An Iterative Technique for Solving theN-electronHamiltonian: The Hartree-Fock method

 Tony Hyun Kim

 

 Abstract

 The problem of electron motion in an arbitrary field of nuclei is an importantquantum mechanical problem finding applications in many diverse fields. Fromthe variational principle we derive a procedure, called the Hartree-Fock (HF)approximation, to obtain the many-particle wavefunction describing such a sys-tem. Here, the central physical concept is that of electron indistinguishability:while the antisymmetry requirement greatly complexifies our task, it also of-fers a symmetry that we can exploit. After obtaining the HF equations, wethen formulate the procedure in a way suited for practical implementation ona computer by introducing a set of spatial basis functions. An example imple-mentation is provided, allowing for calculations on the simplest heteronuclearstructure: the helium hydride ion. We conclude with a discussion of derivingphysical information from the HF solution.

 

https://web.stanford.edu/~kimth/www-mit/thk_hartreefock.pdf 

 

 A How to access the paper’s programs

 

 All of the numerical computations in this paper were conducted in Matlab, and allscripts can be found at  :http://web.mit.edu/kimt/www/8.06/paper/program/ There, one finds the following subdirectories:1. “first/”: This section contains the scripts responsible for the results presented inSection 5. The two-element basis set used for the calculation is called “minimalSTO-3G”, where each basis function is a linear combination of three Gaus-sian functions. Many years of work by quantum chemists and applied math-ematicians have gone into developing sophisticated and efficient mathematicalroutines involving Gaussian functions. (In particular, for calculating Eq. 26).2. “hartreeiteration/”: These are the programs used for conducting the first twoiterations of the Hartree method (Section 1.1). There is an app to compute Eq.2, and also to produce Figure 2 of the paper.3. “scrap/”: In the paper, I did not discuss the sophisticated routines involvingGaussian functions. Hence, I attempted to implement a simpler HF solver withstraightforward “brute force” methods for computing the necessary matrices.To my surprise, I found that it took over three minutes to compute (to terribleaccuracy) a single basis function integral (Eq. 26). Since we need to performhundreds of such integrations for a single HF run, this implementation is totallyimpractical, and was eventually “scrapped”.4. “second/”: Over the course of this paper, it was realized that the helium hydrideion produces identical results for the Hartree-Fock method and the Hartreeiteration. (This follows because the exchange term of HF is always zero forN= 2.) Hence, I worked on a four-electron system: namely, two-interactinghydrogen molecules. However, I have not yet established the correctness of thenumerical results, so they were never incorporated into the paper.Finally, to produce Figure 5, I utilized the impressive “Volume Browser” programwritten by Eike Rietsch, obtained from the MATLAB Central File Exchange:http://www.mathworks.com/matlabcentral/fileexchange/loadFile.do?objectId=13526