Symmetry notations

Point group symmetry, notation and representations, and the group theoretical condition for when an integral is zero. 

Two forms of symmetry notation are commonly used. As chemists, you will come across both. The Schoenflies notation is useful for describing the point symmetry of individual molecules and is used by spectroscopists. The Hermann-Mauguin notation can be used to describe the point symmetry of individual molecules but in addition can also describe the relationship of different molecules to one another in space—their so-called space-symmetry—and so is the form most commonly met in crystallography and the solid state. We give here the Schoenflies notation in parentheses after the Hermann-Mauguin notation. 

M. Kasha has evolved still another system of notation in which electronic states are expressed in terms of the initial and final orbitals involved in a transition. This form of description is less precise than the symmetry notation but is very convenient for photochemical purposes, specially for designating energy levels of polyatomic organic compounds. In general, four types of molecules can be identified. 

Spectroscopists usually use notations based upon the symmetry properties of wave functions to describe excited states. Since many molecules have no special symmetry properties, such devices are not really strictly applicable in general. However, symmetry notation applicable to related molecules of high symmetry is often extended to unsymmetrical systems. The procedure has been treated formally by Platt (7) who introduced the concept of local symmetry. ... 

Symmetry Notation.—A state is described in terms of the behavior of the electronic wave function under the symmetry operations of the point group to which the molecule belongs. The characters of the one-electron orbitals are determined by inspection of the character table the product of the characters of the singly occupied orbitals gives the character of the molecular wave function. A superscript is added on the left side of the principal symbol to show the multiplicity of the state. Where appropriate, the subscript letters g (gerade) and u (ungerade) are added to the symbol to show whether or not the molecular wave function is symmetric with respect to inversion through a center of symmetry. 

Table 8.2 Symmetry notation in the Schoenflies and Herman-Maugin systems commonly used in...

Symmetry of crystal field Symmetry notation Crystal field states Mineral examples... 

FIGURE 11. (a) Possible silicocene conformers and their symmetry notation, (b) (Hel) PE spectrum (6-10 eV) of bis(j)5-pentamethylcyclopentadienyl)silicon with Koopmans assignment, IE = gMNDO for tjje most j)5d conformer and (c) comparison of radical cation states with analogous Ge and Sn pentamethylcyclopentadienyl sandwiches... 

Table 3-1. Symmetry Notations of the Crystallographic and a Few Limiting Groups...

FIG. 11. Energy level diagram for the cluster modelling the Cu(lOO) surface, the bare clusters Cut, Cus and Cug evaluated with the standard MO-LCAO method to the left and the calculations for jellium to the right were done using the spherical jeUium model [72,74]. The MO-LCAO results include contribution from the 3d levels while the jellium model only include the free 4s electrons. The one-electron orbitals are characterized by the symmetry notation corresponding to C v symmetry. Since the calculation for jellium were performed within the LSD scheme the one electron levels are split. 

The calculation of Mitroy started by calculating the Hartree—Fock approximation to the ground state 3s where we denote the states by the orbitals of the two active electrons in the configuration with the largest coefficient, in addition to the symmetry notation. The calculation used the analytic method with the basis set of Clementi and Roetti (1974) augmented by further Slater-type orbitals in order to give flexibility for the description of unoccupied orbitals. The total energy calculated by this method was —199.614 61, which should be compared with the result of a numerical Hartree—Fock calculation, —199.614 64. 

Table 19-13. Theoretical and experimental term energies/cm (D411 symmetry notation) of the 5 = 1 form of trans-Fe(0)(NCCH3)TMC + (I) and LFDFT values for the Fe(0)(NCCH3)(NH3)/+ (II) model complex...

Figure 2 shows several all-trans and mono-czs isomers of -carotene which have been identified. The ail-trans isomer is unique in that it contains a stretched conj ugated system with C21, synunetry. The symmetry notations of nAJ, nAj, nB and nB are appropriate for this structure (Tavan and Schulten,... 

Symmetric-ridge reconstruction, 240 Symmetry of bands, 145, 442f, 483 Symmetry-breaking transition, on surface, 234 Symmetry forbidden reactions, 39 Symmetry notation for bands, 145 Symmetry points, in Briltouin Zone, 73... 

Table 5.1 Intramolecular normal vibrations in anthracene crystals vibrational modes, symmetry notation and wavenumbers. Where two different wavenumbers are given for a single normal vibration, a Davydov splitting was observed. From [1-3]. Non-planar (out-of-plane) vibrations are marked with. ...

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