Direct search for wave operator by A Genetic Algorithm (GA): Route to few eigenvalues of a Hamiltonian Rahul Sharma and S. P. Bhattacharyya Abstract— A Genetic Algorithm is invoked to search out the wave operator leading to the determination of a few eigenvalues and eigenvectors of a specially designed real symmetric matrix (Durand matrix) that simulates a Hamiltonian supporting bound states coupled to continuum. The performance is compared with that of a standard iter- ative method for different partition sizes, and parallelizability of the GA-based approach is tested. In many cases the GA-based approach smoothly converges while the standard iterative schemes diverge. I. I NTRODUCTION The calculation of the energy levels of atoms and molecules are routinely done now-a-days by using large but finite basis sets to represent the Hamiltonian and diagonal- izing it numerically in the N-dimensional space spanned by the basis. Frequently, one requires information about only a few of the lowest energy levels (m < N ). In such situation it has been considered prudent to work with a modified Hamiltonian matrix in the m-dimensional space which generates the m exact eigenvalues and the projection of the m-exact eigenvectors on this lower dimensional space. The eigenvectors in the full space are then recovered by forming the appropriate wave operator and solving the non- linear equation it satisfies. One of the ways of finding the wave-operator is through an energy-dependent partitioning of the matrix eigenvalue problem [1 − 4] while an equivalent energy-independent partitioning method also exists [5 − 7]. The latter leads to a non-linear matrix equation which must be solved to obtain the wave operator and the eigenvalues and eigenvectors. The present communication suggests a direct search method for finding the wave operator by Genetic Algorithms [8 − 9]. Stochastic Diagonalization by Monte Carlo [10 − 11] or by Genetic Algorithms has received some attention in the recent times [12 − 13] although the deterministic methods are still the methods of choice for large scale eigen problems [14 − 17]. However, if the reduced problem has a built-in non-linearity, the stochastic methods could be of use. Since finding the wave-operator involves solution of a non-linear matrix equation, the Genetic or Evolutionary algorithms are expected to be useful for finding the solution matrix. The main objective of this communication is to demonstrate that non-deterministic methods like the GA can explore and exploit the information present in the search space under extreme situations where other methods could fail to find out Rahul Sharma and S. P. Bhattacharyya are with the Department of Physical Chemistry, Indian Association for the Cultivation of Science, Jadavpur, Calcutta 700 032, India. (for correspondence at email address pcspb@mahendra.iacs.res.in). the eigenvalues through the wave operator route and home on to the eigenvalue and the corresponding eigenvector, without any preconditioning of the initial population. The technique used is easily parallelized and scalable to higher dimensions. II. THE METHOD Let H be the real-symmetric Hamiltonian matrix (N × N ) which is transformed into a partitioned matrix-eigenvalue problem in n a and n b dimensional subspaces, respectively, leading to two sets of equations: H aa C a + H ab C b = EC a (1) H ba C a + H bb C b = EC b (2) The energy-dependent effective Hamiltonian (H a eff ) that generates n a exact eigenvalues can be obtained by eliminat- ing C b from the two equations by using [1], [2], [3], [4] C b = {(E. b − H bb ) -1 H ba }C a = WC a (3) where W is the wave operator (here energy-dependent) Using equation(3) in equation(1) we get H a eff C a = {H aa + H ab (E. b − H bb ) -1 H ba }C a = EC a (4) The energy-dependence can be removed by setting W = Z ba , whence H a eff C a = {H aa + H ab Z ba }C a = EC a (5) and finding out the non-linear equation that Z ba satisfies. It is straightforward to show that Z ba satisfies equation (6) H a eff C a =(H aa + H ab Z ba )C a = EC a (6) which in turn demands that Z ba be so chosen that [1], [5], [6] X = H ba + H bb Z ba − Z ba (H aa + H ab Z ba )=0 (7) We propose that equation (7) (non-linear matrix equation) can be solved by GA directly, by allowing it to search out the appropriate Z ba matrix. Once such a Z ba has been found, we may go back to the effective Hamiltonian of equation (5) and diagonalize it to obtain the n a desired eigenvalues and the a-space projections (C a ) of the corresponding eigenvectors in the N -dimensional space. The b-space projections of the eigenvectors are found by using equation (8) C b = Z ba C a (8) 3812 1-4244-1340-0/07/$25.00 c 2007 IEEE