Phys 498CQM Lecture # 11

Matrix methods in computational quantum mechanics
Useful for atoms, molecules, crystals
Example: Gaussian basis functions

Outline

1. Representing the wavefunction in a basis
   • Vector in Hilbert space
   • Norms
   • Orthonormal bases
   • transformations of basis - unitary transformations
   • Overlap matrices for non-orthonormal bases
2. Matrix representation of operators
   • unitary transformation matrices
   • Hermitian matrices for physical observables
3. Examples of Hamiltonian in often used bases
   • Plane waves
     • Fourier analysis - complete - orthonormal
     • Eigenstates for free particles
     • Kinetic energy operator
     • Potential energy - convolution in Fourier space
   • Gaussians
     • Localized - complete - eigenstates for harmonic oscillator
     • Non-orthogonal and overcomplete if expanded around multiple sites and/or multiple widths
     • All needed matrix elements are analytic
     • Detailed derivations of Gaussian matrix elements in Thijssen, 4.8
     • Analytic advantages NOT so elegant for DFT calculations
   • GTOs and STOs
     • GTO - sum of Gaussians
     • STO- "fully contracted", i.e., one sum of Gaussians for an atomic-like orbital
     • STO-NG - standard notation for N Gausians in an STO
   • Numerical atomic orbitals
     Localized - realistic - difficult to use
4. Finding the energy eigenstates
   • Hamiltonian matrix for simple potential - Kohn-Sham Eqs.
   • Determinants
   • Eigenvalues and eigenvectors
   • Variational theorems
   • McDonald's theorem for each eigenvalue
   • Hartree-Fock more complex - Roothan Eqs.
     (Referred to as the "Pople-Nesbet" equations for the Gaussian basis in Thijssen)
5. Computer good for representation of finite matrices
   • Appropriate for discrete spectrum of eigenstates
   • By using conservation laws, we can often block diagonalize the Hamiltonian, leaving discrete spectra for each block
6. Algorithms appropriate for computation
   • See "Numerical Recipes", ch. 11; Koonin, ch 5
   • Inversion; Determinants
   • Eigenvalues of tridiagonal matrix - O(N)
   • Reducing matrices to tridiagonal form - O(N³)
   • Eigenvectors - inverse iteration
   • Libraries - e.g., Netlib
     • lapack, eispack
     • examples in lapack: dsyevx; cheevx
   • Special approaches - Lanczos - later
7. Greens Functions in matrix form
   • Inverse of Hamiltonian matrix (with energy shift)
   • Diagonal representation in eigenstates of energy

Note Gaussian web site: http://www.gaussian.com/