Accurate Prediction of Nonadiabatic Transitions through Avoided Crossings

Abstract

In the case of one nuclear degree of freedom, we derive an explicit formula for the nuclear wave function transmitted through an avoided crossing, and show that it agrees to high accuracy with precise numerical calculations. DOI: 10.1103/PhysRevLett.103.213001 PACS numbers: 31.70.Hq, 31.50.Gh, 33.50.Hv The time-dependent Born-Oppenheimer approximation (BOA) is at the basis of our understanding of dynamics of molecules. The small ratio " 2 of electronic and nuclear mass allows one to replace the electronic degrees of freedom with an effective potential, and to separate the nuclear dynamics according to the electronic energy surfaces. This dramatically reduces the complexity of the problem.Despite its success, situations where the BOA fails are of great interest in quantum chemistry. These occur when electronic energy levels are not well separated for a given nuclear configuration. Important applications include photodissociation of diatomic molecules like NaI [1], or the reception of light in the retina [2]. Two basic types of failure occur: conical crossings as appearing in [2], and avoided crossings. The latter are typical for systems with one nuclear degree of freedom [1,3], and are the topic of this Letter. We give an explicit formula, cf., (11), for the transmitted wave function at a generic avoided crossing, using only data that is local in time and space. An algorithm for calculating this wave function is then straightforward.The importance of nonadiabatic transitions has led to many efforts to understand and predict them. A simplification of the problem is to replace the nuclear degree of freedom by a classical trajectory. It is both ancient [4] and well understood [5,6], and leads to the famous LandauZener (LZ) formula for the transition probabilities between electronic levels. This formula lies at the basis of several surface hopping models, such as [7][8][9]. While these and other [10] trajectory based methods yield reasonably good transition probabilities, their accuracy in predicting the shape of the transmitted wave function is limited [7]. An improvement to the LZ-transition rates, based on the full quantum scattering theory of the problem, is achieved by the Zhu-Nakamura theory [11]. Again, only transition rates are treated, and not the full transmitted wave function. Hagedorn and Joye [12] derive rigorous asymptotic formulas for the transmitted wave function, in the limit of " ! 0. However, these formulas are difficult to apply in practice and are neither local in time nor space.We consider a two level system with a Hamiltonian with one effective degree of freedom: H ¼ Àð1=2Þ" 2 @ 2 x I þ VðxÞ, where VðxÞ ¼ XðxÞ x þ ZðxÞ z þ dðxÞI is the real-symmetric potential energy matrix in the diabatic representation. I is the 2 Â 2 unit matrix, and x , z are the Pauli matrices. Units are such that @ ¼ 1 and the electron mass m el ¼ 1. " 2 is the ratio of electron and reduced nuclear mass.The time-dependent Schrödinger equation is given by i"@ t c ðx; tÞ ¼ Hc ðx; tÞ, with c 2 L 2 ðR; C 2 Þ, in the time ...

Non-adiabatic transitions near avoided crossings: theory and numerics

“…This "superadiabatic" approach is successful as long as ǫ is not too small, since the optimal number of terms in the Born-Oppenheimer expansion behaves like C/ǫ 2 . Also, the numerical results of [1,2] appear to be very good for values of ǫ that are not too small, but are not consistent with the rigorous results of [12] in the ǫ → 0 limit.…”

“…We now consider the time-dependent Schrödinger equation (3) with the matrix potential (1). While the theoretical investigations consider the asymptotics as ǫ → 0, concrete chemical systems have a given fixed ǫ.…”

“…Recently, a numerical technique [1,2] has been developed that is motivated by optimal truncation of the time-dependent Born-Oppenheimer expansion. The Born-Oppenheimer expansion concentrates on a single electron energy surface, and it is asymptotic, but not convergent.…”

“…The idea of optimal truncation is to take the expansion to the order that minimizes the error for any given ǫ. The central idea of [1] and [2] is to study nonadiabatic transitions between the optimally truncated wave functions. This "superadiabatic" approach is successful as long as ǫ is not too small, since the optimal number of terms in the Born-Oppenheimer expansion behaves like C/ǫ 2 .…”

Nonadiabatic Transitions Through Tilted Avoided Crossings

“…Under the reasonable assumption that the matrix elements X and Z are analytic functions of x at least close to the real axis, then so is ρ 2 . We write ρ(q) 2 = δ 2 + g(q) 2 , where g is analytic and g(0) = 0. Since g 2 is quadratic at 0, a Stokes line (a curve with Im(ρ) = 0) crosses the real axis perpendicularly, and, for small δ, extends into the complex plane to two complex Downloaded 11/27/14 to 142.157.65.132.…”