Wolynes's theory for tunneling in dissipative systems was constructed for parabolic barriers. Its recent generalization to anharmonic potentials is further developed, most notably by employing the second order vibrational perturbation theory expression for the action, derived by Miller and coworkers. With this construct there is no need to know the full potential energy surface involved in the hopping, it is sufficient to know well and barrier frequencies, barrier heights, friction coefficients and the fourth order derivative of the potential at the barrier top. The resulting theory is applied to model the experimentally measured hopping rates of H and D atoms on a Pt(111) surface and H atom hopping over a barrier on the Ru(0001) surface. In all cases, the results indicate that the barrier frequency is substantially larger than the well frequency. The comparison with experiment sheds light on the information content of the measured data.

Almost a century has passed since the publication of the seminal Brillouin, Wentzel, and Kramers (BWK) papers on the semiclassical quantization of vibrations, yet the BWK semiclassical quantization formula does not lead to the correct zero point energy estimate of the energy except for a few special cases. In this Letter, a simple energy shift is introduced into the expression for the action, whose magnitude is determined by second order vibrational perturbation theory, removing this deficiency. The resulting modified semiclassical quantization formula, when appropriately expanded, is shown to be identical to second order vibrational perturbation theory. It improves the resulting energy eigenvalues for the symmetric Rosen-Morse potential and is shown to provide rather accurate energy estimates for resonance energies in a cubic potential, better than those predicted by the standard semiclassical BWK expression or second order perturbation theory.

The vacuum field of an optical cavity can potentially modify chemical reactivity and other dynamical properties via vibrational strong coupling (VSC). This intriguing finding has inspired numerous studies, but the underlying mechanisms remain unresolved. While many theoretical efforts focus on solvent or nuclear fluctuations, the tunneling overlap in non-adiabatic processes is usually assumed unperturbed by the cavity field. This paper presents a rigorous calculation of the tunneling splitting and associated ground-state shift resulting from the non-adiabatic coupling between two degenerate, harmonic diabatic surfaces in the ground vibrational state manifold under VSC. Based on this calculation, the tunneling splitting is suppressed by the cavity field for a single-molecule or a few-molecule system, but this cavity-induced effect is neither resonant nor cooperative and vanishes in the thermodynamic limit. This prediction demonstrates the many facets of VSC-induced phenomena and sheds new light on cavity-modified non-adiabatic processes, including charge transfer, Förster resonance energy transfer, energy relaxation, and conical intersection.

The computation of tunneling splitting in physical systems, such as molecular systems, qubits, and more, is still quite challenging because of its very small magnitude, as compared with the typical vibrational spacing between doublet levels. Thus, it is important to understand and develop methods that can reproduce the physics of the tunneling splitting in a simple and direct way. Herring's formula is probably the most popular expression in this sense. To shed light on the implications of this formula, which was first proposed by Herring [C. Herring, Rev. Mod. Phys. 34, 631 (1962)], we investigate the connection between the two-state approximation, as employed for nonadiabatic-induced tunneling splitting, and the Herring formula, which is relevant to adiabatic tunneling splitting. We show that the two-state approximation and the Herring formula, which may be derived as a weak value of the flux operator and is a derivative result, are identical for a symmetric double well potential. This unveils the physics underlying Herring's formula and provides further justification for the two-state approximation for the nonadiabatic tunneling splitting estimate. We conclude that, when Herring's formula is used with approximate eigenfunctions, it will not be accurate when the two-state approximation is not valid, i.e., when the energy splitting is comparable to the eigenenergy spacing. More generally, Herring's formula will fail when the two-state approximation fails. We also show how the identity between the integral two-state approximation form and Herring's formula may be used to obtain analytic expressions for some nontrivial integrals.

A multidimensional version of the modification to vibrational perturbation theory is developed in this article. The modifications to the action are of two types: one is by shifting the energy scale with the VPT2 zero point energy E0 (mVPT2) and the other is by shifting the action by a constant VPT2-based action ΔS and is denoted mYF. These modifications give a continuous "modified" action over the whole energy range. The multidimensional versions of the mVPT2 and mYF theories have been applied to the collinear H + H2 and D + H2 reactions to calculate thermal reaction rates. The results show that the rates computed using the mVPT2 theory are marginally better than those computed by the mYF theory. The corresponding kinetic isotopic effects have also been computed. Both the theories account for the correct ℏ2 limit at high temperature and not the parabolic barrier limit as in various other theories. The mVPT2 and mYF theories also improve upon the thermal rates in the low temperature limit due to the shifting of the action by the zero point energy shift E0. The resulting theory is more accurate than the ring polymer molecular dynamics based approximation over the whole temperature range probed. The results presented here indicate that the multidimensional version of the modified VPT2 theory may be the recommended method for computing thermal tunneling rates in multidimensional systems.

Estimating tunneling splittings is a long-standing quantum mechanical challenge for theoretical methods. Sometimes splittings are so small, i.e., within a fraction of a wavenumber, pushing the limits of experimental detection and computational precision. Currently, most computational methods are able, at best, to obtain only ground-state tunneling splittings, either for symmetric or asymmetric potentials. In this Letter, we introduce a unified theoretical approach, based on a two-state approximation that can be equally applied to symmetric and asymmetric diabatic potential crossing and for excited states, providing reliable estimates even for states near the energy crossing. The method opens the door to analytic approximations for the tunneling splitting of model potential systems. It provides a framework for the introduction of vibrational perturbation theory to the estimation of nonadiabatic tunneling splittings. It also provides new insight into the semiclassical theory, leading to an instanton based steepest descent expression applicable also to excited states. Numerical tests on model systems are promising, providing the groundwork for implementation to future multidimensional applications.

The central topic of this letter is to show that light-matter hybridization not only gives rise to novel dynamic responses but can also modify intermolecular interactions and induce new structural order. Using the van der Waals (vdW) system in an optical cavity as an example, we predict the effects of quantum interference and collectivity in cavity-induced many-body dispersion forces. Specifically, the leading order correction due to cavity-induced quantum fluctuations leads to 3-body and 4-body vdW interactions, which can align intermolecular vectors and are not pairwise additive. In addition, the cavity-induced dipole leads to a single-molecule energy shift that aligns individual molecules, and a pairwise interaction that scales as R-3 instead of the standard R-6 distance scaling. The coefficients of all these cavity-induced interactions depend on the cavity frequency and are renormalized by the effective Rabi frequency, which in turn depends on the particle density. Finally, we study the interaction of the vdW system in a cavity with an external object and find a significant enhancement in the interaction range due to modified distance scaling laws. These theoretical predictions suggest the possibility of cavity-induced nematic or smectic order and may provide an essential clue to understand intriguing phenomena observed in optical cavities, such as strongly modified ground-state reactivity, ion transport and charge mobility.

The uniform semiclassical expression for the energy-dependent transmission probability through a barrier has been a staple of reaction rate theory for almost 90 years. Yet, when using the classical Euclidean action, the transmission probability is identical to 1/2 when the energy equals the barrier height since the Euclidean action vanishes at this energy. This result is generally incorrect. It also leads to an inaccurate estimate of the leading order term in an ℏ2n expansion of the thermal transmission coefficient. The central result of this paper is that adding an ℏ2 dependent correction to the uniform semiclassical expression, whether as a constant action or as a shift in the energy scale, not only corrects this inaccuracy but also leads to a theory that is more accurate than the previous one for almost any energy. Shifting the energy scale is a generalization of the vibrational perturbation theory 2 (VPT2) and is much more accurate than the "standard" VPT2 theory, especially when the potential is asymmetric. Shifting the action by a constant is a generalization of a result obtained by Yasumori and Fueki (YF) only for the Eckart barrier. The resulting modified VPT2 and YF semiclassical theories are applied to the symmetric and asymmetric Eckart barrier, a Gaussian barrier, and a tanh barrier. The one-dimensional theories are also generalized to many-dimensional systems. Their effect on the thermal instanton theory is discussed.

The combination of vibrational perturbation theory with the replacement of the harmonic oscillator quantization condition along the reaction coordinate with an imaginary action to be used in the uniform semiclassical approximation for the transmission probability has been shown in recent years to be a practical method for obtaining thermal reaction rates. To date, this theory has been developed systematically only up to second order in perturbation theory. Although it gives the correct leading order term in an (h) over bar (2) expansion, its accuracy at lower temperatures, where tunneling becomes important, is not clear. In this paper, we develop the theory to fourth order in the action. This demands developing the quantum perturbation theory up to sixth order. Remarkably, we find that the fourth order theory gives the correct (h) over bar (2) term in the expansion of the exact thermal rate. The relative magnitude of the fourth order correction as compared to the second order term objectively indicates the accuracy of the second order theory. We also extend the previous modified second order theory to the fourth order case, creating an (h) over bar (2) modified potential for this purpose. The resulting theory is tested on the standard examples-symmetric and asymmetric Eckart potentials and a Gaussian potential. The modified fourth order theory is remarkably accurate for the asymmetric Eckart potential. (c) 2024 Author(s).

Reaction rate theory has been at the center of physical chemistry for well over one hundred years. The evolution of the theory is not only of historical interest. Reliable and accurate computation of reaction rates remains a challenge to this very day, especially in view of the development of quantum chemistry methods, which predict the relevant force fields. It is still not possible to compute the numerically exact rate on the fly when the system has more than at most a few dozen anharmonic degrees of freedom, so one must consider various approximate methods, not only from the practical point of view of constructing numerical algorithms but also on conceptual and formal levels. In this Perspective, I present some of the recent analytical results concerning leading order terms in an ℏ2m series expansion of the exact rate and their implications on various approximate theories. A second aspect has to do with the crossover temperature between tunneling and thermal activation. Using a uniform semiclassical transmission probability rather than the "primitive" semiclassical theory leads to the conclusion that there is no divergence problem associated with a "crossover temperature." If one defines a semiclassical crossover temperature as the point at which the tunneling energy of the instanton equals the barrier height, then it is a factor of two higher than its previous estimate based on the "primitive" semiclassical approximation. In the low temperature tunneling regime, the uniform semiclassical theory as well as the "primitive" semiclassical theory were based on the classical Euclidean action of a periodic orbit on the inverted potential. The uniform semiclassical theory wrongly predicts that the "half-point," which is the energy at which the transmission probability equals 1/2, for any barrier potential, is always the barrier energy. We describe here how augmenting the Euclidean action with constant terms of order ℏ2 can significantly improve the accuracy of the semiclassical theory and correct this deficiency. This also leads to a deep connection with and improvement of vibrational perturbation theory. The uniform semiclassical theory also enables an extension of the quantum version of Kramers' turnover theory to temperatures below the "crossover temperature." The implications of these recent advances on various approximate methods used to date are discussed at length, leading to the conclusion that reaction rate theory will continue to challenge us both on conceptual and practical levels for years to come.

The modified version of second and fourth order vibrational perturbation theory, whereby the Euclidean action for tunneling is computed on the inverted potential at a shifted energy that is ℏ2 dependent, is applied to a symmetric double well quartic potential. The mean energies of the doublets in each well are also computed using vibrational perturbation theory. Results show that the modified vibrational perturbation theory significantly improves the estimates of tunneling splitting energies both for the ground state and for excited state doublets.

Understanding quantum tunneling and above-barrier reflection effects on unimolecular and bimolecular reaction rate constants remains challenging to this very day. In many applications, especially when considering moderate-to-high temperatures, the "standard" procedure is to use the parabolic barrier approximation. Recent work has shown though that this may be insufficient, and one cannot ignore anharmonicity. In this work, we study the analytic theory, including anharmonicity obtained when expanding the thermal rate up to order (sic)(4). Such theories need high-order derivatives of the potential at the barrier top. We show that such derivatives are computed straightforwardly for six different reactions. We suggest a straightforward methodology for assessing whether the parabolic barrier approximation is valid and show that when the reaction asymmetry is large, this may lead to significant quantum above-barrier reflection and transmission coefficients, which are less than unity.

The quantum version of Kramers turnover theory is generalized beyond the parabolic barrier approximation. The result is a uniform instanton-based quantum Kramers turnover theory that does not display any divergence at what is known as the crossover temperature. The theory is analyzed using a model of a particle trapped in a cubic potential. As the temperature is lowered, the maximum in the Kramers turnover curve moves to lower friction values. When the temperature is sufficiently low, the quantum rate at low friction becomes almost independent of the friction strength.

Using semiclassical methods, an analytical approach to describe grazing incidence scattering of fast atoms (GIFAD) from surfaces is described. First, we consider a model with a surface corrugated in the scattering plane, which includes the surface normal and the incidence direction. The treatment uses a realistic, Morse potential, within a perturbation approach, and correctly reproduces the basic GIFAD phenomenology, whereby the scattering is directed primarily in the specular direction. Second, we treat the more general case of scattering from a surface corrugated in two-dimensions. Using time averaging along the direction of fast motion in the incidence direction, we derive a time dependent potential for the GIFAD scattering away from a low index direction. The results correctly describe the observation that diffraction is seen only when the scattering plane is aligned close to a low-index direction in the surface plane. For the case of helium scattering from LiF(001) we demonstrate that the resulting theoretical predictions agree well with experiment and show that the analysis provides new information on the scattering time and the length scale of the interaction. The analysis also gives insights into the validity of the axial surface channeling approximation (ASCA) and shows that within first order perturbation theory, along a low-index direction, the full 3-dimensional problem can be represented accurately by an equivalent 2-dimensional problem with a potential averaged along the third dimension. In contrast, away from low-index directions, the effective 2-dimensional potential in the projectile frame is time-dependent.

The uniform semiclassical expression for the energy-dependent transmission probability through a barrier has been a staple of reaction rate theory for almost 90 years. Yet, when using the classical Euclidean action, the transmission probability is identical to 1/2 when the energy equals the barrier height since the Euclidean action vanishes at this energy. This result is generally incorrect. It also leads to an inaccurate estimate of the leading order term in an (h) over bar (2n) expansion of the thermal transmission coefficient. The central result of this paper is that adding an PLANCK CONSTANT (h) over bar (2) dependent correction to the uniform semiclassical expression, whether as a constant action or as a shift in the energy scale, not only corrects this inaccuracy but also leads to a theory that is more accurate than the previous one for almost any energy. Shifting the energy scale is a generalization of the vibrational perturbation theory 2 (VPT2) and is much more accurate than the "standard" VPT2 theory, especially when the potential is asymmetric. Shifting the action by a constant is a generalization of a result obtained by Yasumori and Fueki (YF) only for the Eckart barrier. The resulting modified VPT2 and YF semiclassical theories are applied to the symmetric and asymmetric Eckart barrier, a Gaussian barrier, and a tanh barrier. The one-dimensional theories are also generalized to many-dimensional systems. Their effect on the thermal instanton theory is discussed.

The combination of vibrational perturbation theory with the replacement of the harmonic oscillator quantization condition along the reaction coordinate with an imaginary action to be used in the uniform semiclassical approximation for the transmission probability has been shown in recent years to be a practical method for obtaining thermal reaction rates. To date, this theory has been developed systematically only up to second order in perturbation theory. Although it gives the correct leading order term in an ℏ2 expansion, its accuracy at lower temperatures, where tunneling becomes important, is not clear. In this paper, we develop the theory to fourth order in the action. This demands developing the quantum perturbation theory up to sixth order. Remarkably, we find that the fourth order theory gives the correct ℏ4 term in the expansion of the exact thermal rate. The relative magnitude of the fourth order correction as compared to the second order term objectively indicates the accuracy of the second order theory. We also extend the previous modified second order theory to the fourth order case, creating an ℏ2 modified potential for this purpose. The resulting theory is tested on the standard examples—symmetric and asymmetric Eckart potentials and a Gaussian potential. The modified fourth order theory is remarkably accurate for the asymmetric Eckart potential.

In this Reply, we show that criticisms of perturbation theory for grazing-incidence fast-atom diffraction (GIFAD) are ill-founded. We show explicitly that our formulation (W. Allison, S. Miret-Artés and E. Pollak, Phys. Chem. Chem. Phys., 2022, 24, 15851) provides a similar precision in describing the observed phenomena as ab initio potentials. Since that is the main criterion to distinguish between methods, it seems reasonable to conclude that the perturbation approach using a Morse-type potential reproduces the essential aspects of the dynamics correctly. In addition we expand on the historical context and summarize the physical insights provided by our methods.

The instanton expression for the thermal transmission probability through a one-dimensional barrier is derived by using the uniform semiclassical energy-dependent transmission coefficient of Kemble. The resulting theory does not diverge at the "crossover temperature" but changes smoothly. The temperature-dependent energy of the instanton is the same as the barrier height when ℏβω‡ = π and not 2π as in the "standard" instanton theory. The concept of a crossover temperature between tunneling and thermal activation, based on the divergence of the instanton rate, is obsolete. The theory is improved by assuring that at high energy when the energy-dependent transmission coefficient approaches unity the integrand decays exponentially as dictated by the Boltzmann factor and not as a Gaussian. This ensures that at sufficiently high temperatures the uniform theory reduces to the classical. Application to Eckart barriers demonstrates that the uniform theory provides a good estimate of the numerically exact result over the whole temperature range.