Two-center

Boranes are typical species with electron-deficient bonds, where a chemical bond has more centers than electrons. The smallest molecule showing this property is diborane. Each of the two B-H-B bonds (shown in Figure 2-60a) contains only two electrons, while the molecular orbital extends over three atoms. A correct representation has to represent the delocalization of the two electrons over three atom centers as shown in Figure 2-60b. Figure 2-60c shows another type of electron-deficient bond. In boron cage compounds, boron-boron bonds share their electron pair with the unoccupied atom orbital of a third boron atom [86]. These types of bonds cannot be accommodated in a single VB model of two-electron/ two-centered bonds. 

However, the CNDO method showed systematic weaknesses that were directly attributable to the approximations outlined above, so that it was superseded by the intermediate m lect of diatomic differential overlap (INDO) method, introduced by Pople, Beveridge, and Dobosh in 1967 [13]. The approximation outlined in Eq. (50) proved to be too severe and was replaced by individual values for the possible different types of interaction between two AOs. These individual values, often designated Cgg, Ggp, Gpp and in the literature, can be adjusted to give better agreement with experiment than was possible for CNDO. However, in INDO the two-center terms remain of the same type as those given in Eqs. (51) and (52) (again, there are many variations). This approximation leads to systematic weaknesses, for instance in treating interactions between lone pairs. 

The two-center two-eleciroii. one-center two-electron, iwo-center one-electron, one-center one-electron, and core-core repulsion integrals involved in the above equations are discussed below. 

The ZDO approximation is made only for two-center integrals one-center coulomb Za,b = and exchange... 

Trivalent ( classical carbenium ions contain an sp -hybridized electron-deficient carbon atom, which tends to be planar in the absence of constraining skeletal rigidity or steric interference. The carbenium carbon contains six valence electrons thus it is highly electron deficient. The structure of trivalent carbocations can always be adequately described by using only two-electron two-center bonds (Lewis valence bond structures). CH3 is the parent for trivalent ions. 

Penta- (or higher) coordinate ( nonclassical carbonium ions contain five or (higher) coordinate carbon atoms. They cannot be described by two-electron two-center single bonds alone but also neces-... 

A molecular orbital is a linear combination of basis functions. Normalization requires that the integral of a molecular orbital squared is equal to 1. The square of a molecular orbital gives many terms, some of which are the square of a basis function and others are products of basis functions, which yield the overlap when integrated. Thus, the orbital integral is actually a sum of integrals over one or two center basis functions. 

The first three types of integrals involve one or two centers. The fourth type of integral involves up to four centers. 

One center overlaps are all zero so that the above describes only two center contributions. 

All non-zero integrals over atomic orbitals on the two centers are set equal, as in CNDO/INDO, to an averaged y. Thus, (s sj s s ) = (s s I PbPb) = (PaPa I PbPb) = The two-center Coulomb integrals, rather than being calculated from first principles using s orbitals as in CNDO/INDO, are approximated by an Ohno-Klop-man [K.Ohno, Theor. Chim. Acta, 2, 219 (1964) G. Klopman,... 

The two-center one-electron integral Hj y, sometimes called the resonance integral, is approximated in MINDO/3 by using the overlap integral, Sj y, in a related but slightly different manner to... 

The sum A is over all the atoms in the quantum region and B is over all the atoms in the classical region. The two-electron and two-center Coulomb integral, Y b, is computed in MINDO/3 by... 

The MNDO method has 22 unique two-center two-electron integrals for each pair of heavy (non-hydrogen) atoms in their local atomic frame. They are ... 

The two-center two-electron repulsion integrals ( AV Arr) represents the energy of interaction between the charge distributions at atom Aand at atom B. Classically, they are equal to the sum over all interactions between the multipole moments of the two charge contributions, where the subscripts I and m specify the order and orientation of the multipole. MNDO uses the classical model in calculating these two-center two-electron interactions. 

Because the repulsion interaction energy of two point charges is inversely proportional to the distance separating the two charges, Dewar and co-workers, for example, represent the (ssiss) two-center two-electron integral by ... 

There are some boundary conditions which can be used to fix parameters and Ag. For example, when the distance between nucleus A and nucleus B approaches zero, i.e., R g = 0.0, the value of the two-electron two-center integral should approach that of the corresponding monocentric integral. The MNDO nomenclature for these monocentric integrals is. 

Using the above asymptotic forms of the two-center two-electron integrals, the parameters and Ag can be derived. Certainly, parameter A is different for different orbitals even though they reside on the same atom. Dewar used AM to represent the parameter A obtain ed via G s, AD to represen t th e param eter A obtain ed via Hj,p, and AQ to represent the parameter A obtained from Hpp. 

Both PSI and PSII are necessary for photosynthesis, but the systems do not operate in the implied temporal sequence. There is also considerable pooling of electrons in intermediates between the two photosystems, and the indicated photoacts seldom occur in unison. The terms PSI and PSII have come to represent two distinct, but interacting reaction centers in photosynthetic membranes (36,37) the two centers are considered in combination with the proteins and electron-transfer processes specific to the separate centers. 

The valence theory (4) includes both types of three-center bonds shown as well as normal two-center, B—B and B—H, bonds. For example, one resonance stmcture of pentaborane(9) is given in projection in Figure 6. An octet of electrons about each boron atom is attained only if three-center bonds are used in addition to two-center bonds. In many cases involving boron hydrides the valence stmcture can be deduced. First, the total number of orbitals and valence electrons available for bonding are determined. Next, the B—H and B—H—B bonds are accounted for. Finally, the remaining orbitals and valence electrons are used in framework bonding. Alternative placements of hydrogen atoms require different valence stmctures. 

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