Character vector

One can assign any number, character, vector, matrix or a more general object (see below), or function to another object. 

Place the text end at the location specified by x and y Place legend given by character vector leg at location x and y... 

An extremely useful mode of calculation with characters arises from considering the characters of any representation as a vector. Thus the four character vectors of C2v are ... 

Then if T is the character vector of any irreducihle representation of any group ... 

The mutual orthogonality of the character vectors is reminiscent of the axes of a Cartesian coordinate system, and suggests the valuable idea that the character vectors of a group form a basis for the symmetry. Any vector can be resolved into components of different symmetry types. The projection of any vector onto any symmetry species is calculable. So we have returned to the geometrical point of view ... 

The four numbers on the right-hand sides of these equations, (1,1,1,1), are the character vector for the totally-symmetric representation Ai. Accordingly, we say that Az transforms as Ai in C2v... 

Similar analysis assigns a symmetry species to rotations about the three coordinate axes Rx, Ry and Rz- These may be visualized as directed loops around the axes. A symmetry operation that reverses the sense of the rotation is said to take the rotation into its negative. Thus in C2v, ERx = +Rx, C2RX = —Rx, and ayRx = —Rx and a yRx = +Rx- Remember that the molecular plane is the y-z plane. The character vector is (1 —1 —1 1) ... 

To compute the character vector of the representation generated, apply the same rule that was used to compute the characters of Ftot- Under E, all three... 

The character vector Tvibrationai = (3113) may now be reduced as before, giving Tvibrationai = 2Ai + B2 as expected. 

The distribution of the molecular orbitals can be derived from the patterns of symmetry of the atomic orbitals from which the molecular orbitals are constructed. The orbitals occupied by valence electrons form a basis for a representation of the symmetry group of the molecule. Linear combination of these basis orbitals into molecular orbitals of definite symmetry species is equivalent to reduction of this representation. Therefore analysis of the character vector of the valence-orbital representation reveals the numbers of molecular orbitals... 

It is no accident that the coefficients of the operators E and i in the projection operators are the same as the character vectors in the table of irreducible representations of Q ... 

Problem 11-28. Prove the statement Pn Pf2 = 0 from the ortho-gonahty properties of the character vectors. 

A brief summary of the mathematical notation adopted throughout this text is in order. Scalar quantities, whether constants or variables, are represented by italic characters. Vectors and matrices are represented by boldface characters (individual matrix elements are scalar, however, and thus are represented by italic characters that are indexed by subscript(s) identifying the particular element). Quantum mechanical operators are represented by italic characters if diey have scalar expectation values and boldface characters if their expectation values are vectors or matrices (or if they are typically constructed as matrices for computational purposes). The only deliberate exception to the above rules is that quantities represented by Greek characters typically are made neither italic nor boldface, irrespective of their scalar or vector/matrix nature. 

In general, when a Dirac character multiplies a Dirac character, as in eq. (A2.2.8), it produces a linear combination of Dirac characters (vectors). But for a particular linear combination of Dirac characters... 

Since the GIFT vectorizing algorithm is based on very general principles, vectors identified as character vectors (see classification of components, below) are analyzed, and errors detected in these vectors are corrected to obtain a proper representation of bond structure. This cleanup stage corrects two types of defects ... 

The absolute squares of the characters in a given row (times the weighting factor for each class if the abbreviated form is used) equals the order of the group. (Character vectors are normalized) This results directly from Eq. (13-2). 

The character vectors are orthogonal (again, using weighting factors, if appropriate). 

Furthermore, no matter how F is disguised by a similarity transformation, its character vector is unchanged. Now, suppose you were given the representation F, disguised through some similarity transformation so as to be nonblock diagonal and asked to tell which irreducible representations were present. How could you do it One way would be to find the similarity transformation that would return the representation to block... 

Hence, each representation must be one dimensional, therefore having character +1 or —1 for every operation. One of these must be +1 everywhere ( 1). The others must all have +1 in the first column and —1 in two of the other three columns. The result is as given in Appendix 13 for C2v b) Twelve operations gives order 12. Six classes means six representations. This must mean two 2x2 and four 1x1. There is but one unique set of orthonormal character vectors that fit this framework. Check against the Z)3h character table. 

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