Correlation of orbitals

Similar correlations of orbital interactions with substituent effects were also found in additions of alkenes to substituted carbenes and of N2 to transition metal complexes (see Zollinger, 1983 b, 1990). 

Correlation of orbitals between bent and linear geometries... 

Figure 11.10 Symmetry classification and correlation of orbitals for the disrotatory closure of butadiene. Closure with electron pairs remaining in their original levels would lead to the excited state indicated by the orbital occupancy on the right.

Since cr and cr" are maintained throughout the reaction, there must be a continuous correlation of orbitals of the same symmetry type. Therefore, orbitals of like symmetry correlate with one another and they can be connected. This, the crucial idea of the Woodward-Hoffmann method, is shown in the central part of the diagram. 

In a previous section of this chapter a general discussion of the theoretical background of the molecular orbital method was presented. There are correlations of orbitals between large and small internuclear distances and also between different geometrical conformations of the final molecule. 

Calculations on triatomic molecules, because of their theoretical and experimental significance, will be mentioned below in some detail. Herzberg s Polyatomics [6] gives diagrams of the correlation of orbitals between large and small internuclear distances in linear AH2 molecules and in linear ABa molecules [6]. Walsh, in a classic series of papers [97], gave correlations of molecular orbitals between linear and bent AH2 molecules and linear and... 

A Walsh diagram for the correlation of orbitals between nonplanar and planar XH3 is depicted in Herzberg [6]. 

Woodward and Roald Hoffmann, Nobel prize winners for their work, formulated the theoretical rules involving the correlation of orbital symmetry, which govern the Diels-Alder and other electrocyclic reactions. 

Isotope shifts in a spectral line arises not only through differences in the charge distribution, but also from differences in nuclear mass, M. The normal mass shift is simply obtained by replacing the electron mass rr e by the reduced mass fi — me/(l - - m lM), whereas the so-called specific mass shift (SMS) arising from a correlation of orbital momenta through the motion of the... 

Fig. 10.25. Correlation of orbitals of cyclobutene with the conrotatory transition state and the product, 1,3-butadiene. Energies (in eV) are from HF/6-31G(d) computations. Reproduced from J. Am. Chem. Soc., 125, 5072 (2003), by permission of the American Chemical Society.

Qualitatively speaking, x(l)j measures the quantum correlation of orbital i with all remaining orbitals contained in the active space, while s(2)y measures this correlation of the pair of orbitals i andj embedded in the complement of orbitals in the active space. The quantum entanglement among two orbitals embedded in... 

Fig. 5. Correlation of orbitals of linear XYg molecules to those of the separated atoms, including d orbitals. (Reproduced with permission of Litton Educational Publishing.)...

For these reasons, in the MCSCF method the number of CSFs is usually kept to a small to moderate number (e.g. a few to several thousand) chosen to describe essential correlations (i.e. configuration crossings, near degeneracies, proper dissociation, etc, all of which are often tenned non-dynamicaI correlations) and important dynamical correlations (those electron-pair correlations of angular, radial, left-right, etc nature that are important when low-lying virtual orbitals are present). 

The Woodward-Hoffmann method [52], which assumes conservation of orbital symmetry, is another variant of the same idea. In it, the emphasis is put on the symmetries of molecular orbitals. Longuet-Higgins and Abramson [53] noted the necessity of state-to-state correlation, rather than the orbital correlation, which is not rigorously justified (see also, [30,44]). However, the orbital symmetry conservation rules appear to be very useful for most themial reactions. 

Another approach is spin-coupled valence bond theory, which divides the electrons into two sets core electrons, which are described by doubly occupied orthogonal orbitals, and active electrons, which occupy singly occupied non-orthogonal orbitals. Both types of orbital are expressed in the usual way as a linear combination of basis functions. The overall wavefunction is completed by two spin fimctions one that describes the coupling of the spins of the core electrons and one that deals with the active electrons. The choice of spin function for these active electrons is a key component of the theory [Gerratt ef al. 1997]. One of the distinctive features of this theory is that a considerable amount of chemically significant electronic correlation is incorporated into the wavefunction, giving an accuracy comparable to CASSCF. An additional benefit is that the orbitals tend to be... 

The orbitals from which electrons are removed and those into which electrons are excited can be restricted to focus attention on correlations among certain orbitals. For example, if excitations out of core electrons are excluded, one computes a total energy that contains no correlation corrections for these core orbitals. Often it is possible to so limit the nature of the orbital excitations to focus on the energetic quantities of interest (e.g., the CC bond breaking in ethane requires correlation of the acc orbital but the 1 s Carbon core orbitals and the CH bond orbitals may be treated in a non-correlated manner). 

Unfortunately, these methods require more technical sophistication on the part of the user. This is because there is no completely automated way to choose which configurations are in the calculation (called the active space). The user must determine which molecular orbitals to use. In choosing which orbitals to include, the user should ensure that the bonding and corresponding antibonding orbitals are correlated. The orbitals that will yield the most correlation... 

An MCSCF calculation in which all combinations of the active space orbitals are included is called a complete active space self-consistent held (CASSCF) calculation. This type of calculation is popular because it gives the maximum correlation in the valence region. The smallest MCSCF calculations are two-conhguration SCF (TCSCF) calculations. The generalized valence bond (GVB) method is a small MCSCF including a pair of orbitals for each molecular bond. 

In this formulation, the electron density is expressed as a linear combination of basis functions similar in mathematical form to HF orbitals. A determinant is then formed from these functions, called Kohn-Sham orbitals. It is the electron density from this determinant of orbitals that is used to compute the energy. This procedure is necessary because Fermion systems can only have electron densities that arise from an antisymmetric wave function. There has been some debate over the interpretation of Kohn-Sham orbitals. It is certain that they are not mathematically equivalent to either HF orbitals or natural orbitals from correlated calculations. However, Kohn-Sham orbitals do describe the behavior of electrons in a molecule, just as the other orbitals mentioned do. DFT orbital eigenvalues do not match the energies obtained from photoelectron spectroscopy experiments as well as HF orbital energies do. The questions still being debated are how to assign similarities and how to physically interpret the differences. 

In subsequent studies attempting to find a correlation of physicochemical properties and antimicrobial activity, other parameters have been employed, such as Hammett O values, electronic distribution calculated by molecular orbital methods, spectral characteristics, and hydrophobicity constants. No new insight on the role of physiochemical properties of the sulfonamides has resulted. Acid dissociation appears to play a predominant role, since it affects aqueous solubiUty, partition coefficient and transport across membranes, protein binding, tubular secretion, and reabsorption in the kidneys. An exhaustive discussion of these studies has been provided (10). 

When the orbitals have been classified with respect to symmetry, they can be arranged according to energy and the correlation lines can be drawn as in Fig. 11.10. From the orbital correlation diagram, it can be concluded that the thermal concerted cycloadditon reaction between butadiene and ethylene is allowed. All bonding levels of the reactants correlate with product ground-state orbitals. Extension of orbital correlation analysis to cycloaddition reactions involving other numbers of n electrons leads to the conclusion that the suprafacial-suprafacial addition is allowed for systems with 4n + 2 n electrons but forbidden for systems with 4n 7t electrons. 

Fig. 13.4. Orbital correlation of energy states involved in the photochemical butadiene-to-cyclobutene conversion.

A common interpretation of the interaction of chalcogens with nucleophiles considers donation of electron density from a lone pair on the donor atom into the o- (E-X) orbital (Figure 15.1). As the degree of covalency increases, a hypervalent three-centre four-electron bond is formed. Real systems fall somewhere between secondary interactions and hypervalent (three centre - four electron) bonds. The two extremes can be distinguished by the correlation of X-E and E D distances.In the hypervalent case both bond distances decrease simultaneously, whereas in the secondary bond the distances are anticorrelated. This concept has been applied in a study of selenoquinones 15.17 (R = Ph, Me) with short Se 0 contacts,for... 

Naively it may be expected that the correlation between pairs of electrons belonging to the same spatial MO would be the major part of the electron correlation. However, as the size of the molecule increases, the number of electron pairs belonging to different spatial MOs grows faster than those belonging to the same MO. Consider for example the valence orbitals for CH4. There are four intraorbital electron pairs of opposite spin, but there are 12 interorbital pairs of opposite spin, and 12 interorbital pairs of the same spin. A typical value for the intraorbital pair correlation of a single bond is 20kcal/ mol, while that of an interorbital pair (where the two MO are spatially close, as in CH4) is 1 kcal/mol. The interpair correlation is therefore often comparable to the intrapair contribution. 

---