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Part 1 Improving Self-consistent Field Convergence energy increase in other iterations. In short, the identification of µ from the overlap requirement a orb min orb min = A appears to be a good and secure way to control the step sizes in the optimization. orb a min 1.0 0.8 orb A min = 0.975 orb a min 1.0 0.8 orb A min = 0.98 0.6 0.6 0.4 0.2 0.0 A 0 2 4 6 8 10 µ 0.4 0.2 0.0 A 0 2 4 6 8 10 µ 40.0 20.0 RH ∆E HF 40.0 20.0 RH ∆E LDA ∆E / a.u. 0.0 -20.0 -40.0 RH ∆E 0 2 4 6 8 10 µ Fig. 1.4 HF/6-31G, iteration 7. (A) The overlap orb RH a min and (B) the changes in the HF energy ∆ E HF RH and in the RH energy model ∆ E as a function of the level shift µ. B ∆E / a.u. 0.0 -20.0 -40.0 ∆E RH 0 2 4 6 8 10 µ Fig. 1.5 LDA/6-31G, iteration 7. (A) The overlap orb a min and (B) the changes in the LDA energy RH RH ∆ E LDA and in the RH energy model ∆ E as a function of the level shift µ. B 1.4.1.3 DIIS and Dynamically Level Shifted RH For accelerating the SCF convergence, DIIS is a simple and in general very successful scheme. We would expect to get an even better performance and improve the stability of the scheme if DIIS was combined with a dynamically level shifted RH step like TRRH instead of the standard RH with no control of the step. To investigate how a combination of DIIS and TRRH performs, we carried out a number of DIIS-TRRH optimizations. A typical example is seen in Fig. 1.7 and an extraordinary example is seen in Fig. 1.8. Fig. 1.6 Cd 2+ complexed with an imidazole ring. 16
Development of SCF Optimization Algorithms Error in energy / E h 1.E+02 1.E+00 1.E-02 1.E-04 1.E-06 1.E-08 DIIS DIIS-TRRH TRSCF 0 5 10 15 20 25 Iteration Fig. 1.7 LDA/STO-3G calculations with a H1-core start guess on the cadmium complex in Fig. 1.6. Error in energy / E h 1.E+02 1.E+00 1.E-02 1.E-04 1.E-06 TRSCF DIIS-TRRH DIIS 0 5 10 15 20 25 30 Iteration Fig. 1.8 LDA/STO-3G calculations with a Hückel start guess on the zinc complex in Fig. 1.3. Somewhat surprisingly the calculations rarely converge with the DIIS-TRRH method. To understand this behavior, we note that, in the global region, the TRRH method typically produces gradients that do not change much, even though large changes may occur in the energy. In such cases, the DIIS method may stall, not being able to identify a good combination of density matrices. This behavior is illustrated in Table 1-1, where the gradient norm and Kohn–Sham energy of the first six iterations of the cadmium complex calculations in Fig. 1.7 are listed. Table 1-1. The Gradient norm ||g||=||4(SDF-FDS)|| in the first six iterations of the cadmium complex calculations of Fig. 1.7. DIIS DIIS-TRRH TRSCF It. 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 The TRSCF and DIIS-TRRH gradients stay almost the same during these iterations, stalling the DIIS-TRRH optimization but not the TRSCF optimization, whose energy decreases in each iteration. In the pure DIIS optimization, by contrast, the gradient changes significantly from iteration to iteration; at the same time, the energy decreases at each iteration except the second and fifth, where also the gradient norms increase. Eventually, DIIS enters the local region with its rapid rate of convergence although we note a sudden, large increase in the energy in iterations 10 and 11. However, these changes are accompanied with large increases in the gradient norm, allowing DIIS to recover safely. 17
Page 1: PhD Thesis Optimization of Densitie
Page 5 and 6: Contents Preface ..................
Page 7: Summary............................
Page 11: List of Publications This thesis in
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Part 2 Atomic Orbital Based Respons
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Part 3 Benchmarking for Radicals ar
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Part 3 Benchmarking for Radicals le
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Part 3 Benchmarking for Radicals As
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Part 3 Benchmarking for Radicals R
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Summary The developments in compute
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Appendix A The Derivatives of the D
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Thus, the density element D µν is
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References 1 2 3 4 5 6 7 8 9 C. C.
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65 T. Helgaker and P. Jørgensen, T
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JOURNAL OF CHEMICAL PHYSICS VOLUME
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Part 1 The Trust-region Self-consis
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Part 3 A Coupled Cluster and Full C
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