Authors of successful density-functionals have some of the most cited articles in both the physics and chemistry literature since 1999 or earlier. This can be confirmed by examining CAS Science Spotlight for chemistry and S. Redner's article for physics. (Thanks to John Perdew for pointing out Redner's article.)

My goal for the following table is to include every major modern functional and at least the historically more important functionals. Should you find an error in this table or would like to draw my attention to an important omission, please contact me at Mark.Casida@UJF-Grenoble.FR. ^aThe names of functionals are confusing at best. Here, I have tried to use the most common names in the literature. Usually this means just the author's initials, perhaps supplemented with a year if the author has more than one functional, or (in the case of hybrid functionals) the number of fitting parameters used in combining Hartree-Fock exchange with the GGA functional. Especially popular functionals have been put in italics.

^bClick on the name of a functional to go to the full reference.

In reverse chronological order:

[B2PLYP]

S. Grimme, J. Chem. Phys. 124, 034108 (2006).

"Semiempirical hybrid density functional with perturbative second-order correlation"

COMMENT: Portions of both Hartree-Fock exchange and MP2 correlation are included in the recipe for this functional.

ABSTRACT : A new hybrid density functional for general chemistry applications is proposed. It is based on a

mixing of standard generalized gradient approximations (GGAs) for exchange by Becke (B) and for

correlation by Lee, Yang, and Parr (LYP) with Hartree-Fock (HF) exchange and a perturbative

second-order correlation part (PT2) that is obtained from the Kohn-Sham (GGA) orbitals and

eigenvalues. This virtual orbital-dependent functional contains only two global parameters that

describe the mixture of HF and GGA exchange .ax. and of the PT2 and GGA correlation (c),

respectively. The parameters are obtained in a least-squares-fit procedure to the G2/97 set of heat

of formations. Opposed to conventional hybrid functionals, the optimum ax is found to be quite large

(53% with c=27%) which at least in part explains the success for many problematic molecular

systems compared to conventional approaches. The performance of the new functional termed

B2-PLYP is assessed by the G2/97 standard benchmark set, a second test suite of atoms, molecules,

and reactions that are considered as electronically very difficult (including transition-metal

compounds, weakly bonded complexes, and reaction barriers) and comparisons with other hybrid

functionals of GGA and meta-GGA types. According to many realistic tests, B2-PLYP can be

regarded as the best general purpose density functional for molecules (e.g., a mean absolute

deviation for the two test sets of only 1.8 and 3.2 kcal/mol compared to about 3 and 5 kcal/mol,

respectively, for the best other density functionals). Very importantly, also the maximum and

minium errors (outliers) are strongly reduced (by about 10–20 kcal/mol). Furthermore, very good

results are obtained for transition state barriers but unlike previous attempts at such a good

description, this definitely comes not at the expense of equilibrium properties. Preliminary

calculations of the equilibrium bond lengths and harmonic vibrational frequencies for diatomic

molecules and transition-metal complexes also show very promising results. The uniformity with

which B2-PLYP improves for a wide range of chemical systems emphasizes the need of .virtual.

orbital-dependent terms that describe nonlocal electron correlation in accurate exchange-correlation

functionals. From a practical point of view, the new functional seems to be very robust and it is thus

suggested as an efficient quantum chemical method of general purpose.

reprint

This was followed up by an interesting application to TDDFT:

S. Grimme and F. Neese, J. Chem. Phys. 127, 154116 (2007).

"Double-hybrid density functional theory for excited electronic states

of molecules"

ABSTRACT : Double-hybrid density functionals are based on a mixing of standard generalized gradient

approximations (GGAs) for exchange and correlation with Hartree-Fock (HF) exchange and a

perturbative second-order correlation part (PT2) that is obtained from the Kohn-Sham (GGA)

orbitals and eigenvalues. This virtual orbital-dependent functional (dubbed B2PLYP) contains only

two empirical parameters that describe the mixture of HF and GGA exchange (a_x) and of the PT2

and GGA correlation (a_c), respectively. Extensive testing has recently demonstrated the outstanding

accuracy of this approach for various ground state problems in general chemistry applications. The

method is extended here without any further empirical adjustments to electronically excited states in

the framework of time-dependent density functional theory (TD-DFT) or the closely related

Tamm-Dancoff approximation (TDA-DFT). In complete analogy to the ground state treatment, a

scaled second-order perturbation correction to configuration interaction with singles (CIS(D)) wave

functions developed some years ago by Head-Gordon et al. (Chem. Phys. Lett. 219, 21 (1994)) is

computed on the basis of density functional data and added to the TD(A)-DFT/GGA excitation

energy. The method is implemented by applying the resolution of the identity approximation and the

efficiency of the code is discussed. Extensive tests for a wide variety of molecules and excited states

(of singlet, triplet, and doublet multiplicities)including electronic spectra are presented. In general,

rather accurate excitation energies (deviations from reference data typically <0.2 eV) are obtained

that are mostly better than those from standard functionals. Still, systematic errors are obtained for

Rydberg (too low on average by about 0.3 eV) and charge-transfer transitions but due to the

relatively large ax parameter (0.53), B2PLYP outperforms most other functionals in this respect.

Compared to conventional HF-based CIS(D), the method is more robust in electronically complex

situations due to the implicit account of static correlation effects by the GGA parts. The (D)

correction often works in the right direction and compensates for the overestimation of the transition

energy at the TD level due to the elevated fraction of HF exchange in the hybrid GGA part. Finally,

the limitations of the method are discussed for challenging systems such as transition metal

complexes, cyanine dyes, and multireference cases.

reprint

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[CAM-B3LYP]

T. Yanai, D.P. Tew, and N.C. Handy, Chem. Phys. Lett. 393, 51 (2004).

"A new hybrid exchange-correlation functional using the Coulomb-attenuating method (CAM-B3LYP)"

The Coulomb attentuating method proposes to split the electron repulsion into a short-range part to be treated by DFT and a long-range part to be treated by HF. Thus 1/r12 = [(1-[alpha+beta erf(mu r12)])/r12]_{DFT} + [(alpha + beta erf(mu r12))/r12]_{HF} . This helps to overcome a number of problems in DFT having to do with long-range interactions.

ABSTRACT : A new hybrid exchange-correlation functional named CAM-B3LYP is proposed. It combines the hybrid qualities of B3LYP and the long-range correction presented by Tawada et al. [J. Chem. Phys., in press]. We demonstrate that CAM-B3LYP yields atomization energies of similar quality to those from B3LYP, while also performing well for charge transfer excitations in a dipeptide model, which B3LYP underestimates enormously. The CAM-B3LYP functional comprises of 0.19 Hartree-Fock (HF) plus 0.81 Becke 1988 (B88) exchange interaction at short-range, and 0.65 HF plus 0.35 B88 at long-range. The intermediate region is smoothly described through the standard error function with parameter 0.33.

This was followed up by

T. Yanai, R.J. Harrison, and N.C. Handy, Mol. Phys. 103, 413 (2005).

"Multiresolution quantum chemistry in multiwavelet bases: time-dependent density functional theory with asymptotically corrected potentials in local density and generalized gradient approximations"

ABSTRACT : A multiresolution solver for fully numerical linear response calculations of excitation states via the time-dependent Hartree-Fock and density functional theory (TD-HF/DFT) is presented. The linear response method Yanai et al. previously reported [J. Chem. Phys., submitted] was limited to the Tamm-Dancoff approximation and could only use the Hartree-Fock exchange and the local-spin density approximation (LSDA) with a crude asymptotic correction. The present development enables us to perform full TD-HF/DFT calculations employing generalized gradient approximation (GGA) exchange-correlation potentials as well as hybrid ones. The linear response of TD-HF/DFT is computed by means of iteratively solving the coupled integral equations with the Green's functions. In this study, Tozer and Handy's asymptotic correction (AC) is applied to existing DFT exchange-correlations, and is found numerically stable and efficient. Furthermore, the new hybrid exchange-correlation functional CAM-B3LYP, which was recently proposed by Yanai et al. [Chem. Phys. Lett. 393, 51 (2004)], is implemented. The implementation requires a new separated representation of the integral kernel for the Coulomb-attenuated potential. We demonstrate linear response calculations free of basis set error for the excited states of Be, N2, C2H4 and C6H6 using LSDA, HCTH, CAM-B3LYP and PBE0 exchange-correlation functionals. The mean absolute errors of the C6H6 calculations with HCTH and CAM-B3LYP are 0.12 and 0.18 eV, respectively. The second derivative of exchange-correlation functionals is represented fully numerically at O(N) computation cost.

and by

M.J.G. Peach, T. Helgaker, P. Salek, T.W. Keal, O.B. Lutnæs, D.J. Tozer, and N.C. Handy, Phys. Chem. Chem. Phys. 8, 558 (2006).

"Assessment of a Coulomb-attenuated exchange-correlation energy functional"

ABSTRACT : The recently proposed CAM-B3LYP exchange-correlation energy functional, based on a partitioning of the r₁₂ operator in the exchange interaction into long- and short-range components, is assessed for the determination of molecular thermochemistry, structures, and second order response properties. Rydberg and charge transfer excitation energies and static electronic polarisabilities are notably improved over the standard B3LYP functional; classical reaction barriers also improve. Ionisation potentials, bond lengths, NMR shielding constants and indirect spin-spin coupling constants are comparable with the two functionals. CAM-B3LYP atomisation energies and diatomic harmonic vibrational wavenumbers are less accurate than those of B3LYP. Future research directions are outlined.

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[TPSSh]

V.N. Staroverov, G.E. Scuseria, J. Tao, and J.P. Perdew, J. Chem. Phys. 119, 12129 (2003).

"Comparative assessment of a new nonempirical density functional: Molecules and hydrogen-bonded complexes"

ABSTRACT : A comprehensive study is undertaken to assess the nonempirical meta-generalized gradient approximation (MGGA) of Tao, Perdew, Staroverov, and Scuseria (TPSS) against 14 common exchange-correlation energy functionals. Principal results are presented in the form of statistical summaries of deviations from experiment for the G3/99 test set (223 enthalpies of formation, 86 ionization potentials, 58 electron affinities, 8 proton affinities) and three additional test sets involving 96 bond lengths, 82 harmonic vibrational frequencies, and 10 hydrogen-bonded complexes, all computed using the 6-311 + + G(3df,3pd) basis. The TPSS functional matches, or exceeds in accuracy all prior nonempirical constructions and, unlike semiempirical functionals, consistently provides a high-quality description of diverse systems and properties. The computational cost of self-consistent MGGA is comparable to that of ordinary GGA, and exact exchange (unavailable in some codes) is not required. A one-parameter global hybrid version of the TPSS functional is introduced and shown to give further improvement for most properties.

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[TPSS]

J. Tao, J.P. Perdew, V.N. Staroverov, and G.E. Scuseria, Phys. Rev. Lett. 91, 146401 (2003).

"Climbing the density functional ladder: Nonempirical meta-generalized gradient approximation designed for molecules and solids" and preprint

ABSTRACT : The electron density, its gradient, and the Kohn-Sham orbital kinetic energy density are the local ingredients of a meta-generalized gradient approximation (meta-GGA). We construct a meta-GGA density functional for the exchange-correlation energy that satisfies exact constraints without empirical parameters. The exchange and correlation terms respect two paradigms: one- or two-electron densities and slowly varying densities, and so describe both molecules and solids with high accuracy, as shown by extensive numerical tests. This functional completes the third rung of "Jacob's ladder" of approximations, above the local spin density and GGA rungs.

abstract and preprint

J.P. Perdew, J. Tao, V.N. Staroverov, and G.E. Scuseria, J. Chem. Phys. 120, 6898 (2004).

"Meta-generalized gradient approximation: Explanation of a realistic nonempirical density functional"

Note: The second article is a more detailed explanation of the functional proposed in the first article.

ABSTRACT : 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 Fxc 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.

abstract and preprint

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[mPBE]

C. Adamo and V. Barone, J. Chem. Phys. 116, 5933 (2002).

"Physically motivated density functionals with improved performances: The modified Perdew-Burke-Ernzerhof model"

ABSTRACT:In this paper we propose a modification of the exchange functional introduced by Perdew, Burke, and Ernzerhof, which significantly enlarges the original field of applications. This modification is obtained by a series expansion of the functional, which introduces one additional parameter, but retains all the asymptotic and scaling properties of the original model. The results obtained for structural, thermodynamic, kinetic, and spectroscopic properties are satisfactory and not far from those delivered by the most reliable functionals including heavy parametrization. The way in which the functional is derived and the lack of empirical parameters fitted to specific properties makes the new exchange functional widely applicable, for both quantum chemistry and for condensed-matter physics.

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[GRAC]

M. Grüning, O.V. Gritsenko, S.J.A. van Gisbergen, and E.J. Baerends, J. Chem. Phys. 114, 652 (2001).

"Shape corrections to exchange-correlation potentials by gradient regulated seamless conection of model potentials for inner and outer region"

ABSTRACT: Shape corrections to the standard approximate Kohn-Sham exchange-correlation (xc) potentials are considered with the aim to improve the excitation energies (especially for higher excitations) calculated with time-dependent density functional perturbation theory. A scheme of gradient-regulated connection (GRAC) of inner to outer parts of a model potential is developed. Asymptotic corrections based either on the potential of Fermi and Amaldi or van Leeuwen and Baerends (LB) are seamlessly connected to the (shifted) xc potential of Becke and Perdew (BP) with the GRAC procedure, and are employed to calculate the vertical excitation energies of the prototype molecules N2, CO, CH2O, C2H4, C5NH5, C6H6, Li2, Na2, K2. The results are compared with those of the alternative interpolation scheme of Tozer and Handy as well as with the results of the potential obtained with the statistical averaging of (model) orbital potentials. Various asymptotically corrected potentials produce high quality excitation energies, which in quite a few cases approach the benchmark accuracy of 0.1 eV for the electronic spectra. Based on these results, the potential BP-GRAC-LB is proposed for molecular response calculations, which is a smooth potential and a genuine "local" density functional with an analytical representation.

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[OPTX]

N.C. Handy and A.J. Cohen, Mol. Phys. 99, 403 (2001).

"Left-right correlation energy"

GGA

ABSTRACT: We first attempt to determine a local exchange functional Ex[rho] which accurately reproduces the Hartree-Fock (HF) energies of the 18 first and second row atoms. Ex[rho] is determined from rho and |grad rho|, and we find that we can improve significantly upon Becke's original generalized gradient approximation functional (commonly called B88X) by allowing the coefficient of the Dirac exchange term to be optimized (it is argued that molecules do not behave like the uniform electron gas). We call this new two parameter exchange functional OPTX. We find that neither grad² rho or tau - sum | grad phi_i |² improve the fit to these atomic energies. These exchange functionals include not only exchange, but also left-right correlation. It is therefore proposed that this functional provides a definition for exchange energy plus left-right correlation energy when used in Kohn-Sham (KS) calculations. We call this energy the Kohn-Sham exchange (or KSX) energy. It is shown that for nearly all molecules studied these KSX energies are lower than the corresponding HF energies, thus giving values for the non-dynamic correlation energy. At stretched geometries, the KSX energies are always lower than the HF energies, and often substantially so. Furthermore all bond lengths from the KSX calculations are longer than HF bond lengths and experimental bond lengths, which again demonstrates the inclusion of left-right correlation effects in the functional. For these reasons we prefer to split the correlation energy into two parts: left-right correlation energy and dynamic correlation energy, arguing that the usage of the words 'non-dynamic' or 'static' or 'near-degeneracy' is less meaningful. We recognize that this definition of KSX is not precise, because the definition of a local Ex \[„] can never be precise. We also recognize that these ideas are not new, but we think that their importance has been insufficiently recognized in functional determination. When we include third row atoms in our analysis, we are unable to find a local exchange functional which is a substantial improvement over B88X for the reproduction of HF energies. This must arise from the effects of the core orbitals, and therefore we do not consider that this detracts from the improved accuracy of OPTX. We report some MCSCF calculations constructed from bonding-antibonding configurations, from which we attempt to calculate ab initio left-right correlation. There is only moderate agreement between the two approaches. Finally we combine the OPTX functional with established correlation functionals (LYP, P86, P91) to form OLYP, OP86 and OP91; OLYP is a great improvement on BLYP for both energy and structure, and OP86, OP91 are an improvement over BP86, BP91 for structure. The importance of the exchange functional for molecular structure is therefore underlined.

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[LC]

H. Iikura, T. Tsuneda, T. Yanai, and K. Hirao, J. Chem. Phys. 115, 3540 (2001).

"A long-range correction scheme for generalized-gradient-approximation exchange functionals"

The LC functional proposes to split the electron repulsion into a short-range part to be treated by DFT and a long-range part to be treated by HF. Thus 1/r12 = [(1-erf(mu r12))/r12]_{DFT} + [erf(mu r12)/r12]_{HF} . This helps to overcome a number of problems in DFT having to do with long-range interactions.

ABSTRACT: We propose a new long-range correction scheme that combines generalized-gradient-approximation (GGA) exchange functionals in density-functional theory (DFT) with the ab initio Hartree-Fock exchange integral by using the standard error function. To develop this scheme, we suggest a new technique that constructs an approximate first-order density matrix that corresponds to a GGA exchange functional. The calculated results of the long-range correction scheme are found to support a previous argument that the lack of the long-range interactions in conventional exchange functionals may be responsible for the underestimation of 4s-3d interconfigurational energies of the first-row transition metals and for the overestimation of the longitudinal polarizabilities of pi-conjugated polyenes in DFT calculations.

This was followed up by

Y. Tawada, T. Tsuneda, S. Yanagisawa, T. Yanai, and K. Hirao, J. Chem. Phys. 120, 8425 (2004).

"A long-range-corrected time-dependent density functional theory"

ABSTRACT: We apply the long-range correction (LC) scheme for exchange functionals of density functional theory to time-dependent density functional theory (TDDFT) and examine its efficiency in dealing with the serious problems of TDDFT, i.e., the underestimations of Rydberg excitation energies, oscillator strengths, and charge-transfer excitation energies. By calculating vertical excitation energies of typical molecules, it was found that LC-TDDFT gives accurate excitation energies, within an error of 0.5 eV, and reasonable oscillator strengths, while TDDFT employing a pure functional provides 1.5 eV lower excitation energies and two orders of magnitude lower oscillator strengths for the Rydberg excitations. It was also found that LC-TDDFT clearly reproduces the correct asymptotic behavior of the charge-transfer excitation energy of ethylene-tetrafluoroethylene dimer for the long intramolecular distance, unlike a conventional far-nucleus asymptotic correction scheme. It is, therefore, presumed that poor TDDFT results for pure functionals may be due to their lack of a long-range orbital-orbital interaction.

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[PCS00]

E.Proynov, H. Chermette, and D. R. Salahub, J. Chem. Phys. 113, 10013 (2000).

"New tau dependent Correlation Functional combined with a modified Becke exchange"

meta-GGA

ABSTRACT: A new correlation functional is derived within the Kohn-Sham (KS) Density Functional Theory (DFT) involving the electron kinetic energy density and the Laplacian of the electron density as key nonlocal variables. The derivation is based on a direct resolution of the adiabatic connection formula and using an analogy with the local thermodynamic approach in DFT, following the Lap3 theory developed previously. Compared to the latter, the new functional involves higher order -dependent energy terms in a form suggesting a possible resummation procedure that could be used for further development. It is combined with the nonlocal exchange functional of Becke, by modifying the latter in an empirical fashion to achieve better synchronization between the two energy components. The resulting exchange-correlation scheme (named "Bm1") is validated on several test systems known as difficult for DFT, at least at the Local Spin Density and Generalized Gradient Approximation levels. The recent nonempirical hybrid scheme PBE1PBE ("PBE0") is included in the comparative tests as a parameter-free benchmark for the hybrid HF-KS DFT approach. Improved results for relative energies, activation barriers and equilibrium geometries are obtained with the Bm1 functional, particularly concerning aromatic compounds, systems with weak hydrogen bonds, proton transfer processes and transition-metal carbonyls.

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[B00]

A.D. Becke, J. Chem. Phys. 112, 4020 (2000).

"Simulation of delocalized exchange by local density functionals"

An xc meta-GGA which looks promising for small systems

abstract and reprint

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[WGC99]

Y.A. Wang, N. Govind, and E.A. Carter Phys. Rev. B 60, 16 350 (1999).

"Orbital-free kinetic-energy density functionals with a density-dependent kernel"

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[PBE0]

C. Adamo and V. Barone, J. Chem. Phys. 110, 6158 (1999).

"Toward reliable density functional methods without adjustable parameters: The PBE0 model"

abstract and reprint

ABSTRACT : We present an analysis of the performances of a parameter free density functional model (PBE0) obtained combining the so called PBE generalized gradient functional with a predefined amount of exact exchange. The results obtained for structural, thermodynamic, kinetic and spectroscopic (magnetic, infrared and electronic) properties are satisfactory and not far from those delivered by the most reliable functionals including heavy parameterization. The way in which the functional is derived and the lack of empirical parameters fitted to specific properties make the PBE0 model a widely applicable method for both quantum chemistry and condensed matter physics.

See also:

M. Ernzerhof and G.E. Scuseria, J. Chem. Phys. 110, 5029 (1999).

"Assessment of the Perdew-Burke-Ernzerhof exchange-correlation functional"

abstract and reprint

hybrid functional

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[PKZB]

J.P. Perdew, S. Kurth, A. Zupan, P. Blaha, Phys. Rev. Lett. 82, 2544 (1999); Erratum Phys. Rev. Lett. 82 5179 (1999).

"Accurate Density Functional with Correct Formal Properties: A Step Beyond the Generalized Gradient Approximation"

abstract and reprint

A reference for tests of PKZB and many other functionals in atoms, molecules and solids is: S. Kurth, J.P. Perdew and P. Blaha, Int. J. Quantum Chem. 75, 889 (1999).

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[SAOP]

O.V. Gritsenko, P.R.T. Schipper, and E.J. Baerends, Chem. Phys. Lett. 302, 199 (1999).

"Approximation of the exchange-correlation Kohn-Sham potential with a statistical average of different orbital model potentials"

abstract and reprint

An orbital-dependent potential designed to behave like a full OEP calculation

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[RPBE]

B. Hammer, L. B. Hansen and J. K. Nørskov, Phys. Rev. B 59, 7413 (1999).

"Improved adsorption energetics within density-functional theory using revised Perdew-Burke-Ernzerhof functionals"

abstract and reprint

ABSTRACT : A simple formulation of a generalized gradient approximation for the exchange and correlation energy of electrons has been proposed by Perdew, Burke, and Ernzerhof (PBE) [Phys. Rev. Lett. 77, 3865 (1996)]. Subsequently Zhang and Yang [Phys. Rev. Lett. 80, 890 (1998)] have shown that a slight revision of the PBE functional systematically improves the atomization energies for a large database of small molecules. In the present work, we show that the Zhang and Yang functional (revPBE) also improves the chemisorption energetics of atoms and molecules on transition-metal surfaces. Our test systems comprise atomic and molecular adsorption of oxygen, CO, and NO on Ni(100), Ni(111), Rh(100), Pd(100), and Pd(111) surfaces. As the revPBE functional may locally violate the Lieb-Oxford criterion, we further develop an alternative revision of the PBE functional, RPBE, which gives the same improvement of the chemisorption energies as the revPBE functional at the same time as it fulfills the Lieb-Oxford criterion locally.

GGA

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[tauPBE]

M. Ernzerhof and G.E. Scuseria, J. Chem. Phys. 111, 911 (1999).

"Kinetic energy density dependent approximations to the exchange energy"

abstract and preprint

metaGGA

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[EDF1]

R.D. Adamson, P.M.W. Gill and J.A. Pople, Chem. Phys. Lett. 284, 6 (1998).

"Empirical Density Functionals"

abstract and reprint

A frankly semiempirical basis-set dependent fit to G2 data.

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[mPW1,3PW]

C. Adamo and V. Barone, J. Chem. Phys. 108, 664 (1998).

"Exchange functionals with improved long-range behavior and adiabatic connection methods without adjustable parameters: The mPW and mPW1PW models"

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[ZW98]

Y. Zhang and W. Yang, J. Chem. Phys. 109, 2604 (1998).

"A challenge for density functionals: Self-interaction error increases for systems with a noninteger number of electrons"

abstract and reprint

not a new functional, but points out a problem with the old ones

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[GCRA98]

O.V. Gritsenko, N.A. Cordero, A. Rubio and J.A. Alonso, Chem. Phys. Lett. 296, 307 (1998).

"Gradient correction to the exchange pair-correlation function of the weighted spin-density approximation in the density functional formalism"

WDA+gradient correction

abstract and article

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[VSXC]

T.V. Voorhis and G.E. Scuseria, J. Chem. Phys. 109, 400 (1998).

"A novel form for the exchange-correlation energy functional"

one of the first meta-GGAs

ABSTRACT: A new approximate form for the exchange-correlation energy functional is developed. The form is based on the density matrix expansion (DME) for the exchange functional [R. M. Koehl, G. K. Odom, and G. E. Scuseria, Mol. Phys. 87, 835 (1996)]. The nonlocal portion of the correlation energy is assumed to have the same general form as that derived for exchange, while the local portion is taken to be that of the uniform electron gas. The resulting formula does not resort to the use of exact-exchange mixing. A Kohn-Sham implementation of this functional is constructed and the parameters within the functional are adjusted to minimize the difference between the theoretical and the experimental data for a large set of atomic and molecular systems. The results are found to compare favorably with existing functionals, even those which include exact-exchange mixing.

J.C. Sancho-Garcia and J. Cornil, J. Chem. Phys. 121, 3096 (2004) have noted a serious problem with this functional for torsion potentials in pi-conjugated molecules.

"Assessment of recently developed exchange-correlation functionals for the description of torsion potentials in pi-conjugated molecules"

ABSTRACT: Newly developed exchange-correlation functionals in density functional theory (DFT) have been applied to describe conjugation effects in organic molecules. The performance of the various approaches is assessed through the calculation of torsion energy profiles and their critical comparison with available experimental data. Our results indicate that the OPTX-B95 exchange-correlation functional as well as its corresponding hybrid versions perform better than the well-established BLYP or B3LYP schemes when dealing with -conjugated molecules. In contrast, the recently introduced VSXC functional is not as reliable as other DFT methods for the systems examined here.

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[SB98b]

H.L. Schmider and A. Becke, J. Chem. Phys. 109, 8188 (1998).

"Density functionals from the extended G2 test set: Second-order gradient corrections"

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[SB98a]

H.L. Schmider and A. Becke, J. Chem. Phys. 108, 9624 (1998).

"Optimized density functionals from the extended G2 test set"

abstract and reprint

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[B98]

A. Becke, J. Chem. Phys. 109, 2092 (1998).

"A new inhomogeneity parameter in density-functional theory"

abstract and article

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[HCTH]

F.A. Hamprecht, A.J. Cohen, D.J. Tozer, and N.C. Handy, J. Chem. Phys. 109, 6264 (1998).

"Development and assessment of new exchange-correlation functionals"

abstract and reprint

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[CCS98,AC]

M.E. Casida, K.C. Casida, and D.R. Salahub, Int. J. Quant. Chem. 70, 933 (1998). (International Journal of Quantum Chemistry, Quantum Chemistry Symposium No. 32, Proceedings of the International Symposium on Atomic, Molecular, and Condensed Matter Theory)

"Excited-state potential energy curves from time-dependent density-functional theory: A cross-section of formaldehyde's ¹A₁ manifold"

This basic idea of "shift and graft" to correct unoccupied orbitals is very important for time-dependent density functional theory. It has been subsequently implemented in different variations:

• D.J. Tozer and N.C. Handy, J. Chem. Phys. 109, 10180 (1998).
  • "Improving virtual Kohn-Sham orbitals and eigenvalues: Application to excitation energies and static polarizabilities"
  • abstract and reprint
  • HF asymptotic behavior is grafted onto HTCH GGA.
• D.J. Tozer, R.D. Amos, N.C. Handy, B.O. Roos, and L. Serrano-Andrés, Mol. Phys. 97, 859 (1999).
  • "Does density functional theory contribute to the understanding of excited states of unsaturated organic compounds?"
  • Classical coulombic term based on Becke's Voronoi polyhedra is grafted onto HTCH GGA.

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[revPBE]

Y. Zhang and W. Yang, Phys. Rev. Lett. 80, 890 (1998).

"Comment on 'Generalized Gradient Approximation Made Simple'"

abstract and reprint

A modification of PBE that has been used successfully with surface calculations

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[K2-BVWN]

S.A. Kafafi, J. Phys. Chem. A 102, 10404 (1998).

"Novel Density Functional Methodology for the Compuation of Accurate Electronic Thermodynamic Properties of Molecular Systems and Improved Long-Range Behavior"

A hybrid functional. Interesting results but the theoretical discussion is sometimes unclear (or incorrect). For example, the article seems to claim (i) N³ scaling whereas the presence Hartree-Fock exchange implies N⁴ scaling, and (ii) -1/r asymptotic behavior of Vxc whereas the functional actually given has a formal asymptotic behavior of -(2/3)*0.375/r . In addition, the use of element-dependent empirical corrections seems to deviate considerably from the overall DFT philosophy of having a unique functional for the entire periodic table.

See also Kafafi's letter to the Computational Chemistry List regarding the theoretical justification for his functional.

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[HFS-BVWN]

S.A. Kafafi and E.R.H. El-Gharkawy, J. Phys. Chem. A 102, 3202 (1998).

"A Simple Coupling Scheme between Hartree-Fock and Local Spin-Density Functional Theories"

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[FT98]

Micheal Filatov and Walter Thiel, Phys. Rev. A 57, 189 (1998).

"Exchange-correlation density functional beyond the gradient approximation"

According to the authors, FT98 performs very similarly to FT97, both of which perform similarly to the B3LYP functional.

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[FT97]

M. Filatov and W. Thiel, Mol. Phys. 91, 847 (1997).

"A new gradient-corrected exchange-correlation density functional"

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[B97]

A.D. Becke, J. Chem. Phys. 107, 8554 (1997).

"Density-functional thermochemistry. V. Systematic optimization of exchange-correlation functionals"

abstract and reprint

Further development of the idea:

• H.L. Schmider and A. Becke, J. Chem. Phys. 109, 8188 (1998).
  • "Density functionals from the extended G2 test set: Second-order gradient corrections"
  • abstract and reprint
• H.L. Schmider and A. Becke, J. Chem. Phys. 108, 9624 (1998).
  • "Optimized density functionals from the extended G2 test set"
  • abstract and reprint

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[W97]

T.A. Wesolowski J. Chem. Phys. 106, 8516 (1997).

"Density functional theory with approximate kinetic energy functionals applied to hydrogen bonds"

GGA

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[PBE]

J.P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77, 3865 (1996); Erratum: Phys. Rev. Lett. 78, 1386 (1997).

"Generalized gradient approximation made simple"

abstract and reprint

Constructed nonempirically, using e.g. coordinate scaling constraints. Exact for the uniform electron gas and very useful for solids.

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[LRC95x]

A. Lembarki, F. Rogemond, and H. Chermette, Phys. Rev. A 52, 3704 (1995).

"Gradient corrected exchange potential with correct asymptotic behavior and corresponding energy functional obtained from the virial theorem''

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[ACM]

J. Baker, M. Muir, and J. Andzelm, J. Chem. Phys. 102, 2063 (1995).

"A study of some organic reactions using density functional theory"

abstract and reprint

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[PVS95c, LAP]

E. Proynov, A. Vela and D. R. Salahub, Chem. Phys. Lett. 230, 419 (1994); Erratum, Chem. Phys. Lett. 234, 462 (1995).

"Nonlocal Correlation functional Involving the Laplacian of the density."

E.I. Proynov, E. Ruiz, A. Vela, and D.R. Salahub, Int. J. Quant. Chem. Symp. 29, 61 (1995).

"Determining and Extending the Domain of Exchange and Correlation Functionals."

These correspond to the correlation option LAP in deMon-KS version 3. It is a nonlocal generalization of the PVS94 functional, involving the kinetic energy density and the Laplacian of the electron density as ingredients of nonlocality. Three different input lines are recommended, corresponding to several different type of parametrizations being under validation. These different parametrizations reflect a different relative strength of the antiparallel spin correlation compared to the parallel spin one (the one beyond the Kohn Sham-exchange):

BLAP3:

POTENTIAL NONLOCAL BECKE LAP 0.197 1.276 1.477 0.04

PLAP3:

POTENTIAL NONLOCAL PD86 LAP 0.197 1.26 1.48 0.01

PLAP4:

POTENTIAL NONLOCAL PD86 LAP 0.197 1.2557 1.48 0.003

[LB94xc]

R. van Leeuwen and E.J. Baerends, Phys. Rev. A 49 (1994) 2421.

"Exchange-correlation potential with correct asymptotic behavior."

This is a generalized gradient correction which gives the correct asymptotic behavior for the exchange-correlation potential.

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[PVS94xc]

E.I. Proynov and D.R. Salahub, Phys. Rev. B 49 (1994) 7874. `` A Simple but Efficient Correlation Functional from a Pair Correlation Function.'' ; E.I.Proynov, A. Vela and D.R. Salahub, Phys. Rev. A 50 (1994) 3766. `` A gradientless exchange-correlation functional beyond the LSD Approximation.''

This is an alternative to the VWN option programmed in deMon1p2 and deMon3. The corresponding keyword in deMon3 is

POTENTIAL LOCAL PVS1 PVS1

This is a scheme which leads to considerable improvement over VWN with respect to binding energies and geometries of molecules, often comparable with the GGA results, with a required computer time about the same as VWN.

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[B3LYP]

The B3LYP functional is Gaussian Inc. variation on the B3P hybrid functional. The original published description is apparently that in Gaussian NEWS, v. 5, no. 2, summer 1994, p. 2. ``Becke3LYP Method References and General Citation Guidelines.'' The functional has the form Exc = (1-a_0)Ex(LDA) + a_0 Ex(HF) + a_x Ex(B88x) + a_c Ec(LYP88c) + (1-a_c) Ec(VWN80c) , where a_0 = 0.20, a_x = 0.72, and a_c = 0.81 .

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[B93xc, B3P]

A.D. Becke, J. Chem. Phys. 98 (1993) 5648.

``Density-functional thermochemistry. III. The role of exact exchange.''

This is a 3 parameter hybrid functional mixing the exact Hartree-Fock exchange with the VWN80c, B88x, and PW91c functionals. Exc = Ex(LDA) + Ec(VWN80c) + a_0 [Ex(HF)-Ex(LDA)] + a_x Ex(B88x) + a_c Ec(P91c)

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[B93xc, 1/2&1/2]

A.D. Becke, J. Chem. Phys. 98 (1993) 1372.

``A new mixing of Hartree-Fock and local density functional theories.''

This is the original hybrid functional: Exc = (1/2) Ex(HF) + (1/2) Exc(LDA).

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[PW91xc, PW91]

J.P. Perdew, J.A. Chevary, S.H. Vosko, K.A. Jackson, M.R. Pederson, D.J. Singh, and C. Fiolhais, Phys. Rev. B 46, 6671 (1992); Erratum, Phys. Rev. B 48, 4978 (1993).

"Atoms, molecules, solids, and surfaces: Applications of the generalized gradient approximation for exchange and correlation"

abstract

GGA, constructed non-empirically, using e.g. coordinate scaling constraints. Exact for the uniform gas, and very useful for solids. Note that this GGA is very unusual in that it was widely incorporated in many DFT programs significantly before any report appeared in the literature.

J.P. Perdew, K. Burke, and Y. Wang, Phys. Rev. B 54, 16533 (1996); Erratum, Phys. Rev. B 57, 14999 (1998).

"Generalized gradient approximation for the exchange-correlation hole of a many-electron system"

abstract

GGA, constructed non-empirically, exact for the uniform gas, and very useful for solids. Note that this GGA is very unusual in that it was widely incorporated in many DFT programs significantly before any report appeared in the literature.

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[KLI]

J.B. Krieger, Y. Li, and G.J. Iafrate, Phys. Lett. A 146, 256 (1990).

"Derivation and application of an accurate Kohn-Sham potential with integer discontinuity"

See also:

• R.T. Sharp and G.K. Horton, Phys. Rev. 90, 317 (1953).
  • "A Variational Approach to the Unipotential Many-Electron Problem"
• J.B. Krieger, Y. Li, and G.J. Iafrate, Phys. Lett A 148, 470 (1990).
  • "Exact relations in the optimized effective potential method employing an arbitrary E_xc[{psi_i,sigma}]"
• J.B. Krieger, Y. Li, and G.J. Iafrate, Phys. Rev. A 46, 5453 (1992).
  • "Systematic approximations to the optimized effective potential: Application to orbital-density-functional theory"
• J.B. Krieger, Y. Li, and G.J. Iafrate, Int. J. Quantum Chem. 41, 489 (1992).
  • "Accurate local spin-polarized exchange potential: Reconciliation of generalized Slater and Kohn-Sham methods"
• Y. Li, J.B. Krieger, and G.J. Iafrate, Chem. Phys. Lett. 191, 38 (1992).
  • "Negative ions as described by Kohn-Sham exchange-only theory"
• Y. Li, J.B. Krieger, and G.J. Iafrate, Phys. Rev. A 47, 165 (1993).
  • "Self-consistent calculations of atomic properties using self-interaction-free exchange-only Kohn-Sham potentials"
• Y.-H. Kim, M. Städele, and R.M. Martin, Phys. Rev. A 60, 3633 (1999).
  • "Density-functional study of small molecules within the Krieger-Li-Iafrate approximation"
  • abstract and preprint
• P. Süle, S. Kurth, and V. Van Doren, J. Chem. Phys. 112, 7355 (2000).
  • "Optimized effective potential method for polymers"
  • abstract and reprint

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[WL90]

L.C. Wilson and M. Levy, Phys. Rev. B 41, 12930 (1990).

"Nonlocal Wigner-like correlation-energy density functional through coordinate scaling"

abstract

Simple functional constructed to satisfy certain coordinate scaling conditions.

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[LYP88c,LYP]

C. Lee, W. Yang, and R.G. Parr, Phys. Rev. B 37, 785 (1988).

"Development of the Colle-Salvetti correlation energy formula into a functional of the electron density"

GGA

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[BR89]

A.D. Becke and M.R. Roussel, Phys. Rev. A 39, 3761 (1989).

"Exchange holes in inhomogeneous systems: A coordinate-space model"

metaGGA, exchange-only

abstract

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[B88x, B]

A.D. Becke, Phys. Rev. A 38, 3098 (1988).

"Density-functional exchange-energy approximation with correct asymptotic behavior."

This is a generalized gradient approximation for the exchange energy which has the correct asymptotic behavior for the exchange energy density. It is sometimes confused with Becke's 1986 functional. A formula for the B88x potential Vx may be found in

L. Fan and T. Ziegler, J. Chem. Phys. 94, 6057 (1991).

"The influence of self-consistency on nonlocal density functional calculations."

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[DK87]

A.E. DePristo and J.D. Kress, J. Chem. Phys. 86, 1425 (1987).

"Rational function representation for accurate exchange energy functionals"

abstract

GGA

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[P86c, P]

J.P. Perdew, Phys. Rev. B 33, 8822 (1986).

``Density-functional approximation for the correlation energy of the inhomogeneous electron gas.''

This is a generalized gradient approximation for the correlation energy.

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[PW86x, P]

J.P. Perdew and Y. Wang, Phys. Rev. B 33, 8800 (1986).

"Accurate and simple density functional for the electronic exchange energy: Generalized gradient approximation"

COMMENT: This is a generalized gradient approximation for the exchange energy.

ABSTRACT: The electronic exchange energy as a functional of the density may be approximated as E_x[n]=A_x∫d³rn^4/3F(s), where s=|∇n|/2k_Fn, k_F=(3π²n)^1/3,

and F(s)=(1+1.296s²+14s⁴+0.2s⁶)^1/15. The basis for this approximation is the gradient expansion of the exchange hole, with real-space cutoffs chosen to

guarantee that the hole is negative everywhere and represents a deficit of one electron. Unlike the previously publsihed version of it, this functional is

simple enough to be applied routinely in self-consistent calculations for atoms, molecules, and solids. Calculated exchange energies for atoms fall within

1% of Hartree-Fock values. Significant improvements over other simple functionals are also found in the exchange contributions to the valence-shell

removal energy of an atom and to the surface energy of jellium within the infinite barrier model.

reprint

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[LM83]

D.C. Langreth and M.J. Mel, Phys. Rev. B 28, 1809 (1983).

``Beyond the local-density approximation in calculations of ground-state electronic properties''

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[PZ81xc, SIC]

J.P. Perdew and A. Zunger, Phys. Rev. B 23, 5048 (1981).

"Self-interaction correction to density-functional approximations for many-electron systems."

abstract

Also contains a parameterization of the LDA. An efficient and unitarily invariant implementation of the Perdew-Zunger SIC is frought with difficulties. These now appear to be resolved --- see especially:

• M.R. Pederson, R.A. Heaton, and C.C. Lin, J. Chem. Phys. 80, 1972 (1984).
  • "Local-density Hartree-Fock theory of electronic states of molecules with self-interaction correction"
  • Introduces two key ideas: (1) the use of a unified orbital Hamiltonian and (2) the need for an additional unitary tranformation to fully minimize the energy.
• S. Goedecker and C.J. Umriger, Phys. Rev. A 55, 1765 (1997).
  • "Critical assessment of the self-interaction-corrected local-density-functional method and its algorithmic implementation"
  • abstract and reprint
  • Analytic first derivatives permitting, for the first time, automatic geometry optimizations with the SIC-LDA

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[LP80]

D.C. Langreth and J.P. Perdew, Phys. Rev. B 21, 5469 (1980).

``Theory of nonuniform electronic systems. I. Analysis of the gradient approximation and a generalization that works''

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[SMW80]

C.C. Shih, D.R. Murphy, and W.-P. Wang, J. Chem. Phys. 73, 1340 (1980).

???

LDA exchange plus first gradient correction with semiempirically determined coefficient (beta), hence X_alpha,beta

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[LDA]

The local density approximation has a long history bringing it ever closer to numerical perfection:

• W. Kohn and L.J. Sham, Phys. Rev. 140, A1133 (1965).
  • "Self-consistent equations including exchange and correlation effects"
  • Defines the LDA.
• E. Wigner, Phys. Rev. 46, 1002 (1934).
  • "On the interaction of electrons in metals"
  • E. Wigner, Trans. Faraday Soc. 34, 678 (1938).
  • "Effects of the electron interaction on the energy levels of electrons in metals"
  • Early example of parameterization of correlation energy for LDA. Possibly the last time we had a simple formula, even if not terribly accurate. Wigner's formula has been corrected by Pines to read
  • Ec = -0.88/(Rs+7.8) Ry
  • in
  • D. Pines, Solid State Physics 1,367 (1955).
  • "Electron interaction in metals"
• S.H. Vosko, L. Wilk, and M. Nusair, Can. J. Phys. 58, 1200 (1980).
  • "Accurate spin-dependent electron liquid correlation energies for local spin density calculations: a critical analysis."
• J.P. Perdew and A. Zunger, Phys. Rev. B 23, 5048 (1981).
  • "Self-interaction correction to density-functional approximations for many-electron systems."
  • abstract
  • Contains a parameterization of the LDA.
• J. P. Perdew and Y. Wang, Phys.Rev. B 45, 13244 (1992).
  • "Accurate and simple analytic representation of the electron-gas correlation energy"
  • abstract
  • This is the LDA parameterization to be used with the PW91 and PBE GGAs.

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[DFT]

The founding formal papers of modern density-functional theory (in chronological order):

• P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 (1964).
  • "Inhomogenous electron gas"
  • First paper, laying the foundations for modern DFT.
• W. Kohn and L.J. Sham, Phys. Rev. 140, A1133 (1965).
  • "Self-consistent equations including exchange and correlation effects"
  • Defines the Kohn-Sham variant of DFT and gives the modern definition of the LDA.
• L.J. Sham and W. Kohn, Phys. Rev. 145, 561 (1966).
  • "One-particle properties of an inhomogeneous interacting electron gas"
  • Early attempt at a rigorous formulation of DFT for excited states
• M. Levy, Proc. Natl. Acad. Sci.(USA) 76, 6062 (1979).
  • "Universal variational functionals of electron densities, first order density matrices, and natural spin orbitals and solution of the v-representability problem"
  • M. Levy, Phys. Rev. A 26, 1200 (1982).
  • "Electron densities in search of Hamiltonians"
  • abstract
  • Extension of DFT to include degeneracies via introduction of the constrained search formalism.

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[Xalpha]

J.C. Slater, The Self-Consistent Field for Molecules and Solids, (McGraw-Hill, New York, 1974)

The X-alpha method was a very successful early attempt to approximate Hartree-Fock with a local potential. When the semiempirical parameter is set at alpha=2/3, we recover the exchange-only LDA. See also:

• K. Schwarz, Phys. Rev. B 5, 2466 (1972).
  • "Optimization of the Statistical Exchange Parameter alpha for the Free Atoms H through Nb"
  • Optimization of the parameter alpha in the Xalpha method.
• D.R. Salahub, S.H. Lambson, and R.P. Messmer, Chem. Phys. Lett. 85, 430 (1982).
  • "Is there correlation in Xalpha? Analysis of Hartree-Fock and LCAO Xalpha calculations for O₃"
  • Unlike Hartree-Fock where exchange interactions can be localized in different regions for different orbitals, every Xalpha orbital feels the same exchange potential. Thus Xalpha acts to keep electrons of the same spin apart (electrons of opposite spin paired) more than in Hartree-Fock and in so doing tends to (accidently) mimic some of the effects of electron correlation.

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[ALT81]

K. Aashamar, T.M. Luke, and J.D. Talman, J. Phys. B: Atom. Molec. Phys. 14, 803 (1981).

"A multi-configuration optimised central potential method for atomic structure calculations II: application to magnesium"

First correlated (i.e. MCSCF) OEP

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[ALT79]

K. Aashamar, T.M. Luke, and J.D. Talman, J. Phys. B: Atom. Molec. Phys. 12, 3455 (1979).

"A multi-configuration optimised central potential method for atomic structure calculations: application to carbon"

First correlated (i.e. MCSCF) OEP

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[PT78b]

M.M. Pant and J.D. Talman, Phys. Lett. 68A, 1819 (1978).

"Atomic properties for Ne, Ar, Kr from an optimised potential model for atoms"

exchange-only OEP

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[PT78a]

M.M. Pant and J.D. Talman, Phys. Rev. A 17, 1819 (1978).

"Impulse Compton profiles calculated in an optimized potential model for atoms"

exchange-only OEP

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[TS76]

J.D. Talman and W.F. Shadwick, Phys. Rev. A 14, 36 (1976).

"Optimized effective atomic central potential"

First calculations with the exchange-only optimized effective potential.

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[SH55]

R.T. Sharp and G.K. Horton, Phys. Rev. 90, 317 (1953).

"A Variational Approach to the Unipotential Many-Electron Problem"

The original, but very short, paper explaining how to calculate the exchange-only optimized effective potential.

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[S50]

J.C. Slater, Phys. Rev. 81, 385 (1950).

"A Simplification of the Hartree-Fock Method"

To my knowledge, the first attempt to define a local exchange potential.

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[W]

C.F. Weizsäcker, Z. Phys. 96, 451 (1935).

The original gradient-corrected kinetic energy functional

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[TF]

L.H. Thomas, Proc. Cambridge Philos. Soc. 23, 542 (1927).

E. Fermi, Rend. Accad, Lincei 6, 602 (1927).

E. Fermi, Z. Phys. 48, 73 (1928).

The original local density kinetic energy functional

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JOURNALS

• J. Chem. Phys. Go there.
• Phys. Rev. A Go there.
• Phys. Rev. A Go there.
• Phys. Rev. Lett. Go there.
• Phys. Rev. archives for older papers. Go there.

ACKNOWLEDGEMENTS

I would like to thank Dale A. Braden, Henry Chermette, Steeve Chretien, Helio Duarte, Ira N. Levine, Mel Levy, John Perdew, Emil Proynov, Andreas Savin, and Weitao Yang for helpful comments on this page. All errors are (of course) my own.

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