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The operator approach for representing the symmetry of many-electron systems (series I)

Jianhua Wang

Published Apr 1, 2025• 0 citations
Publisher

Abstract

This work reports on a new and effective method that simplifies the many-electron problem. Instead of a wave function, an operator is used to describe the symmetry of the many-electron system through a transformation. It is proven that the two approaches are physically equivalent, but the operator formalism works much more effectively than the wave function formalism. In the operator formalism, the wave function of an n-electron system can be separated into its spatial and spin parts, and the symmetries of identical particles can be correctly described. For example, it can provide the correct spin and spatial symmetry of many-electron atoms or molecules. For many-electron atoms, one can obtain the simultaneous eigenfunction of L̂2 and Ŝ2. Consequently, the spectroscopic terms of a configuration can be obtained, for example, the hydrogen atom in an energy formula with four quantum numbers (n, l, ml, and ms) for each electron in an atom, including the Hartree–Fock theory. We theoretically analyze and discuss the atomic structure of many-electron atoms using this formula.

DOI
10.1063/5.0168829

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This work reports on a new and effective method that simplifies the many-electron problem. Instead of a wave function, an operator is used to describe the symmetry of the many-electron system through a transformation. It is proven that the two approaches are physically equivalent, but the operator formalism works much more effectively than the wave function formalism. In the operator formalism, the wave function of an n-electron system can be separated into its spatial and spin parts, and the symmetries of identical particles can be correctly described. For example, it can provide the correct spin and spatial symmetry of many-electron atoms or molecules. For many-electron atoms, one can obtain the simultaneous eigenfunction of L̂2 and Ŝ2. Consequently, the spectroscopic terms of a configuration can be obtained, for example, the hydrogen atom in an energy formula with four quantum numbers (n, l, ml, and ms) for each electron in an atom, including the Hartree–Fock theory. We theoretically analyze and discuss the atomic structure of many-electron atoms using this formula.
Extending the one-particle Schrodinger equation from one to three spatial dimensions enabled us to describe electron states in the hydrogen atom and in other hydrogen-like systems. This chapter generalizes the Schrodinger equation from one to two and then to many particles bound as a single system. For all particles of half-integral spin, the total wave function, as for electrons, is always antisymmetric with respect to interchange, whereas with all particles of integral spin the total wave function is always symmetric. The chemical and spectroscopic characteristics of an element depend on the details of the quantum state of its outermost electrons. The chemical valence depends upon the extent to which the total number of electrons in an atom represents an excess or a deficit with respect to the more stable configuration of a completed shell. The Schrödinger equation for helium contains no explicit mention of the spin states of the electrons.
A system of NNN antisymmetric particles, moving under the influence of a fixed potential and their mutual many-particle interactions, is investigated in the ordinary Hartree-Fock scheme, having the total wave function approximated by a single Slater determinant. It is shown that all the density matrices of various orders, the wave function, and the entire physical situation depends only on a fundamental invariant $\ensuremath{\rho}({x}_{1}, {x}_{2})$, which is identical with the first-order density matrix. The Hartree-Fock equations are expressed in terms of this quantity.The Hartree-Fock equations are also solved by expanding the eigenfunctions in a given complete set, and applications to the MO-LCAO theory of the electronic structure of molecules, and crystals are given. It is shown that, in this scheme, the entire physical situation depends on a charge- and bond-order matrix $R(\ensuremath{\nu}\ensuremath{\mu})$ with respect to the ordinary atomic spin-orbitals involved. The Hartree-Fock equations for this matrix are investigated.Finally, the ionized and excited states are investigated, and it is shown that the Hartree-Fock scheme has a high degree of physical visuality also in case of many-particle interactions. The excitation energy of the system is the difference (${{\ensuremath{\omega}}_{j}}^{\ensuremath{'}}\ensuremath{-}{\ensuremath{\omega}}_{i}$) between two "spin-orbital energies," being eigenvalues to the effective Hamiltonians associated with the two states under consideration.
In this paper we analyze a recently proposed approach for the construction of antisymmetric functions for atomic and molecular systems. It is based on the assumption that the main problems with Hartree-Fock wavefunctions stem from their lack of proper permutation symmetry. This alternative building approach is based on products of a space times a spin function with opposite permutation symmetry. The main argument for devising such factors is that the eigenfunctions of the non-relativistic Hamiltonian are either symmetric or antisymmetric with respect to the transposition of the variables of a pair of electrons. However, since the eigenfunctions of the non-relativistic Hamiltonian are basis for the irreducible representations of the symmetric group they are not necessarily symmetric or antisymmetric, except in the trivial case of two electrons. We carry out a simple and straightforward general analysis of the symmetry of the eigenfunctions of the non-relativistic Hamiltonian and illustrate our conclusions by means of two exactly-solvable models of N=2N=2N=2 and N=3N=3N=3 identical interacting particles.
For n identical particles of electrons,the spatial configurations ( l f +l g - )with the given permutation symmetry are expressed in terms of n λ standard Young tableaux T [λ] i .The spatial wavefunction  1 of the n electron system related T [λ] 1 is Changed to the non perturbation function  1e .From the hypothesis of equal probability for spatial exchange between electrons with the opposite spins,the operator q +e related to T [λ] 1 is determined,the analytic forms of the non perturbation wavefunctions with the given permutation symmetry are derived,and the formulas for the symmetric spatial wavefunctions of identical electrons are obtained in terms of the Standard Young Tableaux.
The concept of an effective Hamiltonian in Fock space is introduced. It is based on the division of the entire one-particle space into subspaces of ‘‘active’’ and ‘‘inactive’’ orbitals. The effective Fock space Hamiltonian has—for active model states—the same eigenvalues as the full Hamiltonian. The theory outlined in this context differs from that of paper I mainly in a different definition of the ‘‘diagonal part’’ of an operator, and in the fact that the ‘‘quasidegenerate case’’ applies throughout. The separation theorem, and as a consequence the connected diagram theorem, is shown to hold, in a more limited sense though, even for those normalizations where it did not in the context of universal wave and energy operators. Unlike in the theory of the ‘‘universal’’ operators of paper I the Fock space and n-particle Hilbert space approaches with analogous normalizations are no longer equivalent. In particular, the Primas normalization with a fully Lie-algebraic structure does not lead to a connected diagram expansion if it is formulated in n-particle Hilbert space, only so in a Fock space formulation. In n-particle Hilbert space with the present definition of the diagonal part of an operator the normalizations b (‘‘canonical’’) and c (‘‘Primas’’) happen to agree. As an alternative to the construction of the wave and energy operator W and L by perturbation theory the nonperturbative approach is presented as a generalization of the coupled-cluster method, in detail both in the intermediate and in the unitary normalization. In the unitary variant only a linear system for σ (the logarithm of the wave operator) has to be solved in order to get L correct through fifth order in perturbation theory with important contributions of higher orders included. A generalization of the Hartree–Fock method to Fock space theory is outlined, which guarantees stationarity of all (active) eigenstates with respect to one-particle transformations. A generalized electron pair theory is also defined. An analysis of the necessary computational steps shows that the nonperturbative approaches do not require significantly more computational effort than perturbation theory to the corresponding order. As a numerical example the H2 molecule in a small basis is discussed.
There are many examples in theoretical physics where a fundamental quantity can be considered a quadratic form ρ=∑iρi=∣Ψ∣2 and the corresponding linear form Ψ=∑iψi is highly relevant for the physical problem under study. This, in particular, is the case of the density and the wave function in quantum mechanics. In the study of N-identical-fermion systems we have the additional feature that Ψ is a function of the 3N configuration space coordinates and ρ is defined in three-dimensional real space. For many-electron systems in the ground state the wave function and the Hamiltonian are to be expressed in terms of the configuration space (CS), a replica of real space for each electron. Here we present a geometric formulation of the CS, of the wave function, of the density, and of the Hamiltonian to compute the electronic structure of the system. Then, using the new geometric notation and the indistinguishability and equivalence of the electrons, we obtain an alternative computational method for the ground state of the system. We present the method and discuss its usefulness and relation to other approaches.