Connections between coupled cluster and generalized valence bond theories

Perspective on Coupled-cluster Theory. The evolution toward simplicity in quantum chemistry

Coupled-cluster theory has revolutionized quantum chemistry. It has provided the framework to effectively solve the problem of electron correlation, the main focus of the field for over 60 years. This has enabled ab initio quantum chemistry to provide predictive quality results for most quantities of interest that are obtainable from first-principle calculations. The best that one can do in a basis is the 'full CI,' the exact solution of the non-relativistic Schrödinger equation or, if need be, the relativistic Dirac equation. With due regard to converging the basis set and adequate consideration of higher clusters and relativity in a calculation, virtually predictive results can be obtained. But in addition to its numerical performance, coupled-cluster theory also offers a conceptually new, many-body foundation for the theory that should be appreciated by all practitioners. The latter is emphasized in this perspective, leading to the 'evolution toward simplicity' in the title. The ultimate theory will benefit from the several features that are uniquely exact in coupled-cluster theory and its equation-of-motion (EOM-CC) extensions.

Excited states with pair coupled cluster doubles tailored coupled cluster theory

It is known that some non-dynamic effects of electron correlation can be included in coupled cluster theory using a tailoring technique that separates the effects of non-dynamic and dynamic correlations. Recently, the simple pCCD (pair coupled cluster doubles) wavefunction was shown to provide good results for some non-dynamic correlation problems, such as bond-breaking, in a spin-adapted way with no active space selection. In this paper, we report a study of excited states using "tailored coupled cluster singles and doubles," to attempt to use pCCD as a kernel for more complete coupled-cluster singles and doubles (CCSD) results for excited states. Several excited states are explored from those primarily due to single excitations to those dominated by doubly excited states and from singlet-triplet splittings for some diradical states. For the first two situations, tailored pCCD-TCCSD offers no improvement over equation of motion-CCSD. However, when we explore the singlet-triplet gap of diradical molecules that are manifestly multi-reference, a pCCD kernel provides improved results, particularly with generalized valence bond orbitals.

Geminal-based electronic structure methods in quantum chemistry. Toward a geminal model chemistry

In this review, we discuss the recent progress in developing geminal-based theories for challenging problems in quantum chemistry. Specifically, we focus on the antisymmetrized geminal power, generalized valence bond, antisymmetrized product of strongly orthogonal geminals, singlet-type orthogonal geminals, the antisymmetric product of 1-reference orbital geminal, also known as the pair coupled cluster doubles ansatz, and geminals constructed from Richardson-Gaudin states. Furthermore, we review various corrections to account for the missing dynamical correlation effects in geminal models and possible extensions to target electronically excited states and open-shell species. Finally, we discuss some numerical examples and present-day challenges for geminal-based models, including a quantitative and qualitative analysis of wave functions, and software availability.

Evaluation of full valence correlation energies and gradients

Complete-active-space self-consistent field (CASSCF) wave functions are central to understanding strongly correlated molecules as they capture the entirety of electronic interactions within a subset of the orbital space. The most interesting case for CASSCF is the full valence limit, where all bonding and an equal number of virtual orbitals are included in the active space, and no approximation is made in selecting the important valence orbitals or electrons. While conventional algorithms require exponential computational time to evaluate full valence CASSCF, this article shows that the method of increments can do the same with polynomial effort, in a new method denoted iCASSCF. The method of increments can also provide density matrices and other necessary ingredients for the construction of the nuclear gradient. These goals are met through a many-body expansion that breaks the problem into smaller pieces that are subsequently reassembled to form close approximations of conventional CAS results. Practical demonstrations on a number of medium-sized molecules, with up to 116 valence electrons correlated in 116 orbitals, show the power of this methodology.

Geminal perturbation theory based on the unrestricted Hartree-Fock wavefunction

A perturbative correction exploiting natural orbitals and the pair function structure of the unrestricted Hartree-Fock (UHF) wavefunction is devised. The method offers a simple framework for describing multireference systems where static correlation is captured by UHF. The UHF wavefunction is built of two-electron fragments (geminals), involving both singlet and triplet (m_s = 0) parts. At order zero of the perturbative treatment, configuration interaction coefficients of UHF geminals are relaxed. The zero order Hamiltonian is of the Dyall-type, including explicit two-electron interaction within geminals and leading to a formal 6th power scaling. Adopting an effective one-electron zero order Hamiltonian term for the subset of virtual orbitals reduces scaling of the correction step to 4th power. Formal properties of the proposed schemes are discussed. Energetic data and natural occupation numbers of illustrative test systems are used to assess the new approach. The cases where the wavefunction becomes essentially spin pure at the level of reference show good performance. Spin contamination remaining at order zero is found to undermine the perturbative correction.

Correction to constrained coupled cluster doubles models based on the second coupled cluster central moment

We develop a correction for the coupled cluster version of the perfect pairing (PP) model. The correction is based on finding modified values of the PP amplitudes such that the second coupled cluster central moment defined in the space of all valence single and double substitutions vanishes and, subject to this constraint, minimizing the deviation between the modified and unmodified PP amplitudes with respect to a chosen metric. We discuss how this correction can be generalized to other constrained doubles models, such as local correlation and active-space models. While the correction is not strictly size consistent and retains some of the deficiencies of the PP model, numerical results indicate that much of the missing active-space coupled cluster singles and doubles correlation energy is recovered.

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.

Multireference correlation in long molecules with the quadratic scaling density matrix renormalization group

We have devised a local ab initio density matrix renormalization group algorithm to describe multireference correlations in large systems. For long molecules that are extended in one of their spatial dimensions, we can obtain an exact characterization of correlation, in the given basis, with a cost that scales only quadratically with the size of the system. The reduced scaling is achieved solely through integral screening and without the construction of correlation domains. We demonstrate the scaling, convergence, and robustness of the algorithm in polyenes and hydrogen chains. We converge to exact correlation energies (in the sense of full configuration interaction, with 1-10 microE(h) precision) in all cases and correlate up to 100 electrons in 100 active orbitals. We further use our algorithm to obtain exact energies for the metal-insulator transition in hydrogen chains and compare and contrast our results with those from conventional quantum chemical methods.

Second-order correction to perfect pairing: An inexpensive electronic structure method for the treatment of strong electron-electron correlations

We have formulated a second-order perturbative correction for perfect-pairing wave functions [PP2] based on similarity-transformed perturbation techniques in coupled cluster theory. The perfect-pairing approximation is used to obtain a simple reference wave function which can qualitatively describe bond breaking, diradicals, and other highly correlated systems, and the perturbative correction accounts for the dynamical correlation. An efficient implementation of this correction using the resolution of the identity approximation enables PP2 to be computed at a cost only a few times larger than that of canonical MP2 for systems with hundreds of active electrons and tens of heavy atoms. PP2 significantly improves on MP2 predictions in various systems with a challenging electronic structure.

Advances in methods and algorithms in a modern quantum chemistry program package

Advances in theory and algorithms for electronic structure calculations must be incorporated into program packages to enable them to become routinely used by the broader chemical community. This work reviews advances made over the past five years or so that constitute the major improvements contained in a new release of the Q-Chem quantum chemistry package, together with illustrative timings and applications. Specific developments discussed include fast methods for density functional theory calculations, linear scaling evaluation of energies, NMR chemical shifts and electric properties, fast auxiliary basis function methods for correlated energies and gradients, equation-of-motion coupled cluster methods for ground and excited states, geminal wavefunctions, embedding methods and techniques for exploring potential energy surfaces.

Unrestricted Perfect Pairing: The Simplest Wave-Function-Based Model Chemistry beyond Mean Field

The perfect pairing (PP) approximation from generalized valence bond theory is formulated in an unrestricted fashion for both closed- and open-shell systems using a coupled cluster ansatz. In the model chemistry proposed here, active electron pairs are correlated, but the unpaired or radical electrons remain uncorrelated, leading to a linear number of decoupled cluster amplitudes which can be solved for analytically. The alpha and beta spatial orbitals are variationally optimized independently. This minimal treatment of electron-electron correlation noticeably improves upon symmetry-breaking problems and other pathologies in Hartree-Fock (HF) theory and may be computed using the resolution of the identity approximation at only a factor of several times more effort than HF itself. PP also generally predicts improved molecular structures over HF. This compact, correlated wave function potentially provides a useful starting point for dynamical correlation corrections.

Using molecular similarity to construct accurate semiempirical electronic structure theories

Ab initio electronic structure methods give accurate results for small systems, but do not scale well to large systems. Chemical insight tells us that molecular functional groups will behave approximately the same way in all molecules, large or small. This molecular similarity is exploited in semiempirical methods, which couple simple electronic structure theories with parameters for the transferable characteristics of functional groups. We propose that high-level calculations on small molecules provide a rich source of parametrization data. In principle, we can select a functional group, generate a large amount of ab initio data on the group in various small-molecule environments, and "mine" this data to build a sophisticated model for the group's behavior in large environments. This work details such a model for electron correlation: a semiempirical, subsystem-based correlation functional that predicts a subsystem's two-electron density matrix as a functional of its one-electron density matrix. This model is demonstrated on two small systems: chains of linear, minimal-basis (H-H)(5), treated as a sum of four overlapping (H-H)(2) subsystems; and the aldehyde group of a set of HOC-R molecules. The results provide an initial demonstration of the feasibility of the approach.

Geminal model chemistry II. Perturbative corrections

We introduce and investigate a chemical model based on perturbative corrections to the product of singlet-type strongly orthogonal geminals wave function. Two specific points are addressed (i) Overall chemical accuracy of such a model with perturbative corrections at a leading order; (ii) Quality of strong orthogonality approximation of geminals in diverse chemical systems. We use the Epstein-Nesbet form of perturbation theory and show that its known shortcomings disappear when it is used with the reference Hamiltonian based on strongly orthogonal geminals. Application of this model to various chemical systems reveals that strongly orthogonal geminals are well suited for chemical models, with dispersion interactions between the geminals being the dominant effect missing in the reference wave functions.

Design and application of a multicoefficient correlation method for dispersion interactions

A new multicoefficient correlation method (MCCM) is presented for the determination of accurate van der Waals interactions. The method utilizes a novel parametrization strategy that simultaneously fits to very high-level binding, Hartree-Fock and correlation energies of homo- and heteronuclear rare gas dimers of He, Ne, and Ar. The decomposition of the energy into Hartree-Fock and correlation components leads to a more transferable model. The method is applied to the krypton dimer system, rare gas-water interactions, and three-body interactions of rare gas trimers He3, Ne3, and Ar3. For the latter, a very high-level method that corrects the rare-gas two-body interactions to the total binding energy is introduced. A comparison with high-level CCSD(T) calculations using large basis sets demonstrates the MCCM method is transferable to a variety of systems not considered in the parametrization. The method allows dispersion interactions of larger systems to be studied reliably at a fraction of the computational cost, and offers a new tool for applications to rare-gas clusters, and the development of dispersion parameters for molecular simulation force fields and new semiempirical quantum models.