Page 280 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

10.2 Nuclear motion in diatomic molecules 271

M A M B
ì ˆ (10:26)
M A ‡ M B
:
The cross terms in = X = R cancel each other.
For the diatomic molecule, equations (10.1), (10.3), (10.5), and (10.25)
combine to give
" #
" 2 " 2 Z A Z B e9 2
2
^
2
ÿ = ÿ = ‡ ‡ H e ÿ E tot Ø tot ˆ 0 (10:27)
X
R
2M 2ì R
where R r AB is the magnitude of the vector R and where now the laplacian
^
2
operator = in H e of equation (10.4) refers to the position of electron i relative
i
to the center of mass. The interparticle distances r AB ˆ R, r Ai , r Bi , and r ij are
independent of the choice of reference coordinate system and do not change as
a result of the transformation from external to internal coordinates. If we write
Ø tot as the product
Ø tot ˆ Ö(X)Ø(R, r)
and E tot as the sum
E tot ˆ E cm ‡ E
then the differential equation (10.27) separates into two independent differen-
tial equations
" 2
2
ÿ = Ö(X) ˆ E cm Ö(X) (10:28a)
X
2M
and
" #
" 2 Z A Z B e9 2
^
2
ÿ = ‡ ‡ H e ÿ E Ø(R, r) ˆ 0 (10:28b)
R
2ì R
Equation (10.28b) describes the internal motions of the two nuclei and the
electrons relative to the center of mass. Our next goal is to solve this equation
using the method described in Section 10.1. Equation (10.28a), on the other
hand, describes the translational motion of the center of mass of the molecule
and is not considered any further here.

Electronic motion and the nuclear potential function
The ®rst step in the solution of equation (10.28b) is to hold the two nuclei ®xed
2
in space, so that the operator = drops out. Equation (10.28b) then takes the
R
form of (10.6). Since the diatomic molecule has axial symmetry, the eigenfunc-
^
tions and eigenvalues of H e in equation (10.6) depend only on the ®xed value
R of the internuclear distance, so that we may write them as ø k (r, R) and
å k (R). If equation (10.6) is solved repeatedly to obtain the ground-state energy
å 0 (R) for many values of the parameter R, then a curve of the general form