The 2-matrix of the spin-polarized electron gas: Contraction sum rules and spectral resolutions

Annalen der Physik (Leipzig)

Volumn 13, Issue 3, 2004, Pages 124-148

  Ziesche, P ^a   Tasnadi F ^b,c  

^a MAX PLANCK INSTITUT FÜR PHYSIK KOMPLEXER SYSTEME   (Germany)

^b IFW DRESDEN   (Germany)

^c UNIVERSITY OF DEBRECEN   (Hungary)

Author keywords

Cumulant expansion; Density matrices; Geminals; Spin polarized electron gas; Sum rules

Indexed keywords

APPROXIMATION THEORY; CARRIER CONCENTRATION; EIGENVALUES AND EIGENFUNCTIONS; FOURIER TRANSFORM INFRARED SPECTROSCOPY; FUNCTIONS; LIGHT POLARIZATION; MATRIX ALGEBRA; PARAMETER ESTIMATION; SHRINKAGE; SPECTRUM ANALYSIS;

CUMULANT EXPANSION; DENSITY MATRICES; GEMINALS; SPECTRAL RESOLUTIONS; SPIN POLARIZATION; SPIN POLARIZED ELECTRON GAS; SUM RULES;

ELECTRON GAS;

EID: 1642296105     PISSN: 00033804     EISSN: None     Source Type: Journal    
DOI: 10.1002/andp.200310068     Document Type: Article

References (78)

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39. The general parameter theorem ∂E/∂λ = (∂Ĥ/∂λ) for bound states of Ĥ is due to P. Güttinger, Z. Phys. 73, 169 (1932), cf. his eq. (11). It is later given by W. Pauli, in: Handbuch der Physik, Band XXIV, Teil 1, edited by H. Geiger and K. Scheel (Springer, Berlin, 1933), p. 162; reprinted in: Handbuch der Physik, Band V, Teil 1, edited by S. Flügge (Springer, Berlin, 1958), p. 83. λ can be any parameter (coupling constant, nuclear coordinate, . . .). For a coupling constant A and in lowest order of perturbation theory, the theorem was given already by E. Schrödinger, Ann. Phys. (Leipzig) [4], 80, 437 (1926), cf. his eq. (7''). The theorem is also implicitly contained in eq. (28) of the paper by M. Born and V.Fock, Z. Phys. 51, 165 (1928). -Hellmann and later Feynman explicitly referred to the special case of nuclear coordinates within the Born-Oppenheimer approximation, leading to the 'Hellmann-Feynman' forces upon nuclei: H. Hellmann, Einführung in die Quantenchemie (Deuticke, Leipzig, 1937), pp. 61, 285 (the original russian version is of 23 October 1936: G. Gel'man, Quantenchemie (ONTI, Moscow and Leningrad), p. 428) and R. P. Feynman, Phys. Rev. 56, 340 (1939). Hellmann stressed, that the forces calculated in this way sensitively depend on the electron density used therein. For variationally determined densities cf. S. T. Epstein, The Variation Method in Quantum Chemistry (Academic Press, New York, 1974). At degeneracies the theorem is discussed by O. E. Alon and L. S. Cederbaum, Phys. Rev. B 68, 033105 (2003). The Hellmann-Feynman theorem has been generalized to Gamow states by P. Ziesche, K. Kunze, and B. Milek, J. Phys. A 20, 2859 (1987).
40. The general parameter theorem ∂E/∂λ = (∂Ĥ/∂λ) for bound states of Ĥ is due to P. Güttinger, Z. Phys. 73, 169 (1932), cf. his eq. (11). It is later given by W. Pauli, in: Handbuch der Physik, Band XXIV, Teil 1, edited by H. Geiger and K. Scheel (Springer, Berlin, 1933), p. 162; reprinted in: Handbuch der Physik, Band V, Teil 1, edited by S. Flügge (Springer, Berlin, 1958), p. 83. λ can be any parameter (coupling constant, nuclear coordinate, . . .). For a coupling constant A and in lowest order of perturbation theory, the theorem was given already by E. Schrödinger, Ann. Phys. (Leipzig) [4], 80, 437 (1926), cf. his eq. (7''). The theorem is also implicitly contained in eq. (28) of the paper by M. Born and V.Fock, Z. Phys. 51, 165 (1928). -Hellmann and later Feynman explicitly referred to the special case of nuclear coordinates within the Born-Oppenheimer approximation, leading to the 'Hellmann-Feynman' forces upon nuclei: H. Hellmann, Einführung in die Quantenchemie (Deuticke, Leipzig, 1937), pp. 61, 285 (the original russian version is of 23 October 1936: G. Gel'man, Quantenchemie (ONTI, Moscow and Leningrad), p. 428) and R. P. Feynman, Phys. Rev. 56, 340 (1939). Hellmann stressed, that the forces calculated in this way sensitively depend on the electron density used therein. For variationally determined densities cf. S. T. Epstein, The Variation Method in Quantum Chemistry (Academic Press, New York, 1974). At degeneracies the theorem is discussed by O. E. Alon and L. S. Cederbaum, Phys. Rev. B 68, 033105 (2003). The Hellmann-Feynman theorem has been generalized to Gamow states by P. Ziesche, K. Kunze, and B. Milek, J. Phys. A 20, 2859 (1987).
41. The general parameter theorem ∂E/∂λ = (∂Ĥ/∂λ) for bound states of Ĥ is due to P. Güttinger, Z. Phys. 73, 169 (1932), cf. his eq. (11). It is later given by W. Pauli, in: Handbuch der Physik, Band XXIV, Teil 1, edited by H. Geiger and K. Scheel (Springer, Berlin, 1933), p. 162; reprinted in: Handbuch der Physik, Band V, Teil 1, edited by S. Flügge (Springer, Berlin, 1958), p. 83. λ can be any parameter (coupling constant, nuclear coordinate, . . .). For a coupling constant A and in lowest order of perturbation theory, the theorem was given already by E. Schrödinger, Ann. Phys. (Leipzig) [4], 80, 437 (1926), cf. his eq. (7''). The theorem is also implicitly contained in eq. (28) of the paper by M. Born and V.Fock, Z. Phys. 51, 165 (1928). -Hellmann and later Feynman explicitly referred to the special case of nuclear coordinates within the Born-Oppenheimer approximation, leading to the 'Hellmann-Feynman' forces upon nuclei: H. Hellmann, Einführung in die Quantenchemie (Deuticke, Leipzig, 1937), pp. 61, 285 (the original russian version is of 23 October 1936: G. Gel'man, Quantenchemie (ONTI, Moscow and Leningrad), p. 428) and R. P. Feynman, Phys. Rev. 56, 340 (1939). Hellmann stressed, that the forces calculated in this way sensitively depend on the electron density used therein. For variationally determined densities cf. S. T. Epstein, The Variation Method in Quantum Chemistry (Academic Press, New York, 1974). At degeneracies the theorem is discussed by O. E. Alon and L. S. Cederbaum, Phys. Rev. B 68, 033105 (2003). The Hellmann-Feynman theorem has been generalized to Gamow states by P. Ziesche, K. Kunze, and B. Milek, J. Phys. A 20, 2859 (1987).
42. The general parameter theorem ∂E/∂λ = (∂Ĥ/∂λ) for bound states of Ĥ is due to P. Güttinger, Z. Phys. 73, 169 (1932), cf. his eq. (11). It is later given by W. Pauli, in: Handbuch der Physik, Band XXIV, Teil 1, edited by H. Geiger and K. Scheel (Springer, Berlin, 1933), p. 162; reprinted in: Handbuch der Physik, Band V, Teil 1, edited by S. Flügge (Springer, Berlin, 1958), p. 83. λ can be any parameter (coupling constant, nuclear coordinate, . . .). For a coupling constant A and in lowest order of perturbation theory, the theorem was given already by E. Schrödinger, Ann. Phys. (Leipzig) [4], 80, 437 (1926), cf. his eq. (7''). The theorem is also implicitly contained in eq. (28) of the paper by M. Born and V.Fock, Z. Phys. 51, 165 (1928). -Hellmann and later Feynman explicitly referred to the special case of nuclear coordinates within the Born-Oppenheimer approximation, leading to the 'Hellmann-Feynman' forces upon nuclei: H. Hellmann, Einführung in die Quantenchemie (Deuticke, Leipzig, 1937), pp. 61, 285 (the original russian version is of 23 October 1936: G. Gel'man, Quantenchemie (ONTI, Moscow and Leningrad), p. 428) and R. P. Feynman, Phys. Rev. 56, 340 (1939). Hellmann stressed, that the forces calculated in this way sensitively depend on the electron density used therein. For variationally determined densities cf. S. T. Epstein, The Variation Method in Quantum Chemistry (Academic Press, New York, 1974). At degeneracies the theorem is discussed by O. E. Alon and L. S. Cederbaum, Phys. Rev. B 68, 033105 (2003). The Hellmann-Feynman theorem has been generalized to Gamow states by P. Ziesche, K. Kunze, and B. Milek, J. Phys. A 20, 2859 (1987).
43. The general parameter theorem ∂E/∂λ = (∂Ĥ/∂λ) for bound states of Ĥ is due to P. Güttinger, Z. Phys. 73, 169 (1932), cf. his eq. (11). It is later given by W. Pauli, in: Handbuch der Physik, Band XXIV, Teil 1, edited by H. Geiger and K. Scheel (Springer, Berlin, 1933), p. 162; reprinted in: Handbuch der Physik, Band V, Teil 1, edited by S. Flügge (Springer, Berlin, 1958), p. 83. λ can be any parameter (coupling constant, nuclear coordinate, . . .). For a coupling constant A and in lowest order of perturbation theory, the theorem was given already by E. Schrödinger, Ann. Phys. (Leipzig) [4], 80, 437 (1926), cf. his eq. (7''). The theorem is also implicitly contained in eq. (28) of the paper by M. Born and V.Fock, Z. Phys. 51, 165 (1928). -Hellmann and later Feynman explicitly referred to the special case of nuclear coordinates within the Born-Oppenheimer approximation, leading to the 'Hellmann-Feynman' forces upon nuclei: H. Hellmann, Einführung in die Quantenchemie (Deuticke, Leipzig, 1937), pp. 61, 285 (the original russian version is of 23 October 1936: G. Gel'man, Quantenchemie (ONTI, Moscow and Leningrad), p. 428) and R. P. Feynman, Phys. Rev. 56, 340 (1939). Hellmann stressed, that the forces calculated in this way sensitively depend on the electron density used therein. For variationally determined densities cf. S. T. Epstein, The Variation Method in Quantum Chemistry (Academic Press, New York, 1974). At degeneracies the theorem is discussed by O. E. Alon and L. S. Cederbaum, Phys. Rev. B 68, 033105 (2003). The Hellmann-Feynman theorem has been generalized to Gamow states by P. Ziesche, K. Kunze, and B. Milek, J. Phys. A 20, 2859 (1987).
44. The general parameter theorem ∂E/∂λ = (∂Ĥ/∂λ) for bound states of Ĥ is due to P. Güttinger, Z. Phys. 73, 169 (1932), cf. his eq. (11). It is later given by W. Pauli, in: Handbuch der Physik, Band XXIV, Teil 1, edited by H. Geiger and K. Scheel (Springer, Berlin, 1933), p. 162; reprinted in: Handbuch der Physik, Band V, Teil 1, edited by S. Flügge (Springer, Berlin, 1958), p. 83. λ can be any parameter (coupling constant, nuclear coordinate, . . .). For a coupling constant A and in lowest order of perturbation theory, the theorem was given already by E. Schrödinger, Ann. Phys. (Leipzig) [4], 80, 437 (1926), cf. his eq. (7''). The theorem is also implicitly contained in eq. (28) of the paper by M. Born and V.Fock, Z. Phys. 51, 165 (1928). -Hellmann and later Feynman explicitly referred to the special case of nuclear coordinates within the Born-Oppenheimer approximation, leading to the 'Hellmann-Feynman' forces upon nuclei: H. Hellmann, Einführung in die Quantenchemie (Deuticke, Leipzig, 1937), pp. 61, 285 (the original russian version is of 23 October 1936: G. Gel'man, Quantenchemie (ONTI, Moscow and Leningrad), p. 428) and R. P. Feynman, Phys. Rev. 56, 340 (1939). Hellmann stressed, that the forces calculated in this way sensitively depend on the electron density used therein. For variationally determined densities cf. S. T. Epstein, The Variation Method in Quantum Chemistry (Academic Press, New York, 1974). At degeneracies the theorem is discussed by O. E. Alon and L. S. Cederbaum, Phys. Rev. B 68, 033105 (2003). The Hellmann-Feynman theorem has been generalized to Gamow states by P. Ziesche, K. Kunze, and B. Milek, J. Phys. A 20, 2859 (1987).
45. The general parameter theorem ∂E/∂λ = (∂Ĥ/∂λ) for bound states of Ĥ is due to P. Güttinger, Z. Phys. 73, 169 (1932), cf. his eq. (11). It is later given by W. Pauli, in: Handbuch der Physik, Band XXIV, Teil 1, edited by H. Geiger and K. Scheel (Springer, Berlin, 1933), p. 162; reprinted in: Handbuch der Physik, Band V, Teil 1, edited by S. Flügge (Springer, Berlin, 1958), p. 83. λ can be any parameter (coupling constant, nuclear coordinate, . . .). For a coupling constant A and in lowest order of perturbation theory, the theorem was given already by E. Schrödinger, Ann. Phys. (Leipzig) [4], 80, 437 (1926), cf. his eq. (7''). The theorem is also implicitly contained in eq. (28) of the paper by M. Born and V.Fock, Z. Phys. 51, 165 (1928). -Hellmann and later Feynman explicitly referred to the special case of nuclear coordinates within the Born-Oppenheimer approximation, leading to the 'Hellmann-Feynman' forces upon nuclei: H. Hellmann, Einführung in die Quantenchemie (Deuticke, Leipzig, 1937), pp. 61, 285 (the original russian version is of 23 October 1936: G. Gel'man, Quantenchemie (ONTI, Moscow and Leningrad), p. 428) and R. P. Feynman, Phys. Rev. 56, 340 (1939). Hellmann stressed, that the forces calculated in this way sensitively depend on the electron density used therein. For variationally determined densities cf. S. T. Epstein, The Variation Method in Quantum Chemistry (Academic Press, New York, 1974). At degeneracies the theorem is discussed by O. E. Alon and L. S. Cederbaum, Phys. Rev. B 68, 033105 (2003). The Hellmann-Feynman theorem has been generalized to Gamow states by P. Ziesche, K. Kunze, and B. Milek, J. Phys. A 20, 2859 (1987).
46. The general parameter theorem ∂E/∂λ = (∂Ĥ/∂λ) for bound states of Ĥ is due to P. Güttinger, Z. Phys. 73, 169 (1932), cf. his eq. (11). It is later given by W. Pauli, in: Handbuch der Physik, Band XXIV, Teil 1, edited by H. Geiger and K. Scheel (Springer, Berlin, 1933), p. 162; reprinted in: Handbuch der Physik, Band V, Teil 1, edited by S. Flügge (Springer, Berlin, 1958), p. 83. λ can be any parameter (coupling constant, nuclear coordinate, . . .). For a coupling constant A and in lowest order of perturbation theory, the theorem was given already by E. Schrödinger, Ann. Phys. (Leipzig) [4], 80, 437 (1926), cf. his eq. (7''). The theorem is also implicitly contained in eq. (28) of the paper by M. Born and V.Fock, Z. Phys. 51, 165 (1928). -Hellmann and later Feynman explicitly referred to the special case of nuclear coordinates within the Born-Oppenheimer approximation, leading to the 'Hellmann-Feynman' forces upon nuclei: H. Hellmann, Einführung in die Quantenchemie (Deuticke, Leipzig, 1937), pp. 61, 285 (the original russian version is of 23 October 1936: G. Gel'man, Quantenchemie (ONTI, Moscow and Leningrad), p. 428) and R. P. Feynman, Phys. Rev. 56, 340 (1939). Hellmann stressed, that the forces calculated in this way sensitively depend on the electron density used therein. For variationally determined densities cf. S. T. Epstein, The Variation Method in Quantum Chemistry (Academic Press, New York, 1974). At degeneracies the theorem is discussed by O. E. Alon and L. S. Cederbaum, Phys. Rev. B 68, 033105 (2003). The Hellmann-Feynman theorem has been generalized to Gamow states by P. Ziesche, K. Kunze, and B. Milek, J. Phys. A 20, 2859 (1987).
47. The general parameter theorem ∂E/∂λ = (∂Ĥ/∂λ) for bound states of Ĥ is due to P. Güttinger, Z. Phys. 73, 169 (1932), cf. his eq. (11). It is later given by W. Pauli, in: Handbuch der Physik, Band XXIV, Teil 1, edited by H. Geiger and K. Scheel (Springer, Berlin, 1933), p. 162; reprinted in: Handbuch der Physik, Band V, Teil 1, edited by S. Flügge (Springer, Berlin, 1958), p. 83. λ can be any parameter (coupling constant, nuclear coordinate, . . .). For a coupling constant A and in lowest order of perturbation theory, the theorem was given already by E. Schrödinger, Ann. Phys. (Leipzig) [4], 80, 437 (1926), cf. his eq. (7''). The theorem is also implicitly contained in eq. (28) of the paper by M. Born and V.Fock, Z. Phys. 51, 165 (1928). -Hellmann and later Feynman explicitly referred to the special case of nuclear coordinates within the Born-Oppenheimer approximation, leading to the 'Hellmann-Feynman' forces upon nuclei: H. Hellmann, Einführung in die Quantenchemie (Deuticke, Leipzig, 1937), pp. 61, 285 (the original russian version is of 23 October 1936: G. Gel'man, Quantenchemie (ONTI, Moscow and Leningrad), p. 428) and R. P. Feynman, Phys. Rev. 56, 340 (1939). Hellmann stressed, that the forces calculated in this way sensitively depend on the electron density used therein. For variationally determined densities cf. S. T. Epstein, The Variation Method in Quantum Chemistry (Academic Press, New York, 1974). At degeneracies the theorem is discussed by O. E. Alon and L. S. Cederbaum, Phys. Rev. B 68, 033105 (2003). The Hellmann-Feynman theorem has been generalized to Gamow states by P. Ziesche, K. Kunze, and B. Milek, J. Phys. A 20, 2859 (1987).
48. The general parameter theorem ∂E/∂λ = (∂Ĥ/∂λ) for bound states of Ĥ is due to P. Güttinger, Z. Phys. 73, 169 (1932), cf. his eq. (11). It is later given by W. Pauli, in: Handbuch der Physik, Band XXIV, Teil 1, edited by H. Geiger and K. Scheel (Springer, Berlin, 1933), p. 162; reprinted in: Handbuch der Physik, Band V, Teil 1, edited by S. Flügge (Springer, Berlin, 1958), p. 83. λ can be any parameter (coupling constant, nuclear coordinate, . . .). For a coupling constant A and in lowest order of perturbation theory, the theorem was given already by E. Schrödinger, Ann. Phys. (Leipzig) [4], 80, 437 (1926), cf. his eq. (7''). The theorem is also implicitly contained in eq. (28) of the paper by M. Born and V.Fock, Z. Phys. 51, 165 (1928). -Hellmann and later Feynman explicitly referred to the special case of nuclear coordinates within the Born-Oppenheimer approximation, leading to the 'Hellmann-Feynman' forces upon nuclei: H. Hellmann, Einführung in die Quantenchemie (Deuticke, Leipzig, 1937), pp. 61, 285 (the original russian version is of 23 October 1936: G. Gel'man, Quantenchemie (ONTI, Moscow and Leningrad), p. 428) and R. P. Feynman, Phys. Rev. 56, 340 (1939). Hellmann stressed, that the forces calculated in this way sensitively depend on the electron density used therein. For variationally determined densities cf. S. T. Epstein, The Variation Method in Quantum Chemistry (Academic Press, New York, 1974). At degeneracies the theorem is discussed by O. E. Alon and L. S. Cederbaum, Phys. Rev. B 68, 033105 (2003). The Hellmann-Feynman theorem has been generalized to Gamow states by P. Ziesche, K. Kunze, and B. Milek, J. Phys. A 20, 2859 (1987).
49. The general parameter theorem ∂E/∂λ = (∂Ĥ/∂λ) for bound states of Ĥ is due to P. Güttinger, Z. Phys. 73, 169 (1932), cf. his eq. (11). It is later given by W. Pauli, in: Handbuch der Physik, Band XXIV, Teil 1, edited by H. Geiger and K. Scheel (Springer, Berlin, 1933), p. 162; reprinted in: Handbuch der Physik, Band V, Teil 1, edited by S. Flügge (Springer, Berlin, 1958), p. 83. λ can be any parameter (coupling constant, nuclear coordinate, . . .). For a coupling constant A and in lowest order of perturbation theory, the theorem was given already by E. Schrödinger, Ann. Phys. (Leipzig) [4], 80, 437 (1926), cf. his eq. (7''). The theorem is also implicitly contained in eq. (28) of the paper by M. Born and V.Fock, Z. Phys. 51, 165 (1928). -Hellmann and later Feynman explicitly referred to the special case of nuclear coordinates within the Born-Oppenheimer approximation, leading to the 'Hellmann-Feynman' forces upon nuclei: H. Hellmann, Einführung in die Quantenchemie (Deuticke, Leipzig, 1937), pp. 61, 285 (the original russian version is of 23 October 1936: G. Gel'man, Quantenchemie (ONTI, Moscow and Leningrad), p. 428) and R. P. Feynman, Phys. Rev. 56, 340 (1939). Hellmann stressed, that the forces calculated in this way sensitively depend on the electron density used therein. For variationally determined densities cf. S. T. Epstein, The Variation Method in Quantum Chemistry (Academic Press, New York, 1974). At degeneracies the theorem is discussed by O. E. Alon and L. S. Cederbaum, Phys. Rev. B 68, 033105 (2003). The Hellmann-Feynman theorem has been generalized to Gamow states by P. Ziesche, K. Kunze, and B. Milek, J. Phys. A 20, 2859 (1987).
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