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074103-16 Thøgersen et al. J. Chem. Phys. 123, 074103 2005 TABLE VI. The gradient norm g=4SDF−FDS in the first six iterations of the cadmium complex calculations seen in Fig. 9. DIIS DIIS-TRRH TRSCF Iteration E KS g E KS g E KS g 1 −5597.0 7.8 −5597.0 7.8 −5597.0 7.8 2 −5502.3 14.9 −5598.4 7.2 −5598.3 7.1 3 −5602.1 9.7 −5600.3 8.5 −5603.7 9.3 4 −5628.5 2.1 −5599.9 7.7 −5611.1 9.1 5 −5627.4 3.5 −5599.9 7.8 −5616.8 7.7 6 −5628.8 0.8 −5600.2 8.1 −5622.7 7.5 conv no conv conv In the previous implementation of the TRSCF algorithm, the focus was on the optimization of the Hartree–Fock energy. As the Kohn–Sham energy is nonquadratic in the density matrix, the local DSM energy model has been generalized and is now expanded about the current-density matrix D 0 in the subspace of the density matrices D i of the previous iterations. To satisfy the idempotency condition, the energy model function is parametrized in terms of a purified averaged density matrix. The local energy function is correct to second order in D i −D 0 and can be set up solely in terms of the density matrices and Kohn–Sham matrices of the previous iterations. In the Hartree–Fock theory, the new local energy model is identical to the one previously used in TRSCF optimizations. The EDIIS function is discussed in the context of the proposed model. In the Hartree–Fock theory, the EDIIS function is obtained from our proposed energy function by neglecting terms that result from the purification of the density matrix; the EDIIS function therefore does not reproduce the Hartree–Fock gradient at the expansion point. In the DFT, the EDIIS function is inappropriate for other reasons as well. A rederivation of the original DIIS algorithm is also performed to understand when it can safely be applied. In particular, it is shown that the DIIS method may be viewed as a quasi-Newton method, thus explaining its fast local convergence. In the global region, its behavior is less predictable, although we note that its gradient-norm minimization mechanism usually allows it to recover safely from sudden, large increases in the total energy brought on by the Roothaan– Hall iterations. The TRSCF scheme is tested both in a computationally demanding, robust line-search implementation TRSCF-LS, and in our standard implementation, where only the Fock/ Kohn–Sham matrices of previous iterations are used. Our test calculations indicate not only that the TRSCF-LS method is a highly stable and robust method, but also that the standard TRSCF implementation converges rapidly in most cases, with little degradation relative to the TRSCF-LS scheme. Relative to these schemes, the DIIS method is somewhat more erratic since it makes no use of Hessian information and therefore cannot predict reliably what directions will reduce the total energy. For example, in situations where the energy changes in the course of the iterations but the gradient does not, the DIIS algorithm is unable to identify the density matrix with the lowest energy and may diverge. Nevertheless, the DIIS method handles most optimizations amazingly well, which is particularly impressive in view of its very simplicity; never has so few lines of code done so much good for so many calculations. In general, however, it is outperformed by the TRSCF method, which introduces Hessian information at little extra cost, and is well founded in the global as well as local regions of the optimization. The current formulation of TRSCF requires a few diagonalizations in each TRRH step, and to obtain linear scaling these diagonalizations should be avoided. An even more efficient algorithm may be obtained if the Roothaan–Hall and DSM steps are integrated in such a manner that the information from the previous density matrices are directly used in the Roothaan–Hall optimization step. Work along these lines is in progress. ACKNOWLEDGMENTS We thank Peter Taylor, Ditte Jørgensen, and Stephan Sauer for providing some of the test examples. This work has been supported by the Danish Natural Research Council. We also acknowledge support from the Danish Center for Scientific Computing DCSC. 1 C. C. J. Roothaan, Rev. Mod. Phys. 23, 691951. 2 G. G. Hall, Proc. R. Soc. London A205, 541 1951. 3 P. Pulay, Chem. Phys. Lett. 73, 393 1980; J. Comput. Chem. 3, 556 1982. 4 K. N. Kudin, G. E. Scuseria, and E. Cancès, J. Chem. Phys. 116, 8255 2002. 5 G. Karlström, Chem. Phys. Lett. 67, 348 1979. 6 L. Thøgersen, J. Olsen, D. Yeager, P. Jørgensen, P. Sałek, and T. Helgaker, J. Chem. Phys. 121, 162004. 7 R. Fletcher, Practical Methods of Optimization, 2nd ed. Wiley, New York, 1987. 8 V. R. Saunders and I. H. Hillier, Int. J. Quantum Chem. 7, 6991973. 9 J. B. Francisco, J. M. Martínez, and L. Martínez, J. Chem. Phys. 121, 22 2004. 10 W. Koch and M. C. Holthausen, A Chemist’s Guide to Density Functional Theory Wiley-VCH, Weinheim, 2000. 11 T. Helgaker, P. Jørgensen, and J. Olsen, Molecular Electronic Structure Theory Wiley & Son, ltd., Chichester, 2000. Downloaded 23 Aug 2005 to 130.225.22.254. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp
074103-17 Trust-region self-consistent-field J. Chem. Phys. 123, 074103 2005 12 R. Seeger and J. A. Pople, J. Chem. Phys. 65, 265 1976. 13 G. B. Bacskay, Chem. Phys. 61, 385 1981; J. Phys. France 35, 639 1982. 14 P. Jørgensen, P. Swanstrøm, and D. Yeager, J. Chem. Phys. 78, 347 1983. 15 R. McWeeny, Rev. Mod. Phys. 32, 335 1960. 16 X. P. Li, W. Nunes, and D. Vanderbilt, Phys. Rev. B 47, 10891 1993. 17 J. M. Millam and G. E. Scuseria, J. Chem. Phys. 106, 5569 1997. 18 C. Ochsenfeld and M. Head-Gordon, Chem. Phys. Lett. 270, 3391997. 19 X. Li, J. M. Millam, G. E. Scuseria, M. J. Frisch, and H. B. Schlegel, J. Chem. Phys. 119, 7651 2003. 20 T. Helgaker, H. J. Jensen, P. Jørgensen et al., DALTON, a molecular electronic structure program, Release 2.0, 2004; http://www.kjemi.uio.no/ software/dalton 21 A. Schäfer, H. Horn, and R. Ahlrichs, J. Chem. Phys. 97, 2571 1992. 22 T. H. Dunning, J. Chem. Phys. 90, 10071989. Downloaded 23 Aug 2005 to 130.225.22.254. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp
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PhD Thesis Optimization of Densitie
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Contents Preface ..................
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Summary............................
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List of Publications This thesis in
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Part 1 Improving Self-consistent Fi
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