Canonical orbitals conditions

While the canonical orbitals of a system are unique, aside from degeneracies due to multidimensional representations, this is not always the case for localized orbitals, and there may be several sets of localized orbitals in a particular molecule. This situation is related to the fact that the localization sum of Eq. (28) may have several relative maxima under suitable conditions. If one of these maxima is considerably higher than the others, then the corresponding set of molecular orbitals would have to be considered as the localized set. In some cases, however, the two maxima are equal in value, so that there exist two sets of localized orbitals with equal degree of localization. 23) In such a case there... 

The original formula (eqn (3)) was later extended by Angyan et al i by releasing the condition of integer occupancy of canonical orbitals and replacing the doubly occupied orbitals by fractionally occupied orbitals e.g. natural orbitals resulting from the diagonalization of the correlated first order density matrix). In such a case the formula in eqn (3) transforms to... 

Although we may keep the redundant parameters fixed (equal to zero) during the optimization of the Hartree-Fock state, we are also free to vary them so as to satisfy additional requirements on the solution - that is, requirements that do not follow from the variational conditions. In canonical Hartree-Fock theory (discussed in Section 10.3), the redundant rotations are used to generate a set of orbitals (the canonical orbitals) that diagonalize an effective one-electron Hamiltonian (the Fock operator). This use of the redundant parameters does not in any way affect the final electronic state but leads to a set of MOs with special properties. 

The next five chapters are each devoted to the study of one particular computational model of ab initio electronic-structure theory Chapter 10 is devoted to the Hartree-Fock model. Important topics discussed are the parametrization of the wave function, stationary conditions, the calculation of the electronic gradient, first- and second-order methods of optimization, the self-consistent field method, direct (integral-driven) techniques, canonical orbitals, Koopmans theorem, and size-extensivity. Also discussed is the direct optimization of the one-electron density, in which the construction of molecular orbitals is avoided, as required for calculations whose cost scales linearly with the size of the system. 

Let us note that the two conditions ej,=0 and < Pj >=0 (i 7 j) can be satisfied only with canonical SCF orbitals. Thus, in fact, the present theory can be applied only in such cases. However it has been demonstrated (12) that in most systems, the strongly occupied MCSCF orbitals and the SCF orbitals are extremely close one to the others. Therefore, in practice, the present theory also applies to the strongly occupied MCSCF orbitals. 

These conditions determine a unique set of molecular orbitals, the canonical molecular orbitals, (CMO s), . Inserting the conditions (25) in the SCF Eqs. (17), one sees that the CMO s are solutions of the canonical Hartree-Fock equations 10)... 

Species such as 5 and 6 are called benzynes (sometimes dehydrobenzenes), or more generally, arynes, and the mechanism is known as the benzyne mechanism. Benzynes are very reactive. Neither benzyne nor any other aryne has yet been isolated under ordinary conditions,34 but benzyne has been isolated in an argon matrix at 8 K,35 where its ir spectrum could be observed. In addition, benzynes can be trapped e.g., they undergo the Diels-Alder reaction (see 5-47). It should be noted that the extra pair of electrons does not affect the aromaticity. The original sextet still functions as a closed ring, and the two additional electrons are merely located in a tt orbital that covers only two carbons. Benzynes do not have a formal triple bond, since two canonical forms (A and B) contribute to the hybrid. 

It is well known that Eq. (11) has an infinity of solutions differing only by a unitary transformation. If we have n orbitals to determine, there are n2 Lagrangian multipliers. The orthonormality condition introduces n(n + l)/2 constraints, leaving n(n - l)/2 arbitrary values. Putting these remaining parameters to zero, we suppress the off-diagonal elements and uniquely define Eqs. (10) in their canonical form ... 

The carbocation of Figure 5.19 can be written either as one of the two canonical forms (5.26) and (5.27) or using an electron smear (5.28) as shown in Figure 5.20. The smear is closer to reality since the molecular orbitals of the molecule will be distributed across the three carbons of the allylic cation. However, it will be able to react as if the positive charge were localised at either end. For each individual reaction, the nature of the other reactive species, the reaction conditions and the nature of the product will determine which way round the system will react. 

The transformation of Eq. (16.4) is not really a quasiparticle transformation, since it just reflects a transformation of the underlying orbital space If creates an electron on orbital Xi than creates one on = X/t ikXk- The canonical condition of Eq. (16.5) means that the transformation matrix A is unitary. In Sect. 13 we have also seen non-unitary basis set transformations of the form of Eq. (16.4), for which Eq. (16.5) does not hold, and which do not leave the anticommutation rules invariant. 

Finally, Eq. (11) contains a set of coefficients that need to be determined. In the original work, only the diagonal terms c were used, but this procedure is not invariant with respect to the choice of the occupied orbitals (e.g. canonical or localized). Orbital invariance requires to also consider the off-diagonal elements. As we will discuss below, one can also make use of the cusp conditions to determine the coefficients, as was first suggested by Ten-no. This procedure is also orbital invariant and we will come back to this at the end of this section. 

It turns out that the Hartree-Fock state is invariant to unitary transformations among the occupied spin orbitals. The spin orbitals of the Hartree-Fock state are therefore not uniquely determined by the stationary condition (5.4.3) and the canonical spin orbitals are therefore just one possible choice of spin orbitals for the optimized N-particle state. Indeed, any set of energy-optimized spin orbitals decomposes the Fock matrix into two noninteracting blocks - one for the occupied spin orbitals and another for the unoccupied spin orbitals. When these subblocks are diagonalized, the canonical spin orbitals are obtained. 

When constructed from an arbitrary set of orbitals, the Fock matrix is nondiagonal. In the special canonical representation, however, the orbitals satisfy the canonical conditions... 

At this point, it is instructive to compare the two sets of conditions we have set up for the closed-shell wave function - the variational conditions and the canonical conditions. As discussed in Section 3.2, we may, from a given set of spin orbitals, generate any other set of spin orbitals according to the expressions... 

In either case, the number of orbital-rotation parameters is equal to the number of conditions. In setting up the variational conditions (10.3.31), the invariances of the wave function have been exploited to minimize the number of orbital-rotation parameters and there ate only OV parameters present in k (where O and V are the numbers of occupied and virtual orbitals, respectively). In setting up the canonical conditions, on the other hand, the invariances of the wave function... 

Since f is s)mmetric, we may extend (10.6.13) to give also a set of mthoncamal virtual MOs that ate orthogonal to the occupied MOs. These virtual orbitals satisfy the same canonical conditions as the occupied MOs (10.6.12) ... 

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