The Electronic Structure of Smoothly Deformed Crystals: Wannier Functions and the Cauchy–Born Rule

The electronic structure of a smoothly deformed crystal is analyzed for the case when the effective Hamiltonian is a given function of the nuclei by considering the regime when the scale of the deformation is much larger than the lattice parameter. Wannier functions are defined by projecting the Wannier functions for the undeformed crystal to the space spanned by the wave functions of the deformed crystal. The exponential decay of such Wannier functions is proved for the case when the undeformed crystal is an insulator. The celebrated Cauchy–Born rule for crystal lattices is extended to the present situation for electronic structure analysis.

[1]  G. Wannier.  The Structure of Electronic Excitation Levels in Insulating Crystals , 1937 .

[2]  Walter Kohn,et al.  Analytic Properties of Bloch Waves and Wannier Functions , 1959 .

[3]  J. D. Cloizeaux,et al.  Energy Bands and Projection Operators in a Crystal: Analytic and Asymptotic Properties , 1964 .

[4]  J. D. Cloizeaux.  Analytical Properties of n-Dimensional Energy Bands and Wannier Functions , 1964 .

[5]  Tosio Kato.  Perturbation theory for linear operators , 1966 .

[6]  L. Thomas,et al.  Asymptotic behaviour of eigenfunctions for multiparticle Schrödinger operators , 1973 .

[7]  S. Kivelson,et al.  Wannier functions in one-dimensional disordered systems: Application to fractionally charged solitons , 1982 .

[8]  Barry Simon,et al.  Semiclassical analysis of low lying eigenvalues. I. Non-degenerate minima : asymptotic expansions , 1983 .

[9]  G. Nenciu.  Existence of the exponentially localised Wannier functions , 1983 .

[10]  Barry Simon,et al.  Semiclassical analysis of low lying eigenvalues, II. Tunneling* , 1984 .

[11]  R. Parr.  Density-functional theory of atoms and molecules , 1989 .

[12]  R. Dreizler,et al.  Density-Functional Theory , 1990 .

[13]  G. Nenciu,et al.  The Existence of Generalised Wannier Functions for One-Dimensional Systems , 1998 .

[14]  X. Blanc,et al.  From Molecular Models¶to Continuum Mechanics , 2002 .

[15]  Gianluca Panati.  Triviality of Bloch and Bloch–Dirac Bundles , 2006 .

[16]  Weinan E,et al.  Cauchy–Born Rule and the Stability of Crystalline Solids: Static Problems , 2007 .

[17]  W. E,et al.  The Elastic Continuum Limit of the Tight Binding Model* , 2007 .

[18]  E Wei-nan,et al.  Cauchy-Born Rule and the Stability of Crystalline Solids: Dynamic Problems , 2007 .

[19]  M. Orio,et al.  Density functional theory , 2009, Photosynthesis Research.

[20]  E Weinan,et al.  The Kohn-Sham Equation for Deformed Crystals , 2012 .