Improving the performances of the collocation method for numerically solving linear differential equations of the wavefunctions in large atomic systems SEVER SPANULESCU Department of Physics Hyperion University of Bucharest Calea Calarasilor nr. 169, Bucharest ROMANIA MIRCEA MOLDOVAN Department of Physics UMF Targu Mures Gh. Marinescu, nr.38, Targu Mures ROMANIA T Abstract: - The influence of analytical treatment of terms in collocation method applied to the boundary condition equations of Hartree-Fock type is presented. The explicit form of matrix elements for monomial and Chebyshev trial functions is given and some methods for increasing the speed in these time critical problems are proposed. The numerical tests indicate the possibility of decreasing the amount of computing effort up to 10 times with the proposed technique while maintaining the precision Key-Words: - Self consistent field, differential equation, Collocation, Trial functions, Evanescence 1 Introduction Solving the Hartree-Fockk (HF) equations for many atoms systems is very computing-demanding, as a large number of differential equations (typically several hundred) have to be dealt in an iterative way until the self consistency is achieved. These types of calculations are very important in quantum chemistry, science of materials, molecular biology, etc. That is why a great interest is shown for developing high performance methods for integrating the differential equations of the self consistent field, and new techniques are necessary for various atoms configurations taking into account the limited power of the available computers. It is known that for each electron of each atom, the wave function satisfy the Schrödinger equation with a special potential that have a coulombian part due to the nucleus and an electron interaction part. The classical Hartree-Fock equations (HF) may be written as [1]: 2 ( ') 1 () ' () 2 ( ') ( ') ' () ' i j j i j j i ss j i i j Z d r d δ ∗ Ψ ⎡ ⎤ − ∇+ Ψ + Ψ ⎢ ⎥ − ⎣ ⎦ Ψ Ψ − Ψ − ∑ ∫ ∑ ∫ r r r r r r r r r r r ' () i ε = Ψ r r (1) where is the one electron wave function and the three terms stand for the nuclear interaction, electron repulsion interaction and, due to spin, the exchange interaction. () i Ψ r Another important term is due to the correlation effect of the electrons kinematics and it may be included in various ways in the framework of the Density Functional Theory (DFT) [2]. These approaches reveal some rather sophisticated potentials and exchange-correlation terms. Thus, the exchange term may have several forms, as: exact HF exchange, Slater local exchange functional Becke[3], Perdew-Wang[4,5,6], Vosko-Wilk-Nusair (VWN)[7], Zunger[8], Lee-Yang-Parr[9]. Concerning the numerical methods, the algorithms and the mathematics used in these calculations, there are many popular techniques that may be considered, each of them with their pros and cons[10]. Here the possibilities are also diversified, as the form of the self consistent equations may be purely differential or integro-differential, due to the exchange and correlation terms. Some methods, as the Multi-configurational self-consistent field, are even more computer expensive as they use linear combinations of Slater determinants to approximate the wavefunction [12]. The two electrons integrals are usually calculated by several methods: Lebedev and Gauss- Legendre Angular Quadrature Schemes combined with Linear Scaling Methods [13], The differential form is often treated by the Numerov's fifth order method, which is robust and accurate but is not self starting and require some initial iterations, as many other point by point methods of high order. A notable exception should be the forth order Runge-Kutta method but it is not well suited for boundary conditions equations as the HF ones. Some shooting method must accompany the point by point methods and, although this provides the eigenvalue of the equation (which Proceedings of the 13th WSEAS International Conference on APPLIED MATHEMATICS (MATH'08) ISSN: 1790-2769 148 ISBN: 978-960-474-034-5