Reviews in Computational Chemistry

Table 15 Direct Product Representations for Space and Spin Parts of Singlet and Triplet Wave Functions in C2v Symmetry

Note that there is no contribution from the near-degenerate first excited singlet state because states of equal spatial symmetries do not interact via SOC in C2v and higher point groups.

The evaluation of the transition moment is straightforward now. Even though the energy difference between a3B1 and 11A2 is small and the Tx level is strongly perturbed, the dipole transition to the ground state is forbidden as long as the molecule remains planar because of the dipole selection rules: this transition would require an operator of A2 symmetry, but x^, y^, and ^z transform like B1, B2, and A1, respectively. The transition may gain some intensity due to second-order spin–vibronic interactions, however.

A careful inspection of the spin-perturbation coefficients a2 and c1 shows that their absolute values are equal but they differ in sign, that is, a2 ¼ c1. It is

was shown to make the major contribution to the transition probability

b1 þg ! X3 g in O2.182,183

Again expressing W in units of reciprocal seconds ðs 1Þ, E in reciprocal centimeters ðcm 1Þ, and mmag in atomic units (h), the probability of a magnetic

Nonradiative Transitions

Bound electronic states exhibit a discrete spectrum of rovibrational eigenstates below the dissociation energy. The interaction between discrete levels of two bound electronic states may lead to perturbations in their rovibrational spectra and to nonradiative transitions between the two potentials. In the case of an intersystem crossing, this process is often followed by a radiative depletion. Above the dissociation energy and for unbound states, the energy is not quantized, that is, the spectrum is continuous. The coupling of a bound state to the vibrational continuum of another electronic state leads to predissociation.

Apart from the selection rules for the electronic coupling matrix element, spin-forbidden and spin-allowed nonradiative transitions are treated completely analogously. Nonradiative transitions caused by spin–orbit interaction are mostly calculated in the basis of pure spin Born–Oppenheimer states. With respect to spin–orbit coupling, this implies a diabatic behavior, meaning that curve crossings may occur in this approach. The nuclear Schro¨ dinger equation is first solved separately for each electronic state, and the rovibronic states are spin–orbit coupled then in a second step.

In principle, static or dynamic approaches can be employed to describe the nonradiative transition between two states, although dynamic approaches are generally preferable. In a time-independent formalism, the coupling manifests itself in a mixing between vibrational levels of the involved electronic states. The relative probability of finding the molecule in a particular state is proportional to its squared expansion coefficient. In the language of timedependent perturbation theory, the coupling is expressed as a propagation back and forth between vibronic states. The equivalence between the static and dynamic picture holds as long as there are no other fast depletion mechanisms. In particular, the radiative lifetimes of the interacting states have to be long enough for an equilibration. If one of the states is short-lived, the nonradiative transition occurs in essence only in one direction. The theory and computational treatment of the coupling between multidimensional potential energy surfaces have recently been reviewed by Ko¨ ppel and Domcke.184

Here, we confine the discussion to cases that can be reduced to one dimension. As far as bound-continuum interactions are concerned, we restrict ourselves to weak interactions. This condition is mostly fulfilled for

188 Spin–Orbit Coupling in Molecules

spin-forbidden processes. Otherwise the full apparatus of scattering theory has to be invoked, which is far beyond the scope of this chapter.

Bound–Bound Interactions

Perturbations in the spectrum involving a pair of vibrationally bound states occur only occasionally, for instance, when the vibrational ladders of the electronic states approximately match in energy. The mixing coefficients can be calculated from the potential energy curves of the states involved and the electronic coupling matrix element for various values of the vibrational coordinate. As noted previously, the Schro¨ dinger equation for the vibrational motion is solved separately for each of the bound states. Their solutions are taken as the basis for a matrix representation of the interaction Hamiltonian. To ease the integration over the vibrational coordinate Q, the electronic coupling matrix element X is often approximated by an analytic function of Q, mostly a polynomial or a cubic spline function. The matrix elements of the perturbation matrix are then calculated in a way similar to the vibrational averaging of spectroscopic parameters described in the previous section on the Comparison of Fine-Structure Splittings with Experiment. Instead of employing the same vibronic level on both sides, now

is evaluated, where XðQÞ is the off-diagonal electronic coupling, and vi and v0j belong to different electronic states. Diagonalization of this matrix directly yields energies and mixing coefficients of the levels.

In practical applications, one often combines experimental and theoretical information. Potential energy curves, deduced from experiment with high accuracy, if available, are employed together with theoretically determined Q dependent coupling matrix elements. As the degree of mixing between vibronic states strongly depends on their energetic separation, even uncertainties in the electronic excitation energy of some 10 wavenumbers—which is a small error by present-day standards—are often sufficient to change the order-of- magnitude for a transition rate. Coupling matrix elements, on the other hand, can be computed with high accuracy, whereas they are not really available from experimental data.

From Eq. [239], it is apparent that the size of a particular Hij is not only determined by the magnitude of the electronic coupling matrix element but also by the overlap of the vibrational wave functions vi and v0j. Squared overlap integrals of the type jhwvi ðQÞjwv0j ðQÞiQj2 are frequently called Franck–Con- don (FC) factors. In contrast to radiative processes, FC factors for nonradiative transitions become particularly unfavorable if two states differing considerably in their electronic energies exhibit similar shapes and equilibrium coordinates of their potential curves. Due to the near-degeneracy requirement, an upper state vibrational wave function, with just a few nodes

Figure 22 Schematic drawing of nuclear wave functions with 30 (to the left) and 3 (to the right) vibrational quanta. The little dots indicate the vibrational wave function amplitude at the classical turning points of the potentials. Portions of the potential energy curves are shown as thick lines. The horizontal lines indicate the energy of the vibronic (vibrational plus electronic) states.

(Figure 22, right), has to interact with a highly oscillatory vibrational wave function of the ground state (Figure 22, left). As a consequence, low vibrational levels of excited triplet states are often metastable with respect to nonradiative depletion despite a considerable electronic coupling matrix element. An example is the a3 state of carbon monoxide. Although its v ¼ 3 level is nearly degenerate with the v0 ¼ 30 level of the X1 þ electronic ground state and the spin–orbit matrix element amounts to about 35 cm 1, there is hardly any coupling between the states.185

In the case of a curve crossing, FC factors profit from the fact that the amplitudes of highly excited states are large at the turning points. Levels that lie energetically close to the intersection will therefore have a nonnegligible overlap. Even weak spin–orbit matrix elements (as occur, e.g., in organic molecules) are often sufficient to make an intersystem crossing an effective depletion channel for excited triplet states.

Bound–Continuum Interactions

Continuum wave functions are spatially extended and are not normalizable in the usual spatial sense. Instead an energy normalization is chosen.186 In the weak interaction approximation, the Wentzel–Rice formula can be

applied to calculate the predissociation line width and thereby the lifetime for the nonradiative decay t ¼ h= .187–190

Here XðQÞ represents the electronic matrix element of any operator coupling the potentials under consideration. The Wentzel–Rice approximation restricts the coupling of a bound initial vibrational state wvi to a single continuum state with energy E ¼ Ei. Schematically, the interaction between a bound

190 Spin–Orbit Coupling in Molecules

ψd ψk

ψ0

Figure 23 Predissociation of the v ¼ 2 vibrational level of the bound electronic state d by a vibrational continuum wave function of the dissociative electronic state k after radiative excitation (arrows) from the electronic ground state 0. The circle around the potential curve crossing point indicates an area of large overlap between the vibrational wave functions.

vibrational wavefunction and a continuum wave function of equal energy is depicted in Figure 23.

If the electronic matrix element does not vary significantly with the vibrational coordinate Q, nuclear and electronic factors can be separated in the sense of a Franck–Condon approximation. The line width may then be estimated by

where hXðQxÞi is the electronic coupling matrix element at the crossing point. The variation of FC factors (FCFs) with vibrational quantum number and the variation’s dependence on the relative slopes of the potential energy

curves has been discussed in detail by Murrell and Taylor191 for the Schu- mann–Runge bands of O2. Assuming a constant electronic coupling, the line widths depend on the FCFs only. Murrell and Taylor chose three model cases similar to the ones sketched in Figure 24. In case of a potential energy curve crossing of type (a), only one or two vibrational states with energies close to the energy at the crossing point have a marked overlap with the continuum.191

Figure 24 Model cases for the potential energy curve crossing between a bound and dissociative state. (a) The potential energy curves cross approximately at right angles. This is often the case when the dissociative state intersects the bound state on its outer limb, i.e., at bond distances longer than its equilibrium internuclear separation. (b) Bound and dissociative state exhibit similar slopes and cross on the inner limb of the bound state. (c) The dissociative state crosses the bound potential both on the inner and outer limbs.

A Franck–Condon-type approximation might be appropriate then, even if hXðQÞi varies strongly with the vibrational coordinate.192,193 In contrast,

many bound vibrational levels will be predissociated in case (b). The FCFs will change slowly from one vibrational level to the next one. A strong variation of the electronic interaction matrix element precludes the possibility of applying the FC approximation. Case (c) is in between. The FCFs do not vary as strongly with vibrational quantum number as in case (a), but their distribution shows marked maxima in the proximity of the two crossing points. An extension to crossings of multidimensional curves interacting via spin– orbit coupling was discussed by Scho¨ n and Ko¨ ppel.194

In practical applications, the continuum is often approximated by a dis-

crete spectrum. To this end, one conveniently introduces a potential wall at long internuclear separations and solves for the artifically bound states.171,172

Alternatively, basis set expansion techniques can be employed.195,196 In either case, the density of states depends on external conditions, that is, the size of the box or the number of basis functions. This dependence on external conditions has to be accounted for by the energy normalization. Instead of employing a single continuum wave function with proper energy E in Eq. [240], one samples over the discrete levels with energy Ej

and determines ðEÞjE¼Ei by interpolation.192,195 In this expression, the mean energy separation between neighboring states has been used for normalization.

Even for states of equal multiplicity spin–orbit coupling may be the ratedetermining process for predissociation. The b4 g state of Oþ2 (see Figure 25)

Summary and Outlook 193

SUMMARY AND OUTLOOK

Spin–orbit interaction plays an important role in many areas of chemistry, physics, and biology. It causes two essential effects. The first one is a splitting of multiplet levels, referred to as zero-field splitting or fine-structure splitting. The energetic separation between the multiplet components increases strongly with growing nuclear charge. In heavy element compounds, spin– orbit splittings are of a size comparable to excitation energies between different electron configurations. Second, spin–orbit coupling mixes electronic states of different multiplicities, thus allowing radiative transitions (phosphorescence) and nonradiative transitions (intersystem crossing) between them. Phosphorescence and intersystem crossing rates determine, for instance, the lifetime of the first electronically excited triplet state of an organic molecule. The availibility of molecules in this state plays a key role in photochemical reactions, photosynthesis, the photobiological activity of drugs, luminescence, and so on.

The first spin–orbit coupling Hamiltonian was derived by Pauli in the 1920s. As was shown later, the Breit-Pauli Hamiltonian is not bounded from below. Strictly, it may only be employed in first-order perturbation theory. Neither the Douglas–Kroll transformed no-pair Hamiltonians [Eq. 105] nor the zero-order regular approximation spin–orbit (ZORA) operators suffer from being unbounded. They are suitable for applications in which heavy elements are involved and can safely be utilized in variational procedures such as the spin–orbit configuration interaction approach. To ease the computational effort connected with the evaluation of two-electron spin–orbit integrals, several efficient one-electron operators have been devised. Among these, the most reliable appear to be the all-electron mean-field Hamiltonian and semilocal model Hamiltonians parameterized to be used with effective core potentials.

The tensorial structure of the spin–orbit operators can be exploited to reduce the number of matrix elements that have to be evaluated explicitly. According to the Wigner–Eckart theorem, it is sufficient to determine a single (nonzero) matrix element for each pair of multiplet wave functions; the matrix element for any other pair of multiplet components can then be obtained by multiplying the reduced matrix element with a constant. These vector coupling coefficients, products of 3j symbols and a phase factor, depend solely on the symmetry of the problem, not on the particular molecule. Furthermore, selection rules can be derived from the tensorial structure; for example, within an LS coupling scheme, electronic states may interact via spin–orbit coupling only if their spin quantum numbers S and S0 are equal or differ by 1, i.e., S ¼ S0 or

S ¼ S0 1.

Symmetry considerations are instrumental in a qualitative discussion of spin–orbit effects. Qualitatively, a phenomenological Hamiltonian of the form

utilized. The operators L and S form a basis for a matrix representation of the

usual molecular point group. The same is true for the spatial and spin wave