Ab Initio Valence Bond Theory

Valence bond(VB)theory is one of the most important methods in quantum chemistry,since it provides distinct physical pictures corresponding to classical chemical concepts,and discovers intrinsic nature of chemical bonds.Therefore,VB theory was widely applied in early quantum chemistry.Due to the use of non-orthogonal orbitals,however,the ability of VB method is limited to some extent.Therefore,the development of VB theory was slow,while molecular theory became the mainstream in computational chemistry.The resuscitation of VB theory began from 80s of last century,when computational techniques innovated and varieties of ab initio VB methods and algorithms subsequently came out.Based on the previous works of our research group,this thesis mainly focuses on the development of valence bond self-consistent field(VBSCF)method and algorithms and VB methods combined with density functional theory(DFT).The thesis is organized as follows:In Chapter 1,we briefly introduce the development of VB theory and previous works of our research group.In Chapter 2,the concept of seniority number(SN)is introduced in VB theory to truncate the wave function.Since SN has one to one correspondence with ionicity,VB wave functions truncated by SN present explicit chemical significance.At the lowest truncation level,a VB wave function consists of configuration state functions with the largest possible SN,while configuration state functions with smaller SN are considered at higher seniority-truncated levels.Though it’s also convenient to apply SN in molecular theory,we find that the introduction of SN in VB theory exhibits superior performances than in molecular theory,as shown in the calculations of potential energy curves of H8,N2 and C2 molecules and singlet-triplet energy gaps of typical diradical molecules.In Chapter 3,we extent Malmqvist’s algorithm and develop a novel determinant tensor transformation algorithm for non-orthogonal orbital based multi-determinantal wave functions truncated by SN.In our algorithm,the biorthogonal property between covariant and contravariant tensors is ultilized to reduce computational costs in VBSCF method.In traditional Malmqvist’s algoritlhm,a covariant wave function expanded in complete active space is transformed to its equivalent contravariant form,thus a simplified calculation with non-orthogonal basis could be performed.However,it’s sometimes no need to construct an equivalent contravariant wave function,especially for the compact seniority-truncated wave function,which might consider majority of static correlation at low truncation levels.Given the fact that we just concern about expectation values associated with t-body operators in most computations,our novel algorithm shows superior performances than Malmqvist’s algorithm,in both memory and computation amounts,by considering various restrictions during determinant tensor transformation.Thus,we lay the foundation of non-orthogonal based multi-configurational self-consistent field(MCSCF)method in large active space.In Chapter 4,a novel non-orthogonal based MCSCF method,named snVBSCF,is presented.In snVBSCF,a wave function is truncated by SN so that the memory bottleneck of storing a vectors in CASSCF is overcome and calculations in larger active space become available.By using our algorithm presented in Chapter 3,the process of determinant tensor transformation,which is the most memory demanding in snVBSCF method,could be accomplished with lower computational costs compared with Malmqvist’s algorithm.snVBSCF is proved to be capable of handling calculations in(22,22)active space.Test calculations show that snVBSCF could consider majority of static correlation with low computation demands by using the following scheme:the orbitals are optimized at low seniority truncation level,and a CI calculation is performed by considering VB structures of higher ionicity without orbital optimization.In a word,snVBSCF method provides an efficient way to approach CASSCF method.In Chapter 5,we propose a Hamiltonian matrix correction based density functional valence bond method,which is names as hc-DFVB,to incorporate dynamical correlation in VB calculation.In this method,each Hamiltonian matrix element obtained from VBSCF calculation is corrected with dynamical correlation via density functional.Compared with our previous DFVB method,we are able to obtain correlation for each individual VB structure in hc-DFVB,which means it’s possible to apply hc-DFVB to calculations of electronic coupling.Furthermore,hc-DFVB is not limited to the use of pure correlation functionals,but more density functionals of higher levels are available in hc-DFVB method.Test calculations reveal that the use of meta-GGA type of functionals,such as M06 or revTPSS,gives higher accuaracy.Finally,potential energy curves of diatomic molecules obtained from hc-DFVB are more accurate,and the method is size-consistent for single bond dissociation.In Chapter 6,in order to fundamentally solve double-counting problem in multi-configurational density functional theory,we propose A-DFVB method,which is a VB based multi-configurational density functional method with a single variable hybrid parameter.In A-DFVB,the electronic interaction term is divided into two parts,namely long-range and short-range interaction respectively.The long-range part is obtained from multi-configurational wave function,while the short-range part is considered via density functionals.The hybrid parameter,λ,is a variable and related to free valence in VB theory.By using separation techniques of inactive and active parts and the determinant tensor transformation algorithm proposed in Chapter 3,λ-DFVB could be efficiently applied to calculations in large active space.Test calculations show that λ-DFVB gives qualitatively right description for potential energy curve of chromium dimmer.