Ground-state wave function observability

The classical formalism quantifies the above observations by assuming that both the ground-state wave functions and the excited state wave function can be written in terms of antisymmetrized product wave functions in which the basis functions are the presumed known wave functions of the isolated molecules. The requirements of translational symmetry lead to an excited state wave function in which product wave functions representing localized excitations are combined linearly, each being modulated by a phase factor exp (ik / ,) where k is the exciton wave vector and Rt describes the location of the ith lattice site. When there are several molecules in the unit cell, the crystal symmetry imposes further transformation properties on the wave function of the excited state. Using group theory, appropriate linear combinations of the localized excitations may be found and then these are combined with the phase factor representing translational symmetry to obtain the crystal wave function for the excited state. The application of perturbation theory then leads to the E/k dependence for the exciton. It is found that the crystal absorption spectrum differs from that of the free molecule as follows ... 

The Kekuld wave function also yields considerable insight into the structure of buckminsterfullerene. Since buckminsterfullerene is a Clar sextet molecule, it has the special Kekule structure which places double bonds on all 30 of the 6-6 edges. Since the pi electrons tend to localize to these positions, this Kekule function should be the most important single contributor to the ground state wave function, while Kekuld functions with many double bonds in 6-5 positions should be less important. Fig. 3 shows the absolute value of the wave function coefficient for each Kekuld structure as a function of the number of double bonds in 6-5 positions. The expected decreasing trend is clearly observed, but something else stands out as well. The Kekule functions divide into two classes, separated by the solid line in the figure. This separation is a reflection of the nonaltemant character of a fiillerene. 

One set of quantities often evaluated from the ground-state wave function that are not quantum-mechanical observables are the various components of the Mulliken population analysis (Mulliken, 1955, 1962). For example, we could define the net Mulliken charge on an atom A as ... 

More generally, the Hohenberg-Kohn theorem of SDFT states that in the presence of a magnetic field B r) that couples only to the electron spin (via the familiar Zeeman term), the ground-state wave function and all ground-state observables are unique functionals of n and m or, equivalently, of n- and. In the particular field-free case, the SDFT HK theorem still holds and continues to be useful, e.g., for systems with spontaneous polarization. Almost the entire... 

For detailed discussion of theoretical treatments of photoionization cross sections, the reader is referred to more general accounts6,11). Here we are concerned with the band intensities observed in a conventional P.E. experiment and their variation with photon energy, in so far as this data can aid assignment of P.E. spectra and give information on the nature of the ground state wave-function. 

A nearly universal feature of EDA complexation is the presence of new absorption bands in the electronic spectrum of the complex that are not found in the spectrum of uncomplexed donor or acceptor [137-140]. These spectral bands are observed even in cases where no other evidence of complexation exists, i.e., where Keda is too small to measure. The charge-transfer resonance theory of Mulliken [141] was originally formulated to account for these striking spectral features. According to Mulliken, the ground-state wave function for the complex can be formulated as... 

Following Slater, some approximate He ground state wave functions are plotted against the interelectronic coordinate ri2 in Figure 3. The two electrons are assumed to move on a sphere with radius n = r2 = 0.5 no, and the wave functions are evaluated for different positions of the two electrons on this sphere. The above choice implies that s = 1 and r = 0 in (12). It is observed that the wave functions have a nonzero first derivative (cusp) with respect to ri2 at the origin, where ri2 = 0, Moreover, the slope of the functions at ri2 = 0 is very close to 5 (the dotted line in Figure 3),... 

In equation (A.8), is the wave function which describes the distribution of particles in the system. It may be the exact wave function [the solution to equation (A.l)] or a reasonable approximate wave function. For most molecules, the ground electronic state wave function is real, and in writing the expectation value in the form of equation (A.8), we have made this simplifying (though not necessary) assumption. The electronic energy is an observable of the system, and the corresponding operator is the Hamiltonian operator. Therefore, one may obtain an estimate for the energy even if one does not know the exact wave function but only an approximate one, P, that is,... 

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