Rotational 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... 

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. 

Wigner rotation/adiabatic-to-diabatic transformation matrices, 92 Electronic structure theory, electron nuclear dynamics (END) structure and properties, 326-327 theoretical background, 324-325 time-dependent variational principle (TDVP), general nuclear dynamics, 334-337 Electronic wave function, permutational symmetry, 680-682 Electron nuclear dynamics (END) degenerate states chemistry, xii-xiii direct molecular dynamics, structure and properties, 327 molecular systems, 337-351 final-state analysis, 342-349 intramolecular electron transfer,... 

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... 

VIII. PERMUTATIONAL SYMMETRY OF ROTATIONAL WAVE FUNCTION... 

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... 

Second-quantization formalism was introduced into the theory of many-electron atoms by Judd [12]. This formalism enables one to give a simple and elegant description of both the rotation symmetry of a system and its permutational symmetry the tensorial properties of wave functions are translated to electron creation and annihilation operators, and the Pauli exclusion principle stems automatically from the anticommutation relations between these operators. 

The main effect is already taken into account if symmetry numbers are included in the densities of states. The symmetry number is a correction to the density of states that allows for the fact that indistinguishable atoms occupy symmetry-related positions and these atoms have to obey the constraints of the Pauli principle (i.e. the wave function must have a definite symmetry with respect to any permutation), whereas the classical density of states contains no such constraint. The density of states is reduced by a factor that is equal to the dimension of the rotational subgroup of the molecule. When a molecule is distorted, its symmetry is reduced, and so its symmetry number changes by a proportion that is equivalent to the number of indistinguishable ways in which the distortion may be produced. For example, the rotational subgroup of the methane molecule is T, whose dimension is 12, whereas the rotational subgroup of a distorted molecule in which one bond is stretched is C3, whose dimension is 3. The ratio of these symmetry numbers, 4, is the number of ways in which the distortion can occur, i.e. the reaction path degeneracy. 

What happens with the interaction between the rotational and spin symmetries once the system is characterized as being defined by at least different spinors Wigner and von Neumann [10] combined both types of symmetries with the permutation aspect [11]. They intuitively reached the idea using atomic spectroscopy that the H operator has to be constructed by two terms H, resulting from the spatial motion of the single electron only (and the electromagnetic interaction with the field of the atomic core), and (//2), which has to visualize the electron spin. For simplicity, we can consider the eigenvalue problem of the spinless wave function i r without the second term as... 

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... 

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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