Complete set of functions

Up to this point we are still dealing with undetermined quantities, energy and wave funetion corrections at each order. The first-order equation is one equation with two unknowns. Since the solutions to the unperturbed Schrddinger equation generates a complete set of functions, the unknown first-order correction to the wave function can be expanded in these functions. This is known as Rayleigh-Schrddinger perturbation theory, and the equation in (4.32) becomes... 

Suppose the members of a complete set of functions tpi are simultaneously eigenfunctions of two hermitian operators A and B with eigenvalues a,- and j3i, respectively... 

Thus, the functions tpi are eigenfunctions of the commutator A, S] with eigenvalues equal to zero. An operator that gives zero when applied to any member of a complete set of functions is itself zero, so that A and B commute. We have just shown that if the operators A and B have a complete set of simultaneous eigenfunctions, then A and B commute. 

The spherical harmonic functions constitute a complete set of functions in the spherical point group. A product of two spherical harmonics such as ytyj must therefore be a linear combination of spherical harmonic functions. An example of such an expression is... 

However, it should be emphasized that when these recursions are applied to a set of s-type Gaussian fimctions centred on some point, they generate a complete set of fimctions of s-symmetry, which should be supplemented by complete sets of functions of higher symmetry centred on the same point and/or complete sets of functions centred on other points. 

Next, each component of the eigenstate /k is expanded in terms of the unperturbed eigenstates (as they can be because the form a complete set of functions) ... 

The problem with these equations is that they correspond to infinite different Hamiltonians so that the solutions for different electronic quantum numbers are incommensurate. To do away with these objections, use instead the complete set of functions rendering the kinetic energy operator Kn diagonal. The set, within normalization factors, is fk(Q) exp(ik Q) k is a vector in nuclear reciprocal space. Including the system in a box of volume V, the reciprocal vectors are discrete, ki, and the functions f (Q) = (1/Vv) exp(iki Q) form an orthonormal set with the completeness relation 8(Q-Q ) = Si fi(Q) fi(Q )- The direct product set ( )j(q)fki(Q) is complete. The matrix elements of eq. (8) reads ... 

The two most common of the methods of weighted residuals are the Galerkin method and collocation. In the Galerkin method, the weighting functions are chosen to be the trial functions, which must be selected as members of a complete set of functions. (A set of functions is complete if any function of a given class can be expanded in terms of the set.) Also according to Finlayson (1972),... 

Because of the spin-spin coupling term, (8.41) is not separable into the sum of Hamiltonians for the individual nuclei, and the corresponding Schrodinger equation is not separable. To deal with the problem, we shall use the method of expanding the unknown wave functions in terms of a known complete set of functions. The eigenvalues and eigenfunctions (eigenvectors) are obtained as the solutions of (2.68) [or (2.77)] and (2.67). 

The knowledge of the complete set of functions plus that of the intcrmoleculuf forces would permit the compulation of all equilibrium properties of the liquid, and indeed if the inlcrmolecular forces are the suni of pair forces, only a knowledge of />. ai all P. 7 values is necessary. An adequate although numerically difficult, theory of the transport properties also exists, using the equilibrium functions. /> . At present, only qualitative success is obtained in the completely a priori use of the equations. 

The complete set of function operators R S forms a group isomorphous with the... 

The complete solution of the eigenvalue problem would require that we solve equation (2.127) for the infinite complete set of functions coupled equations (2.131) in a self-consistent manner with the infinite number of functions V ", This is clearly an impossible task and our procedure is to find approximate solutions to which can then be improved by perturbation theory. 

In the very same way as the Bom-Oppenheimer approximation allows the definition of a potential energy surface for a Van der Waals molecule, it enables, too, the conceit of an interaction tensor field. This is a field dependent on the relative coordinates of the monomers and transforming as a tensor under rotation of the complex as a ole. (The potential energy surface is an example of a rank zero interaction tensor field). In the case of tensor fields it is also convenient to base the theory on irreducible tensors and to use an e7q>ansion in terms of a complete set of functions of the five angular coordinates describing a Van der Waals dimer. 

We now introduce a biorthonormal complete set of functions defined in the q subspace... 

It may be mentioned that in some cases it is convenient to make use of complete sets of functions which are not mutually orthogonal. An arbitrary function can be expanded in terms of the functions of such a set the determination of the values of the coefficients is, however, not so simple as for orthogonal functions. An example of an expansion of this type occurs in Section 24. 

It is interesting to notice that the condition that the dynamical quantity f be represented by a diagonal matrix f in the x representation can be expressed as a differential equation. In order for V to be a diagonal matrix, must equal 0 for m not equal to n and a constant value, / - >, say, for m = n. This means that on expanding /oP.Xi> in terms of the complete set of functions X only the one term fn-n-Xn will occur that is, that... 

Perturbation theory can be formulated in terms of a complete set of functions Wo,..., Wt, which may be eigenfunctions of Ho (although this is not necessary and may in fact yield a poor convergence). If an external perturbation is apphed, for example an electric or magnetic field, a second perturbation operator must be added to Ho and the Hamiltonian becomes... 

A choice of basis set implies a partitioning of the Hamiltonian, H = Hel -I- Hso + Tn(R) + Hrot, into two parts a part, H ° which is fully diagonal in the selected basis set, and a residual part, H(1b The basis sets associated with the various Hund s cases reflect different choices of the parts of H that are included in fP°). Although in principle the eigenvalues of H are unaffected by the choice of basis, as long as this basis set forms a complete set of functions, one basis set is usually more convenient to use or better suited than the others for a particular problem. Convenience is a function of both the nature of the computational method and the relative sizes of electronic, spin-orbit, vibrational, and rotational energies. The angular momentum basis sets, from which Hund s cases (a)-(e) bases derive, are... 

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