A density matrix approach to the convergence of the self-consistent field iteration

Parikshit Upadhyaya; Elias Jarlebring; Emanuel H. Rubensson

doi:10.3934/naco.2020018

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2021, Volume 11, Issue 1: 99-115. Doi: 10.3934/naco.2020018

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A density matrix approach to the convergence of the self-consistent field iteration

1.
Lindstedtsvägen 25, Department of Mathematics, SeRC - Swedish e-Science research center, Royal Institute of Technology, SE-11428 Stockholm, Sweden
2.
Division of Scientific Computing, Department of Information Technology, , Uppsala University, Box 337, SE-75105 Uppsala, Sweden

^* Corresponding author
^* Corresponding author

Received: November 2018

Revised: December 2019

Early access: March 2020

Published: March 2021

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Abstract

In this paper, we present a local convergence analysis of the self-consistent field (SCF) iteration using the density matrix as the state of a fixed-point iteration. Conditions for local convergence are formulated in terms of the spectral radius of the Jacobian of a fixed-point map. The relationship between convergence and certain properties of the problem is explored by deriving upper bounds expressed in terms of higher gaps. This gives more information regarding how the gaps between eigenvalues of the problem affect the convergence, and hence these bounds are more insightful on the convergence behaviour than standard convergence results. We also provide a detailed analysis to describe the difference between the bounds and the exact convergence factor for an illustrative example. Finally we present numerical examples and compare the exact value of the convergence factor with the observed behaviour of SCF, along with our new bounds and the characterization using the higher gaps. We provide heuristic convergence factor estimates in situations where the bounds fail to well capture the convergence.

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Mathematics Subject Classification: Primary: 65F15, 65H17.

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Figure 1. Schematic illustration of elements of $ \Omega_3 $ as indices of $ R $ for the real-valued problem in subsection 4.1 with $ n = 7, p = 3, \alpha = 10.0 $

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Figure 2. Convergence factor and bounds for the illustrative example

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Figure 3. Norm of $ \mathcal{L}(x_1x_2^H) $ and $ \mathcal{L}(x_2x_3^H) $

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Figure 4. Variance of $ \delta_1 $ and $ \delta_2 $ with $ \epsilon $

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Figure 5. Complex-valued problem for $ n = 30 $ ((a),(b) and (d)), $ p = 15, \alpha = 40.0 $ ((a),(b) and (c))

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Figure 6. Real-valued problem for $ n = 60 $, $ \alpha = 5.0 $, $ p = 25 $

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Figure 7. Water molecule problem with $ n = 13, p = 5 $

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