Herman-Kluk-Like Semi-Classical Initial-Value Representation for Boltzmann Operator

TL;DR

The paper develops an Herman–Kluk–like semi-classical initial-value representation (IVR) for the imaginary-time Boltzmann operator, addressing divergences that arise from a naive real-time to imaginary-time transform. It presents a compact HK-like integral with a real Gaussian integrand, $K_{\tau}(\tilde{x},x)=A\int d\mathbf{p}\,d\mathbf{q}\,\left[D_{\tau}\,e^{-(S_{\tau}+B_{\tau}+C_{\tau})/\hbar}\right]$, and defines the action $S_{\tau}$, boundary factors $B_{\tau},C_{\tau}$, and a phase-space pre-factor $D_{\tau}$ computed from the stability of the classical flow. The derivation, built around a function $F_D$ with an initial condition $F_D(0)=1$, shows that in the semi-classical limit $F_D=D_{\tau}$, recovering the desired IVR; the approach is exact for free particles and harmonic oscillators and applicable to potentials with bounded force or long-range harmonic behavior. Numerical examples for an anharmonic system demonstrate high accuracy in many regimes, with limitations at very low temperatures when quantum-classical scales compete. The method offers a practical tool for finite-temperature quantum dynamics and thermodynamics in multi-dimensional systems, complementing existing Gaussian-series and Mandelshtam-type approaches.

Abstract

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.

Figures (5)

• Figure 1: Schematic diagrams of some typical potentials (red solid lines) of types (a) or (b) of Sec. \ref{['introduction']}, for a one-dimensional system with coordinate $q$. All these potentials are smooth functions of $q$. (1-3): Typical potentials of type (a). Specifically, the potential in (1) approaches to the same constant (dashed line) in the limits $q\rightarrow +\infty$ and $-\infty$, the potential in (2) approaches to the different constants (dashed lines) in these two limits, and the potential in (3) exhibits oscillatory behavior over the entire $q$-axis, with finite upper and lower bounds (dashed lines). (4): A typical potential of type (b), which approaches to a harmonic potential (dashed line) in the limits $q\rightarrow\pm\infty$.
• Figure 2: The potential $V(q)$ of Eq. (\ref{['vq']}), and matrix elements of the Boltzmann operator, for cases with $L=10l_0$ and $q_0=6l_0$. (a): The potential $V(q)$. (b): Enlarged view of the region around $q=0$ in (a). Here we also show the ground-state energy $E_0$ (green dashed line), the first excited-state energy $E_1$ (blue dotted line), and the second excited-state energy $E_2$ (purple solid line), which are given by numerical diagonalization of the Hamiltonian. Note that $E_0$ and $E_1$ lie very close to each other. (c-e): The diagonal elements $K(x,x)=\langle x|e^{-\hat{H}/(k_{B}T)}|x\rangle$ (in units of $1/l_0$). (f-h): The non-diagonal elements $K(-x,x)=\langle -x|e^{-\hat{H}/(k_{B}T)}|x\rangle$ (in units of $1/l_0$). Here we show the results with temperature $T=10\hbar\omega/k_B$ (c, f), $T=\hbar\omega/k_B$ (d, g) and $T=0.5\hbar\omega/k_B$ (e, h). For each temperature, we illustrate the results $K_{\rm Exac}$ from exact diagonlization of the Hamiltonian (black dotted line) and the results $K_{\rm IVR}$ given by our HK-like IVR (blue dots).
• Figure 3: Same as Fig. (\ref{['x4']}), but for $L=10l_0$ and $q_0=3l_0$.
• Figure 4: $K_{\rm Exac}(\tilde{x}, x)$ (in units of $1/l_0$), $K_{\rm IVR}(\tilde{x}, x)$ (in units of $1/l_0$), and the relative error of our HK-like IVR approach which is defined as ${|K_{\rm Exact}(\tilde{x}, x) - K_{\rm IVR}(\tilde{x}, x)|}/{|K_{\rm Exact}(\tilde{x}, x)|}$. Here we show the results for cases with $L=10l_0$, $q_0=6l_0$ (a-i) and $L=10l_0$, $q_0=3l_0$ (j-r), for temperatures $T=10\hbar\omega/k_B$, $T=\hbar\omega/k_B$, and $T=0.5\hbar\omega/k_B$. We do not show the results in the white regions, since in these regions both $K_{\rm Exact}$ and $K_{\rm IVR}$ are below 1% of the maximum value of $K_{\rm Exact}$ across the entire domain (denoted as $K^{\rm max}_{\rm Exact}$). Moreover, in the four squares enclosed by black lines in (r), the relative error is between $4 \times 10^{-2}$ and $8 \times 10^{-2}$, while both $K_{\rm Exact}$ and $K_{\rm IVR}$ are below 3% of $K^{\rm max}_{\rm Exact}$.
• Figure 5: $K_{\rm IVR}(-q_0, q_0)$ and $K_{\rm Exact}(-q_0, q_0)$(a), and the ratio $q_0/l_T$(b), as functions of temperature $T$, for systems with parameter (ii). Inset of (b): the ratio $L/l_T$.