34
2 Computational Methods
2.13.4 Hybrid DFT
Another improvement is to mix a portion of exact exchange from Hartree–Fock theory
(HF with the rest of the exchange-correlation energy from other DFT exchanges
defining E
XC as
E
XC
= c HF E
X
HF + c DFT E
XC
DFT
(2.37)
The parameters determining the weight of each individual functional are typically specified by fitting the functional’s predictions to experimental or accurately
calculated thermochemical data.
The most popular example is B3LYP which stands for Becke, 3-parameter, Lee–
Yang–Parr (Lee et al. 1988).
2.13.5 Double-Hybrid Density Functionals (DHDFT)
Such functionals as B2-PLYP (sometimes also called B2PLYP) expand the DFT
exchange-correlation energies into four terms:
E
XC
= c x E
X
HF + (1 − c x )E
X
DFT + (1 − a c )E
C
DFT + a c E
C
MP2
(2.38)
The first two terms describe the exchange energy as a mix of terms derived from
GGA functionals and exact exchange HF. Likewise, the correlation energy is a sum
of terms derived from GGA functionals and the correlation energy E
C
MP2 calculated
with second-order perturbation theory (MP2). It is important to recognize that all four
terms are derived from the same Kohn–Sham orbitals. The first DHDFT method of
general applicability B2PLYP was proposed by Grimme (2006a). These functionals
are better to take into account van der Waals forces. The convergence of the double
hybrid can be more difficult than with simpler functionals. The running time is
roughly twice of a B3LYP job. Furthermore, the double hybrids need bigger basis
sets.
One problem common to all these functionals is the treatment of dispersion interactions. The easiest and most popular solution is the D3 model of Grimme (2006b)
which uses precomputed reference data for dispersion coefficients, and the concept of
“fractional coordination number” to mimic the environment of an atom in a molecule.
The functional B3LYP is at present one of the most popular approximations in
chemistry and it is used in a considerable number of papers. There are several reasons
for this success. First, it accounts for electron correlation, and yet the computational
cost is of the same order as the HF method (i.e., proportional to n
4 , n being the
size of the system, whereas, for the MP2 method, the cost is proportional to n
5 ). One
further advantage is that the basis set convergence is much faster than in conventional
correlated methods: A basis set of triple-zeta quality is sufficient in most cases and,
2 Computational Methods
2.13.4 Hybrid DFT
Another improvement is to mix a portion of exact exchange from Hartree–Fock theory
(HF with the rest of the exchange-correlation energy from other DFT exchanges
defining E
XC as
E
XC
= c HF E
X
HF + c DFT E
XC
DFT
(2.37)
The parameters determining the weight of each individual functional are typically specified by fitting the functional’s predictions to experimental or accurately
calculated thermochemical data.
The most popular example is B3LYP which stands for Becke, 3-parameter, Lee–
Yang–Parr (Lee et al. 1988).
2.13.5 Double-Hybrid Density Functionals (DHDFT)
Such functionals as B2-PLYP (sometimes also called B2PLYP) expand the DFT
exchange-correlation energies into four terms:
E
XC
= c x E
X
HF + (1 − c x )E
X
DFT + (1 − a c )E
C
DFT + a c E
C
MP2
(2.38)
The first two terms describe the exchange energy as a mix of terms derived from
GGA functionals and exact exchange HF. Likewise, the correlation energy is a sum
of terms derived from GGA functionals and the correlation energy E
C
MP2 calculated
with second-order perturbation theory (MP2). It is important to recognize that all four
terms are derived from the same Kohn–Sham orbitals. The first DHDFT method of
general applicability B2PLYP was proposed by Grimme (2006a). These functionals
are better to take into account van der Waals forces. The convergence of the double
hybrid can be more difficult than with simpler functionals. The running time is
roughly twice of a B3LYP job. Furthermore, the double hybrids need bigger basis
sets.
One problem common to all these functionals is the treatment of dispersion interactions. The easiest and most popular solution is the D3 model of Grimme (2006b)
which uses precomputed reference data for dispersion coefficients, and the concept of
“fractional coordination number” to mimic the environment of an atom in a molecule.
The functional B3LYP is at present one of the most popular approximations in
chemistry and it is used in a considerable number of papers. There are several reasons
for this success. First, it accounts for electron correlation, and yet the computational
cost is of the same order as the HF method (i.e., proportional to n
4 , n being the
size of the system, whereas, for the MP2 method, the cost is proportional to n
5 ). One
further advantage is that the basis set convergence is much faster than in conventional
correlated methods: A basis set of triple-zeta quality is sufficient in most cases and,