Projected electron density function

Collins, Streirwieser, and McKelvey A7 developed a method of reducing the four-dimensional function p(r) into a three-dimensional projected electron density function P(x,z). When the atomic orbitals are expressed as a combination of Gaussian functions defined by Eq. [45],... 

Figure 4.7. Fourier projection of an electron density function [JAM 67]...

The discussion above is intended to emphasize that a systematic selection of density basis functions can be accomplished by projection analysis. Also note that these x-ray population coefficients are not the elements of a density matrix from a quantum chemical calculation as has been suggested.23 To compare the x-ray results, one can project the theoretical one-electron density function by least squares into the same basis functions that were used in the x-ray analysis. 

Using the combination of main-frame CDC 6400 and Tektronix computations, a number of phenomena were studied with electron density functions, and especially with projection plots. Particularly useful were plots of difference functions in which the electron distributions of isoelectronic systems were compared directly. In such applications, we noted that a corresponding difference plot of the electron density itself in any given plane is not meaningful since the number of electrons may change that is, from one compound to the next the electron density can shift from one plane to elsewhere. In the projection plot the total number of electrons remains the same for both species and the integral of an isoelectronic difference function must sum to zero. Some examples of the kinds of problems studied are the vm transition of formaldehyde, substituent effects in substituted benzenes, and polarization... 

The atom-centered models do not account explicitly for the two-center density terms in Eq. (3.7). This is less of a limitation than might be expected, because the density in the bonds projects quite efficiently in the atomic functions, provided they are sufficiently diffuse. While the two-center density can readily be included in the calculation of a molecular scattering factor based on a theoretical density, simultaneous least-squares adjustment of one- and two-center population parameters leads to large correlations (Jones et al. 1972). It is, in principle, possible to reduce such correlations by introducing quantum-mechanical constraints, such as the requirement that the electron density corresponds to an antisymmetrized wave function (Massa and Clinton 1972, Frishberg and Massa 1981, Massa et al. 1985). No practical method for this purpose has been developed at this time. 

Calculations of the Patterson function may be carried out in exactly the same way as those of electron densities. Bragg s optical method may also be used indeed, in general it may be applied more readily to the formation of vector maps, since (the signs of the jF2 coefficients being all positive) the question of phase adjustment does not arise. The optical method has been shown to give a correct vector map for the 6 projection of haemoglobin. ... 

The mathematical term functional, which is akin to function, is explained in Section 7.2.3.1. To the chemist, the main advantage of DFT is that in about the same time needed for an HF calculation one can often obtain results of about the same quality as from MP2 calculations (cf. e.g. Sections 5.5.1 and 5.5.2). Chemical applications of DFT are but one aspect of an ambitious project to recast conventional quantum mechanics, i.e. wave mechanics, in a form in which the electron density, and only the electron density, plays the key role [5]. It is noteworthy that the 1998 Nobel Prize in chemistry was awarded to John Pople (Section 5.3.3), largely for his role in developing practical wavefunction-based methods, and Walter Kohn,1 for the development of density functional methods [6]. The wave-function is the quantum mechanical analogue of the analytically intractable multibody problem (n-body problem) in astronomy [7], and indeed electron-electron interaction, electron correlation, is at the heart of the major problems encountered in... 

From here it is obvious that for the Hartree-Fock approximation the parametrization of the energy not referring directly to the wave function is nevertheless possible (the Hartree-Fock density is a projection operator and it can be directly written using say eq. (1.107)), but the cost is fixing x = 0 with the consequences of this. (We can say that the works devoted to foundations of DFT basically reduce to developing a more or less widely applicable form of the two-electron density.)... 

If structure and shape are not intrinsic properties of free molecules and only emerge in response to environmental pressure the interpretation of crystallographic structures becomes less obvious. The electron-density transform (1) may well be the three-dimensional projection of a four-dimensional periodic function that fluctuates with time. The possibility that crystallography looks at a time-averaged projection with the appearance of of a rigid arrangement cannot be discounted. 

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.

Abstract X-ray spectroscopy provides a number of experimental techniques that give an atom-specific projection of the electronic structure. When applied to surface adsorbates in combination with theoretical density functional spectrum simulations, it becomes an extremely powerful tool to analyze in detail the surface chemical bond. This is of great relevance to heterogeneous catalysis as discussed in depth for a number of example systems taken from the five categories of bonding types (i) atomic radical, (ii) diatomics with unsaturated n-systems (Blyholder model), (iii) unsaturated hydrocarbons (Dewar-Chatt-Duncanson model), (iv) lone-pair interactions, and (v) saturated hydrocarbons (physisorption). 

Figure 12.3 shows a schematic illustration of the resulting electron density of states projected onto the adatom in the Newns-Anderson model [17, 18] for two different cases. In this model, the interaction strength between the adatom wave function of one specific electronic level and the metal states is often denoted the hopping matrix element. When the hopping matrix element is much smaller than the bandwidth of the metal states, in this case the i-electrons, the interaction leads... 

In the case of well-ordered crystals, It Is possible to deduce their atomic structures by appropriate manipulation of diffraction Intensities. In the case of x-ray scattering by liquids, direct use of measured intensities yields, at best, very limited structural Information (radial distribution functions). For ordered liquids, however, it is possible to posit structural models and to calculate what their scattering Intensities would be so that it is more productive to conduct the comparisons in diffraction space. To this end, it is possible to devise a point model to represent the spatial repetition of the constituent units in the ordered array and to compare its scattering maxima to the observed ones (6,9). More sophisticated analyses (10-12) make use of the complete electron densities (or projections onto the chain axis z), usually by calculating their Patterson functions P(z) since the scattering intensity function is its Fourier transform. 

If the selfconvolution of the electron density of a molecule (projected onto z) is denoted M(z) and the point-distribution function prescribing how the molecules are repeated along z by D(z), the desired Patterson function is their convolution... 

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