Ground-state, generally wavefunction

To illustrate the convergence of the FCI principal expansion with respect to short-range electron correlation, we have in Fig. 1.1 plotted the ground-state He wavefunction with both electrons fixed at a distance of 0.5 ao from the nucleus, as a function of the angle 0i2 between the position vectors ri and r2 of the two electrons. The thick grey lines correspond to the exact nonrelativistic wavefunction, whereas the FCI wavefunctions are plotted using black lines. Clearly, the description of the Coulomb cusp and more generally the Coulomb hole is poor in the orbital approximation. In particular, no matter how many terms we include in the FCI wavefunction, we will not be able to describe the nondifferentiability of the wavefunction at the point of coalescence. 

Return now to the generalized phosphor MXO4 A. Let us further suppose that the activator has a transition of Sq => Pi for excitation. We now know that the ground state electronic wavefunctions are symmetrical, whereas the Pi state will have a vectored orientation due to the nature of the p-electron orbital, as shown in the following diagram ... 

Generally the binding of electron donors (D) to electron acceptors (A) in EDA complexes can be conveniently described (Mulliken, 1952a Mulliken and Person, 1969 Hanna and Lippert, 1973) in qualitative valence-bond terms as in (33), where the ground-state wavefunction ifes mainly contains... 

The ground-state wavefunction will be antisymmetric in the spin coordinates of the two electrons and symmetric in their spatial coordinates. It will also have zero orbital angular momentum (an S state) the most general S state can be shown to depend only on the interparticle distances ri, r2, and ri2 [11]. We construct it from a basis of functions of the form... 

General p-particle A -representability conditions on the 2-RDM are derivable from metric (or overlap) matrices. From the ground-state wavefunction lih) and a set of p-particle operators. of basis functions can be defined. 

Here, the pt are the permanent dipoles of molecules i = 1 and 2, and the ptj( r, i 2, Rij) are the dipoles induced by molecule i in molecule j the are the vectors pointing from the center of molecule i to the center of molecule j and the r, are the (intramolecular) vibrational coordinates. In general, these dipoles are given in the adiabatic approximation where electronic and nuclear wavefunctions appear as factors of the total wavefunction, 0(rf r) ( ). Dipole operators pop are defined as usual so that their expectation values shown above can be computed from the wavefunctions. For the induced dipole component, the dipole operator is defined with respect to the center of mass of the pair so that the induced dipole moments py do not depend on the center of mass coordinates. For bigger systems the total dipole moment may be expressed in the form of a simple generalization of Eq. 4.4. In general, the molecules will be assumed to be in a electronic ground state which is chemically inert. 

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