Optical Absorption Spectra Calculated Using Linear-Scaling Density-Functional Theory by Laura Ratcliff

Author:Laura Ratcliff
Language: eng
Format: epub
Publisher: Springer International Publishing, Heidelberg

4.2.1.1 Stationary Perturbations

Starting from a system for which all the eigenvalues and eigenvectors are known, defined by the equation:

(4.1)

some change is made to the Hamiltonian such that . Assuming this change is indeed small, the new Hamiltonian can be expressed in terms of a small parameter , so that . The new eigenvalue equation therefore becomes:

(4.2)

This allows the new energies and eigenstates to be written as the following power series of :

(4.3)

(4.4)

This expansion can then be used to calculate corrections to different orders of , by subsituting the above expressions into Eq. 4.2 and equating powers of . For the non-degenerate case, the first and second order expressions for the eigenvalues are:

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