Symmetry matrix representation

Here, Xr(R) is the eharaeter belonging to symmetry E for the symmetry operation R. Applying this projeetor to a determinental flinetion of the form ( )i( )j generates a sum of determinants with eoeffieients determined by the matrix representations Ri ... 

Within a variational treatment, the relative eontributions of the spin-and spaee-symmetry adapted CSFs are determined by solving a seeular problem for the eigenvalues (Ei) and eigenveetors (C ) of the matrix representation H of the full many-eleetron Hamiltonian H within this CSF basis ... 

Symmetry tools are used to eombine these M objeets into M new objeets eaeh of whieh belongs to a speeifie symmetry of the point group. Beeause the hamiltonian (eleetronie in the m.o. ease and vibration/rotation in the latter ease) eommutes with the symmetry operations of the point group, the matrix representation of H within the symmetry adapted basis will be "bloek diagonal". That is, objeets of different symmetry will not interaet only interaetions among those of the same symmetry need be eonsidered. 

We ean likewise write matrix representations for eaeh of the symmetry operations of the C3v point group ... 

The Spin adapted Reduced Hamiltonian SRH) is the contraetion to a p-electron space of the matrix representation of the Hamiltonian Operator, 2 , in the N-electron space for a given Spin Symmetry [17,18,25,28], The basis for the matrix representation are the eigenfunctions of the operator. The block matrix which is contracted is that which corresponds to the spin symmetry selected In this way, the spin adaptation of the contracted matrix is insnred. 

For any symmetry operator T = T 0) (rewritten r when operating on the domain of basis functions x)) for instance, the rotation-reflexion about the z-axis, with matrix representation... 

It should be noted that the trace of a matrix that represents a given geo] operation is equal to 2 cos y 1, the choice of signs is appropriate to or improper operations. Furthermore, it should be noted that the aim direction of rotation has no effect on the value of the trace, as a inverse sense corresponds only to a change in sign of the element sin y. TE se operations and their matrix representations will be employed in the following chapter, where the theory of groups is applied to the analysis of molecular symmetry. 

It is reasonable to hope to assemble a complete set of representations to provide a full and non-redundant description of the symmetry species compatible with a point group The problem is that there are far too many representations of any group. On the one hand, matrices in representations derived from expressing symmetry operations in terms of coordinates - as in problem 5-18 - depend on the coordinate system. Thus there are an infinite number of matrix representations of C2v equivalent to example 7, derivable in different coordinate systems. These add no new information, but it is not necessarily easy to recognize that they are related. Even in the cases of representations not derived from geometric models via coordinate systems, an infinite number of other representations are derivable by similarity transformations. 

It is also possible to construct matrix representations by considering the effeot that the symmetry operations of a point group have on one or more sets of base veotors. We will consider two cases, both using the point group as an example (1) the set of base veotors eu e, and e introduced in 5-2 (2) three sets of mutually perpendioular base vectors, each located at the foot of a symmetric tripod. 

Inspection of this result shows that it is not equivalent to the partner function Eq. 3.15, and that although it and the result of Eq. 3.17 are linearly independent they are not orthogonad. Explicit Schmidt orthogonadization of the result of Eq. 3.18 to that of Eq. 3.17 yields the same functions as those obtained from the full matrix projection and shift operators. However, without knowledge of the full matrix representations we cannot identify these character projection results with specific rows of the e irrep. In fact, in generad the results of character projection will not yield basis functions that can be identified with symmetry species. 

As was discussed in Chapter 2, the need to have full matrix representations available to obtain basis functions adapted to symmetry species is something of a handicap. Although character projection itself is not adequate for this task, Hurley has shown how the use of a sequence of character projectors for a chain of subgroups of the full point group can generate fully symmetry-adapted functions. Further discussion of this approach is beyond the scope of the present course, but interested readers may care to refer to the originad literature [6]. 

In 1989, however, a practical resolution of this problem was derived independently by Haser and by Almlof [16]. They obtain the necessary matrix representation information by utilizing the reducible representation matrices obtained from symmetry transformations on the AO basis as in Eq. 5.1. Full details of their procedure is beyond the scope of this course, but, as would be expected, it has many similarities to the non-totally symmetric operators discussed in the previous section. 

Example 4.1-1 Find a matrix representation of the symmetry group C3v which consists of the symmetry operators associated with a regular triangular-based pyramid (see Section 2.2). 

From Table 2.3, we see that crecrt = Cf, so that multiplication of the matrix representations does indeed give the same result as binary combination of the group elements (symmetry operators) in this example. 

Explain why the point group D2 = E C2z C2x C2y is an Abelian group. How many IRs are there in D2 Find the matrix representation based on (e e2 e31 for each of the four symmetry operators R e D2. The Jones symbols for R 1 were determined in Problem 3.8. Use this information to write down the characters of the IRs and their bases from the set of functions z xy. Because there are three equivalent C2 axes, the IRs are designated A, B1 B2, B3. Assign the bases Rx, Ry, Rz to these IRs. Using the result given in Problem 4.1 for the characters of a DP representation, find the IRs based on the quadratic functions x2, y2, z2, xy, yz, zx. 

It will be more economical in the first two sections to label the coordinates of a point P by xi x2 x3. Symmetry operations transform points in space so that under a proper or improper rotation A, P(xi x2 x3) is transformed into P (xi x x3 ). The matrix representation of this... 

Write down the matrix representation of eq. (16.2.1). Hence find the coordinates (x y 7 ) of a general point (x y z) after the following symmetry operations ... 

Second, we need to define a matrix representation between the atomic term functions for each symmetry operation, that depends upon three Euler angles [73]... 

The familiar set of the three t2g orbitals in an octahedral complex constitutes a three-dimensional shell. Classical ligand field theory has drawn attention to the fact that the matrix representation of the angular momentum operator t in a p-orbital basis is equal to the matrix of — if in the basis of the three d-orbitals with t2g symmetry [2,3]. This correspondence implies that, under a d-only assumption, l2 g electrons can be treated as pseudo-p electrons, yielding an interesting isomorphism between (t2g)" states and atomic (p)" multiplets. We will discuss this relationship later on in more detail. 

The correct form of the inner product is often just mentioned in passing when complex rotation is discussed, and then usually only for a rotation of an originally real matrix representation of the Hamiltonian. A clearer understanding can be obtained by going back to matrix algebra where the form of the inner product is a direct consequence of the symmetry of the matrix. 

The expressions eqs. (1.197), (1.199), (1.200), (1.201) are completely general. From them it is clear that the reduced density matrices are much more economical tools for representing the electronic structure than the wave functions. The two-electron density (more demanding quantity of the two) depends only on two pairs of electronic variables (either continuous or discrete) instead of N electronic variables required by the wave function representation. The one-electron density is even simpler since it depends only on one pair of such coordinates. That means that in the density matrix representation only about (2M)4 numbers are necessary to describe the system (in fact - less due to antisymmetry), whereas the description in terms of the wave function requires, as we know n 2m-n) numbers (FCI expansion amplitudes). However, the density matrices are rarely used directly in quantum chemistry procedures. The reason is the serious problem which appears when one is trying to construct the adequate representation for the left hand sides of the above definitions without addressing any wave functions in the right hand sides. This is known as the (V-representability problem, unsolved until now [51] for the two-electron density matrices. The second is that the symmetry conditions for the electronic states are much easier formulated and controlled in terms of the wave functions (Density matrices are the entities of the second power with respect to the wave functions so their symmetries are described by the second tensor powers of those of the wave functions). 

Symmetry considerations are instrumental in a qualitative discussion of spin-orbjt effects. Qualitatively, a phenomenological Hamiltonian of the form Aso Z matrix representation of the usual molecular point group. The same is true for the spatial and spin wave... 

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