Noncrossing theorem for the ground ensembles of systems with noninteger particle numbers

Physical Review A - Atomic, Molecular, and Optical Physics

Volumn 74, Issue 2, 2006, Pages

  Sagvolden, Espen ^a  

^a Tulane University   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

BAND STRUCTURE; DERIVATIVES; GROUND STATE; KINETIC ENERGY; THEOREM PROVING;

CONVEX FUNCTION; HOHENBERG-KOHN THEOREM; NONCROSSING THEOREM; NONINTEGER PARTICLE NUMBERS;

ELECTRONS;

EID: 33747373348     PISSN: 10502947     EISSN: 10941622     Source Type: Journal    
DOI: 10.1103/PhysRevA.74.022506     Document Type: Article

References (31)

1. P. Hohenberg and W. Kohn, Phys. Rev. PRVBAK 0096-8269 10.1103/PhysRev.136.B864 136, B864 (1964).
2. M. Levy, Proc. Natl. Acad. Sci. U.S.A. PNASA6 0027-8424 10.1073/pnas.76.12.6062 76, 6062 (1979).
3. E. H. Lieb, in Physics as Natural Philosophy: Essays in Honor of Laszlo Tisza on his 75th Birthday, edited by, H. Feshbach, and, A. Shimony, (MIT Press, Cambridge, MA, 1982);
4. E. H. Lieb, Int. J. Quantum Chem. IJQCB2 0020-7608 23, 243 (1983);
5. in Density Functional Methods in Physics, edited by, R. M. Dreizler, and, J. da Providencia, (Plenum, New York, 1985).
6. S. M. Valone, J. Chem. Phys. JCPSA6 0021-9606 10.1063/1.440656 73, 4653 (1980).
7. J. P. Perdew, R. G. Parr, M. Levy, and J. L. Balduz, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.49.1691 49, 1691 (1982).
8. J. P. Perdew, in Density Functional Methods in Physics, edited by, R. M. Dreizler, and, J. da Providencia, (Plenum, New York, 1985).
9. J. P. Perdew and S. Kurth, in Density Functionals: Theory and Application, edited by, D. Joubert, Lecture Notes in Physics, Vol. 500 (Springer, Berlin, 1998).
10. M. Levy, Phys. Rev. A PLRAAN 1050-2947 10.1103/PhysRevA.26.1200 26, 1200 (1982).
11. H. Englisch and R. Englisch, Physica A PHYADX 0378-4371 10.1016/0378-4371(83)90254-6 121, 253 (1983).
12. N. D. Mermin, Phys. Rev. PRVAAH 0096-8250 10.1103/PhysRev.137.A1441 137, A1441 (1965). Mermin extends the Hohenberg-Kohn theorem to grand canonical ensembles at T ≠ 0. He uses the property that for every external potential and chemical potential there exists a unique ensemble Γ, 0 such that Ω [Γ, 0] <Ω [Γ] for all Γ, ≠ Γ, 0. From this he constructs a proof by reduction ad absurdum, which is practically identical to the one presented in this text. However, his proof cannot be used in our context. Granted, our search for the ground ensemble is equivalent to searching for the ensemble yielding the lowest grand potential at T=0 K. However, at T=0 K there no longer exists a unique ensemble minimizing the grand potential. Rather, all ensembles with the same particle number and the same energy expectation value create the same value of the grand potential (the opposite would, of course, contradict the observation that minimizing the grand potential and energy are equvalent at T=0 K). Therefore, one has to use a more direct, albeit straightforward, argument presented in Eq. 3.
13. T. C. Li and P. Q. Tong, Phys. Rev. A PLRAAN 1050-2947 10.1103/PhysRevA.31.1950 31, 1950 (1985).
14. F. E. Zahariev and Y. A. Wang, Phys. Rev. A PLRAAN 1050-2947 10.1103/PhysRevA.70.042503 70, 042503 (2004). This proof requires that the ground ensemble contains only J - and (J+1) -particle states. As discussed in Ref. and Sec. 2, this is true if the ground-state energy is a strictly convex function of the particle number; however, it cannot be true in a general case. Since some applications of the noncrossing theorem developed in Sec. 2 involve noninteracting systems, where the above convexity is not as strict, and since the strict convexity even of electronic ground-state energies has never been proven (as discussed in Ref.), it is desirable to take the time to develop a proof that does not depend on convexity.
15. R. G. Parr and W. Yang, Density-Functional Theory of Atoms and Molecules (Oxford University Press, New York, 1989).
16. U. von Barth and L. Hedin, J. Phys. Chem. JPCHAX 0022-3654 5, 1629 (1972).
17. K. Capelle and G. Vignale, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.86.5546 86, 5546 (2001).
18. K. Capelle and G. Vignale, Phys. Rev. B PRBMDO 0163-1829 10.1103/PhysRevB.65.113106 65, 113106 (2002).
19. H. Eschrig and W. E. Pickett, Solid State Commun. SSCOA4 0038-1098 10.1016/S0038-1098(01)00053-9 118, 123 (2001).
20. C. A. Ullrich, Phys. Rev. B PRBMDO 0163-1829 10.1103/PhysRevB.72.073102 72, 073102 (2005).
21. N. I. Gidopoulos, in Progress in Theoretical Chemistry and Physics, edited by, N. I. Gidopoulos, and, S. Wilson, (Kluwer, Dordrecht, 2003), Vol. 14.
22. N. I. Gidopoulos, e-print cond-mat/0510199.
23. J. P. Perdew and Y. Wang, Phys. Rev. B PRBMDO 0163-1829 10.1103/PhysRevB.45.13244 45, 13244 (1992).
24. C. F. von Weizsäcker, Z. Phys. ZEPYAA 0044-3328 10.1007/BF01337700 96, 431 (1935).
25. L. H. Thomas, Proc. Cambridge Philos. Soc. PCPSA4 0068-6735 23, 542 (1927);
26. E. Fermi, Rend. Accad. Naz. Lincei ZZZZZZ 0392-7881 6 602 (1927).
27. D. A. Kirzhnitz, Sov. Phys. JETP SPHJAR 0038-5646 5 64 (1957).
28. C. Herring, Phys. Rev. A PLRAAN 1050-2947 10.1103/PhysRevA.34.2614 34, 2614 (1986);
29. C. Herring and M. Chopra, Phys. Rev. A PLRAAN 1050-2947 10.1103/PhysRevA.37.31 37, 31 (1988).
30. Y. A. Wang, N. Govind, and E. A. Carter, Phys. Rev. B PRBMDO 0163-1829 10.1103/PhysRevB.60.16350 60, 16350 (1999).
31. L. A. Constantin, J. P. Perdew, and L. Pollack (unpublished).