Abstract
We propose a bilinear sampling algorithm in the Green's function Monte Carlo for expectation values of operators that do not commute with the Hamiltonian and for differences between eigenvalues of different Hamiltonians. The integral representations of the Schrödinger equations are transformed into two equations whose solution has the formψ a(x) t(x, y)ψb(y), where ψa and ψb are the wavefunctions for the two related systems andt(x, y) is a kernel chosen to couplex andy. The Monte Carlo process, with random walkers on the enlarged configuration spacex ⊗y, solves these equations by generating densities whose asymptotic form is the above bilinear distribution. With such a distribution, exact Monte Carlo estimators can be obtained for the expectation values of quantum operators and for energy differences. We present results of these methods applied to several test problems, including a model integral equation, and the hydrogen atom.
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Zhang, S., Kalos, M.H. Bilinear quantum Monte Carlo: Expectations and energy differences. J Stat Phys 70, 515–533 (1993). https://doi.org/10.1007/BF01053583
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DOI: https://doi.org/10.1007/BF01053583