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J. Chem. Phys., Vol. 121, No. 1, 1 July 2004 Self-consistent field method 21 n D¯ c i D i . i1 37 Ideally, this averaged density should also fulfill the conditions Eqs. 3–5. The symmetry condition Eq. 3 is trivially satisfied since the averaged density Eq. 37 is a linear combination of symmetric density matrices. The trace condition Eq. 4 is also easily taken care of by imposing the restriction n i1 c i 1 38 on the expansion coefficients n Tr D¯S c i Tr D i S N i1 2 . 39 By contrast, the idempotency condition Eq. 5 cannot be imposed on the averaged density matrix. However, the idempotency may be significantly improved if, instead of working with D¯, we work with the purified density matrix 6 D˜ 3D¯SD¯2D¯SD¯SD¯, 40 as proposed by Nunes and Vanderbilt. 7 The electronic energy may be expressed in terms of the purified average density matrix as ED˜ 2 TrhD˜ Tr D˜ GD˜ . 41 FIG. 2. For the ninth iteration of the rhodium calculation described in Sec. III we have displayed as a function of the level-shift parameter ; a the HOMO-LUMO gap ai , where min 0, b the overlap a between the old and new density matrices, where a min is the smallest accepted overlap and c the change in the model energy E RH , the actual energy E RH SCF and the P P predicted energy E SCF . opt is found at the minimum of E SCF (). gives a minimum at 0. Clearly, 0 should be avoided in the calculation since it would lead to an increase in the SCF energy. Instead, the value of the level-shift parameter P that corresponds to the minimum of E SCF denoted by opt ) is chosen for the calculation of the next density matrix. This procedure may be summarized as follows. If min 0 and a(0)a min , then we calculate the predicted energies P P P E SCF (0) and E SCF () with 0. If E SCF (0) P E SCF (), then we use D0. Otherwise, we estimate the P minimum opt of E SCF () by an inexact line search and use the density matrix D( opt ) at this minimum. B. Density-subspace minimization 1. The DSM energy function Let us assume that we have carried out n RH iterations and that we have kept all previous density matrices D i and the corresponding Fock matrices F i . We would now like to construct an optimal density as a linear combination of the densities from these iterations according to Eq. 9, We note that the purified density is correct to first order in the expansion coefficients c i and that E(D˜ ) thus contains errors through second order in c i . To determine the best average density matrix Eq. 37, we shall minimize Eq. 41 with respect to the expansion coefficients c i subject to the condition Eq. 38. One problem we encounter when minimizing Eq. 41 is that new Fock matrices F(D˜ ) need to be evaluated. To avoid this problem, we shall use an approximate form of Eq. 41. Since the purified density matrix D˜ is close to the original density matrix D¯, we can write it as D˜ D¯, 42 where is the correction term. Inserting Eq. 42 into Eq. 41, we obtain E2 TrhD¯Tr D¯GD¯2 Trh 2 TrGD¯Tr G. 43 Since is small, we may ignore the term quadratic in and arrive at the density-subspace minimization DSM energy function E DSM c2 TrhD¯Tr D¯GD¯2 Trh2 TrGD¯ ED¯2 TrFD¯D˜ D¯. 44 Since is first order in the expansion coefficients c i , the DSM energy differs from the true energy to second and Downloaded 25 Jun 2004 to 130.225.22.254. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp
22 J. Chem. Phys., Vol. 121, No. 1, 1 July 2004 Thøgersen et al. higher orders in c i . The first contribution to the DSM energy function may for example be evaluated using the energy expression of the EDIIS algorithm, 5 Lc,,E 0 c T g 1 2 c T Hcc T 1 1 2 c T Mch 2 , 52 ED¯ i c i E SCF D i 1 2 ij c i c j TrF i F j D i D j . 45 Using Eq. 40, we find that the second contribution may be evaluated as where 1 is a column vector with elements equal to 1. Differentiating this Lagrangian and setting the derivatives equal to zero, we obtain the equations L c gHcMc10, 53 2TrFD¯D˜ D¯2 ij c i c j Tr F i D j L cT 10, 54 6 ijk c i c j c k Tr F i D j SD k L 1 2 c T Mch 2 0. 55 4 ijkl c i c j c k c l Tr F i D j SD k SD l . 46 All contributions to the DSM energy function are therefore easily calculated from the previous density and Fock/KS matrices. 2. The trust-region DSM minimization We minimize the DSM energy functional by the trustregion method. 12 We thus consider the second-order Taylor expansion of the DSM energy in Eq. 44 about c 0 . Introducing the step vector ccc 0 , we obtain 47 E DSM (2) cE 0 c T g 1 2 c T Hc, 48 where the energy, gradient, and Hessian at the expansion point are given by E 0 Ec 0 , g Ec c cc 0 , H 2 Ec c 2 cc 0 . 49 As starting point c 0 , we choose the density matrix with the lowest energy E SCF (D i ), usually from the last RH iteration. The trace condition Eq. 38 imply n i1 c i 0. 50 We also introduce a trust region of radius h for E DSM (2) (c) and require that steps are always taken inside or to the boundary of this region. To determine a step to the boundary, we restrict the step to have the length h in the S metric norm of Eq. 34, c S 2 ij c i M ij c j h 2 . 51 Introducing the undetermined multipliers and for the trace and step-size constraints, we arrive at the following Lagrangian for minimization on the boundary of the trust region: The optimization of the Lagrangian thus corresponds to the solution of the following set of linear equations: HM 1 T 1 0 c g 0 , 56 where the multiplier is iteratively adjusted until the step is to the boundary of the trust region Eq. 55. The step-length restriction may be lifted by setting 0, as needed for steps inside the trust region. To understand the behavior of the step-length function, we consider first the generalized eigenvalue problem H 1 v 1 T 0 M 0 0 T v , 57 where 0 is a column vector with zero elements, is a small positive constant, and the eigenvector is normalized such that v T v 2 1. 58 We first note that, for a finite , v0. Next, carrying out block multiplications in Eq. 57, we obtain Hv1Mv, 1 T v, 59 60 which upon elimination of from the first equation yields the relation Hv1 T v1 2 Mv. 61 Since (1 T v)1 is finite, we conclude that, as tends to zero, the eigenvalue tends to either plus or minus infinity 1/2 . Next, substituting these values of into Eq. 60, we find that v tends to the zero vector with elements proportional to 1/2 and that , because of the normalization Eq. 58, tends to 1. In short, the eigenvalue problem Eq. 57 with 0 has two eigenvalues , whose eigenvectors have zero elements except for the last element, which is equal to 1. Finally, invoking the Hylleraas–Undheim interlace theorem, 10,11 we conclude that the remaining n1 finite eigenvalues of Eq. 57 bisects the n eigenvalues of the reduced eigenvalue problem HvMv. 62 Downloaded 25 Jun 2004 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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Page 70 and 71: Part 1 Improving Self-consistent Fi
Page 72 and 73: Part 2 Atomic Orbital Based Respons
Page 90 and 91: Part 3 Benchmarking for Radicals ar
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Page 94 and 95: Part 3 Benchmarking for Radicals As
Page 96 and 97: Part 3 Benchmarking for Radicals R
Page 99: Summary The developments in compute
Page 103: Appendix A The Derivatives of the D
Page 106 and 107: Thus, the density element D µν is
Page 109 and 110: References 1 2 3 4 5 6 7 8 9 C. C.
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Page 147: Part 3 A Coupled Cluster and Full C
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