Multiscale Modelling in Computational Heterogeneous Catalysis - Multiscale Molecular Methods in Applied Chemistry - page 79

cluster models of increasing size which allows extrapolation to the periodic MP2
limit. Basis set truncation errors (BSSE) are estimated by extrapolation of the MP2
energy to the CBS limit. Contributions from higher-order correlation effects are
accounted for by CCSD(T) coupled cluster calculations. The sum of all contribu-
tions provides the “final estimates” for adsorption energies and energy barriers.
Dispersion contributes significantly to the PES. As a result, the MP2:DFT potential
energy profile is shifted downward compared to the PBE profile. More importantly
this shift is not the same for reactants and transition structures due to different self-
interaction errors.
Other recent developments in electronic structure theory were reviewed by
Sherrill [ 62 ] and Huang et al. [ 195 ].
Only a very brief description of some quantum chemical correlation approaches
could be given. For details see [ 5 , 6 ].
Over the last 10 years the DFT became more and more popular because of its
high computational efficiency and good accuracy [ 2 , 3 ]. The basis for DFT is the
proof of Hohenberg and Kohn [ 63 ] that the ground state electronic energy is
determined completely by the electron density

. In other words, there exists a
one-to-one correspondence between the electron density of a system and the energy.
Within DFT all aspects of the electronic structure of the system of interacting
electrons in an “external” potential V ext (r) generated by atom cores are completely
determined by the electronic charge density

r

. In DFT, the total energy is decom-
posed into three contributions, a kinetic energy, a Coulomb energy due to classical
electrostatic interactions among all charged particles in the system and an
exchange-correlation energy term that captures all many-body interactions. Unfor-
tunately the exact expressions that should be used for the many-body exchange and
correlation interactions are unknown. The local density approximation (LDA)
turned out to be computationally convenient and very accurate. The LDA assumes
that the density locally can be treated as a uniform electron gas, or equivalently that
the density is a slowly varying function. An improvement is the local spin density
approximation (LSDA) which is useful in cases where the

r

-spin densities
are not equal. Kohn and Sham [ 1 ] developed a self-consistent system including
exchange-correlation effects. For some problems further improvements have to be
made. The most common approach is the generalised gradient approximation
(GGA). These approximations depend upon the gradient of the electron density at
each point in space and not just on its value. These gradient corrections are typically
divided into separate exchange and correlation contributions. A variety of gradient
corrections have been proposed in the literature. A very popular one is the B3LYP
functional [ 64 ]. In most DFT programs for calculating the electronic structure of
molecules the Kohn-Sham orbitals (KS) are expressed as a linear combination of
atomic-centred basis functions:

a - and b

X

K

c i ð r Þ¼ c ni f n : (15)

n¼ 1