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Discrete variable representation in electronic structure theory: Quadrature grids for least-squares tensor hypercontraction
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10.1063/1.4802773
/content/aip/journal/jcp/138/19/10.1063/1.4802773
http://aip.metastore.ingenta.com/content/aip/journal/jcp/138/19/10.1063/1.4802773

Figures

Image of FIG. 1.
FIG. 1.

GQ scheme for the Hermite functions ψ to ψ. (Top) orthonormal spectral basis set {ψ()}. (Bottom) DVR function set {ξ()}, Because this is a Gaussian quadrature, the DVR basis is a true interpolating set, with the exact property ξ( ) = δξ( ). These interpolation nodes correspond to the zeros of the next basis function outside of the set, ψ().

Image of FIG. 2.
FIG. 2.

Conceptual representation of the Gaussian quadrature or discrete variable representation scheme, in this case ξ() and its Dirac delta partner, corresponding to the Hermite functions from ψ to ψ. The eigenfunctions and eigenvalues of the position operator are determined in the orthonormal basis. The eigenfunctions are finite spectral approximations to Dirac delta functions, and the eigenvalues provide the discretization of the otherwise continuous position operator. To move from spectral space to physical space, the DVR functions are replaced in a one-to-one fashion by weighted Dirac delta functions placed at the eigenvalues of the finite position operator.

Image of FIG. 3.
FIG. 3.

DVR scheme for the particle-in-a-box wavefunctions ψ to ψ. (Top) orthonormal spectral basis set {ψ()}. (Bottom) DVR function set {ξ()}. Because this is a DVR and not a GQ, the DVR basis is a not a true interpolating set, and only exhibits the approximate interpolation property ξ( ) ≈ δξ( ). These interpolation nodes correspond approximately to the zeros of the next basis function outside of the set, ψ().

Image of FIG. 4.
FIG. 4.

Error of DVR quadrature scheme for the particle-in-a-box wavefunctions ψ to ψ. Depicted is the quadrature error in the overlap matrix ( ), computed as . Note that the exact overlap matrix is the identity .

Image of FIG. 5.
FIG. 5.

R-DVR scheme for Ne atom with a cc-pVDZ-RI DVR auxiliary basis set. (Top panel) canonically orthogonalized radial basis {ψ(ρ)}. (Bottom panel) DVR function set {ξ(ρ)}. Note that, in both frames, the square-root of the inner-product weight is rolled into the functions as , for clarity. Because this is a DVR and not a GQ, the DVR basis is a not a true interpolating set, and only exhibits the approximate interpolation property property ξ) ≈ δξ).

Image of FIG. 6.
FIG. 6.

Comparison of physical-space grids obtained with various generation methods for the HO molecule. (Top panel) Becke grid with 19/11-node Treutler/Ahlrichs radial quadratures (on heavy/hydrogen atoms, respectively) and a pruned 7th order Lebedev-Laikov spherical quadrature, for a total of 442 nodes. (Middle panel) R-DVR grid within an aug-cc-pVTZ-RI DVR auxiliary basis, with a pruned 7th order Levedev-Laikov spherical quadrature, for a total of 486 nodes. (Bottom panel) F-DVR grid within an aug-cc-pVTZ-RI DVR auxiliary basis, with a total of 198 nodes. Gridpoints are colored according to the magnitude of the quadrature weight on a scale from red to blue in order of increasing weight.

Image of FIG. 7.
FIG. 7.

As in Figure 6 , for the anthracene molecule. (Top panel) Becke grid with 19/11-node Treutler/Ahlrichs radial quadratures (on heavy/hydrogen atoms, respectively) and a pruned 7th order Lebedev-Laikov spherical quadrature, for a total of 3992 nodes. (Middle panel) R-DVR grid within an aug-cc-pVTZ-RI DVR auxilairy basis, with a pruned 7th order Levedev-Laikov spherical quadrature, for a total of 4300 nodes. (Bottom panel) Full DVR grid within an aug-cc-pVTZ-RI DVR auxiliary basis, with a total of 1944 nodes.

Image of FIG. 8.
FIG. 8.

Full DVR grid for (HO) within an aug-cc-pVTZ-RI DVR auxiliary basis. This grid contains 594 nodes.

Image of FIG. 9.
FIG. 9.

As in Figure 8 , for the (Ala) helix within an aug-cc-pVTZ-RI DVR auxiliary basis. This grid contains 2478 nodes.

Image of FIG. 10.
FIG. 10.

Energy error (kcal/mol) compared to conventional MP2 for DF-MP2 (blue bars), LS-THC-DF-MP2/Becke (turquoise bars), LS-THC-DF-MP2/R-DVR (yellow bars), and LS-THC-DF-MP2/F-DVR (red bars) for linear alkanes CH, with up to 20 carbon atoms. See Table I for size parameters.

Image of FIG. 11.
FIG. 11.

As in Figure 10 , for linear alkenes CH, with up to 20 carbon atoms. See Table I for size parameters.

Image of FIG. 12.
FIG. 12.

As in Figure 10 , for linear acenes from benzene ( = 1) to pentacene ( = 5). See Table I for size parameters.

Image of FIG. 13.
FIG. 13.

As in Figure 10 , for helical alanine polypeptides, from (Ala) to (Ala). See Table I for size parameters.

Image of FIG. 14.
FIG. 14.

As in Figure 10 , for water clusters from HO to (HO). See Table I for size parameters.

Tables

Generic image for table

R-DVR grid generation.

Generic image for table

F-DVR grid generation.

Generic image for table
Table I.

Number of primary basis functions , DF auxiliary basis functions , and LS-THC grid points , for the largest system of each class encountered in this study. The primary basis set is cc-pVDZ, the DF auxiliary basis set is cc-pVDZ-RI, and the DVR auxiliary basis set is aug-cc-pVTZ-RI. The various grids are constructed as discussed in Sec. III A .

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/content/aip/journal/jcp/138/19/10.1063/1.4802773
2013-05-21
2014-04-16
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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Discrete variable representation in electronic structure theory: Quadrature grids for least-squares tensor hypercontraction
http://aip.metastore.ingenta.com/content/aip/journal/jcp/138/19/10.1063/1.4802773
10.1063/1.4802773
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