32
2 Bonds
The indices 1 and 2 denote the two electrons of atoms. x 1 and x 2 are the displacements of the electrons.
Both harmonic oscillators have a resonance frequency ω 0 =
√
C/m, and the zero-point energy is
ω 0 /2.
Taking into account the Coulomb interaction of the four charges, an additional term H 1 arises
H 1 =
e
2
R
+
e
2
R + x 1 + x 2
−
e
2
R + x 1
−
e
2
R − x 2
≈ −
2e
2
R 3 x 1 x 2 .
(2.14)
The approximation is valid for small amplitudes x i R. A separation of variables can be achieved
by transformation to the normal modes
x s =
x 1 + x 2
√
2
, x a =
x 1 − x 2
√
2
.
(2.15)
Then we find
H = H 0 + H 1
=
1
2m
p
2
s +
1
2
C −
2e
2
R 3
x
2
s
+
1
2m
p
2
a +
1
2
C −
2e
2
R 3
x
2
a
.
(2.16)
This equation is the Hamiltonian of two decoupled harmonic oscillators with the normal frequencies
ω ± =
C ±
2e 2
R 3
/m ≈ ω 0
1 ±
1
2
2e
2
C R 3
−
1
8
2e
2
C R 3
2
+ . . .
.
(2.17)
The coupled system thus has a lower (zero-point) energy than the uncoupled. The energy difference
per atom is (in lowest order) proportional to R
−6 .
U = ω 0 −
1
2
(ω + − ω − ) ≈ −ω 0
1
8
2e
2
C R 3
2
= −
A
R 6 .
(2.18)
The interaction is a true quantum-mechanical effect, i.e. the reduction of the zero-point energy of
coupled oscillators.
2.7 Hamilton Operator of the Solid
The total energy of the solid, including kinetic and potential terms, is
H =
i
p
2
i
2m i
+
j
P
2
j
2M j
+
1
2
j, j
Z j Z j e
2
4ππ 0 |R j − R j |
+
1
2
i,i
e
2
4ππ 0 |r i − r i |
−
i, j
Z j e
2
4ππ 0 |R j − r i |
,
(2.19)