Overlap populations, molecular orbitals

In each of the B-H-B bridges, only two electrons bond the three atoms together by having the orbitals on the boron atoms simultaneously overlap the hydrogen Is orbital. A bond of this type is known as a two-electron three-center bond. In terms of molecular orbitals, the bonding can be described as the combination of two boron orbitals and one hydrogen orbital to produce three molecular orbitals, of which only the one of lowest energy is populated ... 

The calculation of the indices requires the overlap matrix S of atomic orbitals and the first-order density (or population) matrix P (in open-shell systems in addition the spin density matrix Ps). The summations refer to all atomic orbitals /jl centered on atom A, etc. These matrices are all computed during the Hartree-Fock iteration that determines the molecular orbitals. As a result, the three indices can be obtained... 

The different shift mechanisms may be understood in more detail by considering the effect of the magnetic field on the populations and energies of the different crystal orbitals (Figure 7a). Transfer of electron density via the 90° interaction arises due to a direct delocalization of spin density due to overlap between the half-filled tzg. oxygen jt, and empty Li 2s atomic orbitals (the delocalization mechanism. Figure 7b).This overlap is responsible for the formation of the tzg (antibonding) molecular orbital in a molecule or the tzg crystal orbital (or band) in a solid. No shift occurs for the 180° interaction from this mechanism as the eg orbitals are empty. 

The orbital interaction which is responsible for the adsorption of molecular oxygen on the Ag(110) surface is mainly between silver dxz and 7t 0-0- The interaction between the silver surface and molecular oxygen leads to a donation of 1.55 electrons into the 7t o o from surface because the Fermi level of the surface is located higher in energy than n oo In Figure 1 is shown the contribution for the adsorbed molecular oxygen in 2 to the total density of states (DOS) and also the Ag-O and 0-0 crystal orbital overlap population (COOP) curves. 

We shall explore how the orbital symmetries and energies of these small molecular fragments contribute to the observed properties of these compounds by using semiempirical electronic structure calculations. Our tools for analysis include an arsenal of computational indices, e.g. overlap populations and Mulliken populations.4 The remainder of this introduction... 

Mulliken population analysis is in the general spirit of the scheme used in the simple Hiickel method, but allows for several basis functions on an atom and does not require the overlap matrix to be a unit matrix. In ab initio theory each molecular orbital has a wavefunction ijj (Section 5.2.3.6.1) ... 

Molecular orbital calculations have also provided theoretical justification for these stereoelectronic effects in tetracovalent and pentacovalent phosphorus species (2-7). As has been shown in molecular orbital calculations on the X -P-X2 (X = 0,N) structural fragments, the X.-P bond is strengthened (as indicated by an increase in the Mulliken overlap population) while the P-X3 bond is weakened when the X atom lone pair is app to the P-X3 bond. Thus, in the g,t conformation of dimethyl phosphate (Structure ll the overlap population for the trans P-0 bond is. 017 electron lower than the overlap population for the gauche P-0 bond. As shown for g,t dimethyl phosphate one lone pair (shaded in 1) on the gauche bond oxygen is app to the trans bond, while no lone pairs on the trans bond oxygen are app to the gauche bond. Thus, the weakest X.-P bond has one app lone pair and no lone pairs on X. app to the P=X2 bond. 1... 

Molecular orbitals are characterized by energies and amplitudes expressing the distribution of electron density over the nuclear framework (1-3). In the linear combination of atomic orbital (LCAO) approximation, the latter are expressed in terms of AO coefficients which in turn can be processed using the Mulliken approach into atomic and overlap populations. These in turn are related to relative charge distribution and atom-atom bonding interactions. Although in principle all occupied MOs are required to describe an observable molecular property, in fact certain aspects of structure and reactivity correlate rather well with the nature of selected filled and unfilled MOs. In particular, the properties of the highest occupied MO (HOMO) and lowest unoccupied MO (LUMO) permit the rationalization of trends in structural and reaction properties (28). A qualitative predictor of stability or, alternatively, a predictor of electron... 

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