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Part 1 Improving Self-consistent Field Convergence where λ – the damping factor - is a scalar chosen between zero and one. The iterative sequence is then continued with D damp as the new density. Hartree found that this scheme could force convergence in problematic cases. To get an idea of the effect of the damping factor, we consider a block-diagonal Fock matrix in the MO basis F MO ⎛ εo Fov ⎞ = ⎜ ⎟ , (1.10) ⎝Fvo εv ⎠ where ‘o’ denotes occupied, ‘v’ virtual and [ε o ] ij = δ ij ε i and [ε v ] ab = δ ab ε a . The change in electronic energy from the first order variation of the occupied orbitals through first-order perturbation theory is then given as virtual occupied 2 ( 1) −Fai SCF 4 a i εa − εi ∆ E = ∑ ∑ . (1.11) ( ) If this first order term is negative and sufficiently small such that the higher order contributions are insignificant, then a decrease in the electronic energy is seen. If the MOs obey the aufbau principle, then all ε i < ε a and it is clear that the term is negative as desired. The Hartree damping of Eq. (1.9) roughly corresponds to multiplying the numerator of Eq. (1.11) by the factor λ, which is positive and less than one virtual occupied 2 ( 1) −λFai SCF 4 a i εa − εi ∆ E = ∑ ∑ , (1.12) ( ) thus giving the opportunity to obtain a negative first order change of arbitrarily small magnitude, making the higher order terms insignificant. Though this would seem promising, the aufbau principle is seldom obeyed all through the optimization. If λ could be freely chosen, the damping technique would lead to an extrapolation scheme in the densities. Since SCF generates an iterative sequence where each step only depends upon the preceding, it was natural to apply the mathematical extrapolation methods (e.g. the Aitken extrapolation 28 procedures) on SCF to improve in particular the convergence rate close to the minimum. When the individual MO expansion coefficients are chosen as the extrapolated parameters, as Winter and Dunning Jr. 29 suggested, unphysical result may be obtained, though they can be corrected at the end of the calculation. Nielsen used instead the density matrix as the extrapolated parameter 30 and an eigenvalue extrapolation instead of the Aitken method. This led to a scheme more similar to Hartree damping, but with λ found within the eigenvalue extrapolation scheme. 8
A Survey of Methods for Improving SCF Convergence Different approaches have been taken to dynamically find the damping factor λ. Zerner and Hehenberger 31 found it based on an extrapolation of the Mulliken gross population. Karlström 32 expressed the electronic energy in the damped density E(D damp ) and used the first derivative with respect to λ, to choose in each iteration the λ that minimized the electronic energy. None of these schemes were very successful solving the convergence problems. They all had some particular problematic cases they could handle better than the predecessors, but in general they did not catch on. Pulay then suggested in the early 1980s to use the norm of a linear combination of error vectors e i from the individual iterations, where the vanishing of the error vector is a necessary and sufficient condition for SCF convergence. The norm is then optimized with respect to the coefficients c i n e ( c) = ∑ ciei , (1.13) where n is the number of previous iterations, and the coefficients are restricted to add up to 1 n i= 1 i= 1 ∑ ci = 1. (1.14) The resulting coefficients are used to construct a favorable linear combination of the previous Fock matrices n F = ∑ ciF i , (1.15) i= 1 which is diagonalized to obtain a new density, and so the iterative procedure is reestablished. This was the first density subspace minimization scheme that deliberately exploited the information obtained in the previous iterations and he named the approach DIIS 33 for “Direct Inversion in the Iterative Subspace”. For the special case of two matrices, the DIIS density corresponds to the damped density of Eq. (1.9), but with no restrictions on λ. A decade later the DIIS algorithm was a standard option in most ab initio programs and had effectively solved a number of the convergence problems. The orbital rotation gradient was typically used as the error vector for wave function optimizations, and Sellers pointed out 34 that the DIIS algorithm exploits the second-order information contained in a set of gradients to obtain quadratic convergence behavior. Some numerical problems were seen though, where numerical instabilities appeared because of linear dependencies in the space of error vectors. Sellers introduced the C2-DIIS method 34 , which is similar to DIIS except the restriction is on the squares of the coefficients n 2 ∑ ci = 1 , (1.16) i= 1 9
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
Page 14 and 15: Part 1 Improving Self-consistent Fi
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Part 1 Improving Self-consistent Fi
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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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18 J. Chem. Phys., Vol. 121, No. 1,
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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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