Fourier transform wavefunctions

Application of a Fourier transformation to (n,, rN) produces the wavefunction (pi, , pjv) in momentum space. We have used a tilde to indicate that this Fourier-transformed wavefunction does not necessarily correspond to the optimal wavefunction (pi, , pN) within the momentum orbit The latter satisfies the extremum condition of the variational minimization of the energy functional S[tt(p) W] subject to the normalization condition / d3pir(p) = N. 

In both the diatom-diatom and atom-triatom reactions, the energy-dependent scattering wavefunction is obtained by a Fourier transform of the propagated wavepacket ... 

Now it becomes apparent why it was useful to replace the delta function by its Fourier transform. The wavefUnctions Xin are products of harmonic oscillator functions, the Hamiltonians Hi and H/ are sums of harmonic oscillator terms. Therefore the terms in the brackets factorize in the form ... 

Let us consider an N-particle wavefunction pj( i,..., r y) (where for simplicity we disregard spin) associated with the one-particle density Pi(r), and let us obtain the Fourier transform of this wavefunction by means of ... 

In short, and are related by a 3A-dimensional, norm-preserving, Fourier transform. If the r-space wavefunction is constructed from one-electron functions, then there is an isomorphism [2] between and <1>. In particular, if the wavefunction tk can be written in terms of spin-orbitals / (x) as a single Slater determinant... 

The r-space and p-space representations of the ( th-order density matrices, whether spin-traced or not, are related [127] by a fif -dimensional Fourier transform because the parent wavefunctions are related by a 3A -dimensional Fourier transform. Substitution of Eq. (5.1) in Eq. (5.8), and integration over the momentum variables, leads to the following explicit spin-traced relationship ... 

Figure 5.1. The hierarchy of r- and -space wavefunctions and >) and density matrices and and the connections between them. Two-headed arrows with a beside them signify reversible Fourier transformations, whereas normal arrows signify irreversible contractions.

Although the r- and p-space representations of wavefunctions and density matrices are related by Fourier transformation, Eqs. (5.19) and (5.20) show that the densities are not so related. This is easily understood for a one-electron system where the r-space density is just the squared magnitude of the orbital and the p-space density is the squared magnitude of the Fourier transform of the orbital. The operations of Fourier transformation and taking the absolute value squared do not commute, and so the p-space density is not the Fourier transform of its r-space counterpart. In this section, we examine exactly what the Fourier transforms of these densities are. 

Given a wavefunction built from a one-particle basis set, it is convenient [9,197,198] to proceed with the computation of the six-dimensional Fourier transform in Eq. (5.15) by inserting the spectral expansion [123,126]... 

Showing that (p) is the proper fourier transform of TUx) suggests that the fomier integral theorem should hold for the two wavefunctions P(x) and F(p) we have obtained, e.g. 

Another approach to the problem of rare gas scattering is to replace the spatial wavefunctions of Eq. (11.4) with their Fourier transforms, the momentum space wavefunctions. These wavefunctions represent the velocity distributions of the electron in the Rydberg states. Proceeding along these lines, we rewrite Eq. (11.4)... 

The dominant error term is third order in At. The initial wavefunction (Qx,Qy,t) at t = 0 is normally the lowest energy eigenfunction of the initial state of the spectroscopic transition. The value of the wavefunction at incremental time intervals At is calculated by using Eq. (7) for each point on the (Qx,Qy) grid. The autocorrelation function is then calculated at each time interval and the resulting < (t> is Fourier transformed according to Eq. (2) to give the emission spectrum. 

Raman Spectroscopy The time-dependent picture of Raman spectroscopy is similar to that of electronic spectroscopy (6). Again the initial wavepacket propagates on the upper excited electronic state potential surface. However, the quantity of interest is the overlap of the time-dependent wavepacket with the final Raman state 4>f, i.e. < f (t)>. Here iff corresponds to the vibrational wavefunction with one quantum of excitation. The Raman scattering amplitude in the frequency domain is the half Fourier transform of the overlap in the frequency domain,... 

However, the total dissociation wavefunction is useful in order to visualize the overall dissociation path in the upper electronic state as illustrated in Figure 2.3(a) for the two-dimensional model system. The variation of the center of the wavefunction with r intriguingly illustrates the substantial vibrational excitation of the product in this case. As we will demonstrate in Chapter 5, I tot closely resembles a swarm of classical trajectories launched in the vicinity of the ground-state equilibrium. Furthermore, we will prove in Chapter 4 that the total dissociation function is the Fourier transform of the evolving wavepacket in the time-dependent formulation of photodissociation. The evolving wavepacket, the swarm of classical trajectories, and the total dissociation wavefunction all lead to the same general picture of the dissociation process. 

The time-independent total dissociation wavefunction tot Ef) is the Fourier transform of the time-dependent wavepacket y (t). 

The time-independent and the time-dependent approaches are completely equivalent. Equation (4.11) documents this correspondence in the clearest way the time-dependent wavepacket f(t), which contains the stationary states for all energies, and the time-independent wavefunction tot(Ef), which embraces the entire history of the fragmentation process, are related to each other by a Fourier transformation between the time and the energy domains,... 

Computational advances which have led to pseudopotential methods becoming efficient are Fast Fourier Transfoms and the approach of Car and Parrinello in which wavefunction coefficients are treated as d5mamical variables. Parts of the calculations on periodic systems are easier to calculate in real space and others in reciprocal space. Fourier transforms are needed to convert between the two spaces. The introduction of Fast Fourier Transforms substantially reduced the computational costs of this conversion. The key feature of the Car-Parrinello approach is the use of minimisation rather than diagonalisation of the Kohn-Sham equations to reach the ground state. 

It has been shown that, under appropriate conditions, the momentum distribution uj py for an individtial electronic state is directly meastired by electron-momentum spectroscopy (ref. 29). Figure 3.8 compares the experimental momentum distribution for the hydrogen atom ground state with the function calculated by the Fourier transform of the hydrogen Is orbital. In general, the electron-momentum spectroscopy results serve to evaluate wavefunctions at various levels of theory for a variety of atomic (and molecular) systems. 

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