Excitation energies and energy gaps

Thomas-Fermi theory) requires a Frechet derivative for the kinetic energy, and cannot exist for more than two electrons [288], 

This proposition has been tested in the exact-exchange limit of the implied linear-response theory [329], The TDFT exchange response kernel disagrees qualitatively with the corresponding expression in Dirac s TDHF theory [79,289]. This can be taken as evidence that an exact local exchange potential does not exist in the form of a Frechet derivative of the exchange energy functional in TDFT theory. 

Time-dependent OFT implies matrix equations for excitation energies tico, 

The simplest internally consistent approximation to an excitation energy is obtained by limiting the summation to the diagonal term j, b = i, a. The second line vanishes because of antisymmetry, (i i f hxc ad) = 0, and the first line reduces to a single equation, 

Neglecting correlation response, this implies a well-known formula for zeroeth-order hole-particle excitation energies [261, 149], hon = ca — e, — (ai u ai). The two-electron integral here depends strongly on orbital localization. Since the lower 

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Here, grs is a parameter that is quantified either from experimental data, or is calculated by an ab initio method as one-half of the singlet-triplet excitation energy gap of the r—s bond. In terms of the qualitative theory in Chapter 3, grs is therefore identical to the key quantity —2(3 5 - This empirical quantity incorporates the effect of the ionic components of the bond, albeit in an implicit way. (c) The Hamiltonian matrix element between two determinants differing by one spin permutation between orbitals r and s is equal to grs. Only close neighbor grs elements are taken into account all other off-diagonal matrix elements are set to zero. An example of a Hamiltonian matrix is illustrated in Scheme 8.1 for 1,3-butadiene. 

Winter has investigated on what makes a photol3rtic reaction change its path from homolysis to heterolysis. He found that a destabilized ground state and a stabilized excited state can lead to a favorable, nearby conical intersection. Furthermore, he found that excited energy gap for a carbocation (see the black arrows in Fig. l) can be taken as a simple, easy-to-calculate probe for recognizing such situations, skipping expensive calculations. ... 

The Boltzmann equation (Equation 18.2) shows that, under equilibrium conditions, the ratio of the number (n) of ground-state molecules (A ) to those in an excited state (A ) depends on the energy gap E between the states, the Boltzmann constant k (1.38 x 10" J-K" ), and the absolute temperature T(K). 

Many other measures of solvent polarity have been developed. One of the most useful is based on shifts in the absorption spectrum of a reference dye. The positions of absorption bands are, in general, sensitive to solvent polarity because the electronic distribution, and therefore the polarity, of the excited state is different from that of the ground state. The shift in the absorption maximum reflects the effect of solvent on the energy gap between the ground-state and excited-state molecules. An empirical solvent polarity measure called y(30) is based on this concept. Some values of this measure for common solvents are given in Table 4.12 along with the dielectric constants for the solvents. It can be seen that there is a rather different order of polarity given by these two quantities. 

As seen from Fig. 2(a) the energy gap between 0> and 1> is too large to allow the observed population of the first excited state. Thus we assume some orbital reduction to fit the observed behavior. The values found are k = 0.91 and E( l>) = 424 cm-1 and yield the curve (b). 

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