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3
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84952274313
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The correlation energy is defined as the difference between the Hartree—Fook energy and the exact, nonrelativistic, Born—Oppenheimer energy. It is so named since it accounts for the electronic correlation missing in the Hartree—Fock model.
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6
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84952274316
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Nevertheless, very good predictions for a variety of molecular properties have been achieved (see e.g., Ref. 1). For example, relatively accurate values for bond strengths are obtained with the CI method by subtracting the energy obtained for the molecule from the sum of the energies obtained for the two fragments which result when the bond is broken. Significantly, however, the large mathematical uncertainty inherent in this procedure—e.g., due to the nature of the convergence, and hence to the mathematically unknown degree of cancellation in the separate errors—is generally not systematically improvable.
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25
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84952274315
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The computational complexity of our code goes as [formula omitted] where the cubic term comes from inverting the Slater matrix order N times (once for each electron moved), and the quadratic term comes from computing pairwise interactions. Generally [formula omitted] For N sufficiently large, the [formula omitted] term would ultimately dominate, although the algorithm is effectively [formula omitted] in the range of N we have treated. In large systems, where [formula omitted] would begin to dominate, suitable modifications can be made, by use of sparse matrix algorithms, to eliminate this term. The computational complexity then goes as [formula omitted] However, [formula omitted] making this modification costly at small N.
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28
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84952274311
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In fact, the constancy of the local energy can be used as a quantltative measure of the accuracy of any proposed [formula omitted]
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29
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84952274312
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The cusp condition is a requirement on a wave function Ψ that the leading singularity in V (R), when two particles come together, cancels when evaluating the energy [formula omitted] This leads to the conditions that, for two electrons [formula omitted] for an electron and a nucleus. Thus, e.g., for opposite spins at small [formula omitted] [formula omitted] implying that the coefficient a in [formula omitted] equals [formula omitted]
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31
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84952274309
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If the trial function contains only a single Slater determinant, the full Slater matrix can always be block diagonalized into spin up and spin down submatrices by relabeling the coordinates.
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42
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84952274310
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Because of the boundary condition imposed on Φ by this approximation, the expansion of Eq. (3) must be in terms of eigenfunctions of H within the separate volume elements. Thus, the spectrum of eigenvalues [formula omitted] will not be exactly that of the true Fermion problem unless the nodes are correct. Inparticular, [formula omitted] of Eqs. (4) and (7) is replaced by [formula omitted] in volume [formula omitted]
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43
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84952274307
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The proof given here is an expanded version of the proof given in Ref. 16.
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44
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84952274308
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For any given total spin, the particular spin configuration [formula omitted] is unimportant since the eleetrons can be simply relabeled.
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45
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84952274300
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The use of a Green’s function here is not to be confused with the (GFMC) method of Kalos described in Ref. 13.
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Green’s Function Monte Carlo
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59
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84952274299
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Thus, energy is in hartrees, length in bohr, charge in units of e, and the diffusion constant [formula omitted]
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