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Proceedings of the Physical Society

Construction of S2 Eigenfunctions by the Method of Spin Operators I: General Theory

F Berencz, Reuben Pauncz

Published Feb 1, 1958• 3 citations
Publisher

Abstract

A general spin operator, a function of the so-called step-up and step-down operators, is suggested which when operating on the eigenfunctions of the total Sz operator related to maximal, minimal and zero projections of the total spin creates each eigenfunction of the total S2 corresponding to its different eigenvalues.
DOI
10.1088/0370-1328/71/2/301

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Construction of S2 Eigenfunctions by the Method of Spin Operators II: Six-electron Systems

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In the first part of this paper a spin operator was introduced which when operating on the eigenfunctions of the total S z operator related to maximal, minimal and zero projections of total spin creates each eigenfunction of the total S 2 corresponding to its different eigenvalues. In this second part a brief discussion of the proposed operator technique developed for the problem of six-electron system is presented.

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Theoretical modeling of many-electron spin systems

Bryan Conner

The study of quantum models for magnetism requires the diagonalization of large matrices. For electrons occupying a lattice of sixteen sites, the dimension of these matrices is 65,536 x 65,536. I have written a program in Mathematica to exploit symmetries of the model and render the diagonalization of the Hamiltonian matrix tractable. Specifically, the program utilizes four different operators: translational in the x and y directions (Tx and Ty), total spin in one direction (Sz), and total spin squared (S2). As symmetries, all of these operators commute with the Hamiltonian and therefore the eigenstates of the Hamiltonian are simultaneously eigenstates of these operators. In practice, the eigenstates of these symmetry operators are easier to discover then those of the Hamiltonian. The most difficult symmetry to exploit is the S2 symmetry. In order to generate this symmetry, I reference Clebsch-Gordan coefficient tables. These tables exist to aid in the addition of angular momentum under the laws of quantum mechanics. The tables are generally used to couple two spins. When adding sixteen spins, the coefficients are multiplied together iteratively, adding one spin at a time up to the desired number. The products of these coefficients are then employed to construct all 65,536 eigenstates of the S2 operator. Combined with the remaining symmetry operators, these states can be used to put the Hamiltonian into a simpler, block diagonal form that is readily diagonalized.

•Jan 1, 2006•0 citations
We develop a systematic framework for the spin adaptation of the cumulants of p-particle reduced density matrices (RDMs), with explicit constructions for p = 1 to 3. These spin-adapted cumulants enable rigorous treatment of both Ŝz and Ŝ2 symmetries in quantum systems, providing a foundation for spin-resolved electronic structure methods. We show that complete spin adaptation-referred to as completeS-representability-can be enforced by constraining the variances of Ŝz and Ŝ2, which require the 2-RDM and 4-RDM, respectively. Importantly, the cumulants of RDMs scale linearly with system size-size-extensive-making them a natural object for incorporating spin symmetries in scalable electronic structure theories. The developed formalism is applicable to density-based methods, one-particle RDM functional theories, and two-particle RDM methods. We further extend the approach to spin-orbit-coupled systems via total angular momentum adaptation. Beyond spin, the framework enables the adaptation of RDM theories to additional symmetries through the construction of suitable irreducible tensor operators.
The study of quantum models for magnetism requires the diagonalization of large matrices. For electrons occupying a lattice of sixteen sites, the dimension of these matrices is 65,536 x 65,536. I have written a program in Mathematica to exploit symmetries of the model and render the diagonalization of the Hamiltonian matrix tractable. Specifically, the program utilizes four different operators: translational in the x and y directions (Tx and Ty), total spin in one direction (Sz), and total spin squared (S2). As symmetries, all of these operators commute with the Hamiltonian and therefore the eigenstates of the Hamiltonian are simultaneously eigenstates of these operators. In practice, the eigenstates of these symmetry operators are easier to discover then those of the Hamiltonian. The most difficult symmetry to exploit is the S2 symmetry. In order to generate this symmetry, I reference Clebsch-Gordan coefficient tables. These tables exist to aid in the addition of angular momentum under the laws of quantum mechanics. The tables are generally used to couple two spins. When adding sixteen spins, the coefficients are multiplied together iteratively, adding one spin at a time up to the desired number. The products of these coefficients are then employed to construct all 65,536 eigenstates of the S2 operator. Combined with the remaining symmetry operators, these states can be used to put the Hamiltonian into a simpler, block diagonal form that is readily diagonalized.