Cornell University

Department of Physics

Phys 480/680, Astro 690 February 25, 2006

Computational Physics, Spring 2006

Homework Assignment # 5

(Due Wed, March 15)

Agenda and readings for the week of February 27 and all of March:

Goal: Building upon the components you have made, you will perform your own full density-functional-theory calculations of atoms. You will reproduce the accepted values to at least as much precision as published by the US National Institute of Standards and Technology, the micro-hartree level!!!

Readings marked NR are from Numerical Recipes: The Art of Scientific Computing, 2$^{\mathrm nd}$ edition (in C). Readings marked LN are from the course lecture notes to be found at http://www.muchomas.lassp.cornell.edu/P480.

Recap of last week:

• Lec 9, 02/21 (Tue) Numerical interpolation; Numerical stability of solution of ODE's, shooting to a fitting point; Y$_{lm}$'s and Non-spherical states (higher angular momenta).
  Readings: NR 17.2, 3.0-3.2.

• Lec 10, 02/23 (Thu): Non-spherical states cont'd, nomenclature for (nl)-shells in atoms; normalization conventions and definitions for our calculations; calculation of total energy within density functional theory; exchange-correlation energy per particle, electron gas parameter r$_s$; Functional form for exchange-correlation energy.
  Readings: Sections ``Introduction,'' ``Procedure'' (you can skip the relativistic part), ``Results,'' and ``The LDA approximation,'' from
  http://physics.nist.gov/PhysRefData/DFTdata/contents.html

• Lab 3, 02/23 (Thu): Effectiveness of root finders; convergence of eigenvalues with N.

Preview of coming weeks:

• Lec 11, 02/28: Solution of many non-linear equations in many unknowns, mixing method, convergence rates, condition numbers; Newton-Raphson and Broyden approaches; motivation for a ``modified'' Broyden's method for large systems of equations
  Readings: NR 9.6-9.7

• Lec 12, 03/02 Modified Broyden method

• 03/07 Lec 14: Recap modified Broyden method; Introduction to numerical linear algebra; Benefits of LAPACK; Solution of linear systems through LU decomposition; Crout's algorithm for LU decomposition
  Reading: NR 2.0, 2.3

• 03/09 Lec 15: Crout's Algorithm with pivoting
  Reading: NR 2.3

• 03/09 Lab 4: ``Hot-wired'' solver for atoms within Hartree theory, reproducing analytic Hydrogenic solutions to Schrödinger's equation ($-(1/2)Z^2/n^2$).

• 03/14 LECTURE CANCELLED: APS Meeting
• 03/16 Lec 16 (Meet in Rock B3!!!): Inner loop tricks
• 03/16 Lab 5: Profiling and real-time optimization of full atomic solver: using gprof; avoiding malloc; inlining optimizations.

• 03/20-03/24 SPRING BREAK

• 03/28 Lec 17 (Meet early, 8:00am!!!) Cache optimization, avoiding cache misses, BLAS1-3, blocked matrix multiplies; Automated empirical optimization of software (AEOS): automatically tuned linear algebra software (ATLAS), fastest Fourier Transform in the West (FFTW). Solution of Poisson's equation in d>1 dimensions; expressive software; solution to Poisson's equation in a single line of code; choice of plane-wave (complex exponential) basis for periodic boundary conditions.

• 03/30 Lec 18 (Meet early again if needed!!!): Form of operators and transforms in a plane-wave basis; octave/matlab notations; standard ordering of indices for Discrete Fourier Transforms (DFTs).

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