Symmetric spin factor

The first three functions correspond to a triplet term S = 1). The fourth corresponds to a singlet term S = 0). The symmetric spin factors correspond to the triplet, and the antisymmetric spin factor corresponds to the singlet. 

The three values of Mi, correspond to L = 1 with no states leftover, so that only P terms occur. Each symmetric spin factor combines with each one of the three antisymmetric space factors to give the nine states of the P term, and the antisymmetric spin factor combines with each one of the symmetric space factors to give the three states of the P term. 

For all diatomic molecules, with the exception of hydrogen below 300 K and of deuterium below 200 K, a considerable simplification is possible for temperatures above the very lowest. In the first place, the nuclear spin factor may be ignored for the present (see, however, 24j), since it is independent of temperature and makes no contribution to the heat capacity. The consequence of the nuclei being identical is then allowed for by introducing a s]rmmetry number a, giving the number of equivalent epatial orienta-turns that a tnolecule can occupy as a result of simple rotation. The value of F is 2 for symmetrical diatomic molecules, and for unsymmetrical molecules... 

To see the relevance of the inclusion of spin it is only necessary to recall the two-electron case. Since, in this simple case separation of the spatial and spin factors in the wavefunction into a simple product is possible, it is quite possible to have a wavefunction which is symmetric with respect to exchange of the spatial coordinates provided that it is multiplied by a spin factor which is antisymmetric with respect to exchange. Unless we explicitly include a coordinate for electron spin in our wavefunctions we cannot satisfy the Pauli principle. This leads us into the strange position that, even though the Hamiltonian contains no spin-dependent terms, the total energy of the system will, in general, depend on the spin state of the molecule. 

Here, the space-spin function is a one-term product of a spatial part and a spin factor. The overall function is antisymmetric, the spin factor is antisymmetric with respect to exchange of spin coordinates thus the spatial part must be symmetric with respect to exchange of spatial coordinates. Having used this formalism to ensure that the total wavefunction is antisymmetric, we are finished with spin completely. This result is clearly independent of the specific form of the HL trial function and ... 

To further demonstrate that the spin factor does not affect the energy, we shall assume we are doing a variational calculation for the He ground state using the trial function (j) = /(ri, t-2, ri2)2 / [a(l)/3(2) - /3(l)a(2)], where / is a normalized function symmetric in the coordinates of the two electrons. The variational integral is... 

The functions (13.86) and (13.88) are symmetric with respect to exchange. They therefore go with the antisymmetric two-electron spin factor (11.60), which has 5 = O.Thus (13.86) and (13.88) are the spatial factors in the wave functions for the two states of the doubly degenerate A term. The antisymmetric functions (13.87) and (13.89) must go with the symmetric two-electron spin functions (11.57), (11.58), and (11.59), giving the six states of the A term. These states all have the same energy (if we neglect spin-orbit interaction). 

From our previous discussion, we know that the ground state of H2 is a 2 state with the antisymmetric spin factor (11.60) and a symmetric spatial factor. Hence must be the ground state. The Heitler-London ground-state wave function is... 

For one- and two-electron systems, the wave function is a product of the spatial and spin factors. A normalized spin factor for two-electron systems - [a( )f (2) — f ( )a 2) guarantees that the state in question is a singlet (see Appendix Q available at booksite.elsevier.com/978-0-444-59436-5 p. el 33). Since we will only manipulate the spatial part of the wave function, the spin is the default. Since the total wave function has to be antisymmetric, and the spin function is antisynunetric, the spatial fimction should be symmetric-and it is. 

The number of states can be determined by counting the number of ways of arranging two spins up-up, down-down, up-down, and down-up. As shown in Eq. (18.4-2), the up-down and down-up arrangements combine in two ways, corresponding to symmetric and antisymmetric spin factors. 

Both of these space factors correspond to Mp = 1. The symmetric space factor combines with the antisymmetric spin factor, leading to one state with S = 0. The three triplet spin factors are all symmetric, so the antisymmetric space factor can combine with any of these, leading to three states, with Ms equal to 1,0, and —1, corresponding to 5 = 1. The 2p0 orbital combines with the li orbital in exactly the same way as the 2pl orbital except that Mp = 0 for these wave functions. The 2p,— orbital combines with the li orbital in exactly the same way as the 2p orbital except that Mp = — 1. This gives a total of 12 states. 

This is essentially the quantum-mechanical generalization of Pauli s exclusion principle. The connection is easily made in the case N = 2, where the wavefunction may be written as a product of space and spin factors for if two electrons are put into the same orbital with the same spins (i.e. into the same spin-orbital) the wavefunction can only be symmetric (cf. IPl and above), in violation of the antisymmetry requirement. Two electrons cannot therefore (in an IPM description) occupy the same state or— in Pauli s statement—possess identical sets of quantum numbers. The generality of the principle, which applies for any number of electrons in any kind of system and even when interaction is admitted, will be discussed further in Chapter 3. For more than two electrons the wavefunction has no simple symmetry for interchange of space or spin variables separately, exchange of particles implies exchange of space and spin variables together and (1.2.27) applies to this case only. 

Set up excited-state wavefunctions for Hj, assuming one electron in each of the MOs in (1.3.5) with spins opposed and the spin factor (i) symmetric and (ii) antisymmetric. Show that the energy expectation value will be unchanged on identifying the two spins in case (i), indicating a triplet state. 

In discussing the helium atom (Section 1.2) the antisymmetry requirement on the electronic wavefunction was easily satisfied for with only two electrons the function would be written as a product of space and spin factors, one of which had to be antisymmetric, the other symmetric, lliis is possible even for an exact eigenfunction of the Hamiltonian (1.2.1), as well as for an orbital product. The construction of an antisymmetric many-electron function is less easy. We have seen in Section 1.2 that for a general permutation (involving both space and spin variables) an antisymmetric function has the property... 

In evaluating the elements of H and M, the special form of the spatial function may now be recognized. By assuming that electrons 1 and 2 occupy the first orbital, 3 and 4 the second, and so on, we impose a symmetry on the spatial function 0. If 0 is symmetric under transposition (12), it will be necessary to ensure that the spin factor is anrisymmetric under (12) this must be so for each doubly occupied orbital, and the first g columns of any Young tableau describing an associated spin eigenfunction will thus read... 

Thus the promoting vibrations reduce the Franck-Condon factor itself, which is not reflected in the spin-boson model (5.55), (5.67). As an illustration, three-dimensional trajectories for various interrelations between symmetric (Ws) and antisymmetric (oja) vibration frequencies, and odo are shown in fig. 33. 

Now consider homonuclear diatomic molecules. To satisfy the Pauli principle, the overall molecular wave function must have the following behavior with respect to interchange of the spin and spatial coordinates of the identical nuclei it must be symmetric if the nuclei are bosons (/ = 0,1,2,...) it must be antisymmetric if the nuclei are fermions (/ = h >.. ) We therefore investigate the behavior of the factors in p with... 

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