Antisymmetric state

As was shown in the preceding discussion (see also Sections VIII and IX), the rovibronic wave functions for a homonuclear diatomic molecule under the permutation of identical nuclei are symmetric for even J rotational quantum numbers in + and Xu electronic states antisymmetric for odd J values in + and electronic states symmetric for odd J values in Xj and X+ electronic states and antisymmetric for even J values in X and X+ electronic states. Note that the vibrational ground state is symmetric under permutation of the two nuclei. The most restrictive result arises therefore when the nuclear spin quantum number of the individual nuclei is 0. In this case, the nuclear spin function is always symmetric with respect to interchange of the identical nuclei, and hence only totally symmetric rovibronic states are allowed since the total wave function must be symmetric for bosonic systems. For example, the 12C nucleus has zero nuclear spin, and hence the rotational levels with odd values of J do not exist for the ground electronic state ( X+) of 12C2. 

One must agree that the precise recipe implied by Van Vleck s and Sherman s language is daunting. The use of characters of the irreducible representations in dealing with spin state-antisymmetrization problems does not appear to lead to any very useful results. Prom today s perspective, however, it is known that some irreducible representation matrix elements (not just the characters) are fairly simple, and when applications are written for large computers, the systematization provided by the group methods is useful. 

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. 

If an electronic state corresponds to occupation of a doubly degenerate orbital, say of symmetry r(e), the full square V (e) x T(e) gives the full set of possible symmetries of the space parts of the electronic wavefunction. The square divides into symmetric and antisymmetric components the symmetric component [T(e) ] is associated with singlet states (permutation-ally symmetric space part, permutationally antisymmetric spin part), and the antisymmetric component T(e) with triplet states (antisymmetric space part, symmetric spin part). When the two electrons are in orbitals belonging to different degenerate e pairs, the full set of symmetries in T e) x T e) is accessible as both singlets and triplets. 

As was shown in the preceding discussion (see also Sections VIII and IX), the rovibronic wave functions for a homonuclear diatomic molecule under the permutation of identical nuclei are symmetric for even J rotational quantum numbers in and electronic states antisymmetric for odd J values in and electronic states symmetric for odd J values in S7 and electronic... 

The asymmetric reconstruction, the buckling of the ionic state Contrary to the covalent state, the ionic component should be stabilized by an asymmetric distortion if this one accommodates a positive charge on one side and a negative charge on the other one. The asymmetry allows to mix the first excited state (antisymmetric and ionic) with the upper component of the doubly excited state (symmetric and ionic) and thus to localize the electrons on a single-atom dimer. 

Symmetric combinations of atomic or molecular orbitals produce bonding states Antisymmetric combinations produce antibonding states. 

The fact that allowed fennion states have to be antisymmetric, i.e., changed m sign by any odd pemuitation of the fennions, leads to an interesting result concerning the allowed states. Let us write a state wavefiinction for a system of n noninteracting fennions as... 

It is beyond the scope of these introductory notes to treat individual problems in fine detail, but it is interesting to close the discussion by considering certain, geometric phase related, symmetry effects associated with systems of identical particles. The following account summarizes results from Mead and Truhlar [10] for three such particles. We know, for example, that the fermion statistics for H atoms require that the vibrational-rotational states on the ground electronic energy surface of NH3 must be antisymmetric with respect to binary exchange... 

Next, we address some simple eases, begining with honronuclear diatomic molecules in E electronic states. The rotational wave functions are in this case the well-known spherical haimonics for even J values, Xr( ) symmetric under permutation of the identical nuclei for odd J values, Xr(R) is antisymmetric under the same pemrutation. A similar statement applies for any type molecule. 

Finally, let us consider molecules with identical nuclei that are subject to C (n > 2) rotations. For C2v molecules in which the C2 rotation exchanges two nuclei of half-integer spin, the nuclear statistical weights of the symmetric and antisymmetric rotational levels will be one and three, respectively. For molecules where C2 exchanges two spinless nuclei, one-half of the rotational levels (odd or even J values, depending on the vibrational and electronic states)... 

As discussed in preceding sections, FI and have nuclear spin 5, which may have drastic consequences on the vibrational spectra of the corresponding trimeric species. In fact, the nuclear spin functions can only have A, (quartet state) and E (doublet) symmetries. Since the total wave function must be antisymmetric, Ai rovibronic states are therefore not allowed. Thus, for 7 = 0, only resonance states of A2 and E symmetries exist, with calculated states of Ai symmetry being purely mathematical states. Similarly, only -symmetric pseudobound states are allowed for 7 = 0. Indeed, even when vibronic coupling is taken into account, only A and E vibronic states have physical significance. Table XVII-XIX summarize the symmetry properties of the wave functions for H3 and its isotopomers. 

Because of the quantum mechanical Uncertainty Principle, quantum m echanics methods treat electrons as indistinguishable particles, This leads to the Paiili Exclusion Pnn ciple, which states that the many-electron wave function—which depends on the coordinates of all the electrons—must change sign whenever two electrons interchange positions. That IS, the wave function must be antisymmetric with respect to pair-wise permutations of the electron coordinates. 

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