A new Lanczos method for electronic structure calculations

Abstract

In the heart of most electronic structure simulation programs, there is a routine to find the solution of eigenvalue problems. Solving these eigenvalue problems usually dominates the computer time used for the whole simulation[6]. Because of their physical properties, these eigenvalue problems are always symmetric real or Hermitian. The dimensions of the matrices involved are usually very large and a large number of eigenvalues and their corresponding eigenvectors are needed to compute the desired physical quantities. To solve this type of problems, we introduce a variant of the Lanczos method called the thick-restart Lanczos method. In material science, this method is most appropriate for non-selfconsistent cases where the eigenvalue problems are linear and the number of required eigenvalues is relatively small compared to the size of the matrix.The Lanczos method is very simple and yet effective in finding eigenvalues. It is also well suited for parallel computing. There are two common ways of implementing the Lanczos method depending on whether the Lanczos vectors are stored. When the Lanczos vectors are not stored, they may lose orthogonality and the Lanczos method may generate spurious eigenvalues [2, 10]. Though spurious eigenvalues can be effectively identified, we still prefer not to deal with the spurious eigenvalues. When the Lanczos vectors are stored, the loss of orthogonality problem can be corrected by re-orthogonalization [4, 5, 7]. No spurious eigenvalue is generated in this case. However, because each Lanczos step generates one vector, a large amount of computer memory may be required to store all the Lanczos vectors. To limit the maximum amount of memory used, we typically restart the Lanczos algorithm after a certain number of steps. The restarted versions usually use considerably more matrix-vector multiplications than the non-restarted version. In recent years, newly developed restarting strategies have significantly reduced the number of matrix-vector multiplications used. The two most successful ones are the implicit restarting technique [1, 3, 8] and the dynamic thick-restart technique [9, 12]. For symmetric or Hermitian eigenvalue problems, these two schemes are equivalent. Because the thick-restart scheme is easier to implement and it is slightly more flexible than the implicit restarted scheme [9, 12], the new method described here uses the thick-restart scheme. Other thick-restart eigenvalue methods, e.g., the thick-restart Davidson method, can be applied on symmetric eigenvalue problems as well. Compared to them, the main advantage of the new scheme is that it uses less arithmetic operations by taking full advantage of the symmetry of the matrix [13].