Presentation is loading. Please wait.

Presentation is loading. Please wait.

The following slides show you how to treat the Coulomb interaction in a many particle Hamiltonian. As the Coulomb interaction diverges for the case where.

Published byΧρυσάωρ Αγγελίδου Modified over 5 years ago

Similar presentations

Presentation on theme: "The following slides show you how to treat the Coulomb interaction in a many particle Hamiltonian. As the Coulomb interaction diverges for the case where."— Presentation transcript:

1 The following slides show you how to treat the Coulomb interaction in a many particle Hamiltonian. As the Coulomb interaction diverges for the case where the two electrons have the same position one needs to be careful on how to integrate. As our basis functions are atom centered radial functions times spherical harmonics (or linear combinations thereof) we can expand the Coulomb interaction on spherical harmonics as well and treat the angular part analytical. The radial part are the Slater integrals. An excellent text book introducing this topic is Ballhausen – Ligand field theory chapter 1 and 2.

2 Coulomb repulsion and Slater Integrals
Maurits W. Haverkort Institute for theoretical physics – Heidelberg University

3 The Coulomb Integral is nasty: The integrant diverges at r1=r2
Coulomb Hamiltonian: In order to create the Hamiltonian as a matrix we need to evaluate the following integral Solution by Slater: Expand the operator on Spherical Harmonics. Solve the angular part analytical and the Radial integral numerical (Slater Integrals.) Also works in solids. (Spherical Harmonics are not eigen-states, but still a valid basis set.

4 Coulomb interaction – Slater Integrals
Expansion on renormalized Spherical Harmonics with Useful expansion because our basis functions are (close to) spherical

5 Coulomb interaction – Slater Integrals
Integral to calculate Expansion on renormalized Spherical Harmonics

6 Coulomb interaction – Slater Integrals
Radial part: Slater integrals Angular part: Analytical solution

7 Coulomb interaction – Slater Integrals
Graphical representation

8 Coulomb interaction – Slater Integrals

9 Coulomb interaction – Slater Integrals
Triangular equations

10 Coulomb interaction – Slater Integrals
Parity

11 Coulomb interaction – Slater Integrals
d - electrons

12 Coulomb interaction – Slater Integrals
f - electrons

13 Coulomb interaction – Slater Integrals
Core (p) valence (d) interaction – direct term

14 Coulomb interaction – Slater Integrals
Core (p) valence (d) interaction – exchange term

15

Download ppt "The following slides show you how to treat the Coulomb interaction in a many particle Hamiltonian. As the Coulomb interaction diverges for the case where."

Similar presentations

Basis Sets for Molecular Orbital Calculations

Basis Sets for Molecular Orbital Calculations
Example 1 Matrix Solution of Linear Systems Chapter 7.2 Use matrix row operations to solve the system of equations  2009 PBLPathways.

Example 1 Matrix Solution of Linear Systems Chapter 7.2 Use matrix row operations to solve the system of equations  2009 PBLPathways.
0 Jack SimonsJack Simons, Henry Eyring Scientist and Professor Chemistry Department University of Utah Electronic Structure Theory Session 7.

0 Jack SimonsJack Simons, Henry Eyring Scientist and Professor Chemistry Department University of Utah Electronic Structure Theory Session 7.
I.What to recall about motion in a central potential II.Example and solution of the radial equation for a particle trapped within radius “a” III.The spherical.

I.What to recall about motion in a central potential II.Example and solution of the radial equation for a particle trapped within radius “a” III.The spherical.
Molecular Bonding Molecular Schrödinger equation

Molecular Bonding Molecular Schrödinger equation
CHE-30042 Inorganic, Physical & Solid State Chemistry Advanced Quantum Chemistry: lecture 4 Rob Jackson LJ1.16, 01782 733042

CHE Inorganic, Physical & Solid State Chemistry Advanced Quantum Chemistry: lecture 4 Rob Jackson LJ1.16,
Molecular Quantum Mechanics

Molecular Quantum Mechanics
Introduction to Molecular Orbitals

Introduction to Molecular Orbitals
Chapter 3 Electronic Structures

Chapter 3 Electronic Structures
1 Cold molecules Mike Tarbutt. 2 Outline Lecture 1 – The electronic, vibrational and rotational structure of molecules. Lecture 2 – Transitions in molecules.

1 Cold molecules Mike Tarbutt. 2 Outline Lecture 1 – The electronic, vibrational and rotational structure of molecules. Lecture 2 – Transitions in molecules.
Helium Nikki Meshkat Physics 138.

Helium Nikki Meshkat Physics 138.
Single Particle Energies

Single Particle Energies
Degenerate Electron Gas Unrealistic model: Uniform positive charge density n L Very large volume V=L 3 Eventually take limit: System is neutral Positive.

Degenerate Electron Gas Unrealistic model: Uniform positive charge density n L Very large volume V=L 3 Eventually take limit: System is neutral Positive.
CHE/ME 109 Heat Transfer in Electronics LECTURE 11 – ONE DIMENSIONAL NUMERICAL MODELS.

CHE/ME 109 Heat Transfer in Electronics LECTURE 11 – ONE DIMENSIONAL NUMERICAL MODELS.
P460 - 3D S.E.1 3D Schrodinger Equation Simply substitute momentum operator do particle in box and H atom added dimensions give more quantum numbers. Can.

P D S.E.1 3D Schrodinger Equation Simply substitute momentum operator do particle in box and H atom added dimensions give more quantum numbers. Can.
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley.

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley.
Ground State of the He Atom – 1s State First order perturbation theory Neglecting nuclear motion 1 - electron 1 2 - electron 2 r 1 - distance of 1 to nucleus.

Ground State of the He Atom – 1s State First order perturbation theory Neglecting nuclear motion 1 - electron electron 2 r 1 - distance of 1 to nucleus.
Section 8.3 – Systems of Linear Equations - Determinants Using Determinants to Solve Systems of Equations A determinant is a value that is obtained from.

Section 8.3 – Systems of Linear Equations - Determinants Using Determinants to Solve Systems of Equations A determinant is a value that is obtained from.
Lecture 17 Hydrogenic atom (c) So Hirata, Department of Chemistry, University of Illinois at Urbana-Champaign. This material has been developed and made.

Lecture 17 Hydrogenic atom (c) So Hirata, Department of Chemistry, University of Illinois at Urbana-Champaign. This material has been developed and made.
Physics 250-06 “Advanced Electronic Structure” Pseudopotentials Contents: 1. Plane Wave Representation 2. Solution for Weak Periodic Potential 3. Solution.

Physics “Advanced Electronic Structure” Pseudopotentials Contents: 1. Plane Wave Representation 2. Solution for Weak Periodic Potential 3. Solution.

Similar presentations

Ads by Google