Perspective : Ring-polymer instanton theory

Since the earliest explorations of quantum mechanics, it has been a topic of great interest that quantum tunneling allows particles to penetrate classically insurmountable barriers. Instanton theory provides a simple description of these processes in terms of dominant tunneling pathways. Using a ring-polymer discretization, an efficient computational method is obtained for applying this theory to compute reaction rates and tunneling splittings in molecular systems. Unlike other quantum-dynamics approaches, the method scales well with the number of degrees of freedom, and for many polyatomic systems, the method may provide the most accurate predictions which can be practically computed. Instanton theory thus has the capability to produce useful data for many fields of low-temperature chemistry including spectroscopy, atmospheric and astrochemistry, as well as surface science. There is however still room for improvement in the efficiency of the numerical algorithms, and new theories are under development for describing tunneling in nonadiabatic transitions.


I. INTRODUCTION
The most widely used approach for predicting and interpreting the rate of a chemical reaction is probably transitionstate theory (TST). 1,2It defines a particularly simple formula based on quantities available from a limited knowledge of the potential-energy surface (PES).The TST approximation to the rate constant is where V ‡ is the barrier height, β = 1/k B T is the reciprocal temperature, and A TST is a prefactor including contributions from the partition functions within a harmonic approximation.Not only does TST provide us with a numerical method for obtaining the rate of a chemical reaction, but even more importantly it also defines our understanding of how a chemical reaction proceeds, namely, by overcoming an energy barrier.When multiple mechanisms are possible, the one with the smallest barrier is expected to dominate.However, as described by Wigner, 3 TST is based on classical assumptions, and certain quantum effects of the true dynamics are ignored.One of the most important effects neglected by TST is quantum tunneling.
Even when the energy is lower than the potential barrier, molecules can rearrange via a tunneling mechanism. 4his can lead to a dramatic effect on the rate of a chemical reaction. 5,6Nuclear tunneling is most prevalent at low temperature, particularly for the transfer of hydrogen atoms, which being so light strongly manifest their quantum nature.Rates of these processes are observed to be much larger than would be expected from classical TST, and the trends are observed to be non-Arrhenius.Another indicator of nuclear tunneling is the presence of a strong kinetic isotope effect on substituting the tunneling atom.a) Electronic mail: jeremy.richardson@phys.chem.ethz.chQuantum tunneling effects have been measured in atmospheric and astrochemistry, 7,8 as well as organic chemistry at low temperatures. 9In some cases, they can change the mechanism for the reaction and hence the selectivity of the products. 10On surfaces, one can directly observe tunneling events such as the hopping of atoms between sites 11 or the rearrangement of a molecular cluster. 12In isolation, molecules and molecular clusters also rearrange using tunneling mechanisms which can be observed using high-resolution spectroscopy. 13,14Tunneling also clearly occurs in some enzymecatalysed reactions, 15 even if it is not the dominant factor leading to the speed-up of the reaction. 16f we are to explain and predict how these chemical reactions proceed, we cannot base our understanding on TST.It is not possible to reintroduce the quantum effects as a one-dimensional correction factor to classical TST in a rigorous way because all degrees of freedom are coupled together. 17Instead, a more general formulation of rate theory is needed which does not ignore these quantum effects.Such a formulation is provided by instanton theory. 18

II. INSTANTON RATE THEORY
In classical mechanics, it is forbidden for a particle to cross a barrier if its energy is lower than that of the barrier height.The tunneling process is thus essentially nonclassical but arises naturally from the wave-function approach to quantum mechanics. 4While in principle wave-function approaches give exact results, they are practically impossible to apply to polyatomic systems.A more useful approach would be a simple extension of classical mechanics to allow for tunneling events.This has the dual advantage of being easier to apply in practice as well as being simpler to interpret.
Miller 18 showed that the necessary extension is provided by semiclassical trajectories, such as the one represented in FIG. 1. Representation of a tunneling process described by a semiclassical trajectory.Configurations on the left of the potential-energy barrier, V (x), are defined as "reactants," and configurations on the right are "products."Because the total energy, E, is less than the barrier height, V ‡ , the trajectory is required to pass through a classically forbidden region marked in red.In this region, the trajectory proceeds in imaginary time.
Fig. 1.Based on the definition of the energy of a classical particle, E = p 2 /2m + V (x), one can see that if E is lower than the potential energy of the barrier, V (x), then the kinetic energy term, p 2 /2m, must be negative.This implies that the momentum, p, would have to be imaginary and that therefore particles tunneling through barriers can be considered to be moving in imaginary time.The semiclassical trajectories then obey Newton's equation of motion for imaginary-time dynamics, which has solutions equivalent to standard classical dynamics in an upside-down potential.
Semiclassical instanton rate theory is a quantummechanical generalization of transition-state theory which utilizes imaginary-time dynamics to describe tunneling.It can be derived from first principles [18][19][20][21] starting from the definition of the exact rate constant 22 in a path-integral representation 23 and taking a number of asymptotic approximations.Its connection to quantum-mechanical scattering theory puts the use of instanton theory for chemical reaction rates on a rigorous footing.However, it can also be shown 24 that the resulting formulation is equivalent to Coleman's expression for the rate constant using a derivation based on the "Im F" premise. 25Coleman was primarily interested in the T → 0 limit of quantum field theory and only later was the "Im F" approach extended to treat finite-temperature problems using a different functional form above and below the crossover temperature. 26he theory is based on a periodic classical orbit of imaginary-time length β , known as the instanton, which traverses the barrier region.The approximation to the reaction rate constant is given by where the Euclidean action, measured along the instanton trajectory, x(τ), is and the prefactor, A inst , is a measure of fluctuations around the instanton path.These formulae are easily generalized for multidimensional systems by taking √ mx to be a mass-weighted vector of the Cartesian coordinates of all atoms.
Instanton theory thus provides a simple and rigorous formula for predicting rates of chemical reactions including tunneling effects.It is clear that the most important term is the exponent which depends only on the action along the instanton trajectory.If two mechanisms are competing, the one with the smaller value of S will dominate the product, and not necessarily, as would be suggested by TST, the mechanism with the lower barrier.
Using the fact that the energy E = 1 ∂S ∂β is constant for a classical trajectory, we can also write the action as dx is an integral around the orbit.Using this relationship gives an equivalent formulation of the rate, and makes it clear how instanton theory is a generalization of TST.In both cases, there is a Boltzmann factor, but due to quantum tunneling, the energy, E, of the instanton may be less than the barrier height, V ‡ .There are two competing factors which determine the instanton energy: the Boltzmann factor, e −βE , favours low energies, but the reaction probability, e −W (E)/ , is larger closer to the barrier top.E has to be chosen such that the instanton is a periodic orbit of a given length, and its value can be found by minimizing W (E) + β E. 20 For this reason, the instanton is said to define the dominant or optimal tunneling pathway.
The reaction probability, e −W (E)/ , is recognized to be the same as that given by the one-dimensional WKB approximation. 6This is unsurprising as both the instanton and WKB approaches are asymptotic semiclassical approximations.However, unlike WKB, instanton theory is based on path integrals and is thus much more easily extended to multidimensional systems.In general, the instanton will take a curved path through many degrees of freedom of a molecular system.As discussed by Marcus and Coltrin, 27 the optimal tunneling path is thus equal neither to the minimum-energy pathway (MEP) nor to the direct straight-line path.Instead the path typically "cuts the corner," 28 as shown in Fig. 2, and unlike the MEP does not pass through the transition state.Alternative methods based on the small-or large-curvature tunneling (SCT or LCT) approximations 29 apply the one-dimensional WKB formula along a tunneling path chosen a priori.Instanton theory on the other hand uniquely defines and uses the optimal tunneling pathway.Because of modern techniques for locating instantons discussed in Sec.II A, obtaining the correct corner-cutting path is possible even for large polyatomic systems.
The optimal tunneling pathway defined by the instanton is temperature dependent.At high temperatures, the instanton is able to sustain a high energy and therefore shrinks close to the transition state.At temperatures higher than a particular crossover temperature, T c , the instanton orbit collapses to the barrier top.Above T c , a semiclassical analysis shows that a good prediction for the rate constant can be obtained using classical TST with shallow-tunneling corrections. 20These are commonly based on a parabolic approximation to the barrier. 30t temperatures lower than T c , the instanton becomes delocalized and can stretch across the barrier at low energy.This describes a deep-tunneling mechanism and typically has a rate which is orders of magnitude faster than that predicted by TST.
The crossover temperature can be estimated using k B T c = ωb /2π, where ωb is the magnitude of the imaginary frequency at the barrier top. 26If the magnitude of the imaginary frequency is greater than about 1300 cm −1 , then deep tunneling is likely to be important even at room temperature.As a frequency analysis of the transition state is anyway performed as part of the TST calculation, a simple guide for predicting whether deep tunneling may be important is immediately available.
The ratio between the instanton rate and TST defines the tunneling factor.This factor increases dramatically with decreasing temperature, but nonetheless the instanton rate still typically decreases in this range.However, for exothermic unimolecular reactions, S has a low-temperature limit, and at some point, the rate becomes independent of temperature.This occurs because at low temperature, only the lowest reactant state is occupied and the reaction can only proceed via tunneling decay of this state.For bimolecular reactions with scattering boundary conditions, no such limit exists and lowering the temperature continues to lower the rate.These two cases are both correctly described by instanton theory.
The fluctuations around the instanton pathway are only included up to second order, which implies that modes perpendicular to the tunneling pathway are effectively treated by a harmonic approximation.However, because of the semiclassical approximations, the most important part of the rate formula-the exponential-is treated fully anharmonically.Therefore the results of the instanton approach should be considered as order-of-magnitude predictions for the rate.Indeed, when exact results are available for comparison, the method is consistently observed to be in good agreement.

A. Ring-polymer instanton theory
In its original form, instanton theory is not a practical computational method for studying tunneling in molecular systems.Analytic solutions are known only for a very limited number of special cases, 5 and so in order to treat more realistic systems, various approximations have been suggested based on a Taylor series expansion around stationary points of the PES. 31,32However, in this subsection, we discuss the ring-polymer version of instanton theory, 33,34 which does offer a practical method without formally introducing any further approximations.
As discussed above, instanton pathways are imaginarytime classical trajectories and equal to solutions of Newton's equations of motion on an upside-down potential.Historically the pathways were located by computationally solving the equations of motion and searching for boundary conditions leading to the required periodic orbit solution. 28,35This method, known as "shooting," is a simple procedure for onedimensional systems but quickly becomes practically impossible to perform as the number of degrees of freedom increases beyond two.This is because when searching for an unstable orbit in a multidimensional potential, the final position at the end of the trajectory changes exponentially strongly as the initial condition is varied.
The disadvantages of the shooting method for solving general double-ended boundary problems are clearly outlined in Numerical Recipes, 36 and the alternative method of "relaxation" is recommended instead.To apply this approach to the problem of locating instantons, the path is discretized into a set of points with the periodic boundary condition implicitly built in.All points are then optimized simultaneously until the required solution is found.According to the variation principle, the first variation of S is zero for a classical trajectory, and thus discretized trajectories can be obtained by searching for stationary points of the action.It turns out that when searching for instantons in multidimensional systems, the relaxation method is far more efficient than the shooting method and leads to a practical computational approach for applying instanton theory to molecular systems.
8][39] However, the most common discretization scheme is that of the ring polymer. 40,41Here the path is represented by N "beads," x i , each representing an image of the full multidimensional system and specifying the Cartesian coordinates of all atoms.The definition of the ring polymer can be derived either from finite differences and a trapezium-rule integration of Eq. ( 3) or directly from the Boltzmann operator.
There is one simple trick which can be used to reduce the computational cost of representing the ring-polymer instanton.As the instanton trajectory is known to fold back on itself in order to complete its periodic orbit, it is only necessary to optimize one half of the trajectory.Therefore, only half of the path, x(τ) for τ ∈ [0, β /2], is discretized into beads x i for i ∈ {1, . .., N/2}, where we are assuming that N is even. 33The half-ring-polymer potential is given by (5)  where β N = β/N.The stationary point of this function is denoted as x and defines a discrete representation of the instanton.The action of this instanton is given by S/ = 2 β N U N/2 (x).
It turns out that the optimal tunneling pathway defined by instanton rate theory is a first-order saddle point. 253][44] A number of methods have been tested and it was found that the Newton-Raphson solver required the fewest iterations to optimize the instanton. 45This method requires knowledge of the Hessians (second-derivative matrices of the PES) at each bead but uses update formulae to avoid computing them at each optimization step.A particular advantage of this method over other quasi-Newton eigenvectorfollowing optimizers is that it can take advantage of the banded nature of the half-ring-polymer Hessian. 21Extensions of the nudged-elastic band (NEB) algorithm have also been developed for optimizing instantons which do not require Hessians at all. 46,47However, note that even in this case, Hessians are still required to compute the fluctuation terms in the prefactor.
The prefactor, A inst , can be formulated from a product of the non-zero eigenvalues of the ring-polymer Hessian matrix computed at the instanton configuration. 48As the instanton is a saddle point, one eigenvalue is negative but contributes its absolute value to the product.Another eigenmode describes the permutation of beads around the ring polymer and has a zero eigenvalue.This is factored out, along with any other zero eigenvalues corresponding to translations and rotations.These are treated instead using their partition functions within the rigid-rotor approximation.In Ref.21, other equivalent formulations of the instanton rate are presented, some of which avoid the storage and diagonalization of the ring-polymer Hessian altogether, which may be important for applications to particularly large systems.
Ring-polymer instanton rate theory provides the simplest extension for TST to rigorously describe deep tunneling in nonseparable multidimensional systems.It has proved itself to be a remarkably efficient and accurate method, and as with TST, the instanton approach can be combined with accurate but computationally expensive electronic-structure calculations.Applications include the study of gas-phase collisions, 48,49 kinetic isotope effects, 50,51 molecular rearrangements in matrix isolation, 52 surface processes, 46,53,54 and enzyme catalysis. 55These applications include systems from 3 to about 100 atoms, and there is little to halt the progress of the method there.
A representation of the ring-polymer instanton for the prototypical polyatomic H + CH 4 reaction can be seen in Fig. 3.The rate constants predicted by the instanton method for this system are in fairly good agreement (∼35% error at 250 K) 51 with benchmark quantum dynamics results 56 using the same parametrized PES. 57However, as the rate depends exponentially on the shape of the barrier, the PES itself may be a significant cause of error for both these approaches.To test this assertion, instanton calculations have also been performed using on-the-fly electronic-structure [RCCSD(T)-F12a/cc-pVTZ] calculations. 48,51At low temperatures, the calculations based on the parametrized PES underpredicted the ab initio rate by more than a factor of two.This shows that in order to obtain accurate results, the most important factor is the quality of the PES itself and that the semiclassical approximation inherent in the instanton method is of a lesser concern.In many cases, we expect that an instanton calculation utilizing an ab initio PES will give more accurate predictions than even an exact solution of the nuclear Schrödinger equation on an approximate parametrized PES.

B. Implementation of ab initio instanton theory
The ring-polymer instanton approach leads to a relatively efficient computational method because it requires only a small amount of knowledge of the PES localized along the instanton orbit.In contrast to this, methods which run trajectory, path-integral, or wave-packet dynamics often need the pre-calculation of a global potential defined over a multidimensional space or at least a statistically large number of single-point calculations.
Nonetheless, the instanton method is still less efficient than a TST calculation, which requires only one Hessian for each stationary point.For a typical low-temperature instanton calculation, about 100 beads are needed for convergence and therefore the standard method requires at least 100 Hessians computed on the PES.As this is typically the computational bottleneck, the ring-polymer instanton method probably needs to reduce its computational requirements to below 10 Hessians in order to become practical for use with high-level ab initio electronic-structure calculations.
As all ring-polymer beads are found close together in a small region of the PES, this is achievable by computing the Hessian at only a few beads and interpolating between them to obtain a well-controlled approximation for the others. 58Fitting methods such as Gaussian process regression or neural networks can accurately describe the local PES based on only a few points.These have been successfully applied to computing minimum-energy pathways 59 and recently also to instanton calculations. 60,61echniques developed for increasing the efficiency of path-integral molecular dynamics could conceivably also be employed in instanton calculations.In particular, the use of higher-order discretization schemes 62 and ring-polymer contraction 63,64 has not yet been fully explored.

C. Connection to other path-integral approaches
Instanton rate theory is related to a number of other pathintegral-based rate theories, and the connection with instanton theory can explain when they give good approximations and when they fail.For instance, instanton theory is equivalent to the harmonic limit of one definition of quantum transitionstate theory, 33,65 which is itself (apart from a difference in the prefactor) equivalent to ring-polymer transition-state theory (RPTST) with an optimal dividing surface. 34,66Like the instanton approach, these two methods are just as accurate for describing tunneling through asymmetric barriers as for symmetric barriers.Centroid quantum transition-state theory 67 is equivalent to RPTST for symmetric systems only, but as shown by Refs.34 and 65, fails for asymmetric systems as the optimal dividing surface should include higher ring-polymer modes.A similar problem with asymmetric systems is also seen in the so-called "quantum instanton" method. 68,69ing-polymer molecular dynamics (RPMD) commonly also uses a centroid dividing surface for rate calculations.However, it takes into account the recrossing of trajectories to correct the rate for the suboptimal dividing surface and thus gives good results for asymmetric as well as symmetric barriers. 70he trajectories also describe dynamical recrossing effects to go beyond the TST approximation as in classical rate theory.Like RPMD, instanton theory can also describe the effects of Kramers' strong-friction regime 71 by coupling to bath modes in the same way as classical TST. 72However, as it is derived in terms only of imaginary-time trajectories, instanton theory effectively ignores the dynamics of the system outside the barrier region and cannot therefore describe the transition rate of a double-well potential in Kramers' weak-friction regime. 73The dynamical recrossing of RPMD offers the necessary extension and can thus describe Kramers' turnover. 74For these reasons, RPMD is the most widely applicable of these path-integral rate methods, and the only one which is valid in liquid systems, for which finding the optimum dividing surface would be impossible and the instanton approximation breaks down.
6][77] However, the instanton method may offer a good alternative for the many low-temperature tunneling reactions occurring in the gas phase, on surfaces, or in solids, where distinct optimal pathways describe the mechanism and dynamical recrossing is unimportant.In principle, the instanton method has the advantage of not requiring a well-sampled ensemble of paths, but only a single optimal trajectory.It will therefore surely become the method of choice for many of the reactions which do not occur in liquids.

III. TUNNELING SPLITTINGS
So far we have discussed an instanton theory for computing rate constants.A rate constant describes the exponential decay of the population of the reactants caused by incoherent quantum dynamics.This exponential decay occurs only for systems with a continuum of product states such as scattering problems or rearrangements of molecules coupled to a large environment.However, if there are only a small finite number of accessible states, the quantum dynamics of a molecular system become coherent, there is no exponential decay, and a rate constant cannot be defined.Different approaches are needed for this regime, and the study of the transition from incoherent to coherent dynamics is an active field of study. 73ven though we can no longer define a rate constant, there is a well-defined quantity which characterizes the coherent tunneling dynamics of a degenerate rearrangement in the zerotemperature limit, namely, the tunneling splitting.This is a consequence of mixing between the vibrational states of the degenerate potential wells leading to a splitting pattern which can sometimes be observed by high-resolution spectroscopic methods. 13 subtly different, but strongly related, instanton theory also exists for computing tunneling splittings. 25,38As in the previous case, the splitting is related to the exponential of the action of an imaginary-time trajectory.The ring-polymer instanton approach can again be used to provide an efficient method for numerically evaluating this theory for polyatomic systems. 78The major difference in the computational implementation is related to the fact that at zero temperature, the instanton is no longer a saddle point on the ring-polymer surface, but a minimum.Because of this, different optimization algorithms can be used.The limited-memory BFGS algorithm 79 is generally regarded to be the best, partly because it neither needs to compute Hessians nor store large matrices in computer memory.
In order to obtain results in the zero-temperature limit, the imaginary-time length of the instanton trajectory must be increased until convergence of the results.This can result in very long pathways which require a large number of ringpolymer beads.Most of the beads settle close to the bottom of the wells, and only a minority describe the pathway over the barrier, as can be seen in Fig. 2.This is in fact the origin of the name "instanton"; the trajectory is such that it is almost stationary at the beginning and end, but somewhere in the middle the tunneling event happens very quickly, almost in an instant.For this reason, the tunneling splitting calculations typically require more single-point calculations of the PES than a rate calculation.The method is still extremely fast to perform on fitted potential-energy surfaces, but in order to run the method with expensive onthe-fly ab initio potentials, a more efficient method would be required.
A number of suggestions have been made to redistribute the beads closer to the part of the trajectory which is moving and to avoid them clustering in the wells.One idea is to redefine the discretization in a non-uniform way. 80However, this requires a priori knowledge of the pathway in order to choose the discretization in an optimal way.A more promising approach optimizes the abbreviated action function, W (E), based on the Hamilton-Jacobi formulation of classical mechanics.This integral can be discretized in coordinate-space rather than in imaginary-time such that the beads can be evenly spaced. 81An extension of the NEB algorithm has been successfully applied to locate the instanton in this way and compute the tunneling splittings. 82,83These methods surely will provide the most efficient approaches for further studies of tunneling splittings in systems using on-the-fly potentials.
Instanton theory gives fairly accurate results for small tunneling splittings in systems with a high barrier between the wells.However, if the barrier height is so small that it becomes similar to the harmonic zero-point energy of the wells, then the semiclassical approximation breaks down and the tunneling splitting can be overestimated by about a factor of two. 78nstanton theory thus complements the diffusion Monte-Carlo (DMC) method, 84 which is accurate for computing large tunneling splittings but can be dominated by statistical errors for calculations of a small splitting.
One of the major advantages of the instanton approach over DMC is that it can be easily applied to systems with more than two degenerate wells. 85For these systems, DMC would need to define a nodal surface, which can be an extremely complicated function and is not known a priori.The implementation of the ring-polymer instanton approach, however, is exactly equivalent to the double-well case, and one simply needs to compute a tunneling matrix element for each of the possible rearrangement pathways.These matrix elements are combined into an adjacency matrix using a graph theoretical description of the connectivity of the wells.In a similar way to Hückel theory, 86 the eigenvalues of this matrix give the energy levels and hence the tunneling splitting pattern.
Two of the best studied benchmark systems exhibiting tunneling splittings are malonaldehyde and the formic acid dimer.It has been experimentally determined that malonaldehyde has a splitting of 21.6 cm −1 due to a degenerate intramolecular proton transfer, 87 whereas the formic acid dimer's splitting of 0.016 cm −1 is much smaller as it is due to a double proton transfer. 88Ring-polymer instanton theory has been used to study both of these systems and obtains splittings of 25 cm −1 and 0.014 cm −1 which are both within 20% of the experimental values. 82,89This excellent result is due to a high level of accuracy not only of instanton theory but also of the potentialenergy surfaces fitted to high-level ab initio calculations. 90,91ote that even small errors in the surface will affect the action, S, and thus lead to exponentially larger errors in the predicted splitting.The accuracy of the PES is therefore of the utmost importance.
It has also become clear from instanton studies that it is necessary to include all nuclear degrees of freedom in the problem.Even in planar systems such as malonaldehyde and the formic acid dimer where the instanton trajectory lies completely in the plane, the out-of-plane modes cannot be ignored without introducing a factor-of-two error into the predicted splitting. 89,92ing-polymer instanton theory has been used to calculate the tunneling splitting patterns of various water clusters. 85,93,94he results are in good agreement with the experiments except for the splitting due to the flip motion in the water trimer, which has a very low barrier and is therefore overestimated.Most importantly, it has been possible to identify the mechanisms responsible for the observed splitting patterns.For example, the water hexamer prism has two different mechanisms shown in Fig. 4, one which breaks one hydrogen bond and one which simultaneously breaks two. 94,95Surprisingly it turns out that both mechanisms have similar tunneling probabilities.Note that it would not have been possible to know the tunneling pathways a priori as they lie far from the minimum-energy paths.The predictions were confirmed by the experimental measurements and explained the observed doublet-of-triplets splitting pattern. 94,96

IV. FURTHER DEVELOPMENTS
In addition to further improving the efficiency of the method, developments are underway to improve the accuracy of the instanton approximation and to derive novel semiclassical theories for studying new problems in molecular systems.
It is well known 26,73 that the standard instanton approach fails to predict the rate accurately near the crossover temperature.8][99] This results in different expressions being used in different temperature regimes, and it is not always obvious where one formula should take over from the other.
An alternative approach traces the cause of the problem to a steepest-descent approximation for an integral over energy. 20he solution suggested is to use an approximation to the microcanonical rate over a range of energies which is weighted by a thermal distribution and integrated numerically to give a single unified formula for reaction rates at all temperatures of interest.This approach is necessary at all temperatures for barriers with a particularly broad-top shape where the steepest-descent approximation always fails. 100 The derivation of a rigorous microcanonical instanton theory is thus required.Such an approach was first suggested by Miller and co-workers 28 and is now under further development. 20,101he advantage brought by this new theory would not just be the elimination of the crossover-temperature problem but would provide an accurate approximation for microcanonical rates, which is much needed in the fields of atmospheric and astrochemistry.Because of low-pressure environments, reaction networks are not thermalized on the time scale of the overall reaction, and so they are simulated with masterequation solvers which require microcanonical rates as input.A practical microcanonical instanton method has the potential to provide accurate data for these important fields of study.
Further development of instanton rate theory to the application of nonadiabatic reactions is also possible.In particular, a ring-polymer instanton method has been developed for electron-transfer reactions in the golden-rule limit. 102The formulae reduce to Marcus theory in the classical limit for the case that the free-energy curves are harmonic.However, both anharmonic and tunneling effects are accurately treated by the new theory. 103eactions such as proton-coupled electron transfer are also nonadiabatic but do not occur in the golden-rule limit.For these, a more general approach is needed which correctly gives the limiting cases in both the strong-and weak-coupling regimes.Some suggestions have been given as to how instanton theory could be extended to describe these reactions, 104 but as they are based on the "Im F" premise, they do not appear to give a reasonable classical high-temperature limit.Further work is needed to benchmark these approaches and to apply them to realistic problems.
Finally, in order to go beyond the steepest-descent approximation, one would like to develop path-integral molecular dynamics approaches, like RPMD, which effectively sample around the instanton and make no harmonic approximations for the fluctuations.A method has also been developed for calculating tunneling splittings in this way. 105Work is in progress to find a similar extension of nonadiabatic instanton theory, which would be able to predict rates for nonadiabatic reactions in solution.In this way, based on an instanton study, 106 kinetically constrained RPMD 107 was designed to correct the failure of a direct RPMD simulation of electron transfer to describe the Marcus turnover curve. 108hus the future of instanton theory is bright, both in providing methodology for various applications as well as in guiding the development of new theories.

FIG. 2 .
FIG.2.The instanton pathway (inst.)follows a different trajectory from the minimum-energy path (MEP) or the large-curvature tunneling approximation (LCT).Blue circles along the instanton pathway show the location of the ring-polymer beads, and black contours show the underlying two-dimensional potential-energy surface.

FIG. 3 .
FIG. 3. Representation of the ring-polymer instanton describing proton tunneling in the gas-phase H + CH 4 reaction at 200 K.All atoms take part to some extent in the tunneling process and become delocalized as they pass through the potential barrier.Reproduced with permission from Beyer et al., J. Phys.Chem.Lett.7, 4374 (2016).Copyright 2016 Author(s), licensed under a Creative Commons Attribution 4.0 License.

FIG. 4 .
FIG.4.The two tunneling pathways calculated for the water hexamer prism (left) showing concerted hydrogenbond breaking.The instanton method predicts the correct splitting pattern (a doublet of triplets), as observed in the experimental spectrum (right).Adapted from Ref.94.