American Institute of Mathematical Sciences

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

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

Parikshit Upadhyaya 1,, , Elias Jarlebring 1, and  Emanuel H. Rubensson 2,

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 2021 Early access  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.

Keywords: Self-consistent field iteration, convergence analysis, nonlinear eigenvalue problems, eigenvector nonlinearity, electronic structure calculations, iterative methods.
Mathematics Subject Classification: Primary: 65F15, 65H17.
Citation: Parikshit Upadhyaya, Elias Jarlebring, Emanuel H. Rubensson. A density matrix approach to the convergence of the self-consistent field iteration. Numerical Algebra, Control and Optimization, 2021, 11 (1) : 99-115. doi: 10.3934/naco.2020018
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.

[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.

[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.

[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.

[5]

T. Helgaker, P. Jorgensen and J. Olsen, Molecular Electronic-Structure Theory, John Wiley and Sons, 2000. doi: 10.1002/9781119019572.

[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.

[7]

N. Higham, Functions of Matrices, Society for Industrial and Applied Mathematics, 2008. doi: 10.1137/1.9780898717778.

[8]

T. Kato, Perturbation Theory for Linear Operators, 2nd edition, Springer-Verlag, 1995. doi: 10.1007/978-3-642-66282-9.

[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.

[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.

[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.

[12]

C. E. McCulloch, Symmetric matrix derivatives with applications, J. Amer. Stat. Assoc., 77 (1982), 679-682.  doi: 10.2307/2287736.

[13]

A. Messiah, Quantum Mechanics, Dover Publications, 1999.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[22]

A. Szabo and N. S. Ostlund, Modern Quantum Chemistry: Introduction to Advanced Electronic Structure Theory, Dover Publications, 1996.

[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.

[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.

[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.

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.

[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.

[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.

[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.

[5]

T. Helgaker, P. Jorgensen and J. Olsen, Molecular Electronic-Structure Theory, John Wiley and Sons, 2000. doi: 10.1002/9781119019572.

[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.

[7]

N. Higham, Functions of Matrices, Society for Industrial and Applied Mathematics, 2008. doi: 10.1137/1.9780898717778.

[8]

T. Kato, Perturbation Theory for Linear Operators, 2nd edition, Springer-Verlag, 1995. doi: 10.1007/978-3-642-66282-9.

[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.

[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.

[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.

[12]

C. E. McCulloch, Symmetric matrix derivatives with applications, J. Amer. Stat. Assoc., 77 (1982), 679-682.  doi: 10.2307/2287736.

[13]

A. Messiah, Quantum Mechanics, Dover Publications, 1999.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[22]

A. Szabo and N. S. Ostlund, Modern Quantum Chemistry: Introduction to Advanced Electronic Structure Theory, Dover Publications, 1996.

[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.

[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.

[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.

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 $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  Convergence factor and bounds for the illustrative example
Figure Options

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

[1]

Shi Jin, Christof Sparber, Zhennan Zhou. On the classical limit of a time-dependent self-consistent field system: Analysis and computation. Kinetic and Related Models, 2017, 10 (1) : 263-298. doi: 10.3934/krm.2017011
[2]

Vladimir S. Gerdjikov, Georgi Grahovski, Rossen Ivanov. On the integrability of KdV hierarchy with self-consistent sources. Communications on Pure and Applied Analysis, 2012, 11 (4) : 1439-1452. doi: 10.3934/cpaa.2012.11.1439
[3]

Fanni M. Sélley. A self-consistent dynamical system with multiple absolutely continuous invariant measures. Journal of Computational Dynamics, 2021, 8 (1) : 9-32. doi: 10.3934/jcd.2021002
[4]

Yulan Wang. Global solvability in a two-dimensional self-consistent chemotaxis-Navier-Stokes system. Discrete and Continuous Dynamical Systems - S, 2020, 13 (2) : 329-349. doi: 10.3934/dcdss.2020019
[5]

Yan Tang. Convergence analysis of a new iterative algorithm for solving split variational inclusion problems. Journal of Industrial and Management Optimization, 2020, 16 (2) : 945-964. doi: 10.3934/jimo.2018187
[6]

David Bourne, Howard Elman, John E. Osborn. A Non-Self-Adjoint Quadratic Eigenvalue Problem Describing a Fluid-Solid Interaction Part II: Analysis of Convergence. Communications on Pure and Applied Analysis, 2009, 8 (1) : 143-160. doi: 10.3934/cpaa.2009.8.143
[7]

Nikolaos S. Papageorgiou, Vicenţiu D. Rădulescu, Dušan D. Repovš. Perturbations of nonlinear eigenvalue problems. Communications on Pure and Applied Analysis, 2019, 18 (3) : 1403-1431. doi: 10.3934/cpaa.2019068
[8]

Yanxing Cui, Chuanlong Wang, Ruiping Wen. On the convergence of generalized parallel multisplitting iterative methods for semidefinite linear systems. Numerical Algebra, Control and Optimization, 2012, 2 (4) : 863-873. doi: 10.3934/naco.2012.2.863
[9]

Alain Miranville, Costică Moroşanu. Analysis of an iterative scheme of fractional steps type associated to the nonlinear phase-field equation with non-homogeneous dynamic boundary conditions. Discrete and Continuous Dynamical Systems - S, 2016, 9 (2) : 537-556. doi: 10.3934/dcdss.2016011
[10]

Sanming Liu, Zhijie Wang, Chongyang Liu. On convergence analysis of dual proximal-gradient methods with approximate gradient for a class of nonsmooth convex minimization problems. Journal of Industrial and Management Optimization, 2016, 12 (1) : 389-402. doi: 10.3934/jimo.2016.12.389
[11]

Markus Haltmeier, Antonio Leitão, Otmar Scherzer. Kaczmarz methods for regularizing nonlinear ill-posed equations I: convergence analysis. Inverse Problems and Imaging, 2007, 1 (2) : 289-298. doi: 10.3934/ipi.2007.1.289
[12]

Lili Ju, Wei Leng, Zhu Wang, Shuai Yuan. Numerical investigation of ensemble methods with block iterative solvers for evolution problems. Discrete and Continuous Dynamical Systems - B, 2020, 25 (12) : 4905-4923. doi: 10.3934/dcdsb.2020132
[13]

Dang Van Hieu, Le Dung Muu, Pham Kim Quy. New iterative regularization methods for solving split variational inclusion problems. Journal of Industrial and Management Optimization, 2021  doi: 10.3934/jimo.2021185
[14]

Kanishka Perera, Andrzej Szulkin. p-Laplacian problems where the nonlinearity crosses an eigenvalue. Discrete and Continuous Dynamical Systems, 2005, 13 (3) : 743-753. doi: 10.3934/dcds.2005.13.743
[15]

Alexander Mielke. Weak-convergence methods for Hamiltonian multiscale problems. Discrete and Continuous Dynamical Systems, 2008, 20 (1) : 53-79. doi: 10.3934/dcds.2008.20.53
[16]

Björn Sandstede, Arnd Scheel. Evans function and blow-up methods in critical eigenvalue problems. Discrete and Continuous Dynamical Systems, 2004, 10 (4) : 941-964. doi: 10.3934/dcds.2004.10.941
[17]

Qun Lin, Hehu Xie. Recent results on lower bounds of eigenvalue problems by nonconforming finite element methods. Inverse Problems and Imaging, 2013, 7 (3) : 795-811. doi: 10.3934/ipi.2013.7.795
[18]

Leszek Gasiński, Nikolaos S. Papageorgiou. Multiple solutions for a class of nonlinear Neumann eigenvalue problems. Communications on Pure and Applied Analysis, 2014, 13 (4) : 1491-1512. doi: 10.3934/cpaa.2014.13.1491
[19]

Jana Kopfová. Nonlinear semigroup methods in problems with hysteresis. Conference Publications, 2007, 2007 (Special) : 580-589. doi: 10.3934/proc.2007.2007.580
[20]

Zhiwen Zhao. Asymptotic analysis for the electric field concentration with geometry of the core-shell structure. Communications on Pure and Applied Analysis, 2022, 21 (4) : 1109-1137. doi: 10.3934/cpaa.2022012

 Impact Factor: 

Tools

Metrics

  • PDF downloads (343)
  • HTML views (514)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy