Self‐Consistent Procedures for Generalized Valence Bond Wavefunctions. Applications H3, BH, H2O, C2H6, and O2

Single Reference Treatment of Strongly Correlated H4 and H10 Isomers with Richardson-Gaudin States

Richardson-Gaudin (RG) states are employed as a variational wave function ansatz for strongly correlated isomers of H4 and H10. In each case, a single RG state describes the seniority-zero sector quite well. Simple natural orbital functionals offer a cheap and reasonable approximation of the outstanding weak correlation in the seniority-zero sector, while systematic improvement is achieved by performing a configuration interaction in terms of RG states.

Reduced density matrices/static correlation functions of Richardson–Gaudin states without rapidities

Seniority-zero geminal wavefunctions are known to capture bond-breaking correlation. Among this class of wavefunctions, Richardson–Gaudin states stand out as they are eigenvectors of a model Hamiltonian. This provides a clear physical picture, clean expressions for reduced density matrix (RDM) elements, and systematic improvement (with a complete set of eigenvectors). Known expressions for the RDM elements require the computation of rapidities, which are obtained by first solving for the so-called eigenvalue based variables (EBV) and then root-finding a Lagrange interpolation polynomial. In this paper, we obtain expressions for the RDM elements directly in terms of the EBV. The final expressions can be computed at the same cost as the rapidity expressions. Therefore, except, in particular, circumstances, it is entirely unnecessary to compute rapidities at all. The RDM elements require numerically inverting a matrix, and while this is usually undesirable, we demonstrate that it is stable, except when there is degeneracy in the single-particle energies. In such cases, a different construction would be required.

Density matrices of seniority-zero geminal wavefunctions

Scalar products and density matrix elements of closed-shell pair geminal wavefunctions are evaluated directly in terms of the pair amplitudes, resulting in an analog of Wick’s theorem for fermions or bosons. This expression is, in general, intractable, but it is shown how it becomes feasible in three distinct ways for Richardson–Gaudin (RG) states, the antisymmetrized geminal power, and the antisymmetrized product of strongly orthogonal geminals. Dissociation curves for hydrogen chains are computed with off-shell RG states and the antisymmetrized product of interacting geminals. Both are near exact, suggesting that the incorrect results observed with ground state RG states (a local maximum rather than smooth dissociation) may be fixable using a different RG state.

N-Body Reduced Density Matrix-Based Valence Bond Theory and Its Applications in Diabatic Electronic-Structure Computations

Valence bond (VB) theory, as a helpful complement to the more popular molecular orbital theory, is a fundamental electronic-structure theory that aims at interpreting molecular structure and chemical reactions in a lucid way. Both theoretical and experimental chemists have shown great interest in VB theory because of its capability of providing intuitive insight into the nature of chemical bonding and the mechanism of chemical reaction in a clear and comprehensible language rooted in Lewis structure. Therefore, there is a great call for the renaissance of VB theory. Nevertheless, this is possible only after a series of methods and algorithms were developed and efficiently implemented in user-friendly programs so as to serve computational chemists for general applications. In the past three decades, we have devoted a great amount of scientific enthusiasm toward this goal. In this Account, we will concisely summarize and briefly but insightfully discuss recent developments in ab initio VB theory, especially the N-body reduced density matrices (RDM)-based approach and its applications in diabatic electronic-structure computations, which is very useful for the vivid interpretation of many fundamental chemical processes such as electron and energy transfers. Furthermore, because of the fundamentally important role that the diabatic state plays in electron and energy transfers, which are two frontier research topics in both molecular and biochemical sciences, there are a broad range of applications that VB theory can handle.We start by briefly reviewing the general feature of ab initio VB wave functions. In particular, we focus on the multistructural ab initio VB theory that uses strictly localized orbitals, including the fundamental VB self-consistent field (VBSCF) and two post-SCF methods, VBCI and VBPT2, that use the VBSCF wave function as reference. We then allot a section to describing the recent developments of the RDM-based VB approach in the second quantization language. In this section, the enhanced Wick theorem is first outlined, followed by a brief discussion of its applications in evaluating VBSCF energy gradients and a Hessian with respect to the orbital expansion coefficients, together with a short review of the implementation of an automatic formula and code generator (AFCG) designed for many-body methods with nonorthogonal orbitals. Then, we introduce the application of the RDM-based approach in implementing the post-SCF method that addresses dynamic electronic correlation via perturbation theory, viz., the icVBPT2 method that adopts an internal contraction technique naturally. We finish this section by incorporating VB theory with the concept of seniority number, in which the tensor analysis technique is carefully exploited with the RDM-based approach, resulting in significant improvements in both the number of the active electrons/orbitals and in the speedup of the computational efficiency, thus pushing VB theory to its new limit. With these achievements available, we present the applications of VB theory in diabatic electronic-structure computations by using the intuitive insight rendered by VB theory. Therefore, we believe that there is a bright future in VB theory with true opportunities and new challenges coexisting both for theoretical developments and computational applications.

Multiconfiguration Pair-Density Functional Theory

Kohn-Sham density functional theory with the available exchange-correlation functionals is less accurate for strongly correlated systems, which require a multiconfigurational description as a zero-order function, than for weakly correlated systems, and available functionals of the spin densities do not accurately predict energies for many strongly correlated systems when one uses multiconfigurational wave functions with spin symmetry. Furthermore, adding a correlation functional to a multiconfigurational reference energy can lead to double counting of electron correlation. Multiconfiguration pair-density functional theory (MC-PDFT) overcomes both obstacles, the second by calculating the quantum mechanical part of the electronic energy entirely by a functional, and the first by using a functional of the total density and the on-top pair density rather than the spin densities. This allows one to calculate the energy of strongly correlated systems efficiently with a pair-density functional and a suitable multiconfigurational reference function. This article reviews MC-PDFT and related background information.

The electronic structure of carbones revealed: insights from valence bond theory

In this contribution, we studied the OC-C bond in carbon suboxide and related allene compounds using the valence bond method. The nature of this bond has been the subject of debate, whether it is a regular, electron sharing bond or a dative bond. We compared the nature of this bond in carbon suboxide with the gold-CO bond in Au(CO)₂⁺, which is a typical dative bond, and we studied its charge-shift bond character. We found that the C-CO bond in carbon suboxide is unique in the sense that it cannot be assigned as either a dative or electron sharing bond, but it is an admixture of electron sharing and dative components, together with a high contribution of ionic character. These findings provide a clear basis for distinguishing the commonly found dative bonds between ligands and transition metals and the present case of what may be described as coordinative bonding to carbon.

Multireference Electron Correlation Methods: Journeys along Potential Energy Surfaces

Multireference electron correlation methods describe static and dynamical electron correlation in a balanced way and, therefore, can yield accurate and predictive results even when single-reference methods or multiconfigurational self-consistent field theory fails. One of their most prominent applications in quantum chemistry is the exploration of potential energy surfaces. This includes the optimization of molecular geometries, such as equilibrium geometries and conical intersections and on-the-fly photodynamics simulations, both of which depend heavily on the ability of the method to properly explore the potential energy surface. Because such applications require nuclear gradients and derivative couplings, the availability of analytical nuclear gradients greatly enhances the scope of quantum chemical methods. This review focuses on the developments and advances made in the past two decades. A detailed account of the analytical nuclear gradient and derivative coupling theories is presented. Emphasis is given to the software infrastructure that allows one to make use of these methods. Notable applications of multireference electron correlation methods to chemistry, including geometry optimizations and on-the-fly dynamics, are summarized at the end followed by a discussion of future prospects.

Numerical and Theoretical Aspects of the DMRG-TCC Method Exemplified by the Nitrogen Dimer

In this article, we investigate the numerical and theoretical aspects of the coupled-cluster method tailored by matrix-product states. We investigate formal properties of the used method, such as energy size consistency and the equivalence of linked and unlinked formulation. The existing mathematical analysis is here elaborated in a quantum chemical framework. In particular, we highlight the use of what we have defined as a complete active space-external space gap describing the basis splitting between the complete active space and the external part generalizing the concept of a HOMO–LUMO gap. Furthermore, the behavior of the energy error for an optimal basis splitting, i.e., an active space choice minimizing the density matrix renormalization group-tailored coupled-cluster singles doubles error, is discussed. We show numerical investigations on the robustness with respect to the bond dimensions of the single orbital entropy and the mutual information, which are quantities that are used to choose a complete active space. Moreover, the dependence of the ground-state energy error on the complete active space has been analyzed numerically in order to find an optimal split between the complete active space and external space by minimizing the density matrix renormalization group-tailored coupled-cluster error.

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.

Pair 2-electron reduced density matrix theory using localized orbitals

Full configuration interaction (FCI) restricted to a pairing space yields size-extensive correlation energies but its cost scales exponentially with molecular size. Restricting the variational two-electron reduced-density-matrix (2-RDM) method to represent the same pairing space yields an accurate lower bound to the pair FCI energy at a mean-field-like computational scaling of O(r³) where r is the number of orbitals. In this paper, we show that localized molecular orbitals can be employed to generate an efficient, approximately size-extensive pair 2-RDM method. The use of localized orbitals eliminates the substantial cost of optimizing iteratively the orbitals defining the pairing space without compromising accuracy. In contrast to the localized orbitals, the use of canonical Hartree-Fock molecular orbitals is shown to be both inaccurate and non-size-extensive. The pair 2-RDM has the flexibility to describe the spectra of one-electron RDM occupation numbers from all quantum states that are invariant to time-reversal symmetry. Applications are made to hydrogen chains and their dissociation, n-acene from naphthalene through octacene, and cadmium telluride 2-, 3-, and 4-unit polymers. For the hydrogen chains, the pair 2-RDM method recovers the majority of the energy obtained from similar calculations that iteratively optimize the orbitals. The localized-orbital pair 2-RDM method with its mean-field-like computational scaling and its ability to describe multi-reference correlation has important applications to a range of strongly correlated phenomena in chemistry and physics.

Computational Kinetics by Variational Transition-State Theory with Semiclassical Multidimensional Tunneling: Direct Dynamics Rate Constants for the Abstraction of H from CH3OH by Triplet Oxygen Atoms

Rate constants and the product branching ratio for hydrogen abstraction from CH3OH by O(3P) were computed with multistructural variational transition-state theory including microcanonically optimized multidimensional tunneling. Benchmark calculations of the forward and reverse classical barrier heights and the reaction energetics have been carried out by using coupled cluster theory and multireference calculations to select the most reliable density functional method for direct dynamics computations of the rate constants. The dynamics calculations included the anharmonicity of the zero-point energies and partition functions, with specific-reaction-parameter scaling factors for reactants and transition states, and multistructural torsional anharmonicity was included for the torsion around the C-O bond in methanol and in the transition states. The resulting rate constants are presented over a wider range than they are available from experiment, but in the temperature range where experiments are available, they agree well with experimental values, which is encouraging for their reliability over the wider temperature range and for future computations of oxygen atom reaction rates. In contrast to a previous computational prediction, the branching ratio predicted by the present work shows that the formation of CH2OH + OH is the dominant channel over the whole range of temperature from 250 to 2000 K.

Multiconfiguration Pair-Density Functional Theory: A New Way To Treat Strongly Correlated Systems

The electronic energy of a system provides the Born-Oppenheimer potential energy for internuclear motion and thus determines molecular structure and spectra, bond energies, conformational energies, reaction barrier heights, and vibrational frequencies. The development of more efficient and more accurate ways to calculate the electronic energy of systems with inherently multiconfigurational electronic structure is essential for many applications, including transition metal and actinide chemistry, systems with partially broken bonds, many transition states, and most electronically excited states. Inherently multiconfigurational systems are called strongly correlated systems or multireference systems, where the latter name refers to the need for using more than one ("multiple") configuration state function to provide a good zero-order reference wave function. This Account describes multiconfiguration pair-density functional theory (MC-PDFT), which was developed as a way to combine the advantages of wave function theory (WFT) and density functional theory (DFT) to provide a better treatment of strongly correlated systems. First we review background material: the widely used Kohn-Sham DFT (which uses only a single Slater determinant as reference wave function), multiconfiguration WFT methods that treat inherently multiconfigurational systems based on an active space, and previous attempts to combine multiconfiguration WFT with DFT. Then we review the formulation of MC-PDFT. It is a generalization of Kohn-Sham DFT in that the electron kinetic energy and classical electrostatic energy are calculated from a reference wave function, while the rest of the energy is obtained from a density functional. However, there are two main differences with respent to Kohn-Sham DFT: (i) The reference wave function is multiconfigurational rather than being a single Slater determinant. (ii) The density functional is a function of the total density and the on-top pair density rather than being a function of the spin-up and spin-down densities. In work carried out so far, the multiconfigurational wave function is a multiconfiguration self-consistent-field wave function. The new formulation has the advantage that the reference wave function has the correct spatial and spin symmetry and can describe bond dissociation (of both single and multiple bonds) and electronic excitations in a formally and physically correct way. We then review the formulation of density functionals in terms of the on-top pair density. Finally we review successful applications of the theory to bond energies and bond dissociation potential energy curves of main-group and transition metal bonds, to barrier heights (including pericyclic reactions), to proton affinities, to the hydrogen bond energy of water dimer, to ground- and excited-state charge transfer, to valence and Rydberg excitations of molecules, and to singlet-triplet splittings of radicals. We find that that MC-PDFT can give accurate results not only with complete-active-space multiconfiguration wave functions but also with generalized-active-space multiconfiguration wave functions, which are practical for larger numbers of active electrons and active orbitals than are complete-active-space wave functions. The separated-pair approximation, which is a special case of generalized active space self-consistent-field theory, is especially promising. MC-PDFT, because it requires much less computer time and storage than pure WFT methods, has the potential to open larger and more complex strongly correlated systems to accurate simulation.

Long-range interactions from the many-pair expansion: A different avenue to dispersion in DFT

One of the several problems that plague majority of density functional theory calculations is their inability to properly account for long-range correlations giving rise to dispersion forces. The recently proposed many-pair expansion (MPE) [T. Zhu et al., Phys. Rev. B 93, 201108(R) (2016)] is a hierarchy of approximations that systematically corrects any deficiencies of an approximate functional to finally converge to the exact energy. This is achieved by decomposing the total density into a sum of two-electron densities and accounting for successive two-, four-, six-,… electron interactions. Here, we show that already low orders of MPE expansion recover the dispersion energy accurately. To this end, we employ the Pariser-Parr-Pople Hamiltonian and study the behavior of long-range interactions in trans-polyacetylene as well as stacks of ethylene and benzene molecules. We also show how convergence of the expansion is affected by electron conjugation and the choice of the density partitioning.

A minimalistic approach to static and dynamic electron correlations: Amending generalized valence bond method with extended random phase approximation correlation correction

A perfect-pairing generalized valence bond (GVB) approximation is known to be one of the simplest approximations, which allows one to capture the essence of static correlation in molecular systems. In spite of its attractive feature of being relatively computationally efficient, this approximation misses a large portion of dynamic correlation and does not offer sufficient accuracy to be generally useful for studying electronic structure of molecules. We propose to correct the GVB model and alleviate some of its deficiencies by amending it with the correlation energy correction derived from the recently formulated extended random phase approximation (ERPA). On the examples of systems of diverse electronic structures, we show that the resulting ERPA-GVB method greatly improves upon the GVB model. ERPA-GVB recovers most of the electron correlation and it yields energy barrier heights of excellent accuracy. Thanks to a balanced treatment of static and dynamic correlation, ERPA-GVB stays reliable when one moves from systems dominated by dynamic electron correlation to those for which the static correlation comes into play.

Separated-pair approximation and separated-pair pair-density functional theory

Multi-configuration pair-density functional theory (MC-PDFT) has proved to be a powerful way to combine the capabilities of multi-configuration self-consistent-field theory to represent the an electronic wave function with a highly efficient way to include dynamic correlation energy by density functional theory. All applications reported previously involved complete active space self-consistent-field (CASSCF) theory for the reference wave function. For treating large systems efficiently, it is necessary to ask whether good accuracy is retained when using less complete configuration interaction spaces. To answer this question, we present here calculations employing MC-PDFT with the separated pair (SP) approximation, which is a special case (defined in this article) of generalized active space self-consistent-field (GASSCF) theory in which no more than two orbitals are included in any GAS subspace and in which inter-subspace excitations are excluded. This special case of MC-PDFT will be called SP-PDFT. In SP-PDFT, the electronic kinetic energy and the classical Coulomb energy, the electronic density and its gradient, and the on-top pair density and its gradient are obtained from an SP approximation wave function; the electronic energy is then calculated from the first two of these quantities and an on-top density functional of the last four. The accuracy of the SP-PDFT method for predicting the structural properties and bond dissociation energies of twelve diatomic molecules and two triatomic molecules is compared to the SP approximation itself and to CASSCF, MC-PDFT based on CASSCF, CASSCF followed by second order perturbation theory (CASPT2), and Kohn-Sham density functional theory with the PBE exchange-correlation potential. We show that SP-PDFT reproduces the accuracy of MC-PDFT based on the corresponding CASSCF wave function for predicting C-H bond dissociation energies, the reaction barriers of pericyclic reactions and the properties of open-shell singlet systems, all at only a small fraction of the computational cost.

Reduced density matrix embedding. General formalism and inter-domain correlation functional

An embedding method for a one-electron reduced density matrix (1-RDM) is proposed.

Multireference Model Chemistries for Thermochemical Kinetics

By combining the generalized valence bond ansatz of correlated participating orbitals (CPO) with the complete-active-space prescription for selecting configurations and with the use of multireference second order perturbation theory (MRMP2) for including dynamical correlation, we define three levels of multireference (MR) theoretical model chemistries for electronic structure calculations of chemical reaction energies and barrier heights. The three levels differ in their choice of which orbitals are considered to be participating; the choices are called nominal (nom-CPO), moderate (mod-CPO), and extended (ext-CPO). Combining any of these three choices with a method for treatment of dynamical correlation energy and a one-electron basis set yields a theoretical model chemistry. Unlike the full-valence choice of active orbitals, the CPO choices lead to active spaces that contain the orbitals needed to include important static correlation effects on chemical reactions but do not increase with the size of the nonparticipating portion of the system, and hence they remain viable computational options even for many large and complex reacting systems. The accuracies of the new levels, combined with the MG3S basis set (a partially augmented, multiply polarized valence triple-ζ basis with appropriately tight d functions for 3p-block elements) and with the fully augmented correlation-consistent aug-cc-pVTZ basis set, are assessed against a previously presented database of barrier heights for diverse reaction types. We find that nom-CPO level captures the bulk of the static correlation energy, and MRMP2/nom-CPO calculations have an average error of only 1.4 kcal/mol in barrier heights, which may be compared to 5.0 kcal/mol for single-reference MP2 theory, 2.5 kcal/mol for CCSD, and 4.1 and 1.0 kcal/mol for the B3LYP and M06-2X density functionals, respectively. The accuracy of MRMP2/CPO for transition structure bond lengths and donor-acceptor distances is excellent, with a mean unsigned error of only 0.007 Å as compared to 0.018 Å for CCSD, 0.019 Å for M06-2X, and 0.039 Å for MP2 and B3LYP. We also introduce a new multireference diagnostic, called the M diagnostic, that allows one to measure the importance of static correlation in a given reagent or transition state.

Molcas 8: New capabilities for multiconfigurational quantum chemical calculations across the periodic table

In this report, we summarize and describe the recent unique updates and additions to the Molcas quantum chemistry program suite as contained in release version 8. These updates include natural and spin orbitals for studies of magnetic properties, local and linear scaling methods for the Douglas-Kroll-Hess transformation, the generalized active space concept in MCSCF methods, a combination of multiconfigurational wave functions with density functional theory in the MC-PDFT method, additional methods for computation of magnetic properties, methods for diabatization, analytical gradients of state average complete active space SCF in association with density fitting, methods for constrained fragment optimization, large-scale parallel multireference configuration interaction including analytic gradients via the interface to the Columbus package, and approximations of the CASPT2 method to be used for computations of large systems. In addition, the report includes the description of a computational machinery for nonlinear optical spectroscopy through an interface to the QM/MM package Cobramm. Further, a module to run molecular dynamics simulations is added, two surface hopping algorithms are included to enable nonadiabatic calculations, and the DQ method for diabatization is added. Finally, we report on the subject of improvements with respects to alternative file options and parallelization.

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.

Compact wavefunctions from compressed imaginary time evolution

Compact wavefunctions built through compressed imaginary time evolution enable more efficient modeling of quantum systems.

How much tetraradical character is present in the Si₆Ge₉ cluster?

This study discusses in detail the supposedly tetraradicaloid nature of a spirobis(pentagerma[1.1.1]propellane) derivative recently reported by Ito et al. (J. Am. Chem. Soc., 2013, 135, 6770). The electronic structure properties of the Si6Ge9 cluster are computationally explored by means of the composition of the ground state wavefunction, excitation energies to low-lying singlet, triplet and quintet states, and magnetic couplings between radical centers. Two main conclusions can be extracted from the obtained results regarding the radical character of spriobis(pentagerma[1.1.1]propellane): (i) the ground state of the Si6Ge9 cluster presents a rather small amount of effective unpaired electrons, which might be related to its chemical stability and (ii) there is in fact a perceptible tetraradical character within the small overall radical nature of the molecule. The proposed description do not contradict the conclusions drawn by the introductory work of Ito et al., but it provides a more detailed and precise interpretation of radical character of the molecule.