Orbital density difference

Figure 3. Orbital density difference plot Ap in one of the coordinate planes. Full lines Ap > 0 dashed lines Ap < 0 dotted lines Ap = 0. Ap at the different isodensity lines equals 0.16, 0.08, 0.04,. .. 0.0025.

Similar diagrams can be obtained for MLCT-states. In the 2t + 9t. case, the formal orbital density difference does corres-... 

A UHF wave function may also be a necessary description when the effects of spin polarization are required. As discussed in Differences Between INDO and UNDO, a Restricted Hartree-Fock description will not properly describe a situation such as the methyl radical. The unpaired electron in this molecule occupies a p-orbital with a node in the plane of the molecule. When an RHF description is used (all the s orbitals have paired electrons), then no spin density exists anywhere in the s system. With a UHF description, however, the spin-up electron in the p-orbital interacts differently with spin-up and spin-down electrons in the s system and the s-orbitals become spatially separate for spin-up and spin-down electrons with resultant spin density in the s system. 

The basis functions are normally the same as used in wave mechanics for expanding the HF orbitals, see Chapter 5 for details. Although there is no guarantee that the exponents and contraction coefficients determined by the variational procedure for wave functions are also optimum for DFT orbitals, the difference is presumably small since the electron densities derived by both methods are very similar. ... 

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. 

As Figure 10-19 shows, bonds that form from the side-by-side overlap of atomic p orbitals have different electron density profdes than a bonds. A p orbital has zero electron density—a node—in a plane passing through the nucleus, so bonds that form from side-by-side overlap have no electron density directly on the bond axis. High electron density exists between the bonded atoms, but it is concentrated above and below the bond axis. A bond of this type is called a pi ( r) bond, and a bonding orbital that describes a ttbond is a tt orbital. 

Figure 3). This relationship determines how much electron density is allowed to be transferred from the dihydrogen moiety to the electron-deficient AH fragment. A double logarithmic representation of the orbital energy difference and Dq supports this trend (Figure 11). 

The general Jacobian problem associated with the transformation of a density Pi(r) into a density p2(r) (where these densities differ from that of the free-electron gas) was discussed by Moser in 1965 [58]. This work was not performed in the framework of orbital transformations - which might have interested chemists, nor was it done in the context of density functional theory - which might have attracted the attention of physicists. It was a paper written for mathematicians and, as such, it remained unknown to the quantum chemistry community. In the discussion that follows, we use the more accessible reformulation of Bokanowski and Grebert (1995) [65] which relies heavily on the work of Zumbach and Maschke (1983) [61]. Let us define as ifjy = the space of... 

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... 

Figure 2.29 shows charge density differences for CO adsorbed on Ni compared to CO in gas phase [3]. The results for the total density are rather similar to what has been observed elsewhere [58,66,67] and there interpreted in terms of the simple frontier orbital picture illustrated in the upper part of Figure 2.17. The left part of Figure 2.29 shows the total difference in the charge density and it is very similar to...

Electron density difference matrices that correspond to the transition energies in the EP2 approximation may be used to obtain a virtual orbital space of reduced rank [27] that introduces only minor deviations with respect to results produced with the full, original set of virtual orbitals. This quasiparticle virtual orbital selection (QVOS) process provides an improved choice of a reduced virtual space for a given EADE and can be used to speed up computations with higher order approximations, such as P3 or OVGF. Numerical tests show the superior accuracy and efficiency of this approach compared to the usual practice of omission of virtual orbitals with the highest energies [27],... 

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