E -orbital

For Woodward-Hoffm an allowed thermal reactions (such as the con rotatory ring opening of cyclobulan e), orbital symmetry is conserved and there is no change in orbital occupancy. Hven though bonds are made and broken, you can use the RHFwave fun etion. 

If the mini her of electrons, N, is even, yon can haven dosed shell (as shown ) where the occupied orbitals each contain two electron s. For an odd n nrn her of electron s, at least on e orbital rn ust be singly occupied. In the example, three orbitals are occupied by-electron s and two orbitals arc nn occupied. Th e h ighest occupied nioleciilar orbital (HOMO is t[r), and the lowest unoccupied molecular orbital (LUMO) is The example above is a singlet, a state oh total spin S=0. Exciting one electron from the HOMO to the LUMO orbital would give one ol the I ollowing excited states ... 

Solving the previous matrix equation tor the coefficients C describing the LCAO expan sion of th e orbitals and orbital energies n requires a matrix dia.i>onaliz(ition. If the overlap matrix were a... 

The diagoital matrix elemeitts for an atomic orbital can be shown to bear a very close assoeialion wiLli the energy needed to ion i/ e (remove) an electron from th e orbital. Valen ce state ion-i /ation eti ergies (VSlKs) can be determ in ed from experiment and extended HUekel theory equates to the eon espon din g VSIH, with opposite sign ... 

In a second example, the three CH bonds, three CH antibonds, CO bond and antibond, and three 0-atom non-bonding orbitals of the methoxy radical H3C-O also cluster into ai and e orbitals as shown below. In these cases, point group symmetry allows one to identify degeneracies that may not have been apparent from the structure of the orbital interactions alone. 

For the ground state of BH3 you would fill the bottom level (B-H bonding), af and e orbitals, with 6 eleetrons. 

For a 7C% 1 configuration in which two non-equivalent n orbitals (i.e., orbitals whieh are of n symmetry but whieh are not both members of the same degenerate set an example would be the n and 7i orbitals in the B2 moleeule) are oeeupied, the above analysis must be expanded by ineluding determinants of the form 7iia7i ia, ... 

This redueible representation (the oeeupaney of two e orbitals in the anion gives rise to more than one state, so the direet produet e x e eontains more than one symmetry eomponent) ean be deeomposed into pure symmetry eomponents (labels F are used to denote the irredueible symmetries) by using the deeomposition formula given in Appendix... 

To make further progress, one needs to evaluate the Rik matrix elements for the partieular orbitals given above and to then use these explieit values in the above equations. The matrix representations for the two e orbitals ean easily be formed and are as follows ... 

In summary, doubly exeited CSFs are often employed to permit polarized orbital pair formation and henee to allow for eleetron eorrelations. Singly exeited CSFs are ineluded to permit orbital relaxation (i.e., orbital reoptimization) to oeeur. 

These equations state that the three Ish orbitals can be combined to give one Ai orbital and, since E is degenerate, one pair of E orbitals, as established above. With knowledge of the ni, the symmetry-adapted orbitals can be formed by allowing the projectors... 

In octahedral complexes, the e -orbitals (dz< and dx2 -yi) lie higher in energy than the t2 -orbitals (dxy, dyz, and dzx). The opposite is true in a tetrahedral complex, for which the ligand field splitting is smaller. 

The subsets of d orbitals in Fig. 3-4 may also be labelled according to their symmetry properties. The d ildxi y2 pair are labelled and the d yldxMyz trio as t2g. These are group-theoretical symbols describing how these functions transform under various symmetry operations. For our purposes, it is sufficient merely to recognize that the letters a ox b describe orbitally i.e. spatially) singly degenerate species, e refers to an orbital doublet and t to an orbital triplet. Lower case letters are used for one-electron wavefunctions (i.e. orbitals). The g subscript refers to the behaviour of... 

Some prefer to write 4,ei = - % A>ct in order to emphasize the inversion of t and e orbital subsets. However, if A and Z, e, are defined as the orbital splittings, it is probably best to omit the sign. 

Immediately, therefore, we have one insight into the spectrochemical series by noting that both bonding and antibonding energy shifts of eg and e orbitals are... 

Steric repulsions come from two orbital-four electron interactions between two occupied orbitals. Facially selective reactions do occur in sterically unbiased systems, and these facial selectivities can be interpreted in terms of unsymmetrical K faces. Particular emphasis has been placed on the dissymmetrization of the orbital extension, i.e., orbital distortions [1, 2]. The orbital distortions are described in (Chapter Orbital Mixing Rules by Inagaki in this volume). Here, we review the effects of unsymmetrization of the orbitals due to phase environment in the vicinity of the reaction centers [3]. 

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