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Part 1 Improving Self-consistent Field Convergence Table 1-4 The lowest Hessian eigenvalues for the RH energy model and SCF energy at convergence of the calculations in Fig. 1.27 and Fig. 1.28. The deviation is found as ( ⎡ (2) ⎤ ⎡ (2) ⎤ ) (2) RH SCF 100% ⎡ ⎤ ⎣ E ⎦ − ⎣ E ⎦ ⋅ ⎣ E SCF ⎦ . (2) SCF (2) RH min min min cadmium complex zinc complex HF LDA HF LDA ⎡ ⎣ E ⎤ ⎦ min 0.557 0.017 1.000 0.290 ⎡ ⎤ ⎣ E ⎦ min 1.112 0.014 1.621 0.281 Deviation 100% -21% 62% -2% As expected, the lowest Hessian eigenvalue for the RH energy model, that is the HOMO-LUMO gap, is much smaller for LDA than for HF, but surprisingly it is seen that the Hessian prediction in the RH energy model for LDA is much better than the one for HF. Of course this is only the lowest eigenvalue, and we have not studied the corresponding eigenvector. We know for sure that the size of the orbital rotation parameters κ ai decreases during the optimization and should be very small at convergence, where only small adjustments to the density are made. It is thus difficult to imagine that terms of third and higher order in κ should be the reason for the larger errors in the DSM energy model for LDA compared to HF. This is a matter we will investigate further in the future since it is not understood at the moment. The importance of the higher order terms should be examined directly to understand how they affect the errors, and the Hessian should be studied more carefully introducing information about the direction of the eigenvalues. However, it can still be concluded from Fig. 1.27 and Fig. 1.28 that the RH energy model is poorer for LDA than for HF optimizations. 1.5.2 The Quality of the TRDSM Energy Model The TRDSM energy model of Section 1.4.2.2 is formulated in a general manner and is as applicable to DFT theory as to HF theory. Still, the model will be poorer for DFT than for HF because of the general exchange-correlation term appearing in the DFT energy. For the DSM energy model there are in general four possible sources of errors: 1. The purified density D still has an idempotency error. 2. The term 1 T [2] 2 δ 0 δ D E D in E( D ) , Eq. (1.50), is neglected. 3. E( D ) , Eq. (1.50), is truncated after second order. 4. ( 2 ) 0 + E D in Eq. (1.50) is approximated by 2 F + . 42
The Quality of the Energy Models for HF and DFT Let us take a closer look at the errors one by one. In ref. 39 a general order analysis of the purified density D used in the parameterization of the DSM energy is given, and the results are summarized in Table 1-5. Table 1-5. Comparison of the properties of the unpurified density D and the purified density D . c is the density expansion coefficients and κ is the orbital rotation parameters that change D 0 to another density in the subspace D i . D Differences D+ = D− D0 = ( c κ ) O 2 Dδ = D − D = O ( c κ ) Idempotency error 2 4 DSD − D = O ( c κ ) DSD − D = O ( c 2 κ ) Trace error Tr DS − N / 2 = 0 2 4 Tr DS − N / 2 = O ( c κ ) In the D column, the order of the idempotency correction D δ and the idempotency error for D are found. These are the same for DFT and HF; the idempotency error is of order c 2 ||κ|| 4 , and since D δ is of the order c||κ|| 2 , the error connected to the neglect of the term second order in D δ , will be of order c 2 ||κ|| 4 as well. The third possible source of errors is the truncation of the energy E( D ) after second order in the density. Since the Hartree-Fock energy is quadratic in the density, this truncation leads to no errors for HF, but for DFT there will be an error of order ||D + || 3 and from the first column in Table 1-5 it is seen that it can be written as an error of order c 3 ||κ|| 3 , since D + is of the order c||κ||. Also since the (3) HF energy is quadratic in the density, no third derivative E 0 exists and thus the Taylor expansion ( 2 ) used to find E0 D+ = 2F + is terminated for HF, but for DFT terms of order ||D + || 2 are neglected. ( 2 ) Since E0 D + is multiplied by D + in the energy function Eq. (1.50), this gives an error for DFT of the order ||D + || 3 or as before c 3 ||κ|| 3 . The sizes of the introduced errors are summarized in Table 1-6. Table 1-6. Comparison of the errors introduced in the DSM energy model for HF and DFT respectively. D 1 Idempotency error DSD − D 2 Neglected term 3 Truncation of ( ) 4 Approximation of ( ) error in HF error in DFT ( 2 4 O c κ ) 2 4 O ( c κ ) 1 T [2] D 2 δ E0 D 2 4 2 4 δ O ( c κ ) O ( c κ ) E D 0 3 3 O ( c κ ) 2 E0 D + 0 3 3 O ( c κ ) Depending on the sizes of c and ||κ|| respectively, the error for DFT will be of same or lower order than the one for HF. To inspect whether or not the DSM energy is a poorer model for DFT than for HF, a number of calculations have been carried out, and the sizes of ||D δ || and ||D + || for the DSM step in each iteration are examined. Since D δ is of the order c||κ|| 2 and D + is of the order c||κ||, the 43
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PhD Thesis Optimization of Densitie
Page 5 and 6: Contents Preface ..................
Page 7: Summary............................
Page 11: List of Publications This thesis in
Page 14 and 15: 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
Page 92 and 93: Part 3 Benchmarking for Radicals le
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
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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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074103-2 Thøgersen et al. J. Chem.
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074103-10 Thøgersen et al. J. Chem
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Part 3 A Coupled Cluster and Full C
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