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074103-12 Thøgersen et al. J. Chem. Phys. 123, 074103 2005 and accept a favorable configuration shift. A configuration shift may be recognized when an a orb min profile has an abrupt change where on the right-hand side a orb min is close to 1 and on the left-hand side a orb min is close to 0. To maintain the high degree of control characteristic of the TRSCF method, the energy of the new configuration is checked before the shift is accepted, at the cost of an additional Kohn–Sham matrix build. As seen from Fig. 4a, this check is well worth the effort, saving more than ten iterations, and thus it is made an integrated part of our TRSCF implementation. VI. THE DIIS METHOD VIEWED AS A QUASI-NEWTON METHOD Since its introduction by Pulay in 1980, the DIIS method has been extensively and successfully used to accelerate the convergence of SCF optimizations. We here present a rederivation of the DIIS method to demonstrate that, in the iterative subspace of density matrices, it is equivalent to a quasi- Newton method. From this observation, we conclude that, in the local region of the SCF optimization, the DIIS steps can be used safely and will lead to fast convergence. The convergence of the DIIS algorithm in the global region is also discussed and is much more unpredictable. We assume that, in the course of the SCF optimization, we have determined a set of n+1 AO density matrices D 0 ,D 1 ,D 2 ,...,D n and the associated Kohn–Sham or Fock matrices FD 0 ,FD 1 ,FD 2 ,...,FD n . Since the electronic gradient gD is given by 11 gD =4SDFD − FDDS, 79 we also have available the corresponding gradients gD 0 ,gD 1 ,gD 2 ,...,gD n . We now wish to determine a corrected density matrix, n D¯ = D 0 + c i D i0 , D i0 = D i − D 0 , 80 i=1 that minimizes the norm of the gradient gD¯ . For this purpose, we parameterize the density matrix in terms of an antisymmetric matrix X=−X T and the current density matrix D 0 as 11 DX = exp− XSD 0 expSX. 81 With each old density matrix D i , we now associate an antisymmetric matrix X i such that D i = exp− X i SD 0 expSX i = D 0 + D 0 ,X i S + OX 2 i . 82 Introducing the averaged antisymmetric matrix, n X¯ = c i X i , i=1 we obtain 83 n DX¯ = D 0 + c i D 0 ,X i S + OX¯ 2 , i=1 84 where we have used the S-commutator expansion of DX¯ analogeous to Eq. 82. Our task is hence to determine X¯ in Eq. 83 such that DX¯ minimizes the gradient norm gDX¯ . In passing, we note that, whereas D¯ is not in general idempotent and therefore not a valid density matrix, DX¯ is a valid, idempotent density matrix for all choices of c i . Expanding the gradient in Eq. 79 about the currentdensity matrix D 0 , we obtain gDX¯ = gD 0 + HD 0 X¯ + OX¯ 2 , 85 where HD is the Jacobian matrix. Neglecting the higherorder terms, our task is therefore to minimize the norm of the gradient, n gc = gD 0 + c i HD 0 X i , 86 i=1 with respect to the elements of c. For an estimate of HD 0 X i , we truncate the expansion, gD i = gD 0 + HD 0 X i + OX i 2 , and obtain the quasi-Newton condition, gD i − gD 0 = HD 0 X i . Inserting this condition into Eq. 86, we obtain n gc = gD 0 + i=1 n c i gD i − gD 0 = c i gD i , i=0 87 88 89 where we have introduced the parameter c 0 =1− n i=1 c i . The minimization of gc=gc may therefore be carried out as a least-squares minimization of gc in Eq. 89 subject to the constraint n c i =1. 90 i=0 If we consider gD i as an error vector for the density matrix D i , this procedure becomes identical to the DIIS method. From Eq. 86 we also see that DIIS may be viewed as a minimization of the residual for the Newton equation in the subspace of the density matrix differences D i −D 0 , i=1, n, where the quasi-Newton condition is used to set up the subspace equations. Since the quasi-Newton steps are reliable only in the local region of the optimization, we conclude that the DIIS method can be used safely only in this region, when the electronic Hessian is positive definite. The optimal combination of the density matrices is obtained in the DIIS method, by carrying out a least-squares minimization of the gradient norm subject to the constraint in Eq. 90. However, since a small gradient norm in the global region does not necessarily imply a low Kohn–Sham energy, the DIIS convergence may be unpredictable. Furthermore, we may encounter regions where the gradient norms are Downloaded 23 Aug 2005 to 130.225.22.254. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp
074103-13 Trust-region self-consistent-field J. Chem. Phys. 123, 074103 2005 ethylenediamine tetra-acetic acid EDTA. Next, in Sec. VII B, we consider the calculations on five different systems. All calculations have been carried out with a local version of the DALTON program package. 20 Unless otherwise indicated, the starting orbitals have been obtained by diagonalization of the one-electron Hamiltonian. FIG. 5. The convergence of different algorithms in a LDA/6-31G computation with core Hamiltonian start guess for the zinc complex depicted in the lower left corner. The algorithms being QC-SCF , DIIS , TRSCF , and TRSCF-LS . similar but the energies different. The DIIS method may then diverge, not being able to identify the density matrix of lowest energy, as illustrated in Sec. VII. VII. APPLICATIONS to In this section, we give numerical examples to illustrate the convergence characteristics of the Kohn–Sham TRSCF calculations, comparing with the DIIS and QC-SCF calculations. Comparisons are also made with the TRSCF-LS technique, where the TRRH-LS and TRDSM-LS line-search methods of Secs. III A and IV D are combined to set up an expensive but highly robust method, in which the lowest Kohn–Sham energy is identified by a line search at each step. In Sec. VII A, we discuss the calculations on the zinc complex in Fig. 5, where Zn 2+ is complexated with ethylenediamine-N, N -disuccinic acid EDDS, an isomer A. Calculations on the zinc complex In Fig. 5, we have plotted the error in the Kohn–Sham energy at each iteration of LDA/6-31G calculations on the zinc complex. The standard TRSCF method performs almost as well as the very smooth but much more expensive TRSCF-LS method , giving a somewhat higher energy between iterations 13 and 22. By contrast, the DIIS method shows no sign of converging; after 100 iterations, the Kohn–Sham gradient norm is still about 20. Whereas the smooth TRSCF convergence arises because Hessian information is used to ensure downhill TRRH and TRDSM steps at each iteration, no such information is employed in the DIIS method. Finally, the QC-SCF method converges but exceedingly slow—even after 90 iterations it has not reached the quadratically convergent local region! The difficulties experienced with the QC-SCF method illustrate clearly that the use of Hessian information by itself is no guarantee of fast convergence. More details about the TRSCF zinc complex calculation are given in Tables I–V and in Figs. 1–3 and 6, partly discussed in Secs. III F and IV D. In Table IV, we have listed the changes in the Kohn–Sham energy generated separately in the TRRH E RH KS and TRDSM E DSM KS steps at each SCF iteration, and likewise the norms of the changes in the TABLE IV. Convergence details for the TRSCF calculation on the zinc complex in Fig. 5. Energies given in a.u. DSM Iteration E KS E KS RH E KS 2 D¯ n−D n S DSM D n+1 −D¯ n S 2 2 −8.366 865 0.000 000 −8.366 865 0.000 000 0.197 607 3 −68.378 567 −48.255 717 −20.122 850 6.129 310 1.141 536 4 −137.038 420 −105.996 850 −31.041 569 12.140 844 1.265 250 5 −70.415 468 −43.136 482 −27.278 985 1.594 214 1.031 844 6 −41.492 416 −26.390 457 −15.101 958 3.162 679 1.467 802 7 −25.430 533 −14.755 377 −10.675 155 1.468 925 1.364 944 8 −14.460 409 −7.711 220 −6.749 189 1.525 848 1.249 827 9 −8.470 594 −5.289 340 −3.181 254 1.569 113 1.040 337 10 −2.289 664 −2.684 359 0.394 694 1.614 867 0.817 844 11 −2.730 543 −1.053 899 −1.676 644 1.002 744 1.060 298 12 −2.798 537 −1.054 903 −1.743 634 0.867 238 0.632 009 13 −1.061 335 −0.658 907 −0.402 427 0.729 574 0.410 434 14 −0.670 565 −0.293 889 −0.376 675 0.324 338 0.351 715 15 −0.424 253 −0.213 251 −0.211 002 0.201 434 0.203 170 16 −0.074 945 −0.104 012 0.029 066 0.242 928 0.302 723 17 −0.090 241 −0.100 694 0.010 452 0.071 203 0.175 917 18 0.000 195 −0.043 180 0.043 376 0.022 758 0.126 709 19 −0.044 797 −0.057 441 0.012 644 0.047 885 0.032 787 20 −0.015 396 −0.016 501 0.001 104 0.011 897 0.002 976 21 −0.001 118 −0.001 471 0.000 352 0.001 164 0.000 668 22 −0.000 368 −0.000 428 0.000 059 0.000 308 0.000 111 23 −0.000 066 −0.000 076 0.000 010 0.000 050 0.000 019 24 −0.000 011 −0.000 012 0.000 000 0.000 009 0.000 001 25 −0.000 000 −0.000 000 0.000 000 0.000 000 0.000 000 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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Part 2 Atomic Orbital Based Respons
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