Two-state calculations

In [66], we have reported inelastic and reactive transition probabilities. Here, we only present the reactive case. Five different types of probabilities will be shown for each transition (a) Probabilities due to a full tri-state calculation carried out within the diabatic representation (b) Probabilities due to a two-state calculation (for which T] = 0) performed within the diabatic representation (c) Probabilities due to a single-state extended BO equation for the N = 3 case (to, = 2) (d) Probabilities due to a single-state extended BO equation for the N = 2 case (coy =1) (e) Probabilities due to a single-state ordinary BO equation when coy = 0. 

The values due to the two separate calculations are of the same quality we usually get from (pure) two-state calculations, that is, veiy close to 1.0 but two comments have to be made in this respect (1) The quality of the numbers are different in the two calculations The reason might be connected with the fact that in the second case the circle surrounds an area about three times larger than in the first case. This fact seems to indicate that the deviations are due noise caused by CIs belonging to neighbor states [e.g., the (1,2) and the (4,5) CIs]. (2) We would like to remind the reader that the diagonal element in case of the two-state system was only (—)0.39 [73] [instead of (—)1.0] so that incorporating the third state led, indeed, to a significant improvement. 

The two-state model was implemented by treating the two conformers as a pair of non-interacting molecules contributing to the spin relaxation properties in proportion to their Boltzmann factors. The overall energy to be minimized for the two-state calculation is defined as ... 

Two-state calculations based on expression 6 rely on the validity of molecular mechanics energies to a much higher degree than the one-state calculation. Therefore, it is reasonable to start calculations with initial structures near the molecular mechanics minima. Several starting structures were generated for this purpose. A and C ... 

In the last section, our calculation used only the function of Eq. (2.9), what is now called the covalent bonding function. According to our discussion of linear variation functions, we should see an improvement in the energy if we perform a two-state calculation that also includes the ionic function. 

Table 2.1. Numerical values for overlap, kinetic energy, nuclear attraction, and electron repulsion matrix elements in the two-state calculation.

G. R. Fleming An interesting feature of the Redfield theory calculations is that attempts to stop coherence transfer by increasing the dephasing rate also increases the coherence transfer rate. In addition, two-state calculations [M. Jean and G. R. Fleming, J. Chem. Phys. 103, 2092 (1995)] show that the coherence transfer can survive reasonable amounts of anharmonicity. It appears to be quite robust. 

Note Differential elastic and excitation transfer cross sections have been measured for He(2 S) + Nc and for He(23S) + Ne for energies between 25 and 370 meV (1). Some of the data are shown in Fig. 52. It was possible to measure the differential excitation cross sections for the triplet system, too. A semiclassical two-state calculation was performed for the pumping transition of the red line of the HeNe-laser Hc(2 S)+ Nc— Hc + Ne(5S, lPt), which is the dominant transition for not too high energies (2). A satisfactory fit is obtained to the elastic and inelastic differential cross sections simultaneously, as well as to the known rate constant for excitation transfer. The Hc(215)+ Ne potential curve shows some mild structure, much less pronounced than those shown in Fig. 36. The excitation transfer for the triplet system goes almost certainly over two separate curve crossings. This explains easily the 80 meV threshold for this exothermic process as well as its small cross section, which is only 10% of that of the triplet system. 

Figure 24. Excitation probability of the Kx L shell in Ne + Kr collisions versus the inverse projectile velocity. The curve labeled NM(2) follows from two-state calculations using Nikitin s model. The curves labeled SHM(2) and SHM(3) represent model calculations using matrix elements (18) for two and three states, respectively. The curve labeled VSM(3) refers to three-state calculations by Fritsch and Wille using the variable screening model. ...

Figure 29. Vacancy sharing probabilities for the K and L shells in the systems B + Ar, C + Ar, N + Ar, and O + Ar as a function of the inverse projectile velocity. The data refer to the excitation of the lower-lying levels, i.e., the Ar i-shell orbitals for B -I- Ar and the K-shell orbitals of the lighter particle in the other systems. The dots refer to experimental results by Reed et al and the solid lines follow from three-state model calculations on the basis of the SHM matrix elements (18). The dashed line represents two-state calculations for O H- Ar by means of Nikitin s model." ...

The same is true in L-sharing systems, where always the 2s level of the lighter partner is located between the 2p energy levels. Indeed, for Si + Ar and S + Ar an order of magnitude difference is observed between the results of the two-state calculation and the corresponding four-state calculation. Here, definitely an analysis with more than two states is required. 

Semiempirical quantum mechanical calculations of first hyperpolarizabilities, /3, for such dipolar chromophores have proved quite successful in predicting trends in structure-function relationships [7,8]. Marder et al. [33-37], in particular, have been successful in using simple (two-level) calculations to guide synthetic efforts aimed at optimizing molecular hyperpolarizabilities. Recall that within the framework of a two-state calculation, the hyperpolarizability, jS, depends on the transition dipole, /Ag, the difference between the dipole moments of the ground and excited state, /i-ec /Agg and the optical gap, Explicitly,... 

Harvey et al. proposed a hybrid method for calculating the effective gradients (Eqs. [98] and [99]) by approximating the difference gradients at lower level of theory from the evaluation of the energy term. For the approximation to work, the force fields of the two states calculated at a higher level of theory must be equal to those calculated at a lower level of theory. In cases that Harvey et al. have examined, they have shown that the approximation is reliable. 

---