We investigate pfaffian trial wave functions with singlet and triplet pairorbitals by quantum Monte Carlo methods. We present mathematical identities andthe key algebraic properties necessary for efficient evaluation of pfaffians.Following upon our previous study \cite{pfaffianprl}, we explore thepossibilities of expanding the wave function in linear combinations ofpfaffians. We observe that molecular systems require much larger expansionsthan atomic systems and linear combinations of a few pfaffians lead to rathersmall gains in correlation energy. We also test the wave function based onfully-antisymmetrized product of independent pair orbitals. Despite itsseemingly large variational potential, we do not observe additional gains incorrelation energy. We find that pfaffians lead to substantial improvements infermion nodes when compared to Hartree-Fock wave functions and exhibit theminimal number of two nodal domains in agreement with recent results on fermionnodes 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 backflowcorrelations \cite{drummond_bf,rios_bf} with both determinantal and pfaffianbased wave functions.
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