Linear variation method formulation

As discussed in Chapter 1, an equivalent formulation of the linear variation method is simply to write... 

Chapter 9 Matrix Formulation of the Linear Variation Method... 

Since linear variation of hardness is not always the case, equation (5.7) is approximate. But Glazov and Vigdorovich consider that the production of many very complex solid solution systems does require some method if only rough, for hardness analysis of such systems. They formulate the additivity principle for multicomponent systems as follows the numerical increase in hardness of multi-component solid solutions equals the sum of hardness increments in bi-component solutions... 

That is, the method of linear variations is identical to the matrix formulation of the Schrodinger equation. Another way of viewing this result is that only solutions to eq. (12) are energetically stable with respect to variations in the linear expansion coefficients. 

As we shall see, the most common use of the variation method is not to find a set of linear parameters in the determinantal expansion of the wavefunction but to model the electronic structure and optimise the parameters contained in the mathematical formulation of that model. 

The success of our earlier sample variational calculation conceals a haphazard approach. In order for the variational method to be useful to a broad range of chemical systems, we need a systematic way of constructing and optimizing the guess wavefunctions. We attain this by choosing a basis set of the one-electron orbitals formulated in Tables 3.1 and 3.2, and then adding these together to form a linear combination of atomic orbitals (LCAO) ... 

In Chapter 7 we developed a method for performing linear variational calculations. The method requires solving a determinantal equation for its roots, and then solving a set of simultaneous homogeneous equations for coefficients. This procedure is not the most efficient for programmed solution by computer. In this chapter we describe the matrix formulation for the linear variation procedure. Not only is this the basis for many quantum-chemical computer programs, but it also provides a convenient framework for formulating the various quantum-chemical methods we shall encounter in future chapters. 

For formulations A and B, one general procedure is to solve the laminar flow equations which are linear and use the solution as the initial guesses for the nonlinear equations. Variations of this procedure have been used by Bending and Hutchison (B5), Wood and Charles (Wll), and Jeppson and Tavallaee (J2) in conjunction with the linearization method. 

A dilution factor may be incorporated into this calculation if the sample is first extracted and then diluted in order to bring it into the working range of the instrument. This approach to quantitation does not address the linearity of the method but since the variation in the composition of formulations should be within 10% of the stated amount there is some justification for using it. The precision of the method is readily addressed by carrying out repeat preparations of sample and calibration solutions. 

Due to their higher flexibility and accuracy, Finite Elements Methods (FEMs) [5] are often preferred to numerical methods their basic concept consists first of all in establishing a weak variational formulation of the mathematical problem the second step is to introduce the concept of shape functions that are defined into small sub-regions of the domain (see also Chapter 3). Finally, the variational equations are discretised and form a linear system where the unknowns are the coefficients in the linear combination. 

It should be noted that ii t) in formulation (32) can be replaced by time-invariant parameters 6 by parameterizing the variation with time variation explicitly, e.g., by using a piecewise linear function of time. This allows standard finite-dimensional optimization methods to be applied and is the approach adopted in our work. 

---