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High-Speed Pseudo-Orthogonalization for the Car-Parrinello Method

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Tomoo AOYAMA^{a}*, Takatoshi HIGUCHI^{b} and Umpei NAGASHIMA^{c}

^{a}Faculty of Engineering, University of Miyazaki

Gakuen-Kibanadai, Miyazaki 889-2192, Japan

^{b}Science and Technology, Mizuho Information & Research Institute, Inc.

2-3 Kanda-Nishikicho, Chiyoda-ku, Tokyo 101-8443, Japan

^{c}Grid Technology Research Center, National Institute for Advanced Industrial Science and Technology

1-1-1 Umezono, Tsukuba, Ibaraki 305-8568, Japan

(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 1500^{th} 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

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