Best Overlap Orbitals

We have derived the Hartree-Fock method from the requirement that the orbitals should be those that give the lowest energy expectation value of a Slater determinant. In a Cl expansion where only singly substituted Slater determinants are used, all coefficients for the latter are equal to zero according to Brillouin s theorem. However, if doubly substituted Slater determinants (and higher) are introduced in the total wave function, the coefficients for the singly substituted Slater determinants are no longer equal to zero. 

It is possible to choose spin orbitals where the coefficients of the singly substituted determinant are equal to zero, even if doubly and higher substitutions are introduced. In a Cl type expansion using these orbitals, the Slater determinant with the latter spin orbitals has the maximum possible overlap with the true wave function. The new spin orbitals are therefore called best overlap orbitals. In a Slater determinant O, these orbitals minimize 

The best overlap orbitals are not the same as the natural spin orbital, but are probably very close (choosing the best unitary transformation of the best overlap orbitals). In a Cl expansion, the singly substituted determinants disappear if the best overlap orbitals of the same Cl problem are used. The best overlap orbitals are also called Brueckner orbitals (BO), after the American physicist Keith Brueckner, or are referred to as correlation corrected orbitals. 

The problem with the best overlap orbitals is that the total wave function has to be calculated before the correlation corrected orbitals can be obtained. This problem may be circumvented by calculating correlation potentials for each correlated pair of electrons and subsequently sum the correlation potentials for aU pairs. 

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The admixture of singly excited configurations (with respect to the SCF determinant) via their interaction with doubly excited ones is exactly equivalent to a replacement of Hartree-Fock by Brueckner (best-overlap) orbitals. Expectation values of simple one-electron operators like x, r, (3z — r )j2 etc., have been calculated (35) for H2 in its... 

In this context, it is pertinent to recall that in many cases one can obtain the so-called best overlap orbitals [64] of DODS type which are produced by the given many-electron wave function. These orbitals were considered in [65] where they were identified with spin-polarized Brueckner orbitals. However, they exist if and only if the so-called nonsinglet Brueckner instabihty conditions are satisfied. At last, if the correct spin-projected determinant is involved in the consideration, then it is always possible to construct the best overlap orbitals of DODS type for the given exact or approximate state vector ). These orbitals were recently introduced [62] and named the spin-polarized extended Bmeckner (SPEB) orbitals. By construction, they maximize T). 

Backside attack may be favored in order to facilitate transfer of nonbonding electrons from the nucleophile into the electrophile s lowest-unoccupied molecular orbital (LUMO). Efficient electron transfer requires maximal overlap of the LUMO and the donor orbital (usually a nonbonded electron pair on the nucleophile). Examine the LUMO of methyl bromide. How would a nucleophile have to approach in order to obtain the best overlap Is your answer more consistent with preferential backside or frontside attack ... 

Woodward and Hoffmann pointed out that the Diels-Alder reaction involved bonding overlap of the highest-occupied molecular orbital (HOMO) on the diene and the lowest-unoccupied molecular orbital (LUMO) on the dienophile. Display the HOMO for 2-methoxybutadiene. Where is it localized Display the LUMO for acrylonitrile. Where is it localized Orient the two fragments such that the HOMO and LUMO best overlap (A clearer picture is provided by examining-the HOMO map for 2-methoxybutadiene and the LUMO map for acrylonitrile.) Which product should result ... 

With the radical 29, even though loss of an equatorial hydrogen should be sterically less hindered and is favored thermodynamically (by relief of 1,3 interactions of the axial methyl), there is an 8-fold preference for loss of the axial hydrogen (at 100 ( i. The selectivity observed in the disproportionation of this and other substituted cyclohexyl radicals led Beckwith18 to propose that disproportionation is subject to stereoelectronic control which results in preferential breaking of the C-H bond which has best overlap with the orbital bearing the unpaired spin. 

In order to discuss the density of states of metal surface atoms, we need to take a closer look at those orbitals that have a distinct orientation, the d-orbitals. Let us take an fee crystal and see how the d-orbitals combine to form bands. Figure A.7 schematically shows the shape of the d-orbitals. For clarity only the lobes in the (yz) plane are shown, while the toroidal component of the dz2 orbital has been left out. If these orbitals are placed within the fee structure as in Fig. A.7, one readily sees that the best overlap between nearest neighbors in the (yz) or (100) plane occurs between the dyz orbitals, whereas the orbitals along the y and z axes of the cube overlap to a much smaller extent. This is true in the other planes of an fee metal as well, the dtv, dxz, and d orbitals overlap more than the d,2 and the dx2.v2... 

The transition state for conversion of 2 to 3 is particularly reasonable because it combines some of the geometry of both the reactants and the products and therefore gives the best overlap of the reacting orbitals necessary for the formation of the 7r bond. This is shown more explicitly below.9... 

The protonation leads specifically to the trans-decalin system, though reduction could apparently give rise to two stereoisomeric products. The guiding principle appears to be that protonation of the intermediate allylic anion 12 takes place axially, orthogonal to the plane of the double bond, and to the most stable conformation of the carbanion which allows the best sp3-orbital overlap on the /Tcarbon with the -orbital system of the double bond. 

Answer The preferred trajectory will have the best frontier orbital overlap. 

Having similar energies is not the only criterion for good interaction between two atomic orbitals. It also matters how the orbitals overlap. We have seen that p orbitals overlap better in an end-on fashion (forming a C bond) than they do side-on (forming a n bond). Another factor is the size of the atomic orbitals. For best overlap, the orbitals should be the same size—a 2p orbital overlaps much better with another 2p orbital than it does with a 3p or 4p orbital. 

When we consider the carbonyl group as an electrophile, we must look at antibonding orbitals too. The only one that concerns us is the relatively low-energy k orbital of the C-0 double bond (the LUMO). This orbital is biased towards the carbon to compensate for the opposite bias in the filled Jt orbital. How do we know this if there are no electrons in il Simply because nucleophiles, whether charged or not, attack carbonyl groups at the carbon atom. They get the best overlap with the larger orbital component of the ic orbital. 

In an E2 elimination, the new 7t bond is formed by overlap of the C-H a bond with the C-X a antibonding orbital. The two orbitals have to lie in the same plane for best overlap, and now there are two conformations that allow this. One has H and X syn-periplanar, the other anti-periplanar. The anti-periplanar conformation is more stable because it is staggered (the syn-periplanar conformation is eclipsed) but, more importantly, only in the anti-periplanar conformation are the bonds (and therefore the orbitals) truly parallel. 

Answer. The HOMO of I- is a filled 5p AO whereas the LUMO of I2 is the highest lying a-antibonding MO shown in Figure 1.3. Best overlap of the donor and acceptor orbitals will be achieved in a linear structure. As [13] is ahomonuclear compound this HOMO-LUMO analysis cannot be pushed too far and we will defer presentation of the MOs of a species like [L] until we develop a model for three-center bonds later in the chapter. 

Each of the electrons involved in delocalization must have some overlap with the other electrons. This means that if the orbitals are oriented at a 90° angle, there will be no overlap. The best overlap will occur when the orbitals are oriented at a 0° angle. 

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