Orbital contribution

The mixing has nothing to do with the possibility of any molecules populating the term, which is typically 12,000 cm above the ground state term. The population of such a term is of the order -12000/200 room temperature kT 200 cm at 300 K), which is absolutely negligible. The mixing arises because a description of the molecular Hamiltonian in terms of Eq. (5.14) is incomplete and should be replaced with Eq. (5.15). 

The extent of the mixing - the magnitude of c in (5.13) - is proportional to the cause of the mixing and inversely proportional to the energy separation of the 

Reminder The one-electron spin-orbit coupling coefficient, is intrinsically positive. The many-electron spin-orbit parameter X is defined by 

Some useful spin-orbit coupling coefficients  

A similar expression has been derived for cubic-field complex ions having an E ground term (Eq. 5.17). 

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The first theoretical handling of the weak R-T combined with the spin-orbit coupling was carried out by Pople [71]. It represents a generalization of the perturbative approaches by Renner and PL-H. The basis functions are assumed as products of (42) with the eigenfunctions of the spin operator conesponding to values E = 1/2. The spin-orbit contribution to the model Hamiltonian was taken in the phenomenological form (16). It was assumed that both interactions are small compared to the bending vibrational frequency and that both the... 

All of these trends can be accommodated by the orbital hybridization model The bond angles are characteristic for the sp sp and sp hybridization states of carbon and don t require additional comment The bond distances bond strengths and acidities are related to the s character m the orbitals used for bonding s Character is a simple concept being nothing more than the percentage of the hybrid orbital contributed by an s orbital Thus an sp orbital has one quarter s character and three quarters p an sp orbital has one third s and two thirds p and an sp orbital one half s and one half p We then use this information to analyze how various qualities of the hybrid orbital reflect those of its s and p contributors... 

All filled orbitals contribute zero total orbital and total electron spin momenta. Therefore, in deriving the states arising from the configuration we need consider only the two equivalent Ad electrons. According to Table 7.2, the terms arising are... 

In accordance with the one-dimensional periodic orbit theory, any orbit contributing to g E) is supposedly constructed from closed classical orbits in the well and subbarrier imaginary-time trajectories. These two classes of trajectories are bordering on the turning points. For the present model the classical motion in the well is separable, and the harmonic approximation for classical motion is quite reasonable for more realistic potentials, if only relatively low energy levels are involved. 

The atomic orbital contributions for each atom in the molecule are given for each molecular orbital, numbered in order of increasing energy (the MO s energy is given in the row labeled EIGENVALUES preceding the orbital coefficients). The symmetry of the orbital and whether it is an occupied orbital or a virtual (unoccupied) orbital appears immediately under the orbital number. 

Fe " complexes in general have magnetic moments at room temperature which are close to 5.92 BM if they are high-spin and somewhat in excess of 2BM (due to orbital contribution) if they are low-spin. A number of complexes, however, were prepared in 1931 by L. Cambi and found to have moments intermediate between these extremes. They are the iron(lll)-A,A-dialkyldithiocarbamates, [Fe(S2CNR2)3], in which the ligands are ... 

Eg term. A magnetic moment of around 5.5 BM (i.e. 4.90 BM- -orbital contribution) is expected for pure octahedral symmetry but, in practice, distortions produce values in the range 5.2-5.4BM. Similarly, in the electronic spectrum, the expected single band due to the Eg t ge g) T2g t ge ) transition is broadened... 

The T ground term of the tetrahedral ion is expected to lead to a temperature-dependent orbital contribution to the magnetic moment, whereas the A ground term of the octahedral ion is not, though mixing of the excited T2g(F) term into the AigiF) ground term is expected to raise its moment to ... 

Since the coiTelation between opposite spins has both intra- and inter-orbital contributions, it will be larger than the correlation between electrons having the same spin. The Pauli principle (or equivalently the antisymmetry of the wave function) has the consequence that there is no intraorbital conelation from electron pairs with the same spin. The opposite spin correlation is sometimes called the Coulomb correlation, while the same spin correlation is called the Fermi correlation, i.e. the Coulomb correlation is the largest contribution. Another way of looking at electron correlation is in terms of the electron density. In the immediate vicinity of an electron, here is a reduced probability of finding another electron. For electrons of opposite spin, this is often referred to as the Coulomb hole, the corresponding phenomenon for electrons of the same spin is the Fermi hole. 

Sometimes several different types of localized group orbitals contribute significantly to the molecular orbital. The molecular orbital is then labeled by listing all the different types of group orbitals which have a large amplitude in the overall molecular orbital. Thus... 

The problem could be stated from another point of view. In an isostructural series the uranium and neptunium compounds tend to be itinerant electron magnets or band magnets (like iron) and their orbital contribution is at least partially quenched. For much heavier actinides we know that the compounds will make local moment magnets with orbital contributions. It is quite possible that in between these two clear cut forms of magnetism that the intermediate case could be dominated by fluctuations, and no recognizable form of magnetism would occur. To state that the... 

The molecular orbital energy-level diagrams of heteronuclear diatomic molecules are much harder to predict qualitatitvely and we have to calculate each one explicitly because the atomic orbitals contribute differently to each one. Figure 3.35 shows the calculated scheme typically found for CO and NO. We can use this diagram to state the electron configuration by using the same procedure as for homonuclear diatomic molecules. 

Another important polyatomic molecule is benzene, C6f I6, the parent of the aromatic compounds. In the molecular orbital description of benzene, all thirty C2s-, C2p-, and Hls-orbitals contribute to molecular orbitals spreading over all twelve atoms (six C plus six H). The orbitals in the plane of the ring (the C2s-, C2px, and ( 2/ -orbitals on each carbon atom and all six Hls-orbitals) form delocalized o-orbitals that bind the C atoms together and link the H atoms to the C atoms. The six C2pz-orbitals, which are perpendicular to the ring, contribute to six delocalized tt-orbitals that spread all the way around the ring. However, chemists... 

These two formulae describe orbital contributions to ground A or E terms that arise by so-called second-order spin-orbit coupling with appropriate excited... 

For the cubic-field (octahedral or tetrahedral) subshells g(g), there is spatial degeneracy for e(g) but not for eQy Nevertheless, neither of these configurations give rise to an orbital contribution to the magnetic moment. The conditions for orbital contributions to arise in strong-field configurations are that the orbitals must be... 

Aromatic sextets can also be present in five- and seven-membered rings. If a five-membered ring has two double bonds and the fifth atom possesses an unshared pair of electrons, the ring has five p orbitals that can overlap to create five new orbitals— three bonding and two antibonding (Fig. 2.6). There are six electrons for these orbitals the four p orbitals of the double bonds each contribute one and the filled orbital contributes the other two. The six electrons occupy the bonding orbitals and constitute an aromatic sextet. The heterocyclic compounds pyrrole, thiophene, and... 

Among the compounds that form complexes with silver and other metals are benzene (represented as in 9) and cyclooctatetraene. When the metal involved has a coordination number >1, more than one donor molecule participates. In many cases, this extra electron density comes from CO groups, which in these eomplexes are called carbonyl groups. Thus, benzene-chromium tricarbonyl (10) is a stable compound. Three arrows are shown, since all three aromatic bonding orbitals contribute some electron density to the metal. Metallocenes (p. 53) may be considered a special case of this type of complex, although the bonding in metallocenes is much stronger. 

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