Abstract
We investigate hole transfer in open carbynes, i.e., carbon atomic nanowires, using Real-Time Time-Dependent Density Functional Theory (RT-TDDFT). The nanowire is made of N carbon atoms. We use the functional B3LYP and the basis sets 3-21G, 6-31G*, cc-pVDZ, cc-pVTZ, cc-pVQZ. We also utilize a few Tight-Binding (TB) wire models, a very simple model with all sites equivalent and transfer integrals given by the Harrison expression (TBI) as well as a model with modified initial and final sites (TBImod) to take into account the presence of one or two or three hydrogen atoms at the edge sites. To achieve similar site occupations in cumulenes with those obtained by converged RT-TDDFT, TBImod is sufficient. However, to achieve similar frequency content of charge and dipole moment oscillations and similar coherent transfer rates, the TBImod transfer integrals have to be multiplied by a factor of four (TBImodt4times). An explanation for this is given. Full geometry optimization at the B3LYP/6-31G* level of theory shows that in cumulenes bond length alternation (BLA) is not strictly zero and is not constant, although it is symmetrical relative to the molecule center. BLA in cumulenic cases is much smaller than in polyynic cases, so, although not strictly, the separation to cumulenes and polyynes, approximately, holds. Vibrational analysis confirms that for N even all cumulenes with coplanar methylene end groups are stable, for N odd all cumulenes with perpendicular methylene end groups are stable, and the number of hydrogen atoms at the end groups is clearly seen in all cumulenic and polyynic cases. We calculate and discuss the Density Functional Theory (DFT) ground state energy of neutral molecules, the CDFT (Constrained DFT) “ground state energy” of molecules with a hole at one end group, energy spectra, density of states, energy gap, charge and dipole moment oscillations, mean over time probabilities to find the hole at each site, coherent transfer rates, and frequency content, in general. We also compare RT-TDDFT with TB results.
Highlights
To achieve, using a simple TB wire model, the same mean over time probabilities to find the hole at each site with RT-TDDFT, and fast frequency content of charge and dipole moment oscillations as well as coherent transfer rates, we have to multiply the TBImod transfer integrals by a factor of four [17]
For each of the two ppπ channels made of 2p x and 2py orbitals, respectively, the transfer integral between two consecutive sites i and j would be −0.63A. These channels occur at a higher energy compared to the previous channel, that is E2p C, they will somehow have a higher weight relative to the spspσ channel, whereas, if we add up all these contributions we get −4.45A, which is very close to −4A, the necessary transfer integral between intermediate sites to bring the TB wire results very close to the RT-TDDFT results in terms of frequency content of charge and dipole moment oscillations as well as in terms of coherent transfer rates
The summary is that TBI and TBImod can not follow the fast dynamics of RT-TDDFT and in order to obtain by TB similar frequency content with that of RT-TDDFT, we have to employ TBImodt4times
Summary
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]. 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]. TBImod (and TBI) predicted charge oscillations that were approximately four times slower than the RT-TDDFT ones.
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