Canonical Tensor Products as a Generalization of Gaussian-type Orbitals

Abstract We propose a possible generalization of Gaussian-type orbital (GTO) bases by means of canonical tensor products. The present work focus on the application of tensor products as an alternative to conventional GTO based density fitting schemes. Tensor product approximation leads to highly nonlinear optimization problems which require sophisticated algorithms. We give a brief description of the optimization problem and algorithm. The present work extends our previous paper [S. R. Chinnamsetty, M. Espig, B. N. Khoromskij, W. Hackbusch and H.-J. Flad, J. Chem. Phys. 127 (2007), 084110], where we discussed tensor product approximations of the electron density and the Hartree potential, to orbital products which are required for the exchange part of Hartree-Fock and in post Hartree-Fock methods. We provide a detailed error analysis for the Coulomb and exchange terms in Hartree-Fock calculations. Furthermore, a comparison is given between all-electron and pseudopotential calculations.

[1]  Eugene E. Tyrtyshnikov,et al.  Tucker Dimensionality Reduction of Three-Dimensional Arrays in Linear Time , 2008, SIAM J. Matrix Anal. Appl..

[2]  Joos Vandewalle,et al.  Computation of the Canonical Decomposition by Means of a Simultaneous Generalized Schur Decomposition , 2005, SIAM J. Matrix Anal. Appl..

[3]  W. Hackbusch,et al.  A New Scheme for the Tensor Representation , 2009 .

[4]  Henrik Koch,et al.  Method specific Cholesky decomposition: coulomb and exchange energies. , 2008, The Journal of chemical physics.

[5]  Boris N. Khoromskij.  On tensor approximation of Green iterations for Kohn-Sham equations , 2008 .

[6]  Marco Häser,et al.  Auxiliary basis sets to approximate Coulomb potentials , 1995 .

[7]  Boris N. Khoromskij,et al.  Tensor decomposition in electronic structure calculations on 3D Cartesian grids , 2009, J. Comput. Phys..

[8]  Frederick R. Manby,et al.  Fast linear scaling second-order Møller-Plesset perturbation theory (MP2) using local and density fitting approximations , 2003 .

[9]  Andrew G. Glen,et al.  APPL , 2001 .

[10]  Joos Vandewalle,et al.  A Multilinear Singular Value Decomposition , 2000, SIAM J. Matrix Anal. Appl..

[11]  J. Almlöf,et al.  Integral approximations for LCAO-SCF calculations , 1993 .

[12]  Georg Hetzer,et al.  Low-order scaling local electron correlation methods. I. Linear scaling local MP2 , 1999 .

[13]  John R. Sabin,et al.  On some approximations in applications of Xα theory , 1979 .

[14]  Philippe Y. Ayala,et al.  Linear scaling coupled cluster and perturbation theories in the atomic orbital basis , 1999 .

[15]  Evert Jan Baerends,et al.  Self-consistent molecular Hartree—Fock—Slater calculations I. The computational procedure , 1973 .

[16]  N. H. Beebe,et al.  Simplifications in the generation and transformation of two‐electron integrals in molecular calculations , 1977 .

[17]  Boris N. Khoromskij,et al.  Verification of the cross 3D algorithm on quantum chemistry data , 2008 .

[18]  Frederick R. Manby,et al.  The Poisson equation in density fitting for the Kohn-Sham Coulomb problem , 2001 .

[19]  C. Loan.  The ubiquitous Kronecker product , 2000 .

[20]  Daniel T. Mainz,et al.  New pseudospectral algorithms for electronic structure calculations: Length scale separation and analytical two‐electron integral corrections , 1994 .

[21]  C Filippi,et al.  Energy-consistent pseudopotentials for quantum Monte Carlo calculations. , 2007, The Journal of chemical physics.

[22]  Roland Lindh,et al.  Unbiased auxiliary basis sets for accurate two-electron integral approximations. , 2007, The Journal of chemical physics.

[23]  BORIS N. KHOROMSKIJ,et al.  Tensor-Product Approximation to Multidimensional Integral Operators and Green's Functions , 2008, SIAM J. Matrix Anal. Appl..

[24]  Rick A. Kendall,et al.  The impact of the resolution of the identity approximate integral method on modern ab initio algorithm development , 1997 .

[25]  Florian Weigend,et al.  A fully direct RI-HF algorithm: Implementation, optimised auxiliary basis sets, demonstration of accuracy and efficiency , 2002 .

[26]  Frederick R. Manby,et al.  Fast Hartree–Fock theory using local density fitting approximations , 2004 .

[27]  Kevin Barraclough,et al.  I and i , 2001, BMJ : British Medical Journal.

[28]  Eugene E. Tyrtyshnikov,et al.  Breaking the Curse of Dimensionality, Or How to Use SVD in Many Dimensions , 2009, SIAM J. Sci. Comput..

[29]  M. Ratner.  Molecular electronic-structure theory , 2000 .

[30]  Reinhold Schneider,et al.  Esaim: Mathematical Modelling and Numerical Analysis Best N -term Approximation in Electronic Structure Calculations I. One-electron Reduced Density Matrix , 2022 .

[31]  Martin W. Feyereisen,et al.  Use of approximate integrals in ab initio theory. An application in MP2 energy calculations , 1993 .

[32]  Boris N. Khoromskij,et al.  Mathematik in den Naturwissenschaften Leipzig Tensor-Product Approximation to Operators and Functions in High Dimensions , 2007 .

[33]  Dietrich Braess,et al.  Asymptotics for the approximation of wave functions by exponential sums , 1995 .

[34]  李幼升,et al.  Ph , 1989 .

[35]  F. Weigend,et al.  RI-MP2: first derivatives and global consistency , 1997 .

[36]  Frederick R. Manby,et al.  Linear scaling local coupled cluster theory with density fitting. Part I: 4-external integrals , 2003 .

[37]  Rasmus Bro,et al.  Multi-way Analysis with Applications in the Chemical Sciences , 2004 .

[38]  Boris N. Khoromskij,et al.  Multigrid Accelerated Tensor Approximation of Function Related Multidimensional Arrays , 2009, SIAM J. Sci. Comput..

[39]  I. V. Oseledets,et al.  Minimization methods for approximating tensors and their comparison , 2006 .

[40]  Vin de Silva,et al.  Tensor rank and the ill-posedness of the best low-rank approximation problem , 2006, math/0607647.

[41]  T. H. Dunning.  Gaussian basis sets for use in correlated molecular calculations. I. The atoms boron through neon and hydrogen , 1989 .

[42]  B. Khoromskij,et al.  Low rank Tucker-type tensor approximation to classical potentials , 2007 .

[43]  Ronald H. Felton,et al.  A new computational approach to Slater’s SCF–Xα equation , 1975 .

[44]  Aaas News,et al.  Book Reviews , 1893, Buffalo Medical and Surgical Journal.

[45]  C. M. Reeves,et al.  Automatic Fundamental Calculations of Molecular Structure , 1956, Nature.

[46]  Holger Patzelt,et al.  RI-MP2: optimized auxiliary basis sets and demonstration of efficiency , 1998 .

[47]  P. Knowles,et al.  Poisson equation in the Kohn-Sham Coulomb problem. , 2001, Physical review letters.

[48]  Vladimir Maz’ya,et al.  Scattered data approximate approximations , 2007 .

[49]  H. Preuss.  Bemerkungen zum Self-consistent-field-Verfahren und zur Methode der Konfigurationenwechselwirkung in der Quantenchemie , 1956 .

[50]  Werner Kutzelnigg,et al.  Theory of the expansion of wave functions in a gaussian basis , 1994 .

[51]  I. V. Oseledets,et al.  Tensors inside of matrices give logarithmic complexity , 2009 .

[52]  D. Bernholdt,et al.  Large-scale correlated electronic structure calculations: the RI-MP2 method on parallel computers , 1996 .

[53]  C. Van Alsenoy,et al.  Ab initio calculations on large molecules: The multiplicative integral approximation , 1988 .

[54]  S. F. Boys.  Electronic wave functions - I. A general method of calculation for the stationary states of any molecular system , 1950, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[55]  Trygve Helgaker,et al.  Basis-set convergence of the energy in molecular Hartree–Fock calculations , 1999 .

[56]  Michael Dolg,et al.  Ab initio energy-adjusted pseudopotentials for elements of groups 13-17 , 1993 .

[57]  S. Ten-no,et al.  Three-center expansion of electron repulsion integrals with linear combination of atomic electron distributions , 1995 .

[58]  Wolfgang Hackbusch,et al.  Tensor product approximation with optimal rank in quantum chemistry. , 2007, The Journal of chemical physics.