Page 279 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

270 Molecular structure
!
ÿ" 2 = 2 = 2
^ A ‡ B (10:23)
T Q ˆ
2 M A M B
The laplacian operators in equation (10.23) refer to the spaced-®xed coordi-
nates Q á with components Q xá , Q yá , Q zá , so that
@ 2 @ 2 @ 2
2
= ˆ 2 ‡ 2 ‡ 2 , á ˆ A, B
á
@Q @Q @Q
xá yá zá
However, these operators change their form when the reference coordinate
system is transformed from space ®xed to center of mass.
To transform these laplacian operators to the coordinates X and R, with
components X x , X y , X z and R x, R y , R z , respectively, we note that
@ @X x @ @R x @ M A @ @
ˆ ‡ ˆ ÿ
@Q xA @Q xA @X x @Q xA @R x M @X x @R x
@ @X x @ @R x @ M B @ @
ˆ ‡ ˆ ‡
@Q xB @Q xB @X x @Q xB @R x M @X x @R x
from which it follows that
@ 2 M A 2 @ 2 @ 2 2M A @ 2
ˆ ‡ ÿ
@Q 2 M @X 2 @R 2 M @X x @R x
xA x x
2
@ 2 M B @ 2 @ 2 2M B @ 2
ˆ ‡ ‡
@Q 2 M @X 2 @R 2 M @X x @R x
xB x x
Analogous expressions apply for Q yA, Q yB , Q zA , and Q zB . Therefore, in terms of
2
2
the coordinates X and R, the operators = and = are
A B
2

2 M A 2 2 2M A :
= ˆ M = ‡ = ÿ M = X = R (10:24a)
A
R
X
2
2 M B 2 2 2M B :
= ˆ = ‡ = ‡ = X = R (10:24b)
B X R
M M
:
2
:
2
where = ˆ = X = X and = ˆ = R = R are the laplacian operators for the
X R
vectors X and R and where = X and = R are the gradient operators. When the
^
transformations (10.24) are substituted into (10.23), the operator T Q becomes
ÿ" 2 1 1
2
^ = ‡ = 2 (10:25)
T Q ˆ
2 M X ì R
where ì is the reduced mass of the two nuclei
1 1 1
ˆ ‡
ì M A M B
or