# zbMATH — the first resource for mathematics

Symmetry breaking in the periodic Thomas-Fermi-Dirac-von Weizsäcker model. (English) Zbl 1420.35376
The paper addresses a density-functional theory for the electron gas interacting with a spatially periodic ionic lattice. It is proved that, if strength $$c$$ of the term which accounts for the exchange-correlation energy in the system’s Hamiltonian exceeds a certain critical value, the ground state changes the periodicity of its electron density undergoes doubling, in comparison with the period of the ionic lattice. In that case, an effective nonlinear Schrödinger equation for the real stationary electron wave function takes the following form: $-\Delta Q+ Q^{7/3} - cQ^{5/3} = \mu Q,$ where $$\mu$$ is the chemical potential.

##### MSC:
 35Q55 NLS equations (nonlinear Schrödinger equations) 35Q40 PDEs in connection with quantum mechanics 81V55 Molecular physics
PROFESS
Full Text:
##### References:
 [1] Bahouri, H.; Gérard, P., High frequency approximation of solutions to critical nonlinear wave equations, Am. J. Math., 121, 31-175, (1999) · Zbl 0919.35089 [2] Bellazzini, J., Ghimenti, M.: Symmetry breaking for Schrödinger-Poisson-Slater energy? arXiv:1601.05626 (2016) [3] Berestycki, H.; Lions, P-L, Nonlinear scalar field equations. I. existence of a ground state, Arch. Ration. Mech. Anal., 82, 313-345, (1983) · Zbl 0533.35029 [4] Bokanowski, O.; Grebert, B.; Mauser, NJ, Local density approximations for the energy of a periodic Coulomb model, Math. Models Methods Appl. Sci., 13, 1185-1217, (2003) · Zbl 1053.82033 [5] Bokanowski, O.; Mauser, NJ, Local approximation for the Hartree-Fock exchange potential: a deformation approach, Math. Models Methods Appl. Sci., 9, 941-961, (1999) · Zbl 0956.81097 [6] Brothers, JE; Ziemer, WP, Minimal rearrangements of Sobolev functions, J. Reine Angew. Math., 384, 153-179, (1988) · Zbl 0633.46030 [7] Catto, I., Le Bris, C., Lions, P.-L.: The Mathematical Theory of Thermodynamic Limits: Thomas-Fermi Type Models. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York (1998) · Zbl 0938.81001 [8] Chen, M.; Xia, J.; Huang, C.; Dieterich, JM; Hung, L.; Shin, I.; Carter, EA, Introducing PROFESS 3.0: an advanced program for orbital-free density functional theory molecular dynamics simulations, Comput. Phys. Commun., 190, 228-230, (2015) · Zbl 1344.81010 [9] Dirac, PA, Note on exchange phenomena in the Thomas atom, Proc. Camb. Philos. Soc., 26, 376-385, (1930) · JFM 56.0751.04 [10] Frank, RL; Lenzmann, E., Uniqueness of non-linear ground states for fractional Laplacians in $$\mathbb{R}$$, Acta Math., 210, 261-318, (2013) · Zbl 1307.35315 [11] Frank, RL; Lenzmann, E.; Silvestre, L., Uniqueness of radial solutions for the fractional Laplacian, Commun. Pure Appl. Math., 69, 1671-1726, (2016) · Zbl 1365.35206 [12] Frank, RL; Lieb, EH, Possible lattice distortions in the Hubbard model for graphene, Phys. Rev. Lett., 107, 066801, (2011) [13] Friesecke, G., Pair correlations and exchange phenomena in the free electron gas, Commun. Math. Phys., 184, 143-171, (1997) · Zbl 0874.60094 [14] Fröhlich, H., On the theory of superconductivity: the one-dimensional case, Proc. R. Soc. Lond. A, 223, 296-305, (1954) · Zbl 0055.44105 [15] Garcia Arroyo, M., Séré, E.: Existence of kink solutions in a discrete model of the polyacetylene molecule. Working paper or preprint (2012) [16] Gérard, P., Description du défaut de compacité de l’injection de Sobolev, ESAIM Control Optim. Calc. Var., 3, 213-233, (1998) · Zbl 0907.46027 [17] Gidas, B.; Ni, WM; Nirenberg, L., Symmetry of positive solutions of nonlinear elliptic equations in $${\mathbb{R}}^N$$, Adv. Math. Suppl. Stud. A, 7, 369-402, (1981) · Zbl 0469.35052 [18] Graf, GM; Solovej, JP, A correlation estimate with applications to quantum systems with Coulomb interactions, Rev. Math. Phys., 6, 977-997, (1994) · Zbl 0843.47041 [19] Grillakis, M.; Shatah, J.; Strauss, W., Stability theory of solitary waves in the presence ofsymmetry. I, J. Funct. Anal., 74, 160-197, (1987) · Zbl 0656.35122 [20] Hmidi, T.; Keraani, S., Blowup theory for the critical nonlinear Schrödinger equations revisited, Int. Math. Res. Not., 2005, 2815-2828, (2005) · Zbl 1126.35067 [21] Johnson, RA, Empirical potentials and their use in the calculation of energies of point defects in metals, J. Phys. F Met. Phys., 3, 295, (1973) [22] Kennedy, T.; Lieb, EH, An itinerant electron model with crystalline or magnetic long range order, Phys. A, 138, 320-358, (1986) · Zbl 1002.82508 [23] Kennedy, T.; Lieb, EH, Proof of the Peierls instability in one dimension, Phys. Rev. Lett., 59, 1309-1312, (1987) [24] Killip, R.; Oh, T.; Pocovnicu, O.; Vişan, M., Solitons and scattering for the cubic-quintic nonlinear Schrödinger equation on $$\mathbb{R}^3$$, Arch. Ration. Mech. Anal., 225, 469-548, (2017) · Zbl 1367.35158 [25] Killip, R., Vişan, M.: Nonlinear Schrödinger Equations at Critical Regularity. Lecture Notes for the Summer School of Clay Mathematics Institute (2008) [26] Kin-Lic Chan, G.; Handy, NC, Optimized Lieb-Oxford bound for the exchange-correlation energy, Phys. Rev. A, 59, 3075-3077, (1999) [27] Le Bris, C.: Quelques problèmes mathématiques en chimie quantique moléculaire. Ph.D. thesis, École Polytechnique (1993) [28] Lenzmann, E., Uniqueness of ground states for pseudorelativistic Hartree equations, Anal. PDE, 2, 1-27, (2009) · Zbl 1183.35266 [29] Levy, M.; Perdew, JP, Tight bound and convexity constraint on the exchange-correlation-energy functional in the low-density limit, and other formal tests of generalized-gradient approximations, Phys. Rev. B, 48, 11638-11645, (1993) [30] Lewin, M.: Variational methods in quantum mechanics. Unpublished lecture notes (University of Cergy-Pontoise) (2010) [31] Lewin, M.; Lieb, EH, Improved Lieb-Oxford exchange-correlation inequality with a gradient correction, Phys. Rev. A, 91, 022507, (2015) [32] Lewin, M.; Rota Nodari, S., Uniqueness and non-degeneracy for a nuclear nonlinear Schrödinger equation, Nonlinear Differ. Equ. Appl., 22, 673-698, (2015) · Zbl 1325.35179 [33] Li, C., Monotonicity and symmetry of solutions of fully nonlinear elliptic equations on unbounded domains, Commun. Partial Differ. Equ., 16, 585-615, (1991) · Zbl 0741.35014 [34] Lieb, E.H.: Existence and uniqueness of the minimizing solution of Choquard’s nonlinear equation. Stud. Appl. Math. 57, 93-105 (1976/1977) · Zbl 0369.35022 [35] Lieb, EH, A lower bound for Coulomb energies, Phys. Lett. A, 70, 444-446, (1979) [36] Lieb, EH, Thomas-Fermi and related theories of atoms and molecules, Rev. Mod. Phys., 53, 603-641, (1981) · Zbl 1049.81679 [37] Lieb, EH, A model for crystallization: a variation on the Hubbard model, Phys. A, 140, 240-250, (1986) · Zbl 0686.58045 [38] Lieb, E.H., Loss, M.: Analysis. Vol. 14 of Graduate Studies in Mathematics, 2nd edn. American Mathematical Society, Providence, RI (2001) · Zbl 0966.26002 [39] Lieb, E.H., Nachtergaele, B.: Dimerization in ring-shaped molecules: the stability of the Peierls instability. In: XIth International Congress of Mathematical Physics (Paris, 1994), pp. 423-431. International Press, Cambridge, MA (1995) · Zbl 1052.81684 [40] Lieb, EH; Nachtergaele, B., Stability of the Peierls instability for ring-shaped molecules, Phys. Rev. B, 51, 4777, (1995) · Zbl 1052.81684 [41] Lieb, EH; Nachtergaele, B., Bond alternation in ring-shaped molecules: the stability of the Peierls instability, Int. J. Quantum Chem., 58, 699-706, (1996) [42] Lieb, EH; Oxford, S., Improved lower bound on the indirect Coulomb energy, Int. J. Quantum Chem., 19, 427-439, (1980) [43] Lieb, E.H., Seiringer, R.: The Stability of Matter in Quantum Mechanics. Cambridge University Press, Cambridge (2010) · Zbl 1179.81004 [44] Lieb, EH; Simon, B., The Thomas-Fermi theory of atoms, molecules and solids, Adv. Math., 23, 22-116, (1977) · Zbl 0938.81568 [45] Lions, P-L, The concentration-compactness principle in the calculus of variations. the locally compact case. II, Ann. Inst. Henri Poincaré Anal. Non Linéaire, 1, 223-283, (1984) · Zbl 0704.49004 [46] Lions, P-L, Solutions of Hartree-Fock equations for Coulomb systems, Commun. Math. Phys., 109, 33-97, (1987) · Zbl 0618.35111 [47] Morrey, CB, On the analyticity of the solutions of analytic non-linear elliptic systems of partial differential equations. I. analyticity in the interior, Am. J. Math., 80, 198-218, (1958) · Zbl 0081.09402 [48] Nam, PT; Bosch, H., Nonexistence in Thomas-Fermi-Dirac-von Weizsäcker theory with small nuclear charges, Math. Phys. Anal. Geom., 20, 6, (2017) · Zbl 1413.35390 [49] Peierls, R.E.: Quantum Theory of Solids. Clarendon Press, Oxford (1955) · Zbl 0068.23207 [50] Perdew, JP; Ziesche, P. (ed.); Eschrig, H. (ed.), Unified theory of exchange and correlation beyond the local density approximation, 11-20, (1991), Berlin [51] Perdew, JP; Burke, K.; Ernzerhof, M., Generalized gradient approximation made simple, Phys. Rev. Lett., 77, 3865-3868, (1996) [52] Perdew, JP; Wang, Y., Accurate and simple analytic representation of the electron-gas correlation energy, Phys. Rev. B, 45, 13244-13249, (1992) [53] Prodan, E., Symmetry breaking in the self-consistent Kohn-Sham equations, J. Phys. A, 38, 5647-5657, (2005) · Zbl 1073.81691 [54] Prodan, E.; Nordlander, P., Hartree approximation. III. symmetry breaking, J. Math. Phys., 42, 3424-3438, (2001) · Zbl 1029.82005 [55] Ricaud, J., On uniqueness and non-degeneracy of anisotropic polarons, Nonlinearity, 29, 1507-1536, (2016) · Zbl 1338.35380 [56] Ricaud, J.: Symétrie et brisure de symétrie pour certains problèmes non linéaires. Ph.D. thesis, Université de Cergy-Pontoise (2017) [57] Seiringer, R., A correlation estimate for quantum many-body systems at positive temperature, Rev. Math. Phys., 18, 233-253, (2006) · Zbl 1102.82003 [58] Serrin, J.; Tang, M., Uniqueness of ground states for quasilinear elliptic equations, Indiana Univ. Math. J., 49, 897-923, (2000) · Zbl 0979.35049 [59] Sherrill, CD; Lee, MS; Head-Gordon, M., On the performance of density functional theory for symmetry-breaking problems, Chem. Phys. Lett., 302, 425-430, (1999) [60] Tod, P.; Moroz, IM, An analytical approach to the Schrödinger-Newton equations, Nonlinearity, 12, 201-216, (1999) · Zbl 0942.35077 [61] Weinstein, MI, Modulational stability of ground states of nonlinear Schrödinger equations, SIAM J. Math. Anal., 16, 472-491, (1985) · Zbl 0583.35028
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.