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Register a new userExact exchange matrix of periodic Hartree-Fock theory for all-electron simulations
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Authors
Qiming Sun
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This work presents an algorithm to evaluate Coulomb and exchange matrices in Fock operator using range separation techniques at various aspects. This algorithm is particularly favorable for the scenario of (1) all-electron calculations or (2) computing exchange matrix for a large number of $mathbf{k}$-point samples. An all electron Hartree-Fock calculation with 110k basis functions is demonstrated in this work.
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A recently developed semiclassical approximation to exchange in one dimension is shown to be almost exact, with essentially no computational cost. The variational stability of this approximation is tested, and its far greater accuracy relative to local density functional calculations demonstrated. Even a fully orbital-free potential-functional calculation (no orbitals of any kind) yields little error relative to exact exchange, for more than one orbital.
The Hartree-Fock problem was recently recast as a semidefinite optimization over the space of rank-constrained two-body reduced-density matrices (RDMs) [Phys. Rev. A 89, 010502(R) (2014)]. This formulation of the problem transfers the non-convexity of the Hartree-Fock energy functional to the rank constraint on the two-body RDM. We consider an equivalent optimization over the space of positive semidefinite one-electron RDMs (1-RDMs) that retains the non-convexity of the Hartree-Fock energy expression. The optimized 1-RDM satisfies ensemble $N$-representability conditions, and ensemble spin-state conditions may be imposed as well. The spin-state conditions place additional linear and nonlinear constraints on the 1-RDM. We apply this RDM-based approach to several molecular systems and explore its spatial (point group) and spin ($S^2$ and $S_3$) symmetry breaking properties. When imposing $S^2$ and $S_3$ symmetry but relaxing point group symmetry, the procedure often locates spatial-symmetry-broken solutions that are difficult to identify using standard algorithms. For example, the RDM-based approach yields a smooth, spatial-symmetry-broken potential energy curve for the well-known Be--H$_2$ insertion pathway. We also demonstrate numerically that, upon relaxation of $S^2$ and $S_3$ symmetry constraints, the RDM-based approach is equivalent to real-valued generalized Hartree-Fock theory.
Quantum computational chemistry is a potential application of quantum computers that is expected to effectively solve several quantum-chemistry problems, particularly the electronic structure problem. Quantum computational chemistry can be compared to the conventional computational devices. This review comprehensively investigates the applications and overview of quantum computational chemistry, including a review of the Hartree-Fock method for quantum information scientists. Quantum algorithms, quantum phase estimation, and variational quantum eigensolver, have been applied to the post-Hartree-Fock method.
The evaluation of exact (Hartree--Fock, HF) exchange operator is a crucial ingredient for the accurate description of electronic structure in periodic systems through ab initio and hybrid density functional approaches. An efficient formulation of periodic HF exchange in LCAO representation presented here is based on the concentric atomic density fitting (CADF) approximation, a domain-free local density fitting approach in which the product of two atomic orbitals (AOs) is approximated using a linear combination of fitting basis functions centered at the same nuclei as the AOs in that product. Significant reduction in the computational cost of exact exchange is demonstrated relative to the conventional approach due to avoiding the need to evaluate four-center two-electron integrals, with sub-millihartree/atom errors in absolute Hartree-Fock energies and good cancellation of fitting errors in relative energies. Novel aspects of the evaluation of the Coulomb contribution to the Fock operator, such as the use of real two-center multipole expansions and spheropole-compensated unit cell densities are also described.
We study the decomposition of the Coulomb integrals of periodic systems into a tensor contraction of six matrices of which only two are distinct. We find that the Coulomb integrals can be well approximated in this form already with small matrices compared to the number of real space grid points. The cost of computing the matrices scales as O(N^4) using a regularized form of the alternating least squares algorithm. The studied factorization of the Coulomb integrals can be exploited to reduce the scaling of the computational cost of expensive tensor contractions appearing in the amplitude equations of coupled cluster methods with respect to system size. We apply the developed methodologies to calculate the adsorption energy of a single water molecule on a hexagonal boron nitride monolayer in a plane wave basis set and periodic boundary conditions.
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