Momentum-space electron densities

Allan, N.L. and Cooper, D. Momentum-Space Electron Densities and Quantum Molecular Similarity. 173, 85-111 (1995). 

Momentum-Space Electron Densities and Quantum Molecular Similarity... 

The measured differential cross-sections depend on the rotationally and vibrationally averaged values of

. Allan et al. have discussed the interpretation of (e, 2e) experiments in terms of spin-coupled orbitals and have presented momentum-space electron density maps for individual orbitals for LiH, NH and the 7t electrons in CH2CHX(X = F,C1). The nonorthogonality of the spatial orbitals allows them to assume properties very different from those of MO theory this leads to striking differences in momentum space. 

Momentum-space concepts are not, in general, familiar to the chemist and so we outline first the calculation of momentum-space electron densities, p p), from ab initio wavefunctions. The form of pip) for different molecules is discussed, using as examples (i) the ground state of H2, (ii) bond formation in BH", and (iii) the n-orbitals in large conjugated polyenes. 

The momentum-space electron density, p p), is just as straightforward to evaluate as its analogue in position spac, p(r). The wavefiinction in momentum space, (/ ), is simply the Fourier transform of the r-space wavefunction (r) ... 

Kaijser and Smith [17] have presented analytic forms for many of the Slater-type orbitals and to Gaussian-type orbitals in both spherical harmonic and Cartesian form. The total momentum-space electron density p(p) is given by... 

Rg. - . Isometric views of the electron density for H. The intemuclear axis is the z-axis, with the nuclei at (0,0, + 0.7) bohn a j sition-space electron density in the plane x = 0, extending to 1.5 atomic units in both the y and z directions b momentum-space electron density in the plane p, = 0. extending to 1 atomic units in both the p, and Pj directions, with p = 0 at the centre of the plot c momentum-space electron density in the plane p, = 0, extending to 6 atomic units in both the p, and p, directions the peak around p = 0 (at the centre of the plot) has been removed in order to highlight the nodal planes in p(p) (Eq. (15)) at p = + hIRab (i.e. Pj + 2.2 atomic units)... 

In summary, our unconventional approach to molecular similarity, based on indices derived from momentum-space electron densities, appears to show considerable promise for a wide range of applications. Work on a wide range of applications, both biological and non-biological, is currently in progress and the results will be reported in due course. 

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