Matrix elements of unitary group generators in many-fermion correlation problem. III. Green-Gould approach
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Matrix elements of unitary group generators in many‑fermion correlation problem. III. Green‑Gould approach Josef Paldus1 Received: 10 August 2020 / Accepted: 8 September 2020 © Springer Nature Switzerland AG 2020
Abstract The third part of our survey series concerning the evaluation of matrix elements (MEs) of the unitary group generators and of their products in the electronic Gel’fand–Tsetlin basis of the two-column irreps of U(n)—which are essential in the unitary group approach (UGA) to the many-electron correlation problem as handled by the configuration interaction (CI) and the coupled cluster (CC) approaches— relies on what we refer to as the Green-Gould (G-G) approach. In addition to the CI and CC methods, the G-G formalism proved to be very helpful in a number of other tasks, particularly in handling of the spin-dependent operators, the density matrices, or partitioned basis sets adapted to a chosen group chain. Keywords Unitary group approach (UGA) · Many-electron correlation problem · UGA generator matrix elements (MEs) · Spin-dependent MEs · Generator polynomial identities · Green-Gould approach
1 Introduction This Part III of our series [1, 2] concerning the evaluation of matrix elements (MEs) of the unitary group generators and of their products within the scope of the unitary group approach (UGA) to the correlation problem of many-electron systems [3–11] is devoted to the ingenious formalism developed by Gould and Chandler [12–15] by relying on an earlier general developments by Green and Bracken [16, 17] and extended by Gould [18–20]. The basics of the Gould–Chandler (G-C) UGA devoted to a spinfree or a spin-independent case, as described by the ab-initio or semiempirical spinindependent Hamiltonians (cf., e.g., Eq. (8) of Ref. [1] (referred to in the following as Part I), was outlined in three papers by Gould and Chandler [12–14]. Unfortunately, an * Josef Paldus [email protected] http://www.math.uwaterloo.ca/~paldus/ 1
Department of Applied Mathematics, University of Waterloo, Waterloo, ON N2L 3G1, Canada
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oversight slipped up into their developments early on, which subsequently required an extensive 14 page erratum [15]. Although, happily, this oversight did not influence the final expressions for the elementary generator MEs, it nonetheless required a correction. Thus, one purpose of this overview is to present G-C formalism in its definite and somewhat simplified form for the benefit of potential readers. The powerful Green–Gould (G-G) formalism was later exploited in a number of other developments, be it the case of partitioned systems [21, 22], the reduced density matrix (RDM) formalism [23, 24], or in the handling of the spin-dependent case [25–31], and in the complete active space configuration interaction (CAS CI) method [32, 33]. We also note here applications to the parastatistics and the Clifford algebra UGA [34] as well as to the para-Fermi algebras [35]. For obvious reasons we will restrict our overview to expoundi
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