Multiscale Modelling in Computational Heterogeneous Catalysis - Multiscale Molecular Methods in Applied Chemistry - page 73

catalysis. Compilations of density functional theory (DFT) applications to catalysis
may be found in the books by van Santen and Neurock [ 40 ] and van Santen
et al. (eds.) [ 41 ].

2 Computational Approaches

Some of the most widely used computational approaches will be briefly described
below, namely some quantum chemical methods, classical simulations by Monte
Carlo and Molecular Dynamics techniques and a few mesoscale methods.

2.1 Quantum Chemistry

In this section a recapitulatory description of the most common ab initio and density
functional approaches will be presented. Ab initio methods calculate the electron
properties of atoms and molecules at the absolute temperature ( T ¼ 0). The starting

point is in most cases the non-relativistic Schr € odinger equation

H c ¼ E c; (1)

where H is the Hamiltonian operator which is given by

X X X X X X

h 2
2 m e r

h 2
2 m k r

e 2 Z k
r ik þ

e 2
r ij þ

e 2 Z k Z l
r kl

i k

H ¼ ; (2)

i k i k i < j k < l

where i and j run over electrons, k and l run over nuclei, h is Planck's constant

divided by 2 p , m e is the mass of the electron, m k is the mass of the nucleus k , r

2 is
the Laplacian operator, e is the charge on the electron, Z is an atomic number and
r ab is the distance between particles (electrons or nuclei) a and b . The wave function
c

is thus a function of 3 n coordinates where n is the total number of particles
(electrons and nuclei).
The Born-Oppenheimer approximation leads to the electronic Schr € odinger

equation

H el c el ¼ E el c el : (3)

PES can only be computed by employing the Born-Oppenheimer approxima-
tion. The variational principle leads to an upper bound of the electronic energy
Ð c el H

c el d r

Ð c

el d r

E 0 : (4)

2