By David A. Mazziotti
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J. Coleman and R. M. ), Queen’s Press, Kingston, Ontario, 1969, p. 76. 13. R. M. Erdahl and M. Rosina, The B condition is implied by the G condition, in Reduced Density Operators with Applications to Physical and Chemical Systems II (R. M. ), Queen’s Papers in Pure and Applied Mathematics No. 40, Queen’s University, Kingston, Ontario, 1974, p. 36. historical introductionreferences 17 14. M. Rosina, (a) Direct variational calculation of the two body density matrix; (b) On the unique representation of the two body density matrices corresponding to the AGP wave function; (c) The characterization of the exposed points of a convex set bounded by matrix nonnegativity conditions; (d) Hermitian operator method for calculations within the particle hole space; in Reduced Density Operators with Applications to Physical and Chemical Systems II (R.
An AGP wavefunction [48, 49] is generated from wedge products involving a single geminal gð1; 2Þ: ÉAGP ¼ gð1; 2Þ ^ gð3; 4Þ ^ Á Á Á ^ gðN À 1; NÞ ð75Þ Coleman  has shown that the Hamiltonians bðgÞ 2 BN2 , which have an AGP ground-state wavefunction, are given by bðgÞ ¼ 2 I À ðN À 2Þ 1 DðgÞ ^ 1 I À ðN À 1ÞggÃ ð76Þ DðgÞ ¼ L12 ½ggÃ ð77Þ where 1 Erdahl and Rosina  have demonstrated that the set of bðgÞ Hamiltonians is contained in the convex set deﬁned by the extreme reduced Hamiltonians from 38 david a.
Partial 3-Positivity Conditions Two different partial 3-positivity conditions have been proposed: (i) the lifting conditions of Mazziotti [21, 33], and (ii) the T1 =T2 conditions of Erdahl [27, 34, 38]. The T1 =T2 conditions have been implemented for molecules by Zhao et al.  and Mazziotti . 1. Lifting Conditions The lifted 3-RDMs  are deﬁned by taking the expectation values of particle ayk ^ ak (or 1 À ^nk ¼ ^ak ^ayk ) over the space (or hole) projection operators ^ nk ¼ ^ spanned by the basis functions in the three metric matrices for 2-positivity: 2 D Dijkl ¼ hÈD ayi ^ ayj ^ al ^ak jÉi ij jÈkl i ¼ hÉj^ ð27Þ 2 Q Qijkl ¼ hÈQ ai ^ aj a^yl ^ayk jÉi ij jÈkl i ¼ hÉj^ ð28Þ 2 Gik lj hÉj^ ayi ^ ak a^yj ^al jÉi ð29Þ ¼ G hÈG ik jÈlj i ¼ Q G where jÈD kl i, jÈkl i, and jÈkl i, are ðN À 2Þ-, ðN þ 2Þ-, and N-particle basis functions, respectively.