Triplet-Pair Character of the $2^1A_g$ Dark State of Polyenes

Abstract

We define and calculate the triplet-pair population of the $2^1A_g$ dark state in polyenes, as predicted by the $π$-electron Pariser-Parr-Pople (PPP) model, for chains of 8 to 14 C-atoms and Coulomb interaction parameter between 4-14 eV. Our definition of the triplet-pair population is motivated by a two-particle model of the $2^1A_g$ state. We use DMRG to solve the PPP model and we exploit the MPS representation of the DMRG wavefunction to compute the triplet-pair population. Using our results for short chain sizes, we predict a finite-size scaling value of the triplet-pair population of ca. 75% for realistic Coulomb interactions for polyene chains. Our results agree with other theoretical work on the doubly-excited character of polyenes, and represents further evidence that the $2^1A_g$ state is predominantly triplet-pair in character - with implications for singlet fission mechanisms in polyenes.

Figures (5)

• Figure 1: The structure of the DMRG blocks. The first and last blocks are described by the set of block states $\{|\alpha_n\rangle \}$ and $\{|\beta_{n+2}\rangle \}$ respectively, which correspond to linear combinations of tensor products of site states, $|\sigma_m\rangle$. Above the block diagram is a representation of the respective segments of a real chain that these block states describe. The site states describe the occupation of the carbon $p_z$ orbitals, i.e., $|\sigma\rangle =\{ |0\rangle, |\uparrow\rangle, |\downarrow\rangle,|\uparrow \downarrow \rangle\}$.
• Figure 2: The definition of our two-particle (or triplet-pair) basis, described in Section \ref{['Se:2.3']} (and Section \ref{['Se:2.4']}). $\times$ represents a site (or dimer). For a chain of $N_d$ sites (or dimers), we define a two-particle (or triplet-pair) state as the tensor product between a single-particle (or triplet) state $|T_j(m)\rangle$ occupying the left subchain of $m$ sites (or dimers) and a single-particle (or triplet) state $|T_1(N_d-m)\rangle$ occupying the right subchain of $(N_d-m)$ sites (or dimers). $1 \le m \le (N_d - 1)$ and $1 \le j \le m$, such that $\sum_{m=1}^{N_d-1} m = N_d(N_d-1)/2$.
• Figure 3: (a) The triplet-pair population, $P_{TT}$, of the $2^1A_g$ state (defined in Eq. (\ref{['TT_weight']})) as a function of the Coulomb interaction parameter, $U$. (b) The extrapolation of the finite-size-scaling (FSS) values of $P_{TT}$ for the six different Coulomb parameters versus $N^{-1}$, where $N$ is the number of C-atoms. The data was fitted to the function $P_{TT} = aN^{-\alpha} + c$. The inset in (a) shows the FSS values of $P_{TT}$ as a function of $U$. Also shown are the exact values of $P_{TT}$ for a 4 C-atom chain, as described in Appendix \ref{['AppB:Huckel']}.
• Figure 4: The triplet-pair binding energy, $\Delta E_{TT}$, (defined in Eq. (\ref{['Eq:14']})) as a function of $N^{-1}$, where $N$ is the number of C-atoms, for different values of the Coulomb parameter, $U$.
• Figure 5: The triplet-pair population of the first and second lowest energy dark states, $2^1A_g^+$ and $1^1B_u^+$, in the limit of infinite chains. Also shown is the triplet-pair population calculated for the lowest energy quintet state, $1^5A_g^+$.