Mathematical Models in First-Principles Calculations for Materials Science

  • Hajime Kobayashi
Part of the Mathematics for Industry book series (MFI, volume 5)


The mathematical models used in first-principles calculations aresummarized, and the uses of mathematics in various industries are introduced. The different approximations used to obtain the Hartree-Fock equation from the Schrödinger equation for multi-atom systems are summarized, and difficulties in solving the Hartree-Fock equation in a self-consistent way are presented. Novel algorithms are needed in order to reduce computational costs of large systems.


First-principles calculations Hartree-Fock molecular orbital SCF DFT computer simulation materials science electron correlation 


  1. 1.
    G.B. Bacskay, A quadratically convergent Hartee-Fock (QC-SCF) method. Application to closed shell systems. Chem. Phys. 61, 385 (1981)CrossRefGoogle Scholar
  2. 2.
    C.J. Cerjan, W.H. Miller, On finding transition states. J. Chem. Phys. 75, 2800 (1981)CrossRefGoogle Scholar
  3. 3.
    P.A.M. Dirac, Quantum mechanics of many-electron systems. Proc. Roy. Soc. A123, 714 (1929)CrossRefGoogle Scholar
  4. 4.
    Ö. Farkas, H.B. Schlegel, Methods for optimizing large molecules. II. Quadratic search. J. Chem. Phys. 111, 10806 (1999)CrossRefGoogle Scholar
  5. 5.
    J.B. Foresman, Æ. Frisch, Exploring Chemistry with Electronic Structure Methods, 2nd edn. (Gaussian Inc, Pittsburgh, 1996)Google Scholar
  6. 6.
    K.N. Kudin, G.E. Scuseria, A black-box self-consistent field convergence algorithm: one step closer. J. Chem. Phys. 116, 8255 (2002)CrossRefGoogle Scholar
  7. 7.
    T. Nakajima, Ryoshi kagaku (Shokabo, Japan, 2009) (in Japanese)Google Scholar
  8. 8.
    S. Obara, A. Saika, Efficient recursive computation of molecular integrals over Cartesian Gaussian function. J. Chem. Phys. 84, 3963 (1986)CrossRefGoogle Scholar
  9. 9.
    P. Pulay, Improved SCF convergence acceleration. J. Comput. Chem. 3, 556 (1982)CrossRefGoogle Scholar
  10. 10.
    H.B. Schlegel, Optimization of equilibrium geometries and transition structures. J. Comput. Chem. 3, 214 (1982)CrossRefGoogle Scholar
  11. 11.
    R. Seeger, J.A. Pople, Self-consistent molecular orbital methods. XVI. Numerically stable direct energy minimization procedures for solution of Hartree-Fock equations. J. Chem. Phys. 65, 265 (1976)CrossRefGoogle Scholar
  12. 12.
    A. Szabo, N.S. Ostlund, Modern Quantum Theory (Macmillan, New York, 1982)Google Scholar

Copyright information

© Springer Japan 2014

Authors and Affiliations

  1. 1.Advanced Materials LaboratoriesSony CorporationAtsugiJapan

Personalised recommendations