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
              becke97-1, becke-2, becke-3
   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 
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 
   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 a 
forthcoming paper,
   F.Zahariev, S.Sok, M.S.Gordon (to be submitted)

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 strengths.  TPA calculations
   a) should not use the Tamm/Dancoff approximation,
   b) may include solvent effects by EFP or PCM,
   c) make sense only at a single geometry: use the 
ordinary TD-DFT program to optimize excited state 
geometries, if needed.

Solvation effects on the excited state energies can be 
added 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

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)
    "Perspecitve on density functional theory"
     Kieron Burke  J.Chem.Phys. 136, 150901/1-9(2012)

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:
     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 (

          --- 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 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 

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 
   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 
    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 on.

          --- 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)
  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)

  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:

    Y.Tawada, T.Tsuneda, S.Yanagisawa, Y.Yanai, K.Hirao
        J.Chem.Phys. 120, 8425-8433(2004)

    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",
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)

created on 6/21/2013