Open Shell Methods

So far, we have considered only the restricted Hartree-Fock method. For open shell systems, an unrestricted method, capable of treating unpaired electrons, is needed. For this case, the alpha and beta electrons are in different orbitals, resulting in two sets of molecular orbital expansion coefficients  

264 Exploring Chemistry with Electronic Structure Methods 

The two sets of coefficients result in two sets of Fock matrices (and their associated density matrices), and ultimately to a solution producing two sets of orbitals. These separate orbitals produce proper dissociation to separate atoms, correct delocalized orbitals for resonant systems, and other attributes characteristic of open shell systems. However, the eigenfunctions are not pure spin states, but contain some amount of spin contamination from higher states (for example, doublets are contaminated to some degree by functions corresponding to quartets and higher states). 

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If spin contamination is small, continue to use unrestricted methods, preferably with spin-annihilated wave functions and spin projected energies. Do not use spin projection with DFT methods. When the amount of spin contamination is more significant, use restricted open-shell methods. If all else fails, use highly correlated methods. 

From the point of view of the present state of computational possibilities, an extension of open-shell methods to a and a + rr electronic systems is rather tempting. This extension is easy for the method of Longuet-Higgins and Pople e.g., the CNDO/2 method is amenable to radicals having one nondegenerate open shell if in the original terms for F matrix elements (66),... 

CNDO/2 with DelBene-Jaffe parametrization [for geometry, see (91)] open-shell method of Longuet-Higgins and Pople eq. (90). 

The values of the ionization potentials and electron affinities calculated by the open-shell method are given in parentheses. 

The theory will not be based on any infinite series expansion so that where necessary closed form methods such as the Via coordinate or open shell method, etc., can be used. It should be recognized, however, that there is no particular advantage in generalizingi-i the HyUeraas method directly by putting in the ff/s for all the electron correlations. The many t3q)es of correlation effects existing even within the same S5retem necessitate different means of evaluation for each once the separation is achieved. 

The optical spectra of radicals provide important data for testing open-shell methods. Until about 1960 open-shell MO studies were rather rare. An intensive development of open-shell MO theory started in the early sixties and was followed by chemical apphcations and systematic studies in the late sixties. At present it is possible to state that the physicochemical properties of radicals are predicted with an accuracy comparable to that attained for closed-shell molecules. This is important not only from the viewpoint of the electronic spectra, which can hardly be interpreted without MO theory, but also from the viewpoint of the general theory of reactivity, since radicals and excited states offer the means to overcome spin and S5unmetry restrictions in certain chemical reactions. 

With radicals there is no convenient method like the Hartree-Fock-Roothaan procedure commonly used for closed-shell systems. In contrast, the open-shell theory is typical of a number of methods suggested which differ in accuracy from the viewpoint of true SCF theory, in range of applicability, complexity, and computing feasibility. A critical survey of open-shell SCF methods reported by Berthier D covers the literature up to 1962. We shall not duplicate that review here we propose rather to note some features of open-shell methods relevant to their computation feasibility and to mention procedures published after 1962. The unrestricted treatments that assume different space orbitals for different spins will be disregarded here because the restricted wave functions... 

When discussing open-shell methods, it is convenient to use the formalism of the well-known Roothaan procedure 3). Hence, consider an open-shell configuration for which the total energy can be expressed as... 

Fig. 8,5. Some restrictions imposed on the GHF method in computational practice the RHF, the UHF, and the ROHF (Restricted Open Shell) methods, (a) RHF we frace the same (real) orbitals for the elearon pair (opposite s[ s) producing a pair of spinorbitals by using the same orbital, (b) UHF we relieve this restriction for oibitals (still being real), (c) ROHF we keep the double occupancy for inner shells as for the RHF method, while for the valence shell, we use the UHF-type splitting of orbitals.

There was a rich choice of available open-shell methods in the early 1960s [1] but their applications were rare and the results met with a differing degree of success. Obviously, a systematic examination was lacking. 

We decided to undertake this with the aim of formulating a generally applicable computational scheme for radicals which would be a natural extension of the PPP and semiempirical all-valence-electron methods for closed-shell molecules. We used for this purpose the self-consistent-field open-shell methods of Longuet-Higgins and Pople [2] and of Roothaan [3], we derived all expressions necessary for CI-S calculations [4, 5], and we tested the semiempirical open-shell PPP-type and INDO/S calculations systematically for various classes of radicals. 

Semiempirical calculations have been carried out by an unparameterized SCF-MO method with integral approximations [5], various versions of the CNDO [37 to 42] and INDO [6, 38, 43 to 45] methods, the MNDO [46, 47] and MINDO [48] methods, the extended Hiickel method [3, 4, 49, 50] (presumably also [51 ]), a Pariser-Parr-Pople-type open-shell method [49] (presumably also [51]), and a simple MO approach [52]. Besides some other molecular properties, the charge distribution (atomic charges and/or overlap populations) [5, 38,40,41,43,49 to 51] and the spin density distribution (and thus, the hyperfine coupling constants, compare above and p. 241) [3 to 6, 46, 48] have been the subjects of many of these studies. 

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