P H YSI CAL REVIEW VOLUA1I.: 124, NUMBER 1 OCTOBER 1, 1961

Oscillator Frequency and Vibrational Quanta in the Hydrogen Molecule*

J. P. AUzrRAv mn J. W. CooLEvInstitute of Mathematical Sciences, New Fork University, New Fork, Nem Fork

(Received May 22, 1961)

Theoretical values of the oscillator frequency and vibrational quanta in the electronic ground state of H&

are obtained and compared with the corresponding experimental data. Agreement between the two sets ofvalues is found to be of the order of 1 part in 10', This is somewhat larger than the estimate (2 parts in 10')of the experimental error.

TABI.E I. Vibrational quanta T +1, p—1', p in the electronic

ground state of Ho (in wave numbers).

Experimental

4161'392636953468b3242

Theoretical

41663930370234773253

a Stoicheff values (see reference 1).~ Herzberg and Howe values (see reference 2).*The work presented in this paper was supported by the U. S.

Atomic Energy Commission Computing and Applied Mathe-matics Center, Institute of Mathematical Sciences, ¹wYorkUniversity. The calculations were performed on the IBM-704 Com-puter at the Institute of Mathematical Sciences.

' B. P. Stoiche8, Can. J. Phys. 35, 730 (1957).' G. Herzberg arid L. L. Howe, Can. J. Phys. 37, 635 (1959).' J. W. Cooley, Math. of Comp. (to be published). See also J.P. Aui7ray and J, W. Cooley, Phys. Rev. 122, 1203 (1961).

HE oscillator frequency co, in the electronic groundstate of H2 has been determined experimentally

by Stoiche6, ' and Herzberg and Howe, ' using therelation

T +1,0 ~o, o ooo—oo.*.(s+s)+~.y.(s+s)'—.. ., (1)where v is the vibrational quantum number, T,+&,0

—T, , o

are the measured vibrational quanta, and x, and y, aresmall constants arising from the anharmonicity of theinternuclear potential. The present authors have ob-tained theoretical values of the vibrational quantawhich may be compared directly with the experimentaldata. The vibrational-rotational energies T„g weredetermined by solving by an accurate numerical pro-cedure' the radial Schrodinger equation with the po-tential (in atomic units)

V (r) =E(r)+1/r+ J(J+1)/2pr', (2)

where r is the internuclear distance, E!(r) is the elec-tronic energy, 4 J is the rotational quantum number, and

p is the reduced nuclear mass.The vibrational quanta so obtained, converted to

wave numbers' and corrected for the effects of inter-actions between the electronic and nuclear motions, 'are given in Table I for v=0 to 4, along with the cor-responding experimental data for comparison. Th eagreement between the two sets of values is seen to beof the order of 1 part in 10' for e small. From the formof Eq. (1) this suggests that the experimental andtheoretical co, must also agree within about 1 part in 10'.This discrepancy is somewhat larger than Herzberg'sestimate (2 parts in 10') of the uncertainty in the ex-perimental value and is probably due, in part, to smallerrors in the curvature of the theoretical potential (2),as well as to an underestimation of the effects of inter-actions between the electronic and nuclear motions'and the neglect of relativistic eff'ects.

It should be noted that in the present treatment mo

corrections for the anharmonicity of the internuclearpotential were required. '

' The best available theoretical E(r), (W. Kolos and C. C. J.Roothaan, Revs. Modern Phys. 32, 203 (1960)j, was used in thiscalculation.' Using 1 a.u. =219 474.62 cm '.

'The correction was made by multiplying the theoreticalvalues by the factor 1+8=0.99929 /see J.H. Van Vleck, J. Chem.Phys. 4, 327 (1936)j. This correction really applies to cu, butT +i, o T, o =oo. for s small Lsee E—q. (1)j.

7 See reference 6.' Such a correction would have been required however, if onehad wanted to compare the experimental co, with the theoreticalvalue obtained from the relation sr, = (y/p)&, where 7 is the curva-ture of V(r) at the equilibrium internuclear distance r, This cor-.rection was considered by Dunham LPhys. Rev. 41, 721 (1932)g.See also the discussion of this point in reference 1.

137