Generating Representations from Basis Functions

The reader may wonder how one goes about discovering nontrivial representations like Fs. A convenient way to do this will now be described, and we will show at this stage the connection between quantum mechanics and the group theory of symmetry 

Consider the molecule shown in (III). This molecule has but one nontrivial symmetry element—a point of inversion. According to our flowchart, this places it in the C,- point group. The only symmetry operations here are E and i. Now consider two functions, /i and /2. Let f be located on one end of the molecule. For instance, let /i be Isp, a Is AO centered on the fluorine atom on the left side of the molecule. Let /2 be a similar function on the other side of the molecule, Ispj. Now let us see what happens to these functions when they are acted upon by our symmetry operations E and i  

We see that f and fi are interchanged by inversion. Let us try to find numbers to reprejenf these results. Clearly, replacing by-hi will give the correct result. However, to obtain the effect of operation by i we need to interchange f and fi. We cannot achieve this by multiplying by a number, since f and /2 are linearly independent. If, however, we rewrite the effect of i as 

Now that our 2x2 representation has been diagonalized to two 1x1 blocks, we can rewrite our representations as shown in Table 13-12. 

Our first choice of basis generated a reducible representation. What bases would we need to generate the 1 x 1 representations Fi and F2 To generate Fi, we need a basis function that turns into itself when it is inverted. One possibility is f = Isp + Isp. If we sketch this function (Fig. 13-8a), it is evident that it is regenerated unchanged by inversion. The mathematical demonstration is 

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Section 13-8 Generating Representations from Basis Functions... 

Thus far, we have defined representations and shown how they may be generated from basis functions. We have distinguished between reducible and irreducible representations and have indicated that there is an unlimited number of equivalent representations corresponding to any given two- or higher-dimensional representation. An example of a pair of equivalent, reducible, two-dimensional representations, derived in Section 13-11, is given in Table 13-17. Equivalent representations are related through unitary transformations, which are a special kind of similarity transformation (see Chapter 9), and two matrices that differ only by a similarity transformation have the same... 

Methods are introduced for generating many-electron Sturmian basis sets using the actual external potential experienced by an N-electron system, i.e. the attractive potential of the nuclei. When such basis sets are employed, very few basis functions are needed for an accurate representation of the system the kinetic energy term disappears from the secular equation solution of the secular equation provides automatically an optimal basis set and a solution to the many-electron problem is found directly, including electron correlation, and without the self-consistent field approximation. In the case of molecules, the momentum-space hyperspherical harmonic methods of Fock, Shibuya and Wulfman are shown to be very well suited to the construction of many-electron Sturmian basis functions. 

The factor lj/h, where h is the order of the group and b 18 the dimension of the y th irreducible representation, has been included in (9.67) for convenience. Application of this procedure to the functions / gives us (unnormalized) symmetry-adapted functions g,. This procedure is applicable to generating sets of functions that form bases for irreducible representations from any set of functions that form a basis for a reducible representation. The proof of the procedure (9.67) for one-dimensional representations is outlined in Problem 9.22 we omit its general proof.5 Symmetry-adapted functions produced by (9.67) that belong to the same irreducible representation are not, in general, orthogonal. 

In order to apply the representation theory of so(2,1) to physical problems we need to obtain realizations of the so(2, 1) generators in either coordinate or momentum space. For our purposes the realizations in three-dimensional coordinate space are more suitable so we shall only consider them (for N-dimensional realizations, see Cizek and Paldus, 1977, and references therein). First we shall show how to build realizations in terms of the radial distance and momentum operators, r, pr. These realizations are sufficiently general to express the radial parts of the Hamiltonians we shall consider linearly in the so(2,1) generators. Then we shall obtain the corresponding realizations of the so(2,1) unirreps which are bounded from below. The basis functions of the representation space are simply related to associated Laguerre polynomials. For finding the eigenvalue spectra it is not essential to obtain these explicit realizations of the basis functions, since all matrix elements can... 

There are a number of factors which contribute to the lack of consistency among current DFT programs. For example, many different basis representations of the KS orbitals are employed, including plane waves, Slater-type orbitals, numerically tabulated atomic orbitals, numerical functions generated from muffin-tin potentials, and delta functions. Gaussian basis functions, ubiquitous in the ab initio realm, were introduced into KS calculations in 1974 by Sambe and... 

Illustration of a microchip incorporating CE separation and contactless conductivity detection, (a) Schematic representation of the microchip including a microchip holder, b electrode plate, c microchip, d screws and clip and e Faraday shield, (b) Anti-parallel detection electrode configuration including f detection electrodes, g basis laminate and h electrode extensions used to connect the signal from the function generator and to the detector circuitry [8]... 

Let us see how we may generate a segmented basis set from the primitive functions in Table 8.2. We would like to combine these functions into a smaller set of contracted GTOs that reproduces as accurately as possible the Hartree-Fock orbitals and which at the same time has the flexibility to describe the changes that occur in the valence region upon formation of chemical bonds. Clearly, some compromises must be made since an accurate representation of the Hartree-Fock orbitals requires the use of generally contracted GTOs. 

The orthogonal characteristic polynomials or eigenpolynomials Qn(u) play one of the central roles in spectral analysis since they form a basis due to the completeness relation (163). They can be computed either via the Lanczos recursion (84) or from the power series representation (114). The latter method generates the expansion coefficients q , -r through the recursion (117). Alternatively, these coefficients can be deduced from the Lanczos recursion (97) for the rth derivative Q /r(0) since we have qni r = (l/r )Q r(0) as in Eq. (122). The polynomial set Qn(u) is the basis comprised of scalar functions in the Lanczos vector space C from Eq. (135). In Eq. (135), the definition (142) of the inner product implies that the polynomials Qn(u) and Qm(u) are orthogonal to each other (for n= m) with respect to the complex weight function dk, as per (166). The completeness (163) of the set Q (u) enables expansion of every function f(u) e C in a series in terms of the... 

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