Valence Bond Self-consistent Field Research Articles

Exploring the Bonding Nature of Iron(IV)-Oxo Species through Valence Bond Calculations and Electron Density Analysis.

Compound I (Cpd I) plays a pivotal role in substrate transformations within the catalytic cycle of cytochrome P450 enzymes (P450s). A key constituent of Cpd I is the iron(IV)-oxo unit, a structural motif also found in other heme enzymes and nonheme enzymes. In this study, we performed ab initio valence bond (VB) calculations, employing the valence bond self-consistent field (VBSCF) and breathing orbital valence bond (BOVB) methods, to unveil the bonding nature of this vital "Fe(IV)═O″ unit in bioinorganic chemistry. Comparisons were drawn with the triplet O2 molecule, which shares some electronic characteristics with iron(IV)-oxo. Additionally, Cpd I models of horseradish peroxidase (HRP) and catalase (CAT) were analyzed to assess the proximal ligand effect on the electronic structure of iron(IV)-oxo. Our VB analysis underscores the significant role of noncovalent resonance effects in shaping the iron(IV)-oxo bonding. The resonance stabilization within the π and σ frameworks occurs to comparable degrees, with additional stabilization resulting from resonance between VB structures from these frameworks. Furthermore, we elucidated the substantial influence of proximal and equatorial ligands in modulating the relative significance of different VB structures. Notably, in the presence of these ligands, iron(IV)-oxo is better described as iron(III)-oxyl or iron(II)-oxygen, displaying weak covalent character but enhanced by resonance effects. Although both species exhibit diradicaloid characters, resonance stabilization in iron(IV)-oxo is weaker than in O2. Further exploration using the Laplacian of electron density shows that, unlike O2, which exhibits a charge concentration region between its two oxygen atoms, iron(IV)-oxo species display a charge depletion region.

Synergistic Charge Transfer Effect in Ferrous Heme-CO Bonding within Cytochrome P450.

We conducted ab initio valence bond (VB) calculations employing the valence bond self-consistent field (VBSCF) and breathing orbital valence bond (BOVB) methods to investigate the nature of the coordination bonding between ferrous heme and carbon monoxide (CO) within cytochrome P450. These calculations revealed the significant influence exerted by both proximal and equatorial ligands on the π-backdonation effect from the heme to the CO. Moreover, our VB calculations unveiled a phenomenon of synergistic charge transfer (sCT). In the case of ferrous heme-CO bonding, the significant stabilization in this sCT arises from cooperative resonance between the VB structures associated with σ donation and π backdonation. Unlike many other ligands, CO possesses the unique ability to establish two mutually perpendicular π-backdonation orbital interaction pairs, leading to an intensified stabilization attributed to σ-π resonance. Furthermore, while of a smaller energy magnitude, sCT due to one π-π pair is also present, contributing to the differential stabilization of ferrous heme-CO bonding.

Hybrid Density Functional Valence Bond Method with Multistate Treatment.

Recently, a hybrid density functional valence bond (VB) method, λ-DFVB(U), has been proposed and shown to give accuracy that is comparable to that of CASPT2 in calculations of atomization energies, atomic excitation energies, and reaction barriers, while its computational cost is approximately the same as the valence bond self-consistent-field (VBSCF) method. However, the interaction between electronic states is not included in λ-DFVB(U) since the last step of λ-DFVB(U) is not a diagonalization of the Hamiltonian matrix on the electronic state basis. Therefore, λ-DFVB(U) gives the wrong topology of the potential energy surfaces (PESs) near the conical intersection region. In the present paper, we propose a novel hybrid density functional VB method with multistate treatment, named λ-DFVB(MS), in which an effective Hamiltonian matrix is constructed on the basis of the diabatic states obtained by the valence-bond-based compression approach for the diabatization scheme, and the interaction between electronic states can be included through the diagonalization of the effective Hamiltonian matrix. Test calculations show that λ-DFVB(MS) gives the correct topology of the PESs near the conical intersection region. We also show that the VBSCF wave function with selected VB structures can be applied as a reference in λ-DFVB(MS).

λ-DFVB(U): A hybrid density functional valence bond method based on unpaired electron density.

In this paper, a hybrid density functional valence bond method based on unpaired electron density, called λ-DFVB(U), is presented, which is a combination of the valence bond self-consistent field (VBSCF) method and Kohn-Sham density functional theory. In λ-DFVB(U), the double-counting error of electron correlation is mitigated by a linear decomposition of the electron-electron interaction using a parameter λ, which is a function of an index based on the number of effectively unpaired electrons. In addition, λ-DFVB(U) is based on the approximation that correlation functionals in KS-DFT only cover dynamic correlation and exchange functionals mimic some amount of static correlation. Furthermore, effective spin densities constructed from unpaired density are used to address the symmetry dilemma problem in λ-DFVB(U). The method is applied to test calculations of atomization energies, atomic excitation energies, and reaction barriers. It is shown that the accuracy of λ-DFVB(U) is comparable to that of CASPT2, while its computational cost is approximately the same as VBSCF.

Partial Double Metal-Carbon Bonding Model in Transition Metal Methyl Compounds.

The chemical bond between a transition metal and a methyl group (M-CH3) is typically defined as a single covalent bond, which is of fundamental significance and general interest in understanding the structural properties and reactivity of transition metal alkyl compounds. Herein, we demonstrate that the M-CH3 bonding involves varying σ and π components and thus should be best described in terms of the partial double M═CH3 bond. The often-neglected π bonding stems from an occupied π-symmetric orbital of the methyl group comprising all three C-H σ bonds (but one C-H' contributes more than the other two) and a vacant low-lying metal d(π) orbital, and is associated with the intramolecular C-H'···M agostic effect (i.e., an acute M-C-H' angle and a short H'···M distance), whose origin is still controversial. We quantify the geometric and energetic impacts of the π interaction involved in the M-CH3 bond by explicitly computing the intramolecular πCH' → dM interaction with the ab initio valence bond (VB) theory. Our computations of the ligand-free [TiCH3]3+ and a series of metallocene catalysts provide a direct proof for the presence of the π bonding in M-CH3 bonds, which is the cause for the agostic effect. The partial double M═CH3 bonding model is not only validated by a range of bonding analyses including VB self-consistent field (VBSCF)-based energy decomposition and quantum theory of atoms in molecules (QTAIM) but also authenticated by the specific activity of double M═CH3 bonds in the C-H activation and olefin insertion. More importantly, the σ bond gradually switches from a classical covalent bond to a novel charge-shift bond with the π bonding becoming increasingly significant. We anticipate that the recognition of the π interaction between electrophilic metal centers and C-H bonds can benefit the understanding of the nature of metal-carbon bonds in transition metal ethyl, alkyl, and carbene compounds.

A Valence-Bond-Based Multiconfigurational Density Functional Theory: The λ-DFVB Method Revisited.

A recently developed valence-bond-based multireference density functional theory, named λ-DFVB, is revisited in this paper. λ-DFVB remedies the double-counting error of electron correlation by decomposing the electron–electron interactions into the wave function term and density functional term with a variable parameter λ. The λ value is defined as a function of the free valence index in our previous scheme, denoted as λ-DFVB(K) in this paper. Here we revisit the λ-DFVB method and present a new scheme based on natural orbital occupation numbers (NOONs) for parameter λ, named λ-DFVB(IS), to simplify the process of λ-DFVB calculation. In λ-DFVB(IS), the parameter λ is defined as a function of NOONs, which are straightforwardly determined from the many-electron wave function of the molecule. Furthermore, λ-DFVB(IS) does not involve further self-consistent field calculation after performing the valence bond self-consistent field (VBSCF) calculation, and thus, the computational effort in λ-DFVB(IS) is approximately the same as the VBSCF method, greatly reduced from λ-DFVB(K). The performance of λ-DFVB(IS) was investigated on a broader range of molecular properties, including equilibrium bond lengths and dissociation energies, atomization energies, atomic excitation energies, and chemical reaction barriers. The computational results show that λ-DFVB(IS) is more robust without losing accuracy and comparable in accuracy to high-level multireference wave function methods, such as CASPT2.

Ab initio valence bond theory: A brief history, recent developments, and near future.

This Perspective presents a survey of several issues in ab initio valence bond (VB) theory with a primary focus on recent advances made by the Xiamen VB group, including a brief review of the earlier history of the ab initio VB methods, in-depth discussion of algorithms for nonorthogonal orbital optimization in the VB self-consistent field method and VB methods incorporating dynamic electron correlation, along with a concise overview of VB methods for complex systems and VB models for chemical bonding and reactivity, and an outlook of opportunities and challenges for the near future of the VB theory.

Novel implementation of seniority number truncated valence bond methods with applications to H22 chain.

Both the efficiency and capability of the seniority number truncated valence bond (VB) methods are highly improved by using our recently proposed extension of Malmqvist's algorithm for the reciprocal transformation of many-electron bases constructed by nonorthogonal orbitals [Zhou, Chen, and Wu, J. Chem. Phys. 149(4), 044111 (2018)] and by the adoption of the direct technique in solving the generalized eigenvalue problem. Due to the compactness of the wave function that benefited from seniority number restriction, the memory need and computational cost for energy evaluation and orbital optimization in valence bond self-consistent field calculation are largely reduced. The last obstacle in nonorthogonal orbital based ab initio VB calculation is thus removed. Consequently, we can accomplish seniority number truncated VB calculation at the same computational scaling as that of the most general configuration selected multiconfigurational self-consistent field with a memory cost much less than the corresponding complete active space self-consistent field (CASSCF). Test on Hn string molecules shows that the seniority number truncated VB calculation maintains the majority of static correlation by using a more compact wave function than CASSCF.

λ-Density Functional Valence Bond: A Valence Bond-Based Multiconfigurational Density Functional Theory With a Single Variable Hybrid Parameter.

A new valence bond (VB)-based multireference density functional theory (MRDFT) method, named λ-DFVB, is presented in this paper. The method follows the idea of the hybrid multireference density functional method theory proposed by Sharkas et al. (2012). λ-DFVB combines the valence bond self-consistent field (VBSCF) method with Kohn–Sham density functional theory (KS-DFT) by decomposing the electron–electron interactions with a hybrid parameter λ. Different from the Toulouse's scheme, the hybrid parameter λ in λ-DFVB is variable, defined as a function of a multireference character of a molecular system. Furthermore, the EC correlation energy of a leading determinant is introduced to ensure size consistency at the dissociation limit. Satisfactory results of test calculations, including potential energy surfaces, bond dissociation energies, reaction barriers, and singlet–triplet energy gaps, show the potential capability of λ-DFVB for molecular systems with strong correlation.

Performance of the VBSCF method for pericyclic and π bond shift reactions.

The performance of the valence bond self-consistent field (VBSCF) method was investigated in this paper by predicting the activation barriers and reaction energies in pericyclic and π bond shift reactions for hydrocarbons. The benchmarking set includes 3 electrocyclic reactions, 3 sigmatropic shifts, 3 cycloadditions, 2 cycloreversions, and 7 π bond shift reactions, where the first 11 reactions are taken from Houk's standard set (J. Phys. Chem. A 2003, 107, 11445). Computational results reveal that the VB(CI) method, which performs VBSCF calculations first with full covalent structures and then includes all mono- and di-ionic structures to compute the total energy without further orbital optimization, matches the accuracy of the complete active space SCF (CASSCF) method very well. The mean absolute error values (the deviations from the CASSCF data) are 0.01 kcal/mol for the reaction energy, and 0.26 and 0.32 kcal/mol for the activation energy with the 6-31G and 6-31G(d) basis sets, respectively. © 2018 Wiley Periodicals, Inc.

Valence Bond Theory Reveals Hidden Delocalized Diradical Character of Polyenes.

The nature of the electronic-structure of polyenes, their delocalization features, and potential diradicaloid characters constitute a fundamental problem in chemistry. To address this problem, we used valence bond self-consistent field (VBSCF) calculations and modeling of polyenes, C2nH2n+2 (n = 2-10). The theoretical treatment shows that starting with n = 5, the polyene's wave function is mainly a shifting 1,4-diradicaloid, a character that increases as the chain length increases, while the contribution of the fundamental Lewis structure with alternating double and single bonds (1) decays quite fast and becomes minor relative to the diradicaloid pack. We show how, nevertheless, it is this wave function that predicts that polyenes will still exhibit alternating short/long CC bonds like the fundamental structure 1. Furthermore, despite the decay of the VB contribution of 1, it remains the single structure with the largest weight among all the individual structures. The mixing of all the 1,4-diradicaloid structures into 1 follows perturbation theory rules, with the result that the delocalization energy due to this mixing is additive and behaves as a linear function of the number of the double bonds, ΔEdel = -6.9 × n (kcal mol-1). The VB modeling shows that while the conjugation stabilizes structure 1, this stabilization energy is energetically overridden by the Pauli repulsion between two adjacent double bonds. Nevertheless, unsubstituted polyenes remain planar; this observation is addressed. Potential manifestations of the diradicaloid nature of polyenes are discussed, and it is concluded that the diradicaloid character is clearly not a well-defined physical property as in real diradicals. Thus, we went full circle to realize that our philosophical question may not be strictly resolved. The localized/delocalized properties of polyenes seem to define a "chemical duality principle". This duality of molecular wave functions is a ubiquitous beguiling phenomenon.

Bonding in B2 and B2+: Insights from full configuration interaction and valence bond studies

Full Configuration Interaction (Full-CI) and Valence Bond Self-Consistent Field (VBSCF) methods have been used to study the electronic structure and bonding in B2 and B2+ molecules. The bonding analysis based on these calculations shows that the B2 molecule is stabilised due to the formation of a double σ bond, one strong σ-bond together with one second weaker σ-bond, and two weak π bonds. Upon ionization one π electron is removed from the system and B2+ is formed, which has a one electron σ bond, instead of a π bond. It has been shown that a few carefully chosen VB configurations are enough to describe the bonding; with these structures, geometrical parameters as well as dissociation energies of these unusual molecular species are in agreement with full-CI results.

Hamiltonian Matrix Correction Based Density Functional Valence Bond Method.

In this work, a valence bond type multireference density functional theory (MRDFT) method, called the Hamiltonian matrix correction based density functional valence bond method (hc-DFVB), is presented. In hc-DFVB, the static electronic correlation is considered by the valence bond self-consistent field (VBSCF) strategy, while the dynamic correlation energy is taken into account by Kohn-Sham density functional theory (KS-DFT). Different from our previous version of DFVB (J. Chem. Theory Comput. 2012, 8, 1608), hc-DFVB corrects the dynamic correlation energy with a Hamiltonian correction matrix, improving the functional adaptability and computational accuracy. The method was tested for various physical and chemical properties, including spectroscopic constants, bond dissociation energies, reaction barriers, and singlet-triplet gaps. The accuracy of hc-DFVB matches that of KS-DFT and high level molecular orbital (MO) methods quite well. Furthermore, hc-DFVB keeps the advantages of VB methods, which are able to provide clear interpretations and chemical insights with compact wave functions.

The application of cholesky decomposition in valence bond calculation.

The Cholesky decomposition (CD) technique, used to approximate the two-electron repulsion integrals (ERIs), is applied to the valence bond self-consistent field (VBSCF) method. Test calculations on ethylene, C2 n H2 n +2 , and C2 n H4 n -2 molecules (n = 1-7) show that the performance of the VBSCF method is much improved using the CD technique, and thus, the integral transformation from basis functions to VB orbitals is no longer the bottleneck in VBSCF calculations. The errors of the CD-based ERIs and of the total energy are controlled by the CD threshold, for which a value of 10(-6) ensures to control the total energy error within 10(-6) Hartree. © 2016 Wiley Periodicals, Inc.

Structure–Property Relationships for Exciton and Charge Reorganization Energies of Dipolar Organic Semiconductors: A Combined Valence Bond Self-Consistent Field and Time-Dependent Hartree-Fock and DFT Study of Merocyanine Dyes

We present an analysis of the optoelectronic properties of merocyanine dyes by means of valence bond self-consistent field (VBSCF), time-dependent Hartree-Fock (TD-HF), density functional theory (DFT), and high-level ab initio calculations. The electronic structure of merocyanines can be described as a superposition of two resonance structures, a neutral one and a zwitterionic one. Calculated valence bond (VB) weights for these resonance structures demonstrate the importance of strong accepting groups when increasing the weight of the zwitterionic structures of different merocyanines. The dependence of exciton and charge reorganization energies on VB weights’ composition is analyzed, demonstrating that the special case of equal contributions of both structures, the so-called cyanine limit, goes along with minimal exciton and charge reorganization energies. For the latter, it is shown that the external (outer-sphere) reorganization energy plays a crucial role. Furthermore, a careful investigation of the ex...

Nonorthogonal orbital based N-body reduced density matrices and their applications to valence bond theory. IV. The automatic implementation of the Hessian based VBSCF method.

In this paper, the Hessian matrix of valence bond (VB) self-consistent field (VBSCF) energy with respect to orbitals are evaluated by applying the nonorthogonal orbital based N-body reduced density matrices, which was presented in Paper I. To this end, an automatic formula/code generator (AFCG) is developed; with which the matrix elements between internally contracted excited configurations of VB wave function and the corresponding codes are generated automatically. Compared to the tedious manual formula deducing and implementing, AFCG is much more convenient and efficient, and enables us to avoid troublesome debugging. With the help of AFCG, the Hessian-based Newton-Raphson algorithm is implemented for the VBSCF orbital optimization. Test calculations indicate that the Newton-Raphson algorithm converges quadratically and has much better convergence behavior than the gradient-based LBFGS algorithms. Furthermore, a combined approach with LBFGS and Newton-Raphson algorithms is applied to reduce the total CPU time of the calculation.

Nonorthogonal orbital based n-body reduced density matrices and their applications to valence bond theory. III. Second-order perturbation theory using valence bond self-consistent field function as reference.

Using the formulas and techniques developed in Papers I and II of this series, the recently developed second-order perturbation theory based on a valence bond self-consistent field reference function (VBPT2) has been extended by using the internally contracted correction wave function. This ansatz strongly reduces the size of the interaction space compared to the uncontracted wave function and thus improves the capability of the VBPT2 method dramatically. Test calculations show that internally contracted VBPT2 using only a small number of reference valence bond functions, can give results as accuracy as the VBPT2 method and other more sophisticated methods such as full configuration interaction and multireference configuration interaction.

The effect of diffuse basis functions on valence bond structural weights

Structural weights and bond dissociation energies have been determined for H-F, H-X, and F-X molecules (-X = -OH, -NH2, and -CH3) at the valence bond self-consistent field (VBSCF) and breathing orbital valence bond (BOVB) levels of theory with the aug-cc-pVDZ and 6-31++G(d,p) basis sets. At the BOVB level, the aug-cc-pVDZ basis set yields a counterintuitive ordering of ionic structural weights when the initial heavy atom s-type basis functions are included. For H-F, H-OH, and F-X, the ordering follows chemical intuition when these basis functions are not included. These counterintuitive weights are shown to be a result of the diffuse polarisation function on one VB fragment being spatially located, in part, on the other VB fragment. Except in the case of F-CH3, this problem is corrected with the 6-31++G(d,p) basis set. The initial heavy atom s-type functions are shown to make an important contribution to the VB orbitals and bond dissociation energies and, therefore, should not be excluded. It is recommended to not use diffuse basis sets in valence bond calculations unless absolutely necessary. If diffuse basis sets are needed, the 6-31++G(d,p) basis set should be used with caution and the structural weights checked against VBSCF values which have been shown to follow the expected ordering in all cases.

Nonorthogonal orbital based N-body reduced density matrices and their applications to valence bond theory. II. An efficient algorithm for matrix elements and analytical energy gradients in VBSCF method

In this paper, by applying the reduced density matrix (RDM) approach for nonorthogonal orbitals developed in the first paper of this series, efficient algorithms for matrix elements between VB structures and energy gradients in valence bond self-consistent field (VBSCF) method were presented. Both algorithms scale only as nm(4) for integral transformation and d(2)n(β)(2) for VB matrix elements and 3-RDM evaluation, while the computational costs of other procedures are negligible, where n, m, d, and n(β )are the numbers of variable occupied active orbitals, basis functions, determinants, and active β electrons, respectively. Using tensor properties of the energy gradients with respect to the orbital coefficients presented in the first paper of this series, a partial orthogonal auxiliary orbital set was introduced to reduce the computational cost of VBSCF calculation in which orbitals are flexibly defined. Test calculations on the Diels-Alder reaction of butadiene and ethylene have shown that the novel algorithm is very efficient for VBSCF calculations.

Nonorthogonal orbital based N-body reduced density matrices and their applications to valence bond theory. I. Hamiltonian matrix elements between internally contracted excited valence bond wave functions

In this series, the n-body reduced density matrix (n-RDM) approach for nonorthogonal orbitals and their applications to ab initio valence bond (VB) methods are presented. As the first paper of this series, Hamiltonian matrix elements between internally contracted VB wave functions are explicitly provided by means of nonorthogonal orbital based RDM approach. To this end, a more generalized Wick's theorem, called enhanced Wick's theorem, is presented both in arithmetical and in graphical forms, by which the deduction of expressions for the matrix elements between internally contracted VB wave functions is dramatically simplified, and the matrix elements are finally expressed in terms of tensor contractions of electronic integrals and n-RDMs of the reference VB self-consistent field wave function. A string-based algorithm is developed for the purpose of evaluating n-RDMs in an efficient way. Using the techniques presented in this paper, one is able to develop new methods and efficient algorithms for nonorthogonal orbital based many-electron theory much easier than by use of the first quantized formulism.