Wave packet dynamics in the optimal superadiabatic approximation

Abstract

We explain the concept of superadiabatic representations and show how in the context of electronically non-adiabatic transitions they lead to an explicit formula that can be used to predict transitions at avoided crossings. Based on this formula, we present a simple method for computing wave packet dynamics across avoided crossings. Only knowledge of the adiabatic potential energy surfaces near the avoided crossing is required for the computation. In particular, this means that no diabatization procedure is necessary, the adiabatic electronic energies can be computed on the fly, and they only need to be computed to higher accuracy when an avoided crossing is detected. We test the quality of our method on the paradigmatic example of photo-dissociation of NaI, finding very good agreement with results of exact wave packet calculations.

Nonadiabatic Transitions in Multiple Dimensions

“…Applications of the one dimensional formula have been widely successful on a variety of examples. In [9], the formula is used to accurately approximate the transmitted wavepacket for sodium iodide. Tilted avoided crossings have also been examined, and a formula developed which is dependent on the value n, so the optimal superadiabatic representation must be calculated.…”

“…In contrast, in [10] and [7], a formula is derived to accurately approximate the full transmitted wavepacket, in one dimension, using only decoupled dynamics. The formula has been applied to a variety of examples with accurate results, including the transmitted wavepacket due to photo-dissociation of sodium iodide [9].…”

Multiple Superadiabatic Transitions and Landau-Zener Formulas

“…These results, which are restricted to the semiclassical regime where the nuclei move classically, were later made rigorous 40,41 . It was later shown that, through the use of such superadiabatic representations (which are generalisations of the well-known adiabatic representation), it is possible to derive a formula for the transmitted wavepacket, including phase, at an avoided crossing [42][43][44][45] . The associated algorithm requires only the quantum evolution of wavepackets on single energy surfaces.…”

“…Whilst this is still computationally demanding if one wants to solve the full Schrödinger equation, there are approximate methods, such as Hagedorn wavepackets 36,46,47 or standard quantum chemistry techniques such as MCTDH 48 , which make small relative errors and are computationally much more tractable. The associated algorithm has so far been applied to single transitions through avoided crossings [42][43][44] , and to multiple transitions of a single crossing in the case of the photodissociation of NaI 45 . The main goals here are to extend the methodology to multiple transitions through different avoided crossings and to systematically study the effects of making various approximations that lead to a LZ-like transition probability.…”

“…However, in some regimes, for example when the LZ formula is accurate, we can bypass this requirement; (iii) The scheme is, in principle, restricted to wavepackets that are semiclassical near the avoided crossing. However, due to the linearity of the Schrödinger equation, and as demonstrated in 45 , it is possible to 'slice' the wavepacket at the crossing. However, this may be more problematic in higher dimensions; (iv) As it stands, the method is restricted to 1D.…”