Kummer variety, geometry of N-representability, and phase transitions

Physical Review A - Atomic, Molecular, and Optical Physics

Volumn 66, Issue 2, 2002, Pages

  Coleman, A John ^a  

^a QUEEN S UNIVERSITY   (Canada)

Author keywords

[No Author keywords available]

Indexed keywords

APPROXIMATION THEORY; ELECTRON ENERGY LEVELS; FERMIONS; GROUND STATE; HAMILTONIANS; PHASE TRANSITIONS; POLYNOMIALS; PROBABILITY DENSITY FUNCTION;

BOSONS; FULL CONFIGURATION INTERACTION; KUMMER VARIETY;

QUANTUM THEORY;

EID: 0036702015     PISSN: 10502947     EISSN: None     Source Type: Journal    
DOI: 10.1103/PhysRevA.66.022503     Document Type: Article

References (34)

1. A. J. Coleman and V. I. Yukalov, Reduced Density Matrices (Springer, New York, 2000).
2. H. Kummer, J. Math. Phys. 8, 2063 (1967).
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9. A. S. Eddington, Fundamental Theory (Cambridge University Press, Cambridge, England, 1989), pp. 19, 247, and 250.
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11. Phase Transitions and Self-organization in Electronic and Molecular Networks, edited by J. C. Phillips and M. F. Thorpe (Kluwer Academic, Dordrecht, 2001), Chap. 2.
12. A. J. Coleman and V. I. Yukalov, Reduced Density Matrices (Ref. [1]), Sec. 5.2.
13. A. J. Coleman and V. I. Yukalov, Reduced Density Matrices (Ref. [1]), p. 80.
14. A. S. Eddington, The Philosophy of Physical Science (Cambridge University Press, Cambridge, England, 1939)
15. Fundamental Theory (Cambridge University Press, Cambridge, England, 1946)
16. bibliography in the valuable biography A. V. Douglas, The Life of Arthur Stanley Eddington (Nelson, New York, 1956). There are wonderful flashes of insight scattered through the huge corpus of his writings.
17. A. N. Whitehead, Principle of Relativity (Cambridge University Press, Cambridge, England, 1922)
18. Process and Reality (Macmillan, New York, 1929). Neither of those two books is easy. The latter is probably the greatest work on metaphysics since Aristotle. Whitehead wrote several other relevant books. He brought to metaphysics a profound knowledge of mathematics, relativity, and quantum theory unique among philosophers.
19. A. Beste, K. Runge, and R. Bartlett, Chem. Phys. Lett. 355, 263 (2002).
20. In addition to references at the end of Chapter 7 of Ref. [1], more recent information about the VNM approach may be found in the following papers. C. Valdemoro, L. M. Tel and E. Perez-Romero, Phys. Rev. A 61, 032507 (2000)
21. D. R. Alcoba and C. Valdemoro, ibid. 64, 062105 (2001)
22. M. Nakata, M. Ehara, K. Yasuda, and H. Nakatsuji, J. Chem. Phys. 112, 8772 (2000)
23. D. A. Mazziotti, Phys. Rev. E 65, 026704 (2002)
24. J. Chem. Phys. 116, 1239 (2002).
25. For the AGP ansatz, in addition to Chapter 4 of Ref. [1], the reader may consult the following. A. J. Coleman, Int. J. Quantum Chem. 63, 23 (1997).
26. D. A. Mazziotti, J. Chem. Phys. 112, 10 125 (2000)
27. Chem. Phys. Lett. 338, 323 (2001).
28. If CSE is accurate, how accurate is it? While there is no "theorem" to answer this question, there certainly is considerable numerical evidence that the CSE or VNM approach meets Yukalov's criteria for a satisfactory theory. I personally believe it does. However, the evidence for this at present pertains chiefly to systems for which the one-particle interaction dominates over two-particles forces. Garrod et al., J. Math. Phys. 16, 923 (1975) found very accurate results for the ground-state energy for beryllium. In the papers of Nakatsuji and colleagues and in the early work of Mazziotti it appeared that variational and CSE methods work well for systems with relatively weak interaction between electrons. In problems of nuclear structure which involved only two-particle forces, even by invoking all known N-representability conditions, results were of quite poor accuracy. This left a doubt in our minds that CSE in its initial formulation would enable us to handle problems in which interaction forces were dominant. In that version of CSE positivity of the 2-matrix is among the N-representability conditions, with the hierarchy truncated by setting the 2-cumulant to zero. The role of cumulants was noted by Mazziotti.
29. The importance of cumulants for RDM theory was also noted, for example, by Kutzeling et al., J. Chem. Phys. 107, 432 (1997)
30. 110, 2800 (1999). Returning to our original hope of finding the ground state by variation, Erdahl added to the traditional P,D,Q conditions, the positivity of the 3-matrix.
31. The results, expounded in Beiyan Jin, Doctoral thesis, Queen's University, 1998
32. R. M. Erdahl and B. Jin, J. Mol. Struct.: THEOCHEM 527, 207 (2000);
33. in Many-Electron Densities and Reduced Density Matrices, edited by Jerzy Cioslowski (Kluwer Academic, Dordrecht, 2000), Chap. 4, pp. 57-84, suggested that several orders of magnitude of accuracy had been gained by imposing this one additional condition. Mazziotti had previously entertained the possibility of demanding the positivity of the 4-matrix. Combining forces, they produced a remarkable paper,
34. David A. Mazziotti and Robert M. Erdahl, Phys. Rev. A 63, 042113 (2001), in which they found lower bounds for the GS energy of the Lipkin model for three values of N and for both moderate and strong interaction between the particles. These results were three to five orders of magnitude better than those given by other currently available methods. Though not exactly the VNM approach, since it is a variational method, it invokved essential ideas from VNM. However, so far we have not obtained a reliable method of finding the energy of excited states.