Transformation properties of atomic orbitals

The symmetry group to which an A(B, C,. . . ) molecule belongs is determined by the arrangement of the pendent atoms. The A atom, being unique, must lie on all planes and axes of symmetry. The orbitals that atom A uses in forming the A—(B, C,. . . ) bonds must therefore be discussed and classified in terms of the set of symmetry operations generated by these axes and planes—that is, in terms of the overall symmetry of the molecule. Thus, our first order of business is to examine the wave functions for AOs and consider their transformation (symmetry) properties under the various operations which constitute the point group of the A(B, C,. . . ) molecule. 

The wave functions for the electron in the hydrogen atom are all products of two functions. First there is the radial function R(n, r), which depends on the principal quantum number n and the coordinate r. Then there is the 

Their product, the complete orbital wave function, is then also normalized to unity. 

Because no symmetry operation can alter the value of R(n, r), we need not consider the radial wave functions any further. Symmetry operations do alter the angular wave functions, however, and so we shall now examine them in more detail. It should be noted that, since A(0, 0) does not depend on n, the angular wave functions for all s, all / , all d, and so on, orbitals of a given type are the same regardless of the principal quantum number of the shell to which they belong. Table 8.1 lists the angular wave functions for sy p, d, and / orbitals. 

In an example worked out at the end of Section 5.1 it was noted in passing that the p orbital with an angular dependence on sin 0 cos 0 was called a px orbital because the function sin 0 cos 0 has the same transformation properties as does the Cartesian coordinate x. At this point we shall discuss the transformation properties and hence the notation for the various orbitals more 

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Transformation properties of atomic orbitals 11-3. Hybrid orbitals for c-bonding systems 11 -i- Hybrid orbitals for >r-bonding systems 11-5. The mathematical form of hybrid orbitals... 

The transformation properties of the orbitals of the carbon atom and the group orbitals of the four hydrogen atoms, with respect to the Td point group, are given in Table 6.2. 

As a result of the fact that the polynomial subscript to an orbital symbol tells us that the orbital transforms in the same way as the subscript, we can immediately determine the transformation properties of any orbital on an atom lying at the center of the coordinate system by looking up its subscript in the appropriate column on the right of a character table, if it is a p or d orbital. An s orbital always transforms according to the totally symmetric... 

For the metal atom, iron, the valence shell orbitals are the five 3d orbitals, the 4s orbital, and the three 4p orbitals. The transformation properties of these orbitals may be ascertained immediately by inspection of the character table for D5J, the results being... 

We now examine the transformation properties of the orbitals on N and on the two O atomic nuclei (see Table 6-5). The energy-level... 

Transition metal clusters also have Axy and atomic orbitals, which are classified as 5-type in TSH theory. To represent the transformation properties of these orbitals, we use second derivatives of the spherical harmonics, that is, tensor spherical harmonics - hence the name of the theory. As for the vector surface harmonics, there are again both odd and even 5 cluster orbitals, denoted by L and L, respectively. Usually, both sets are completely filled in transition metal clusters, and we will not consider their properties in any detail in this review. However, the cases of partial occupation are important and have been described in previous articles. ... 

Tie first consideration is that the total wavefunction and the molecular properties calculated rom it should be the same when a transformed basis set is used. We have already encoun-ered this requirement in our discussion of the transformation of the Roothaan-Hall quations to an orthogonal set. To reiterate suppose a molecular orbital is written as a inear combination of atomic orbitals ... 

Ab initio molecular orbital theory is concerned with predicting the properties of atomic and molecular systems. It is based upon the fundamental laws of quantum mechanics and uses a variety of mathematical transformation and approximation techniques to solve the fundamental equations. This appendix provides an introductory overview of the theory underlying ab initio electronic structure methods. The final section provides a similar overview of the theory underlying Density Functional Theory methods. 

Q Look up the transformation properties of the 2s and 2p orbitals of the nitrogen atom in the character tables of the C3v point group in Appendix 1 to confirm the content of Table 6.1. Carry out the procedure for classifying the Is orbitals of the three hydrogen atoms as group orbitals in the pyramidal molecule. 

Let us now consider a delocalized description of the bonding in water. First we investigate the transformation properties of the various atomic orbitals, because we want to form as general a set of molecular orbitals as possible. 

We shall now find the transformation properties of the 3d, 4s, and 4p orbitals of the central atom in octahedral symmetry. The symmetry operations are given in Figure 8-3. 

Two examples of polyatomic calculations, on H20 and NH3, are outlined and explained in detail. In both cases the analysis starts from an assumed molecular structure of known symmetry. The transformation properties of the atomic orbitals on each atomic centre, under the symmetry operations of the group, are examined next. The atomic orbitals are defined as Is, 2s, 2pxi 2py and 2pz. Nothing can be more explicit - these are the occupied atomic orbitals of a many-electron atom. This configuration violates the exclusion principle9. Although the quantum numbers may not be needed,... 

In the present paper the angular overlap model is elucidated by discussing it in the fight of the transformation properties of the involved atomic orbitals under the three-dimensional rotation group. 

Various methods (described in Chapter 4) can be used to determine the symmetry of atomic orbitals in the point group of a molecule, i. e., to determine the irreducible representation of the molecular point group to which the atomic orbitals belong. There are two possibilities depending on the position of the atoms in the molecule. For a central atom (like O in H20 or N in NH3), the coordinate system can always be chosen in such a way that the central atom lies at the intersection of all symmetry elements of the group. Consequently, each atomic orbital of this central atom will transform as one or another irreducible representation of the symmetry group. These atomic orbitals will have the same symmetry properties as those basis functions in the third and fourth areas of the character table which are indicated in their subscripts. For all other atoms, so-called group orbitals or symmetry-adapted linear combinations (SALCs) must be formed from like orbitals. Several examples below will illustrate how this is done. 

Suppose that G is the group of symmetry operations of a polyhedron or polygon, with vertices corresponding to the atomic positions in a particular molecular structure. The division of the structure into orbits, as sets of vertices equivalent under the actions of the group symmetry operations and the calculation of associated permutation representations/characters were described in Chapter 2. In this chapter, the identity between the permutation representa-tion/character on the labels of the vertices of an orbit and the a representation/character on sets of local s-orbitals or a-oriented local functions is exploited to constmct the characters of the representations that follow from the transformation properties of higher order local functions. 

The theorems described in section 3.2 provide for the ready calculation of the r , and Fs reducible characters, generated by the transformation properties of s, p and d-atomic orbitals distributed over the vertices of the structure orbits of the various point groups, which decompose into the direct sums of irreducible components listed in Tables 3.1 to 3.4. Application of the theorems requires the identification of sufficient numbers of central harmonics to act as basis functions for the irreducible components of the regular orbits of these molecular point groups. 

The actual sign ("phase") of the molecular orbital at any given point r of the 3D space has no direct physical significance in fact, any unitary transformation of the MO s of an LCAO (linear combination of atomic orbitals) wavefunction leads to an equivalent description. Consequently, in order to provide a valid basis for comparisons, additonal constraints and conventions are often used when comparing MO s. The orbitals are often selected according to some extremum condition, for example, by taking the most localized [256-260] or the most delocalized [259,260] orbitals. Localized orbitals are often used for the interpretation of local molecular properties and processes [256-260]. The shapes of contour surfaces of localized orbitals are often correlated with local molecular shape properties. On the other hand, the shapes of the contour surfaces of the most delocalized orbitals may provide information on reactivity and on various decomposition reaction channels of molecules [259,260]. 

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