An efficient solver for large structured eigenvalue problems in relativistic quantum chemistry

ABSTRACT

We report an efficient program for computing the eigenvalues and symmetry-adapted eigenvectors of very large quaternionic (or Hermitian skew-Hamiltonian) matrices, using which structure-preserving diagonalisation of matrices of dimension N > 10, 000 is now routine on a single computer node. Such matrices appear frequently in relativistic quantum chemistry owing to the time-reversal symmetry. The implementation is based on a blocked version of the Paige–Van Loan algorithm, which allows us to use the Level 3 BLAS subroutines for most of the computations. Taking advantage of the symmetry, the program is faster by up to a factor of 2 than state-of-the-art implementations of complex Hermitian diagonalisation; diagonalising a 12, 800 × 12, 800 matrix took 42.8 (9.5) and 85.6 (12.6) minutes with 1 CPU core (16 CPU cores) using our symmetry-adapted solver and Intel Math Kernel Library's ZHEEV that is not structure-preserving, respectively. The source code is publicly available under the FreeBSD licence.

Keywords:

• Relativistic
• diagonalisation
• quaternion

Acknowledgments

This paper is part of the special issue in honour of the 60th birthday of Hans Jørgen Aa. Jensen. The author thanks Jack Poulson for helpful suggestions and Terry Loring for useful comments.

Disclosure statement

No potential conflict of interest was reported by the author.

Additional information

Funding

This work has been supported by the NSF CAREER Award (CHE-1351598). The author is an Alfred P. Sloan Fellow.