Molecular Elements of Computers - Molecular Computing: Origins and Promises - page 52

Fig. 3.2 Coordinate system
for the quantum-chemical
calculation of the
hydrogen atom

Upon moving the origin of coordinates to the center of mass, this equation splits
into two ordinary differential equations. The first one describes the motion of a
particle with the mass M + m , while the second one describes electronic character-
istics of the hydrogen atom:

( " # )

2 2 2

2

Ze 2
r

ℏ

∂

x 2 þ ∂ y 2 þ ∂

Ψ n ¼ E n Ψ n :

2 ʼ z 2

∂ ∂ ∂

Here the coordinates x , y , z are measured from the center of mass, and ʼ is the

reduced mass of the hydrogen atom:

1
ʼ

1
M þ

1
m :

¼

A remarkable property of the hydrogen atom is its high spherical symmetry. By
going over to spherical coordinates with the origin at the nucleus of the hydrogen
atom, an exact solution of the Schr¨dinger equation can be obtained. It is defined by
three quantum numbers n , l , m , with electronic energy levels depending only on the
principal quantum number n :

A
n 2 , A ¼

Z 2
2

E n ¼

Ψ nlm ¼ const R n ðÞY l

me 4 ,

ð ˑ; ˆ Þ:

ℏ

Thus, the energy levels of the hydrogen atom are degenerate. Their degree of
degeneracy is determined by the quantum numbers l , m . In accordance with the
solution of the Schr¨dinger equation, restrictions are imposed on the quantum
numbers:
l n 1, an integer which can take values 0, 1, 2,

n 1, if l ¼ 0, the
electronic state is called s -state, with l ¼ 1— p -state, with l ¼ 2— d -state, etc.
A quantum number “ m ” may take integer values from l to + l . It determines the
degree of degeneracy of each l -sublevel, with l ¼ 0 corresponding to no degener-
acy, l ¼ 1 to threefold degeneracy, and l ¼ 2 to fivefold degeneracy.

...