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Fixed‐node quantum Monte Carlo for moleculesa) b)
1.H. F. Schaefer, Electronic Structure of Atoms and Molecules (Addison‐Wesley, Readlng, Mass., 1972).
2.R. J. Bartlett, Annu. Rev. Phys. Chem. 32, 359 (1981) and references therein.
3.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.
4.B. J. Rosenberg and I. Shavitt, J. Chem. Phys. 63, 2162 (1975);
4.W. Meyer, Int. J. Quantum Chem. Symp. No. 5, 341 (1971).
5.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.
6.N. Metropolis and S. M. Ulam, J. Am. Stat. Assoc. 44, 247 (1949).
7.M. H. Kalos, Phys. Rev. 128, 1791 (1962);
7.M. H. Kalos, J. Comp. Phys. 2, 257 (1967).
8.W. L. McMillan, Phys. Rev. A 138, 442 (1965).
9.R. C. Grimm and R. G. Storer, J. Comp. Phys. 7, 134 (1971).
10.M. H. Kalos, D. Levesque, and L. Verlet, Phys. Rev. A 9, 2178 (1974).
11.J. B. Anderson, J. Chem. Phys. 63, 1499 (1975);
11.J. B. Anderson, 65, 4121 (1976)., J. Chem. Phys.
12.D. M. Ceperley, G. V. Chester, and M. H. Kalos, Phys. Rev. B 16, 3081 (1977).
13.D. M. Ceperley and M. H. Kalos, in Monte Carlo Methods in Statistical Physics, edlted by K. Binder (Springer, Berlin, 1979), pp. 145–197.
14.J. B. Anderson, J. Chem. Phys. 73, 3897 (1980);
14.F. Mentch and J. B. Anderson, J. Chem. Phys. 74, 6307 (1981).
15.D. M. Ceperley and B. J. Alder, Phys. Rev. Lett. 45, 566 (1980).
16.D. M. Ceperley, The Stochastic Solution of the Many‐Body Schrödinger Equation for Fermions in Recent Progress in Many‐Body Theories, edited by J. G. Zabolitzky, M. de Llano, M. Fortes, and J. W. Clark (Springer, Berlin, 1981), pp. 262–269.
17.B. J. Alder, D. M. Ceperley, and P. J. Reynolds, J. Phys. Chem. 86, 1200 (1982).
18.J. W. Moskowitz and M. H. Kalos, Int. J. Quantum Chem. 20, 1107 (1981);
18.J. W. Moskowitz, K. E. Schmidt, M. A. Lee, and M. H. Kalos, J. Chem. Phys. 76, 1064 (1982).
19.J. M. Hammersley and D. C. Handscomb, Monte Carlo Methods (Chapman and Hall, London, 1964), pp. 57–59.
20.The computational complexity of our code goes as 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 For N sufficiently large, the term would ultimately dominate, although the algorithm is effectively in the range of N we have treated. In large systems, where 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 However, making this modification costly at small N.
21.R. Jastrow, Phys. Rev. 98, 1479 (1955);
21.R. B. Dingle, Philos. Mag. 40, 573 (1949).
22.In fact, the constancy of the local energy can be used as a quantltative measure of the accuracy of any proposed
23.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 This leads to the conditions that, for two electrons for an electron and a nucleus. Thus, e.g., for opposite spins at small implying that the coefficient a in equals
24.See e.g., D. M. Ceperley, M. H. Kalos, and J. L. Lebowitz, Macromolecules 14, 1472 (1981).
25.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.
26.Good results have also been obtained with other forms for the pair‐correlation function. See e.g., C. C. J. Roothaan and A. W. Weiss, Rev. Mod. Phys. 32, 194 (1960);
26.W. Kolos and C. C. J. Roothaan, Rev. Mod. Phys. 32, 205 (1960);
26.and W. A. Lester, Jr. and M. Krauss, J. Chem. Phys. 41, 1407 (1964);
26.W. A. Lester, Jr. and M. Krauss, J. Chem. Phys. 44, 207 (1966)., J. Chem. Phys.
27.E. A. Hylleraas, Z. Phys. 65, 209 (1930);
27.H. R. Hasse, Proc. Cambridge Philos. Soc. 26, 542 (1930);
27.J. O. Hirschfelder, C. F. Curtiss, and R. B. Bird, Molecular Theory of Gases and Liquids (Wiley, New York, 1954), pp. 942–946.
28.D. M. Ceperley and B. J. Alder, Physica B 108, 875 (1981).
29.R. J. White and F. H. Stillinger, Phys. Rev. A 3, 1521 (1971);
29.D. J. Klein and H. M. Pickett, J. Chem. Phys. 64, 4811 (1976).
30.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 will not be exactly that of the true Fermion problem unless the nodes are correct. Inparticular, of Eqs. (4) and (7) is replaced by in volume
31.The proof given here is an expanded version of the proof given in Ref. 16.
32.For any given total spin, the particular spin configuration is unimportant since the eleetrons can be simply relabeled.
33.The use of a Green’s function here is not to be confused with the “Green’s Function Monte Carlo” (GFMC) method of Kalos described in Ref. 13.
34.W. K. Hastings, Biometrika 57, 97 (1970).
35.W. Kolos and C. C. J. Roothaan, Rev. Mod. Phys. 32, 219 (1960).
36.P. E. Cade and W. M. Huo, J. Chem. Phys. 47, 614 (1967).
37.G. Das and A. C. Wahl, J. Chem. Phys. 44, 87 (1966).
38.B. Liu, J. Chem. Phys. 58, 1925 (1973).
39.W. Kolos and L. Wolniewicz, J. Chem. Phys. 41, 3663 (1964);
39.W. Kolos and L. Wolniewicz, J. Chem. Phys. 43, 2429 (1965); , J. Chem. Phys.
39.W. Kolos and L. Wolniewicz, J. Chem. Phys. 49, 404 (1968)., J. Chem. Phys.
40.W. Meyer and P. Rosmus, J. Chem. Phys. 63, 2356 (1975).
41.G. Das, J. Chem. Phys. 46, 1568 (1967).
42.D. D. Konowalow and M. L. Olson, J. Chem. Phys. 71, 450 (1979).
43.G. C. Lie and E. Clementi, J. Chem. Phys. 60, 1275 (1974).
44.G. C. Lie and E. Clementi, J. Chem. Phys. 60, 1288 (1974).
45.Thus, energy is in hartrees, length in bohr, charge in units of e, and the diffusion constant
46.S. Aung, R. M. Pitzer, and S. I. Chan, J. Chem. Phys. 49, 2071 (1968).
47.J. W. Moskowitz, K. E. Schmidt, M. A. Lee, and M. H. Kalos, preprint (1982).
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