Performance and Scope of Perturbative Corrections to Random-Phase Approximation Energies.

It has been suspected since the early days of the random-phase approximation (RPA) that corrections to RPA correlation energies result mostly from short-range correlation effects and are thus amenable to perturbation theory. Here we test this hypothesis by analyzing formal and numerical results for the most common beyond-RPA perturbative corrections, including the bare second-order exchange (SOX), second-order screened exchange (SOSEX), and approximate exchange kernel (AXK) methods. Our analysis is facilitated by efficient and robust algorithms based on the resolution-of-the-identity (RI) approximation and numerical frequency integration, which enable benchmark beyond-RPA calculations on medium- and large-size molecules with size-independent accuracy. The AXK method systematically improves upon RPA, SOX, and SOSEX for reaction barrier heights, reaction energies, and noncovalent interaction energies of main-group compounds. The improved accuracy of AXK compared with SOX and SOSEX is attributed to stronger screening of bare SOX in AXK. For reactions involving transition-metal compounds, particularly 3d transition-metal dimers, the AXK correction is too small and can even have the wrong sign. These observations are rationalized by a measure α̅ of the effective coupling strength for beyond-RPA correlation. When the effective coupling strength increases beyond a critical α̅ value of approximately 0.5, the RPA errors increase rapidly and perturbative corrections become unreliable. Thus, perturbation theory can systematically correct RPA but only for systems and properties qualitatively well captured by RPA, as indicated by small α̅ values.

[1]  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 .

[2]  John P. Perdew,et al.  The exchange-correlation energy of a metallic surface , 1975 .

[3]  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.

[4]  John P. Perdew,et al.  Density functional for short-range correlation: Accuracy of the random-phase approximation for isoelectronic energy changes , 2000 .

[5]  Trygve Helgaker,et al.  Basis-set convergence of correlated calculations on water , 1997 .

[6]  János G. Ángyán,et al.  Correlation Energy Expressions from the Adiabatic-Connection Fluctuation-Dissipation Theorem Approach. , 2011, Journal of chemical theory and computation.

[7]  G. Scuseria,et al.  Comparative assessment of a new nonempirical density functional: Molecules and hydrogen-bonded complexes , 2003 .

[8]  A. Fetter,et al.  Quantum Theory of Many-Particle Systems , 1971 .

[9]  David,et al.  Gaussian basis sets for use in correlated molecular calculations . Ill . The atoms aluminum through argon , 1999 .

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

[11]  Gustavo E. Scuseria,et al.  Renormalized Second-order Perturbation Theory for The Electron Correlation Energy: Concept, Implementation, and Benchmarks , 2012, 1212.3674.

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

[13]  B. Lundqvist,et al.  Exchange and correlation in atoms, molecules, and solids by the spin-density-functional formalism , 1976 .

[14]  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.

[15]  Andreas Savin,et al.  Adiabatic-connection fluctuation-dissipation density-functional theory based on range separation. , 2008, Physical review letters.

[16]  Alfonso Baldereschi,et al.  Mean-value point and dielectric properties of semiconductors and insulators , 1978 .

[17]  Weitao Yang,et al.  Many-electron self-interaction error in approximate density functionals. , 2006, The Journal of chemical physics.

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

[19]  Sree Ganesh Balasubramani,et al.  Random-Phase Approximation Methods. , 2017, Annual review of physical chemistry.

[20]  L. Dagum,et al.  OpenMP: an industry standard API for shared-memory programming , 1998 .

[21]  Trygve Helgaker,et al.  Basis-set convergence in correlated calculations on Ne, N2, and H2O , 1998 .

[22]  A. D. McLACHLAN,et al.  Time-Dependent Hartree—Fock Theory for Molecules , 1964 .

[23]  John P. Perdew,et al.  Exchange-correlation energy of a metallic surface: Wave-vector analysis , 1977 .

[24]  Vamsee K Voora,et al.  Variational generalized Kohn-Sham approach combining the random-phase-approximation and Green's-function methods , 2019, Physical Review A.

[25]  Andreas Hansen,et al.  A look at the density functional theory zoo with the advanced GMTKN55 database for general main group thermochemistry, kinetics and noncovalent interactions. , 2017, Physical chemistry chemical physics : PCCP.

[26]  T. Van Voorhis,et al.  Fluctuation-dissipation theorem density-functional theory. , 2005, The Journal of chemical physics.

[27]  Benjamin G. Janesko,et al.  Long-range-corrected hybrid density functionals including random phase approximation correlation: application to noncovalent interactions. , 2009, The Journal of chemical physics.

[28]  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.

[29]  C. Ochsenfeld,et al.  Efficient calculation of beyond RPA correlation energies in the dielectric matrix formalism. , 2018, The Journal of chemical physics.

[30]  K. Burke,et al.  Rationale for mixing exact exchange with density functional approximations , 1996 .

[31]  Stefan Grimme,et al.  A General Database for Main Group Thermochemistry, Kinetics, and Noncovalent Interactions - Assessment of Common and Reparameterized (meta-)GGA Density Functionals. , 2010, Journal of chemical theory and computation.

[32]  A. Savin,et al.  A Systematic Failing of Current Density Functionals: Overestimation of Two-Center Three-Electron Bonding Energies , 1998 .

[33]  Marco Häser,et al.  Improvements on the direct SCF method , 1989 .

[34]  F. Weigend,et al.  Gaussian basis sets of quadruple zeta valence quality for atoms H–Kr , 2003 .

[35]  Georg Kresse,et al.  Accurate bulk properties from approximate many-body techniques. , 2009, Physical review letters.

[36]  Georg Jansen,et al.  Dielectric Matrix Formulation of Correlation Energies in the Random Phase Approximation: Inclusion of Exchange Effects. , 2016, Journal of chemical theory and computation.

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

[38]  J. Perdew,et al.  A simple but fully nonlocal correction to the random phase approximation. , 2011, The Journal of chemical physics.

[39]  K. Hirao,et al.  Transition state barrier height for the reaction H2CO→H2+CO studied by multireference Mo/ller–Plesset perturbation theory , 1997 .

[40]  Pair distribution function of the spin-polarized electron gas: A first-principles analytic model for all uniform densities , 2002, cond-mat/0206147.

[41]  Sara K. Avrantinis,et al.  Comparison of Relative Activation Energies Obtained by Density Functional Theory and the Random Phase Approximation for Several Claisen Rearrangements. , 2016, The journal of physical chemistry. B.

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

[43]  Eunji Sim,et al.  Understanding and reducing errors in density functional calculations. , 2012, Physical review letters.

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

[45]  John P. Perdew,et al.  DENSITY-FUNCTIONAL CORRECTION OF RANDOM-PHASE-APPROXIMATION CORRELATION WITH RESULTS FOR JELLIUM SURFACE ENERGIES , 1999 .

[46]  G. Scuseria,et al.  Climbing the density functional ladder: nonempirical meta-generalized gradient approximation designed for molecules and solids. , 2003, Physical review letters.

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

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

[49]  Weitao Yang,et al.  A challenge for density functionals: Self-interaction error increases for systems with a noninteger number of electrons , 1998 .

[50]  Adrienn Ruzsinszky,et al.  The RPA Atomization Energy Puzzle. , 2010, Journal of chemical theory and computation.

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

[52]  Filipp Furche,et al.  The performance of semilocal and hybrid density functionals in 3d transition-metal chemistry. , 2006, The Journal of chemical physics.

[53]  A. Prykarpatsky,et al.  The Correlation Energy of Electron Gas at High Density , 2002 .

[54]  M. Sierka,et al.  Turbomole , 2014 .

[55]  D. Tew.  Communication: Quasi-robust local density fitting. , 2018, The Journal of chemical physics.

[56]  Dario Rocca,et al.  Improving the Efficiency of Beyond-RPA Methods within the Dielectric Matrix Formulation: Algorithms and Applications to the A24 and S22 Test Sets. , 2017, Journal of chemical theory and computation.

[57]  Gustavo E. Scuseria,et al.  The connection between self-interaction and static correlation: a random phase approximation perspective , 2010 .

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

[59]  J. Almlöf,et al.  Principles for a direct SCF approach to LICAO–MOab‐initio calculations , 1982 .

[60]  Dongho Kim,et al.  Synthesis of highly twisted and fully π-conjugated porphyrinic oligomers. , 2015, Journal of the American Chemical Society.

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

[62]  K. Burke,et al.  Describing static correlation in bond dissociation by Kohn-Sham density functional theory. , 2004, The Journal of chemical physics.

[63]  P. E. Trevisanutto,et al.  Optical Spectra of Solids Obtained by Time-Dependent Density-Functional Theory with the Jellium-with-Gap-Model Exchange-Correlation Kernel , 2012, 1210.7149.

[64]  D. Pines.  A COLLECTIVE DESCRIPTION OF ELECTRON INTERACTIONS: IV. ELECTRON INTERACTION IN METALS , 1953 .

[65]  Evert Jan Baerends,et al.  Self-consistent molecular hartree—fock—slater calculations. IV. On electron densities, spectroscopic constants and proton affinities of some small molecules , 1976 .

[66]  Weitao Yang,et al.  Insights into Current Limitations of Density Functional Theory , 2008, Science.

[67]  Jun Wang,et al.  SUCCESSFUL TEST OF A SEAMLESS VAN DER WAALS DENSITY FUNCTIONAL , 1999 .

[68]  Matthias Scheffler,et al.  Random-phase approximation and its applications in computational chemistry and materials science , 2012, Journal of Materials Science.

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

[70]  Simple dynamic exchange-correlation kernel of a uniform electron gas , 2007, 0706.1919.

[71]  Weitao Yang,et al.  Delocalization errors in density functionals and implications for main-group thermochemistry. , 2008, The Journal of chemical physics.

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

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

[74]  Thomas M Henderson,et al.  Long-range-corrected hybrids including random phase approximation correlation. , 2009, The Journal of chemical physics.

[75]  Wang,et al.  Correlation hole of the spin-polarized electron gas, with exact small-wave-vector and high-density scaling. , 1991, Physical review. B, Condensed matter.

[76]  Thomas Olsen,et al.  Extending the random-phase approximation for electronic correlation energies: The renormalized adiabatic local density approximation , 2012, 1208.0419.

[77]  D. Freeman.  Coupled-cluster expansion applied to the electron gas: Inclusion of ring and exchange effects , 1977 .

[78]  L. Hedin,et al.  A local exchange-correlation potential for the spin polarized case. i , 1972 .

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

[80]  J. Ángyán,et al.  Improving the accuracy of ground-state correlation energies within a plane-wave basis set: The electron-hole exchange kernel. , 2016, The Journal of chemical physics.

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

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

[83]  Nathan J. DeYonker,et al.  Multireference Character for 3d Transition-Metal-Containing Molecules. , 2012, Journal of chemical theory and computation.

[84]  Trygve Helgaker,et al.  Spin flipping in ring-coupled-cluster-doubles theory , 2011 .

[85]  L. Overman,et al.  Diastereoselective Coupling of Chiral Acetonide Trisubstituted Radicals with Alkenes. , 2016, Chemistry.

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

[87]  Charles L. Lawson,et al.  Basic Linear Algebra Subprograms for Fortran Usage , 1979, TOMS.

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

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

[90]  R. Lata,et al.  SOME ESTIMATES OF NORMS OF RANDOM MATRICES , 2004 .

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

[92]  Markus Reiher,et al.  The Delicate Balance of Static and Dynamic Electron Correlation. , 2016, Journal of chemical theory and computation.

[93]  K. Thygesen,et al.  Accurate Ground State Energies of Solids and Molecules from Time Dependent Density Functional Theory , 2014, 1405.0745.

[94]  J F Dobson,et al.  Cohesive properties and asymptotics of the dispersion interaction in graphite by the random phase approximation. , 2010, Physical review letters.

[95]  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.

[96]  Harbhajan Singh.  Electron correlations at metallic densities , 1975 .

[97]  Adrienn Ruzsinszky,et al.  Density functionals that are one- and two- are not always many-electron self-interaction-free, as shown for H2+, He2+, LiH+, and Ne2+. , 2007, The Journal of chemical physics.

[98]  Correlation energies of inhomogeneous many-electron systems , 2001, cond-mat/0111447.

[99]  Weitao Yang,et al.  Challenges for density functional theory. , 2012, Chemical reviews.

[100]  L. Reining,et al.  Electronic excitations: density-functional versus many-body Green's-function approaches , 2002 .

[101]  Donald G Truhlar,et al.  Benchmark database of barrier heights for heavy atom transfer, nucleophilic substitution, association, and unimolecular reactions and its use to test theoretical methods. , 2005, The journal of physical chemistry. A.

[102]  Igor Ying Zhang,et al.  Insight into organic reactions from the direct random phase approximation and its corrections. , 2015, The Journal of chemical physics.

[103]  D. Truhlar,et al.  The DBH24/08 Database and Its Use to Assess Electronic Structure Model Chemistries for Chemical Reaction Barrier Heights. , 2009, Journal of chemical theory and computation.

[104]  H. Bethe,et al.  A Relativistic equation for bound state problems , 1951 .

[105]  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.

[106]  G. Galli,et al.  Large scale GW calculations. , 2015, Journal of chemical theory and computation.

[107]  Kirk A. Peterson,et al.  Explicitly correlated composite thermochemistry of transition metal species. , 2013, The Journal of chemical physics.

[108]  Filipp Furche,et al.  Communication: Random phase approximation renormalized many-body perturbation theory. , 2013, The Journal of chemical physics.

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

[110]  J. Dobson,et al.  Energy-optimized local exchange-correlation kernel for the electron gas: Application to van der Waals forces , 2000 .