Generalized Valence Bond Perfect Pairing Research Articles

Bivariational principle for an antisymmetrized product of nonorthogonal geminals appropriate for strong electron correlation

We develop a bivariational principle for an antisymmetric product of nonorthogonal geminals. Special cases reduce to the antisymmetric product of strongly-orthogonal geminals (APSG), the generalized valence bond-perfect pairing (GVB-PP), and the antisymmetrized geminal power (AGP) wavefunctions. The presented method employs wavefunctions of the same type as Richardson-Gaudin (RG) states, but which are not eigenvectors of a model Hamiltonian which would allow for more freedom in the mean-field. The general idea is to work with the same state in a primal picture in terms of pairs, and in a dual picture in terms of pair-holes. This leads to an asymmetric energy expression which may be optimized bivariationally, and is strictly variational when the two representations agree. The general approach may be useful in other contexts, such as for computationally feasible variational coupled-cluster methods.

Symmetry-Adapted Perturbation Theory Based on Multiconfigurational Wave Function Description of Monomers.

We present a formulation of the multiconfigurational (MC) wave function symmetry-adapted perturbation theory (SAPT). The method is applicable to noncovalent interactions between monomers which require a multiconfigurational description, in particular when the interacting system is strongly correlated or in an electronically excited state. SAPT(MC) is based on one- and two-particle reduced density matrices of the monomers and assumes the single-exchange approximation for the exchange energy contributions. Second-order terms are expressed through response properties from extended random phase approximation (ERPA). The dispersion components of SAPT(MC) have been introduced in our previous works [HapkaM.J. Chem. Theory Comput.2019, 15, 1016−102730525591; HapkaM.J. Chem. Theory Comput.2019, 15, 6712–672331670950]. SAPT(MC) is applied either with generalized valence bond perfect pairing (GVB) or with complete active space self-consistent field (CASSCF) treatment of the monomers. We discuss two model multireference systems: the H2 ··· H2 dimer in out-of-equilibrium geometries and interaction between the argon atom and excited state of ethylene. Using the C2H4* ··· Ar complex as an example, we examine second-order terms arising from negative transitions in the linear response function of an excited monomer. We demonstrate that the negative-transition terms must be accounted for to ensure qualitative prediction of induction and dispersion energies and develop a procedure allowing for their computation. Factors limiting the accuracy of SAPT(MC) are discussed in comparison with other second-order SAPT schemes on a data set of small single-reference dimers.

Second-Order Exchange-Dispersion Energy Based on a Multireference Description of Monomers.

We present a method for calculation of the second-order exchange-dispersion energy in the framework of the symmetry-adapted perturbation theory (SAPT) for weakly interacting monomers described with multiconfigurational wave functions. The proposed formalism is based on response properties obtained from extended random phase approximation (ERPA) equations and assumes the single-exchange (S2) approximation. The approach is applicable to closed shell systems where static correlation cannot be neglected or to systems in nondegenerate excited states. We examine the new method in combination with either generalized valence bond perfect pairing (GVB) or complete active space self-consistent field (CASSCF) description of the interacting monomers. For model multireference dimers in ground states (H2···H2, Be···Be, He···H2), exchange-dispersion energies are reproduced accurately. For the interaction between the excited hydrogen molecule and the helium atom we found unacceptably large errors which is attributed to the neglect of diagonal double excitations in the employed approximation to the linear response function.

Generalized Valence Bond Perfect-Pairing Made Versatile Through Electron-Pairs Embedding.

We present an electron-pairs-based method employing a generalized valence bond perfect-pairing (GVB-PP) ansatz that provides a uniformly accurate description of systems where various types of electron correlation play a role and the GVB-PP wave function is a suitable reference. In the proposed EERPA-GVB approach, a GVB-PP energy is amended by adding correlation among electron pairs. The latter is achieved by embedding single pairs or couples of pairs in the environment of the other electron fragments and separately accounting for intra- and interfragment correlation effects. For this purpose, we employ truncated extended random phase approximation equations. Application of EERPA-GVB to systems governed by both short-range (energy barriers) and long-range (molecular interactions) correlation effects proves the good accuracy of the method. Moreover, EERPA-GVB is shown to cure a notorious problem of uncorrelated electron-pair models, namely, spatial symmetry breaking in aromatic molecules, using the example of benzene. We have also successfully applied EERPA-GVB to a challenging problem of a phase transition of the boron chain system, where the correlation changes its character along the reaction path. The accuracy and versatility of EERPA-GVB are accompanied by its attractively low computational cost. By truncation of the extended RPA equations and consideration of only at most two-fragment correlation contributions, the cost of computing the EERPA correlation energy is reduced to scale only quadratically with the number of pairs of electrons.

Second-Order Dispersion Energy Based on Multireference Description of Monomers.

We propose a method for calculating a second-order dispersion energy for weakly interacting multireference systems in arbitrary electronic states. It is based on response properties obtained from extended random phase approximation equations. The introduced formalism is general and requires only one- and two-particle reduced density matrices of monomers. We combine the new method with either generalized valence bond perfect pairing (GVB) or complete active space (CAS) self-consistent field description of the interacting systems. In addition to a general scheme, three approximations, leading to significant reduction of the computational cost, are developed by exploiting Dyall partitioning of the monomer Hamiltonians. For model multireference systems (H2···H2 and Be···Be) the method is accurate, unlike its single-reference-based counterpart. Neither GVB nor CAS description of single-reference monomers improves the dispersion energy with respect to the Hartree-Fock-based results.

Electron Correlation from the Adiabatic Connection for Multireference Wave Functions

An adiabatic connection (AC) formula for the electron correlation energy is derived for a broad class of multireference wave functions. The AC expression recovers dynamic correlation energy and assures a balanced treatment of the correlation energy. Coupling the AC formalism with the extended random phase approximation allows one to find the correlation energy only from reference one- and two-electron reduced density matrices. If the generalized valence bond perfect pairing model is employed a simple closed-form expression for the approximate AC formula is obtained. This results in the overall M^{5} scaling of the computation cost making the method one of the most efficient multireference approaches accounting for dynamic electron correlation also for the strongly correlated systems.

Excitation energies from time-dependent generalized valence bond method

A generalized valence bond perfect-pairing (GVB-PP) wavefunction has been extensively used in computational chemistry methods due to its multiconfigurational character, which captures in an inexpensive way static electron correlation. GVB-PP has been mostly applied to ground states and not much is known about its performance in predicting electronic spectra of molecules. Here, we present the formalism based on the time-dependent linear response theory which provides excitation energies from the GVB-PP ground-state wavefunction. The accuracy of the excitation energy dissociation curves parallels that of the TD-HF method around ground-state equilibrium geometries of the investigated molecules. For stretched-bond molecules, when TD-HF breaks down, the proposed TD-GVB method remains reliable. TD-GVB may therefore serve as a useful and inexpensive tool for exploring potential energy surfaces of excited states of molecules.

Nature of the chemical bond and origin of the inverted dipole moment in boron fluoride: a generalized valence bond approach.

The generalized product function energy partitioning (GPF-EP) method has been applied to investigate the nature of the chemical bond and the origin of the inverted dipole moment of the BF molecule. The calculations were carried out with GPF wave functions treating all of the core electrons as a single Hartree-Fock group and the valence electrons at the generalized valence bond perfect-pairing (GVB-PP) or full GVB levels, with the cc-pVTZ basis set. The results show that the chemical structure of both X (1)Σ(+) and a (3)Π states is composed of a single bond. The lower dissociation energy of the excited state is attributed to a stabilizing intraatomic singlet coupling involving the B 2sp-like lobe orbitals after bond dissociation. An increase of electron density on the B atom caused by the reorientation of the boron 2sp-like lobe orbitals is identified as the main responsible effect for the electric dipole inversion in the ground state of BF. Finally, it is shown that π back-bonding from fluorine to boron plays a minor role in the electron density displacement to the bonding region in both states. Moreover, this effect is associated with changes in the quasi-classical component of the electron density only and does not contribute to covalency in either of the states. Therefore, at least for the case of the BF molecule, the term back-bonding is misleading, since it does not contribute to the bond formation.

Restricted Hartree Fock using complex-valued orbitals: a long-known but neglected tool in electronic structure theory.

Restricted Hartree Fock using complex-valued orbitals (cRHF) is studied. We introduce an orbital pairing theorem, with which we obtain a concise connection between cRHF and real-valued RHF, and use it to uncover the close relationship between cRHF, unrestricted Hartree Fock, and generalized valence bond perfect pairing. This enables an intuition for cRHF, contrasting with the generally unintuitive nature of complex orbitals. We also describe an efficient computer implementation of cRHF and its corresponding stability analysis. By applying cRHF to the Be + H2 insertion reaction, a Woodward-Hoffmann violating reaction, and a symmetry-driven conical intersection, we demonstrate in genuine molecular systems that cRHF is capable of removing certain potential energy surface singularities that plague real-valued RHF and related methods. This complements earlier work that showed this capability in a model system. We also describe how cRHF is the preferred RHF method for certain radicaloid systems like singlet oxygen and antiaromatic molecules. For singlet O2, we show that standard methods fail even at the equilibrium geometry. An implication of this work is that, regardless of their individual efficacies, cRHF solutions to the HF equations are fairly commonplace.

Description of Polar Chemical Bonds from the Quantum Mechanical Interference Perspective.

The Generalized Product Function Energy Partitioning (GPF-EP) method has been applied to a set of molecules, AH (A = Li, Be, B, C, N, O, F), CO and LiF with quite different dipole moments, in order to investigate the role played by the quantum interference effect in the formation of polar chemical bonds. The calculations were carried out with GPF wave functions treating all the core electrons as a single Hartree-Fock group and the bonding electrons at the Generalized Valence Bond Perfect-Pairing (GVB-PP) level, with the cc-pVTZ basis set. The results of the energy partitioning into interference and quasi-classical contributions along the respective Potential Energy Surfaces (PES) show that the main contribution to the depth of the potential wells comes from the interference term, which is an indication that all the molecules mentioned above form typical covalent bonds. In all cases, the stabilization promoted by the interference term comes from the kinetic contribution, in agreement with previous results. The analysis of the effect of quantum interference on the electron density reveals that while polarization effects (quasi-classical) tend to displace electronic density from the most polarizable atom toward the less polarizable one, interference (quantum effects) counteracts by displacing electronic density to the bond region, giving rise to the right electronic density and dipole moment.

ORMAS-SCF Study of Silicon (100) Surface Clusters†

An occupation restricted multiple active space (ORMAS) study of clusters that represent the silicon(100) surface (up to nine surface dimers) is discussed. The accuracy of three different active orbital ORMAS partition schemes for Si(100) surface clusters are compared. In addition to ORMAS-SCF calculations, generalized valence bond-perfect pairing (GVB-PP) properties are generated for comparison purposes. The ability of ORMAS to generate a reliable multiconfigurational zeroth-order wave function is systematically tested and when possible is compared to the full complete active space self-consistent field (CASSCF) method. This provides good benchmarks for the accuracy of ORMAS compared to CASSCF. It is demonstrated that ORMAS consistently provides a high degree of accuracy with a significantly reduced computational effort relative to a CASSCF calculation. For the largest cluster, for which a full CASSCF calculation is not possible, ORMAS predicts that the Si(100) surface dimers are symmetric.

Ab initio study of the reactions: Ethylene → H + vinyl, propylene → methyl + vinyl, and propylene → H + propen-2-yl

The first C–H bond dissociation energy ( D 0 ) of ethylene is calculated using the Generalized-Valence-Bond-Perfect Pairing (GVB-PP) and coupled cluster (CCSD and CCSD(T)) methods. The CCSD(T) results indicate that it is close to 460 kJ mol −1. The GVB-PP method is used to investigate the relative energies and structural behavior along the reaction coordinates for propylene → methyl + vinyl and for propylene → H + propen-2-yl. The D 0 obtained using the GVB-PP, CCSD and CCSD(T) methods are compared to earlier theoretical literature values.

Penalty functions for combining coupled-cluster and perturbation amplitudes in local correlation methods with optimized orbitals

Local active space correlation models based on the coupled-cluster doubles (CCD) model like Generalized Valence Bond Perfect Pairing (GVB-PP) and Imperfect Pairing (IP) are attractive methods for treating electron correlation, because they are computationally inexpensive and can describe strong correlations. However, they suffer from symmetry-breaking (SB) in systems with multiple resonance structures, which arises due to neglected correlations. We investigate the extent to which these problems can be removed by using second-order perturbation theory (PT) for weak correlations coupling three different electron pairs, and (infinite-order) coupled-cluster (CC) theory for stronger correlations involving electrons in only one or two pairs. The resulting Three-Pair Corrected Imperfect Pairing (TIP) method is explored here, and it is shown that to robustly combine CC and PT it is necessary to modify several aspects of the basic method. Most importantly, a penalty function term is introduced to ensure the PT amplitudes remain small. Comparison against CC treatment of the three-pair correlations suggests penalty terms will be beneficial for any hybrid CC/PT method that includes orbital optimization. The TIP method greatly reduces SB in aromatic hydrocarbons and recovers a significantly higher fraction of the valence electron correlation energy than IP.

An approximate diatomics in molecules formulation of generalized valence bond theory

The slow computational speed of the generalized valence bond perfect pairing method (GVB-PP) has been an impediment to its routine use. We have addressed this problem by employing a diatomics in molecules Hamiltonian derived from a second quantization perturbation approach. This results in all three- and four-centered two-electron integrals being dropped from the traditional GVB-PP calculation. For moderate sized molecules, as for example C20 computed with a double zeta + polarization basis, there is on average a fifty-fold decrease in computational times. In this article, we present the theory behind our approach and analyze the accuracy and speed of this approximate GVB-PP method for several cases where density functional methods have produced ambivalent results.

The localizability of valence space electron–electron correlations in pair-based coupled cluster models

Inspired by the success of very restrictive local active space approaches like Generalized Valence Bond Perfect Pairing (GVB-PP) and Imperfect Pairing (IP), we investigate the localizability of valence correlation in the context of electron pair models. In particular, the role of two-center local excitations in local active space coupled cluster doubles models is examined in the context of a variety of chemical applications. The full two spatial center pairing model is found to recover the majority of the correlation effects missed by GVB-PP and IP, recovering up to 95% of the untruncated valence correlation energy, and could provide an inexpensive reference wave function for the treatment of additional electron–electron correlations. All of these models improve noticeably upon the Hartree–Fock wave function and can be computed for very large systems due to their only cubic computational cost. Their main weakness is a tendency to exhibit symmetry breaking in systems with multiple resonance structures.

Properties of the phosphorus oxide radical, PO, its cation and anion in their ground electronic states: comparison of theoretical and experimental data

Experimental and theoretical data for the phosphorus oxide radical (PO), its cation (PO+) and anion (PO−) in their electronic ground states are reviewed. The internuclear distances, fundamental vibrational frequencies, bond orders, partial atomic charges, free valences, dipole moments, dissociation energies, ionization potential and electron affinity are discussed. The literature data are augmented by the results of a theoretical study including computations using restricted Hartree–Fock closed- and open-shell, generalized valence bond–perfect pairing, Møller–Plesset perturbation theory, complete active space self-consistent field, coupled-cluster with single and double substitutions up to the level augmented by a perturbative estimate of triple excitations and density functional theory methods.

Quadratic Coupled-Cluster Doubles: Implementation and Assessment of Perfect Pairing Optimized Geometries

We present orbital amplitude and orbital gradient equations for the quadratic coupled-cluster doubles (QCCD) theory. These expressions are size extensive and can be solved in O(N 6 ) time with the use of O(N 4 ) intermediates, which are efficiently defined. The optimized orbital formalism is naturally suited for excitations in only a valence space to describe nondynamical correlation. This also minimizes the additional cost relative to conventional CCD. The resulting valence QCCD is used to study the effects of correlation on equilibrium geometries of a range of small molecules in the perfect pairing active space. This enables comparison with generalized valence bond perfect pairing calculations on the same systems. Additionally, the use of valence QCCD as a supplement to complete active space (CAS) methods is illustrated with calculations of the transition structure for addition of hydrogen to trans-diazene (N 2 H 2 ).

Application and development of multiconfigurational localized perturbation theory

Generalization of localized perturbation theory, which results with a method able to span the spin space correctly, is presented. This generalization is achieved by using a multiconfigurational (MC) wave function as the reference. This is the most comprehensive expansion used within MC–LMP2 approach to date, with, however, low computational cost [computational scaling with system size (N) of the new method is O(N3)]. Recently, we have reported the successful Jaguar2 (J2) model for calculating atomization energies. Within the MC–LMP2 framework, the J2 model for calculating heats of formation is based on the generalized valence bond–perfect pairing (GVB–PP) wave function. The J2 model was applied only to closed shell cases because of the perfect pairing (PP) restriction in the reference function. In order to describe other systems, the PP restriction needs to be lifted. This work describes efforts in that direction. The PP restriction can be lifted by a restricted configuration interaction (RCI) procedure applied to the GVB–PP wave function. In this paper, the equations describing the application of LMP2 theory to self-consistent RCI wave function are derived and explained. The RCI wave function is a “true” MC expansion as opposed to the GVB–PP, which uses only a single spin eigenfunction (SEF). We also present the self-consistent (SC) optimization of the RCI wave function. The SC–RCI–LMP2 is the first MC–LMP2 method where the spin space is spanned in the reference. This is important for describing the nondynamical correlation (near degeneracy) effects associated, for example, with bond breaking processes. The SC–RCI–LMP2 is an efficient method applicable to large systems; it is shown to reproduce the potential energy surfaces calculated by the complete active space–second order perturbation (CAS–SCF–PT2) method. This is demonstrated, for the first time, on some widely used test cases.

Is GVB-CI superior to CASSCF?

Presently the most reliable approach for the study of reaction pathways where chemical bonds are broken and formed is to carry out CASSCF calculations followed by corresponding multireference perturbation or CI treatments. The latter step generally relaxes the “antibonding character” of the CASSCF results. In this study we demonstrate that similar results can be well approximated by using a less optimized MCSCF method and not performing the multireference perturbation or CI step at all. This is accomplished by performing a complete CI calculation within the active orbital space of the generalized valence bond perfect pairing (GVB-PP) model. The local bond/antibond character of the orbital space of the GVB-PP method also allows development of a fast, but robust, Bethe–Goldstone algorithm, which reconstructs the CI energy to an accuracy of a few tenths of a millihartree for most types of bond breaking cases found in chemical reactions. This algorithm executes at a speed proportional to N where Np is the number of localized electron pairs in the active space. ©1999 John Wiley & Sons, Inc. J Comput Chem 20: 999–1008, 1999

Ab initio study of C—H bond breaking in olefins. II. GVB computations on propene ↔ H + cis- or trans-propen-1-yl

The Generalized-Valence-Bond-Perfect-Pairing (GVB-PP) method has been used to investigate the structural behaviour, energy, and dipole moment along the reaction coordinates for propene ↔ H + cis- or trans-propen-1-yl. Geometry optimizations were carried out at the GVB(9)/STO-3G level (complete valence shell) for the minimum energy propene structure (complete optimization) and for numerous structures up to r(C—H) = 10 Å (only the elongated C—H distance kept fixed). The dissociation curves are smooth, without a maximum, and yield predicted dissociation energies of propene to H + cis-propen-1-yl and H + trans-propen-1-yl of 555.8 and 554.8 kJ mol−1, respectively. These values are within several percent of those obtained for C—H bond rupture in ethylene using GVB and MCSCF methods with the same basis set. They are obviously too high but they confirm that removal of a hydrogen atom from the CH2 moiety in propene requires about the same energy as removal of a hydrogen atom from ethylene. GVB(7)/6-31G//GVB(9)/STO-3G computations lower the predicted dissociation energies of propene ↔ H + cis-propen-1-yl and H + trans-propen-1-yl to 448.2 and 448.6 kJ mol−1, respectively.The reduced energy concept (ER = (E∞ − Er)/De) is applied to the reaction coordinates. Linear behaviour for In ER versus bond length is observed at long bond distances. At r(C—H) = 3 Å, the values of the slopes, d(ln ER)/dr(C—H), which are related to the effective Morse constant B are −3.73 and −3.74 (GVB(9)/STO-3G) and −2.75 and −2.81 (GVB(7)/6-31 G//GVB(9)/STO-3G) for the H + cis- and H + tras-propen-1-yl reaction coordinates, respectively.