# Hole Transfer in Open Carbynes

^{*}

## Abstract

**:**

## 1. Introduction

_{3}– or CH

_{2}– or CH–.

_{n}– end groups (TBImod) reproduced approximately the magnitude of the energy gap in the polyynic case. The DFT calculations showed that, due to the presence of end groups, there exists a cumulenic energy gap, too, smaller than the polyynic one [17]. The Real-Time Time-Dependent Density Functional Theory (RT-TDDFT) calculations showed that the mean over time probabilities to find the hole at various sites (site occupations) converged with increasing the size of the basis set. TBImod agreed with the mean over time probabilities (site occupations) RT-TDDFT predicted, for cumulenic molecules. The site occupations of polyynic sl (i.e., starting with shorter bond length) and of polyynic ls (i.e., starting with longer bond length) molecules were different than the cumulenic ones, and the simplistic TBImod model could qualitatively explain the RT-TDDFT trends [17]. However, TBImod (and TBI) predicted charge oscillations that were approximately four times slower than the RT-TDDFT ones. A simple Fast Fourier Transform (FFT) analysis of dipole moment oscillations, which are independent of the population analysis used, confirmed that fact. Similarly faster were found the coherent transfer rates k predicted by RT-TDDFT compared to those predicted by TBImod (and TBI) [17]. $k\left(N\right)$ or $lnk(lnN)$ converged increasing the basis set. TBImod was, as explained, slower but followed the trend. The trends in the behaviour of $k\left(N\right)$ or $lnk(lnN)$ as predicted by RT-TDDFT could be qualitatively explained by TBImod, although oscillations in RT-TDDFT were always faster [17]. We are expecting experiments to obtain coherent transfer rates in carbynes, probably using time-resolved spectroscopy.

## 2. Bond Lengths-Structures-Vibrational Analysis

_{2}– groups in perpendicular configuration and then optimizing hydrogen atoms, results in CH

_{2}– groups in coplanar configuration. Therefore, for N even, we only include cumulenic co molecules in our RT-TDDFT simulations. For N odd (even), the ground-state molecule is that with perpendicular (coplanar) end groups [23]. Polyynic ls molecules exist with eclipsed and staggered methyl groups for N even, with negligibly different ground state energy; in panel (d) we show the staggered configuration.

_{3}–. The negative values in panel Figure 4a for the $N=7$ molecule with coplanar methylene groups and in panel Figure 5c for the $N=2$ ls molecule with eclipsed methyl groups are associated with imaginary eigenfrequencies. As explained in Ref. [26]: “If you have optimized to a transition state, or to a higher order saddle point, then there will be some negative frequencies which may be listed before the “zero frequency” modes. (Frequencies which are printed out as negative are really imaginary; the minus sign is simply a flag to indicate that this is an imaginary frequency)”.

## 3. Tight-Binding Wire Model Variants

## 4. Real-Time Time-Dependent Density Functional Theory

_{2}or CH

_{3}group), where we increased the charge by $+1$, creating a hole). For example, for the cumulenic $N=5$ molecule, if we obtained at the beginning from DFT the charges $+0.02,-0.01,-0.02,-0.01,+0.02$, at CH

_{2}, C, C, C, CH

_{2}, respectively, then the CDFT constraints were $+1.02,-0.01,-0.02,-0.01,+0.02$, at CH

_{2}, C, C, C, CH

_{2}, respectively.

## 5. Results

#### 5.1. DFT Ground-State Energy

#### 5.2. CDFT “Ground-State” Energy with a Hole at the First Site

_{2}or CH

_{3}) by CDFT affects the “ground-state” energy, depicted in Figure 7 in a similar way as the ground state energy of the neutral molecules is depicted in Figure 6. A molecule with a hole has larger energy than the respective neutral molecule, as expected. This is why neutral molecules exist. CDFT evaluates the excited state energy in accord with its constraint. Therefore, the term “ground-state” is excessive here. The insets illustrate that, for odd N, the creation of a hole brings the two cumulenic molecules with coplanar or perpendicular methylene groups much closer in energy (actually, it seems that for $N>3$ cumulenic co has slightly lower energy), the polyynic sl molecule (creation of a hole at CH) has slightly higher energy, and the creation of a hole at a polyynic ls molecule (creation of a hole at CH

_{3}) has even higher energy. These differences diminish increasing N, as expected.

#### 5.3. Eigenenergies, Density of States, and Energy Gap

#### 5.4. Charge Oscillations

_{2}site contains eight electrons and a CH

_{3}site contains nine electrons. We place the hole initially (time zero) at the first site, as always in this article. In Figure 12 we show charge oscillations obtained by RT-TDDFT, for $N=8$, for cumulenic molecules with coplanar methylene groups and for polyynic molecules starting with short or long bonds, at the B3LYP/cc-pVTZ level of theory.

#### 5.5. Mean over Time Probabilities

#### 5.6. Coherent Transfer Rates

#### 5.7. Electric Dipole Moment

_{2}–) and even N sl polyynes (both end groups CH–), and of the order of 0.1 a.u. for other polyynes. In the right columns of Figure 17 and Figure 18 we present the corresponding FFT of each ${\mathcal{P}}_{z}$, as obtained simply by MATLAB, without any further elaboration. The time step in RT-TDDFT was 0.5 a.u.; we covered 1000 a.u. ≈ 25 fs with ≈ 2000 points.

#### 5.8. Frequency Content

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

BLA | bond length alternation |

CDFT | Constrained Density Functional Theory |

DFT | Density Functional Theory |

DOS | density of states |

FFT | Fast Fourier Transform |

RT-TDDFT | Real-Time Time-Dependent Density Functional Theory |

TB | Tight Binding |

TBI | TB wire model |

TBImod | a crude modification of TB wire model |

TBImodt4times | TBImod with four times greater transfer parameters |

TDDFT | time-dependent DFT |

TDKS | time-dependent Kohn-Sham |

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**Figure 1.**Illustration of cumulenic and polyynic carbynes. By co (pe) we denote a molecule with coplanar (perpendicular) methylene groups and by sl (ls) we denote a molecule with short-long-... (long-short-...) sequence of bonds. (

**a**) $N=6$ cumulenic co, (

**b**’) $N=7$ cumulenic co, (

**b**) $N=7$ cumulenic pe, (

**c**) $N=6$ polyynic sl, (

**d**) $N=6$ polyynic ls, (

**e**) $N=7$ polyynic sl, and (

**f**) $N=7$ polyynic ls. To study charge transfer, we place a hole initially at the first site, which is made of the first carbon and one or two or three hydrogens. Then, we follow its temporal and spatial evolution. In a simple picture, the first and the last carbons have in (

**a**), (

**b**’), (

**b**) $s{p}^{2}$, in (

**c**) $sp$, in (

**d**) $s{p}^{3}$, in (

**e**) $sp$ and $s{p}^{3}$, and in (

**f**) $s{p}^{3}$ and $sp$ hybridizations.

**Figure 2.**Bond lengths of cumulenic molecules. B3LYP/6-31G* level of theory, without any constraint on the position of atoms. (

**a**) $N=7$, (

**b**) $N=8$, (

**c**) $N=9$, (

**d**) $N=10$, (

**e**) N odd, (

**f**) N even.

**Figure 3.**Bond lengths of polyynic molecules. B3LYP/6-31G* level of theory, without any constraint on the position of atoms. (

**a**) $N=7$, (

**b**) $N=8$, (

**c**) $N=9$, (

**d**) $N=10$, (

**e**) N odd, (

**f**) N even.

**Figure 4.**Vibrational analysis of cumulenic molecules. B3LYP/6-31G* level of theory, without any constraint on the position of atoms. (

**a**) $N=7$, (

**b**) $N=8$, (

**c**) $N=9$, (

**d**) $N=10$, (

**e**) N odd (pe), (

**f**) N even.

**Figure 5.**Vibrational analysis of polyynic molecules. B3LYP/6-31G* level of theory, without any constraint on the position of atoms. (

**a**) even sl, (

**b**) even lss, (

**c**) even lse, (

**d**) odd.

**Figure 6.**Ground state energy of neutral molecules, ${E}_{\mathrm{GS}}$, B3LYP/cc-pVTZ level of theory. N is the number of carbon atoms and n is the number of all atoms, cu co (cu pe) denotes cumulenic molecules with coplanar (perpendicular) methylene groups, po sl (po ls) denotes polyynic molecules starting with short (long) bonds. (

**Left**) ${E}_{\mathrm{GS}}/N$ as a function of N. (

**Right**) ${E}_{\mathrm{GS}}/n$ as a function of N. The insets emphasize that, for odd N, the cu pe (cu co) molecule has the lowest (highest) ${E}_{\mathrm{GS}}$.

**Figure 7.**“Ground state” energy, ${E}_{\mathrm{GS}}$, of molecules with a hole created at its first site, B3LYP/cc-pVTZ level of theory. N is the number of carbon atoms, n is the number of all atoms, cu co and cu pe denote cumulenic molecules with coplanar and perpendicular methylene groups, respectively; po sl and po ls denote polyynic molecules starting with short and long bonds, respectively. The insets emphasize that, for odd N, the cu pe and cu co molecules come closer in energy (it seems that for $N>3$ cu co has slightly lower energy), po sl has higher energy and po ls still higher.

**Figure 8.**An example, for $N=99$ and $N=100$, of the formation of energy gap between occupied and empty eigenstates of neutral molecules, as obtained by our DFT simulations, at the B3LYP/cc-pVTZ level of theory.

**Figure 9.**The eigenenergies of the Highest Occupied Molecular Orbital (HOMO), the Lowest Unoccupied Molecular Orbital (LUMO) and the energy gap (inset) between them, as functions of the number of carbon atoms in the chain, N, as obtained by our DFT simulations, at the B3LYP/cc-pVTZ level of theory. The points shown correspond to N = 11, 12, 49, 50, 99, 100, 149, 150, 199, 200.

**Figure 10.**Density of states, $g\left(E\right)$, per number of carbon atoms, N, for the three Tight-Binding (TB) variants used in this work. (

**a**) cumulenic TBI, (

**b**) cumulenic TBImod, (

**c**) cumulenic TBImodt4times, (

**d**) polyynic TBI, (

**e**) polyynic TBImod, (

**f**) polyynic TBImodt4times.

**Figure 11.**Charge oscillations obtained by RT-TDDFT, at the B3LYP/cc-pVTZ level of theory, $N=7$, for cumulenic (cu) molecules with coplanar (co) and perpendicular (pe) methylene groups as well as polyynic (po) molecules starting with short (sl) or long (ls) bonds. (

**a**) cu co, (

**b**) cu pe, (

**c**) po sl, (

**d**) po ls.

**Figure 12.**Charge oscillations obtained by RT-TDDFT, at the B3LYP/cc-pVTZ level of theory, $N=8$, for cumulenic (cu) molecules with coplanar (co) methylene groups as well as polyynic (po) molecules starting with short (sl) or long (ls) bonds. (

**a**) cu co, (

**b**) po sl, (

**c**) po ls.

**Figure 13.**Charge oscillations obtained by the TB variants (

**a**) TBI, (

**b**) TBImod, (

**c**) TBImodt4times, for cumulenic molecules with $N=7$ (left column) and $N=8$ (right column).

**Figure 14.**Site occupations, i.e., mean over time probabilities to find the hole at site j, for initial placement of the hole at the first site, for $N=$ 7, obtained by Real-Time Time-Dependent Density Functional Theory (RT-TDDFT) [3-21G (pink left triangles), 6-31G* (red up triangles), cc-pVDZ (orange right triangles), cc-pVTZ (gray down triangles), cc-pVQZ (dark gray hexagons)] and the functional B3LYP, as well as by TB wire model variants [TBI (black squares), TBImod (green pentagons), TBImodt4times (purple circles)]. (

**a**) cumulenic molecules with coplanar methylene groups (cu co), (

**b**) cc-pVTZ/B3LYP for cu co molecules versus polyynic molecules starting with short or long bonds (po sl or po ls). Half-filled down triangles for po sl (blue filled right) and po ls (purple filled left). (

**c**) cu co versus cumulenic molecules with perpendicular methylene groups (cu pe) for the 3 larger basis sets. Dotted lines are guides to the eyes.

**Figure 15.**Site occupations, i.e., mean over time probabilities to find the hole at site j, for initial placement of the hole at the first site, for $N=$ 8, obtained by RT-TDDFT [3-21G (pink left triangles), 6-31G* (red up triangles), cc-pVDZ (orange right triangles), cc-pVTZ (gray down triangles), cc-pVQZ (dark gray hexagons)] and functional B3LYP as well as by TB wire model variants [TBI (black squares), TBImod (green pentagons), TBImodt4times (purple circles)]. (

**a**) cumulenic molecules with coplanar methylene groups (cu co), (

**b**) cc-pVTZ/B3LYP for cu co molecules versus polyynic molecules starting with short or long bonds (po sl or po ls). Half-filled down triangles correspond to po sl (blue filled right) and po ls (purple filled left). Dotted lines are guides to the eyes.

**Figure 16.**Transfer rates along carbyne wires as obtained by RT-TDDFT at the B3LYP/cc-pVTZ level of theory as well as by TBI, TBImod and TBImodt4times: cu co (dark gray hexagons with dot), cu pe (dark gray hexagons with cross), po sl (blue right half-filled down triangles), po ls (purple left half-filled down triangles), TBI (black squares), TBImod (olive pentagons), TBImodt4times (magenta circles). (

**a**) $k\left(N\right)$, (

**b**) $lnk(lnN)$.

**Figure 17.**Left column: Dipole moment oscillations along the z-axis, ${\mathcal{P}}_{z}$, obtained by RT-TDDFT at the B3LYP/cc-pVTZ level of theory, for $N=7$, for cumulenic (cu) molecules with coplanar (co) and perpendicular (pe) methylene groups as well as for polyynic (po) molecules starting with short (sl) or long (ls) bonds. Right column: The corresponding Fast Fourier Transform (FFT) amplitudes obtained by MATLAB. (

**a**) cu co, (

**b**) cu pe, (

**c**) po sl, (

**d**) po ls.

**Figure 18.**Left column: Dipole moment oscillations along the z-axis, ${\mathcal{P}}_{z}$, obtained by RT-TDDFT at the B3LYP/cc-pVTZ level of theory, for $N=8$, for cumulenic (cu) molecules with coplanar (co) methylene groups as well as for polyynic (po) molecules starting with short (sl) or long (ls) bonds. Right column: The corresponding FFT amplitudes obtained by MATLAB without any further elaboration. (

**a**) cu co, (

**b**) po sl, (

**c**) po ls.

**Figure 19.**Left column: Dipole moment ($\mathcal{P}$) oscillations obtained by the TB variants (

**a**) TBI, (

**b**) TBImod, (

**c**) TBImodt4times, for $N=7$. Right column: The corresponding FFT amplitudes obtained by MATLAB without any further elaboration.

**Figure 20.**Left column: Dipole moment ($\mathcal{P}$) oscillations obtained by the TB variants (

**a**) TBI, (

**b**) TBImod, (

**c**) TBImodt4times, for $N=8$. Right column: The corresponding FFT amplitudes obtained by MATLAB without any further elaboration.

d | d | d | d | d | |||||
---|---|---|---|---|---|---|---|---|---|

$s{p}^{3}$-$s{p}^{3}$ | 154 [19] | $s{p}^{3}$-$sp$ | 146 [19] | C−C | 154 [20,21] | benzene | 140 [19] | polyynic long | 130.1 [22] |

$s{p}^{3}$-$s{p}^{2}$ | 150 [19] | $s{p}^{2}$-$sp$ | 143 [19] | C=C | 134 [20,21] | alkene | 134 [19] | cumulenic | 128.2 [22] |

$s{p}^{2}$-$s{p}^{2}$ | 147 [19] | $sp$-$sp$ | 137 [19] | C≡C | 120 [20,21] | alkyne | 120 [19] | polyynic short | 126.5 [22] |

**Table 2.**n is the number of atoms, N is the number of carbon atoms. The number of modes, $m=3n$, from which three are translational modes (TM). Linear and nonlinear molecules have two and three rotational modes (RM), respectively; therefore, the number of vibrational modes (VM) is $3n-5$ and $3n-6$, respectively.

Type | N | n | TM | RM | VM |
---|---|---|---|---|---|

cumulenes | even, odd | $N+4$ | 3 | 3 | $3n-6$ |

polyynes sl | even | $N+2$ | 3 | 2 | $3n-5$ |

polyynes sl | odd | $N+4$ | 3 | 3 | $3n-6$ |

polyynes ls | even | $N+6$ | 3 | 3 | $3n-6$ |

polyynes ls | odd | $N+4$ | 3 | 3 | $3n-6$ |

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**MDPI and ACS Style**

Simserides, C.; Morphis, A.; Lambropoulos, K. Hole Transfer in Open Carbynes. *Materials* **2020**, *13*, 3979.
https://doi.org/10.3390/ma13183979

**AMA Style**

Simserides C, Morphis A, Lambropoulos K. Hole Transfer in Open Carbynes. *Materials*. 2020; 13(18):3979.
https://doi.org/10.3390/ma13183979

**Chicago/Turabian Style**

Simserides, Constantinos, Andreas Morphis, and Konstantinos Lambropoulos. 2020. "Hole Transfer in Open Carbynes" *Materials* 13, no. 18: 3979.
https://doi.org/10.3390/ma13183979