Analytical First-Order Molecular Properties and Forces within the Adiabatic Connection Random Phase Approximation.

The random phase approximation (RPA) is an increasingly popular method for computing molecular ground-state correlation energies within the adiabatic connection fluctuation-dissipation theorem framework of density functional theory. We present an efficient analytical implementation of first-order RPA molecular properties and nuclear forces using the resolution-of-the-identity (RI) approximation and imaginary frequency integration. The centerpiece of our approach is a variational RPA energy Lagrangian invariant under unitary transformations of occupied and virtual reference orbitals, respectively. Its construction requires the solution of a single coupled-perturbed Kohn-Sham equation independent of the number of perturbations. Energy gradients with respect to nuclear displacements and other first-order properties such as one-particle densities or dipole moments are obtained from partial derivatives of the Lagrangian. Our RPA energy gradient implementation exhibits the same [Formula: see text] scaling with system size N as a single-point RPA energy calculation. In typical applications, the cost for computing the entire gradient vector with respect to nuclear displacements is ∼5 times that of a single-point RPA energy calculation. Derivatives of the quadrature nodes and weights used for frequency integration are essential for RPA gradients with an accuracy consistent with RPA energies and can be included in our approach. The quality of RPA equilibrium structures is assessed by comparison to accurate theoretical and experimental data for covalent main group compounds, weakly bonded dimers, and transition metal complexes. RPA outperforms semilocal functionals as well as second-order Møller-Plesset (MP2) theory, which fails badly for the transition metal compounds. Dipole moments of polarizable molecules and weakly bound dimers show a similar trend. RPA harmonic vibrational frequencies are nearly of coupled cluster singles, doubles, and perturbative triples quality for a set of main group compounds. Compared to the ring-coupled cluster based implementation of Rekkedal et al. [J. Chem. Phys. 2013, 139, 081101.], our method scales better by two powers of N and supports a semilocal Kohn-Sham reference. The latter is essential for the good performance of RPA in small-gap systems.

[1]  Martin Head-Gordon,et al.  A method for two-electron Gaussian integral and integral derivative evaluation using recurrence relations , 1988 .

[2]  Kirk A Peterson,et al.  Systematically convergent basis sets for transition metals. I. All-electron correlation consistent basis sets for the 3d elements Sc-Zn. , 2005, The Journal of chemical physics.

[3]  Trygve Helgaker,et al.  Configuration-interaction energy derivatives in a fully variational formulation , 1989 .

[4]  J. Olsen,et al.  Coupled-cluster connected quadruples and quintuples corrections to the harmonic vibrational frequencies and equilibrium bond distances of HF, N(2), F(2), and CO. , 2004, The Journal of chemical physics.

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

[6]  Christof Hättig,et al.  Geometry optimizations with the coupled-cluster model CC2 using the resolution-of-the-identity approximation , 2003 .

[7]  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.

[8]  P. Taylor,et al.  An accurate ab initio quartic force field for ammonia , 1992 .

[9]  Nicholas C. Handy,et al.  Exchange functionals and potentials , 1996 .

[10]  V. Barone,et al.  Toward reliable density functional methods without adjustable parameters: The PBE0 model , 1999 .

[11]  L. Coudert,et al.  The torsional–rotational spectrum and structure of the formaldehyde dimer , 1990 .

[12]  M. Nekovee,et al.  Recent Progress in the Computational Many-Body Theory of Metal Surfaces , 2000 .

[13]  Guntram Rauhut,et al.  Transferable Scaling Factors for Density Functional Derived Vibrational Force Fields , 1995 .

[14]  T. Helgaker,et al.  Communication: Analytic gradients in the random-phase approximation. , 2013, The Journal of chemical physics.

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

[16]  Vidar R. Jensen,et al.  The accuracy of DFT-optimized geometries of functional transition metal compounds: a validation study of catalysts for olefin metathesis and other reactions in the homogeneous phase. , 2012, Dalton transactions.

[17]  J. Kussmann,et al.  Efficient distance-including integral screening in linear-scaling Møller-Plesset perturbation theory. , 2013, The Journal of chemical physics.

[18]  Trygve Helgaker,et al.  On the evaluation of derivatives of Gaussian integrals , 1992 .

[19]  R. Ahlrichs,et al.  Efficient molecular numerical integration schemes , 1995 .

[20]  Andreas Görling,et al.  Random-phase approximation correlation methods for molecules and solids , 2011 .

[21]  Georg Kresse,et al.  Assessment of correlation energies based on the random-phase approximation , 2011, 1111.0173.

[22]  A. Dalgarno,et al.  A perturbation calculation of properties of the helium iso-electronic sequence , 1958, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[23]  A. Tkatchenko,et al.  Resolution-of-identity approach to Hartree–Fock, hybrid density functionals, RPA, MP2 and GW with numeric atom-centered orbital basis functions , 2012, 1201.0655.

[24]  Filipp Furche,et al.  On the density matrix based approach to time-dependent density functional response theory , 2001 .

[25]  R. Mcweeny,et al.  The density matrix in self-consistent field theory I. Iterative construction of the density matrix , 1956, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[26]  M. Head‐Gordon,et al.  A fifth-order perturbation comparison of electron correlation theories , 1989 .

[27]  Luigi Cavallo,et al.  On the accuracy of DFT methods in reproducing ligand substitution energies for transition metal complexes in solution: the role of dispersive interactions. , 2012, Chemphyschem : a European journal of chemical physics and physical chemistry.

[28]  A. Görling,et al.  Efficient self-consistent treatment of electron correlation within the random phase approximation. , 2013, The Journal of chemical physics.

[29]  Christian Ochsenfeld,et al.  An atomic orbital-based reformulation of energy gradients in second-order Møller-Plesset perturbation theory. , 2008, The Journal of chemical physics.

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

[31]  E. Engel,et al.  Random-phase-approximation-based correlation energy functionals: benchmark results for atoms. , 2007, The Journal of chemical physics.

[32]  B. Beagley,et al.  A re-evaluation of the molecular structure of iron pentacarbonyl , 1974 .

[33]  Jan Almlöf,et al.  Laplace transform techniques in Mo/ller–Plesset perturbation theory , 1992 .

[34]  V. Barone,et al.  Accurate static polarizabilities by density functional theory: assessment of the PBE0 model , 1999 .

[35]  Peter Pulay,et al.  CAN (SEMI) LOCAL DENSITY FUNCTIONAL THEORY ACCOUNT FOR THE LONDON DISPERSION FORCES , 1994 .

[36]  E K U Gross,et al.  Correlation potentials for molecular bond dissociation within the self-consistent random phase approximation. , 2011, The Journal of chemical physics.

[37]  A. Krasheninnikov,et al.  van der Waals bonding in layered compounds from advanced density-functional first-principles calculations. , 2012, Physical review letters.

[38]  Robert L. Cook,et al.  Molecular force field and structure of hydrogen sulfide: recent microwave results , 1975 .

[39]  Joachim Sauer,et al.  Gas-phase infrared spectrum of the protonated water dimer: molecular dynamics simulation and accuracy of the potential energy surface. , 2005, Chemphyschem : a European journal of chemical physics and physical chemistry.

[40]  Thomas M Henderson,et al.  The ground state correlation energy of the random phase approximation from a ring coupled cluster doubles approach. , 2008, The Journal of chemical physics.

[41]  F. Weigend,et al.  Balanced basis sets of split valence, triple zeta valence and quadruple zeta valence quality for H to Rn: Design and assessment of accuracy. , 2005, Physical chemistry chemical physics : PCCP.

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

[43]  G. Herzberg.  Molecular Spectra and Molecular Structure IV. Constants of Diatomic Molecules , 1939 .

[44]  T. Ziegler.  A Chronicle About the Development of Electronic Structure Theories for Transition Metal Complexes , 2011 .

[45]  Matthias Scheffler,et al.  Exploring the random phase approximation: Application to CO adsorbed on Cu(111) , 2009 .

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

[47]  Georg Kresse,et al.  Making the random phase approximation to electronic correlation accurate. , 2009, The Journal of chemical physics.

[48]  Pavel Hobza,et al.  Describing Noncovalent Interactions beyond the Common Approximations: How Accurate Is the "Gold Standard," CCSD(T) at the Complete Basis Set Limit? , 2013, Journal of chemical theory and computation.

[49]  D. Truhlar,et al.  Applications and validations of the Minnesota density functionals , 2011 .

[50]  R. Grubbs,et al.  Mechanism and activity of ruthenium olefin metathesis catalysts. , 2001, Journal of the American Chemical Society.

[51]  Milan Uhlár,et al.  A study of H2S ··· NO+ complex , 2005 .

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

[53]  K. Asmis.  Structure characterization of metal oxide clusters by vibrational spectroscopy: possibilities and prospects. , 2012, Physical chemistry chemical physics : PCCP.

[54]  Per E M Siegbahn,et al.  Significant van der Waals Effects in Transition Metal Complexes. , 2010, Journal of chemical theory and computation.

[55]  J. Xue,et al.  Characterization of reactive intermediates generated during photolysis of 4-acetoxy-4-aryl-2,5-cyclohexadienones: oxenium ions and aryloxy radicals. , 2008, Journal of the American Chemical Society.

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

[57]  L. Radom,et al.  An evaluation of harmonic vibrational frequency scale factors. , 2007, The journal of physical chemistry. A.

[58]  Dmitrij Rappoport,et al.  Density functional methods for excited states: equilibrium structure and electronic spectra , 2005 .

[59]  H. Eshuis,et al.  Electron correlation methods based on the random phase approximation , 2012, Theoretical Chemistry Accounts.

[60]  H. Eshuis,et al.  A parameter-free density functional that works for noncovalent interactions , 2011 .

[61]  Georg Kresse,et al.  Assessing the quality of the random phase approximation for lattice constants and atomization energies of solids , 2010 .

[62]  Robert H. Crabtree,et al.  The organometallic chemistry of the transition metals , 1992 .

[63]  Trygve Helgaker,et al.  A numerically stable procedure for calculating Møller-Plesset energy derivatives, derived using the theory of Lagrangians , 1989 .

[64]  B. Roos,et al.  A simple method for the evaluation of the second-order-perturbation energy from external double-excitations with a CASSCF reference wavefunction , 1982 .

[65]  Björn O. Roos,et al.  Second-order perturbation theory with a complete active space self-consistent field reference function , 1992 .

[66]  Wolfram Koch,et al.  Relativistic Effects in Cationic Gold(I) Complexes: A Comparative Study of ab Initio Pseudopotential and Density Functional Methods , 1995 .

[67]  Hans Peter Lüthi,et al.  TOWARDS THE ACCURATE COMPUTATION OF PROPERTIES OF TRANSITION METAL COMPOUNDS : THE BINDING ENERGY OF FERROCENE , 1996 .

[68]  R. J. Boyd,et al.  An assessment of theoretical methods for the study of transition metal carbonyl complexes: [Cl2Rh(CO)2]− and [Cl2Rh(CO)]− as case studies , 2000 .

[69]  Isaiah Shavitt,et al.  Many-Body Methods in Chemistry and Physics: MBPT and Coupled-Cluster Theory , 2009 .

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

[71]  P. Jørgensen,et al.  Accuracy of spectroscopic constants of diatomic molecules from ab initio calculations , 2003 .

[72]  M. Bühl,et al.  Noncovalent interactions in a transition-metal triphenylphosphine complex: a density functional case study. , 2009, Inorganic chemistry.

[73]  R. L. Kuczkowski,et al.  Dipole moments of furan-argon and pyrrole-argon , 1990 .

[74]  G. Frenking,et al.  Structures and Bond Energies of the Transition Metal Hexacarbonyls M(CO)6 (M = Cr, Mo, W). A Theoretical Study , 1994 .

[75]  A. Becke.  Density-functional thermochemistry. III. The role of exact exchange , 1993 .

[76]  Michael Bühl,et al.  Geometries of Transition-Metal Complexes from Density-Functional Theory. , 2006, Journal of chemical theory and computation.

[77]  E. K. Plyler,et al.  Infrared Spectrum of Hydrogen Sulfide , 1956 .

[78]  R. Parr.  Density-functional theory of atoms and molecules , 1989 .

[79]  M. Plesset,et al.  Note on an Approximation Treatment for Many-Electron Systems , 1934 .

[80]  Burke,et al.  Generalized Gradient Approximation Made Simple. , 1996, Physical review letters.

[81]  Christof Hättig,et al.  Optimization of auxiliary basis sets for RI-MP2 and RI-CC2 calculations: Core–valence and quintuple-ζ basis sets for H to Ar and QZVPP basis sets for Li to Kr , 2005 .

[82]  Parr,et al.  Development of the Colle-Salvetti correlation-energy formula into a functional of the electron density. , 1988, Physical review. B, Condensed matter.

[83]  Xavier Gonze,et al.  Accurate density functionals: Approaches using the adiabatic-connection fluctuation-dissipation theorem , 2002 .

[84]  S. Grimme,et al.  A consistent and accurate ab initio parametrization of density functional dispersion correction (DFT-D) for the 94 elements H-Pu. , 2010, The Journal of chemical physics.

[85]  Poul Jørgensen,et al.  Geometrical derivatives of energy surfaces and molecular properties , 1986 .

[86]  Filipp Furche,et al.  Basis set convergence of molecular correlation energy differences within the random phase approximation. , 2012, The Journal of chemical physics.

[87]  J. Olsen,et al.  Molecular equilibrium structures from experimental rotational constants and calculated vibration–rotation interaction constants , 2002 .

[88]  Correlation potential in density functional theory at the GWA level: Spherical atoms , 2007, cond-mat/0703819.

[89]  Robert J. Buenker,et al.  Individualized configuration selection in CI calculations with subsequent energy extrapolation , 1974 .

[90]  Hans W. Horn,et al.  ELECTRONIC STRUCTURE CALCULATIONS ON WORKSTATION COMPUTERS: THE PROGRAM SYSTEM TURBOMOLE , 1989 .

[91]  Dmitrij Rappoport,et al.  Property-optimized gaussian basis sets for molecular response calculations. , 2010, The Journal of chemical physics.

[92]  D. Harding,et al.  Incipient chemical bond formation of Xe to a cationic silicon cluster: Vibrational spectroscopy and structure of the Si4Xe+ complex , 2013 .

[93]  Julian Yarkony,et al.  Fast computation of molecular random phase approximation correlation energies using resolution of the identity and imaginary frequency integration. , 2010, The Journal of chemical physics.

[94]  Donald G Truhlar,et al.  Density functionals with broad applicability in chemistry. , 2008, Accounts of chemical research.

[95]  Andreas Görling,et al.  Random phase approximation correlation energies with exact Kohn–Sham exchange , 2010 .

[96]  Jan Almlöf,et al.  Elimination of energy denominators in Møller—Plesset perturbation theory by a Laplace transform approach , 1991 .

[97]  A. Haaland,et al.  Organometallic compounds studied by gas-phase electron diffraction. , 1975, Topics in current chemistry.

[98]  Marco Häser,et al.  Møller-Plesset (MP2) perturbation theory for large molecules , 1993 .

[99]  Stefan Grimme,et al.  Semiempirical GGA‐type density functional constructed with a long‐range dispersion correction , 2006, J. Comput. Chem..

[100]  Ian M. Mills,et al.  Anharmonic force constant calculations , 1972 .

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

[102]  Martin Head-Gordon,et al.  Scaled opposite-spin second order Møller-Plesset correlation energy: an economical electronic structure method. , 2004, The Journal of chemical physics.

[103]  Gustavo E Scuseria,et al.  Theoretical study of CeO2 and Ce2O3 using a screened hybrid density functional. , 2006, The Journal of chemical physics.

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

[105]  R. Sternheimer,et al.  ELECTRONIC POLARIZABILITIES OF IONS FROM THE HARTREE-FOCK WAVE FUNCTIONS , 1954 .

[106]  Carsten A. Ullrich,et al.  Time-Dependent Density-Functional Theory: Concepts and Applications , 2012 .

[107]  T. Iijima,et al.  Nickel tetracarbonyl, Ni(CO)4. I. Molecular structure by gaseous electron diffraction. II. Refinement of quadratic force field , 1979 .

[108]  Filipp Furche,et al.  An efficient implementation of second analytical derivatives for density functional methods , 2002 .

[109]  Benny G. Johnson,et al.  Kohn—Sham density-functional theory within a finite basis set , 1992 .

[110]  J. Boyd.  Exponentially convergent Fourier-Chebshev quadrature schemes on bounded and infinite intervals , 1987 .

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

[112]  M. Canagaratna,et al.  Determination of the dipole moment of H3N-SO3 in the gas phase , 1997 .

[113]  John S. Muenter,et al.  THE STRUCTURE OF WATER DIMER FROM MOLECULAR BEAM ELECTRIC RESONANCE SPECTROSCOPY: PARTIALLY DEUTERATED DIMERS , 1977 .

[114]  Filipp Furche,et al.  Adiabatic time-dependent density functional methods for excited state properties , 2002 .

[115]  Filipp Furche,et al.  Molecular tests of the random phase approximation to the exchange-correlation energy functional , 2001 .

[116]  A. Savin,et al.  Range-separated density-functional theory with the random-phase approximation: Detailed formalism and illustrative applications , 2010, 1006.2061.

[117]  M. Hellgren,et al.  Correlation energy functional and potential from time-dependent exact-exchange theory. , 2009, The Journal of chemical physics.