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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
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,
in case you need to reproduce numbers from older versions.
Since April 2008, the default is
The default for the more accurate meta-GGA functionals was
changed in February 2012 to a more accurate grid
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
Very accurate (converged) results come from using the "army
Turn to the next page to see numerical results.

A numerical demonstration of grid accuracies can be
obtained from ethanol, DFTTYP=BECKE:
sg1=.true.                -154.105070   0.010837    11
nrad=96 nthe=12 nphi=24   -154.104863   0.010724    56
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"
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:
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
J.Chem.Phys. 120, 8425-8433(2004)
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
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)
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
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",
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
E.J.Baerends, O.V.Gritsenko
J.Phys.Chem.A 101, 5383-5403(1997)

Some philosophy about designing functionals at each rung of
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)

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,
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)
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.
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 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.
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:
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

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:

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

---- long-range corrected functionals:

LC-BLYP, LC-BOP, LC-BVWN:
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.Chem.Phys. 128, 084106/1-15(2004)
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,

---- "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):
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
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)
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)
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)

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