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7. Valone SM (1980) Consequences of extending 1 matrix energy functionals from pure-state
representable to all ensemble representable 1 matrices. J Chem Phys 73(3):1344–1349.
doi:10.1063/1.440249
8. Valone SM (1980) A one-to-one mapping between one-particle densities and some normalparticle ensembles. J Chem Phys 73(9):4653–4655. doi:10.1063/1.440656
9. Nguyen-Dang TT, Lude~ na EV, Tal Y (1985) Variation of the energy functional of the
reduced first-order density operator. J Mol Struct (THEOCHEM) 120:247–264.
doi:10.1016/0166-1280(85)85114-9
10. Donnelly RA (1979) Fundamental difference between energy functionals based on 1st-order
and on 2nd-order density matrices. J Chem Phys 71(7):2874–2879. doi:10.1063/1.438678
11. Donnelly RA, Parr RG (1978) Elementary properties of an energy functional of first-order
reduced density matrix. J Chem Phys 69(10):4431–4439. doi:10.1063/1.436433
12. L€ owdin PO (1955) Quantum theory of many-particle systems. I. Physical interpretations by
means of density matrices, natural spin-orbitals, and convergence problems in the method of
configurational interaction. Phys Rev 97(6):1474–1489. doi:10.1103/PhysRev.97.1474
13. Hohenberg P, Kohn W (1964) Inhomogeneous electron gas. Phys Rev B 136(3B):B864–
B871. doi:10.1103/PhysRev.136.B864
14. Lieb EH (1981) Variational principle for many-fermion systems. Phys Rev Lett 46
(7):457–459. doi:10.1103/PhysRevLett.46.457
15. Yasuda K (2001) Correlation energy functional in the density-matrix functional theory. Phys
Rev A 63:032517. doi:10.1103/PhysRevA.63.032517
16. Cioslowski J (2005) New constraints upon the electron–electron repulsion energy functional
of the one-electron reduced density matrix. J Chem Phys 123:164106
17. Cioslowski J, Pernal K, Ziesche P (2002) Systematic construction of approximate one-matrix
functionals for the electron–electron repulsion energy. J Chem Phys 117(21):9560–9566.
doi:10.1063/1.1516804
18. Mu ¨ller AMK (1984) Explicit approximate relation between reduced two-particle and
one-particle density-matrices. Phys Lett A 105(9):446–452. doi:10.1016/0375-9601(84)
91034-X
19. Buijse MA, Baerends EJ (2002) An approximate exchange-correlation hole density as a
functional of the natural orbitals. Mol Phys 100(4):401–421. doi:10.1080/
00268970110070243
20. Buijse MA (1991) Ph.D. thesis, Electron correlation. Fermi and Coulomb holes, dynamical
and nondynamical correlation. Vrije Universiteit, Amsterdam
21. Frank RL, Lieb EH, Seiringer R, Siedentop H (2007) Muller’s exchange-correlation energy in
density-matrix-functional theory. Phys Rev A 76(5):052517. doi:10.1103/PhysRevA.76.
052517
22. Staroverov VN, Scuseria GE (2002) Assessment of simple exchange-correlation energy
functionals of the one-particle density matrix. J Chem Phys 117(6):2489–2495.
doi:10.1063/1.1491395
23. Herbert JM, Harriman JE (2003) Self-interaction in natural orbital functional theory. Chem
Phys Lett 382(1–2):142–149. doi:10.1016/j.cplett.2003.10.057
24. Herbert JM, Harriman JE (2003) N-representability and variational stability in natural orbital
functional theory. J Chem Phys 118(24):10835–10846. doi:10.1063/1.1574787
25. Cohen AJ, Baerends EJ (2002) Variational density matrix functional calculations for the
corrected Hartree and corrected Hartree–Fock functionals. Chem Phys Lett 364
(3–4):409–419. doi:10.1016/S0009-2614(02)01345-3
26. Cioslowski J, Pernal K (2001) Response properties and stability conditions in density matrix
functional theory. J Chem Phys 115(13):5784–5790. doi:10.1063/1.1383292
176
K. Pernal and K.J.H. Giesbertz
7. Valone SM (1980) Consequences of extending 1 matrix energy functionals from pure-state
representable to all ensemble representable 1 matrices. J Chem Phys 73(3):1344–1349.
doi:10.1063/1.440249
8. Valone SM (1980) A one-to-one mapping between one-particle densities and some normalparticle ensembles. J Chem Phys 73(9):4653–4655. doi:10.1063/1.440656
9. Nguyen-Dang TT, Lude~ na EV, Tal Y (1985) Variation of the energy functional of the
reduced first-order density operator. J Mol Struct (THEOCHEM) 120:247–264.
doi:10.1016/0166-1280(85)85114-9
10. Donnelly RA (1979) Fundamental difference between energy functionals based on 1st-order
and on 2nd-order density matrices. J Chem Phys 71(7):2874–2879. doi:10.1063/1.438678
11. Donnelly RA, Parr RG (1978) Elementary properties of an energy functional of first-order
reduced density matrix. J Chem Phys 69(10):4431–4439. doi:10.1063/1.436433
12. L€ owdin PO (1955) Quantum theory of many-particle systems. I. Physical interpretations by
means of density matrices, natural spin-orbitals, and convergence problems in the method of
configurational interaction. Phys Rev 97(6):1474–1489. doi:10.1103/PhysRev.97.1474
13. Hohenberg P, Kohn W (1964) Inhomogeneous electron gas. Phys Rev B 136(3B):B864–
B871. doi:10.1103/PhysRev.136.B864
14. Lieb EH (1981) Variational principle for many-fermion systems. Phys Rev Lett 46
(7):457–459. doi:10.1103/PhysRevLett.46.457
15. Yasuda K (2001) Correlation energy functional in the density-matrix functional theory. Phys
Rev A 63:032517. doi:10.1103/PhysRevA.63.032517
16. Cioslowski J (2005) New constraints upon the electron–electron repulsion energy functional
of the one-electron reduced density matrix. J Chem Phys 123:164106
17. Cioslowski J, Pernal K, Ziesche P (2002) Systematic construction of approximate one-matrix
functionals for the electron–electron repulsion energy. J Chem Phys 117(21):9560–9566.
doi:10.1063/1.1516804
18. Mu ¨ller AMK (1984) Explicit approximate relation between reduced two-particle and
one-particle density-matrices. Phys Lett A 105(9):446–452. doi:10.1016/0375-9601(84)
91034-X
19. Buijse MA, Baerends EJ (2002) An approximate exchange-correlation hole density as a
functional of the natural orbitals. Mol Phys 100(4):401–421. doi:10.1080/
00268970110070243
20. Buijse MA (1991) Ph.D. thesis, Electron correlation. Fermi and Coulomb holes, dynamical
and nondynamical correlation. Vrije Universiteit, Amsterdam
21. Frank RL, Lieb EH, Seiringer R, Siedentop H (2007) Muller’s exchange-correlation energy in
density-matrix-functional theory. Phys Rev A 76(5):052517. doi:10.1103/PhysRevA.76.
052517
22. Staroverov VN, Scuseria GE (2002) Assessment of simple exchange-correlation energy
functionals of the one-particle density matrix. J Chem Phys 117(6):2489–2495.
doi:10.1063/1.1491395
23. Herbert JM, Harriman JE (2003) Self-interaction in natural orbital functional theory. Chem
Phys Lett 382(1–2):142–149. doi:10.1016/j.cplett.2003.10.057
24. Herbert JM, Harriman JE (2003) N-representability and variational stability in natural orbital
functional theory. J Chem Phys 118(24):10835–10846. doi:10.1063/1.1574787
25. Cohen AJ, Baerends EJ (2002) Variational density matrix functional calculations for the
corrected Hartree and corrected Hartree–Fock functionals. Chem Phys Lett 364
(3–4):409–419. doi:10.1016/S0009-2614(02)01345-3
26. Cioslowski J, Pernal K (2001) Response properties and stability conditions in density matrix
functional theory. J Chem Phys 115(13):5784–5790. doi:10.1063/1.1383292
176
K. Pernal and K.J.H. Giesbertz