Introduction to ab initio methods I Kirill Gokhberg

„We‘ve got fascinating results!“ „... this is ICD in the Ar dimer...“

Anatomy of an ICD process Ar 2 GS PEC Initial vibrational WP Optical excitation. Dipole TM needed! Core-Excited state PEC Satellite Ar +* Ar PEC Final Ar + Ar + PEC Resonant Auger partial rates ICD rates  (R) Nuclear dynamics in ICD states ICD electron (and KER) spectra

H e describes only electronic motion (nuclei fixed). H describes the motion of N electrons and M nuclei. + relativistic terms if needed. Electron interaction term

1. Bound electronic states  PEC or PES Used to obtain the „properties“ :TM, etc Kirill&Andreas 2. Resonance (bound-in-continuum) states  Premysl Electronic widths 3. Nuclear dynamics (in local approximation) time dependent formulation + coupling to final states when computing ICD spectra Nicolas

N-electron SE cannot be solved exactly! Approximate solutions are found numerically and two questions should be answered before starting the work. 1.What electronic structure method should we use? 2.How do we represent the respective Hilbert space accurately enough? or What basis set should we select?

Choosing a method Independent particle (mean-field) methods – an electron moves in an average field of (N-1) other electrons. Typical example - Hartree-Fock (HF) approximation. Correlated methods – a motion of an electron is influenced by (correlated with) the motion of (N-1) other electrons at each instant. Examples – configuration interaction (CI), propagator (ADC), many-body perturbation theory (MBPT), coupled cluster (CC),... methods. Hartree-Fock solution is used as an input for the correlated methods.

Independent-electron wave function spin-orbital spatial orbital or molecular orbital (MO) spin function Slater determinant (ground state) Electrons with anti-parallel spins are uncorrelated. Electrons with parallel spins are exchange correlated.

Hartree-Fock approximation HF equations Fock operator Total electronic energy Coulomb and Exchange integrals Coulomb and exchange operators Orbital energy

Restricted vs. Unrestricted HF

HF recovers up to 99% of the total electronic energy in the ground state However, the energy differences of interest in chemical and spectroscopic processes are a fraction of one percent of the total energy. For example HF cannot describe binding between two rare-gas atoms

Ar dimer electronic ground state E HF =28670 eV, E int [CCSD(T)]=11.5 meV

Koopmans‘ theorem Negative of the HF orbital energies of the occupied MOs are the electron binding energies (ionization potentials).

Photoelectron spectrum of H 2 O „Breakdown of the MO picture“ due to the intra-molecular correlation

Double ionisation threshold Breakdown of monomer lines due to ICD driven by inter-molecular correlation

1.MOs and orbital energies serve as input for the correlated methods. 2.HF approximation furnishes us with a vivid picture of many electron system with electrons stacked on shelves called molecular orbitals. 3.HF solutions are of some, albeit limited, use for computing electronic decay rates. 4.The HF approximation usually does not deliver ground state PES in acceptable quality. It generally fails to produce excited state PES at all. 5.It fails to reproduce correlation driven phenomena.

Choosing the basis set Proper behaviour at r→0 and r→∞. Small number of STO basis functions are sufficient to represent a MO. Allow for very efficient computation of four-centre two-electron integals:

cc-pVDZ basis set for Ne Contracted Gaussian function Primitive Gaussian function Contraction coefficient Contraction exponent Polarisation basis function Adding diffuse functions Valence basis function Core basis function

RHF calculation of Ne

Augmenting a basis set

Introducing electron correlation

Configuration Interaction scheme − creation operator, − annihilation operator Excitation (N electrons) Ionisation (N-1 electrons) Double ionisation (N-2 electrons) E 0, E 1,... and corresponding wavefunctions Diagonalisation More on correlation! Andreas‘s lecture tomorrow.