Seminar: Eli Pollak

The Simons Center for Computational Physical Chemistry at NYU regularly hosts visiting scholars to discuss their work.  Join us on December 8th for a presentation by Eli Pollak of the Weizmann Institute:

Various aspects of the theory of quantum tunneling

All Simons Center seminars are held in Waverly 540.  Refreshments will be served at 10:45, and the seminar begins promptly at 11:00 AM ET.      

Or join via Zoom:  https://nyu.zoom.us/j/99318701420?pwd=eGVvSzlKWFRlV0ZldnJJbjhYVUtEQT09

Abstract:

Quantum tunneling was discovered by Friedrich Hund in 1927¹. The semiclassical theory of tunneling was developed by Wigner in 1932² and Kemble in 1935³. Yet, even after close to a century, quantum tunneling is an enigma. Analytic results are few and far between. Modern algorithms for computing tunneling rates are incorrect even at high temperatures. The semiclassical low temperature theory of deep tunneling, initiated by Miller⁴ in terms of instanton trajectories – periodic orbits moving on the inverted potential, suffered from a divergence at the so called crossover temperature between tunneling and thermal activation. Kemble’s uniform semiclassical theory predicted that the transmission probability through a barrier will always be ½ when the energy equals the barrier energy. This is not the case for simple models such as a square barrier or Eckart potentials, yet this “half point” problem remained unanswered.

Tunneling rates have been measured in many chemical reactions and surface diffusion. Heavy atom tunneling is an active area of experimental research, tunneling splittings in many symmetric molecules have been measured with high spectroscopic accuracy. It is thus not surprising that tunneling remains an intriguing topic to this very day.

In this talk I will describe our recent solutions to these challenges. An analytic theory for the tunneling probability up to order ħ4 will be presented⁵. The divergence inherent to instanton theory will be removed⁶, with the conclusion that the crossover temperature between tunneling and thermal activation is twice as high as the one considered by Affleck⁷. The half point puzzle will be solved.⁸ Some recent applications and implications of the resulting modern theory of tunneling will be discussed, not the least of which is a critique of present-day numerical algorithms for the computation of tunneling rates⁹.

The theory of tunneling splitting has challenged us ever since Hund discovered tunneling in double well potentials in 1927. There is a difference between tunneling splitting as obtained on a single (adiabatic) electronic state to non-adiabatic induced coupling between two (or more) electronic states which then induces tunneling splitting. We have recently made some progress on these topics¹⁰, by introducing a simple two-state approximation that can be equally applied to symmetric and asymmetric diabatic potential crossing and for excited states. This leads 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 method applicable also to excited states. It also calls into questions predictions based on single adiabatic potentials. The two state approximation will be used¹¹ for tunneling splitting on a single surface, to show that it is identical to the well known Herring approximation¹². Finally, it will also be used to study the effect of a cavity on tunneling splitting¹³.

1 F. Hund, Z. Phys. 40, 742 (1927), Z. Phys. 43, 805 (1927).

2 E. Wigner, Z. Phys. Chem. 19B, 203 (132).

3 E.C. Kemble, Phys. Rev. 48, 549 (1935).

4 W. H. Miller, J. Chem. Phys. 62, 1899 (1975).

5 E. Pollak and J. Cao, J. Chem. Phys. 157, 074109 (2022), Phys. Rev. A 107, 022203 (2023), 109 039901 (2024).

6 S. Upadhyayula and E. Pollak, J. Phys. Chem. Lett. 14, 9892 (2023).

7 I. Affleck, Phys. Rev. Lett. 46, 388 (1981).

8 S. Upadhyayula and E. Pollak, J. Phys. Chem. A. 128, 3434-3448 (2024); E. Pollak and S. Upadhyayula, J.

Chem. Phys. 160, 184110 (2024).

9 E. Pollak, A personal perspective of the present status and future challenges facing thermal reaction rate

theory, J. Chem. Phys. 160, 150902 (2024).

10 L. Raso, M. Ceotto and E. Pollak, preprint, J. Phys. Chem. Lett., in press.

11 E. Pollak and M. Ceotto, Theory of tunneling splitting in symmetric double well systems: Equivalence of the

two-state approximation and the Herring formula, Phys. Rev. A, 112 032225 (2025).

12 C. Herring, Rev. Mod. Phys. 34, 631 (1962).

13 E. Pollak and J. Cao, The effect of a cavity on diabatic tunneling in a symmetric double well potential,

preprint, submitted to J. Chem. Phys..