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Part 1 Improving Self-consistent Field Convergence In the C-shift scheme seen in Eq. (1.31) the changes introduced are controlled compared to the previous density, whereas in the d orth -shift scheme the changes are controlled compared to the subspace of all the previous densities. This scheme is thus less restrictive than the C-shift scheme, but it seems that the C-shift scheme is too restrictive, ignoring the stability gained from the subspace information. To compare the overall effect of the two level shift schemes on the SCF convergence, calculations are given in Fig. 1.13 and Fig. 1.14, for HF and LDA, respectively. The HF calculations are on CrC with bond distance 2.00Å in the STO-3G basis and the LDA calculations are on the zinc complex seen in Fig. 1.3 in the 6-31G basis, both cases for which DIIS diverges. The starting orbitals have been obtained by diagonalization of the one-electron Hamiltonian (H1-core start guess). Error in energy / E h 1.E+02 1.E+00 1.E-02 1.E-04 1.E-06 TRSCF d orth -shift DIIS TRSCF C-shift Error in energy / E h 1.E+02 1.E+00 1.E-02 1.E-04 1.E-06 TRSCF d orth -shift DIIS TRSCF C-shift 1.E-08 0 4 8 12 16 Iteration Fig. 1.13 SCF convergence for HF/STO-3G calculations on CrC. 1.E-08 0 8 16 24 32 Iteration Fig. 1.14 SCF convergence for LDA/6-31G calculations on the zinc complex in Fig. 1.3. The only difference in the “TRSCF/d orth -shift” and the “TRSCF/C-shift” optimizations is the way the level shift is found in the TRRH step. Since DIIS diverges, the examples display the stability of the TRSCF algorithm, and the ability of the two level shifting schemes to handle problematic cases. In all examples studied so far, both problematic and simple, the d orth -shift has proven as good as or better than the C-shift. The cost of the level shift search process is similar in the two schemes; the matrix M in Eq. (1.37) is updated in each iteration as a part of TRDSM and is then reused for the d orth -shift scheme in TRRH. In Table 1-3 The SCF energy change in each iteration is divided in the part of the change obtained from the RH and DSM step, respectively, and it is seen how the RH step is now allowed to accept larger changes in the density, but still in a controlled manner, thus leading to larger decreases in the energy and improved convergence. 22
Development of SCF Optimization Algorithms Table 1-3. The SCF energy change for each RH and DSM step in the TRSCF calculations in Fig. 1.13. C-shift d orth -shift It. RH DSM RH DSM ∆ E HF ∆ E HF ∆ E HF ∆ E HF 2 -1.1768 0.0000 -1.3976 0.0000 3 -1.8964 -3.8998 -4.1319 -4.5865 4 -1.6764 -1.9603 -1.8021 -1.0448 5 -0.3655 -1.7543 -0.2103 -0.1200 6 -0.1881 -0.1624 -0.0111 -0.0463 7 -0.0932 -0.1505 -0.0036 -0.0037 8 0.0065 -0.0212 -0.0001 -0.0008 9 -0.0039 -0.0154 10 0.0002 -0.0009 1.4.1.6 The Trace Purification Scheme The dynamic level shift scheme described in the previous section has no reference to the MO basis. This opens the possibility to replace the diagonalizations in the TRRH step with some alternative scheme without affecting the overall result. There have been many suggestions as to how the diagonalization can be replaced by a linear scaling algorithm 47 . The trace purification (TP) scheme 19,48 , however, is a simple and useful approach and it has thus been implemented in our SCF program in a local version of DALTON 38,49 . The trace purification scheme was originally formulated for tight binding theory by Palser and Manolopoulos 19 and later improved by Niklasson 48 , and is linear scaling when formulated in an orthogonal basis. The scheme uses the trace and idempotency properties of the density to iteratively find the new density from a suitable start guess constructed from the Fock matrix. Since the SCF optimization is formulated in the non-orthogonal AO basis to avoid the delocalized MO basis, it is necessary to transform the matrices to an orthogonal basis. This is done by a Cholesky decomposition 50 of the AO overlap matrix S T S = LL , (1.39) where L then is used to transform the Fock matrix to an orthogonal basis orth -1 −T F = L FL . (1.40) The density resulting from the trace purification scheme will also be in the orthogonal basis and should be transformed back as −T orth -1 D = L D L . (1.41) Since the AO overlap matrix does not change during the optimization, the Cholesky decomposition and the inversion of L can be done once and for all in the beginning of the calculation. 23
Page 1: 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
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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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