Mulliken symbol

Mulliken symbols The designators, arising from group theory, of the electronic states of an ion in a crystal field. A and B are singly degenerate, E doubly degenerate, T triply degenerate states. Thus a D state of a free ion shows E and Tj states in an octahedral field. 

It should be noticed that lower case Mulliken symbols are used to indicate the irreducible representations of orbitals. The upper case Mulliken symbols are reserved for the description of the symmetry properties of electronic states. 

Area. II. We have previously designated the /th representation, or its set of characters, by the symbol T, in a fairly arbitrary way. Although this practice is still to be found in some places and is common in older literature, most books and papers—in fact, virtually all those by English-speaking authors— now use the kind of symbols found in the C3v table above and all tables in Appendix II. This nomenclature was proposed by R. S. Mulliken, and the symbols are normally called Mulliken symbols. Their meanings are as follows ... 

We can also assign Mulliken symbols to each representation. The first two, being symmetric with respect to C4 are A j and A2 while the next two are Bx and B2. The unique two-dimensional representation is called E. We have now developed the table to the extent shown below. 

An s orbital, because it b spherical, will always be symmetric (i.e, it will remain unchanged) with respect to all operations of a point group. Thus it will always belong to a representation for which all characters are equal to I (a "totally symmetric" representation), although this is not explicitly indicated in character tables. The totally symmetric representation for a point group always appears first in its character table and has an A designation (A i. Aa,Ale> etc.). When these or any other Mulliken symbols are used to label orhitals or other one-electron functions, the convention is to use the lower case a1( a, etc. 

Bethe symbol and Mulliken symbol for the irreducible representation of S4 point group... 

In our calculations, the Mulliken symbols are used for MO, and Bethe symbols are used for many-electron wave functions. The correspondence between Bethe and Mulliken symbols is shown in table 6. 

Area II. In this area, we have the Mulliken symbols for the representations. The meaning of these symbols carry is summarized in Table 6.3.2. 

Consider now the symbols used for the names of the irreducible representations. These are the so-called Mulliken symbols, and their meaning is described below, along with other Mulliken symbols collected in Table 4-6. 

The character tables usually consist of four main areas (sometimes three if the last two are merged), as is seen in Table 4-5 for the C3v and in Table 4-7 for the C2h point group. The first area contains the symbol of the group (in the upper left corner) and the Mulliken symbols referring to the dimensionality of the representations and their relationship to various symmetry operations. The second area contains the classes of symmetry operations (in the upper row) and the characters of the irreducible representations of the group. 

Figure 1.7 The worksheet for the calculation of the permutation character and its direct sum components, listed as Mulliken symbols from orbit lists. This display is accessed from the Characters from Orbits command button of the window shown in Figure 1.5.

Figure 1.8 Calculation of the permutation character generated on the regular orbit of the Iji point group geometry by the actions of the symmetry operation and its decomposition into the direct sum components identified by their Mulliken symbols. Note the appearance of extra option command buttons on the right of the display.

The worksheet displayed by the GT Calculator files by the action of the Reduce a Character command button, Figure 1.5, is shown in Figure 1.12. This worksheet takes as input the character, F, normally a reducible representation, i.e. a set of traces of the matrix representatives of the operators of the group and returns the direct sum components of this character, identified by Mulliken symbols. This input is entered in the red-bordered cells and the direct sum components are returned as numbers of Mulliken symbols in the last row of the display. 

The Direct Sum command button in all the GT Calculator files leads to the worksheet display shown in Figure 1.14. Direct Sums, as reducible characters, are returned on input of the appropriate integer numbers of Mulliken symbols for the particular point group and starting the calculation using the Calculate command button. A Print command button becomes available as one of the actions of the Calculate command button. 

The initial display returned by the Symmetric Powers command button is shown in Figure 1.23. Input to initiate a calculation is in the form of a direct sum characterized by the numbers of irreducible components identified by their Mulliken symbols. 

Table Al.l The icosahedral harmonics, fashioned to be basis functions for all the irreducible representations of the regular orbit cage of Ih point symmetry. The polynomial functions are written in Elert s notation, as described in Chapter 1 and their irreducible properties under Ih and I are identified, in columns 1 and 2 of the table, using Mulliken symbols.

In general discussions label of the irreducible representation is T. To differentiate between the symmetry of electronic states and vibrations, irreps of point groups Ghs and Gls we add subscript and superscript, e.g. etc. In particular examples Mulliken symbols are used, e.g. Ai,... 

The character table of a point group defines the symmetry properties of a (wave) function as either 1 for symmetric or —1 for antisymmetric with respect to each symmetry operation/ The first row lists all the symmetry operations of the point group and the first column lists the Mulliken symbols of all possible irreducible representations, the symmetry transformation properties that are allowed for wavefunctions. As an example, the character table for the D2h point group is given in Table 4.1. The character tables of all relevant point groups are given in many textbooks.134,273-275 The last column shows the transformation properties of the axes x, y and z, which are used to determine electronic dipole and transition moments (Section 4.5). 

In Tables 2 and 4, a fundamental or overtone is denoted by the symbol nV, where b is the Mulliken symbol for the irreducible representation of the mode and a is the number of the mode starting with 1 for the highest frequency, 2 for the second highest, etc. The n is omitted for fundamentals, equals 2 for first overtones, 3 for second overtones, etc. 

To finish this section, we provide a few comments about the Mulliken symbols that are used for irreducible representations, though without full details. One-dimensional representations are indicated by the letters A or B, whereas the letters E and T are used for two- and... 

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