Functional <i>N</i>-Representability in 2-Matrix, 1-Matrix, and Density Functional Theories — Oak Academic Publishing
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Functional <i>N</i>-Representability in 2-Matrix, 1-Matrix, and Density Functional Theories
Grupo Ecuatorianopara el Estudio Experimental y Teórico de Nanosistemas (ETNano), Universidad San Francisco de Quito (USFQ), Edificion Newton, Oficina N102C iego de Robles y VíaInteroceánica, Quito, Ecuador
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Grupo Ecuatorianopara el Estudio Experimental y Teórico de Nanosistemas (ETNano), Universidad San Francisco de Quito (USFQ), Edificion Newton, Oficina N102C iego de Robles y VíaInteroceánica, Quito, Ecuador
,
Grupo Ecuatorianopara el Estudio Experimental y Teórico de Nanosistemas (ETNano), Universidad San Francisco de Quito (USFQ), Edificion Newton, Oficina N102C iego de Robles y VíaInteroceánica, Quito, Ecuador
1 Grupo Ecuatorianopara el Estudio Experimental y Teórico de Nanosistemas (ETNano), Universidad San Francisco de Quito (USFQ), Edificion Newton, Oficina N102C iego de Robles y VíaInteroceánica, Quito, Ecuador
2 Grupo Ecuatorianopara el Estudio Experimental y Teórico de Nanosistemas (ETNano), Universidad San Francisco de Quito (USFQ), Edificion Newton, Oficina N102C iego de Robles y VíaInteroceánica, Quito, Ecuador
3 Grupo Ecuatorianopara el Estudio Experimental y Teórico de Nanosistemas (ETNano), Universidad San Francisco de Quito (USFQ), Edificion Newton, Oficina N102C iego de Robles y VíaInteroceánica, Quito, Ecuador
The N -representability conditions on the reduced second-order reduced density matrix ( 2-RDM ) , impose restrictions not only in the context of reduced density matrix theory ( RDMT ) , but also on functionals advanced in one-matrix theory such as natural orbital functional theory ( NOFT ) , and on functionals depending on the one-electron density such as those of density functional theory ( DFT ) . We review some aspects of the applications of these N -representability conditions in these theories and present some conclusions.
Keywords<i>N</i>-RepresentabilityDFTD1-Matrix TheoryD2-Matrix Theory
D. A. Mazziotti, “Two-Electron Reduced Density Matrix as the Basic Variable in Many-Electron Quantum Chemistry and Physics,” Chemical Review, Vol. 112, No. 1, 2012, pp. 244-262. doi:10.1021/cr2000493
M. Nakata, M. Fukuda and K. Fujisawa, “Variational Approach to Electronic Structure Calculations on Second-Order Reduced Density Matrices and the N-Representability Problem,” Lecture Note Series, Vol. 9, 2012, pp. 1-32.
P. A. M. Dirac, “Quantum Mechanics of Many Electron Systems,” Proceedings of the Royal Society A, Vol. 123, No. 792, 1929, pp. 714-735
P.-O. Lowdin, “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,” Physical Review, Vol. 97, No. 6, 1955, pp. 1474-1489. doi:10.1103/PhysRev.97.1474
R. McWeeny, “Some Recent Advances en Density Matrix Theory,” Reviews of Modern Physics, Vol. 32, No. 2, 1960, pp. 335-369. doi:10.1103/RevModPhys.32.335
P. A. M. Dirac, “Note on Exchange Phenomena in the Thomas Atom,” Proceedings of Cambridge Philosophy Society, Vol. 26, No. 3, 1930, pp. 376-385. doi:10.1017/S0305004100016108
P. A. M. Dirac, “Note on the Interpretation of the Density Matrix in the Many-Electron Problem,” Proceedings of Cambridge Philosophical Society, Vol. 27, No. 2, 1931, pp. 240-243. doi:10.1017/S0305004100010343
K. Husimi, “Some Formal Properties of the Density Matrix,” Proceedings of Physical Mathematical Society of Japan, Vol. 22, 1940, pp. 264-314.
J. E. Mayer, “Electron Correlation,” Physical Review, Vol. 100, No. 6, 1955, pp. 1579-1586.
R. H. Tredgold, “Density Matrix and the Many-Body Problem,” Physical Review, Vol. 105, No. 5, 1957, pp. 1421-1423. doi:10.1103/PhysRev.105.1421
A. J. Coleman, “Structure of Fermion Density Matrices,” Reviews of Modern Physics, Vol. 35, No. 3, 1963, pp. 668-687. doi:10.1103/RevModPhys.35.668
A. J. Coleman, “The Convex Structure of Electrons,” International Journal of Quantum Chemistry, Vol. 11, No. 6, 1977, pp. 907-916. doi:10.1002/qua.560110604
A. J. Coleman, “Reduced Density Operators and the N-Particle Problem,” International Journal of Quantum Chemistry, Vol. 13, No. 1, 1978, pp. 67-82. doi:10.1002/qua.560130106
C. Garrod and J. K. Percus, “Reduction of the N-Particle Variational Problem,” Journal of Mathematical Physics, Vol. 5, No. 12, 1964, pp. 1756-1776. doi:10.1063/1.1704098
E. R. Davidson, “Reduced Density Matrices in Quantum Chemistry,” Academic Press, London, 1976.
J. Cioslowski, “Many-Electron Densities and Reduced Density Matrices,” Kluwer, New York, 2000.
A. J. Coleman and V. I. Yukalov, “Reduced Density Matrices: Coulson’s Challenge,” Springer-Verlag, New York, 2000. doi:10.1007/978-3-642-58304-9
L. Cohen and C. Frishberg, “Hartree-Fock Density Matrix Equation,” Physical Review A, Vol. 13, No. 3, 1976, pp. 4234-4238. doi:10.1103/PhysRevA.13.927
H. Nakatsuji, “Equation for the Direct Determination of the Density Matrix,” Physical Review A, Vol. 14, No. 1, 1976, pp. 41-50. doi:10.1103/PhysRevA.14.41
D. R. Alcoba and C. Valdemoro, “Family of Modified-Contracted Schrodinger Equations,” Physical Review A, Vol. 64, No. 6, 2001, Article ID: 062105.
J. E. Harriman, “Limitation on the Density-Equation Approach to Many-Electron Problems,” Physical Review A, Vol. 19, No. 5, 1979, pp. 1893-1895. doi:10.1103/PhysRevA.19.1893
W. Kutzelnigg, “Generalized K-Particle Brillouin Conditions and Their Use for the Constructlon of Correlated Electronic Wavefunctions,” Chemical Physics Letters, Vol. 64, No. 2, 1979, pp. 383-387. doi:10.1016/0009-2614(79)80537-0
C. Valdemoro, “Approximating the Second-Order Reduced Density Matrix in Terms of the First-Order One,” Physical Review A, Vol. 45, No. 7, 1992, pp. 4462-4467. doi:10.1103/PhysRevA.45.4462
F. Colmenero and C. Valdemoro, “Approximating q-Order Reduced Density Matrices in Terms of the Lower-Order Ones. II. Applications,” Physical Review A, Vol. 47, No. 2, 1993, pp. 979-987. doi:10.1103/PhysRevA.47.979
H. Nakatsuji and K. Yasuda, “Direct Determination of the Quantum-Mechanical Density Matrix Using the Density Equation,” Physical Review Letters, Vol. 76, No. 7, 1996, pp. 1039-1042. doi:10.1103/PhysRevLett.76.1039
D. A. Mazziotti, “Contracted Schrodinger Equation: Determining Quantum Energies and Two-Particle Density Matrices without Wave Functions,” Physical Review A, Vol. 57, No. 7, 1998, pp. 4219-4243. doi:10.1103/PhysRevA.57.4219
D. A. Mazziotti, “Approximate Solution for Electron Correlation through the Use of Schwinger Probes,” Chemical Physics Letters, Vol. 289, No. 6, 1998, pp. 419-427. doi:10.1016/S0009-2614(98)00470-9
D. A. Mazziotti, “Complete Reconstruction of Reduced Density Matrices,” Chemical Physics Letters, Vol. 326, No. 3-4, 2000, pp. 212-218. doi:10.1016/S0009-2614(00)00773-9
D. A. Mazziotti, “Pursuit of N-Representability for the Contracted Schrodinger Equation through Density-Matrix Reconstruction,” Physical Review A, Vol. 60, No. 5, 1999, pp. 3618-3626. doi:10.1103/PhysRevA.60.3618
W. Kutzelnigg and D. Mukherjee, “Direct Determination of the Cumulants of the Reduced Density Matrices,” Chemical Physics Letters, Vol. 317, No. 6, 2000, pp. 567-574. doi:10.1016/S0009-2614(99)01410-4
M. Nooijen, M. Wladyslawski and A. Hazra, “Cumulant Approach to the Direct Calculation of Reduced Density Matrices: A Critical Analysis,” Journal of Chemical Physics, Vol. 118, No. 11, 2003, pp. 4832-4848. doi:10.1063/1.1545779
D. R. Alcoba and C. Valdemoro, “The Correlation Contracted Schrdinger Equation: An Accurate Solution of the G-Particle-Hole Hypervirial,” International Journal of Quantum Chemistry, Vol. 109, No. 14, 2009, pp. 3178-3190. doi:10.1002/qua.21943
D. A. Mazziotti, “Parametrization of the Two-Electron Reduced Density Matrix for Its Direct Calculation without the Many-Electron Wave Function: Generalizations and Applications,” Physical Review A, Vol. 81, No. 6, 2010, Article ID: 062515.
L. Vandenbergue and S. Boyd. “Semidefinite Programming,” SIAM Review, Vol. 38, No. 1, 1996, pp. 49-50. doi:10.1137/1038003
M. J. Todd, “Semidefinite Optimization,” Acta Numerals, Vol. 10, 2001, pp. 515-560. doi:10.1017/S0962492901000071
E. A. Yildirim and M. J. Todd, “Sensitivity Analysis in Linear Programming and Semidefinite Programming Using Interior-Point Methods,” Mathematical Programming, Vol. 90, No. 2, 2001, pp. 229-261. doi:10.1007/PL00011423
S. Boyd and L. Vandenberghe, “Convex Optimization,” Cambridge University Press, Cambridge, 2009.
S. Pironio, M. Navascues and A. Acio, “Convergent Relaxations of Polynomial Optimization Problems with Noncommuting Variables,” SIAM Journal of Optimization, Vol. 20, No. 5, 2010, pp. 2157-2180. doi:10.1137/090760155
M. Nakata, H. Nakatsuji, M. Ehara, M. Fukuda, K. Nakata and K. Fujisawa, “Variational Calculations of Fermion Second-Order Reduced Density Matrices by Semidefinite Programming Algorithm,” Journal of Chemical Physics, Vol. 114, No. 19, 2001, pp. 8282-8292. doi:10.1063/1.1360199
D. A. Mazziotti and R. M. Erdahl, “Uncertainty Relations and Reduced Density Matrices: Mapping Many-Body Quantum Mechanics onto Four Particles,” Physical Review A, Vol. 63, No. 4, 2001, Article ID: 042113.
S. Burer and R. D. C. Monteiro, “A Nonlinear Programming Algorithm for Solving Semidefinite Programs via Low-Rank Factorization,” Mathematical Programming, Vol. 95, 2003, pp. 329-357. doi:10.1007/s10107-002-0352-8
D. A. Mazziotti, “Realization of Quantum Chemistry without Wave Functions through First-Order Semidefinite Programming,” Physical Review Letters, Vol. 93, No. 21, 2004, Article ID: 213001.
D. A. Mazziotti, “Exactness of Wave Functions from Two-Body Exponential Transformations in Many-Body Quantum Theory,” Physical Review A, Vol. 69, No. 1, 2004, Article ID: 012507.
M. Fukuda, B. J. Braams, M. Nakata, M. L. Overton, J. K. Percus, M. Yamashita and Z. Zhao, “Large-Scale Semidefinite Programs in Electronic Structure Calculation,” Mathematical Programming, Vol. 109, 2007, pp. 553-580. doi:10.1007/s10107-006-0027-y
M. Fukuda, M. Nakata and M. Yamashita, “Semidefinite Programming: Formulations and Primal-Dual Interior-Point Methods,” Advanced Chemical Physics, Vol. 134, 2007, pp. 103-118.
M. Nakata, B. J. Braams, K. Fujisawa, M. Fukuda, J. K. Percus, M. Yamashita and Z. Zhao, “Variational Calculation of Second-Order Reduced Density Matrices by Strong N-Representability Conditions and an Accurate Semidefinite Programming Solver,” Journal of Chemical Physics, Vol. 128, No. 16, 2008, Article ID: 164113.
M. Yamashita, K. Fujisawa, M. Fukuda, K. Nakata and M. Nakata, “Algorithm 925: Parallel Solver for Semidefinite Programming Problem Having Sparse Schur Complement Matrix,” ACM Transactions on Mathematical Software, Vol. 39, No. 1, 2012, pp. 1-22. doi:10.1145/2382585.2382591
T. Baumgratz and M. B. Plenio, “Lower Bounds for Ground States of Condensed Matter Systems,” New Journal of Physics, Vol. 14, Article ID: 023027.
D. A. Mazziotti, “Variational Minimisation of Atomic and Molecular Ground-State Energies via the Two-Particle Reduced Density Matrix,” Physical Review A, Vol. 65, No. 6, 2002Article ID: 062511.
D. A. Mazziotti, “Variational Method for Solving the Contracted Schrodinger Equation through a Projection of the N-Particle Power Method onto the Two-Particle Space,” Journal of Chemical Physics, Vol. 116, No. 4, 2002, pp. 1239-1249. doi:10.1063/1.1430257
G. Guidofalvi and D. A. Mazziotti, “Boson Correlation Energies via Variational Minimization with the Two-Particle Reduced Density Matrix: Exact N-Representability Conditions for Harmonic Interactions,” Physical Review A, Vol. 69, No. 4, 2004, Article ID: 042511.
K. Yasuda, “Uniqueness of the Solution of the Contracted Schrodinger Equation,” Physical Review A, Vol. 65, No. 5, 2002, Article ID: 052121.
M. Nakata, M. Ehara and H. Nakatsuji, “Fundamental World of Quantum Chemistry,” Kluwer Academic Publishers, New York, 2003.
D. A. Mazziotti, “Variational Two-Electron Reduced Density Matrix Theory for Many-Electron Atoms and Molecules: Implementation of the Spin- and Symmetry-Adapted T2 Condition through First-Order Semidefinite Programming,” Physical Review A, Vol. 72, No. 3, 2005, Article ID: 032510.
G. Guidofalvi and D. A. Mazziotti, “Computation of Quantum Phase Transitions by Reduced-Density-Matrix Mechanics,” Physical Review A, Vol. 74, No. 1, 2006, Article ID: 012501.
D. R. Alcoba, C. Valdemoro, L. M. Tel and E. Pérez-Romero, “Controlling the N- and S-Representability of the Second-Order Reduced Density Matrix: The Doublet-State Case,” Physical Review A, Vol. 77, No. 4, 2008, Article ID: 042508.
Z. Zhao, B. J. Braams, M. Fukuda, M. L. Overton and J. K. Percus, “The Reduced Density Matrix Method for Electronic Structure Calculations and the Role of Three-Index Representability,” Journal of Chemical Physics, Vol. 120, No. 5, 2004, pp. 2095-2125. doi:10.1063/1.1636721
M. Nakata, B. J. Braams, K. Fujisawa, M. Fukuda, J. K. Percus, M. Yamashita and Z. Zhao, “Variational Calculation of Second-Order Reduced Density Matrices by Strong N-Representability Conditions and an Accurate Semidefinite Programming Solver,” Journal of Chemical Physics, Vol. 128, No. 16, 2008, Article ID: 164113.
D. A. Mazziotti, “Structure of Fermionic Density Matrices: Complete N-Representability Conditions,” Physical Review Letters, Vol. 108, No. 26, 2012, Article ID: 263002.
D. A. Mazziotti, “Significant Conditions for the Two-Ele- ctron Reduced Density Matrix from the Constructive Solution of N-Representability,” Physical Review A, Vol. 85, No. 6, 2012, Article ID: 062507.
M. Levy, “Universal Variational Functionals of Electron Densities, First-Order Density Matrices, and Natural Spin-Orbitals and Solution of the V-Representability Problem,” Proceedings of National Academy Science of the USA, Vol. 76, No. 12, 1979, pp. 6062-6065. doi:10.1073/pnas.76.12.6062
T. L. Gilbert, “Hohenberg-Kohn Theorem for Nonlocal Externai Potentiais,” Physical Review B, Vol. 12, No. 6, 1975, pp. 2111-2120. doi:10.1103/PhysRevB.12.2111
R. A. Donnelly and R. G. Parr, “Elementary Properties of an Energy Functional of the First Order Reduced Density Matrix,” Journal of Chemical Physics, Vol. 69, No. 10, 1978, pp. 4431-4439. doi:10.1063/1.436433
R. A. Donnelly, “On a Fundamental Difference between Energy Functionals Based on First and on Second Order Density Matrices,” Journal of Chemical Physics, Vol. 71, No. 7, 1979, pp. 2874-2879. doi:10.1063/1.438678
S. M. Valone, “Consequences of Extending 1matrix Energy Functionals from Pure State Representable to All Ensemble Representable 1 Matrices,” Journal of Chemical Physics, Vol. 73, No. 3, 1980, pp. 1344-1349. doi:10.1063/1.440249
T. T. Nguyen-Dang, E. V. Ludena and Y. Tal, “Variation of the Energy Functional of the Reduced 1st-Order Density Operator,” Journal of Molecular Structures (Theochem), Vol. 120, 1985, pp. 247-264. doi:10.1016/0166-1280(85)85114-9
E. V. Ludena and A. Sierraalta, “Necessary Conditions for the Mapping of Gamma-into ρ,” Physical Review A, Vol. 32, No. 1, 1985, pp. 19-25. doi:10.1103/PhysRevA.32.19
E. V. Ludena, “Density Matrices and Density Functionals,” Reidel, Dordrecht, 1987. doi:10.1007/978-94-009-3855-7_15
M. Piris, “Natural Orbital Functional Theory,” In: D. A. Mazziotti, Ed., Reduced Density-Matrix Mechanics with Applications to Many-Electron Atoms and Molecules, John Wiley and Sons, New York, 2007, p. 387. doi:10.1002/9780470106600.ch14
A. M. K. Müller, “Explicit Approximate Relation between Reduced Two- and One-Particle Density Matrices,” Physical Letter A, Vol. 105, No. 9, 1984, pp. 446-452. doi:10.1016/0375-9601(84)91034-X
S. Goedecker and C. J. Umrigar, “A Natural Orbital Functional for the Many-Electron Problem,” Physical Review Letters, Vol. 81, No. 4, 1998, pp. 866-870. doi:10.1103/PhysRevLett.81.866
G. Csanyi and T. A. Arias, “Tensor Product Expansions for Correlation in Quantum Many-Body Systems,” Physical Review B, Vol. 61, No. 11, 2000, pp. 7348-7352. doi:10.1103/PhysRevB.61.7348
E. J. Baerends, “Exact Exchange-Correlation Treatment of Dissociated H2 in Density Functional Theory,” Physical Review Letters, Vol. 87, No. 13, 2001, Article ID: 133004.
M. A. Buijse and E. J. Baerends, “An Approximate Exchange-Correlation Hole Density as a Functional of the Natural Orbitals,” Molecular Physics, Vol. 100, No. 4, 2002, pp. 401-421. doi:10.1103/PhysRevB.61.7348
O. Gritsenko, K. Pernal and E. J. Baerends, “An Improved Density Matrix Functional by Physically Motivated Repulsive Corrections,” Journal of Chemical Physics, Vol. 122, No. 20, 2005, Article ID: 204102.
S. Sharma, J. K. Dowhurst, N. N. Lathiotakis and E. K. U. Gross, “Reduced Density Matrix Functional for Many-Electron Systems,” Physical Review B, Vol. 78, No. 20, 2008, Article ID: 201103.
N. N. Lathiotakis, N. Helbig and E. K. U. Gross, “Performance of One-Body Reduced Density-Matrix Functionals for the Homogeneous Electron Gas,” Physical Review B, Vol. 75, No. 19, 2007, Article ID: 195120.
N. N. Lathiotakis and M. A. L. Marques, “Benchmark Calculations for Reduced Density-Matrix Functional Theory,” Journal of Chemical Physics, Vol. 128, No. 18, 2008, Article ID: 184103.
N. N. Lathiotakis, N. Helbig, A. Zacarias and E. K. U. Gross, “A Functional of the One-Body-Reduced Density Matrix Derived from the Homogeneous Electron Gas: Performance for Finite Systems,” Journal of Chemical Physics, Vol. 130, No. 6, 2009, Article ID: 064109.
N. N. Lathiotakis, N. I. Guidopoulos and N. Helbig, “Size Consistency of Explicit Functionals of the Natural Orbitals in Reduced Density Matrix Functional Theory,” Journal of Chemical Physics, Vol. 132, No. 8, 2010, Article ID: 084105.
R. L. Frank, E. H. Lieb, R. Seiringer and H. Siedentrop, “Müller’s Exchange-Correlation Energy in Density-Matrix-Functional Theory,” Physical Review A, Vol. 76, No. 5, 2007, Article ID: 052517.
M. Piris, “A New Approach for the Two-Electron Cumulant in Natural Orbital Functional Theory,” International Journal of Quantum Chemistry, Vol. 106, No. 5, 2006, pp. 1093-1104. doi:10.1002/qua.20858
M. Piris, “Natural Orbital Functional Theory: Molecules and Polymers,” Recent Research Development of Quantum Chemistry, Vol. 4, 2004, pp. 1-26.
M. Piris, J. M. Matxain, X. Lopez and J. M. Ugalde, “Communication: The Role of the Positivity N-Representability Conditions in Natural Orbital Functional Theory,” Journal of Chemical Physics, Vol. 133, No. 11, 2010, Article ID: 111101.
M. Piris, X. Lopez, F. Ruizpérez, J. M. Matxain and J. M. Ugalde, “A Natural Orbital Functional for Multiconfigurational States,” Journal of Chemical Physics, Vol. 134, No. 16, 2011, Article ID: 164102.
J. M. Matxain, M. Piris, J. Uranga, X. Lopez, G. Merino and J. M. Ugalde, “The Nature of Chemical Bonds from PNOF5 Calculations,” Chemical Physics, Vol. 13, No. 9, 2012, pp. 2297-2303. doi:10.1002/cphc.201200205
M. Piris, “A Natural Orbital Functional Based on an Explicit Approach of the Two-Electron Cumulant,” International Journal of Quantum Chemistry, Vol. 113, No. 5, 2013, pp. 620-630. doi:10.1002/qua.24020
K. Pernal, “The Equivalence of the Piris Natural Orbital Functional 5 (PNOF5) and the Antisymmetrized Product of Strongly Orthogonal Geminal Theory,” Computational and Theoretical Chemistry, Vol. 1003, 2013, pp. 127-129. doi:10.1016/j.comptc.2012.08.022
E. V. Ludena, F. Illas and A. Ramrez-Sols, “On the N-Representability and Universality of F[ρ] in the Hohenberg-Kohn-Sham Version of Density Functional Theory,” International Journal of Modern Physics, Vol. 22, No. 25-26, 2008, pp. 4642-4654. doi:10.1142/S0217979208050395
P.-O. Lowdin, “Density Matrices and Density Functionals,” Reidel, Dordrecht, 1987. doi:10.1007/978-94-009-3855-7_3
R. McWeeny, “Density-Functions and Density Functionals,” Philosophical Magazine Part B, Vol. 69, No. 5, 1994, pp. 727-735. doi:10.1080/01418639408240141
E. V. Ludena and J. Keller, “The Importance of Pure-State N-Representability in the Derivation of Extended Kohn-Sham Equations,” Advaced Quantum Chemistry, Vol. 21, 1990, pp. 46-67.
E. S. Kryachko and E. V. Ludena, “Formulation of N-Representable and V-Representable Density-Functional Theory,” Physical Review A, Vol. 43, No. 5, 1991, pp. 2179-2193. doi:10.1103/PhysRevA.43.2179
E. S. Kryachko and E. V. Ludena, “The N-Representability Problem and the Local-Scaling Version of Density Functional Theory,” Condensed Matter Theories, Vol. 7, pp. 229-241. doi:10.1007/978-1-4615-3352-8_22
E. V. Ludena, V. V. Karasiev, A. Artemyev and D. Gomez, “Many-Electron Densities and Reduced Density Ma- trices,” Kluwer, New York, 2000. doi:10.1007/978-1-4615-4211-7_10
O. Bokanowski, “New N-Representability Results Involving Symmetry and Application to the Density-Functional Theory Formalism,” Journal of Mathematical Chemistry, Vol. 26, No. 4, 1999, pp. 271-296. doi:10.1023/A:1019106516542
P. W. Ayers and S. Liu, “Necessary and Sufficient Conditions for the N-Representability of Density Functionals,” Physical Review A, Vol. 75, No. 2, 2007, Article ID: 022514.
P. W. Ayers, R. Cuevas-Saavedra and D. Chakraborty, “A Variational Principle for the Electron Density Using the Exchange Hole & Its Implications for N-Representability,” Physical Letters A, Vol. 376, No. 6-7, 2012, pp. 839-844. doi:10.1016/j.physleta.2012.01.028
W. Kurlancheek and M. Head-Gordon, “Violations of N-Representability from Spin-Unrestricted Orbitals in M?ller-Plesset Perturbation Theory and Related Double-Hybrid Density Functional Theory,” Molecular Physics, Vol. 107, No. 8-12, 2009, pp. 1223-1232. doi:10.1080/00268970902835637
C. Lee, W. Yang and R. G. Parr, “Development of the Colle-Salvetti Correlation-Energy Formula into a Functional of the Elcetron Density,” Physical Review B, Vol. 37, No. 2, 1988, pp. 785-789. doi:10.1103/PhysRevB.37.785
R. Colle and O. Salvetti, “Approximate Calculation of the Correlation Energy or the Closed Shells,” Theoretica Chimica Acta, Vol. 37, No. 4, 1975, pp. 329-334. doi:10.1007/BF01028401
B. Miehlich, A. Savin, H. Stoll and H. Preuss, “Results Obtained with the Correlation-Energy Density Functionals of Becke and Lee, Yang and Parr,” Chemical Physics Letters, Vol. 157, No. 3, 1989, pp. 200-206. doi:10.1016/0009-2614(89)87234-3
R. C. Morrison, “The Non-N-Representability of the Colle-Salvetti Second-Order Reduced Density Matrix,” International Journal of Quantum Chemistry, Vol. 46, No. 4, 1993, pp. 583-587. doi:10.1002/qua.560460406
S. Caratzoulas, “Gaussian Resummation Approximation of the Reference Spin-Reduced Second-Order Density Matrix in the Colle-Salvetti Model for Electron Correlation,” Physical Review A, Vol. 63, No. 6, 2001, Article ID: 062506.
S. Ragot, “Assessment of an Analytical Density Matrix Derived from a Modified Colle-Salvetti Approach to the Electron Gas,” Journal of Chemical Physics, Vol. 132, No. 6, 2010, Article ID: 064104.