Representation orbital characters

Clearly, for a one-dimensional representation the character and the full matrix are the same thing. Hence, the incomplete projection operator is complete in these cases, and will provide the appropriate SALC unambiguously and automatically. Let us illustrate by asking what SALCs can be formed by the Is orbitals of the four hydrogen atoms in ethylene. 

This means that the four MOs that will be equivalent to the set of four a orbitals must be chosen so as to include one orbital of Ax symmetry and a set of three orbitals belonging to the T2 representation. The character table also tells us that AOs of atom A falling into these categories are as follows ... 

Atomic basis functions on B are straightforward to classify. Evidently, an s type function on B will be totally symmetric — an a orbital. A quick inspection of the D3h character table will show that a p set on B, which transforms like the three Cartesian directions, spans the reducible representation a 2 e. Functions centred on the F atoms require more effort. Since the operations in the classes containing C3 and S3 move all three F atoms, their character is necessarily zero for any functions centred on the F atoms. Consider first a set of s functions on each F. These span a reducible representation with character... 

The numbers in the table, the characters, detail the effect of the symmetry operation at the top of the colurrm on each representation labelled at the front of the row. The mirror plane that contains the H2O molecule, a (xz), leaves an orbital of bi symmetry unchanged while a Ci operation on the same basis changes the sign of the wavefimction (orbital representations are always written in the lower case). An orbital is said to span an irreducible representation when its response upon operation by each symmetry element reproduces the same characters in the row for that irreducible representation. For atoms that fall on the central point of the point group, the character table lists the atomic orbital subscripts (e.g. x, y, z as p , Pj, p ) at the end of the row of the irreducible representation that the orbital spans. A central s orbital always spans the totally synunetric representation (aU characters = 1). For the central oxygen atom in H2O, the 2s orbital spans ai and the 2px, 2py, and 2p span the bi, b2, and ai representations, respectively (see (25)). If two or more atoms are synunetry equivalent such as the H atoms in H2O, the orbitals must be combined to form symmetry adapted hnear combinations (SALCs) before mixing with fimctions from other atoms. A handy mathematical tool, the projection operator, derives the functions that form the SALCs for the hydrogen atoms. 

In what follows, the symmetry of atomic or molecular orbitals will be represented with lower-case letters (ai, ti , Og, tig, etc.), while uppercase letters will be reserved for the different irreducible representations in character tables, for the symmetry of electronic states and for spectroscopic terms. 

We now turn to the representation. The character is still zero for the C3 and S3 axes, and still —1 for a Cz-axis (6-36). Reflection in the ah plane changes each orbital into its opposite (6-37, / = —3), whereas a ay plane maintains one orbital and interchanges the two others (6-38,... 

The hydrogen atoms determine the symmetry of the molecnle, and their li orbitals form the basis of a redncible representation. The characters for each operation for the hydrogen atom Is orbitals are readily obtained, F. The sum of the contributions to the character (1, 0, or -1, as described previonsly) for each symmetry operation is the character for that operation in the representation. The complete list for aU operations of the group is the reducible representation for the atomic orbitals ... 

By inspection, Fjo = 0 - This is a general result s orbitals located on atoms, through which all the symmetry elements of the point group pass, always transform as the totally symmetric representation. The characters of the representation given by the oxygen 2p orbital (F o) are generated in the same way (4.6). Here, of course, a contribution of — I is found whenever a given sends one of these functions to minus itself. 

By means of group theory, we can find directly the molecular orbitals which form bases for irreducible representations of the symmetry group of the benzene molecule. The procedure is identical with that followed in finding the proper linear combinations of bond eigenfimctions. In place of the set of five bond eigenfunctions we use the set of six atomic orbitals as the basis for the reducible representation. The character of this representation is... 

As a first example, we examine the occupied surface state found close to Bf at the Y point of the Cu(llO) and Ag(llO) surfaces [37]. The surface state occurs in the L21 — Li band gap (see Figure 5.16 for the position of the surface-state band in the projected bulk band gap at Y and Figure 5.17 for the location of the Y point with respect to the bulk Brillouin zone). Energetically, the surface state lies very close to the L2/ point of the bulk band structure. Thus, we can consider it as a state split from the L2/ bulk band. The orbital composition of the L2/ band can be found in tabulations, for example, in Ref. [38], and is given as 1/V (x + y + z). This function is a representation of a p orbital oriented in the [111] direction. Note that the coordinates refer to the orientation of the bulk Brillouin zone as shown in Figure 5.17. We can transform these coordinates into surface coordinates for the (110) surface Zs = l/v (x + y) is oriented in the [110] direction, Xj = l/-s/2 (—x + y) in the [110] direction, and yj = z in the [001] direction. This yields the orbital character of the L2/ band in terms of the surface coordinates 1/V3 (y + V2zs). 

To conclude, three remarks may be added first, the L3 — L2/ band gap projecting onto the S point is expected to support a Bi and an Ai surface state. The former has orbital representation Zsys and is identical to the one derived from the above consideration of the Wp point The Ai state has the orbital representation Zg — x. Second, the present discussion neglects the spin-orbit interaction. The latter wiU scramble the orbital representations to some extent Third, if one tries to explore not just the orbital character of a surface state with respect to the topmost surface atoms, but rather to reconstmct the orbital composition of the complete surface-state Bloch function (including the contributions from the deeper layers) one has to take into account also the translational symmetry. This is formally more complicated, as indicated in Ref [36], but again, once the appropriate operations have been carried out, the result is intuitively clear. In case of degenerate irreducible representations, transfer projection operators have to be applied, but this is a straightforward generahzation of the method outlined above. 

These six matrices form another representation of the group. In this basis, each character is equal to unity. The representation formed by allowing the six symmetry operations to act on the Is N-atom orbital is clearly not the same as that formed when the same six operations acted on the (8]s[,S 1,82,83) basis. We now need to learn how to further analyze the information content of a specific representation of the group formed when the symmetry operations act on any specific set of objects. 

A planar molecule of point group 03b is shown in Fig. 5. The sigma orbitals i, <72 and (73 represented there will be taken as the basis set Application of the method developed in Section 8.9 yields the characters of the reducible representation given in Table 14. With the use of the magic formula (Eq. (37)] the structure of the reduced representation is of the form Ta — A, ... 

Consider the trans isomer of butadiene. Both the symmetry operations that define the group < 2h and the characters of the representation r are given in Table 3. The reduction of this representation leads to Tn =2Bg 2Aa. Thus, two linear combinations of the atomic orbitals can be constructed of symmetry Bg and two others of symmetry A. Their use will factor the secular determinant into two 2x2 blocks, as described in the following paragraph. 

The possible wave functions for the molecular orbitals for molecules are those constructed from the irreducible representations of the groups giving the symmetry of the molecule. These are readily found in the character table for the appropriate point group. For water, which has the point group C2 , the character table (see Table 5.4) shows that only A1 A2, B1 and B2 representations occur for a molecule having C2 symmetry. 

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