Chemistry Reference
In-Depth Information
principles the origin of the quantum numbers (n,
, and m) that char-
acterize these orbitals, and which arise from the regularity conditions
imposed upon the mathematical solutions. We shall move next to
consideration of STOs and GTOs, giving some general definitions and
some simple one-centre one-electron integrals which will be needed in
Chapter 4.
'
3.2 HYDROGEN-LIKE ATOMIC ORBITALS
HAOs are obtained as exact solutions of the Schroedinger eigenvalue
equation for the atomic one-electron system:
1
2
r
Z
r
H
c ¼
E
c
H
¼
2
ð
3
:
1
Þ
whereZ is the nuclear charge, giving for Z
¼
1
;
2
;
3
;
4
; ...
the isoelectronic
series H, He
þ
,Li
2
þ
,Be
3
þ
,
. In what follows we shall go briefly through
the solution of the eigenvalue equation, Equation 3.1. Solution of the
second-order partial differential equation embodied in (3.1) necessarily
involves the steps outlined in Sections 3.2.1-3.2.5.
...
3.2.1 Choice of an Appropriate Coordinate System
The spherical symmetry of the potential energy V(r) suggests use of the
spherical coordinates
ð
r
; u; wÞ
(see Figure 1.1). We have seen that in this
case the Laplacian operator
r
2
r
and into an angular part which depends on the square of the angular
momentum operator L
2
separates into a radial Laplacian
r
2
. In this way, it is possible to separate radial from
angular equations if we put
cð
r
; u; wÞ¼
R
ð
r
Þ
Y
ðu; wÞ
ð
3
:
2
Þ
Upon substitution in (3.1) we obtain the following two separate
differential equations:
r
2
R
¼
0
d
2
R
dr
2
þ
2
r
dR
dr
þ
2
Z
r
E
þ
ð
3
:
3
Þ
