NBS MONOGRAPH 115 The Calculation of Rotational Energy Levels and Rotational I Line Intensities in Diatomic Molecules U.S. RTMENT OF MMERCE National Bureau - NATIONAL BUREAU OF STANDARDS ' The National Bureau of Standards was established by an act of Congress March 3, 1901 . Today, in addition to serving as the Nation's central measurement laboratory, the Bureau is a principal focal point in the Federal Government for assuring maximum application of the physical and engineering sciences to the advancement of technology in industry and commerce. To this end the Bureau conducts research and provides central national services in four broad program areas. These are: (1) basic measurements and standards, (2) materials measurements and standards, (3) technological measurements and standards, and (4) transfer of technology. 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' Located at 5285 Port Royal Road, Springfield, Virginia 22151. UNITED STATES DEPARTMENT OF COMMERCE Maurice H. Stans. Secretary NATIONAL BUREAU OF STANDARDS • Lewis M. Branscomb, Director The Calculation of Rotational Energy Levels and Rotational Line Intensities in Diatomic Molecules Jon T. Hougen Institute for Basic Standards National Bureau of Standards Washington, D.C. 20234 y.S 1 National Bureau of Standards Monograph 115, Nat. Bur. Stand. (U.S.), Monogr. 115. 52 pages (June 1970) CODEN: NBSMA Issued June 1970 For sale by the Superintendent of Documents, U.S. Government Printinfi Office Washington, D.C. 20402 (Order by SO Catalog No. C13.44:n5). Price 55 cents mmfil BUREAU OF STANDARDS SEP 1 2 197^ Freiace i, / i This monograph describes procedures for making quantum mechanical calculatioiis of rota- tional energy levels and rotational line intensities in diatomic molecules. The procedures are illustrated by sample calculations. A familiarity with the material of this monograph should enable a practicing electronic spectroscopist to carry out, though in a rather mechanical way, his own theoretical calculations for molecules under experimental investigation. A true understanding of the procedures described below can only be acquired by studying the theoretical reference material cited. These references are not exhaustive; they represent sources which the author finds convenient and instructive. Unfortunately, the material in the references is sometimes presented in a notation different' from that used here. In addition, it is sometimes slightly too general or slightly too specific to apply directly to diatomic molecules. The material of this monograph is aimed at electronic spectroscopists who have had the equivalent of one semester of graduate-level quantum mechanics. Finally, I should like to point out, that the procedures described here for carrying out calcula- tions are not new. Neither do they represent all possible correct procedures. They do represent, however, a unified approach to the problem, which, in the opinion of the author, is easier to under- stand than the original literature cited, and less likely to lead to error than some of the alternative calculation procedures. Note added in proof: The reader is referred to the book Rotational Structure in the Spectra of Diatomic Molecules [28] by I. Kovacs, which just appeared and which represents another author's discussion of much of the material in this monograph. Library of Congress Catalog Card No. 76-604235 II Contents Page Preface II 1. Calculation of rotational energy levels 1 1.1. Hund's coupling cases (a), (b), (c), and (d) 1 1.2. General approach to the calculations 2 1.3. Nonrotating-molecule Hamiltonian 3 1.4. Nonrotating-molecule basis set 1.5. Nonrotating-molecule matrix elements 6 1.6. Rotating-molecule Hamiltonian 7 1.7. Rotating-molecule basis set 8 1.8. Rotating-molecule matrix elements 8 1.9. Example: The Hill and Van Vleck expression for ^11 states 9 1.10. Example: The Schlapp expression for ^2 states 13 2. Symmetry properties of the rotational energy levels 15 2.1. Geometric symmetry operations 16 2.2. Permutation-inversion symmetry operations 18 2.3. The symmetry operation o"„ 19 2.4. Example: Parities of the rotational levels in a ' state 21 2.5. Example: Parities of the rotational levels in a ^2"^ state 22 2.6. The symmetry operation i 22 2.7. The symmetry operation C2 : 22 2.8. Example: Symmetry properties of the rotational levels in a 'Ha state 23 2.9. Relations between matrix elements 23 2.10. Example: Li = L2-L| 25 2.11. The time inversion operation 0 25 3. Calculation of rotational line intensities 27 3.1. Laboratory-fixed components of the electric dipole moment operator fi 28 3.2. Molecule-fixed components of ft 29 3.3. The direction cosine matrix a 31 3.4. Example: Honl-London intensity expressions for a 'D — '2"^ transition 32 — 3.5. Example: Rotational intensity distribution in a^S" 'S"*" transition 34 3.6. Intensity calculations when closed-form expressions cannot be obtained. Example: Rotational intensity distribution in a''A — transition 37 4. Perturbations 41 4.1. General remarks 41 4.2. Homogeneous and heterogeneous perturbations 42 4.3. Example: 'H — heterogeneous perturbation 42 4.4. Example: — 'H homogeneous perturbation 44 4.5. Van Vleck transformations 45 4.6. Example: A-type doubling in a 'H state 45 4.7. Centrifugal distortion corrections to rotational energy levels 46 5. References 49 III ; The Calculation of Rotational Energy Levels and Rotational Line Intensities in Diatomic Molecules Jon T. Hougen Procedures are described, in this pedagogical monograph, for