Vibrational wave function, permutational

IT. Total Molecular Wave Functdon TIT. Group Theoretical Considerations TV. Permutational Symmetry of Total Wave Function V. Permutational Symmetry of Nuclear Spin Function VT. Permutational Symmetry of Electronic Wave Function VIT. Permutational Symmetry of Rovibronic and Vibronic Wave Functions VIIT. Permutational Symmetry of Rotational Wave Function IX. Permutational Symmetry of Vibrational Wave Function X. Case Studies Lis and Other Systems... 

As discussed above, the permutational symmetry of the total wave function requires the proper combination of its various contributions. These are summarized in Tables V-Xn for all isotopomers of Lis. Note that the conclusions hold provided that the various wave functions have the appropriate symmetries. If, for some reason, one of the components fails to meet such a requirement, then the symmetry of the total wave function will fail too. For example, even if the vibrational wave functions are properly assigned, the total wave... 

Planar molecules, permutational symmetry electronic wave function, 681-682 rotational wave function, 685-687 vibrational wave function, 687-692... 

As was shown in the preceding discussion (see also Sections Vin 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 E electronic states antisymmeUic for odd J values in and E elecbonic states symmetric for odd J values in E and E electronic states and antisymmeteic for even J values in Ej and E+ electeonic states. Note that the vibrational ground state is symmetric under pemrutation 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 symmeUic rovibronic states are allowed since the total wave function must be symmetric for bosonic systems. For example, the nucleus has zero nuclear spin, and hence the rotational levels with odd values of J do not exist for the ground electronic state f EJ") of Cr. 

According to the argument presented above, any molecule must be described by wavefunctions that are antisymmetric with respect to the exchange of any two identical particles. For a homonuclear diatomic molecule, for example, thepossibility of permutation of the two identical nuclei must be considered. Although both the translational and vibrational wavefunctions are symmetric under such a permutation, die parity of the rotational wavefunction depends on the value of 7, the rotational quantum number. It can be shown that the wave-function is symmetric if J is even and antisymmetric if J is odd The overall... 

Specifically, the various papers working within both the adiabatic and the Condon approximations, and using the (frequent) assumption of harmonic vibrations, can still differ in how many and what type (optical, acoustic, or local) modes they consider and in how they approximate the four separate integrals on the right-hand side of Eq. (40). And the choice of modes applies to both the ground and the excited states (so does the choice of electronic wave functions, but this choice is implicit in the evaluation of the electronic integrals.) It is this choice regarding the two states that was emphasized in connection with Fig. 15 (Section 10b). It can be seen that even within the stated approximations (adiabatic, Condon, harmonic) there is an appreciable number of permutations and combinations. 

In H2, the two nuclear spins of the H atoms can be combined in two different ways, resulting in the molecular species with the total nuclear spin of the two hydrogens of / = 1 or / = 0. In the ground electronic and vibrational state, each H2 molecule can be characterized by a combination of a certain rotational state with a certain nuclear spin state. As protons are fermions, Pauli s principle requires that the total wave function of a molecule should be antisymmetric with respect to the permutation of the two nuclei. The rotational states with even values of the rotational quantum number / are symmetric with respect to such permutations, including the rotational state with the lowest energy, which has / = 0. Such rotational states can be combined only with the antisymmetric nuclear spin state with 1 = 0, and such combinations correspond to parahydrogen (PH2). All rotational states with odd / values are antisymmetric and are only allowed in combination with the symmetric (/ = 1) nuclear spin states. These combinations correspond to orthohydrogen (oH2). At the same time, H2 molecules with even-even and odd-odd combinations of the two quantum numbers do not exist. As a result. 

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