Abstract
Due to the high dimensionality of the spaces where the problems are set, adapted discretization basis are often advocated in complex physical problems (Navier–Stokes equations, solid mecanics, ab initio electronic structure computations) to express the solution in terms of solution of similar (but easier to solve) problems. However, very few mathematical studies have been undertaken to asses the numerical properties of these approximations. Within this context, we will present in this paper an overview of the tools required to develop more rigorous reduced basis approaches for quantum chemistry: a posteriori numerical analysis and fast exponential decay of the n-width of the solution set.
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REFERENCES
Kohn, W., and Sham, L. J. (1965). Self-consistent equations including exchange and correlation effects. Phys. Rev. 140, A1133.
Hohenberg, P., and Kohn, W. (1964). Inhomogeneous electron gas. Phys. Rev. 136, B864.
Strain, M. C., Scuseria, G. E., and Frisch, M. J. (1996). Achieving linear scaling for the electronic quantum Coulomb problem. Science 271, 51.
Hehre, W. H., Radom, L., Schleyer, P. v. R., and Pople, J. A. (1986). Ab Initio Molecular Orbital Theory, Wiley, New York.
Parr, R. G., and Yang, W. (1989). Density-Functional Theory of Atoms and Molecules, Oxford Science, Oxford.
Frisch, M. J., Trucks, G. W., Schlegel, H. B., Scuseria, G. E., Robb, M. A., Cheeseman, J. R., Zakrzewski, V. G., Montgomery, Jr., J. A., Stratmann, R. E., Burant, J. C., Dapprich, S., Millam, J. M., Daniels, A. D., Kudin, K. N., Strain, M. C., Farkas, O., Tomasi, J., Barone, V., Cossi, M., Cammi, R., Mennucci, B., Pomelli, C., Adamo, C., Clifford, S., Ochterski, J., Petersson, G. A., Ayala, P. Y., Cui, Q., Morokuma, K., Malick, D. K., Rabuck, A. D., Raghavachari, K., Foresman, J. B., Cioslowski, J., Ortiz, J. V., Baboul, A. G., S tefanov, B. B., Liu, G., Liashenko, A., Piskorz, P., Komaromi, I., Gomperts, R., Martin, R. L., Fox, D. J., Keith, T., Al-Laham, M. A., Peng, C. Y., Nanayakkara, A., Gonzalez, C., Challacombe, M., Gill, P. M. W., Johnson, B., Chen, W., Wong, M. W., Andres, J. L., Gonzalez, C., Head-Gordon, M., Replogle, E. S., and Pople, J. A. (1998). Gaussian 98, Revision A.7, Gaussian, Inc., Pittsburgh PA.
Cancès, E., and Le Bris, C. (2000). On the convergence of SCF algorithms for the Hartree-Fock equations M2AN. Math. Model. Numer. Anal. 34(4), 749–774.
Szabo, A., and Ostlund, N. S. (1996). Modern Quantum Chemistry, Dover, pp. 410–416.
Pinkus, A. (1985). n-Widths in Approximation Theory, Springer-Verlag, Berlin.
Maday, Y., and Turinici, G. (2001). Error bars and quadratically convergent methods for the numerical simulation of the Hartree-Fock equations. Numerische Matematik, in print.
Machiels, L., Maday, Y., Oliveira, I. B., Patera, A. T., and Rovas, D. V. (2000). Output bounds for reduced-basis approximations of symmetric positive definite eigenvalue problems. C. R. Acad. Sci. Paris Sér. I Math. 331(2), 153–158.
Machiels, L., Maday, Y., and Patera, A. T. (2001). Output bounds for reduced-order approximations of elliptic partial differential equations. Comput. Methods Appl. Mech. Engrg. 190, 3413–3426.
Machiels, L., Peraire, J., and Patera, A. T. (2000). A posteriori finite element output bounds for the incompressible Navier–Stokes equations; application to a natural convection problem. J. Comput. Phys., to appear.
Maday, Y., Patera, A. T., and Peraire, J. (1999). A general formulation for a posteriori bounds for output functionals of partial differential equations; application to the eigenvalue problem. C. R. Acad. Sci. Paris Sér. I Math. 328, 828.
Paraschivoiu, M., and Patera, A. T. (1998). A hierarchical duality approach to bounds for the outputs of partial differential equations. Comput. Methods Appl. Mech. Engrg. 158(3/4), 407.
Patera, A. T. Research Group website, Massachusetts Institute of Technology, http:// augustine.mit.edu/
Barrault, M., Cancès, E., LeBris, C., Maday, Y., and Turinici, G. Reduced basis approaches for the Hartree-Fock equations, work in progress.
Truhlar, D. G. (1998). Basis-set extrapolation. Chem. Phys. Lett. 294(1–3), 45–48.
Chuang, Y.-Y., and Truhlar, D. G. (1999). Geometry optimization with an infinite basis set. J. Phys. Chem. A 103(6), 651–652.
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Cancès, E., LeBris, C., Maday, Y. et al. Towards Reduced Basis Approaches in ab initio Electronic Structure Computations. Journal of Scientific Computing 17, 461–469 (2002). https://doi.org/10.1023/A:1015150025426
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DOI: https://doi.org/10.1023/A:1015150025426