Rotational wave functions, symmetry

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... 

The permutational symmetry of the rotational wave function is determined by the rotational angular momentum J, which is the resultant of the electronic spin S, elecbonic orbital L, and nuclear orbital N angular momenta. We will now examine the permutational symmetry of the rotational wave functions. Two important remarks should first be made. The first refers to the 7 = 0 rotational... 

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

VIII. Permutational Symmetry of Rotational Wave Function... 

However, as mentioned above, T c)3) will be orthogonal to all the k states, and T ) is nonzero. This implies that the number of total states of the same eigenvalue E is (k + 1), which contradicts our initial hypothesis. Thus, we conclude that k must be even, and hence proved the generalized Kramers theorem for total angular momentum. The implication is that we can use double groups as a powerful means to study the molecular systems including the rotational spectra of molecules. In analyses of the symmetry of the rotational wave function for molecules, the three-dimensional (3D) rotation group SO(3) will be used. 

VIII. PERMUTATIONAL SYMMETRY OF ROTATIONAL WAVE FUNCTION... 

Proceeding in the spirit above it seems reasonable to inquire why s is equal to the number of equivalent rotations, rather than to the total number of symmetry operations for the molecule of interest. Rotational partition functions of the diatomic molecule were discussed immediately above. It was pointed out that symmetry requirements mandate that homonuclear diatomics occupy rotational states with either even or odd values of the rotational quantum number J depending on the nuclear spin quantum number I. Heteronuclear diatomics populate both even and odd J states. Similar behaviors are expected for polyatomic molecules but the analysis of polyatomic rotational wave functions is far more complex than it is for diatomics. Moreover the spacing between polyatomic rotational energy levels is small compared to kT and classical analysis is appropriate. These factors appreciated there is little motivation to study the quantum rules applying to individual rotational states of polyatomic molecules. 

The spherical-top Hamiltonian is Htot= P2/1I. The ellipsoid of inertia is a sphere. The spherical-top rotational wave functions can thus be classified according to the symmetry species of %h. (The angular-momentum operator P2 =P2 + Pf+P2 remains unchanged no matter how the abc axes are rotated.)... 

Symmetry of Rotational Wave Functions for trans and cis Paths of Ammonia Dimer Interconversion... 

The origins of symmetry induced nuclear polarization can be summarized as follows as mentioned above molecular dihydrogen is composed of two species, para-H2, which is characterized by the product of a symmetric rotational wave-function and an antisymmetric nuclear spin wave function and ortho-H2, which is characterized by an antisymmetric rotational and one of the symmetric nuclear spin wavefunctions. In thermal equilibrium at room temperature each of the three ortho-states and the single para-state have practically all equal probability. In contrast, at temperatures below liquid nitrogen mainly the energetically lower para-state is populated. Therefore, an enrichment of the para-state and even the separation of the two species can be easily achieved at low temperatures as their interconversion is a rather slow process. Pure para-H2 is stable even in liquid solutions and para-H2 enriched hydrogen can be stored and used subsequently for hydrogenation reactions [54]. 

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