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Paired Orbitals for Different Spins equations
Igor Zilberberg, Sergey Ph. Ruzankin
TL;DR
The paper addresses the need for alternatives to standard spin-polarized Hartree–Fock or Kohn–Sham equations that maintain biorthogonal spin pairing. It develops PODS equations using Adams-Gilbert-type operators, yielding non-canonical unrestricted equations suited to paired orbitals and broken-symmetry treatment.
Problem
Standard spin-polarized Hartree–Fock or Kohn–Sham equations do not directly provide equations that fix non-canonical orbitals as biorthogonal spin-paired sets.
Method
PODS equations are derived as non-canonical unrestricted equations using Adams-Gilbert-type operators and separate non-canonicalization operators constructed from spin densities.
Results
The resulting equations fix biorthogonal paired orbitals, with eigenvalues representing squared α–β overlaps or Hartree–Fock/Kohn–Sham orbital energies split by a pairing operator.
Takeaways & Limitations
PODS equations provide a tool for controlling paired magnetic channels and treating broken-symmetry solutions in antiferromagnetic systems.
Abstract
from arXiv · showhide
Eigenvalue-type equations for Lowdin-Amos-Hall spin-paired (corresponding) orbitals are developed to provide an alternative to the standard spin-polarized Hartree-Fock or Kohn-Sham equations. Obtained equations are non-canonical unrestricted Hartree-Fock-type equations in which non-canonical orbitals are fixed to be biorthogonal spin-paired orbitals. To derive paired orbitals for different spins (PODS) equations there has been applied Adams-Gilbert localizing operator approach. PODS equations are especially useful for treatment of the broken-symmetry solutions for antiferromagnetic materials.
I. INTRODUCTION
The paper develops PODS equations to fix biorthogonal spin-paired orbitals within non-canonical unrestricted equations, motivated by controlling magnetic orbital channels and analyzing broken-symmetry systems.
- Non-canonical orbitals: Orbital rotations preserve the determinant, spin-density operators, and total energy, allowing non-canonical orbitals to provide structural insight.This freedom can also support orbital localization on atomic centers.
- Spin-paired orbitals: Paired orbitals, introduced by Löwdin, Amos, and Hall, form a non-canonical orbital representation obtained by SVD of α–β orbital overlaps.The occupied paired sets are biorthogonal, with analogous orthogonality conditions for virtual orbitals.
- Broken-symmetry applications: For antiferromagnetically coupled systems, paired orbitals provide channels for coupling and support interpretations through spin contamination and S^2-expansion configurations.The expansion can assign structures in terms of idealized covalent and charge-transfer configurations.
- PODS equations: The paper introduces PODS equations as non-canonical equations that additionally require the α and β orbital sets to be biorthogonal.The equations are intended to control orbital magnetic channels during self-consistent iterations.
- PODS equations: Adams-Gilbert-type effective operators are constructed from spin densities, while alternative non-canonicalization operators diagonalize α–β orbital overlap.The paper presents two operator choices with corresponding equation sets.
- PODS equations: The resulting eigenvalues encode paired-orbital overlaps or Hartree–Fock/Kohn–Sham orbital energies split by the pairing operator.The first formulation yields squared overlaps between paired α and β orbitals.
II. PAIRED ORBITAL ADAMS-GILBERT-LIKE EQUATIONS
The Adams-Gilbert-like construction defines Hermitian effective operators whose eigenvectors are non-canonical orbitals constrained to form biorthogonal spin pairs. Their eigenvalue structure distinguishes paired, unpaired, and degenerate orbitals during self-consistent iterations.
- The effective operators are Hermitian, so their eigenvectors provide non-canonical Hartree-Fock-type orbitals.The operators combine Fock and density-dependent terms, with arbitrary Hermitian operators entering the construction.
- An additional diagonal-overlap constraint fixes the α and β orbitals as spin-paired biorthogonal sets.Diagonalizing the relevant matrix pairs the α and β occupied orbitals, with analogous treatment for virtual orbitals.
- For systems with more α than β electrons, a set of eigenvalues equals exactly 1 and corresponds to truly unpaired α electrons.These orbitals are the only source of differences between the α and β eigenvalues.
- Degenerate unoccupied α and β orbitals are unpaired and have zero eigenvalues.Occupied orbitals alone contribute to the α and β densities used for the total energy.
- Self-consistency requires distinguishing paired, unpaired, and degenerate orbital groups at every iteration.Ordering orbitals by eigenvalues can cause unwanted mixing near squared overlap values of 0.5; shifting operators avoid this while preserving vectors, densities, and total energy.
III. EDMISTON-RUEDENBERG-LIKE EQUATIONS
The Edmiston-Ruedenberg-like formulation uses non-local operators to select a particular non-canonical Hartree-Fock solution and interpret pairing eigenvalues relative to canonical orbital energies.
- The Edmiston-Ruedenberg approach replaces off-diagonal Fock-matrix elements with matrix elements of a non-local operator to fix a particular non-canonical solution.The same localization idea is applied to the paired-orbital equations.
- A corresponding operator is introduced to construct the effective pairing-operator matrix within this localization-based formulation.
- In a canonical UHF or UKS basis, diagonal elements of the effective operators correspond to the α and β orbital energies.The trace of each matrix equals the sum of its orbital eigenvalues, and block-diagonal structure yields analogous occupied and unoccupied sum rules.
- The effective pairing eigenvalues can be interpreted as α and β one-electron energies split by fields proportional to the opposite-spin density.The splitting preserves the center of gravity of the one-electron levels.
IV. CONCLUDING REMARKS
The work develops PODS unrestricted equations for Hartree-Fock and Kohn-Sham theory, producing biorthogonal paired orbitals with equal paired-spin eigenvalues. The approach has unresolved practical and initialization limitations.
- PODS equations provide modified unrestricted equations applicable to both Hartree-Fock and Kohn-Sham theory.
- PODS orbitals for either spin are biorthogonal and paired, unlike standard unrestricted solutions.
- Paired α and β PODS orbitals have equal eigenvalues.
- The PODS operators and their eigenvalues are not uniquely defined, and two versions have been developed.
- It remains unclear which PODS version is more practical computationally.
- When the initial α and β orbital sets are equivalent, established brokenization recipes must be applied to obtain broken-symmetry solutions.
APPENDIX
The appendix constructs spin-paired orbitals by diagonalizing overlap-related Hermitian matrices and transforming orbitals so corresponding α and β orbitals are paired.
- APPENDIX: The overlap integrals form a rectangular O matrix whose O†O product is Hermitian, positive definite, and rank-limited.O is built from overlaps between occupied α and β orbitals.
- APPENDIX: Unitary diagonalization of O†O and OO† supplies transformed orbital representations for the two spin spaces.Both products are Hermitian and can be diagonalized by unitary matrices.
- APPENDIX: The corresponding matrix elements in occupied β and α orbital bases are expressed through sums over spin-orbital overlaps.These expressions provide the matrix elements used in the diagonalization procedure.
- APPENDIX: The transformed occupied orbitals are paired, with their overlap structure constrained to be diagonal and overlaps chosen non-negative.The pairing is identified with the stated orbital transformation and overlap condition.
- APPENDIX: Analogous diagonalization is applied to matrix elements involving the unoccupied β and α orbitals.The appendix states that the α and β unoccupied-orbital operator matrices are treated similarly.