Chemical Bonding in Molecules and Solids
TEXTBOOK READING
:
BLB-10
, Chapters 8 and 9, pp. 300-397.
The basis for chemical bonding models in molecules and solids also relies on solving the
Schrödinger wave equation.
For such structures, this is an extremely complicated problem, and
cannot be solved exactly because of the many particles making up a molecule, i.e., protons,
neutrons (in the nuclei) and electrons.
To see how complicated the problem can be, even for a
very simple molecule, let’s look at the complete Schrödinger wave equation for the H
2
molecule.
In this equation, we have H atom #1 (proton 1 + electron 1) and H atom #2 (proton 2 + electron
2); the masses of the protons are
M
, the coordinates are
R
i
= (
X
i
,
Y
i
,
Z
i
); the masses of the
electrons are
m
, the coordinates are
r
i
= (
x
i
,
y
i
,
z
i
).
22
2
2
2
2
2
2
2
2
2
2
11
1
2
222
2
2
2
2
2
2
2
111222
2
21
1122
2
12 21
1
2
8
8
hdddddd
M
d
Xd
Yd
Zd
Z
m
d
xd
yd
zd
z
ee
e
e
π
⎡⎤
⎛⎞
−+
+
+
+
+
⎢⎥
⎜⎟
⎝⎠
+
+
+
+
+−+
−−
−
−++
−
⎣⎦
RR
r
R
rR
rR rR
rr
()
(
)
1
212
12
1
,; ,
,
QQ
Q
E
Ψ=
Ψ
rrRR
The expression in [ ]’s is the kinetic energy + potential energy operator and includes terms for
the nuclei and the electrons.
In order, they are: (i) kinetic energy of the nuclei; (ii) kinetic energy
of the electrons; (iii) repulsion between nuclei
p
1
and
p
2
; (iv) attractions between electron and
proton within each atom (
e
1
and
p
1
;
e
2
and
p
2
); (v) attractions between electron and proton
between atoms (
e
1
and
p
2
;
e
2
and
p
1
); and (vi) repulsion between electrons
e
1
and
e
2
.
The
solutions to this equation are the wavefunctions, which are given with respect to the coordinates
of the electrons and protons, and the energies, which depend solely on the positions of the nuclei
(protons).
There are quantum numbers {
Q
}, but these are no longer {
n
,
l
,
m
}.
To simplify this equation, we use the fact that the protons are 2000
×
more massive than the
electrons, so that the kinetic energies of the protons (speeds) are much smaller than those of the
electrons, and are considered to be zero, i.e., stationary nuclei at
R
1
and
R
2
.
This is called the
Born-Oppenheimer approximation
.
The new equation to solve now involves just the coordinates
of the electrons, while the coordinates of the nuclei are fixed.
(
)
2222
2
2
2
2
2
2
2
1
2
11 2 2
1
2
8
,,
Q
m
d
z
E
e
+
+
+
+
−
Ψ
−
+
+
−
R R