Locally Self-Consistent Multiple Scattering (LSMS) Method
The locally self-consistent multiple scattering (LSMS) method is an order-N approach to the calculation of the electronic structure of large systems within the local density approximation (LDA). It is based on the observation that a good approximation to the electron density and the density of states on a particular atom within a large system, and thereby the total energy of that system, can be obtained by considering only the electronic multiple scattering processes in a finite spatial region centered at that atom.
The cluster of M atoms included in the region is referred to as the local interaction zone (LIZ) of the atom (Blue atoms in the first figure). Every atom in the system is considered to be at the center of its own LIZ, and one can use this fact to obtain an approximation to the electron density and DOS associated with every atom in the system. The potential to be used in the next iteration of the self-consistent field (SCF) calculation is then reconstructed by solving Poisson's equation for a crystal electron density made up of a sum of the single site densities.
Given the above algorithm, the electronic structure problem is reduced to that of calculating the single particle Green's function at the central atom of a finite cluster of sites. Clearly, this method is highly scalable on a massively parallel processing (MPP) supercomputer since each compute node can be assigned the calculation of the scattering matrix elements, the electron density, and the density of states for the atoms mapped onto it. A schematic representation of the mapping of the LSMS algorithm onto a MPP platform is displayed in the second figure, where, in the interests of clarity, a one atom per node implementation scheme is assumed. Consider, for example, the atom i of an N-atom system and assume, although this is not necessary, that the LIZ of the atom i is confined to its nearest neighbor shell (atoms j,k,l,m in the illustration). The LSMS algorithm proceeds as follows. An initial guess of the potential and electron density for the atom i, and the positions of all atoms in the system, is loaded onto the node i. The node determines which other nodes are within its LIZ. It then calculates the t-matrix, a single site scattering matrix, corresponding to its own potential and requests and receives the t-matrices of atoms (nodes) j, k, l, and m. In addition it sends its own t-matrix to the nodes for which it is one of the atoms (nodes) in the remote atoms' LIZ. At this point, the node i has sufficient information to calculate the real space scattering path matrix elements and hence the local electron density for the atom it is associated with. Since all nodes execute this procedure, the total electron density is now known and the potential for the next SCF iteration can be constructed. Calculation of the Fermi energy and the Madelung potential apart, this is essentially a local process. Calculating the Madelung potential requires knowledge of the boundary conditions of the large cell, as well as the electric multipole moments on the remote sites which have to be exchanged among nodes. The complete process is then repeated until charge self-consistency is achieved. Since the problem of solving the LDA equations for a N-atom system is decomposed into N linked locally self-consistent sub-problems, the total computing time increases only linearly with N.
Currently the LSMS method is being used to perform large cell simulations of disordered alloys, bulk amorphous metals , and magnetic inhomogeneities in disordered alloys. It has been used to understand the nature of charge correlations and magnetic moment correlations in random alloys. In the former case a new relationship has been discovered between charge transfer and the Madelung contribution to the total energy of random alloys that has clarified an area of recent controversy. Large cell spin-polarized calculations of disordered NiCu alloys have been used to understand the nature of magnetic moment inhomogeneities in these alloys and to provide the first quantitative theory of the results of neutron scattering experiments of the magnetic scattering cross-section performed at the High Flux Isotopes Reactor at ORNL. The LSMS method has also been applied to disordered alloys with non-collinear magnetic structures, in which magnetic moments associated with individual atoms in the alloy are not aligned along the global magnetization direction. This type of magnetic structure can be found in systems like Cr, MnAu2, FeCr multilayers, and Fe-rich NiFe alloys.
The LSMS method has been implemented on many systems and on heterogeneous networks of computing resources using virtual parallelism packages such as PVM (Parallel Virtual Machine). The LSMS code is portable and can be installed on PSC systems by request.