Electronic States and Band Spectrum Structure in Diatomic Molecules III. Intensity Relations

Robert S. Mulliken

doi:10.1103/PhysRev.29.391

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Electronic States and Band Spectrum Structure in Diatomic Molecules III. Intensity Relations

Robert S. Mulliken

Washington Square College, New York University,

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Phys. Rev. 29, 391 – Published 1 March, 1927

DOI: https://doi.org/10.1103/PhysRev.29.391

Abstract

The correspondence principle predicts definite intensity relations for $P$ , $Q$ and $R$ band lines in molecules having a rotational energy term $F (j) = B (j^{2} - σ^{2}) + \dots$ , provided $σ$ is an electronic quantum number correlated with a precession about the internuclear axis (along which the angular momentum $\frac{σ h}{2 π}$ is directed). Hönl and London have used the summation rule method to obtain exact equations for each of three possible cases $Δ σ = 0, \pm 1$ ; these equations are recast here in more convenient form. According to recent work of Dennison, these equations very probably remain valid in the new quantum mechanics. Theoretical curves for the cases $Δ σ = + 1 and - 1$ are given.

Various electronic emission bands have been interpreted by the writer as corresponding to electronic transitions ${}^{1}S \to {}^{1}S$ (CuH type), ${}^{1}P \to {}^{1}S$ (AlH, ${He}_{2}$ "series" bands), ${}^{1}D \to {}^{1}P$ ( ${He}_{2}$ $λ 5733$ ), ${}^{1}S \to {}^{1}P$ ( ${He}_{2}$ $λ 4546$ , $λ 6400$ , and CO Ångstrom bands), with $σ = 0$ for ${}^{1}S$ , $σ = 1$ for ${}^{1}P$ , and $σ = 2$ for ${}^{1}D$ states. Comparison of the predictions of the above theory with the available intensity data on the bands mentioned, assuming in each case thermal equilibrium at a suitable (arbitrarily chosen) temperature, gives in every case qualitative agreement with the predicted distribution for that particular case; usually the agreement appears to be complete within the often comparatively large uncertainties of the data. Oscillation-rotation absorption bands corresponding to $σ^{'} = σ^{''} = 0$ fall under the theory as a special case, and the data on these bands (in particular the quantitative data of Kemble and Bourgin on HCl) agree with the theory. Intensity relations in ${}^{2}S \to {}^{2}S$ (violet CN type) and ${}^{2}P \to {}^{2}S$ (ZnH type) transitions are briefly discussed; the intensity relations resemble those for ${}^{1}S \to {}^{1}S$ and ${}^{1}P \to {}^{1}S$ transitions, but no satisfactory theory is as yet available.

The close agreement between theory and experiment for spectra interpreted as due to transitions involving ${}^{1}S$ , ${}^{1}P$ , or ${}^{1}D$ states seems to make conclusive the already strong evidence for these interpretations. Furthermore (in connection with the observed selection $Δ σ = 0, \pm 1$ ) it shows that ${}^{1}S$ , ${}^{1}P$ , and ${}^{1}D$ molecular states all belong to a class in which $σ$ is an electronic quantum number correlated with a precession about the internuclear axis.

In ${}^{1}S \to {}^{1}P$ , ${}^{1}P \to {}^{1}S$ , and ${}^{1}D \to {}^{1}P$ transitions, there exist in practice (cf. II of this series) two rotational states ( $A$ and $B$ ) for each value of $j$ . The agreements noted were obtained by disregarding the subdivision into $A$ and $B$ substates; the fact that agreement was obtained in this way shows incidentally that the a priori probabilities of $A$ and $B$ sub-states must be equal.

Another phenomenon not contemplated by the theory, but present in all the ${He}_{2}$ bands, is that alternate lines are missing in each branch. The cause of this is obscure, but the evidence for its existence is conclusive and so justifies the procedure, which is necessary to obtain the agreement described above between theory and experiment, of inserting in Eqs. (5)-(6) a factor zero for alternate lines in each branch.

References (40)

A. Kratzer, Naturwiss. 27, 577 (1923) Ann. d. Physik 71, 72 (1923)
E. C. Kemble, Proc. Nat. Acad. Sci. 10, 274 (1924) Phys. Rev. 25, 1 (1925)
E. C. Kemble, Zeits. f. Physik 35, 286 (1925)
R. S. Mulliken, Proc. Nat. Acad. Sci. 12, 144 (1926)
R. S. Mulliken, Proc. Nat. Acad. Sci. 12, 151 (1926)
R. S. Mulliken, Phys. Rev. 28, 481 (1926)
R. S. Mulliken, Phys. Rev. 28, 1202 (1926)
A. Sommerfeld, Atombau und Spektrallinien, 4th Edition, Chapter 9, (1924)
H. Hönl and F. London, Zeits. f. Physik 33, 803 (1925)
G H. Dieke, Zeits. f. Physik 33, 161 (1925)
L. Mensing, Zeits. f. Physik 36, 814 (1926)
E. Fues, Ann. der Physik 80, 367 (1926)
J. R. Oppenheimer, Proc. Camb. Phil. Soc. 23, Part 3, p. 327 (1926)
Kemble, Fowler, and Dieke, [7], p. 1206
Omitted endnote
R. Frerichs, Zeits. f. Physik 20, 170 (1923)
Omitted endnote
F. Hund, Zeits. f. Physik 36, 657 (1926)
Omitted endnote
Hönl and London. [9]
Omitted endnote
Omitted endnote
D. M. Dennison, Phys. Rev. 28, 329 (1926)
R. H. Fowler, Phil. Mag. 50, 1079 (1925)
R. S. Mulliken, Phys. Rev. 26, 21-2 and 333 (1925)
R. T. Birge, Astrophys. J. 55, 273 (1922)
J. H. Van Vleck, Phys. Rev. 28, 980 (1926)
Omitted endnote
R. S. Mulliken, Proc. Nat. Acad. Sci. 12, 158 (1926)
Omitted endnote
Curtis and Long's (Proc. Roy. Soc. 108A, 513, 1925)
W. E. Curtis, Proc. Roy. Soc. 103A, 315 (1923)
G. Eriksson and E. Hulthén, Zeits. f. Physik 34, 775 (1925). p. 777
W. Mörikofer, Dissertation Basel, 1925
Omitted endnote
W. E. Curtis, Proc. Roy. Soc. 101A, 38 (1922)
Omitted endnote
J. C. McLennan, H. G. Smith, and C. A. Lea, Proc. Roy. Soc. 113A, 183 (1926)
Omitted endnote
Omitted endnote

[c1] A. Kratzer, Naturwiss. 27, 577 (1923) Ann. d. Physik 71, 72 (1923)

[c2] E. C. Kemble, Proc. Nat. Acad. Sci. 10, 274 (1924) Phys. Rev. 25, 1 (1925)

[c3] E. C. Kemble, Zeits. f. Physik 35, 286 (1925)

[c4] R. S. Mulliken, Proc. Nat. Acad. Sci. 12, 144 (1926)

[c5] R. S. Mulliken, Proc. Nat. Acad. Sci. 12, 151 (1926)

[c6] R. S. Mulliken, Phys. Rev. 28, 481 (1926)

[c7] R. S. Mulliken, Phys. Rev. 28, 1202 (1926)

[c8] A. Sommerfeld, Atombau und Spektrallinien, 4th Edition, Chapter 9, (1924)

[c9] H. Hönl and F. London, Zeits. f. Physik 33, 803 (1925)

[c10] G H. Dieke, Zeits. f. Physik 33, 161 (1925)

[c11] L. Mensing, Zeits. f. Physik 36, 814 (1926)

[c12] E. Fues, Ann. der Physik 80, 367 (1926)

[c13] J. R. Oppenheimer, Proc. Camb. Phil. Soc. 23, Part 3, p. 327 (1926)

[c14] Kemble, Fowler, and Dieke, [7], p. 1206

[c15] Omitted endnote

[c16] R. Frerichs, Zeits. f. Physik 20, 170 (1923)

[c17] Omitted endnote

[c18] F. Hund, Zeits. f. Physik 36, 657 (1926)

[c19] Omitted endnote

[c20] Hönl and London. [9]

[c21] Omitted endnote

[c22] Omitted endnote

[c23] D. M. Dennison, Phys. Rev. 28, 329 (1926)

[c24] R. H. Fowler, Phil. Mag. 50, 1079 (1925)

[c25] R. S. Mulliken, Phys. Rev. 26, 21-2 and 333 (1925)

[c26] R. T. Birge, Astrophys. J. 55, 273 (1922)

[c27] J. H. Van Vleck, Phys. Rev. 28, 980 (1926)

[c28] Omitted endnote

[c29] R. S. Mulliken, Proc. Nat. Acad. Sci. 12, 158 (1926)

[c30] Omitted endnote

[c31] Curtis and Long's (Proc. Roy. Soc. 108A, 513, 1925)

[c32] W. E. Curtis, Proc. Roy. Soc. 103A, 315 (1923)

[c33] G. Eriksson and E. Hulthén, Zeits. f. Physik 34, 775 (1925). p. 777

[c34] W. Mörikofer, Dissertation Basel, 1925

[c35] Omitted endnote

[c36] W. E. Curtis, Proc. Roy. Soc. 101A, 38 (1922)

[c37] Omitted endnote

[c38] J. C. McLennan, H. G. Smith, and C. A. Lea, Proc. Roy. Soc. 113A, 183 (1926)

[c39] Omitted endnote

[c40] Omitted endnote

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Electronic States and Band Spectrum Structure in Diatomic Molecules III. Intensity Relations

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