Spin functions a and

The spin functions a and P which accompany each orbital in lsalsP2sa2sP have been eliminated by carrying out the spin integrations as discussed above. Because H contains no spin operators, this is straightforward and amounts to keeping integrals <( i I f I ( j > only if ( )i and ( )j are of the same spin and integrals... 

It can easily be shown that the HF approximation discussed in Chapter 1 does include the Fermi-correlation, but completely neglects the Coulomb part. To demonstrate this, we analyze the Hartree-Fock pair density for a two-electron system with the two spatial orbitals ()> and < )2 and spin functions a and o2... 

But this is not the full story. The Hamiltonian operator employed is a spin-free operator and does not work on the spin functions a and p. H commutes therefore with the spin operators Sz and S ... 

The fact that there are only two kinds of spin function (a and (1), leads to the conclusion that two electrons at most may occupy a given molecular orbital. Were a third electron to occupy the orbital, two different rows in the determinant would be the same which, according to the properties of determinants, would cause it to vanish (the value of the determinant would be zero). Thus, the notion that electrons are paired is really an artifact of the Hartree-Fock approximation. 

Here 1, 2, represent the electrons. P is the operation of permuting the electrons among the spin-orbit functions, for example, interchanging 1 and 2 between afi and ba. There are (2n) of these operations in the permutation group 2n is the number of electrons for n bonds. The symbol (—l)p is 1 if P involves an even number of interchanges of pairs of electrons and — 1 if it involves an odd number. The function in the brackets satisfies the Pauli exclusion principle. R represents the 2 operations of interchanging the spin functions a and 0 for orbitals (such as a and 6) that are bonded together. 

It is convenient to consider the quantum number ms as the variable in the spin functions a and / a = a(ms) and p = fi(ms). Since ms takes on only two values, rather than a continuous range of values, we use sums rather than integrals to express the orthonormality of a and / , and the Hermitian... 

Fig. 5.2 A Slater determinant is made from spin orbitals derived from the occupied spatial molecular orbitals and two spin functions, a and P...

For atomic systems, it is often said that each electron is defined by four quantum numbers n, l, me, and ms. Actually, there is a fifth quantum number, s, which has the value of 1 /2 for all electrons. Quantum number ms can be either 1/2 or -1/2, corresponding to spin function a (spin up) and p (spin down), respectively. Spin functions a and p form an orthonormal set,... 

Let us set up a 2D unitary matrix representation for the transformation of the spin functions a and (1 in Civ. So far, we have established only a relation between 0(3)+ and SU(2). The matrix representations of reflections or improper rotations do not belong to 0(3)+ because their determinants have a value of -1. To find out how a and p behave under reflections, we notice that any reflection in a plane can be thought of as a rotation through n about an axis perpendicular to that plane followed by the inversion operation. For instance, 6XZ may be constructed as xz = Cz(y) i. Herein, it is not necessarily required... 

The spin functions a and (3 (i.e., the components of a spin doublet) belong to the Ej/2 irrep of C2v- But what about singlet and triplet spin functions For this purpose, we look at the action of the symmetry operators on typical two-electron spin functions such as aa and ap. The results, displayed in Table 6, are easily verified. 

In the restricted Hartree-Fock (RHF) method, two restrictions are placed on the molecular orbitals u< in equation (11). The first is chat each ui transform according to one of the irreducible representations of the point group of the molecule. The second restriction is that the space functions u come in identical pairs one with spin function a and the other with spin function /S. These are called, respectively, the symmetry and equivalence restrictions.190... 

Since the spin functions a and p are orthonormal, one gets directly... 

In the structure /, electrons i and 2 mutually neutralize their spins, therefore the projection of the spin angular momentum of one is + J (spin function a) and of the other — (spin function J9). The spin function of the two electrons in the case of the bond will be, as we have seen,... 

Multiplying each MO with one of the spin functions a and p yields the spin orbitals % j) = and V /O ) = Here, the space... 

The spin-orbitals, themselves, are the product of a spin function, a, and a space function, . The spin function can take either of two values, a or /3, while the space function is expanded as a linear combination of basis functions,... 

In the last chapter we have seen that a good approximation to the wave function for a system of atoms at a considerable distance from one another is obtained by using single-electron orbital functions wa(l), etc., belonging to the individual atoms, and combining them with the electron-spin functions a and /3 in the form of a determinant such as that of Equation 44-3. Such a function is antisymmetric in the electrons, as required by Pauli s principle, and would be an exact solution of the wave equation for the system if the interactions between the electrons and those between the electrons of one atom and the nuclei of the other atoms could be neglected. Such determinantal... 

When the d-orbitals are combined with the spin functions a and p, the total degeneracy is 10. In a useful notation we may write... 

This set is often described as an n basis set where the quantum members n, if have been suppressed since n = 3, = 2 in all cases of the present discussion the spin functions a and p are indicated by -f and — respectively. 

The spin eigenfunctions of a system consisting of two coupled 7r-electrons i = 1 and 2 are, in the absence of an applied magnetic field (Bo = 0), given by the following four linearly-independent, orthonormal linear combinations of the products of the single-electron spin functions a() and fii) with the spin quantum number s=yi ... 

Each molecular orbitalis used twice, successively associated with each of the spin functions a and 3. 

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