Abstract
We study the question of existence and uniqueness for the finite temperature Kohn-Sham equations. For finite volumes, a unique soluion is shown to exists if the effective potential satisfies a set of general conditions and the coupling constant is smaller than a certain value. For periodic background potentials, this value is proven to be volume independent. In this case, the finite volume solutions are shown to converge as the thermodynamic limit is considered. The local density approximation is shown to satisfy the general conditions mentioned above.
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REFERENCES
H. Eschrig, The Fundamentals of Density Functional Theory (Teubner Verlagsgesellschaft, Stuttgart, 1996).
D. Hohenberg and W. Kohn, Phys. Rev. B 136:864(1964).
N. D. Mermin, Phys. Rev. A 137:1441(1965).
W. Kohn and L. J. Sham, Phys. Rev. 140:1133(1965).
M. V. Stoitsov and I. Z. Petkov, Ann. Physics 184:121(1988).
I. S. O. Bokanowski and H. Zidani, Nonlinear Anal. 41:33(2000).
H. C. Kaiser and J. Rehberg, Z. Angew. Math. Phys. 50:423(1999).
I. Catto, C. L. Bris, and P. Lions, The Mathematical Theory of Thermodynamics Limits: Thomas-Fermi Type Models (Oxford University Press, Oxford, 1998).
I. Catto, C. L. Bris, and P. L. Lions, Ann. Inst. H. Poincare 18:687(2001).
E. Prodan and P. Nordlander, J. Math. Phys. 42:3390(2001).
E. Prodan and P. Nordlander, J. Math. Phys. 42:3407(2001).
E. Prodan and P. Nordlander, J. Math. Phys. 42:3424(2001).
V. Bach, E. H. Lieb, M. Loss, and J. P. Solovej, Phys. Rev. Lett. 72:2981(1994).
E. H. Lieb, in Density Functional Methods in Physics, R. Dreizler and J. D. Providencia, eds. (Plenum Press, New York, 1985), pp. 31-80.
G. D. Mahan, Many-Particle Physics (Plenum, New York, 1990).
E. Lieb and B. Simon, Commun. Math. Phys. 53:185(1977).
P. Lions, Commun. Math. Phys. 109:33(1987).
B. Simon, Trace Ideals and Their Applications (Cambridge University Press, New York, 1979).
M. Reed and B. Simon, Methods of Modern Mathematical Physics (Academic Press, New York, 1978), Vol. IV.
M. Reed and B. Simon, Methods of Modern Mathematical Physics (Academic Press, New York, 1972), Vol. I.
J. Perdew and Y. Wang, Phys. Rev. B 45:13244(1992).
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Prodan, E., Nordlander, P. On the Kohn-Sham Equations with Periodic Background Potentials. Journal of Statistical Physics 111, 967–992 (2003). https://doi.org/10.1023/A:1022810601639
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DOI: https://doi.org/10.1023/A:1022810601639