Excitation energy, average

The first term in the brackets is the expectation value of the square of the dipole moment operator (i.e. the second moment) and the second term is the square of the expectation value of the dipole moment operator. This expression defines the sum over states model. A subjective choice of the average excitation energy As has to be made. 

A more balanced description requires MCSCF based methods where the orbitals are optimized for each particular state, or optimized for a suitable average of the desired states (state averaged MCSCF). It should be noted that such excited state MCSCF solutions correspond to saddle points in the parameter space for the wave function, and second-order optimization techniques are therefore almost mandatory. In order to obtain accurate excitation energies it is normally necessarily to also include dynamical Correlation, for example by using the CASPT2 method. 

The excitation energy and dynamic properties are evaluated from the time-averaged derivatives of the corresponding time-dependent energy functionals [11, 23-25]. However, a more straightforward way to define dynamic properties is through an expectation value of the corresponding properties over a state / ... 

The quantity L(0) = In I, where I is the mean excitation potential introduced by Bethe, which controls the stopping of fast particles (see Sect. 2.3.4) L(2) = In K, where K is the average excitation energy, which also enters into the expression for Lamb shift (Bethe, 1947). Various oscillator sum rules have been verified for He and other rare gases to a high degree of accuracy. Their validity is now believed to such an extent that doubtful measurements of photoabsorption and electron-impact cross sections are sometimes altered or corrected so as to satisfy these. 

Platzman (1967) has emphasized that most direct ionizations in molecules leave the positive ions in an excited state. Based on crude DOSD, he estimated that in water the average positive ion will have about 8 eV excitation energy. Later, the less approximate calculation of Pimblott and Mozumder (1991) reduced that figure to about 4 eV The chemical role of this excitation energy is unknown, although it may have some effect in the radiolysis of highly concentrated solutions. 

Platzman (1967) estimated that in the radiolysis of water the positive ion is left, on average, with an excitation energy of -8 eV this estimate was later lowered to 4 eV by Pimblott and Mozumder (1991). In any case, the chemical consequences of such excess energy of the positive ion is unknown, and it will be assumed that, at least in the condensed phase, the positive ion is ther-malized locally. 

Calculated and Observed Carbon-13 Chemical Shifts in Azines, Jaffe and Bene CNDO-MO s, Average Excitation Energy (AEE) and Different Excitation Energy... 

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