J. Phys. zyxwvutsrq A: Math. Gen. zyxwvutsr 22 (1989) 2427-2439. Printed in the UK Block diagonalisation of Hermitian matrices L S Cederbaum, J Schirmer and H-D Meyer Theoretische Chemie, Institut fur Physikalische Chemie, Universitat Heidelberg, D-6900 Heidelberg, Federal Republic of Germany Received 3 June 1988, in final form 10 March 1989 Abstract. Block diagonalisation of the Hamiltonian by an unitary transformation is an important theoretical tool, e.g., for deriving the effective Hamiltonian of the quasidegenerate perturbation theory or for determining diabatic molecular electronic states. There are infinitely many different unitary transformations which bring a given Hermitian matrix into block diagonal form. It is, therefore, important to investigate under which conditions the transformation becomes unique. The explicit construction of such a transformation and its properties is discussed in detail. An illustrative example is presented. The non- Hermitian case is briefly discussed as well. 1. Introduction Much attention has been paid to the diagonalisation of matrices. Most of the work has been devoted to applications in numerous fields and to the development of efficient numerical procedures zyxwvu [ 1,2]. Recently interest has arisen in the transformation which block diagonalises a matrix. By a block diagonal matrix we mean a matrix which consists of square matrices (blocks) along its diagonal and is zero elsewhere. In the following we briefly discuss a few examples where block diagonalisation of matrices is of interest. The Born-Oppenheimer approximation leads to a basic concept for molecules, liquids and solids. It allows for the introduction of (adiabatic) electronic states and of nuclear vibrations in these states. Although valid in many cases, this approximation may fail in particular if the energies of two or more electronic states are close to each other. The Jahn-Teller [3] and Renner-Teller [4] effects are well known examples for such a failure. In these and other situations it is convenient and extremely useful to introduce so-called diabatic [ 51 electronic states which simplify the treatment and more naturally reflect the physics of the problem. Diabatic states can be obtained from the adiabatic ones by an orthogonal transformation which block diagonalises the Hamiltonian of the system (see zyxwv [6,7] and references therein). Another interesting example is the construction of the effective interaction between ‘particles’. Using a block diagonalisation procedure the interaction of the one-particle states can be replaced by the effective interaction of only those one-particle states which are relevant for the problem under consideration. Such an effective Hamiltonian approach may simplify the problem, give additional insight and, most importantly, lead to effective interaction elements which can be transferable to other related systems z [8]. Furthermore, the method may be used to justify and possibly extend semiempirical approaches [9] which have been successfully applied to molecules and solids. Closely related to the effective Hamiltonian approach is a branch of perturbation theory called quasidegenerate perturbation theory [ 10-131. Here, a matrix representation of the 0305-4470/89/ 132427 + 13%02.50 @ 1989 IOP Publishing Ltd 2427