Orbital densities

These densities are identieal to one another and do not deseribe polarized orbital densities. Therefore, the Cl wavefunetion whieh mixes the two eonfigurations with like sign, when analyzed in terms of orbital pairs, plaees the eleetrons into orbitals ([) +=(2s ia2px,y,or z)... 

HOMO and LUMO energies FMO reactivity indices Refractivity Total energy Ionization potential Electron affinity Energy of protonation Orbital populations Frontier orbital densities Superdelocalizabilities... 

Once the job is completed, the UniChem GUI can be used to visualize results. It can be used to visualize common three-dimensional properties, such as electron density, orbital densities, electrostatic potentials, and spin density. It supports both the visualization of three-dimensional surfaces and colorized or contoured two-dimensional planes. There is a lot of control over colors, rendering quality, and the like. The final image can be printed or saved in several file formats. 

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. 

The necessity for going beyond the HF approximation is the fact that electrons are further apart than described by the product of their orbital densities, i.e. their motions are con-elated. This arises from the electron-electron repulsion operator, which is a sum of ten-ns of the type... 

In an octahedral crystal field, for example, these electron densities acquire different energies in exactly the same way as do those of the J-orbital densities. We find, therefore, that a free-ion D term splits into T2, and Eg terms in an octahedral environment. The symbols T2, and Eg have the same meanings as t2g and eg, discussed in Section 3.2, except that we use upper-case letters to indicate that, like their parent free-ion D term, they are generally many-electron wavefunctions. Of course we must remember that a term is properly described by both orbital- and spin-quantum numbers. So we more properly conclude that a free-ion term splits into -I- T 2gin octahedral symmetry. Notice that the crystal-field splitting has no effect upon the spin-degeneracy. This is because the crystal field is defined completely by its ordinary (x, y, z) spatial functionality the crystal field has no spin properties. 

Orbital pictures have an advantage over electron density plots in that they can indicate the three-dimensional nature of orbitals. One type of orbital picture is a two-dimensional color pattern in which the density of color represents electron density. Figure 7-19Z> shows such an orbital density picture of the 2s orbital. This two-dimensional pattern of color density shows a cross-sectional slice through the middle of the orbital. 

Orbital density pictures are probably the most comprehensive views we can draw, but they require much time and care. An electron contour drawing provides a simplified orbital picture. In this representation, we draw a contour surface that encloses almost all the electron density. Commonly, almost all means 90%. Thus, the electron density is high inside the contour surface but very low outside the surface. Figure 7-19c shows a contour drawing of the 2s orbital. 

Figure 3. Comparison of the measured momentum distributions of the outermost valence orbital for wafer [6-8] with spherically averaged orbital densities from Hartree-Fock limit and correlated wave functions [6].

The base-catalyzed allyl rearrangement of olefins can be treated by the LU orbital density criteria 132>. The LU orbital remarkably localizes at the hydrogen atoms attached to the / carbon to the double bond in... 

Gallant, R. T. and A. St-Amant. 1996. Linear scaling for the charge density fitting procedure of the linear combination of Gaussian-type orbitals density functional method. Chem. Phys. Lett. 256, 569. 

From the preceding discussion, it is quite clear that p(r) is indeed a fundamental quantum mechanical entity of no less significance than the wave function and that p(r) generates numerous attractive and transparent models of chemical behavior. How does one calculate p(r) One way would of course be to calculate it from the normalized occupied orbital densities, viz.,... 

The type ir 2 can occur in molecules which have adjacent joint atoms and in which the main orbital density is in a joint-joint bond. It extends somewhat... 

Brown concludes, therefore, that the frontier orbital density distribution Cff determines the position of attack. 

Generation of orbital densities for free and confined atoms through... 

The multiplying factor in equation (29) is chosen so that the Dirichlet boundary condition is satisfied for each orbital density, i.e., pk(r—R)—0. 

Given a confinement radius, R, the orbital density parameters are allowed to evolve freely until the total energy given by equation (19) becomes a minimum subject to the normalization constraint indicated in equations (27) and (30). An auxiliary condition for the free-case (R 00) is the fulfillment of the Virial... 

There have been discussions whether the amide addition at C-6 is charge-controlled or orbital-controlled. Charge density calculations in 4-phenylpyrimidine (MNDO method) predict that the addition of the amide ion would preferably take place at position 2 (95UP1) this, however, does not agree with the experimental results. Therefore, the conclusion seems justified that the addition is not charge-controlled. Frontier orbital calculations, using the SCF-PPP method, show that the frontier orbital densities in the LUMO of pyrimidine are zero at C-2 and C-5, making these positions... 

Using this equation we find the following asymptotic behavior of the Kohn-Sham orbital densities ... 

Here, Y(j is the 15-element column vector of the angular part of the < (df)0(dj) orbital products, Py is the row vector of the 15 unique elements of the symmetric 5x5 matrix of the coefficients in Eq. (10.3), and Pimp is the row vector containing the coefficients of the 15 spherical harmonic density functions d,mp with / = 0, 2, or 4. Density functions with other / values do not contribute to the d-orbital density. 

In the symmetry-adapted formulation, the 43- term no longer occurs because the d-orbital density contains a vertical mirror plane even if such a plane is absent in the point group. This is illustrated as follows. Point groups without vertical mirror planes differ from those with vertical mirror planes by the occurrence of both dlm+ and d(m functions, with m being restricted to n, the order of the rotation axis. But the coordinate system can be rotated around the main symmetry axis such that P4 becomes zero. As proof, we write the (p dependence as... 

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