Baer M., Billing G.D. (eds.) - The role of degenerate states in chemistry (Adv.Chem.Phys. special issue, Wiley, 2002)

I.INTRODUCTION

Conical intersections are known to play a key role in nonrelativistic, spinconserving electronically nonadiabatic processes [1]. If we include the spin– orbit interaction we introduce new nonadiabatic pathways and unexpected complications. By coupling states of different spin-multiplicity the spin–orbit interaction gives rise to spin-nonconserving transitions while making conical intersections out of intersections that otherwise would not be. However, the spin– orbit interaction produces a more subtle but no less significant effect when the molecule in question has an odd number of electrons. Let Z be the dimension of the branching space, the space in which the conical topography is evinced. More precisely, the branching space is the smallest space in whose orthogonal complement the degeneracy is lifted only at quadratic or higher order in displacements, if it is lifted at all. The orthogonal complement of the branching space is referred to as the seam. The dimension of the seam is W ¼ Nint Z, where Nint is the number of internal coordinates. Here Z ¼ 2 in the nonrelativistic case and when the spin–orbit interaction is included, provided the molecule has an even number of electrons. However, for a molecule with an odd number of electrons, an odd electron molecule, the dimension of the branching space is 5, in general, or 3 when the system is restricted to Cs symmetry [2].

To study the ramifications of this change, the locus of the seam should be known. However, locating points on this seam of conical intersection is a challenging undertaking. To better understand the difficulty of the task, consider the history of the accidental same-symmetry intersection. For the nonrelativistic Coulomb Hamiltonian, conical intersections can be classified using the point group symmetry of the intersecting states. Intersections are symmetry-required, accidental symmetry-allowed, or accidental same-symmetry, according to whether the electronic states in question carry a multidimensional irreducible representation, distinct one-dimensional irreducible representations, or the same irreducible representation of the spatial point group. While the existence of same-symmetry conical intersections was firmly established over 70 years ago [3], until approximately a decade ago [4,5], virtually all conical intersections based on ab initio wave functions were determined with the help of symmetry. This is a consequence, in part, of the fact that to locate a single point of conical intersection for a same symmetry intersection a two-dimensional branching plane must be searched, whereas for an accidental symmetry-allowed intersection only an one-dimension search is required. Indeed, it was only in the last decade, after the introduction of efficient algorithms [6,7] for locating samesymmetry intersections, that their true significance began to emerge. The situation for odd electron molecules when the spin–orbit interaction is included in the Hamiltonian is similar, but even more extreme, since a five- (or three-) dimensional branching space must be searched.

Contrary to this gloomy assessment, it is rapidly becoming possible to describe non-adiabatic processes driven by conical intersections, for which the spin–orbit interaction cannot be neglected, on the same footing that has been so useful in the nonrelativistic case. An effective algorithm for locating points of conical intersection for odd electron molecules has been developed [8] and an analytic representation of the energies and derivative couplings in the vicinity of these points of conical intersection has been determined [9,10] based on degenerate perturbation theory [11,12]. These advances, in addition to providing conceptual insights, will lead to a more rigorous approach to nonadiabatic dynamics whose computational utility increases with the size of the spin–orbit interaction.

In this chapter, recent advances in the theory of conical intersections for molecules with an odd number of electrons are reviewed. Section II presents the mathematical basis for these developments, which exploits a degenerate perturbation theory previously used to describe conical intersections in nonrelativistic systems [11,12] and Mead’s analysis of the noncrossing rule in molecules with an odd number of electrons [2]. Section III presents numerical illustrations of the ideas developed in Section II. Section IV summarizes and discusses directions for future work.

II.THEORY

A. The Electronic Hamiltonian

In this work, relativistic effects are included in the no-pair or large component only approximation [13]. The total electronic Hamiltonian is Heðr; RÞ ¼ H0ðr; RÞ þ Hsoðr; RÞ, where H0ðr; RÞ is the nonrelativistic Coulomb Hamiltonian and Hsoðr; RÞ is a spin–orbit Hamiltonian. The relativistic (nonrelativistic) eigenstates, ei ð 0I Þ, are eigenfunctions of Heðr; RÞðH0ðr; RÞÞ. Lower (upper)

case letters will be used to denote eigenfunctions He ðH0Þ. A point of conical intersection of states i; j of He½I; J of H0& will be denoted Rx; ij½Rx; IJ &.

B.Time-Reversal Symmetry

Table I summarizes the differences in the dimension of the branching space. The origin of these differences is the behavior of the wave functions under

TABLE I

Z the Dimension of Branching Space

aZ ¼ 3 when Cs symmetry is present.

time-reversal symmetry [14,15]. The time-reversal operator, T is an antiunitary operator, that is, hfjci ¼ hTfjTci. In addition T2 ¼ þ1ð 1Þ if the number of electrons is even (odd). For a molecule with an odd number of electrons

hfjTfi ¼ hTfjT2fi ¼ hTfjfi ¼ hfjTfi so that hfjTfi ¼ 0 ð1aÞ

that is, f and Tf are orthogonal and degenerate, since T commutes with He. This degeneracy owing to time-reversal symmetry is referred to as Kramers’ degeneracy [16]. For a molecule with an even number of electrons, f and Tf are linearly dependent. With the choice f ¼ Tf

so that hfjHeci is real valued. For an odd electron system

so that, for example, for f ¼ c; hfjHeTfi ¼ 0.

A set of functions will be referred to as time-reversal adapted, provided that for each f in the set Tf is also in the set.

We are now in a position to explain the results of Table I. As a consequence of the degeneracy of f and Tf, at a conical intersection there are four degen-

erate functions ei ; ej and T ei eTi; T ej eTj. By using Eq. (1c), an otherwise arbitrary Hermitian matrix in this four function time-reversal adapted

basis has the form

When Cs symmetry is present, ek and T ek k ¼ i; j can be chosen to transform according to the e0 and e00 irreducible representations of the Cs double group (see Table II) so that HiTje ðRÞ ¼ 0 by symmetry. In this case, there are only three conditions for a degeneracy Hjie ðRÞ ¼ 0 and Hije ðRÞ ¼ 0; He is block diagonal

and clearly evinces Kramers’ degeneracy. Finally, for the even electron case the T ei are linearly dependent and only one of the diagonal blocks survives,

and Hije is real valued, so only two conditions need be satisfied.

This analysis is heuristic in the sense that the Hilbert spaces in question are in general of large, if not infinite, dimension while we have focused on spaces of dimension four or two. A form of degenerate perturbation theory [3] can be used to demonstrate that the preceding analysis is essentially correct and, to provide the means for locating and characterizing conical intersections.

C.Perturbation Theory

ei is expanded in a basis of time-reversal adapted configuration state functions [8] (TRA–CSFs, we)

NXCSF

The di dr; i þ idi; i are the solution of the electronic Schro¨dinger equation in the TRA–CSF basis

where He ¼ He; r þ iHe; i. Near Rx; ij the eigenvalue problem in Eq. (4) can be simplified with the use of a crude adiabatic basis

NXCSF

HeðRÞ ¼ HeðRx; ijÞ þ rHeðRx; ijÞ dR þ 1=2dR rrHeðRx; ijÞ dR ð6Þ

where dR ¼ R Rx; ij. Reexpressing this result in the crude adiabatic basis gives

where y denotes the complex conjugate transpose,

EeðRx; ijÞkl ¼ dklEleðRx; ijÞ ð7dÞ

a single (double) bar under a quantity denotes a vector (matrix) of matrices, so that

The ek are expanded in the crude adiabatic basis

XX

where Q is spanned by the degenerate functions at Rx;ij and P is its orthogonal complement. To describe the vicinity of a conical intersection we require the

first-order contributions in dR to Eq. (4b). To accomplish this, we expand Eei ðRÞ; nðRÞ; NðRÞ in powers of dR, giving

In Eq. (11b), we observed that since the crude adiabatic basis is used Nð0Þk ¼ 0, for keQ. Therefore the degeneracy is lifted at first order in the Q-space only, which is therefore used to identified the branching space. The first-order result is

Equation (12) and the qualifying equalities, Eqs. (8) and (9) are the lynchpin for the remainder of this work.

D.Perturbation Theory, Time-Reversal Symmetry,

and Conical Intersections

To procede further, it is essential to distinguish between even and odd electron systems. While Eq. (12) is formally independent of the dimension of Q, in the

Equation (13) and definitions (8), (9), and (11) enable a description of the energy near, and the singular part of the derivative coupling at, Rx;ij.

E.Conical Intersections: Location

where

EijðR0Þ ¼ Re½ðdiðR0Þ þ djðR0ÞÞyHeðR0ÞðdiðR0Þ djðR0ÞÞ& ¼ V1ðR0Þ

ð14dÞ

Equations (14a)–(14d) form the basis for our algorithm for locating conical intersections. However, these equations determine only five (or three when Cs symmetry can be imposed) internal nuclear coordinates. Determination of any remaining internal degrees of freedom requires additional constraints. We employ the approach used in our algorithm for determining seams of conical intersection for the nonrelativistic Hamiltonian [7] where geometrical constraints KiðRÞ ¼ 0, and/or minimization of the energy of the crossing, provide the