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arXiv:1308.5297 · The Journal of Chemical Physics

The Jastrow antisymmetric geminal power in Hilbert space: Theory, benchmarking, and application to a novel transition state

Eric Neuscamman

Published Nov 18, 2013• 73 citations
arXivHTMLPDF

Abstract

The Jastrow-modified antisymmetric geminal power (JAGP) ansatz in Hilbert space successfully overcomes two key failings of other pairing theories, namely, a lack of inter-pair correlations and a lack of multiple resonance structures, while maintaining a polynomially scaling cost, variational energies, and size consistency. Here, we present efficient quantum Monte Carlo algorithms that evaluate and optimize the JAGP energy for a cost that scales as the fifth power of the system size. We demonstrate the JAGP's ability to describe both static and dynamic correlation by applying it to bond stretching in H2O, C2, and N2 as well as to a novel, multi-reference transition state of ethene. JAGP's accuracy in these systems outperforms even the most sophisticated single-reference methods and approaches that of exponentially scaling active space methods.

DOI
10.1063/1.4829835

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The Jastrow-modified antisymmetric geminal power (JAGP) ansatz in Hilbert space successfully overcomes two key failings of other pairing theories, namely, a lack of inter-pair correlations and a lack of multiple resonance structures, while maintaining a polynomially scaling cost, variational energies, and size consistency. Here, we present efficient quantum Monte Carlo algorithms that evaluate and optimize the JAGP energy for a cost that scales as the fifth power of the system size. We demonstrate the JAGP's ability to describe both static and dynamic correlation by applying it to bond stretching in H2O, C2, and N2 as well as to a novel, multi-reference transition state of ethene. JAGP's accuracy in these systems outperforms even the most sophisticated single-reference methods and approaches that of exponentially scaling active space methods.
We present a novel specialization of the variational Monte Carlo linear method for the optimization of the recently introduced cluster Jastrow antisymmetric geminal power ansatz, achieving a lower-order polynomial cost scaling than would be possible with a naive application of the linear method and greatly improving optimization performance relative to that of the previously employed quasi-Newton approach. We test the methodology on highly multireference triple-bond stretches, achieving accuracy superior to those of the traditional coupled cluster theory and multireference perturbation theory in both the typical example of N2 and the transition-metal-oxide example of [ScO](+).
Single-reference methods such as Hartree-Fock-based coupled cluster theory are well known for their accuracy and efficiency for weakly correlated systems. For strongly correlated systems, more sophisticated methods are needed. Recent studies have revealed the potential of the antisymmetrized geminal power (AGP) as an excellent initial reference for the strong correlation problem. While these studies improved on AGP by linear correlators, we explore some non-linear exponential Ansätze in this paper. We investigate two approaches in particular. Similar to Wahlen-Strothman et al. [Phys. Rev. B 91, 041114(R) (2015)], we show that the similarity transformed Hamiltonian with a Hilbert-space Jastrow operator is summable to all orders and can be solved over AGP by projecting the Schrödinger equation. The second approach is based on approximating the unitary pair-hopper Ansatz recently proposed for application on a quantum computer. We report benchmark numerical calculations against the ground state of the pairing Hamiltonian for both of these approaches.
We present a variational function that targets excited states directly based on their position in the energy spectrum, along with a Monte Carlo method for its evaluation and minimization whose cost scales polynomially for a wide class of approximate wave functions. Being compatible with both real and Fock space and open and periodic boundary conditions, the method has the potential to impact many areas of chemistry, physics, and materials science. Initial tests on doubly excited states show that using this method, the Hilbert space Jastrow antisymmetric geminal power ansatz can deliver order-of-magnitude improvements in accuracy relative to equation of motion coupled cluster theory, while a very modest real space multi-Slater Jastrow expansion can achieve accuracies within 0.1 eV of the best theoretical benchmarks for the carbon dimer.
We show that for both single-Slater-Jastrow and Jastrow geminal power wave functions the formal cost scaling of Hilbert space variational Monte Carlo can be reduced from fifth to fourth order in the system size, thus bringing it in line with the long-standing scaling of its real space counterpart. While traditional quantum chemistry methods can reduce costs related to the two-electron integral tensor through various tensor decomposition methods, we show that such approaches are ineffective in the presence of Hilbert space Jastrow factors. Instead, we develop a simple semi-stochastic approach that can take similar advantage of the near-sparsity of this four-index tensor. Through demonstrations on alkanes of increasing length, we show that accuracy and overall statistical uncertainty are not meaningfully affected and that a total cost crossover is reached as early as 50 electrons when using a minimal basis. Further study will be needed to assess where the crossover occurs in more compact molecular geometries and larger basis sets and to explore how in that context the crossover can be accelerated.
We study the ionization energy, electron affinity, and the π → π(∗) ((1)La) excitation energy of the anthracene molecule, by means of variational quantum Monte Carlo (QMC) methods based on a Jastrow correlated antisymmetrized geminal power (JAGP) wave function, developed on molecular orbitals (MOs). The MO-based JAGP ansatz allows one to rigorously treat electron transitions, such as the HOMO → LUMO one, which underlies the (1)La excited state. We present a QMC optimization scheme able to preserve the rank of the antisymmetrized geminal power matrix, thanks to a constrained minimization with projectors built upon symmetry selected MOs. We show that this approach leads to stable energy minimization and geometry relaxation of both ground and excited states, performed consistently within the correlated QMC framework. Geometry optimization of excited states is needed to make a reliable and direct comparison with experimental adiabatic excitation energies. This is particularly important in π-conjugated and polycyclic aromatic hydrocarbons, where there is a strong interplay between low-lying energy excitations and structural modifications, playing a functional role in many photochemical processes. Anthracene is an ideal benchmark to test these effects. Its geometry relaxation energies upon electron excitation are of up to 0.3 eV in the neutral (1)La excited state, while they are of the order of 0.1 eV in electron addition and removal processes. Significant modifications of the ground state bond length alternation are revealed in the QMC excited state geometry optimizations. Our QMC study yields benchmark results for both geometries and energies, with values below chemical accuracy if compared to experiments, once zero point energy effects are taken into account.
We introduce an efficient method to construct optimal and system adaptive basis sets for use in electronic structure and quantum Monte Carlo calculations. The method is based on an embedding scheme in which a reference atom is singled out from its environment, while the entire system (atom and environment) is described by a Slater determinant or its antisymmetrized geminal power (AGP) extension. The embedding procedure described here allows for the systematic and consistent contraction of the primitive basis set into geminal embedded orbitals (GEOs), with a dramatic reduction of the number of variational parameters necessary to represent the many-body wave function, for a chosen target accuracy. Within the variational Monte Carlo method, the Slater or AGP part is determined by a variational minimization of the energy of the whole system in presence of a flexible and accurate Jastrow factor, representing most of the dynamical electronic correlation. The resulting GEO basis set opens the way for a fully controlled optimization of many-body wave functions in electronic structure calculation of bulk materials, namely, containing a large number of electrons and atoms. We present applications on the water molecule, the volume collapse transition in cerium, and the high-pressure liquid hydrogen.