ANCHORING THE POTENTIAL ENERGY SURFACES OF HOMOGENOUS AND HETEROGENOUS DIMERS OF FORMALDEHYDE AND THIOFOMALDEHYDE
by Christina Michelle Holy
A thesis submitted to the faculty of The University of Mississippi in partial fulfillment of the requirements of the Sally McDonnell Barksdale Honors College
Oxford May 2014
Approved by
______Advisor: Professor Gregory Tschumper
______Reader: Associate Professor Nathan Hammer
______Reader: Professor Susan Pedigo
© 2014 Christina Michelle Holy ALL RIGHTS RESERVED
ii
ACKNOWLEDGEMENTS
Many people in the Department of Chemistry and Biochemistry here at the University of Mississippi were of great assistance during the work presented in this manuscript. I would like to thank the Mississippi Center for Supercomputing Research for the use of their computational facilities and the research group of Dr. Gregory Tschumper for their guidance and computational assistance. Specifically, I would like to thank Eric Van Dornshuld for his continual support and advisement throughout this research project. Lastly, I would like to thank my advisor, Dr. Gregory Tschumper, for allowing me to work in his laboratory as well as providing me with a worthwhile project to fulfill this thesis requirement.
iii ABSTRACT
This work characterizes five stationary points of the formaldehyde dimer,
(CH2O)2, two of which are minima, seven newly-identified stationary points of the formaldehyde/thioformaldehyde (mixed) dimer, CH2O/CH2S, four of which are minima, and five newly-identified stationary points of the thioformaldehyde dimer, (CH2S)2 , three of which are minima. Full geometry optimizations and corresponding harmonic vibrational frequencies were performed on CH2O and CH2S as well as each of the dimer configurations (Figures 3-5). The computations were carried out with second order
Møller-Plesset perturbation theory (MP2), with the heavy-auc-cc-pVTZ (haTZ) basis set.
Additionally, thirteen density functional theory methods were employed: B3LYP,
B3LYP-D3, B3LYP-D3(BJ), TPSS, TPSS-D3, TPSS-D3(BJ), APF, APF-D, M06-2X,
M06-2X-D3, N12SX, MN12SX, and VSXC, in conjunction with the 6-311+G(2df,2pd) basis set. Six of these functionals are dispersion corrected with either the original D3 damping function (DFT-D3) or the Becke-Johnson damping function [DFT-D3(BJ)].
Binding energies were computed via the supermolecular approach. Single-point energies were also computed for all optimized structures using explicitly correlated MP2-F12 and
CCSD(T)-F12 methods with the haTZ basis set. The (CH2O)2 and CH2O/CH2S global minimum are the same at the MP2, MP2-F12, and CCSD(T) levels of theory. However,
-1 MP2 methods overbind (CH2S)2 by as much as 1.1 kcal mol , effectively altering the energetic ordering of the (CH2S)2 minima relative to the CCSD(T)-F12 energies.
iv TABLE OF CONTENTS
1 Introduction………………………………………………………………………...…1
1.1 Ionic Bonds, Covalent Bonds, and Noncovalent Interactions……………...1
1.2 Overview of Formaldehyde………………………………………………...4
1.3 Overview of Thioformaldehyde…………………………………………….5
2 Theoretical Methods………………………………………………………………8
3 Results and Discussion…………………………………………………………..10
3.1 Structures and Energies…………………………………………………….10
3.1.1 Formaldehyde Dimer…………………………………………….10
3.1.2 Mixed Dimer……………………………………………………..13
3.1.3 Thioformaldehyde Dimer………………………………………...15
3.2 DFT Analysis………………………………………………………………17
3.2.1 Formaldehyde Dimer…………………………………………….17
3.2.2 Mixed Dimer………..……………………………………………20
3.2.3 Thioformaldehyde Dimer ………………………………………..23
4 Conclusions………………………………………………………………………25
5 References………………………………………………………………………..27
v
1 Introduction
1.1 Ionic Bonds, Covalent Bonds, and Noncovalent
Interactions
The joining of two or more atoms forms a chemical compound. A stable compound occurs when the combined atoms have a lower total energy than the separated atoms.
This attraction between atoms is termed a chemical bond. An ionic bond is a type of chemical bond in which one atom loses one or more electrons while another atom gains them in order to fill their electron shell and produce noble gas electron configuration.
The atom that transfers its electron(s) obtains a positive charge whereas the atom that gains the electron(s) obtains a negative charge. These positive and negative ions attract each other, forming the ionic bond.
A covalent bond is another type of chemical bond that involves the sharing of electrons between atoms. It is formed when partially occupied orbitals of interacting atoms overlap and contain a pair of electrons shared by these atoms.1 These bonds are found in inorganic metal complexes, and also have biological relevance. For example amino acids are held together by a special kind of covalent bond referred to as a peptide bond in which the carboxyl group of one amino acid reacts with the amino group of another amino acid. Additionally, the monomeric units of nucleic acids and polysaccharides are joined by covalent bonds. Even the structure of hair is determined by
1 covalent bonds known as disulfide bonds.2 Both ionic and covalent bonds have large bond dissociation energy, indicating a strong bond between the involved atoms.
Unlike ionic and covalent bonds, noncovalent interactions are much weaker than their covalent counterparts and are often referred to as weak attractive interactions.1 For example, an input of about 350 kJ mol-1 of energy is required to break a C—C single bond, and about 410 kJ mol-1 to break a C—H bond, but as little as 4 kJ mol-1 is sufficient to disrupt a typical van der Waals interaction.2 Types of noncovalent interaction include, hydrogen bonds, ionic interactions, dipole-dipole interactions, ion-dipole interactions, and van der Waals interactions.2 In hydrogen bonding, a hydrogen atom bound to an electronegative atom (such as oxygen or nitrogen) interacts with another electronegative atom. This allows the hydrogen atom to form a strong interaction with the loan pair on the neighboring oxygen or nitrogen. In ionic interactions, an electrostatic interaction forms between positively and negatively charged species or ions. A dipole-dipole interaction occurs when two polar molecules align themselves so that their positive and negative poles interact with each other. When an ion interacts with a polar molecule, an ion-dipole interaction forms. Finally, van der Waals interactions arise when two adjacent molecules form temporary dipoles due to shifting electron density, thus creating a temporary attractive force. Noncovalent interactions are responsible for holding together the molecules of supramolecular complexes, and therefore play an important role in determining the structure of liquids, molecular crystals and bio-macromolecules like
DNA and proteins.2-7
Although noncovalent interactions are individually weak compared to covalent bonds, the cumulative effect of many noncovalent interactions can be incredibly significant. For
2 instance, the noncovalent binding of an enzyme to its substrate may involve multiple hydrogen bonds as well as several other types of noncovalent interactions.2 Additionally, the binding of an antigen to a specific antibody and the binding of a hormone or neurotransmitter to its cellular receptor protein depends on the cumulative effects of many weak interactions. Furthermore, the most stable structure (or native state) of any supramolecular complex is typically the one in which weak, noncovalent interactions are maximized. This principle explains the folding of a single polypeptide chain into its secondary, tertiary, and sometimes quaternary structures.3 After the initial amino acid chain is assembled through the process of translation, the polypeptide chain then folds into its secondary structure. The most common types of secondary structures are α helices and β sheets. Both conformations are held together by hydrogen bonds between the peptide bonds, but the α helix maximizes these internal hydrogen bonds, making it the most common conformer. The side chains of the amino acids protrude from the secondary structure and interact via noncovalent forces to stabilize the tertiary structure.
This global folding of a single polypeptide chain often represents the functional form of a protein. However some proteins, like hemoglobin, form quaternary structure in which subunits are held together through noncovalent interactions in order to create the function form.2,3
Due to their importance in both biology and chemistry, noncovalent interactions are an extremely important area of scientific research; however, experimental difficulties often arise when studying these weak interactions. Fortunately, computational chemistry offers a means for scientists to investigate noncovalent interactions and how they
3 influence chemical systems.8 It simulates the behavior of molecules, and as models improve, they reflect more accurately the behavior molecules in the real world.9
1.2 Overview of Formaldehyde
As explained above, noncovalent interactions are extremely important in protein folding and function. Specifically, noncovalent interactions between carbonyl groups have been shown to play an important role in protein structure, folding, function, and stability.4-7 In a cabonyl-carbonly noncovalent interaction, the delocalized lone pair of electrons (n) from the carbonyl oxygen atom interacts with the antibonding orbital of the nearby carbonyl group.4,7 This sort of noncovalent interaction is termed the n→π* interaction.
Formaldehyde (Figure 1) is the simplest compound that contains a carbonyl group. It is a naturally occurring organic compound composed of carbon, hydrogen, and oxygen. Structurally, it is a planar molecule with C2v symmetry and has a molecular formula of CH2O.
Figure 2: Formaldehyde (CH2O)
It was first synthesized in 1859 by the Russian chemist, Aleksandr Mikhailovich
Butlerov. This aliphatic aldehyde has countless uses. For example, morticians use
4 formaldehyde as embalming fluid, and it is used for the production of many synthetic polymers like Bakelite, one of the first commercial synthetic plastics.10
The formaldehyde dimer [i.e. (CH2O)2] is the simplest system that can model carbonyl-carbonyl non-covalent interactions. Over the past few decades, this dimer and its monomeric subunit have been characterized using both experimental and computational techiniques.11-21 Experimentally, the formaldehyde dimer has been observed in argon and nitrogen matrices using infrared (IR)11-13 and Raman10 spectroscopy. In 1997, Ford and Glasser first characterized the potential energy surface
(PES) of the formaldehyde dimer using second-order Møller-Plesset perturbation theory
(MP2) and the 6-31++G** basis set.14 Five stationary points were reported. Structures I,
II, and III (located in Figure 3) were reported to be minima (ni = 0). Furthermore,
Structure I was found to be the global minimum with a binding energy of − 4.40 kcal
-1 mol . Structure III, however, was later found to be a transition state (ni = 1) when the largest basis set was employed.15 In 2013, Dolgonos et. al. reported structure I to be 0.8 kcal mol-1 lower in energy than structure II at the CCSD(T) (i.e. the couple-cluster method that includes all single and double substitutions as well as a perturbative treatment of the connected triple excitations) complete basis set (CBS) limit.21
1.3 Overview of Thioformaldehyde
Thioformaldehyde (Figure 2) is an isovalent analog of formaldehyde made up of carbon, hydrogen, and sulfur with a molecular structure of CH2S. It is the simplest thio- carbonyl compound.
5 Figure 2: Thioformaldehyde (CH2S)
Just like formaldehyde, it is a planar molecule with C2v symmetry. Previous spectroscopic studies on this species have been impeded by the fact that the molecule is unstable in the gas phase under typical laboratory conditions.22 However, in 1970, thioformaldehyde was first identified in the laboratory through the observation of its microwave spectrum.23 Monomeric thioformaldehyde is now thought to be present in interstellar space24 as well as associated with the Hale-Bopp comet.25 Due to modern methods of synthesis, purification, and structural determination used in conjunction with fast reaction techniques, a substantial amount of spectroscopic data has been obtained for the thioformaldehyde monomer.26, 27 Additionally, investigations into the ground state vibrational28-37, ground state rotational23, 30, 31, 35, 37-39, and excited state rotational40-42 properties have previously been performed on monomeric thioformaldehyde. Extensive ab initio work has also targeted the structural, vibrational, and bonding properties of monomeric thioformaldehyde.36, 43-57
Like the carbonyl functional group, thiocarbonyls can establish attractive inter- and intra- molecular non-covalent interactions.46, 49, 51, 52, 55, 56, 58 For instance, sulfur has been shown to participate in hydrogen bonds that are weaker than those formed with oxygen.45 Additionally, the sulfur containing amino acids—cystine and methionine— which were once thought to act simply as hydrophobic moieties in protein folding, are
6 now thought to partake in significant noncovalent interactions, which may influence tertiary protein structure.59-60
Interestingly, neither the homogenous nor the heterogeneous sulfur analogs of the formaldehyde dimer model system have been studied. The (CH2O)2 , (CH2S)2 , and
(CH2O)/(CH2O)2 systems provide a means of comparing the noncovalent interactions between carbonyl and thiocarbonyl groups. Insight into these noncovalent interactions is of biological importance because it may help explain the phenomenon of protein folding.
This document presents the first detailed characterization (i.e. full optimization and frequency calculations) of the thioformaldehyde dimer and formaldehyde/thioformaldehyde (mixed) dimer using both ab initio electronic structure methods and density functional theory (DFT). The stationary points of the formaldehyde dimer are characterized for comparison. Thirteen DFT functionals were used in an attempt to locate a DFT method that reproduced data similar to that obtained using ab initio methods. This is significant because DFT methods are computationally less expensive than ab initio methods, and therefore, locating a DFT method that accurately characterizes the formaldehyde, mixed, and thioformaldehyde dimers with respect to ab initio methods would aid in the future study of these interactions.
7
2 Theoretical Methods
Full geometry optimizations and corresponding harmonic vibrational frequencies were performed on monomeric formaldehyde and monomeric thioformaldehyde as well as each of the dimer configurations found in Figure 3, Figure 4, and Figure 5. The computations were carried out with second order Møller-Plesset perturbation theory
(MP2)61-66 using the analytic gradients and Hessians available in the Gaussian 0960 software package. The heavy-aug-cc-pVTZ correlation consistent basis set68,69 (denoted as haTZ), where non-hydrogen (i.e. heavy) atoms are augmented with diffuse functions
(i.e. cc-pVTZ for H and aug-cc-pVTZ for all other atoms), was employed when performing MP2 computations. Additionally, thirteen density functional theory (DFT) methods were employed, as implemented in Gaussian 09.67 These DFT implementations include B3LYP70, 71, B3LYP-D3, B3LYP-D3(BJ), TPSS72, TPSS-D3, TPSS-D3(BJ),
APF73 , APF-D73 , M06-2X74, M06-2X-D3, N12SX75, MN12SX75, and VSXC.76 The
DFT-D3 scheme employs Grimme’s third-generation dispersion correction with the original D3 damping functions77, while the DFT-D3(BJ) scheme employs the Becke-
Jonhson damping function.78 Note that the TPSS is implemented for both the exchange and correction part of the functional (i.e. TPSSTPSS). The 6-311+G(2df,2pd)79-83 basis set was used when performing all DFT computations. Strict convergence criteria were employed throughout the computations.
8 The binding energies, Ebind, were computed via the supermolecular approach by comparing the energy of each optimized dimer structure to the corresponding monomer energy (Equation 1).
(1) E bind = E dimer − (E monomer 1 + E monomer 2)
Single-point €energy computations were performed for all optimized structures using explicitly correlated MP2-F12 [specifically MP2-F12 3C(FIX)84] and CCSD(T)-
F12 [specifically CCSD(T)-F12a85 with unscaled triples contribution] methods.
MP2−F12 CC−F12 Corresponding binding energies are denoted as E bind and E bind , respectively. All explicitly correlated computations were performed with the haTZ basis set and include the default density fitting (DF) and resolution€ of the€ identity (RI) basis sets implemented in Molpro 2010.1.86
Minimum root means square deviations (RMSD) between un-weighted Cartesian coordinates of the MP2 and DFT optimized geometries were computed with the
SUPERPOSE program offered in the TINKER87 software package.
For all computations, spherical harmonic basis functions (i.e. 5d and 7f) were used rather than their Cartesian counterparts (i.e. 6d and 10f). Residual Cartesian
-5 -1 gradients of the optimized structures were less than 1.77 x 10 Eh a0 . For all DFT computations, a pruned integration grid having 150 radical shells and 974 angular points per shell was employed. The frozen core approximation was used for MP2 and CCSD(T) computations (1s, 2s, 2p-like orbitals on sulfur and 1s-like orbitals on oxygen and carbon).
9
3 Results and Discussion
3.1 Structures and Energies
3.1.1 Formaldehyde Dimer
Five stationary points of the formaldehyde dimer, (CH2O)2, were found using
MP2 electronic structure theory and are shown in Figure 3 along with their point group symmetries. Structure I has Cs symmetry. Structures II and III have C2h symmetry, and
Structures IV and V have C2v symmetry. Structures I-V are characterized by an edge-to- face, edge-to-edge, face-to-face, non-planar head-to-tail, and planar head-to-tail alignment of monomers, respectively. These results are consistent with the previous work carried out by Ford and Glasser9.
The Hessian index (number of imaginary modes of vibration denoted as ni) for the
MP2 optimized (CH2O)2 structures are shown in Table 1. (CH2O)2 Structures I and II represent minima on the MP2 PES because they have zero imaginary modes of vibration
(ni = 0). Structure IV is a transition state (ni = 1), and structures III and V are higher order saddle points (ni > 1). The intermolecular separations of (CH2O)2 with respect to the center of mass (COM) of each monomer (RCOM) are also listed in Table 1 and range from 2.89 Å (Structure I) to 4.35 Å (Structure V).
10
Figure 3: (CH2O) 2 structures and point groups
The MP2, MP2-F12, and CCSD(T)-F12 binding energies of the MP2 (CH2O)2
MP2 MP2−F12 CC−F12 optimized structures ( E bind , E bind , and E bind , respectively), as well as the higher-
CC−F12 order correlation effects (δMP2−F12), defined in Equation 2, are listed in Table 1.
€ € €
€ CC−F12 CC−F12 MP2−F12 δMP2−F12 = E bind − E bind (2)
€
11
Table 1: Number of imaginary vibrational frequencies (ni), intermolecular MP2 separation (RCOM in Å), binding energies ( E bind ), and MP2-F12/haTZ and MP2−F12 CC−F12 CCSD(T)-F12/haTZ binding energies ( E bind and E bind ) of the MP2 optimized structures with the haTZ basis set. All energies are in kcal mol-1.
€ € €
12 MP2 Good agreement is observed between MP2 and MP2-F12 binding energies ( E bind
MP2−F12 -1 and E bind ) with deviations less than a tenth of a kcal mol . Furthermore, higher-order
CC−F12 -1 € correlation effects (δMP2−F12) do not exceed 0.20 kcal mol as seen in Structure II.
€ Structure I is the global minimum lying 1.08 kcal mol-1, 1.06 kcal mol-1, and 0.81 kcal
-1 € MP2 MP2−F12 CC−F12 mol lower in energy than Structure II according to E bind , E bind , and E bind ,
respectively. The CCSD(T)-F12 binding energy difference between Structure I and II
(0.81 kcal mol-1) is in excellent agreement€ with the€ previously €reported CCSD(T) CBS
limit electronic energy difference between Structures I and II of 0.80 kcal mol-1.17
3.1.2 Mixed Dimer
The seven, newly-identified, stationary points of the heterogeneous or “mixed”
formaldehyde/thioformaldehyde dimer, CH2O/CH2S, were found using MP2 electronic
structure theory and are shown in Figure 4 along with their point group symmetries.
Structure Ia, Ib, and II have Cs symmetry while the remaining structures (Structures IVa,
IVb, Va, and Vb) have C2v symmetry. These structures contain similar orientations to the
corresponding (CH2O)2 stationary points; however, due to the different arrangement of
the formaldehyde and thioformaldehyde monomers, many structures have two
conformers (represented a and b). A CH2O/CH2S configuration comparable to (CH2O)2
Structure III was not found.
The number of imaginary modes of vibration (ni) for the MP2 optimized
CH2O/CH2S structures are shown in Table 1. Structures Ia, Ib, II, and Va represent
minima on the MP2 PES. Structures IVa and IVb are transition states, and structure Vb
is a higher order saddle point. The RCOM values for the CH2O/CH2S structures, listed in
Table 1, are slightly larger than those of the (CH2O)2, ranging from 3.12 Å (Structure I)
13
Figure 4: (CH2O) /(CH2S) structures and point groups
to 4.88 Å (Structure Va).
MP2 MP2−F12 CC−F12 CC−F12 E bind , E bind , E bind , andδMP2−F12 for CH2O/CH2S are listed in Table 1. Just
like (CH2O)2, good agreement is observed between MP2 and MP2-F12 binding energies
€ MP2€ MP2€ −F12 € -1 ( E bind and E bind ) with a maximum deviation of 0.12 kcal mol as seen in Structure
CC−F12 IVa. Additionally, higher-order correlation effects (δMP2−F12) grown no larger than 0.37 € € 14 € kcal mol-1 as seen in Structure Ia. Since Structure Ia is more than 1 kcal mol-1 lower in
MP2 MP2−F12 CC−F12 energy than Structure Ib according to E bind , E bind , and E bind , it is considered the
global minimum.
3.1.3 Thioformaldehyde€ € Dimer €
Five newly-identified stationary points of the thioformaldehyde dimer, (CH2S)2,
were found using MP2 electronic structure theory and are shown in Figure 5 along with
their point group symmetries. Just like the formaldehyde dimer, Structure I has Cs
symmetry. Structures II and III have C2h symmetry, and Structures IV and V have C2v
symmetry. Furthermore, structures I-V are characterized by an edge-to-face, edge-to-
edge, face-to-face, non-planar head-to-tail, and planar head-to-tail alignment of
monomers, respectively. The (CH2O)2 and (CH2S)2 structures are qualitatively similar
except Structure III. The (CH2O)2 Structure III is characterized by two anti-parallel
CH2O monomers in a near-stacked, fact-to-face orientation containing two C⋅⋅⋅O
interactions, while the two monomers in the corresponding (CH2S)2 structure slip into an
anti-parallel direction forming a face-to-face alignment of the carbonyl centers. The
number of imaginary modes of vibration (ni) for the MP2 optimized (CH2S)2 structures
are shown in Table 1. Structures I, II, and III are minima, Structure V is a transition
state, and Structure IV is a higher order saddle point on the MP2 PES. The RCOM values
for the (CH2S)2 structures, listed in Table 1, are significantly larger than those of the
(CH2O)2, growing as large as 5.22 Å as seen in Structure V.
MP2 MP2−F12 CC−F12 CC−F12 E bind , E bind , E bind , andδMP2−F12 or (CH2S)2 are also listed in Table 1. Much
like (CH2O)2 and CH2O/CH2S, good agreement is observed between MP2 and MP2-F12
€ € € MP2 € MP2−F12 binding energies ( E bind and E bind ) with deviations exceeding no more than 0.13
15 € €
Figure 5: (CH2S) 2 structures and point groups
-1 CC−F12 kcal mol for Structure IV. However, higher-order correlation effects (δMP2−F12) grow as large as 1.09 kcal mol-1 for Structure I. According to explicitly correlated MP2-F12
MP2−F12 € -1 computations ( E bind ), Structure I is the global minimum lying 0.67 kcal mol lower
CC−F12 in energy than Structure II. In contrast, the CCSD(T)-F12 binding energies ( E bind ) show€ Structure II to be the global minimum lying 0.30 kcal mol-1 lower in energy than
Structure I. These contrasting global minima along with the rather€ large higher-order
16 correlation effects suggests that the MP2 and CCSD(T) PES could be qualitatively different for the thioformaldehyde dimer.
3.2 DFT Analysis
3.2.1 Formaldehyde Dimer
Five stationary points of (CH2O)2 have been characterized with thirteen DFT methods (Table 2).
Table 2: Number of imaginary vibrational frequencies (ni) of the DFT/6- 311+G(2df,2pd) optimized (CH2O) 2 structures as well as the number of structures with a different number of imaginary modes from the MP2 reference structures (δni).
In accordance with MP2 computations, all DFT functionals characterized
Structures I and II as minima (ni = 0) on the DFT PES . Structure IV appears to be a transition state (ni = 1); however, both APF and VSXC characterize it as a minima on the
DFT PES. All DFT functionals indicate that Structure III is a higher order saddle point
17 (ni > 1). Seven DFT functionals (B3LYP-D3, TPSS-D3, B3LYP, TPSS, APF, N12SX, and VSXC) characterize Structure V as a higher order saddle point, while the remaining functionals consider it a transition state. The last column in Table 2 (δni) indicates the number of DFT optimized structures that produce different imaginary vibrational frequencies compared to the MP2 optimized structures for each DFT method. Only six functionals characterize each series of stationary points in accordance with the MP2 level of theory (δni = 0). These functionals include B3LYP-D3(BJ), TPSS-D3(BJ), M06-2X-
D3, APF-D, M06-2X, and MN12SX.
The average absolute deviations (AADs) and maximum absolute deviations
(MADs) of the Cartesian root means square deviations (RMSD) (in Å) and the difference in intermolecular separations (ΔRCOM in Å) of the DFT optimized (CH2O)2 structures with respect to the MP2 reference structures are listed Table 3. Additionally, deviations
DFT of the DFT binding energies ( ΔE bind ), relative energies (ΔErel), and CCSD(T)-F12
CC−F12 binding energies ( ΔE bind ) of the DFT optimized geometries are shown in Table 3 with respect to the CCSD(T)€ -F12 binding energies of the MP2 optimized geometries.
All€ of the RMSD AADs lie within 0.10 Å, and the MADs are relatively small, the largest being 0.23 Å for the VSXC functional. The ΔRCOM AADs are within 0.20 Å for every DFT functional, and the corresponding MADs grow no larger than 0.34 Å as seen in TPSS. Both RMSD and ΔRCOM values show how greatly the DFT optimized geometries differ from the MP2 optimized geometries. Since these values stay relatively low, it is clear that the DFT functionals did a good job computing the geometry of the formaldehyde dimers. Specifically, B3LYP-D3(BJ), B3LYP-D3, and N12SX perform
18 Table 3: The average (AADs) and maximum (MADs) absolute deviations of the root mean squared deviation (RMSD), the difference in intermolecular separation DFT (ΔRCOM ), deviations in DFT/6-311+G(2df,2pd) binding energies ( ΔE bind ), deviations in relative energies (ΔErel), and deviations in CCSD(T)-F12/haTZ binding energies CC−F12 ( ΔE bind ) of the DFT/6-311+G(2df,2pd) optimized structures with respect to the -1 MP2/haTZ reference structures. Energy deviations (in kcal mol ). €
€
19 remarkably well with RMSD and ΔRCOM AADs and MADs growing no larger than 0.05
Å.
DFT -1 All AADs of the ΔE bind are roughly within 1 kcal mol except for VSXC, which
DFT -1 has a ΔE bind of 2.79 kcal mol . B3LYP-D3(BJ) performs exceptionally well with a
DFT € -1 ΔE bind AAD of only 0.09 kcal mol . TPSS-D3 and MN12SX also perform well with
€ DFT -1 ΔE bind AADs of 0.14 kcal mol for both functionals. ΔErel indicates how consistent the
€ error in the DFT binding energies was compared to Structure I. All DFT methods have a
€ -1 -1 ΔErel AAD of less than 1 kcal mol except for VSXC (1.76 kcal mol ). N12SX has the
-1 CC−F12 lowest ΔErel AAD of 0.17 kcal mol . The AADs of ΔE bind are all less than a tenth of
a kcal mol-1 except TPSS and VSXC with values of 0.16 kcal mol-1 and 0.27 kcal mol-1,
€ CC−F12 respectively. B3LYP-D3(BJ), B3LYP-D3, APF-D, and N12SX all have ΔE bind AADs
of 0.01 kcal mol-1. These results indicate the effect geometrical deviations have on
CC−F12 € binding energies, because ΔE bind represents the difference between CCSD(T)-F12
binding energies of the DFT optimized geometries with respect to the CCSD(T)-F12
binding energies€ of the MP2 optimized geometries. B3LYP-D3(BJ)’s, B3LYP-D3’s,
CC−F12 -1 APF-D’s, and N12SX’s small ΔE bind AADs of 0.01 kcal mol indicate that differing
geometries of (CH2O)2 had little to know effect on the binding energies when using
those four DFT functionals.€
3.2.2 Mixed Dimer
Seven newly-identified stationary points of the mixed dimer CH2O/CH2S have
been characterized with thirteen DFT methods (Table 4).
20 Table 4: Number of imaginary vibrational frequencies (ni) of the DFT/6- 311+G(2df,2pd) optimized (CH2O) /(CH2S) structures as well as the number of structures with a different number of imaginary modes from the MP2 reference structures (δni).
Structures Ia, Ib, II, and Va are minima on the DFT PES, although MN12SX and
VSXC characterize Structure Va as a transition state. Structure IVa is a transition point according to all the DFT functionals except M06-2X-D3, M06-2X, and VSXC, which characterize it as a minimum. Five functionals—B3LYP-D3, APF-D, B3LYP, AFP, and
N12SX—characterize Structure IVb as a transition state, which is in accordance with the
MP2 findings. Another five functionals—B3LYP-D3(BJ), TPSS-D3(BJ), TPSS-D3,
TPSS, and MN12SX—characterize Structure IVb as a higher order saddle point. The remaining three functionals—M06-2X-D3, M06-2X, and VSXC—characterize the structure as a minima. All functionals characterize Structure Vb as a higher order saddle point except M062X-D3 and M062X, which consider it a transition state. None of the
DFT methods characterize all stationary points in accordance with MP2 (δni ≠ 0).
DFT CC-F12 The AADs and MADs of RMSD, ΔRCOM, ΔE bind , ΔErel, and ΔE bind for
CH2O/CH2S are located in Table 3. RMSD AADs grow no larger than 0.13 Å as seen in € € 21 TPSS, and the MADs grow no larger than 0.25 Å as seen in VSXC. ΔRCOM AADs all lie
within 0.2 Å except for B3LYP and TPSS, which have AADs of 0.21 Å and 0.26 Å,
respectively. Much like the formaldehyde dimer, the small RMSD and ΔRCOM AADs and
MADs values indicate that the DFT functionals (with the exception of B3LYP, TPSS,
and VSXC) did a good job computing the geometry of the mixed dimer structures.
Furthermore, B3LYP-D3(BJ), B3LYP-D3, APFD, and N12SX perform remarkably well
with RMSD and ΔRCOM AADs growing no more than 0.03 Å and MADs growing no
more than 0.05 Å.
DFT -1 For ΔE bind , all AADs are roughly within 1 kcal mol except for VSXC, which
grows as large as 3.17 kcal mol-1. Both M06-2X-D3 and MN12SX have the smallest
€ DFT -1 DFT ΔE bind AAD of 0.30 kcal mol . VSXC also has the largest ΔE bind MAD of 5.02 kcal
-1 DFT -1 mol with the next largest being TPSS-D3, having a ΔE bind MAD of 1.46 kcal mol .
€ € -1 Most DFT methods have an AAD ΔErel of less than 1 kcal mol except for TPSS-D3(BJ),
TPSS-D3, and VSXC with values of 1.51, €1.11, and 2.16, respectively. These three
functionals produce inconsistent binding energies and are therefore potentially bad
methods to use when performing computations on the mixed dimer system. When
CC-F12 looking at the AADs of ΔE bind , TPSS-D3(BJ), TPSS, and VSXC perform the worst yet
again. While all other functionals lie within 0.1 kcal mol-1, these functionals grow as
large as 0.21 kcal€ mol-1, 0.23 kcal mol-1, and 0.23 kcal mol-1, respectively. On the other
hand, B3LYP-D3(BJ), B3LYP-D3, M06-2X-D3, APF-D, M06-2X, and N12SX, have
CC-F12 -1 -1 extremely small ΔE bind AADs of either 0.01 kcal mol or 0.02 kcal mol .
€
22 3.2.3 Thioformaldehyde Dimer
Five newly-identified stationary points of the mixed dimer (CH2S)2 have been characterized with thirteen DFT methods (Table 5).
Table 5: Number of imaginary vibrational frequencies (ni) of the DFT/6- 311+G(2df,2pd) optimized (CH2S) 2 structures as well as the number of structures with a different number of imaginary modes from the MP2 reference structures (δni).
Structures I and II are minima on the DFT PES, although VSXC characterize
Structure II as a transition point. Only dispersion corrected functionals and M06-2X characterize Structure III as a minima. The remaining six functionals characterize it as a transition state. Five functionals (B3LYP-D3(BJ), B3LYP-D3, M06-2X-D3, M06-2X, and VSXC) characterize Structure IV as a minima, two functionals (TPSS-D3(BJ) and
TPSS-D3) characterize it as a transition state, and the remaining six functionals characterize it as higher order saddle point. B3LYP-D3 and M06-2X-D3 characterize
Structure V as a minima. APFD, B3LYP, TPSS, APF, N12SX, and VSXC characterize it
23 as a transition state, while the remaining functionals characterize it as a higher order
saddle point. None of the DFT methods characterized all five stationary points in
accordance with MP2 (δni ≠ 0).
DFT DFT The AADs and MADs of RMSD, ΔRCOM, ΔE bind , ΔErel, and ΔE bind for (CH2S)2
are located in Table 3. All RMSD AADs lie within 0.10 Å except B3LYP, TPSS, and
APF whose AADs are 0.20 Å, 0.19 Å, and€ 0.15 Å, respectively.€ The MADs grow as
large as 0.43 Å as seen in B3LYP. For ΔRCOM, B3LYP, TPSS, and APF are again the
major outliers. All ΔRCOM AADs lie around 0.10 Å for every DFT functional except
B3LYP, TPSS, and APF, which have values of 0.41 Å, 0.40 Å, and 0.31 Å, respectively.
Thee rather large RMSD and ΔRCOM AADs and MADs values indicate that B3LYP,
TPSS, and APF performed poorly when computing the geometry of the thioformaldehyde
dimer structures.
DFT -1 ΔE bind AADs are roughly within 1 kcal mol except for VSXC, which grows as
-1 -1 large as 3.46 kcal mol . B3LYP offers the lowest ΔErel AAD of 0.34 kcal mol , while
€ -1 TPSS-D3(BJ) provides the highest ΔErel AAD of 3.76 kcal mol . Seven of the DFT
CC-F12 -1 functionals all have ΔE bind AADs of 0.05 kcal mol or less. These functionals include
B3LYP-D3(BJ), B3LYP-D3, M06-2X-D3, APF-D, M06-2X, N12SX, and MN12SX.
This suggests€ that small geometrical differences of (CH2S)2 had little to know effect on
the binding energies when using these seven DFT functionals. On the other hand, the
remaining functionals all lie above 0.20 kcal mol-1 and grow as large as 0.52 kcal mol-1 as
seen in TPSS.
24
4 Conclusions
Five stationary points of the formaldehyde dimer, seven newly-identified stationary points of the formaldehyde/thioformaldehyde (mixed) dimer, and five newly- identified stationary points of the thioformaldehyde dimer were characterized with MP2 electronic structure theory. (CH2O)2 Structures I and II, CH2O/CH2S Structures Ia, Ib, II, and Va, and (CH2S)2 Structures I, II, and III are all minima (ni = 0), on the MP2 potential energy surface (PES). Deviations between the MP2 and MP2-F12 binding energies of the MP2 optimized structures grow to be no larger than 0.13 kcal mol-1. Higher order correlation effects of (CH2O)2 and CH2O/CH2S are small, growing to be no more than 0.2 and 0.37 kcal mol-1, respectively. However, higher order correlation effects grow to be as
-1 large as 1.1 kcal mol for (CH2S)2. According to both MP2-F12 and CCSD(T)-F12 binding energies Structure I for (CH2O)2 and Structure Ia for CH2O/CH2S are the global minima. However, according to the MP2-F12 binding energy for (CH2S)2 , Structure I is the global minimum (0.67 kcal mol-1 lower in energy than Structure II), while the
CCSD(T)-F12 binding energy indicates Structure II to be the global minimum (0.3 kcal mol-1 lower in energy than Structure I). This suggests that the MP2 and CCSD(T) PES could be qualitatively different for (CH2S)2.
Every stationary point was fully characterized with thirteen DFT methods. All DFT functionals characterized (CH2O)2 Structures I and II and CH2O/CH2S Structures Ia, Ib, and II as minima on the DFT PES. All DFT functionals except VSXC characterized
25 (CH2S)2 Structures I and II as minima, and only seven functionals characterized (CH2S)2
Structure III as a minimum. For the formaldehyde dimer, only six functionals (B3LYP-
D3(BJ), TPSS-D3(BJ), M06-2X-D3, APF-D, M06-2X, and MN12SX) predicted the same number of imaginary frequencies as the MP2 level of theory. However, for CH2O/CH2S and (CH2S)2 , none of the DFT methods predicted the same number of imaginary modes of vibrations as MP2. This indicates that DFT functionals were not successful in characterizing the modes of imaginary vibrational frequency in accordance with MP2 for these dimer systems. B3LYP, TPSS, and VSXC DFT functionals consistently performed poorly when computing the geometry of (CH2O)2 , CH2O/CH2S, and (CH2S)2 structures, while, B3LYP-D3(BJ), B3LYP-D3, APFD, and N12SX typically performed
DFT well. When computing the DFT binding energies ( E bind ) of the DFT optimized geometries, B3LYP, TPSS, APF, and VSXC perform badly compared to CCSD(T)-F12 binding energies of the MP2 optimized geometries€ for all three dimer systems. TPSS-
DFT D3(BJ) and TPSS-D3 perform poorly when computing E bind for the thioformaldehyde structures. There is no clear set of functionals that consistently produces good DFT binding energies compared to CCSD(T)-F12€ binding energies of the MP2 optimized geometries.
Studying the homogenous and heterogeneous sulfur analogs of the formaldehyde dimer model system provided a means of comparing the noncovalent interactions between carbonyl and thiocarbonyl groups. The research found within this document may help shed light on the noncovalent interactions involved in protein folding, and may specifically aid in studying the interactions between the sulfur containing amino acids: cystine and methionine.
26
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