Linear variation method matrix

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. 

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... 

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... 

The linear variation method is the most commonly used method to find approximate molecular wave functions, and matrix algebra gives the most computationally efficient method to solve the equations of the linear variation method. If the functions /i, in the linear variation function [Pg.228]

Thus the energy of this wave function is below the HF energy by the pair correlation energy. To obtain the best possible energy for the above pair-function, we use the linear variation method. Thus we construct the matrix representation of the Hamiltonian in the subspace spanned by q> and all double excitations involving Xa Xb lowest eigenvalue... 

This problem deals with the matrix solution of the linear-variation method when the basis functions are nonorthogonal. (a) If / in = 2"=i Cif is not an orthononnal set, we take linear combinations of the functions /J to get a new set of functions g that are orthonormal. We have 1, 2,. . ., n, where the coefficients are constants and... 

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

The reactance matrix K is ai Q. Exact solutions require the matrix to be of rank n0, implying n linearly independent null-vectors as solutions of the homogeneous equations ma = 0. Because this algebraic condition is not satisfied in general by approximate wave functions, a variational method is needed in order to specify in some sense an optimal approximate solution matrix a. 

We apply the variational method to the total energy in order to obtain linear equations for the vectors c . From a set of trial vectors C/ that satisfy the normalisation constraint (5.41) we compute the density matrix... 

Since the variational theorem proves that the energy of a Cl wavefunction is always an upper bound to the exact energy, one might start simply from the linear expansion (1) and attempt to minimize the energy by varying the Cl coefficients subject to the constraint that they remain normalized. It is easy to show63 that this method of linear variations, or the Ritz method,73 yields the matrix equation... 

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. 

Inserting Eq. (84) in Eq. (82) and then inserting this expression for the density matrix into the variational principle yield the linear-scaling method proposed by Kim et al. [42] ... 

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