Adapted Su-Schrieffer-Heeger Hamiltonian for polypyrrole

A generic class of Hamiltonians based on the Su-Schrieffer-Heeger (SSH) model is introduced to address the material-specific properties of conducting polymers beyond polyacetylene. Two physical parameters are incorporated into the original SSH model Hamiltonian, one being the scaling parameter $\gamma$ accounts for the modified electron-phonon coupling strength of aromatic rings and the other parameter $\epsilon$ representing the attractive cores of heterogeneous atoms, such as NH, S, and O. Their values are uniquely determined by two independent measurements, such as the experimental band gap of bulk polypyrrole (PPy) and the dimerization amplitude of pyrrole monomer computed by ab initio coupled cluster methods as chosen in this work. With the optimized values of $\left\{\gamma =1.46,\epsilon =4.27\text{eV}\right\}$, the adapted SSH (aSSH) Hamiltonian accurately reproduces the band gaps, molecular geometries, quasiparticle energies and wave functions, order parameters, discretized phonons around nonlinear polarons, and infrared and Raman spectra of both neutral and $p$-doped PPy chains of all lengths. It is expected that the generic formalism of the aSSH Hamiltonian, equipped with their corresponding $\left\{\gamma ,\epsilon \right\}$ values, would be applicable to other conducting polymers, such as polythiophene, polyfuran, poly-($p$-phenylene-vinylene), polyaniline, and their functional derivatives.

Figure 1

(Color online) Chemical structure of PPy. The position vector of the $i\text{th}$ atomic site is ${\mathbf{r}}_i$ and the bond length between the $i\text{th}$ and $j\text{th}$ sites is $b_{ij}=\left|{\mathbf{r}}_j-{\mathbf{r}}_i\right|$. The $\alpha$ and $\beta$ carbon sites are the C sites directly connected to the NH sites (such as 1, 4, 6, and 9) and the CH sites two sites away from NH (such as 2, 3, 7, and 8), respectively. The bridge $\left(\alpha \text{−}\alpha \right)$ bond, such as $b_{46}$, is the C-C bond that connects two neighboring pyrrole rings via two $\alpha \text{-C}$ sites.

Figure 2

(Color online) The band gap $E_g$ and bond-length dimerization amplitude $\Delta b_{\alpha \beta -\beta \beta }$ as functions of the electron-phonon scaling parameters $\gamma$ and the NH on-site energy parameter $\epsilon$. (a) Contour plot of $E_g$ for $N=7$, with the pluses indicating the targeted experimental value $E_g=3.25\text{eV}$. (b) Contour plot of $\Delta b_{\beta \beta -\alpha \beta }$ for $N=1$, with the crosses indicating the CCSD result of $\Delta b_{\beta \beta -\alpha \beta }=0.051\text{Å}$. The optimized parameter set $\left\{\gamma =1.46,\epsilon =4.27\text{eV}\right\}$ which gives $\left\{E_g=3.25\text{eV},\Delta b_{\beta \beta -\alpha \beta }=0.051\text{Å}\right\}$ is highlighted as the circles in both (a) and (b). The dashed line in (b), $\Delta b_{\beta \beta -\alpha \beta }=0\text{Å}$, indicates the phase boundary of the quinoid and aromatic structures.

Figure 3

(Color online) Bond lengths of the (a) pyrrole monomer and (b) dimer computed by CCSD and the $N=20$ oligomer (shown only the central ring) (c) computed by HF. Bond lengths of the pyrrole (d) monomer, (e) dimer, and $N=20$ oligomer (shown only the central ring) (f) computed by the aSSH Hamiltonian. The three bond-length fitting targets, obtained by CCSD, are highlighted in red bold. All bond lengths are in angstrom. The basis sets for CCSD and HF are $6\text{−}31++\text{G}\left(\text{d},\text{p}\right)$ and 3-21G, respectively.

Figure 4

(Color online) The quasiparticle energy band gap $E_g$ and the bond-length dimerization amplitude $\Delta b_{\beta \beta -\alpha \beta }$ of PPy as a function of $N$ computed by the aSSH model with $\left\{\gamma =1.46,\epsilon =4.27\text{eV}\right\}$. The DFT, TDDFT, and experimental data 1–5 are taken from Refs. 51, 52, 40, 53, 39 ($E_g=3.2\text{eV}$ with an unknown $N$, assumed to be 7), Refs. 47, 46, respectively. The HF, CCSD, QCISD, and MP4 are computed using GAUSSIAN-03 (Ref. 37) with the $6\text{−}31++\text{G}\left(\text{d},\text{p}\right)$ basis set.

Figure 5

(Color online) The complete sets of filled and subsets of unfilled $\pi$ orbitals of the neutral PPy of (a) $N=1$, (b) $N=2$, and $N=20$ (c) for aSSH and (d) for HF. In (a) and (b), the aSSH results are shown on the left and HF results on the right, with the aSSH $E_{\text{HOMO}}\left(N=1\right)=-1.69\text{eV}$, $E_{\text{LUMO}}\left(N=1\right)=3.97\text{eV}$, $E_{\text{HOMO}}\left(N=2\right)=-1.04\text{eV}$, and $E_{\text{LUMO}}\left(N=2\right)=3.18\text{eV}$ and the HF $E_{\text{HOMO}}\left(N=1\right)=-8.16\text{eV}$, $E_{\text{LUMO}}\left(N=1\right)=4.14\text{eV}$, $E_{\text{HOMO}}\left(N=2\right)=-7.02\text{eV}$, and $E_{\text{LUMO}}\left(N=2\right)=2.99\text{eV}$. The red and green wave function lobes represent two different phases of the corresponding quasiparticle wave functions and their sizes are proportional to the wave-function amplitudes on the corresponding atomic sites. The HF wave function isosurface values are $0.8{\text{Å}}^{-3/2}$, $0.6{\text{Å}}^{-3/2}$, and $0.2{\text{Å}}^{-3/2}$ for the monomer, dimer, and $N=20$ cases, respectively. For HF results, the $6\text{−}31\text{G}++\left(\text{d},\text{p}\right)$ basis set is used for monomer and dimer, and the 3-21G basis sets for $N=20$.

Figure 6

(Color online) The bond-length alternation order parameter (a)–(d) $\eta \left(m\right)$, (e)–(h) DOS, and the corresponding polaron wave functions of an $N=20$ PPy chain computed by the aSSH Hamiltonian. (a) and (e) are the neutral $N=20$ PPy reference. (b) and (f) contain one polaron ${\text{P}}^{+}$. (c) and (g) contain one bipolaron ${\text{P}}^{2+}$. A bipolaron lattice, $3\left({\text{P}}^{2+}\right)$, starts to form above 30% doping, (d) and (h). The order parameters computed by DFT-PBE and HF, both using the 3-21G basis set in GAUSSIAN-03 (Ref. 37), are shown as delocalization versus localization comparisons.

Figure 7

(Color online) The first-order IR intensity $I^{\text{IR}}$, Raman-scattering activity $I^{\text{Ram}}$, and the 37 (or 17 nondegenerate) localized C-C vibrational modes $Q_k\left(i\right)={\left(-1\right)}^i\Delta b_i$ are plotted for an $N=100$ PPy chain, where $i$ labels the number order of C-C bonds. Explicitly, the bond pairs of $⟨5m+1,5m+2⟩$, $⟨5m+2,5m+3⟩$, $⟨5m+3,5m+4⟩$, and $⟨5m+4,5m+6⟩$ defined by Fig. 1 are labeled as the bonds of number $4m+1$, $4m+2$, $4m+3$, and $4m+4$, respectively; $\Delta b_i$ measures the change in the $i\text{th}$ C-C bond length according to the vibrational mode $Q_k$. No differences in the IR or Raman spectra are found for $N=20$; however the $N=100$ vibrational modes show clearer localizations. Only the localized regions are plotted for all the vibrational modes except for the Goldstone mode g1. The vibrational mode indices are labeled according to the primary motions; for instance, $2\times \left(\text{g}{1}_{\beta \beta }\right)846{\text{cm}}^{-1}$ indicates that this mode is the g1 mode involving primarily $\beta \text{−}\beta$ bonds with a vibrational frequency of $846{\text{cm}}^{-1}$ and the degeneracy is 2. Vibrational modes are plotted in colors, red for $\alpha \text{−}\alpha$ (bridge) bonds, blue for $\alpha \text{−}\beta$ and $\beta \text{−}\alpha$ bonds, and black for $\beta \text{−}\beta$ bonds.