Similarity transformation of the electronic Schrödinger equation via Jastrow factorization

Abstract

By expressing the electronic wavefunction in an explicitly-correlated (Jastrow-factorised) form, a similarity-transformed effective Hamiltonian can be derived. The effective Hamiltonian is non-Hermitian and contains three-body interactions. The resulting ground-state eigenvalue problem can be solved projectively using a stochastic configuration-interaction formalism. Our approach permits use of highly flexible Jastrow functions, which we show to be effective in achieving extremely high accuracy, even with small basis sets. Results are presented for the total energies and ionisation potentials of the first-row atoms, achieving accuracy within a mH of the basis-set limit, using modest basis sets and computational effort. PACS numbers: Valid PACS appear hereMethods aiming to obtain high-accuracy solutions to the electronic Schrödinger equation must tackle two essential components of the problem, namely providing highly flexible expansions capable of resolving nonanalytic features of the wavefunction, including the Kato cusps[1] at electron coalesence points, as well as treatment of many-electron correlation at medium and long range. The combination of these two facets of the problem leads to overwhelming computational complexity, requiring large basis sets and high-order correlation methods, approximations to which can result in a significant loss of accuracy. The goal of achieving "chemical" accuracy remains extremely challenging for all but the simplest systems.In Fock space approaches, including the majority of quantum chemical methodologies based on configurational expansions, the first-quantised Schrödinger Hamiltonian is replaced by a second-quantised form, expressed in a one-electron basis. The passage from first quantisation to second is invoked primarily to impose antisymmetry on the solutions, via fermionic creation and annihilation operators of the orbital basis. However, this formulation loses the ability to explicitly include electron pair variables (such as electron-electron distances) into the wavefunction, which has long been known [2] to be crucial in obtaining an efficient description of electron correlation. Correlation effects are then indirectly obtained via superpositions of Slater determinants over the Fock space, as in configuration interaction, coupled-cluster and tensor-decomposition methods. These are computationally costly methods, especially with large basis sets. In quantum chemistry, explicitly-correlated methods usually proceed via the R12 formalism of Kutzelnigg [3], and its more modern F12 variants [4], in combination with perturbation theory [5] or coupled-cluster theory [6]. These methods augment the Fock-space (configurational) wavefunctions with strongly-orthogonal geminal terms with fixed amplitudes, imposing a first-order cusp condition. This approximation is suitable for systems whose ground state wavefunction is dominated by a single determinant. The inclusion of explicit correlation in strongly correlated, multi-determinantal, wavefunctions remains an open challenge.In this...

Jastrow-type Decomposition in Quantum Chemistry for Low-Depth Quantum Circuits

“…[56][57][58] Another approach is to cast the dynamical correlation into the active space Hamiltonian by similarity transformations in the first or second quantization, e.g. Ref [59,60] and Ref [61][62][63][64], respectively, and many others though it is not possible to list them all here. The k-uCJ ansatz is particularly suitable for this approach because the similarity transformed Hamiltonians usually possess more than two-body interactions and it is straightforward to extend the k-uCJ ansatz to more than two-body Jastrow factor forms.…”

Incremental treatments of the full configuration interaction problem

“…In addition, the use of alternative and approximative FCI solvers as a means to allow for larger (and faster) CASCI calculations remains to be explored within MBE‐FCI. Finally, in the spirit of recent work seeking to revitalize the idea of transcorrelation, 49–54 a natural, albeit non‐trivial extension of the current generation of MBE‐FCI will be to allow for expansions to spawn from a similarity‐transformed Hamiltonian, akin to what is found in equation‐of‐motion coupled cluster theory 55 . Its appealing traits as a correlated zeroth‐order formulation of electron correlation aside, the non‐Hermiticity of the theory will pose entirely new conceptual as well as technical challenges.…”

Transcorrelated coupled cluster methods

“…68 The integrals are calculated as outlined in Ref. 64 and imported using an FCIDUMP-type interface.…”

“…The results are compared to experimental numbers back-corrected for relativistic effects. 64,70 First, we compare the absolute atomic energies to TC-FCIQMC and experimental numbers, Table I. Evidently, transcorrelation drastically improves the basis set convergence.…”

“…This idea goes back to the pioneering work of Boys and Handy and is termed transcorrelation, 58 and it has been recently demonstrated that a combination of the transcorrelation with a stochastic full configuration interaction quantum Monte-Carlo (FCIQMC) [59][60][61] yields very promising results for weakly and strongly correlated systems. [62][63][64][65] Since the Hamiltonian itself is transformed, the transcorrelation approach is not specific for low-order methods and can be used together with any excitation orders. The transcorrelated Hamiltonian has been applied before to linearized coupled cluster with singles and doubles, 66 and for the uniform electron gas for coupled cluster and distinguishable cluster with singles and doubles, 67 however with some approximations to the three-body terms in the transformed Hamiltonian and with a simpler correlation factor.…”