Simplified LCAO Method for the Periodic Potential Problem

The LCAO, or Bloch, or tight binding, approximation for solids is discussed as an interpolation method, to be used in connection with more accurate calculations made by the cellular or orthogonalized plane-wave methods. It is proposed that the various integrals be obtained as disposable constants, so that the tight binding method will agree with accurate calculations at symmetry points in the Brillouin zone for which these calculations have been made, and that the LCAO method then be used for making calculations throughout the Brillouin zone. A general discussion of the method is given, including tables of matrix components of energy for simple cubic, face-centered and body-centered cubic, and diamond structures. Applications are given to the results of Fletcher and Wohlfarth on Ni, and Howarth on Cu, as illustrations of the fcc case. In discussing the bcc case, the splitting of the energy bands in chromium by an antiferromagnetic alternating potential is worked out, as well as a distribution of energy states for the case of no antiferromagnetism. For diamond, comparisons are made with the calculations of Herman, using the orthogonalized plane-wave method. The case of such crystals as InSb is discussed, and it is shown that their properties fit in with the energy band picture.

[1]  H. Bethe,et al.  Termaufspaltung in Kristallen , 1929 .

[2]  F. Bloch.  Über die Quantenmechanik der Elektronen in Kristallgittern , 1929 .

[3]  A. Wilson,et al.  The Theory of Electronic Semi-Conductors , 1931 .

[4]  N. Mott,et al.  A Theory of the Form of the X-Ray Emission Bands of Metals , 1934 .

[5]  George E. Kimball,et al.  The Electronic Structure of Diamond , 1935 .

[6]  E. Wigner,et al.  Theory of Brillouin Zones and Symmetry Properties of Wave Functions in Crystals , 1936 .

[7]  N. Mott,et al.  The Electronic Specific Heat and X-Ray Absorption of Metals, and Some Other Properties Related to Electron Bands , 1937 .

[8]  Linus Pauling,et al.  The Nature of the Interatomic Forces in Metals , 1938 .

[9]  C. Herring.  Character tables for two space groups , 1942 .

[10]  John Edward Lennard-Jones,et al.  The determination of molecular orbitals , 1949, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[11]  W. Shockley.  Energy Band Structures in Semiconductors , 1950 .

[12]  G. G. Hall.  The molecular orbital theory of chemical valency. VI. Properties of equivalent orbitals , 1950, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[13]  J. Pople,et al.  The molecular orbital theory of chemical valency. IV. The significance of equivalent orbitals , 1950, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[14]  P. Löwdin.  On the Non‐Orthogonality Problem Connected with the Use of Atomic Wave Functions in the Theory of Molecules and Crystals , 1950 .

[15]  Clarence Zener,et al.  Interaction Between the d Shells in the Transition Metals , 1951 .

[16]  John C. Slater,et al.  Magnetic Effects and the Hartree-Fock Equation , 1951 .

[17]  E. Wohlfarth,et al.  Calculation of the density of states curve for the 3d electrons in nickel , 1951 .

[18]  F. Herman.  Electronic Structure of the Diamond Crystal , 1952 .

[19]  K. Ganzhorn.  Notizen: Quantenmechanik der kubisch raumzentrierten Strukturen der Übergangsmetalle , 1952 .

[20]  G. Fletcher.  Density of States Curve for the 3d Electrons in Nickel , 1952 .

[21]  G. G. Hall.  XXVIII. The electronic structure of diamond , 1952 .

[22]  L. Pauling,et al.  A Theory of Ferromagnetism. , 1953, Proceedings of the National Academy of Sciences of the United States of America.

[23]  The Electronic Structure of some Body-Centred Cubic Metals , 1953 .

[24]  G. L. Pearson,et al.  The Magnetoresistance Effect in InSb , 1953 .

[25]  G. G. Hall.  The Electronic Structure of Diamond , 1953 .

[26]  J. Callaway,et al.  Electronic Structure of the Germanium Crystal , 1953 .

[27]  D. Howarth.  Electronic eigenvalues of copper , 1953, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[28]  M. K. Wilkinson,et al.  NEUTRON DIFFRACTION STUDIES OF VARIOUS TRANSITION ELEMENTS , 1953 .