Table 2:  Thermal rate constants for the overdamped spin–boson model in Eq. (5.5) with $\hbar=1$, $\Lambda=60$, $\epsilon=0$, $\gamma=32\Omega$ and $\Omega=2$. All results are divided by $\Delta^{2}$.  $\beta$ | $1$ | $2$ | $4$
---|---|---|---
$k_{2}$ | $7.895\text{\times}{10}^{-08}$ | $4.929\text{\times}{10}^{-14}$ | $2.781\text{\times}{10}^{-26}$
$k_{2}^{\mathrm{sc}}$ | $7.893\text{\times}{10}^{-08}$ | $4.928\text{\times}{10}^{-14}$ | $2.781\text{\times}{10}^{-26}$
$\frac{\tilde{k}_{{4\mathrm{A}}}}{\tilde{k}_{2}}$ | $2.309$ | $1.701$ | $0.789$
$\frac{\tilde{k}_{{4\mathrm{A}}}^{\mathrm{sc}}}{\tilde{k}_{2}^{\mathrm{sc}}}$ | $0.399$ | $0.557$ | $0.762$
$\frac{\tilde{k}_{{4\mathrm{B}}}}{\tilde{k}_{2}}$ | $0.121$ | $0.443$ | $1.289$
$\frac{\tilde{k}_{{4\mathrm{B}}}^{\mathrm{sc}}}{\tilde{k}_{2}^{\mathrm{sc}}}$ | $0.123$ | $0.469$ | $1.424$
$\frac{\widetilde{Z}_{\mathrm{r},2}}{Z_{\mathrm{r},0}}$ | $0.016$ | $0.033$ | $0.066$
$\frac{\widetilde{Z}_{\mathrm{r},2}^{\mathrm{sc}}}{Z_{\mathrm{r},0}^{\mathrm{sc}}}$ | $0.016$ | $0.033$ | $0.066$
$\frac{k_{4}}{k_{2}}$ | $-2.204$ | $-1.291$ | $0.433$
$\frac{k_{4}^{\mathrm{sc}}}{k_{2}^{\mathrm{sc}}}$ | $-0.292$ | $-0.121$ | $0.595$

It was mentioned in the introduction that the methods developed in this paper
are not generally aimed at condensed-phase systems and do not attempt to
describe rate processes involving diffusive motion along the reaction
coordinate. Hence it was reasonable to base the preceding discussion on an
underdamped spin–boson model ($\gamma=\Omega$). Even so, it is instructive to
also analyse the predictions for a strongly overdamped system, which may shed
light on the limitations of the method and give insight into the underlying
physics. In Table 2 we list the results for a symmetric spin–boson model at a
range of temperatures, with $\Omega=2$, $\gamma=32\Omega$, and the remaining
parameters kept the same as in Table 1. At $\beta=1$,
$\tilde{k}_{2}^{\mathrm{sc}}$, $\tilde{k}_{{4\mathrm{B}}}^{\mathrm{sc}}$ and
$\widetilde{Z}_{\mathrm{r},2}^{\mathrm{sc}}$ are all in excellent agreement
with their quantum counterparts, whereas
$\tilde{k}_{{4\mathrm{A}}}^{\mathrm{sc}}$ underestimates the corresponding
quantum value by about an order of magnitude, completely compromising the
total rate, $k_{4}$. The same pattern (not shown) is found at the other values
of $\epsilon$ and $\Omega$ listed in Table 1. This suggests that the
assumptions underlying our derivation of
$\tilde{k}_{{4\mathrm{A}}}^{\mathrm{sc}}$ do not hold in the high-friction
regime. We can narrow the issue down to the behaviour of
$L_{{4\mathrm{A}}}(t_{0},t_{1})$, previously defined in Eq. (4.4a). In Fig. 9
we compare this function to its semiclassical approximation

$\displaystyle L_{{4\mathrm{A}}}^{\mathrm{sc}}(t_{0},t_{1})$
$\displaystyle=\frac{\sqrt{2\pi\hbar}}{2\hbar^{4}}\left(\frac{\partial^{2}{\phi_{{4\mathrm{A}}}}}{\partial{t}^{2}}\right)^{\\!\\!-1/2}\\!\\!\mathrm{e}^{-\phi_{{4\mathrm{A}}}/\hbar}$
(5.7)
$\displaystyle\qquad{}\times\exp\left(-\frac{d_{0}t_{0}^{2}+d_{1}t_{1}^{2}+d_{01}t_{0}^{2}t_{1}^{2}}{2\hbar}\right),$

with $\phi_{{4\mathrm{A}}}$ defined as in Eq. (D.4a), and both it and its
derivative evaluated at the stationary point.

Figure 9: Plots of $L_{{4\mathrm{A}}}(t_{0},t_{1})/L_{{4\mathrm{A}}}(0,0)$
[Eq. (4.4a)] calculated either by numerical integration of the quantum
spin–boson $c_{{4\mathrm{A}}}$ time-correlation function [Eqs. (D.1) and
(D.4a)] or using the semiclassical approximation [Eq. (5.7)]. Quantum and
semiclassical results are shown in black and red respectively. The
calculations were done for $\epsilon=0$, $\beta=1$, $\Omega=2$, and
$\Lambda=60$. On the left-hand side $\gamma=\Omega$, and on right-hand side
$\gamma=32\Omega$. Panels (a–b) show contour plots of the quantum results.
Panels (c–d) compare the exact and semiclassical
$L_{{4\mathrm{A}}}(t_{0},t_{1})$ along the diagonal $t_{0}=t_{1}=s$. Panels
(e–f) do the same for the $y$-axis $t_{0}=0$, $t_{1}=s$.

The comparison allows us to identify two distinct factors that cause the
accuracy of the semiclassical approximation to deteriorate. Firstly, from the
plots of $L_{{4\mathrm{A}}}(s,s)$ in Fig. 9(c–d) we see that the approximation
in Eq. (4.27) significantly overestimates the true value of $d_{01}$ at high
friction. At least in principle this can be fixed by basing the approximation
on a more sophisticated analytically soluble model. Alternatively one could
evaluate $d_{01}$ numerically, which requires implementing the necessary high-
order action derivatives333In practice evaluating the exact $d_{01}$ would
involve calculating terms containing third- and fourth-order derivatives of
the potential energy, which would be impractical. However we expect that in
most cases a very accurate approximation to $d_{01}$ would result even if such
terms were omitted. in addition to those calculated during GR instanton
optimisation.[21] We may pursue one or both of these strategies in future
work.

Nevertheless a refined estimate of $d_{01}$ does not bring the semiclassical
$\tilde{k}_{{4\mathrm{A}}}^{\mathrm{sc}}$ into alignment with the quantum
results due to a second feature, illustrated in Fig. 9(e–f). There we plot
$L_{{4\mathrm{A}}}(0,s)$, whose semiclassical approximation is proportional to
$\exp(-d_{1}s^{2}/2\hbar)$ and contains no contributions from the quartic
$d_{01}$ term. It is apparent that the quantum and semiclassical values agree
at short times (indicating that $d_{1}$ is calculated correctly), but whereas
the semiclassical function is shaped like a Gaussian, the quantum expression
behaves like a decaying exponential at long times. The emergence of this “fat”
exponential tail is almost certainly due to the motion along the reaction
coordinate becoming diffusive.[56] This implies that the system spends more
time in the barrier region and is more likely to undergo an unreactive
transition of the kind described in Sec. IV.1, resulting in a larger
$\tilde{k}_{{4\mathrm{A}}}$ and a diminished overall rate constant.

That the semiclassical approximation fails to capture the long-time behaviour
of $L_{{4\mathrm{A}}}(t_{0},t_{1})$, and hence the true magnitude of
$\tilde{k}_{{4\mathrm{A}}}$, is not a problem of the instanton method _per se_
, but rather a breakdown of steepest-descent in time that affects all
transition-state-theory (TST) methods.[46] To avoid this, one has to define a
more suitable dividing surface/projection operator $\hat{h}$, leading to an
altogether different set of instantons. Thankfully we need not implement so
radical a change, since the primary target of our semiclassical rate theory is
gas-phase reactions in the deep-tunnelling regime, where instanton rate
theories are typically most useful. In this regime, the reaction rate is
unlikely to have significant contributions from diffusive over-the-barrier
motion, even if it happens to be a feature at higher temperatures. Instead the
rate is dominated by nuclear tunnelling contributions, which are captured
accurately by our theory, as shown by the low-temperature results in Table 2.
As $\beta$ increases, the $\tilde{k}_{{4\mathrm{A}}}$ term decreases relative
to the other fourth-order contributions and becomes better approximated by the
semiclassical expression. At the lowest temperature considered ($\beta=4$),
the overall agreement between the semiclassical and the quantum rates is
comparable to that in Table 1, even though the system is heavily overdamped.
We expect that a similar level of accuracy will be seen for more chemically
realistic potentials.

Our results so far suggest that Eq. (3.3) evaluated semiclassically provides a
reliable indicator of whether a reaction is in the golden-rule limit (as long
as the rate is not dominated by diffusive motion through the barrier region).
The GR rate, $k_{2}$, should be an accurate approximation to the exact rate if
$\lvert k_{4}/k_{2}\rvert\ll 1$. When this condition is not satisfied, we may
use $k_{4}$ to correct the GR approximation, but first we have to address the
convergence of the series in Eq. (2.8), which cannot in general be guaranteed.
Slowly convergent and divergent series are a common feature of asymptotic
analysis, and there exist several series acceleration techniques that can
yield high-accuracy approximations to the true values of functions represented
in this fashion, given a limited number of terms. One widely used approach is
Padé summation, which represents the target function as a ratio of two
polynomials (see 82, Chapter 8.3 in Ref.). This ratio is called a Padé
approximant. To quote from Ref. 82, “Padé approximants often work quite well,
even beyond their proven range of applicability.” In the spirit of this quote,
we do not attempt to prove that Padé summation of Eq. (2.8) converges to the
exact result, and use it as a heuristic means of extending the range of
coupling strengths at which our new theory gives reasonable predictions. For
the series truncated at $\mathcal{O}(\lambda^{4})$, the diagonal Padé
approximant is

$\lambda^{2}k_{2}+\lambda^{4}k_{4}\sim\frac{\lambda^{2}k_{2}}{1-\lambda^{2}k_{4}/k_{2}}.$
(5.8)

Given that $k_{4}/k_{2}$ can be either positive or negative, it is possible
that the proposed approximant is singular for some real $\lambda$. We want to
avoid this, as the presence of such a singularity degrades the accuracy of the
approximation, even compared to straightforward series summation. Instead we
can apply the technique to a _subset_ of all terms, chosen so as to give a
non-singular expression. We have decided to only include
$-\tilde{k}_{{4\mathrm{A}}}^{\mathrm{sc}}/\widetilde{Z}_{\mathrm{r},0}^{\mathrm{sc}}$
into the subset, as this term is always negative, and its Padé summation with
$k_{2}^{\mathrm{sc}}$ results in a non-singular function for all real
$\lambda$. The remaining contribution,
$\tilde{k}_{{4\mathrm{B}}}^{\mathrm{sc}}/\widetilde{Z}_{\mathrm{r},0}^{\mathrm{sc}}-k_{2}^{\mathrm{sc}}\widetilde{Z}_{\mathrm{r},2}^{\mathrm{sc}}/\widetilde{Z}_{\mathrm{r},0}^{\mathrm{sc}}$,
is usually positive, even if the diabatic coupling is not vanishingly small
around the equilibrium reactant/product geometry. This follows from the
dependence of $\tilde{k}_{{4\mathrm{B}}}^{\mathrm{sc}}$ and
$\widetilde{Z}_{\mathrm{r},2}^{\mathrm{sc}}$ on the energy separation between
the two diabatic surfaces. For the systems considered in this work, the best
results were achieved when this remaining contribution was added to the
$\tilde{k}_{{4\mathrm{A}}}^{\mathrm{sc}}$ Padé approximant in the usual
manner,444 Quite possibly there are cases when it is better to also include
fourth-order components other than $\tilde{k}_{{4\mathrm{A}}}^{\mathrm{sc}}$
into the Padé approximant, _e.g._ , for systems in the Marcus inverted regime,
where $\tilde{k}_{{4\mathrm{B}}}^{\mathrm{sc}}$ appears in the expansion of
the full rate with a minus sign. yielding

$\left[k_{2+4}^{\mathrm{sc}}\right]_{\text{Pad\'{e}}}=k_{2}^{\mathrm{sc}}\Bigg{(}\bigg{[}1+\frac{\tilde{k}_{{4\mathrm{A}}}^{\mathrm{sc}}}{\tilde{k}_{2}^{\mathrm{sc}}}\bigg{]}^{-1}\\!\\!+\frac{\tilde{k}_{{4\mathrm{B}}}^{\mathrm{sc}}}{\tilde{k}_{2}^{\mathrm{sc}}}-\frac{\widetilde{Z}_{\mathrm{r},2}^{\mathrm{sc}}}{\widetilde{Z}_{\mathrm{r},0}^{\mathrm{sc}}}\Bigg{)}.$
(5.9)

This choice is not unique, and we do not undertake to explore all the
possibilities at this stage. A rigorous justification of the current approach
(or indeed an alternative) can only emerge once the theory of further high-
order terms ($k_{6},k_{8},$ _etc._) is developed. For the present we note that
(a) Eqs. (3.3) and (5.9) by construction agree at small $\Delta$; (b) Eq.
(5.9) makes remarkably accurate numerical predictions at intermediate $\Delta$
for the models considered in this section; and (c) our Padé-summed term is
analogous to the nonadiabatic rate derived in Ref. 43 for a reduced model of
electron transfer in solution.

Figure 10: Exact nonadiabatic rates calculated with HEOM[55] (black circles),
compared against three semiclassical (SC) approximations: golden-rule
$k_{2}^{\mathrm{sc}}$ on its own, the sum
$k_{2}^{\mathrm{sc}}+k_{4}^{\mathrm{sc}}$, and the Padé approximation in Eq.
(5.9). Also shown are the output of Eq. (5.9) evaluated with quantum rate
constants (QM) and rates calculated with the Zusman equation [Eq. (70) in Ref.
55]. The semiclassical approximation is almost exactly on top of the quantum
result. Top row panels (a–b) correspond to symmetric systems, $\epsilon=0$,
and bottom-row panels (c–d) to asymmetric systems, $\epsilon=15\beta^{-1}$.
Plots on the left are for the high-temperature regime
($\beta\hbar\Omega=0.5$), and plots on the right are for low temperature
($\beta\hbar\Omega=4$). In all cases $\hbar=1$, $\Omega=\gamma$ and
$\beta\Lambda=60$.

In Fig. 10 we compare the new expression to the exact nonadiabatic rate
constants that were taken directly from Ref. 55. Shown alongside are the plain
GR rate $k_{2}^{\mathrm{sc}}$, the standard sum
$k_{2}^{\mathrm{sc}}+k_{4}^{\mathrm{sc}}$, and the output of the Zusman
equation.[40, 56, 44] The latter is accurate at high temperature and weak
coupling, and is included as a classical benchmark to illustrate the magnitude
of nuclear quantum effects. In all cases the corrected expression (be it the
sum or Padé) remains accurate up to larger $\Delta$ than $k_{2}$ on its own.
At high temperatures [panels (a) and (c) of Fig. 10], where the correction is
dominated by the $k_{{4\mathrm{A}}}$ term, there is a clear benefit to using
the Padé approximant. At low temperature [panels (b) and (d)] the difference
between the two corrected expressions becomes marginal, as $k_{{4\mathrm{A}}}$
is relatively small. Overall, Eq. (5.9) is at least as good or better than the
more naïve expression for the corrected rate, and so we recommend its use at
all temperatures. The expression correctly predicts both the sign and
magnitude of the deviation of $k_{2}$ from the exact rate at small coupling,
making it a reliable and computationally inexpensive means of testing whether
Fermi’s golden rule is right for the problem.

Given that our goal was to derive a correction to GR in the weak-coupling
limit, at large $\Delta$ our theory cannot rival the accuracy of the
interpolation formula by Lawrence and co-workers.[55] It does, however, go a
long way towards fixing the rate at intermediate coupling. Furthermore, the
new theory predicts the correct high-temperature dependence of $k_{4}$ [Eq.
(4.13)] for reactions whose rate is not solvent controlled (_i.e._ , away from
the high-friction regime). As mentioned in Sec. IV.1, Appendix E shows that
Landau–Zener theory[47, 48] is consistent with our prediction. In contrast,
the Zusman equation, and hence the interpolation formula, are not—a
discrepancy due to Zusman’s underlying assumption of strong solvent
friction.[40, 56, 44] Our approach thus offers insight into nonadiabatic rate
processes at some temperatures and frictions that could not be accurately
described by previously existing (semi)classical theories.

##  VI Discussion

In this paper we have presented a rigorous derivation of the second term,
$k_{4}$, in the perturbation series expansion of the nonadiabatic rate
constant $k$ in powers of the diabatic coupling $\hat{\Delta}$ (the first term
being the golden-rule rate constant $k_{2}$). Our expression consists of three
components, each of which is well approximated by semiclassical instanton
methods. The first component, $\tilde{k}_{{4\mathrm{A}}}^{\mathrm{sc}}$,
corresponds to a four-bounce instanton path. It dominates $k_{4}$ at high
temperatures, decreasing the overall rate, and accounts for recrossing
transitions between the two diabats, which can either take place via
tunnelling or during the passage of the system through the barrier region. The
second component, $\tilde{k}_{{4\mathrm{B}}}^{\mathrm{sc}}$ corresponds to a
two-bounce instanton path. It dominates $k_{4}$ at low temperatures and
accounts for the enhanced tunnelling probability at stronger diabatic
coupling, increasing the overall rate. The final component,
$\widetilde{Z}_{\mathrm{r},2}$, derives from the minima of the reactant and
product wells. Along with $\tilde{k}_{{4\mathrm{B}}}^{\mathrm{sc}}$, it
dominates at low temperature, but decreases the rate instead. This term
largely accounts for changes in the potential energy around the reactant
minimum.

Like the golden-rule rate $\tilde{k}_{2}^{\mathrm{sc}}$, the
$\tilde{k}_{{4\mathrm{A}}}^{\mathrm{sc}}$ and
$\tilde{k}_{{4\mathrm{B}}}^{\mathrm{sc}}$ components are exact for two-state
systems comprised of linear diabats. As far as we are aware, this is the first
time that these expressions are reported in the literature, and hence the
first time that the high-temperature asymptotics of $k_{4}$ are rigorously
derived for a system of one-dimensional diabatic potentials. By extension, we
expect that the high-temperature rates for multidimensional systems with
moderate friction along the reaction coordinate are more accurately described
by our theory than by, _e.g._ , the Zusman equation[40, 56, 44] or uniform
rate expressions[41, 42, 46] derived from the Holstein nonadiabatic
transmission coefficient[88] (based in turn on Landau–Zener theory).[47, 48]
It should, however, be noted that the latter is consistent with our new
theory, as discussed in Appendix E, albeit the underlying assumptions of the
Holstein expression mean it can only give the asymptotic form of
$k_{4}(\beta\to 0)$ up to a temperature-independent constant. Our theory goes
beyond LZ and Holstein expressions: not only does it fully establish the
high-$T$ asymptotics of $k_{4}$, it also captures the mechanistic transition
from over-the-barrier to tunnelling processes, and can therefore be applied at
low temperatures.

Calculations of $k_{4}$ in a one-dimensional predissociation model have shown
that our semiclassical expressions readily generalise to anharmonic systems,
with the model in Sec. V.1 displaying deviations from the quantum ratio
$k_{4}/k_{2}$ that are no greater than $0.06\Delta^{2}$. Analogous
calculations for a spin–boson model have shown that the semiclassical rates
are also accurate in multidimensional systems. Problems only arise when both
the temperature and friction are high, at which point the motion of the nuclei
through the barrier region becomes diffusive,[56] and recrossing events quite
likely. In line with our physical interpretation, this affects the
$\tilde{k}_{{4\mathrm{A}}}^{\mathrm{sc}}$ term, causing it to considerably
underestimate the value of its quantum counterpart. One could attempt to fix
the problem either by redefining the product projection operator (which would
lead to a different semiclassical theory) or by employing our knowledge of the
exact $c_{{4\mathrm{A}}}$ time correlation function for the spin–boson model.
Neither is pursued in this work, as our intended domain of application is low-
temperature reactions in the gas phase, which are not likely to be rate-
limited by diffusion along the reaction coordinate. Nevertheless, in future
work it could be interesting to study the spin–boson $c_{{4\mathrm{A}}}$
correlation function more closely, as such investigations could lead to a
generalisation of the Zusman formula[40, 56, 44] that would rigorously account
for nuclear quantum statistics.

For spin–boson models at moderate friction our theory captures the behaviour
of the exact rate for a wider range of diabatic coupling strengths than the GR
rate on its own, and correctly predicts the sign and magnitude of the error
introduced by making the golden-rule approximation. One can even use our
fourth-order expression to correct the GR approximation when its deviation
from the exact rate is not too large (up to a factor of 2 in cases we have
tested so far). Crucially, few additional calculations are necessary to
evaluate the new rate constant. Its $k_{{4\mathrm{A}}}$ component derives from
a GR instanton at $\beta/2$, _i.e._ , at twice the temperature of the
simulation. Since instanton optimisation typically involves starting at high
temperature and cooling the system down, it is easy to arrange for $\beta/2$
to be visited along the way. All the quantities needed to evaluate
$\tilde{k}_{{4\mathrm{A}}}^{\mathrm{sc}}$ are in that case directly available
from the output of the optimisation.

The $k_{{4\mathrm{B}}}$ component derives from a GR instanton at precisely the
simulation temperature. In addition to the output of the GR optimisation, the
calculation of $\tilde{k}_{{4\mathrm{B}}}^{\mathrm{sc}}$ requires the
potential energy and gradient of the upper diabatic state ($V_{1}$ on the
reactant side and $V_{0}$ on the product side), as well as the diabatic
coupling $\Delta$ along the instanton trajectory. However no instanton re-
optimisation is required, and the additional quantities need only be evaluated
on a relatively sparse grid. The computational overhead thus remains marginal
compared to the initial GR optimisation.

The final (partition function) component, rather than deriving from a
delocalised instanton structure, comes from just two points—the minima of the
reactant and product diabats. Consequently this term should also add little to
the overall computation time. Moreover, this term is expected to often be
insignificant in practice, as it is proportional to the square of the diabatic
coupling at the reactant/product minimum, which may be negligible.

On the whole, lengthy derivation notwithstanding, one can obtain
$k_{4}^{\mathrm{sc}}$ in a simple post-processing step to a GR instanton
optimisation, involving only a few additional electronic structure
calculations. It has all the makings of a practical method that, like GR
instanton theory, can be applied to _ab initio_ simulations of real
molecules.[28] Furthermore, the derivation presented in this paper suggests
how one might obtain further high-order terms:
$k_{6}^{\mathrm{sc}},\,k_{8}^{\mathrm{sc}},$ and so on. Based on the Green’s
function formalism we expect $k_{6}^{\mathrm{sc}}$ to comprise contributions
from three instantons. The first two are likely simple generalisations of the
terms encountered so far: a six-bounce 6A term and a two-bounce 6C term, now
with _two_ points sampling the upper diabat. The remaining 6B term is expected
to combine these elements, resulting in a four-bounce path with a single point
sampling the upper diabat. In addition to that, $k_{6}^{\mathrm{sc}}$ is
expected to contain contributions derived from the minima of the diabatic
PESs, likely to be insignificant in practice due to the small magnitude of the
diabatic coupling at these points. We see no reason why the methods presented
in this paper cannot be applied to the derivation of all these components.

Of course it is likely that, beyond a certain point, there is little merit to
calculating increasingly higher-order terms. Since all of them are associated
with golden-rule, rather than adiabatic, instantons, this may not be the most
accurate or efficient way of describing nonadiabatic rates in the
Born–Oppenheimer limit. An alternative would be to apply the ideas developed
in this paper to deriving a correction to the _adiabatic_ rate constant, and
to then connect the two limits using a suitable interpolation formula, not
unlike Ref. 55. Apart from encompassing the full range of coupling strengths,
this approach would offer new mechanistic insight into nonadiabatic reactions,
derived from the instantons that give rise to the correction terms.

Looking beyond semiclassical instantons, our theory may also find applications
in the development of path-integral sampling approaches to quantum TST
simulations. Existing sampling-based methods include ring-polymer molecular
dynamics (RPMD)[57, 58, 59] and quantum instanton rates[96, 97, 98] for
adiabatic calculations, and Wolynes theory[60, 75] and golden-rule quantum
TST[99, 100] for the weak-coupling limit. More general methods have also been
proposed for arbitrary coupling strengths,[101, 102, 103, 104, 105, 106, 107,
108, 109, 65] although unlike RPMD in the Born–Oppenheimer case,[4] none has
been connected to a rigorous instanton theory. [110, 99, 111] New path-
integral rate theories may thus be inspired by the theoretical developments in
this work. Our approach should also help address the long-standing issue of
tackling the Marcus inverted regime. While it was initially assumed that our
two-state system is in the normal regime, the analytical formulae we have
derived for linear diabats are in fact general.555The only difference is that
in the inverted regime both $\tilde{k}_{{4\mathrm{A}}}$ and
$\tilde{k}_{{4\mathrm{B}}}$ reduce the total rate, so technically Eq. (B.3)
should be multiplied by $\operatorname{sgn}(\kappa_{0}\kappa_{1})$. This
suggests that $k_{4}^{\mathrm{sc}}$ can be reformulated to encompass the
inverted regime for arbitrary potentials, much like has been done for golden-
rule instantons.[25, 24] We intend to pursue a similar generalisation of our
fourth-order semiclassical rates in future work.

###### Acknowledgements.

The authors acknowledge financial support by the Swiss National Science
Foundation through SNSF Project 175696 and thank Joseph E. Lawrence for
helpful discussions.

##  Appendix A  Asymptotic expansion of L4B

Here we derive the expressions in Eq. (4.39). To this end, it is convenient to
perform steepest descent integration of $S_{{4\mathrm{B}}}$ in stages,
treating the difference coordinate $\mathsf{x}_{-}$ first, followed by all the
remaining variables. We begin by expressing Eq. (IV.2) in terms of the new
position variables in Eq. (4.36) and use the short-time form of the action for
the segment of length $\tau_{1}$. The remaining terms are Taylor expanded to
second order about $\tau_{1}=0$ and $\mathsf{x}_{-}=\mathsf{0}$, yielding

$\displaystyle S_{{4\mathrm{B}}}\sim
S^{\prime}_{0}+S^{\prime\prime}_{0}+S_{1}+\frac{m\lVert\mathsf{x}_{-}\rVert^{2}}{2\tau_{1}}+\tau_{1}V_{1}(\mathsf{x})$
$\displaystyle{}+\bigg{(}\frac{\partial{S^{\prime}_{0}}}{\partial{\mathsf{x}}}-\frac{\partial{S^{\prime\prime}_{0}}}{\partial{\mathsf{x}}}\bigg{)}\\!\cdot\\!\bigg{(}\frac{\mathsf{x}_{-}\\!}{2}\bigg{)}-\frac{\partial{S_{1}}}{\partial{\tau}}\tau_{1}$
(A.1)
$\displaystyle{}+\frac{1}{2}\bigg{(}\frac{\mathsf{x}_{-}\\!}{2}\bigg{)}\\!\cdot\\!\left[\frac{\partial^{2}{(S^{\prime}_{0}+S^{\prime\prime}_{0})}}{\partial{\mathsf{x}\mkern
1.0mu}\partial{\mathsf{x}}}\right]\\!\cdot\\!\bigg{(}\frac{\mathsf{x}_{-}\\!}{2}\bigg{)}+\frac{1}{2}\frac{\partial^{2}{S_{1}}}{\partial{\tau}^{2}}\tau_{1}^{2},$

with $S_{0}^{\prime},S_{0}^{\prime\prime},$ and $S_{1}$ defined in Eq. (4.37).
Substituting this expansion for $S_{{4\mathrm{B}}}$, we evaluate the integral

$\int_{-\infty}^{\infty}\\!\mathrm{e}^{-S_{{4\mathrm{B}}}/\hbar}\,\mathrm{d}\mathsf{x}_{-}\sim
A\,\mathrm{e}^{-S_{{4\mathrm{B}}}^{\mathrm{eff}}/\hbar},$ (A.2)

where

$\displaystyle S_{{4\mathrm{B}}}^{\mathrm{eff}}$ $\displaystyle\sim
S_{0}^{\prime}+S_{0}^{\prime\prime}+S_{1}$ (A.3)
$\displaystyle+\tau_{1}\left\\{V_{1}(\mathsf{x})-\frac{\lVert\bar{\mathsf{p}}\rVert^{2}}{2m}-\frac{\partial{S_{1}}}{\partial{\tau}}\right\\}$
$\displaystyle+\frac{1}{2}\tau_{1}^{2}\left\\{\frac{\bar{\mathsf{p}}}{2m}\cdot\frac{\partial^{2}{(S_{0}^{\prime}+S_{0}^{\prime\prime})}}{\partial{\mathsf{x}}\partial{\mathsf{x}}}\cdot\frac{\bar{\mathsf{p}}}{2m}+\frac{\partial^{2}{S_{1}}}{\partial{\tau_{1}}^{2}}\right\\}$

and we define

$\bar{\mathsf{p}}=\frac{1}{2}\frac{\partial{(S_{0}^{\prime\prime}-S_{0}^{\prime})}}{\partial{\mathsf{x}}}.$
(A.4)

The symbol $A$ denotes a prefactor whose derivation can be omitted. We are in
a position to do so because our final result is known to have as its prefactor
the semiclassical limit of $L_{{4\mathrm{B}}}(\tau_{0},0)$, which we have
already given in Eq. (4.35).

The new, effective action is now expanded to second order in the fluctuations
$\delta\bm{v}$ about the point $\bm{v}$ that satisfies Eq. (4.38). At this
point $\mathsf{x}^{\prime},\,\mathsf{x}^{\prime\prime}\\!,$ and $\tau$ assume
the values that minimise the GR instanton action; $\mathsf{x}$ is the
coordinate along the GR instanton after travelling for imaginary time
$\tau_{0}$ on the reactant diabat, away from $\mathsf{x}^{\prime\prime}$. A
few simplifications can now be made. Equation (2.26) relates the derivative of
the stationary action to the energy of the corresponding classical trajectory,
which we write as

$\displaystyle-\frac{\partial{S_{0}^{\prime}}}{\partial{\tau_{0}}}$
$\displaystyle=-\frac{1}{2m}\left\lVert\frac{\partial{S_{0}^{\prime}}}{\partial{\mathsf{x}}}\right\rVert^{2}+V_{0}(\mathsf{x})$
(A.5a)
$\displaystyle\frac{\partial{S_{0}^{\prime\prime}}}{\partial{\tau_{0}}}$
$\displaystyle=-\frac{1}{2m}\left\lVert\frac{\partial{S_{0}^{\prime\prime}}}{\partial{\mathsf{x}}}\right\rVert^{2}+V_{0}(\mathsf{x}).$
(A.5b)

Differentiating the difference of these two equations with respect to either
$\mathsf{x}$ or $\tau_{0}$ and using the fact that at the stationary $\bm{v}$

$\bar{\mathsf{p}}=\frac{\partial{S_{0}^{\prime\prime}}}{\partial{\mathsf{x}}}=-\frac{\partial{S_{0}^{\prime}}}{\partial{\mathsf{x}}},$
(A.6)

we can show that

$\frac{\bar{\mathsf{p}}}{2m}\cdot\frac{\partial^{2}{(S_{0}^{\prime}+S_{0}^{\prime\prime})}}{\partial{\mathsf{x}}\partial{\mathsf{x}}}\cdot\frac{\bar{\mathsf{p}}}{2m}=\frac{1}{4}\frac{\partial^{2}{(S_{0}^{\prime}+S_{0}^{\prime\prime})}}{\partial{\tau_{0}}^{2}}.$
(A.7)

Furthermore, considering the factor multiplying $\tau_{1}$ in Eq. (A.3)
together with Eqs. (A.5a) and (A.6), we can write

$\displaystyle-\frac{\lVert\bar{\mathsf{p}}\rVert^{2}}{2m}-\frac{\partial{S_{1}}}{\partial{\tau}}$
$\displaystyle=V_{1}(\mathsf{x})-V_{0}(\mathsf{x})-\frac{\partial{S_{0}^{\prime}}}{\partial{\tau_{0}}}-\frac{\partial{S_{1}}}{\partial{\tau}}$
(A.8) $\displaystyle=V_{1}(\mathsf{x})-V_{0}(\mathsf{x}),$

where the last equality follows from the relation

$\frac{\partial{S_{0}^{\prime}}}{\partial{\tau_{0}}}=\frac{\partial{S_{0}^{\prime}}}{\partial{\tau}}=-\frac{\partial{S_{1}}}{\partial{\tau}}$
(A.9)

at the stationary $\bm{v}$. Last, to calculate the
$\mathcal{O}(\tau_{1}\delta\bm{v})$ coefficient, we need the derivative of the
same factor with respect to $\bm{v}$. In particular, consider

$\frac{1}{2m}\frac{\partial{\lVert\bar{\mathsf{p}}\rVert^{2}}}{\partial{\bm{v}}}=\frac{1}{2}\frac{\partial^{2}{(S_{0}^{\prime\prime}-S_{0}^{\prime})}}{\partial{\bm{v}}\partial{\mathsf{x}}}\cdot\left[\frac{1}{2m}\frac{\partial{(S_{0}^{\prime\prime}-S_{0}^{\prime})}}{\partial{\mathsf{x}}}\right].$
(A.10)

According to Eq. (A.6) this can be rewritten as

$\displaystyle\frac{1}{2}\left\\{\frac{\partial^{2}{S_{0}^{\prime\prime}}}{\partial{\bm{v}}\partial{\mathsf{x}}}\cdot\left[\frac{1}{m}\frac{\partial{S_{0}^{\prime\prime}}}{\partial{\mathsf{x}}}\right]+\frac{\partial^{2}{S_{0}^{\prime}}}{\partial{\bm{v}}\partial{\mathsf{x}}}\cdot\left[\frac{1}{m}\frac{\partial{S_{0}^{\prime}}}{\partial{\mathsf{x}}}\right]\right\\}$
(A.11)
$\displaystyle\qquad{}=\frac{1}{2}\frac{\partial{}}{\partial{\bm{v}}}\left\\{\frac{1}{2m}\left\lVert\frac{\partial{S_{0}^{\prime\prime}}}{\partial{\mathsf{x}}}\right\rVert^{2}+\frac{1}{2m}\left\lVert\frac{\partial{S_{0}^{\prime}}}{\partial{\mathsf{x}}}\right\rVert^{2}\right\\}.$

From Eq. (A.5) it then follows that

$\frac{1}{2m}\frac{\partial{\lVert\bar{\mathsf{p}}\rVert^{2}}}{\partial{\bm{v}}}=\frac{\partial{V_{0}(\mathsf{x})}}{\partial{\bm{v}}}-\frac{1}{2}\frac{\partial^{2}{(S_{0}^{\prime\prime}-S_{0}^{\prime})}}{\partial{\bm{v}}\partial{\tau_{0}}}.$
(A.12)

Combining all these results we have the following terms in the Taylor series
expansion of $S_{{4\mathrm{B}}}^{\mathrm{eff}}$:

$\displaystyle\mathcal{O}(1)\ :$ $\displaystyle\
S_{0}^{\prime}+S_{0}^{\prime\prime}+S_{1}\equiv S_{2}(\beta),$ (A.13a)
$\displaystyle\mathcal{O}(\tau_{1})\ :$ $\displaystyle\
\tau_{1}\left[V_{1}(\mathsf{x})-V_{0}(\mathsf{x})\right],$ (A.13b)
$\displaystyle\mathcal{O}(\tau_{1}^{2})\ :$ $\displaystyle\
\frac{1}{2}\tau_{1}^{2}\left[\frac{1}{4}\frac{\partial^{2}{(S_{0}^{\prime}+S_{0}^{\prime\prime})}}{\partial{\tau_{0}}^{2}}+\frac{\partial^{2}{S_{1}}}{\partial{\tau}^{2}}\right],$
(A.13c) $\displaystyle\mathcal{O}(\delta\bm{v}\delta\bm{v})\ :$
$\displaystyle\
\frac{1}{2}\delta\bm{v}\cdot\mathsf{\Phi}_{0}\\!\cdot\delta\bm{v},$ (A.13d)
$\displaystyle\mathcal{O}(\tau_{1}\delta\bm{v})\ :$ $\displaystyle\
\tau_{1}\delta\bm{v}\cdot\mathsf{a},$ (A.13e)

where $\mathsf{\Phi}_{0}$ is defined in Eq. (4.40),
$\left[\mathsf{\Phi}_{0}\right]_{\bm{v}\tau}$ refers to its column of
derivatives with respect to $\tau$, and

$\mathsf{a}=\frac{\partial{[V_{1}(\mathsf{x})-V_{0}(\mathsf{x})]}}{\partial{\bm{v}}}+\frac{1}{2}\frac{\partial^{2}{(S_{0}^{\prime}+S_{0}^{\prime\prime})}}{\partial{\bm{v}}\partial{\tau_{0}}}-\left[\mathsf{\Phi}_{0}\right]_{\bm{v}\tau}.$
(A.14)

The exponential of this truncated Taylor series can now be integrated over
$\delta\bm{v}$ to get another effective action, now depending on $\tau_{n}$
only. This has the same $\mathcal{O}(1)$ and $\mathcal{O}(\tau_{1})$ terms as
in Eq. (A.13), and a quadratic term

$\frac{1}{2}\tau_{1}^{2}\left[\frac{1}{4}\frac{\partial^{2}{(S_{0}^{\prime}+S_{0}^{\prime\prime})}}{\partial{\tau_{0}}^{2}}+\frac{\partial^{2}{S_{1}}}{\partial{\tau}^{2}}-\mathsf{a}\cdot\mathsf{\Phi}_{0}^{-1}\\!\\!\cdot\mathsf{a}\right].$
(A.15)

The latter can be simplified by noting that

$\mathsf{\Phi}_{0}^{-1}\cdot\left[\mathsf{\Phi}_{0}\right]_{\bm{v}\tau}=\left(1,\,0,\,\ldots,\,0\right)^{\mkern
0.0mu\top}$ (A.16)

and

$\displaystyle\frac{\partial^{2}{(S_{0}^{\prime}+S_{0}^{\prime\prime})}}{\partial{\tau_{0}}^{2}}-\frac{\partial^{2}{(S_{0}^{\prime}+S_{0}^{\prime\prime})}}{\partial{\tau_{0}}\partial{\bm{v}}}\cdot\mathsf{\Phi}_{0}^{-1}\cdot\frac{\partial^{2}{(S_{0}^{\prime}+S_{0}^{\prime\prime})}}{\partial{\bm{v}}\partial{\tau_{0}}}$
$\displaystyle{}=\frac{\mathrm{d}^{2}{\left(S_{0}^{\prime}+S_{0}^{\prime\prime}+S_{1}\right)}}{\mathrm{d}{\tau_{0}}^{2}}=0.$
(A.17)

The final relation follows because $S_{0}^{\prime}+S_{0}^{\prime\prime}+S_{1}$
is equal to the GR instanton stationary action regardless of the value of
$\tau_{0}$. Therefore a full derivative of this expression with respect to
$\tau_{0}$ necessarily evaluates to zero. After this simplification, the
bracketed terms in Eqs. (A.13b) and (A.15) can be identified with $\alpha_{1}$
and $-\gamma_{1}$ respectively, leading to the final result in Eq. (4.39).

##  Appendix B Exact _k̃_ 4B for linear diabats

For a general potential, evaluating Eq. (4.42) exactly is not mathematically
justified, since the integrand is an asymptotic approximation to the exact
quantum $L_{{4\mathrm{B}}}$ that is only valid for $\tau_{1}\to 0$. Linear
diabats are a notable exception, since in this case the exact
$L_{{4\mathrm{B}}}$ is precisely of the form in Eq. (4.34). Hence for these
systems it is meaningful to evaluate the expression in Eq. (4.42) as is,
yielding

$\displaystyle
j^{\mathrm{lin}}_{0}(\tau_{0})=\sqrt{\frac{2\hbar}{\gamma_{1}}}\bigg{[}\mathrm{daw}(\lambda_{1})$
(B.1)
$\displaystyle\qquad\qquad{}-\exp\\!\left(\frac{\phi_{1}^{2}}{4\lambda_{1}^{2}}-\phi_{1}\right)\mathrm{daw}\\!\left(\lambda_{1}-\frac{\phi_{1}}{2\lambda_{1}}\right)\bigg{]},$

where
$\phi_{1}=\tfrac{\alpha_{1}}{\hbar}\mkern-1.0mu\tfrac{\tau_{1}^{*}}{\tau_{0}^{*}}\tau_{0}$,
$\lambda_{1}=\alpha_{1}/\sqrt{2\hbar\gamma_{1}}$, and

$\mathrm{daw}(x)\equiv\mathrm{e}^{-x^{2}}\\!\int_{0}^{x}\mathrm{e}^{t^{2}}\,\mathrm{d}t$
(B.2)

is the Dawson function. Substituting this and the analogously defined
$j^{\mathrm{lin}}_{1}$ into Eq. (4.45) instead of $j_{n}^{\mathrm{sc}}$ gives
the exact $\tilde{k}_{{4\mathrm{B}}}$ for linear diabats with constant
coupling $\Delta$. Alternatively we can evaluate Eq. (4.33) directly, without
splitting the $\mathcal{A}_{\tau\tau}$ domain along the diagonal. Both
approaches lead to the same expression,

$\displaystyle\tilde{k}_{{4\mathrm{B}}}^{\mathrm{lin}}(\beta)=\tilde{k}_{2}^{\mathrm{lin}}(\beta)\frac{\Delta^{\\!2}\mkern-1.0mu\sqrt{\rule{0.0pt}{8.99994pt}8\beta
m}}{\hbar\lvert\kappa_{0}-\kappa_{1}\rvert}\int_{0}^{\frac{\pi}{2}}\\!\\!\\!\mathrm{daw}\bigg{(}\mkern-1.0mu\frac{\widetilde{\beta}^{\frac{3}{2}}\sin\theta}{4}\mkern-1.0mu\bigg{)}\mathrm{d}\theta,$
(B.3)

with $\tilde{k}_{2}^{\mathrm{lin}}$ and $\widetilde{\beta}$ defined in Eqs.
(4.8) and (4.9). It can be shown that the rate calculated using
$j_{n}^{\mathrm{sc}}$ tends to the exact rate for a system of linear diabats
both as $\beta\to 0$ and $\beta\to\infty$. Furthermore, for any combination of
parameters the semiclassical approximation is never in error by more than
$2.8\%$, with the largest deviation seen at $\widetilde{\beta}\approx 3.5$. In
all instances the semiclassical approximation underestimates the quantum rate
constant.

##  Appendix C Stationary action for the scattering
Morse potential

The abbreviated action is defined as

$W(x^{\prime\prime},x^{\prime},E)=\int_{x^{\prime}}^{x^{\prime\prime}}\mathsf{p}(x,E)\cdot\mathrm{d}\mathsf{x},$
(C.1)

where $\mathsf{p}$ is the imaginary-time momentum whose magnitude is
$p(x,E)=\sqrt{2m[V(x)-E]}$. For the scattering Morse potential in Eq. (5.1b),
this integral can be evaluated analytically. One must distinguish two types of
trajectories: those going directly from $x^{\prime}$ to $x^{\prime\prime}$,
with action

$W_{\mathrm{d}}(x^{\prime\prime}\mkern-2.0mu,x^{\prime},E)=\left\lvert\frac{p^{\prime}-p^{\prime\prime}\\!}{\alpha}-\frac{\mu}{\alpha}\tan^{-1}\\!\left[\frac{\mu
p^{\prime}-\mu
p^{\prime\prime}}{\mu^{2}+p^{\prime}p^{\prime\prime}}\right]\right\rvert,$
(C.2)

and those passing through a turning point, with

$W_{\mathrm{b}}(x^{\prime\prime}\mkern-2.0mu,x^{\prime},E)=\frac{p^{\prime}+p^{\prime\prime}\\!}{\lvert\alpha\rvert}-\frac{\mu}{\lvert\alpha\rvert}\tan^{-1}\\!\left[\frac{\mu
p^{\prime}+\mu p^{\prime\prime}}{\mu^{2}-p^{\prime}p^{\prime\prime}}\right].$
(C.3)

Here $p^{\prime}\equiv p(x^{\prime},E)$, $p^{\prime\prime}\equiv
p(x^{\prime\prime},E)$, $\mu=\sqrt{2m(E-\epsilon)}$ and $\tan^{-1}(y/x)$
refers to the principal value of the argument of $x+\mathrm{i}y$. The
abbreviated action satisfies

$S(x^{\prime\prime},x^{\prime},\tau)=W(x^{\prime\prime},x^{\prime},E)+E\tau,$
(C.4)

where for a given $\tau$ the energy is such that

$\frac{\partial{W}}{\partial{E}}=-\tau.$ (C.5)

The first-order derivatives of $S$ are given by

$\frac{\partial{S}}{\partial{x^{\prime}}}=\frac{\partial{W}}{\partial{x^{\prime}}},\qquad\frac{\partial{S}}{\partial{x^{\prime\prime}}}=\frac{\partial{W}}{\partial{x^{\prime\prime}}},\qquad\frac{\partial{S}}{\partial{\tau}}=E,$
(C.6)

and the second-order derivatives are

$\displaystyle\frac{\partial^{2}{S}}{\partial{\tau}^{2}}$
$\displaystyle=-\left(\frac{\partial^{2}{W}}{\partial{E}^{2}}\right)^{\\!\\!-1},$
(C.7a)
$\displaystyle\frac{\partial^{2}{S}}{\partial{x^{\prime}}\partial{\tau}}$
$\displaystyle=-\frac{\partial^{2}{W}}{\partial{x^{\prime}}\partial{E}}\left(\frac{\partial^{2}{W}}{\partial{E}^{2}}\right)^{\\!\\!-1},$
(C.7b)
$\displaystyle\frac{\partial^{2}{S}}{\partial{x^{\prime}}\partial{x^{\prime\prime}}}$
$\displaystyle=\frac{\partial^{2}{W}}{\partial{x^{\prime}}\partial{x^{\prime\prime}}}-\frac{\partial^{2}{W}}{\partial{x^{\prime}}\partial{E}}\left(\frac{\partial^{2}{W}}{\partial{E}^{2}}\right)^{\\!\\!-1}\\!\\!\frac{\partial^{2}{W}}{\partial{E}\partial{x^{\prime\prime}}},$
(C.7c)

along with all possible variations with $x^{\prime}$ and $x^{\prime\prime}$
variously interchanged. This gives all the information necessary to calculate
$\tilde{k}_{2}^{\mathrm{sc}},\,\tilde{k}_{{4\mathrm{A}}}^{\mathrm{sc}}$ and
$\tilde{k}_{{4\mathrm{B}}}^{\mathrm{sc}}$. The only other, minor modification
is the change of integration variable in Eq. (4.45) from imaginary time to
position,

$\int_{0}^{\tau_{n}^{*}}j_{n}(\tau_{n})\,\mathrm{d}\tau_{n}\to\int_{x_{<}}^{x_{>}}\\!\\!\frac{mj_{n}(\tau_{n})}{\sqrt{2m[V_{n}(x)-E]}}\,\mathrm{d}x,$
(C.8)

where $x_{<}$ and $x_{>}$ are the smaller and the larger of $x^{\ddagger}$
(hopping point) and $x_{\mathrm{b}}$ (turning point), and

$\tau_{n}=-\frac{\partial{W(x,\,x_{<},E)}}{\partial{E}}.$ (C.9)

The integral over position gives the same result as the integral over
imaginary time, but is the simpler alternative to implement for a one-
dimensional system.

##  Appendix D Exact flux-correlation functions for the spin–boson model

In Eq. (3.14) we give the perturbation series expansion of
$\operatorname{Tr}[\mathrm{e}^{-\beta\hat{H}}\ket{0}\\!\bra{0}]$ for the
Hamiltonian in Eq. (2.2). The terms in this series are nested imaginary-time
integrals, whose integrands are precisely the multi-time flux-correlation
functions that define the rate constants $k_{2\nu}$ in the perturbation
expansion of the total rate. Weiss, in Chapter 19.1 of Ref. 86, gives the
analytic form of these correlation functions for the spin–boson model
described in Sec. V.2, namely

$\displaystyle\frac{c_{2\nu}(u_{1\ldots
2\nu})}{Z_{0}(\beta)}=\Delta^{2\nu}\mathrm{e}^{-\phi_{2\nu}(u_{1\ldots
2\nu})/\hbar}$ (D.1a)
$\displaystyle\phi_{2\nu}={-\epsilon}\sum_{j=1}^{\nu}s_{j}+\sum_{j=2}^{2\nu}\sum_{i=1}^{j-1}(-1)^{i+j}Q(u_{j}-u_{i}),$
(D.1b)

where

$\displaystyle
Q(u)=\int_{0}^{\infty}\frac{4J(\omega)}{\pi\omega^{2}}\left[\frac{1-\cosh(\omega
u)}{\tanh\big{(}\frac{\beta\hbar\omega}{2}\big{)}}+\sinh(\omega
u)\right]\mathrm{d}\omega,$ (D.2)

and $s_{j}=u_{2j}-u_{2j-1}$. Obtaining the time-correlation function that
corresponds to a particular component of the $k_{2\nu}$ rate constants amounts
to finding the appropriate variable transformation from $u_{1\ldots 2\nu}$ to
$\\{z,z_{0\ldots 2\nu-2}\\}$. Note that because the exponent in Eq. (D.1)
depends only on the differences of $u_{j}$, one of the original variables can
be eliminated prior to the transformation. For $\nu=1$ this procedure yields
the familiar expression used to define the GR rate,[93, 25, 89]

$\phi_{2}=Q(z)-\epsilon z.$ (D.3)

For $\nu=2$ we get

$\displaystyle\phi_{{4\mathrm{A}}}$
$\displaystyle=Q\Big{(}\tfrac{\beta\hbar-z}{2}+z_{0}\Big{)}+Q\Big{(}\tfrac{z}{2}+z_{1}\Big{)}$
(D.4a)
$\displaystyle+Q\Big{(}\tfrac{\beta\hbar-z}{2}-z_{0}\Big{)}+Q\Big{(}\tfrac{z}{2}-z_{1}\Big{)}$
$\displaystyle-Q\Big{(}\tfrac{\beta\hbar}{2}-z_{0}-z_{1}\Big{)}-Q\Big{(}\tfrac{\beta\hbar}{2}-z_{0}+z_{1}\Big{)}-\epsilon
z,$ $\displaystyle\phi_{{4\mathrm{B}}}$ $\displaystyle=Q\left(\beta\hbar-z-
z_{0}\right)+Q\left(z_{1}\right)\vphantom{\Big{(}\Big{)}}$ (D.4b)
$\displaystyle+Q\left(z_{0}\right)+Q\left(z-z_{1}\right)\vphantom{\Big{(}\Big{)}}$
$\displaystyle-Q\left(z+z_{0}-z_{1}\right)-Q\left(z_{0}+z_{1}\right)-\epsilon
z.\vphantom{\Big{(}\Big{)}}$

Continuing in the same fashion, one can generate further multi-time
correlation functions, although the expressions quickly become cumbersome, as
$\phi_{2\nu}+\epsilon z$ contains a total of $\nu(2\nu-1)$ terms.

##  Appendix E Nonadiabatic rates in the high-T limit

As $\beta\to 0$, the $\mathcal{O}(\Delta^{4})$ correction to the GR rate
becomes dominated by the $\tilde{k}_{{4\mathrm{A}}}$ contribution, whose high-
temperature form is given in Eq. (4.13). Combining this with the high-
temperature limit of the GR rate

$k_{2}^{\mathrm{ht}}(\beta)=\sqrt{\frac{2\pi
m}{\beta\hbar^{2}}}\frac{\Delta^{2}}{\hbar\lvert\kappa_{0}-\kappa_{1}\rvert}\frac{Z^{\ddagger}}{Z_{\mathrm{r},0}}\,\mathrm{e}^{-\beta
V^{\ddagger}},$ (E.1)

gives

$\displaystyle k(\beta)$ $\displaystyle\sim\frac{Z^{\ddagger}\mkern
1.0mu\mathrm{e}^{-\beta
V^{\ddagger}}}{Z_{\mathrm{r},0}}\Bigg{[}\sqrt{\frac{2\pi
m}{\beta\hbar^{2}}}\frac{\Delta^{2}}{\hbar\lvert\kappa_{0}-\kappa_{1}\rvert}-{}$
(E.2) $\displaystyle\quad\frac{\pi
m\Delta^{4}}{\hbar^{3}(\kappa_{0}-\kappa_{1})^{2}}\\!\left\\{\ln\\!\left[\frac{128m(\kappa_{0}-\kappa_{1})^{2}}{\beta^{3}\hbar^{2}\kappa_{0}^{2}\kappa_{1}^{2}}\right]-\gamma\right\\}\Bigg{]}.$

We know that this expansion is correct for a linear model, and we have
confirmed that it gives accurate rates in nonlinear and multidimensional
systems, except at high friction, where the steepest-descent approximation to
the integral over $c_{{4\mathrm{A}}}(t_{0},t_{1},z)$ breaks down. We
especially want to emphasize this, since the logarithmic dependence on $\beta$
at $\mathcal{O}(\Delta^{4})$ seems unusual when compared, for example, to the
Zusman equation

$k_{\mathrm{Zus}}=\frac{k_{2}^{\mathrm{ht}}\mkern
1.0muk_{\mathrm{ad}{,0}}}{k_{2}^{\mathrm{ht}}+k_{\mathrm{ad}{,0}}},$ (E.3)

which gives the classical limit of the spin–boson rate at small $\Delta$ and
large friction. Here $k_{\mathrm{ad}{,0}}$ is the classical rate on the lower
(cusped) adiabatic surface666 The expression used by us is derived from
transition-state theory and is not strictly valid at high friction. In that
regime it would be more appropriate to use Kramers theory (see Sec. 14.4.2 of
Ref. 46), which simply re-scales the TST rate by a temperature-independent
factor. Hence using this alternative rate expression would have no effect on
our line of argument. in the limit as $\Delta\to 0$,

$k_{\mathrm{ad}{,\Delta}}=\frac{1}{2\pi\beta\hbar}\frac{Z^{\ddagger}\mkern
1.0mu\mathrm{e}^{-\beta
V^{\ddagger}}}{Z_{\mathrm{r},0}}\exp\\!\bigg{[}2\beta\Delta\frac{\sqrt{-\kappa_{0}\kappa_{1}}}{\kappa_{0}-\kappa_{1}}\\!\bigg{]},$
(E.4)

and hence

$\displaystyle k_{\mathrm{Zus}}(\beta)$ $\displaystyle\sim
k_{2}^{\mathrm{ht}}(\beta)-\frac{Z^{\ddagger}\mkern 1.0mu\mathrm{e}^{-\beta
V^{\ddagger}}\\!}{Z_{\mathrm{r},0}}\frac{4\pi^{2}m\Delta^{4}}{\hbar^{3}(\kappa_{0}-\kappa_{1})^{2}},$
(E.5)

which is clearly different from Eq. (E.2). Ultimately this should not come as
a surprise, since $k_{\mathrm{Zus}}$ is derived assuming high friction,[40,
56] and is therefore valid in a different limit to our semiclassical rate
theory.

The interpolation formula of Ref. 55 also fails to recover Eq. (E.2) as
$\beta\to 0$. The key change here compared to Eq. (E.3) is that the
$k_{\mathrm{ad}{,0}}$ in the denominator is replaced with
$k_{\mathrm{ad}{,\Delta}}$. This results in a modification to the
$\mathcal{O}(\Delta^{4})$ term in Eq. (E.5) that is of order $\beta\Delta$ and
is therefore negligible in this limit. Additionally, the interpolation formula
contains an unphysical term of order $\Delta^{3}$, although this term is
proportional to $k_{2}^{\mathrm{ht}}\times\beta\Delta$ and is therefore also
negligible in this limit. In short, the Zusman equation and the interpolation
formula agree to leading order in $\beta\Delta$ but differ from the exact
result for the one-dimensional linear system.

The temperature dependence of our semiclassical rate constants can also be
compared to that of thermal rates derived from the cumulative reaction
probability

$P_{\mathrm{H}}(v)=\frac{2P_{\mathrm{LZ}}(v)}{1+P_{\mathrm{LZ}}(v)},$ (E.6)

given in the form suggested by Holstein.[88] Here $v$ is the velocity of the
system along the reaction coordinate in the vicinity of the hopping point, and

$P_{\mathrm{LZ}}(v)=1-\exp\left[-\frac{2\pi\Delta^{2}}{\hbar
v\lvert\kappa_{0}-\kappa_{1}\rvert}\right]$ (E.7)

is the Landau–Zener transmission coefficient.[47, 48] The latter approximates
the nonadiabatic transmission probability for a single passage through the
hopping point in a linear system, and is valid at all coupling strengths
$\Delta$, but only for sufficiently large $v$. $P_{\mathrm{H}}(v)$ has been
proposed as the starting point for uniform approximations to classical
nonadiabatic rates, via

$kZ_{\mathrm{r},0}=\frac{\mathrm{e}^{-\beta
V^{\ddagger}}}{2\pi\hbar}\int_{0}^{\infty}mvP_{\mathrm{H}}(v)\mkern
1.0mu\mathrm{e}^{-\beta mv^{2}/2}\,\mathrm{d}v$ (E.8)

or similar expressions,[49, 41, 46, 89, 33] where for simplicity we have taken
the system to be one-dimensional. This is in part motivated by the observation
that replacing $P_{\mathrm{H}}(v)$ in Eq. (E.8) with its $\Delta\to 0$ limit
correctly recovers the classical GR rate. However the next term in the weak-
coupling expansion of $P_{\mathrm{H}}(v)$, which should give the
$\mathcal{O}(\Delta^{4})$ behaviour, causes the integral to diverge. This
happens because $P_{\mathrm{LZ}}(v)$ is only valid at high velocities. At
$\mathcal{O}(\Delta^{2})$, the (spurious) contribution from $v\to 0$ as
$\beta\to 0$ is small enough that Eq. (E.8) can be integrated all the way from
0. At $\mathcal{O}(\Delta^{4})$ this is no longer the case, and a non-zero
bound has to be introduced. Hence the correct $\mathcal{O}(\Delta^{4})$
behaviour cannot be obtained using $P_{\mathrm{H}}(v)$ on its own—for this we
would also require the transmission coefficient at small velocities, which is
not available in closed form.[50] Nevertheless, Eq. (E.8) gives us qualitative
insight. First, we rewrite it as

$\displaystyle k$ $\displaystyle=k_{2}^{\mathrm{ht}}-\frac{\mathrm{e}^{-\beta
V^{\ddagger}}}{Z_{\mathrm{r},0}}\frac{3\pi
m\Delta^{4}}{\hbar^{3}(\kappa_{0}-\kappa_{1})^{2}}\int_{w_{0}}^{\infty}\frac{\mathrm{e}^{-w}}{w}\,\mathrm{d}w+\mathcal{O}(\Delta^{6})$
$\displaystyle\sim k_{2}^{\mathrm{ht}}-\frac{\mathrm{e}^{-\beta
V^{\ddagger}}}{Z_{\mathrm{r},0}}\frac{\pi
m\Delta^{4}}{\hbar^{3}(\kappa_{0}-\kappa_{1})^{2}}\left\\{\ln\\!\left[\frac{1}{w_{0}^{3}}\right]-3\gamma\right\\},$
(E.9)

where $w=\beta mv^{2}/2$ and we have introduced a lower bound $0<w_{0}\ll 1$.
Next we note that the exact transmission coefficient for any linear system can
be expressed in terms of just two parameters.[114] One can then use
dimensional arguments to show that $w_{0}$ has to be proportional to the
reduced temperature $\widetilde{\beta}$ that we defined previously in Eq.
(4.9). It is no coincidence that the same parameter also appears in the exact
$\tilde{k}_{2}$, $\tilde{k}_{{4\mathrm{A}}}$ and $\tilde{k}_{{4\mathrm{B}}}$
for the linear system [Eqs. (4.8), (4.12) and (B.3)].

Landau–Zener theory cannot give us the proportionality constant in
$w_{0}=\widetilde{\beta}/\alpha$, as this requires knowing the small-$v$
behaviour of $P(v)$, which falls outside the range of validity of Eq. (E.7).
By the same token, this approach fails to account for transitions at energies
below $w_{0}/\beta+V^{\ddagger}$. However the latter are expected to be small
for $\beta\to 0$, and indeed Eq. (E.9) reproduces the leading-order behaviour
of $k_{4}$ in this limit, correctly predicting the
$\ln(1/\widetilde{\beta}^{3})$ term inside the curly brackets. The asymptotic
behaviour of $k_{4}$ is therefore determined up to an additive constant, and
setting $\alpha=4\,\mathrm{e}^{2\gamma/3}$ happens to recover Eq. (E.2)
exactly, bringing LZ into agreement with our theory. This suggests that
expressions like Eqs. (E.6) and (E.8) can inform future work on rigorous
uniform nonadiabatic rate theories, valid across the entire spectrum of
diabatic coupling strengths.

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  * Note [1] For clarity we only show the lower triangle.
  * Note [2] This is also the more consistent treatment, since at temperatures low enough that $d_{01}<0$, the corresponding contribution to the steepest-descent integral is small and no longer dominates the other quartic terms ($t^{4}$ and $t_{n}^{4}$), which are neglected. There is therefore no reason to give the $d_{01}$ contribution privileged treatment and include it into the final result.
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  * Note [3] In practice evaluating the exact $d_{01}$ would involve calculating terms containing third- and fourth-order derivatives of the potential energy, which would be impractical. However we expect that in most cases a very accurate approximation to $d_{01}$ would result even if such terms were omitted.
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