Many-electron theory based on a similarity transformation and a condensate reference system
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Many‑electron theory based on a similarity transformation and a condensate reference system Alexander Quandt1 Received: 14 July 2020 / Accepted: 4 November 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract A general method based on a similarity transformation is presented, which treats many-fermion and many-boson systems in an exact fashion. It can be shown that under such a similarity transformation, the many-electron problem will split into an interacting condensate-like bosonic reference system, and a non-Hermitian eigenvalue problem of independent (quasi-) electrons. We discuss several elementary examples to show that the present approach is compatible with the canonical quantum mechanical formalism and that the resulting splitting offers several useful simplifications as compared to other quantum chemical approaches. Keywords Methods of electronic structure calculations · Many-electron systems · Non-Hermitian eigenvalue problem · Fermionic wave functions
1 Introduction Quantum chemistry is marked by a plethora of rather sophisticated ab initio methods and approaches to determine the properties of complex many-electron systems [1]. The central task is always the determination of antisymmetric manyelectron wave functions 𝜓n (x1 , … xn ) , which correspond to the ground state or an excited state of a many-electron Hamiltonian H . At the forefront of quantum chemistry in terms of accuracy are the configuration interaction (CI) methods [2, 3], where the antisymmetric many-electron wave functions are systematically expanded in a large set of Slater determinants. But the unfortunate scaling of the CI methods with the size of a given electronic system prevents these methods from being applied to more than just some small molecules. A somewhat different approach is density functional theory (DFT). Here, the key idea is to shift all the complex correlations contained in a many-electron wave function into density-dependent mean-field terms that are part of an approximate DFT Hamiltonian. The corresponding one-particle density is usually constructed from a reference wave function that is just a single Slater determinant, and which * Alexander Quandt [email protected] 1
School of Physics, University of the Witwatersrand, Private Bag 3, Johannesburg 2050, South Africa
is related to the ground state of a non-interacting reference system [4]. But the resulting one-particle density is supposed to contain the full information necessary to reconstruct the physical observables, which is equivalent to the knowledge of the real many-electron wave function of the system. Based on this particular approach, DFT is computationally inexpensive compared to other ab initio methods [1]. And it has been successfully applied to large molecules, clusters and solids [5], where it generates predictive results, but not with the same numerical accuracies that are provided by CI and similar methods. The question remains if two widely different approaches like CI and DFT already mark the two
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