Expansion orbital/shell

So far, we have considered only the restricted Hartree-Fock method. For open shell systems, an unrestricted method, capable of treating unpaired electrons, is needed. For this case, the alpha and beta electrons are in different orbitals, resulting in two sets of molecular orbital expansion coefficients ... 

Atoms of these elements have empty J-orbitals in the valence shell. Another factor—possibly the main factor—in determining whether more atoms than allowed by the octet rule can bond to a central atom is the size of that atom. A P atom is big enough for as many as six Cl atoms to fit comfortably around it, and PC15 is a common laboratory chemical. An N atom, though, is too small, and NC15 is unknown. A compound that contains an atom with more atoms attached to it than is permitted by the octet rule is called a hypcrvalent compound. This name leaves open the question of whether the additional bonds are due to valence-shell expansion or simply to the size of the central atom. 

A hybridization scheme is adopted to match the electron arrangement of the molecule. Valence-shell expansion requires the use of d-orbitals. 

Ag+ with 1.2 A ) [99]. Spin-orbit coupling is neglected in our analysis because the results shown in Table 4.2 are from scalar relativistic Douglas-Kroll calculations. Because of the additional shell expansion of the 5ds/2 orbital due to spin-orbit coupling, we expect a further increase of the polarizability of Au. Table 4.3 also... 

The electrostatic energy is calculated using the distributed multipolar expansion introduced by Stone [39,40], with the expansion carried out through octopoles. The expansion centers are taken to be the atom centers and the bond midpoints. So, for water, there are five expansion points (three at the atom centers and two at the O-H bond midpoints), while in benzene there are 24 expansion points. The induction or polarization term is represented by the interaction of the induced dipole on one fragment with the static multipolar field on another fragment, expressed in terms of the distributed localized molecular orbital (LMO) dipole polarizabilities. That is, the number of polarizability points is equal to the number of bonds and lone pairs in the molecule. One can opt to include inner shells as well, but this is usually not useful. The induced dipoles are iterated to self-consistency, so some many body effects are included. 

Nakatsuji H, Hirao K (1978) Cluster expansion of the wavefunction. symmetry-adapted-cluster expansion, its variational determination, and extension of open-shell orbital theory. J Chem Phys 68 2053... 

The 2p orbital radius may be considered anomalously small (of the same order as the 2s orbital radius) because there is no inner shell of the same angular symmetry that exerts outward steric pressure due to the Pauli exclusion principle. (A similar exception causes the first transition series to appear anomalous compared with later lanthanides, since 3d orbitals form the innermost d shell.) The 2p -> 3p expansion therefore appears to be relatively more pronounced than 2s —> 3s expansion. 

It is noteworthy that Rydberg orbital occupancies on the central atom (rY, final column of Table 3.29) are relatively negligible (0.01-0.03e), showing that d-orbital participation or other expansion of the valence shell is a relatively insignificant feature of hyperbonded species. However, the case of HLiH- is somewhat paradoxical in this respect. The cationic central Li is found to use conventional sp linear hybrids to form the hydride bonds, and thus seems to represent a genuine case of expansion of the valence shell (i.e., to the 2p subshell) to form two bonding hybrids. However, the two hydride bonds are both so strongly polarized toward H (93%) as to have practically no contribution from Li orbitals, so the actual occupancy of extra-valent 2pu orbitals ( 0.03< ) remains quite small in this case. 

Burdett (35-38) has extended the AOM by the introduction of a quartic term in the expansion of the perturbation determinant as a power series in the overlap integral Sx. In the conventional AOM, only the quadratic term (proportional to Sx) is considered. In closed-shell systems, the sum of the energies of the relevant orbitals is independent of angular variations in the molecular geometry if only the quadratic term is used. This is no longer true if the quartic term is included, and it is possible to rationalise many stereochemical observations. 

Note that the energy contribution depends only on the principal quantum number n. Therefore, each of the n2 orbitals that constitute the shell n contributes the same amount of energy, justifying the use of the principal expansion. Summing the energy contributions from all orbitals, we obtain... 

Shell corrections can also be evaluated without recourse to an expansion in powers of v, but existing calculations such as Refs. [13,14] are based on specific models for the target atom and, unlike equation (19), do not end up in expressions that would allow to identify the physical origin of various contributions. It is clear, however, that orbital motion cannot be the sole cause of shell corrections The fact that the Bethe logarithm turns negative at 2mv /I< 1 cannot be due to the neglect of orbital motion but must be of a purely mathematical nature. Unfortunately, the uncertainty principle makes it impossible to eliminate orbital motion in an atom from the beginning. 

The number No of occupied valence SCF orbitals in a molecule is typically less than the total number Nmb of orbitals in the minimal valence basis sets of all atoms. The full valence MCSCF wavefunction is the optimal expansion in terms of all configurations that can be generated from N b molecular orbitals. Closely related is the full MCSCF wavefunction of all configurations that can be generated from Ne orbitals, where Nc is the number of valence electrons, i.e. each occupied valence orbital has a correlating orbital, as first postulated by Boys (48) and also presumed in perfect pairing models (49,50), We shall call these two types of frill spaces FORS 1 and FORS 2. In both, the inner shell remains closed. 

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