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Appendix C Symbols, conversions, and atomic units This appendix provides a tabulation of all symbols used throughout this thesis, as well as conversion units that may useful for reference while comparing results. Tabel C.6 lists the most common energy units and their conversion factors. All operators in this thesis are denoted as Ô, all matrices are in bold-face, M, and all vector quantities appear with a vector symbol, ~v. Symbols in each table are listed alphabetically. Atomic units were used in this thesis during the simulations of rotational wave packets (see Section 4.4.1). Since the eigenvalues of the angular momentum operators are quantized according to ~, atomic units are convenient in discussions of angular momentum since ~ receives the numerical value 1. The mass of the electron, me , and the charge of the proton, e0 , and permittivity 4π²0 are also set to 1 in atomic units. Table C.7 lists several common physical quantities in atomic units. 162 Symbols, conversions, and atomic units QUANTITY SYMBOL Energies exact ground state nonrelativistic energy correlation energy exact nonrelativistic electronic energy Hartree-Fock energy ith orbital energy ith exact electronic eigenvalue ith eigenvalue of Hamiltonian nth order perturbation energy terms Jth rigid rotor energy kth unperturbed wave function energy vibrational bound-state eigenenergy total energy classical kinetic energy discretized kinetic energy effective Hartree-Fock potential discretized potential energy Angular momenta total angular momentum projection of J on body-fixed z axis electronic orbital angular momentum projection of L on body-fixed z axis projection of J on space-fixed Z axis Λ+Σ nuclear angular momentum spin (intrinsic) angular momentum projection of S on body-fixed z axis E0 Ecorr Eel EHF εi Ei Ei (n) Ei EJ Ek Eυ Etot T Tk VHF V(r) J K L Λ M Ω R S Σ Table C.1: Symbols of energies and angular momenta QUANTITY Constants Bohr radius speed of light dielectric constant Planck’s constant Boltzmann constant SYMBOL a0 c ²0 ~ kB Table C.5: Symbols of constants 163 QUANTITY SYMBOL Functions spin-up (↑) function spin-down (↓) function spin orbital ith canonical Fock spin orbital time-dependent electric field time-dependent electric field frequency-dependent electric field basis functions exact wave function for electronic ground state exact electronic wave function exact time-dependent nuclear wave function exact total wave function sum of exact total time-dependent wave functions gobbler function error kets of momentum space rotational partition function kets of coordinate space autocorrelation function electric field envelope function absorption cross-section integral over Euler angles Boltzmann weighting spherical harmonic spatial orbital Hartree-Fock wave function jth vibrational eigenvector of discretized space α($) β($) χ χi ~ E(t) f (t)(≡ E(t)) F(ω) φ Φ0 Φel Φnuc (t) Φtot (t) Ψtot (t) G(ri ) O(∆t) |pi Qrot (T ) |ri S(t) s(t) σ(ω) Wdip (φ, θ, χ) wJ (T ) YJM (θ, φ) ψ Ψ0 Ψυj Table C.2: Symbols of functions 164 Symbols, conversions, and atomic units QUANTITY SYMBOL Operators identity operator Fock operator one-electron Hamiltonian Hamiltonian 1̂ f̂ ĥ Ĥ perturbation Hamiltonian zeroth-order Hamiltonian electronic Hamiltonian Hartree-Fock Hamiltonian Ĥ Ĥ0 Ĥel ĤHF nuclear Hamiltonian in effective electronic field rigid rotor Hamiltonian total angular momentum operator one-electron coulomb operator one-electron exchange operator Ĥnuc Ĥrr Ĵ Ĵb K̂b 0 el 2 Legendrian operator dipole moment operator Laplacian operator momentum operator position operator two-electron potential energy operator kinetic energy operator potential energy operator time-dependent potential energy operator time-dependent external potential energy time evolution operator Matrices polarizability tensor expansion coefficients matrix orbital energy matrix Fock matrix inertia tensor rotation matrix overlap matrix Table C.3: Symbols of operators and matrices Λ̂ ~ µ ˆ2 ∇ p̂ r̂ r̂−1 12 T̂ V̂ V̂(t) V̂ ext (t) Û α C ε F I R S 165 QUANTITY SYMBOL Variables spin-orbit interaction term rotational constant expansion coefficients rotational wave function expansion coefficients Clebsch-Gordan coefficients expansion coefficients of rotation matrix R electric field polarization electric field strength electric field carrier envelope phase Euler angles of rotation gobbler parameters collinear bond angle maximum intensity force constant discretized wave number number of spatial orbitals grid length generalized particle mass electron mass nuclear mass permanent dipole moment reduced mass main quantum number generalized rotation axis number of electrons number of particles number of grid discretizations A B c CJ Ω M C J DM 0 M ~ε E0 ϕ φ,θ,χ g, g0 γ Imax k ∆k K L = N ∆r m me mnuc µ0 µ n ~n N N N 166 Symbols, conversions, and atomic units vibrational frequency radiation wavelength electric field carrier frequency discretized momentum variable generalized rotation angle charge spatial coordinate equilibrium internuclear distance discretized position variable i–j interelectron distance i–A nuclear-electron distance A–B internuclear distance Gaussian pulse width rotational period time initial time pulse duration discretized time variable temperature vibrational quantum number body-fixed Cartesian coordinates space-fixed Cartesian coordinates spatial and spin coordinate spin coordinate of α and β spin functions atomic number perturbation Table C.4: Symbols of variables ν λ ω ∆p ξ Q ~r r0 ∆r rij riA RAB σ τrot t t0 tp ∆t T υ x, y, z X, Y, Z ~x $ ZA ζ 167 Joule 1 Joule =1 1 kJ·mol−1 =1.661×10−21 1 eV =1.602×10−19 1 au =4.359×10−18 1 cm−1 =1.986×10−23 1 Hz =6.626×10−34 kJ·mol−1 eV au cm−1 Hz 6.022×1020 6.242× 1018 2.2939×1017 5.035×1022 1.509×1033 1 1.036×10−2 3.089×10−4 83.60 2.506×1012 96.48 1 3.675×10−2 8065 2.418×1014 2625 27.21 1 2.195×105 6.580×1015 1.196×10−2 1.240×10−4 4.556×10−6 1 2.998×1010 3.990×10−13 4.136×10−15 1.520×10−16 3.336×10−11 1 Table C.6: Conversion factors for energy units, adapted from Ref. [176] Quantity mass charge angular momentum permittivity length energy time electric dipole electric potential electric field strength SI units Atomic Units 9.109 534×10−31 kg me = 1 −19 1.602 189 2×10 C |e| = 1 −34 1.054 588×10 Js ~=1 1.113 65 ×10−10 κ = 4πε0 = 1 −11 5.291 772 49×10 m κ~2 /me2 = a0 = 1 (bohr) 27.211 4 eV e2 /κa0 = 1 (hartree) 2.418 9×10−17 s κ2 ~3 /me4 = 1 8.478 36 ×10−30 C m(=2.54 Debye) ea0 = 1 27.11 V e/κa0 = 1 5.142 208 ×1011 V/m e/κa0 2 = 1 Table C.7: Conversion from SI units to atomic units, adapted from Ref. [176] 168 Symbols, conversions, and atomic units
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