Symmetry-adapted linear combinations basis functions

Finally, new material and more rigorous methods have been introduced in several places. The major examples are (1) the explicit presentation of projection operators, and (2) an outline of the F and G matrix treatment of molecular vibrations. Although projection operators may seem a trifle forbidding at the outset, their potency and convenience and the nearly universal relevance of the symmetry-adapted linear combinations (SALC s) of basis functions which they generate justify the effort of learning about them. The student who does so frees himself forever from the tyranny and uncertainty of intuitive and seat-of-the-pants approaches. A new chapter which develops and illustrates projection operators has therefore been added, and many changes in the subsequent exposition have necessarily been made. 

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. 

To test your understanding of the MO model for a typical octahedral coordination complex, construct an appropriate, qualitative MO diagram for Oh SHg (a model for known SF6). Hint first calculate the total number of MOs you should end up with from the number of available basis functions (AOs). Second, compare the valence AO functions of S with those of a transition metal (refer to Figure 1.9 and realize that, for a coordinate system with the H atoms on the x, y and z axes, the AO functions of the central atom and the symmetry-adapted linear combinations of ligand functions transform as s, aig p, tiu djey d dy, t2g dx2-y2 dz2, eg in the Oh point group). Now count the number of filled MOs and the number of S-H bonding interactions. 

The usually well-localised nature of the orbitals appearing in VB wavefunction makes spatial symmetry more difficult to use than in the MO case. In MO theory, symmetry can be introduced and utilised at the orbital level Each delocalised MO can be constructed as a symmetry-adapted linear combination (SALQ of basis functions, which is straightforward to implement in program code and can be exploited to achieve substantial computational savings. As a rule, the individual localised orbitals from VB wavefunctions are not S5mimetry-adapted, but transform into one another under the symmetry operations of the molecular point group. The use of symmetry of this type normally requires prior knowledge of the orbital shapes and positions and is very difficult to handle without human intervention. 

Strictly speaking the densities of the electronic cloud on the sites of the atomic nuclei, the so-called on-site density, need not be different for different components of a degenerate wave function. A simple counter-example is a orbital level in a cubic cluster. Let a,) denote a a-type atomic orbital on a given site i. The symmetry adapted linear combinations (SALC s) of these basis orbitals are given by (see Fig. 1) ... 

These same programs can also be used to transform any localized atomic basis into a symmetry-adapted linear-combination-of-atomic-orbitals (LCAO) basis. For methods that begin with an atomic basis set and include multi-center overlap, however, symmetry adaption can actually increase the complexity of the calculation as will be shown. Nevertheless, even with localized basis functions, symmetry can be used to significantly reduce the cost of the calculation if it is carefully applied. How to do so is the subject of the next section. On the other hand, sometimes one chooses not to symmetrize 14c to approximate real physical effects that cannot yet be treated exactly within density functional theory. 

When the EOM excitation basis operators (al,a, al,ala a,...] or EOM ionization basis operators [a, a al,a,...] act on the ground-state wave function (either approximate or exact), the many-electron basis wave functions (9/10> that are formed are linear combinations of many determinants. (Since 0> is taken to include correlation effects, it is a linear combination of many determinants.) This is in contrast to the basis of configurations employed in Cl calculations, which are generally symmetry-adapted linear combinations of only a few determinants. Since in large-scale EOM or Cl calculations, it is necessary to truncate the basis set of operators or configurations employed, respectively, the EOM method often includes the effects of many more excited-state or ionized-state configura-... 

Now to the problem in hand the use of symmetry-adapted linear combinations of the basis functions in the SCF procedure. The method is now obvious, it is simply one more additional step in the chain of transformations from the raw basis to the actual working basis. Deferring the actual technique of calculation of the elements of the transformation between the raw basis and the symmetry-adapted basis until later in this chapter, we assume that the matrix U effects this transformation. The matrix is actually unitary or, more commonly in the real case, orthogonal so that, unlike the monomial/harmonic transformation above, it does not disturb the orthogonality properties of the basis and it is, of course, square. 

Let us assume then that we have a given raw basis and wish to transform it into a working basis which is in the form of an orthogonalised, symmetry-adapted linear combination of cubic harmonic functions. The steps are clear ... 

There is a better way to write out all these orbitals, making use of the translational symmetry. If we have a lattice whose points are labelled by an index n = 0, 1, 2, 3, 4 as shown in 6, and if on each lattice point there is a basis function (a H 1s orbital), xo> Xi> X2 3tc., then the appropriate symmetry adapted linear combinations (remember translation is just as good a symmetry operation as any other one we know) are given in... 

Use the different sets of AOs on the ligands as basis sets to generate reducible representations known as symmetry-adapted linear combinations (SALCs). Any basis function that is unchanged by a given symmetry operation will contribute I to the character, any basis function that transforms into the opposite of itself will contribute — I, and any basis function that is transformed into a basis function on a different li d will be an off-diagonal element and contribute 0 to the character. 

These functions are shown graphically in Figure 6.4. Applying translational symmetry constraints gives rise to what a chemist would refer to as symmetry adapted linear combinations of the basis functions, Xb and Xab. In the language of electronic band structure, these wavefunctions are called Bloch functions. The Bloch functions, %, are periodic waves delocalize throughout the crystal and can be mathematically expressed as follows ... 

Collective functions formed from the basis which belong to a particular irreducible representation are referred to as symmetry adapted linear combinations (SALCs). In essence, each SALC can be thought of as a sum over all the basis functions ... 

The phrase symmetry adapted basis functions refers to those linear combinations of basis functions (on several atoms) that transform like the particular irreducible representation of the appropriate point group. Molecular symmetry is used at various points in these calculations twenty years ago I would have had to write several chapters on molecular symmetry, point groups, constructing symmetry-adapted combinations of basis functions, factoring a Hamiltonian matrix using symmetry and related topics. The point is that twenty... 

Prove that if the irreducible representation r, is one dimensional, then the function g of (9.67) is a symmetry-adapted function that transforms according to r,. Start by writing the basis function / as some linear combination of symmetry-adapted functions g, apply Or to this equation then multiply by x (R) and sum over R. 

If the TV-particle basis were a complete set of JV-electron functions, the use of the variational approach would introduce no error, because the true wave function could be expanded exactly in such a basis. However, such a basis would be of infinite dimension, creating practical difficulties. In practice, therefore, we must work with incomplete IV-particle basis sets. This is one of our major practical approximations. In addition, we have not addressed the question of how to construct the W-particle basis. There are no doubt many physically motivated possibilities, including functions that explicitly involve the interelectronic coordinates. However, any useful choice of function must allow for practical evaluation of the JV-electron integrals of Eq. 1.7 (and Eq. 1.8 if the functions are nonorthogonal). This rules out many of the physically motivated choices that are known, as well as many other possibilities. Almost universally, the iV-particle basis functions are taken as linear combinations of products of one-electron functions — orbitals. Such linear combinations are usually antisymmetrized to account for the permutational symmetry of the wave function, and may be spin- and symmetry-adapted, as discussed elsewhere ... 

By forming suitable linear combinations of basis functions (symmetry adapted functions), many one- and two-electron integrals need not be calculated as they are known lu be exactly zero due to symmetry. Furthermore, the Fock (in an HP calculation)... 

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