Spin-Spin and Spin-Orbit Coupling Studies of Small Species and Magnetic System

Spin-Spin and Spin-Orbit coupling studies of small species and magnetic system

Sathya S R R Perumal Licentiate Dissertation

Deparment of Theoretical Chemistry

Kungliga Tekniska Hogskolan¨

Stockholm 2010 TRITA-BIO Report 2010:7 ISSN 1654-2312 ISBN 978-91-7415-607-2 Printed by Universitetservice US AB Stockholm 2010 i

ACKNOWLEDGMENT

The preparation of this licentiate thesis would not have been possible without the support of Prof. Hans Agren.˚ I personally thank him for all the comfort he oﬀered me during my stay in Dept. of Theoretical Chemistry, KTH, Stockholm and introducing me to the fascinating science of Quantum Chemistry.

I am indebted to Prof. Boris Minaev for his vital advice and teaching. I always admire his unprecedented passion for science and stimulating discussion.

I appreciate Prof. Olav Vahtras for his scientiﬁc inputs and review during the group meetings. I also thank him for all the assistance he rendered me then and there.

I express my gratitude to Prof. Ramasesha, SSCU, IISc, Bangalore, India, for stressing me to take this thought-provoking assignment that helped me to evolve as an individual.

I regard the timely assistance and comfort from Prof. Per-Ake˚ Nygren, Director of Research studies, School of Biotechnology, KTH.

I greatly adore and owe to my parents for their endless sacriﬁce regardless of their challenging situation to give a best possible education and moral values to build me as a human being. Really these souls are unsung heroes! ii

CONTRIBUTION

I declare that all calculations for the project carried out for this licentiate thesis were done by me. For Paper I the interpretation of the theoretical results, literature survey of the earlier methods and write up for the manuscript were accomplished by me. All the coupling calculations for Pa- per II were done by me along with response properties and I also wrote the corresponding part of it. And I assure this with the best of my knowledge. iii

ABSTRACT

The spin of an electron often misleadingly interpreted as the classical rotation of a particle. The quantum spin distinguishes itself from classical rotation by possessing quantized states and can be detected by its magnetic moment. The properties of spin and its collective behavior with other fundamental properties are fascinating in basic sciences. In many aspects it oﬀers scope for designing new materials by manipulating the ensembles of spin. In recent years attention towards high density storage devices has driven the attention to the fundamental level were quantum physics rules. To understand better design of molecule based storage materials, studies on spin degrees of freedom and their coupling properties can not be neglected.

To account for many body effect of two or more electrons consistent with relativity, an approximation like the Breit Hamiltonian(BH) is used in mod- ern quantum chemical calculations, which is successful in explaining the split in the spectra and corresponding properties associated with it. Often different tactics are involved for a specific level of computations. For example the multi-configurational practice is different from the functional based calculations, and it depends on the size of the system to choose between resources and accuracy. As the coupling terms offers extra burden of calculating the integrals it is literally challenging.

One can readily employ approximations as it suits best for the application oriented device computations. The possible methods available in the literature are presented in chapter 2. The theoretical implementations of coupling for the multi-reference and density functional method are discussed in detail. The multi-reference method precedes the density functional method in terms of accuracy and generalizations, however it is ineﬃcient in dealing very large systems involving many transition elements, which is vital for molecule based magnets as they often possess open shell manifolds. On the other hand existing density functional method exercise perturbations techniques which is extremely specialized for a speciﬁc system - highly coupled spins.

The importance of spin-spin coupling(SSC) in organic radical-Oxyallyl(OXA) was systematically studied with diﬀerent basis sets and compared with a similar isoelectronic radical(TMM). The method of spin-spin coupling implementations are also emphasized. Similar coupling studies were carried iv out for the species HCP and NCN along with spin-orbit coupling(SOC). The splitting of the triplet states are in good agreement with experiments. v

PUBLICATION LIST

1. Spin-Spin coupling of Oxyallyl- A comparative studies with TMM, Sathya S R R Perumal, B. Minaev, O. Vahtras and H. Agren,˚ to be submitted to Molecular Structure, Theochem

2. Calculation of Zero-Field splitting and spin selectivity in the singlet- triplet transitions of small molecules, Sathya S R R Perumal, B. Mi- naev, O. Vahtras and H. Agren,˚ to be submitted to International Jour- nal of Quantum Chemistry

Other work not related to this thesis

1. Mono transition element doped clusters, Sathya S R R Perumal, manuscript vi Contents

1 Introduction 1

2 Theoretical Methods 5 2.1 Breit-PauliHamiltonian ...... 5 2.2 Mean-FieldApproximation ...... 6 2.3 Eﬀective Core Potential Approximation ...... 7 2.4 DandEparameters ...... 7 2.5 Perturbationstechnique ...... 8 2.5.1 Pederson and Khanna Method ...... 8 2.5.2 NeeseAnalyticalMethod ...... 11 2.6 Spin-SpinCoupling...... 12

3 Results and Discussion 15 3.1 SpinsinBiradical ...... 15 3.1.1 Trimethylenemethane ...... 15 3.1.2 Oxyallyl...... 17 3.2 Spin coupling in low lying states of HCP and NCN ...... 20

4 Conclusion 27

vii viii CONTENTS Chapter 1

Introduction

Atomic and molecular properties are computed by including relativistic effects as expectation values of corresponding operators.[4] The relativistic effects are an essential part of any theoretical calculations that attempt to account for application oriented devices like molecular magnets. Spin and orbital angular momenta are coupled by the Breit-Pauli Hamiltonian, which lifts the degeneracy resulting in a splitting of the spectrum. Many re- markable phenomena are associated with this splitting both in gaseous and condensed phase material science. Fig[1.1] shows the shift in the spectrum after introducing the spin-orbit operator in a typical configuration interaction reference. Such splitting of degeneracy occurs even in the absence of any external field, accordingly this splitting of the spectra is called Zero- Field Splitting(ZFS). However a minor contribution comes from classical spin-spin coupling and contributes to ZFS to the first order,[22] along with the second-order spin-orbit coupling.

Recent interest in miniaturization of information storage devices at the molecular scale has driven the attention to design molecules for enhanced magnetic properties. These novel magnetic objects can be isolated or as- semblies of molecules.[16] The discipline Molecular Magnetism explores at this nanoscale for physical realization of such polynuclear molecular complexes. Since then the ﬁrst synthesized low temperature single-molecule magnet(SMM) [Mn12O12(CH3COO)16(H2O)4] complex[33], commonly referred as Mn12, in 1993 there has been a number of interesting reports in various perspective to improve the fundamental issues on it for a real- izable practical applications.[10, 6] These ensemble of spin causes uniaxial anisotropy- a measure that determines the energy barrier, regarded as ten-

1 2 CHAPTER 1. INTRODUCTION

Figure 1.1: Spin-Orbit Eﬀect [39]

dency to align the angular momentum in a certain direction and hence large energy barrier to hold the magnetization over an extended period of time at low temperature. Experimental physical characterization of molecular magnets includes SQUID, high resolution EPR, magnetic circular dichroism (MCD).

Understanding the scheme of coupling is a vital part of the theoretical model of molecular magnets. The coupling scheme of the quantized spin and orbital angular degrees of freedom are decomposed into spin-spin and spin-orbital contributions in the relativistic Hamiltonian. The precise computation of spin-orbit coupling is one of the challenges for theoretical methods. Computational cost can be reduced eﬀectively, if only a few electrons are taken into consideration and the interaction with others are approxi- mated. A multi-conﬁgurational approach enables one explicitly to specify all the magnetic sub-levels of each multiplicity and gives quite good values for ZFS for small systems. In order to evaluate the spin-orbit coupling one needs to compute the one-electron and two-electron parts. The calculation of the two-electron contribution is often expensive. Instead approximations 3

Figure 1.2: Fe8 SMM [1]

are used in obtaining the corresponding contributions for spin-orbit matrix element. The successfully employed approximations in the literature are discussed in chapter 2.

Spin-spin computations using ab-initio methods has been reported recently.[37]. A multi-configurational self-consistent field(MCSCF) implementation of spin- spin coupling was done in our group and coded in the ab-initio quantum chemistry package DALTON [13] and hence successfully applied to small organic molecules.[37, 20]. It is demonstrated that classical dipole- dipole interactions has significant contributions for ZFS, whereas the spin- orbit coupling term is negligible for such system. Other workers obtained the spin-spin coupling tensor using a DFT based approach in which the McWeeny and Mizuno[23] derivation was adapted for the single Kohn-Sham determinant.[30]. Elsewhere, a version of that kind was implemented in the ORCA package as well.[25]. The spin-spin contributions to the ZFS tensor for organic triplets, carbenes and biradicals are presented both in density functional and ab-initio methods by Sinnecker and Neese using ORCA. [34]. For the MCSCF framework, a spin-spin mean-field approximation was employed and the results are in reasonable agreement with experiments. 4 CHAPTER 1. INTRODUCTION Chapter 2

Theoretical Methods

2.1 Breit-Pauli Hamiltonian

In quantum theory extra terms are added into the non-relativistic Hamil- tonian H0 to account for all the types of interactions to describe quantita- tively the molecular properties. Relativistic corrections due to variation of mass with velocity, classical relativistic correction for retardation of electro- magnetic field, Dirac term, dipole-orbit and dipole-dipole couplings are all added for a realistic explanation of experimentally observed spectra. As this thesis is intended to study the coupling effect for splittings we focus only on the corresponding operator over the quantum state. The relativistic terms are included by averaging the BP Hamiltonian to the non-relativistic wave functions for light and atomic and molecular systems[4]. Spin-orbit coupling are described by the Breit-Pauli Hamiltonian written as N N α2 Z α2 1 HˆBP = (ˆriN pî) sî (ˆrij pî) (ˆsi + 2ˆsj) (2.1) 2 r3 × · − 2 r × · i iN i j ij X X= The α is the fine structure constant. ri and pi defines position and momentum of the electron with respect to its nucleus. Zi is the nuclear charge of atom. The Eq(2.1) has two terms recognized as 1. Spin-orbit interactions of each electron i in the electrostatic field of nucleus Zi and 2. Spin-orbit interactions of electron i in the coloumbic field of electron j 3. The coupling between electron j and the orbital angular momentum of electron i.

5 6 CHAPTER 2. THEORETICAL METHODS

In principle, the HBP operator is evaluated through all centers of the system. However multi-center coupling would be expensive computationally. Practically Eq.(2.1) is expressed in tensor form for matrix operations over the coupling of spin-orbitals

HˆBP = Tisi + Tij (si + 2sj) (2.2) i ij X X where Ti and Tij are one and two electron operator respectively The expectation value of Hbp becomes

Hbp = k T k σ(k) (2.3) h i h | 1| i Xk + kl T kl (σ(k) + 2σ(l)) + kl T lk (σ(k) + 2σ(l))δ h | 12| i h | 12| i σ(k)σ(l) Xk,l Xk,l σ are spin functions of electron.

The direct evaluation of microscopic Breit-Pauli Hamiltonian operator over the quantum states of a chemical system is challenging. The two- electron integrals contribute to non-vanishing matrix elements even between single reference states(HF). Commonly referred to as spin-other-orbit interactions, since the interactions are between spin angular momentum of an electron and orbital momentum of other orbits. And the one-electron part has a signiﬁcant contribution in deﬁning the value of HBP .

2.2 Mean-Field Approximation

Mean-field approximations by Hess offers a simplified and hence popular way to address two-electron contributions. The multi-centre integrals are neglected considering only the one-centre two-electron part. Combined with one-electron part this results in an effective mean field operator. The Mean- field Hamiltonian is

The spin-orbit coupling dependencies in this scheme falls with third power of electron-nucleus and electron-electron distance, resulting in computation of multi-center two-electron integrals. There has been number of reports employing this procedure both for heavy and lighter elements.[21] The mean-ﬁeld approximation for spin-orbit terms are implemented in quantum- chemistry packages, MOLCAS.[2]. Partial two-electron integral evaluations is used in the package GAMESS-US[12], the spin-orbit coupling calculations were done using this package.

2.3 Eﬀective Core Potential Approximation

In this approximation the inner core shell orbitals are replaced by an effective potential. However the active orbitals in the inner shells do not have a proper shape in the implementation, resulting in lower accuracy for spin-orbit computation for heavier elements. There has been some methods to tackle this difficulty. Effective one-electron operator is dealt with in relativistic atomic calculations, or spin-orbit operator used with pseudo- orbitals.[8, 19]. In some other, the shape of the core is preserved by employing model potential[32]. Marian et al.[21] used both electron basis set and EPS set, while transforming back to ECP set when the integrals are done in electron basis set. Koseki et al[18] interchange two-electron integrals with effective fictitious nuclear potential Zeff greatly reducing the problem to deal with only one-electron part. The effective nuclear potential charge for the spin-orbit coupling comes from semi-empirical methods. GAMESS - US package has an option for ECP approximation while computing spin- Orbit[12].

2.4 D and E parameters

We parametrize the splitting of the degeneracy to realize the couplings. The Hamiltonian with external ﬁeld B~

Hˆspin = βbB~ gS~ + S~DS~ (2.5) where βb is the Bohr magneton, B~ is the external magnetic flux density, S~ is the effective spin operator, g and D are electronic g-tensor and Zero field splitting tensor respectively. With proper choice of co-ordinate system D diagonalizes, reducing the Zero field splitting Hamiltonian as

2 1 2 2 HZF S = S S(S + 1) + E S S (2.6) z − 3 x − y 8 CHAPTER 2. THEORETICAL METHODS expressed in a parameter form

1 D = Dzz (Dxx + Dyy) (2.7) − 2 1 E = (Dxx Dyy) (2.8) 2 − In a proper coordinate system axes x, y, z are chosen such that 0 E/D ≤ ≤ 1/3. Also the D-tensor is not necessarily traceless [5]

Dxx + Dyy + Dzz = 0 (2.9) 6 For S = 1, triplet system the degenerate spin functions splits with Eigen values 1 Ex = D E (2.10) 3 − 1 E = D + E (2.11) y 3 2 Ez = D (2.12) −3 The Breit-Pauli contributions for the D tensors arises from second-order perturbation theory

i k k j so ψ0 Hso ψn ψn Hso ψ0 Dij = h | | ih | | i (2.13) En E0 Xn,k − The above Eq(2.13) runs over diﬀerent multiplicities. i.e) between singlet and corresponding axes of triplet states, if we do splitting calculations between singlet and triplet states of a molecular orbitals. Implementation of that kind is coded in package like MOLCAS, GAMESS, though partial two- electron integral evaluation method is considered in the latter package. It should be noted that in MOLCAS, B Roos extended CASSCF by including the relativistic Hamiltonian to consider heavy elements like Uranium[9]

2.5 Perturbations technique

2.5.1 Pederson and Khanna Method Density functional theory based computations have a very reasonable balance between accuracy and cost, making it popular for condensed and 2.5. PERTURBATIONS TECHNIQUE 9 gaseous phase ﬁrst principle studies. Recently Magnetic Anisotropy En- ergy(MAE) determination in DFT calculations is quite successful in highly coupled transition metal inorganic complexes. This thesis brieﬂy illustrates a couple of widely employed approaches. Pederson and Khanna[29] have developed a very simple method built upon a perturbative treatment of spin-orbit coupling of the spin Hamiltonian. For a non-spherical multi center nuclear systems the perturbation energy due to spin-orbit coupling term in terms of Cartesian system is given by.

1 U (r, p, S)= S p Φ (r) (2.14) −2c2 · ×

The perturbation operates on the basis formed by single-particle wave functions, expressed as is Ψis(r)= Cj,σφj(r)χσ (2.15) j X,σ is where the co-eﬃcients Cj,σ are from eﬀective diagonalizing the Hamiltonian matrix, χσ is either majority or minority spin spinor, φj(r) is a spatial basis function. Resulting matrix elements are of the form

Note that Vy, Vz are obtained with cyclic permutations. The spin-orbit coupling matrix element Eq(2.19) only depends on how accurate the Coulomb potential and gradient of each basis function and quite ideal for implementation with Gaussian-type orbitals, slater type orbitals and plane waves. Now we could obtain magnetic anisotropy for arbitrary symmetry axis. In the absence of any external ﬁeld the second order perturbation change to the total energy is ′ ′ ′ σσ σσ σ σ ∆2 = Mij Si Sj (2.20) ′ ij Xσσ X 10 CHAPTER 2. THEORETICAL METHODS where σ are sum over spin degree of freedom, i,j rungs over the Cartesian ′ σσ coordinates. The Mij are matrix elements

′ φlσ Vi φ ′ φ ′ Vj φlσ M σσ = h | | kσ ih kσ | | i (2.21) ij ′ − ǫlσ ǫkσ Xkl −

φlσ, φ ′ are occupied and unoccupied states with energies ǫlσ, ǫ ′ respec- kσ ′ lσ σσ tively. Si explicitly depends on the orientations of the axis of quantization. The second order shift in energy can be written as

∆2 = γxy Sx Sy (2.22) xy h ih i X where γxy is the anisotropy tensor, which diagonalizing one could readily obtain the principal axes and anisotropy energies. After diagonalization the second order perturbation energy for the spin Hamiltonian takes the form

1 ∆ = (γ + γ + γ ) S (S + 1) (2.23) 2 3 xx yy zz 1 1 + [γzz (γxx + γyy)] 3 − 2 2 1 2 2 [3S S (S + 1)] + (γxx γyy) S S × z − 2 − x − y Sx, Sy, Sx are rotated according to the eigenvectors of the anisotropy tensor. The spin Hamiltonian above suggests that (2S + 1) micro-states are split due to anisotropy. Writing it in a parameter form

H = DS2 + E S2 S2 (2.24) z x − y Thus once anisotropy tensors γij are known, D and E values can be reduced out of it. The 2.24 lifts the degeneracy due to spin-orbit coupling ﬁeld in the system.

Qualitatively the anisotropy energies is defined as the barrier between the highest spin state and the lowest one. The description mentioned here does not take into account the quantum tunneling effects, which overlaps between these quantum states through the quantum phenomenon called tunneling. Such effects are obvious in experimental detections leaving the theoretical predictions of barrier relatively bit higher. There has been number of implementations for the perturbative approach.[31, 3]. The Pederson and 2.5. PERTURBATIONS TECHNIQUE 11

Khanna approach gives quite impressive results for highly coupled multi- nuclear SMM, while for mononuclear systems there is an underestimation of the D value by a factor of two. Revikain recently reported overesti- mation of Pederson and Khanna perturbative approach in predicting ZFS for Mn(II) complexes and concluded same prefactors for diﬀerent excitation classes in summation could be the reason. Recently Neese proposed a coupled-perturbed(CP) coupling method in which the prefactors are included correctly. Also the linear response equations for SOC perturbation were are also obtained.[26]

2.5.2 Neese Analytical Method

Using spin-orbit mean field approximations the spin-orbit coupling operator is expressed as effective reduced one-electron operator. The sum- overstates are only the excited states with spin ∆S = 0, 1 for a ground ± state spin S system, which contribute to the splitting tensor D. According Ref.[27] the spin states ∆S = 0 and ∆S = 1 are referred as same-spin and ± spin-flip states. The equation based on Pederson and Khanna for same-spin and spin-flip in spin-unrestricted Kohn and Sham formalism is given by

0 −αα −ββ DK,L = DK,L + DK,L (2.25) 1 α α L;0 1 β β L;0 = ψ hK,so ψ U + ψ hK,so ψ U 4S2 h i | | a i aαiα 4S2 h i | | a i aβ iβ i ,a i ,a Xα α Xβ β −1 αβ DK,L = DK,L 1 α β L;−1 = ψ hK,so ψ U −4S2 h i | | a i aβ iα i ,a Xα β +1 αβ DK,L = DK,L (2.26) 1 β α L;+1 = ψ hK,so ψ U −4S2 h i | | a i aαiβ i ,a Xβ α

σ ψ’s are spin-unrestricted Kohn-Sham orbitals (ψp = µ cµpφν) with corresponding orbital energy,ǫ. K, L run over the Cartesian coordinates. The U P is the second order correction due to SOC coupling perturbation as given by equation(2.21). Neese adopted ﬁrst order wave function coeﬃcients in the 12 CHAPTER 2. THEORETICAL METHODS framework of linear response theory. The perturbed wave functions are

ψα,L;m(r) = U L;m ψα(r)+ U L;m ψβ(r) (2.27) i aαiα a aβ iα a a a Xα Xβ ψβ,L;m(r) = U L;m ψα(r)+ U L;m ψβ(r) (2.28) i aαiβ a aβ iβ a a a Xα Xβ Expressing D-tensor in terms of 2.27

0 1 L;0 α α L;0 β β D = µ hk;so ν U c c U c c (2.29) K,L 4S2 h | | i  aαiα µi µa aβ iβ µi µa µν i ,a i ,a X Xα α Xα β   −1 1 L;−1 α β L;−1 β α D = µ hk;so ν U c c U c c K,L 2S(2S 1) h | | i  aβ iα µi µa aαiβ µi µa µν i ,a i ,a − X Xα β Xβ α   +1 1 L;+1 α β L;+1 β α D = µ hk;so ν U c c U c c K,L 2(S + 1)(2S + 1) h | | i  aβ iα µi µa aαiβ µi µa µν i ,a i ,a X Xα β Xβ α   Neese explicitly included the prefactors for diﬀerent spin excitations by re- formulating the PK equation and claims Hartree-Fock exchange can be prop- erly included in the predictions of ZFS, thus employing hybrid functionals as well, Coupled-Perturbed approach has been tested for transition metal and concludes inclusion of prefactors does not guarantee accurate results for functional based ZFS computation. However substantially reducing the error by treating spin-ﬂip with proper prefactors. Additionally spin-spin interactions are also accounted with McWeeny and Mizuono implementations.

2.6 Spin-Spin Coupling

Two interacting charged particles with intrinsic angular momentum can be deﬁned mathematically by dipole-dipole approximations

2 α ~si ~sj 3 (~si ~rij) (~sj ~rij) Hˆss = · · · (2.30) 2 r3 − r5 i,j " ij ij # X where α is the ﬁne structure constant, the above equation can be recognized as classical dipole-dipole interactions. Many diﬀerent tactics are followed for evaluating Hso as we discussed in detail earlier. The Hss evaluations are based on the implementation with 2.6. SPIN-SPIN COUPLING 13

MCSCF from the formalism of second quantization, as pairs of conﬁgura- tion are computed on the basis diﬀerence in excitation. The two-electron integrals for the spin dipole-dipole interactions is

2 α ∗ x1kx2i 3~r1 ~r2 dkl,pqrs = dV dV φp(~r )φr(~r ) − · φq(~r )φs(~r ) (2.31) 2 1 2 1 2 r5 2 2 Z 12 which is summed over spin-projected ﬁeld operators Sk Sl gives the × pqrs contribution of spin dipole-dipole coupling tensor. One has to remind that the spin-spin interactions is consistent with selection rule ∆S = 0, 1 and ± the calculations of Ms = S is through Wigner-Eckart theorem instead of di- 6 rect evaluation of coupling elements. Now all the possible spin-spin coupling contributions leads to evaluation of the ZFS tensor

Dkl = dkl,pqrsqpqrs (2.32) pqrs X and the quintet density is calculated from

SS (2szsz sxsx sysy)pqrs SS q = h | − − | i (2.33) pqrs 3S2 S(S + 1) − Evaluation of 2.33 considers summation over partially occupied orbitals only. The earlier benchmark of this implementation is reported in the reference[24, 20, 20, 37] 14 CHAPTER 2. THEORETICAL METHODS Chapter 3

Results and Discussion

3.1 Spins in Biradical

3.1.1 Trimethylenemethane Trimetylenemethane(TMM) is one of the simplest forms of biradical. It has been under experimental and theoretical studies for long time.[7]. Many report the singlet-triplet splittings in the literature and vibrational frequencies of triplet state were theoretically predicted.[35] With high symmetric(D3h) structure the ground state is characterized by degenerate π triplet state. The whole structure is a good example of the Jahn-Teller eﬀect, in which forces are not balanced by the charge distribution. The triplet ground state wave functions are 1 ψ3 = (φ φ φ φ1) (αβ + βα) (3.1) 1 2 1 2 − 2 3 1 ψ = (φ1φ2 φ2φ1) (αα) 2 √2 − 3 1 ψ = (φ1φ2 φ2φ1) (ββ) 3 √2 − where φ’s are spatial wave functions of the electrons, α, β are usual notations of spins. The degenerate triplet system splits due to spin-spin coupling as shown in the ﬁgure. The splittings between the triplet states are known experimentally as D = 0.020cm−1 with E = 0.[17] Although there has been lots of calculations on this system, we attempt here to explain only the spin-spin coupling which dominates, such interactions are treated in MCSCF framework using the McWeeny and Mizuono formula.

15 16 CHAPTER 3. RESULTS AND DISCUSSION

Figure 3.1: Triplet π ground state

Eq(3.1)The degenerate magnetic spin sublevels (2S + 1), split in the absence of external magnetic field, leading to an explanation of the splitting in terms of parameters. More details are about reducing into parameter form and the implementation of coding is discussed in detail in chapter 2. The triplet ground state configuration is ′2 ′4 ′2 ′2 ′4 ′2 ′4 ′2 ′4 ′2 ′′1 ′′1 1a1 1e 2a1 3a1 2e 4a1 3e1 1a2 4e1 1a2 1ex 1ey (3.2) ′′1 ′′1 where ex and ey are degenerate highest occupied π orbitals in D3h full point group of the molecule. In order to handle such high symmetry molecule in most computational chemistry packages one needs to map the high symmetry functions to lower symmetry. We reduce the symmetry to C2v and hence the degenerate ground 3 state is represented as B2. All the calculations are carried out in this reduced symmetry and we choose the active spaces constrained to this representation. It is a well known fact that although organic molecules can be reason- ably described by single determinant wave functions for its equilibrium state, those with predominant John-Teller effect could not be well described without correlated methods. Hence multi-configurational studies with better 3.1. SPINS IN BIRADICAL 17

Table 3.1: MCSCF energy and splitting parameter of TMM

Trimethylene Methane Basis (2,2) (4,4) (10,10) Energy D Energy D Energy D

cc-pVDZ -154.880041 0.05460 -154.918629 0.02974 -154.933248 0.03323 cc-pVTZ -154.922044 0.05382 -154.960090 0.02900 -155.007611 0.02200 aug-cc-pVDZ -154.885816 0.05313 -154.923628 0.02924 -154.970954 0.02219 ano-1 -154.908096 0.05292 -154.946381 0.02911 -154.993668 0.02234 ano-2 -154.929526 0.05330 -154.96ll92 0.02877 -155.014940 0.02178

correlated basis are in need to have better geometries and accuracy. This fact is quite evident in the calculation presented here. As we improve the number conﬁgurations and diﬀuse functions of the basis, we get good accuracy. Table. 3.1

3.1.2 Oxyallyl

The oxyally diradical has a similar structure to that of TMM and is a derivative of it obtained by replacing a single methylene group with an oxygen atom. For a long time the oxyally ground state has been subject of study by both experiments and theory[14]. Oxyally plays an important role in the ring opening reactions of cyclopropanone. Previous theoretical studies shows that singlet oxyally is formed as intermediate with opening of cyclopropanone and 30 kcal/mol more energy than the closed ring of ′′ ′′ cyclopropanone. The triplet ex and ey degeneracy of tmm is lifted with the substitution of an oxygen atom as shown in the ﬁgure. The previous 1 calculations predicts a singlet ground state A1 with not much diﬀerence 3 with closely lying excited triplet state B2.[28] The π bond between the carbon and oxygen is thought to be the reason for the singlet resonance state. Recently there was an experimental report proving the gap between singlet state with triplet state separated by only (55meV ). [15]. We do obtain a result of that kind with very low lying excited states as presented in Table.3.1.2 18 CHAPTER 3. RESULTS AND DISCUSSION

Figure 3.2: Splits of allyl

The electronic conﬁgurations of singlet and triplet states are

2 2 2 2 ...1b1b1 + ...... 1b1a2

1 for singlet A1 2 1 1 ...1b1b12a2

3 for triplet B2

(3.3)

The photoelectron spectrum of the oxyally anion obtained in reference [15] explains the relative energies of the lowest singlet and triplet state of the oxyally anion with vibrations of each state. The triplet states vibrations are placed at a regular spacing of 405 cm−1 with a broader peak corresponding to singlet state. The position of the peaks were supported by both by density functional theory and multireference configurations with perturbations corrections calculations- to account for the dynamic electronic correlations effectively. In this report we study the role of this biradical spins in the splitting of the spectra of these closely lying states and to better understand the role of spin operators in biradicals. When we operate with the spin-orbit Hamiltonian on converged triplet state and singlet state, we observed that the operator does not affect the degeneracy of triplet ground state with zero contributions at it’s ground state geometry and its relaxed geometry. 3.1. SPINS IN BIRADICAL 19 4876 -0.062688893 -0.00220 -0.06425 -0.00321 843118 -0.05907897324 -0.00244 -0.06333 -0.00297 6 -0.854873 -0.06386 -0.00287 Oxyallyl D E Energy D E Energy D E Table 3.2: Splitting parameter with energies for OXA 1 Energy 000000 . 190 − a cc-pVDZ -0.714652 -0.09565 -0.00803 -0.75850 -0.04928 -0.00927 -0. Basis (2,2) (4,4) (10,10) cc-pVTZaug-cc-pVDZ -0.726669ano-1 -0.768153 -0.10007ano-2 -0.09960 -0.00653 -0.00695 -0.76882 -0.81086 -0.05271 -0.756478 -0.05195 -0.0086 -0.10010 -0.779415 -0.00851 -0.00657 -0.10066 -0. -0.79910 -0.00652 -0.05227 -0.82183 -0.00858 -0.05249 -0.88 -0.00834 -0.90 20 CHAPTER 3. RESULTS AND DISCUSSION

Figure 3.3: Oxyally Triplet homo orbitals

Interestingly replacing the methylene group by an oxygen atom makes a stronger bond (C=O) by placing the alpha electrons in the terminal carbon atom to be well separated along with Coulomb repulsion. This is evident that there is reversal of sign in the D parameter as reported here in Table 3.1.2, unlike in TMM where a positive sign of D indicating a spin orbitals distribute in a ring. These can be attributed to repulsion of the electrons and hence reducing the spin angular momentum coupling energy. The small E is due to two fold symmetry of the point group as it gives ﬁnite diﬀerence 2 2 between Sx and Sy operator of spin.

3.2 Spin coupling in low lying states of HCP and NCN

Methinophosphide, ﬁrst synthesized by Gier[11], is the only isolated compound involving phosphorous bonded to a neighboring atom by a triple bond with carbon. Later Tyler obtained the singlet ground state geometry using microwave conﬁrming the previous studies reported by Gier.[36] The exten- sive ultraviolet spectrum studies were reported by Johns et. al. identifying 3.2. SPIN COUPLING IN LOW LYING STATES OF HCP AND NCN 21 seven electronic transitions in the corresponding region.

The conﬁgurations that forms the singlet linear ground state of HCP X˜ 1Σ+ described as [1 7]σ2[1 2]π4 (3.4) − − The low-lying states of 1Σ+,1Σ−,1∆, 3Σ+, 3Σ−, 3∆ originates from the excitations of 7σ and 2π to the anti-bonding molecular orbitals 8σ and 3π, precisely can be drawn out of the following conﬁgurations [1 7](2π)33(π)1, [1 7](2π)32(π)2 − − for singlet and triplet state respectively. The low lying triplet and singlet states are presented in the Fig. 3.2. As one can infer that the order of these states are as follows

1Σ+, 3Σ+, 3∆, 3Σ−, 1Σ−, 1∆

The Fig. 3.2, Fig 3.2 shows the spin-spin coupling dependence of C-P and C-N length for the corresponding ﬁrst excited triplet state 3Σ+ of HCP 3 − and ground state Σg of NCN molecule, indicating almost linear dependence in the intermediate region for both cases.

The SOC between these states are calculated and presented for HCP .3.3. −1 3 + The λso = 0.5cm for the ﬁrst excited states Σ at the equilibrium − distance and is comparable with λss. The relaxed geometry SOC values are also presented in the table. and indicates strong dependence of bond length. 4 2 2 2 The electronic conﬁguration of NCN radical is given by ..1πu4σg .3σu1πg 3 − It resembles the O2 molecule with triplet ground state of X Σg and low 1 1 + lying a ∆g and b Σg . The energy gaps were 1.21 eV (a-X) and 1.93 eV (b- X) and vertical transition energies of 1.21 and 2.21 eV . Low lying singlets and triplet states of NCN are given in the table 3.5 with corresponding optimized geometries.

−1 The SSC contributions for the ground state is λss = 0.654cm and SOC part turns out to be λso = 0.124cm 1. The matrix element −

3 − 1 + −1 X Σ Hso b Σ = 66.56cm (3.5) h g,0| | g i −1 The total splitting calculated λtot = λss + λso = 0.778cm is in good −1 agreement with experimental value of λexp = 0.794cm from the reference. 22 CHAPTER 3. RESULTS AND DISCUSSION

[38]. We also calculated magnetic phosphorescence of for otherwise spin forbidden electronic dipole transitions and it is discussed in the paper.

Table 3.3: Excited states of HCP with relativistic eﬀect ∆E in cm−1 States Ω 1.3 A˚ 1.4 A˚ 1.5 A˚ ∆En En(a.u.) ∆En En(a.u.) ∆En En(a.u.) X1Σ+ 0 0 -379.23782 0 -379.25187 0 -379.24095 3Σ+ 1 36790.7 -379.07019 28855.8 -379.12045 21783.9 -379.14696 3Σ+ 0 36792.0 -379.07018 28847.1 -379.12045 21785.2 -379.14690 3∆ 1 45908.8 -379.02865 38147.6 -379.07807 31292.8 -379.09837 3∆ 2 45910.6 -379.02864 38149.7 -379.07805 31293.3 -379.09837 3∆ 3 45949.7 -379.02846 38170.2 -379.07796 31291.3 -379.09832 3Σ− 0 50499.7 -379.00773 42721.9 -379.05722 35852.3 -379.07760 3Σ− 1 50500.9 -379.00772 42723.0 -379.05722 35853.3 -379.07759 1Σ− 0 51155.9 -379.00473 44936.5 -379.04713 39546.4 -379.06076 1∆ 2 52970.7 -378.99647 47001.2 -379.03773 41833.4 -379.05035 3.2. SPIN COUPLING IN LOW LYING STATES OF HCP AND NCN 23

ZFS spin-spin contribution

-0.86

-0.88 -1

-0.9 in cm ss D

-0.92

1.4 1.6 1.8 2 RCP

Figure 3.4: Spin-Spin coupling as a functions of C-P distance

0.3

1Σ+ 3Σ+ 3∆ − 0.2 3Σ 1Σ− E 1∆

0.1

0 1.4 1.6 1.8 RC-P

Figure 3.5: Low lying states of HCP 24 CHAPTER 3. RESULTS AND DISCUSSION

Spin-Spin coupling of ground state NCN

1.5

1.45 3Σ - g

1.4 ss D

1.35

1.3

1.2 1.4 1.6 1.8 2 RC-N

Figure 3.6: Spin-Spin coupling as a functions of C-N distance

3Σ - g 1∆ g 1Σ + g 0.1 E

0.05

0 1.2 1.3 1.4 1.5 RC-N

Figure 3.7: NCN low lying states 3.2. SPIN COUPLING IN LOW LYING STATES OF HCP AND NCN 25

Table 3.4: Excited states of NCN with relativistic eﬀect. ∆E cm−1 States Ω 1.2 A˚ 1.24 A˚ 1.3 A˚ ∆En En(a.u) ∆En En(a.u.) ∆En En(a.u.) 3 − X Σg 0 0 -146.85715 0 -146.86344 0 -146.85186 3 − X Σg 1 0.249 -146.85715 0.248 -146.86344 0.245 -146.85186 1 ∆g 2 9889.0 -146.81209 9831.9 -146.81865 9758.7 -146.80740 1 + Σg 0 17943.7 -146.77540 17885.6 -146.78195 17821.7 -146.77066 1 − Σu 0 34043.5 -146.70204 31604.5 -146.71944 26148.1 -146.73272 3 ∆u 3 38512.9 -146.68167 36485.9 -146.69720 32120.5 -146.70551 1 ∆g 2 38563.9 -146.68144 36536.7 -146.69697 32170.4 -146.70528 1 ∆g 1 38615.1 -146.68121 36537.5 -146.69674 32220.3 -146.70506 3 + Σu 1 40469.6 -146.67276 38484.7 -146.68808 26148.1 -146.73272 3 + Σu 0 40469.7 -146.67271 38490.3 -146.68807 26148.1 -146.73272 3 − Σu 0 41723.7 -146.66705 39073.7 -146.668541 34277.6 -146.69568 3 − Σu 1 41723.8 -146.66705 39073.9 -146.668541 34277.8 -146.69568 1 + Σu 0 54040.4 -146.61093 51162.4 -146.63054 44786.8 -146.64780 26 CHAPTER 3. RESULTS AND DISCUSSION

Table 3.5: Optimized low lying states of NCN

Linear States Representation(D2h) -Energy(a.u)-146.00000 RC−P A˚

∆g Ag 0.82034 1.2455 B1g 0.82034 1.2455 ∆g B1g 0.82034 1.2455 + Σg Ag 0.78358 1.2455 − Σu Au 0.73583 1.3079 Singlets + Σu B1u 0.65569 1.3251 B2u 0.64281 1.2421 Πu B3u 0.64281 1.2421 B2g 0.60481 1.2266 Πg B3g 0.60481 1.2266 Σg Ag 0.47749 1.2455

− Σg B1g 0.86756 1.2485 Au 0.70947 1.2950 ∆u B1u 0.70947 1.2950 − Σu Au 0.70143 1.2950 Σ+ B 0.70039 1.2950 Triplets u 1u B2u 0.68830 1.2411 Πu B3u 0.68830 1.2411 B2g 0.63933 1.2101 Πg B3g 0.63933 1.2101 + Σg Ag 0.55000 1.4221 − Σg B1g 0.53616 1.2485 Au 0.52254 1.2950 ∆u B1u 0.52254 1.2950 B2g 0.49832 1.2101 Πg B3g 0.49832 1.2101 B2u 0.46935 1.2411 Πu B3u 0.46935 1.2411 Chapter 4

Conclusion

Both spin-spin and spin-orbit coupling in the relativistic Hamiltonian were explored at multi-configurational and density functional theory level, for better understanding of splitting of quantum chemical states. This study pro- vide insight that spin-orbit coupling contributions in small organic molecules is negligible and hence spin-spin operator plays a significant role in splitting the degeneracy. Better approximations through perturbation techniques are emphasized for modeling practically important devices like molecular magnets. The implementation of the spin-spin coupling operator for practical evaluations through multiconfigurational methods had been outlined and the scheme is used to determine the splits in the triplet spectra of oxyallyl and compared with the TMM biradical. Particularly the states of the HCP and NCN are subject to the spin- orbit operator resulting in small contributions along with spin-spin coupling, to the splitting. Although the splittings are very small they are resonably accounted for and consistent with the methodological implementation of the coupling operator in practical evalutations of quantum chemical systems.

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