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Part 1 Improving Self-consistent Field Convergence F orth R λ min Estimate and for F orth λ max 0 orth ( λ max I − F ) = ( λ − λ ) max min 1 x n +1 = 2x n - x n 2 n = n + 1 Tr Rn > N yes R n+ 1 = R 2 n no 2 n+ 1 = 2 n − n R R R x n +1 no Tr Rn N ε + 1 − < yes D orth = R n+1 Fig. 1.15 Flow diagram for the trace purification (TP) scheme. N is the number of electrons. 0 x n +1 = x n 2 0 x n 1 Fig. 1.16 The purifying polynomials used in the trace purification scheme. The orange line is the McWeeny purification polynomial x n+1 = 3x n 2 – 2x n 3 . The trace purification is carried out by the Niklasson model with second order purification polynomials, and is schematized in Fig. 1.15. The initial density guess R 0 is obtained by normalizing the Fock matrix such that it only has eigenvalues between 0 and 1. To do this, the bounds for the Fock eigenvalues, λ min and λ max , must be found. They can be estimated using Gerschgorin’s theorem or the Lanczos algorithm for eigenvalues 51 with only a small extra computational cost. R is then iteratively purified, and the purification function applied in each iteration is chosen based on the trace of the matrix R, always keeping the direction towards the correct trace condition. The purification functions are sketched in Fig. 1.16 including the McWenny purification function 8 . One of the functions used in the scheme has a stationary point for x = 1 and the other has a stationary point for x = 0; depending of the function chosen we thus go towards a larger or smaller trace. When R fulfils the trace and/or idempotency conditions Eq. (1.2) of the one electron density within some threshold ε, the new density D orth = R has been found and the density to use in the next TRSCF iteration can be evaluated from Eq. (1.41). The number of purification iterations required to obtain a new density depends on the threshold ε. For the test calculations carried out so far, the threshold has been an error of 10 -7 in the trace, and the number of iterations ranges from 30 to 70 for a single RH step, with the typical number being closer to 30 than 70. Still, it is less expensive than the diagonalization as soon as more than a couple 24
Development of SCF Optimization Algorithms of thousand basis functions are needed. The scaling of the TRRH step in general and the trace purification scheme in particular is illustrated and discussed in Section 1.7.1. 1.4.2 Density Subspace Minimization The DIIS scheme seems to have been the overall most successful of all the suggestions on how to improve SCF convergence described in Section 1.3. DIIS was the first scheme to take advantage of the information contained in the densities and Fock matrices of the previous iterations, and this made the difference. This is also exploited in the EDIIS scheme by Kudin et. al. 37 in which an energy model is optimized with respect to the linear combination of previous densities. The density subspace minimization presented in this section is an improvement to EDIIS with a smaller idempotency error in the density, the correct gradient compared to SCF, and thus better convergence properties in both the local and global region of the optimization. 1.4.2.1 The Trust Region DSM Parameterization After a sequence of Roothaan-Hall iterations, we have determined a set of density matrices D i and a corresponding set of Fock matrices F i = F(D i ). An improved density D and Fock matrix F should now be found as a linear combination of the previous n + 1 stored matrices. Taking D 0 as the reference density matrix, the improved density matrix can be written n = 0 +∑ ci i= 0 D D D , (1.42) which, ideally, should satisfy the symmetry, trace and idempotency conditions Eq. (1.2) of a valid one-electron density matrix. Whereas the symmetry condition is trivially satisfied for any such linear combination, the trace condition holds only for combinations that satisfy the constraint n i= 0 i ∑ ci = 0 , (1.43) leading to a set of n + 1 constrained parameters c i with 0 ≤ i ≤ n. Alternatively, an unconstrained set of n parameters c i with 1 ≤ i ≤ n can be used, with c 0 defined so that the trace condition is fulfilled: c 0 n =−∑ c . (1.44) i= 1 i In terms of these independent parameters, the density matrix D becomes where we have introduced the notation D = D0 + D + , (1.45) 25
Page 1: PhD Thesis Optimization of Densitie
Page 5 and 6: Contents Preface ..................
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Page 11: List of Publications This thesis in
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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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Part 1 The Trust-region Self-consis
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Part 3 A Coupled Cluster and Full C
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