Extended Born-Oppenheimer equation for a three-state system

Abstract

We present explicit forms of nonadiabatic coupling (NAC) elements of nuclear Schrodinger equation (SE) for a coupled three-state electronic manifold in terms of mixing angles of real electronic basis functions. If the adiabatic-diabatic transformation (ADT) angles are the mixing angles of electronic bases, ADT matrix transforms away the NAC terms and brings diabatic form of SE. ADT and NAC matrices are shown to satisfy a curl condition with nonzero divergence. We have demonstrated that the formulation of extended Born-Oppenheimer (EBO) equation from any three-state BO system is possible only when there exists a coordinate-independent ratio of the gradients for each pair of mixing angles. On the contrary, since such relations among the mixing angles lead to zero curl, we explore its validity analytically around conical intersection(s) and support numerically considering two nuclear-coordinate-dependent three surface BO models. Numerical calculations are performed by using newly derived diabatic and EBO equations and expected transition probabilities are obtained.

Topological effects for nonsymmetrical configurations: The C2H2+ as a case study

“…It is noticed that in all four cases both 12 and 45 have values that oscillate in the vicinity of ϳ 0.5 rad −1 , whereas the other -matrix elements, namely, 13 , 23 , 34 , and 35 have, on the average, much smaller values. Among those we see that the NACTs related to two upper states, i.e., 34 and 35 , are much smaller than the corresponding ones related to the lower states, i.e., 23 and 34 .…”

“…͑2͒ It is well known that topological effects are formed in the vicinity of a point of degeneracy between two adjacent states ͑points of degeneracy are better known as the points of conical/parabolic intersections͒. However, two states may not be enough to form topological effects for larger regions in configuration space 17,25,27,34,35 ͑see also Ref. 33, Chaps.…”

“…[17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35] The two types differ also in two additional ways: ͑a͒ the BO-RT nonadiabatic coupling terms ͑NACTs͒ are formed by states of opposite symmetry and the resulting topological ͑Berry͒ phase ␣ is equal to 2 . [2][3][4]7 ͑b͒ The BO-JT NACTs are formed by states of same symmetry and the resulting JT phase ␣ is equal to ͑Refs.…”

A rigorous approach to the formulation of extended Born‐Oppenheimer equation for a three‐state system

“…Moreover, it predicts that EBO equations can provide meaningful solution only when eigenvalues of NAC matrix are gauge-invariant. Sarkar et al [23] has performed a generalized BO treatment of any three coupled electronic states and formulated an approximate EBO in terms of electronic basis functions angles. The same article also finds that calculated results obtained by using diabatic and approximate EBO equations for various chosen forms of NAC elements either have excellent or good agreement as the eigenvalues of NAC matrix are either exactly or approximately gauge invariant, respectively.…”

“…Since the necessary condition to derive the EBO equations is the existence of a relation among the ADT angles implicating zero Curls (apart from Curl conditions being satisfied) at least around the CIs, we briefly present the analytical proof for the validity of such relations considering nuclear coordinates dependent ADT angles. The major aim of this article is to formulate a rigorous EBO equation in terms of ADT angles and to explore it's necessity with respect to the approximate one [23]. In this context, we wish to construct two models by choosing various sets on the form of the ADT angles vis-a-vis NAC terms and thereby, to find out the validity of the rigorous EBO equation with respect to the diabatic one and its' advantage, if any, with respect to the approximate EBO equation [23].…”