Analytical Gradients of Hartree-Fock Exchange with Density Fitting Approximations.

We extend the local exchange (LK) algorithm [Aquilante, F.; Pedersen, T. B.; Lindh, R. J. Chem. Phys.2007, 126, 194106] to the calculation of analytical gradients with density fitting. We discuss the features of the screening procedure and demonstrate the possible advantages of using this formulation, which is easily interfaced to a standard integral-direct gradient code. With auxiliary basis sets obtained from Cholesky decomposition of atomic or molecular integral blocks with a decomposition threshold of 10(-4)Eh, typical errors due to the density fitting in bond lengths, bond angles, and dihedral angles are 0.1 pm, 0.1°, and 0.5°, respectively. The overall speedup of geometry optimizations is about 1 order of magnitude for atomic natural-orbital-type basis sets but much less pronounced for correlation-consistent basis sets.

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

[2]  Florian Weigend,et al.  Hartree–Fock exchange fitting basis sets for H to Rn † , 2008, J. Comput. Chem..

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

[4]  Henry Krakauer,et al.  Assessing weak hydrogen binding on Ca+ centers: an accurate many-body study with large basis sets. , 2011, The Journal of chemical physics.

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

[6]  Thomas Bondo Pedersen,et al.  Reduced scaling in electronic structure calculations using Cholesky decompositions , 2003 .

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

[8]  Walter Thiel,et al.  Benchmarks for electronically excited states: CASPT2, CC2, CCSD, and CC3. , 2008, The Journal of chemical physics.

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

[10]  Roland Lindh,et al.  Main group atoms and dimers studied with a new relativistic ANO basis set , 2004 .

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

[12]  Roland Lindh,et al.  Atomic Cholesky decompositions: a route to unbiased auxiliary basis sets for density fitting approximation with tunable accuracy and efficiency. , 2009, The Journal of chemical physics.

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

[14]  B. Roos,et al.  Bond length and bond order in one of the shortest Cr-Cr bonds. , 2008, Inorganic chemistry.

[15]  Pavel Hobza,et al.  On the Structure and Geometry of Biomolecular Binding Motifs (Hydrogen-Bonding, Stacking, X-H···π): WFT and DFT Calculations. , 2010, Journal of chemical theory and computation.

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

[17]  L. Gagliardi,et al.  Strong correlation treated via effective hamiltonians and perturbation theory. , 2011, The Journal of chemical physics.

[18]  Francesco Aquilante,et al.  Systematic truncation of the virtual space in multiconfigurational perturbation theory. , 2009, The Journal of chemical physics.

[19]  Florian Weigend,et al.  Auxiliary basis sets for main row atoms and transition metals and their use to approximate Coulomb potentials , 1997 .

[20]  Thomas Bondo Pedersen,et al.  Polarizability and optical rotation calculated from the approximate coupled cluster singles and doubles CC2 linear response theory using Cholesky decompositions. , 2004, The Journal of chemical physics.

[21]  R. Lindh,et al.  Low-cost evaluation of the exchange Fock matrix from Cholesky and density fitting representations of the electron repulsion integrals. , 2007, The Journal of chemical physics.

[22]  T. Crawford,et al.  Gas phase optical rotation calculated from coupled cluster theory with zero-point vibrational corrections from density funcional theory. , 2009, Chirality.

[23]  E. Carter,et al.  Density fitting of two-electron integrals in local multireference single and double excitation configuration interaction calculations , 2010 .

[24]  Francesco Aquilante,et al.  Fast noniterative orbital localization for large molecules. , 2006, The Journal of chemical physics.

[25]  Roland Lindh,et al.  Accurate ab initio density fitting for multiconfigurational self-consistent field methods. , 2008, The Journal of chemical physics.

[26]  F. Weigend.  Accurate Coulomb-fitting basis sets for H to Rn. , 2006, Physical chemistry chemical physics : PCCP.

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

[28]  K. Pierloot,et al.  Performance of CASPT2 and DFT for Relative Spin-State Energetics of Heme Models. , 2010, Journal of chemical theory and computation.

[29]  FRANCESCO AQUILANTE,et al.  MOLCAS 7: The Next Generation , 2010, J. Comput. Chem..

[30]  M. Olivucci,et al.  Matrix isolation and computational studies of the CF2I radical , 2010 .

[31]  J. Leszczynski,et al.  QM/MM Studies of Light-responsive Biological Systems , 2021, Challenges and Advances in Computational Chemistry and Physics.

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

[33]  Jonas Boström,et al.  Ab Initio Density Fitting: Accuracy Assessment of Auxiliary Basis Sets from Cholesky Decompositions. , 2009, Journal of chemical theory and computation.

[34]  Emily A Carter,et al.  Cholesky decomposition within local multireference singles and doubles configuration interaction. , 2010, The Journal of chemical physics.

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

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

[37]  Francesco Aquilante,et al.  Quartic scaling evaluation of canonical scaled opposite spin second-order Møller Plesset correlation energy using Cholesky decompositions , 2007 .

[38]  Daniel Svozil,et al.  Reference MP2/CBS and CCSD(T) quantum-chemical calculations on stacked adenine dimers. Comparison with DFT-D, MP2.5, SCS(MI)-MP2, M06-2X, CBS(SCS-D) and force field descriptions. , 2010, Physical chemistry chemical physics : PCCP.

[39]  L. Cederbaum,et al.  On the Cholesky decomposition for electron propagator methods: General aspects and application on C(60). , 2009, The Journal of chemical physics.

[40]  P. Neogrády,et al.  Three- and four-body nonadditivities in nucleic acid tetramers: a CCSD(T) study. , 2010, Physical chemistry chemical physics : PCCP.

[41]  C. Cramer,et al.  Second-order perturbation theory with complete and restricted active space reference functions applied to oligomeric unsaturated hydrocarbons. , 2009, Physical chemistry chemical physics : PCCP.

[42]  Yuji Mochizuki,et al.  Acceleration of fragment molecular orbital calculations with Cholesky decomposition approach , 2010 .

[43]  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.

[44]  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 .

[45]  K. Pierloot,et al.  Copper corroles: the question of noninnocence. , 2010, Inorganic chemistry.

[46]  Jon Baker,et al.  The location of transition states: A comparison of Cartesian, Z‐matrix, and natural internal coordinates , 1996 .

[47]  Florian Weigend,et al.  Approximated electron repulsion integrals: Cholesky decomposition versus resolution of the identity methods. , 2009, The Journal of chemical physics.

[48]  Roland Lindh,et al.  Analytic derivatives for the Cholesky representation of the two-electron integrals. , 2008, The Journal of chemical physics.

[49]  Roland Lindh,et al.  Density fitting with auxiliary basis sets from Cholesky decompositions , 2009 .

[50]  F. Aquilante.  New Approaches to Large-Scale Electronic Structure Calculations , 2007 .

[51]  Luis Serrano-Andrés,et al.  Multiconfigurational Second-Order Perturbation Theory Restricted Active Space (RASPT2) Method for Electronic Excited States: A Benchmark Study. , 2011, Journal of chemical theory and computation.

[52]  Francesco Aquilante,et al.  Cholesky Decomposition-Based Multiconfiguration Second-Order Perturbation Theory (CD-CASPT2): Application to the Spin-State Energetics of Co(III)(diiminato)(NPh). , 2008, Journal of chemical theory and computation.

[53]  C. Cramer,et al.  What Active Space Adequately Describes Oxygen Activation by a Late Transition Metal? CASPT2 and RASPT2 Applied to Intermediates from the Reaction of O2 with a Cu(I)-α-Ketocarboxylate. , 2009, Journal of chemical theory and computation.

[54]  Pavel Hobza,et al.  Parallelized implementation of the CCSD(T) method in MOLCAS using optimized virtual orbitals space and Cholesky decomposed two-electron integrals , 2011 .

[55]  B. Roos,et al.  On the analysis of the Cr-Cr multiple bond in several classes of dichromium compounds. , 2010, Inorganic chemistry.

[56]  C. de Graaf,et al.  Study of the light-induced spin crossover process of the [Fe(II)(bpy)3]2+ complex. , 2010, Chemistry.

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

[58]  C. David Sherrill,et al.  Density fitting and Cholesky decomposition approximations in symmetry-adapted perturbation theory: Implementation and application to probe the nature of π-π interactions in linear acenes , 2010 .

[59]  Roland Lindh,et al.  Multiconfiguration second‐order perturbation theory approach to strong electron correlation in chemistry and photochemistry , 2012 .

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

[61]  Paul G. Mezey,et al.  A fast intrinsic localization procedure applicable for ab initio and semiempirical linear combination of atomic orbital wave functions , 1989 .

[62]  J. Simons,et al.  Application of cholesky-like matrix decomposition methods to the evaluation of atomic orbital integrals and integral derivatives , 1989 .

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

[64]  M. Schütz,et al.  Density fitted, local Hartree–Fock treatment of NMR chemical shifts using London atomic orbitals , 2010 .

[65]  Andreas Heßelmann,et al.  Accurate Intermolecular Interaction Energies from a Combination of MP2 and TDDFT Response Theory. , 2010, Journal of chemical theory and computation.

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

[67]  Lorenz S. Cederbaum,et al.  Accurate Quantum Chemistry in Single Precision Arithmetic: Correlation Energy. , 2011, Journal of chemical theory and computation.

[68]  Francesco Aquilante,et al.  Calculation of protein-ligand interaction energies by a fragmentation approach combining high-level quantum chemistry with classical many-body effects. , 2009, The journal of physical chemistry. B.

[69]  F. Manby,et al.  Local and density fitting approximations within the short-range/long-range hybrid scheme: application to large non-bonded complexes. , 2008, Physical chemistry chemical physics : PCCP.

[70]  H. Koch,et al.  Origin invariant calculation of optical rotation without recourse to London orbitals , 2004 .

[71]  Jonas Boström,et al.  Calibration of Cholesky Auxiliary Basis Sets for Multiconfigurational Perturbation Theory Calculations of Excitation Energies. , 2010, Journal of chemical theory and computation.