A Toy Model of Chemical Reaction Networks - TBI - Universität Wien

3.1. EXTENDED HÜCKEL THEORY 15
is the distance between the nuclei or electrons i and j. e − and nuc indicate
sums running over all electrons and nuclei, respectively. V n , the electrostatic
nuclear repulsion, does not depend on electron coordinates and is thus an
additive constant to the final energy. For more clarity, it is omitted from the
following equations.
Hartree proposed the orbital approximation, in which the wave function
is built from Slater determinants,
∣ ∣∣∣∣∣∣∣∣ φ 1 (1) φ 2 (1) · · · φ N (1)
Ψ SD = √ 1 φ 1 (2) φ 2 (2) · · · φ N (2)
. N! . . .. . (3.5)
.
φ 1 (N) φ 2 (N) · · · φ N (N) ∣
The φ are one-electron wave functions, called molecular orbitals (MO) or
spin orbitals. The energy of a single Slater determinant will be used lateron
for the variational principle. It may be written as
∫
E el = Ψ SD Ĥ el Ψ SD d τ (3.6)
and decomposed into one-electron or core integrals ∫ φ i (i)ĥiφ i (i) dτ, twoelectron
Coulomb integrals ∫ φ i (i)φ j (j)ĝ ij φ i (i)φ j (j) dτ, and two-electron exchange
integrals ∫ φ i (i)φ j (j)ĝ ij φ j (i)φ i (j) dτ.
Semi-empirical methods, Extended Hückel Theory [62, 65] for instance,
now further approximate the wave function by building it from one
single Slater determinant, thus neglecting electron correlation.
First, using the basis set or LCAO approximation, each φ is expanded
as a linear combination of atomic orbitals χ (LCAO) :
φ i = ∑ j
C ij χ j . (3.7)
The definition of the electron density ρ in LCAO will be needed in sect. 3.4
and will be shortly presented here. The electron density or the probability
of finding an electron in a MO φ i , occupied by n i electrons, at the position
defined by the position vector r is
ρ i (r) = n i φ 2 i (r) . (3.8)