Orbital total electron density

The Total Electron Density Distribution and Molecular Orbitals... 

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. 

In addition to total energy and gradient, HyperChem can use quantum mechanical methods to calculate several other properties. The properties include the dipole moment, total electron density, total spin density, electrostatic potential, heats of formation, orbital energy levels, vibrational normal modes and frequencies, infrared spectrum intensities, and ultraviolet-visible spectrum frequencies and intensities. The HyperChem log file includes energy, gradient, and dipole values, while HIN files store atomic charge values. 

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. 

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. 

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)... 

The bent-bond model can be expressed in orbital terms by assuming that the two components of the double bond are formed from sp3 hybrids on the carbon atoms (Figure 3.19) That this model and the ct-tt model are alternative and approximate, but equivalent, descriptions of the same total electron density distribution can be shown by converting one into the other by taking linear combinations of the orbitals, as shown in Figure 3.20. But neither form of the orbital model can predict the observed deviations from the ideal angles of 109° and 120°. 

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. 

Extension of this method for correcting the energies of approximate wave functions to systems containing more electrons and orbitals would be very useful. But difficulties quickly arise. The interelectronic effects become complicated because of exchange and correlation. More importantly, in DFT, it is only the highest occupied orbital whose energy is equal to the electronic chemical potential. This potential is valid for the total electron density. 

Figure 1. An Fe(H20)62+-Fe(HgO)6s+ complex at the traditional inner-sphere contact distance with the inner-sphere complexes (Th symmetry) oriented to give overall S6 symmetry. This geometry is favorable for transfer of an electron between t g-5d atomic orbitals (AO s, which have

Evidently, the LSD and GGA approximations are working, but not in the way the standard spin-density functional theory would lead us to expect. In Ref [36], a nearly-exact alternative theory, to which LSD and GGA are also approximations, is constructed, which yields an alternative physical interpretation in the absence of a strong external magnetic field. In this theory, Hf(r) and rti(r) are not the physical spin densities, but are only intermediate objects (like the Kohn-Sham orbitals or Fermi surface) used to construct two physical predictions the total electron density n(r) from... 

Electron distributions are ascertained by means of a Mulliken population analysis of the ir-electron atomic populations (in the case of complete n-delocalization, each atom in the 67r-electron five-membered ring would have 1.20 7r-electrons associated with it) by determining the spatial extent of the localized orbitals of both the out-of-plane lone pair of the heteroatom and the C=C double bonds as well as through comparing the total electron density plots in planes parallel to the molecular plane. 

Equation (11.8) reads The average of the expectation values of r — for the various valence AOs of atom I, weighted by the rations of the orbital populations to the total atomic population of atom I equals the inverse of the — / distance. For all their their simplicity, Eqs. (11.7) and (11.8) cannot be tested numerically by direct calculation. The reason is linked to the difficulty of partitioning the total electron density into atomic contributions, in spite of an important conceptual step forward due to Parr [219]. A practical step in the same direction is in the construction of suitable in situ valence atomic orbitals (VAO) from accurate ab initio computations [143], as advocated long ago by Mulliken [220] and discussed by Del Re [221]. As will be seen, such in situ VAOs do provide useful information, but they are of no help in solving the additional problem of assigning suitable populations to the orbitals and of dividing overlap populations into atomic contributions. In view of this situation, we take Eqs. (11.5) and (11.8) as statements whose validity rests on experimental evidence, at least for saturated hydrocarbons. 

Another quantity of some utility is the so-called local ionization potential, I(r). This is defined as the sum over orbital electron densities, pi(r) times absolute orbital energies, e i, and divided by the total electron density, p(r). 

The results of Hiickel molecular orbital (HMO) calculations of the 7t-electron distribution in 287 indicate that N-1 is the most electron-rich center. Hiickel MO calculations for protonated species 288 and 289 indicate that 288 is more stable (7t-electron binding energies are 13.353 and 13.297 B). The all-valence electron CNDO/2 calculation for 287 yields virtually identical total electron densities (75CJC119). 

It is convenient to separate the total electron density at each atom into a- and 71-components. It is likely to be the 7t-density that will be important in reactions with nucleophiles, since in an orbitally controlled reaction (Chapter 1) the donor orbital of the incoming nucleophile will initially interact with the lowest vacant 7i -orbital. The overall pattern of charge alternation is repeated in both the 7t- and the a-electron densities, and nucleophiles are expected to attack at the 2- or 4-positions. This is exactly the pattern that is seen in... 

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