Pfaffian pairing and backflow wavefunctions for electronic structure quantum Monte Carlo methods

We investigate pfaffian trial wavefunctions with singlet and triplet pair orbitals by quantum Monte Carlo methods. We present mathematical identities and the key algebraic properties necessary for efficient evaluation of pfaffians. Following upon our previous study [Bajdich et al., Phys. Rev. Lett. 96, 130201 (2006)], we explore the possibilities of expanding the wavefunction in linear combinations of pfaffians. We observe that molecular systems require much larger expansions than atomic systems and linear combinations of a few pfaffians lead to rather small gains in correlation energy. We also test the wavefunction based on fully antisymmetrized product of independent pair orbitals. Despite its seemingly large variational potential, we do not observe additional gains in correlation energy. We find that pfaffians lead to substantial improvements in fermion nodes when compared to Hartree-Fock wavefunctions and exhibit the minimal number of two nodal domains in agreement with recent results on fermion nodes topology. We analyze the nodal structure differences of Hartree-Fock, pfaffian, and essentially exact large-scale configuration interaction wavefunctions. Finally, we combine the recently proposed form of backflow correlations [Drummond et al., J. Phys. Chem. 124, 22401 (2006); Rios et al., Phys. Rev. E. 74, 066701 (2006)] with both determinantal and pfaffian based wavefunctions.

Figure 1

(Color online) A three-dimensional cut through the fermion node hypersurface of oxygen atom obtained by scanning the wavefunction with a spin-up and -down (singlet) pair of electrons at equal positions, while keeping the rest of electrons at a given VMC snapshot positions (small green spheres). Nucleus is depicted in the center of the cube by the blue sphere. The three colors (from left to right) show nodes of: Hartree-Fock (red and/or dark gray), multipfaffian nodes (orange and/or medium gray), and the nodes of the CI wave function (yellow and/or light gray) in two different views (upper and lower rows). The CI nodal surface is very close to the exact one (see text). The HF node clearly divides the space into four nodal cells while pfaffian and CI wavefunctions partitioning leads to the minimal number of two nodal cells. The changes in the nodal topology occur on the appreciable spatial scale of the order of $1\mathrm{a.u.}$

Figure 2

(Color online) Upper figure: Percentages of correlation energy from VMC and DMC methods versus a number of backflow parameters with backflow correlated Slater-Jastrow (SJBF), pfaffian-Jastrow (PFBF), and CI-Jastrow (CIBF) trial wavefunctions for C atom (2B, electron-nucleus and electron-electron terms; 3B, all electron-electron-nucleus terms; 23B for all terms together). Lines connecting the points serve only as a guide to the eye. Lower figure: Variance of the local energy versus a number of backflow parameters.

Figure 3

(Color online) Same as Fig. 2 but for C dimer.