Page 284 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

10.2 Nuclear motion in diatomic molecules 275

after multiplication by the variable R.
The potential function U(q) in equation (10.36) may be expanded according
to (10.30). The factor (R e ‡ q) ÿ2 in the second term on the left-hand side may
also be expanded in terms of the variable q as follows
!
1 1 1 2q 3q 2
ˆ ˆ 1 ÿ ‡ ÿ (10:37)
(R e ‡ q) 2 q 2 R 2 R e R 2
R 2 1 ‡ e e
e
R e
where the expansion (A.3) is used. For small values of the ratio q=R e , equation
(10.37) gives the approximation R R e .
If we retain only the ®rst two terms in the expansion (10.30) and let R be
approximated by R e , equation (10.36) becomes
2
2
ÿ" d S(q)
2
1
‡ ( kq ÿ W)S(q) ˆ 0 (10:38)
2ì dq 2 2
where
W E í ÿ U(0) ÿ J(J ‡ 1)B e (10:39)
2
2
2
B e " =2ìR ˆ " =2I (10:40)
e
2
The quantity I (ˆ ìR ) is the moment of inertia for the diatomic molecule
e
with the internuclear distance ®xed at R e and B e is known as the rotational
constant (see Section 5.4).
È
Equation (10.38) is recognized as the Schrodinger equation (4.13) for the
one-dimensional harmonic oscillator. In order for equation (10.38) to have the
same eigenfunctions and eigenvalues as equation (4.13), the function S(q) must
have the same asymptotic behavior as ø(x) in (4.13). As the internuclear
distance R approaches in®nity, the relative distance variable q also approaches
in®nity and the functions F(R) and S(q) ˆ RF(R) must approach zero in order
for the nuclear wave functions to be well-behaved. As R ! 0, which is
equivalent to q !ÿR e , the potential U(q) becomes in®nitely large, so that
F(R) and S(q) rapidly approach zero. Thus, the function S(q) approaches zero
as q !ÿR e and as R !1. The harmonic-oscillator eigenfunctions ø(x)
decrease rapidly in value as jxj increases from x ˆ 0 and approach zero as
x ! 1. They have essentially vanished at the value of x corresponding to
q ˆÿR e . Consequently, the functions S(q) in equation (10.38) and ø(x)in
(4.13) have the same asymptotic behavior and the eigenfunctions and eigenva-
lues of (10.38) are those of the harmonic oscillator. The eigenfunctions S n (q)
are the harmonic-oscillator eigenfunctions given by equation (4.41) with x
replaced by q and the mass m replaced by the reduced mass ì. The
eigenvalues, according to equation (4.30), are