Representations from basis functions

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

Basis functions for the irreducible representations of Dj may be obtained directly from basis functions in 0. A table showing the correspondence between the two groups is given in Table 8 in which the signs that have been chosen constitute one of several phase combinations which are consistent with the transformation properties. 

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... 

The functions put into the determinant do not need to be individual GTO functions, called Gaussian primitives. They can be a weighted sum of basis functions on the same atom or different atoms. Sums of functions on the same atom are often used to make the calculation run faster, as discussed in Chapter 10. Sums of basis functions on different atoms are used to give the orbital a particular symmetry. For example, a water molecule with symmetry will have orbitals that transform as A, A2, B, B2, which are the irreducible representations of the C2t point group. The resulting orbitals that use functions from multiple atoms are called molecular orbitals. This is done to make the calculation run much faster. Any overlap integral over orbitals of different symmetry does not need to be computed because it is zero by symmetry. 

Using GTO bases, it cannot be expected that the variational representations of the electron waves are snfficiently accnrate far ontside the so-called molecular region , i.e. the rather limited region of space where the potential clearly deviates from the asymptotic Conlomb form. Therefore the phaseshifts of the pwc basis states cannot be obtained from the analysis of their long-range behaviour, as was done in previous works with the STOCOS bases. In the present approach, this analysis may be avoided since the K-matrix techniqne allows to determine, by equation [3] below, the phase-shift difference between the eigenfunctions of Hp and the auxiliary basis functions... 

For the analytical representation the signals have to be transformed from time functions into conventional measuring functions. These are characterized by analytical quantities on abscissa and ordinate axes where the values of them may be relativized in some cases (e.g. MS). Such a transformation of quantities is mostly carried out on the basis of instrument-internal adjustment and calibration. 

The method of many-electron Sturmian basis functions is applied to molecnles. The basis potential is chosen to be the attractive Conlomb potential of the nnclei in the molecnle. When such basis functions are used, the kinetic energy term vanishes from the many-electron secular equation, the matrix representation of the nnclear attraction potential is diagonal, the Slater exponents are automatically optimized, convergence is rapid, and a solution to the many-electron Schrodinger eqeuation, including correlation, is obtained directly, without the use ofthe self-consistent field approximation. 

General p-particle A -representability conditions on the 2-RDM are derivable from metric (or overlap) matrices. From the ground-state wavefunction lih) and a set of p-particle operators. of basis functions can be defined. 

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. 

Projection operators are a technique for constructing linear combinations of basis functions that transform according to irreducible representations of a group. Projection operators can be used to form molecular orbitals from a basis set of atomic orbitals, or to form normal modes of vibration from a basis of displacement vectors. With projection operators we can revisit a number of topics considered previously but which can now be treated in a uniform way. 

We end this chapter with an example of the determination of the irreducible representations produced by certain basis functions using the rules and theorems which we have developed. It is at this point that we are ready, at last, to produce results of genuine chemical interest from the sole knowledge of the point group to which a molecule belongs. 

We are now in a position to show that two representations with a one-to-one correspondence in characters for each operation, are necessarily equivalent (see 7-3). If we consider two different nonequivalent irreducible representations then, since the characters are orthogonal (eqn (7-3.4)), there cannot be a one-to-one correspondence. If we consider two different reducible representations T° and Tb then, by eqn (7-4.2), if the characters are the same, the reduction will also be the same, that is the number of times occurs in P (a ) will, by the formula, be the same as the number of times T occurs in Fb. The reduced matrices can therefore be brought to the same form by reordering the basis functions of either Ta or Tb. The reduced matrices are therefore equivalent and necessarily Ta and Tb from whence the reduced matrices came (via a similarity transformation) must also be equivalent. Hence, we have proved our proposition. 

Let us now consider the n-dimensional reducible representation r d which is produced from the function space whose basis functions are Qi> 9i> - Qn> d let us assume that in the reduction of I 1 no irreducible representation of the point group occurs more than once. One way of looking at the reduction is to see it as a change of basis functions from gl9 9i> gn to... 

The functions(8 = 1, 2,... ft) must be functions which belong to the space which produces r since they are simply linear combinations of the basis functions which define that space. If we choose one of the irreducible representations, say T 4, and apply the corresponding projection operator P to gt, we obtain from eqns (7-6.9) and (7-6.10),... 

It is always possible to form a new, and in general reducible, representation r of a given point group from any two given representations T and r of the group. This is done by forming a new function space for which the basis functions are all possible products of the basis functions of T and T Let the basis functions of T and t9 be... 

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. 

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