# American Institute of Mathematical Sciences

March  2021, 11(1): 99-115. doi: 10.3934/naco.2020018

## 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

Received  November 2018 Revised  December 2019 Published  March 2020

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.

Citation: Parikshit Upadhyaya, Elias Jarlebring, Emanuel H. Rubensson. A density matrix approach to the convergence of the self-consistent field iteration. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 99-115. doi: 10.3934/naco.2020018
##### References:
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show all references

##### References:
 [1] Z. Bai, D. Lu and B. Vandereycken, Robust Rayleigh quotient minimization and nonlinear eigenvalue problems, , SIAM J. Sci. Comput., 40 (2018), A3495–A3522. doi: 10.1137/18M1167681.  Google Scholar [2] D. R. Bowler and T. Mizayaki, $\mathcal{O}$($n$) methods in electronic structure calculations, , Rep. Prog. Phys., 75 (2012), 036503, http://iopscience.iop.org/article/10.1088/0034-4885/75/3/036503. doi: 10.1088/0034-4885/75/3/036503.  Google Scholar [3] Y. Cai, L.-H. Zhang, Z. Bai and R.-C. Li, On an eigenvector-dependent nonlinear eigenvalue problem, SIAM J. Matrix Anal. Appl., 39 (2018), 1360-1382.  doi: 10.1137/17M115935X.  Google Scholar [4] E. Cancés and C. L. Bris, On the convergence of SCF algorithms for the Hartree-Fock equations, M2AN, Math. Model. Numer. Anal., 34 (2000), 749-774.  doi: 10.1051/m2an:2000102.  Google Scholar [5] T. Helgaker, P. Jorgensen and J. Olsen, Molecular Electronic-Structure Theory, John Wiley and Sons, 2000. doi: 10.1002/9781119019572.  Google Scholar [6] H. V. Henderson and S. R. Searle, Vec and vech operators for matrices, with some uses in jacobians and multivariate statistics, Can J. Stat., 7 (1979), 65-81.  doi: 10.2307/3315017.  Google Scholar [7] N. Higham, Functions of Matrices, Society for Industrial and Applied Mathematics, 2008. doi: 10.1137/1.9780898717778.  Google Scholar [8] T. Kato, Perturbation Theory for Linear Operators, 2nd edition, Springer-Verlag, 1995. doi: 10.1007/978-3-642-66282-9.  Google Scholar [9] A. Levitt, Convergence of gradient-based algorithms for the Hartree-Fock equations, ESAIM: Math. Model. Numer. Anal., 46 (2012), 1321-1336.  doi: 10.1051/m2an/2012008.  Google Scholar [10] X. Liu, X. Wang, Z. Wen and Y. Yuan, On the convergence of the self-consistent field iteration in Kohn-Sham density functional theory, SIAM J. Matrix Anal. Appl., 35 (2014), 546-558.  doi: 10.1137/130911032.  Google Scholar [11] X. Liu, Z. Wen, X. Wang, M. Ulbrich and Y. Yuan, On the analysis of the discretized Kohn-Sham density functional theory, SIAM J. Numer. Anal., 53 (2015), 1758-1785.  doi: 10.1137/140957962.  Google Scholar [12] C. E. McCulloch, Symmetric matrix derivatives with applications, J. Amer. Stat. Assoc., 77 (1982), 679-682.  doi: 10.2307/2287736.  Google Scholar [13] A. Messiah, Quantum Mechanics, Dover Publications, 1999. Google Scholar [14] T. T. Ngo, M. Bellalij and Y. Saad, The trace ratio optimization problem for dimensionality reduction, SIAM J. Matrix Anal. Appl., 31 (2010), 2950-2971.  doi: 10.1137/090776603.  Google Scholar [15] P. Pulay, Convergence acceleration of iterative sequences. The case of scf iteration, Chem. Phys. Lett., 73 (1979), 393-398.  doi: 10.1016/0009-2614(80)80396-4.  Google Scholar [16] T. Rohwedder and R. Schneider, An analysis for the DIIS acceleration method used in quantum chemistry calculations, J. Math. Chem., 49 (2011), 1889-1914.  doi: 10.1007/s10910-011-9863-y.  Google Scholar [17] E. Rudberg, Quantum Chemistry for Large Scale Systems, PhD thesis, Royal Institute of Technology, 2007, Available at http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-4561. Google Scholar [18] E. Rudberg, E. H. Rubensson and P. Saƚek, Kohn-Sham density functional theory electronic structure calculations with linearly scaling computational time and memory usage, J. Chem. Theory Comput., 7 (2011), 340-350.  doi: 10.1021/ct100611z.  Google Scholar [19] E. Rudberg, E. H. Rubensson, P. Saƚek and A. Kruchinina, Ergo: An open-source program for linear-scaling electronic structure calculations, SoftwareX, 7 (2018), 107-111.  doi: 10.1016/j.softx.2018.03.005.  Google Scholar [20] Y. Saad, J. T. Chelikowsky and S. M. Shontz, Numerical methods for electronic structure calculations of materials, SIAM Rev., 52 (2010), 3-54.  doi: 10.1137/060651653.  Google Scholar [21] R. E. Stanton, Intrinsic convergence in closed-shell SCF calculations. A general criterion, J. Chem. Phys., 75 (1981), 5416-5422.  doi: 10.1063/1.441942.  Google Scholar [22] A. Szabo and N. S. Ostlund, Modern Quantum Chemistry: Introduction to Advanced Electronic Structure Theory, Dover Publications, 1996. Google Scholar [23] C. Yang, W. Gao and J. C. Meza, On the convergence of the self-consistent field iteration for a class of nonlinear eigenvalue problems, SIAM J. Matrix Anal. Appl., 30 (2009), 1773-1788.  doi: 10.1137/080716293.  Google Scholar [24] L.-H. Zhang, L.-Z. Liao and M. K. Ng, Fast algorithms for the generalized Foley Sammon discriminant analysis, SIAM J. Matrix Anal. Appl., 31 (2010), 1584-1605.  doi: 10.1137/080720863.  Google Scholar [25] L.-H. Zhang, W. H. Yang and L.-Z. Liao, A note on the trace quotient problem, Opt. Lett., 8 (2014), 1637-1645.  doi: 10.1007/s11590-013-0680-z.  Google Scholar
Schematic illustration of elements of $\Omega_3$ as indices of $R$ for the real-valued problem in subsection 4.1 with $n = 7, p = 3, \alpha = 10.0$
Convergence factor and bounds for the illustrative example
Norm of $\mathcal{L}(x_1x_2^H)$ and $\mathcal{L}(x_2x_3^H)$
Variance of $\delta_1$ and $\delta_2$ with $\epsilon$
Complex-valued problem for $n = 30$ ((a),(b) and (d)), $p = 15, \alpha = 40.0$ ((a),(b) and (c))
Real-valued problem for $n = 60$, $\alpha = 5.0$, $p = 25$
Water molecule problem with $n = 13, p = 5$
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