Page 283 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

274 Molecular structure
" #

1 @ @ 1 1 @ @ 1 @ 2
2 2
= ˆ R @R R @R ‡ R 2 sin è @è sin è @è ˆ sin è @j 2
R
2
2

1 @ @ 1
L
ˆ R 2 ÿ ^ 2 (10:32)
2
2
R @R @R " R 2
^ 2
where L is the square of the orbital angular momentum operator given by
2
equation (5.32). With = expressed in spherical polar coordinates, equation
R
(10.31) becomes
" #
" 2 @ @ 1
L ‡ U(R) ÷ í (R, è, j) ˆ E í ÷ í (R, è, j)
ÿ R 2 ‡ ^ 2
2
2ìR @R @R 2ìR 2
(10:33)
The operator in square brackets on the left-hand side of equation (10.33)
^
^ 2
^ 2
commutes with the operator L and with the operator L z in (5.31c), because L
^
^
^ 2
commutes with itself as well as with L z and neither L nor L z contain the
variable R. Consequently, the three operators have simultaneous eigenfunc-
tions. From the argument presented in Section 6.2, the nuclear wave function
÷ í (R, è, j) has the form
÷ í (R, è, j) ˆ F(R)Y Jm (è, j) (10:34)
where F(R) is a function of only the internuclear distance R, and Y Jm (è, j) are
the spherical harmonics, which satisfy the eigenvalue equation
^ 2 2
L Y Jm (è, j) ˆ J(J ‡ 1)" Y Jm (è, j)
J ˆ 0, 1, 2, ... ; m ˆÿJ, ÿJ ‡ 1, ... ,0, ... , J ÿ 1, J
It is customary to use the index J for the rotational quantum number. Equation
(10.33) then becomes
" #
" 2 d d J(J ‡ 1)" 2
ÿ R 2 ‡ ‡ U(R) ÿ E í F(R) ˆ 0 (10:35)
2
2ìR dR dR 2ìR 2
where we have divided through by Y Jm (è, j).
We next replace the independent variable R in equation (10.35) by q as
de®ned in equation (10.29). Equation (10.35) has a more useful form if we also
make the substitution S(q) RF(R). Since dq=dR ˆ 1, we have

2
dF(R) 1 dS(q) 1 d 2 dF(R) d S(q)
ˆ ÿ S(q), R ˆ R
dR R dq R 2 dR dR dq 2
and equation (10.35) becomes
" #
2
" d 2 J(J ‡ 1)" 2
ÿ ‡ ‡ U(q) ÿ E í S(q) ˆ 0 (10:36)
2ì dq 2 2ì(R e ‡ q) 2