The Electronic Adiabatic-to-Diabatic ... - ACS Publications

Chapter 18

Downloaded by UNIV MASSACHUSETTS AMHERST on September 14, 2012 | http://pubs.acs.org Publication Date: August 14, 2002 | doi: 10.1021/bk-2002-0828.ch018

The Electronic Adiabatic-to-Diabatic Transformation Matrix and the Irreducible Representation of the Rotation Group Michael Baer Department of Applied Physics, Soreq NRC, Yavne 81800, Israel

In this publication we consider the electronically multi-fold degeneracy with the aim of revealing the connection between the adiabatic-to-diabatic transformation matrices and Wigner's irreducible representation of the rotation group. To form the connection we constructed simplified models of two, three and four states, all (electronically) degenerate at a single point, and we employed the relevant non-adiabatic coupling matrices. We found that once these matrices are properly quantized (Baer, M. Chem. Phys. 259,123,2000) the adiabatic-to-diabatic transformation matrices and Wigner's d -rotation matrices are related via a similarity transformation. j

I. Introduction During the last few years major efforts were made to understand the features of the non-adiabatic coupling terms (NACTs) and their role in the theory of molecular physics (7-9). The NACTs owe their existence to the BornOppenheimer (BO) assumption which says that the fast moving electrons can be treated separately from the (assumed) slowly moving nuclei (10,11). The B O treatment seems to be somewhat artificial and somewhat too mathematical but

© 2002 American Chemical Society

In Low-Lying Potential Energy Surfaces; Hoffmann, M., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 2002.

361


362 nevertheless it justified itself in endless experiments in spectroscopy and scattering processes. Accepting the B O separation leads to an unavoidable encounter with these unique unresolved mathematical functions. The N A C T s are characterized by two features: They are vectors (in contrast to potentials that are scalars) and they can become singular (in contrast to potentials which cannot). If arranged in matrices they acquire a third interesting feature, namely, the matrices are antisymmetric. It could be that the NACTs, compared to potential energy surfaces, are less physical and more mathematical objects. This belief can, eventually, be justified if the unique feature of the N A C T s namely their being singular were only rarely encountered in ab-initio treatments. In fact what happens is that singular N A C T s are found all over configuration space (CS) (9,72), in large quantities, much more than ever anticipated. The fact that they are so numerous and are expected to play a dominant role in any study related to excited states calls for a better knowledge of these entities and a deeper understanding of their abilities. The ordinary way to get acquainted with objects like the N A C T s is to derive them from first principles, via ab-initio calculations (13-16), and probe their spatial structure - somewhat reminiscent of the way potential energy surfaces (PES) are studied. However, in contrast to PESs, this way is, by far, not enough. The fact that the N A C T s are so frequently singular in addition to being vectors calls for more mathematics oriented approaches in order to understand their role in molecular physics. A very important methodology in this respect is to assume ad-hoc models and to look for the features these N A C T s have to fulfill in order to be, indeed, appropriate for molecular systems (17-23). During the last decade we followed both courses but our main interest was pointing towards the physical-mathematical features of the N A C T s (17,18,20,24-26). In this process we revealed the necessity of being able to form sub-Hilbert spaces (SHS) in the given region of interest in CS and the fact that the non-adiabatic coupling matrix ( N A C M ) has to be quantized for this SHS i f they are supposed to have physical significance (20,25). This quantization requirement enhanced the recognition that the N A C M s are closely related, much more than anticipated, to the angular momentum operators. In the present article this approach is pursued. In particular we show, for special cases, that the adiabatic-to-diabatic transformation (ADT) matrix is closely related to Wigner's d rotation matrices that form the irreducible representation of the rotation group (27,28). j

The model we consider for this purpose is a case of Ν surfaces all degenerate at one single point in CS where the N A C M t(s) is of the following form (20): X(s) = gt(s)


(1)

363 Here g is a N x N anti-symmetric constant matrix, t(s) is a vector and s is a point in CS. In what follows we consider integrals along contours. In particular we shall be interested in the integral:

Y(S) = JdB'.tiO

(2)


0 carried out along Γ where both points, i.e. s=0 and s itself are on Γ. Here ds is the vectorial differential length along the contour and the dot stands for a scalar product. Next we introduce the angle α defined as the value of the above integral for a closed contour Γ. Thus: CC = j ds-t(s)

(3)

T

In what follows we shall be interested only in contours that surround the above mentioned point of degeneracy. Next it is assumed that for each chosen contour the elements of the g-matrix are arranged in such a way that the vector t(s) is guaranteed to be normalized as follows: α(Γ) = 2 π

(4)

This implies that for each such a contour the g-matrix may have a different set of constant elements (although the elements of the t(s)-matrix will be unchanged). In the present article we derive conditions to be fulfilled by the g-matrix elements in order for the x(s)-matrix to be a matrix of physical meaning. This we do in three steps: (a) first for the general case, next (b) for the T(s)-matrix defined in Eq. (1), and finally (c) for the three special cases, namely: the twodegenerate case (N=2), the tri-degenerate case (N=3) and the tetra-degenerate case (N=4). Whereas the first two cases were already worked out by us on various occasions (17,18,20), the third case is not only new but also presents new interesting features different from the ones obtained in these first two studies.

II. Theoretical Background In their treatment of the mixed systems of nuclei and electrons Born and Oppenheimer derived the Schroedinger equation (SE) for the nuclei which in present day notation can be written as (20,25,29): 1 2

(V +τ ) Ψ+(α-£)ψ=0

2m


K

)

364 where V is the usual (mass-scaled) gradient operator, Ε is the total energy, Ψ is a column matrix which contains the nuclear functions {ψί; i=l,2...}, u is a diagonal matrix which contains the adiabatic potentials and τ is a matrix which contains the above mentioned NACTs:


(6) This derivation holds for a complete Hilbert space but it can be shown to hold also for a SHS of finite dimension M (20,25) if and only if certain conditions are fulfilled (26). From Eq. (5) it is noticed that τ, like the grad operator itself, is a momentum operator. With respect to that, a relevant question is: What happens if one integrates over τ along a closed contour Γ? In other words, is there any demand to be associated with the following differential dA:

The Electronic Adiabatic-to-Diabatic ... - ACS Publications

Recommend Documents