We present a method to construct pseudo-BCS wave functions from the one-body density matrix.The resulting many-body wave function, which can be produced for any fermion systems, including those with purely repulsive interactions, has the form of a number-projected BCS form, or antisymmetrized geminal power (AGP).Such wave functions provide a better ansatz for correlated fermion systems than a single Slater determinant, and often better than a linear combination of Slater determinants (for example from a truncated active space calculation).We describe a procedure to build such a wave function conveniently from a given reduced density matrix of the system, rather than from a mean-field solution (which gives a Slater determinant for repulsive interactions).The pseudo-BCS wave function thus obtained reproduces the density matrix or minimizes the difference between the input and resulting density matrices.One application of the pseudo-BCS wave function is in auxiliary-field quantum Monte Carlo (AFQMC) calculations as the trial wave function to control the sign/phase problem.AFQMC is often among the most accurate general methods for correlated fermion systems.We show that the pseudo-BCS form further reduces the constraint bias and leads to improved accuracy compared to the usual Slater determinant trial wave functions, using the two-dimensional Hubbard model as an example.Furthermore, the pseudo-BCS trial wave function allows a new systematically improvable self-consistent approach, with pseudo-BCS trial wave function iteratively generated by AFQMC via the one-body density matrix.
Unlike the situation for the 1d Dirac delta derivative Schrodinger pseudo potential (SPP) and the 2d Dirac delta SPP, where the indeterminacy originates from a lack of scale in the first and both a lack of scale as well as the wave function not being well defined at the support of the generalized function SPP; the obstruction in 3d Euclidean space for the Schrodinger equation with the Dirac delta as a SPP only comes from the wave function (the
L2 bound sate solution) being singular at the compact point support of the Dirac delta function (measure). The problem is solved here in a completely mathematically rigorous manner with no recourse to renormalization nor regularization. The method involves a distributionally generalized version of the Schrodinger theory as developed by the author, which regards the formal symbol "
Hψ" as an element of the space of distributions, the topological dual vector space to the space of smooth functions with compact support. Two main facts come to light. The first is the bound state energy of such a system can be calculated in a well-posed context, the value of which agrees with both the mathematical and theoretical physics literature. The second is that there is then a rigorous distributional version of the Hellmann-Feynman theorem.