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10.2 Nuclear motion in diatomic molecules 273
Oppenheimer and the adiabatic approximations are essentially identical. Since
we are interested here in only the ground electronic state, we drop the subscript
on U 0 (R) from this point on for the sake of simplicity.
The functional form of U(R) differs from one diatomic molecule to another.
Accordingly, we wish to ®nd a general form which can be used for all
molecules. Under the assumption that the internuclear distance R does not
¯uctuate very much from its equilibrium value R e so that U(R) does not
deviate greatly from its minimum value, we may expand the potential U(R)in
a Taylor's series about the equilibrium distance R e
1
(1) (2) 2
U U(R e ) U (R e )(R ÿ R e ) U (R e )(R ÿ R e )
2!
1 1
3
4
(4)
(3)
U (R e )(R ÿ R e ) U (R e )(R ÿ R e )
3! 4!
where
l
(l) d U(R)
U (R e ) , l 1, 2, ...
dR l
RR e
(1)
The ®rst derivative U (R e ) vanishes because the potential U(R) is a minimum
at the distance R e . The second derivative U (2) (R e ) is called the force constant
for the diatomic molecule (see Section 4.1) and is given the symbol k. We also
introduce the relative distance variable q, de®ned as
q R ÿ R e (10:29)
With these substitutions, the potential takes the form
4
1
3
1
(3)
2
1
U(q) U(0) kq U (0)q U (4) (0)q (10:30)
2 6 24
Nuclear motion
The nuclear equation (10.21) when applied to the ground electronic state of a
diatomic molecule is
^
[T Q U(R)]÷ í (R) E í ÷ í (R) (10:31)
(1) (1)
where the superscript and one subscript on ÷ (R) and on E 0í are omitted for
0í
simplicity. In solving this differential equation, the relative coordinate vector R
is best expressed in spherical polar coordinates R, è, j. The coordinate R is the
magnitude of the vector R and is the scalar distance between the two nuclei.
The angles è and j give the orientation of the internuclear axis relative to the
2
external coordinate axes. The laplacian operator = is then given by (A.61) as
R
