ABSTRACT

An expression is derived which relates the distribution of vibrational levels near the dissociation limit $D$ of a given diatomic species to the nature of the long‐range interatomic potential, in the region where the latter may be approximated by ${\mathrm{D\hspace{0.17em}-\hspace{0.17em}C}}_{n}{\mathrm{\hspace{0.17em}/\hspace{0.17em}R}}^{n}$. Fitting experimental energies directly to this relationship yields values of $D$, $n$, and ${C}_{n}$. This procedure requires a knowledge of the relative energies and relative vibrational numbering for at least four rotationless levels lying near the dissociation limit. However, it requires no information on the rotational constants or on the number and energies of the deeply bound levels. $D$ can be evaluated with a much smaller uncertainty than heretofore obtainable from Birge–Sponer extrapolations. The formula predicts the energies of all vibrational levels lying above the highest one measured, with uncertainties no larger than that of the binding energy of the highest level. The validity of the method is tested with model potentials, and its usefulness is demonstrated by application to the precise data of Douglas, Mo/ller, and Stoicheff for the ${B}^{3}{\Pi}_{\text{0u}}^{+}$ state of Cl

_{2}.- 1. R. T. Birge and H. Sponer, Phys. Rev.
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**25**, 707 (1929).**Google Scholar****Crossref** - 3. For a recent review see A. G. Gaydon,
*Dissociation Energies*(Chapman and Hall Ltd., London, 1968), 3rd. ed.**Google Scholar** - 4. G. Herzberg,
*Molecular Structure and Molecular Spectra: I. Spectra of Diatomic Molecules*(D. Van Nostrand Co., Inc., Toronto, 1950), 2nd ed.**Google Scholar** - 5.Note that while $\mathrm{\Delta G(\upsilon )}$ is Herzberg’s ${\mathrm{\Delta G}}_{\upsilon},$ $\mathrm{\Delta G(\upsilon +}{\scriptscriptstyle \frac{1}{2}}\mathrm{)\equiv dE(\upsilon +}{\scriptscriptstyle \frac{1}{2}}\mathrm{)/d\upsilon}$ is not identical to the “observable” vibrational level spacing

(see p. 98 in Ref. 4).$${\mathrm{\Delta G}}_{\text{\upsilon +1/2}}\mathrm{\hspace{0.17em}=\hspace{0.17em}}{\displaystyle {\int}_{\upsilon}^{\text{\upsilon +1}}}\mathrm{\Delta G(\upsilon )d\upsilon}$$ - 6. (a) H. Harrison and R. B. Bernstein, J. Chem. Phys.
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(b) H. Harrison and R. B. Bernstein, Erratum**47**, 1884 (1967)., J. Chem. Phys. ,**Google Scholar****Scitation** - 7. See the review by E. A. Mason and L. Monchick, Advan. Chem. Phys.
**12**, 329 (1967).**Google Scholar** - 8. The parameters of the LJ (12, 6) potential were chosen to allow for 24 bound states. In the notation of Ref. 6, this corresponds to ${B}_{z}{\text{\hspace{0.17em}=\hspace{0.17em}2\mu D}}_{e}{R}_{e}^{2}{\mathrm{/\hslash}}^{2}\text{\hspace{0.17em}=\hspace{0.17em}10\hspace{0.17em}000,}$ where ${D}_{e}$ is the well depth, and ${R}_{e}$ the position of the potential minimum. Eigenvalues were calculated numerically and are accurate to ${10}^{\text{\u22127}}{\mathrm{\hspace{0.17em}D}}_{e}.$ This was done using a slightly modified form of the Cooley‐Cashion program: J. W. Cooley, Math. Computation
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J. K. Cashion, J. Chem. Phys.**39**, 1872 (1963).**Google Scholar****Scitation** - 9. I. S. Gradshteyn and I. M. Ryzhik,
*Table of Integrals, Series and Products*(Academic Press Inc., New York, 1965), Sec. 3.251, p. 295.**Google Scholar** - 10. M. Abramowitz and I. Stegun, Natl. Bur. Std. (U.S.), Appl. Math. Ser.
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also*Handbook of Mathematical Functions*(Dover Publications, Inc., New York, 1965).**Google Scholar** - 11. This is because a direct fit of experimental data to Eq. (4) requires a prior numerical smoothing of the data to obtain accurate values of the derivatives $\mathrm{dE(\upsilon )/d\upsilon .}$
**Google Scholar** - 12. It is interesting to note that for $\text{n\hspace{0.17em}=\hspace{0.17em}4}$ (ion‐induced dipole forces) Eq. (6) is simply a quartic in υ, and for $\text{n\hspace{0.17em}=\hspace{0.17em}6}$ (induced dipole‐induced dipole, London dispersion forces) it is cubic.
**Google Scholar** - 13. By comparing Eq. (1) for $\mathrm{E(\upsilon )\hspace{0.17em}=\hspace{0.17em}D}$ and
*E*(υ) at a slightly smaller υ, W. C. Stwalley (private communication, 1969) independently obtained a result for $\text{n\hspace{0.17em}=\hspace{0.17em}6}$ which, upon generalization for any $\text{n>2,}$ may be cast into the useful form of Eq. (6). However, his factor equivalent to the present ${K}_{n}$ is slightly less general, and his approach (unlike the present one) cannot be applied to cases with $\text{n\u2a7d2.}$**Google Scholar**

While seeking a “natural” analytic expression to describe the vibrational spectrum of ${\mathrm{H}}_{2},$ C. L. Beckel [J. Chem. Phys.**39**, 90 (1963)] proposed empirical formulas somewhat similar in form to Eq. (6).**Google Scholar****Scitation** - 14. P. M. Morse and H. Feshbach,
*Methods of Theoretical Physics*(McGraw‐Hill Book Co., New York, 1953), Vol. 2, Sec. 12.3.**Google Scholar** - 15. For pure inverse‐power potentials with $\text{n>2,}$ there are a finite number of levels within any finite neighborhood of the dissociation limit, but there are an infinite number of discrete levels below it, extending down to infinite binding energy. For potentials with $\text{n<2,}$ there exists a lowest level bound by a finite energy, while there are an infinite number of levels within any finite neighborhood of
*D*. For $\text{n\hspace{0.17em}=\hspace{0.17em}2,}$ the levels extend down to infinite binding energy, and there are an infinite number of levels in any finite neighborhood of*D*.**Google Scholar** - 16.Using the Langer WKB modification [i.e., replacing $\text{J(J+1)}$ by $\mathrm{(J+}{\scriptscriptstyle \frac{1}{2}}{)}^{2}$] would require replacing Eq. (3a) by

. For $\text{n\hspace{0.17em}=\hspace{0.17em}2}$ this just means that ${C}_{2}$ in Eq. (7) becomes ${\mathrm{[C}}_{2}{\mathrm{-(\hslash}}^{2}\text{/8\mu )],}$ but for $\text{n\u22602,}$ the integral arising from Eq. (2) is no longer analytically soluble. However, for realistic systems the Langer correction is fortunately very small. R. E. Langer, Phys. Rev.$${\mathrm{V(R)\hspace{0.17em}=\hspace{0.17em}D-(C}}_{n}{\mathrm{/R}}^{n}{\mathrm{)+(\hslash}}^{2}{\text{/2\mu )(1/4R}}^{2})$$ **51**, 669 (1937).**Google Scholar****Crossref** - 17. Within the context of the present approach, potentials with exponential long‐range tails (such as the Morse potential) correspond qualitatively to inverse‐power potentials with very large
*n*. The purely attractive exponential potential has both a discrete lowest level and a finite number of bound states within any finite neighborhood of*D*.**Google Scholar** - 18. A linear B–S plot for levels near the dissociation limit of a potential will be considered as an indication that the potential in the given region is effectively exponential in form.
**Google Scholar** - 19. Care should be taken to avoid confusion between the well depth ${D}_{e}$ and
*D*, the position of the dissociation limit.**Google Scholar** - 20. See the discussion of intermolecular forces in (a) J. O. Hirschfelder, C. F. Curtiss, and R. B. Bird,
*Molecular Theory of Gases and Liquids*(John Wiley & Sons, Inc., New York, 1964).**Google Scholar**

(b) J. O. Hirschfelder and W. J. Meath, Advan. Chem. Phys.**12**, 3 (1967).**Google Scholar** - 21. Note that in the case where some of the dominant terms in Eq. (13) are repulsive (i.e., their ${C}_{m}\text{<0}$), some of these weighting factors will have differing signs, and the resulting value of
*n*may then lie outside the range of the*m*’s of the contributing terms. If the lowest inverse‐power term is repulsive while the higher power terms are attractive, this gives rise to a potential maximum at large*R*. This appears to be the case for the ${}^{3}{\Pi}_{\text{0g}}^{+}$ state of ${{I}}_{2};$ R. J. LeRoy, J. Chem. Phys.**52**, 2678 (1970).**Google Scholar****Scitation** - 22. In this context a potential is “well behaved” if it has no potential maximum and no nonadiabatic perturbation.
**Google Scholar** - 23. Of course, both errors approach zero for levels approaching
*D*.**Google Scholar** - 24. (a) See, e.g., the discussion by J. K. Cashion, J. Chem. Phys.
**48**, 94 (1968); see also Appendix A.**Google Scholar****Scitation**

(b) A. S. Dickinson (private communication, 1968). ,**Google Scholar** - 25. See Appendix B for a summary of the theoretical $\xf1$ values for a wide variety of cases.
**Google Scholar** - 26. Nonlinear least‐squares regression computer programs for fitting arbitrary analytic functions are available at most computing centers. The present calculations used the University of Wisconsin Computing Center subroutine GASAUS for such fits.
**Google Scholar** - 27. Primes denote differentiation with respect to υ; e.g., $\mathrm{E\prime (\upsilon )\hspace{0.17em}=\hspace{0.17em}dE(\upsilon )/d\upsilon .}$
**Google Scholar** - 28. Parameter values obtained from Eqs. (15) and (16) should, in principle, be just as reliable as those obtained from Eq. (6). However, the former approach requires a prior smoothing of the data to obtain accurate values of the derivatives $\mathrm{E\prime (\upsilon )}$ and $\mathrm{E\u2033(\upsilon ),}$
^{27}and in practice this introduces some error. Experience has shown that while trial parameter values from Eqs. (15) and (16) are satisfactory, they are measurably improved by four‐parameter fittings to Eq. (6).^{26},**Google Scholar** - 29. In all of the results presented, an initial fit of the data to Eqs. (15) and (16) yielded trial parameter values which were used to initiate the general nonlinear fit to Eq. (6).
^{26,30},**Google Scholar** - 30. R. J. LeRoy and R. B. Bernstein, (a) Wisc. Theoret. Chem. Inst. Tech. Rept. WIS‐TCI‐369, 1970. The report contains in an appendix FORTRAN listings of the programs used for carrying out fits to Eq. (6), (15), and (16).
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(b) See also, “Dissociation Energies of Diatomic Molecules from Vibrational Spacings of Higher Levels: Applications to the Halogens,” Chem. Phys. Letters (to be published).**Google Scholar** - 31.This is mainly because of the problem of averaging the estimates of
*D*and ${K}_{n}$ obtained at different values of υ, to yield a mutually consistent set of parameters. It is interesting that analogous to Eq. (17)

, but because of the above problem, this expression is less reliable than is Eq. (15).$${\text{n\hspace{0.17em}=\hspace{0.17em}[4[E\u2033(\upsilon )]}}^{2}\text{/E\u2032(\upsilon )E\u2034(\upsilon )]\u22122}$$ - 32. Since the derivatives are obtained from the highest 11 energies only, they cannot be accurate at the end points, so only the 9 points shown on Fig. 2 are reliable.
**Google Scholar** - 33. Since the input data (level energies) are never completely error free, a given fit should always utilize at least one level more than the number of free parameters being fitted. If there is significant experimental uncertainty in the energies (e.g., more than a few percent of the level spacings), a redundancy of more than one level may be required to yield meaningful values of the parameters.
**Google Scholar** - 34. In the application of this method to the ${\mathrm{B\hspace{0.17em}}}^{3}{\Pi}_{\text{0u}}^{+}$ state of ${{I}}_{2},$
^{30}the experimental uncertainty introduces considerable imprecision into the four‐parameter fits, so that*n*could not be directly determined within required accuracy of better than $\text{\xb11.}$**Google Scholar** - 35. D. E. Stogryn and J. O. Hirschfelder, J. Chem. Phys.
**31**, 1531 (1959). These authors derived an analytic expression [their Eq. (89)] for the exact first‐order WKB value of ${\upsilon}_{D}$ (which omits the effect of the Langer correction^{16}). A more exact value of the numerical constant in their Eq. (92) is 1.6826.**Google Scholar****Scitation**,**ISI** - 36. A. E. Douglas, Chr. Kn. Møller, and B. P. Stoicheff, Can. J. Phys.
**41**, 1174 (1963).**Google Scholar****Crossref** - 37. The experimental data for this system are for the most common isotope ${}^{\text{35,35}}{\mathrm{Cl}}_{2};$ all energies are expressed relative to the $\text{\upsilon \u2033\hspace{0.17em}=\hspace{0.17em}0,}$ $\text{J\u2033\hspace{0.17em}=\hspace{0.17em}0}$ level of its ground electronic state.
**Google Scholar** - 38. T. Y. Chang, Mol. Phys.
**13**, 487 (1967); see also the discussion in Appendix B.**Google Scholar****Crossref** - 39. M. A. Byrne, W. G. Richards, and J. A. Horsley, Mol. Phys.
**12**, 273 (1967).**Google Scholar****Crossref** - 40. In choosing these values it is assumed that the “hook” at the end of the $\text{n\hspace{0.17em}=\hspace{0.17em}5}$ curves in Fig. 5 is significant, illustrating the decrease of the error term for levels farther into the asymptotic $\mathrm{(n\hspace{0.17em}=\hspace{0.17em}\xf1)}$ region. The indicated uncertainties (including the error bars in Figs. 5 and 6) correspond to a statistical confidence limit of 95%.
**Google Scholar** - 41. It has been shown by J. K. Knipp [Phys. Rev.
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that ${C}_{5}$ coefficients may be expressed as a product of an angular factor and ${\mathrm{[\u3008r}}_{A}^{2}{\mathrm{\u3009\u3008r}}_{B}^{2}\mathrm{\u3009],}$ the product of the expectation values for the square of the electron radii in the unfilled valence shells on interacting atoms A and B. Knipp presented values of the angular factors and approximate expectation values for a few systems, and T. Y. Chang [Rev. Mod. Phys.**39**, 911 (1967)] extended these results considerably. ,**Google Scholar****Crossref**

Recently C. F. Fischer [Can. J. Phys.**46**, 2336 (1968)] has reported Hartree‐Fock values of ${\mathrm{\u3008r}}^{2}\u3009$ for all shells of atoms from He to Rn. ,**Google Scholar****Crossref** - 42. The erratic nature of the curve in Fig. 6 is due to the influence of small errors in the experimental energies on the fitted values of the parameters; the corresponding values of
*n*, ${C}_{n},$ and ${\upsilon}_{D}$ show similar behavior. Including more levels in each fit dampens these oscillations.**Google Scholar** - 43. Holding
*D*fixed dampens the “noise” due to experimental uncertainty,^{42}yielding a more reliable segmented potential.**Google Scholar** - 44. J. A. C. Todd, W. G. Richards, and M. A. Byrne, Trans. Faraday Soc.
**63**, 2081 (1967).**Google Scholar****Crossref** - 45. For a related discussion of the quasibound states, see A. S. Dickinson and R. B. Bernstein, “Some Properties of Bound and Quasibound States for Various Interatomic Potential Functions,” Mol. Phys. (to be published).
**Google Scholar** - 46. While the present method is expected to give values of ${C}_{n}$ which are slightly small (see Sec. II.C), there is reason to suspect that the theoretical ${C}_{5}$ value used for comparison
^{41}may be somewhat too large. M. T. Marron (private communication, 1969) points out that Fischer’s^{41}values of ${\mathrm{\u3008r}}^{2}\u3009$ are based on Hartree‐Fock wavefunctions which do not have correct asymptotic tails and that correcting for this may decrease ${\mathrm{\u3008r}}^{2}\mathrm{\u3009,}$ and hence the theoretical ${C}_{2}.$**Google Scholar** - 47. For a few systems, such as isotopic hydrogen and most hydrides, the inverse‐power long‐range forces are relatively weak, so that the B–S plot shows negative or zero curvature even for the very highest levels.
**Google Scholar** - 48. R. B. Bernstein, Phys. Rev. Letters
**16**, 385 (1966).**Google Scholar****Crossref** - 49. J. A. Horsley and W. G. Richards, J. Chim. Phys.
**66**, 41 (1969).**Google Scholar****Crossref** - 50. See, for example, H. Pauly and J. P. Toennies, in
*Atomic and Electron Physics: Atomic Interactions, Part A*, L. Marton, B. Bederson, and W. L. Fite, Eds. (Academic Press Inc., New York, 1968), Vol. 7, Chap. 3.1, p. 227.**Google Scholar** - 51. Although all of the cases thus far considered correspond to $\text{\xf1\hspace{0.17em}=\hspace{0.17em}5}$ or 6, the present method should be even more successful for systems with smaller $\xf1$ (e.g., $\text{\xf1\hspace{0.17em}=\hspace{0.17em}4,}$ for molecules which dissociate to ion+neutral) because of the relatively higher density of levels near
*D*.**Google Scholar** - 52. The present work utilized the corrected tables reported in Ref. 6b. These are available as Document No. 9499 in the ADI Auxiliary Publications Project, Photoduplication Service, Library of Congress, Washington, D.C. 20540.
**Google Scholar** - 53. Comparison of the φ values
^{6,52}for $\text{\theta \hspace{0.17em}=\hspace{0.17em}0}$ and ${\text{\theta \hspace{0.17em}=\hspace{0.17em}10}}^{\text{\u22124}}$ shows that this introduces negligible error.**Google Scholar** - 54. This was done by piecewise fitting of third‐order polynomials in φ. Despite the rather large gaps between the tabulated points for large φ, this is expected to be fairly accurate since the eigenvalue distribution for the highest levels of an ${R}^{\text{\u22126}}$‐tailed potential is expected to be cubic in υ (i.e., in φ).
^{12},**Google Scholar** - 55. Although the exact ${\upsilon}_{D}$ is infinite for the pure ${R}^{\text{\u22126}}$ attractive potential, there are a finite number of levels within any finite interval about
*D*.^{15}Hence the quantities ${\mathrm{(\upsilon}}_{D}\mathrm{-\upsilon )}$ and Curve A in Fig. 8 are significant in the semiclassical (WKB) approximation.**Google Scholar** - 56. G. W. King and J. H. Van Vleck, Phys. Rev.
**55**, 1165 (1939).**Google Scholar****Crossref** - 57. H. Margenau, Rev. Mod. Phys.
**11**, 1 (1939).**Google Scholar****Crossref** - 58. This conclusion is partly based on Chang’s conclusion
^{41}that for the ${0}_{g}^{+}$ states of ${\mathrm{O}}_{2}$ and ${\mathrm{Cu}}_{2},$ these effects do not dominate the interaction until $\text{R>60\hspace{0.17em}}\mathrm{a.u.}$**Google Scholar** - 59. This case is, however, relatively uncommon; Hirschfelder and Meath
^{20b}point out that only an excited H atom can have a permanent dipole moment.**Google Scholar**

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