Projection operators group theoretical

Wilhin a group theoretical analysis, this is normally accomplished with projection operators. For a discussion of (his method, see the group iheory texts listed in Footnote I of Chapter 3. 

These atomic orbitals are four-component Dirac spinors. The symmetriza-tion coefficients are obtained by the use of group theoretical projection operators [21]... 

From a quantum-theoretical point of view only the state property is observable, and this is determined by the overall symmetry of the molecule under consideration. In planar molecules the ir-a separation is physically relevant, as it is connected with state properties which are associated with projection operators for the antisymmetric (a") and symmetric (o ) representation of the symmetry group C. In nonplanar allenes only for molecules of Q symmetry, for example, monosubstituted compounds RHC=C=CH2, there exists a correspondingly physically relevant and unique iv a")-a a ) classifieation (24). 

In order to determine the mathematical form of the four sp hybrid orbitals, we can apply the projection operator method. Application of the projection operator for the A IRR on the cp basis function yields k[(p +02+ < 3+< 4] which Is identical to the result obtained in Equation (10.4) after the resulting wavefunction has been normalized and the appropriate s and p AOs have been substituted for the basis functions cp -cp. Likewise, application of the projection operator for the Tj IRR yields (after normalization) the three wavefunctions given by Equations (I0.5)-(I0.7). Thus, application of group theoretical methods to the methane molecule can determine not only which type of hybridization will occur but also the mathematical forms of the resulting hybrid orbitals. 

A comparative vibrational analysis of the CH- and NiF-stretching modes in ethylene and NiFj" respectively illustrates the distinction between the characters of the irreps of commutative and non-commutative symmetry point groups. It also allows the introduction of two particularly useful group theoretical terms direct sum and projection operator. 

The representations provided in the basis of degenerate eigenfunctions are usually irreducible and can be chosen to be unitary matrices (in fact usually orthogonal matrices if the functions are real functions). In practice, what one is usually faced with is a collection of functions which have arbitrary (but known) transformation properties and what one actually wants to do is to adapt these functions so that they actually transform like the true eigenfunctions of the problem. This can be done by means of the group theoretical projection operator. 

A.C. Hurley, R.D. Harcourt and P.R. Taylor, Israel J. Chem 19, 215 (1980) use group theoretical projection operators to generate symmetry-adapted wavefunctions for O2. 

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