Diagonalisation

We next solve the secular equation F — I = 0 to obtain the eigenvalues and eigenvectors o the matrix F. This step is usually performed using matrix diagonalisation, as outlined ii Section 1.10.3. If the Hessian is defined in terms of Cartesian coordinates then six of thes( eigenvalues will be zero as they correspond to translational and rotational motion of th( entire system. The frequency of each normal mode is then calculated from the eigenvalue using the relationship ... 

A key feature of the Car-Parrinello proposal was the use of molecular dynamics a simulated annealing to search for the values of the basis set coefficients that minimise I electronic energy. In this sense, their approach provides an alternative to the traditioi matrix diagonalisation methods. In the Car-Parrinello scheme, equations of motion ... 

This calculation has been made here using the 4s basis set (which includes no polarisation p gaussian orbitals). The energy obtained in this way is very good it reproduces the energy obtained by diagonalisation viz. -0.59088 H cf the Table 1) with an error equal to 0.02 eV. 

Concerning the expansion coefficients, the most significant comparison concerns the values of the two orbitals the one obtained by the fitting process just described and the one obtained by diagonalising the matrix of the hamiltonian in the gaussian basis. In fact we have found that the difference between these two orbitals never exceeds 3% in the internal region as well as in the external region. 

We have checked, using as a test case, that the description of the optimum orbital of the molecular system is then complete in the sense that it allows (assuming that the orbital energy is known) to construct by a fit process an optimum orbital which is very close to the one obtained by a diagonalisation process in a gaussian basis. 

The adiabatic states are obtained by diagonalisation and then submitted... 

Later, the occupied-occupied block will be diagonalised using different effective potentials to get eigenvalues. 

The abelian nature of the group allows one to block-diagonalise the matrix SG into n blocks of dimension n each, independent of the coin C. The spectrum of the quantum graph is then given by the spectra of the sub-matrices... 

The overwhelming number of decompositions of the form (20) will of course have no common symmetry, that is, it is not possible to block-diagonalise the pi s simultaneously. We therefore do not expect any... 

L. Thylen, and C. M. Lee, Beam propagation method based on matrix diagonalisation, Jowrna/ of the Optical Society of America A 9, 142-146 (1992). 

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