Singly-occupied orbitals

Semiempirical programs often use the half-electron approximation for radical calculations. The half-electron method is a mathematical technique for treating a singly occupied orbital in an RHF calculation. This results in consistent total energy at the expense of having an approximate wave function and orbital energies. Since a single-determinant calculation is used, there is no spin contamination. 

The first line indicates that the syrrumetry could not be determined for this state (the symmetry itself is given as Sym). We will need to determine it ourselves. Molecular symmetry in excited states is related to how the orbitals transform with respect to the ground state. From group theory, we know that the overall symmetry is a function of symmetry products for the orbitals, and that only singly-occupied orbitals are... 

For diatomics with ten valence electrons, pole strengths lie between 0.86 and 0.89. DOs are dominated by a single occupied orbital in all cases. In the normalized DO for the state of AlO, there are other contributions with coefficients near 0.02. For the states of BO and AlO, certain operators have U elements that are approximately 0.1. Recent experimental work has produced a revised figure, 2.508 0.008 eV, for the electron affinity of BO [42] and the entry in Table III is in excellent agreement. Similar agreement occurs for the electron affinities of CN, AlO and AIS. 

Radicals and excited states have an orbital occupied by one electron. The interaction of the singly occupied orbital with a vacant orbital (Scheme 15) and with a singly occupied orbital (Scheme 16) leads to the stabilization. The stabilized orbitals occupy one and two electrons, respectively. There are no electrons in the destabilized orbital. For the interaction with a doubly occupied orbital there are two electrons in the stabilized orbital and one electron in the destabilized orbital (Scheme 17). Although the destabilization of the out-of-phase combined orbital is greater than the stabilization of the in-phase combination, there is one more electron in the stabilized orbital. Net stabilization is then expected. 

Scheme 15 The stabilization by the interaction between a singly occupied orbital and a vacant orbital...

The well-established prototypes of 1,3-diradicals, trimethylenemethane (TMM, 1) and trimethylene (TM, 2) are selected to illustrate our idea. The diradical is assumed to have two singly occupied orbitals, p and q, of nearly the same energy on two radical centers (P and Q in Fig. 4). As addressed in Sect. 2, it is possible for... 

The F matrix elements in eqs. (15) and (16) are formally the same as for closed-shell systems, the only difference being the definition of the density matrix in eq. (17), where the singly occupied orbital (m) has also to be taken into account. The total electronic energy (not including core-core repulsions) is given by... 

These methods can give us useful information on radicals in a manner similar to that for closed-shell systems, provided the exploitation is correct. Of course, in expressions for total energy, bond orders, etc., a singly occupied orbital must be taken into account. One should be aware of areas where the simple methods give qualitatively incorrect pictures. The HMO method, for example, cannot estimate negative spin densities or disproportionation equilibria. On the other hand, esr spectra of thousands of radicals and radical ions have been interpreted successfully with HMO. On the basis of HMO orbital energies and MO symmetry... 

Finally, the expression for the electron affinity for the uptake of one electron into the singly occupied orbital (m) should be mentioned (106) ... 

There are 39 doubly and one singly occupied orbitals in YbF of which the 25th occupied orbital of YbF corresponds to the 5s occupied spin orbital of Yb. As the contribution of the 5s and 5p orbitals of Yb to Wi is quite significant [43,48,49], these orbitals are included in the Cl space. The occupied orbitals above the 25th are also included in the active space based on energy considerations. (Note that... 

There are 32 doubly and one singly occupied orbitals in BaF of which the 25th occupied orbital of BaF corresponds to the 5s occupied spin orbital of Ba. 

For example, Di has coupled singly occupied orbitals on Ci and C2 (leading to a Ci—C2 7t bond) and a doubly occupied orbital on C3, corresponding to the lone pair of the first resonance structure. 

Table II shows the same results for the nitrosyl ferrous heme complex at Fe-NO 1.743 and 2.143 A. The singly occupied orbital (79) has primarily NO n and a character with both d and d 2 contributions. Although this orbital is deeply buried, as can be seen from its energy, it is shown between the doubly occupied and virtual orbitals for convenience. For comparison, the ground-state molecular orbitals of carbonyl-(44) and oxyheme (45) complexes are shown in Table III.

A natural starting point for modem valence bond applications lies in the optimization of the spin-coupled wavefunction [28], which consists of a single covalent configuration of N singly occupied orbitals ... 

Afterward, we have to find approximations for 2-RDM spin components D". Let us now focus on high-spin cases only, such as doublet, triplet, quartet... spins for one, two, three. .. unpaired electrons outside the closed shells. Accordingly, singly occupied orbitals will always have the same spin (Al p = 0 or = 0) so the trace of the one-matrix, Eq. (6), becomes... 

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