(Received: September 11, 2006; Accepted for publication: December 14, 2006; Published on Web: April 4, 2007)
An approximate orthogonalization method called "pseudo-orthogonalization" was derived here from the Schmidt orthogonalization in a partial space. Precision of the orthogonalization is decimal 2 digits when the partial space of 1/3 of the Schmidt orthogonalization is used. The speed-up ratio is 3.3 with 1000 vectors of 1500th dimension. Although the pseudo-orthogonalization is high-speed, the precision is inadequate for general numerical calculations. Two possible approaches to improve this precision were considered: (a) introducing a band structure in the vectors, and (b) restricting the sign of the vector elements. These approaches improved the precision by a factor of 2.1 and 1.5, respectively, without increasing the CPU time.
To demonstrate the applicability of this high-speed pseudo-orthogonalization for large-scale calculations, we coded a complex-form of it into the SCF part of the Car-Parrinello method. In calculations of the total energy for bulk-silicon, the difference in total energy calculated using the pseudo-orthogonalization and that using the Schmidt orthogonalization was less than O(-5) [a.u.]. Based on this accuracy, this pseudo-orthogonalization can be used to speed-up the Car-Parrinello method when it is applied to large-scale calculations.
Keywords: Schmidt orthogonalization, Car-Parrinello method, Orthogonal space