MIT OpenCourseWare http://ocw.mit.edu 5.80 Small-Molecule Spectroscopy and Dynamics Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.Lecture # 8 Supplement Contents 1. Excerpts from The Spectra and Dynamics of Diatomic Molecules . . . . . . . . . . . 1 2. Electronic Configurations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1. Excerpts from The Spectra and Dynamics of Diatomic Molecules by Professor Robert W. Field and Hel´ `ene Lefebvre-Brion Elsevier Academic Press 2004 1Chapter 3 Terms Neglected in the Born-Oppenheimer Approximation Contents 3.1 The Born-Oppenheimer Approximation ...... 89 3.1.1 Potential Energy Curves . . . . . . . . . . . . . . . . 90 3.1.2 Terms Neglected in the Born-Oppenheimer Approximation . . . . . . . . . . . . . . . . . . . . . 92 3.1.2.1 Electrostatic and Nonadiabatic Part of H .92 3.1.2.1.1 Crossing or Diabatic Curves . . . 93 3.1.2.1.2 Noncrossing or Adiabatic Curves . 94 3.1.2.2 The Spin Part of H ............. 94 3.1.2.3 Rotational Part of H ............ 96 3.2 Basis Functions . . . . ................. 99 3.2.1 Hund’s Cases . . . . . . . . . . . . . . . . . . . . . . 101 3.2.1.1 Definition of Basis Sets . . . . . . . . . . . 103 3.2.1.2 Quantum Numbers, Level Patterns, and the Effects of Terms Excluded from H(0). . 113 3.2.1.3 Intermediate Case Situations . . . . . . . . 126 3.2.1.3.1 Introduction . . . . . . . . . . . . 126 3.2.1.3.2 Examples . . . . . . . . . . . . . . 127 3.2.1.4 Transformations Between Hund’s Case Ba- sis Sets . . . . . . . . . . . . . . . . . . . . 130 3.2.1.5 Spectroscopic vs. Dynamical Hund’s Cases 136 3.2.1.6 Relationship between Noncommuting Terms in H and the Most Appropriate Hund’s Case137 3.2.2 Symmetry Properties . . . . . . . . . . . . . . . . . . 138� � 88 Terms Neglected in the Born-Oppenheimer Approximation 3.2.2.1 Symmetry Properties of Hund’s Case (a) Basis Functions . . . . . . . . . . . . . . . 138 3.2.2.2 Symmetry Properties of non-Hund’s Case (a) Basis Functions . . . . . . . . . . . . . 146 3.2.3 Molecular Electronic Wavefunctions . . . . . . . . . 148 3.2.4 Matrix Elements between Electronic Wavefunctions 156 3.3 Electrostatic Perturbations ............. 161 3.3.1 Diabatic Curves . . . . . . . . . . . . . . . . . . . . 163 3.3.2 Approximate Representation of the Diabatic Elec- tronic Wavefunction . . . . . . . . . . . . . . . . . . 165 3.3.3 Adiabatic Curves . . . . . . . . . . . . . . . . . . . . 168 3.3.4 Choice between the Diabatic and Adiabatic Models . 172 3.3.5 Electromagnetic Field-Dressed Diabatic and Adia- batic Potential Energy Curves . . . . . . . . . . . . 177 3.4 Spin Part of the Hamiltonian . . . . . . ...... 180 3.4.1 The Spin-Orbit Operator . . . . . . . . . . . . . . . 181 3.4.2 Expression of Spin-Orbit Matrix Elements in Terms of One-Electron Molecular Spin-Orbit Parameters . 183 3.4.2.1 Matrix Elements of the lzi szi Term . . . 183· 3.4.2.1.1 Diagonal Matrix Elements . . . . 183 3.4.2.1.2 Off-Diagonal Matrix Elements . . 187 l+ i s−+ l− +3.4.2.2 Matrix Elements of the i i si Part of HSO .................... 190 3.4.3 The Spin-Rotation Operator . . . . . . . . . . . . . 191 3.4.4 The Spin-Spin Operator . . . . . . . . . . . . . . . . 196 3.4.4.1 Diagonal Matrix Elements of HSS: Calcu- lation of the Direct Spin-Spin Parameter . 196 3.4.4.2 Calculation of Second-Order Spin-Orbit Ef- fects . . . . . . . . . . . . . . . . . . . . . . 198 3.4.4.2.1 π2 Configuration . . . . . . . . . . 201 3.4.4.2.2 π3π (or π3π3 and ππ) Configu- rations . . . . . . . . . . . . . . . 201 3.4.4.3 Off-Diagonal Matrix Elements . . . . . . . 202 3.4.5 Tensorial Operators . . . . . . . . . . . . . . . . . . 203 3.5 Rotational Perturbations . . ............. 210 3.5.1 Spin-Electronic Homogeneous Perturbations . . . . . 210 3.5.2 The S-Uncoupling Operator . . . . . . . . . . . . . . 212 3.5.3 The L-Uncoupling Operator . . . . . . . . . . . . . . 213 3.5.4 2Π ∼ 2Σ+ Interaction . . . . . . . . . . . . . . . . . 217 3.6 References ........................ 22789 3.1 The Born-Oppenheimer Approximation 3.1 The Born-Oppenheimer Approximation The exact Hamiltonian H for a diatomic molecule, with the electronic coordi-nates expressed in the molecule-fixed axis system, is rather difficult to derive. Bunker (1968) provides a detailed derivation as well as a review of the coordi-nate conventions, implicit approximations, and errors in previous discussions of the exact diatomic molecule Hamiltonian. Our goal is to find the exact solutions, ψT i (T = Total), of the Schr¨odinger equation, HψT i = ET i ψT i , (3.1.1) which correspond to the observed (exact) EiT energy levels. H is the nonrela-tivistic Hamiltonian, which may be approximated by a sum of three operators, H = TN (R, θ, φ)+ Te(r)+ V (r, R), (3.1.2) where TN is the nuclear kinetic energy, Te is the electron kinetic energy, V is the electrostatic potential energy for the nuclei and electrons (including e−−e−,e−−N and N − N interactions), R is the internuclear distance, θ and φ specify the orientation of the internuclear axis (molecule-fixed coordinate system) relative to the laboratory coordinate system (see Section 2.3.3 and Fig. 2.4), and r represents all electron coordinates in the molecule-fixed system. The nuclear kinetic energy operator is given by �� � � � � R2 2µR2 ∂R ∂R sin θ ∂θ ∂θ sin2 θ ∂φ2TN (R, θ, φ)= −�2 ∂ ∂ +1 ∂ sin θ∂ +1 ∂2 , (3.1.3a) where MAMB µ = MA + MB is the nuclear reduced mass, with MA and MB the masses of atoms A and B. TN can be divided into vibrational and rotational terms, TN (R, θ, φ)= TN (R)+ HROT (R, θ, φ). (3.1.3b) The electron kinetic energy operator is 2Te(r)= −�2 � …