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Part 1 Improving Self-consistent Field Convergence Error in energy / E h 1.E+02 1.E+00 1.E-02 1.E-04 1.E-06 1.E-08 TRSCF Improved TRDSM 0 5 10 15 20 Iteration Fig. 1.22 Convergence for the cadmium complex in Fig. 1.6, both for TRSCF with no improvements, and for TRSCF where E is used in TRDSM. DSM imp ∆E / E h 1.0 0.0 -1.0 -2.0 -3.0 -4.0 DSM ∆E DSM ∆E HF DSM ∆E imp 0 2 4 6 8 10 12 α Fig. 1.23 TRDSM line search for iteration 7 in the TRSCF optimization Fig. 1.22. For different α in DSM Eq. (1.67), the changes in E HF , E DSM DSM and E imp compared to E HF (D 0 ) are found. It is seen in Fig. 1.23 that the improved DSM energy describes the HF energy better than the standard DSM energy does, just as expected. As the step moves away from the expansion point, the part of the energy which cannot be described in the old densities grows and both the DSM energy models become poor. The improvements presented in this section add complexity to the TRDSM algorithm, even though the computational cost is not significant. As seen in Fig. 1.22 and Fig. 1.23, the improvements to the TRSCF calculation are minor. The overall gain does not justify the extra complexity added to the TRDSM algorithm. 1.4.3 Energy Minimization Exploiting the Density Subspace Section 1.3.1 describes how different approaches have been taken to avoid the diagonalization in the Roothaan-Hall step. Replacing the standard diagonalization of the Fock matrix can be done for the purpose of improving either the convergence properties or the scaling of the algorithm or for both reasons. With the purpose of improving both, a newly developed scheme is presented in this section, in which an energy minimization replaces the standard diagonalization in the SCF optimization. When the RH energy model is minimized, the density subspace information used with great success in TRDSM is ignored. The novel idea is thus to exploit the valuable information saved in the density subspace of the previous densities to construct an improved RH energy model and minimize this model instead of the RH model. This makes the TRDSM step redundant since a density subspace minimization now is included in the RH energy model minimization. The Hessian update methods 40,53 , in which an approximate Hessian is updated in each iteration and an approximate Newton step is taken, exploit some of the same ideas, but they are all based on 32
Development of SCF Optimization Algorithms approximate second order energy expansions in the orbital rotation parameters and therefore do not include the third and higher order terms included in the RH energy. In the following subsections the improved RH energy model and its minimization will be described. The SCF convergence of a test case is then displayed, in which the new energy minimization approach is compared to standard DIIS and the TRSCF schemes. As the scheme has not yet been extended to DFT, this section will only consider HF theory and calculations. 1.4.3.1 The Augmented RH Energy model If the Hartree-Fock energy, Eq. (1.1), is expanded through second order around some reference density D 0 E ( D) = E ( D ) + 2TrF( D )( D− D ) + Tr( D−D ) G( D−D ) , (1.75) HF HF 0 0 0 0 0 the first two terms are recognized as E RH (D) from Eq. (1.22) plus the terms of zeroth order E HF (D 0 ) and - E RH (D 0 ) ( ) ( ) ( ) RH RH E ( D) = E ( D) + E ( D ) − E ( D ) + Tr D−D G D−D . (1.76) HF HF 0 0 0 0 In a standard RH step, the energy function to minimize is the RH energy, neglecting the last term which contains the Hessian information, because it is too expensive to evaluate. Since Hessian information is very valuable to an optimization, the scheme presented in this section will replace the diagonalization in the RH step by an energy minimization of an augmented RH (ARH) energy model, where as much Hessian information as possible is included without directly evaluating new Fock matrices. This is done by exploiting the information contained in the density and Fock matrices of the previous iterations. As previously exploited, a density or density difference, in this case ∆ = D – D 0 , can be split in a part that can be described in the subspace of the n + 1 previous densities ∆ and an unknown part orthogonal to the space ⊥ ∆ ∆ is expanded in the previous densities D i as D− D = ∆ = ∆ + ∆ 0 n i= 0 ⊥ . (1.77) ∆ = ∑ωiDi , (1.78) where n is the number of previously stored densities and the expansion coefficients ω i are determined in a least-squares manner ω n i = −1 ⎡⎣ ⎤⎦ Tr ij j= 0 j , Mij = Tr i j ∑ M D S∆S D SD S . (1.79) 33
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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Page 72 and 73: Part 2 Atomic Orbital Based Respons
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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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