Symmetry breaking and the generation of spin ordered magnetic states in density functional theory due to Dirac exchange for a hydrogen molecule. (English) Zbl 1506.37112
Summary: We study symmetry breaking in the mean field solutions to the electronic structure problem for the 2 electron hydrogen molecule within the Kohn Sham (KS) local spin density functional theory with Dirac exchange (the XLDA model). This simplified model shows behavior related to that of the (KS) spin density functional theory (SDFT) predictions in condensed matter and molecular systems. The KS solutions to the constrained SDFT variation problem undergo spontaneous symmetry breaking leading to the formation of spin ordered states as the relative strength of the non-convex exchange term increases. Numerically, we observe that with increases in the internuclear bond length, the molecular ground state changes from a paramagnetic state (spin delocalized) to an antiferromagnetic (spin localized) ground state and a symmetric delocalized (spin delocalized) excited state. We further characterize the limiting behavior of the minimizer when the strength of the exchange term goes to infinity both analytically and numerically. This leads to further bifurcations and highly localized states with varying character. Finite element numerical results provide support for the formal conjectures. Various solution classes are found to be numerically stable. However, for changes in the \(R\) parameter, numerical Hessian calculations demonstrate that these are stationary but not stable solutions.
MSC:
| 37N15 | Dynamical systems in solid mechanics |
| 65P30 | Numerical bifurcation problems |
| 81R40 | Symmetry breaking in quantum theory |
| 81V10 | Electromagnetic interaction; quantum electrodynamics |
| 81V55 | Molecular physics |
References:
| [1] | Anantharaman, A., Cancès, E.: Existence of minimizers for Kohn-Sham models in quantum chemistry. Annales de l’Institut Henri Poincare (C) Non Linear Anal. 26(6), 2425-2455 (2009) · Zbl 1186.81138 |
| [2] | Aprà, E.; Bylaska, EJ; De Jong, WA; Govind, N.; Kowalski, K.; Straatsma, TP; Valiev, M.; Hubertus, JJ; van Dam, YA; Anchell, J., NWChem: past, present, and future, J. Chem. Phys., 152, 18 (2020) · doi:10.1063/5.0004997 |
| [3] | Axelsson, O.; Alan Barker, V., Finite Element Solution of Boundary Value Problems: Theory and Computation (2001), New York: SIAM, New York · Zbl 0981.65130 · doi:10.1137/1.9780898719253 |
| [4] | Bank, RE; Dupont, T., An optimal order process for solving finite element equations, Math. Comput., 36, 153, 35-51 (1981) · Zbl 0466.65059 · doi:10.1090/S0025-5718-1981-0595040-2 |
| [5] | Barca, GMJ; Gilbert, ATB; Gill, PMW, Hartree-Fock description of excited states of H2, J. Chem. Phys., 141, 11 (2014) · doi:10.1063/1.4896182 |
| [6] | Benguria, R.; Brézis, H.; Lieb, EH, The Thomas-Fermi-von Weizsäcker theory of atoms and molecules, Commun. Math. Phys., 79, 2, 167-180 (1981) · Zbl 0478.49035 · doi:10.1007/BF01942059 |
| [7] | Braess, D., Finite Elements: Theory, Fast Solvers and Applications in Solid Mechanics (2001), Cambridge: Cambridge University Press, Cambridge · Zbl 0976.65099 |
| [8] | Bramble, JH, Multigrid Methods (1993), London: CRC Press, London · Zbl 0786.65094 |
| [9] | Bramble, JH; Pasciak, JE, New convergence estimates for multigrid algorithms, Math. Comput., 49, 180, 311-329 (1987) · Zbl 0659.65098 · doi:10.1090/S0025-5718-1987-0906174-X |
| [10] | Brandt, A., Algebraic multigrid theory: the symmetric case, Appl. Math. Comput., 19, 1, 23-56 (1986) · Zbl 0616.65037 |
| [11] | Brandt, A., McCormick, S., Ruge, J.: Algebraic multigrid (amg) for sparse matrix equations. Spars. Appl. 257 (1985) · Zbl 0548.65014 |
| [12] | Brenner, SC; Ridgway Scott, L., The Mathematical Theory of Finite Element Methods (2008), Berlin: Springer, Berlin · Zbl 1135.65042 · doi:10.1007/978-0-387-75934-0 |
| [13] | Briggs, WL; Henson, VE; McCormick, SF, A Multigrid Tutorial (2000), New Delhi: SIAM, New Delhi · Zbl 0958.65128 · doi:10.1137/1.9780898719505 |
| [14] | Bylaska, E.J., Kohn, S.R., Baden, S.B., Edelman, A., Kawai, R., Ong, M.E.G., Weare, J.H.: Scalable parallel numerical methods and software tools for material design. In: Proceedings of the Seventh SIAM Conference on Parallel Processing for Scientific Computing, pp. 219-224 (1995) · Zbl 0836.65132 |
| [15] | Bylaska, EJ; Holst, M.; Weare, JH, Adaptive finite element method for solving the exact Kohn-Sham equation of density functional theory, J. Chem. Theory Comput., 5, 4, 937-948 (2009) · doi:10.1021/ct800350j |
| [16] | Chen, Y.; Bylaska, E.; Weare, J.; Kubicki, JD, First principles estimation of geochemically important transition metal oxide properties, Molecular Modeling of Geochemical Reactions, Chapter 4 (2016), New York: Wiley, New York |
| [17] | Cohen, AJ; Mori-Sánchez, P.; Yang, W., Insights into current limitations of density functional theory, Science, 321, 792-794 (2008) · doi:10.1126/science.1158722 |
| [18] | Cohen, AJ; Mori-Sánchez, P.; Yang, W., Challenges for density functional theory, Chem. Rev., 112, 289-320 (2012) · doi:10.1021/cr200107z |
| [19] | Cohen-Tannoudji, C.; Diu, B.; Laloe, F., Quantum Mechanics (1991), New York: Wiley, New York · Zbl 1440.81003 |
| [20] | Cox, PA, Transition Metal Oxides (1992), Oxford: Clearendon Press, Oxford |
| [21] | Dupont, T.; Hoffman, J.; Johnson, C.; Kirby, RC; Larson, MG; Logg, A.; Ridgway, SL, The FEniCS Project (2003), Gothenburg: Chalmers Finite Element Centre, Chalmers University of Technology, Gothenburg |
| [22] | Finite element basis functions. http://hplgit.github.io |
| [23] | Foresman, J.B., Frisch, A.: Exploring Chemistry with Electronic Structure Methods. Gaussian (1996) |
| [24] | Frank, RL; Lieb, EH; Seiringer, R.; Thomas, LE, Bipolaron and n-polaron binding energies, Phys. Rev. Lett., 104, 21 (2010) · doi:10.1103/PhysRevLett.104.210402 |
| [25] | Frank, RL; Lieb, EH; Seiringer, R.; Thomas, LE, Stability and absence of binding for multi-polaron systems, Publications mathématiques de l’IHÉS, 113, 1, 39-67 (2011) · Zbl 1227.82083 · doi:10.1007/s10240-011-0031-5 |
| [26] | Golub, GH, Matrix Computations (1996), Baltimore: Johns Hopkins Press, Baltimore · Zbl 0865.65009 |
| [27] | Gontier, D., Existence of minimizers for Kohn-Sham within the local spin density approximation, Nonlinearity, 28, 1, 57 (2015) · Zbl 1307.35243 · doi:10.1088/0951-7715/28/1/57 |
| [28] | Gontier, D., Hainzl, C., Lewin, M.: Lower bound on the Hartree-Fock energy of the electron gas. arXiv:1811.12461 (2018) |
| [29] | Gontier, D., Lewin, M.: Spin symmetry breaking in the translation-invariant Hartree-Fock uniform electron gas. arXiv:1812.07679 (2018) · Zbl 1426.82061 |
| [30] | Griesemer, M.; Hantsch, F., Unique solutions to Hartree-Fock equations for closed shell atoms, Arch. Ration. Mech. Anal., 203, 3, 883-900 (2012) · Zbl 1256.35101 · doi:10.1007/s00205-011-0464-5 |
| [31] | Hackbusch, W., Multi-Grid Methods and Applications (1985), Berlin: Springer, Berlin · Zbl 0595.65106 · doi:10.1007/978-3-662-02427-0 |
| [32] | Harrison, RJ; Fann, GI; Yanai, T.; Gan, Z.; Beylkin, G., Multiresolution quantum chemistry: basic theory and initial applications, J. Chem. Phys., 121, 23, 11587-11598 (2004) · doi:10.1063/1.1791051 |
| [33] | Hehre, WJ; Stewart, RF; Pople, JA, self-consistent molecular-orbital methods. i. use of gaussian expansions of slater-type atomic orbitals, J. Chem. Phys., 51, 6, 2657-2664 (1969) · doi:10.1063/1.1672392 |
| [34] | Hypre library. https://computation.llnl.gov/projects/hypre-scalable-linear-solvers-multigrid-methods |
| [35] | Hu, H.: Electronic Structure Models: Solution Theory, Linear Scaling Methods, and Stability Analysis. UCSD Ph.D. Thesis (2014) |
| [36] | Kendall, RA; Aprà, E.; Bernholdt, DE; Bylaska, EJ; Dupuis, M.; Fann, GI; Harrison, RJ; Jialin, J.; Nichols, JA; Nieplocha, J., High performance computational chemistry: an overview of nwchem a distributed parallel application, Comput. Phys. Commun., 128, 1, 260-283 (2000) · Zbl 1002.81571 · doi:10.1016/S0010-4655(00)00065-5 |
| [37] | Kirr, E.; Kevrekidis, PG; Pelinovsky, DE, Symmetry-breaking bifurcation in the nonlinear Schrödinger equation with symmetric potentials, Commun. Math. Phys., 308, 3, 795-844 (2011) · Zbl 1235.34128 · doi:10.1007/s00220-011-1361-3 |
| [38] | Kohn, S.R., Weare, J.H., Ong, M.E.G., Baden, S.B.: Parallel adaptive mesh refinement for electronic structure calculations. In: Proceedings of the Eighth SIAM Conference on Parallel Processing for Scientific Computing (1997) · Zbl 0942.65126 |
| [39] | Le Bris, C., Some results on the Thomas-Fermi-Dirac-von Weizsäcker model, Differ. Integral Eq., 6, 2, 337-353 (1993) · Zbl 0828.49024 |
| [40] | Lenzmann, E., Uniqueness of ground states for pseudorelativistic Hartree equations, Anal. PDE, 2, 1, 1-27 (2009) · Zbl 1183.35266 · doi:10.2140/apde.2009.2.1 |
| [41] | Lieb, EH, Thomas-Fermi and related theories of atoms and molecules, Rev. Mod. Phys., 53, 4, 603 (1981) · Zbl 1049.81679 · doi:10.1103/RevModPhys.53.603 |
| [42] | Lieb, EH; Loss, M., Analysis (2001), New York: American Mathematical Society, New York · Zbl 0966.26002 |
| [43] | Lieb, EH; Simon, B., The Hartree-Fock theory for Coulomb systems, Commun. Math. Phys., 53, 3, 185-194 (1977) · doi:10.1007/BF01609845 |
| [44] | Lions, P-L, Solutions of Hartree-Fock equations for Coulomb systems, Commun. Math. Phys., 109, 1, 33-97 (1987) · Zbl 0618.35111 · doi:10.1007/BF01205672 |
| [45] | Logg, A., Wells, G.N.D.: Automated finite element computing. ACM Trans. Math. Softw. (TOMS) 37(issue 2, article 20) (2010) · Zbl 1364.65254 |
| [46] | Logg, A.; Mardel, K-A; Wells, GN, Automated Solution of Differential Equations by the Finite Element Method: The FEniCS Book (2012), Berlin: Springer, Berlin · Zbl 1247.65105 · doi:10.1007/978-3-642-23099-8 |
| [47] | McCormick, SF, Multigrid Methods (1987), New Delhi: SIAM, New Delhi · Zbl 0659.65094 · doi:10.1137/1.9781611971057 |
| [48] | Oliver, GL; Perdew, JP, Spin-density gradient expansion for kinetic energy, Phys. Rev. A, 20, 2, 397-403 (1979) · doi:10.1103/PhysRevA.20.397 |
| [49] | Parr, RG, Quantum Theory of Molecular Electronic Structure (1972), San Francisco: W. A. Benjamin, San Francisco |
| [50] | Parr, RG; Yang, W., Density-Functional Theory of Atoms and Molecules (1989), Oxford: Oxford Science Publications, Oxford |
| [51] | Peng, H.; Perdew, JP, Synergy of van der waals and self-interaction corrections in transition metal monoxides, Phys. Rev. B, 96, 100101, 1-5 (2017) |
| [52] | Pozun, ZD; Henkelman, G., Hybrid density functional theory band structure engineering in hematite, J. Chem. Phys., 134, 224706-1-9 (2011) · doi:10.1063/1.3598947 |
| [53] | Reed, M.; Simon, B., Analysis of Operators. Methods of Modern Mathematical Physics (1978), New York: Academic Press, New York · Zbl 0401.47001 |
| [54] | Ricaud, J.: Symétrie et brisure de symétrie pour certains problèmes non linéaires. Ph.D. thesis, Université de Cergy Pontoise (2017) |
| [55] | Ricaud, J., Symmetry breaking in the periodic Thomas-Fermi-Dirac-von Weizsäcker model, Ann. Henri Poincaré, 19, 10, 3129-3177 (2018) · Zbl 1420.35376 · doi:10.1007/s00023-018-0711-5 |
| [56] | Rollmann, G.; Rohrbach, A.; Entel, P.; Hafner, J., First-principle calculations of the structure and magnetic properties of hematite, Phys. Rev. B, 69, 165107, 1-12 (2004) |
| [57] | Ruiz, D., On the Schrödinger-Poisson-Slater system: behavior of minimizers, radial and nonradial cases, Arch. Ration. Mech. Anal., 198, 1, 349-368 (2010) · Zbl 1235.35232 · doi:10.1007/s00205-010-0299-5 |
| [58] | Ruskai, MB; Stillinger, FH, Binding limit in the Hartree approximation, J. Math. Phys., 25, 6, 2099-2103 (1984) · doi:10.1063/1.526367 |
| [59] | Saad, Y.; Schultz, MH, Gmres: a generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. Stat. Comput., 7, 3, 856-869 (1986) · Zbl 0599.65018 · doi:10.1137/0907058 |
| [60] | Sulem, C.; Sulem, P-L, The Nonlinear Schrödinger Equation: Self-Focusing and Wave Collapse (1999), Berlin: Springer, Berlin · Zbl 0928.35157 |
| [61] | Szabo, A.; Ostlund, NS, Modern Quantum Chemistry (1989), London: Dover Publications, London |
| [62] | Tatebe, O.: The multigrid preconditioned conjugate gradient method. In: The Sixth Copper Mountain Conference on Multigrid Methods, Part 2, NASA. Langley Research Center (1993) · Zbl 0939.65548 |
| [63] | Valiev, M.; Bylaska, EJ; Govind, N.; Kowalski, K.; Straatsma, TP; Van Dam, HJJ; Wang, D.; Nieplocha, J.; Aprà, E.; Windus, TL, Nwchem: a comprehensive and scalable open-source solution for large scale molecular simulations, Comput. Phys. Commun., 181, 9, 1477-1489 (2010) · Zbl 1216.81179 · doi:10.1016/j.cpc.2010.04.018 |
| [64] | Van Emden, H.; Yang, UM, Boomer amg: a parallel algebraic multigrid solver and preconditioner, Appl. Numer. Math., 41, 155-177 (2002) · Zbl 0995.65128 · doi:10.1016/S0168-9274(01)00115-5 |
| [65] | Weinstein, MI, Lyapunov stability of ground states of nonlinear dispersive evolution equations, Commun. Pure Appl. Math., 39, 1, 51-67 (1986) · Zbl 0594.35005 · doi:10.1002/cpa.3160390103 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
