Rank reduced coupled cluster theory. I. Ground state energies and wavefunctions

Abstract

We propose a compression of the opposite-spin coupled cluster doubles amplitudes of the form τijab≡UiaVTVWUjbW, where UiaV are the nV-highest magnitude eigenvectors of the MP2 or MP3 doubles amplitudes. Together with a corresponding parameterization of the opposite-spin coupled cluster Lagrange multipliers of the form λabij≡UiaVLVWUjbW, this yields a fully self-consistent parameterization of reduced-rank coupled cluster equations in terms of the Lagrangian L0TVW,LVW. Making this Lagrangian stationary with respect to the LVW parameters yields a perfectly determined set of equations for the TVW equations and coupled cluster energy. These equations can be solved using a Lyapunov equation for the first-order amplitude updates. We test this “rank-reduced coupled cluster” method for coupled cluster singles and doubles in medium sized molecules and find that substantial compression of the T^2 amplitudes is possible with acceptable accuracy.

Near-exact CCSDT energetics from rank-reduced formalism supplemented by non-iterative corrections

“…The second approximation is the neglect of the L SVD 3 component in Eq. (35). We found that this term is numerically negligible in most cases, especially for smaller SVD subspaces.…”

“…However, these virtues come at a price of a rather steep scaling of the computational costs of the calculations with the system size. Therefore, many approaches such as optimized virtual orbital space, [9][10][11][12] frozen natural orbitals, [13][14][15][16] orbital-specific virtuals, [17][18][19][20] and local correlation treatments based on local pair natural orbitals [21][22][23][24][25] were proposed in the literature to alleviate this problem. Recently, a new idea to reduce the cost of CC (and related) methods has emerged which draws inspiration from the field of applied mathematics 26 and employs tensor decomposition techniques to the wavefunction parameters.…”

“…Recently, a new idea to reduce the cost of CC (and related) methods has emerged which draws inspiration from the field of applied mathematics 26 and employs tensor decomposition techniques to the wavefunction parameters. [27][28][29][30][31][32][33][34][35][36][37][38] These techniques rely on representing the information contained in the cluster excitation amplitudes by a combination of lower-rank quantities. While the basic idea is simple enough, the real difficulty lies in selecting a suitable excitation subspace and then solving the CC equations within this subspace without "unpacking" the compressed quantities to their initial rank at any stage of the calculations.…”

“…However, this may no longer be possible in higher orders. The equation (35) constitutes the backbone of our formalism. However, for pragmatic reasons we introduce two additional approximations.…”

Extension of Frozen Natural Orbital Approximation to Open-Shell References: Theory, Implementation, and Application to Single-Molecule Magnets

“…The sparsity of twoelectron repulsion integrals (ERI) can be utilized through the approximate decomposition over auxiliary basis sets obtained by density fitting, as in the resolution-of-identity (RI) methods [13][14][15][16][17][18][19][20][21], or by Cholesky decomposition of ERIs [22][23][24][25][26][27][28][29][30][31], or using alternative schemes such as pseudospectral decomposition [32][33][34], chain-of-spheres exchange [35][36][37], and tensor hypercontraction [38][39][40][41][42]. Decomposition of ampliomtudes can also be used to reduce computational costs [43][44][45][46]. A number of strategies exploit the physical decay of Coulomb interaction with a distance by using localized orbitals [47][48][49][50].…”