Local density approximations for the energy of a periodic Coulomb model

Mathematical Models and Methods in Applied Sciences

Volumn 13, Issue 8, 2003, Pages 1185-1217

  Bokanowski, Olivier ^a   Grebert, Benoît ^b   Mauser, Norbert J ^c  

^a UNIVERSITÉ PIERRE ET MARIE CURIE   (France)

^b UNIVERSITÉ DE NANTES   (France)

^c UNIVERSITY OF VIENNA   (Austria)

Author keywords

Deformations; High density limit; Local density functional theory; Local scaling transformations; Thomas Fermi von Weizs cker model; Xalpha method

Indexed keywords

EID: 0242643001     PISSN: 02182025     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0218202503002866     Document Type: Article

References (36)

1. V. Bach, Accuracy of mean field approximations for atoms and molecules, Comm. Math. Phys. 155 (1993) 295-310.
2. O. Bokanowski and B. Grébert, A decomposition theorem for wave functions in molecular quantum chemistry, Math. Models Methods Appl. sci. 6 (1996) 437-466.
3. O. Bokanowski and B. Grébert, Deformations of density functions in molecular quantum chemistry, J. Math. Phys. 37 (1996) 1553-1573.
4. O. Bokanowski and B. Grébert, Utilization of deformations in molecular quantum chemistry and application to density-functional theory, Int. J. Quantum Chem. 68 (1998) 221-231.
5. O. Bokanowski, B. Grébert and N. J. Mauser, Approximations de l'énergie cinétique en fonction de la densité pour un modèle de Coulomb périodique, C. R. Acad. Sci. Math. Phys. 329 (1999) 85-90.
6. O. Bokanowski, B. Grébert and N. J. Mauser, Rigorous derivation of the "Xα" exchange potential: A deformation approach, in Density Functional and Ab Initio Theories Applied to Atoms, Molecules and Solids J. Mol Struct. (Theochem), Vol. 501-502 (2000), pp. 47-58.
7. O. Bokanowski, B. Grébert and N. J. Mauser, in Mathematical Models and Methods for Ab Initio Quantum Chemistry, eds. M. Defranceschi and C. Le Bris, Lecture Notes in Chemistry, Vol. 74 (Springer, 2000), pp. 121-131.
8. O. Bokanowski and N. J. Mauser, Local approximation for the Hartree-Fock exchange potential: A deformation approach, Math. Models Methods Appl. Sci. 9 (1999) 941-961.
9. I. Catto, C. Le Bris and P. L. Lions, Limite thermodynamique pour les modèles de type Thomas-Fermi, C. R. Acad. Sci. 322 (1996) 357-364.
10. I. Catto, C. Le Bris and P. L. Lions, Sur la limite thermodynamique pour les modèles de type Hartrse et Hartree-Fock, C. R. Acad. Sci. 327 (1998) 259-266.
11. P. A. M. Dirac, Note on exchange phenomena in the Thomas-Fermi atom, Proc. Cambridge Philos. Soc. 26 (1931) 376-385.
12. B. Dacorogna and J. Moser, On a partial differential equation involving the Jacobian determinant, Ann. Inst. Henri Poincaré, Analyse non linéaire 7 (1990) 1-26.
13. L. De Santis and R. Resta, N-representability and density-functional construction in curvilinear coordinates, Solid State Comm. 106 (1998) 763.
14. G. Friesecke, Pair correlation and exchange phenomena in the free electron gas, Comm. Math. Phys. 184 (1997) 143-171.
15. C. Fefferman and L. A. Seco, On the Dirac and Schwinger corrections to the ground-state energy of an atom, Adv. Maths. 107 (1994) 1-185.
16. P. Gèrard, P. A. Markowich, N. J. Mauser and F. Poupaud, Homogenization limits and Wigner transforms, Comm. Pure Appl. Math. 50 (1997) 323-379.
17. G. M. Graf and J. P. Solovej, A correlation estimate with application to quantum systems with Coulomb interactions, Rev. Math. Phys. 6 (1994) 977-997.
18. C. S. Herz, On the number of lattice points in a convex set, Amer. J. Math. 84 (1962) 126-133.
19. E. Hlawka, Über Integrale auf konvexen Körpern, Monatsh. Math. 54 (1950) 1-36.
20. E. V. Ludeña and E. S. Kryachko, Formulation of N- and v-representable density-functional theory. I. Groundstates, Phys. Rev. A43 (1991) 2179-2193.
21. O. A Ladyzenskaja and N. N. Uralceva, Linejnye I Kvazilinejnye Uravnenija Ellipticeskogo Tipa (Russian) Nauka Moscu, 1964- French translation: Equations aux dérivées partielles de type elliptique (Dunod, 1968).
22. O. A Ladyzenskaja and N. N. Uralceva, Linejnye I Kvazilinejnye Uravnenija Ellipticeskogo Tipa (Russian) Nauka Moscu, 1964. French translation: Equations aux dérivées partielles de type elliptique (Dunod, 1968).
23. E. H. Lieb, Thomas-Fermi and related theory of atoms and molecules, Rev. Mod. Phys. 53 (1981) 603-641.
24. E. H. Lieb, Density functional for Coulomb systems, Int. J. Quantum Chem. 24 (1983) 243.
25. E. H. Lieb and B. Simon, The Thomas-Fermi theory of atoms, molecules and solids, Adv. Math. 23 (1977) 22-116.
26. E. H. Lieb and B. Simon, The Hartree-Fock theory for Coulomb systems, Comm. Math. Phys. 53 (1977) 185-194.
27. N. H. March and W. H. Young, Variational methods based on the density matrix, Proc. Phys. Soc. 72 (1958) 182-192.
28. P. A. Markowich, N. J. Mauser and F. Poupaud, A Wigner function approach to (semi)classical limits: Electrons in a periodic potential, J. Math. Phys. 35 (1994) 1066-1094.
29. P. L. Lions, Solution of Hartree-Fock equations for Coulomb systems, Comm. Math. Phys. 109 (1987) 33-97.
30. R. G. Parr and W. Yang, Density Functional Theory of Atoms and Molecules (Oxford Univ. Press, 1989).
31. I. Petkov, M. Stoitsov and E. Kryachko, Method of local-scaling transformation and DFT in quantum chemistry, part I, Int. J. Quantum. Chem. 29 (1986) 149-161.
32. M. M. Skriganov, Geometric and Arithmetic Methods in the Spectral Theory of Multidimensional Periodic Operators, Proc. Steklov Instit. of Math. (1987) issue 2.
33. J. C. Slater, A simplification of the Hartree-Fock method, Phys. Rev. 81 (1951) 385-390.
34. I. M. Vinogradov, Special Variants of the Method of Trigonometric Sums (Nauka, 1976), English transl. in his Selected Works (Springer-Verlag, 1985).
35. I. M. Vinogradov, Special Variants of the Method of Trigonometric Sums (Nauka, 1976), English transl. in his Selected Works (Springer-Verlag, 1985).
36. Dong Yé, Prescribing the Jacobian determinant in Sobolev spaces, Ann. Inst. Poincaré, Anal. Non Linéaire 11 (1994) 275-296.