Antisymmetric spin functions

The study of the two-electron systems was greatly simplified by the fact that the total wave function could be factorized into a space part and a spin part according to Eq. III. I. ForiV = 3, 4,. . , such a separation of space and spin is no longer possible, and an explicit treatment of the spin is actually needed in considering correlation effects. This question of the connection between space and spin in an antisymmetric spin function is a rather complicated problem, which has been brought to a simple solution first during the last few years. 

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. 

Table 7.7. Symmetric and antisymmetric spin functions for two electrons in two different orbitals.

It is also possible to derive this result by incorporating the condition (2.5) below from the beginning by using the factorization of a two-electron function into a symmetrical spatial function and an antisymmetrical spin function, see equ. (1.16).) The expression in the braces indicates that the two electrons in the final state have opposite spins, i.e., the photoprocess reaches a singlet final state. This can be easily understood, because in LS-coupling spin-orbit effects are absent, and the photon operator does not act on the spin. Therefore, the selection rule... 

The first combination is symmetric, and the second is antisymmetric with respect to permutation. In the ground state of the helium atom both electrons occupy the same spatial orbital (Is), so that they must have the antisymmetric spin function for the total wave function to be antisymmetric they therefore form a singlet spin state. In order to find a helium atom with a triplet spin state (so-called spins parallel), the spatial part of the wave function must be antisymmetric with respect to interchange. 

The permutation operator P 2 which interchanges the electrons leaves the A = 2 functions unchanged they are symmetric and form the degenerate components of a Ag state which, since it must be combined with the antisymmetric spin function... 

The symmetric combination corresponds to a + state, and since it must be combined with the antisymmetric spin function, we have a 1 + state, the highest energy state arising from the ground electron configuration. The antisymmetric combination in (6.75) is a state, and combined with the symmetric triplet spin functions forms the ground state, 3 g. We discuss permutation and inversion symmetry in much more detail later in this chapter. 

This assignment accords with the singlet state (antisymmetric spin function) observed for the H2 molecule, as the product of spatial and spin functions would then be antisymmetric as required. 

The symmetric and antisymmetric squares have special prominence in molecular spectroscopy as they give information about some of the simplest open-shell electronic states. A closed-shell configuration has a totally symmetric space function, arising from multiplication of all occupied orbital symmetries, one per electron. The required antisymmetry of the space/spin wavefunction as a whole is satisfied by the exchange-antisymmetric spin function, which returns Fq as the term symbol. In open-shell molecules belonging to a group without... 

The plus superscript in the first term symbol is rather tricky, but don t worry about it. However, if you insist... two-electron singlet spin state has antisymmetric spin function, thus must have symmetric orbital function like Tlx- -7ty )tIx (2) which doesn t change sign upon transformation (p —Singlet oxygen and other active oxygen species are involved in lipid metabolism. 

The first, t/ j, incorporates the function we need for the chemical bond. The antisymmetric spin function implies that the spins of the two electrons in the bonding pair have opposite orientations hence, their magnetic moments cancel one another. For this reason, the majority of molecules have no net magnetic moment. The possession of a magnetic moment by a molecule indicates that one or more of the electrons in the molecule are unpaired. 

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 ... 

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. 

Deuterium proves to have its ortho states (higher statistical weight, antisymmetrical spin function) associated with even numbers of rotational quanta. Thus its total wave function is symmetrical, and it is said to follow the Bose-Einstein statistics. Moreover, the nuclear spin appears to have the value 1 instead of the value as with hydrogen. 

Furthermore, the electrons possess spin, and, as already shown, symmetrical and antisymmetrical spin functions, cr and c, are possible. By the principle that the total wave function must be antisymmetrical (p. 191) we see that the combinations must be or Xa< s-... 

We now include spin in the He zeroth-order ground-state wave function. The function ls(l)ls (2) is symmetric with respect to exchange. According to the Pauli principle, the overall wave function including spin must be antisymmetric with respect to interchange of the two electrons. Hence we must multiply the symmetric space function ls(l)ls(2) by an antisymmetric spin function. Tliere is only one antisymmetric... 

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). 

In order to make the overall wavefunction antisymmetric, this spatial wavefunction has to be multiplied by the antisymmetric spin function, a( )P(2) - a 2)P(l). The wavefunction to be used in calculations is therefore ... 

We now use (10.37) to prove that it is impossible to construct an antisymmetric spin function for three electrons. The functions/, g, and h may each be either a or /3. If we take... 

We showed in Section 5-2 that two space functions having proper space symmetry could be written for the configuration ls2s. One was symmetric (Eq. 5-15) and one was antisymmetric (Eq. 5-16). Now we find that spin functions must be included in our wavefunctions, and in a way that makes the final result antisymmetric when space and spin coordinates are interchanged. We can accomplish this by multiplying the symmetric space function by an antisymmetric spin function, calling the result i/ s,a-Thus,... 

According to the Pauli principle (Further information 9.3), the overall wave-function of the molecule (the wavefunction including spin) must change sign when we interchange the labels 1 and 2. Therefore, we must multiply v -b(2.1) by an antisymmetric spin function of the form shown in Further information 9.3. There is only one choice ... 

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