Symmetric spin functions

The quantum numbers listed are for the eigenvalues of the total-spin operators S2 and Sz, where the total spin S is defined as S = S, + S2. Since electrons are fermions, the symmetric two-electron spatial function (1.249) must be multiplied by the antisymmetric spin function (1.254) to give an overall wave function that is antisymmetric the antisymmetric spatial function (1.250) must be multiplied by one of the symmetric spin functions (1.251H1.253). 

As an example, consider H2. The nuclear spin of H is and we have three symmetric nuclear spin functions and one antisymmetric function. The symmetric spin functions are of the form (1.251)—(1.253), and correspond to the two nuclear spins being parallel. Designating the quantum number of the vector sum of the two nuclear spins as 7, we have 7= 1 for the symmetric spin functions. The antisymmetric spin function has the form (1.254), and corresponds to 7 0. The ground electronic state of H2 is a 2 state, and the nuclei are fermions hence the symmetric (7=1) nuclear spin functions go with the J= 1,3,5,... rotational levels, whereas the 7=0 spin function goes with the7=0,2,4,... levels. 

Write down the nine nuclear spin functions of D2. Show that the three antisymmetric spin functions are eigenfunctions of the operator for the square of the magnitude of the total nuclear spin with the eigenvalue 2ft2. Find the corresponding eigenvalues for the symmetric spin functions. 

Other things being equal, for repulsive electronic correlations the ground state of a2, bi pairs is likely to be the triplet state, 3Bi, by Hund s rule. This principle energetically favours the most antisymmetrical space function and symmetrical spin function for a pair by minimising the short range Coulomb repulsion between electrons and is found to be widely applicable [40] in molecular systems. 

Those for the latter, called triplet states, are obtained by multiplying the antisymmetric orbital wave functions by the three symmetric spin functions.1 The spin-orbit interactions which we have neglected cause some of the triplet levels to be split into three adjacent levels. Transitions from a triplet to a singlet level can result only from a perturbation involving the electron spins, and since interaction of electron spins is small for light atoms, these transitions are infrequent no spectral line resulting from such a transition has been observed for helium. 

Since the nuclear spin quantum number for the proton is and there are two protons, it follows that the spins can add to give a total spin quantum number of 1, or they can subtract to give a net spin quantum number of 0. The first case corresponds to a triplet (three symmetric spin functions) and the second corresponds to a singlet (one antisymmetric spin function). The functions are the same as those for the electron pair ... 

For the two-helium excited state electron configurations, we can write three product two-electron symmetric spin functions,... 

The symmetrical spin functions have been seen to be triplet and the antisymmetrical to be singlet. Furthermore, the singlet states include the lowest or ground state, where presumably the two electrons have their orbital quantum numbers equal. Thus the antisymmetrical spin combination is associated with the symmetrical orbital wave function to give a total wave function which is antisymmetrical. This result verifies the principle of the antisymmetrical character of electron wave functions. 

Now since the Xgymm ii ust be associated with Santisymm vice versa, it follows that the even values of J correspond to the single antisymmetrical spin functions and the odd values of J to the triple symmetrical spin functions. 

Now consider the excited states of helium. We found the lowest emted state to have the zeroth-order spatial wave function 2 [ls(l)25(2) - 2s(l)ls(2)] [Eq. (9.105)]. Since this spatial function is antisynunetric, we must multiply it by a symmetric spin function. We can use any one of the three symmetric two-electron spin functions, so instead of the nondegenerate level previously found, we have a triply degenerate level with the three zeroth-order wave functions... 

Of course, our error is failure to consider spin and the Pauli principle. Tlie hypothetical zeroth-order wave function ls(l)ls(2)ls(3) is symmetric with respect to interchange of any two electrons. If we are to satisfy the Pauli principle, we must multiply this symmetric space function by an antisymmetric spin function. It is easy to construct completely symmetric spin functions for three electrons, such as a(l)a(2)a[(3). However, try as we may, it is impossible to construct a completely antisymmetric spin function for three electrons. 

The two-electron spin eigenfunctions consist of the symmetric functions a(l)o (2), /3(l)/3(2), and [a(l)/3(2) -I- j8(l)a(2)]/ V and the antisymmetric function [a(l)/3(2) —/3(l)a(2)]/ V. For the helium atom, each stationary state wave function is the product of a symmetric spatial function and an antisymmetric spin function or an antisymmetric spatial function and a symmetric spin function. Some approximate helium-atom wave functions are Eqs. (10.26) to (10.30). 

The 57/n-bimane has a substantial permanent dipole moment, fi = lAbD [12], but Si has B2 symmetry it is polarized along x, at right angles to the dipole axis, so fluorescence - though allowed - is not an overwhelmigly favored process. Intersystem crossing from Si to Ti is strictly forbidden in the planar 5yn-bimane, because both have the same space symmetry, and conservation of overall symmetry would require crossing to a triplet component with a totally symmetric spin function that cannot exist in C2V Rx is totally symmetric in so production of Tx(a") is weakly allowed in the non-planar 5yn-bimane, but would hardly be expected to compete with fluorescence and internal conversion to Sq. 

The spin wavefunction that goes with the symmetric spatial wavefunction lAsym must again be anti> 3 for spatially symmetric ground state. But for lAanti th spin wavefunction may be any of the three symmetric spin functions shown in Eq. 4.36. The number of spin wavefunctions that accompany a particular spatial state is called the spin multiplicity of the state. A state with only one spin wavefunction is called singlet, one with three spin wavefunctions is called triplet, and so on. 

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