Topological Analysis of Magnetically Induced Current Densities in Strong Magnetic Fields Using Stagnation Graphs
Abstract
:1. Introduction
2. Theoretical Background
2.1. Magnetically Induced Current Densities
2.2. Topological Characteristics
3. Computational Methods
3.1. Selecting an Appropriate Objective Function
3.2. Optimization Algorithm
Algorithm 1 Trust Region optimization 

3.3. Initial Point Selection
4. Results
4.1. CH${}_{4}$
4.2. C${}_{2}$H${}_{2}$
4.3. C${}_{2}$H${}_{4}$
5. Discussion
6. Conclusions
Supplementary Materials
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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$(\mathit{r},\mathit{s})$  i  ${\mathit{\eta}}_{\mathit{i}}$ Conditions  Magnetotropicity  Classification 

$(3,\pm 1)$  ${\eta}_{1},{\eta}_{2},{\eta}_{3}\in \mathbb{R}$  isolated singularity: saddle  
$(3,\pm 1)$  ${\eta}_{3}\in \mathbb{R},{\eta}_{1}={\eta}_{2}^{\ast}$  $\mathbf{B}\xb7(\nabla \times \mathbf{j})>0$  isolated singularity: paramagnetic focus  
$(3,\pm 1)$  ${\eta}_{3}\in \mathbb{R},{\eta}_{1}={\eta}_{2}^{\ast}$  $\mathbf{B}\xb7(\nabla \times \mathbf{j})<0$  isolated singularity: diamagnetic focus  
$(2,0)$  $1$  ${\eta}_{1},{\eta}_{2}\in \mathbb{R},{\eta}_{3}=0$  stagnation line: saddle  
$(2,0)$  $+1$  $\mathfrak{Re}\left({\eta}_{1}\right),\mathfrak{Re}\left({\eta}_{2}\right)=0,{\eta}_{3}=0$  $\mathbf{B}\xb7(\nabla \times \mathbf{j})>0$  stagnation line: paramagnetic vortex 
$(2,0)$  $+1$  $\mathfrak{Re}\left({\eta}_{1}\right),\mathfrak{Re}\left({\eta}_{2}\right)=0,{\eta}_{3}=0$  $\mathbf{B}\xb7(\nabla \times \mathbf{j})<0$  stagnation line: diamagnetic vortex 
$(0,0)$  ${\eta}_{1},{\eta}_{2},{\eta}_{3}=0$  branching point 
$\left\mathit{B}\right$ / ${\mathit{B}}_{0}$  Energy / ${\mathit{E}}_{\mathbf{h}}$  R${}_{\mathbf{C}}\mathbf{H}$ / bohr  ∢ H–C–H / Degree  Point Group 

0.00  −40.541554  2.0638  109.5  T${}_{\mathrm{d}}$ 
0.05  −40.536608  2.0631, 2.0632  109.4, 109.5  C${}_{3}$ 
0.10  −40.522124  2.0614, 2.0615  109.3, 109.6  C${}_{3}$ 
0.15  −40.499129  2.0585, 2.0604  109.2, 109.8  C${}_{3}$ 
0.20  −40.469243  2.0543, 2.0620  108.9, 110.0  C${}_{3}$ 
$\left\mathit{B}\right$ / ${\mathit{B}}_{0}$  Energy / ${\mathit{E}}_{\mathbf{h}}$  R${}_{\mathbf{C}}\mathbf{C}$ / bohr  R${}_{\mathbf{C}}\mathbf{H}$ / bohr  ∢ H–C–C / Degree  Point Group 

0.00  −77.374420  2.2727  2.0136  180.0  D${}_{\infty \mathrm{h}}$ 
0.05  −77.368853  2.2722  2.0123  180.0  C${}_{2\mathrm{h}}$ 
0.10  −77.352224  2.2703  2.0085  180.0  C${}_{2\mathrm{h}}$ 
0.15  −77.378285  2.5974  2.0679  118.5  C${}_{\mathrm{i}}$ 
0.20  −77.421742  2.6106  2.0658  115.8  C${}_{\mathrm{i}}$ 
$\left\mathit{B}\right$ / ${\mathit{B}}_{0}$  Energy / ${\mathit{E}}_{h}$  R${}_{\mathbf{C}}\mathbf{C}$ / bohr  R${}_{\mathbf{C}}\mathbf{H}$ / bohr  ∢ H–C–H / Degree  Point Group 

0.00  −78.633815  2.5150  2.0532  116.5  D${}_{2\mathrm{h}}$ 
0.05  −78.628954  2.5146  2.0523  116.2  C${}_{2\mathrm{h}}$ 
0.10  −78.614244  2.5133  2.0497  115.2  C${}_{2\mathrm{h}}$ 
0.15  −78.589425  2.5110  2.0457  113.7  C${}_{2\mathrm{h}}$ 
0.20  −78.554391  2.5084  2.0407  111.5  C${}_{2\mathrm{h}}$ 
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Irons, T.J.P.; Garner, A.; Teale, A.M. Topological Analysis of Magnetically Induced Current Densities in Strong Magnetic Fields Using Stagnation Graphs. Chemistry 2021, 3, 916934. https://doi.org/10.3390/chemistry3030067
Irons TJP, Garner A, Teale AM. Topological Analysis of Magnetically Induced Current Densities in Strong Magnetic Fields Using Stagnation Graphs. Chemistry. 2021; 3(3):916934. https://doi.org/10.3390/chemistry3030067
Chicago/Turabian StyleIrons, Tom J. P., Adam Garner, and Andrew M. Teale. 2021. "Topological Analysis of Magnetically Induced Current Densities in Strong Magnetic Fields Using Stagnation Graphs" Chemistry 3, no. 3: 916934. https://doi.org/10.3390/chemistry3030067