J. Chem. Phys., Vol. 121, No. 1, 1 July 2004 Self-consistent field method 27 In Fig. 6b, we have plotted the error of the energy in H 2 O optimizations starting with the orbitals that diagonalize the one-electron Hamiltonian. In this case, convergence to 10 10 is reached in 13 iterations with the TRSCF method and in 18 iterations with the DIIS method. The main reason for the better performance of the TRSCF algorithm is that, in the global region, it gives a significant energy lowering in each step, whereas the DIIS algorithm shows a much less systematic behavior. IV. CONCLUSION A conventional SCF optimization consists of a sequence of iterations, each of which begins with a Roothaan–Hall RH diagonalization step, where a Fock/KS matrix is diagonalized to obtain an improved density matrix, followed by an averaging step, where the optimal density matrix is determined in the subspace of the density matrices of the previous RH diagonalization steps. In this paper, we have introduced a trust-region SCF TRSCF algorithm, where improvements have been made to both the diagonalization and the averaging steps. In both steps, local energy model functions are constructed which have the same gradient as the true energy function E SCF but approximate Hessians. Recognizing the locality of these energy functions, trust regions are introduced as regions where they represent a good approximation to E SCF and only steps inside these trust regions are allowed. For the density-subspace minimization step, an energy function is constructed and minimized with respect to the coefficients of the linear combination of the previous density matrices. Its functional form is based on a purified averaged density matrix that is idempotent to first order. The advantages of this model compared to EDIIS is the built-in density purification, which helps to avoid problems arising from non-idempotency. In addition, information about the Hessian is extracted and used, leading to a monotonic and stable convergence. The RH diagonalization step corresponds to a minimization of an energy function E RH that represents the sum of the orbital energies of the occupied MOs. Since this very simple energy function is a local model function for E SCF , large steps cannot be trusted. To generate steps to the boundary of the trust region, level-shifted RH equations are solved where the level shifts are determined in a systematic and general manner, leading to a decrease in the model energy at each iteration. If sufficiently small steps are taken, a similar decrease is obtained in the SCF energy. In the TRSCF algorithm a few diagonalizations are required in each SCF iteration to obtain solutions for the levelshifted RH equations in order to determine the optimal density matrix. The number of diagonalizations may be reduced in the local SCF region solving RH equations with zero level shift with little consequence for the convergence. In the local SCF region one may also safely use the DIIS algorithm if desired. The advantages of the TRSCF algorithm are demonstrated by calculations on a rhodium complex and on a water molecule with stretched bonds. In the rhodium-complex optimization, the TRSCF algorithm converges monotonically and fast, with a significant decrease in the energy in both the RH part and DSM part at each iteration. By contrast, convergence is not obtained with the DIIS method for this complex. For the simpler water molecule, the TRSCF and DIIS methods behave in a more similar manner, the TRSCF method converging slightly faster than the DIIS method when the initial orbitals are obtained by diagonalizing the one-electron Hamiltonian. With the Hückel guess, the water convergence is essentially obtained in the same number of steps for the TRSCF and DIIS methods. In short, it appears that the TRSCF algorithm, and its use of local energy model functions to obtain significant reductions in E SCF in each iteration, constitutes a significant step towards a black-box optimization of SCF wave functions. ACKNOWLEDGMENTS This work has been supported by the Danish Natural Research Council Grant No. 21-02-0467 and the Carlsbergfondet. We also acknowledge support from the Danish Center for Scientific Computing DCSC. D.Y. acknowledges support from the Robert A. Welch Foundation, Grant No. A-770. 1 C. C. J. Roothaan, Rev. Mod. Phys. 23, 691951. 2 G. G. Hall, Proc. R. Soc. London, Ser. A 205, 5411951. 3 T. Helgaker, P. Jørgensen, and J. Olsen, Molecular Electronic Structure Theory Wiley, Chichester, 2000. 4 P. Pulay, Chem. Phys. Lett. 73, 393 1980; J. Comput. Chem. 3, 556 1982. 5 K. N. Kudin, G. E. Scuseria, and E. Cances, J. Chem. Phys. 116, 8255 2002. 6 R. McWeeny, Rev. Mod. Phys. 32, 335 1960. 7 R. W. Nunes and D. Vanderbilt, Phys. Rev. B 50, 176111994. 8 A. D. Daniels and G. E. Scuseria, Phys. Chem. Chem. Phys. 2, 2173 2000. 9 C. Ochsenfeld and M. Head-Gordon, Chem. Phys. Lett. 270, 3991997. 10 E. A. Hylleraas and B. Undheim, Z. Phys. 65, 759 1930. 11 J. K. L. MacDonald, Phys. Rev. 43, 830 1933. 12 R. Fletcher, Practical Methods of Optimization, 2nd ed. Wiley, New York, 1987. 13 G. B. Bacskay, Chem. Phys. 61, 385 1981. 14 H. J. Aa. Jensen and P. Jørgensen, J. Chem. Phys. 80, 1204 1984. 15 T. H. Dunning, J. Chem. Phys. 90, 1007 1989. 16 A. Schafer, H. Horn, and R. Ahlrichs, J. Chem. Phys. 97, 25711992. 17 DALTON, a molecular electronic structure program, Release 1.2 2001, written by T. Helgaker, H. J. Aa. Jensen, P. Jørgensen et al. http:// www.kjemi.uio.no/software/dalton. Downloaded 25 Jun 2004 to 130.225.22.254. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp
Part 1 The Trust-region Self-consistent Field Method in Kohn-Sham Density-functional Theory, L. Thøgersen, J. Olsen, A. Köhn, P. Jørgensen, P. Sałek, and T. Helgaker, J. Chem. Phys. 123, 074103 (2005)
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