Atomic orbitals coefficients

AEis the interaction energy q and qg are charges on atoms A and B, separated by Rab, on different molecules, r and s are molecular orbitals on the two different molecules, p and v label the atomic orbitals that contribute to these molecular orbitals, with coefficients Cjjj. and Cyg. Hj y is the matrix element between atomic orbitals p and V, which is a measure of the energy of their interaction, roughly proportional to their overlap. The energies of the molecular orbitals are and e. ... 

Mathematically, the molecular orbitals are treated as linear combinations of atomic orbitals, so that the wave function, is expressed as a sum of individual atomic orbitals multiplied by appropriate weighting factors (atomic coefficients) ... 

In die HMO approximation, the n-electron wave function is expressed as a linear combination of the atomic orbitals (for the case in which the plane of the molecule coincides with the x-y plane). Minimizing the total rt-electron energy with respect to the coefficients leads to a series of equations from which the atomic coefficients can be extracted. Although the mathematical operations involved in solving the equation are not... 

The Huckel molecular orbitals for acenaphthylene are shown below. The atomic coefficients for the orbital which is the LUMO in the neutral compound and the... 

The first paper of the frontier-electron theory pointed out that the electrophilic aromatic substitution in aromatic hydrocarbons should take place at the position of the greatest density of electrons in the highest occupied (HO) molecular orbital (MO). The second paper disclosed that the nucleophilic replacement should occur at the carbon atom where the lowest unoccupied (LU) MO exhibited the maximum density of extension. These particular MO s were called "frontier MO s . In homolytic replacements, both HO and LU.were shown to serve as the frontier MO s. In these papers the "partial" density of 2 pn electron, in the HO (or LU) MO, at a certain carbon atom was simply interpreted by the square of the atomic orbital (AO) coefficient in these particular MO s which were represented by a linear combination (LC) of 2 pn AO s in the frame of the Huckel approximation. These partial densities were named frontier-electron densities . 

In this equation the Cg represent the respective atomic coefficients for the orbitals of the nucleophile (HOMO) and those of the different atoms in the electron-acceptor (LUMO and LUMO + 1) ( homo E lumo) and (T homo umo + 1) represent the energy gaps between the HOMO and LUMO (LUMO + 1) levels of nucleophile and acceptor. 

This is a quantity which can be easily constructed given a set of molecular orbitals (the coefficients C ) and a precalculated set of atomic orbital integrals. At this point, the Hartree-Fock equations have been reduced to a matrix eigenvector problem, FC = SCe, but not in a computationally convenient form. Following the analysis leading to equation 84, we first define the transformed Fock matrix as... 

The spin-orbit splitting coefficient v f = - Asf, proportional to the spin-orbit coupling constant, is used in place of the free-atom spin-orbit splitting parameter I. 

Note that, upon examination of the AMI linear combination of atomic orbitals (LCAO) coefficients for the two aluminum on poly(/>-phenylenevinylene) systems, the highest occupied molecular orbitals are localized in character. The HOMO and HOMO-1 levels are almost totally localized to the aluminum atoms and to the carbon atoms within the moieties to which the aluminum atoms are attached. The conjugation within the frontier orbitals is thus totally lost... 

The Patemo-Buchi reaction on furan has been studied from a theoretical point of view [42]. The HOMO of furan has an energy of 0.32 eV, and the atomic coefficients of this orbital (Fig. 3.3) are in agreement with an attack of the carbonyl oxygen on the a carbon. The structures of the possible adducts deriving from the attack of benzaldehyde in the a position of furan are depicted in Figure 3.4. The exo isomer is more stable (0.5 kcal mol 1) than the other isomer, in agreement with the experimental results. 

Technically, the simultaneous optimization of orbitals and coefficients for a multistructure VB wave function can be done with the VBSCF method due to Balint-Kurti and van Lenthe [21,22], The VBSCF method has the same format as the classical VB method with an important difference. While the classical VB method uses orbitals that are optimized for the separate atoms, the VBSCF method uses a variational optimization of the atomic orbitals in the molecular wave function. In this manner the atomic orbitals adapt themselves to the molecular environment with a resulting significant improvement in the total energy and other computed properties. 

Calculations performed at the HF/3-21G level indicated smaller energy gaps between the HOMOs of the aforementioned electron-rich dienophiles and the LUMOs of the quinone ketals, as can be expected for inverse electron-demand Diels-Alder reactions under FMO control [141]. Regiochemical controls observed with quinone ketals such as 76a were well corroborated by the relative magnitudes of the atomic coefficients of the frontier orbitals. The highest coefficients at C-5 of the quinone ketal LUMO and at C-2 of the electron-rich alkenes would indeed promote bond formation between these centers. The results of calculations on other quinone ketals were, however, rather vague [141]. 

The basis set is chosen such that around the Zth Dyson pole, the electron propagator matrix is diagonal. Consequently, the eigenstate /) can be presented as a linear expansion over atomic orbitals with coefficients ck(l) that depend on the number of the Dyson pole, Z, and the number of the atomic orbital, k, so that... 

More effective stabilization is provided by genuine conjugation with 7t or lone-pair electrons. The ailyl cation has a filled (bonding) orbital containing two electrons delocalized over all three atoms and an important empty orbital with coefficients on the end atoms only. It s this orbital that is attacked by nucleophiles and so it s the end carbon atoms that are attacked by nucleophiles. The normal curly arrow picture tells us the same thing, molecular orbitals... 

There is no difficulty in defining an effective charge in terms of the LCAO electronic structure even when the softening is large. Ultimately the band slates can be written as linear combinations of atomic orbitals, with coefficients, as... 

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