How to improve the accuracy of semiclassical and quasiclassical dynamics with and without generalized quantum master equations

Abstract

Semi- and quasi-classical (SC) theories can handle arbitrary interatomic interactions and are thus well-suited to predict quantum dynamics in condensed phases that encode energy and charge transport, spectroscopic responses, and chemical reactivity. However, SC theories can be computationally expensive and inaccurate. When combined with generalized quantum master equations (GQMEs), the resulting SC-GQMEs have been observed to enhance the efficiency and accuracy of SC dynamics. Yet, while the mechanism responsible for improved efficiency is clear, the underlying improved accuracy remains elusive. What is worse, SC-GQMEs can yield unphysical dynamics in challenging parameter regimes -- a shortcoming that might be avoided if the mechanism of accuracy improvement were understood. Here, we uncover this mechanism. We leverage short-time analyses to prove that exact, "left-handed" time-derivatives delay the onset of SC inaccuracy, and show that their numerical integration yields dynamics with improved accuracy, even without the GQME. We find, however, that these derivatives are a double-edged sword: while offering greater short-time accuracy, they become unphysical in challenging parameter regimes. Because short-lived memory kernels can leverage short-time accuracy while circumventing long-time instability, we develop a protocol to unambiguously determine the memory kernel cutoff, even in challenging regimes where previous treatments had failed. Our insights into accuracy improvement and kernel cutoff protocol can be expected to apply to complex systems that go beyond simple models.

Figures (9)

• Figure 1: Accuracy improvement in the SC-GQME and its relation to left- and right-handed derivatives. Nonequilibrium population dynamics in the spin–boson model subject to initial condition $\rho(0) = \rho_B\ket{1}\bra{1}$ with parameters $\epsilon = \Delta$, $\beta = 5.0\Delta^{-1}$, $\omega_c = 2.0\Delta$, $\eta = 0.2\Delta$ with Ohmic spectral density. Panels compare exact (black) and LSC dynamics (gray) with (A) $\mathcal{K}^{(3b)}(t)$ as the generator of the SC-GQME, (B) $\mathcal{K}^{(3f)}(t)$ as the generator of the SC-GQME. Panels (C) and (D) show the dynamics obtained by integrating $[\dot{\mathcal{C}}^L(t)]_{\rm LSC}$ and $[\dot{\mathcal{C}}^R(t)]_{\rm LSC}$, respectively.
• Figure 2: Schematic of how left-handed derivatives can delay the onset of inaccuracy of semiclassical dynamics.Left: Taylor expansion of a representative correlation function, $\mathcal{C}(t)$, where $\chi_n$ represents the $n^{\text{th}}$ static coefficient. Green checks indicate agreement with the exact static coefficient while an orange (red) diamond suggests minor (major) disagreement with the exact static coefficient. Right: graphical representation of the static coefficients of the same representative correlation function, $\mathcal{C}(t)$. Top right: the static moments of $[\mathcal{C}^R(t)]$ are identical to $\mathcal{C}(t)$ and have minor disagreement from exact dynamics at $4^{\rm th}$ order (yellow region). Larger disagreements arise at higher orders (red region). Bottom right: the static moments of $[\mathcal{C}^L(t)]$ match the exact quantum dynamics through $4^{\rm th}$ order, but minor disagreements arise at $5^{\rm th}$ order.
• Figure 3: Extent to which left-handed derivatives can improve the accuracy of semiclassical dynamics. Comparison of exact (black) and LSC (gray) population dynamics to integrating $[\mathcal{C}^L(t)]_{\rm LSC}$ and $[\mathcal{C}^{2L}(t)]$. All panels correspond to the spin–boson model with $\epsilon = \Delta$, $\beta = 5.0\Delta^{-1}$, $\omega_c = \Delta$, an Ohmic spectral density, and varying system–bath coupling, $\eta$, subject to initial condition $\rho(0) = \rho_B\ket{1}\bra{1}$. Red shaded regions denote unphysical negative populations.
• Figure 4: A simple shift allows left-handed derivatives to conserve population. Conservation of total population, $h$, in (A) $[\mathcal{C}^L(t)]_{\rm LSC}$ versus $[\bar{\mathcal{C}}^L(t)]_{\rm LSC}$ and (B) $[\mathcal{C}^{2L}(t)]_{\rm LSC}$ versus $[\bar{\mathcal{C}}^{2L}(t)]_{\rm LSC}$ for a spin-boson model with parameters $\epsilon = \Delta$, $\beta = 5.0\Delta^{-1}$, $\omega_c = 2.0\Delta$, $\eta = 0.2\Delta$ subject to initial condition $\rho(0) = \rho_B\ket{1}\bra{1}$.
• Figure 5: Resolving apparent discrepancies between the SC-GQME and the left-handed derivative. Nonequilibrium population dynamics for the spin-boson model with $\epsilon = \Delta$, $\beta = 5.0\Delta^{-1}$, $\omega_c =\Delta$, an Ohmic spectral density, and subject to initial condition $\rho(0) = \rho_B\ket{1}\bra{1}$. (A) $\eta = 0.4\Delta$. (B) $\eta = \Delta$. Red shaded regions denote nonphysical negative populations. Note that the dynamics obtained from the single-accuracy SC-GQME built on the shifted left-handed derivative, $[\bar{\mathcal{C}}^L(t)]_{{\rm LSC}}$-SC-GQME (dashed cyan line), and the direct integration of $[\bar{\mathcal{C}}^L(t)]_{{\rm LSC}}$ (dashed red line) agree, in contrast to the more strongly oscillatory dynamics predicted with the traditional SC-GQME (dashed dark blue line), which is built on mixed-accuracy inputs to the auxiliary kernels. Use of single-accuracy constructions validates the use of equivalence proofsnoneq1whencanonewin for GQME dynamics.