Computational Quantum Mechanics

Welcome to CSI 783 crosslisted with PHYS/CHEM 736

Instructor: Estela Blaisten-Barojas

Spring, 2006

This is one of the core science courses of the Computational Physics and Computational Materials and Chemical Science, Space Science disciplines in the doctoral program in Computational Sciences and Informatics (CSI). The course is also an alternative core course of the Masters in Physics and Astronomy and it is an elective course of the Masters in Chemistry and the PhD in Physical Sciences.

This server will expand as the semester progresses.

Textbooks and Course Materials

The textbook for this course is either

  • Quantum Chemistry by J. Lowe and K. Peterson, Academic Press 2005 (third edition), or
  • Quantum Chemistry by Levine, Prentice Hall 1998 (fifth edition).

    • Extra references can be found in:

  • Introductory Quantum Chemistry by Frank Pilar, Mc Graw-Hill, second edition 1990.
  • Quantum Mechanics by L. Schiff, Mc Graw-Hill Co, 1985.
  • Introduction to the Structure of Matter by J. J. Berehm, J. Wiley, 1988.
  • Numerical Recipes by Press, Flannery, Teukolsky, Vetterling, Cambridge Press.
  • Atoms and Molecules by Karplus and Porter, Benjamin, 1986.

    • Theme: Molecular Physics and Quantum Chemistry

    Structure of the course:

  • This is an introductory course for doctoral students that includes computational techniques largely used by the science research community. Each student will be responsible to cover the material in the textbook, do the assigned homework exercises, and develop a certain number of short computational projects. There will be a mid-term and a final exam.

  • Lectures will supplement the textbook by providing an introduction to the physical concepts involved in various research areas where numerical methods are fundamental.

  • Projects will be assigned as homework from material discussed in the lectures. A report will be due for each project. Every report is to be structured according to the following suggested scheme:

    1) Introduction to the subject 2) Model and methods 3) Results and any extra finding of your own. 4) Conclusions and observations 5) Bibliography 6) Appendices

    Evaluation:

  • - 35% acquired through the computational projects
  • - 25% homework exercises
  • - 20% mid-term exam
  • - 20% final exam

    Tentative contents:

  • - 1. Review of Systems with spherically symmetric potentials The central-field problem. Separation of the central-field Schroedinger equation. Solution of the equation in q and f. The spherical harmonics in real form. Solution of the radial equation for the Coulomb potential. The Laguerre polynomials. The wave functions of the one- electron atom. Orbital probability distributions functions. Selections rules for the one-electron atoms. Electron spin. The rigid rotator. Numerical solution of the Schroedinger equation for a potential in one dimension. Birge -Spooner plot to calculate the dissociation energy

  • - 2.Introduction to the many-electron atom- approximate solutions to Schroedinger equation The non-relativistic atomic hamiltonian operator. Atomic units. The independent-particle model. The effect of electron repulsions on atomic energies. The mass-polarization effect. Scaling and the virial theorem. The variation method. Basis functions for electronic calculations. Perturbation theory for non-degenerate states. Perturbation theory for degenerate states. Review of matrix algebra. Numerical manipulation of linear nonhomogeneous simultaneous equations. Numerical matrix inversion and diagonalization. The Gauss-Seidel iterative method.

  • - 3. Electron spin and many-electron systems: quantum states of atoms Indistinguishability of identical particles: fermions and bosons. The antisymmetry principle. Spin angular momentum and their operators. Two-electron wavefunctions. The helium atom revisited. Orbital angular momentum in many-electron atoms. Spin-orbit coupling. Atomic term symbols and energy level diagrams for atoms. Selections rules: an introduction to interaction between matter and radiation-- Fermi's golden rule as a result of first order time-dependent perturbation theory for transition rates. Exponential decay. Emission and absorption of light. Numerology of curve fitting as applied to spectroscopic problems.

  • - 4. Algebra of many-electron calculations:The Hartree-Fock (HF) self- consistent field method. Construction of determinant eigenfunctions of S2. Manipulation of determinants in many electron calculations. The method of Configuration Interaction. Optimized orbitals. Koopman's theorem. The aufbau principle. Electron correlation energy. Density matrices and analysis in the HF approximation. Natural orbitals. Roothaan' equations and the matrix solution of the HF equations. Open-shell HF calculations. Multivariate least square analysis. Gradient calculations.

  • - 5. Topics on molecular structure. The Huckel approximation. The Born-Oppenheimer approximation. Solution of the nuclear equation. Selection rules for the molecular electronic transitions. Molecular HF calculations. Electronic transitions in molecules. The MO-LCAO approximation. The hydrogen molecule. Heteronuclear diatomic molecules. Linear polyatomic molecules. Molecular CI. The valence Bond method. Molecular perturbation calculations. Molecular symmetry. Elements of point groups: irreducible and reducible representations, direct-product, symmetry projection-operators. Point symmetry and molecular spectra. Numerical applications of the Huckel method and calculation of spectroscopic transition in linear molecules. Two numerical examples of SCF calculations using GAUSSIAN and maybe other package.

  • - 6. Topics on solid state structure. Nearly free electron model. Density functional theory. Origin of the energy gap. Wave function of electron in a periodic potential. Bloch functions. Crystal momentum of an electron. Reduce and periodic zone schemes. Approximate solutions near a zone boundary. Metals and insulators. Construction of the Fermi surfaces. Electrons, holes and open orbits. Effective mass of electrons in crystals. Wavefunctions for zero wavevector.

    Students are: chapter 1

    Estela Blaisten-Barojas, blaisten-at-gmu.edu