Fast local-MP2 method with density-fitting for crystals. I. Theory and algorithms

When solving the M\o{}ller-Plesset second order perturbation theory (MP2) equations for periodic systems using a local-correlation approach [J. Chem. Phys. 122 (2005) 094113], the computational bottleneck is represented by the evaluation of the two-electron Coulomb interaction integrals between product distributions, each involving a Wannier function and a projected atomic orbital. While for distant product distributions a multipolar approximation performs very efficiently, the four index transformation for close-by distributions, which by far constitutes the bottleneck of correlated electronic structure calculations of crystals, can be avoided through the use of density fitting techniques. An adaptation of that scheme to translationally periodic systems is described, based on Fourier transformation techniques. The formulas and algorithms adopted allow the point symmetry of the crystal to be exploited. Problems related to the possible divergency of lattice sums of integrals involving fitting functions are identified and eliminated through the use of Poisson transformed fitting functions and of dipole-corrected product distributions. The iterative scheme for solving the linear local MP2 (LMP2) equations is revisited. Prescreening in the evaluation of the residual matrix is introduced, which significantly lowers the scaling of the LMP2 equations.

[1]  Beate Paulus,et al.  Convergence of the ab initio many-body expansion for the cohesive energy of solid mercury , 2004 .

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

[3]  Š. Varga,et al.  Density fitting of Coulomb integrals in electronic structure calculations of solids , 2005 .

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

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

[6]  Martin Schütz,et al.  Low-order scaling local electron correlation methods. V. Connected triples beyond (T): Linear scaling local CCSDT-1b , 2002 .

[7]  Alistair P. Rendell,et al.  COUPLED-CLUSTER THEORY EMPLOYING APPROXIMATE INTEGRALS : AN APPROACH TO AVOID THE INPUT/OUTPUT AND STORAGE BOTTLENECKS , 1994 .

[8]  R. Orlando,et al.  Ab Initio Quantum Simulation in Solid State Chemistry , 2005 .

[9]  H. Monkhorst,et al.  Hartree-Fock density of states for extended systems , 1979 .

[10]  Frederick R Manby,et al.  Explicitly correlated local second-order perturbation theory with a frozen geminal correlation factor. , 2006, The Journal of chemical physics.

[11]  Christian Ochsenfeld,et al.  Multipole-based integral estimates for the rigorous description of distance dependence in two-electron integrals. , 2005, The Journal of chemical physics.

[12]  Stoll,et al.  Correlation energy of diamond. , 1992, Physical review. B, Condensed matter.

[13]  Peter Pulay,et al.  Localizability of dynamic electron correlation , 1983 .

[14]  Hans-Joachim Werner,et al.  Local treatment of electron correlation in coupled cluster theory , 1996 .

[15]  Peter Pulay,et al.  Orbital-invariant formulation and second-order gradient evaluation in Møller-Plesset perturbation theory , 1986 .

[16]  Hans-Joachim Werner,et al.  Low-order scaling local electron correlation methods. IV. Linear scaling local coupled-cluster (LCCSD) , 2001 .

[17]  P. Fulde,et al.  A local approach to the computation of correlation energies of molecules , 1977 .

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

[19]  Guntram Rauhut,et al.  Analytical energy gradients for local second-order Mo/ller–Plesset perturbation theory , 1998 .

[20]  Martin Schütz,et al.  Low-order scaling local electron correlation methods. III. Linear scaling local perturbative triples correction (T) , 2000 .

[21]  Frederick R Manby,et al.  Analytical energy gradients for local second-order Møller-Plesset perturbation theory using density fitting approximations. , 2004, The Journal of chemical physics.

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

[23]  Christian Ochsenfeld,et al.  Rigorous integral screening for electron correlation methods. , 2005, The Journal of chemical physics.

[24]  Peter Pulay,et al.  Fourth‐order Mo/ller–Plessett perturbation theory in the local correlation treatment. I. Method , 1987 .

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

[26]  J. Mintmire,et al.  Fitting the Coulomb potential variationally in linear-combination-of-atomic-orbitals density-functional calculations , 1982 .

[27]  Georg Hetzer,et al.  Low-order scaling local correlation methods II: Splitting the Coulomb operator in linear scaling local second-order Møller–Plesset perturbation theory , 2000 .

[28]  Philippe Y. Ayala,et al.  Atomic orbital Laplace-transformed second-order Møller–Plesset theory for periodic systems , 2001 .

[29]  Tatiana Korona,et al.  Local CC2 electronic excitation energies for large molecules with density fitting. , 2006, The Journal of chemical physics.

[30]  Rodney J. Bartlett,et al.  Second‐order many‐body perturbation‐theory calculations in extended systems , 1996 .

[31]  Cesare Pisani,et al.  Symmetry-adapted Localized Wannier Functions Suitable for Periodic Local Correlation Methods , 2006 .

[32]  Frederick R. Manby,et al.  Fast local-MP2 method with density-fitting for crystals. II. Test calculations and application to the carbon dioxide crystal , 2007 .

[33]  Brett I. Dunlap,et al.  Robust and variational fitting , 2000 .

[34]  Peter Pulay,et al.  Local configuration interaction: An efficient approach for larger molecules , 1985 .

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

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

[37]  R. Parr,et al.  Discontinuous approximate molecular electronic wave‐functions , 1977 .

[38]  R Dovesi,et al.  Local-MP2 electron correlation method for nonconducting crystals. , 2005, The Journal of chemical physics.

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

[40]  G. Friesecke,et al.  Decay behavior of least-squares coefficients in auxiliary basis expansions. , 2005, The Journal of chemical physics.

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

[42]  E. Baerends,et al.  Precise density-functional method for periodic structures. , 1991, Physical review. B, Condensed matter.

[43]  Shigeru Obara,et al.  Efficient recursive computation of molecular integrals over Cartesian Gaussian functions , 1986 .

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

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

[46]  J. Connolly,et al.  On first‐row diatomic molecules and local density models , 1979 .

[47]  F. Weigend,et al.  Efficient use of the correlation consistent basis sets in resolution of the identity MP2 calculations , 2002 .

[48]  Martin Schütz,et al.  A new, fast, semi-direct implementation of linear scaling local coupled cluster theory , 2002 .

[49]  Jozef Noga,et al.  Density fitting of two-electron integrals in extended systems with translational periodicity: the Coulomb problem. , 2006, The Journal of chemical physics.

[50]  Cesare Pisani,et al.  On the Prospective Use of the One-Electron Density Matrix as a Test of the Quality of Post-Hartree–Fock Schemes for Crystals , 2006 .

[51]  Denis Usvyat,et al.  Nonorthogonal ultralocalized functions and fitted Wannier functions for local electron correlation methods for solids , 2005 .

[52]  S. Prager,et al.  USE OF ELECTROSTATIC VARIATION PRINCIPLES IN MOLECULAR ENERGY CALCULATIONS , 1962 .

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

[54]  J. L. Whitten,et al.  Coulombic potential energy integrals and approximations , 1973 .