Molecular momentum density electron number densities

As a second illustration of the use of the general formula (2.14), we take a perfect gas and consider the fluctuations of the number of molecules in a group of G cells in the molecular phase space. Two important physical problems are special cases of this. In the first place, the G cells may include all those, irrespective of momentum, which lie in a certain region of coordinate space. Then the fluctuation is that of the number of molecules in a certain volume, leading immediately to the fluctuation in density. Or in the second place, we may be considering the number of molecules striking a certain surface per second and the fluctuation of this number. In this case, the G cells include all those whose molecules will strike the surface in a second, as for example the cells contained in prisms similar to those shown in Fig. IV-2. Such a fluctuation is important in the theory of the shot effect, or the fluctuation of the number of electrons emitted thermionically from an element of surface of a heated conductor, per second we assume that the number emitted can be computed from the number striking the surface from inside the metal. 

Figure 3.8. Molecular orbitals for the oxygen atom, with indication of their quantum numbers (main, orbital angular momentum and projection along axis of quantisation). Shown is the oxygen nucleus and the electron density (where it has fallen to 0.0004 it is identical for each pair of two spin projections), but with two different shades used for positive and negative parts of the wavefunction. The calculation uses density functional theory (B3LYP) and a Gaussian basis of 9 functions formed out of 19 primitive Gaussian functions (see text for further discussion). The first four orbitals (on the left) are filled in the ground state, while the remaining ones are imoccupied.

For many-electron molecules, the Hartree-Fock wavefunction that is computed by conventional electronic structure packages, such as GAUSSIAN, can be expanded from singleparticle molecular orbitals, i /i(r), that are themselves constructed from atom-centered gaussians that are functions of coordinate-space variables. The phase information that is contained in the molecular orbitals is necessary to define the wavefunction in momentum-space. In other words, the density in coordinate-space cannot be Fourier transformed into the density in momentum-space. Rather, within the context of molecular orbital theory, the electron density in momentum space is obtained by a Fourier-Dirac transformation of all of the v /i (r) s, followed by reduction of the phase information, weighting by the orbital occupation numbers. 

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