Closed-shell molecular orbitals

The following presentation is limited to closed-shell molecular orbital wave-functions. The first section discusses the unique ability of molecular orbital theory to make chemical comparisons. The second section contains a discussion of the underlying basic concepts. The next two sections describe characteristics of canonical and localized orbitals. The fifth section examines illustrative examples from the field of diatomic molecules, and the last section demonstrates how the approach can be valuable even for the delocalized electrons in aromatic ir-systems. All localized orbitals considered here are based on the self-energy criterion, since only for these do the authors possess detailed information of the type illustrated. We plan to give elsewhere a survey of work involving other types of localization criteria. 

While the localized orbital concept was first formulated for closed shell molecular orbital wavefunctions, the method is by no means so limited. For a restricted open shell Hartree-Fock wavefunction, the subset of doubly occupied orbitals may be transformed in a manner analogous to that described above. Similarly, the sets of a and orbitals in an unrestricted Hartree-Fock wavefunction are separately invariant, and the set of active orbitals in a FORS (fully... 

A single-determinant wave-function of closed shell molecular systems is invariant against any unitary transformation of the molecular orbitals apart from a phase factor. The transformation can be chosen in order to obtain LMOs. Starting from CMOs a number of localization procedures have been proposed to get LMOs the most commonly used methods are as given by the authors of (Edmiston et ah, 1963) and (Boys, 1966), while the procedures provided by (Pipek etal, 1989) and (Saebo etal., 1993) are also of interest. It could be stated that all the methods yield comparable results. Each LMO densities are found to be relatively concentrated in some spatial region. They are, furthermore, expected to be determined mainly by that part of the molecule which occupies that given region and its nearby environment rather than by the whole system. 

Unlike HE molecular orbitals, the natural orbitals are not restricted to a low-level approximation, hut are rigorously defined for any theoretical level, up to and including the exact P. As eigenfunctions of a physical (Hermitian) operator, the NOs automatically form a complete orthonormal set, able to describe every nuance of the exact P and associated density distribution, whereas the occupied MOs are seriously mcomplete without augmentation by virtual MOs. Furthermore, the occupancies , of NOs are not restricted to integer values (as are those of MOs), but can vary continuously within the limits imposed by tbe Pauli exclusion principle, namely, for closed-shell spatial orbitals. 

To summarize, we have found that the IPs and EAs of a closed-shell molecular system may be identified with the negative orbital energies of the canonical orbitals ... 

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 ... 

A closed-shell means that every occupied molecular orbital contains exactly two electrons. 

Huckel realized that his molecular orbital analysis of conjugated systems could be extended beyond neutral hydrocarbons He pointed out that cycloheptatrienyl cation also called tropyhum ion contained a completely conjugated closed shell six tt electron sys tern analogous to that of benzene... 

If the number of electrons, N, is even, you can have a closed shell (as shown) where the occupied orbitals each contain two electrons. For an odd number of electrons, at least one orbital must be singly occupied. In the example, three orbitals are occupied by electrons and two orbitals are unoccupied. The highest occupied molecular orbital (HOMO) is /3, and the lowest unoccupied molecular orbital (LUMO) is 11/4. The example above is a singlet, a state of total spin S=0. Exciting one electron from the HOMO to the LUMO orbital would give one of the following excited states ... 

The Roothaan equations are the basic equations for closed-shell RHF molecular orbitals, and the Pople-Nesbet equations are the basic equations for open-shell UHF molecular orbitals. The Pople-Nesbet equations are essentially just the generalization of the Roothaan equations to the case where the spatials /j and /jP, as shown previously, are not defined to be identical but are solved independently. 

Here we give the molecule specification in Cartesian coordinates. The route section specifies a single point energy calculation at the Hartree-Fock level, using the 6-31G(d) basis set. We ve specified a restricted Hartree-Fock calculation (via the R prepended to the HF procedure keyword) because this is a closed shell system. We ve also requested that information about the molecular orbitals be included in the output with Pop=Reg. 

The simplest antisymmetric function that is a combination of molecular orbitals is a determinant. Before forming it, however, we need to account for a factor we ve neglected so far electron spin. Electrons can have spin up i+Vi) or down (-V2). Equation 20 assumes that each molecular orbital holds only one electron. However, most calculations are closed shell calculations, using doubly occupied orbitals, holding two electrons of opposite spin. For the moment, we will limit our discussion to this case. 

We can now build a closed shell wavefunction by defining n/2 molecular orbitals for a system with n electrons, and then assigning electrons to these orbitals in pairs of opposite spin ... 

You probably noted that the original papers were couched in terms of HF-LCAO theory. From Chapter 6, the defining equation for a Hamiltonian matrix element (in the usual doubly occupied molecular orbital, closed-shell case) is... 

Electron propagator theory generates a one-electron picture of electronic structure that includes electron correlation. One-electron energies may be obtained reliably for closed-shell molecules with the P3 method and more complex correlation effects can be treated with renormalized reference states and orbitals. To each electron binding energy, there corresponds a Dyson orbital that is a correlated generalization of a canonical molecular orbital. Electron propagator theory enables interpretation of precise ab initio calculations in terms of one-electron concepts. 

Figure 3. Molecular-orbital diagrams as obtained by the ROHF method. Dashed lines indicate MOs dominated by the metal d-orbitals, the solid lines stand for doubly occupied or virtual ligand orbitals. Orbitals which are close in energy are presented as degenerate the average deviation from degeneracy is approximately 0.01 a.u. In the case of a septet state (S=3), the singly occupied open-shell orbitals come from a separate Fock operator and their orbital energies do not relate to ionization potentials as do the doubly occupied MOs (i.e. Koopmann s approximation). For these reasons, the open-shell orbitals appear well below the doubly occupied metal orbitals. Doubly occupying these gives rise to excited states, see text.

A radical has a singly occupied molecular orbital (SOMO). This is the frontier orbital. The SOMO interacts with HOMO and the LUMO of closed-shell molecules to stabilize the transition state (Scheme 27). The radical can be a donor toward a monomer with low LUMO or an acceptor toward one with high HOMO. 

The Hartree-Fock equations for the /-th element of a set containing occ occupied molecular orbitals i in a closed shell system with n = 2occ electrons are [8]... 

On the other hand, the XPS data near the Fermi level provide us the valuable information about the band structures of nanoparticles. XPS spectra near the Fermi level of the PVP-protected Pd nanoparticles, Pd-core/ Ni-shell (Ni/Pd = 15/561, 38/561) bimetallic nanoparticles, and bulk Ni powder were investigated by Teranishi et al. [126]. The XPS spectra of the nanoparticles become close to the spectral profile of bulk Ni, as the amount of the deposited Ni increases. The change of the XPS spectrum near the Fermi level, i.e., the density of states, may be related to the variation of the band or molecular orbit structure. Therefore, the band structures of the Pd/Ni nanoparticles at Ni/Pd >38/561 are close to that of the bulk Ni, which greatly influence the magnetic property of the Pd/Ni nanoparticles. 

---