# Partitioning Hückel–London Currents into Cycle Contributions

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

#### Graph Theoretical Background

## 2. The Hückel–London Model as a Superposition of Cycle Contributions

## 3. A Pairing Theorem for HL Currents

**Theorem**

**1.**

**Proof.**

**Corollary**

**1.**

**Corollary**

**2.**

**Corollary**

**3.**

## 4. Implementation of the Aihara Method

#### 4.1. Generating All Cycles of a Planar Graph

#### 4.2. Efficient Computation of Necessary Derivatives

`p[i]`$=x-{\alpha}_{i+1}$ and

`q[i]`$=x-{\beta}_{i+1}$. These are used to compute derivatives instead of computing characteristic polynomials explicitly.

`e`val_deriv differentiates $p\left(x\right)/q\left(x\right)$

`power`times, where the argument x at which to evaluate the derivative has already been chosen and the vectors have been pre-computed. If

`power`is 0 then the answer is the product of the values

`p[0]`to

`p[`${d}_{p}$-1

`]`divided by the product of the values

`q[0]`to

`q[`${d}_{q}$-1

`]`. Otherwise the answer is computed from: ${f}^{\prime}\left(x\right)={\sum}_{i=1}^{{d}_{p}}\{p\left(x\right)[-i]/q\left(x\right)\}-{\sum}_{j=1}^{{d}_{q}}\{p\left(x\right)/\left[q\left(x\right)(x-i{\beta}_{j})\right]\}$.

`MAX_DEG`is the maximum degree of any polynomial.

`double eval_deriv(int power, int dp, double p[MAX_DEG], int dq, double q[MAX_DEG])`

`{`

`double r[MAX_DEG];`

`double ans, top, bottom;`

`int limit, pos, i, j;`

`// When power is 0, stop taking derivatives and evaluate.`

`if (power == 0)`

`{`

`if (dp < dq) limit = dq;`

`else limit = dp;`

`ans = 1;`

`// The answer is the product of the p values divided by the product of the q values.`

`for (i = 0; i < limit; i++)`

`{`

`if (i < dp) top = p[i];`

`else top = 1;`

`if (i < dq) bottom = q[i];`

`else bottom = 1;`

`ans* = (top/bottom);`

`}`

`return(ans);`

`}`

`ans = 0;`

`// Compute qp’ / q^2 = p’/q.`

`// Ignore if dp = 0 since a polynomial of degree 0 has a derivative of 0.`

`if (dp > 0)`

`{`

`// If dp = 1 then the polynomial is x-a0 and the derivative of this is 1.`

`if (dp == 1)`

`{`

`r[0] = 1;`

`ans+ = eval_deriv(power-1, dp-1, r, dq, q);`

`}`

`else // dp > 1.`

`{`

`for (i = 0; i < dp; i++)`

`{`

`// Compute p(x)[-i]:`

`pos = 0;`

`for (j=0; j < dp; j++)`

`{`

`if (i != j)`

`{`

`r[pos] = p[j];`

`pos++;`

`}`

`}`

`ans+ = eval_deriv(power-1, dp-1, r, dq, q);`

`}`

`}`

`}`

`// Now subtract off p q’ / q^2`

`for (i = 0; i < dq; i++)`

`r[i] = q[i];`

`for (i = 0; i < dq; i++)`

`{`

`r[dq] = q[i];`

`ans -= eval_deriv(power-1, dp, p, dq+1, r);`

`}`

`return(ans);`

`}`

## 5. Some Examples of the Aihara Model

#### 5.1. The Basic Case: Benzene

#### 5.2. An Analytical Example: The HL Current in Anthracene

#### 5.3. A Numerical Example: An Non-Kekulean Case

**I**) in Figure 2a.

## 6. A New Cycle Current Model

#### 6.1. Conjugated-Circuit Models of Current

#### 6.2. A Cycle-Current Model for All Benzenoids

#### 6.3. Testing the Model

## 7. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**A non-Kekulean benzenoid, I. (

**a**) Labelling of faces. (

**b**) Distribution of coefficients in the unique non-bonding Hückel molecular orbital. For the normalised orbital, multiply all entries by $1/\sqrt{22}$.

**Table 1.**Cycles and corresponding polynomials ${P}_{G\u2013C}\left(x\right)$ in anthracene. Bold lines represent edges in C; removal of bold and dashed lines yields the graph $G\u2013C$.

Cycle | Cycle Diagram | ${\mathit{P}}_{\mathit{G}-\mathit{C}}\left(\mathit{x}\right)$ |
---|---|---|

${C}_{1}$ | $\left({x}^{4}-2{x}^{3}-2{x}^{2}+3x+1\right)\left({x}^{4}+2{x}^{3}-2{x}^{2}-3x+1\right)$ | |

${C}_{2}$ | $\left({x}^{4}-2{x}^{3}-2{x}^{2}+3x+1\right)\left({x}^{4}+2{x}^{3}-2{x}^{2}-3x+1\right)$ | |

${C}_{3}$ | ${\left({x}^{2}+x-1\right)}^{2}{\left({x}^{2}-x-1\right)}^{2}$ | |

${C}_{4}$ | $\left({x}^{2}+x-1\right)\left({x}^{2}-x-1\right)$ | |

${C}_{5}$ | $\left({x}^{2}+x-1\right)\left({x}^{2}-x-1\right)$ | |

${C}_{6}$ | 1 |

**Table 2.**Circuit resonance energy (CRE) values, ${A}_{C}$, calculated using Equation (2) for cycles of anthracene. Cycles are labelled as shown in Table 1.

CRE | Formula | Value |
---|---|---|

${A}_{1}={A}_{2}$ | $\left(\frac{53+38\sqrt{2}}{2128+1512\sqrt{2}}+\frac{19}{252}+\frac{-83\sqrt{2}}{392}+\frac{13}{36}+\frac{53-38\sqrt{2}}{1512\sqrt{2}-2128}\right)=\left(\frac{55}{126}-\frac{12\sqrt{2}}{49}\right)$ | $\approx 0.0902$ |

${A}_{3}$ | $\left(\frac{153+108\sqrt{2}}{2128+1512\sqrt{2}}+\frac{-25}{252}+\frac{-113\sqrt{2}}{392}+\frac{17}{36}+\frac{153-108\sqrt{2}}{1512\sqrt{2}-2128}\right)=\left(\frac{47}{126}-\frac{43\sqrt{2}}{196}\right)$ | $\approx 0.0628$ |

${A}_{4}={A}_{5}$ | $\left(\frac{9+6\sqrt{2}}{2128+1512\sqrt{2}}+\frac{-5}{252}+\frac{85\sqrt{2}}{392}+\frac{-11}{36}+\frac{9-6\sqrt{2}}{1512\sqrt{2}-2128}\right)=\left(\frac{25\sqrt{2}}{98}-\frac{41}{126}\right)$ | $\approx 0.0354$ |

${A}_{6}$ | $\left(\frac{1}{2128+1512\sqrt{2}}+\frac{-1}{252}+\frac{-57\sqrt{2}}{392}+\frac{5}{36}+\frac{1}{1512\sqrt{2}-2128}\right)=\left(\frac{17}{126}-\frac{15\sqrt{2}}{196}\right)$ | $\approx 0.0267$ |

Cycle Current | Area, ${\mathit{S}}_{\mathit{C}}$ | Formula | Value |
---|---|---|---|

${J}_{1}={J}_{2}$ | 1 | $\left(\frac{55}{28}-\frac{54\sqrt{2}}{49}\right)$ | ≈0.4058 |

${J}_{3}$ | 1 | $\left(\frac{47}{28}-\frac{387\sqrt{2}}{392}\right)$ | ≈0.2824 |

${J}_{4}={J}_{5}$ | 2 | $\left(\frac{225\sqrt{2}}{98}-\frac{41}{14}\right)$ | ≈0.3183 |

${J}_{6}$ | 3 | $\left(\frac{51}{28}-\frac{405\sqrt{2}}{392}\right)$ | ≈0.3603 |

Face | Contribution | ${\widehat{\mathit{J}}}_{\mathit{F}}$ |
---|---|---|

Terminal hexagon | ${J}_{1}+{J}_{4}+{J}_{6}={J}_{2}+{J}_{5}+{J}_{6}$ | $\left(\frac{6}{7}+\frac{9\sqrt{2}}{56}\right)$$\approx 1.0844$ |

Central hexagon | ${J}_{3}+{J}_{4}+{J}_{5}+{J}_{6}$ | $\left(\frac{18\sqrt{2}}{7}-\frac{33}{14}\right)$$\approx 1.2794$ |

**Table 5.**Cycle contributions to HL current in the non-Kekulean benzenoid

**I**. D and P stand for diatropic and paratropic contributions, respectively.

Cycle | Size | ${\mathit{S}}_{\mathit{c}}$ | Composition | ${\mathit{J}}_{\mathit{C}}$ | Tropicity |
---|---|---|---|---|---|

${C}_{1}$ | 6 | 1 | F${}_{1}$ | $+0.0795$ | D |

${C}_{2}$ | 6 | 1 | F${}_{2}$ ≅ F${}_{2}^{\prime}$ | $+0.0852$ | D |

${C}_{3}$ | 6 | 1 | F${}_{3}$ ≅ F${}_{3}^{\prime}$ | $+0.2386$ | D |

${C}_{4}$ | 10 | 2 | F${}_{1}$ + F${}_{2}$ ≅ F${}_{1}$ + F${}_{2}^{\prime}$ | $+0.0795$ | D |

${C}_{5}$ | 10 | 2 | F${}_{2}$ + F${}_{2}^{\prime}$ | $+0.0227$ | D |

${C}_{6}$ | 10 | 2 | F${}_{2}$ + F${}_{3}$ ≅ F${}_{2}^{\prime}$ + F${}_{3}^{\prime}$ | $+0.1705$ | D |

${C}_{7}$ | 12 | 3 | F${}_{1}$ + F${}_{2}$ + F${}_{2}^{\prime}$ | $-0.0170$ | P |

${C}_{8}$ | 14 | 3 | F${}_{1}$ + F${}_{2}$ + F${}_{3}$ ≅ F${}_{1}$ + F${}_{2}^{\prime}$ + F${}_{3}^{\prime}$ | $+0.1193$ | D |

${C}_{9}$ | 14 | 3 | F${}_{2}$ + F${}_{2}^{\prime}$ + F${}_{3}$ ≅ F${}_{2}$ + F${}_{2}^{\prime}$ + F${}_{3}^{\prime}$ | $+0.0341$ | D |

${C}_{10}$ | 16 | 4 | F${}_{1}$ + F${}_{2}$ + F${}_{2}^{\prime}$ + F${}_{3}$ ≅ F${}_{1}$ + F${}_{2}$ + F${}_{2}^{\prime}$ + F${}_{3}^{\prime}$ | $-0.0227$ | P |

${C}_{11}$ | 18 | 4 | F${}_{2}$ + F${}_{2}^{\prime}$ + F${}_{3}$ + F${}_{3}^{\prime}$ | $+0.0455$ | D |

${C}_{12}$ | 20 | 5 | F${}_{1}$ + F${}_{2}$ + F${}_{2}^{\prime}$ + F${}_{3}$ + F${}_{3}^{\prime}$ | $-0.0284$ | P |

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**MDPI and ACS Style**

Myrvold, W.; Fowler, P.W.; Clarke, J.
Partitioning Hückel–London Currents into Cycle Contributions. *Chemistry* **2021**, *3*, 1138-1156.
https://doi.org/10.3390/chemistry3040083

**AMA Style**

Myrvold W, Fowler PW, Clarke J.
Partitioning Hückel–London Currents into Cycle Contributions. *Chemistry*. 2021; 3(4):1138-1156.
https://doi.org/10.3390/chemistry3040083

**Chicago/Turabian Style**

Myrvold, Wendy, Patrick W. Fowler, and Joseph Clarke.
2021. "Partitioning Hückel–London Currents into Cycle Contributions" *Chemistry* 3, no. 4: 1138-1156.
https://doi.org/10.3390/chemistry3040083