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Optimized t-expansion method for the Rabi Hamiltonian

A polemic arose recently about the applicability of the $t$-expansion method to the calculation of the ground state energy $E_0$ of the Rabi model. For specific choices of the trial function and very large number of involved connected moments, the $t$-expansion results are rather poor and exhibit considerable oscillations. In this letter, we formulate the $t$-expansion method for trial functions containing two free parameters which capture two exactly solvable limits of the Rabi Hamiltonian. At each order of the $t$-series, $E_0$ is assumed to be stationary with respect to the free parameters. A high accuracy of $E_0$ estimates is achieved for small numbers (5 or 6) of involved connected moments, the relative error being smaller than $10^{-4}$ (0.01%) within the whole parameter space of the Rabi Hamiltonian. A special symmetrization of the trial function enables us to calculate also the first excited energy $E_1$, with the relative error smaller than $10^{-2}$ (1%).