Canonical orbitals

There are three cases The original p orbital may have contained two, one, or no electrons. Since the original double bond contributes two electrons, the total number of electrons accommodated by the new orbitals is four, three, or two. A typical example of the first situation is vinyl chloride, CH2—CH—CI. Although the p orbital of the chlorine atom is filled, it still overlaps with the double bond. The four electrons occupy the two molecular orbitals of lowest energies. This is our first example of resonance involving overlap between unfilled orbitals and a filled orbital. Canonical forms for vinyl chloride are... 

All of the delocalization discussed so far involves n electrons. Another type, called hyperconjugation, involves <7 electrons. When a carbon attached to at least one hydrogen is attached to an unsaturated atom or one with an unshared orbital, canonical forms such as 110 can be drawn. In such canonical forms, there is no bond at all between the carbon and hydrogen. The effect of 110 on the actual molecule is that the electrons in the C—H bond are closer to the carbon than they would be if 110 did not contribute at all. 

As noted in Sect. 4, a unitary transformation (/> —> ( = leaves both the density n(r) and the total energy invariant. Any unitary transformation of the Kohn-Sham orbitals is thus a valid set of orbitals. Canonical orbitals are a special set of such orbitals which diagonalize the Kohn-Sham Hamiltonian. Localized orbitals on the other hand are obtained by finding the unitary transformation U so as to optimize the expectation value of a two electrons operator Q ... 

For such a function, variational optimization of the spin orbitals to make the expectation value ( F // T ) stationary produces [30] the canonical FIF equations... 

The sum over eoulomb and exehange interaetions in the Foek operator runs only over those spin-orbitals that are oeeupied in the trial F. Beeause a unitary transformation among the orbitals that appear in F leaves the determinant unehanged (this is a property of determinants- det (UA) = det (U) det (A) = 1 det (A), if U is a unitary matrix), it is possible to ehoose sueh a unitary transformation to make the 8i j matrix diagonal. Upon so doing, one is left with the so-ealled canonical Hartree-Fock equations ... 

Molecular orbitals are not unique. The same exact wave function could be expressed an infinite number of ways with different, but equivalent orbitals. Two commonly used sets of orbitals are localized orbitals and symmetry-adapted orbitals (also called canonical orbitals). Localized orbitals are sometimes used because they look very much like a chemist s qualitative models of molecular bonds, lone-pair electrons, core electrons, and the like. Symmetry-adapted orbitals are more commonly used because they allow the calculation to be executed much more quickly for high-symmetry molecules. Localized orbitals can give the fastest calculations for very large molecules without symmetry due to many long-distance interactions becoming negligible. 

The equations may be simplified by choosing a unitary transformation (Chapter 13) which makes the matrix of Lagrange multipliers diagonal, i.e. Ay 0 and A This special set of molecular orbitals (f> ) are called canonical MOs, and they transform eq. (3.40) mto a set of pseudo-eigenvalue equations. 

The orbital energies can be considered as matrix elements of the Fock operator with the MOs (dropping the prime notation and letting 0 be the canonical orbitals). The total energy can be written either as eq. (3.32) or in terms of MO energies (using the definition of F in eqs. (3.36) and (3.42)). 

The canonical MOs are convenient for the physical interpretation of the Lagrange multipliers. Consider the energy of a system with one electron removed from orbital number Ic, and assume that the MOs are identical for the two systems (eq. (3.32)). 

Since the singly excited determinants effectively relax the orbitals in a CCSD calculation, non-canonical HF orbitals can also be used in coupled cluster methods. This allows for example the use of open-shell singlet states (which require two Slater determinants) as reference for a coupled cluster calculation. 

We may again chose a unitary transfonnation which makes tlie matrix of the Lagrange multiplier diagonal, producing a set of canonical Kohn-Sham (KS) orbitals. The resulting pseudo-eigenvalue equations are known as the Kohn-Sham equations. 

For computational purposes it is convenient to work with canonical MOs, i.e. those which make the matrix of Lagrange multipliers diagonal, and which are eigenfunctions of the Fock operator at convergence (eq, (3.41)). This corresponds to a specific choice of a unitary transformation of the occupied MOs. Once the SCF procedure has converged, however, we may chose other sets of orbitals by forming linear combinations of the canonical MOs. The total wave function, and thus all observable properties, are independent of such a rotation of the MOs. 

The traditional view of molecular bonds is that they are due to an increased probability of finding electrons between two nuclei, as compared to a sum of the contributions of the pure atomic orbitals. The canonical MOs are delocalized over the whole molecule and do not readily reflect this. There is, furthermore, little similarity between MOs for systems which by chemical measures should be similar, such as a series of alkanes. The canonical MOs therefore do not reflect the concept of functional groups. 

This review demonstrates that representatives of all four major classes of heterocyclic mesomeric betaines were isolated from natural sources. The profound differences in the electronic structures of these distinct classes can be realized by a closer look at the canonical formulae, the frontier orbital profile, the isoconjugate relationships, physico-organic properties, and the... 

The canonical molecular orbitals of any molecule can by obtained by computer calculations. All MO methods involve the diagonalization of a secular matrix. It can be said that by moving from AOs to FOs to BOs basis sets one proceeds through the various stages of this diagonalization process, as the number of non-zero off-diagonal overlap matrix elements decreases. 

An example Six electrons with only spin degeneracy. Placing the six orbits in a ring in the order abcdef, the five structures forming a canonical set are those given in Fig. 2. It is seen that... 

A canonical set of structures for a system with more orbits than electrons is obtained by arranging all the orbits (including phantom orbits for 5>0) in a ring and then drawing non-intersecting bonds to a number determined by the number of electrons and the multiplicity. If two electrons occupy the same orbit, forming an unshared pair, a loop is drawn with its ends at the orbit. 

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