Variation method linear

3 APPROXIMATE METHODS OF QUANTUM MECHANICS 1.3.1 Linear variation method 

Here we introduced a column vector c of the expansion coefficients c( then c+ is the row vector of the coefficients c H is a Hamiltonian matrix with elements 

The variational principle states that the mean energy value will be an upper estimate of the lowest eigenvalue E0 of the Hamiltonian hence 

The problem can be formulated as an evaluation of the coefficients ct which minimise the mean energy value (E). The above energy function should be minimised at the boundary condition 

In such a case the method of Lagrangian multipliers is feasible such a Lagrangian multiplier e is chosen so that the functional 

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Variational methods, in particular the linear variational method, are the most widely used approximation techniques in quantum chemistry. To implement such a method one needs to know the Hamiltonian H whose energy levels are sought and one needs to construct a trial wavefunction in which some flexibility exists (e.g., as in the linear variational method where the Cj coefficients can be varied). In Section 6 this tool will be used to develop several of the most commonly used and powerful molecular orbital methods in chemistry. 

The virial ratio is, as we noted above, 1.3366 for the separate-atom AO basis MO calculation, i.e. not 1.0. Now within the confines of the linear variation method (the usual LCAO approach) there is no remaining degree of freedom to use in order to constrain the virial ratio to its formally correct value (or indeed to impose any other constraint). Thus imposing the correct virial ratio on the linear variation method is, in this case, not possible without simultaneously destroying the symmetry of the wave function. Only by optimising the non-linear parameters can we improve the virial ratio as the above results show. Even at this most elementary level, the imposition of various formally correct constraints on the wave function is seem to generate contradictions. 

If the basis set used is finite and incomplete, solution of the secular equation yields approximate, rather than exact, eigenvalues. An example is the linear variation method note that (2.78) and (1.190) have the same form, except that (1.190) uses an incomplete basis set. An important application of the linear variation method is the Hartree-Fock-Roothaan secular equation (1.298) here, basis AOs centered on different nuclei are nonorthogonal. Ab initio and semiempirical SCF methods use matrix-diagonalization procedures to solve the Roothaan equations. 

In terms of the linear variation method the secular equation is obeyed H 2... 

Among the classes of the trial wave functions, those employing the form of the linear combination of the functions taken from some predefined basis set lead to the most powerful technique known as the linear variational method. It is constructed as follows. First a set of M normalized functions dy, each satisfying the boundary conditions of the problem, is selected. The functions dy are called the basis functions of the problem. They must be chosen to be linearly independent. However we do not assume that the set of fdy is complete so that any T can be exactly represented as an expansion over it (in contrast with exact expansion eq. (1.36)) neither is it assumed that the functions of the basis set are orthogonal. A priori they do not have any relation to the Hamiltonian under study - only boundary conditions must be fulfilled. Then the trial wave function (D is taken as a linear combination of the basis functions dyy... 

The most direct way to represent the electronic structure is to refer to the electronic wave function dependent on the coordinates and spin projections of N electrons. To apply the linear variational method in this context one has to introduce the complete set of basis functions k for this problem. The complication is to guarantee the necessary symmetry properties (antisymmetry under transpositions of the sets of coordinates referring to any two electrons). This is done as follows. 

Thus the basis of Slater determinants can be used as a basis in a linear variational method eq. (1.42) when the Hamiltonian dependent or acting on coordinates of N electrons is to be studied. The problem with this theorem is that for most known choices of the basis of spin-orbitals used for constructing the Slater determinants of eq. (1.137) the series in eq. (1.138) is very slow convergent. We shall address this problem later. 

The correspond to different electron configurations. In configuration interaction o is the Hartree-Fock function (or an approximation to it in a truncated basis set) and the other 4>t are constructed from virtual orbitals which are the by-product of the Hartree-Fock calculation. The coefficients Ci are found by the linear-variation method. Unfortunately, the so constructed are usually an inadequate basis for the part of the wavefunction not represented by [Pg.5]

T introduces true n-particle correlation, and products like T T etc., arising out of the expansion eq.(3.2 ), generate simultaneous presence of k-particle amd m-particle correlations in a (k+m)-fold excited determinants etc. The truncation of T == T then corresponds to the pair—correlation model of Sinanoglu, while incorporating higher excited states with several disjoint pair excitations induced through the powers T - The amplitudes for T may be called linked or connected clusters for n electrons. The difficulty of a linear variation method such as Cl lies in its inability to realize the cluster expansion structure eq.(3.2),in a simple and practicable manner. 

Applying the linear variational method, the energies of the MO s are the roots of the 4 X 4 secular equation... 

The optimized coefficients Cai for the MO s a are then found by the linear variational method introduced in Section 11.4. Specifically this leads to the n simultaneous equations for the coefficients Caj for each MO (pa-... 

Historically, the most common technique used has been the linear variation method. In this procedure, the wave functions are expressed as linear combinations of harmonic-oscillator basis functions... 

The above classification of asymmetric potential functions is convenient for comparison of different molecules or as a systematic basis for making an initial fit to experimental data. However, when the Schrodinger equation is being solved by the linear variation method with harmonic-oscillator basis functions, it may not provide the best choice of origin for the basis function. For example, a better choice in the case of an asymmetric double-minimum oscillator, where accurate solutions are required in both wells, would be somewhere between the two wells. Systematic variation of the parameters may still be made as outlined above, but the origin should be translated before the Hamiltonian matrix is set up. The equations given earlier... 

Table 4.1 compares the observed frequencies with those calculated by a least squares adjustment of the potential constants using the linear variation method as incorporated in a computer program written by Ueda and Shimanouchi12). The potential... 

The linear variational method is a useful and accurate tool for solving problems in different fields, such as quantum chemistry, molecular physics and solid state physics, among others [200]. 

The linear variation method and the perturbation method for stationary states belong to the most important approximate methods of quantum mechanics. The latter exists in several variants. 

There is an artifact which may occur when using the linear expansion technique which does not arise in the general functional variation method. If we use a basis set which contains no functions of the same symmetry as the exact ground state, we cannot reach that ground state. This leads, paradoxically, to an extension of the linear variation method. 

The linear variation method introduced in Chapter 1 is the most straightforward way of generating the coefficients Co and CJ since, during the calculation of the matrices hf and we must evaluate all the G and K matrices which are required to evaluate all the elements of the matrix H with elements... 

Again wc must note that in the limit of a large number of points on the grid and a particular choice of the weighting function w(ffc), this equation becomes identical to the linear variation method. 

As we noted much earlier in Chapter 3, making the LCAO expansion independently of the linear variation method is usually invalid in the sense that the operation of an arbitrary differential operator (h, say) on a linear expansion wiU... 

However, there is a stationary variation principle of precisely the type employed in the quantum chemical linear variation method. In the derivation of the Roothaan equations based on finite basis set expansions of Schrodinger wavefimctions, one insists only that the Rayleigh quotient be stationary with respect to the variational parameters, and then assumes that the variational principle guarantees an absolute minimum. In the corresponding linear equations based on the Dirac equation, the stationary condition is imposed, but no further assumption is made about the nature of the stationary point. 

Figure 16 The linear variational method applied to the particle in the slanted box.

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