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International Journal of Quantum Chemistry

Scale‐adaptive tensor algebra for local many‐body methods of electronic structure theory

Dmitry I. Lyakh

Published Jul 7, 2014• 4 citations
Publisher

Abstract

While the formalism of multiresolution analysis, based on wavelets and adaptive integral representations of operators, is actively progressing in electronic structure theory (mostly on the independent‐particle level and, recently, second‐order perturbation theory), the concepts of multiresolution and adaptivity can also be utilized within the traditional formulation of correlated (many‐particle) theory based on second quantization and the corresponding (generally nonorthogonal) tensor algebra. In this article, we present a formalism called scale‐adaptive tensor algebra, which introduces an adaptive representation of tensors of many‐body operators via the local adjustment of the basis set quality. Given a series of locally supported fragment bases of a progressively lower quality, we formulate the explicit rules for tensor algebra operations dealing with adaptively resolved tensor operands. The formalism suggested is expected to enhance the applicability of certain local correlated many‐body methods of electronic structure theory, for example, those directly based on atomic orbitals (or any other localized basis functions in general). © 2014 Wiley Periodicals, Inc.

DOI
10.1002/qua.24732

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While the formalism of multiresolution analysis, based on wavelets and adaptive integral representations of operators, is actively progressing in electronic structure theory (mostly on the independent‐particle level and, recently, second‐order perturbation theory), the concepts of multiresolution and adaptivity can also be utilized within the traditional formulation of correlated (many‐particle) theory based on second quantization and the corresponding (generally nonorthogonal) tensor algebra. In this article, we present a formalism called scale‐adaptive tensor algebra, which introduces an adaptive representation of tensors of many‐body operators via the local adjustment of the basis set quality. Given a series of locally supported fragment bases of a progressively lower quality, we formulate the explicit rules for tensor algebra operations dealing with adaptively resolved tensor operands. The formalism suggested is expected to enhance the applicability of certain local correlated many‐body methods of electronic structure theory, for example, those directly based on atomic orbitals (or any other localized basis functions in general). © 2014 Wiley Periodicals, Inc.
A novel reduced-scaling, general-order coupled-cluster approach is formulated by exploiting hierarchical representations of many-body tensors, combined with the recently suggested formalism of scale-adaptive tensor algebra. Inspired by the hierarchical techniques from the renormalisation group approach, H/H2-matrix algebra and fast multipole method, the computational scaling reduction in our formalism is achieved via coarsening of quantum many-body interactions at larger interaction scales, thus imposing a hierarchical structure on many-body tensors of coupled-cluster theory. In our approach, the interaction scale can be defined on any appropriate Euclidean domain (spatial domain, momentum-space domain, energy domain, etc.). We show that the hierarchically resolved many-body tensors can reduce the storage requirements to O(N), where N is the number of simulated quantum particles. Subsequently, we prove that any connected many-body diagram consisting of a finite number of arbitrary-order tensors, e.g. an arbitrary coupled-cluster diagram, can be evaluated in O(NlogN) floating-point operations. On top of that, we suggest an additional approximation to further reduce the computational complexity of higher order coupled-cluster equations, i.e. equations involving higher than double excitations, which otherwise would introduce a large prefactor into formal O(NlogN) scaling.
We apply tensor methods to formulate theories of electron correlation in nonorthogonal basis sets. The resulting equations are manifestly invariant to nonorthogonal basis transformations, between functions spanning either the occupied or virtual subspaces of the one-particle Hilbert space. The tensor approach is readily employed in either first or second quantization. As examples, second-order Mo/ller–Plesset perturbation theory, and coupled cluster theory with single and double substitutions, including noniterative triples, are recast using the tensor formalism. This gives equations which are invariant to larger classes of transformations than existing expressions. Procedures for truncating these equations are discussed.
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Multiresolution (or wavelet) analysis offers a strictly local basis set for a systematic introduction of new details into Hilbert space operators. Using this tool we have previously developed an expansion method for density matrices. The set of density operators providing a given electron density plays an essential role in density functional theory, in the minimization of energy expectation values with the constraint that the electron density is fixed. In this contribution, using multiresolution analysis, we present an excellent quality density matrix expansion yielding a prescribed electron density, and compare it to other known methods. Due to the strictly local nature of the applied basis functions, our construction has the specific advantage that the resulting density matrix is correlated and N-representable in the infinite resolution limit. As a further consequence of this scheme we can conclude that the deviation of the exact kinetic energy functional from the Weizsäcker term is not a necessary consequence of the particle statistics.
Abstract This paper lays the algebraic foundation of a unitary group approach to many‐body theory. We define a set of second quantized spin‐shift operators which are used to construct spin‐adapted many‐electron configuration functions. We investigate the particle‐hole transformation, normal ordering, and contraction of spin shifts. This gives an orbital Wick's theorem reflecting the permutational structure of the states spanning the irreducible representations of the spatial unitary group. We study the spinless Hamiltonian, and the relationship between spin shifts and unitary group generators.
In electronic structure theory, variational computing offers a valuable paradigm for the approximation of electronic ground states. However, for historical reasons, this principle is mostly restricted to model chemistries in predefined fixed basis sets. Especially in quantum computation, these model chemistries are far from an accurate description of the initial electronic Hamiltonian. This work demonstrates a fully variational approach to the electronic structure problem by variationally optimizing the orbitals representing the second quantized Hamiltonian alongside a quantum circuit that generates the many-electron wavefunction. To this end, the orbitals are represented within an adaptive multi-wavelet format, guaranteeing numerical precision. We then showcase explicit numerical protocols and highlight the quantum circuit’s effects on determining the optimal orbital basis.