MacKinnon - Computational Physics (523159), страница 12

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Anyway, the better the trial wavefunction, the closer to thetrue ground state energy the variational estimate should be.In this project you are given a possible trial wavefunction,z;; ;(4.41) ¿ e 0 Z¿ e 0 ÿ ¿ Zÿ’0 [Use the variational Monte Carlo technique to calculate variational estimates of the true ground state energy.Before you start programming, you will need the analytic expressions for the local energy. This involvessome nasty algebra, but the answer is,”“r rrx zz 9(4.42)r rr rT0L$ ÷ $ T T š T T 0 0 š T T 0 [0 0 0×T0 0 $ Trr .where r and r are the coordinates of the 2 atoms relative to the nucleus and rThe Monte Carlo moves can be made by generating random numbers (use a library routine to do this)and adding them to the electron coordinates.

I suggest that you update all six electron position coordinateseach move, and so you will need six random numbers each time. The acceptedlore is that the Metropolis algorithm is most efficient when the step size is chosen to keep the acceptanceprobability close to 0.5. However, the method should work in principle no matter what the step size andyou should try a few different step sizes to confirm that this is indeed the case. The starting positions ofthe two electrons can be chosen randomly, but remember that the Metropolis algorithm only samples theprobability distribution exactly in the limit as the number of moves tends to infinity. You will thereforehave to throw away the results from the moves near the beginning of the run and only start accumulatingthe values of the local energy once things have settled down.

You should experiment to find out how manymoves you need to throw away.–• , where is the total numberThe statistical errors in Monte Carlo calculations decrease likeof moves after the initial equilibration period. The errors therefore improve only slowly as the length ofthe run is increased. You will not be able (and should not attempt) to attain great accuracy. However, youshould think hard about the magnitude of the statistical errors involved. Calculating the variance of thevalues in the list of energies accumulated during the random walk is easy and you should certainly do it."<K£T : T :\­ T : K 0 : 0 :\­ 0 &74.8.3 The PhysicsWhat are your best estimates of the ground state energy and the corresponding statistical error? Can yousee any physics behind the form of the trial wavefunction?BibliographyLapack Numerical Library n.d.*ftp://unix.hensa.ac.uk/pub/netlib/lapack/Metropolis N, Rosenbluth A W, Rosenbluth M N, Teller A H & Teller E 1953 J.

Chem. Phys. 21, 1087.Numerical Algorithms Group n.d.*http://www.nag.co.uk:70/Potter D 1973 Computational Physics Wiley Chichester. (Out of print, but several copies in the library).Press W H, Flannery B P, Teukolsky S A & Vettering W T 1992 Numerical Recipes: The Art of ScientificComputing Cambridge University Press Cambridge. (Full text available on the Web, except C++version.).*http://www.nr.com/Wilkinson J H 1964 The Algebraic Eigenvalue Problem Clarendon Press Oxford.Wolfram S 1991 Mathematica — A System for Doing Mathematics by Computer Addison-Wesley RedwoodCity, California.

ISBN 0 201 51502 4.47.

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