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In-Depth Information
(
"
#
ℏ
"
#
2
X
α
2
2
2
2
m
X
i
2
2
2
2
2
1
M
α
ℏ
∂
þ
∂
þ
∂
∂
x
i
þ
∂
y
i
þ
∂
X
2
α
Y
2
α
Z
2
α
z
i
∂
∂
∂
∂
∂
∂
X
iα
r
iα
þ
X
αʲ
þ
X
ij
Z
α
e
2
z
α
Z
ʲ
e
2
r
αʲ
e
2
r
ij
gΨ
n
¼
E
n
Ψ
n
:
Here
X
α
,
Y
α
,
Z
α
and
x
i
,
y
i
,
z
i
denote spatial coordinates of the nuclei (
α
) and of the
electrons
i
) of the molecule.
M
α
,
m
—masses of the atomic nuclei and electrons.
r
αi
,
r
αʲ
,
r
ij
—distances between the nucleus and the electrons, between two
nuclei, and between two electrons, respectively.
ℏ
—Planck's constant.
The physical meaning of this equation is simple. The first and the second terms
on the right side correspond to the kinetic energy of the nuclei and electrons of the
molecular system. Subsequent terms describe electrostatic interactions between
nuclei and electrons as well as interactions of nuclei and electrons with each
other. A quantum system is characterized by discrete energy levels
E
n
and wave
functions
Ψ
n
or each energy state. The square modulus of this function reflects the
probability of finding nuclei and electrons in the corresponding point in space.
Let us consider molecules with discrete energy levels
E
n
, with their electronic
states described by the wave functions
Ψ
n
. The square modulus of the wave
function determines the probability of finding electrons in space.
The Schr¨dinger equation was derived under some very serious assumptions.
First of all, it does not take into account the electron spin, which has to be
introduced based on physical considerations.
The Schr¨dinger equation is a partial differential equation for which exact
solutions are in general unknown. Nevertheless, experience shows that approximate
ab initio numerical methods (i.e., methods not making use of additional experi-
mental information) developed so far allow for calculating molecular characteris-
tics with an error not exceeding the experimental error.
For some fairly simple systems the Schr¨dinger equation does have an exact
solution. One of them is the problem of the hydrogen atom, which plays a funda-
mental role in describing atomic and molecular structure. The Schr¨dinger equation
for the hydrogen atom (one nucleus and one electron) is written in the laboratory
coordinate system (see Fig.
3.2
)as
(
"
#
"
#
)
2
2
2
2
2
2
2
2
Ze
2
r
ℏ
∂
X
l
þ
∂
Y
l
þ
∂
ℏ
∂
x
l
þ
∂
y
l
þ
∂
Ψ
ln
¼
E
ln
Ψ
ln
,
Z
l
z
l
2
M
2
m
∂
∂
∂
∂
∂
∂
where the index
l
indicates that these distances are measured from the origin of
coordinates (see Fig.
3.2
) and the distance
r
from the nucleus of an atom to the
electron.
