Energy molecular orbital

It would appear that identical particle pemuitation groups are not of help in providing distinguishing syimnetry labels on molecular energy levels as are the other groups we have considered. However, they do provide very usefiil restrictions on the way we can build up the complete molecular wavefiinction from basis fiinctions. Molecular wavefiinctions are usually built up from basis fiinctions that are products of electronic and nuclear parts. Each of these parts is fiirther built up from products of separate uncoupled coordinate (or orbital) and spin basis fiinctions. Wlien we combine these separate fiinctions, the final overall product states must confonn to the pemuitation syimnetry mles that we stated above. This leads to restrictions in the way that we can combine the uncoupled basis fiinctions. 

You can order the molecular orbitals that arc a solution to etjtia-tion (47) accordin g to th eir en ergy, Klectron s popii late the orbitals, with the lowest energy orbitals first. normal, closed-shell, Restricted Hartree hock (RHK) description has a nia.xirnuin of Lw o electrons in each molecular orbital, one with electron spin up and one w ith electron spin down, as sliowm ... 

In our treatment of molecular systems we first show how to determine the energy for a given iva efunction, and then demonstrate how to calculate the wavefunction for a specific nuclear geometry. In the most popular kind of quantum mechanical calculations performed on molecules each molecular spin orbital is expressed as a linear combination of atomic orhilals (the LCAO approach ). Thus each molecular orbital can be written as a summation of the following form ... 

These absorptions are ascribed to n-n transitions, that is, transitions of an electron from the highest occupied n molecular orbital (HOMO) to the lowest unoccupied n molecular orbital (LUMO). One can decide which orbitals are the HOMO and LUMO by filling electrons into the molecular energy level diagram from the bottom up, two electrons to each molecular orbital. The number of electrons is the number of sp carbon atoms contributing to the n system of a neuhal polyalkene, two for each double bond. In ethylene, there is only one occupied MO and one unoccupied MO. The occupied orbital in ethylene is p below the energy level represented by ot, and the unoccupied orbital is p above it. The separation between the only possibilities for the HOMO and LUMO is 2.00p. 

The UHE wave function can also apply to singlet molecules. Usually, the results are the same as for the faster RHEmethod. That is, electrons prefer to pair, with an alpha electron sharing a molecular space orbital with a beta electron. Use the UHE method for singlet states only to avoid potential energy discontinuities when a covalent bond is broken and electrons can unpair (see Bond Breaking on page 46). 

Another aspect of qualitative application of MO theory is the analysis of interactions of the orbitals in reacting molecules. As molecules approach one another and reaction proceeds, there is a mutual perturbation of the orbitals. This process continues until the reaction is complete and the new product (or intermediate in a multistep reaction) is formed. PMO theory incorporates the concept of frontier orbital control. This concept proposes that the most important interactions will be between a particular pair of orbitals. These orbitals are the highest filled oihital of one reactant (the HOMO, highest occupied molecular oihital) and the lowest unfilled (LUMO, lowest unoccupied molecular oihital) orbital of the other reactant. The basis for concentrating attention on these two orbitals is that they will be the closest in energy of the interacting orbitals. A basic postulate of PMO... 

FIGURE 11.4 The u molecular orbitals of benzene arranged in order of increasing energy and showing nodal surfaces. The six u electrons of benzene occupy the three lowest energy orbitals, all of which are bonding. 

Molecular energies and structures Energies and structures of transition states Bond and reaction energies Molecular orbitals Multipole moments... 

Figure 19.17 Schematic molecular energy level diagram for CO. The Is orbitals have been omitted as they contribute nothing to the bonding. A more sophisticated treatment would allow some mixing of the 2s and 2p, orbitals in the bonding direction (z) as implied by the orbital diagram in Fig. 19.18.

There is a nice point as to what we mean by the experimental energy. All the calculations so far have been based on non-relativistic quantum mechanics. A measure of the importance of relativistic effects for a given atom is afforded by its spin-orbit coupling parameter. This parameter can be easily determined from spectroscopic studies, and it is certainly not zero for first-row atoms. We should strictly compare the HF limit to an experimental energy that refers to a non-relativistic molecule. This is a moot point we can neither calculate molecular energies at the HF limit, nor can we easily make measurements that allow for these relativistic effects. 

Having just seen a resonance description of benzene, let s now look at the alternative molecular orbital description. We can construct -tt molecular orbitals for benzene just as we did for 1,3-butadiene in Section 14.1. If six p atomic orbitals combine in a cyclic manner, six benzene molecular orbitals result, as shown in Figure 15.3. The three low-energy molecular orbitals, denoted bonding combinations, and the three high-energy orbitals are antibonding. 

Figure 15.10 Energy levels of the six benzene 77 molecular orbitals. There is a single, lowest-energy orbital, above which the orbitals come in degenerate pairs.

The total molecular energy is invariant to all transformations involving basis orbitals, just as any physical event is invariant under any transformation of coordinates. But just as the proper choice of coordinates helps in visualizing physical events, so the choice of the proper orbital basis is helpful in visualizing molecular properties. 

Figure 7.5 Bonding and antibonding molecular orbitals for the H2 molecule. Antibonding orbitais are higher energy orbitals than bonding orbitals.

Molecular orbital The orbitals for electrons in a molecule. Molecular orbitals are calculated by combining the wave functions of the highest-energy orbitals of the atoms in the molecule. 

Molecular Energetics. Molecular energies can be computed in a variety of ways including empirical fixed valence potentials, full force field potentials, and semi-empirical molecular orbital techniques (CNDO-2, INDO, MINDO-3, MNDO, PCILO). 

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