Molecular orbitals normalization condition

The first approximation we ll consider comes from the interpretation of as a probability density for the electrons within the system. Molecular orbital theory decomposes t(/ into a combination of molecular orbitals <()j, (jij,. To fulfill some of the conditions on we discussed previously, we choose a normalized, orthogonal set of molecular orbitals ... 

Let us consider lithium as an example. In the usual treatment of this metal a set of molecular orbitals is formulated, each of which is a Bloch function built from the 2s orbitals of the atoms, or, in the more refined cell treatment, from 2s orbitals that are slightly perturbed to satisfy the boundary conditions for the cells. These molecular orbitals correspond to electron energies that constitute a Brillouin zone, and the normal state of the metal is that in which half of the orbitals, the more stable ones, are occupied by two electrons apiece, with opposed spins. 

Of these three diatomic moiecuies, only N2 exists under normal conditions. Boron and carbon form soiid networks rather than isolated diatomic molecules. However, molecular orbital theory predicts that B2 and C2 are stable molecules under the right conditions, and in fact both molecules can be generated in the gas phase by vaporizing solid boron or soiid carbon in the form of graphite. 

Wave functions for the orbitals of molecules are calculated by linear combinations of all wave functions of all atoms involved. The total number of orbitals remains unaltered, i.e. the total number of contributing atomic orbitals must be equal to the number of molecular orbitals. Furthermore, certain conditions have to be obeyed in the calculation these include linear independence of the molecular orbital functions and normalization. In the following we will designate wave functions of atoms by % and wave functions of molecules by y/. We obtain the wave functions of an H2 molecule by linear combination of the Is functions X and of the two hydrogen atoms ... 

A remarkable number of organic compounds luminesce when subjected to consecutive oxidation-reduction (or reduction-oxidation) in aprotic solvents1-17 under conditions where anion radicals are oxidized or cation radicals are reduced. In many instances, the emission is identical with that of the normal solution fluorescence of the compound employed. In these instances the redox process has served to produce neutral molecules in an excited electronic state. These consecutive processes which result in emission are not special examples of oxidative chemiluminescence, but are more properly classified as electron transfer luminescence in solution since the sequence oxidation-reduction can be as effective as reduction-oxidation.8,10,12 A simple molecular orbital diagram, although it is a zeroth-order approximation of what might be involved under some conditions, provides a useful starting... 

Solution of the secular equation amounts to finding the roots of an iVth order equation in E. The N roots are the energies of the N molecular orbitals the forms of the orbitals in terms of the basis atomic orbitals 9are found by substituting each value of E, in turn, back into Equations A2.13 and solving for the c s using the additional condition that each MO tf)t is to be normalized,... 

The last term can be neglected owing to the combined effect of the small coefficient (cL)2 and the small integral jxf i a Using the normalization condition for molecular orbitals... 

E, and E2 are the only energies which an electron belonging to the diatomic molecule A2 can have. Each energy level E is associated with a molecular orbital 4 whose coefficients may be obtained by setting E = E in Equation (2.9) and solving these equations, taking into account the normalization condition ... 

In any nonbonding molecular orbital (NBMO), the sum of the coefficients of the atoms s adjacent to a given atom r is zero. Hence the NBMO coefficients can be calculated very easily. The benzyl radical provides a nice example. All of the non-starred atoms have coefficients of zero. We give the para atom an arbitrary coefficient a. The sum of the coefficients of the atoms adjacent to the meta carbons must be zero, so the ortho coefficients are —a. For the sum of the coefficients around the ipso carbon to cancel, the benzylic carbon coefficient must be 2a. The value for a is given by the normalization condition ... 

The terms added to W correspond to the condition of the overall normalization of the wavefunction, and the orthonormalization of the molecular orbitals ... 

In order to construct the -independent matrices and we first need to normalize the molecular orbitals. The normalization condition is given by... 

A homonuclear diatomic molecule is one in which both nuclei are the same, for example H2 and N2. In the first row of the Periodic Table, H2 is the only example. From the second row we have N2, 02 and F2, which are stable under normal conditions of temperature and pressure. We looked at N2 in the previous Section. Here we shall consider the molecular orbital description of 02, and use it as an example of how we can use the theory to explain and/or predict properties of molecules. 

Draw an orbital energy-level diagram for OF, showing only those orbitals made from the 2p atomic orbitals on both atoms. Sketch the occupied molecular orbital that is highest in energy. What is the bond order of OF Why is this molecule not observed under normal conditions (F 2p orbitals are lower in energy than O 2p orbitals.)... 

Solve the Fock matrix eigenvalue equations given above to obtain the orbital energies and an improved occupied molecular orbital. In so doing, note that the normalization condition <0i i> = 1 = ]SCi gives the needed normalization condition for the expansion coefTicients of the 0i in the atomic orbital basis. 

Here, the B, and A states account for the a bonding to the copper, the Bj state represents the in-plane n bonding, ana the states the out-of-plane n bondjng. The quantities a, a, and [i are the corresponding molecular orbital coefficients. In addition. ff = (l/ /3)(]/2 p +. ) are, sp -hybridized directed orbitals centered on the i-th ligand. The normalization condition for the ground state gives... 

Aromatic hydrocarbons are, in general, oxidized through loss of an electron from the highest occupied jt-molecular orbitals generating a Jt-delocalised radical cation. The electron-transfer process is generally irreversible in acetonitrile under normal conditions because the radical cation reacts rapidly with nucleophile species present as impurities, including residual water [7, 8]. A second irreversible oxidation wave is seen at more positive potentials due to the formation of a transitory dication. Dichloromethane and mixtures of dichloromethane with fluorosulphonic acid or trifiuoroacetic acid are better solvents in which to demonstrate the reversible one-electron oxidation of aromatic species [7, 8]. 

Also, as shown in this example and similar to the Diels-Alder reaction, the normal or traditional mode of reaction is for the lowest unoccupied molecular orbital (LUMO) of the electron-deficient enophile to react with the highest occupied molecular orbital (HOMO) of the electron-rich ene. However, there are some carbon and hydrogen containing systems that have been proposed to proceed via radical or stepwise pathways. For example, allenyl alkynes 7 and 8 were thermally cyclized using PhMe under relatively mild conditions to provide 10 and 11 in good yield. The authors suggested a radical mechanism for this transformation via the fulvene biradical intermediate 9. ... 

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