Symbols, conversions, and atomic units

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1 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, m e, and the charge of the proton, e, and permittivity 4πɛ 0 are also set to 1 in atomic units. Table C.7 lists several common physical quantities in atomic units.

2 162 Symbols, conversions, and atomic units Energies exact ground state nonrelativistic energy E 0 correlation energy E corr exact nonrelativistic electronic energy E el Hartree-Fock energy E HF ith orbital energy ε i ith exact electronic eigenvalue E i ith eigenvalue of Hamiltonian E i 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 E (n) i E J E k E υ E tot T T k V HF V(r) Table C.1: Symbols of energies and angular momenta J K L Λ M Ω R S Σ Constants Bohr radius a 0 speed of light c dielectric constant ɛ 0 Planck s constant Boltzmann constant k B Table C.5: Symbols of constants

3 163 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 α(ϖ) β(ϖ) χ χ i E(t) f(t)( E(t)) F(ω) φ exact wave function for electronic ground state Φ 0 exact electronic wave function Φ el exact time-dependent nuclear wave function Φ nuc (t) exact total wave function sum of exact total time-dependent wave functions Φ tot (t) Ψ tot (t) gobbler function G(r i ) error kets of momentum space O( t) rotational partition function Q rot (T ) kets of coordinate space autocorrelation function electric field envelope function absorption cross-section p r S(t) s(t) σ(ω) integral over Euler angles W dip (φ, θ, χ) Boltzmann weighting w J (T ) spherical harmonic spatial orbital YJ M (θ, φ) ψ Hartree-Fock wave function Ψ 0 jth vibrational eigenvector of discretized space Ψ υ j Table C.2: Symbols of functions

4 164 Symbols, conversions, and atomic units Operators identity operator Fock operator one-electron Hamiltonian Hamiltonian Ĥ perturbation Hamiltonian Ĥ zeroth-order Hamiltonian Ĥ 0 electronic Hamiltonian Ĥ el Hartree-Fock Hamiltonian Ĥ HF nuclear Hamiltonian in effective electronic field rigid rotor Hamiltonian total angular momentum operator one-electron coulomb operator one-electron exchange operator Legendrian operator ˆ1 ˆf ĥ Ĥ el nuc Ĥ rr dipole moment operator µ 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 Ĵ Ĵ b ˆKb ˆΛ2 ˆ 2 ˆp ˆr ˆr 1 12 ˆT ˆV ˆV(t) ˆVext (t) Û α C ε F I R S

5 165 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 A B c C J Ω M C D J M M ε electric field strength E 0 electric field carrier envelope phase ϕ Euler angles of rotation φ,θ,χ gobbler parameters g, g 0 collinear bond angle γ maximum intensity force constant discretized wave number number of spatial orbitals grid length generalized particle mass electron mass nuclear mass I max k k K L=N r m m e m nuc permanent dipole moment µ 0 reduced mass µ main quantum number generalized rotation axis number of electrons number of particles number of grid discretizations n n N N N

6 166 Symbols, conversions, and atomic units vibrational frequency radiation wavelength electric field carrier frequency discretized momentum variable generalized rotation angle charge spatial coordinate ν λ ω p ξ Q r equilibrium internuclear distance r 0 discretized position variable r i j interelectron distance i A nuclear-electron distance A B internuclear distance Gaussian pulse width rotational period time r ij r ia R AB σ τ rot initial time t 0 pulse duration t p discretized time variable t 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 t T υ x, y, z X, Y, Z x ϖ Z A ζ

7 167 Joule kj mol 1 ev au cm 1 Hz 1 Joule = kj mol 1 = ev = au = cm 1 = Hz = Table C.6: Conversion factors for energy units, adapted from Ref. [176] Quantity SI units Atomic Units mass kg m e =1 charge C e =1 angular momentum J s =1 permittivity κ=4πε 0 =1 length m κ 2 /me 2 =a 0 =1 (bohr) energy ev e 2 /κa 0 =1 (hartree) time s κ 2 3 /me 4 =1 electric dipole C m(=2.54 Debye) ea 0 =1 electric potential V e/κa 0 =1 electric field strength V/m e/κa 2 0 =1 Table C.7: Conversion from SI units to atomic units, adapted from Ref. [176]

8 168 Symbols, conversions, and atomic units

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