Density Functional Theory There are actually two DFT programs in GAMESS, one using the typical grid quadrature for integration of functionals, and one using resolution of the identity to avoid the need or grids. The default METHOD=GRID program is discussed below, following a short description of METHOD=GRIDFREE. The final section is references to various functionals, and other topics of interest. DFTTYP keywords Let's begin with a translation table to NWchem's input: GAMESS NWchem's XC keyword Slater slater Gill gill96 SVWN slater vwn_5 (SVWN1RPA=slater vwn_1_rpa, etc) Becke becke88 BVWN becke88 vwn_5 BLYP becke88 lyp B97 becke97 B97-1,B97-2,B97-3 becke97-1, becke-2, becke-3 HCTH93,HCTH120,HCTH147,HCTH407 hcth,hcth120,hcth147,hcth407 B98 becke98 B3LYP HFexch 0.20 slater 0.80 \ becke88 nonlocal 0.72 \ lyp 0.81 vwn_5 0.19 B3LYPV1R b3lyp or, if you like to type: HFexch 0.20 slater 0.80 \ becke88 nonlocal 0.72 \ lyp 0.81 vwn_1_rpa 0.19 B3P86 HFexch 0.20 slater 0.80 \ becke88 nonlocal 0.72 \ vwn_5 0.19 perdew81 0.81 perdew86 0.81 X3LYP HFexch 0.218 slater 0.782 \ becke88 nonlocal 0.542 \ xperdew91 nonlocal 0.167 \ lyp 0.871 vwn_1_rpa 0.129 PW91 xperdew91 perdew91 B3PW91 HFexch 0.20 slater 0.80 \ becke88 nonlocal 0.72 \ perdew91 0.81 pw91lda 1.00 PBE xpbe96 cpbe96 PBE0 pbe0 revPBE revpbe cpbe96 VS98 vs98 M06 m06 (and similarly for M05-2X, etc.) PKZB xpkzb99 cpkzb99 TPSS xtpss03 cptss03 TPSSh xctpssh Note that B3LYP in GAMESS is based in part on the VWN5 electron gas correlation functional. Since there are five formulae with two possible parameterizations mentioned in the VWN paper about local correlation, other programs may use other choices, and therefore generate different B3LYP energies. For example, NWChem's manual says it uses the "VWN 1 functional with RPA parameters as opposed to the prescribed Monte Carlo parameters" as its default. Should you wish to use this parameterization of the VWN1 formula in a B3LYP hybrid, simply choose "DFTTYP=B3LYPV1R". grid-free DFT The grid-free code is a research tool into the use of the resolution of the identity to simplify evaluation of integrals over functionals, rather than quadrature grids. This trades the use of finite grids and their associated errors for the use of a finite basis set used to expand the identity, with an associated truncation error. The present choice of auxiliary basis sets was obtained by tests on small 2nd row molecules like NH3 and N2, and hence the built in bases for the 3rd row are not as well developed. Auxiliary bases for the remaining elements do not exist at the present time. The grid-free Becke/6-31G(d) energy at a C1 AM1 geometry for ethanol is -154.084592, while the result from a run using the "army grade grid" is -154.105052. So, the error using the AUX3 RI basis is about 5 milliHartree per 2nd row atom (the H's must account for some of the error too). The energy values are probably OK, the differences noted should by and large cancel when comparing different geometries. The grid-free gradient code contains some numerical inaccuracies, possibly due to the manner in which the RI is implemented for the gradient. Computed gradients consequently may not be very reliable. For example, a Becke/6-31G(d) geometry optimization of water started from the EXAM08 geometry behaves as: FINAL E= -76.0439853638, RMS GRADIENT = .0200293 FINAL E= -76.0413274662, RMS GRADIENT = .0231574 FINAL E= -76.0455283912, RMS GRADIENT = .0045887 FINAL E= -76.0457360477, RMS GRADIENT = .0009356 FINAL E= -76.0457239113, RMS GRADIENT = .0001222 FINAL E= -76.0457216186, RMS GRADIENT = .0000577 FINAL E= -76.0457202264, RMS GRADIENT = .0000018 FINAL E= -76.0457202253, RMS GRADIENT = .0000001 Examination shows that the point on the PES where the gradient is zero is not where the energy is lowest, in fact the 4th geometry is the lowest encountered. The behavior for Becke/6-31G(d) ethanol is as follows: FINAL E= -154.0845920132, RMS GRADIENT = .0135540 FINAL E= -154.0933138447, RMS GRADIENT = .0052778 FINAL E= -154.0885472996, RMS GRADIENT = .0009306 FINAL E= -154.0886268185, RMS GRADIENT = .0002043 FINAL E= -154.0886352947, RMS GRADIENT = .0000795 FINAL E= -154.0885599794, RMS GRADIENT = .0000342 FINAL E= -154.0885514829, RMS GRADIENT = .0000679 FINAL E= -154.0884955093, RMS GRADIENT = .0000205 FINAL E= -154.0886438244, RMS GRADIENT = .0000330 FINAL E= -154.0886596883, RMS GRADIENT = .0000325 FINAL E= -154.0886094081, RMS GRADIENT = .0000120 FINAL E= -154.0886054003, RMS GRADIENT = .0000109 FINAL E= -154.0885939751, RMS GRADIENT = .0000152 FINAL E= -154.0886711482, RMS GRADIENT = .0000439 FINAL E= -154.0886972557, RMS GRADIENT = .0000230 with similar fluctuations through a total of 50 steps without locating a zero gradient. Note that the second energy above is substantially below all later points, so geometry optimizations with the grid-free DFT gradient code are at this time unsatisfactory. DFT with grids METHOD=GRID (the default for DFT) produces good energy and gradient quantities. Its energy errors should usually be less than 10 microHartree/atom, using the default grid. The default grid was changed on April 11, 2008 to use Lebedev angular grids. This changes all results obtained prior to that date using the original polar coordinate angular grid. The old grids can still be used, $dft nrad=96 nthe=12 nphi=24 $end $tddft nrad=24 nthe=8 nphi=16 $end in case you need to reproduce numbers from older versions. Since April 2008, the default is $dft nrad=96 nleb=302 $end $tddft nrad=48 nleb=110 $end The default for the more accurate meta-GGA functionals was changed in February 2012 to a more accurate grid $dft nrad=99 nleb=590 $end $tddft nrad=48 nleb=110 $end but all GGA and LDA functionals remain 96/302 and 48/110. The 96/302 grid settings produce root mean square gradient vectors accurate to about 0.00010, which matches the default value for OPTTOL in $STATPT. The "standard grid- one" contains many fewer points, $dft sg1=.true. $end $tddft sg1=.true. $end so SG1 will produce nuclear gradients accurate only to about 5 times OPTTOL, namely 0.00050 or so. SG1 is a very fast grid, and will provide substantial speedups if SG1 is used for the early steps of geometry optimizations. Rather high quality results, meaning an OPTTOL near 0.00001 can be used, may be obtained by $dft nrad=96 nleb=590 $end Very accurate (converged) results come from using the "army grade" grid, $dft nrad=96 nleb=1202 $end Turn to the next page to see numerical results. A numerical demonstration of grid accuracies can be obtained from ethanol, DFTTYP=BECKE: energy RMS grad. CPU sg1=.true. -154.105070 0.010837 11 nrad=96 nthe=12 nphi=24 -154.104863 0.010724 56 nrad=96 nleb=302 -154.105042 0.010704 58 nrad=96 nleb=590 -154.105051 0.0107349 108 nrad=96 nleb=1202 -154.105052 0.0107353 214 Note that the energies are a function of the grid size, just as they are a function of the basis used, so you must only compare runs that use the same grid size (and of course the same basis set). The default grid (and the 590 point grid) will give nuclear gradients which are accurate enough to lead to satisfactory geometry optimizations. This means that DFT frequencies obtained by numerical differentiation should also be OK. RUNTYP=ENERGY, GRADIENT, HESSIAN, and their chemical combinations for OPTIMIZE, SADPOINT, IRC, DRC, VSCF, RAMAN, and FFIELD should all work. The grid DFT program uses symmetry during the numerical quadrature in two ways. First, the integration runs only over grid points placed around the symmetry unique atoms. Your run should be done in the full non-Abelian group, so that grid points as well as the usual integrals and the SCF steps can exploit full molecular symmetry. Symmetry is turned off during any TD-DFT stages, since excited states often have different symmetry than the ground state, but will be used in the ground state DFT. Secondly, for polar coordinate angular grids only, "octant symmetry" is implemented using an appropriate Abelian subgroup of the full group. The grid evaluation automatically uses an appropriate subgroup to reduce the number of grid points for atoms that lie on symmetry axes or planes. For example, in Cs, atoms lying in the xy plane will be integrated only over the upper hemisphere of their grid points. Octant symmetry is not used for any of these: a) if a non-standard axis orientation is input in $DATA b) if the angular grid size (NTHE,NTHE0,NPHI,NPI0) is not a multiple of the octant symmetry factors, such as NTHE=15 in C2v. The permissible values depend on the group, but NTHE a multiple of 2 and NPHI a multiple of 4 is generally safe. Time Dependent Density Functional Theory (TD-DFT) Two review articles are available, "Single-Reference ab Initio Methods for the Calculation of Excited States of Large Molecules" A.Dreuw, M.Head-Gordon Chem.Rev. 105, 4009-4037(2005) "Excited states from time-dependent density functional theory" P.Elliott, F.Furche, K.Burke Rev.Comp.Chem. 26, 91-166(2009) The following article is very informative: S.Hirata, M.Head-Gordon Chem.Phys.Lett. 314, 291-299(1999) It also explains the Tamm/Dancoff approximation which connects TD-DFT to CIS. TD-DFT requires higher functional derivatives of the exchange correlation energy with respect to the density: 2nd derivatives to do TD-DFT excitation energies, and 3rd derivatives to do TD-DFT nuclear gradients. Consequently, some of the functionals permit only excitation energies. To use metaGGAs in TD-DFT, the above functional derivatives involve a non-trivial differentiation of the kinetic energy tau's density dependence. The latter is the subject of F.Zahariev, S.S.Leang, M.S.Gordon J.Chem.Phys. 138, 244108/1-11(2013) The TD-DFT nuclear gradient implementation in GAMESS is M.Chiba, T.Tsuneda, K.Hirao J.Chem.Phys. 124, 144106/1-11(2006) and the long-range correction (useful in Rydberg and/or charge transfer states is Y.Tawada, T.Tsuneda, S.Yanagisawa, Y.Yanai, K.Hirao J.Chem.Phys. 120, 8425-8433(2004) See also K.A.Nguyen, P.N.Day, R.Pachter Int.J.Quantum Chem. 110, 2247-2255(2010) The "lambda diagnostic" is described by M.J.G.Peach, P.Benfield, T.Helgaker, D.J.Tozer J.Chem.Phys. 128, 044118/1-8(2008) This is a criterion for separating valence states from charge transfer and Rydberg states. Note that it is possible to do TD-HF excitation energies, by requesting TDDFT=EXCITE, but leaving DFTTYP=NONE. * * * * * Two-photon absorption (TPA) cross-sections are computed by first evaluating the excitation energy to some desired number of excited states. The oscillator strengths for each state give some idea of the intensity of one-photon absorption (OPA cross-section). Then, the TPA cross- sections to these same excited states are evaluated. Note that Franck-Condon factors are not included in either OPA or TPA oscillator strengths. Beta hyperpolarizabilities (see BETA in $TDDFT) are computed by the TDDFT code, but the computation of excited states is skipped. TPA and BETA calculations a) should not use the Tamm/Dancoff approximation, b) may include solvent effects by EFP, c) make sense only at a single geometry: use the ordinary TD-DFT program to optimize excited state geometries, if needed. TPA and BETA calculations are described in F.Zahariev, M.S.Gordon J.Chem.Phys. 140, 18A523/1-10(2014) * * * * * Solvation effects on the excited state energies can be modeled by PCM or EFP or both, with nuclear gradients. PCM + TD-DFT gradient: Y.Wang, H.Li J.Chem.Phys. 133, 034108/1-11(2010) EFP1 + TD-DFT energy: S.Yoo, F.Zahariev, S.Sok, M.S.Gordon J.Chem.Phys. 129, 144112/1-8(2008) EFP1 + TD-DFT gradient: N.Minezawa, N.De Silva, F.Zahariev, M.S.Gordon J.Chem.Phys. 134, 05411(2011) POL5P + TD-DFT gradient (similar polarizable solvent): D.Si, H.Li J.Chem.Phys. 133, 144112(2010) * * * * * In some cases, a more balanced description of the states might be obtained if the orbitals are optimized for a reference with unpaired electrons. This is possible with spin-flip methods, see TDDFT=SPNFLP. For example, in C2H4, one might optimize the orbitals for the triplet state (pi)1(pi*)1, but be interested in the energies of the three singlets and one triplet states N=(pi)2, T=(pi)1(pi*)1, V=(pi)1(pi*)1, and Z=(pi*)2. Using the T state as the reference optimizes the shape of both pi and pi*, since both are occupied. Flipping one of the two unpaired alpha spins in the T reference will access all four valence states (recall that a triplet state with Ms=0 is perfectly OK, namely ab+ba). For more information, see Y.Shao, M.Head-Gordon, A.I.Krylov J.Chem.Phys. 118,4807(2003) F.Wang, T.Ziegler J.Chem.Phys. 121, 12191(2004) J.Chem.Phys. 122, 074109(2005) O.Vahtras, Z.Rinkevicius J.Chem.Phys. 126, 114101(2007) Z.Rinkevicius, H.Agren Chem.Phys.Lett. 491, 132(2010) Z.Rinkevicius, O.Vahtras, H.Agren J.Chem.Phys. 113, 114101(2010 M.Huix-Rotllant, B.Natarajan, A.Ipatov, C.M.Wawire, T.Deutsch, M.E.Casida Phys.Chem.Chem.Phys. 12, 12811(2010) The penalty constrained optimization procedure was used to find ethylene's conical intersections by N.Minezawa, M.S.Gordon J.Phys.Chem.A 113, 12749(2009) references for DFT An excellent overview of DFT can be found in Chapter 6 of Frank Jensen's book. Two other monographs are "Density Functional Theory of Atoms and Molecules" R.G.Parr, W.Yang Oxford Scientific, 1989 "A Chemist's Guide to Density Functional Theory" W.Koch, M.C.Holthausen Wiley-VCH, 2001 If you would like to understand the "theory" of Density Functional Theory, see Kieron Burke's online book "The ABC of DFT", at http://dft.uci.edu/dftbook.html A delightful and thought provoking paper on the relationship of DFT to conventional wavefunction theory: "Obituary: Density Functional Theory (1927-1993)" P.M.W.Gill Aust.J.Chem. 54, 661-662(2001) You may also enjoy "Fourteen easy lessons in Density Functional Theory", John Perdew and Adrienn Ruzsinszky Int. J. Quantum Chem. 110, 2801-2807(2010) "Perspective on density functional theory" Kieron Burke J.Chem.Phys. 136, 150901/1-9(2012) "Perspective: fifty years of density-functional theory in chemical physics" A.D.Becke J.Chem.Phys. 140, 18A301/1-18(2014) A paper comparing DFT's approach to correlation to traditional quantum chemistry methods: E.J.Baerends, O.V.Gritsenko J.Phys.Chem.A 101, 5383-5403(1997) Some philosophy about designing functionals at each rung of DFT's "Jacob's ladder": J.P.Perdew, A.Ruzsinszky, J.Tao, V.N.Staroverov, G.E.Scuseria, G.I.Csonka J.Chem.Phys. 123, 062201/1-9(2005) On hybridization: J.P.Perdew, M.Ernzerhof, K.Burke J.Chem.Phys. 105, 9982-9985(1996) G.I.Csonka, J.P.Perdew, A.Ruzsinszky J.Chem.Theory Comput. 6, 3688-3703(2010) Some reading on the grid-free approach to density functional theory is: Y.C.Zheng, J.Almlof Chem.Phys.Lett. 214, 397-401(1996) Y.C.Zheng, J.Almlof J.Mol.Struct.(Theochem) 288, 277(1996) K.Glaesemann, M.S.Gordon J.Chem.Phys. 108, 9959-9969(1998) K.Glaesemann, M.S.Gordon J.Chem.Phys. 110, 6580-6582(1999) K.Glaesemann, M.S.Gordon J.Chem.Phys. 112, 10738-10745(2000) References about gridding: A.D.Becke J.Chem.Phys. 88, 2547-2553(1988) C.W.Murray, N.C.Handy, G.L.Laming Mol.Phys. 78, 997-1014(1993) P.M.W.Gill, B.G.Johnson, J.A.Pople Chem.Phys.Lett. 209, 506-512(1993) A.A.Jarecki, E.R.Davidson Chem.Phys.Lett. 300, 44-52(1999) R.Lindh, P.-A.Malmqvist, L.Gagliardi Theoret.Chem.Acc. 106, 178-187(2001) S.-H.Chien, P.M.W.Gill J.Comput.Chem. 27, 730-739(2006) J.Grafenstein, D.Izotov, D.Cremer J.Chem.Phys. 127, 164113/1-7(2007) Gill's 1993 paper is the reference for SG1=.TRUE. Handy's 1993 paper is a reference for polar coordinates. Lebedev grids may be referenced as V.I.Lebedev, D.N.Laikov Doklady Math. 59, 477-481(1999) GAMESS uses Christoph van Wuellen's FORTRAN translation of these grids, originally coded in C by Laikov (www.ccl.net). --- exchange functionals Slater exchange: J.C.Slater Phys.Rev. 81, 385-390(1951) XALPHA is Slater with alpha=0.70 BECKE (often called B88) exchange: A.D.Becke Phys.Rev. A38, 3098-3100(1988) GILL (often called G96) exchange: P.M.W.Gill Mol.Phys. 89, 433-445(1996) OPTX exchange: N.C.Handy, A.J.Cohen Mol.Phys. 99, 403-412(2001) Depristo/Kress exchange: A.E.DePristo, J.E.Kress J.Chem.Phys. 86, 1425-1428(1987) --- correlation functionals VWN local correlation: S.H.Vosko, L.Wilk, M.Nusair Can.J.Phys. 58, 1200-1211(1980) This paper has five formulae in it, and since the 5th is a good quality fit, it states "since formula 5 is easiest to implement in LSDA calculations, we recommend its use". PZ81 correlation: J.P.Perdew, A.Zunger Phys.Rev.B 23, 5048-5079(1981) P86 GGA correlation: J.P.Perdew Phys.Rev.B 33, 8822(1986) PW local correlation (used in PW91): J.P.Perdew, Y.Wang Phys.Rev.B 45, 13244-13249(1992) LYP correlation: C.Lee, W.Yang, R.G.Parr Phys.Rev. B37, 785-789(1988) For practical purposes this is always used in a transformed way, involving the square of the density gradient: B.Miehlich, A.Savin, H.Stoll, H.Preuss Chem.Phys.Lett. 157, 200-206(1989) OP (One-parameter Progressive) correlation: T.Tsuneda, K.Hirao Chem.Phys.Lett. 268, 510-520(1997) T.Tsuneda, T.Suzumura, K.Hirao J.Chem.Phys. 110, 10664-10678(1999) --- exchange/correlation functionals PW91 exchange/correlation: J.P.Perdew, J.A.Chevray, S.H.Vosko, K.A.Jackson, M.R.Pederson, D.J.Singh, C.Fiolhais Phys.Rev. B46, 6671-6687(1992) EDF1 - empirical density functional #1, a tweaked BLYP developed for use with 6-31+G(d) basis sets, R.D.Adamson, P.M.W.Gill, J.A.Pople Chem.Phys.Lett. 284, 6-11(1998) MOHLYP - metal optimized OPTX exchange, half LYP correlation N.E.Schultz, Y.Zhao, D.G.Truhlar J.Phys.Chem.A 109, 11127-11143(2005) See also comp.chem.umn.edu/info/MOHLYP_reference.pdf for information about the related functional MOHLYP2. PBE exchange/correlation functional: J.P.Perdew, K.Burke, M.Ernzerhof Phys.Rev.Lett. 77, 3865-8(1996); Err. 78,1396(1997) revPBE (revised PBE exchange, but see RPBE below): Y.Zhang, W.Yang Phys.Rev.Lett. 80, 890(1998) RPBE (a different revision of PBE exchange): B.Hammer, L.B.Hansen, J.K.Norskov Phys.Rev.B 59, 7413-7421(1999) This revision retains the same increase in accuracy for atomization energies that revPBE affords, while rigorously preserving the correct Lieb-Oxford limit, unlike revPBE. PBEsol (modified PBE parameters, for solid properties): J.P.Perdew, A.Ruzsinszky, G.I.Csonka, O.A.Vydrov, G.E.Scuseria, L.A.Constantin, Z.Zhou, K.Burke Phys.Rev.Lett. 100, 136406/1-7(2008) The next two occur in the grid-free program only, various WIGNER exchange/correlation functionals: Q.Zhao, R.G.Parr Phys.Rev. A46, 5320-5323(1992) CAMA/CAMB exchange/correlation functionals: G.J.Laming, V.Termath, N.C.Handy J.Chem.Phys. 99. 8765-8773(1993) --- dispersion corrections: dispersionless Density Functional Theory (dlDF) K.Pernal, R.Podeszwa, K.Patkowski, K.Szalewicz Phys.Rev.Lett. 103, 263201/1-4(2009) This approach recognizes that density functionals may be optimized to reproduce interaction energies from which the dispersion energy has been subtracted. dlDF adjusts the M05-2X parameterization to accomplish this for a training set. A -D correction specific to dlDF was developed (see the paper's supplementary material) to address the now cleanly separated dispersion energy. The dldf-D correction term is available in the form of a Python script at the Szalewicz web site. Usage of dlDF by itself is not sensible. Local Response Dispersion (LRD) T.Sato, H.Nakai J.Chem.Phys. 131, 224104/1-12(2009) T.Sato, H.Nakai J.Chem.Phys. 132, 194101/1-9(2010) This computes dispersion energies using C6/C8 parameters evaluated from the final electron density of the molecule's DFT calculation. empirical dispersion correction (DC): This is developed in three successive versions by Grimme 1: S.Grimme J.Comput.Chem. 25, 1463-1473(2004) 2: S.Grimme J.Comput.Chem. 27, 1787-1799(2006) 3: S.Grimme, J.Antony, S.Ehrlich, H.Krieg J.Chem.Phys. 132, 154104/1-19(2010) which are applied to different functionals with different parameterizations of the correction. Setting DC=.TRUE. thus converts functionals such as BLYP/B3LYP/PBE/BP86/TPSS to BLYP-D, B3LYP-D, and so forth. See the papers for more details. The analytic calculation of hessians has been implemented into GAMESS for these corrections: H.Nakata, D.G.Fedorov, S.Yokojima, K.Kitaura, S.Nakamura J.Chem.Theory Comput. 10, 3689-3698(2014) A functional where the input keyword contains already the -D, namely B97-D, consists of a revamping of the B97 functional to remove its hybridization with HF exchange and reparameterization, as well as adding the dispersion correction: S.Grimme J.Comput.Chem. 27, 1787-1799(2006) A somewhat different form for the dispersion correction is used in the wB97-D functional. Selection of DFTTYP=B97-D or wB97-D does not require setting DC, as the D is already present in the name entered for DFTTYP. --- hybrids with HF exchange B3PW91 hybrid: A.D.Becke J.Chem.Phys. 98, 5648-5642(1993) B3LYP hybrid: A.D.Becke J.Chem.Phys. 98, 5648-5642(1993) P.J.Stephens, F.J.Devlin, C.F.Chablowski, M.J.Frisch J.Phys.Chem. 98, 11623-11627(1994) R.H.Hertwig, W.Koch Chem.Phys.Lett. 268, 345-351(1997) The first paper is actually on B3PW91 hybridization, and optimizes the mixing of five functionals with PW91 as the correlation GGA. The second paper then proposed use of LYP in place of PW91, without reoptimizing the mixing ratios of the hybrid. The final paper discusses the controversy surrounding which VWN functional is used in the hybrid. GAMESS uses VWN5 in its B3LYP hybrid, but see also B3LYPV1R to use the RPA parameterized VWN1 formula. B97 hybrid: A.D.Becke J.Chem.Phys. 107, 8554-8560(1997) B97-1 hybrid, a reparameterization of B97: F.A.Hamprecht, A.J.Cohen, D.J.Tozer, N.C.Handy J.Chem.Phys. 109, 6264-6271(1998) B97-2 hybrid, a reparameterization of B97: P.J.Wilson, T.J.Bradley, D.J.Tozer J.Chem.Phys. 115, 9233-9242(2001) B97-3 hybrid, a reparameterization of B97: T.W.Keal, D.J.Tozer J.Chem.Phys. 123, 121103-1/4(2005) B97-K and BMK hybrids, K=kinetics: A.D.Boese, J.M.L.Martin J.Chem.Phys. 121, 3405-3416(2004) HCTH93, HCTH120, HCTH147, and HCTH407 use training sets with the indicated number of atoms and molecules used to adjust the B97 functional: HCTH93 is defined in the B97-1 paper. HCTH120 and HCTH147: A.D.Boese, N.L.Doltsinis, N.C.Handy, M.Sprik J.Chem.Phys. 112, 1670-1678(2000) HCTH407: A.D.Boese, N.C.Handy J.Chem.Phys. 114, 5497-5503(2001) B98, Becke's reparameterization of B97: A.D. Becke J.Chem.Phys. 108, 9624-9631(1998) ...bringing to an end "the B97 family". X3LYP hybrid: X.Xu, Q.Zhang, R.P.Muller, W.A.Goddard J.Chem.Phys. 122, 014105/1-14(2005) PBE0 hybrid: C.Adamo, V.Barone J.Chem.Phys. 110, 6158-6170(1999) in the grid free program only, HALF exchange: This is programmed as 50% HF plus 50% B88 exchange. BHHLYP exchange/correlation: This is 50% HF plus 50% B88, with LYP correlation. Note: neither is the HALF-AND-HALF exchange/correlation: A.D.Becke J.Chem.Phys. 98, 1372-1377(1993) which he defined as 50% HF + 50% SVWN. --- meta-GGA functionals These are pure DFT meta-GGAs, unless the description explicitly says it is a hybrid! PKZB (a prototype of the TPSS family): J.P.Perdew, S.Kurth, A.Zupan, P.Blaha Phys.Rev.Lett. 82, 2544-2547(1999) tHCTH and tHCTHhyb=15% HF exchange: A.D.Boese, N.C.Handy J.Chem.Phys. 116, 9559-9569(2002) TPSS: J.P.Perdew, J.Tao, V.N.Staroverov, G.E.Scuseria Phys.Rev.Lett. 91, 146401/1-4(2003) J.P.Perdew, J.Tao, V.N.Staroverov, G.E.Scuseria J.Chem.Phys. 120, 6898-6911(2004) TPSSm, a modified TPSS improving atomization energies: J.P.Perdew, A.Ruzsinszky, J.Tao, G.I.Csonka, G.E.Scuseria Phys.Rev.A 76, 042506/1-6(2007) TPSSh, a 10% hybrid using TPSS: V.N.Staroverov, G.E.Scuseria, J.Tao, J.P.Perdew J.Chem.Phys. 119, 12129-12137(2003), erratum is J.Chem.Phys. 121, 11507(2004) revTPSS, "workhorse functional for CMP and QC" J.P.Perdew, A.Ruzsinsky, G.I.Csonka, L.A.Constantin, J.Sun Phys.Rev.Lett. 103, 026403/1-4(2009) VS98 (whose form is the prototype of the M06 family): T.V.Voorhis, G.E.Scuseria J.Chem.Phys. 109, 400-410(1998) U.Minnesota xc family: M05: Y.Zhao, N.E.Schultz, D.G.Truhlar J.Chem.Phys. 123, 161103/1-4(2005) M05-2X: Y.Zhao, D.G.Truhlar J.Comput.Chem.Theory Comput. 2, 1009-1018(2006) M06: Y.Zhao, D.G.Truhlar Theoret.Chem.Acc. 120,215-241(2008) M06-2X: ibid M06-HF: Y.Zhao, D.G.Truhlar J.Phys.Chem.A 110, 13126-13130(2006) M06-L: Y.Zhao, D.G.Truhlar J.Chem.Phys. 125, 194101/1-18(2006) SOGGA: Y.Zhao, D.G.Truhlar J.Chem.Phys. 128, 184109/1-8(2008) M08-HX and M08-SO: Y.Zhao, D.G.Truhlar J.Chem.Theory Comput. 4, 1849-1868(2008) SOGGA11: R.Peverati, Y.Zhao, D.G.Truhlar J.Phys.Chem.Lett. 2, 1991-1997(2011) SOGGA11-X: R.Peverati, D.G.Truhlar J.Chem.Phys. 135, 191102(2011) M11: R.Peverati, D.G.Truhlar J.Phys.Chem.Lett. 2, 2810-2817(2011) M11-L: R.Peverati, D.G.Truhlar J.Phys.Chem.Lett. 3, 117-124(2012) For reviews, please see the paper for M06, and also Y.Zhao, D.G.Truhlar Acc.Chem.Res. 41, 157-167(2008) These contain recommendations for choosing the one most appropriate to your problem. ---- long-range corrected functionals: LC-BLYP, LC-BOP, LC-BVWN: Y.Tawada, T.Tsuneda, S.Yanagisawa, Y.Yanai, K.Hirao J.Chem.Phys. 120, 8425-8433(2004) CAM-B3LYP: T.Yanai, D.P.Tew, N.C.Handy Chem.Phys.Lett. 393, 51-57(2004) wB97, wB97X, wB97X-D: J.-D. Chai, M.Head-Gordon J.Chem.Phys. 128, 084106/1-15(2004) J.-D. Chai, M.Head-Gordon Phys.Chem.Chem.Phys. 10, 6615-6620(2008) A review on the topic of long range corrections, which are also called 'range separated hybrids', is D.Jacquemin, E.A.Perpete, G.E.Scuseria, I.Ciofini, C.Adamo J.Chem.Theory Comput. 4, 123-135(2008) ---- "double-hybrid" ---- The B2PLYP family is a mixture of B88 and HF exchange, and a mixture of LYP and MP2 correlation: B2-PLYP: S.Grimme J.Chem.Phys. 124, 034108/1-15(2006) B2G-PLYP: A.Karton, A.Tarnopolsky, J.F.Lamere, G.C.Schatz, J.M.L.Martin J.Phys.Chem. A 112, 12868(2008) B2K-PLYP, B2T-PLYP: A.Tarnopolsky, A.Karton, R.Sertchook, D.Vuzman, J.M.L.Martin J.Phys.Chem. A 112, 3(2008) Double hybrids which are also "long range corrected" (and whose parameters depend on the basis set): wB97X-2, wB97X-2L: J.-D. Chai, M.Head-Gordon J.Chem.Phys. 131, 174105/1-13(2009) * * * * * Some of the functionals now present in GAMESS were made using code from the "density functional repository", http://www.cse.clrc.ac.uk/qcg/dft We thank Huub van Dam for his assistance with this, and particularly for providing the VWN1RPA functional. The Minnesota functionals are based on subroutines provided by the Truhlar group at the University of Minnesota. Some functionals, and particularly their high derivatives needed by TDDFT, were created by MAXIMA's algebraic manipulation, along the lines described by P.Salek, A.Hesselmann J.Comput.Chem. 28, 2569-2575(2007) * * * * * The paper of Johnson, Gill, and Pople listed below has a useful summary of formulae, and details about a gradient implementation. A paper on 1st and 2nd derivatives of DFT with respect to nuclear coordinates and applied fields is A.Komornicki, G.Fitzgerald J.Chem.Phys. 98, 1398-1421(1993) and see also P.Deglmann, F.Furche, R.Ahlrichs Chem.Phys.Lett. 362, 511-518(2002). A few of the many papers assessing the accuracy of DFT: B.Miehlich, A.Savin, H.Stoll, H.Preuss Chem.Phys.Lett. 157, 200-206(1989) B.G.Johnson, P.M.W.Gill, J.A.Pople J.Chem.Phys. 98, 5612-5626(1993) N.Oliphant, R.J.Bartlett J.Chem.Phys. 100, 6550-6561(1994) L.A.Curtiss, K.Raghavachari, P.C.Redfern, J.A.Pople J.Chem.Phys. 106, 1063-1079(1997) E.R.Davidson Int.J.Quantum Chem. 69, 241-245(1998) B.J.Lynch, D.G.Truhlar J.Phys.Chem.A 105, 2936-2941(2001) R.A.Pascal J.Phys.Chem.A 105, 9040-9048(2001) A.D.Boese, J.M.L.Martin, N.C.Handy J.Chem.Phys. 119, 3005-3014(2003) Y.Zhao, D.G.Truhlar, J.Phys.Chem.A 109, 5656-5667(2005) K.E.Riley, B.T.Op't Holt, K.M.Merz J.Chem.Theory Comput. 3, 407-433(2007) S.F.Sousa, P.A.Fernandes, M.J.Ramos J.Phys.Chem.A 111, 10439-10452(2007) Boese et al. include basis set comparisons, as well as functional comparisons. The final paper is a review of reviews, and encourages you to think past B3LYP, which after all dates from 1993! Of course there are assessments in many of the functional papers as well. On the accuracy of DFT for large molecule thermochemistry: L.A.Curtiss, K.Ragavachari, P.C.Redfern, J.A.Pople J.Chem.Phys. 112, 7374-7383(2000) P.C.Redfern, P.Zapol, L.A.Curtiss, K.Ragavachari J.Phys.Chem.A 104, 5850-5854(2000) On the accuracy of TD-DFT excitation energies: S.S.Leang, F.Zahariev, M.S.Gordon J.Chem.Phys. 136, 104101/1-12(2012) Spin contamination in DFT: 1. It is empirically observed that the~~values for unrestricted DFT are smaller than for unrestricted HF. 2. GAMESS computes the~~~~quantity in an approximate way, namely it pretend that the Kohn-Shan orbitals can be used to form a determinant (WRONG, WRONG, WRONG, there is no wavefunction in DFT!!!) and then uses the same formula that UHF uses to evaluate that determinant's spin expectation value. See G.J.Laming, N.C.Handy, R.D.Amos Mol.Phys. 80, 1121-1134(1993) J.Baker, A.Scheiner, J.Andzelm Chem.Phys.Lett. 216, 380-388(1993) C.Adamo, V.Barone, A.Fortunelli J.Chem.Phys. 98, 8648-8652(1994) J.A.Pople, P.M.W.Gill, N.C.Handy Int.J.Quantum Chem. 56, 303-305(1995) J.Wang, A.D.Becke, V.H.Smith J.Chem.Phys. 102, 3477-3480(1995) J.M.Wittbrodt, H.B.Schlegel J.Chem.Phys. 105, 6574-6577(1996) J.Grafenstein, D.Cremer Mol.Phys. 99, 981-989(2001) and commentary in Koch & Holthausen, pp 52-54. Orbital energies: The discussion on page 49-50 of Koch and Holthausen shows that although the highest occupied orbital's eigenvalue should be the ionization potential for exact Kohn-Sham calculations, the functionals we actually have greatly underestimate IP values. The 5th reference below shows how inclusion of HF exchange helps this, and provides a linear correction formula for IPs. The first two papers below connect the HOMO eigenvalue to the IP, and the third shows that while the band gap is underestimated by existing functionals, the gap's center is correctly predicted. However, the 5th paper shows that DFT is actually pretty hopeless at predicting these gaps. The 4th paper uses SCF densities to generate exchange-correlation potentials that actually give fairly good IP values: J.F.Janak Phys.Rev.B 18, 7165-7168(1978) M.Levy, J.P.Perdew, V.Sahni Phys.Rev.A 30, 2745-2748(1984) J.P.Perdew, M.Levy Phys.Rev.Lett. 51, 1884-1887(1983) A.Nagy, M.Levy Chem.Phys.Lett. 296, 313-315(1998) G.Zhang, C.B.Musgrave J.Phys.Chem.A 111, 1554-1561(2007)~~