Meta-generalized gradient approximation: explanation of a realistic nonempirical density functional.

Tao, Perdew, Staroverov, and Scuseria (TPSS) have constructed a nonempirical meta-generalized gradient approximation (meta-GGA) [Phys. Rev. Lett. 91, 146401 (2003)] for the exchange-correlation energy, imposing exact constraints relevant to the paradigm densities of condensed matter physics and quantum chemistry. Results of their extensive tests on molecules, solids, and solid surfaces are encouraging, suggesting that this density functional achieves uniform accuracy for diverse properties and systems. In the present work, this functional is explained and details of its construction are presented. In particular, the functional is constructed to yield accurate energies under uniform coordinate scaling to the low-density or strong-interaction limit. Its nonlocality is displayed by plotting the factor F(xc) that gives the enhancement relative to the local density approximation for exchange. We also discuss an apparently harmless order-of-limits problem in the meta-GGA. The performance of this functional is investigated for exchange and correlation energies and shell-removal energies of atoms and ions. Non-self-consistent molecular atomization energies and bond lengths of the TPSS meta-GGA, calculated with GGA orbitals and densities, agree well with those calculated self-consistently. We suggest that satisfaction of additional exact constraints on higher rungs of a ladder of density functional approximations can lead to further progress.

[1]  Jianmin Tao,et al.  Tests of a ladder of density functionals for bulk solids and surfaces , 2004 .

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

[3]  J. Perdew,et al.  Properties of the exchange hole under an appropriate coordinate transformation , 2003 .

[4]  A. Becke.  A real-space model of nondynamical correlation , 2003 .

[5]  Marcel Swart,et al.  Accuracy of geometries: influence of basis set, exchange–correlation potential, inclusion of core electrons, and relativistic corrections , 2003 .

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

[7]  P. Hyldgaard,et al.  Van der Waals density functional for layered structures. , 2003, Physical review letters.

[8]  J. Perdew,et al.  Two avenues to self-interaction correction within Kohn—Sham theory: unitary invariance is the shortcut , 2003, physics/0301039.

[9]  J. Perdew,et al.  Simple iterative construction of the optimized effective potential for orbital functionals, including exact exchange. , 2002, Physical review letters.

[10]  F. Reboredo,et al.  Exact-Exchange Density Functional Theory applied to a strongly inhomogeneous electron gas , 2002, cond-mat/0209006.

[11]  F. Nogueira,et al.  A primer in density functional theory , 2003 .

[12]  R. Armiento,et al.  Subsystem functionals in density-functional theory: Investigating the exchange energy per particle , 2002 .

[13]  A. Daniel Boese,et al.  New exchange-correlation density functionals: The role of the kinetic-energy density , 2002 .

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

[15]  Jianmin Tao,et al.  An accurate MGGA-based hybrid exchange-correlation functional , 2002 .

[16]  M. Grüning,et al.  The failure of generalized gradient approximations (GGAs) and meta-GGAs for the two-center three-electron bonds in He-2(+), (H2O)(2)(+), and (NH3)(2)(+) , 2001 .

[17]  Jianmin Tao,et al.  Exchange energy density of an atom as a functional of the electron density , 2001 .

[18]  N. Handy,et al.  Left-right correlation energy , 2001 .

[19]  M Nekovee,et al.  Quantum Monte Carlo analysis of exchange and correlation in the strongly inhomogeneous electron gas. , 2000, Physical review letters.

[20]  G. Scuseria,et al.  Performance of recently developed kinetic energy density functionals for the calculation of hydrogen binding strengths and hydrogen-bonded structures , 2000 .

[21]  M. Seidl,et al.  Density functionals for the strong-interaction limit , 2000 .

[22]  Perdew,et al.  Simulation of all-order density-functional perturbation theory, using the second order and the strong-correlation limit , 2000, Physical review letters.

[23]  J. Perdew,et al.  Evaluating density functional performance for the quasi-two-dimensional electron gas , 2000 .

[24]  G. Scuseria,et al.  The meta-GGA functional: Thermochemistry with a kinetic energy density dependent exchange-correlation functional , 2000 .

[25]  Walter Kohn,et al.  Nobel Lecture: Electronic structure of matter-wave functions and density functionals , 1999 .

[26]  John P. Perdew,et al.  Accurate Density Functional with Correct Formal Properties: A Step Beyond the Generalized Gradient Approximation , 1999 .

[27]  J. Nørskov,et al.  Improved adsorption energetics within density-functional theory using revised Perdew-Burke-Ernzerhof functionals , 1999 .

[28]  Michael Seidl,et al.  Strictly correlated electrons in density-functional theory , 1999 .

[29]  John P. Perdew,et al.  Molecular and solid‐state tests of density functional approximations: LSD, GGAs, and meta‐GGAs , 1999 .

[30]  K. Burke,et al.  Unambiguous exchange-correlation energy density , 1998 .

[31]  Fred A. Hamprecht,et al.  Development and assessment of new exchange-correlation functionals , 1998 .

[32]  Axel D. Becke,et al.  A new inhomogeneity parameter in density-functional theory , 1998 .

[33]  Gustavo E. Scuseria,et al.  A novel form for the exchange-correlation energy functional , 1998 .

[34]  Edge Electron Gas , 1998, cond-mat/9806154.

[35]  Kieron Burke,et al.  Nonlocality of the density functional for exchange and correlation: Physical origins and chemical consequences , 1998 .

[36]  M. Levy,et al.  CONNECTIONS BETWEEN HIGH-DENSITY SCALING LIMITS OF DFT CORRELATION ENERGIES AND SECOND-ORDER Z-1 QUANTUM CHEMISTRY CORRELATION ENERGY , 1998 .

[37]  E. J. Baerends,et al.  Kohn-Sham potentials and exchange and correlation energy densities from one- and two-electron density matrices for Li2, N2, F2. , 1998 .

[38]  K. Burke,et al.  Perdew, Burke, and Ernzerhof Reply: , 1998 .

[39]  Yingkai Zhang,et al.  Comment on “Generalized Gradient Approximation Made Simple” , 1998 .

[40]  M. Scheffler,et al.  Pseudopotential study of binding properties of solids within generalized gradient approximations: The role of core-valence exchange-correlation. , 1997, cond-mat/9707242.

[41]  Yingkai Zhang,et al.  Describing van der Waals interaction in diatomic molecules with generalized gradient approximations: The role of the exchange functional , 1997 .

[42]  K. Burke,et al.  Generalized Gradient Approximation Made Simple [Phys. Rev. Lett. 77, 3865 (1996)] , 1997 .

[43]  W. Thiel,et al.  A nonlocal correlation energy density functional from a Coulomb hole model , 1997 .

[44]  Svendsen,et al.  Gradient expansion of the exchange energy from second-order density response theory. , 1996, Physical review. B, Condensed matter.

[45]  Wang,et al.  Generalized gradient approximation for the exchange-correlation hole of a many-electron system. , 1996, Physical review. B, Condensed matter.

[46]  R. Ahlrichs,et al.  STABILITY ANALYSIS FOR SOLUTIONS OF THE CLOSED SHELL KOHN-SHAM EQUATION , 1996 .

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

[48]  Burke,et al.  Escaping the symmetry dilemma through a pair-density interpretation of spin-density functional theory. , 1995, Physical review. A, Atomic, molecular, and optical physics.

[49]  D. Salahub,et al.  Determining and extending the domain of exchange and correlation functionals , 1995 .

[50]  M. Frisch,et al.  Ab Initio Calculation of Vibrational Absorption and Circular Dichroism Spectra Using Density Functional Force Fields , 1994 .

[51]  Perdew,et al.  Size-dependent ionization energy of a metallic cluster: Resolution of the classical image-potential paradox. , 1994, Physical review. B, Condensed matter.

[52]  Görling,et al.  Exact Kohn-Sham scheme based on perturbation theory. , 1994, Physical review. A, Atomic, molecular, and optical physics.

[53]  Perdew,et al.  Tight bound and convexity constraint on the exchange-correlation-energy functional in the low-density limit, and other formal tests of generalized-gradient approximations. , 1993, Physical review. B, Condensed matter.

[54]  Singh,et al.  Erratum: Atoms, molecules, solids, and surfaces: Applications of the generalized gradient approximation for exchange and correlation , 1993, Physical review. B, Condensed matter.

[55]  Görling,et al.  Correlation-energy functional and its high-density limit obtained from a coupling-constant perturbation expansion. , 1993, Physical review. B, Condensed matter.

[56]  Davidson,et al.  Ground-state correlation energies for atomic ions with 3 to 18 electrons. , 1993, Physical review. A, Atomic, molecular, and optical physics.

[57]  Wang,et al.  Accurate and simple analytic representation of the electron-gas correlation energy. , 1992, Physical review. B, Condensed matter.

[58]  Benny G. Johnson,et al.  An investigation of the performance of a hybrid of Hartree‐Fock and density functional theory , 1992 .

[59]  Görling,et al.  Requirements for correlation energy density functionals from coordinate transformations. , 1992, Physical review. A, Atomic, molecular, and optical physics.

[60]  Lévy,et al.  Density-functional exchange correlation through coordinate scaling in adiabatic connection and correlation hole. , 1991, Physical review. A, Atomic, molecular, and optical physics.

[61]  Wang,et al.  Spin scaling of the electron-gas correlation energy in the high-density limit. , 1991, Physical review. B, Condensed matter.

[62]  Lévy,et al.  Nonuniform coordinate scaling requirements for exchange-correlation energy. , 1990, Physical review. A, Atomic, molecular, and optical physics.

[63]  A. Becke,et al.  Exchange holes in inhomogeneous systems: A coordinate-space model. , 1989, Physical review. A, General physics.

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

[65]  A. Becke,et al.  Density-functional exchange-energy approximation with correct asymptotic behavior. , 1988, Physical review. A, General physics.

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

[67]  Parr,et al.  Phase-space approach to the exchange-energy functional of density-functional theory. , 1986, Physical review. A, General physics.

[68]  J. Perdew,et al.  Accurate and simple density functional for the electronic exchange energy: Generalized gradient approximation. , 1986, Physical review. B, Condensed matter.

[69]  J. Perdew,et al.  Density-functional approximation for the correlation energy of the inhomogeneous electron gas. , 1986, Physical review. B, Condensed matter.

[70]  J. Perdew,et al.  Accurate density functional for the energy: Real-space cutoff of the gradient expansion for the exchange hole. , 1985, Physical review letters.

[71]  J. Perdew,et al.  Hellmann-Feynman, virial, and scaling requisites for the exact universal density functionals. Shape of the correlation potential and diamagnetic susceptibility for atoms. , 1985, Physical review. A, General physics.

[72]  D. Langreth,et al.  Beyond the local-density approximation in calculations of ground-state electronic properties , 1983 .

[73]  J. Perdew,et al.  Density-Functional Theory for Fractional Particle Number: Derivative Discontinuities of the Energy , 1982 .

[74]  John P. Perdew,et al.  Density-functional theory of the correlation energy in atoms and ions: A simple analytic model and a challenge , 1981 .

[75]  A. Zunger,et al.  Self-interaction correction to density-functional approximations for many-electron systems , 1981 .

[76]  John P. Perdew,et al.  Theory of nonuniform electronic systems. I. Analysis of the gradient approximation and a generalization that works , 1980 .

[77]  G. L. Oliver,et al.  Spin-density gradient expansion for the kinetic energy , 1979 .

[78]  M. Brack,et al.  On the extended Thomas-Fermi approximation to the kinetic energy density , 1976 .

[79]  M. Rasolt,et al.  Exchange and correlation energy of an inhomogeneous electron gas at metallic densities , 1976 .

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

[81]  Renato Colle,et al.  Approximate calculation of the correlation energy for the closed shells , 1975 .

[82]  John C. Slater,et al.  The self-consistent field for molecules and solids , 1974 .

[83]  Shang‐keng Ma,et al.  Correlation Energy of an Electron Gas with a Slowly Varying High Density , 1968 .

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

[85]  Eugene P. Wigner,et al.  Effects of the Electron Interaction on the Energy Levels of Electrons in Metals , 1938 .

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