The Morse Potential in Quantum Chemistry

Johar M. Ashfaque

The Morse potential realistically leads to dissociation, making it more useful than the usual harmonicpotential. The Morse potential is the simplest representative of the potential between two nuclei in whichdissociation is possible.

The form of the Morse potential in terms of the internuclear distance is

D

(1 exp

(r r0

r0

))2where r0 is the equilibrium internuclear distance. It is common to define

x =r r0r0

so that the potential reads

D

(1 ex

)2.

The one-dimensional Schrodinger equation then takes the form

~2

2m

d2

dx2+D

(1 ex

)2 = E

Note. The Taylor expansion of the potential energy operator yields a quadratic leading term.

Define y = x so thatd

dx=

dy

dx

d

dy=

d

dy

which transforms the Schrodinger equation into

~22

2m

d2

dy2+D

(1 ey

)2 = E

which becomesd2

dy2 2mD

~22

(E

D(

1 ey)2)

= 0.

Defining

=

2mD

~and

=E

D

obtaind2

dy2 2

(

(1 ey

)2) = 0.

Next, change variables to make the equation even more tractable. Define = ey which gives

d = eydy

andd

dy= d

d.

1

The transformed Schrodinger equation reads

d

( dd

)d

2(

(

)2) = 0

and expanding the square yields

d

( dd

)d

(2( 1) + 2 2) = 0.For

a = 2( 1), b = 2the equation becomes

2d2

d2+

d

d (a+ b 2) = 0.

One sees immediately that there is going to be a problem here. The ordinary quantum chemistry problemscontain terms arranged in such a way as to guarantee a two term recurrence but th three term recurrencerelations that have been obtained lead to a continued fraction solution in a natural way.

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