On the minimization of the variance of the transcorrelated hamiltonian

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Abstract
One of the difficulties associated with the transcorrelated method is that the hamiltonian C-1HC is not hermitian, and so energies calculated through ⟨ΦC -1 HCΦ⟩/⟨Φ|Φ⟩ are not bounded. Hence, unlike variational methods, when the transcorrelated equations are solved there is no visible monotonic convergence of any quantity. Here it is proposed that the parameters associated with the correlation part of C should be determined through the minimization of the transcorrelated variance U TC=⟨{(C -1 HC-W)Φ}2⟩, the orbital parameters of C and Φ being determined by established methods. No integrals more difficult than 9-dimensional integrals are needed, and if they can be evaluated or approximated in some way, all the parameters of CΦ can be determined through minimization procedures for any molecule. Calculations on helium are reported which certainly demonstrate the applicability of the method. As in other calculations which minimize variance, the associated energies are not quite so accurate as those when the more direct bi-variational equations are solved, but the difference is insignificant compared to the advantage of having a minimization criterion.

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