Semiclassical analysis of the quantum instanton approximation

We explore the relation between the quantum and semiclassical instanton approximations for the reaction rate constant. From the quantum instanton expression, we analyze the contributions to the rate constant in terms of minimum-action paths and find that two such paths dominate the expression. In symmetric systems, these two paths join together to describe the semiclassical instanton periodic orbit. However, for asymmetric barriers, one of the two paths takes an unphysically low energy and dominates the expression, leading to order-of-magnitude errors in the rate predictions. Nevertheless, semiclassical instanton theory remains accurate. We conclude that semiclassical instanton theory can only be obtained directly from the semiclassical limit of the quantum instanton for symmetric systems. We suggest a modification of the quantum instanton approach which avoids sampling this spurious path and thus has a stronger connection to semiclassical instanton theory, giving numerically accurate predictions even for very asymmetric systems in the low temperature limit.


I. INTRODUCTION
The inclusion of nuclear quantum effects in molecular dynamics calculations is a challenging task.One of the main difficulties in calculating exact quantum dynamics is the need to solve the time-dependent Schrödinger equation with many degrees of freedom.For most real molecular systems the dynamics are (currently) computationally impossible to solve exactly, due to the exponentiallyincreasing size of the basis sets required for these calculations.For this reason, approximate quantum dynamics methods based on path integrals are increasingly favoured.
A rigorous formulation of reaction rate theory is provided by the flux-side or flux-flux time correlation function formalism. 1These correlation functions can be evaluated with approximate quantum-dynamics methods, including linearized semiclassical initial value representation (LSC-IVR), 2 centroid molecular dynamics, 3,4 ring-polymer molecular dynamics (RPMD), [5][6][7] and the recently-developed Matsubara dynamics 8,9 and its approximations. 10,11The approximation made by these methods is to neglect quantum coherences, which is not expected to be a cause of significant error for complex systems in thermodynamic equilibrium. 12Thus, solving the system exactly to obtain the long-time dynamics is in many cases unnecessary as the relevant processes can be approximated to a good accuracy at short times.In fact, if the problem is well defined, no time propagation at all is necessary and one can turn to so-called quantum transition state theories (QTSTs), where the expressions involving time correlation functions are approximated using only their properties at zero time. 13,14emiclassical instanton (SCI) theory provides one of a) Electronic mail: christophe.vaillant@epfl.chb) Electronic mail: jiri.vanicek@epfl.ch c Electronic mail: jeremy.richardson@phys.chem.ethz.ch the simplest formulations of a QTST for multidimensional tunneling. 15,16It can be derived rigorously by taking asymptotic ( → 0) approximations to the fluxflux correlation function 17 and is known in a number of formulations, 18 including one obtained from the "ImF" approximation, 19 which can all be shown to be equivalent. 20SCI theory is defined in terms of the dominant tunneling pathway located on the full-dimensional potential-energy surface, and employs a harmonic approximation for fluctuations around this pathway.The ring-polymer instanton approach [21][22][23] provides a computationally efficient algorithm for applying the theory in practice, and can be automated for application to complex systems. 24though the SCI method is very powerful, the harmonic expansion about a single dominant tunneling path can introduce errors, especially for low-frequency anharmonic systems.An alternative method, known as the quantum instanton (QI), 25 was developed as an attempt to correct the quantitative deficiencies of the semiclassical approximation.This method goes beyond the harmonic approximation used in SCI and samples paths using efficient path-integral Monte Carlo or molecular dynamics methods. 261][32][33] However, for large asymmetries (meaning the reaction is highly exothermic or endothermic) the rate constant predicted by the QI method has been observed to give large errors, whereas SCI remains well-behaved. 18,34ere are also a number of other path-integral-based QTSTs developed to go beyond SCI.Unlike in classical mechanics, 35 it is not possible to derive them directly from the flux-side correlation function, as this function tends to zero at zero time if used in its standard form.Instead, QTSTs have been proposed based on a connection to semiclassical instanton theory, including a free-energy version of instanton theory 36 and ring-polymer transition-state theory, which is itself related to the RPMD rate theory. 22More recently, ring-polymer transition-state theory has been rederived from a generalized flux-side correlation function, which yields a good approximation to the rate at zero time. 37,38This provides an extension of the centroid-based QTST of Voth, Chandler and Miller, 39 which does not dominantly sample the instanton and thus fails for asymmetric barriers. 40n summary, it appears that the QTSTs which work best are those with the strongest connection to the SCI.In this article, we examine the relation between the QI and SCI theories, in order to investigate the very plausible conjecture that the SCI is a semiclassical/steepestdescent approximation to the QI; put another way, that the QI is an improvement on the SCI method with more accurate sampling of the paths.Surprisingly, we find that the two methods are not directly connected to each other except in the case of symmetric systems.We perform a semiclassical analysis of the quantities used in the QI method and find that the dominant paths that contribute are not the same as those which define the instanton periodic orbit.This leads to the observed error of QI for systems with large asymmetries and low temperatures.We then suggest an approach for modifying the QI expression to enforce sampling of the instanton orbit, which is seen to greatly improve the results.
The paper is organized as follows.We first give a derivation of the QI method from first principles in Sec.II, followed by a short investigation of the breakdown of the predicted rate for asymmetric systems.Motivated by this breakdown, we analyze the contributions from semiclassical paths in Sec.III, and we show that the SCI expression can only be recovered if the spurious paths are removed.We suggest adding a projection operator to solve this problem and describe an improved approach, which we call the projected quantum instanton method, in Sec.IV.We follow this with a discussion of the numerical results of the various methods in Sec.V, including the semiclassical approximations, which justifies the analysis in terms of semiclassical paths, before concluding in Sec.VI.

II. QUANTUM INSTANTON APPROXIMATION
In the following section we re-examine the derivation of the QI method for calculating reaction rates.The derivation is based on a saddle-point approximation to the time integral of the flux-flux correlation function, and we show that the method breaks down for asymmetric barriers at low temperatures.The goal is to have a self-contained discussion of the QI method before undertaking a more detailed semiclassical analysis.There are two versions of the QI method: the original, which we shall continue to call QI, and a variant, which we call the second-order cumulant expansion (2OCE).
27]30 Our conclusions remain valid for these multidimensional cases.

A. Derivation
We begin by rederiving the QI approximation from the Miller-Schwartz-Tromp expression for the reaction rate constant Here, Q r is the reactant partition function and C ff is the symmetrized flux-flux time correlation function, generalized to two dividing surfaces.The Hamiltonian for a particle in a one-dimensional potential, V (x), is Ĥ = p2 /2m + V (x) and the flux operators are given by where m is the mass and δj = δ(x − x j ) indicates a Dirac delta function centred at the j-th dividing surface located at x = x j .Because both Ĥ and Fj are Hermitian as well as real operators, the correlation function [Eq.( 2)] is a real and even function of t.
The main idea used in deriving the QI expression is to approximate the time integral in Eq. (1) using a steepestdescent (saddle-point) approximation taken around t = 0, where the dot implies differentiation with respect to time.In order to use this approximation, it is necessary that ḟ (0) = 0, which is guaranteed for even functions of t, and also that f (0) < 0. This steepest-descent approximation makes it possible to express the full reaction rate in terms of expressions defined at t = 0, which can then be evaluated exactly with imaginary-time path-integral methods.Although several slightly different forms of the QI approximation exist, the original QI expression [see Eq. ( 14) below] has been derived with one of two approaches: one which uses the energy-integral formulation of the reaction rate (with complicated transformations and two separate steepest-descent integrations) 25 and another which involves multiplying and dividing by the delta-delta correlation function, as the first step. 30,41Both derivations use the same prescription for the dividing surfaces x 1 and x 2 . 25,30,41valuating the trace in Eq. ( 2) in the position representation, we find the well-known expression 41 where "c.c." denotes the complex conjugate, and The delta-delta correlation function ( 6) is then simply For brevity, we will drop the explicit dependence of the time correlation functions on the dividing surfaces in our notation.To apply the steepest-descent approximation (4) to Eq. ( 1), we set which we note obeys the stationarity condition at time t = 0 because Ċff (0) = 0. We are still free to choose the locations of the dividing surfaces, (x 1 , x 2 ), and we should do this in such a way to ensure that C ff (t) has a maximum at t = 0.Many choices for the dividing surfaces have been suggested, but some choices lead to a local minimum at zero time for C ff (t), rather than a maximum. 41The original and most common choice is that the dividing surfaces obey which typically leads to C ff (t) being a maximum at t = 0.In our experience, there is always a solution to these equations with x 1 = x 2 , known as merged dividing surfaces, which describe either a minimum or first-order saddle point of ρ(x 1 , x 2 , 0).In the former case, other solutions exist as saddle points with x 1 = x 2 , leading to what is known as split dividing surfaces.Adopting one of these choices of dividing surfaces such that x 1 and x 2 satisfy Eq. ( 10), we note that C ff (0) = C sp ff (0), where and thus suggest the approximation C ff (t) ≈ C sp ff (t) for t = 0.
We are now faced with a choice for performing the steepest-descent integration in time: we could either perform this approximate time integral on C dd (t) and then take the spatial derivatives, or we could perform the integral directly on C ff (t).In the former approach, the evaluation of the momentum operators will effectively be done before the steepest-descent approximation, whereas the latter will effectively include the momenta in the steepest-descent approximation.Choosing first to perform the integral over time before evaluating the derivatives with respect to the positions of surfaces, the new function for the steepest-descent approximation and its second time derivative (in terms of the energy variance ∆H 2 dd ) are where The final QI rate is then defined as where we have assumed that the spatial derivatives of ∆H dd can be neglected as they are much smaller than the derivatives of C dd (0).Provided that the ratio C ff (t)/C dd (t) is slowly-varying in time, Eq. ( 14) can be generalized for dividing surfaces that do not obey the condition in Eq. ( 10), such that 41 where we have slightly abused the notation and kept the QI subscript for clarity.An alternative approach is derived by employing a saddle-point approximation directly to the time integral of C ff (t).In order to use this approximation, there is no particular requirement on the choice of dividing surfaces, provided C ff (t) is a maximum at t = 0 for that choice.3][44] The steepest-descent function is then where which defines the energy variance ∆H 2 ff .The secondorder cumulant expansion (2OCE) expression that results from this procedure is very similar to Eq. ( 15), namely We will show that both of these quantum instanton approximations have similar properties and break down for asymmetric systems for similar reasons.

B. Behaviour of quantum instanton for asymmetric systems
To test the approximations, it is useful to apply the QI and 2OCE methods to a simple 1D Eckart barrier, where, choosing the same parameters as in Ref. 25, V 0 = 0.425 eV, a = 1.36 a −1 0 , the mass is 1060 m e (a 0 is the Bohr radius and m e denotes the electron mass).α is the dimensionless asymmetry parameter controlling the degree of exothermicity between the reactants and products such that the barrier is symmetric for α = 1.We investigated a variety of different temperatures and asymmetries, and found that the QI method shows pathological behaviour for low temperatures and large asymmetries.We will demonstrate this behaviour throughout the paper, unless otherwise stated, with a particular choice of T = 100 K and α = 1.425, for which the QI prediction is an order of magnitude too large compared to the exact rate.
It would be reasonable to think that the QI method could be improved if a more suitable choice of dividing surfaces were found.We will therefore consider a range of choices for the dividing surfaces, listed in Table I, with the above choice of parameters, showing examples of both split saddle points and a merged minimum of C dd (0) but only a merged saddle point of C ff (0). 45The positions of the dividing surfaces are also indicated in Fig. 1, showing the behavior of the propagator and the flux-flux correlation function as functions of the dividing surface locations.It is possible to choose dividing surfaces that are not at special points, 41 but we will show that it is not generally possible to find dividing surfaces that fix all the problems of the QI and 2OCE methods without needing to evaluate quantities at t = 0.
To summarize the previous section, the QI and 2OCE methods essentially approximate the time dependence of the relevant time correlation functions as gaussians.The functions C dd (t) and C ff (t) are shown in Fig. 2, where the surfaces are chosen to be the split surfaces of ρ if they exist, or the merged surfaces of ρ otherwise.For increasing asymmetry α it is clear that the functions become less and less gaussian (a similar situation to that discussed in Refs.46 and 47 for electron-transfer rates).This will  I.lead to a significant error in the prediction of the rate constant.In fact, we find that none of the choices considered for the dividing surfaces make much difference in the QI prediction for the rate, as the QI prediction still deviates significantly from the exact value.In order to gain insight into this behaviour, in the next section we analyze the correlation functions in terms of semiclassical paths and discover the cause of the breakdown of the approximation.I.In this way, the same surfaces are used for both C ff and C dd , although similar behaviour is also seen for other choices of dividing surfaces.

III. SEMICLASSICAL ANALYSIS
In this section, we will examine the various terms in the QI and 2OCE approximations using a semiclassical analysis, which gives the → 0 limit of a quantummechanical expression in terms of minimum-action paths.We will identify the dominant contributions that cause the observed deviation from the semiclassical instanton expression and determine what improvements would be necessary to fix these problems.

A. Semiclassical contributions to quantum instanton quantities
All terms in the QI and 2OCE rate expressions are defined at t = 0 and thus all of our analysis can be performed using the imaginary-time quantum propagator, ρ(x 1 , x 2 , 0), and its derivatives.Its semiclassical limit is the imaginary-time van Vleck propagator, which involves a sum over all minimum-action paths which travel from x 1 to x 2 in imaginary time β /2.These are clearly the paths which dominate in a path-integral Monte Carlo evaluation of the quantities, in which paths are weighted by the exponential of minus the action.The minimumaction paths follow imaginary-time classical trajectories, which are equivalent to Newtonian trajectories on the upside-down potential-energy surface. 48In fact, there are two such paths which contribute the most, one which bounces to the left, and one to the right.Note that no minimum-action paths travel directly from one dividing surface to the other 49 and that we neglect paths which bounce more than once as these have larger actions and thus make exponentially smaller contributions.The semiclassical limit of the quantum propagator is thus given by with K and K r being the semiclassical contributions from the left and right paths, given by where γ ∈ { , r}.These propagators are functions of x 1 , x 2 and τ γ , and in this case both have imaginary-time length τ γ = β /2.The paths are written in terms of the classical Euclidean action (the action in imaginary time) where X γ (τ ) are the imaginary-time-dependent left and right paths from x 1 to x 2 via a "bounce".Each action is related to an eikonal W γ (also known as the reduced action) by the Legendre transform 50 and where x is integrated along the relevant path X γ .The energy, E γ , is chosen to solve ∂ Wγ ∂Eγ = −τ γ and the path bounces at the point where The two minimum-action paths are shown for the A, B, and D dividing surfaces in Fig. 3.It is important to notice that the two paths do not join together to form the semiclassical instanton solution, which is an imaginarytime periodic orbit with constant energy.Because the reaction is exothermic, for all choices of dividing surfaces the left-bouncing path (green) follows half of the instanton periodic orbit but the right-bouncing path (red) is spurious and has a much lower energy, E r .This remains true even when using dividing surfaces placed at the turning points of the instanton trajectory (D), as advocated in the appendix of Ref. 25.In all these cases, the second half of the instanton periodic orbit cut at the location of the dividing surfaces is a first-order saddle point of the action, as it passes through a conjugate point, 51 which is why the minimum-action path must follow a different classical trajectory.In fact, we found that for this system below about 142 K there is no way to split the instanton into two trajectories of equal imaginary time without encountering a conjugate point.The picture which we show contradicts the ideas explained in the appendix of the original QI paper 25 and, as we will show, the spurious path is the crux of the observed error of the QI method in asymmetric systems.
We use this semiclassical analysis in terms of minimum-action paths to find the dominant contributions to the various quantities used in the QI and 2OCE approximations.For instance, the semiclassical limit of C dd (0) is simply The semiclassical limits of the spatial first derivatives and mixed second derivative of the imaginary-time propagator are where p γ j = + 2m[V (x j ) − E γ ] for j ∈ {1, 2}.Note that we take the positive root such that p γ j is always a positive scalar and is therefore the magnitude of the momentum (without the direction).It is also clear from Fig. 3 that, contrary to what was previously thought, 25,30 choosing the dividing surfaces according to choice A in Table I does not correspond to placing the dividing surfaces at the turning points of the semiclassical instanton orbit.
Using Eqs. ( 6), ( 20) and ( 26), we find the semiclassical limit of C ff (0) to be It is important to note that the terms proportional to K 2 and K 2 r , which appear in C dd (0), completely cancel out in the semiclassical limit of C ff (0) as a consequence of the flux operators.
Using the Cauchy-Riemann equations, the real time derivative can be written ∂ ∂t = i ∂ ∂τ +i ∂ ∂τr .The real-time derivatives (denoted by dots above quantities, as above) of C dd can then also be evaluated from the imaginarytime derivatives of the semiclassical propagator, with the first and second derivatives given by ρ  I, respectively.The choice C gives qualitatively the same picture as B. The actions of each semiclassical path are also indicated.In (c), the green line follows one half of the instanton periodic orbit.All quantities are given in atomic units.gives the energy variance of C dd as

and ρ
A similar analysis gives ∆H ff in terms of the semiclassical quantities: The final expressions for the semiclassical limits of the QI and 2OCE rates are then simply obtained by inserting either Eqs.( 27) and (30) into Eq.( 15) for the QI result, or Eqs. ( 27) and (31) into Eq.( 18) for the 2OCE result.Note that for less strongly asymmetric systems or at higher temperatures (above 142 K in this case), it is possible to find a new definition for the dividing surface for which the two minimum-action paths have the same energy and thus describe the instanton orbit, as shown in Fig. 4.However, despite the fact that these two paths together describe the correct periodic orbit, we find that the QI approximation nonetheless overpredicts the rate by many orders of magnitude.This is because the leftbouncing path has a much smaller action than the rightbouncing path.As a result, the semiclassical analysis of C ff (0) given in Eq. ( 27) is no longer a good estimate of the quantum-mechanical value.The breakdown of the semiclassical analysis is due to the cancellation of the term proportional to K 2 being only valid to first order in and cannot be neglected when compared with the much smaller K K r term.This problem is avoided by choosing dividing surfaces according to the standard prescriptions in Table I, for which K and K r are of a similar order of magnitude and the semiclassical analysis is valid.
For a completely symmetric system at any temperature it is always possible to split the instanton in the middle, thereby generating two minimum-action paths of equal energy with K = K r and therefore join together to form the correct periodic orbit.In the following, we shall use this fact to explain the success of the QI method for symmetric systems.

B. Connection to the semiclassical instanton theory
The semiclassical limits to the QI and 2OCE approximations derived above show that the predicted rates do not just depend on the instanton path, but instead have contributions from the spurious path, X r .However, the SCI result for the rate depends only on the instanton path. 18The implication is that, contrary to the original conjecture, the SCI method is not a simple semiclassical approximation to the QI or 2OCE methods.In fact, it can be seen from both the above expressions and Fig. 3 that the contributions from the unphysical right-hand path will dominate in many situations and lead to erroneous rates.The only exception, when using the standard dividing surfaces satisfying Eq. ( 10), is for perfectly symmetric barriers where the two paths combine into the instanton periodic orbit.
These problems would go away if ρ were dominated by the semiclassical paths which together make up the instanton periodic orbit.This is of course already the case for a symmetric system.Here we will show that if this were generally the case then the SCI rate formula would be recovered as the asymptotic limit of the QI expression.
If the two paths together describe a periodic orbit, then we can use E = E r and hence p 1 = p r 1 = p 1 and p 2 = p r 2 = p 2 .The flux-flux correlation function [Eq.( 27)] thus simplifies to and In order to study the QI approximation based on ∆H dd , we define dividing surfaces which obey Eq. (10) in the semiclassical limit [see also Eq. ( 26)], which implies that K = K r .The semiclassical limit of the energy variance ∆H dd [Eq.(30)] becomes identical to that of ∆H ff given in Eq. (33).
Eqs. ( 32) and ( 33) can thus be used to obtain the semiclassical limit of the QI and 2OCE approximations.Using the known expressions (for the 1D case) 50 and Eq. ( 32) becomes Using 17 where E is the energy of the instanton orbit, we obtain which matches Miller's original SCI expression 15 in the one-dimensional case, bearing in mind that the total action of the instanton periodic orbit is given by S + S r .This equation is equal to the ImF instanton expression, 19 which can be written equivalently in a number of equivalent ways 18,20 including the ring-polymer formulation. 22hus, if the right-hand and left-hand paths combine to form the instanton and the dividing surfaces are chosen to satisfy Eq. ( 10), the QI and 2OCE expressions reduce to the SCI rate in the semiclassical limit.In general, however, the two paths only join exactly into the instanton periodic orbit for a symmetric system, hence the QI method is a particularly accurate method for symmetric barriers.For asymmetric barriers, the QI rate cannot be directly connected to the SCI expression.

IV. PROJECTED QI
In this section, we suggest a modification of the quantum instanton approach to avoid sampling the spurious paths, which we will call the Projected Quantum Instanton (PQI) method.This defines a novel method in the spirit of the quantum instanton but which should give accurate rate predictions even for strongly asymmetric systems in the low-temperature limit.
In order to fix the QI method for asymmetric systems, it will be necessary to require that the two minimumaction paths have matching energies and thus combine into the instanton periodic orbit.For systems with enough asymmetry, it was impossible to choose dividing surfaces such that both minimum-action paths of imaginary-time β /2 had the same energy.Therefore, we will need to relax the requirement that the left-and right-bouncing paths have equal imaginary-time lengths and will therefore also need a projection scheme which can categorize any general path into left and right sets.This can be achieved by defining the projected propagator where the projection operators are written in terms of the Heaviside step function, θ, as Paths projected in this way can thus be categorized depending on whether their central point is to the right or left of the point x 0 .The location of x 0 is somewhat arbitrary for the following arguments as long as it appears between the instanton turning points.This way, Û (−iτ ) will be dominated by the left-bouncing minimum-action path and Ûr (−iτ r ) by the right-bouncing minimumaction path.The projection operator we have used is the simplest choice to implement in our simulations.However, we note that there are many other possible definitions we could have used which also pick out the correct semiclassical pathways.Note that the multidimensional extension of the approach follows directly by projecting along a reaction coordinate.We will neglect contributions from pairs of paths which either both bounce to the left or to the right, as these give zero contribution in the semiclassical analysis of C ff (0) [see Eq. ( 27)]. 17,18Let us therefore propose that the rate can be computed as where which is a general (non-symmetric and complex) correlation function form, with τ + τ r = β , allowing the imaginary time to be different for the two paths.This extra flexibility allows us to choose an appropriate dividing surface for which both the left-and right-bouncing paths can be minimum-action paths and also join together to describe the instanton.This idea follows a similar approach to that used in a first-principles derivation of the SCI method. 17,18It is absolutely necessary to project onto left and right paths when using a nonsymmetric split of the imaginary time to avoid finding the spurious minimum-action paths of a right-bouncing path in time τ and vice versa.Note that the factor of 2 in Eq. ( 43) accounts for the alternative ordering of the projection operators, which integrates to the same result.The final expression for the PQI rate is the 2nd-order cumulant expansion of where the projected matrix elements are given by That is, the PQI expression is defined identically to the 2OCE approximation except that C ff (t) is replaced by C P ff (t) throughout.By following a similar approach as in the previous section we obtain the semiclassical limit of C P ff (0) equal to that of Eq. ( 32), and of ∆H 2 ff given by Eq. (33).The only difference with the previous analysis is that in PQI the minimum-action paths really do describe the instanton periodic orbit and no extra assumptions need be made.One can therefore show that the semiclassical limit of the rate expression reduces to SCI using the results of Sec.III B. This result is actually unsurprising, as the derivation of SCI follows similar lines of reasoning in Ref. 18.
In principle one can use any choice of dividing surfaces for which the two halves of the instanton periodic orbit are both minimum-action paths.We chose the location of the barrier maximum for both the dividing surfaces as well as x 0 , and found that this choice obeyed the rule in each case tested.One should then optimize τ (keeping τ + τ r = β fixed) until ĊP ff (0) = 0.However, the value of τ obtained directly from SCI was found to be an excellent approximation to this optimal value.
We note that the steps used to derive the PQI method share some similarity to Wolynes' nonadiabatic quantumtransition theory, 52 in which the projections are performed automatically by the two electronic states.The idea for deriving new rate theories by ensuring that the instanton is the dominant path contributing to the rate has also been used in previous work on QTST 36,37,53 and most recently in Ref. 54 for nonadiabatic rates.
This new method is also applicable to multidimensional problems and can be efficiently computed using path-integral Monte Carlo approach with a simple extension to the standard methodology.The central bead plays the role of the x variable in Eq. ( 40) and, for example, only paths for which x < x 0 should contribute to ρ .In this work, however, we implement the projection by integrating numerically over the allowed range of x and evaluate the imaginary-time propagator as described in the appendix.

V. RESULTS AND DISCUSSION
In order to validate our semiclassical analysis of the quantum instanton and cumulant expansion, it is useful to compute the numerical values of the various terms making up these approximations and to compare the values obtained from quantum mechanics (using the eigenfunctions for the Eckart barrier discussed in Appendix A) and from the semiclassical approximation.The results of the quantum and semiclassical calculations for the same system as that used in Figs. 1 and 3 are compared in Table II.The semiclassical limit is seen to be very close to the quantum values for each of these choices of dividing surfaces, being at most a factor of two different.The discrepancy is larger for other dividing surface choices, such as the one made in Fig. 4, which leads to K K r .The good agreement is also seen in Table III, where the QI and 2OCE rate predictions are compared using both quantum and semiclassical calculations.These approximations are however an order of magnitude larger than the exact rate constant, showing that the source of the error is in the QI or 2OCE rate formulae themselves.This justifies our use of a semiclassical evaluation of the relevant quantities for analysing the QI results.The PQI method predicts a rate of 4.34 × 10 −12 a.u.without optimization of the imaginary-time split (i.e. using the semiclassical ratio of τ r /τ ≈ 0.258), but when it is optimized to τ r /τ ≈ 0.298, the rate prediction is 4.85 × 10 −12 a.u., which is even closer to the exact result (c.f.caption Table III).
The example system used in Fig. 3 shows at least an order of magnitude error only at low enough temperatures.However, we stress that the error seen in the QI method depends on both the asymmetry and the temperature, such that for more asymmetric systems, even high temperatures can result in very large errors.For a more general outlook, Fig. 5 shows the relative error between the exact rate and the rates calculated using the four methods (QI, 2OCE, SCI, and PQI) as a function of asymmetry, α. 55 To generate these results, we have generally favoured A-type dividing surfaces, although it would have been possible to choose surfaces for the 2OCE calculation that minimize the contributions of higher order terms. 42Nonetheless our semiclassical analysis implies that no dividing surface choice will significantly improve these results.While the SCI rates tend to a small constant relative error in the deep tunneling regime, 18,56 the error in the QI rates increases exponentially with asymmetry.We attribute this growing error in the QI rate to the increasing contribution of the right-hand path in Fig. 3.In principle, 2OCE could be improved by going to a higher order expansion; however, in practice, very high orders will typically be required to cancel the spurious dominant contribution at zero time.In contrast, the PQI results in Fig. 5 show a substantial improvement over both the QI and 2OCE results and is at least as accurate as SCI.In certain cases, especially close to the cross-over temperature T c = ω b /2πk B (where ω b is the absolute value of the imaginary barrier frequency), 18 the accuracy of the SCI approximation breaks down.Above the cross-over temperature the PQI method still gives relatively accurate predictions for the rate, although the error is slightly larger than below T c .For the results in Fig. 5, the value of τ is not optimized but simply chosen to match the imaginary-time for the left bounce in the instanton trajectory.We note that the PQI and 2OCE results do not match exactly at α = 1, and the minor discrepancy here is a measure of the accuracy of the approximation in Eq. (42).
These promising results show that the new PQI method is a substantial improvement over the traditional QI and 2OCE methods.Future extensions of PQI for use with path-integral methods are therefore likely to fulfill the initial promise of QI as an extension to SCI, suitable for accurately predicting reaction rates in highly anharmonic and asymmetric systems.

VI. CONCLUSIONS
In conclusion, we have presented a semiclassical analysis of the quantum instanton method.We have shown that the dominant contribution to the QI expression can arise from spurious paths, which leads to very large errors for very asymmetric barriers, especially at low temperatures.Consequently, despite conventional wisdom suggesting otherwise, no choice of dividing surface can remove this problem.We justify our analysis by showing that semiclassical evaluation of the relevant quantities yields very similar numerical values to those from exact quantum mechanics for the asymmetric Eckart barrier.The major discrepancy from the exact rate therefore lies with the QI and 2OCE approximations themselves.
From our analysis we conclude that the SCI rate is the semiclassical limit of the QI rate only when the barrier is perfectly symmetric.Consequently, the QI method provides an accurate quantum transition-state theory for symmetric barriers.However, we find that SCI is much more accurate than QI at describing tunneling in asymmetric systems, although the SCI clearly cannot describe anharmonicity as well as the QI does.Our proposed new method, PQI, fixes the inherent path sampling problems with the QI and 2OCE methods mentioned above, as well as taking into account anharmonicities that SCI neglects.There exists some similarity between the PQI and theories employed to study the golden-rule limit of electrontransfer reactions 52,54 and we hope that this study will inspire future development into nonadiabatic rate theories.It remains to be seen whether a path integral application of PQI will become a competitive method for reaction rate calculations of complex systems.RPMD rate theory, 7 which also dominantly samples the instanton path even for asymmetric systems, 22 also includes classical recrossing effects neglected by SCI, QI, and PQI, and thus remains the method of choice for reaction rate calculations when these recrossing effects dominate.

FIG. 1 .
FIG. 1. Contour plots of ρ(x1, x2, 0) and C ff (0) for the asymmetric barrier at 100 K and α = 1.425, plotted as functions of the dividing surface locations.The saddle points of ρ (and therefore C dd , which is simply the square of ρ) which define the split surfaces are shown as blue crosses, the semiclassical instanton turning points are shown as red circles, and the merged surfaces locations are shown by the green squares.These coordinates are given in TableI.

C
FIG. 2. (a) Flux-flux and (b) delta-delta time correlation functions as functions of time for several values of the asymmetry parameter α.Darker lines indicate larger values of α, with α = 1, 2, 3, and 4. For each value of α, the dividing surfaces are reoptimized with the split saddle points on C dd chosen if they exist, corresponding to A-type surfaces in TableI.In this way, the same surfaces are used for both C ff and C dd , although similar behaviour is also seen for other choices of dividing surfaces.

)FIG. 3 .
FIG.3.The two semiclassical minimum-action trajectories are shown plotted at their values of the energy for three different choices of dividing surfaces (indicated by dashed lines).The green trajectory starts at one of the dividing surfaces with initial momentum in the negative direction, bounces against the barrier on the left and returns to the other dividing surface with positive momentum.The red trajectory travels in the opposite direction and bounces against the right-hand side of the barrier.The dividing surfaces are chosen at (a) the saddle points of the quantum-mechanical ρ(x1, x2, 0), (b) the merged-surface minimum of ρ(x1, x2, 0), and (c) the semiclassical turning points.These choices correspond to A, B, and D in TableI, respectively.The choice C gives qualitatively the same picture as B. The actions of each semiclassical path are also indicated.In (c), the green line follows one half of the instanton periodic orbit.All quantities are given in atomic units.

FIG. 5 .
FIG.5.Dependence of relative errors of various approximations for the rate constant on the asymmetry parameter α of the Eckart barrier for two different temperatures.The error is measured with the ratio k/kexact (on a logarithmic scale for clarity), where kexact and k are, respectively, the exact and approximate rate constants.The different approximations shown are the quantum instanton with A-type dividing surfaces (thin blue line), the second-order cumulant expansion with A-type surfaces (thin orange dashed line), the semiclassical instanton (green dashed line), and the projected quantum instanton evaluated with dividing surfaces at the barrier maximum (thick red line).The SCI result is only plotted when below the crossover temperature.

TABLE I
. Definition and positions of dividing surfaces for the asymmetric Eckart barrier with α = 1.425 and T = 100 K.The remaining parameters are defined in the main text.

TABLE II .
Numerical values (in a.u.) of various quantities used in the QI expressions are calculated using quantum-mechanical (QM) and semiclassical (SC) methods for the specified dividing surfaces.In each case, the semiclassical version is within a factor of two of the quantum result, confirming that our semiclassical analysis of the quantum instanton is valid.

TABLE III .
Prediction of rates (in a.u.) using different methods, where k2OCE is the rate from the cumulant expansion, and kQI is the rate from the QI method.The parameters of the potential are the same as in TableI.The exact rate is 4.63 × 10 −12 a.u. and the semiclassical instanton rate is 3.93 × 10 −12 a.u.For comparison, the classical TST rate is 8.31 × 10 −26 a.u., which confirms that quantum tunneling effects are of significant importance.