Pople-Nesbet unrestricted equations

The Roothaan equations are the basic equations for closed-shell RHF molecular orbitals, and the Pople-Nesbet equations are the basic equations for open-shell UHF molecular orbitals. The Pople-Nesbet equations are essentially just the generalization of the Roothaan equations to the case where the spatials /j and /jP, as shown previously, are not defined to be identical but are solved independently. 

These two cases are sufficient to cover all situations, but many theoretical scientists prefer an RHF description over a UHF descrip- 

The unrestricted approach defines two different sets of spatial molecular orbitals—those that hold electrons with spin up  

The two sets of coefficients, one for spin-up alpha electrons and the other for spin-down beta electrons, are solutions of two coupled matrix eigenvalue problems  

The two equations couple because the alpha Fock matrix depends on both the alpha and the beta solutions, C and cP (and sim ilarly for the beta Fock matrix). The self-consistent dependence of the Fock matrix on molecular orbital coefficients is best represen ted, as before, via the den sity matrices an d pP, wh ich essen -tially state the probability of describing an electron of alpha spin, and the probability of finding one of beta spin  

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Moreover, the Hartree-Fock Eqs. (50) implemented in Eqs. (51) and (52) are known as Roothaan equations [109] and constitute the basis for closed-shell (or restricted Hartree-Fock, RHF) molecular orbitals calculations. Their extension to spin effects provides the equations for the open shell (or unrestricted Hartree-Fock, UHF), which are also known as the Pople-Nesbet Unrestricted equations [118]. 

In this section, then, we first introduce a set of unrestricted spin orbitab to derive the spatial eigenvalue equations of unrestricted Hartree-Fock theory. We then introduce a basis set and generate the unrestricted Pople-Nesbet matrix equations, which are analogous to the restricted Roothaan equations. We then perform some sample calculations to illustrate solutions to the unrestricted equations. Finally, we discuss the dissociation problem and its unrestricted solution. 

We could numerically solve the Pople-Nesbet equations for minimal basis STO-3G H2, just as we have solved them for CH3, N2, and O2. An appropriate unrestricted initial guess would be required if the iterations were to lead to an unrestricted solution rather than to the restricted solution. The transition from a restricted to an unrestricted wave function will be more transparent, however, if, rather than obtain a numerical solution to the Pople-Nesbet matrix equations, we formulate the problem in an analytical fashion. 

Pople-Nesbet Equations. The set of equations describing the best Unrestricted Single Determinant Wavefunction within the LCAO Approximation. These reduce the Roothaan-Hall Equations for Closed Shell (paired electron) systems. 

Before going on to describe sample unrestricted calculations, an important point should be noted about solutions to the Pople-Nesbet equations for the special case i.e., for the case where a molecule would nor-... 

We have seen that for the ground state of a closed-shell molecule like H2 it appears possible to define unrestricted wave functions which have the qualitatively correct behavior that we expect for the dissociation process. It remains to relate these unrestricted wave functions to solutions of the Hartree-Fock equations. If we solve the Pople-Nesbet equations, will a nonzero value of 0 be obtained To investigate this question, we need to determine the energy as a function of 6. 

To find the values of 8 which solve the Pople-Nesbet equations, ie., to find the values of 8 which make the unrestricted energy stationary, we set... 

In the unrestricted treatment, the eigenvalue problem formulated by Pople and Nesbet (25) resembles closely that of closed-shell treatments.-On the other hand, the variation method in restricted open-shell treatments leads to two systems of SCF equations which have to be connected in one eigenvalue problem (26). This task is not a simple one the solution was done in different ways by Longuet-Higgins and Pople (27), Lefebvre (28), Roothaan (29), McWeeny (30), Huzinaga (31,32), Birss and Fraga (33), and Dewar with co-workers (34). 

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 ... 

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. 

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