Effect of cumulant representations on the standard error of matrix elements for energy and variance minimization. Results are shown for the Ne atom with Hartree–Fock reference wave function and a matrix element with respect to the two basis functions Eq. (41) vs the number of Monte Carlo sample points. (a) Upper part: relative standard error of the matrix element Eq. (35) (solid line), Eq. (36) (dotted line) and Eq. (37) (dashed line) for Newton’s method. Lower part: relative standard error of the matrix element Eq. (38) (solid line), Eq. (39) (dotted line) and Eq. (40) (dashed line) of the JPTE-I equation. (b) Relative standard error of the corresponding matrix element for JPTV-I (30) (solid line), JPTV-II (31) (dotted line) and Newton’s method (32) (dashed line).
Ratios of variational correlation energies for Jastrow factors with respect to the exact ground state (upper part) and fixed-node DMC (lower part) energies. Energy optimized correlation functions from the first iteration step of the JPTE-I , JPTE-II and Newton’s method are compared with fully converged FOPIM based on the JPTE-I equation.
Convergence behavior of JPTE-I (○), JPTE-II (◻), and Newton’s method (◇) for energy optimization with respect to the number of iterations for the (a) Ne atom and (b) .
Convergence of FOPIM based on the JPTE-I equation, in the case of the Ne atom, for an initial reference wave function with prescribed electron-electron cusp (47). For comparison convergence is shown for a Hartree–Fock wave function as an initial guess.
Convergence to the ground state energy of for different bond lengths starting from a RHF reference wave function. Results are shown for FOPIM with JPTE-I, JPTE-II, and Newton’s method. The bond length varies from the equilibrium bond length up to . For each method, results are shown only for those bond lengths at which convergence has been achieved.
Removal of polar configurations by the Jastrow factor for at bond length . (a) RHF and (b) Jastrow wave functions are plotted along the bond axis, i.e., . The nuclei are located at .
Convergence of the standard deviation of the local energy for the Ne atom with different basis sets for the correlation function. Results are shown for FOPIM with JPTV-I (○), JPTV-II (◇), and Newton’s method . Three different basis sets have been studied: (a) 7 term basis, (b) 17 term basis, (c) 92 term basis. The 7 and 17 term bases were taken from Ref. 33. A Hartree–Fock reference wave function has been taken as an initial guess, except for Newton’s method where in (b) and (c) the JPTV-I result from the first iteration step was chosen.
Variational energies (Hartree) and standard deviations of the local energy for first row atoms and the molecules and . Jastrow factors have been optimized by FOPIM for energy and variance minimization based on JPTE-I, JPTV-I, and JPTV-II, respectively. Hartree–Fock orbitals have been used for the Slater determinant. For comparison we have also listed fixed-node DMC and “exact” energies.
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