Additivity in kramers pairs symmetry: Multiplets with up to four unpaired electrons

Bučinský, Lukáš; Malček, Michal; Biskupič, Stanislav

doi:10.1002/qua.25123

Cited by 3 publications

(12 citation statements)

References 48 publications

(72 reference statements)

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“…(61) fixes the relative phase of wave functions Ψ and Ψ. The relations (61) and (62) between Ψ and Ψ have been observed previously for Kramers-restricted Slater determinants [35]. The wave functions Ψ and Ψ are normalized, orthogonal, and share the same eigenvalue (see Appendix C)…”

Section: Time-reversal Generatormentioning

confidence: 64%

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New quantum number for the many-electron Dirac-Coulomb Hamiltonian

2016

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By breaking the spin symmetry in the relativistic domain, a powerful tool in physical sciences was lost. In this work, we examine an alternative of spin symmetry for systems described by the many-electron Dirac-Coulomb Hamiltonian. We show that the square of many-electron operator K+, defined as a sum of individual single-electron time-reversal (TR) operators, is a linear Hermitian operator which commutes with the Dirac-Coulomb Hamiltonian in a finite Fock subspace. In contrast to the square of a standard unitary many-electron TR operator K, the K 2 + has a rich eigenspectrum having potential to substitute spin symmetry in the relativistic domain. We demonstrate that K+ is connected to K through an exponential mapping, in the same way as spin operators are mapped to the spin rotational group. Consequently, we call K+ the generator of the many-electron TR symmetry. By diagonalizing the operator K 2 + in the basis of Kramers-restricted Slater determinants, we introduce the relativistic variant of configuration state functions (CSF), denoted as Kramers CSF. A new quantum number associated with K 2 + has potential to be used in many areas, for instance, (a) to design effective spin Hamiltonians for electron spin resonance spectroscopy of heavy-element containing systems; (b) to increase efficiency of methods for the solution of many-electron problems in relativistic computational chemistry and physics; (c) to define Kramers contamination in unrestricted density functional and Hartree-Fock theory as a relativistic analog of the spin contamination in the nonrelativistic domain.

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Section: Time-reversal Generatormentioning

confidence: 64%

“…In this work, we have shown the connection between the recently proposed time-reversal generator [34,35] K + =…”

Section: Discussionmentioning

confidence: 99%

New quantum number for the many-electron Dirac-Coulomb Hamiltonian

2016

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“…A linear Hermitean operator is again obtained as the square of the

{\overset{K}{true}}_{+}

operator:

{\overset{K}{true}}_{+}^{2} = \sum_{i} {\overset{K}{true}}_{i} \cdot \sum_{j} {\overset{K}{true}}_{j} = \sum_{i, j} {\overset{K}{true}}_{i} {\overset{K}{true}}_{j},

which further simplifies to:

{\overset{K}{true}}_{+}^{2} = - N_{o} + 2 \sum_{i < j} {\overset{K}{true}}_{i} {\overset{K}{true}}_{j},

with N o being the number of unpaired fermions (electrons). Very importantly such a definition allows one to obtain (first phenomenologically in Bucinsky et al) well labeled quantum states with a quantum hierarchy and/or spectrum (degeneracy) as later proved analytically in Komorovsky et al The eigenequation for

{\overset{K}{true}}_{+}^{2}

is the following:

{\overset{K}{true}}_{+}^{2} {normalΨ}_{k} = - k^{2} {normalΨ}_{k},

where k is the quantum number which is closely related to the number of unpaired fermions ( N o ), that is,

k = {- N_{o}, - N_{o} + 2, …., N_{o} - 2, N_{o}}

and the set of

{{normalΨ}_{k}}

eigenfunctions builds the manifold of Kramers configuration space functions (KCSFs). The relation between

\overset{K}{true}

and

{\overset{K}{true}}_{+}

has been established in Komorovsky et al, where it has been shown that

{\overset{K}{true}}_{+}

…”

Section: Introductionmentioning

confidence: 97%

“…Although, Equation establishes an eigenequation of the many‐fermion

{\overset{K}{true}}^{2}

operator, there is no real quantum information available from it (which is so much present in spin algebra). This pitfall has been recently addressed by using additivity in time reversal of many‐fermion systems (

{\overset{K}{true}}_{+}

{\overset{K}{true}}_{+} = \sum_{j} {\overset{K}{true}}_{j} .

…”

Section: Introductionmentioning

confidence: 99%

“…The relation between

\overset{K}{true}

and

{\overset{K}{true}}_{+}

has been established in Komorovsky et al, where it has been shown that

{\overset{K}{true}}_{+}

is the generator of

\overset{K}{true}

and the explicit proof of Equation and the quantum (integer) nature of the

{\overset{K}{true}}_{+}^{2}

eigenvalues is provided. Furthermore, the invariance of

{\overset{K}{true}}_{+}^{2}

KCSFs under

\overset{K}{true}

for even open shell ( N o ) cases has been described analytically, while in odd cases the

\overset{K}{true}

symmetry is obeyed by the

{\overset{K}{true}}_{+}^{2}

KCSFs . Last but not least, a diagonalization method has been adapted in Komorovsky et al to obtain

{\overset{K}{true}}_{+}^{2}

KCSFs.…”

Section: Introductionmentioning

confidence: 99%

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General build up of basis and matrix in the diagonalization approach. Determination of Kramers configuration state functions

Gall

Bučinský

Komorovský

2018

Int J of Quantum Chemistry

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Algorithms to build the b K 1 basis and b K 2 1 matrix representation to obtain the b K 2 1 Kramers configuration space functions (KCSFs) via diagonalization will be formally generalized to an arbitrary number of unpaired (open shell) fermions. Effective build up of the b K 2 1 matrix representation will be outlined (including threading and graphical processing unit parallelism) to subsequently obtain the KCSFs via calling external/numerical library routines for diagonalization. The effective build up of the b K 2 1 matrix representation relays on a binary tree search algorithm to allow evaluation the b K i b K j action on a given b K 1 basis vector. The binary tree search avoids the treatment of zero b K 2 1 matrix elements which leads to an exponential acceleration. The implementation ( b K 1 basis creation, b K 2 1 matrix representation, and b K 2 1 matrix diagonalization) will be done in an all in core and all at once manner, hence the available core memory sets the physical limits in practical applications.Memory limitations, sparsity of the b K 2 1 matrix, general case of n fermions in m spinors, and the application of KCSFs will be put into further perspective.additivity, algorithm, binary tree, diagonalization, time reversal 1 | I N TR ODU C TI ON "Zweideutigkeit" (double degeneracy) of electrons/fermions as denoted by Pauli [1] (1925) is an phenomenological feature which had to be accounted for to allow the appropriate physical descriptions and mathematical formulations in quantum mechanics. This was indeed and with charm achieved by Pauli himself when extending the angular momentum concept to define a spin, [1] canonizing the double degeneracy of electronic states. A spin operator had the proper behavior with respect to commutation with the nonrelativistic Hamiltonian, hence spin defines a constant of motion and/or spin momentum conservation. Spin quantization ( b S 2 and b S z ) is one of the essential symmetries employed in the formulation of principal working equations, for example Hartree-Fock method, in advanced electron correlation methods, and in the treatment of matrix elements and open shell systems. This lead to a thorough development of spin algebra to obtain configuration state functions (CSFs), which describe the eigenfunctions of b S 2 and b S z operators. Nevertheless, spin does not commute with the many-fermion relativistic Hamiltonian, and so the "Zweideutigkeit" (double degeneracy) cannot be resolved using the well exploited spin algebra within the relativistic domain. The relativistic reincarnation of double degeneracy was suggested first by Kramers (1930) [2] who enforced a concept of pairs, denoted Kramers pairs. Two years after Wigner (1932) [3] generalized this concept to time-reversal symmetry, that is, the solution of the Schr€ odinger equation is equivalent (double degenerate) to a time reversal transformation between time t and -t. In the static (time independent) picture, the time reversal transformation operator b K has the following form [3][4][5] at the 4component leve...

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Perspective on Kramers symmetry breaking and restoration in relativistic electronic structure methods for open-shell systems

Kasper

Jenkins

Sun

et al. 2020

The Journal of Chemical Physics

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Without rigorous symmetry constraints, solutions to approximate electronic structure methods may artificially break symmetry. In the case of the relativistic electronic structure, if time-reversal symmetry is not enforced in calculations of molecules not subject to a magnetic field, it is possible to artificially break Kramers degeneracy in open shell systems. This leads to a description of excited states that may be qualitatively incorrect. Despite this, different electronic structure methods to incorporate correlation and excited states can partially restore Kramers degeneracy from a broken symmetry solution. For single-reference techniques, the inclusion of double and possibly triple excitations in the ground state provides much of the needed correction. Formally, however, this imbalanced treatment of the Kramers-paired spaces is a multi-reference problem, and so methods such as complete-active-space methods perform much better at recovering much of the correct symmetry by state averaging. Using multi-reference configuration interaction, any additional corrections can be obtained as the solution approaches the full configuration interaction limit. A recently proposed “Kramers contamination” value is also used to assess the magnitude of symmetry breaking.

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Additivity in kramers pairs symmetry: Multiplets with up to four unpaired electrons

Cited by 3 publications

References 48 publications

New quantum number for the many-electron Dirac-Coulomb Hamiltonian

New quantum number for the many-electron Dirac-Coulomb Hamiltonian

General build up of basis and matrix in the diagonalization approach. Determination of Kramers configuration state functions

Perspective on Kramers symmetry breaking and restoration in relativistic electronic structure methods for open-shell systems

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