G orbitals

A quantum mechanical treatment of molecular systems usually starts with the Bom-Oppenlieimer approximation, i.e., the separation of the electronic and nuclear degrees of freedom. This is a very good approximation for well separated electronic states. The expectation value of the total energy in this case is a fiinction of the nuclear coordinates and the parameters in the electronic wavefunction, e.g., orbital coefficients. The wavefiinction parameters are most often detennined by tire variation theorem the electronic energy is made stationary (in the most important ground-state case it is minimized) with respect to them. The... 

No currently known elements contain electrons in g (< = 4) orbitals in the ground state. If an element is discovered that has electrons in the g orbital, what is the lowest value for n in which these g orbitals could exist What are the possible values of mi How many electrons could a set of g orbitals hold ... 

Finally, a set of 5 g-orbitals, again, generated according to the relations presented above, was added generating a [13s,5p,5d,5f,5g] basis (basis C). The Bethe sum rule calculated [15] using these bases is presented in Fig. 4. As was noted... 

Figure 20-12 summarizes the electrical interactions of an octahedral complex ion. The three orbitals that are more stable are called 2 g orbitals, and the two less stable orbitals are called Sg orbitals. The difference in energy between the two sets is known as the crystal field splitting energy, symbolized by the Greek letter h. 

Whether 2 g or tg 8g is more stable depends on the relative magnitudes of P and Zl. If the energy required to pair electrons in a 2 g orbital is less than the energy required to populate an Sg orbital P < A), the ground state configuration is g Sg. On the other hand, if P > A, the most stable configuration is g g Both possibilities are shown in Figure 20-13 for the four-electron, octahedral case. 

One of the drawbacks of the first iteration, however, is that computation of energy quantities, e.g. orbital and total energies, requires to evaluate the integrals occurring in Eq. 3 on the basis of the ( )il )(p)- Unfortunately, the transcendental functions in terms of which the (]>il Hp) are expressed at the end of the first iteration do not lead to closed form expressions for these integrals and a numerical procedure is therefore needed. This constitutes a barrier to carry out further iterations to improve the orbitals by approaching the HE limit. A compromise has been proposed between a fully numerical scheme and the simple first iteration approach based on the fact that at the end of each iteration the < )j(k)(p) s entail the main qualitative characteristics of the exact solution and most... 

Here 2 fn must be multiplied by 1/3 because there is only 1 unpaired electron in the t%g orbitals instead of 3 as in Mn2+. Table 8 and Fig. 14b show the relationship between i v and total spin transfer. 

Pearson, R. G. Orbital Symmetry Rules for Inorganic Reactions from Perturbation Theory. 41, 75-112(1973). 

The underlying space ofVjlG is the set of closed G -orbits modulo the equivalence relation defined by x y if and only if (3.2) holds. 

The second statement of Theorem 3.1 follows from the following fact the closure of a G -orbit is a union of orbits of smaller dimensions. Hence any orbit contains a closed G -orbit in its closure. (Moreover, it is unique by Theorem 3.3.)... 

The quotient space /r (0)/G is called a symplectic quotient (or Marsden-Weinstein reduction). It has a complex structure and natural Kahler metric (cf. Theorem 3.30) on points where G acts freely. On the other hand, the set of closed G -orbits is the affine algebro-geometric quotient and denoted by YjjG. In fact, it is known that the above identihcation intertwines the complex structures. 

The most striking feature of Figs. 7 and 8, particularly as compared with the hydride plot, is the plunge downfield of the resultant shielding, following the paramagnetic term, across the row of the central atom for the typical elements, and also for the early transition metals (Fig. 8), with ready circulation of fluorine 2p electrons into empty t g orbitals in the complexes. Fluorine is h hly shielded however in the d molecules and ions, and this was discussed in Section III, B as a possible Cornwell effect (62). 

Many additional studies have since been made. A summary of the results for a number of octahedral complexes is given in Table 10.5. The predictions of ligand field theory are clearly borne out by the results, which show pronounced depopulation of the field-destabilized e g orbitals and increased population of the stabilized t2g(eg, ag) orbitals relative to the distribution in the high-spin spherical atom. 

The population of the destabilized e g orbitals is larger for the Co complexes than for the Cr compounds listed, a trend with increasing number of electrons reproduced in the sulfides discussed in the following section. A population of more than two electrons of the e g orbitals implies population of the antibonding metal-ligand orbitals, a state only reached in the Co(II) complex listed in the last column. The total number of d electrons, however, seems to correlate more with the element than with the specific valence state of the element, as there is no systematic difference between the Co(II) amd Co(III) complexes. But the number of available studies is still too small to allow more general conclusions. 

The hyperfine tensor can be used to estimate the spin density on the oxygen atoms. Assuming that the unpaired electron is in an axially symmetrical n g orbital, the axial hyperfine tensor can be separated into an isotropic part (aiso) and an anisotropic traceless tensor in the form... 

A Jahn-Teller distortion should also occur for configuration d. However, in this case the occupied orbital is a t g orbital, for example d, this exerts a repulsion on the ligands on the axes x and y which is only slightly larger than the force exerted along the z axis. The distorting force is usually not sufficient to produce a perceptible effect. Ions like TiF or MoClg show no detectable deviation from octahedral symmetry. 

KTl does not have the NaTl structure because the K+ ions are too large to fit into the interstices of the diamond-like Tl framework. It is a cluster compound KgTl with distorted octahedral Tlg ions. A Tlg ion could be formulated as an electron precise octahedral cluster, with 24 skeleton electrons and four 2c2e bonds per octahedron vertex. The thallium atoms then would have no lone electron pairs, the outside of the octahedron would have nearly no valence electron density, and there would be no reason for the distortion of the octahedron. Taken as a closo cluster with one lone electron pair per Tl atom, it should have two more electrons. If we assume bonding as in the BgHg ion (Fig. 13.11), but occupy the t g orbitals with only four instead of six electrons, we can understand the observed compression of the octahedra as a Jahn-Teller distortion. Clusters of this kind, that have less electrons than expected according to the Wade rules, are known with gallium, indium and thallium. They are called hypoelectronic clusters their skeleton electron numbers often are 2n or 2n — 4. 

Beyond the s, p, d, and/orbitals are the g orbitals, whose shapes are even more complex. 

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