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 $e^{-\hat{H}/(k_B T)}$, with $\hat{H}$,$k_B$, 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., $\langle \tilde{x} | e^{-\hat{H}/(k_B T)} | x \rangle$, or the imaginary-time propagator) could be derived via a straightforward ``real-time $\rightarrow$ 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 $(T \rightarrow \infty)$. In this work, we solve this problem and develop a reasonable HK-like semi-classical IVR for $\langle \tilde{x} | e^{-\hat{H}/(k_B T)} | x \rangle$ 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 $\tilde{x}$, 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.