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Geometrization for Energy Levels of Isotropic Hyperfine Hamiltonian Block and Related Central Spin Problems for an Arbitrarily Complex Set of Spin-1/2 Nuclei

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العنوان: Geometrization for Energy Levels of Isotropic Hyperfine Hamiltonian Block and Related Central Spin Problems for an Arbitrarily Complex Set of Spin-1/2 Nuclei
المؤلفون: Dmitri V. Stass
المصدر: International Journal of Molecular Sciences, Vol 23, Iss 15199, p 15199 (2022)
بيانات النشر: MDPI AG
سنة النشر: 2022
المجموعة: Directory of Open Access Journals: DOAJ Articles
مصطلحات موضوعية: hyperfine interaction, central spin Hamiltonian, nuclear hyperpolarization, eigenvalue, matrix factorization, geometric visualization, Biology (General), QH301-705.5, Chemistry, QD1-999
الوصف: Description of interacting spin systems relies on understanding the spectral properties of the corresponding spin Hamiltonians. However, the eigenvalue problems arising here lead to algebraic problems too complex to be analytically tractable. This is already the case for the simplest nontrivial ( K max − 1 ) block for an isotropic hyperfine Hamiltonian for a radical with spin- 1 2 nuclei, where n nuclei produce an n -th order algebraic equation with n independent parameters. Systems described by such blocks are now physically realizable, e.g., as radicals or radical pairs with polarized nuclear spins, appear as closed subensembles in more general radical settings, and have numerous counterparts in related central spin problems. We provide a simple geometrization of energy levels in this case: given n spin- 1 2 nuclei with arbitrary positive couplings a i , take an n -dimensional hyper-ellipsoid with semiaxes a i , stretch it by a factor of n + 1 along the spatial diagonal ( 1 , 1 , … , 1 ) , read off the semiaxes of thus produced new hyper-ellipsoid q i , augment the set { q i } with q 0 = 0 , and obtain the sought n + 1 energies as E k = − 1 2 q k 2 + 1 4 ∑ i a i . This procedure provides a way of seeing things that can only be solved numerically, giving a useful tool to gain insights that complement the numeric simulations usually inevitable here, and shows an intriguing connection to discrete Fourier transform and spectral properties of standard graphs.
نوع الوثيقة: article in journal/newspaper
اللغة: English
تدمد: 1422-0067
1661-6596
Relation: https://www.mdpi.com/1422-0067/23/23/15199; https://doaj.org/toc/1661-6596; https://doaj.org/toc/1422-0067; https://doaj.org/article/7fdb030ca1324051bddc22ee7850107a
DOI: 10.3390/ijms232315199
الاتاحة: https://doi.org/10.3390/ijms232315199
https://doaj.org/article/7fdb030ca1324051bddc22ee7850107a
رقم الانضمام: edsbas.B0C7BB9B
قاعدة البيانات: BASE
الوصف
تدمد:14220067
16616596
DOI:10.3390/ijms232315199