The Projection Operator Method

The projection operator method for obtaining a picture of the motion represented by each of the irreducible representations begins by considering the effect of each operation in the group on one, or a subset, of the basis vectors for the symmetry-related atoms. 

The two nonzero sums have the same relationship between and 2 as was found earlier normalization, as set out in Section 6.5, would yield the standard forms of Equation (6.35). 

In this example, we have used a basis of two vectors and constructed two SALCs, so the job is done both modes that the basis can represent have been obtained from a single generating vector. We will see below that this is not always the case. 

Problem 6.11 Show that using 2 as the generating vector would lead to conclusions equivalent to those drawn from Table 6.10. Remember that the SALCs only tell us about 

For the general case, the projection method automates this process through the 

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Symmetry properties which have so far been successfully treated by the projection operator method, include translational symmetry in crystals, cyclic systems, spin, orbital and total angular momenta, and further applications are in progress. ... 

Apply the projection-operator method to obtain the molecular orbital expression shown in Fig. 9 and verify the energies. 

The projection-operator technique will be employed in several examples presented in the following chapter and Chapter 12. For. the quantitative interpretation of molecular spectra both electronic and vibrational, molecular symmetry plays an all-important role. The correct linear combinations of electronic wavefunctions, as well as vibrational coordinates, are formed with the aid of the projection-operator method. 

From the above analysis, as shown in the last column of Table 2, it should be obvious that two linear combinations of the internal coordinates Ar can be formed whose symmetry corresponds to the IRs k and B2. They are Arj + Ar2(A]) and Ar — Ar2(B2), as can be verified by inspection of the character table. Tbe choice of the correct linear combinations is in this case simple. However, more generally, it can be made by application of the projection-operator method described in Section 8.11. 

We shall use the projection operator method to derive the Pauli master equation. With the Liouville equation, we separate the Liouville operator into two parts ... 

Use the projection operator method to determine the SALCs required for the formation of M—X a bonds by the eight ligands in an Mo2Xs ion of symmetry D4/i. 

The normal mode displacements are sketched in Figure 9.4. The notation u, v for the degenerate pair of H symmetry and , //, ( for the T2 triplet is standard. Actually, these projections had already been done in Section 6.4, but this example has been worked in full here to illustrate the projection operator method of finding normal modes. 

In applying the projection operator method for the calculation of symmetrized linear combinations of eigenfunctions, we shall need the effect of a space-group function operator (R ) on the FE eigenfunction mk(r), which is... 

Explicit orthogonality constraints can be removed by transformingthe hamiltonian so that it only acts on a specific subspace (e.g. the valence space) of the all-electron space. This can be done formally by the use of the projection operator method (see Huzinaga and Cantu18 or Kahn, Baybutt, and Truhlar20). If the core function is written as in equation (6) a projection operator may be defined for each valence electron p ... 

To describe the dynamics of an isolated system (for example, how the isolated system relaxes to equilibrium), one often follows the dynamics of the population pm(t) of the system. For this purpose, the projection operator method [1,5-7] is employed. A main function of the projection operator D is to divide p t) into two parts... 

The projection operator method splits the density matrix into diagonal part and off-diagonal part. Discuss why this method is powerful for treating non-equilibrium statistical mechanics ... 

Normalization of SALCCS) derived from the projection operator method must account for the doubled contribution of relative to H, and H<, while maintaining the opposite wave function signs for relative to H, and H,. 

The 2p group orbitals also possess Ag and 5i symmetry. The basis for the projection operator method is again the Ag group orbital, with the same signs of the orbital lobes pointing toward the center. 

Use the projection operator method to derive normalized SALCs that define the group orbitals for CO2 based on the 2py orbitals. 

Apply the projection operator method to derive the group orbital SALCs for H2O given in Section 5.4.3. Confirm using the squares of the coefficients that the group orbital wave function equations are normalized and that each H orbital contributes equally to the two group orbitals. 

Apply the projection operator method to derive the group orbital SALCs for BF3 on the basis of the irreducible representations given in Figure 5.31 for sets of 2i, 2p 2py, and 2p orbitals, respectively. Employ a set of three identical orbitals where all have the same bias (i.e., the group orbitals with A/, A2, and Ai" symmetry in Figure 5.31) as a starting point. For each determination, provide a table like that in Section 5.4.4 to tabulate wave function coefficients, their squares, and how these values simultaneously satisfy the normalization requirement and... 

The projection operator method has applications beyond the deduction of group orbital SALCs. Deduce the wave function equations for the six ir molecular orbitals of benzene, using the labels specified for each 2p orbital. First, derive initial SALCs using each representation of the D /, point group some combinations wiU afford zero. Using sketches of the deduced orbitals, symmetry characteristics of the representations, and a coefficient table like that in Section 5.4.4, deduce the SALCs not derived initially from the character table analysis. Provide normalized equations and a sketch for each ir molecular orbital. 

There are a number of ways of finding these linear combinations (see also E.xample 7.5) and in this example we choose the projection operator method. Since = O X C,-, we need only work out the linear combinations for Bi, Bt, and Ei, in D. This is the standard procedure of reducing the computation by one-half. 

GENERATING SYAAMETRY COORDINATES USING THE PROJECTION OPERATOR METHOD... 

TABLE 9.10 Application of the projection operator method using the bond stretch as the basis. 

Use the projection operator method to determine the symmetries and shapes of all the Pt-CI stretching vibrational modes in the octahedral [PtCI ] " ion. 

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. 

Use the great orthogonality theorem to determine the symmetries of the SALCs. The projection operator method can be used to determine the mathematical forms of the SALCs and their corresponding shapes. 

In order to simplify the formation of SALCs, we use the T subgroup to generate the mathematical functions for the SALCs using the projection operator method. The basis set of hydrogen I s AOs is shown in Figure 10.32. 

By this point in time, you should not even need to follow the formal procedure of the projection operator method to determine the symmetries and shapes of the SALCs. The totally symmetric to all of the symmetry operations. The <7 SALC must be antisymmetric with respect to all of the inversion, S, and C2 operations. The jt SALCs will have a nodal plane containing the intemuclear axis, with symmetric with respect to inversion and antisymmetric to inversion. The shapes of the SALCs can be seen in the one-electron MO diagram, where the energies and shapes of the MOs were calculated using Wavefunction s Spartan Student Edition, version 5.0. 

Using the C5 subgroup to determine the shapes of the SALCs with the projection operator method, the a SALC is totally symmetric, having the mathematical form shown in Equation (10.44) and the shape shown in Figure 10.45. 

The Tpi representation reduces to Oju+ 2g + lg + 2u- Using the subgroup and the projection operator method, the following wavefunctions for the MOs are obtained (after normalization) ... 

In order to determine the shapes of the filled MOs, we must first determine the shapes of the six SALCs for the ligands. Using the O sub-rotational group and the projection operator method, the mathematical forms of the six SALCs are derived as follows. Using the cr-donor orbitals lying along the x, —x, y, —y, z, and —z axes as our basis functions, the 0 SALC will be totally symmetric ... 

Using the projection operator method, the results of each IRR acting on , (/>2, and 03 are shown in the table below. 

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