Abstract

The tight binding model is a minimal electronic structure model for molecular modeling and simulation. We show that for a finite temperature model, the total energy in this model can be decomposed into site energies, that is, into contributions from each atomic site whose influence on their environment decays exponentially. This result lays the foundation for a rigorous analysis of QM/MM coupling schemes.

Keywords

  1. QM/MM coupling
  2. crystalline defects
  3. tight binding
  4. strong locality

MSC codes

  1. 65N12
  2. 65N25
  3. 70C20
  4. 81V45

Get full access to this article

View all available purchase options and get full access to this article.

References

1.
R. Baer and M. Head-Gordon, Sparsity of the density matrix in Kohn-Sham density functional theory and an assessment of linear system-size scaling methods, Phys. Rev. Lett., 79 (1997), pp. 3962--3965.
2.
M. Benzi, P. Boito, and N. Razouk, Decay properties of spectral projectors with applications to electronic structure, SIAM Rev., 55 (2013), pp. 3--64, http://dx.doi.org/10.1137/100814019.
3.
N. Bernstein, J.R. Kermode, and G. Csányi, Hybrid atomistic simulation methods for materials systems, Rep. Progr. Phys., 72 (2009), 26051 (25 pages).
4.
J. Bezanson, A. Edelman, S. Karpinski, and V. B. Shah, Julia: A Fresh Approach to Numerical Computing, preprint, http://arxiv.org/abs/1411.1607 arXiv:1411.1607 [cs.MS], 2014.
5.
X. Blanc, C. Le Bris, and P.-L. Lions, From molecular models to continuum mechanics, Arch. Ration. Mech. Anal., 164 (2002), pp. 341--381.
6.
X. Blanc, C. Le Bris, and P.-L. Lions, On the energy of some microscopic stochastic lattices, Part I, Arch. Ration. Mech. Anal., 184 (2007), pp. 303--340.
7.
D.R. Bowler and T. Miyazaki, O(N) methods in electronic structure calculations, Rep. Progr. Phys., 75 (2012), pp. 36503--36546.
8.
E. Cancès and C. Le Bris, Mathematical modeling of point defects in materials science, Math. Models Methods Appl. Sci., 23 (2013), pp. 1795--1859.
9.
E. Cancès, A. Deleurence, and M. Lewin, A new approach to the modelling of local defects in crystals: The reduced Hartree-Fock case, Comm. Math. Phys., 281 (2008), pp. 129--177.
10.
E. Cancès, A. Deleurence, and M. Lewin, Non-perturbative embedding of local defects in crystalline materials, J. Phys. Condens. Mat., 20 (2008), 294213 (6 pages).
11.
E. Cancès and V. Ehrlacher, Local defects are always neutral in the Thomas-Fermi-von Weiszäcker theory of crystals, Arch. Ration. Mech. Anal., 202 (2011), pp. 933--973.
12.
E. Cancès, S. Lahbabi, and M. Lewin, Mean-field models for disordered crystals, J. Math. Pures Appl., 100 (2013), pp. 241--274.
13.
E. Cancès and M. Lewin, The dielectric permittivity of crystals in the reduced Hartree-Fock approximation, Arch. Ration. Mech. Anal., 197 (2010), pp. 139--177.
14.
E. Cancès and N. Mourad, A Mathematical Perspective on Density Functional Perturbation Theory, preprint, http://arxiv.org/abs/1405.1348 arXiv:1405.1348 [math-ph], 2014.
15.
I. Catto, C. Le Bris, and P.-L. Lions, The Mathematical Theory of Thermodynamic Limits: Thomas-Fermi Type Models, Oxford Mathematical Monographs, Clarendon Press, Oxford University Press, New York, 1998.
16.
I. Catto, C. Le Bris, and P.-L. Lions, On the thermodynamic limit for Hartree-Fock type models, Ann. Inst. H. Poincaré Anal. Non Linéaire, 18 (2001), pp. 687--760.
17.
H. Chen, Q. Nazar, and C. Ortner, Variational problems for crystalline defects, in preparation.
18.
H. Chen and C. Ortner, QM/MM methods for crystalline defects. Part 2: Consistent energy and force-mixing, Multiscale Model. Simul., to appear; preprint, http://arxiv.org/abs/1509.06627 arXiv:1509.06627 [math.NA], 2015.
19.
J. Chen and J. Lu, Analysis of the Divide-and-Conquer Method for Electronic Structure Calculations, preprint, http://arxiv.org/abs/1411.0066 arXiv:1411.0066 [math.NA], 2014.
20.
J.M. Combes and L. Thomas, Asymptotic behaviour of eigenfunctions for multiparticle Schrödinger operators, Comm. Math. Phys., 34 (1973), pp. 251--270.
21.
G. Csányi, T. Albaret, G. Moras, M.C. Payne, and A. De Vita, Multiscale hybrid simulation methods for material systems, J. Phys. Condens. Matter, 17 (2005), pp. 691--703.
22.
G. Csányi, T. Albaret, M.C. Payne, and A. De Vita, ``Learn on the fly": A hybrid classical and quantum-mechanical molecular dynamics simulation, Phys. Rev. Lett., 93 (2004), 175503 (4 pages).
23.
W. E and J. Lu, The elastic continuum limit of the tight binding model, Chin. Ann. Math. Ser. B, 28 (2007), pp. 665--676.
24.
W. E and J. Lu, The electronic structure of smoothly deformed crystals: Cauchy-Born rule for the nonlinear tight-binding model, Comm. Pure Appl. Math., 63 (2010), pp. 1432--1468.
25.
W. E and J. Lu, The electronic structure of smoothly deformed crystals: Wannier functions and the Cauchy-Born rule, Arch. Ration. Mech. Anal., 199 (2011), pp. 407--433.
26.
W. E and J. Lu, The Kohn-Sham equation for deformed crystals, Mem. Amer. Math. Soc., 221 (2013), 1040.
27.
V. Ehrlacher, C. Ortner, and A.V. Shapeev, Analysis of Boundary Conditions for Crystal Defect Atomistic Simulations, preprint, http://arxiv.org/abs/1306.5334 arXiv:1306.5334 [math.NA], 2013.
28.
F. Ercolessi, Lecture notes on tight-binding molecular dynamics and tight-binding justification of classical potentials, Lecture notes, Università di Udine, Udine, Italy, 2005.
29.
M. Finnis, Interatomic Forces in Condensed Matter, Oxford University Press, Oxford, UK, 2003.
30.
C.L. Fu and K.M. Ho, First-principles calculation of the equilibrium ground-state properties of transition metals: Applications to Nb and Mo, Phys. Rev. B, 28 (1983), pp. 5480--5486.
31.
J. Gao and D.G. Truhlar, Quantum mechanical methods for enzyme kinetics, Annu. Rev. Phys. Chem., 53 (2002), pp. 467--505.
32.
S. Goedecker, Integral representation of the Fermi distribution and its applications in electronic-structure calculations, Phys. Rev. B, 48 (1993), pp. 17573--17575.
33.
S. Goedecker, Linear scaling electronic structure methods, Rev. Modern Phys., 71 (1999), pp. 1085--1123.
34.
S. Goedecker and M. Teter, Tight-binding electronic-structure calculations and tight-binding molecular dynamics with localized orbitals, Phys. Rev. B, 51 (1995), pp. 9455--9464.
35.
C.M. Goringe, D.R. Bowler, and E. Hernández, Tight-binding modelling of materials, Rep. Progr. Phys., 60 (1997), pp. 1447--1512.
36.
R.A. Horn and C.R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, UK, 1991.
37.
S. Ismail-Beigi and T.A. Arias, Locality of the density matrix in metals, semiconductors, and insulators, Phys. Rev. Lett., 82 (1999), pp. 2127--2130.
38.
J. Kermode, T. Albaret, D. Sherman, N. Bernstein, P. Gumbsch, M.C. Payne, G. Csányi, and A. De Vita, Low-speed fracture instabilities in a brittle crystal, Nature, 455 (2008), pp. 1224--1227.
39.
W. Kohn, Analytic properties of Bloch waves and Wannier functions, Phys. Rev., 115 (1959), pp. 809--821.
40.
M. Kohyama and R. Yamamoto, Tight-binding study of grain boundaries in Si: Energies and atomic structures of twist grain boundaries, Phys. Rev. B, 49 (1994), pp. 17102--17117.
41.
G. Kresse and J. Furthmüller, Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set, Phys. Rev. B, 54 (1996), pp. 11169--11186.
42.
S. Lahbabi, The reduced Hartree-Fock model for short-range quantum crystals with nonlocal defects, Ann. Henri Poincaré, 15 (2014), pp. 1403--1452.
43.
S. Lee, J.D. Dow, and O.F. Sankey, Theory of charge-state splittings of deep levels, Phys. Rev. B, 31 (1985), pp. 3910--3914.
44.
E.H. Lieb and B. Simon, The Thomas-Fermi theory of atoms, molecules and solids, Adv. Math., 23 (1977), pp. 22--116.
45.
L. Lin, J. Lu, L. Ying, and W. E, Pole-based approximation of the Fermi-Dirac function, Chin. Ann. Math. Ser. B, 30 (2009), pp. 729--742.
46.
M. Luskin and C. Ortner, Atomistic-to-continuum coupling, Acta Numer., 22 (2013), pp. 397--508.
47.
P. Motamarria, M.R. Nowakb, K. Leiterc, J. Knapc, and V. Gavini, Higher-order adaptive finite-element methods for Kohn-Sham density functional theory, J. Comput. Phys., 253 (2013), pp. 308--343.
48.
F. Nazar and C. Ortner, Locality of the Thomas-Fermi-von Weizsäcker Equations, preprint, http://arxiv.org/abs/1509.06753 arXiv:1509.06753 [math-ph], 2015.
49.
R.W. Nunes, J. Bennetto, and D. Vanderbilt, Structure, barriers, and relaxation mechanisms of kinks in the $90^\circ$ partial dislocation in silicon, Phys. Rev. Lett., 77 (1996), pp. 1516--1519.
50.
S. Ogata, E. Lidorikis, F. Shimojo, A. Nakano, P. Vashishta, and R.K. Kalia, Hybrid finite-element/molecular-dynamic/electronic-density-functional approach to materials simulations on parallel computers, Comput. Phys. Commun., 138 (2001), pp. 143--154.
51.
C. Ortner and F. Theil, Justification of the Cauchy-Born approximation of elastodynamics, Arch. Ration. Mech. Anal., 207 (2013), pp. 1025--1073.
52.
D.A. Papaconstantopoulos, Handbook of the Band Structure of Elemental Solids, From $Z = 1$ To $Z = 112$, Springer, New York, 2015.
53.
E. Prodan and W. Kohn, Nearsightedness of electronic matter, Proc. Natl. Acad. Sci. USA, 102 (2005), pp. 11635--11638.
54.
J.C. Slater and G.F. Koster, Simplified LCAO method for the periodic potential problem, Phys. Rev., 94 (1954), pp. 1498--1524.
55.
J. Tersoff, New empirical approach for the structure and energy of covalent systems, Phys. Rev. B, 37 (1988), pp. 6991--7000.
56.
C.Z. Wang, C.T. Chan, and K.M. Ho, Tight-binding molecular-dynamics study of defects in silicon, Phys. Rev. Lett., 66 (1991), pp. 189--192.
57.
Y. Wang, G.M. Stocks, W.A. Shelton, D.M.C. Nicholson, Z. Szotek, and W.M. Temmerman, Order-N multiple scattering approach to electronic structure calculations, Phys. Rev. Lett., 75 (1995), pp. 2867--2870.
58.
A. Warshela and M. Levitta, Theoretical studies of enzymic reactions: Dielectric, electrostatic and steric stabilization of the carbonium ion in the reaction of lysozyme, J. Molecular Biol., 103 (1976), pp. 227--249.
59.
H. Yukawa, On the interaction of elementary particles, Proc. Phys. Math. Soc. Japan, 17 (1935), pp. 48--57.

Information & Authors

Information

Published In

cover image Multiscale Modeling & Simulation
Multiscale Modeling & Simulation
Pages: 232 - 264
ISSN (online): 1540-3467

History

Submitted: 21 May 2015
Accepted: 21 December 2015
Published online: 25 February 2016

Keywords

  1. QM/MM coupling
  2. crystalline defects
  3. tight binding
  4. strong locality

MSC codes

  1. 65N12
  2. 65N25
  3. 70C20
  4. 81V45

Authors

Affiliations

Funding Information

European Research Council http://dx.doi.org/10.13039/501100000781 : 335120
Engineering and Physical Sciences Research Council http://dx.doi.org/10.13039/501100000266 : EP/J021377/1
Leverhulme Trust http://dx.doi.org/10.13039/501100000275 : Philip Leverhulme Prize

Metrics & Citations

Metrics

Citations

If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.

Cited By

View Options

View options

PDF

View PDF

Figures

Tables

Media

Share

Share

Copy the content Link

Share with email

Email a colleague

Share on social media