Wave function closed-shell

By including electron correlation in the wave function the UHF method introduces more biradical character into the wave function than RHF. The spin contamination part is also purely biradical in nature, i.e. a UHF treatment in general will overestimate the biradical character. Most singlet states are well described by a closed-shell wave function near the equilibrium geometry, and in those cases it is not possible to generate a UHF solution which has a lower energy than the RHF. There are systems, however, for which this does not hold. An example is the ozone molecule, where two types of resonance structure can be drawn. Figure 4.8. 

The closed-shell wave functions with N >2 can no longer be separated into spatial and spin parts, but are expressed in the following form ... 

The n doubly occupied orbitals in the closed-shell wave function in Eq. 2 are required in order to construct (i.e., compute) the Coulomb and exchange operators in T. Thus, it might seem impossible to use Eq. 3 to find the optimal orbitals for a molecule, because these orbitals must already be known in order to construct the Fock operator in Eq. 3. However, this seemingly impossible task is made possible by using an iterative process. 

This means that the variationally correct yj is a virtual orbital of Hi from the ground-state calculation but, because of the self-terms in H, is not a virtual orbital of H. After integrating over the spin coordinates, the optimum spatial molecular orbital 4>u in the case where we start with a closed-shell wave function, is a solution of... 

As expected, the spin part of > is identical to the spin part of the closed-shell wave function (2.257) since both wave functions are singlets. 

In the final Section 3.8, we leave the restricted closed-shell formalism and derive and illustrate unrestricted open-shell calculations. We do not discuss restricted open-shell calculations. By procedures that are strictly analogous to those used in deriving the Roothaan equations of Section 3.4, we derive the corresponding unrestricted open-shell equations of Pople and Nesbet. To illustrate the formalism and the results of unrestricted calculations, we apply our standard basis sets to a description of the electronic structure and ESR spectra of the methyl radical, the ionization potential of N2, and the orbital structure of the triplet ground state of O2. Finally, we describe in some detail the application of unrestricted wave functions to the improper behavior of restricted closed-shell wave functions upon dissociation. We again use our minimal basis H2 model to make the discussion concrete. 

We continue here with the generalization of our previous results for restricted closed-shell wave functions. If an electron is described by the molecular orbital (r), then the probability of finding that electron in a volume element dr at r is 2(r) dr. The probability distribution function (charge density) is If we have iV" electrons of a spin, then the total charge density... 

Diradicals cannot be described by a closed shell wave function because of the two single electrons on different atoms. One might therefore think of using the unrestricted Hartree-Fock (UHF) method. However the UHF wave function is not an eigenfunction of S. Therefore the restricted Hartree-Fock (RHF) method with a subsequent Cl is the method of choice. The resulting wavefunctions are eigenfunction of S. ... 

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... 

Show that W is positive definite for small k and use this to show that (IOE.2.13) is positive, implying that the stationary point p = 0 is a minimum. Thus, for closed-shell wave functions, p = 0 gives the largest overlap between the rotated and original MOs. ... 

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