Anti-symmetric nuclear spin function

Here, only the deuteron spin projections (z-components) of the nuclear spin fuction ( i) are given in the expression for simplicity. These three anti-symmetric nuclear spin functions belong to B irreducible representation in the C2 symmetry (point group) of CHD2 and are only allowed to couple with the rotational function ( r) at the lowest level of 7 = 0 (even) by the Pauli principle. 

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... 

If the nucleus has an odd mass number, the overall wave function is anti-symmetrical with regard to the nuclei it is the product of all the translational, rotational, vibrational, electronic and nuclear wave functions. With all the translational, vibrational and electronic wave functions being symmetrical, we only have to consider the rotational and nuclear functions, one of which must be symmetrical and the other anti-symmetrical or vice versa. The g(g-l)/2 wave functions with anti-symmetrical nuclear spin must have corresponding symmetrical rotational wave functions, i.e. with even values of j the g(g+l)/2 wave functions with symmetrical nuclear spin must have corresponding rotational wave functions, i.e. with odd values of j. The combined nuclear-rotational partition function will therefore be ... 

This is the phenomenon that explains the abnormal heat capacity values of hydrogen (and deuterium) at low temperatures. At absolute zero, we only have the ortho form with symmetrical nuclear spins. By increasing the temperature, hydrogen gradually transforms into the para form with an anti-symmetrical function the transformation would be complete at approximately 20 K. For all lower temperatures, in the presence of a catalyst, we obtain a mixture with variable proportions of the tw o forms of hydrogen. 

Separability between electronic and nuclear states is fundamental to get a description in terms of a hierarchy of electronic and subsidiary nuclear quantum numbers. Physical quantum states, i.e. wavefiinctions 0(q,Q), are non-separable. On the contrary, there is a special base set of functions Pjt(q,Q) that can be separable in a well defined mode, and used to represent quantum states as linear superpositions over the base of separable molecular states. For the electronic part, the symmetric group offers a way to assign quantum numbers in terms of irreducible representations [17]. Space base functions can hence be either symmetric or anti-symmetric to odd label permutations. The spin part can be treated in a similar fashion [17]. The concept of molecular species can be introduced this is done at a later stage [10]. Molecular states and molecular species are not the same things. The latter belong to classical chemistry, the former are base function in molecular Hilbert space. 

Table 5.5 Allowed combinations of nuclear-spin states and rotational states for H2 and D2 molecules and their ortho-para designations. Anti-symmetric is abbreviated by AS and symmetric by S / is the total molecular nuclear spin and J the rotational quantum number and 4>i are the rotationtil tmd nuclear-spin wave functions, respectively. The table is adapted from [96] by permission of the American Physical Society...

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