Wave function rotational

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... 

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. 

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... 

We must now combine the nuclear wave functions with the rest of the molecular wave function to generate a total wave function which is antisymmetric with respect to exchange of Fermions. For Bosons the total wave function must be symmetric. To do so we write r]r = i rans r]rviB rot r Nuc-spiN and recognize that both the vibrational and translational wave functions are symmetric. Rotational wave functions... 

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 above is a well-understood problem of the BO approximation, and the most accurate calculations of molecular properties takes this into account. A less well understood difference between the physical and chemical pictures is that, in the physical picture, the ground state of any molecule is spherically symmetric. This may be understood in the chemical picture by noting that when all degrees of freedom are taken into account, the total wave function contains the nuclear vibratrional and rotational wave functions as well as the electronic wave function ... 

For a linear polyatomic molecule, there is no rotational angular momentum about the intemuclear axis therefore, the derivatives with respect to chi are absent from (5.40). What are the rotational wave functions for a linear polyatomic molecule ... 

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