Linear scaling conjugate gradient density matrix search as an alternative to diagonalization for first principles electronic structure calculations

Millam, John M.; Scuseria, Gustavo E.

doi:10.1063/1.473579

Cited by 253 publications

(250 citation statements)

References 41 publications

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“…General techniques to overcome the various bottlenecks in DFT calculations have been developed such as the fast multipole method, 1,2 fast quadratures, 3 and linear scaling selfconsistent field optimizers. [4][5][6] Most of these results, together with the exploitation of the nearsightedness of the exact exchange, 7,8 have also made possible a linear scaling formulation of Hartree-Fock ͑HF͒ theory, the cornerstone of any single-reference correlation treatment. Finally, the advent of local correlation methods 9,10 and the atomic orbital ͑AO͒ reformulation of Møller-Plesset ͑MP͒ and coupled-cluster theories 11 have virtually completed this array of developments toward linear scaling implementations of the full arsenal of quantum chemistry.…”

Section: Introductionmentioning

confidence: 99%

Atomic Cholesky decompositions: A route to unbiased auxiliary basis sets for density fitting approximation with tunable accuracy and efficiency

Aquilante

Gagliardi

Pedersen

et al. 2009

The Journal of Chemical Physics

225

274

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Cholesky decomposition of the atomic two-electron integral matrix has recently been proposed as a procedure for automated generation of auxiliary basis sets for the density fitting approximation ͓F. Aquilante et al., J. Chem. Phys. 127, 114107 ͑2007͔͒. In order to increase computational performance while maintaining accuracy, we propose here to reduce the number of primitive Gaussian functions of the contracted auxiliary basis functions by means of a second Cholesky decomposition. Test calculations show that this procedure is most beneficial in conjunction with highly contracted atomic orbital basis sets such as atomic natural orbitals, and that the error resulting from the second decomposition is negligible. We also demonstrate theoretically as well as computationally that the locality of the fitting coefficients can be controlled by means of the decomposition threshold even with the long-ranged Coulomb metric. Cholesky decomposition-based auxiliary basis sets are thus ideally suited for local density fitting approximations.

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Section: Introductionmentioning

confidence: 99%

Atomic Cholesky decompositions: A route to unbiased auxiliary basis sets for density fitting approximation with tunable accuracy and efficiency

Aquilante

Gagliardi

Pedersen

et al. 2009

The Journal of Chemical Physics

225

274

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“…However, transforming back into the atomic orbital basis (as in our AO-CC based formalism 19 ) will yield algorithms that scale near-linearly for sparse enough matrices. 20 The connection between the symplectic eigenvalue problem (Eqn. 1) and its associated Riccati equation (Eqn.…”

mentioning

confidence: 99%

The ground state correlation energy of the random phase approximation from a ring coupled cluster doubles approach

Scuseria¹,

Henderson²,

Sorensen³

2008

The Journal of Chemical Physics

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295

336

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We present an analytic proof demonstrating the equivalence between the Random Phase Approximation (RPA) to the ground state correlation energy and a ring-diagram simplification of the Coupled Cluster Doubles (CCD) equations. In the CCD framework, the RPA equations can be solved in O(N 4 ) computational effort, where N is proportional to the number of basis functions.There has recently been a revival of interest in RPA in the quantum chemistry community. The RPA is popular for calculations of excitation energies both in finite systems 1,2 and in solids, 3,4 and is related to time-dependent density functional theory. 5,6,7 As a technique for describing electronic correlations, RPA has significant advantages, particularly for those interested in density functional theory. It describes dispersion and van der Waals interactions correctly, 8,9 and is exact for long-range correlations. 10 Left-right static correlations seem to be properly described by RPA, 11 and RPA fixes the pathologies of nonlocal Hartree-Fock-type exchange in metallic systems. Readers interested in details about RPA for ground state correlation can refer to the recent paper by Furche 12 where he discusses an interesting simplification to reduce the computational cost of RPA correlation and provides ample background information about RPA. Note that his work focuses on direct RPA, in which the exchange terms are neglected; as discussed later in this communication, this is the form of RPA most useful in the context of density functional theory.A connection between the RPA correlation energy and a ring diagram approximation to CCD was first mentioned by Freeman in his 1977 paper. 13 Very recently, A. Grüneis and G. Kresse reproduced this evidence and found numerical proof of the equivalence between these two approaches. 14 Here, we offer an analytic proof that these two problems yield identical correlation energies. To the best of our knowledge, no such formal proof has been given before.As a method for calculating electronic excitation spectra, RPA requires the solution ofThe matrices A, B, X, and Y are all ov × ov, where o and v are respectively the number of occupied and unoccupied spin-orbitals. The eigenvalue problem above can be completed by noting that if Xi Yi is an eigenvector with eigenvalue ω i , then Yi Xi is also an eigenvector, with eigenvalue −ω i . In the (real) canonical spin-orbital basis we use throughout this letter, we haveHere, ǫ p is a diagonal element of the Fock operator. Indices i, j, k, and l indicate occupied spin-orbitals, while a, b, c, d indicate unoccupied spin-orbitals. For arbitrary spin-orbitals p, q, r, and s, the two-electron integral pq rs is defined bywhere x is a combined space and spin electron coordinate. The RPA correlation energy can be obtained by considering two harmonic excitation energy problems: 12,15 RPA and the Tamm-Dancoff approximation (TDA) thereto, which sets B = 0 and thus solvesIn the quantum chemistry community, TDA is also known as configuration interaction singles (CIS). While TDA includes onl...

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“…However this cubic dependence also means that the method is potentially unstable should any of the eigenvalues of L stray outside the range over which the purification transformation converges. A number of variants of the original LNV method have been proposed [43,44].…”

Section: Li-nunes-vanderbilt Methodsmentioning

confidence: 99%

Density kernel optimization in the ONETEP code

Haynes

Skylaris

Mostofi

et al. 2008

J. Phys.: Condens. Matter

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ONETEP is a linear scaling code for performing first-principles total energy calculations within density-functional theory (DFT). The method is based on the density-matrix formulation of DFT and involves the iterative minimization of the total energy with respect to a set of local orbitals and a density kernel. An overview is given of the kernel optimization methods proposed in the literature and implemented in ONETEP, focusing in particular on the constraints of compatibility, idempotency and normalization that must be applied. A method is proposed for locating the chemical potential which may be useful in applying the normalization constraint and analysing the electronic structure near the Fermi level.

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Linear scaling conjugate gradient density matrix search as an alternative to diagonalization for first principles electronic structure calculations

Cited by 253 publications

References 41 publications

Atomic Cholesky decompositions: A route to unbiased auxiliary basis sets for density fitting approximation with tunable accuracy and efficiency

Atomic Cholesky decompositions: A route to unbiased auxiliary basis sets for density fitting approximation with tunable accuracy and efficiency

The ground state correlation energy of the random phase approximation from a ring coupled cluster doubles approach

Density kernel optimization in the ONETEP code

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