Perturber-triplet separation

In Eq. (43) the energy halfway between the 6sl8s 5o and states has been taken as zero. A px = (5d7d Po, / = 0) - (6sl8s S, 7 = 0) denotes the perturber-triplet separation (cf. Figure 34). The present example requires the solution of a cubic secular equation, in contrast to the unperturbed case [cf. Eq. (24)]. Although general analytic expressions can be derived... 

Purely theoretical data with an accuracy of 0.1 eV have been calculated more recently. Ab initio MO theory corrected by 4th order Moller-Plesset (MP4) perturbation calculations [9], gave Ej = 9.72 and 10.64 (for X and a of PHJ) [1, 10]. A modification of that method (to give also singlet-triplet separations) yielded Ej = 9.77 and 10.64 [11]. A treatment of the electron correlation not only by MP4 but also by quadratic configuration interaction (Cl) [12,13] led to Ej = 9.71 [14]. Older data were obtained by Cl [15,16] and SCF [16,17] calculations. - For orbital energies from SCF calculations, see [17, 18]. 

Theoretically calculated formation enthalpies from ab initio MO methods, including the Moller-Plesset perturbation theory up to the fourth order, are in good agreement with experimental data [1, 2]. A modified procedure (to give singlet-triplet separations) yielded A,h = 260.9 and A H298 = 261.8 [7]. 

Figure 6 Band-to-correlated crossover based on E B-u) = 2E(13BU) of IV-site Hubbard chains with alternating transfer integrals t = 0(1 ) The U = 0 and 21+ points are exact for the infinite chain and dimers, respectively. The dashed Uc(6) line is based on perturbation theory at large 6 and (inset) separate estimates of the singlet and triplet thresholds at small <5[80].

This description of the relative spectral linewidths of the lowest excited toi states applies to the whole family of aromatic hydrocarbons. It also applies to the manifold of triplet jui states. In the case of benzene, Burland, Castro and Robinson 24> and Burland and Castro 25> have used phosphorescence and delayed fluorescence excitation techniques, respectively, to measure the absorption spectrum of the lowest triplet state, 3Biu of ultrapure crystals at 4 K. The origin is located at 29647 cm-1. Unlike all the earlier studies on the lowest singlet triplet absorption spectrum, this was not an 02 perturbation experiment. Here widths of less than 3 cm-1 were obtained. This result should be compared with the much broader bands 150-1 observed for the suspected second triplet ZE i in 5 cm crystals of highly purified benzene 26>. The two triplet states are separated by 7300 cm"1. 

We have derived Eq. (11-36) with mixing of the singlet and triplet states of a radical pair in mind, but it is quite general for the time evolution of two levels separated by an energy under a stationary perturbation. In order to get the dynamic behavior of a radical pair, we should add a diffusion operator D and an operator K for the chemical reaction to Eq. (11-29),... 

The coupling functions 1 and still depend on the molecular vibrational and rotational degrees of freedom as well as the relative molecule-perturber separation, R. Since the experiments imply that the physical origin of the collision-induced intersystem crossing resides in long-range attractive interactions, we may adopt a semiclassical approximation where the quantum-mechanical variables for the relative translation is replaced by a classical trajectory, R(l), for the relative molecule-perturber motion. The internal dynamics is then influenced by the time-dependent interactions f s[ (0] and Fj-j-fR(r)], which are still functions of molecular rotational and vibrational variables. For simplicity and for illustrative purposes we consider only the pair of coupled levels S and T and a pure triplet level T, which represents the molecular state after the collision. Note T may differ in rotational and/or vibrational quantum... 

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