Herman-Kluk-like semi-classical initial-value representation for Boltzmann operator

The coherent-state initial-value representation (IVR) for the semi-classical real-time propagator of a quantum system, developed by Herman and Kluk (HK), is widely used in computational studies of chemical dynamics. On the other hand, the Boltzmann operator e-Ĥ/(kBT), with Ĥ, kB, and T representing the Hamiltonian, Boltzmann constant, and temperature, respectively, plays a crucial role in chemical physics and other branches of quantum physics. One might naturally assume that a semi-classical IVR for the matrix element of this operator in the coordinate representation (i.e., ⟨x̃|e-Ĥ/(kBT)|x⟩, or the imaginary-time propagator) could be derived via a straightforward "real-time → imaginary-time transformation" from the HK IVR of the real-time propagator. However, this is not the case, as such a transformation results in a divergence in the high-temperature limit (T → ∞). In this work, we solve this problem and develop a reasonable HK-like semi-classical IVR for ⟨x̃|e-Ĥ/(kBT)|x⟩, specifically for systems where either the gradient of the potential energy (i.e., the force intensity) has a finite upper bound or the potential becomes harmonic in the long-range limit. The integrand in this IVR is a real Gaussian function of the positions x and x̃, which facilitates its application to realistic problems. Our HK-like IVR is exact for free particles and harmonic oscillators, and its effectiveness for other systems is demonstrated through numerical examples.