Closed-shell systems 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... 

The correlation error is normally defined as the difference between the exact eigenvalue of the non-relativistic Hamiltonian for the molecule and the HF energy. While this definition works well for closed-shell systems, it becomes less meaningful when degeneracies, or near-degeneracies, occur... 

Based on the considerations in this subsection, we shall take the following expression as our definition of the Fock operator of a closed-shell system ... 

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]. 

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. 

The definitions of the perturbation corrections to the interaction energy, as given by Eqs. (8), (9), and (11), involve nonsymmetric operators, like H0, V, and Rq. These operators do not include all electrons in a fully symmetric way, so H0, V, and Rq must act in a larger space them the dimer Hilbert space Hab adapted to a specific irrep of 4. To use these operators we have to consider the space Ha Hb, the tensor product of Hilbert spaces Ha and Hb for the monomers A and B. For the interaction of two closed-shell two-electron systems this tensor product space should be adapted to the irreps of the 2 2 group. In most of the quantum chemical applications the Hilbert spaces Hx, X = A and B, are constructed from one-electron spaces of finite dimension Cx - In the particular case of two-electron systems in singlet states we have Hx — x x, where the symbol denotes the symmetrized tensor product. We define the one-electron space as the space spanned by the union of two atomic bases associated with the monomers in the dimer. The basis of the space includes functions centered on all atoms in the dimer and, consequently, will be referred to as the dimer-centered basis. We assume that the same one-electron space is used to construct the Hilbert spaces Hx 1 X = A and B, i.e., Hx = In such case Ha )Hb can be represented as a direct sum of Hilbert spaces H adapted to the irreps entering the induced product [2] [2] 4, i.e., Ha ( Hb = H[2i] f[3i] f[4]- This means that every function from H , v = [22], [31], or [4], can be expanded in terms of functions from Ha Hb-... 

With the example applications discussed here, it seems to us that it may not be fair at this stage to conclude definitively about the relative performance of SS-MRCEPA(O) and SS-MRCEPA(I) methods. At times CEPA(O) performs better than CEPA(l) and vice versa for the closed-shell case, as we found in more extensive applications, in the case of systems containing only closed-shell configurations [59]. More exhaustive calculations, in particular of the fully developed SS-MRCEPA(I), are needed to come to a definitive conclusion, which is on the way. For the SS-MRPT, we have more extensive applications, not all published yet, which definitely indicate the generally superior performance of the EN partitioning. 

The Roothaan-Hall equations are not applicable to open-shell systems, which contain one or more unpaired electrons. Radicals are, by definition, open-shell systems as are some ground-state molecules such as NO and 02. Two approaches have been devised to treat open-shell systems. The first of these is spin-restricted Hartree-Fock (RHF) theory, which uses combinations of singly and doubly occupied molecular orbitals. The closed-shell approach that we have developed thus far is a special case of RHF theory. The doubly occupied orbitals use the same spatial functions for electrons of both a and spin. The orbital expansion Equation (2.144) is employed together with the variational method to derive the optimal values of the coefficients. The alternative approach is the spin-unrestricted Hartree-Fock (UHF) theory of Pople and Nesbet [Pople and Nesbet 1954], which uses two distinct sets of molecular orbitals one for electrons of a spin and the other for electrons of / spin. Two Fock matrices are involved, one for each type of spin, with elements as follows ... 

As concerns the definition of the prototype molecule, at least two main cases can be distinguished. If the extended system consists of subsystems which are held together by nonbonded interactions (multipolar electrostatic forces, induction and dispersion interactions), it is possible to designate closed shell chemical subsystems without breaking covalent chemical bonds, and the prototype molecules can be the constituent molecules themselves. The situation becomes much more complicated, when the subunits are held together by covalent interactions, chemical bonds. In this case there are no natural frontiers for the representative subsystems, and it is unavoidable that a certain number of chemical bonds are broken. The prototype molecules are formed by saturating the broken (dangling) bonds by some appropriately selected atoms. 

The cage system is treated quantum mechanically. In the original version of the model all valence electrons were included and to allow a natural definition of the cage, orthogonalized atomic hybrid orbitals were used as a basis set [215]. This allows to avoid problems with the saturation of dangling bonds since all hybrids on the same atom may belong to the cage with a wave function obtained by solution of a closed-shell secular equation. 

Considerable progress has been made in this area since the pertinent section of MBBMA was written. At that time only the Mo2 molecule had been well characterized both experimentally and theoretically, although some results were available on other molecules that have only weak bonds in which the d orbitals play essentially no role. In such molecules, of which Fe2, Co2, Ni2 and Cu2 seem typical, the bond is formed by overlap of the valence shell 4s orbitals and the two cF configurations then interact only weakly. However, these d" + d systems give rise to an enormous number of molecular electronic states close in energy to the ground state. These molecules are therefore very difficult to characterize definitively by either theoretical or experimental means. 

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