Born–Oppenheimer Interatomic Forces from Simple, Local Kinetic Energy Density Functionals

Rapid calculation of Born–Oppenheimer (B–O) forces is essential for driving the so-called quantum region of a multi-scale molecular dynamics simulation. The success of density functional theory (DFT) with modern exchange-correlation approximations makes DFT an appealing choice for this role. But conventional Kohn–Sham DFT, even with various linear-scaling implementations, really is not fast enough to meet the challenge of complicated chemo-mechanical phenomena (e.g. stress-induced cracking in the presence of a solvent). Moreover, those schemes involve approximations that are difficult to check practically or to validate formally. A popular alternative, Car-Parrinello dynamics, does not guarantee motion on the B–O surface. Another approach, orbital-free DFT, is appealing but has proven difficult to implement because of the challenge of constructing reliable orbital-free (OF) approximations to the kinetic energy (KE) functional. To be maximally useful for multi-scale simulations, an OF-KE functional must be local (i.e. one-point). This requirement eliminates the two-point functionals designed to have proper linear-response behavior in the weakly inhomogeneous limit. In the face of these difficulties, we demonstrate that there is a way forward. By requiring only that the approximate functional deliver high-quality forces, by exploiting the “conjointness” hypothesis of Lee, Lee, and Parr, by enforcing a basic positivity constraint, and by parameterizing to a carefully selected, small set of molecules we are able to generate a KE functional that does a good job of describing various Hq Sim On clusters as well as CO (providing encouraging evidence of transferability). In addition to that positive result, we discuss several major negative results. First is definitive proof that the conjointness hypothesis is not correct, but nevertheless is useful. The second is the failure of a considerable variety of published KE functionals of the generalized gradient approximation type. Those functionals yield no minimum on the energy surface and give completely incorrect forces. In all cases, the problem can be traced to incorrect behavior of the functionals near the nuclei. Third, the seemingly obvious strategy of direct numerical fitting of OF-KE functional parameters to reproduce the energy surface of selected molecules is unsuccessful. The functionals that result are completely untransferable.

[1]  Lévy,et al.  Exact properties of the Pauli potential for the square root of the electron density and the kinetic energy functional. , 1988, Physical review. A, General physics.

[2]  E. Lieb.  Thomas-fermi and related theories of atoms and molecules , 1981 .

[3]  John P. Perdew,et al.  Exact differential equation for the density and ionization energy of a many-particle system , 1984 .

[4]  Robert G. Parr,et al.  Density Functional Theory of Electronic Structure , 1996 .

[5]  N. Rösch,et al.  A Critical Assessment of Density Functional Theory with Regard to Applications in Organometallic Chemistry , 1999 .

[6]  Wolfram Koch,et al.  A Chemist's Guide to Density Functional Theory , 2000 .

[7]  Mark Earl Casida,et al.  In Recent Advances in Density-Functional Methods , 1995 .

[8]  Vincenzo Barone,et al.  Physically motivated density functionals with improved performances: The modified Perdew–Burke–Ernzerhof model , 2002 .

[9]  J. C. Slater.  Suggestions from Solid‐State Theory Regarding Molecular Calculations , 1965 .

[10]  S. H. Vosko,et al.  Accurate spin-dependent electron liquid correlation energies for local spin density calculations: a critical analysis , 1980 .

[11]  E. V. Ludeña,et al.  Electronic-structure kinetic-energy functional based on atomic local-scaling transformations , 2000 .

[12]  X Blanc,et al.  Nonlinear instability of density-independent orbital-free kinetic-energy functionals. , 2005, The Journal of chemical physics.

[13]  W. Kohn,et al.  Self-Consistent Equations Including Exchange and Correlation Effects , 1965 .

[14]  P. Acharya,et al.  On the functional derivative of the kinetic energy density functional , 1982 .

[15]  A. Becke.  A multicenter numerical integration scheme for polyatomic molecules , 1988 .

[16]  Hans W. Horn,et al.  Fully optimized contracted Gaussian basis sets for atoms Li to Kr , 1992 .

[17]  C. Weizsäcker.  Zur Theorie der Kernmassen , 1935 .

[18]  K. D. Sen,et al.  Reviews of modern quantum chemistry : a celebration of the contributions of Robert G. Parr , 2002 .

[19]  E. V. Ludeña,et al.  Local‐scaling transformation version of density functional theory: Application to atoms and diatomic molecules , 1999 .

[20]  S. Trickey.  Density functional theory of many-fermion systems , 1990 .

[21]  Madden,et al.  Further orbital-free kinetic-energy functionals for ab initio molecular dynamics. , 1996, Physical review. B, Condensed matter.

[22]  B. Alder,et al.  THE GROUND STATE OF THE ELECTRON GAS BY A STOCHASTIC METHOD , 2010 .

[23]  P. García-González,et al.  KINETIC-ENERGY DENSITY FUNCTIONALS BASED ON THE HOMOGENEOUS RESPONSE FUNCTION APPLIED TO ONE-DIMENSIONAL FERMION SYSTEMS , 1998 .

[24]  Kress,et al.  Kinetic-energy functionals via Padé approximations. , 1987, Physical review. A, General physics.

[25]  E. Carter,et al.  Orbital-Free Kinetic-Energy Density Functional Theory , 2002 .

[26]  P. Hohenberg,et al.  Inhomogeneous Electron Gas , 1964 .

[27]  Ranbir Singh,et al.  J. Mol. Struct. (Theochem) , 1996 .

[28]  C. Hodges.  Quantum Corrections to the Thomas–Fermi Approximation—The Kirzhnits Method , 1973 .

[29]  R. O. Jones,et al.  The density functional formalism, its applications and prospects , 1989 .

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

[31]  J. C. Slater.  A Simplification of the Hartree-Fock Method , 1951 .

[32]  Edward Teller,et al.  On the Stability of Molecules in the Thomas-Fermi Theory , 1962 .

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

[34]  D. Chong.  Recent Advances in Density Functional Methods Part III , 2002 .

[35]  P. Salvador,et al.  Energy partitioning for "fuzzy" atoms. , 2004, The Journal of chemical physics.

[36]  P. Fuentealba.  Modified local exchange and kinetic energy functionals for atomic systems , 1997 .

[37]  L. D. Künne,et al.  Recent Developments and Applications of Modern Density Functional Theory , 1998 .

[38]  Thakkar.  Comparison of kinetic-energy density functionals. , 1992, Physical review. A, Atomic, molecular, and optical physics.

[39]  E. Fermi.  Eine statistische Methode zur Bestimmung einiger Eigenschaften des Atoms und ihre Anwendung auf die Theorie des periodischen Systems der Elemente , 1928 .

[40]  J. Perdew.  Generalized gradient approximation for the fermion kinetic energy as a functional of the density , 1992 .

[41]  Efthimios Kaxiras,et al.  Kinetic energy density functionals for non-periodic systems , 2002 .

[42]  Tomasz Adam Wesolowski,et al.  Link between the Kinetic- and Exchange-Energy Functionals in the Generalized Gradient Approximation , 2002 .

[43]  T. Gál.  First-degree homogeneous N-particle noninteracting kinetic-energy density functionals , 2001 .

[44]  Car,et al.  Unified approach for molecular dynamics and density-functional theory. , 1985, Physical review letters.

[45]  Parr,et al.  Conjoint gradient correction to the Hartree-Fock kinetic- and exchange-energy density functionals. , 1991, Physical review. A, Atomic, molecular, and optical physics.

[46]  Cleve Moler,et al.  Mathematical Handbook for Scientists and Engineers , 1961 .

[47]  A. Lembarki,et al.  Obtaining a gradient-corrected kinetic-energy functional from the Perdew-Wang exchange functional. , 1994, Physical review. A, Atomic, molecular, and optical physics.

[48]  Patricio Fuentealba,et al.  Further evidence of the conjoint correction to the local kinetic and exchange energy density functionals , 1995 .

[49]  Wang,et al.  Kinetic-energy functional of the electron density. , 1992, Physical review. B, Condensed matter.

[50]  R. Gáspár,et al.  Über eine Approximation des Hartree-Fockschen Potentials Durch eine Universelle Potentialfunktion , 1954 .

[51]  A. Schäfer,et al.  Fully optimized contracted Gaussian basis sets of triple zeta valence quality for atoms Li to Kr , 1994 .

[52]  Roy G. Gordon,et al.  Tests of nonlocal kinetic energy functionals , 1994 .

[53]  Helmut Eschrig,et al.  The fundamentals of density functional theory , 1996 .

[54]  E. V. Ludeña,et al.  KINETIC ENERGY FUNCTIONALS: HISTORY, CHALLENGES AND PROSPECTS , 2002 .

[55]  M. Levy.  Universal variational functionals of electron densities, first-order density matrices, and natural spin-orbitals and solution of the v-representability problem. , 1979, Proceedings of the National Academy of Sciences of the United States of America.

[56]  C. Herring.  Explicit estimation of ground-state kinetic energies from electron densities. , 1986, Physical review. A, General physics.

[57]  John C. Slater,et al.  Magnetic Effects and the Hartree-Fock Equation , 1951 .

[58]  Á. Nagy,et al.  A method to get an analytical expression for the non-interacting kinetic energy density functional , 2000 .

[59]  B. Y. Tong,et al.  Application of a Self-Consistent Scheme Including Exchange and Correlation Effects to Atoms , 1966 .

[60]  Eduardo V. Ludeña,et al.  Energy Density Functional Theory of Many-Electron Systems , 1990 .

[61]  L. H. Thomas.  The calculation of atomic fields , 1927, Mathematical Proceedings of the Cambridge Philosophical Society.

[62]  Axel D. Becke,et al.  Density Functional Calculations of Molecular Bond Energies , 1986 .