Atom coefficients

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 most easily obtained information from such calculations is the relative orderings of the eneigy levels and the atomic coefficients. Solutions are readily available for a number of frequently encountered delocalized systems, which we will illustrate by referring to some typical examples. Consider, first, linear polyenes of formula C H 2 such as 1,3-butadiene, 1,3,5-hexatriene, and so forth. The energy levels for such compounds are given by the expression... 

While HiickeTs 4n + 2 rule applies only to monocyclic systems, HMO flieory is applicable to many other systems. HMO calculations of fused-ring systems are carried out in much the same way as for monocyclic species and provide energy levels and atomic coefficients for the systems. The incorporation of heteroatoms is also possible. Because of the underlying assumption of orthogonality of the a and n systems of electrons, HMO dieory is restricted to planar molecules. 

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... 

Moreover, it was shown that the presence of Hal Hal interactions between the partially oxidized molecules also contribute to the electronic delocalization. Indeed, the presence of non-zero atomic coefficients on the halogen atoms in the HOMO of EDT-TTF-Br2 or EDT-TTF-I2 [66], together with the short Hal Hal contacts, leads to a sizeable increase of the band dispersion and stabilizes a rare (V structure through the side-by-side arrangement of the inversion-centred dyads connected by Hal- Hal interactions. Both 13 salts are semiconductors with room temperature conductivities around... 

A chemical equation describes a chemical reaction in many ways as an empirical formula describes a chemical compound. The equation describes not only which substances react, but the relative number of moles of each undergoing reaction and the relative number of moles of each product formed. Note especially that it is the mole ratios in which the substances react, not how much is present, that the equation describes. In order to show the quantitative relationships, the equation must be balanced. That is, it must have the same number of atoms of each element used up and produced (except for special equations that describe nuclear reactions). The law of conservation of mass is thus obeyed, and also the "law of conservation of atoms. Coefficients are used before the formulas for elements and compounds to tell how many formula units of that substance are involved in the reaction. A coefficient does not imply any chemical bonding between units of the substance it is placed before. The number of atoms involved in each formula unit is multiplied by the coefficient to get the total number of atoms of each element involved. Later, when equations with individual ions are written (Chap. 9), the net charge on each side of the equation, as well as the numbers of atoms of each element, must be the same to have a balanced equation. The absence of a coefficient in a balanced equation implies a coefficient of 1. 

LUMO Atom coefficients Atom LUMO coefficients... 

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. 

Coefficients multiply a normalized radial functions (not shown), complex spherical harmonics Yj jjj, and spin functions as indicated. Values for the ligand are for a single atom. Coefficients smaller than 0.01 are not shown. 

Remark 1. If all the atomic coefficients /3 are positive we speak (where necessary) of a proper molecular species. If all the /3( are zero we have the null or trivial species, and if any are negative a pseudo-species. ... 

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. 

The same theoretical approach was used to explain the photochemical behavior of 2,3-dihydrofuran. The HOMO of 2,3-dihydrofuran showed at — 0.32 eV is reported in Figure 3.11. The atomic coefficients of the HOMO are in agreement with an attack of the LUMO (the carbonyl oxygen) on the (5-carbon. On the basis of this result, the thermodynamic stability of the possible adducts was tested. The structures of these compounds are reported in Figure 3.12. The endo isomer is more stable for 2 kcal moT 1 than the other isomer, in agreement with experimental results. 

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]. 

Whenever the molecule does not have sufficient symmetry, the determination of the atomic coefficients in the expressions of the various m.o.s... 

This term depends on the average value of r for a 2p electron r is the distance to the nucleus) and on the features (atomic coefficients c and energies j) of the molecular orbitals in which the 2p orbitals of the C atom in question intervene. The average value of r is a function of the electron population of the C orbitals. An excess of electrons leads to an expansion of the 2p orbitals, hence to a smaller absolute value of (ctcc) - As a consequence, the shielding constant gq increases. The expression equivalent to Eq. (12.12) for the para position of benzene derivatives is... 

All non-zero coefficients of fractional parentage in Eq.28 are atomic coefficients... 

---