Application and development of quantum chemical methods. Density functional theory and valence bond theory Fuming Ying Theoretical Chemistry School of Biotechnology Royal Institute of Technology Stockholm 2010 Copyright c 2010 by Fuming Ying TRITA-BIO Report 2010:13 ISSN 1654-2312 ISBN 978-91-7415-633-1 3 Printed by US-AB, Stockholm 2010 i Abstract This thesis deals with two subdiciplines of quantum chemistry. One is the most used electronic structure method today, density functional theory (DFT), and the other one of the least used electronic structure methods, valence bond theory (VB). The work on DFT is based on previous developments in the group in density functional response theory and involves studies of hyperfine cou- pling constants which are measured in electron paramagnetic resonance exper- iments. The method employed is a combination of a restricted-unrestricted ap- proaches which allows for adequate description of spin polarization without spin contamination, and spin-orbit corrections to account for heavy atom ef- fects using degenerate perturbation theory. The work on valence bond theory is a new theoretical approach to higher-order derivatives. The orbital derivatives are complicated by the fact that the wave functions are constructed from de- terminants of non-orthogonal orbitals. An approach based on non-orthogonal second-quantization in bi-orthogonal basis sets leads to straightforward deriva- tions without explicit references to overlap matrices. These formulas are rele- vant for future applications in time-dependent valence bond theory. ii List of publications Paper I. Restricted-unrestricted density functional theory for hyperfine coupling con- stants: vanadium complexes Xing Chen, Fuming Ying, Zilvinas Rinkevicius, Olav Vahtras, Hans Agren,˚ Wei Wu and Zexing Cao. submitted Paper II. An analytical valence bond Hessian, by means of second quantization in biorthonor- mal basis sets Fuming Ying, Olav Vahtras, Zilvinas Rinkevicius and Wei Wu. Manuscript iii My contribution I carried out calculations in paper I and participated in analysis and au- • thoring I derived the Hessian formulas in paper II and participated in authoring • I participated in the preparations of the manuscripts for all papers • iv Acknowledgments I would thank Prof. Yi Luo, Hans Agren˚ and Prof Wei Wu in Xiamen Uni- versity who gave me the opportunity to study in KTH. I appreciate my supervisor Olav Vahtras and my tutor Zilvinas Rinkevicious for their guidance, advice in each project. I still remember the situation when Zilvinas and I discussed the details of DALTON program, just as happened yes- terday. I should also give my gratitude to my colaborator Xing Chen for her help and discussion during the work. I may also say ”Thank you” to Prof. Magareta Bromberg, Aato Laaksonen and Alexander Lyubartsev in SU for the courses and Prof. Poul Jøgensen, Jeppe Olsen and Trygve helgaker for their teaching Quantum Chemistry in the sum- mer school. Thanks to my friends in Department of Theoretical Chemistry, Xiaofei Li, Keyan Lian, Hao Ren, et al. for their kind help in my life in Stockholm. Contents 1 Introduction 1 2 Density functional theory 3 2.1 Densityoptimization .......................... 3 2.2 Variational perturbation theory . 4 2.2.1 The restricted-unrestricted method . 5 3 Valence bond theory 7 3.1 VBWaveFunctionandHamiltonian . 8 3.1.1 VBWaveFunction ....................... 8 3.1.2 Constructing VB Hamiltonian with Determinant Expansion 9 3.2 DifferenttypesofVBOrbitals . 9 3.2.1 LocalizedVBOrbitals . 9 3.2.2 Semi-Localized VB Orbitals . 10 3.3 VBMethods ............................... 11 3.3.1 GeneralizedVB ......................... 11 3.3.2 CASVB.............................. 12 3.3.3 VB Self-Consistent Field . 12 3.3.4 BreathingOrbitalVB. 13 3.3.5 VB Configuration-Interaction . 14 3.3.6 VBPCM ............................. 15 3.4 VariousVBPrograms.......................... 15 3.4.1 TURTLE ............................. 15 3.4.2 VB2000.............................. 16 3.4.3 XMVB .............................. 16 4 Summary of Included Papers 17 4.1 Restricted-unrestricted density functional theory . 17 4.2 The Valence bond Hessian . 18 v Chapter 1 Introduction The work of this thesis deals with two methods of electronic structure theory. The first, density functional theory (DFT), is by far the most important method today with its competitive combination of speed and accuracy. The number of articles based on DFT increase with an exponential rate making a complete overview if not impossible but certainly beyond the scope of this thesis. The second, valence bond (VB) theory, was the most important theory in the infancy of this science. In chapter 2 we provide the theoretical background of DFT for the appli- cations used in paper I. It is customary to introduce an expose of DFT to two pioneering articles, the first by Hohenberg and Kohn[1] who established the mathematical foundation of a ground state theory based on the electron density as a fundamental variable, and the other by Kohn and Sham[2] who introduced a practical single-determinant theory for practical calculations. It was because of this work, which started a tremendous development in computational chem- istry, that Walther Kohn was awarded the Nobel Prize in Chemistry in 1998. In chapter 3 we give a background of valence bond theory. It dates back to G. N. Lewis[3] who introduced concepts such as electron pair bonding and the octet rule, which has influenced the way of thinking for generations of chemists. The theory which gradually lost importance when the rivaling molecular orbital (MO) theory grew in popularity. The latter had its origin in spectroscopy and allowed for efficient computer implementations with predictive power. Recent decades have witnessed a revival of valence bond theory, with new algorithms and new computer implementation. Paper II. is a contribution to this develop- ment. 1 2 CHAPTER 1. INTRODUCTION Chapter 2 Density functional theory The implications of the fundamental theorems of density functional theory are enormous[1]. The ground state energy of a many-electron system is a unique functional of the electron density and that the part of the functional which in- dependent of the external field, the Hohenberg-Kohn functional, is a universal functional independent of the number of electrons. This implies that any many- electron problem can be reduced to an equation in three variables, the coordi- nates of a point in space. The problem is that the equation, or the functional, is not known which has prompted a huge development of approximate function- als which have been parameterized to reproduce a certain class of experimental results. The success of DFT has been due to basis set expansions and the formu- lation of DFT as a single determinant theory[2]. In practice DFT can be viewed as a Hartree-Fock theory with a semi-empirical element which includes electron correlation. In this chapter we review the aspects of DFT that are relevant for paper I; the optimization procedure, the restricted-unrestricted method and perturba- tion theory with application to hyperfine coupling tensors. 2.1 Density optimization The density is optimized for a given energy functional E[ρ]. δE[ρ]=0 (2.1) In the most general case the optimization is carried out with the auxiliary con- dition that the total number of particles is constant dV ρ(r)= N (2.2) Z which leads to a Lagrangian multiplier which is interpreted as the chemical potential[1]. If the density is constructed from a determinantal wave function 3 4 CHAPTER 2. DENSITY FUNCTIONAL THEORY the condition 2.2 is automatically fulfilled. The variation of a functional (func- tion of a function) gives δE δE[ρ(r)] = dV δρ(r) (2.3) Z δρ(r) In practical calculations the density is parameterized for some finite set of pa- rameters λ =(λ1,λ2 ...) such that ∂ρ (r) δρ (r)= λ δλ (2.4) λ ∂λ Salek et al.[4] formulated the optimization in second quantization as follows; the density is the expectation value of a density operator ρλ(r)= λ ρˆ(r) λ (2.5) | | where the density operator projected on a basis set φp is { } ∗ † ρˆ(r)= φp(r) φq(r)apaq (2.6) Xpq When the parameterization is of exponential unitary form, † λ = e−λpqapaq 0 (2.7) | | i.e. the with parameters describing a change from a reference set of orbitals, the transformation is expanded around λ = 0 and the density variation a becomes a commutator ∂ρ † = λ [apaq, ρˆ(r) λ (2.8) ∂λpq | | Defining a Fock operator δE Fˆ = dV ρˆ(r) (2.9) Z δρ(r) the energy optimization reduces to † 0 [a aq, Fˆ] 0 =0 (2.10) | p | which are equivalent to the Kohn-Sham equations[2]. 2.2 Variational perturbation theory If a perturbation V is introduced in the Hamiltonian with a field strength x, H = H0 + xV (2.11) 2.2. VARIATIONAL PERTURBATION THEORY 5 the optimized parameters will be a function of the field strength λ(x). The en- ergy change due to this perturbation can be identified from the power expan- sion dE E[ρλ(x); x]= E[ρλ(0);0]+ x (2.12) dx x=0 λ=λ(0) and in a strictly variational theory we have dE = 0 V 0 (2.13) dx x=0 | | λ=λ(0) 2.2.1 The restricted-unrestricted method The restricted-unrestricted method were developed by Fernandez et al.[5] and later generalized for DFT by Rinkevicius et al.[6] for describing spin polariza- tion in spin-restricted methods. In spin-restricted methods, the alpha- and beta spin-orbitals always have the same spatial parts. For the calculation of a prop- erty such as the hyperfine coupling constant whose main component, the Fermi contact contribution, is a measure of the spin density at the nucleus this is prob- lematic. An approximate wave function models such as MCSCF does not have sufficient flexibility at the nucleus because the core orbitals are normally inac- tive, and to allow for correlation in the core orbitals is computationally very expensive.