Rotational spectroscopy of diatomic molecules

Cover......Page 1
Half-title......Page 3
Series-title......Page 4
Title......Page 5
Copyright......Page 6
Contents......Page 7
Preface......Page 17
Summary of notation......Page 21
Some additional notes......Page 24
Figure acknowledgements......Page 25
1.1. Electromagnetic spectrum......Page 34
1.2. Electromagnetic radiation......Page 36
1.3. Intramolecular nuclear and electronic dynamics......Page 38
1.4. Rotational levels......Page 42
1.5. Historical perspectives......Page 45
1.6.1. Introduction......Page 47
1.6.2. 1Sigma+ states......Page 48
1.6.3. Open shell Sigma states......Page 54
1.6.4. Open shell states with both spin and orbital angular momentum......Page 59
1.7. The effective Hamiltonian......Page 62
1.8. Bibliography......Page 65
Appendix 1.1. Maxwell's equations......Page 66
Appendix 1.2. Electromagnetic radiation......Page 68
References......Page 69
2.1. Introduction......Page 71
2.2.1. Introduction......Page 73
2.2.2. Origin at centre of mass of molecule......Page 74
2.2.3. Origin at centre of mass of nuclei......Page 76
2.3. The total Hamiltonian in field-free space......Page 77
2.4. The nuclear kinetic energy operator......Page 78
2.5.1. Introduction......Page 84
2.5.2. Space transformations......Page 85
2.5.3. Spin transformations......Page 87
2.6. Schrödinger equation for the total wave function......Page 92
2.7. The Born–Oppenheimer and Born adiabatic approximations......Page 93
2.8. Separation of the vibrational and rotational wave equations......Page 94
2.9. The vibrational wave equation......Page 96
2.11. Non-adiabatic terms......Page 100
2.12. Effects of external electric and magnetic fields......Page 101
Appendix 2.1. Derivation of the momentum operator......Page 104
References......Page 105
3.1. The Dirac equation......Page 106
3.2. Solutions of the Dirac equation in field-free space......Page 109
3.3. Electron spin magnetic moment and angular momentum......Page 110
3.4. The Foldy–Wouthuysen transformation......Page 113
3.5. The Foldy–Wouthuysen and Dirac representations for a free particle......Page 118
3.6. Derivation of the many-electron Hamiltonian......Page 122
3.7. Effects of applied static magnetic and electric fields......Page 127
3.8.1. Introduction......Page 130
3.8.2. Lorentz transformation......Page 131
3.8.3. Electromagnetic potentials due to a moving electron......Page 132
3.8.4. Gauge invariance......Page 134
3.8.5. Classical Lagrangian and Hamiltonian......Page 136
3.9.1. Introduction......Page 137
3.9.2. Reduction of the Breit Hamiltonian to non-relativistic form......Page 138
3.10. Electronic interactions in the nuclear Hamiltonian......Page 142
3.11. Transformation of coordinates in the field-free total Hamiltonian......Page 143
3.12. Transformation of coordinates for the Zeeman and Stark terms in the total Hamiltonian......Page 147
3.13. Conclusions......Page 151
Appendix 3.1. Power series expansion of the transformed Hamiltonian......Page 154
References......Page 155
4.1. Nuclear spins and magnetic moments......Page 156
4.2. Derivation of nuclear spin magnetic interactions through the magnetic vector potential......Page 158
4.3.Derivation of nuclear spin interactions from the Breit equation......Page 163
4.4.1. Spherical tensor form of the Hamiltonian operator......Page 164
4.4.2. Cartesian form of the Hamiltonian operator......Page 166
4.4.3. Matrix elements of the quadrupole Hamiltonian......Page 167
4.5. Transformation of coordinates for the nuclear magnetic dipole and electric quadrupole terms......Page 169
References......Page 171
5.1. Introduction......Page 172
5.2.1. Introduction......Page 173
5.2.3. Commutation relations......Page 175
5.2.4. Representations of the rotation group......Page 176
5.2.5. Orbital angular momentum and spherical harmonics......Page 177
5.3.1. Introduction......Page 179
5.3.2. Rotation matrices......Page 181
5.3.4. Symmetric top wave functions......Page 183
5.4.1. Introduction......Page 185
5.4.2. Wigner 3-j symbols......Page 187
5.4.3. Coupling of three or more angular momenta: Racah algebra, Wigner 6-j and 9-j symbols......Page 188
5.4.4. Clebsch–Gordan series......Page 190
5.4.5. Integrals over products of rotation matrices......Page 191
5.5.1. Introduction......Page 192
5.5.4. Matrix elements for composite systems......Page 198
5.5.5. Relationship between operators in space-fixed and molecule-fixed coordinate systems......Page 200
5.5.6. Treatment of the anomalous commutation relationships of rotational angular momenta by spherical tensor methods......Page 201
Appendix 5.1. Summary of standard results from spherical tensor algebra......Page 204
References......Page 208
6.1. Introduction......Page 210
6.2.1. The hydrogen atom......Page 211
6.2.2. Many-electron atoms......Page 214
6.2.3. Russell–Saunders coupling......Page 217
6.2.4. Wave functions for the helium atom......Page 220
6.2.5. Many-electron wave functions: the Hartree–Fock equation......Page 223
6.2.6. Atomic orbital basis set......Page 227
6.2.7. Configuration interaction......Page 229
6.3. Molecular orbital theory......Page 230
6.4. Correlation of molecular and atomic electronic states......Page 236
6.5.1. Introduction......Page 239
6.5.2. Electronic wave function for the…......Page 240
(i) Introduction......Page 245
(ii) The electronic Hamiltonian......Page 246
(iv) Multiple-determinant wave functions: configuration interaction......Page 249
6.6 Corrections to Born–Oppenheimer calculations for…......Page 252
6.7.1. Introduction......Page 257
6.7.2. Hund’s coupling case (a)......Page 258
6.7.3. Hund’s coupling case (b)......Page 259
6.7.5. Hund’s coupling case (d)......Page 261
6.7.6. Hund’s coupling case (e)......Page 262
6.7.7. Intermediate coupling......Page 263
6.7.8. Nuclear spin coupling cases......Page 265
6.8.1. The rigid rotor......Page 266
6.8.2. The harmonic oscillator......Page 268
6.8.3. The anharmonic oscillator......Page 271
6.8.4. The non-rigid rotor......Page 275
6.8.5. The vibrating rotor......Page 276
6.9.1. The space-fixed inversion operator......Page 277
6.9.2. The effect of space-fixed inversion on the Euler angles and on molecule-fixed coordinates......Page 278
6.9.3. The transformation of general Hund’s case (a) and case (b) functions under space-fixed inversion......Page 279
6.10.1. The nuclear permutation operator for a homonuclear diatomic molecule......Page 284
6.10.2. The transformation of general Hund’s case (a) and case (b) functions under nuclear permutation P12......Page 285
6.10.3. Nuclear statistical weights......Page 287
6.11.1. Time-dependent perturbation theory......Page 289
6.11.2. The Einstein transition probabilities......Page 291
6.11.3. Einstein transition probabilities for electric dipole transitions......Page 294
6.11.4. Rotational transition probabilities......Page 296
6.11.5. Vibrational transition probabilities......Page 299
6.11.6. Electronic transition probabilities......Page 300
6.11.7. Magnetic dipole transition probabilities......Page 302
6.12.2. Transit time broadening......Page 306
6.12.3. Doppler broadening......Page 307
6.12.4. Collision broadening......Page 308
6.13.1. Introduction......Page 309
6.13.2. The JWKB semiclassical method......Page 310
6.13.3. Inversion of experimental data to calculate the potential function (RKR)......Page 313
6.14. Long-range near-dissociation interactions......Page 315
6.15. Predissociation......Page 319
Appendix 6.1. Calculation of the Born–Oppenheimer potential for the…......Page 322
References......Page 331
7.1. Introduction......Page 335
7.2. Derivation of the effective Hamiltonian by degenerate perturbation theory: general principles......Page 336
7.3. The Van Vleck and contact transformations......Page 345
7.4.1. Introduction......Page 349
7.4.2. The rotational Hamiltonian......Page 352
7.4.3. Hougen’s isomorphic Hamiltonian......Page 353
7.4.4. Fine structure terms: spin–orbit, spin–spin and spin–rotation operators......Page 356
7.4.5. -doubling terms for a Pi electronic state......Page 361
7.4.6. Nuclear hyperfine terms......Page 364
7.4.7. Higher-order fine structure terms......Page 368
7.5.1. Vibrational averaging and centrifugal distortion corrections......Page 371
7.5.2. The form of the effective Hamiltonian......Page 374
7.5.3. The N2 formulation of the effective Hamiltonian......Page 376
7.5.4. The isotopic dependence of parameters in the effective Hamiltonian......Page 377
7.6. Effective Zeeman Hamiltonian......Page 380
7.7. Indeterminacies: rotational contact transformations......Page 385
7.8.2. Rotational constant......Page 389
7.8.3. Spin–orbit coupling constant, A......Page 390
7.8.4. Spin–spin and spin–rotation parameters, Lambda and Gyama......Page 393
7.8.5. -doubling parameters......Page 395
7.8.6. Magnetic hyperfine interactions......Page 396
7.8.7. Electric quadrupole hyperfine interaction......Page 398
Appendix 7.1. Molecular parameters or constants......Page 401
References......Page 402
8.1. Introduction......Page 404
(i) MOLECULAR BEAM FORMATION......Page 405
(ii) MOLECULAR BEAM DETECTION......Page 406
(iii) STATE SELECTION AND POPULATION TRANSFER......Page 407
(b) Details of the apparatus used......Page 408
(c) Effective Hamiltonian, energy levels and spectroscopic transitions......Page 409
(a) Introduction......Page 423
(b) Semi-classical theory of diamagnetism for a spherically symmetric atom......Page 425
(i) SEMI-CLASSICAL THEORY......Page 426
(ii) MOLECULAR ROTATION......Page 429
(iv) MANETIC FIELD PERTURBATIONS OF THE ELECTRONIC GROUND STATE......Page 436
(v) REPRESENTATION OF THE DIAMAGNETIC SUSCEPTIBILITY IN THE EFFECTIVE HAMILTONIAN......Page 441
(vi) MOLECULAR QUADPUPOLE MOMENTS......Page 442
(d) Nuclear spin effects: magnetic shielding and nuclear spin–rotation interaction......Page 443
8.2.3. Na2 in the X1Sigma+ ground state: optical state selection and detection......Page 449
8.2.4. Other 1Gyama+ molecules......Page 454
(a) Introduction......Page 455
(b) Experimental studies......Page 457
(c) Effective Zeeman Hamiltonian in a case (b) basis......Page 458
(i) SPIN–ROTATION INTERACTION......Page 461
(ii) ELECTRON SPIN–SPIN DIPOLAR INTEERACTION......Page 463
(iii) SPIN–ORBIT INTERACTION......Page 467
(ii) ORBITAL HYPERFINE INTERACTION......Page 473
(iii) ELECTRON SPIN–NUCLEAR SPIN DIPOLAR INTERACTION......Page 474
(a) Introduction......Page 479
(b) Experimental studies and results......Page 480
(c) Zeeman effect......Page 482
(i) EFFECTIVE HAMILTONIAN......Page 485
(ii) MATRIX ELEMENT OF THE EFFECTIVE HAMILTONIAN......Page 486
(iii) CALCULATION OF ENERGY LEVELS AND TRANSITION FREQUENCIES......Page 491
(iv) INTEERPRETATION OF THE MOLECULAR PARAMETERS......Page 494
8.4.1. Principles of electric resonance methods......Page 496
(a) Stark effect......Page 498
(b)‘ Weak’ field coupled basis......Page 503
(i) QUADRUPOLE INTERACTION......Page 509
(iii)19F NUCLEAR SPIN–ROTATION INTERACTION......Page 510
(iv) UCLEAR SPIN DIPOLAR INTERACTIOON......Page 511
(v) ELECTRIC FIELD INTERACTION......Page 512
(d) Interpretation of the molecular constants......Page 514
(b) Effective Hamiltonian, matrix elements and energy levels in the ‘weak’ field case......Page 516
(c) ‘Strong’ magnetic field spectrum......Page 518
8.4.4. Alkaline earth and group IV oxides......Page 520
(b) Effective Hamiltonian and matrix elements......Page 522
(i) STARK INTERACTION......Page 523
(ii) NUCLEAR SPIN–ROTATION TERMS......Page 524
(iii) NUCLEAR SPIN DIPOLAR INTERECTION......Page 525
(c) Calculation of the energy levels and electric resonance spectrum......Page 526
(d) Electric resonance spectrum in the presence of a strong magnetic field......Page 529
8.4.6. HCl in the X1Sigma+ ground state......Page 533
8.5.1. Introduction......Page 541
(a) Introduction......Page 542
(b) Effective Hamiltonian and basis set......Page 544
(c) Matrix elements without nuclear spin in the primitive basis set......Page 546
(d) Matrix elements without nuclear spin in the parity-conserved basis set......Page 548
(e) Matrix elements of the nuclear hyperfine terms in the primitive basis set......Page 550
(f) Matrix elements of the nuclear hyperfine terms in the parity-conserved basis set......Page 555
(g) Values of the molecular hyperfine constants......Page 557
(h) The…doubling constants and frequencies......Page 558
(a) Introduction......Page 559
(b) Theory of the…doubling......Page 560
(c) Nuclear hyperfine interactions......Page 565
(d) Interpretation of the molecular constants......Page 570
(a) Introduction......Page 571
(b) Molecular beam electric resonance and beam maser studies......Page 572
(c) Theory and analysis of the…doubling hyperfine transitions......Page 575
(d) Centrifugal distortion of the…doubling......Page 579
(e) Comparison of OH, OD, SH and SD......Page 581
(f) Electric dipole moment of OH......Page 582
(a) Introduction and experimental results......Page 585
(b) Stark effect......Page 586
(c) Theoretical analysis......Page 589
Appendix 8.1. Nuclear spin dipolar interaction......Page 591
Appendix 8.3. Electron spin–electron spin dipolar interaction......Page 596
Appendix 8.4. Matrix elements of the quadrupole Hamiltonian......Page 601
Appendix 8.5. Magnetic hyperfine Hamiltonian and hyperfine constants......Page 606
References......Page 607
9.2.1. Microwave magnetic resonance......Page 612
9.2.2. Far-infrared laser magnetic resonance......Page 617
(a) Introduction......Page 620
(b) Effective Hamiltonian, matrix elements and assignment......Page 622
(a) Nuclear magnetic and electric hyperfine interactions......Page 624
(b) Parity doubling and Stark effect in 1Delta states......Page 627
9.4.1. Introduction......Page 629
(a) Introduction......Page 630
(b) Effective Hamiltonian......Page 632
(i) RIGID BODY + SPIN–ORBIT COUPLING......Page 634
(ii) MAGNETIC HYPERFINE COUPLING......Page 635
(iii) NUCLEAR ELECTRIC QUADRUPOLE COUPLING......Page 637
(a) Orbital Zeeman interaction......Page 638
(d) Analysis of the ClO spectrum......Page 639
(i) MICROWAVE MAGNETIC RESONANCE......Page 641
(ii) FAR-INFRARED LASER MAGNETIC RESONANCE OF HF+, HCL+ AND HBr+......Page 642
(iii) MICROWAVE MAANETIC RESONANCE SPECTRUM OF NO......Page 644
(i) INTRODUCTION......Page 646
(ii) EXPERIMENTAL STUDIES......Page 647
(iii) THEORY OF THE Lambda-DOUBLING......Page 650
(iv) ZEEMAN EFFECT......Page 653
(b) Far-infrared laser magnetic resonance spectrum of OH......Page 655
(b) Hund’s case (b) behaviour in 2Pi states......Page 657
(c) Experimental studies......Page 661
9.5.2. CN in the X2Sigma+ ground state......Page 666
(ii) SPIN–ROTATION INTERACTION......Page 668
(iv) DIPOLAR HYPERFINE INTERACTION......Page 669
(vi) ZEEMAN INTERACTIONS......Page 670
9.6.1. SO in the X3Sigma- ground state......Page 674
(ii) SPIN–SPIN INTERACTION......Page 676
(iv) ELECTRON SPIN ZEEMAN INTERACTION......Page 678
(vi) ROTATIONAL ZEEMAN INTERACTION......Page 679
9.6.2. SeO in the X3Sigma- ground state......Page 682
9.6.3. NH in the X3Sigma- ground state......Page 685
9.7.1. CO in the a 3Pi state......Page 688
9.8.1. CH in the a4Sigma- state......Page 694
9.9.1. Introduction......Page 698
9.9.2. CrH in the X6Sigma+ ground state......Page 699
9.9.4. CoH in the X3Phi ground state......Page 702
9.9.5. NiH in the X2Delta ground state......Page 707
Appendix 9.1. Evaluation of the reduced matrix element of T3 (S ,S ,S)......Page 711
References......Page 713
10.1.1. Simple absorption spectrograph......Page 716
10.1.2. Microwave radiation sources......Page 718
(a) Stark modulation......Page 721
(c) Source modulation......Page 725
(d) Velocity modulation......Page 732
10.1.4. Superheterodyne detection......Page 734
(a) Introduction......Page 736
(b) Pulsed-nozzle Fourier transform spectrometer......Page 737
(c) Fourier transformation from the time domain to the frequency domain......Page 739
(d) Fabry–Perot cavities......Page 741
(f)Fourier transform far-infrared interferometry......Page 743
(a) Introduction......Page 746
(b) Telescope dishes......Page 747
(c) Receiver/detector systems......Page 748
(d) Nature of discrete line spectra......Page 751
10.1.7. Terahertz (far-infrared) spectrometers......Page 756
10.1.8. Ion beam techniques......Page 761
10.2.1. CO in the X Sigma+ ground state......Page 765
10.2.2. HeH+ in the X 1Sigma+ ground state......Page 769
10.2.3. CuCl and CuBr in their X1Simga+ ground states......Page 771
10.2.4. SO, NF and NCl in their b1Sigma+ states......Page 774
10.2.5. Hydrides (LiH, NaH, KH, CuH, AlH, AgH) in their X1Simga+ ground states......Page 776
10.3.1. CO+ in the X2Sigma+ ground state......Page 778
10.3.2. CN in the X2Simga+ ground state......Page 782
10.4.1. Introduction......Page 785
10.4.2. O2 in its X3Simga…......Page 787
10.4.3. SO, S2 and NiO in their X3Sigma- ground states......Page 792
10.4.4. PF, NCl, NBr and NI in their X3Simga- ground states......Page 796
10.5.1. O2 in its a 1Delta state......Page 809
10.5.2. SO and NCl in their a 1Delta states......Page 812
10.6.1. NO in the X2Pi ground state......Page 815
10.6.2. OH in the X2Pi ground state......Page 821
(a) Observation and assignment of Lambda -doubling and rotational transitions......Page 827
(b) Theoretical analysis and determination of molecular parameters......Page 832
(i) ELECTRONIC STRUCTURE CALCULATIONS......Page 838
(ii) Lambda-DOUBLING COMSTANTS......Page 840
(iii) SPPIN–ROTATION CONSTANT......Page 842
10.6.4. CF, SiF, GeF in their X2Pi ground states......Page 843
10.6.5. Other free radicals with 2Pi ground states......Page 844
(a) Experimental observations......Page 846
(i) VECTOR COUPLING AND QUANTUM NUMBERS......Page 852
(ii) ROTATIONAL HAMILTONIAN AND ROTATIONAL LEVELSS INCASE (c)......Page 853
(iii) ZEEMAN EFFECT IN HUND'5 CASE (c)......Page 854
(iv) ELECTRIC DIPOLE TRANSITION PROBABILITIES......Page 856
(ii) COUPLED-CHANNEL THEORY......Page 857
(iii) COUPLING MATRIX]......Page 859
(v) ZEEMAN EFFECT IN CASE (e)......Page 861
(vi) RESULTS......Page 862
10.7.2. Studies of the HeKr+ ion......Page 865
10.8.1. CO in the a3Pi state......Page 867
10.8.2. SiC in the X3Pi ground state......Page 869
10.8.4. VO and NbO in their X4Sigma- ground states......Page 874
10.8.5. FeF and FeCl in their X 6Delta ground states......Page 878
10.8.6. CrF, CrCl and MnO in their X6Sigma+ ground states......Page 883
10.8.7. FeO in the X5Delta ground state......Page 886
10.8.8. TiCl in the X4Phi ground state......Page 887
10.9. Observation of a pure rotational transition in the H…......Page 889
References......Page 895
11.1. Introduction......Page 903
11.2. Radiofrequency and microwave studies of CN in its excited electronic states......Page 904
11.3.1. Radiofrequency/optical double resonance of CS in its excited A1Pi state......Page 909
11.3.2. Radiofrequency/optical double resonance of OH in its excited A2sIGMA+ state......Page 913
11.3.3. Microwave/optical double resonance of BaO in its ground X1Sigma+ and excited A1Sigma+ states......Page 916
11.4.2. H2 in the G1Sigma+g state......Page 918
11.4.3. H2 in the d 3Piu state......Page 925
11.4.4. H2 in the k 3Piu state......Page 933
11.5.1. Introduction......Page 935
11.6.1. Introduction......Page 939
11.6.2. FeO in the X5Delta ground state......Page 942
11.6.3. CuF in the b3Pi excited state......Page 946
11.6.4. CuO in the X2Pi  ground state......Page 950
11.6.5. ScO in the X2Sigma+ ground state......Page 952
11.6.6. TiO in the X3Delta  ground state and TiN in the X2Sigma+ ground state......Page 955
11.6.7. CrN and MoN in their X4Sigma- ground states......Page 957
11.6.8. NiH in the X2Delta ground state......Page 960
11.6.9. 4d transition metal molecules: YF in the X1Sigma+ ground state, YO and YS in their X2Sigma+ ground states......Page 963
11.7.1. Radiofrequency/optical double resonance of YbF in its X2Sigma+ ground state......Page 969
11.7.2. Radiofrequency/optical double resonance of LaO in its X2Sigma+ and B2Sigma+ states......Page 971
11.8.1. Radiofrequency and microwave/infrared double resonance of HD+ in the X2Sigma+ ground state......Page 975
11.8.2. Radiofrequency/optical double resonance of N+ in the X2Sigma…......Page 986
11.8.3. Microwave/optical double resonance of CO+ in the X2Sigma+ ground state......Page 991
11.9.2. Principles of photo-alignment......Page 993
11.9.3. Experimental methods and results......Page 995
11.9.4. Analysis of the spectra......Page 997
11.9.5. Quantitative interpretation of the molecular parameters......Page 1005
References......Page 1007
Appendix A Values of the fundamental constants......Page 1011
Appendix B Selected set of nuclear properties for naturally occurring isotopes......Page 1012
Appendix C Compilation of Wigner 3-j symbols......Page 1020
Appendix D Compilation of Wigner 6-j symbols......Page 1024
Appendix E Relationships between cgs and SI units......Page 1026
Author index......Page 1027
Subject index......Page 1037