Closed shell, definition

P is the total spinless density matrix (P = P + P ) and P is the spin density matrix (P = p" + P ). For a closed-shell system Mayer s definition of the bond order reduces to ... 

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... 

As an example of the interest to scrutinise the UHF solution, one may quote the Bea problem [19]. The bond is weak but it takes plaee at short interatomic distance and is definitely not the dispersion well which one might expect from two closed shell atoms (and which occurs in Mga and heavier eompounds). Quantum chemical calculations only reproduce this bond when using large basis sets and extensive Cl calculations [20]. It is amazing to notice that the UHF solution gives a qualitatively correct behaviour, and suggests a physical interpretation of this bond since in... 

Since rigorous theoretical treatments of molecular structure have become more and more common in recent years, there exists a definite need for simple connections between such treatments and traditional chemical concepts. One approach to this problem which has proved useful is the method of localized orbitals. It yields a clear picture of a molecule in terms of bonds and lone pairs and is particularly well suited for comparing the electronic structures of different molecules. So far, it has been applied mainly within the closed-shell Hartree-Fock approximation, but it is our feeling that, in the future, localized representations will find more and more widespread use, including applications to wavefunctions other than the closed-shell Hartree-Fock functions. 

In certain problems of torsional isomerism we shall employ the dissection of a molecule A-B into two closed shell fragments A and B. On the other hand, most problems of torsional isomerism which we shall be dealing with in the first part of this work can be treated by employing the dissection shown below. The appropriate definitions are specified in parenthesis. 

One point of debate in defining the magnitude of the captodative effect has been the separation of substituent effects on the radical itself as compared to that on the closed shell reference system. This is, as stated before, a general problem for all definitions of radical stability based on isodesmic reactions such as Eq. 1 [7,74,76], but becomes particularly important in multiply substituted cases. This problem can be approached either through estimating the substituent effects for the closed shell parents separately [77,78], or through the use of isodesmic reactions such as Eq. 5, in which only open shell species are present ... 

The second approach to calculating MCD starts from its definition in terms of the real part of first-order correction to the frequency-dependent polarizability in the presence of a magnetic field (Section II.A.6). This definition can be used to consider all types of MCD linear in the magnetic field (9). Our current implementation is restricted to systems with a closed-shell ground state. We shall therefore only consider the calculation of A and terms by this method. 

The neo classification divides all benzenoids into normal (n), essentially disconnected (e) and non-Kekuleans (o), where the n and e systems cover all the Kekuleans. Cyvin and Gutman [26] have advocated for this classification by saying From the point of view of the enumeration of Kekule structures the classification. . . [neo]. . . seems to be a rather appropriate one [94,87] . However, the distinction between Kekulean (closed-shell, non-radicalic) and non-Kekulean (radicalic) benzenoid hydrocarbons was made long before the explicit definition of the neo classification. This practice started with the first (substantial) enumeration of benzenoids in the chemical context by Balaban and Harary [13]. 

For the solution one starts from the lowest hole-particle sector (0,0) for S and then solves the (0,1) and (1,0) sectors finally reaching the target (1,1) sector. The equation for the(0,0)sector reduces to that for the closed shells. The model spaces for (0,1) and (1,0) sectors are always complete by definition. The energies of these two sectors are those of (N-l) electron states and (N+l) electron states respectively. If we collectively call the q-open and open gp rators as external. then the equations for S are given by... 

While the significance of radicals in biological systems has been appreciated for decades, there is relatively little definitive experimental infonnation on the identity of the radicals and even less on the mechanisms by which they affect the physiology of living systems. The paucity of detailed information is a direct consequence of the fact that most radicals are highly reactive and, therefore, short-lived transient species. Despite the tremendous advances in spectroscopic and laser photolysis techniques, much less is known about radicals than about closed-shell species. The treatment of radicals by theoretical methods is, however, only marginally more difficult than that of closed-shell molecules. It is for these reasons that the numerous applications of quantum chemical techniques to radicals have proven to be complementary to experimental studies. 

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