10
Correction to the Effect of Anharmonicity on Isotopic Exchange Equilibria
Downloaded by UNIV OF GUELPH LIBRARY on February 26, 2013 | http://pubs.acs.org Publication Date: June 1, 1969 | doi: 10.1021/ba-1969-0089.ch010
MAX WOLFSBERG Departments of Chemistry, Brookhaven National Laboratory, Upton, N. Y. 11973 and State University of New York at Stony Brook, Stony Brook, N. Y. 11790
It is demonstrated that the formula for the vibrational energy states of a diatomic molecule should be written as E /hc= G + ω (n + 1/2) - Wexe(n + 1/2) with non-zero G G values are evaluated for some diatomic hydrides and the effect of G on the theoretical calculation of isotopic ex change equilibrium constants is shown. n
2
o
e
o.
o
o
T n calculations of the vibrational contribution to isotope effects on partition functions for diatomic molecules it is usual to employ the expression
A
EJhc = . ( n + i) w
We
x (n +
n = 0, 1, 2 . . .
e
(1)
for the vibrational energy levels. Terms i n higher powers of ( n - f - 1/2) are usually omitted because they tend to be unimportant. Here w is the harmonic frequency of the molecule ( i n cm." ), which depends on the inverse of the square root of the reduced mass //, of the molecule. w x is the so-called first anharmonic correction ( i n cm." ), which has a /x" mass dependence. The vibrational zero-point energy of the molecule is then given by E /hc = fa - |a, x (2) e
1
e
1
0
e
e
e
1
e
When one takes into account anharmonicity in the theoretical calculation of isotopic exchange equilibrium constants, one usually employs only the anharmonic correction to the zero-point energy (— l/4w x ). A formula of the type of Equation 1 can be obtained for a diatomic molecule oscillator subject to the well-known Morse potential V = D { l — e~* ~ ^} where D is the dissociation energy, r — r is the dis placement of the internuclear distance from its equilibrium value r , and ft is a potential parameter (see Reference 3 ) . e
e
ir
r
2
e
e
c
c
185
In Isotope Effects in Chemical Processes; Spindel, W.; Advances in Chemistry; American Chemical Society: Washington, DC, 1969.
186
ISOTOPE EFFECTS IN CHEMICAL PROCESSES
'Perturbation Theory Calculation for the Anharmonic Oscillator The problem considered is the one-dimensional harmonic oscillator perturbed by cubic and quartic potential terms. Thus, the unperturbed Hamiltonian operator is (S /8r ) + ik(r - r )
H=
2
2
e
2
= (-^ /2)(8 /S
