Spin wave functions

Equation (1.48) shows that, for I =, space quantization of nuclear spin angular momentum results in the quantum number Mj taking the values 5 or — 5. The nuclear spin wave function J/ is usually written as a or /i, corresponding to Mj equal to 5 or —5,... 

In general, for a homonuclear diatomic molecule there are (21+ )(/+1) symmetric and (21+ 1)/antisymmetric nuclear spin wave functions therefore... 

If / = 1 for each nucleus, as in H2 and N2, the total wave function must be symmetric to nuclear exchange. There are nine nuclear spin wave functions of which six are symmetric and three antisymmetric to exchange. Figure 5. f 8 illustrates the fact that ortho- ll2 (or N2)... 

The spin part pl can be derived by labelling the electrons 1 and 2 and remembering that, in general, each can have an a or /i spin wave function giving four possible combinations a(l)P(2), P(l)a(2), a(l)a(2) and P(l)P(2). Because the first two are neither symmetric nor antisymmetric to the exchange of electrons, which is equivalent to the exchange of the labels 1 and 2, they must be replaced by linear combinations giving... 

For helium, therefore, the singlet spin wave function of Equation (7.24) can combine only with the orbital wave function of Equation (7.26) giving, for singlet states. 

Equation (7.23) expresses the total electronic wave function as the product of the orbital and spin parts. Since J/g must be antisymmetric to electron exchange the Ig and Ag orbital wave functions of oxygen combine only with the antisymmetric (singlet) spin wave function which is the same as that in Equation (7.24) for helium. Similarly, the Ig orbital wave function combines only with the three symmetric (triplet) spin wave functions which are the same as those in Equation (7.25) for helium. 

In order to include the spin of the two electrons in the wave function, it is assumed that the spin and spatial parts of the wave function can be separated so that the total wave function is the product of a spin and a spatial wave function F — iAspace sp n Since our Hamiltonian for the H2 molecule does not contain any spin-dependent terms, this is a good approximation (NB—the complete Hamiltonian does contain spin-dependent terms, but for hydrogen they are rather small and do not appreciably affect the energetics of chemical bonding). For a two-electron system it turns out that there are four possible spin wave functions they are ... 

Considering first the state with a total spin of 0, we note that since the spin wave function is antisymmetric with respect to interchanging the particle labels, the spatial part of the wave function should be symmetric in order to preserve the overall antisymmetry of the wave function. This leads to the following expression for the wave function ... 

For the triplet spin states it is only necessary to consider one of the three possible spin wave functions and we will take the J wave function. Since... 

To aid our understanding of absorption and emission processes, Eq. (2.1) can be expanded in terms of electronic, vibronic (vibrational components of an electronic transition), and spin wave functions ... 

If any one of these integrals (expectation value equations) is zero, the transition is said to be forbidden. For the electronic and spin wave functions, it is not necessary to evaluate the integral but only to note that an odd function integrated from minus infinity to infinity is zero, while an even function integrated within these limits results in a nonzero value. For example (Figure 2.1),... 

As a consequence of the quantum-mechanical selection rules for resonance energy transfer involving the spin wave functions of the donor and acceptor,... 

J = 1,3,5 — are antisymmetric with respect to the nuclear coordinates. It follows that homonuclear diatomic molecules with anti-symmetric nuclear spin wave functions (nuclei with half-integer I = 1/2, 3/2...) can combine only with symmetric rotational functions (even J = 0,2,4...), while those with symmetric nuclear spin wave functions (even I) can combine only with antisymmetric rotational functions... 

For each of the diatomic examples above, examples which include all possible combinations of symmetric or anti-symmetric nuclear spin wave functions... 

We shall use a and / to stand for nuclear-spin wave functions with... 

In this chapter, we are dealing only with spin wave functions and Hamiltonians. In the complete molecular Hamiltonian, which involves electronic spatial coordinates and momenta in addition to spin coordinates, the contact term in the Hamiltonian would have to be written in a form such that the integral over the electronic spatial wave function J/,... 

Spin effects have their origin in V. If it is assumed that there is no coupling between orbital and spin character, the wavefunctions that appear in equation (17) for V become simple products of orbital and spin components ij/ — i ei spin- Recall, for example, that the spin wave function for a doubly occupied orbital is (l/V2)[a(l)/ (2) - / (l)a(2)] where a and prefer to upwards or downwards spin, the labels are with respect to the two electrons, and 1/-/2 is a normalization constant. 

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