A linear surrogate for optimising functions of an orthogonal matrix with applications in wave function theory

Wang, Zhenling; Head‐Gordon, Martin

doi:10.1080/00268976.2022.2118185

Cited by 3 publications

(3 citation statements)

References 43 publications

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“…In other words, Boys orbitals minimize the sum of the orbital second moments. The optimization can be carried out either by Jacobi sweeps or other techniques. − While the Boys orbitals typically localize well (subject to the magnitude of Δ E HL ), it should be noted that they are not necessarily unique for a given problem.…”

Section: Theorymentioning

confidence: 99%

Local Second-Order Møller–Plesset Theory with a Single Threshold Using Orthogonal Virtual Orbitals: Theory, Implementation, and Assessment

Wang,

Aldossary,

Shi

et al. 2023

J. Chem. Theory Comput.

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It has long been clear that electron correlation methods exhibit unphysical compute scalings with molecular size, which has motivated the development of local correlation methods to discard effectively zero contributions in a controlled way to yield an approximate correlation energy. The ideal local correlation method should have a single numerical threshold that controls the dropping of terms with the ability to have that threshold set small enough so that the correlation energy is reproduced to enough significant figures such that the result is chemically identical. This work reports such a method for the second-order Møller–Plesset (MP2) theory. The theory, implementation, and testing of this local MP2 theory are reported. Thresholds ranging from 10–5 to 10–8 and basis sets ranging from split valence plus polarization through to quadruple-ζ are assessed for local MP2 calculations on a range of molecules, including linear chains and molecules with two- and three-dimensional character. The implementation is shared memory parallel via OpenMP and yields roughly 50% parallel efficiency with 16 cores for a large job. Considerable efforts were made to minimize memory demands, which increased as thresholds were tightened. A variety of relative energy calculations are presented as a function of threshold to provide some guidance to users on how to obtain adequate precision at a low compute cost. It is particularly clear that derivative properties require tighter thresholds in order to achieve an adequate precision.

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Section: Theorymentioning

confidence: 99%

Local Second-Order Møller–Plesset Theory with a Single Threshold Using Orthogonal Virtual Orbitals: Theory, Implementation, and Assessment

Wang,

Aldossary,

Shi

et al. 2023

J. Chem. Theory Comput.

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“…[68][69][70][71] Edmiston-Ruedenberg normally requires an iterative O(N 5 ) scaling because of the integral transformation, which is prohibitive in large molecules. But even though Boys and Pipek-Mezey only cost as iterative O(N 3 ), a very large number of iterations are needed for localizing virtual orbitals, meaning that only occupied Boys and Pipek-Mezey orbitals are suitable for a local MP2 formalism.…”

mentioning

confidence: 99%

Sparsity of the electron repulsion integral tensor using different localized virtual orbital representations in local second-order Møller–Plesset theory

Wang¹,

Head‐Gordon

2023

The Journal of Chemical Physics

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Utilizing localized orbitals, local correlation theory can reduce the unphysically high system-size scaling of post-Hartree-Fock (post-HF) methods to linear scaling in insulating molecules. The sparsity of the 4-index electron repulsion integral (ERI) tensor is central to achieving this reduction. For second-order Møller-Plesset theory (MP2), one of the simplest post-HF methods, only the (ia|jb) ERIs are needed, coupling occupied orbitals i,j, and virtuals a,b. In this paper, we compare the numerical sparsity (called the "ragged list") and another two approaches revealing the low-rank sparsity of the ERI. The ragged list requires only one set of (localized) virtual orbitals, and we find the orthogonal valence virtual-hard virtual (VV-HV) set of virtuals originally proposed by Subotnik et al. gives the sparsest ERI tensor. To further compress the ERI tensor, the pair natural orbital (PNO) type representation uses different sets of virtual orbitals for different occupied orbital pairs, while the occupied-specific virtual (OSV) approach uses different virtuals for each occupied orbital. Our results indicate that while the low-rank PNO representation achieves significant rank reduction, it also requires more memory than the ragged list. The OSV approach requires similar memory to the ragged list, but involves greater algorithmic complexity. An approximation (called the "fixed sparsity pattern") for solving the local MP2 equations using the numerically sparse ERI tensor is proposed and tested to be sufficiently accurate and to have highly controllable error. A low-scaling local MP2 algorithm based on the ragged list and the fixed sparsity pattern is therefore promising.

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“…[66][67][68] Edmiston-Ruedenberg normally requires an iterative O(N 5 ) scaling because of the integral transformation, which is prohibitive in large molecules. But even though Boys and Pipek-Mezey only cost as iterative O(N 3 ), a very large number of iterations are needed for localizing virtual orbitals, meaning that only occupied…”

mentioning

confidence: 99%

Sparsity of the Electron Repulsion Integral Tensor Using Different Localized Virtual Orbital Representations in Local Second Order Møller-Plesset Theory

Wang

Head‐Gordon

2022

Preprint

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Utilizing localized orbitals, local correlation theory can reduce the unphysically high system-size scaling of post-Hartree-Fock (post-HF) methods all the way to linear scaling in insulating molecules. Revealing the numerical sparsity of the 4-index electron repulsion (ERI) tensor is central to achieving linear scaling in local correlation methods. For second order Møller-Plesset theory (MP2), one of the simplest post-HF methods, only the $(ia|jb)$ ERIs are needed, coupling occupied orbitals $i,j$, and virtuals $a,b$. In this paper, we compare the numerical sparsity (called the "ragged list representation") and the low-rank sparsity of the $(ia|jb)$ ERI tensor. The ragged list requires only one set of (localized) virtual orbitals, and we find that the orthogonal valence virtual-hard virtual (VV-HV) set of virtuals originally proposed by Subotnik et al. gives the sparsest ERI tensor relative to 4 other good choices: projected atomic orbitals (PAOs), Boys localized virtuals, Pipek-Mezey localized virtuals, and Cholesky localized virtuals. To further compress the $(ia|jb)$ ERI tensor, the low-rank (pair-natural orbital) representation uses different sets of virtual orbitals for different occupied orbital pairs. Our results indicate that while the low-rank representation achieves significant rank reduction, it also requires more memory than the ragged list representation. An approximation (called the "fixed sparsity pattern") for solving the local MP2 equations using the numerically sparse ERI tensor is proposed and tested to be sufficiently accurate and to have highly controllable error. A low-scaling local MP2 algorithm based on the ragged list and the fixed sparsity pattern using this representation is therefore promising.

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A linear surrogate for optimising functions of an orthogonal matrix with applications in wave function theory

Cited by 3 publications

References 43 publications

Local Second-Order Møller–Plesset Theory with a Single Threshold Using Orthogonal Virtual Orbitals: Theory, Implementation, and Assessment

Local Second-Order Møller–Plesset Theory with a Single Threshold Using Orthogonal Virtual Orbitals: Theory, Implementation, and Assessment

Sparsity of the electron repulsion integral tensor using different localized virtual orbital representations in local second-order Møller–Plesset theory

Sparsity of the Electron Repulsion Integral Tensor Using Different Localized Virtual Orbital Representations in Local Second Order Møller-Plesset Theory

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