The Total Electron Density

By construction, the present density-functional methods are expected to yield not only accurate total energies but also accurate total electron densities. In this section we shall study these. 

This is in fact a general observation. As a further example of this we consider the density of the crystalline, ionic compound a-MnS that was studied by Tappero and Lichanot.57 According to their Hartree-Fock calculations, the material can be considered as consisting of Mn2+ and S2- ions, whereas the density-functional calculations predict the effective charges to be closer to 1.6. 

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Electron density represents the probability of finding an electron at a poin t in space. It is calcii lated from th e elements of th e den sity matrix. The total electron density is the sum of the densities for alpha and beta electrons. In a closed-shell RUE calculation, electron densities are the same for alpha and beta electrons. 

The Total Electron Density Distribution and Molecular Orbitals... 

The total electron density is just the sum of the densities for the two types of electron. The exchange-correlation functional is typically different for the two cases, leading to a set of spin-polarised Kohn-Sham equations ... 

A much less basis set dependent method is to analyze the total electron density. This is called the atoms in molecules (AIM) method. It is designed to examine the small effects due to bonding in the primarily featureless electron density. This is done by examining the gradient and Laplacian of electron density. AIM analysis incorporates a number of graphic analysis techniques as well as population analysis. The population analysis will be discussed here and the graphic techniques in the next chapter. 

FIGURE 13.6 A plot showing two data values. The shape is an isosurface of the total electron density. The color applied to the surface is based on the magnitude of the electrostatic potential at that point in space. 

Wave functions can be visualized as the total electron density, orbital densities, electrostatic potential, atomic densities, or the Laplacian of the electron density. The program computes the data from the basis functions and molecular orbital coefficients. Thus, it does not need a large amount of disk space to store data, but the computation can be time-consuming. Molden can also compute electrostatic charges from the wave function. Several visualization modes are available, including contour plots, three-dimensional isosurfaces, and data slices. 

Davidsou-Fletcher-Powell (DFP) a geometry optimization algorithm De Novo algorithms algorithms that apply artificial intelligence or rational techniques to solving chemical problems density functional theory (DFT) a computational method based on the total electron density... 

Once you have calculated an ab initio or a semi-empirical wave function via a single point calculation, geometry optimization, molecular dynamics or vibrations, you can plot the electrostatic potential surrounding the molecule, the total electronic density, the spin density, one or more molecular orbitals /i, and the electron densities of individual orbitals You can examine orbital energies and select orbitals for plotting from an orbital energy level diagram. 

The total electron density contributed by all the electrons in any molecule is a property that can be visualized and it is possible to imagine an experiment in which it could be observed. It is when we try to break down this electron density into a contribution from each electron that problems arise. The methods employing hybrid orbitals or equivalent orbitals are useful in certain circumsfances such as in rationalizing properties of a localized part of fhe molecule. Flowever, fhe promotion of an electron from one orbifal fo anofher, in an electronic transition, or the complete removal of it, in an ionization process, both obey symmetry selection mles. For this reason the orbitals used to describe the difference befween eifher fwo electronic states of the molecule or an electronic state of the molecule and an electronic state of the positive ion must be MOs which belong to symmetry species of the point group to which the molecule belongs. Such orbitals are called symmetry orbitals and are the only type we shall consider here. 

Since all observable properties depend only on the total electron density, and not the individual MOs, there is no unique choice for H,... 

The success of FMO theory is not because the neglected terms in the second-order perturbation expansion (eq. (15.1)) are especially small an actual calculation will reveal that they completely swamp the HOMO-LUMO contribution. The deeper reason is that the shapes of the HOMO and LUMO resemble features in the total electron density, which determines the reactivity. 

The fact that features in the total electron density are closely related to the shapes of the HOMO and LUMO provides a much better rationale of why FMO theory works as well as it does, than does the perturbation derivation. It should be noted, however, that improvements in the wave function do not necessarily lead to a better performance of the FMO method. Indeed the use of MOs from semi-empirical methods usually works better than data from ab initio wave functions. Furthermore it should be kept in mind that only the HOMO orbital converges to a specific shape and energy as the basis set is... 

The raw output of a molecular structure calculation is a list of the coefficients of the atomic orbitals in each LCAO (linear combination of atomic orbitals) molecular orbital and the energies of the orbitals. The software commonly calculates dipole moments too. Various graphical representations are used to simplify the interpretation of the coefficients. Thus, a typical graphical representation of a molecular orbital uses stylized shapes (spheres for s-orbitals, for instance) to represent the basis set and then scales their size to indicate the value of the coefficient in the LCAO. Different signs of the wavefunctions are typically represented by different colors. The total electron density at any point (the sum of the squares of the occupied wavefunctions evaluated at that point) is commonly represented by an isodensity surface, a surface of constant total electron density. 

Fig. 5.3 Variation of the li, 2s, 3s and contributions to the total electron density at the iron nucleus for a collection of iron complexes. Nonrelativistic B3LYP DFT calculations with the CP (PPP) basis set (taken from [19])...

However, the division of the electron density at the iron nucleus into contributions arising from Is through 4s contributions can be done conveniently at the level of the canonical molecular orbitals. This arises because the iron Is, 2s, and 3s orbitals fall into an orbital energy range where they are well isolated and hence do not mix with any hgand orbital. Hence, the Is, 2s, and 3s contributions are well defined in this way. The 4s contribution then arises typically from several, if not many, molecular orbitals in the valence region that have contributions from the iron s-orbitals. Thus, the difference between the total electron density at the nucleus and... 

To illustrate this point, the contributions of the occupied molecular orbitals to the total electron density at the nucleus are summarized in Table 5.2 for Fep4 (S - 5/2). It is evident from the table that the contributions coming from the orbitals at —6,966 eV must be assigned to the iron Is orbital, those from orbitals at —816 eV to the iron 2s orbital, and those from orbitals at —95 eV to the iron 3s orbital. In this highly symmetric complex, only two valence orbitals contribute to p(0), i.e. the —25 eV contribution from the totally symmetric ligand-group orbital that is derived from the F 2s orbitals and the —1 eV contribution from the totally symmetric... 

Table 5.3 Contributions of -orbitals to the total electron density at the iron nucleus (in a.u. ) as a function of oxidation state and configuration. Calculations were done with the spin-averaged Hartree-Fock method and a large uncontracted Gaussian basis set. (17 1 Ip 5d If)... 

All of the studies published so far have been aiming at the reconstruction of the total electron density in the crystal by redistribution of all electrons, under the constraints imposed by the MaxEnt requirement and the experimental data. After the acceptance of this paper, the authors became aware of valence-only MaxEnt reconstructions contained in the doctoral thesis of Garry Smith [58], The authors usually invoke the MaxEnt principle of Jaynes [23-26], although the underlying connection with the structural model, known under the name of random scatterer model, is seldom explicitly mentioned. 

Most of the relevant features of the charge density distribution can be elegantly elucidated by means of the topological analysis of the total electron density [43] nevertheless, electron density deformation maps are still a very effective tool in charge density studies. This is especially true for all densities that are not specified via a multipole model and whose topological analysis has to be performed from numerical values on a grid. 

Although the electron domain model is, as we shall see, a very useful model, we must remember that it is just that, a model—indeed a very approximate model. We cannot observe the individual domains of electrons but only the total electron density distribution. 

In Eq. (2.48), the summation is clearly taken over the orbitals centered on all atoms other than A, and PBB is the total electron density associated with atom B, i.e., the summation is over all AOs on B. The problem is now to derive suitable expressions for the one-electron elements H in a manner consistent with the neglect of orbital overlap. 

Electron spin resonance, nuclear magnetic resonance, and neutron diffraction methods allow a quantitative determination of the degree of covalence. The reasonance methods utilize the hyperfine interaction between the spin of the transferred electrons and the nuclear spin of the ligands (Stevens, 1953), whereas the neutron diffraction methods use the reduction of spin of the metallic ion as well as the expansion of the form factor [Hubbard and Marshall, 1965). The Mossbauer isomer shift which depends on the total electron density of the nucleus (Walker et al., 1961 Danon, 1966) can be used in the case of Fe. It will be particularly influenced by transfer to the empty 4 s orbitals, but transfer to 3 d orbitals will indirectly influence the 1 s, 2 s, and 3 s electron density at the nucleus. 

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