Final Wave Function

We now have to write an expression for the total wave function that mathematically expresses both antisymmetry at the exchange of electrons and symmetry at the exchange of electron pairs. A general method exists that was mainly developed by the Canadian mathematician J. Coleman. A sum over all geminals (two-electron functions) is formed  

Because of the antisymmetrizer A, the wave function satisfies the Pauli principle. The product of geminal functions (note that there is only a single geminal function) ensures that we may exchange, say (1,2) with (3,4), and get the same wave function back. Whether the wave fnnction appears to follow the Bose-Einstein statistics or the Fermi-Dirac statistics depends entirely on the experiment. 

---

With each random choice of y and its conjugate momentum Py, one can have a separate trajectory with a different final wave function. After a series of calculations, the energy and state resolved cross-sections are obtained. 

The final wave function q/(r) right after the transition is thus given by... 

Now, we assume that the functions, tcoj, j = 1,. .., N are such that these uncoupled equations are gauge invariant, so that the various % values, if calculated within the same boundary conditions, are all identical. Again, in order to determine the boundary conditions of the x function so as to solve Eq. (53), we need to impose boundary conditions on the T functions. We assume that at the given (initial) asymptote all v / values are zero except for the ground-state function /j and for a low enough energy process, we introduce the approximation that the upper electronic states are closed, hence all final wave functions v / are zero except the ground-state function v /. ... 

The strongest absorptions occur when the initial and final wave-functions ( f and / ) most closely resemble one another. 

We now have all the pieces in place to perform an HF calculation—a basis set in which the individual spin orbitals are expanded, the equations that the spin orbitals must satisfy, and a prescription for forming the final wave function once the spin orbitals are known. But there is one crucial complication left to deal with one that also appeared when we discussed the Kohn-Sham equations in Section 1.4. To find the spin orbitals we must solve the singleelectron equations. To define the Hartree potential in the single-electron equations, we must know the electron density. But to know the electron density, we must define the electron wave function, which is found using the individual spin orbitals To break this circle, an HF calculation is an iterative procedure that can be outlined as follows ... 

The symmetries of the initial and the final wave functions and of the electromagnetic radiation operator determine the allowedness or forbiddenness of an electronic transition. The transition moment integrand must be totally symmetric for an allowed transition such that Mmn V0. 

Numerical values of the frequency shift computed from this formula are shown in Fig. 6, where it can be verified that they coincide with the one deduced from the spectral analysis of the final wave function. 

In the procedure just outlined, the final wave function retains the proper symmetry under exchange of state indices or particle exchange. This wave function, described in more detail below, corresponds to a particular partition of the particles into pairs, and each of the pairs is associated with every possible two- particle state that can be formed by and evolves out of an original set of single-particle free and non-interacting states. Denoting by a zero subscript two-particle states in free space, we have the following orthonormality... 

Elere, j and / represent angular momentum quantum numbers of the initial and final wave functions related to the tensor 3 (k and m and m denote the corresponding magnetic quantum numbers. Note that a and a do not mean spin states in this context but stand for all other quantum numbers. 

The Generalized Multistructural Wave Function (GMS) [1,2] is presented as a general variational many-electron method, which encompasses all the variational MO and VB based methods available in the literature. Its mathematical and physico-chemical foundations are settled. It is shown that the GMS wave function can help bringing physico-chemical significance to the classical valence-bond (VB) concept of resonance between chemical structures. The final wave functions are compact, easily interpretable, and numerically accurate. 

Pure case (e) effective g-factors are readily calculated from these expressions, and effective g-factors for the final wave functions, expressed as linear combinations of case (e) functions, are also easy to calculate. 

If we are interested in the energy and angular distributions of the ejected electrons, we choose a pulse which ramps back down to zero over a few additional cycles (a trapezoidal pulse) and then perform an analysis of the final wave function [15]. We define a window operator... 

At the end of each trajectory, we obtain the position Rj. and momentum for the kth atom, and the final wave function li/z) of the system. 

A variety of properties of the collision can be determined from the final wave-function, e.g., excitations, chemical rearrangement, vibrations. 

In analogy with (1.33), we will now introduce some first rough estimates of the initial and final wave functions through the relations ... 

---