Matrix elements of unitary group generators in many-fermion correlation problem. I. tensorial approaches
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Matrix elements of unitary group generators in many‑fermion correlation problem. I. tensorial approaches Josef Paldus1 Received: 10 August 2020 / Accepted: 8 September 2020 © Springer Nature Switzerland AG 2020
Abstract The objective of this series of papers is to survey important techniques for the evaluation of matrix elements (MEs) of unitary group generators and their products in the electronic Gel’fand–Tsetlin basis of two-column irreps of U(n) that are essential to the unitary group approach (UGA) to the many-electron correlation problem as handled by configuration interaction and coupled cluster approaches. Attention is also paid to the MEs of one-body spin-dependent operators and of their relationship to a standard spin-independent UGA formalism. The principal goal is to outline basic principles, concepts, and ideas without getting buried in technical details and thus to help an interested reader to follow the detailed developments in the original literature. In this first instalment we focus on tensorial techniques, particularly those designed specifically for UGA purposes, which exploit the spin-adapted tensorial analogues of the standard creation and annihilation operators of the ubiquitous second-quantization formalism. Subsequent instalments will address techniques based on the graphical methods of spin-algebras and on the Green–Gould polynomial formalism. In the “Appendix A” we then provide a succinct historical outline of the origins of the Lie group and Lie algebra concepts. Keywords Unitary group approach (UGA) · Graphical UGA (GUGA) · Manyelectron correlation problem · UGA generator matrix elements (MEs) · One- and two-body spin-independent MEs · Spin-dependent MEs · Tensor oprators · Spinadapted creation and annihilation vector operators
* 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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Journal of Mathematical Chemistry You should be warned that acquaintance with only one of the approaches will deprive you of techniques and understanding reflected by the other approaches ... (Serge Lang, Algebraic Number Theory, Addison Wesley, Reading, MA, 1970, p. 176.)
Exordium This series of articles has been stimulated by a recent revival of the unitary group approach (UGA) to the spin-adaptation in the many-electron correlation problem [1–9], as represented by the full configuration interaction (FCI) quantum Monte Carlo (QMC) method [10–12]. Although, in the meantime, there has been a number of resourceful exploitations of UGA, and of its graphical version GUGA, within the standard CI approaches [13–22]—culminating in the columbus system of programs [23]—a recent marriage of UGA with the QMC methodology [10–12] certainly offers a brand-new, state-of-the-art approach to the problems of the molecular electronic structure. Needless to say that the ubiquitous limited CI approaches, best represented by the columbus program system, have in the meantime also enjoyed a
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