5210_chap9 - Chapter 9: Slide 1 Chapter 9 Many-Electron...

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Unformatted text preview: Chapter 9: Slide 1 Chapter 9 Many-Electron Atoms Chapter 9: Slide 2 Outline The Hamiltonian for Multielectron Atoms The Hartree-Fock Method The Hartree Method: Helium Hartree-Fock Orbital Energies for Argon Extension to Multielectron Atoms Antisymmetrized Wavefunctions: Slater Determinants Electron Correlation Koopmans Theorem Chapter 9: Slide 3 The Hamiltonian for Multielectron Atoms Atomic Units: 12 2 1 2 2 2 1 2 1 1 2 2 ) ( 2 1 ) ( 2 1 r r r r r H +-- - - = Helium 12 2 2 2 1 2 2 2 2 2 1 2 1 2 4 4 2 4 2 ) ( 2 ) ( 2 r e r e r e r m r m H +-- - - = Z = 2 SI Units: Multielectron Atoms =- = = +- - = N i N i i j ij i N i i r r Z H 1 1 1 1 2 1 2 1 Elect KE Elect- Nuc PE Elect- Elect PE + + + + + + + + = - = 35 34 24 23 13 12 1 1 1 1 1 1 1 1 1 r r r r r r r N i i j ij Chapter 9: Slide 4 Atomic Orbitals In performing quantum mechanical calculations on multielectron atoms, it is usually assumed that each electron is in an atomic orbital, , which can be described as a Linear Combination of Hydrogen-like orbitals, which are called Slater Type Orbitals (STOs). These STOs are usually denoted as i Thus: = i i c The goal of quantum mechanical calculations is to find the values of the c i which minimize the energy (via the Variational Principle). These STOs are also used to characterize the Molecular Orbitals occupied by electrons in molecules. We will discuss these STOs in significantly greater detail when we describe quantum mechanical calculations on polyatomic molecules. Chapter 9: Slide 5 Outline The Hamiltonian for Multielectron Atoms The Hartree-Fock Method The Hartree Method: Helium Hartree-Fock Orbital Energies for Argon Extension to Multielectron Atoms Antisymmetrized Wavefunctions: Slater Determinants Electron Correlation Koopmans Theorem Chapter 9: Slide 6 The Hartree Method: Helium Hartree first developed the theory, but did not consider that electron wavefunctions must be antisymmetric with respect to exchange. Fock then extended the theory to include antisymmetric wavefunctions. We will proceed as follows: 1. Outline Hartree method as applied to Helium 2. Show the results for atoms with >2 electrons 3. Discuss antisymmetric wavefunctions for multielectron atoms (Slater determinants) 4. Show how the Hartree equations are modified to get the the Hartree-Fock equations. Chapter 9: Slide 7 Basic Assumption Each electron is in an orbital, i (e.g. a sum of STOs). The total variational wavefunction is the product of one electron wavefunctions: ) ( ) ( ) , ( 2 2 1 1 2 1 r r r r = Guess initial values the individual atomic orbitals: (This would be an initial set of coefficients in the linear combination of STOs) ) ( and ) ( 2 2 1 1 r r init init Procedure Lets first look at electron #1. Assume that its interaction with the...
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This note was uploaded on 04/11/2011 for the course CHEM 5210 taught by Professor Staff during the Spring '08 term at North Texas.

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5210_chap9 - Chapter 9: Slide 1 Chapter 9 Many-Electron...

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