Electronic energy variation

A quantum mechanical treatment of molecular systems usually starts with the Bom-Oppenlieimer approximation, i.e., the separation of the electronic and nuclear degrees of freedom. This is a very good approximation for well separated electronic states. The expectation value of the total energy in this case is a fiinction of the nuclear coordinates and the parameters in the electronic wavefunction, e.g., orbital coefficients. The wavefiinction parameters are most often detennined by tire variation theorem the electronic energy is made stationary (in the most important ground-state case it is minimized) with respect to them. The... 

Table 2.2 Variation in basis set coefficients and electronic energy for the molecule.

The Fock operator is an effective one-electron energy operator, describing the kinetic energy of an electron, the attraction to all the nuclei and the repulsion to all the other electrons (via the J and K operators). Note that the Fock operator is associated with the variation of the total energy, not the energy itself. The Hamilton operator (3.23) is not a sum of Fock operators. 

The eigenvalues E0, Elt E2,. .. of the Schrodinger equation (Eq. II. 1) form the electronic energies of the system under consideration. It is evident that the solution of Eq. II. 1 must involve considerable mathematical difficulties, and so far, the strongest tool we know for handling this problem is the variation principle. If the wave function W is properly normalized so that... 

The dependence of the currents of m/e 16 and m/e 30 upon sample pressure, using an electron energy of 2.3 e.v., is shown in Figure 8. The linear variation of m/e 16 and the quadratic variation of m/e 30 with pressure, together with the results shown in Figure 7, indicate the occurrence of Reaction 14. 

Top typical saturation curve and variation of mean electron energy with applied field. Middle fraction of the electron swarm exceeding the specific energy at each field strength. Calculated assuming constant collision cross-section and Maxwell-Boltzman distribution. Bottom variation of products typical of involvement of ionic precursors (methane) and excited intermediates (ethane) with applied field strength... 

In the vibrational treatment we assumed, as usually done, that the Born-Oppenheimer separation is possible and that the electronic energy as a function of the internuclear variables can be taken as a potential in the equation of the internal motions of the nuclei. The vibrational anharmonic functions are obtained by means of a variational treatment in the basis of the harmonic solutions for the vibration considered (for more details about the theory see Pauzat et al [20]). 

Finally, we mention that Filatov [10, 11] recently in an interesting new approach discussed the effect of finite nuclei in detail. He suggested calculating the isomer shift from the variation to the total electron energy in dependence of the nuclear charge extension. 

In this chapter we focus attention on the efficiency of ionization, the ionization cross section, and consider some recent experimental measurements and theoretical studies of the ionization process. A sketch of electron impact ionization curves, the variation of the ionization cross section as a function of the electron energy, using CO as an example, are shown in Figure 1. The mass spectrum, collected at the electron energy corresponding to the maximum in the ionization cross section, is also shown, although there will be no further discussion of fragmentation in this... 

Effect of diagonal dynamic disorder (DDD). Fluctuations of the polarization and the local vibrations produce the variation of the positions of the electron energy levels eA(Q) and eB(C ) to meet the requirements of the Franck-Condon principle. 

Using this expression for the total electronic energy, application of the variational principle yields the following set of differential equations to obtain the optimized spatial MOS, 4, for the molecule ... 

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