Theory of tunneling splitting in symmetric double well systems: Equivalence of the two-state approximation and the Herring formula

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.