Optical saturation driven by exciton confinement in molecular-chains: a TDDFT study
We have identified excitonic confinement in one-dimensional molecular chains (i.e. polyacetylene and H) as the main driving force for the saturation of the chain polarizability as a function of the number of molecular units. This conclusion is based on first principles time–dependent density functional theory calculations performed with a new derived exchange–correlation kernel. The failure of simple local and semi–local functionals is shown to be related to the lack of memory effects, spatial ultranonlocality, and self–interaction corrections. These effects get smaller as the gap of the system reduces, in which case such simple approximations do perform better.
pacs:71.35.-y, 31.15.ee, 31.15.ap
The electronic quantum confinement occurring in low–dimensional systems is often responsible for the peculiar spectroscopic properties exhibited by molecules and nanostructures. Exciton confinement explains, for example, the size–dependent color of semiconducting quantum dots used as fluorescent markers in biology Alivisatos (). Highly localized excitons (solitons) play also a fundamental role in describing the process of vibrational energy transfer in complex proteins solitons (). In this context, one–dimensional polymers and molecular chains constitute an excellent playground to analyze the interplay between correlation effects and quantum confinement. For example, in non conducting polymers the longitudinal linear polarizability per monomer unit tends to a constant in the large–N limit resta (). This optical saturation stems from the polarization of the electrons along the chain that tends to counteract the external field and manifest itself in the dependence of the optical absorption spectra on the polymer size. The present Letter aims to provide a consistent description of this saturation within density–functional based schemes.
The commonly used local (LDA) and semilocal (GGA) approximations to Density–Functional–Theory (DFT) and Time–Dependent DFT (TDDFT) that successfully describe the electronic properties of many different physical systems tddft (), fail dramatically for the case of semiconducting one–dimensional molecular chains. ALDA do not describe the main features of the absorption spectra and strongly overestimates the polarizability with respect to quantum chemical calculations toto (); dalskov (); champagne (). The reason for this poor performance of ALDA is still not settled, and it has been traced back to the need of long-range (LR) terms (ultra non–locality) in the exchange-correlation functionals gonze () or to the lack of self–interaction correction (SIC) mori (); burke (). Exact-exchange (EXX) or current-density functional (CDFT) approaches capture some of the effects, but not all: while EXX works fairly well in the case of the finite H chain (reproducing the Hartree–Fock results mori ()) it fails in reproducing the absorption spectrum of the infinite trans–polyacetylene chain rohra (). On the contrary CDFT provides quite good results for finite -conjugated polymers vanfaassen (); vanfaassen2 (), while it breaks down for the H chain.
In TDDFT all the effects beyond the non interacting particles approximation are embodied in the exchange–correlation (xc) kernel , with the xc–potential and the exact ground state electronic density. The recent developments in merging many–body perturbation–theory (MBPT) rmp () and TDDFT sottile (); andrea () open the path to unravel the physical origin of the response properties of one-dimensional systems. Here we show that exciton confinement dictates the evolution (saturation) of the optical response of one–dimensional chains as a function of the chain length. Therefore the failure of EXX, available local or semilocal or CDFT approximations is related to their inability to describe strong excitonic effects in anisotropic low-dimensional systems. We show that the xc–kernel has a hyper-non local behavior and memory dependence that is at least one order of magnitude stronger than in solids. The static and dynamical polarizabilities are both well described in simple H or in more sophisticated trans–polyacetylene molecular chains.
The polarizability of low-dimensional systems is proportional to the spatial average of the reducible polarization function, that can be obtained from the solution of the TDDFT equation
The exchange–correlation part of the kernel as derived in Refs. andrea (); sottile () mimics to a good extent the MBPT results for the polarization function based on the solution of the Bethe-Salpeter (BS) equation for the electron–hole dynamics in the basis of the GW self–energy quasiparticle states rmp (). This kernel is first order in the screened coulomb potential : , where is the non-interacting quasi–particle response function. includes the correct behavior in the long–range regime andrea (); sottile (); botti (). The two constants, and , measure respectively the degree of spatial non locality and memory effects. In spite of its simplicity and in contrast to EXX and CDFT, this fully captures the excitonic features in the absorption spectra of bulk semiconductors, insulators as well as surfacestddft ().
In order to illustrate how the kernel solves the failure of ALDA (and of any semi-local approximations) we consider two Peierls distorted infinite molecular chains method (). In chain A (see Fig. 1) the inter–molecular distance is set to 2.5 a.u. and the intra-molecular distance is 2.0 a.u. (as in previous worksvanfaassen (); champagne (); mori ()). This chain is a semiconductor with an LDA gap of 2.28 eV. In chain B the inter-molecular distance is reduced to 2.05 a.u. and the LDA gap gets very small (0.26 eV). In Fig. 1 we compare the calculated TDDFT absorption spectra for chains A and B in the ALDA and MBPT–. As the electronic density is more homogeneous in chain B than in chain A the ALDA gives an almost indistinguishable dynamical polarizability if compared to the BS calculations (see Fig 1 right panel). For the chain A, however, the ALDA performance worsens considerably, and the main absorption peak is not at all reproduced. The kernel, instead, yields a dynamical polarizability in both chains almost indistinguishable from the BS calculations. Similarly to the case of wide gap insulators andrea () the correct description of excitonic states require the kernel to have non local Fourier components. This means that is a matrix, whose size (a few hundred reciprocal space vectors in the present case) is intimately related to the localisation of the excitonic state andrea ().
The failure of the ALDA can be understood by looking at the spatial dispersion of the excitonic state corresponding to the main absorption peaks shown at the bottom of Fig. 1 for both chains. In chain A, where ALDA does not work, the exciton is confined within few -monomers ( 36 a.u.). Consequently, the excitonic dispersion introduces a characteristic length given by the exciton linear extension. In chain B the exciton is basically spread all over the chain. If we now look at the xc–kernel for the two chains we see very drastic differences: whereas in chain B both and are almost zero, in chain A and eVvolume_dep_note (), which is more than one order of magnitude larger than in solids, where , while eV botti (). This large value of reflects the dynamical renormalization of the excitonic energy, due to memory effects. More importantly the static limit of the long–range part of the total TDDFT kernel, is negative. This means that the spatial non locality of is stronger than the Hartree term solid_vs_chain (). This hyper non locality and the memory effects in cannot be captured by any static, local or semi-local approximation.
Turning now to finite-size effects, we show in Fig. 2 the results for the static polarizability of finite length H chains within different approximations for the xc–kernel. Both ALDA and CDFT yields a very slow saturation of as a function of the chain length and a strong overestimation of the static polarizability compared to accurate quantum chemical coupled cluster results CCSD(T). The TDDFT results obtained using the kernel partially reduce the ALDA and CDFT overestimation, showing a faster saturation. However the agreement with CCSD(T) is not yet satisfactory. We have traced the residual discrepancy to the lack of SIC in the LDA wavefunctions used to construct the xc–kernel. This can be easily corrected by recomputing the kernel, using Hartree–Fock (HF) self–interaction–free wavefunctions to build .
This new kernel yields axial polarizabilities in excellent agreement with the CCSD(T) results. The SIC increases the xc–kernel spatial non locality factor , which translates into further confinement of the excitonic states, compared to the LDA-based results. This can be rationalized by looking at the excitonic wave–functions shown in the bottom of Fig.2. The probability of finding an electron at the end of the chain when the hole is placed in the middle of the bond of the central molecule (that measures excitonic confinement) quickly decreases while increasing the chain length, being almost zero for for . The polarizability follows the same trend, saturating when the excitonic wavefunction dispersion does not change anymore. We should mention that, for small chains, ALDA gives good results as the excitonic wavefunction is spread over the whole molecule, and the electronic homogeneity is not strongly perturbed (like in the chain B case discussed above). It is clear that exciton confinement plays a major role in the determination of the response properties of molecular chains providing microscopic support to the relevance of the ultranonlocality concept in exchange-correlation functionals.
The H chain represents an extreme case of system with few electrons, where confinement is crucial, while electronic screening is negligible (this is the reason why time–dependent HF works fine). For these reason TDDFT kernels that work in the case of H chain may fail in more complex molecular systems where electronic screening becomes more important. This is the case of the trans–polyacetylene (PA) molecular chain. The PA chain has been extensively studied in the past within the BS scheme rohlfing (). It has been showed that the main absorption peak observed experimentally is excitonic, with binding energy. On the contrary all available calculations within TDDFT do not reproduce the experimental results: ALDA spectra (see Fig. 3) is too low in energy and the main absorption peak shape is more similar to a Van Hove singularity than to an isolated excitonic peak. Within an EXX calculation it is possible to obtain a good agreement with the experiment only by neglecting the x–kernel, i.e. in the independent particle approximation rohra (), which is clearly inconsistent. For finite length chains, as in polyethylene, EXX results are, again, very similar to HF mori () that, in the case of PA largely overestimates the quantum chemistry MP2 results (inset of Fig. 3). As shown in Fig. 3, the present kernel, instead, is in very good agreement with the MP2 results in the finite length case and with the experimental absorption peak position in the infinite length case comment (). MP2 calculations toto () predicts the PA chain polarizability to saturate around 15 monomers, in excellent agreement with the excitonic extension obtained with the present kernel (shown in the bottom of Fig. 2 for the bright PA exciton at 1.69 eV). As a result even in the more complex PA chain the polarizability saturation naturally correlates with the excitonic localisation length fully confirming the physical picture that emerged from the previous results for the H chains.
In conclusion we have shown that TDDFT successfully explain the optical saturation in molecular chains in terms of excitonic confinement. The Many-Body based xc–kernel, with the correct long–range tail and including, in the H chains, self–interaction corrections yields static and dynamical polarizabilities in excellent agreement with accurate quantum chemical calculations. We have proved that there exists a close link between the excitonic spatial extension and the axial polarizability in the TDDFT framework, giving a sound interpretation of the severe breakdown of the local–density approximation in anisotropic structures.
The authors thank Myrta Grüning for her contributions at the early stage of this work as well as Rosa Di Felice for fruitful discussions. We acknowledge support by the European Community Network of Excellence Nanoquanta (NMP4-CT-2004-500198), SANES (NMP4-CT-2006-017310), DNA-NANODEVICES (IST-2006-029192), and Spanish MEC (FIS2007-65702-C02-01) projects, Basque Country University (SGIker ARINA), and Barcelona supercomputing Center Mare Nostrum.
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