# Theory of the Dark State of Polyenes and Carotenoids

William Barford<EMAIL_ADDRESS>Department of Chemistry,
Physical and Theoretical Chemistry Laboratory, University of Oxford, Oxford,
OX1 3QZ, United Kingdom

###### Abstract

A theory is developed to describe the singlet dark state (usually labeled
$S_{1}$ or $2A_{g}$) of polyenes and carotenoids. The theory assumes that in
principle this state is a linear combination of a singlet triplet-pair and an
odd-parity charge-transfer exciton. Crucially, these components only couple
when the triplet-pair occupies neighboring dimers, such that an electron
transfer between the triplets creates a nearest-neighbor charge-transfer
excitation. This local coupling stabilises the $2A_{g}$ state and induces a
nearest neighbor attraction between the triplets. In addition, because of the
electron-hole attraction in the exciton, the increased probability that the
electron-hole pair occupies neighboring dimers enhances the triplet-triplet
attraction: the triplet pair is ‘slaved’ to the charge-transfer exciton. The
theory also predicts that as the Coulomb interaction is increased, the
$2A_{g}$ state evolves from a predominately odd-parity charge-transfer exciton
state with a small component of triplet-pair character to a state
predominately composed of a triplet-pair with some exciton character. Above a
critical Coulomb interaction there is a decoupling of the triplet-pair and
charge-transfer exciton subspaces, such that the $2A_{g}$ state becomes
entirely composed of an unbound spin-correlated triplet pair. The predictions
of this theory are qualitatively consistent with high-level density matrix
renormalization group calculations of the Pariser-Parr-Pople (or extended
Hubbard) model.

##  I Introduction

The low-energy singlet dark state of polyenes, usually labeled $S_{1}$ or
$2A_{g}$, has continued to fascinate researchers for over 50 years[1, 2]. Its
intriguing electronic properties are a consequence of electron-electron
interactions and electron-nuclear coupling[3].

The triplet-pair or bimagnon character of the $2^{1}A_{g}$ state was first
predicted theoretically by Schulten and Karplus in 1972[2] and further
elucidated by Tavan and Schulten in 1987[4]. Using the Hubbard-Peierls model,
Hayden and Mele[5] then demonstrated the combined role of electronic
interactions and electron-nuclear coupling in determining the four-soliton
structure of this state for 16 C-atom chains. The four-soliton structure was
further investigated for polyene chains of up to 100 C-atoms by solving the
Pariser-Parr-Pople-Peierls (PPPP) model using the density matrix
renormalization group (DMRG) method[6, 7]. Since a triplet excitation coupled
to the nuclei creates a soliton-antisoliton ($\textrm{S}\bar{\textrm{S}}$)
pair, the four-soliton structure (i.e., a pair of soliton-antisoliton pairs)
is further strong theoretical evidence of the triplet-pair character of the
$2^{1}A_{g}$ state.

Solitons (S) and antisolitons ($\bar{\textrm{S}}$) are associated with domain
walls in the bond alternation of linear polyenes[8, 9]. They are also
associated with spinons, the elementary excitations of a one-dimensional
spin-1/2 quantum antiferromagnet[10, 11]. A spinon is a charge-neutral,
spin-1/2 object, while a triplet excitation is a bound spinon-pair, which is
recognized as a $\textrm{S}\bar{\textrm{S}}$ pair in the bond alternation.
Thus, the $\textrm{S}\bar{\textrm{S}}$ pair separation is a measure of the
internal size of the triplet excitation. Similarly, the
$\textrm{S}\bar{\textrm{S}}$-$\textrm{S}\bar{\textrm{S}}$ separation is a
measure of the triplet-triplet pair separation. Barford et al.[6, 12] showed
that in polyenes the $\textrm{S}\bar{\textrm{S}}$ separation converges to 4
dimers, while the $\textrm{S}\bar{\textrm{S}}$-$\textrm{S}\bar{\textrm{S}}$
separation converges to 6 dimers, indicating that the triplet-pair is bound.
Further numerical evidence that the triplet-pair is bound in the $2^{1}A_{g}$
state was provided by Valentine et al.[12], who showed that its energy
converges to a value of $0.3$ eV lower than the energy of a pair of free-
triplets.

In the other limit of weak electronic correlations, as in light-emitting
polymers, the $2^{1}A_{g}^{-}$ state is predominately a charge-transfer
exciton. The electron-hole wavefunction of this state has odd parity (i.e.,
the wavefunction is odd under an exchange of the electron and hole), and it
lies higher in energy that the optically-allowed $1^{1}B_{u}^{+}$ state (which
is usually labeled $S_{2}$ in polyenes) whose electron-hole wavefunction has
even parity.

The reversal of energies of the $1^{1}B_{u}^{+}$ and $2^{1}A_{g}^{-}$ states
in polyenes is partly a consequence of electronic interactions. As explained
later, electronic interactions simultaneously reduce the excitation energy of
the lowest covalent triplet state and increase the excitation energy of ionic
singlet states, which implies that for large enough interactions a triplet-
pair state will have a lower energy than a singlet ionic state. The triplet-
pair state is further stabilized relative to the ionic state, as electron-
nuclear relaxation is enhanced in covalent states.

The energetic reversal of the bright ($S_{2}$) and dark ($S_{1}$) states has
various photophysical consequences. For example, it explains the non-emissive
properties of linear polyenes, it is responsible for the photoprotection
properties of carotentoids in light harvesting complexes, and – because of its
triplet-pair character – it is thought to be the cause of singlet fission in
polyene-type systems[13, 14, 15, 16, 17]. The electronic states of carotenoids
are reviewed in ref[18, 19, 20], while refs[21, 22, 23] report on recent high-
level ab initio calculations of these states.

As already noted, the $2^{1}A_{g}$ state of polyenes has been extensively
studied via DMRG calculations of the PPPP model. As well as the four-soliton
structure, triplet overlaps and spin-spin correlation functions also reveal
its triplet-pair character[12]. Using a suitable exciton creation
operator[24], Valentine et al.[12] also investigated the excitonic component
of this state. Figure 8 of ref[12] illustrates the odd-parity charge-transfer
exciton wavefunction characteristic of the $2^{1}A_{g}$ state of more weakly
correlated polymers.

Thus, we can conclude that the $2^{1}A_{g}$ state is a linear combination of a
singlet triplet-pair and an odd-parity charge transfer exciton. This suggests
that the mixing of the triplet-pair and charge-transfer exciton subspaces both
stabilizes the $2^{1}A_{g}$ state and causes the strong triplet-pair
attraction. This stabilization is an additional cause of the
$1^{1}B_{u}$/$2^{1}A_{g}$ energy reversal in polyenes. The theory presented in
this paper will explain this property.

The $2^{1}A_{g}^{-}$ state is the lowest energy member of a family of states
with the same elementary excitations, but with different pseudomomentum
quantum numbers (i.e., $2^{1}A_{g}^{-},1^{1}B_{u}^{-},3^{1}A_{g}^{-},\cdots$),
which for convenience we label as the ‘$2A_{g}$ family’. Irrespective of their
even or odd symmetry under a two-fold rotation, these states are all optically
dark. The singlet triplet-pair component of the $2A_{g}$ family is optically
dark for two reasons. First, it is composed of a pair of electronic
excitations and since the dipole operator is a one-electron operator, the
transition dipole moment with the ground state vanishes. Second, each
electronic excitation is a triplet and since the dipole operator commutes with
total spin, the transition dipole moment with the singlet ground state again
vanishes. The exciton component of the $2A_{g}$ family is also optically dark,
because its electron-hole wavefunction has odd parity and thus its transition
dipole moment with the ground state vanishes[3].

In this paper we describe an effective low-energy model of the $2A_{g}$ family
of states in conjugated polyenes that provides a simple, physical explanation
of triplet-pair binding. There are two key approximations in the model. First,
it assumes that polyene chains are comprised of weakly interacting ethylene
dimers. This implies that the ground state is composed of a product of singlet
dimer states. Second, it assumes a reduced basis for the excited states,
composed of singlet triplet-pair excitations and odd-parity electron-hole
excitations. The dimer limit implies that the $\textrm{S}\bar{\textrm{S}}$
pair comprising a triplet excitation are bound on a single dimer. As a
consequence of these approximations, the theory is only able to provide
qualitative predictions for polyene systems, which are nonetheless consistent
with DMRG calculations.

Crucially, the theory predicts that when a pair of triplets occupy neighboring
dimers an electron transfer between the dimers connects the pair of triplets
to the odd-parity charge transfer exciton. This local coupling stabilizes the
$2A_{g}$ state and induces a nearest neighbor attraction between the triplets.
In addition, because of the electron-hole attraction in the exciton, the
increased probability that the electron-hole pair occupies neighboring dimers
enhances the triplet-triplet attraction: the triplet pair is ‘slaved’ to the
charge-transfer exciton. The theory also predicts that as the Coulomb
interaction increases, the charge-transfer exciton becomes energetically less
stable relative to the triplet pair causing a decoupling of the triplet-pair
and exciton subspaces. This decoupling has been observed in computational
studies of the PPPP model[3]. Above this Coulomb interaction the triplet-pair
binding energy vanishes and the $2A_{g}$ state is entirely composed of
triplet-pairs.

Triplet-pair interactions within a dimerized antiferromagnetic chain have been
investigated in refs[25, 11, 26], while stabilization of the $2A_{g}$ state
via configuration interactions was also discussed in ref[27].

The next section describes the model, while the results are presented in
Section III.

##  II Model

###  II.1 Pariser-Parr-Pople model of $\pi$-conjugated polymers

Our starting point for a derivation of a low-energy effective model for the
$2A_{g}$ state is the Pariser-Parr-Pople (PPP) model of $\pi$-conjugated
polymers. This model is defined as

$\displaystyle\hat{H}_{\textrm{PPP}}$ $\displaystyle=$
$\displaystyle-\sum_{m\sigma}t_{m}\left(\hat{c}^{{\dagger}}_{m\sigma}\hat{c}_{m+1\sigma}+\hat{c}^{{\dagger}}_{m+1\sigma}\hat{c}_{m\sigma}\right)$
(1) $\displaystyle+$ $\displaystyle
U\sum_{m}\big{(}\hat{N}_{m\uparrow}-\frac{1}{2}\big{)}\big{(}\hat{N}_{m\downarrow}-\frac{1}{2}\big{)}+\sum_{m}\sum_{n\geq
1}V_{n}\big{(}\hat{N}_{m}-1\big{)}\big{(}\hat{N}_{m+n}-1\big{)},$

where $\hat{c}^{{\dagger}}_{m\sigma}$ ($\hat{c}_{m\sigma}$) creates (destroys)
an electron with spin $\sigma$ in the $p_{z}$ orbital of carbon atom $m$.
$\hat{N}_{m\sigma}$ is the corresponding number operator and
$\hat{N}_{m}=\sum_{\sigma}\hat{N}_{m\sigma}$. Assuming periodic bond
alternation, the nearest neighbor electron transfer integral is

$t_{m}=t_{0}(1+(-1)^{m}\delta),$ (2)

where $\delta$ is the bond alternation parameter. For convenience, we define
$t_{d}=t_{0}(1+\delta)$ and $t_{s}=t_{0}(1-\delta)$ as the double and single
bond transfer integrals, respectively. The Coulomb interaction is represented
by the Ohno potential, i.e.,

$V_{n}=\frac{U}{\left(1+(U\epsilon r_{n}/14.397)\right)^{1/2}},$ (3)

where $U$ is in eV, the separation between atoms $m$ and $m+n$, $r_{n}$, is in
Å and $\epsilon$ is the relative permittivity.

Typical parameter values for $\pi$-conjugated polymers[28] are $U=8$ eV,
$\epsilon=2$, $t_{0}=2.4$ eV. In addition, for polyenes, $\delta=1/12$, so
that $t_{d}=2.6$ eV and $t_{s}=2.2$ eV. In this paper, however, $U$ and
$\delta$ are generally arbitrary parameters.

###  II.2 Effective model of the $2A_{g}$ state

Figure 1: The four electronic eigenstates of the two-electron ethylene dimer,
(a) $|\textrm{GS}\rangle$, (b) $|\textrm{T}\rangle$, (c) $|\textrm{S}\rangle$
and (d) $|\textrm{EX}\rangle$. (e) The basis states $|1\rangle$ and
$|2\rangle$. The eigenstate energies, and $\alpha$ and $\beta$ are given in
Appendix A. Also shown in (b) is the equivalence between the dimer atomic-
orbital representation (i.e., $\phi_{1}$ and $\phi_{2}$ ) and molecular-
orbital representation (i.e., $\psi_{\pm}=(\phi_{1}\pm\phi_{2})/\sqrt{2}$).

The model assumes that carotenoids and polyenes are linear chains of weakly
coupled ethylene dimers, where a dimer is composed of two carbon atoms each
with a single $p_{z}$ orbital. The four electronic states of a dimer with two
$\pi$-electrons are illustrated in Fig. 1. The singlet ground state,
$|\textrm{GS}\rangle=\alpha|1\rangle+\beta|2\rangle$, is a linear combination
of covalent and ionic states (i.e., $|1\rangle$ and $|2\rangle$). As shown by
Eq. (25), for noninteracting electrons, $\alpha=\beta=1/\sqrt{2}$, but as
$U/t_{d}\rightarrow\infty$, $\alpha\rightarrow 1$. For realistic polyene
parameters $\alpha\sim\sqrt{2/3}$.

In general, the lowest energy excitation is the covalent triplet state,
$|\textrm{T}\rangle$, while the lowest singlet excitation is the ionic state,
$|\textrm{S}\rangle$. The excitation energies of these two states as a
function of $U/t_{d}$ are illustrated in Fig. 2. In the noninteracting limit
(i.e., $U=0$) these excitation energies are degenerate at $2t_{d}$. However,
as $U$ increases, the ground state becomes more covalent and the triplet
excitation energy decreases as $4t_{d}^{2}/(U-V_{1})$. In contrast, the
singlet excitation energy increases as $(U-V_{1})$. Appendix A discusses the
dimer solutions in more detail.

Figure 2: The ethylene dimer triplet, $\Delta E_{\textrm{T}}$, and singlet,
$\Delta E_{\textrm{S}}$, excitation energies as a function of $U/t_{d}$. Also
shown for neighboring dimers is the energy of the charge-transfer exciton
relative to a pair of triplets, $\Delta
E=((E_{\textrm{CT}}-\tilde{V}_{1})-E_{\textrm{TT}})$, where $-\tilde{V}_{1}$
is the nearest neighbor electron-hole interaction.

####  II.2.1 Singlet triplet-pair basis

We now make the assumption that the ground state of polyenes can be
approximated as a product of dimer ground states, i.e.,
$|\textrm{GS}\rangle=\Pi_{i}|\textrm{GS}\rangle_{i}$, where the product is
over all dimers and $|\textrm{GS}\rangle_{i}$ is the ground state for dimer
$i$, defined in Eq. (24).

A triplet excitation from the polyene ground state then corresponds to a
triplet excitation on a single dimer (i.e., $|\textrm{T}\rangle$). We denote
this state as $|S=1,M_{S};i\rangle$, where $M_{S}$ is the spin-projection
(i.e., $1$, $0$, or $-1$) and $i$ labels the dimer. A triplet on dimer $i$ can
hop to a neighboring dimer in its singlet ground state by a superexchange
mechanism, i.e., via a virtual charge-transfer state higher in energy by
$(U-V_{1})$. By second order perturbation theory, the superexchange transfer
integral is

$t_{\textrm{TT}}=-\alpha^{2}\frac{t_{s}^{2}}{(U-V_{1})},$ (4)

where $\alpha^{2}$ (defined in Eq. (25)) is the probability that the
neighboring singlet dimer is in the covalent state labeled $|1\rangle$ in Fig.
1.

A triplet-pair excitation corresponds to excitations on separate dimers, $i$
and $j$. A pair of spin-correlated triplets can form an overall singlet,
triplet or quintet state. Here, we concerned with the singlet triplet-pair
state, expressed as

$|i,j;\textrm{TT}\rangle=\frac{1}{\sqrt{3}}\left(|1,1;i\rangle|1,-1;j\rangle-|1,0;i\rangle|1,0;j\rangle+|1,-1;i\rangle|1,1;j\rangle\right)$
(5)

and illustrated schematically in Fig. 3.

Figure 3: Atomic orbital representation of the singlet triplet-pair basis
state,
$|i,j;\textrm{TT}\rangle=\left(|1,1;i\rangle|1,-1;j\rangle-|1,0;i\rangle|1,0;j\rangle+|1,-1;i\rangle|1,1;j\rangle\right)/\sqrt{3}$,
with triplet excitations on dimers $i$ and $j$. The dashed lines $---$
represent $(j-i-1)$ dimers in their singlet ground state. When $j=i+1$ the
state labeled $*$ is connected to the state labeled $\times$ in Fig. 4 via an
electron transfer across the single bond.

Thus, the Hamiltonian that describes the singlet triplet-pair subspace is

$\displaystyle\hat{H}_{\textrm{TT}}$
$\displaystyle=E_{\textrm{TT}}\sum_{i=1}^{N-1}\sum_{j=i+1}^{N}\left|i,j;\textrm{TT}\rangle\langle
i,j;\textrm{TT}\right|$
$\displaystyle+t_{\textrm{TT}}\left[\sum_{i=2}^{N-1}\sum_{j=i+1}^{N}\left(\left|i-1,j;\textrm{TT}\rangle\langle
i,j;\textrm{TT}\right|+\textrm{H.C.}\right)+\sum_{i=1}^{N-2}\sum_{j=i+1}^{N-1}\left(\left|i,j+1;\textrm{TT}\rangle\langle
i,j;\textrm{TT}\right|+\textrm{H.C.}\right)\right].$

The first term on the right-hand-side describes the energy to excite a pair of
triplets on different dimers $i$ and $j$, where

$E_{\textrm{TT}}=2V_{1}-2E_{\textrm{GS}}^{\textrm{dimer}}$ (7)

and $E_{\textrm{GS}}^{\textrm{dimer}}$ is defined in Eq. (21). The second pair
of terms describes the hopping of each triplet onto neighboring singlet
dimers, while avoiding a hop onto the same dimer.

####  II.2.2 Charge-transfer basis

The second kind of excitation from the ground state are electron-hole
excitations. The dimer state $|\textrm{S}\rangle$ shown in Fig. 1 is the basis
state for the tightly bound Frenkel exciton of conjugated polymers[3]. Because
of its even electron-hole parity, however, this state cannot couple with a
singlet triplet-pair state. For an electron-hole excitation to couple to a
triplet-pair state it must have odd electron-hole parity, which implies a
charge-transfer state as illustrated in Fig. 4. The electron in the
antibonding $\psi_{-}$ dimer orbital and a hole in the bonding $\psi_{+}$
dimer orbital hop between neighboring dimers via the transfer integral

$t_{\textrm{CT}}=t_{s}/2,$ (8)

where the factor of $1/2$ arises from the overlap of neighboring dimer
orbitals.

Figure 4: (a) Molecular orbital representation of the odd-parity singlet
charge-transfer basis state, $|i,j;\textrm{CT}\rangle$, with electron-hole
excitations on dimers $i$ and $j$. (b) The equivalence between the dimer
molecular-orbital and atomic-orbital representations. When $j=i+1$ the state
labeled $\times$ is connected to the state labeled $*$ in Fig. 3 via an
electron transfer across the single bond.

The Hamiltonian that describes the odd-parity electron-hole subspace is

$\displaystyle\hat{H}_{\textrm{CT}}$
$\displaystyle=\sum_{i=1}^{N-1}\sum_{j=i+1}^{N}(E_{\textrm{CT}}-\tilde{V}_{(j-i)})\left|i,j;\textrm{CT}\rangle\langle
i,j;\textrm{CT}\right|$
$\displaystyle+t_{\textrm{CT}}\left[\sum_{i=2}^{N-1}\sum_{j=i+1}^{N}\left(\left|i-1,j;\textrm{CT}\rangle\langle
i,j;\textrm{CT}\right|+\textrm{H.C.}\right)+\sum_{i=1}^{N-2}\sum_{j=i+1}^{N-1}\left(\left|i,j+1;\textrm{CT}\rangle\langle
i,j;\textrm{CT}\right|+\textrm{H.C.}\right)\right]$

The first term on the right-hand-side describes the energy to excite an
electron-hole pair on different dimers $i$ and $j$, where

$E_{\textrm{CT}}=(U+2V_{1}-2t_{d}))-2E_{\textrm{GS}}^{\textrm{dimer}}.$ (10)

In addition, there is a Coulomb attraction between the electron-hole pair
$\ell=(j-i)$ dimers apart, given by

$\tilde{V}_{\ell}=(V_{2\ell-1}+2V_{2\ell}+V_{2\ell+1})/4.$ (11)

The final two terms describe the (symmetrized) motion of the electron or hole
between neighboring dimers.

As described in ref[29], in the continuum limit for a $1/r$ Coulomb
interaction there is a Rydberg series of odd-parity bound electron-hole pairs.
The $2A_{g}$ charge-transfer exciton state is the lowest energy member of this
series.111The even-parity excitons alternate in energy with the odd-parity
excitons, while the lowest even-parity (Frenkel) exciton is split off from the
Rydberg series[29, 3].

The relative energies of the triplet-pair and electron-hole pair is
illustrated in Fig. 2. In the noninteracting limit a single triplet is
degenerate with an electron-hole excitation, so $\Delta
E=(E_{\textrm{CT}}-E_{\textrm{TT}})$ is negative. However, as we have seen, as
$U$ increases the triplet energy decreases while the electron-hole energy
increases, causing a reversal of energies at $U\sim 4t_{d}$. It is this
property of covalent and ionic states which partially stabilizes the
$2^{1}A_{g}^{-}$ state relative to the optically active $1^{1}B_{u}^{+}$
Frenkel exciton state in polyenes.

####  II.2.3 Triplet-pair and charge-transfer pair coupling

We now consider the coupling between the singlet triplet-pair and the odd-
parity charge-transfer subspaces. For the special case that a pair of triplet
excitations occupy neighboring dimers, i.e., $|i,i+1;\textrm{TT}\rangle$, an
electron transfer across the single bond between them creates a nearest
neighbor electron-hole pair, i.e., $|i,i+1;\textrm{CT}\rangle$. This may be
understood by examining the basis states labeled $*$ in Fig. 3 and $\times$
Fig. 4. By inspection of $*$ in Fig. 3, we observe that a transfer of the down
electron on site $1$ of dimer $j=i+1$ to site $2$ of dimer $i$ creates a
component of the basis state $\times$ in Fig. 4.

The Hamiltonian describing this process is

$\displaystyle\hat{H}_{\textrm{CT-TT}}=-V_{\textrm{CT-
TT}}\sum_{i=1}^{N-1}\left(\left|i,i+1;\textrm{CT}\rangle\langle
i,i+1;\textrm{TT}\right|+\left|i,i+1;\textrm{TT}\rangle\langle
i,i+1;\textrm{CT}\right|\right),$ (12)

where

$V_{\textrm{CT-TT}}=\sqrt{3}t_{s}/2.$ (13)

Since this term only connects a nearest-neighbor triplet pair with a nearest-
neighbor electron-hole pair, it results in an attraction between the triplet
pair and an additional attraction above the Coulomb interaction for the
electron-hole pair.

####  II.2.4 The $2A_{g}$ state

We express an eigenstate of the full effective Hamiltonian,
$\hat{H}=\hat{H}_{\textrm{TT}}+\hat{H}_{\textrm{CT}}+\hat{H}_{\textrm{CT-
TT}},$ as

$\displaystyle|\Psi\rangle=\sum_{i,j>i}\Psi_{ij}^{\textrm{TT}}|i,j;\textrm{TT}\rangle+\Psi_{ij}^{\textrm{CT}}|i,j;\textrm{CT}\rangle.$
(14)

Since this is a linear combination of basis states from the triplet-pair and
electron-hole pair subbases, and not formed via a direct product of the
subbases, the interaction $V_{\textrm{CT-TT}}$ does not correlate (or
entangle) the triplet-pair and charge-transfer exciton. Instead, it stablizes
the linear combination and causes a triplet-triplet attraction.

Figure 5: A schematic diagram of the relative energy levels of the TT
subspace, the CT exciton subspace, and the virtual triplet-electron-hole
subspace by which a triplet hops to a neighboring dimer in its singlet ground
state (indicated by green arrows). The coupling of the TT and CT exciton
subspaces is indicated by the red-dashed arrow, which only occurs when the
triplet pair occupies neighboring dimers. $BE$ is the CT exciton binding
energy.

Before discussing the predictions of this model, we conclude this section by
illustrating the separation of energy scales via Fig. 5. This shows (for
relevant polyene parameters) the triplet-pair subspace lying lower in energy
than the charge-transfer exciton subspace. Lying higher still in energy is the
virtual triplet-electron-hole subspace, through which the triplets delocalize
via a second-order process.

##  III Results and Discussion

We now describe the solutions of the effective model of the $2A_{g}$ state. We
begin by exploring the parameter regime with arbitrary parameters for a
translationally invariant system of 1000 monomers. We define two energy gaps.
First, the stabilization energy gained by the $2A_{g}$ state caused by the
coupling of the triplet and electron-hole pair subspaces, i.e.,

$\Delta_{\textrm{CT-TT}}=E(V_{\textrm{CT-TT}}=0)-E(V_{\textrm{CT-TT}}).$ (15)

Second, the triplet-triplet pair binding energy, defined as the energy gap
between the band of free triplet-pairs and the $2A_{g}$ state, i.e.,

$\Delta_{\textrm{TT}}=(E_{\textrm{TT}}-4|t_{\textrm{TT}}|)-E(V_{\textrm{CT-
TT}}).$ (16)

The state will be characterized by its triplet-pair weight, i.e.,

$P_{\textrm{TT}}=\sum_{i,j>i}\left|\Psi_{ij}^{\textrm{TT}}\right|^{2},$ (17)

and the mean triplet-pair separation (in monomer units), i.e.,

$L_{\textrm{TT}}=\frac{\sum_{i,j>i}(j-i)\left|\Psi_{ij}^{\textrm{TT}}\right|^{2}}{P_{\textrm{TT}}}.$
(18)

###  III.1 Toy Model Parameters

We begin our investigation of the predictions of the model by considering the
simplest limit of non-interacting electron-hole pairs, i.e., we set the
Coulomb interaction $\tilde{V}_{(j-i)}=0$ in Eq. (II.2.2). In this limit there
is a symmetry between the triplet-pair and the electron-hole pair under the
exchange $E_{\textrm{CT}}\leftrightarrow E_{\textrm{TT}}$ and
$|t_{\textrm{CT}}|\leftrightarrow|t_{\textrm{TT}}|$. In the absence of the
coupling, $V_{\textrm{CT-TT}}$, between the two subspaces the triplet-pair
forms a band of free triplet excitations (subject to a hard-core repulsion)
centered at $E_{\textrm{TT}}$ of width $8|t_{\textrm{TT}}|$. Likewise, there
is a band of unbound electron-hole excitations centered at $E_{\textrm{CT}}$
of width $8|t_{\textrm{CT}}|$.

Turning on the coupling, $V_{\textrm{CT-TT}}$, causes the nearest neighbor
triplet-pair to mix with the nearest neighbor electron-hole pair. At
resonance, i.e., when $E_{\textrm{CT}}=E_{\textrm{TT}}$ and
$|t_{\textrm{CT}}|=|t_{\textrm{TT}}|$, the mixing causes a nearest neighbor
attraction of $V_{\textrm{CT-TT}}$ for both the triplet and electron-hole
pairs. This model then maps onto the well-known one-dimensional lattice model
of spinless fermions interacting with a nearest neighbor attraction,
$V_{\textrm{CT-TT}}$, for which above a critical value of $V_{\textrm{CT-TT}}$
there is a single band of bound states. As shown in refs[31, 32, 33], the
binding energy for the zero-momentum state is

$E_{BE}=V_{\textrm{CT-TT}}+\frac{4t_{\textrm{TT}}^{2}}{V_{\textrm{CT-
TT}}}-4|t_{\textrm{TT}}|,$ (19)

implying a critical value of $V_{\textrm{CT-TT}}$ of $2|t_{\textrm{TT}}|$.

Figure 6 illustrates the solutions when
$|t_{\textrm{CT}}|=|t_{\textrm{TT}}|=1$ eV and $V_{\textrm{CT-TT}}=3$ eV. At
resonance (i.e., $\Delta E=(E_{\textrm{CT}}-E_{\textrm{TT}})=0$), the $2A_{g}$
state, $|\Psi\rangle$, consists of an equal mixture of bound triplet pairs and
bound electron-holes pairs. The mean triplet-triplet and electron-hole
separation in both pairs is ca. 2 dimers. The energy gap between this state
and the bands of free triplets and electron-holes (i.e., $\Delta_{\textrm{CT-
TT}}$) is $|t_{\textrm{TT}}|/3$, in agreement with Eq. (19). At resonance this
energy gap corresponds to the binding energies of both the triplet-pairs and
electron-hole pairs, i.e., $\Delta_{\textrm{CT-
TT}}=\Delta_{\textrm{TT}}=\Delta_{\textrm{CT}}$.

Figure 6: Non-Coulombically interacting electron-hole pairs (i.e., $U=0$).
$\Delta_{\textrm{CT-TT}}$, Eq. (15), is the stabilization energy of the
$2A_{g}$ state caused by the coupling of the triplet-pair and electron-hole
pair; $\Delta_{\textrm{TT}}$, Eq. (16), is the triplet-pair binding energy;
$L_{\textrm{TT}}$, Eq. (17), is the mean triplet-pair separation (scaled by
$0.1$ and in units of dimers); and $P_{\textrm{TT}}$, Eq. (18), is the
triplet-pair weight. $\Delta_{\textrm{TT}}=\Delta_{\textrm{CT-TT}}$ when
$\Delta E\geq 0$. $t_{TT}=-t_{CT}=-1.0$ eV, $V_{CT-TT}=3.0$ eV. $\Delta
E_{\textrm{critical}}=\pm 1.35$ eV, above and below which the triplet-pair and
electron-hole pair subspaces are decoupled.

As the energy of the band of electron-hole excitations increases, i.e., as
$E_{\textrm{CT}}$ increases, the system goes off resonance and the $2A_{g}$
state is predominately composed of bound triplet-pairs; the triplet-pair
binding energy decreases and the mean triplet-pair separation increases. At a
critical value of $\Delta E$, namely $\Delta
E_{\textrm{critical}}^{\textrm{U}}=1.35$ eV, there is a decoupling of the
triplet-pair and electron-hole pair subspaces, and the triplet-pair and the
electron-hole pair both become unbound. For $\Delta E\geq\Delta
E_{\textrm{critical}}^{\textrm{U}}$, $P_{\textrm{TT}}=1$.

Similarly, as $E_{\textrm{CT}}$ decreases from resonance the $2A_{g}$ state is
predominately composed of bound electron-hole pairs. Now the triplet-pair
binding energy, $\Delta_{\textrm{TT}}$, increases and the mean triplet-pair
separation decreases. Again, there is critical value of $\Delta E$, namely
$\Delta E_{\textrm{critical}}^{\textrm{L}}=-1.35$ eV, below which there is a
decoupling of the triplet-pair and electron-hole pair subspaces, and both the
triplet-pair and electron-hole pair become unbound. For $\Delta E\leq\Delta
E_{\textrm{critical}}^{\textrm{L}}$, $P_{\textrm{TT}}=0$.

Figure 7: Coulombically interacting electron-hole pairs ($U=6$ eV).
$t_{TT}=-t_{CT}=-1.0$ eV, $V_{CT-TT}=3.0$ eV. $\Delta
E_{\textrm{critical}}=+3.0$ eV above which the triplet-pair and electron-hole
pair subspaces are decoupled. The charge-transfer exciton binding energy,
$BE_{\textrm{CT}}=0.74$ eV.

We now turn to the physical limit of interacting electron-hole pairs, i.e., we
set the Coulomb interaction $\tilde{V}_{(j-i)}>0$ in Eq. (II.2.2). We set
$U=6$ eV, giving a charge-transfer exciton binding energy of $0.74$ eV. As
illustrated in Fig. 7, the electron-hole pair attraction causes a number of
significant changes to the $\tilde{V}_{(j-i)}=0$ picture. As for Fig. 6, the
results shown are for $|t_{\textrm{CT}}|=|t_{\textrm{TT}}|=1$ eV and
$V_{\textrm{CT-TT}}=3$ eV. At resonance the triplet-pair and charge-transfer
exciton are mutually stabilized by $1.01$ eV. As $1.01$ eV is the excitation
energy to the free triplet-pair band, this is also the triplet-pair binding
energy. This value is considerably enhanced over the $\tilde{V}_{(j-i)}=0$
limit of $0.333$ eV, as the triplets are ‘slaved’ to the Coulombically bound
electron-hole pair. We also note that the triplet contribution is only 33%,
compared to 50% at resonance in the absence of Coulombically bound electron-
hole pairs. Again, as $E_{\textrm{CT}}$ is increased the triplet-pair binding
energy decreases, while the triplet-pair separation and fraction increases.
Above a critical value of $E_{\textrm{CT}}$, given by $\Delta E\gtrsim 3.0$
eV, there is a decoupling of the triplet-pair and electron-hole subspaces and
the triplet-pair unbinds. In contrast, as $E_{\textrm{CT}}$ is decreased there
is no such decoupling of the triplet-pair and electron-hole subspaces. In this
case the triplet-pair becomes more tightly bound, albeit its contribution to
the $2A_{g}$ state becomes less than 10% for $\Delta E<-3.0$ eV.

###  III.2 PPP Model Parameters

Figure 8: Results as a function of the Coulomb repulsion, $U$, using
effective-model parameters derived from the PPP model (see Section II.2). The
bond alternation, $\delta=1/12$. For $U>5.9$ eV the triplet-pair state is more
stable than the charge-transfer exciton. The triplet-pair and charge-transfer
subspaces decouple for $U\geq 6.6$ eV.

The previous section described the predictions for arbitrary parameters. We
now consider the prediction for model parameters derived from the underlying
PPP model, Eq. (1). In this case, as shown by Eq. (4), Eq. (7), Eq. (8), Eq.
(10), Eq. (11) and Eq. (13), the model parameters are determined by the
Coulomb interaction, $U$, and the bond dimerization, $\delta$.

Figure 8 illustrates the results by varying the Coulomb interaction, $U$, and
for a realistic fixed bond alternation for polyenes, i.e., $\delta=1/12$.
Since $(E_{\textrm{CT}}-E_{\textrm{TT}})=(U-2t_{d})$, Fig. 8 qualitatively
resembles Fig. 7, namely as $U$ is increased the triplet-pair band becomes
more stable relative to the charge-transfer exciton. Resonance (i.e.,
$\Delta_{\textrm{TT}}=\Delta_{\textrm{CT-TT}}$) occurs at $U=5.9$ eV, where
$\Delta_{\textrm{TT}}=0.36$ eV and $P_{\textrm{TT}}=0.22$. For $U\geq 6.6$ eV
the triplet-pair and charge-transfer subspaces decouple, implying that the
triplet-pair binding energy vanishes and the $2A_{g}$ state is entirely
composed of triplet-pairs. This prediction of decoupling above a critical $U$
is consistent with DMRG calculations of the PPPP model, as shown in Fig. 7.8
of [3]. As before, as $U$ is reduced the converse does not happen and the
subspaces only decouple at $U=0$.

This model predicts that for a realistic value of $U=8$ eV and $\delta=1/12$,
the triplet-pair and charge-transfer subspaces are decoupled. We explain this
unphysical prediction as follows. First, the model is expected to become more
exact in the dimer limit, i.e., as $\delta\rightarrow 1$. Indeed, when
$\delta=0.16$, resonance occurs at $U=6.4$ eV (where
$\Delta_{\textrm{TT}}=0.31$ eV and $P_{\textrm{TT}}=0.24$) and decoupling of
the subspaces occurs at $U=9.9$ eV. Second, the reduced basis model
underscreens Coulomb interactions, so a smaller value of $U$ is required to
reproduce experimental ionic excitation energies.

Figure 9: Results as a function of the bond alternation, $\delta$, using
effective-model parameters derived from the PPP model (see Section II.2). The
Coulomb repulsion, $U=6$ eV. For $\delta\leq 0.1$ the triplet-pair state is
more stable than the charge-transfer exciton.

We explore the role of $\delta$ further via Fig. 9. Increasing $\delta$
decreases $E_{\textrm{CT}}$ and hence stabilizes the charge-transfer exciton.
It also reduces $t_{\textrm{CT}}$, $|t_{\textrm{TT}}|$ and $V_{\textrm{TT-
CT}}$. Thus, the triplet-pair state is more stable than the charge-transfer
exciton for $\delta\leq 0.1$. In this case, there is no decoupling of the
subspaces.

The results described so far are for the zero-momentum state of a
translationally invariant system (with periodic boundary conditions) and 1000
dimers. We conclude this section by describing our results for carotenoid
oligomers (with open boundary conditions) of between 8 and 13 dimers (i.e., 16
and 26 C-atoms). We set $U=6$ eV and $\delta=1/12$. In this case, compared to
the results shown in Fig. 8, there are some finite-size effects, so the
triplet-pair binding energy is increased to $\sim 0.5$ eV. Its weight in the
$2A_{g}$ state is $\sim 25\%$. The binding energy decreases for higher
pseudomomentum members of the ‘$2A_{g}$’ family, but the triplet-pair weight
remains the same.

##  IV Concluding Remarks

This paper has introduced a theory to describe the singlet dark state (i.e.,
$S_{1}$ or $2A_{g}$) of polyenes and carotenoids. The theory assumes that in
principle this state is a linear combination of a singlet triplet-pair and an
odd-parity charge-transfer exciton. Crucially, these components only couple
when the triplet-pair occupies neighboring dimers, such that an electron
transfer between the triplets creates a nearest-neighbor charge-transfer
excitation. This local coupling stabilises the $2A_{g}$ state and induces a
nearest neighbor attraction between the triplets. In addition, because of the
electron-hole attraction in the exciton, the increased probability that the
electron-hole pair occupies neighboring dimers enhances the triplet-triplet
attraction: the triplet pair is ‘slaved’ to the charge-transfer exciton. The
reduction of the $2A_{g}$ energy is an additional cause for the
$1^{1}B_{u}$/$2^{1}A_{g}$ energy reversal in polyenes.

The theory also predicts that as the Coulomb interaction is increased, the
$2A_{g}$ state evolves from a predominately odd-parity charge-transfer exciton
state with a small component of triplet-pair character to a state
predominately composed of triplet-pairs with some exciton character. Above a
critical Coulomb interaction there is a decoupling of the triplet-pair and
charge-transfer exciton subspaces, such that the $2A_{g}$ state becomes
entirely composed of an unbound spin-correlated triplet pair.

Although the predictions of this theory are qualitatively consistent with
high-level DMRG calculations of the PPP model[12, 3], it is not possible to
make quantitative predictions for polyenes and carotenoids, because of the
approximations of the model. The first approximation is that the dimer limit
is assumed for the ground state and its excitations, implying that the spinons
of a triplet are confined to a dimer. Second, the reduced basis of just the
triplet-pair and odd-parity electron-hole excitation implies that the Coulomb
interactions are underscreened. This means that somewhat smaller values of
Coulomb interactions, $U$, or larger values of bond alternation, $\delta$,
than are realistic for polyenes are required to obtain semiquantitative
agreement with DMRG calculations.

##  Appendix A Dimer Basis States

This Appendix summarizes the exact results of the PPP model for the
eigenstates of the ethylene dimer illustrated in Fig. 1.

The triplet excitation energy is

$\displaystyle\Delta E_{\textrm{T}}$
$\displaystyle=V_{1}-E_{\textrm{GS}}^{\textrm{dimer}}$ (20)
$\displaystyle\rightarrow 2t_{d}\textrm{ as }U/t_{d}\rightarrow 0$
$\displaystyle\rightarrow J_{d}\textrm{ as }U/t_{d}\rightarrow\infty,$

where $J_{d}=4t_{d}^{2}/(U-V_{1})$, the ground state energy is

$\displaystyle E_{\textrm{GS}}^{\textrm{dimer}}=(U+V_{1})/2-\epsilon$ (21)

and

$\epsilon=\frac{1}{2}\left((U-V_{1})^{2}+16t_{d}^{2}\right)^{1/2}.$ (22)

The lowest singlet excitation energy is

$\displaystyle\Delta E_{\textrm{S}}$
$\displaystyle=U-E_{\textrm{GS}}^{\textrm{dimer}}$ (23)
$\displaystyle\rightarrow 2t_{d}\textrm{ as }U/t_{d}\rightarrow 0$
$\displaystyle\rightarrow(U-V_{1})+J_{d}\textrm{ as
}U/t_{d}\rightarrow\infty.$

The ground state is

$|\textrm{GS}\rangle=\alpha|1\rangle+\beta|2\rangle,$ (24)

where the basis kets $|1\rangle$ and $|2\rangle$ are illustrated in Fig. 1,
and

$\displaystyle\alpha$
$\displaystyle=\frac{1}{2}\left(\frac{2\epsilon+(U-V_{1})}{\epsilon}\right)^{1/2}$
(25) $\displaystyle\rightarrow\frac{1}{\sqrt{2}}\textrm{ as
}U/t_{d}\rightarrow 0$ $\displaystyle\rightarrow 1\textrm{ as
}U/t_{d}\rightarrow\infty.$

Also, $\beta=(1-\alpha^{2})^{1/2}$.

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