydrogen-hydrogen bonds play an important role on its stabilization): 3e > 3a > 3d > 3b  3f (for

NICS) and 3e > 3a > 3d > 3f > 3b (for Hƒ). Then, we may assume that the aromaticity (influenced by EWGs or EDGs) is an important stability factor in the tetrahedrane cage besides the intramolecular hydrogen-hydrogen bonds in 3c.

As aforementioned, regarding the set 4a, 5a and 5b, there is no relation between their stability and aromaticity orders. According to NICS, they are not aromatic molecules, but their stability is higher than that in 3a, 3b, 3d and 3f. Probably, the stabilities of 4a, 5a and 5b are influenced by their predecessors’ stabilities. Regarding 4a, its H1 is the least negative, which means that its corresponding silicon cyclobutadiene parent is the least stable of all set of substituted cyclobutadienes. Then, the high instability of predecessor of 4a plays an important role on the large relative stability of 4a.

Figure 6 shows the singly occupied GVB orbitals of sigma C-C bonds, VB(C-C)’, VB (C-C)1 and

VB (C-C)2, of 3a, 3b, 3f and cubane. Table 5 shows the overlap integral between the singly occupied GVB orbitals represented in Figure 6. Tetrahedrane and cubane are platonic hydrocarbons where only cubane was 60 synthetically obtained. The overlap between vicinal GVB orbitals in cubane, VB(C-C)1 – VB(C-C)2, is

0.224 – nearly four times smaller than the overlap between GVB orbitals of each C-C bond, VB (C-C)1 – VB(C-C)’. The overlap between vicinal GVB orbitals in tetrahedrane is higher than that from cubane, 0.279, where we can infer that vicinal C-C bonds tend to increase the electronic repulsion in the tetrahedrane cage, which could partially explain the instability of tetrahedrane with respect to cubane. The electron withdrawing effect of alkyl group can also be noticed by the slightly smaller overlap between vicinal GVB orbitals in 3b (Table 5). Likewise, the higher electron withdrawing effect of the nitro group can also be noticed from overlap of GVB orbital because this value in 3f is similar to that from cubane. However, as aforementioned 3f is more unstable than 3a. Then, the electronic repulsion from vicinal C-C bonds within the cage is not the main instability factor of tetrahedrane cage which means that the main reason for tetrahedrane instability should be attributed to its angle strain.

117

VB(C-C)’

VB(C-C)1

VB(C-C)2 (A) (B) (C)

(D)

Figure 6: Singly occupied generalized valence bond (GVB) orbitals of C-C sigma bond [VB(C-C)1, VB(C-

C)2 and VB(C-C)’] of 3a (A), 3b (B), 3f (C) and cubane (D).

Table 5: Overlap matrix of selected GVB orbitals, VB(C-C)’, VB (C-C)1 and VB (C-C)2, of 3a, 3b, 3f and cubane.

VB(C-C)1 3a 3b 3f Cubane

VB(C-C)2 0.279 0.272 0.224 0.224 VB(C-C)’ 0.828 0.826 0.823 0.810 aVB orbitals of 3a bVB orbitals of 3b cVB orbitals of 3f dVB orbitals of cubane

The Figures 7, 8 and 9 shows the plots of total energy (in Hartree) versus the time of trajectory of 5 fs using atom-centered density matrix propagation (ADMP) method for tetrahedrane and substituted tetrahedranes (3a, 3c and 3e) where the starting structure was previously optimized at ωB97XD/6- 311++G(d,p) level of theory. We used ADMP calculations to test the stability of selected tetrahedrane cages under dynamic simulation. Some corresponding structures from selected points of each trajectory are also depicted in Figures 7, 8 and 9. In Figure 7, we can see that no significant changes occur in 3a structure at the 118 beginning of the trajectory (0.2 to 0.6 fs). However, at about 1.3 fs the C2-H7 bond is broken. At 2.8 fs the C3-H5 bond is broken and C4-H8 bond length increases. Moreover, the C-C bond lengths of three cyclopropane rings of tetrahedrane cage shorten. At about 4.2 fs, the C4-H8 bond is broken and at 5.0 fs the C3-H5 bond is restored. Then, during the ADMP trajectory, 3a undergoes structural changes in its structure but it is not restored to its starting optimized geometry at any point. On the other hand, there is no significant change in the total energy of 3c during the course of its trajectory (Figure 8). Then, 3c has a thermochemical and dynamical relative stability mainly due to the large amount of stabilizing H-H bonds. Likewise, 3e presented a similar behavior (Figure 9), but its relative stability is given by –SiF3 group that acts as σ-EDG, where in 2.5 fs and 4.4 fs occurs a decreasing in two C-Si bond lengths (indicated by red arrows) according to the nature of donating electrons of trifluorosilyl group.

119

Time in Trajectory (femtosec) 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 -783.58070

-783.58072

-783.58074

-783.58076

-783.58078

-783.58080 Total Energy (Hartree) Total -783.58082

-783.58084

Figure 8: Plot of total energy, in Hartree, versus the time of trajectory, in femtoseconds, of 3c using ADMP method. The structure at the end of trajectory is also depicted.

Time in Trajectory (femtosec) 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 -1366.426550 -1366.426555 -1366.426560 -1366.426565 -1366.426570 -1366.426575 -1366.426580 -1366.426585

-1366.426590 Total Energy (Hartree) Total -1366.426595 -1366.426600 -1366.426605

Figure 9: Plot of total energy, in Hartree, versus the time of trajectory, in femtoseconds, of 3e using ADMP method. A couple of structures are depicted at the selected corresponding points of the trajectory.

120

Conclusions

The influence of substituents on the stability of the tetrahedrane cage reveals that strong electron withdrawing groups do not affect significantly the relative stability in the tetrahedrane cage, unlike weak EWG that generate a large instability in tetrahedrane cage. However, substituents that act as sigma electron donating groups (σ-EDG) stabilize the tetrahedrane cage, although π-EDGs leads to disruption of the tetrahedrane cage.

According to NICS and D3BIA and their relations with the corresponding formation enthalpy, the aromaticity is an important factor for the stability of the tetrahedrane cage. The -EDGs increase both aromaticity and stability of the tetrahedrane cage, whereas the EWGs decrease both aromaticity making and cage stability, except for 3c in which its stability is due to the intramolecular hydrogen-hydrogen bonds. Although there is no relation between aromaticity and stability for germanium and silicon tetrahedrane parents 4a, 5a and 5b their NICS and D3BIA values indicate that these molecules are not aromatic.

According to generalized valence bond method, one of the oxygen lone pair has an electronic repulsion with C-C bonds in its tetrahedrane cage and probably when four nitro groups are attached to the tetrahedrane cage the higher electronic repulsion may lead to disruption of corresponding substituted tetrahedrane. Moreover, the GVB analysis indicate that the instability of the tetrahedrane cage is mainly imparted by its high strain energy while the C-C electronic repulsion within the cage has a minor influence.

The ADMP analysis indicates that tetrahedrane 3a is dynamically unstable where some C-H bonds break during the simulation trajectory. Conversely, 3c and 3e remain its structure during the whole trajectory. Therefore, thermodynamic and dynamic results are in accordance for the relatively stable and unstable cage structures.

Acknowledgments

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6. CAPÍTULO 4 – Formatação de artigo a ser submetido para o periódico The Journal of Physical Chemistry B.

Docking, Molecular Dynamics, ONIOM, QTAIM: theoretical investigation of inhibitory effect of Abiraterone on CYP17.

Norberto K. V. Monteiro, Sérgio R. B. Silva and Caio L. Firme*

Institute of Chemistry, Federal University of Rio Grande do Norte, Natal-RN, Brazil, CEP 59078-970.

*Corresponding author. Tel: +55-84-3215-3828; e-mail: [email protected]

ABSTRACT

After skin cancer, prostate cancer (PC) is the most common cancer among men in the western world. Once metastasized, the prostate cancer treatment becomes palliative. The initial treatment is performed with androgen deprivation therapy (ADT) leading to a decrease in the level of serum testosterone (chemical castration). However, resistance to castration-therapy can develop, resulting in men with metastatic castration-resistant prostate cancer (mCRPC). Recently, a novel hormonal agent, Abiraterone acetate (abiraterone) has been shown to increase the survival rate by 4.6 months when compared to placebo in men with mCRPC, by inhibiting the key enzyme for the biosynthesis of androgens [17α-hydroxylase-17,20-lyase (CYP17)]. In this paper, we present a new computational protocol in order to quantify the affinity of abiraterone (ABE) compared to pregnenolone (PREG). Molecular dynamics and molecular docking were used to obtain the CYP17-ABE and CYP17-PREG complexes and to obtain the interaction energies of ligands ABE and PREG. From the ONIOM (B3LYP:AMBER) method, we find that the interaction electronic energy of ABE is 21.38 kcal mol-1 lowest than PREG. The quantum theory of atoms in molecules (QTAIM) method was used to quantify the interactions between ABE (or PREG) and CYP17. The results indicate that ABE has a higher electronic density in their interactions compared to PREG.

Keywords: Prostate cancer; CYP17; Abiraterone; Pregnenolone; Molecular dynamics; ONIOM; QTAIM.

INTRODUCTION

Prostate cancer (PC) is the most common type of cancer in the US and Europe.1 Its development occurs mainly by the action of androgens testosterone (T) and dihydrotestosterone (DHT), which are synthesized by 125 the testis and adrenal glands through the androgen receptor (AR). The androgen action consists of the following steps: (a) the entrance of testosterone in the prostatic cells which is converted to dihydrotestosterone by 5α-reductase; (b) in presence of DHT, AR is released from heat shock protein 90 (HSP90); (c) the AR further undergoes dimerization, phosphorylation and translocation to the nucleus2,3 (Figure 1). The translocated receptor binds to the androgen response elements in the DNA, thereby activating transcription of AR target genes and thus leading to cell proliferation.4,5 In a reciprocal manner, after androgen withdrawal, the prostatic cells cease proliferation and undergo apoptosis.6 Then, there is a dependence of prostatic cells on AR signaling in neoplastic transformation, which is the basis for metastatic PC therapy.

Figure 1. Inside the cell, testosterone is converted to DHT and binds to AR. AR-DHT complex undergoes phosphorylation and is translocated to the nucleus in the form of dimer. Inside the nucleus, AR- dimer binds to DNA coregulators leading to transcription activation.

According to the above description, there is a biological close relationship between testosterone and PC.7 However, epidemiologic studies have shown that the association between testosterone and PC is not clearly understood.7,8 Solania,9 formulated the time-dependency theory which postulates that the endocrine biology of prostatic tissue is dependent on the exposure time at a given concentration of sex steroid. This postulate is a consequence of the results obtained by Solania and collaborators10 where serum testosterone levels could be stable over time in PC patients.

The most common initial treatment of PC is the androgen deprivation therapy (ADT) resulting in temporary regression of disease in most patients. This therapy is accomplished by the chemical castration with luteinizing-hormone releasing-hormone (LHRH) analogs, or surgical castration (orchiectomy).1,11,12 The ADT results in a reduction of about 90% of serum testosterone.13 However, ADT do not affect the 126 production of adrenal and intra-tumoral androgens, which may be relevant in restoring disease14 resulting in patients with metastatic castration-resistant prostate cancer (mCRPC).15 As aforementioned, androgens T and DHT play an important role in development of PC and then, potent and specific compounds that inhibit androgen synthesis may be effective for the treatment of PC.16 A promising target is the 17α-hydroxylase- 17,20-lyase (CYP17) involved in the biosynthesis of T and DHT through of two key reactions, which act sequentially. First, by converting pregnenolone (PREG) to 17α-hydroxypregnenolone and progesterone to 17α-hydroxyprogesterone; and second by converting 17α-hydroxypregnenolone to dehydroepiandrosterona and 17α-hydroxyprogesterone to androstenedione.17,18 Furthermore, Auchus19 described an alternative pathway in the biosynthesis of DHT testosterone-independent wherein CYP17 also acts in this pathway, emphasizing its importance as a therapeutic target, because its inhibition exerts control over androgen synthesis. Ketoconazole is known as antifungal which inhibits nonspecifically the CYP17 and reduces androgen levels, however, its use is associated with several side effects including hepatotoxicity, gastrointestinal toxicity, adrenal insufficiency and their effects on other cytochrome enzymes.14,20,21 Therefore, is very important to search and development of more potent and specific novel CYP17 inhibitors as well as the identification of these inhibitors.

Many experimental and computational studies were reported on inhibitory molecules against CYP17.1,20–24 Recently, several promising agents with different mechanisms of action and therapeutic targets have demonstrated efficacy and four new drugs (cabazitaxel, sipuleucel-T, denosumab, and abiraterone acetate) were approved by FDA for the treatment of patients with mCRPC.25–27

Abiraterone (Figure 2) (ABE) is a high selective irreversible CYP17 inhibitor (IC50, 2 to 4 nmol/L), more potent and selective and less toxic than ketoconazole.12,28,29 Moreover abiraterone reduces serum testosterone levels below 1.0 ng/dl.30,31 In phase 3 trials conducted in 1195 patients, abiraterone increased the survival rate to 15.8 months compared with 11.2 months in the control (placebo).32

Figure 2. Chemical structure of ABE

The recognition of ligands by proteins is a key process of many biological functions and systems that involve the enzymatic catalysis, cell signaling and protein conformational switching.33,34 The binding affinity is a reflection of the molecular recognition. Prediction of the binding affinity are targets of several 127 computational studies.35–38 Molecular dynamics simulations have been used extensively to predict the affinity of protein-ligand interactions.39–41 In the present work, molecular dynamics was used to predict the affinity of the ligands ABE and PREG against CYP17 enzyme.

The main focus of our study is to quantify the affinity of ABE and PRE with the CYP17 enzyme and investigate the specific interactions protein-ligand. We performed the geometric optimization of the CYP17- ABE and CYP17-PREG complexes and CYP17 enzyme with ONIOM (B3LYP/6-311G(d,p):AMBER) method (Figures 10A, 10B and 10C, respectively).

Experimental methods for evaluating the affinity of a specific ligand may require a long working time and a high monetary cost. The objective of the present work was investigate the inhibitory effect of the ABE on CYP17. For this study, we compared the stabilities between CYP17-ABE and CYP17-PREG complexes by using classic molecular dynamics simulations, molecular docking, the ONIOM hybrid method and, QTAIM method. Our comparative study will provide theoretical information for distinguishing inhibitors and natural substrates pointing out the direction of predicting new CYP17 inhibitors.

COMPUTATIONAL DETAILS AND THEORETICAL METHODS

PREPARATION OF THE STRUCTURE AND CLASSIC MOLECULAR DYNAMICS SIMULATIONS

The X-ray crystal structure of human CYP17 in complex with abiraterone was obtained from Protein Data Bank (PDB ID code: 3RUK)42. Protons were added to the protein structure using GROMOS96 43a1 force field43 assuming typical protonation states at pH 7.0. On the other hand, the imidazole side chain of the histidine residue (His) contains two nitrogen atoms that can be protonated (ND1 and NE2) at pH 7.0, called HID, HIE and HIP44 (Figure 3). The protonation state of histidine residue close to the heme group (His373), was systematically varied. The value of the CA-CB-CG-ND1 dihedral angle (χ1) (Figure 3) was also evaluated because an incorrect protonation of a His residue can result in large fluctuations and rotations of the His side chain. Different studies about the analysis of the histidine protonation state have been performed.44–46 In all these studies, the root-mean-square deviation (RMSD) is used to analyze the state of protonation. To determine the proportion of neutral (HID and HIE) and positively charged (HIP) His on CYP17 structure, we used the Henderson-Hasselbalch equation.47 Molecular dynamics (MD) simulations were performed by GROMACS 4.6.5 program48 for each protonation state using GROMOS96 43a1 force field with SPC water molecules.49,50 CYP17 has 26 aspatic acids, 24 glutamic acids, 34 lysines, 20 arginines and 1 positively charged histidine (His160), totaling a +5 charge. The histidine close to the heme group (His373) is hydrogenated in nitrogen NE2 (HIE). Counter ions Na+ and Cl- were added to neutralize charge 128 and maintain the system at a concentration of 150 mM. The system was immersed in a rhombic dodecahedron box and the minimal distance between the protein and the wall of the cell was 1.2 nm. Covalent bonds in the protein were constrained using the LINCS algorithm.51 Electrostatic forces were calculated with the PME52 implementation of the Ewald summation method, and the Lennard–Jones interactions were calculated within a cut-off radius of 1.2 nm. The system was equilibrated according to the following protocol: the solvated CYP17-abiraterone complex geometry was minimized twice by the steepest descent algorithm for 5000 steps with tolerances of 1000 and 500 KJ mol-1 nm-1 followed by two gradient conjugated algorithm for 50000 steps with tolerances of 250 and 100 KJ mol-1 nm-1. In order to adapt the water to the complex due the greatest strain dissipated from the system and couple the system to the heat bath using NVT ensemble at temperature of 310K, several short MD simulations were performed. Initially, 50, 100, 150 and 200-ps molecular dynamics were performed where the non-hydrogen atoms of the complex were kept fixed and the water can move freely. After that, 50, 50 and 37.5-ps MD simulations were performed where the water and abiraterone move freely, following of 25, 50 and 75-ps MD simulations where the heme group is released. Then, two unrestrained MD simulations of 50 and 75-ps were performed. In order to control the pressure coupling of system, 200-ps MD simulation were performed at pressure of 1 bar using isotropic pressure bath with Berendsen pressure coupling algorithm.53 Finally, 50-ns production MD simulation using NVT ensemble was performed for each protonation state in order to assess the stability of the system. The RMSD of C-α of His 373 residue and CYP17 structure was measured with respect to the respective X-ray structure. Snapshots were collected every 10-ns MD simulation and RMSD and dihedral angle χ1 average values of the His 373 residue of each protonation state were analyzed. All data represent five snapshots and were expressed as mean ± SD (standard deviation). Differences between groups were compared by using ANOVA54 and Tukey Test.55 Differences were considered significant when p value was less than 0.05. Statistical data were analyzed by GraphPad Prism 5.0 software.

(CB) (CA) (CG) (ND1)

A B C

Figure 3. The protonation states of histidine at pH 7.0. HID (A) HIE (B) and HIP (C).

129

MOLECULAR DOCKING

The automated docking was conducted in order to find the best conformations and energy ranking of PREG inside the active site of the human CYP17 (previously equilibrated from MD simulation as aforementioned). The docking simulation was carried out using AutoDock Vina 1.1.2.56 The three- dimensional structure of PREG was obtained from Protein Data Bank (http://www.rcsb.org). Polar hydrogens were added and Gasteiger-Marsili partial charges57 were assigned to the PREG and human CYP17 with the AutoDock Tools (ADT).58 All torsions to PREG were allowed to rotate during docking. Grid box with size 38 Å was centered on the iron of the heme group. The AutoDock Vina parameter, Exhaustiveness, that determined how the program searches exhaustively for the lowest energy conformation was set to 16 (the default value is 8). After docking, resulting conformations of PREG were ranked based on the interaction energy. The best conformation of PREG in complex with human CYP17 was chosen with the lowest docked energy and then was submitted to 50-ns MD simulation.

The ABE was also subjected to the same docking methodology as described above for PREG on CYP17. We found that the conformation of the docked ABE was very similar to that found in the CYP17- ABE complex in the Protein Data Bank (PDB ID code: 3RUK).

TOPOLOGY

The topology contains all the necessary information to describe a molecule in a molecular simulation. All this information includes atom types and charges (nonbonded parameters) as well as bonds, angles and dihedrals (bonded parameters).48 In addition, exclusion and constraints parameters are also generated by the force field. For this work, the topology was generated for the protein without ligands (ABE and PREG) through the GROMOS96 43a1 force field. The ligands topology was generated by the PRODRG59 2.5 automated server.

MOLECULAR DYNAMICS FLEXIBLE FITTING (MDFF) METHOD

In order to analyze conformational changes in the three-dimensional structure of CYP17 complexes, before and after the molecular dynamics, the molecular dynamics flexible fitting (MDFF)60 method was applied. The MDFF method consists of a molecular dynamics that includes information from an experimental or simulated electron microscopy (EM) map for fitting an initial model of biomolecule in the target density map. The analysis of MDFF were performed with Visual Molecular Dynamics (VMD)61 software. 130

THE TEMPERATURE FACTOR

The temperature factor (or B-factor) describes the displacement of the atomic positions in three- dimensional protein structures from an average value.62 The B-factor will reflect the flexibility of atoms in relation to their average positions. In other words, B-factor is an important indicator of the flexibility of the protein structure. In this work, we used the B-factor to analyze the regions of the CYP17 complex structures which have undergone flexibility during the course of molecular dynamics.

CALCULATION OF MOLECULAR MECHANICS INTERACTION ENERGIES

The nonbonded interaction energies involves running two MD simulations: one for the ligand in the 푝 푤 protein biding site (퐸푏푖푛푑 ) and the other for ligand in water (퐸푏푖푛푑 ). The binding energies are composed of two separate energy terms: The van der Waals (vdW) interaction energy (VLJ) given by the Lennard –Jones 63–66 (LJ) potential and the electrostatic interaction energy (VC) given by Coulomb potential. The computed interaction energy is the sum of the LJ and Coulomb interaction energies which is given by:

푝 퐸푏푖푛푑 = (〈푉퐿퐽〉푝 + 〈푉퐶〉푝) (1) and

푤 퐸푏푖푛푑 = (〈푉퐿퐽〉푤 + 〈푉퐶 〉푤) (2)

where < > denotes MD averages of the nonbonded interactions between ligand and its receptor binding site or just water. 〈푉퐿퐽〉푝 is the LJ term for ligand-protein interaction; 〈푉퐿퐽〉푤 is the LJ term for ligand-water interaction; 〈푉퐶 〉푝 is the electrostatic term for ligand-protein interaction; and 〈푉퐶 〉푤 is the electrostatic term for ligand-water interaction. For this study, we analyse the nonbonded interaction energies between CYP17 푤 푝 and its ligands ABE and PREG. Two simulations are required to obtain the 퐸푏푖푛푑 and 퐸푏푖푛푑 : one with ligand bound to solvated protein (as described above) and one with the ligand free in solution. The latter was performed with 1.0-ns using NVT ensemble.

QM/MM CALCULATIONS 131

Hybrid methods combine two computational techniques in one calculation and allow the study of large chemical systems.67–69 The QM/MM method is a hybrid technique that combine a quantum mechanical (QM) method with a molecular mechanics (MM) method.70–73 ONIOM74 (N-layer Integrated molecular Orbital molecular Mechanics) is one of the QM/MM methods most employed and allows the combination of different quantum mechanical methods as well as classical methods of molecular mechanics. In general, the region of the system where the chemical process occurs is treated with a more accurate method as well as a QM theory, and the remainder of the system at low level. Many ONIOM studies are focused on biological systems, such as proteins,75–78 treating the active site by a high level method, using often density functional theory (DFT), and the rest of the protein by MM. In a two layer ONIOM calculation, the total energy of the system is obtained from three independent calculations:

퐸푂푁퐼푂푀 = 퐸푚표푑푒푙,푄푀 + 퐸푟푒푎푙,푀푀 − 퐸푚표푑푒푙,푀푀 = 퐸푚표푑푒푙,ℎ푖푔ℎ + 퐸푟푒푎푙,푙표푤 − 퐸푚표푑푒푙,푙표푤 (3)

The real system contains all the atoms (including both QM and MM regions) and is calculated only at the MM level. The model system is the QM region treated at QM level. To obtain the ONIOM energy, the model system also needs to be treated at MM and be subtracted from real system MM energy.79,80 In this work, the ONIOM method was applied in geometric optimization of the complexes (ABE or PREG in complex with human CYP17) from 50-ns production MD simulation. The high layer atoms were treated at the B3LYP/6-311G(d,p) level81–83 and the low layer was treated by means of the AMBER9484 force field implemented in the Gaussian 09 package.85

The Restrained electrostatic potential (RESP)86 atomic charges of the abiraterone and pregnenolone were derived at the HF/6-31G* level by Gaussian03 and AmberTools.87 The heme group, ABE (or PREG) and Cys442 were included in the high layer region, while the remaining residues were in the low layer.

The interaction energy of ABE (or PREG) was calculated as below:

푐표푚푝푙푒푥 푖푛푡 퐸퐶푌푃17+퐴퐵퐸 - 퐸퐶푌푃17 - 퐸퐴퐵퐸 = 퐸퐴퐵퐸 (4)

푐표푚푝푙푒푥 푖푛푡 퐸퐶푌푃17+푃푅퐸퐺 - 퐸퐶푌푃17 - 퐸푃푅퐸퐺 = 퐸푃푅퐸퐺 (5)

푐표푚푝푙푒푥 푐표푚푝푙푒푥 where 퐸퐶푌푃17+퐴퐵퐸 and 퐸퐶푌푃17+푃푅퐸퐺 are the electronic energies of CYP17 in complex with ABE and PREG, respectively. 퐸퐶푌푃17 is the electronic energy of CYP17; 퐸퐴퐵퐸 is the electronic energy of ABE; 퐸푃푅퐸퐺 is the 푖푛푡 푖푛푡 electronic energy of PREG; 퐸퐴퐵퐸 and 퐸푃푅퐸퐺 are the interaction electronic energies of ABE and PREG, respectively. The electronic density of the region enclosed by a cut-off radius of 3Å from ABE or PREG (binding pockets) which was used for QTAIM calculations, was performed at B3LYP/6-31G(d,p) level by means of Gaussian09 package, where all topological information were calculated using the AIM2000 132 software.88 The binding pockets in the CYP17 active site were generated through VMD software and are shown in Figure 4.

The electronic energies of CYP17, ABE, PREG, CYP17-ABE, CYP17-PREG complexes, as well as the electronic interaction energies of ABE and PREG are shown in Table 3. The electronic energy of complexes were obtained after optimization of its geometries. To obtain the electronic energy of the CYP17, the ligands (ABE and PREG) were previously removed from complexes and submitted to the single point calculation at B3LYP/6-311G(d,p) level. Already to obtain ligands electronic energies, these were removed from their complexes and submitted to the single point calculations with the same level of theory described earlier.

QUINTET AND TRIPLET STATE OF HEME

The Density Functional Theory (DFT) method was used in this paper because it is already well employed to porphyrins in general89–93 and B3LYP functional predicts the energy separation of the of FeII- porphyrin of deoxyheme between quintet and triplet states, with the quintet state 12 kJ mol-1 lowest in energy94 and this is agreement with experimental data. A study conducted in 1936 by Pauling and Coryell95 measured the magnetic moments of deoxyhemoglobin and oxyhemoglobin. The results showed that the ferroheme of deoxy state contains four unpaired electrons. In this work, we checked the electronic energy differences of ferroheme of triplet and quintet states using six functionals (B3LYP,96,97 BP86,98,99 BVWN,100 101 102 103–105 M062X, ωB97XD and PBE1PBE ) at 6-311G(d,p) basis set through a simple Fe(NH3)2(NH2)2 complex. We found that the quintet state is about 5.85 kcal mol-1 more stable compared to triplet state using B3LYP functional (Table S1 and Figure S1, Supporting Information). Therefore, we used the quintet spin state (S=2) for heme group of CYP17 at the B3LYP/6-311G(d,p) level on the high layer.

QTAIM RATIONALE

The Quantum Theory of Atoms in Molecules (QTAIM) is a quantum model considered efficient in the study of chemical bond106 and characterization of intra- and intermolecular interactions.107–109

According to the quantum-mechanical concepts of QTAIM, the observable properties of a chemical system are contained in its electron density, ρ. The trajectories of the electron density are obtained from the gradient vector of the electron density (∇ρ)110 and the set of these gradient trajectories form the atomic basins (Ω). The bond critical points [BCP´s, (3,-1)] of the charge density are obtained where ∇ρ = 0111 and its location is performed through the Laplacian of the charge density (∇2ρ) that is given by the trace of the Hessian matrix of the electronic density.112 BCPs may give information about the character of different 133 chemical bonds or chemical interaction, for example, metal-ligand interactions,113–115 covalent bonds,112,116 different kinds of H-bonds,117–119 so-called hydrogen-hydrogen bonds108,120–123 as well as other weak noncovalent interactions.124–126 In addition, several studies show that there is a direct relation between electronic density of a BCP and the interaction energy between two atomic centers forming the bond, where there is a linear relation for small ranges of electronic density values.127–130

Figure 4. Schematic representations of pockets of the CYP17-ABE (A) and CYP17-PREG (B) complexes.

RESULTS

Molecular dynamics was used in order to achieve the stability of the CYP17-ABE complex and to detect the protonation state of His 373. The plots of RMSD of the Cα of the CYP17-ABE complex of each protonation state of His 373 are shown in Figures 5A, 5B and 5C. The RMSD of the Cα atoms of the 134 complex for each protonation state first increased as far as plateau at 5.0 ns, indicating that equilibrium had been reached (Figures 5A, 5B and 5C) wherein the superposition of C-α of CYP17 complexes onto the X- ray coordinates reaching convergence within 50-ns MD simulation. The plateau reached had an average value of 0.39 nm. There was no difference observed between RMSD´s, since the only difference between simulations is the protonation state of the His373.

The Figures 6A and 6B shows the plots of RMSD and χ average values, respectively, of His373 for each protonation state. These average values were obtained every 10-ns snapshots from 50-ns production MD of CYP17-ABE complex.

In order to obtain the CYP17-PREG interaction, and then the appropriate enzyme-substrate binding pose, molecular docking using AutoDock Vina was performed. The binding affinity was characterized by binding energy (ΔG) value (Table 1). The docking poses of PREG within the active site of the CYP17 enzyme are shown in Figures 7A, 7B, 7C and 7D. According to the Autodock Vina binding energies, the best docking pose (pose 1) has a ΔG value of -12.2 kcal mol-1 (Figure 7A) and then was submitted to 50-ns MD simulation (Figure 8) where the plateau was reached at 20 ns with superposition of C-α RMSD value onto the X-ray coordinates of 0.38 nm. The structure obtained after a period of 50 ns simulation was named CYP17-PREG complex.

0.500 0.500

0.400 0.400

0.300 0.300 nm 0.200 nm 0.200

0.100 0.100

0.000 0.000 0.00 10.00 20.00 30.00 40.00 50.00 0.00 10.00 20.00 30.00 40.00 50.00 Time (ns) Time (ns) (A) (B)

0.500

0.400

0.300

nm 0.200

0.100

0.000 0.00 10.00 20.00 30.00 40.00 50.00 Time (ns)

(C)

135

Figure 5. Plots of RMSD (in nm) versus time of simulation (in ns) for CYP17-ABE complex of each protonation state of His 373. HISD (A), HISE (B) and HISH (C).

0.16 1 0.9 0.14 0.8 0.12 0.7 0.1 0.6

0.08 0.5 nm

0.06 (degree) 0.4 χ 0.3 0.04 0.2 0.02 0.1 0 0 HID HIE HIP HID HIE HIP (A) (B)

Figure 6. Plots of RMSD (A) and χ1 (B) of the His373 of each protonation state. The data represent the average of five 10-ns snapshots from 50-ns production MD of CYP17-ABE complex for each protonation state expressed as mean standard deviation. It was considered a p value less than 0.05.

Table 1. Binding energies of CYP17 with PREG. Poses Binding Energy / kcal mol-1 1 -12.2 2 -10.6 3 -10.2 4 -9.5

136

A B

C D

Figure 7. The binding poses of PREG in the active site of the enzyme CYP17.

0.500

0.400

0.300

nm 0.200

0.100

0.000 0.00 10.00 20.00 30.00 40.00 50.00 Time (ns)

Figure 8. Plot of RMSD (in nm) versus time of simulation (in ns) for CYP17-PREG complex

The Figure 9A shows the ABE, heme group and residue CYS442 of the CYP17-ABE. Whereas the Figure 9B shows the PREG, heme group and residue CYP442 of the CYP17-PREG complex. Both structures were obtained after 50-ns MD simulations. We have found that after 50-ns MD simulations for CYP17-ABE and CYP17-PREG complexes, there is no formation of a complex hexacoordinate with octahedral geometry which is imposed by Fe+2 atomic d-orbitals. Conversely, hexacoordinated states are 137 formed after geometric optimization of complexes by ONIOM calculations (Figures 10A and 10B). This leads us to infer that classical MD simulations does not adequately describe the d-orbitals geometry. In addition to the limitations of MD simulations, we can see in Figure 9A and 9B that the porphyrin ring of heme group is nonplanar where the opposite occurs with the quantum mechanical calculations (Figures 10A and 10B).

(C17)

(N22)

(Fe)

(SG)

(SG) A B

Figure 9. ABE, heme group and CYS442 of CYP17-ABE complex (A) and PREG, heme group and CYS442 of CYP17-PREG complex (B).

The Table 2 shows the values of Lennard-Jones potential for protein-ligand (CYP17-ABE/PREG) and ligand-water interactions (〈푉퐿퐽〉푝 and 〈푉퐿퐽〉푤) and Coulomb potential for protein-ligand and ligand-water interactions (〈푉퐶 〉푝 and 〈푉퐶〉푤), as well as 〈푉퐿퐽〉푝 + 〈푉퐶 〉푝 and 〈푉퐿퐽〉푤 + 〈푉퐶 〉푤 are referred as the binding energies between ligand (ABE or PREG) and CYP17 and between ligand and water respectively.

According the Eq. 1 and Eq. 2 and the data of Table 2, the interaction energy for ABE and the CYP17 binding site and for ABE and water are -53.00 kcal mol-1 and -35.50 kcal mol-1 respectively. Likewise, the interaction energies for PREG are -68.90 kcal mol-1 and -60.20 kcal mol-1 (Table 2). The 푝 푤 difference between interaction energies of ligand for enzyme binding site and for water (퐸푏푖푛푑 − 퐸푏푖푛푑) indicates its diffusion trend of the water  enzyme binding site. The more negative energy difference value, the greater the diffusion trend of the water  enzyme binding site.

138

-1 Table 2. Values of Lennard-Jones potential (VLJ) and Coulomb potential (VC), in kcal mol , between CYP17 (p)/water (w) and ligands ABE and PREG.

Interaction ABE / Interaction PREG / Potentials kcal mol-1 kcal mol-1

〈푉퐿퐽〉푝 -52,49 -39,55

〈푉퐶〉푝 -0,51 -29,35

〈푉퐿퐽〉푝 + 〈푉퐶〉푝 -53.00 -68.9

〈푉퐿퐽〉푤 -33,53 -23,90

〈푉퐶 〉푤 -1,52 -36,30

〈푉퐿퐽〉푤 + 〈푉퐶〉푤 -35.5 -60.20

The optimized structures of the QM (high layer) and MM (low layer) of the CYP17-ABE and CYP17-PREG complexes and enzyme CYP17 are shown in Figures 10A, 10B and 10C respectively. The electronic energy values of CYP17, ABE, PREG, CYP17-ABE and CYP17-PREG complexes are shown in Table 3. As expected, the interaction energy results of molecular mechanics and QM/MM calculations for ABE and PREG are not in agreement. This is primarily due to the description of the spin state of the iron heme from ONIOM calculations, which does not occur in molecular mechanics calculations (more detail see section discussion).

Table 3. Values of electronic energy of CYP17 enzyme, ABE, PREG, CYP17-ABE and CYP17-PREG complexes.

Interaction energy / Molecule E / kcal mol-1 e kcal mol-1 ABE -667313.50 -61.75a

PREG -608804.64 -40.37b

CYP17 -2249240.31

CYP17-ABE complex -2916615.57

CYP17-PREG complex -2858085.32 aInteraction energy derived from Eq. (4) aInteraction energy derived from Eq. (5)

139

(Fe)

(C17)

(SG) (N22)

(SG)

C A B

Figure 10. Optimized structures of the QM (highlighted) and MM (gray lines) layers of the CYP17-ABE (A) and CYP17-PREG (B) complexes and CYP17 enzyme (C) at the ONIOM (B3LYP:AMBER) level.

The virial graphs of binding pockets of the abiraterone and pregnenolone in the CYP17 active site are shown in Figures 11A and 11B, respectively. Table 4 shows the charge densities (ρ) of the bond critical points (BCPs) of the bond paths between abiraterone (or pregnenolone) and CYP17 active site.

B A

Figure 11. Virial graphs of binding pockets of CYP17-ABE (A) and CYP17-PREG (B) complexes.

Table 4. Electronic density (ρ), in au., of the critical points between abiraterone (or pregnenolone) and CYP17 active site.

Complex BCP ρ (au.) Complex BCP ρ (au.)

26 0.0003 28 0.0018 CYP17-ABE 31 0.0058 CYP17-PREG 31 0.0006 34 0.0011 36 0.0012 140

42 0.0027 39 0.0007 47 0.0001 49 0.0017 51 0.0001 51 0.0004 53 0.0021 53 0.0005 60 0.0010 56 0.0008 68 0.0017 58 0.0006 72 0.0026 59 0.0005 75 0.0013 62 0.0002 76 0.0014 65 0.0002 81 0.0012 79 0.0073 82 0.0007 82 0.0026 92 0.0010 86 0.0017 97 0.0033 90 0.0010 110 0.0003 101 0.0005 127 0.0017 107 0.0010 149 0.0009 122 0.0009 150 0.0033 126 0.0012 154 0.0008 127 0.0002 163 0.0012 130 0.0059 168 0.0016 131 0.0023 182 0.0013 142 0.0012 195 0.0003 143 0.0018 196 0.0012 155 0.0021 200 0.0012 155 0.0021 204 0.0015 165 0.0010 206 0.0008 174 0.0004 208 0.0020 185 0.0016 212 0.0003 190 0.0010 219 0.0004 199 0.0014 225 0.0022 202 0.0006 231 0.0018 212 0.0015 233 0.0004 213 0.0004 238 0.0012 217 0.0009 246 0.0012 220 0.0001 258 0.0017 232 0.0010 141

286 0.0022 233 0.0005 300 0.0012 237 0.0010 311 0.0012 253 0.0006 323 0.0015 258 0.0006 458 0.0011 287 0.0007 294 0.0007 300 0.0008 362 0.0005 Total 0.0610 0.0560

DISCUSSION

From Henderson-Hasselbalch equation the proportion found between positively charged and neutral His was 1:23. Since CYP17 has fifteen His in its structure, only one histidine was protonated on nitrogens ND1 and NE2, the histidine 160. Molecular dynamics were employed to assess the stability of the CYP- ABE complex and define the protonation state of His373 situated close to the heme group. According to the plot shown in Figure 6A, there is no significant difference between RMSD average values of the three protonation states. The RMSD indicated that there have been no large variations in the positions of C-α of HID, HIE and HIP protonation states compared to X-ray structure. Likewise, the average values of dihedral angle showed no significant differences between protonation states (Figure 6B). Therefore, subsequent theoretical studies involving CYP17 enzyme, its histidine residue close to the heme group (His373), can assume any protonation state.

There are no significant differences between structures before and after the production MD simulation of CYP17-ABE and CYP17-PREG complexes as shown in structural fitting (Figures S2A and S2B, Supporting Information). These data are in accordance with the B-factor62 (that reflects the fluctuation of an atom about its average position) of complexes (Figures S3A and S3B, Supporting Information).

According to the above results, the differences between the interaction energies of the ABE and PREG are -17.50 kcal mol-1 and -8.70 kcal mol-1 respectively. Although the PREG has a more negative interaction energy with the CYP17 binding site (-68.90 kcal mol-1) compared with ABE (-53.00 kcal mol-1), PREG also has a more negative energy interaction energy with water compared to the ABE (-60.20 kcal mol-1 and -35.5 kcal mol-1 respectively). In the other words, ABE has a higher diffusion trend water  enzyme binding site compared to PREG.

According to Eqs. (4) and (5), the electronic interaction energies of ABE and PREG have a value of - 61.75 kcal mol-1 and -40.37 kcal mol-1 respectively (Table 3). In other words, the interaction electronic 142 energy of ABE is 21.38 kcal mol-1 lowest in comparison to the PREG, indicating that the drug for metastatic prostate cancer has highest affinity. This more negative value of interaction electronic energy of ABE is reflected in a smaller distance with respect to iron heme (2.70 Å) compared to PREG (5.80 Å) (Figures 10A and 10B, respectively).

According to the above description, the interaction energy results between ligands (ABE or PREG) and CYP17 using molecular dynamics and ONIOM are not in agreement. PREG interaction energy by MD simulations resulted in a more stable interaction with the CYP17 compared to ABE. In contrast, ONIOM calculations showed that ABE has a higher affinity for the enzyme then PREG. Metalloproteins like Cytochrome P450 are hard to be characterized computationally due to the presence of the metal cation which exerts a strong Coulombic forces acting on structure backbone.131,132 Metals with unpaired electrons of d-shells of atomic orbitals have preferred coordination geometries as well as tetrahedral, octahedral, or square planar. This preference directly influences the position of the amino acids and therefore in the protein tertiary structure. In addition, the electronic structure of a metal is closely related to its coordination geometry. Different electronic configurations may be taken for a metal depending on the nature of its ligand. Therefore, there is a close relationship between electronic structure of a metal and the tertiary structure of the protein.132 For that reason, a classical description of a metal through molecular dynamics would be a failure, because the coordination geometry could undergo changes during the simulation. This is viewed in the distances between ABE or PREG and iron heme after molecular dynamics (Figures 9A and 9B, respectively) and after ONIOM calculations (Figures 10A and 10B, respectively) of CYP17-ABE and CYP17-PREG complexes. After ONIOM calculations, the bond length between aromatic nitrogen of ABE and iron heme (2.70 Å) and the bond length between gamma-sufur (SG) of CYS442 and iron heme (2.40 Å) were shortened compared to bond lengths after molecular dynamics (4.72 Å and 4.61 Å, respectively). The same occurred to the bond lengths between C17 of PREG and iron heme and between SG of CYS442 and iron heme. Furthermore, classical force fields are parameterized for a metal in specific coordination geometry and ligated to a specific ligand. The spin state of iron porphyrins is determined by the coordination geometry and the nature of ligands.133 In heme proteins, iron to be found in different coordination states: four, five or six-coordinate and exhibits formal charge of +2 or +3,134 where unligated ferroheme can have three spin states as a singlet, triplet or quintet (S = 0, 1 and 2, respectively). All these aspects about the metal cations, particularly iron porphyrins, should be approached in terms of electronic structure through the quantum mechanics calculations.135 The study of heme systems through the quantum-mechanical computational methods, as well as Hartree-Fock and density functional theory (DFT) contributed satisfactorily in the interpretation and understanding of the electronic structure of active site of hemoproteins.136,137 Thereby, hybrid quantum mechanical and molecular mechanical (QM/MM) methods can be very effective in the study of hemoproteins, as well ONIOM method. The difference in accuracy 143 between classical and hybrid methods was the main reason for the disagreement between interaction energy results.

The molecular graphs of binding pockets of the ABE and PREG, in Figures 11A and 11B respectively, shows all the interactions between the ligand (ABE or PREG) and the CYP17 active site. There are 43 intramolecular interactions between ABE and active site according to the virial graph of ABE binding pocket where the charge densities (ρ) values of the corresponding critical points are shown in Table 4. On the other hand, PREG binding pocket has 46 intramolecular interactions (Figure 11B and Table 4). The sum of all electronic density of interactions of ABE and PREG binding pockets were 0.0610 au. and 0.0560 au. respectively. This result shows the overall stabilizing effect of the electronic density of interactions of pockets on the ABE binding pocket.

CONCLUSIONS

In the present study, the molecular dynamics revealed that the histidine residue situated close to heme group (His373) in the CYP17 enzyme can display any protonation state since it showed no significant differences both in RMSD average values of Cα as the variation of dihedral angle (χ). With the protonation state of His373 defined, we have obtained the two protein-ligand complexes used in this study: CYP17-ABE and CYP17-PREG complexes.

Although the molecular mechanics interaction energy showed that ABE has a higher diffusion trend (hydrophobicity) water  CYP17 binding site compared to the PREG, classical descriptions cannot be employed for enzymes containing heme group due to the limitations observed in Figures 9A and 9B. The ONIOM hybrid (B3LYP:AMBER) methods showed that ABE has a higher affinity for the CYP17 because it has a more negative interaction energy value compared to PREG.

To investigate the intramolecular interactions of ligands ABE and PREG, we used the quantum theory of atoms and molecules (QTAIM). QTAIM analysis indicates that ABE has a higher electronic density in their interactions with the active site of the CYP17 enzyme compared to PREG.

By using molecular mechanics (Docking and MD simulations) and quantum mechanics calculations (ONIOM and QTAIM), we have investigated the inhibitory effect of Abiraterone through the interaction energies and electronic density. This detailed study proves that the rationality evaluation inhibition performance may grant valuable information about CYP17-ABE recognition providing new insights to design the new homologous CYP17 inhibitors.

144

ACKNOWLEDGMENTS

ASSOCIATED CONTENT

Supporting Information

Table with the electronic energy, bond length Fe-NH2 and bond length Fe-NH3 of triplet and quintet states at B3LYP, BP86, BVWN, M062X, ωB97XD and PBE1PBE methods with the 6-311G(d,p) basis set of the Fe(NH3)2(NH2)2. Optimized geometries of Fe(NH3)2(NH2)2 at B3LYP/6-311G(d,p) level in the triplet and quintet states. Fitting structures of CYP17-ABE and CYP17-PREG complexes. B-factors for the CYP17-ABE and CYP17-PREG complexes. This material is available free of charge via the Internet at http://pubs.acs.org.

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TABLE OF CONTENTS

157

Supporting Information for “Docking, Molecular Dynamics, ONIOM, QTAIM: theoretical investigation of inhibitory effect of Abiraterone on CYP17”.

Bond Bond Electronic Energy length length METHOD Multiplicity (a.u.) Fe-NH2 Fe- (Å) NH3(Å) Triplet -1488.82340086 1.92212 2.02763 B3LYP Quintet -1488.83272142 1.88714 2.35259 Triplet -1489.00912021 1.87322 1.98187 BP86 Quintet -1488.99952264 1.87293 2.31036 Triplet -1492.23892216 1.90792 2.04101 BVWN Quintet -1492.23690248 1.89844 2.41968 Triplet Error - - M062X Quintet Error - - Triplet -1488.74019588 1.87885 2.01482 ωB97XD Quintet -1488.74695927 1.88793 2.31236 Triplet -1488.28486454 1.90899 2.00701 PBE1PBE Quintet Error - -

B3LYP - Triplet B3LYP - Quintet

Figure S1. Optimized geometries of Fe(NH3)2(NH2)2 at B3LYP/6-311G(d,p) level in the triplet and quintet states.

158

A B

Figure S2. Fitting structures of CYP17-ABE (A) and CYP17-PREG (B) complexes before (yellow) and after (red) molecular dynamics simulation. The picture was rendered using VMD.1

A B

Figure S3. B-factors for the CYP17-ABE (A) and CYP17-PREG (B) complexes. The picture was rendered using VMD1, “blue” indicates high, “white” indicates intermediate and “red” low values. High temperature factors imply exibility and low ones rigidity.

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