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Introduction to ab initio methods I

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. Resonant Auger partial rates. Core-Excited state PEC. Nuclear dynamics in ICD states. Satellite Ar +* Ar PEC. ICD rates (R).

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Introduction to ab initio methods I

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  1. Introduction to ab initio methods I Kirill Gokhberg

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

  3. Anatomy of an ICD process Resonant Auger partial rates Core-Excited state PEC Nuclear dynamics in ICD states Satellite Ar+*Ar PEC ICD rates (R) Final Ar+Ar+ PEC Initial vibrational WP Optical excitation. Dipole TM needed! Ar2 GS PEC ICD electron (and KER) spectra

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

  5. 1. Bound electronic states  Kirill&Andreas Used to obtain the „properties“ :TM, etc PEC or PES 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

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

  7. 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.

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

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

  10. Restricted vs. Unrestricted HF

  11. 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

  12. Ar dimer electronic ground state EHF=28670 eV, Eint [CCSD(T)]=11.5 meV

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

  14. Photoelectron spectrum of H2O „Breakdown of the MO picture“ due to the intra-molecular correlation

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

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

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

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

  19. RHF calculation of Ne

  20. Augmenting a basis set

  21. Introducing electron correlation

  22. Configuration Interaction scheme −creation operator, − annihilation operator Excitation (N electrons) Ionisation (N-1 electrons) Double ionisation (N-2 electrons) More on correlation! Diagonalisation Andreas‘s lecture tomorrow. E0, E1, ... and corresponding wavefunctions

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