The spin functions

When there are only two electrons the analysis is much simplified. Even quite elementary textbooks discuss two-electron systems. The simplicity is a consequence of the general nature of what is called the spin-degeneracy problem, which we describe in Chapters 4 and 5. For now we write the total solution for the ESE 4 (1, 2), where the labels 1 and 2 refer to the coordinates (space and spin) of the two electrons. Since the ESE has no reference at all to spin, 4 (1, 2) may be factored into separate spatial and spin functions. For two electrons one has the familiar result that the spin functions are of either the singlet or triplet type. 

If we let Pij represent an operator that interchanges all of the coordinates of the and 7 particles in the function to which it is applied, we see that 

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The sign of the last term depends on the parity of the system. Note that in the first and last term (in fact, determinants), the spin-orbit functions alternate, while in all others there are two pairs of adjacent atoms with the same spin functions. We denote the determinants in which the spin functions alternate as the alternant spin functions (ASF), as they turn out to be important reference terms. 

For molecules with an even number of electrons, the spin function has only single-valued representations just as the spatial wave function. For these molecules, any degenerate spin-orbit state is unstable in the symmetric conformation since there is always a nontotally symmetric normal coordinate along which the potential energy depends linearly. For example, for an - state of a C3 molecule, the spin function has species da and E that upon... 

As a first application, consider the case of a single particle with spin quantum number S. The spin functions will then transform according to the IRREPs of the 3D rotational group SO(3), where a is the rotational vector, written in the operator form as [36]... 

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... 

To use HyperChem for calculations, you specify the total molecular charge and spin multiplicity (see Charge, Spin, and Excited State on page 119). The calculation selects the appropriate many-electron wave function with the correct number of alpha or beta electrons. You don t need to specify the spin function of each orbital. 

Configuration interaction, which is necessary in treatments of excited states and desirable in calculations of spin densities, is more complex with open-shell systems. This is because more types of configurations are formed by one-electron promotions. These configurations (Figure 5) are designated as A, B, Cq, C(3 G is the symbol for a ground state. Configurations C and Cp have the same orbital part but differ in the spin functions. 

Among the many ways to go beyond the usual Restricted Hartree-Fock model in order to introduce some electronic correlation effects into the ground state of an electronic system, the Half-Projected Hartree-Fock scheme, (HPHF) proposed by Smeyers [1,2], has the merit of preserving a conceptual simplicity together with a relatively straigthforward determination. The wave-function is written as a DODS Slater determinant projected on the spin space with S quantum number even or odd. As a result, it takes the form of two DODS Slater determinants, in which all the spin functions are interchanged. The spinorbitals have complete flexibility, and should be determined from applying the variational principle to the projected determinant. 

The Hamiltonian operator for the electric quadrupole interaction, 7/q, given in (4.29), coimects the spin of the nucleus with quantum number I with the EFG. In the simplest case, when the EFG is axial (y = Vyy, i.e. rf = 0), the Schrddinger equation can be solved on the basis of the spin functions I,mi), with magnetic quantum numbers m/ = 7, 7—1,. .., —7. The Hamilton matrix is diagonal, because... 

The basis functions are most commonly chosen such that the spin-function is either a pure spin-up function a(cr) or a pure spin-down function )S(cr). They are defined such that a( ) = / ( j) = 1 and zero for any other argument cr. Since the BO and, consequently, the Fock operator do not contain any spin-dependent terms, the HF equations divide into spin-up and spin-down equations ... 

Remember from basic quantum mechanics that to completely describe an electron its spin needs to be specified in addition to the spatial coordinates. The spin coordinates can only assume the values Vr, the possible values of the spin functions a(s) and fits) are raO/i) = (K- /i) = 1 and a(-V4) = (K /i) = 0. 

The spin functions have the important property that they are orthonormal, i. e., = = 1 and = = 0. For computational convenience, the spin orbitals themselves are usually chosen to be orthonormal also ... 

The spin-independent probability of finding one electron at r, and the other simultaneously at r2 is obtained by integrating over the spins. Since the spin functions are... 

ASl = A (SXIX + SyIy + SZIZ) = ASZIZ + A(S+I- + S-I+) Operating on the spin functions with the extra hyperfine operator then gives ... 

For axially symmetric complexes, the parameter E is zero, and the spin functions S,ms) are eigenfunctions of the spin Hamiltonian ... 

When letting all the spin operators in the Hamiltonian of Equation 7.57 work on the compounded spin functions in Equation 7.58, note that. S, only work on the first part of the spin function, leaving the second part unchanged, and, equivalently, /, works only on the second part leaving the first part unchanged, e.g.,... 

The linear combination is used instead of the unsyinmetrical states / (l)cv(2) and j3 2)a ). It is reasonable to expect that each of these spin states could occur in combination with the ground-state function ij> r) to yield four different levels at the ground state. However, for the helium atom only one ground-state function can be identified experimentally and it is significant to note that only one of the spin functions is anti-symmetrical, i.e. 

There is no theoretical ground for this conclusion, which is a purely empirical result based on a variety of experimental measurements. However, it seems to apply everywhere and to represent a law of Nature, stating that systems consisting of more than one particle of half-integral spin are always represented by anti-symmetric wave functions. It is noted that if the space function is symmetrical, the spin function must be anti-symmetrical to give an anti-symmetrical product. When each of the three symmetrical states is combined with the anti-symmetrical space function this produces what is... 

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 ... 

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