Computational Methods

The electronic structure package GAUSSIAN-94 was used to optimize the molecular geometries, and generate Hartree-Fock wavefunctions (with the 6-3 1 lg** basis set) for all of the molecules discussed below.17 Calculations of perturbed wavefunctions, in the presence of a uniform, external electric field were also done (at the zero-field optimized geometries). Previous research on naphthalene and azulene compared cross-sections of A2n (p) near the origin in momentum-space (out to 1.00 au), from both Hartree-Fock and second-order M0ller-Plesset calculations (also at the 6-31 lg** basis set).18 Similar to studies of the effect of electron correlation on the topological properties of A2p (r),19 there were very slight numerical differences, but no topological differences between the A2n (p) distributions at these two levels of theory.

For many-electron molecules, the Hartree-Fock wavefunction that is computed by conventional electronic structure packages, such as GAUSSIAN, can be expanded from single-particle molecular orbitals, yi (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 y i (r)'s, followed by reduction of the phase information, weighting by the orbital occupation numbers, ni, and finally summation over all orbitals, as shown below.8

This procedure is used to compute the value of the momentum density for a cubic grid of evenly-spaced (0.02 au) points in momentum-space, with max(px) = max(py) = max(pz) = 1 .00 au, and with the origin at the center of the cube. Electrons that are at the origin in momentum-space have no momentum. Kaijser and Smith have shown that the origin is also a center of inversion symmetry for all systems that are in a stationary state, thus only one octant of the cube is unique.20 In our investigation, the entire cube was calculated and visual inspection confirmed a center of inversion at the origin in all cases.

In most computational studies of A2n(r), the Laplacian is computed analytically since the wavefunction, and hence the density is in functional form. The Laplacian of the electron momentum density is defined analogously by,

However, in this work it is computed numerically since the density is not in functional form, as discussed above. The study of fluid dynamics is critical for aerodynamic design, and here too the data is collected and usually simulated, on a grid (often very complex in shape). Fluid dynamicists also routinely examine flow properties in both coordinate-space and momentum-space. The key difference being that for classical fluids the particle trajectories are known (within experimental error). Computational fluid dynamicists can also examine the momentum distribution of all particles that pass through an infinitesimally small volume of coordinate-space, and monitor the evolution of this distribution as the probe position is scanned. For quantum fluids, such as the electronic charge distribution in molecules, the Heisenberg uncertainty principle limits such knowledge to the "fuzzy" Husimi functions employed by Schmider, as discussed above.1221 For the momenta precision implied by the grid spacing in the current work, the uncertainty in the positions of the electrons is several times the molecular dimensions. Nevertheless, we have taken advantage of the advanced visualization techniques in the Flow Analysis Software Toolkit (FAST),22 that was developed for classical fluids by the Numerical Aerodynamic Simulation (NAS) Systems Division at the NASA Ames Research Center. FAST numerically computes the gradient vector field for the n(p) distribution that we obtain via Eq. (1). FAST then computes the divergence of this gradient, also numerically, yielding A2n (p) with high accuracy. FAST allows numerous possibilities for data visualization, including movies. The topological transitions, as well as the overall shape of A2n (p), are sufficiently depicted by simple isovalue surfaces. In the figures presented below, A2n (p) = -0.025 au everywhere on the surface. In all the figures shown, there are no "hidden surfaces", that is A2n (p) < -0.025 au everywhere within the enclosed volumes. This particular value for the surface has no special significance. It was chosen so that no topological features would be obstructed, and it is negative so that the electron flow for all momenta inside the surface is more laminar than it is for momenta outside the surface.

In previous work on clusters of alkali metal atoms,18 in addition to the very pronounced local concentration of charge at the origin in momentum-space (very laminar slow electrons), we have also observed topological features in A2n (p) outside the momentum-space

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