  We extend and benchmark the recently-developed Effective-Hamiltonian (EFFH)
method [PRX Quantum $\bf{4}$, 020307 (2023)] as an approximation to the
equilibrium state ("mean-force Gibbs state") of a quantum system at strong
coupling to a thermal bath. The EFFH method is an approximate framework.
Through a combination of the reaction-coordinate mapping, a polaron
transformation and a controlled truncation, it imprints the system-bath
coupling parameters into the system's Hamiltonian. First, we develop a
$\textit{variational}$ EFFH technique. In this method, system's parameters are
renormalized by both the system-bath coupling parameters (as in the original
EFFH approach) and the bath's temperature. Second, adopting the generalized
spin-boson model, we benchmark the equilibrium state from the EFFH treatment
against numerically-exact simulations and demonstrate a good agreement for both
polarization and coherences using the Brownian spectral function. Third, we
contrast the (normal and variational) EFFH approach with the familiar (normal
and variational) polaron treatment. We show that the two methods predict a
similar structure for the equilibrium state, albeit the EFFH approach offers
the advantage of simpler calculations and closed-form analytical results.
Altogether, we argue that for temperatures comparable to the system's
frequencies, the EFFH methodology provides a good approximation for the
mean-force Gibbs state in the full range of system-bath coupling, from
ultraweak to ultrastrong.
