Density Functional Theory

In density functional theory (DFT) the energy of a system is given as a sum of six components:

E_DFT = E_NN + E_T + E_v + E_coul + E_exch + E_corr

The definitions for the nuclear-nuclear repulsion E_NN, the nuclear-electron attraction E_v, and the classical electron-electron Coulomb repulsion E_coul energies are the same as those used in Hartree-Fock theory. The kinetic energy of the electrons E_T as well as the non-classical electron-electron exchange energies E_exch are, however, different from those used in Hartree-Fock theory. The last term E_corr describes the correlated movement of electrons of different spin and is not accounted for in Hartree-Fock theory. Due to these differences, the exchange energies calculated exactly in Hartree-Fock theory cannot be used in density functional theory.

Various approaches exist to calculate the exchange and correlation energy terms in DFT methods. These approaches differ in using either only the electron density (local methods) or the electron density as well as its gradients (gradient corrected methods or generalized gradient approximation, GGA). Aside from these "pure" DFT methods, another group of hybrid functionals exists, in which mixtures of DFT and Hartree-Fock exchange energies are used.

1) Local methods
The only local exchange functional available in Gaussian is the Slater functional. Combination with the local VWN correlation functional by Vosko, Wilk and Nusair gives the Local Density Approximation (LDA) method. For open shell systems, using unrestricted wavefunctions, this is also referred to as the Local Spin Density Approximation (LSDA). This method is used in Gaussian with either the LSDA or SVWN keyword. A sample input for the frequency calculation for the methanol-1-yl radical using the SVWN functional and the 6-31G(d) basis set is as follows:

#SVWN/6-31G(d) freq

SVWN/6-31G(d) freq methanol-1-yl radical

0 2
C1
O2  1  r2
H3  2  r3  1  a3
H4  1  r4  2  a4  3  d4
H5  1  r5  2  a5  4  d5

r2=1.35454381
r3=0.97675734
r4=1.09713107
r5=1.09208721
a3=108.95266528
a4=119.30335589
a5=113.0329959
d4=-26.58949552
d5=-149.77421707

Please observe that the acrynom VWN is used to indicate different functionals in different software packages. While in Gaussian this keyword refers to functional III described in the VWN-paper (S. J. Vosko, L. Wilk, M. Nusair, Can. J. Phys. 1980, 58, 1200), most other packages refer to functional V in the same paper. This latter functional can also be used in Gaussian with the VWN5 keyword.

2) Gradient corrected methods
The gradient-corrected exchange functionals available in Gaussian include (given with their abbreviations in parenthesis):

Becke88 (B)
Perdew-Wang (PW91)
Modified Perdew-Wang by Barone and Adamo (MPW)
Gill96 (G96)
Perdew-Burke-Ernzerhof (PBE)

and the gradient-corrected correlation functionals are

LYP by Lee, Yang, and Parr (LYP)
Perdew-Wang (PW91)
Perdew 86 (P86)
Becke96 (B96)
Perdew-Burke-Ernzerhof (PBE)

In all cases, the names of these functionals refer to their respective authors and the year of publication. All combinations of exchange and correlation functionals are possible, the keywords being composed of the acronyms for the two functionals. The frequently used BLYP method, for example, combines Becke's 1988 exchange functional with the correlation functional by Lee, Yang, and Parr. An input file for the BLYP frequency calculation of the methanol-1-yl radical is:

#P BLYP/6-31G(d) freq

BLYP/6-31G(d) opt min methanol radical

0 2
C1
O2  1  r2
H3  2  r3  1  a3
H4  1  r4  2  a4  3  d4
H5  1  r5  2  a5  4  d5

r2=1.38575119
r3=0.98034945
r4=1.09658704
r5=1.09123991
a3=108.13230593
a4=118.45265645
a5=112.14316572
d4=-29.40173022
d5=-145.83450795

Another frequently used GGA functional is BP86 composed of the Becke 1988 exchange functional and the Perdew 86 correlation functional. The PW91 functional combines exchange and correlation functionals developed by the same authors in 1991. The keywords used in Gaussian for a particular GGA functional are combinations of the acronyms for exchange and correlation functionals. Use of the PW91 exchange and correlation functionals thus requires the keyword PW91PW91.

3) Hybrid functionals
The basic idea behind the hybrid functionals is to mix exchange energies calculated in an exact (Hartree-Fock-like) manner with those obtained from DFT methods in order to improve performance. Frequently used methods are:

Becke-Half-and-Half-LYP (BHandHLYP) uses a 1:1 mixture of DFT and exact exchange energies:
E_XC = 0.5*E_X(HF) + 0.5*E_X(B88) + E_C(LYP)
Becke-3-LYP (B3LYP) uses a different mixing scheme involving three mixing parameters:
E_XC = 0.2*E_X(HF) + 0.8*E_X(LSDA) + 0.72*DE_X(B88) + 0.81*E_C(LYP) + 0.19*E_C(VWN)
In this latter case, the B88 gradient correction to the local LSDA exchange energies carries its own scaling factor of 0.72. The three scaling factors have been derived through fitting the parameters to a set of thermochemical data (G1 set).
PBE0 uses a 1:3 mixture of DFT and exact exchange energies:
E_XC = 0.25*E_X(HF) + 0.75*E_X(PBE) + E_C(mPW91)
B98 (Becke98) is based on a 10-parameter equation scaling components of exact exchange, GGA exchange and GGA correlation. All parameters have been optimized simultaneously to fit thermochemica data collected in the extended G2 data set:
E_XC = 0.2198*E_X(HF) + E_X + E_C

The Becke-half-and-half-LYP method is used with the BHandHLYP keyword, the very popular Becke-3-LYP method is used with the B3LYP keyword, and the PBE0 method with the PBE1PBE keyword. A sample input file for the B3LYP frequency calculation on the methanol-1-yl radical is:

#P B3LYP/6-31G(d) freq

Becke-3-LYP/6-31G(d) opt min methanol radical

0 2
C1
O2  1  r2
H3  2  r3  1  a3
H4  1  r4  2  a4  3  d4
H5  1  r5  2  a5  4  d5

r2=1.37007444
r3=0.96903862
r4=1.08886325
r5=1.08381987
a3=108.87657926
a4=118.49962766
a5=112.61039767
d4=-29.19863769
d5=-146.73450089

It is important to note that implementations of the B3LYP and BHLYP functionals in different programs (Gaussian, Turbomole, Jaguar, Q-Chem) vary somewhat. Other hybrid functionals available in Gaussian are: B3P86 uses the P86 correlation functional instead of LYP, but retains the three parameters derived for B3LYP; B3PW91 uses the PW91 correlation functional instead of LYP, but retains the three parameters derived for B3LYP; B1B96, a one parameter functional developed by Becke using the Becke 96 correlation functional; MPW1PW91 developed by Barone and Adamo using a modified version of the PW91 exchange functional in combination with the original PW91 correlation functional and a mixing ratio of exact and DFT exchange of 0.25 : 0.75.

4) User-defined models
Calculation of reaction and activation energies in either radical reactions or transition metal mediated reactions often shows that the mixing ratio of HF- and DFT-exchange is not optimal for a given purpose. The definition of user-defined models tailored to a certain purpose is therefore sometimes necessary. This can be achieved by first choosing gradient corrected exchange and correlation functionals from the list of available choices and then specifying the mixing ratios of the various exchange and correlation energy terms. For this latter step Gaussian uses the general formula:

E_XC = P₂E_X^HF + P₁(P₄E_X^Slater + P₃dE_X^non-local) + P₆E_C^local + P₅dE_C^non-local

The values for the parameters P₁ - P₆ are provided through the IOP switches IOP(5/45=xxxxyyyy), IOP(5/46=xxxxyyyy), and IOP(5/47=xxxxyyyy) with P₁ = xxxx/1000 from IOP(5/45) etc. The use of this facility will be illustrated with the MPW1K functional by Truhlar and coworkers (B. J. Lynch, P. L. Fast, M. Harris, D. G. Truhlar, J. Phys. Chem. A 2000, 104, 4811) developed to optimize reaction and activation energies of free radical reactions. This method is very similar to the MPW1PW91 hybrid functional, but uses a mixing ratio of exact and DFT exchange energy terms of 0.428 : 0.572. An input file for the methanol-1-yl radical frequency calculation at the MPW1K/6-31G(d) level is:

#P MPWPW91/6-31G(d) freq
 IOP(5/45=10000428) IOP(5/46=05720572) IOP(5/47=10001000)

MPW1K/6-31G(d) opt min methanol radical

0 2
C1
O2  1  r2
H3  2  r3  1  a3
H4  1  r4  2  a4  3  d4
H5  1  r5  2  a5  4  d5

r2=1.35355619
r3=0.95690342
r4=1.08170148
r5=1.07685321
a3=109.38001654
a4=118.56662303
a5=112.99211737
d4=-29.64928765
d5=-147.42512309

The gradient corrected MPWPW91 functional is chosen here as the basis of the hybrid method and the amount of exact exchange is specified through parameter P₂. The composition of the functional is also described in the output file in a compact format:

 IExCor= 908 DFT=T Ex=mPW+HF Corr=PW91 ScaHFX= 0.4280
 ScaDFX=  0.5720  0.5720  1.0000  1.0000

The value given after ScaHFX corresponds to P₂ while the four values listed after ScaDFX are P₃ - P₆. Please observe that theses lines are only printed when using the extended output option of Gaussian. Before using this facility with a particular version of Gaussian, please make sure that your IOP-directives are read correctly. In some versions of Gaussian, the options P₁ - P₆ are set with IOP(3/76) - IOP(3/78) instead of IOP(5/45) - IOP(5/47). In this latter case, the values of P1 - P6 are given in five-digit format. The keyword line for the PBE0 frequency calculation thus becomes:

#P PBEPBE/6-31G(d) freq
 IOP(3/76=1000002500) IOP(3/77=0750007500) IOP(3/78=1000010000)

Absolute and relative CPU-times for DFT- and Hartree-Fock-calculations depend dramatically on the size of the system under investigation, the computer hardware and the operating system used. The following table contains CPU times (in seconds) for all frequency calculations on the methanol-1-yl radical described above with Gaussian 03, Rev. B.03. The platforms used are the CIP-room Pentium 4/2 GHz computers under LINUX. In all cases the frequency calculation has been performed on the structure optimized at the given theoretical level. The 6-31G(d) basis set has been used in all cases.

method	CPU (P4/2GHz) [sec]	vib(1) [cm^-1]	vib(2) [cm^-1]	vib(3) [cm^-1]
USVWN/6-31G(d)	31	3690	1455	1036
UBLYP/6-31G(d)	69	3599	1463	1042
UB3LYP/6-31G(d)	71	3761	1508	1072
UMPW1K/6-31G(d)	73	3950	1552	1102
UHF/6-31G(d)	11	4123	1627	1156
obsvd.	-	3650	1459	1056

Despite all platform-dependend variations, it is clear that hybrid DFT calculations are similarly laborious as GGA calculations, while local DFT methods are somewhat and HF-calculations much cheaper. In matching calculated (harmonic) gas phase vibrational frequencies with experimentally observed anharmonic values measured in a matrix (R. D. Johnson III, J. W. Hudjens, J. Phys. Chem., 1996, 100, 19874), please remember that uniform scaling is often performed to improve correspondence (see our earlier discussion of this point). The values reported here are unscaled. A general trend visible in this example is the increase of vibrational frequency wavenumbers with an increase in the amount of Hartree-Fock exchange contributions (varying between 0 and 100% from the second entry to the bottom in Table 1).

5) Further technical considerations
Evaluation of the exchange-correlation energies of all density functional methods implemented in Gaussian involves a grid-based numerical integration step. The computational effort required for this step strongly depends on the selected grid size. The larger the number of integration points per atom, the higher is the computational cost and the better the numerical accuracy of the calculation. Several predefined grids are available in Gaussian, which can be used through the keyword

int=(gridsize) or int=(Grid=gridsize)

Alternatively, the grid size can also be specified with IOP settings, using either IOP(5/44=gridsize) or IOP(3/75=gridsize), depending on the version of Gaussian. The following are acceptable values for gridsize:

gridsize	corresponding IOP setting	(radial shells * angular points)	CPU [sec]	E_tot	frequencies (cm^-1)
CoarseGrid	2	(35*110), pruned	33.8	-115.0520127	-110.9325 -0.0015 -0.0009 -0.0008 24.6825 59.6546
SG1Grid	1	(50*194), pruned	45.1	-115.0520371	-30.5802 0.0015 0.0015 0.0016 10.5877 29.2740
finegrid	4	(75*302), pruned	77.1	-115.0520326	-20.7294 -8.3913 0.0009 0.0009 0.0014 11.1426
ultrafine	5	(99*590), pruned	192.7	-115.0520324	-0.0013 -0.0013 0.0011 2.6708 4.0642 14.0308
mmm,nnn	mmmnnn	(mmm*nnn), not pruned	-	-	-

The CPU times and total energies listed here (in sec.) together with the grid specifications are those for the UB3LYP/6-31G(d) frequency calculation on the methanol radical mentioned before on the CIP room Pentium 4 LINUX PCs. It is clear from these results that neither the total electronic energy nor the low-frequency vibrations converge quickly with increasing grid size. Thus, for each application it must be decided, whether the substantial computational effort for a very high grid size is justified. It is also important to note that the energy variation with grid size is significant enough that all calculations for stationary points on the same potential energy surface must be performed with the same grid in order to be comparable!

In addition to the grid selection for integral evaluation using the int=(gridsize) keyword, a particular grid can also be selected for the calculation of second derivatives using the CPHF=gridsize keyword. The available options for gridsize are identical in both cases. If no explicit grid is specified through the CPHF keyword, the following relations exist between the grids used for integral and second derivative calculations:

gridsize used for integral evaluation	grid used for second derivatives
CoarseGrid	CoarseGrid
SG1Grid	CoarseGrid
finegrid	CoarseGrid
ultrafine	SG1Grid
(mmm*nnn)	(mmm*nnn)

These combinations are usually appropriate for most systems. The default grid size for integral evaluation in the most recent versions of Gaussian is finegrid. Better grids are required for kinetic isotope effect calculations, the optimization of stationary points on very flat potential energy surfaces, and IRC-following on flat potential energy surfaces. Increasing the CPHF grid size over what is specified per default through the integral grid selection is rarely helpful. Smaller grids can be quite useful to speed up geometry optimizations of large systems, especially when still far away from the next stationary point.

6) Treating open shell systems
For practical reasons the DFT methods also use a DFT "wavefunction" constructed as an antisymmetrized product of one electron functions (now termed Kohn-Sham orbitals). As in Hartree-Fock theory, electrons of unlike spin can either use the same spatial orbitals (restricted DFT methods) or different spatial orbitals (unrestricted DFT methods). Specification of a restricted open shell wavefunction for the Becke3LYP hybrid functional requires the ROB3LYP keyword, while the unrestricted version is specified using UB3LYP. If no explicit choice is made in the input file, open shell systems are treated with the unrestricted approach by default. It should be noted that unrestricted DFT methods (GGA or hybrid functionals alike) are much less affected by the phenomenon of spin contamination than Hartree-Fock theory and also give much better predictions of ESR spectra of organic radicals. The UB3LYP/6-31G(d) calculation of the allyl radical is achieved with the following input file:

#UBecke3LYP/6-31G(d) scf=(tight)

UB3LYP/6-31G(d) ally radical (C2v)

0 2
H
C,1,r2
C,2,r3,1,a3
C,2,r3,1,a3,3,180.
H,3,r5,2,a5,1,0.
H,4,r5,2,a5,1,0.
H,3,r7,2,a7,1,180.
H,4,r7,2,a7,1,180.

r2=1.09054601
r3=1.38611149
r5=1.08492921
r7=1.08690606
a3=117.45043254
a5=121.64506508
a7=121.12827199

Comparison of the expectation value of the <S²> operator calculated at the UHF/6-31G(d) level (0.9729) with that obtained at UB3LYP/6-31G(d) (0.7818) shows that the spin contamination problem is much reduced with DFT methods. From the construction principle of the hybrid DFT methods it is clear that a larger HF-exchange component will also give rise to larger amounts of spin contamination. Using the allyl radical at the UB3LYP/6-31G(d) structure as an example, the <S²> value amounts to 0.7650 at the BLYP/6-31G(d) level and to 0.8228 at the BHandHLYP/6-31G(d) level. The precise values of the expectation value of the <S²> operator will also depend to some extend on the basis set used.

Another important point to consider in DFT calculations concerns the self interaction error resulting from an artificial interaction of the electron density with itself. While the definition of the exchange and Coulomb energies in Hartree-Fock theory leads to a perfect cancellation of these self interaction terms, the cancellation is incomplete in DFT methods. As nicely explained by Koch and Holthausen in their monograph on DFT methods, this can be easily demonstrated using the hydrogen atom. The exact energy for this one-electron system is -0.50000 Hartree. As there is no second electron in the system, no correlation energies have to be calculated and Hartree-Fock theory is able to provide the exact answer in the complete basis set limit. Using a basis set considered "large" for molecular systems such as the cc-pVQZ basis set, a value of -0.499946 Hartree is indeed obtained. Due to the parameterized exchange and correlation functionals, however, much more negative values can be obtained in particular with hybrid DFT functionals. Using the same basis set, the UB3LYP functional predicts a value of -0.502346 Hartree!!