Chemistry Reference
In-Depth Information
numerous three- and four-electron integrals, and got very good results nevertheless.
The first MP2-R12 calculations on molecules were reported by Klopper and
Kutzelnigg [
50
]. Developments of explicitly correlated wave functions can be
found in a book by Rychlewski [
51
]. There are many extensions of the R
12
method.
Examples are calculations of intermolecular interaction energies [
52
] and an
MP2-R12 method in which each electron pair uses a set of contracted Gaussian-
type Geminals (GTG) with fixed exponents, whose coefficients are optimised
linearly [
53
].
At present no single method can give quantitative results (accuracy of 1 kJ/mol
or less) for practical problems. Therefore, so-called composite methods have been
developed in which the contributions are computed at different levels of theory in
accordance with the accuracy desired. For example, the Gaussian-4 theory (G4)
has been developed for the calculation of energies of molecules containing first- and
third-row main-group atoms [
54
]. Starting points are Becke-Lee-Yang-Parr
B3LYP/6-31G(2df,p) geometries and harmonic vibrational frequencies, scaled by
a factor of 0.9854, for the zero-point vibrational energies (ZPVE). The HF energies
have been extrapolated to the limit of a complete basis, and Møller-Plesset-4/6-31G
(d) single-point calculations are executed. When the MP4/6-31(d) energy E
MP4
is
considered as the base line of G4 theory, then the following energy corrections are
added in the G4 composite scheme:
E
G
4
¼
E
MP
4
þ D
E
HF
þ D
E
CCSDðTÞ
þ D
E
diffuse
þ D
E
polarisation
þ D
E
G
3
L
arg
eXP
þ D
E
HLC
þ D
E
SO
þ D
E
ZPVE
;
(14)
where
D
E
HF
correction for the HF basis set limit
D
E
CCSD(T)
E
(CCSD(T))/6-31G(d)-
E
(MP4/6-31G(d))
D
E
diffuse
corrections for diffuse functions
D
E
polarization
corrections for polarisation functions
D
E
G3LargeXP
correction for larger basis set effects
D
E
HLC
empirical energy correction that depends on the number of occupied
a
b
and
orbitals
D
E
SO
spin-orbit term, taken from experiment or accurate calculations
zero-point vibrational energy
The Gaussian-4 theory was tested on the G3/05 test set [
55
] including 454
energies. The overall average absolute deviation for these energies was found to
be about 3.5 kJ/mol.
There are several other composite approaches, for example, complete basis set
(CBS) models [
56
], focal-point analysis [
57
], multi-coefficient correlation methods
[
58
], “high-accuracy extrapolated ab initio thermo-chemistry” (HEAT) [
59
] and the
Weizmann-4 theory [
60
]. Hansen et al. [
61
] have employed the so-called MP2:DFT
[
193
] scheme for analysing benzene ethylation over H-ZSM-5. Density functional
calculations applying periodic boundary conditions [Perdew-Burke-Ernzerhof
(PBE) functional] were combined with MP2 energy calculations on a series of
D
E
ZPVE
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