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arXiv:1603.06605 · Journal of Chemical Theory and Computation

Improved Optimization for the Cluster Jastrow Antisymmetric Geminal Power and Tests on Triple-Bond Dissociations

Eric Neuscamman

Published Jun 9, 2016• 34 citations
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Abstract

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](+).

DOI
10.1021/acs.jctc.6b00288

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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.
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 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 present a modification to variational Monte Carlo's linear method optimization scheme that addresses a critical memory bottleneck while maintaining compatibility with both the traditional ground state variational principle and our recently introduced variational principle for excited states. For wave function ansatzes with tens of thousands of variables, our modification reduces the required memory per parallel process from tens of gigabytes to hundreds of megabytes, making the methodology a much better fit for modern supercomputer architectures in which data communication and per-process memory consumption are primary concerns. We verify the efficacy of the new optimization scheme in small molecule tests involving both the Hilbert space Jastrow antisymmetric geminal power ansatz and real space multi-Slater Jastrow expansions. Satisfied with its performance, we have added the optimizer to the QMCPACK software package, with which we test a systematically convergent, nonperturbative approach to excitation energies on the example of a Mott-insulating hydrogen ring.
Neural-network quantum states offer a flexible route to compact many-electron wave functions, but their practical accuracy depends strongly on how fermionic antisymmetry, electron correlation, and optimization noise are treated. Here we combine an antisymmetrized geminal power (AGP) determinant with feed-forward neural networks that replace conventional basis-set expansions in the geminal and in two Jastrow-factor constructions. The resulting basis-free Jastrow--AGP ansatz is optimized by variational Monte Carlo and is designed to separate two tasks: the AGP part defines the nodal surface, while the neural-network Jastrow factor recovers dynamical correlation at fixed nodes. This separation makes it possible to distinguish errors associated with dynamical correlation from those caused by static, multireference correlation. Applications to the hydrogen molecule and the rectangular hydrogen tetramer show sub-millihartree accuracy when the AGP nodes are adequate, and expose the residual nodal limitation near the large-radius square geometry of the hydrogen tetramer. These results clarify where neural-network building blocks can improve a compact geminal ansatz and where additional nodal flexibility is required.
We extend our low-scaling variational Monte Carlo (VMC) algorithm to optimize the symmetry-projected Jastrow mean-field (SJMF) wave functions. These wave functions consist of a symmetry-projected product of a Jastrow and a general broken-symmetry mean-field reference. Examples include Jastrow antisymmetrized geminal power, Jastrow-Pfaffians, and resonating valence bond states, among others, all of which can be treated with our algorithm. We will demonstrate using benchmark systems including the nitrogen molecule, a chain of hydrogen atoms, and 2-D Hubbard model that a significant amount of correlation can be obtained by optimizing the energy of the SJMF wave function. This can be achieved at a relatively small cost when multiple symmetries including spin, particle number, and complex conjugation are simultaneously broken and projected. We also show that reduced density matrices can be calculated using the optimized wave functions, which allows us to calculate other observables such as correlation functions and will enable us to embed the VMC algorithm in a complete active-space self-consistent field calculation.
We present a new method for modeling electronically excited states that overcomes a key failing of linear response theory by allowing the underlying ground state ansatz to relax in the presence of an excitation. The method is variational, has a cost similar to ground state variational Monte Carlo, and admits both open and periodic boundary conditions. We present preliminary numerical results showing that, when paired with the Jastrow antisymmetric geminal power ansatz, the variation-after-response formalism delivers accuracies for valence and charge transfer single excitations on par with equation of motion coupled cluster, while surpassing coupled cluster's accuracy for excitations with significant doubly excited character.