Linear-scaling explicitly correlated treatment of solids: periodic local MP2-F12 method.

Theory and implementation of the periodic local MP2-F12 method in the 3*A fixed-amplitude ansatz is presented. The method is formulated in the direct space, employing local representation for the occupied, virtual, and auxiliary orbitals in the form of Wannier functions (WFs), projected atomic orbitals (PAOs), and atom-centered Gaussian-type orbitals, respectively. Local approximations are introduced, restricting the list of the explicitly correlated pairs, as well as occupied, virtual, and auxiliary spaces in the strong orthogonality projector to the pair-specific domains on the basis of spatial proximity of respective orbitals. The 4-index two-electron integrals appearing in the formalism are approximated via the direct-space density fitting technique. In this procedure, the fitting orbital spaces are also restricted to local fit-domains surrounding the fitted densities. The formulation of the method and its implementation exploits the translational symmetry and the site-group symmetries of the WFs. Test calculations are performed on LiH crystal. The results show that the periodic LMP2-F12 method substantially accelerates basis set convergence of the total correlation energy, and even more so the correlation energy differences. The resulting energies are quite insensitive to the resolution-of-the-identity domain sizes and the quality of the auxiliary basis sets. The convergence with the orbital domain size is somewhat slower, but still acceptable. Moreover, inclusion of slightly more diffuse functions, than those usually used in the periodic calculations, improves the convergence of the LMP2-F12 correlation energy with respect to both the size of the PAO-domains and the quality of the orbital basis set. At the same time, the essentially diffuse atomic orbitals from standard molecular basis sets, commonly utilized in molecular MP2-F12 calculations, but problematic in the periodic context, are not necessary for LMP2-F12 treatment of crystals.

[1]  J. VandeVondele,et al.  Second-Order Møller-Plesset Perturbation Theory in the Condensed Phase: An Efficient and Massively Parallel Gaussian and Plane Waves Approach. , 2012, Journal of chemical theory and computation.

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

[3]  Hans-Joachim Werner,et al.  Explicitly correlated RMP2 for high-spin open-shell reference states. , 2008, The Journal of chemical physics.

[4]  Toru Shiozaki,et al.  Communication: Second-order multireference perturbation theory with explicit correlation: CASPT2-F12. , 2010, The Journal of chemical physics.

[5]  M. Schütz,et al.  Second Order Local Møller-Plesset Perturbation Theory for Periodic Systems: the CRYSCOR Code , 2010 .

[6]  Edward F. Valeev,et al.  Second-order Møller-Plesset theory with linear R12 terms (MP2-R12) revisited: auxiliary basis set method and massively parallel implementation. , 2004, The Journal of chemical physics.

[7]  A. Köhn.  Explicitly correlated coupled-cluster theory using cusp conditions. II. Treatment of connected triple excitations. , 2010, The Journal of chemical physics.

[8]  Christof Hättig,et al.  The MP2‐F12 method in the TURBOMOLE program package , 2011, J. Comput. Chem..

[9]  Edward F. Valeev,et al.  Universal perturbative explicitly correlated basis set incompleteness correction. , 2009, The Journal of chemical physics.

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

[11]  M. Hanauer,et al.  Explicitly correlated internally contracted multireference coupled-cluster singles and doubles theory: ic-MRCCSD(F12∗) , 2013 .

[12]  Seiichiro Ten-no,et al.  Initiation of explicitly correlated Slater-type geminal theory , 2004 .

[13]  P. Knowles,et al.  Ab Initio Methods for Electron Correlation in Molecules , 2000 .

[14]  S. Hirata,et al.  Communications: Explicitly correlated second-order Møller-Plesset perturbation method for extended systems. , 2010, The Journal of chemical physics.

[15]  Frederick R. Manby,et al.  Fast local-MP2 method with density-fitting for crystals. I. Theory and algorithms , 2007 .

[16]  E. Hylleraas,et al.  Neue Berechnung der Energie des Heliums im Grundzustande, sowie des tiefsten Terms von Ortho-Helium , 1929 .

[17]  Wim Klopper,et al.  Orbital-invariant formulation of the MP2-R12 method , 1991 .

[18]  J. Noga,et al.  Alternative formulation of the matrix elements in MP2‐R12 theory , 2005 .

[19]  Christof Hättig,et al.  Explicitly Correlated Coupled-Cluster Theory , 2010 .

[20]  M. Schütz,et al.  On the use of the Laplace transform in local correlation methods. , 2008, Physical chemistry chemical physics : PCCP.

[21]  M. Schütz,et al.  Periodic quantum mechanical simulation of the He-MgO(100) interaction potential. , 2011, The Journal of chemical physics.

[22]  Seiichiro Ten-no,et al.  New implementation of second-order Møller-Plesset perturbation theory with an analytic Slater-type geminal. , 2007, The Journal of chemical physics.

[23]  J. Noga,et al.  Explicitly correlated coupled cluster F12 theory with single and double excitations. , 2008, The Journal of chemical physics.

[24]  Jeremiah J. Wilke,et al.  Explicitly correlated atomic orbital basis second order Møller-Plesset theory. , 2013, The Journal of chemical physics.

[25]  Roberto Dovesi,et al.  The structural, electronic and vibrational properties of LiOH and NaOH: an ab initio study , 2004 .

[26]  Cesare Pisani,et al.  Beyond a Hartree–Fock description of crystalline solids: the case of lithium hydride , 2007 .

[27]  S. Ten-no.  A simple F12 geminal correction in multi-reference perturbation theory , 2007 .

[28]  P. J. Bygrave,et al.  The embedded many-body expansion for energetics of molecular crystals. , 2012, The Journal of chemical physics.

[29]  Denis Usvyat,et al.  Local ab initio methods for calculating optical band gaps in periodic systems. I. Periodic density fitted local configuration interaction singles method for polymers. , 2011, The Journal of chemical physics.

[30]  D. Tew,et al.  New correlation factors for explicitly correlated electronic wave functions. , 2005, The Journal of chemical physics.

[31]  J. Noga,et al.  Implementation of the CCSD(T)-F12 method using cusp conditions. , 2008, Physical chemistry chemical physics : PCCP.

[32]  R. Dovesi,et al.  A general method to obtain well localized Wannier functions for composite energy bands in linear combination of atomic orbital periodic calculations , 2001 .

[33]  N. Handy,et al.  Hylleraas-type wavefunction for lithium hydride , 1977 .

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

[35]  W. Kutzelnigg,et al.  MP2-R12 calculations on the relative stability of carbocations , 1990 .

[36]  D. Tew,et al.  Communications: Accurate and efficient approximations to explicitly correlated coupled-cluster singles and doubles, CCSD-F12. , 2010, The Journal of chemical physics.

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

[38]  Gareth W Richings,et al.  Implementation of the full explicitly correlated coupled-cluster singles and doubles model CCSD-F12 with optimally reduced auxiliary basis dependence. , 2008, The Journal of chemical physics.

[39]  Georg Kresse,et al.  Second-order Møller-Plesset perturbation theory applied to extended systems. II. Structural and energetic properties. , 2010, The Journal of chemical physics.

[40]  Peter Pulay,et al.  Comparison of the boys and Pipek–Mezey localizations in the local correlation approach and automatic virtual basis selection , 1993, J. Comput. Chem..

[41]  M. Schütz,et al.  Periodic local Møller-Plesset second order perturbation theory method applied to molecular crystals: study of solid NH3 and CO2 using extended basis sets. , 2010, The Journal of chemical physics.

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

[43]  Wim Klopper,et al.  Computation of some new two-electron Gaussian integrals , 1992 .

[44]  Hans-Joachim Werner,et al.  Systematically convergent basis sets for explicitly correlated wavefunctions: the atoms H, He, B-Ne, and Al-Ar. , 2008, The Journal of chemical physics.

[45]  Edward F. Valeev.  Improving on the resolution of the identity in linear R12 ab initio theories , 2004 .

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

[47]  H. Stoll,et al.  Local correlation methods for solids: Comparison of incremental and periodic correlation calculations for the argon fcc crystal , 2011 .

[48]  M. Schütz,et al.  Geometrical frustration of an argon monolayer adsorbed on the MgO (100) surface: An accurate periodic ab initio study , 2012 .

[49]  P Pulay,et al.  Local Treatment of Electron Correlation , 1993 .

[50]  M. Schütz,et al.  Electron correlation decides the stability of cubic versus hexagonal boron nitride. , 2011 .

[51]  Wim Klopper,et al.  Explicitly correlated second-order Møller–Plesset methods with auxiliary basis sets , 2002 .

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

[53]  Hans-Joachim Werner,et al.  Eliminating the domain error in local explicitly correlated second-order Møller-Plesset perturbation theory. , 2008, The Journal of chemical physics.

[54]  S. F. Boys.  Localized Orbitals and Localized Adjustment Functions , 1966 .

[55]  Cesare Pisani,et al.  Periodic local MP2 method for the study of electronic correlation in crystals: Theory and preliminary applications , 2008, J. Comput. Chem..

[56]  Schuetz Martin,et al.  Density Fitting for correlated calculations in periodic systems , 2010 .

[57]  Frederick R Manby,et al.  Local explicitly correlated second-order perturbation theory for the accurate treatment of large molecules. , 2009, The Journal of chemical physics.

[58]  J. VandeVondele,et al.  Electron Correlation in the Condensed Phase from a Resolution of Identity Approach Based on the Gaussian and Plane Waves Scheme. , 2013, Journal of chemical theory and computation.

[59]  F. Manby,et al.  Explicitly correlated second-order perturbation theory using density fitting and local approximations. , 2006, The Journal of chemical physics.

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

[61]  S. Hirata,et al.  Hybrid coupled-cluster and perturbation method for extended systems of one-dimensional periodicity. , 2011, The Journal of chemical physics.

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

[63]  H. Katagiri.  Equation-of-motion coupled-cluster study on exciton states of polyethylene with periodic boundary condition. , 2005, The Journal of chemical physics.

[64]  M. Gillan,et al.  Calculation of properties of crystalline lithium hydride using correlated wave function theory , 2009 .

[65]  J. Almlöf,et al.  Towards the one‐particle basis set limit of second‐order correlation energies: MP2‐R12 calculations on small Ben and Mgn clusters (n=1–4) , 1993 .

[66]  R. T. Pack,et al.  Cusp Conditions for Molecular Wavefunctions , 1966 .

[67]  George H. Booth,et al.  Natural Orbitals for Wave Function Based Correlated Calculations Using a Plane Wave Basis Set. , 2011, Journal of chemical theory and computation.

[68]  Lorenzo Maschio,et al.  Local ab initio methods for calculating optical bandgaps in periodic systems. II. Periodic density fitted local configuration interaction singles method for solids. , 2012, The Journal of chemical physics.

[69]  T. Helgaker,et al.  Variational and robust density fitting of four-center two-electron integrals in local metrics. , 2008, The Journal of chemical physics.

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

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

[72]  Peter Pulay,et al.  An efficient reformulation of the closed‐shell self‐consistent electron pair theory , 1984 .

[73]  Ali Alavi,et al.  Towards an exact description of electronic wavefunctions in real solids , 2012, Nature.

[74]  Cesare Pisani,et al.  CRYSCOR: a program for the post-Hartree-Fock treatment of periodic systems. , 2012, Physical chemistry chemical physics : PCCP.

[75]  C. Samson,et al.  Benchmarking ethylene and ethane: second-order Møller–Plesset pair energies for localized molecular orbitals , 2004 .

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

[77]  Peter Pulay,et al.  Second-order Møller–Plesset calculations with dual basis sets , 2003 .

[78]  K. Doll,et al.  Approaching the bulk limit with finite cluster calculations using local increments: the case of LiH. , 2012, The Journal of chemical physics.

[79]  Martin Head-Gordon,et al.  Auxiliary basis expansions for large-scale electronic structure calculations. , 2005, Proceedings of the National Academy of Sciences of the United States of America.

[80]  N. Marzari,et al.  Maximally localized generalized Wannier functions for composite energy bands , 1997, cond-mat/9707145.

[81]  Ali Alavi,et al.  Convergence of many-body wave-function expansions using a plane-wave basis: From homogeneous electron gas to solid state systems , 2012, 1202.4990.

[82]  B. Paulus.  The method of increments—a wavefunction-based ab initio correlation method for solids , 2006 .

[83]  J. Noga,et al.  Multireference F12 coupled cluster theory: The Brillouin-Wigner approach with single and double excitations , 2011 .

[84]  Frederick R. Manby,et al.  Density fitting in second-order linear-r12 Møller–Plesset perturbation theory , 2003 .

[85]  W. Kutzelnigg,et al.  Wave functions with terms linear in the interelectronic coordinates to take care of the correlation cusp. III. Second‐order Mo/ller–Plesset (MP2‐R12) calculations on molecules of first row atoms , 1991 .

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

[87]  Hans-Joachim Werner,et al.  A simple and efficient CCSD(T)-F12 approximation. , 2007, The Journal of chemical physics.

[88]  Hans-Joachim Werner,et al.  Simplified CCSD(T)-F12 methods: theory and benchmarks. , 2009, The Journal of chemical physics.

[89]  Werner Kutzelnigg,et al.  r12-Dependent terms in the wave function as closed sums of partial wave amplitudes for large l , 1985 .

[90]  Angela K. Wilson,et al.  Gaussian basis sets for use in correlated molecular calculations. VII. Valence, core-valence, and scalar relativistic basis sets for Li, Be, Na, and Mg , 2011 .

[91]  Toru Shiozaki,et al.  Explicitly correlated multireference configuration interaction: MRCI-F12. , 2011, The Journal of chemical physics.

[92]  A. Erba,et al.  Beyond Wigner's theorems: The role of symmetry equivalences in quantum systems , 2012 .

[93]  Christof Hättig,et al.  Local explicitly correlated second- and third-order Møller-Plesset perturbation theory with pair natural orbitals. , 2011, The Journal of chemical physics.

[94]  D. Tew,et al.  A comparison of linear and nonlinear correlation factors for basis set limit Møller-Plesset second order binding energies and structures of He2, Be2, and Ne2. , 2006, The Journal of chemical physics.

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

[96]  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 .

[97]  Georg Kresse,et al.  Cohesive energy curves for noble gas solids calculated by adiabatic connection fluctuation-dissipation theory , 2008 .

[98]  Edward F. Valeev,et al.  Explicitly correlated combined coupled-cluster and perturbation methods. , 2009, The Journal of chemical physics.

[99]  J. Paier,et al.  Second-order Møller-Plesset perturbation theory applied to extended systems. I. Within the projector-augmented-wave formalism using a plane wave basis set. , 2009, The Journal of chemical physics.

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

[101]  Hans-Joachim Werner,et al.  An explicitly correlated local coupled cluster method for calculations of large molecules close to the basis set limit. , 2011, The Journal of chemical physics.

[102]  Lorenzo Maschio,et al.  Local MP2 with Density Fitting for Periodic Systems: A Parallel Implementation. , 2011, Journal of chemical theory and computation.

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

[104]  B. Civalleri,et al.  Ab initio study of van der Waals and hydrogen-bonded molecular crystals with a periodic local-MP2 method , 2010 .

[105]  Hans-Joachim Werner,et al.  Comparison of explicitly correlated local coupled-cluster methods with various choices of virtual orbitals. , 2012, Physical chemistry chemical physics : PCCP.

[106]  Frederick R Manby,et al.  General orbital invariant MP2-F12 theory. , 2007, The Journal of chemical physics.

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

[108]  A. Erba,et al.  A fundamental connection between symmetry and spatial localization properties of basis sets , 2010 .

[109]  Bartolomeo Civalleri,et al.  Approaching the theoretical limit in periodic local MP2 calculations with atomic-orbital basis sets: the case of LiH. , 2011, The Journal of chemical physics.

[110]  G. Beran,et al.  Accurate Molecular Crystal Lattice Energies from a Fragment QM/MM Approach with On-the-Fly Ab Initio Force Field Parametrization. , 2011, Journal of chemical theory and computation.

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

[112]  L. Maschio,et al.  Fitting of local densities in periodic systems , 2008 .

[113]  Wim Klopper,et al.  Wave functions with terms linear in the interelectronic coordinates to take care of the correlation cusp. I. General theory , 1991 .

[114]  S. Hirata,et al.  Fermi resonance in solid CO2 under pressure. , 2013, The Journal of chemical physics.