Symmetry characters

In Section III.E(2) we discussed the improvement in the con-vergency of expansion III. 18 which could be obtained by introducing a suitably chosen correlation factor g(r12, r13, r23,. . . ), and it is clear that a similar procedure can be carried out also in connection with Eq. III. 142, provided that the factor g does not change the symmetry character of the wave function. In complete analogy with Eq. III. 128, we hence obtain... 

It should be emphasized that whereas the theoretical modelling of An3+ spectra in the condensed phase has reached a high degree of sophistication, the type of modelling of electronic structure of the (IV) and higher-valent actinides discussed here is restricted to very basic interactions and is in an initial state of development. The use of independent experimental methods for establishing the symmetry character of observed transitions is essential to further theoretical interpretation just as it was in the trivalent ion case. 

In dealing with systems containing only two electrons we have not been troubled with the exclusion principle, but have accepted both symmetric and antisymmetric positional eigenfunctions for by multiplying by a spin eigenfunction of the proper symmetry character an antisymmetric total eigenfunction can always be obtained. In the case of two hydrogen atoms there are three... 

In case that the symmetry character of an electron-pair structure and an ionic structure for a molecule are the same, it may be difficult to decide between the two, for the structure may he anywhere between these extremes. The zeroth-order eigenfunction for the two bond electrons for a molecule MX (HF, say, or NaCl) with a single electron-pair bond would be... 

Bischirogenic unsymmetric (S,S)-BisP 6b (cf. (S,S)-BisP -borane 81b) were synthesized from the coupling reaction between synthon 79 and the lithiated (S)-alkylmethylphosphine-boranes 87 in reasonable to quantitative yields with enantioselectivity over 97% (Scheme 15) [94]. These phosphines constitute the unsymmetric version of BisP in that they bear different groups on both phosphorus atoms, breaking the C2-symmetry character of BisP [32,94]. 

The relationship between spin and the symmetry character of the wave function can be established in relativistic quantum theory. In non-relativistic quantum mechanics, however, this relationship must be regarded as a postulate. 

Since the Hamiltonian is symmetric in space coordinates the time-dependent Schrodinger equation prevents a system of identical particles in a symmetric state from passing into an anti-symmetric state. The symmetry character of the eigenfunctions therefore is a property of the particles themselves. Only one eigenfunction corresponds to each eigenfunction and hence there is no exchange degeneracy. 

TABLE 1 Relative energes (in eV) of the lowest 3d" states in bulk TMO (TM=Ni. Co, Mn) obtained by CASSCF and CASPT2. The active space is formed by the TM-3d orbitals and a set of correlating virtuals with the same symmetry character CASPT2 correlates the TM-3s, 3p, 3d electrons and the 0-2s, 2p electrons... 

R. B. Shirts, Correcting two long-standing errors in point group symmetry character tables. J. Chem. Educ. 84, 1882-4 (2007). 

The molecule is plane, the wave functions of the two n electrons of the second C—C bond are perpendicular to this plane and are antisymmetrical with respect to reflection in the plane. The bonding pair of the C—Cl bond is a a-pair as in the C—H bonds. Of the non-bonding electron pairs of the chlorine atom one is a 2s pair which is symmetrical. When we call the bonding pair in its orbit round the chlorine atom a 2p pair, the wave function of one of the non-bonding pairs as a 2py pair also lies in the plane of the molecule but that of the third pair as a 2p pair is perpendicular to it, and these latter electrons are, therefore, also n electrons of the same symmetry character as the pair of n electrons from the C = C bond. There are, therefore, two electron configurations possible, which satisfy the Lewis-Langmuir conditions ... 

Let us consider the two symmetry planes I and II. It is clear that since the charge on atoms left and right of a reflection plane must be equal, this also holds for the squares of the coefficients in the M.O. wave functions (p, thus c22 = c62 etc. This means, however, that the wave functions fall into two groups according as they are symmetrical (s) or antisymmetrical (a) with respect to the reflection plane, that is to say c2 = c6 or c2 = —r6 (p. 126) wave functions of different symmetry character do not combine and so the following groups of four variation functions are produced. [The coefficient in the antisymmetrical case of an atom in the reflection plane is equal to zero because we then have cx == —cx from which follows cx = o]. 

BASIC CONCEPTS OF MOLECULAR SYMMETRY CHARACTER TABLES... 

Basic Concepts of Molecular Symmetry Character Tables 1307... 

The Pauli principle must also apply to the nuclei so that the transposition of the nuclei must lead to an antisymmetric function. Let us now consider the symmetry character of these functions with respect to the nuclei. Transposition of the nuclei in equation 8.10 gives ... 

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