Exact mathematical treatment of the modifications of finite‐dimensional quantum systems

Low rank modification (LRM) is a new mathematical formalism by which one can express eigenvalues and eigenstates of the modified system B in terms of the eigenvalues and eigenstates of the original system A . In this respect, LRM is similar to a standard perturbation expansion, which also expresses eigenvalues and eigenstates of the perturbed system B in terms of the eigenvalues and eigenstates of the unperturbed system A . However, unlike perturbation expansion, LRM produces correct results however strong the “perturbation” of the original system A . LRM is here applied to finite n ‐dimensional systems A and B that are described by generalized n × n eigenvalue equations. In the LRM approach, modified system B is described by a ρ × ρ matrix equation, where ρ is the dimension of the space affected by the “modification” of the original system A . In mathematical terms, ρ is the rank of the operators that describe this modification. In many important cases, ρ << n , which results in the substantial numerical efficiency of the LRM approach. The method is illustrated with two examples. In the first example, LRM is applied to the vibrational isotope effect within the harmonic approximation. In particular, out‐of‐plane vibrations of benzene (H,D)‐isotopomers are analyzed in this way. In the second example, LRM is applied to some problems in a solid state physics. LRM provides a new mathematical treatment of all those features that break translational symmetry of a solid, such as point defects, the surface of a solid, etc. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2009