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

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 scaleadaptive 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).

Efficient electronic structure theory via hierarchical scale-adaptive coupled-cluster formalism: I. Theory and computational complexity analysis

“…Having constructed a specific hierarchical Hilbert space S, defined by some subspace aggregation tree, the underlying aggregated subspaces in general may need to be resolved in a lower-quality basis, thus acquiring reduced dimensionality. Following the concepts of scale-adaptive tensor algebra * (SATA) [29], a rectangular transformation matrix 𝑈 !" (order-2 tensor) will define the mapping from the original subspace to the reduced-dimensional derivative subspace such that 𝑖 < |𝑗| , where |.…”

“…Due to the hierarchical structure of vector spaces over which the tensor is defined, and because of the use of adaptive representations [29], each tensor index will acquire a more complex form in our formalism. Specifically, each tensor index will become a triplet {𝑠, 𝑟, 𝑖} ≡ 𝑠 !:!…”

“…On the other side, there has been a tremendous progress in application of the multiresolution analysis (MRA) and related adaptive techniques in electronic structure theory [24][25][26][27][28], although mostly on the self-consistent-field level in practice. In fact, the pragmatic formalism of scale-adaptive tensor algebra [29] used in this paper was inspired by MRA. Very recently, a more elaborated formalism based on the adaptive resolution of many-body tensors was suggested and implemented [30].…”

“…In the following sections, we assume that the reader is familiar with the structure of the general-order CC equations [6] while the specific details of the adaptive CC theory and scale-adaptive tensor algebra can be found in Refs. [31] and [29], respectively. In order to proceed to the next section, it is enough to know that the general-order CC equations consist of many-body diagrams [38], each many-body diagram representing either a single many-body tensor or a connected contraction of a finite number of many-body tensors.…”

Optimization of numerical orbitals using the Helmholtz kernel

“…Tensorial numerical methods have also been developed by other research groups for expanding functions on 3D grids. 7,8,[18][19][20][21][22][23][24][25][26][27][28] Tensorial based wavelet methods have been developed for calculating electrostatic potentials and solving electronic structure equations using the same approach. 1,2,5,7,11,23,[29][30][31][32][33][34][35][36][37][38][39] 0021-9606/2017/146(8)/084102/6/$30.00…”