Geometrization for Energy Levels of Isotropic Hyperfine Hamiltonian Block and Related Central Spin Problems for an Arbitrarily Complex Set of Spin-1/2 Nuclei
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 <...
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| Format: | Article |
| Language: | English |
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MDPI AG
2022-12-01
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| Series: | International Journal of Molecular Sciences |
| Subjects: | |
| Online Access: | https://www.mdpi.com/1422-0067/23/23/15199 |
| Summary: | 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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mo>(</mo><mrow><msub><mi>K</mi><mrow><mi>max</mi></mrow></msub><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></semantics></math></inline-formula> block for an isotropic hyperfine Hamiltonian for a radical with spin-<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mstyle scriptlevel="+1"><mfrac bevelled="true"><mn>1</mn><mn>2</mn></mfrac></mstyle></mrow></semantics></math></inline-formula> nuclei, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>n</mi></semantics></math></inline-formula> nuclei produce an <i>n</i>-th order algebraic equation with <i>n</i> 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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>n</mi></semantics></math></inline-formula> spin-<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mstyle scriptlevel="+1"><mfrac bevelled="true"><mn>1</mn><mn>2</mn></mfrac></mstyle></mrow></semantics></math></inline-formula> nuclei with arbitrary positive couplings <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>a</mi><mi>i</mi></msub><mo>,</mo></mrow></semantics></math></inline-formula> take an <i>n</i>-dimensional hyper-ellipsoid with semiaxes <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msqrt><mrow><msub><mi>a</mi><mi>i</mi></msub></mrow></msqrt><mo>,</mo></mrow></semantics></math></inline-formula> stretch it by a factor of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msqrt><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msqrt></mrow></semantics></math></inline-formula> along the spatial diagonal <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mo>(</mo><mrow><mn>1</mn><mo>,</mo><mo> </mo><mn>1</mn><mo>,</mo><mo> </mo><mo>…</mo><mo>,</mo><mo> </mo><mn>1</mn></mrow><mo>)</mo></mrow><mo>,</mo></mrow></semantics></math></inline-formula> read off the semiaxes of thus produced new hyper-ellipsoid <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>q</mi><mi>i</mi></msub><mo>,</mo></mrow></semantics></math></inline-formula> augment the set <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mo>{</mo><mrow><msub><mi>q</mi><mi>i</mi></msub></mrow><mo>}</mo></mrow></mrow></semantics></math></inline-formula> with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>q</mi><mn>0</mn></msub><mo>=</mo><mn>0</mn><mo>,</mo></mrow></semantics></math></inline-formula> and obtain the sought <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></semantics></math></inline-formula> energies as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>E</mi><mi>k</mi></msub><mo>=</mo><mo>−</mo><mstyle scriptlevel="+1"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><msubsup><mi>q</mi><mi>k</mi><mn>2</mn></msubsup><mo>+</mo><mstyle scriptlevel="+1"><mfrac><mn>1</mn><mn>4</mn></mfrac></mstyle><mstyle displaystyle="true"><msub><mo>∑</mo><mi>i</mi></msub><mrow><msub><mi>a</mi><mi>i</mi></msub></mrow></mstyle><mo>.</mo></mrow></semantics></math></inline-formula> 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. |
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| ISSN: | 1661-6596 1422-0067 |