Loïc Berger
email: l.berger@ieseg.fr.

Ilke Aydogan

Aurélien Baillon

David Crainich

WHAT IS PARTIAL AMBIGUITY?

Keywords: Ambiguity, model uncertainty, smooth ambiguity aversion, Ellsberg paradox JEL Classification: D81

This paper reflects on the notion of partial ambiguity. Using a framework decomposing ambiguity into distinct layers of analysis, among which are risk and model uncertainty, and allowing for different attitudes toward these layers, I show that partial ambiguity may prove less desirable than full ambiguity, even under ambiguity aversion. This observation poses difficulties for interpreting the notion of partial ambiguity in relation to the partial information available to determine the potential compositions of an ambiguous urn. Two Ellsberg-style thought experiments are described to challenge the meaning of partial ambiguity further, and an alternative interpretation, based on a more ambiguous relation, is discussed.

Introduction

Ambiguity is the term that has emerged in the literature since the seminal paper of [START_REF] Ellsberg | Risk, ambiguity, and the Savage axioms[END_REF] to characterize situations in which the probabilities of uncertain events are unknown. Ambiguity is present in virtually all real-life situations and plays a significant role in most economic problems. It is distinct from the notion of risk, which refers to situations in which probabilities are perfectly known.

In a recent study, [START_REF] Chew | Partial ambiguity[END_REF] extended the decision-making problem originally presented in [START_REF] Ellsberg | Risk, ambiguity, and the Savage axioms[END_REF] by proposing different forms of "partial ambiguity". These authors write (p. 1240):

"As espoused by Knight, the essence of decision making under uncertainty is partial knowledge that may arise from vague evidence, diverse information, or conflicting news. Going beyond pure risk and full ambiguity in the original two-urn paradox, this paper experimentally studies decision making in a richer domain of uncertainty involving intermediate forms of ambiguity."

The interpretation giving rise to such a notion of partial ambiguity implicitly relies on the partial information, or knowledge, available to pin down the potential probability models describing the stochastic phenomenon of interest: from 1, in the pure risk case, to 101, in the full ambiguity case, originated from [START_REF] Ellsberg | Risk, ambiguity, and the Savage axioms[END_REF].

In this paper, I reflect on the meaning of partial ambiguity as an intermediate form of ambiguity. Using a recently developed approach that decomposes ambiguity into distinct layers of analysis, among which are risk and model uncertainty [START_REF] Hansen | Nobel lecture: Uncertainty outside and inside economic models[END_REF][START_REF] Marinacci | Model uncertainty[END_REF][START_REF] Hansen | Ambiguity aversion and model misspecification: An economic perspective[END_REF], and that allows for distinct layers to be treated differently, I show that partial ambiguity may be deemed less desirable than full ambiguity to an ambiguity-averse decision maker (DM).1 This counterintuitive observation challenges the significance of partial ambiguity.

In line with the risk theory literature (for which the notion of partial risk does not exist), I argue that partial ambiguity may not be helpful in characterizing more precisely uncertain situations that are by nature either risky or ambiguous, but hardly admit intermediate forms between them (i.e., either the probability distribution is known or it is unknown). However, if a refinement needs to be made in the classification of different ambiguous situations, I propose that a natural candidate can characterize an ambiguous act as whether it is affected more by ambiguity than another. This classification could be based on the more ambiguous notion proposed by [START_REF] Jewitt | Ordering ambiguous acts[END_REF]. It has the advantage of mirroring the well-established notion of more risky [START_REF] Rothschild | Increasing risk: I. a definition[END_REF] widely used in the risk theory literature. It furthermore allows for establishing a partial ordering among ambiguous situations (within a given class of preferences).

2 Ambiguity and the layers of uncertainty 2.1 Ellsberg paradox [START_REF] Ellsberg | Risk, ambiguity, and the Savage axioms[END_REF] proposed two thought-experiment decision problems, which have been the primary examples motivating research on ambiguity. One of them, known as the two-color problem, goes as follows.2

Thought Experiment [START_REF] Ellsberg | Risk, ambiguity, and the Savage axioms[END_REF] Imagine that you confront two urns containing red and black balls, from one of which a ball will be drawn at random. You can choose the color and the urn on which to bet. With a correct bet, you win $100 (and nothing happens if your bet is incorrect). You have the following information. The first urn, called E100, contains 100 red and black balls, but in a ratio entirely unknown to you. The second urn, called R100, contains exactly 50 red and 50 black balls. On which urn do you prefer to bet?

The four possible event-contingent payment schemes of the problem are presented in Table 1. They depend on the color of the ball drawn from one of the urns. Typically,

E100 R100 Red Black Red Black f $100 $0 g $100 $0 f $0 $100 g $0 $100 Table 1: Ellsberg two-color problem
what is observed is that people are indifferent on the color on which to bet: they express indifference between f and f , and between g and g (which is noted f ∼ f and g ∼ g , respectively); but prefer betting on R100 rather than on E100: they express a preference for g over f and for g over f (noted g f and g f ). Such preferences are, however, incompatible with the existence of subjective probabilities: the ranking f ≺ g would imply that the probability of drawing a red ball in urn E100 is considered to be less than 0.5 (i.e., less than the probability of drawing a red ball in urn R100), while at the same time the ranking f ≺ g would imply that the probability of drawing a black ball in urn E100 is considered to be less than 0.5 (i.e., less than the probability of drawing a black ball in R100). Overall, this would result in having the probabilities of complementary events summing up to less than 1, which is inconsistent. As bets f and f involve ambiguity, whereas bets g and g involve risk, these typically observed preferences indicate ambiguity aversion, that is, a preference for known-probability bets over unknown ones.

Decomposing ambiguity into layers of uncertainty

Following the works of [START_REF] Hansen | Nobel lecture: Uncertainty outside and inside economic models[END_REF]; [START_REF] Marinacci | Model uncertainty[END_REF]; and [START_REF] Hansen | Ambiguity aversion and model misspecification: An economic perspective[END_REF], themselves building on early insights of [START_REF] Arrow | Alternative approaches to the theory of choice in risk-taking situations[END_REF], ambiguity can generally be decomposed into different layers of uncertainty: (i) risk within a model, in which the uncertainty is about the outcomes of a given (probability) model that specifies fully the outcome probabilities; (ii) model uncertainty, in which the uncertainty is about which probability model (among a given set) should be used to assign these probabilities; and

(iii) model misspecification, in which the uncertainty is induced by whether or not the correct model lies among the set under consideration [START_REF] Aydogan | Three layers of uncertainty and the role of model misspecification. iRisk Working Paper[END_REF]. While most uncertain situations in real-life usually encompass the three layers of uncertainty together, [START_REF] Ellsberg | Risk, ambiguity, and the Savage axioms[END_REF] examples are particularly remarkable because they generate ambiguity using only the layers of type (i) and (ii) (indeed, model misspecification is nonexistent in Ellsberg's setup). For example, in the two-color problem described above, the composition of the urn E100 is necessarily one of the 101 compositions physically possible.3

The main distinction between the layers of risk and model uncertainty is the nature of the uncertainty they feature. A risk represents an objectively uncertain prospect. It refers to the physical quantification of uncertainty by means of a probability model (and as such is sometimes referred to as an instance of physical or aleatory uncertainty). It is usually referred to as a lottery, represented by the gamble L = (x 1 , p 1 ; ...; x n , p n ) that yields outcome x i with a well-defined objective probability p i . This probability p i is a measure of randomness. For example, a risk may be characterized by a specific composition of an urn from which a ball is randomly drawn. If the balls in the urn can only have two colors (e.g., red and black), the risk may further be characterized by the binary lottery x p y, yielding

x with probability p and y otherwise.

The uncertainty about which probability model (or risk) should be considered is not of the same nature.

Ambiguity theories

In this section, I shortly outline different approaches economists have developed to deal with ambiguity under the two-layer decomposition. I focus on two approaches that adopt a Bayesian perspective in that they quantify the uncertainty within each layer using a single probability measure.

Consider a finite set S of states of the environment and a set C of consequences.

(Technical details are standard and are omitted here.) Adopting [START_REF] Savage | The Foundations of Statistics[END_REF] approach, an act is defined as a choice with state-contingent consequences, that is a mapping a : S → The (Savage) decision problem under uncertainty is fully characterized by the sets of acts, states, consequences, and models, together with the preference relation, that is, the quintet (A, S, C, M, ).

Subjective expected utility

Under the benchmark approach of subjective expected utility (SEU), a preference satisfying [START_REF] Savage | The Foundations of Statistics[END_REF] axioms is represented by the criterion

U SEU (a) = θ s u(a(s)) m θ (s) µ(θ) = s u(a(s)) m(s).
In these expressions, u is a von Neumann-Morgernstern utility function. It translates economic consequences, measured in monetary terms, into utility levels, and is known to capture risk attitudes. As shown within the bigger parentheses, it is possible, for each model m θ , to compute the expected payoff of a given act. This expected payoff takes an expected utility form: it is computed as the weighted sum of utilities obtained from the consequences in each state of the environment. To address model uncertainty, the DM has a subjective prior probability distribution µ that quantifies her belief about the true parameter θ. Thus, µ(θ) is the DM's subjective belief that m θ is the true model. As the DM does not know which is the correct model, she considers the expected payoff of each possible model and aggregates them out by performing a weighted average according to the relative weights that she associates with each of them (i.e., her prior beliefs µ(θ)). The first representation hereabove is the two-layer version of SEU that has been axiomatized

by [START_REF] Cerreia-Vioglio | Classical subjective expected utility[END_REF]. Under this framework, the layers of risk and model uncertainty are implicitly treated in the same way. The second representation is the original reduced form of [START_REF] Savage | The Foundations of Statistics[END_REF], in which m is the predictive probability induced by the prior µ: m(s) = θ m θ (s) µ(θ) for all states s. In Ellsberg's experiment, for example, a bet on the ambiguous urn E100, giving x = 100 if correct and y = 0 otherwise, yields U SEU = 1 101 100 θ=0 θ 100 u(100) = 1 2 u(100), which is exactly the same as a bet on the risky urn R100.6

Smooth ambiguity

To accommodate the phenomenon of ambiguity aversion highlighted in [START_REF] Ellsberg | Risk, ambiguity, and the Savage axioms[END_REF], the theoretical economic literature has followed different perspectives. One of them, known as the smooth ambiguity approach (Klibanoff et al., 2005, KMM hereinafter), also adopts a Bayesian perspective by having the two layers of uncertainty quantified by a single probability measure. However, in contrast with SEU, a distinct treatment is allowed for the layers of risk and model uncertainty. By letting the function v represent the DM's attitude toward model uncertainty, a natural generalization of the SEU criterion emerges as a version of the smooth ambiguity model of KMM. Under this framework, acts are ranked according to the criterion

U KM M (a) = θ φ s u(a(s)) m θ (s) µ(θ).
A stronger aversion to the layer of model uncertainty than to that of risk leads to ambiguity aversion, modeled by a concave function φ = v • u -1 . Under this criterion, a bet on urn E100 yields U KM M = 1 101 100 θ=0 φ θ 100 u(100) , and it is easy to show that U KM M ≤ φ (U SEU ) under ambiguity aversion. This interpretation of a non-neutral attitude toward ambiguity stems from non-reduction of objectively and subjectively determined probabilistic judgements (i.e., different confidence in such judgments).

Other approaches

The smooth ambiguity approach allows for viewing ambiguity à la Ellsberg as a twostage uncertain prospect, in which the uncertainty in each stage corresponds to a distinct layer. Other approaches also model ambiguity as multiple stages of uncertainty while not necessarily making a distinction between the layers of risk and model uncertainty (see [START_REF] Aydogan | Three layers of uncertainty and the role of model misspecification. iRisk Working Paper[END_REF], for more details). This is for example the case of the approaches proposed by [START_REF] Segal | The ellsberg paradox and risk aversion: An anticipated utility approach[END_REF] (recursive rank dependent utility approach, RRDU) and by [START_REF] Seo | Ambiguity and second-order belief[END_REF]. As these approaches do not distinguish between objectively and subjectively determined probabilistic judgements in the distinct layers, they made no distinction between compound risk (in which there are two stages of risk) and ambiguity à la Ellsberg (in which there are a layer of risk and a layer of model uncertainty). Hence, as non-neutral ambiguity attitudes may result from the violation of reduction of compound lotteries, an elementary rationality condition, these theories assign to ambiguity attitudes a purely descriptive status.

4 Partial ambiguity [START_REF] Chew | Partial ambiguity[END_REF] recently proposed the notion of partial ambiguity to describe the situations "going beyond pure risk and full ambiguity in the original two-urn paradox" (p. 1240). The "intermediate forms of ambiguity" they propose may be described using an urn containing 100 red or black balls. Specifically, by letting n ∈ {0, 1, ..., 50}, partial ambiguity may take the form of:

• Interval Ambiguity, when the number of red (or black) balls is in I n = [50-n; 50+n];

• Disjoint Ambiguity, when the number of red (or black) balls is in D n = [0; n] ∪ [100n; 100]; and

• Two-point Ambiguity, when the number of red (or black) balls is in T n = {50 -n, 50 + n}.

By letting n vary between 0 and 50, these situations span the space of possible urn compositions between the risky urn R100 (e.g., in the case I 0 = T 0 ) in which there is no ambiguity, and Ellsberg's ambiguous urn E100 (e.g., in the case I 50 = D 0 ), which is coined "full ambiguity" [START_REF] Chew | Partial ambiguity[END_REF]Zhong, 2017, p. 1240). Table 2 presents the sets M of possible models (or urn compositions) induced by these three forms of partial ambiguity, for the cases n = 0, n = 25, and n = 50.

To understand why these situations may be seen as involving partial forms of ambiguity, different interpretations may be considered.

Numbers of potential models

The first, natural, way to interpret these situations derives from the number of possible compositions an ambiguous urn may entail. From Table 2, we can observe that: 7

Observation 1 The number of potential models, noted |M |, physically compatible with each form of partial ambiguity is increasing in n (strictly increasing in the cases of I n and D n , and weakly increasing -first increasing, then constant-in the case of T n ).

More specifically, the number of possible models lies between 1, when there is no ambiguity and 101, when there is full ambiguity. Under this interpretation, what makes a situation of partial ambiguity thus corresponds to lying between these boundaries: |M | ∈ (1, 101).

Information and its structure

Amount of information The partial ambiguity situations can also be analyzed in terms of the amount of information they carry. For example, in the case of interval ambiguity, the situations lie between one in which you know the color of each individual ball in the urn (in the no ambiguity situation, I 0 ) and a situation in which you do not know the color of any of the balls (in the full ambiguity situation, I 50 ). More generally, the information available in the different forms of partial ambiguity can be summarized as follows:

• In situations I n , it is known that the urn is composed of at least (50 -n) red balls, and at least (50 -n) black balls,

• In situations D n , it is known that at least (100 -n) balls in the urn have the same color,

7 Remark that an analogous observation is present in Chew, Miao, and Zhong (2017) p. 1241.

• In situations T n , it is known that there are exactly either (50 -n) or (50 + n) red (or black) balls in the urn.

We can thus observe the following.

Observation 2 The amount of information available, within each form of partial ambiguity, is decreasing in n (strictly decreasing in the cases of I n and D n , and weakly decreasing -first decreasing, then constant-in the case of T n ).

Note that this interpretation, in terms of the amount of information, has sometimes been related to the one in terms of the numbers of potential models presented before. This is, for example, the case in the work of Einhorn and Hogarth (1985, p. 435), who wrote "Thus, ambiguity results from the uncertainty associated with specifying which of a set of distributions is appropriate in a given situation. Moreover, the amount of ambiguity is an increasing function of the number of distributions that are not ruled out (or made implausible) by one's knowledge of the situation."

Quality of information Alternatively, the partial ambiguity situations may also be presented in terms of the quality of information they carry, using their information structure.

The DM's information about the likelihoods of the different states (and thus the outcome of the bet) is a priori modeled by a set M . By letting S = {r, b} be the state space, in which the ball drawn from the urn is either red (r) or black (b), the set of models in each uncertain situation can be expressed as M = {m such that P (r) ∈ I} . In this expression, P (r) is the probability that the ball drawn is red, and I ⊆ [0, 1] is a set-theoretic modeling of information. It represents the available information characterizing the chances to make a correct bet. In urn E100 for example, I = 0 100 , 1 100 , ..., 100 100 . Assuming that the DM has information about M , her acts needs to be measurable with respect to M . The set of models can then be partially ordered by the "finer than" and the "coarser than" relations:

Definition. If M ' and M are such that I is a subset of I, then M is said to be finer than M , and M is said to be coarser than M .

In words, this means that better information regarding the structure of ambiguity may be naturally modeled by a smaller set of potential models. As such, a situation is called full ambiguity if its associated set of models is maximal as in E100, and is called partial ambiguity if its associated set of models is finer than in E100. We can then observe the following.

Observation 3

• In situations I n and D n , the set of models becomes coarser as n increases,

• The use of set theory remains silent on the ranking of situations under T n .

Regarding the second point, remark that the use of set theory does not enable us to distinguish T 25 from T 50 , whose associated sets of models M can be ranked by neither the finer nor the coarser relation.

Challenges to partial ambiguity

Rather than characterizing the notion of partial ambiguity in relation to the partial information available and the associated number of possible models, an alternative approach is to focus on how these situations are considered by individuals. Using the smooth ambiguity model discussed in Section 3, and following a simple stochastic dominance argument à la [START_REF] Rothschild | Increasing risk: I. a definition[END_REF], which has been widely used in economics, we can observe that, under a standard symmetry condition on the prior µ, we have Proposition 1.

• In situations I n and T n , a smooth ambiguity-averse DM always strictly prefers a situation with a lower n,

• In situations D n , a smooth ambiguity-averse DM always strictly prefers a situation with a higher n.

The intuition behind these results is simple. Under the smooth ambiguity model, ambiguity aversion can be interpreted as an aversion to mean preserving spreads (MPS) in the space of expected utilities induced by different urn compositions. It is then easy to see that, under a specific, but widely accepted, symmetry condition on the subjective probabilities associated with each model, 8 a MPS pattern emerges as n increases for I n and T n , and as n decreases for D n . In line with a hedging motive against extreme compositions, such MPSs of the possible expected utilities are disliked by ambiguity-averse DMs. Remark also that, in the case of I n , Proposition 1 encompasses the standard result of [START_REF] Ellsberg | Risk, ambiguity, and the Savage axioms[END_REF] that the risky urn R100 is preferred to the ambiguous urn E100 under ambiguity aversion, while in the case of D n , a thought-provoking result arises from comparing partial ambiguity (D n with n < 50) with full ambiguity (D 50 ).

Corollary 1. In situations D n , a smooth ambiguity-averse DM always strictly prefers full ambiguity to partial ambiguity.

8 As noted by [START_REF] Chew | Partial ambiguity[END_REF], uniform, binomial, geometric, and U-shaped priors are all sufficient, because they (1) are symmetric and (2) imply simple spreads as n increases (in the case of In and Tn) or decreases (in the case of Dn). These two properties together are sufficient to guarantee the MPS condition. Remark that these conditions on the prior are less demanding than they appear at first glance. The symmetry in the prior distribution stems from the indifference between betting on a red or black ball. In the same vein, a uniform distribution ensures the same treatment for all compositions of the urn that are physically possible, and a binomial distribution results from associating the same subjective belief to each individual ball being red or black. All these arguments may be justified on the grounds of a general symmetry of information argument: given the information available, there is a priori no reason to believe that one event deserves more weight than another. The prior measure therefore reflects this symmetry, in accordance with the Principle of Insufficient Reason.

challenging the significance of the notion of partial ambiguity. In particular, it implies that more information may either increase (in situations I n ) or decrease (in D n ) utility under ambiguity aversion. Similarly, based on the quality of the partial information available, a finer set of models does not necessarily increase the utility of a bet. As a consequence, an ambiguity-averse DM may be better off (1) when the information provided makes the set of models finer (e.g., in the case of I n ), (2) when it makes the set of models coarser (e.g., in the case of D n ), or (3) when it does not make the set of models finer nor coarser (e.g., in the case of T n ). Said differently, the value of information regarding the composition of the urn is not necessarily positive. 9

This result may not be intuitive at first glance. In the case of I n , it seems relatively straightforward that, starting from Ellsberg urn E100, in which nothing is known, and being told that, out of the 100 balls present in the urn, at least (50 -n) of them are red and at least (50 -n) are black (e.g. going from I 50 to I 25 ), increases the utility of the bet. This information indeed enables the DM to disregard more extreme compositions of the urn. However, in the case of D n , starting from Ellsberg urn E100 and being told that at least (100 -n) balls in the urn have the same color (e.g., going from D 50 to D n ) decreases the utility of the bet under smooth ambiguity aversion. Having more information (and thus being able to restrict the set of potential models) in this case is therefore not valuable.

Ultimately, what matters for decision-making is the quality, rather than the quantity, of information. As noted by Ellsberg (1961, p. 659), "ambiguity may be high even where there is ample quantity of information, where there are questions of reliability and relevance of information, and particularly where there is conflicting opinion and evidence."

In what follows, we propose two thought experiments that further challenge the notion of partial ambiguity. First, consider the situation D 0 = T 50 , which is characterized by the set of models M = 0 100 , 100 100 . As discussed earlier, it might be argued that, in this situation, partial information is available (i.e., the 100 balls have all the same color) compared with the urn E100, for which there is no information. However, does that necessarily imply that such a situation is of partial ambiguity as opposed to the full ambiguity situation E100? Here is an alternative way to present the problem. 9 The value of information may for example be defined as the amount of money that a DM would be willing to pay for information about the composition of the urn (i.e. for a finer set M ) before making a decision. While in general, information is seen as a way to reduce uncertainty -which has thus a positive economic value-it is worth noting that, in some contexts, information may also increase uncertainty, so that individuals may be inclined to avoid it. This is the case in the example provided above, as well as in the context of updating under ambiguity. Indeed, a key concept known as "dilation" of sets of priors, familiar to common models of updating under ambiguity, is that information may increase relevant ambiguity, and thus make individuals worse off [START_REF] Shishkin | Ambiguous information and dilation: An experiment[END_REF]. In that case, the set of relevant priors may indeed become larger (dilate) after information. In a recent experimental study in this context, [START_REF] Kops | A test of information aversion[END_REF] show that information aversion was significantly correlated with ambiguity aversion.

the only true instance of full ambiguity, as it is the only instance in which I = [0, 1]? All ambiguous urns containing finite numbers of balls could indeed be viewed as instances of partial forms of ambiguity lying between such a full ambiguity urn and a risky urn.

Discussion

The two thought experiments presented in the previous section illustrate why a classification of intermediate forms of ambiguity based on the notion of partial ambiguity may be difficult to adopt. First, as an implicit parallelism is made between the notion of partial ambiguity and the partial information available, it might be believed that partial ambiguity conveys a notion of desirability it does not have with respect to the full ambiguity situation. As argued previously, and contrary to what economic intuition would suggest, there may well be situations in which providing more information actually decreases the desirability of the bet, so that full ambiguity is preferred to partial ambiguity under ambiguity aversion. Second, as Thought Experiments 1 and 2 indicate, it might not be consistent to call partial ambiguity a given situation in a specific context in which it is compared with other ambiguous situations, while the same situation (i.e., a situation with the same mathematical properties, which is associated with the same set M of possible models) framed differently may have little, or nothing, to do with partial ambiguity. More fundamentally, partial ambiguity is a relative notion that is defined in opposition to full ambiguity, which has itself no clear meaning.

Overall, it is therefore unclear how the notion of partial ambiguity may be helpful in characterizing further uncertain situations, which typically are either risky (if the probability distribution is objectively known) or ambiguous (if the probability distribution is unknown), but hardly admit "partial" forms between them. However, if one thinks that an extra refinement needs to be made in the classification of different ambiguous situations, a natural candidate could consist in characterizing an ambiguous act as whether it is more affected by ambiguity than another. As Ellsberg (1961, p. 660-661) noted, "ambiguity is a subjective variable, but it should be possible to identify "objectively" some situations likely to present high ambiguity, by noting situations where available information is scanty or obviously unreliable or highly conflicting; or where expressed expectations of different individuals differ widely; or where expressed confidence in estimates tends to be low." Such a classification of ambiguous situations could for example be based on the notion of more ambiguous proposed [START_REF] Jewitt | Ordering ambiguous acts[END_REF]. It allows for establishing a partial ordering among ambiguous situations (within a given class of preferences), and has the advantage of mirroring the well established notion of more risky [START_REF] Rothschild | Increasing risk: I. a definition[END_REF], which has been widely used in the risk theory literature (note that in that literature, the result of a known fair coin coming up Heads or Tails and betting on a coin that has never been tested and is absolutely unknown. Observe that in the latter case, the set of possible models is both continuous and bounded between 0 and 1. the notion of "partial risk" does not exist). Finally, it allows for dealing with ambiguous situations for which partial ambiguity remains silent, such as E1 and E1000.

Following [START_REF] Jewitt | Ordering ambiguous acts[END_REF], the acts of betting on the color of the ball drawn from the ambiguous urns can be ordered in terms of their degree of ambiguity. Formally a more ambiguous relation is defined as follows Definition [START_REF] Jewitt | Ordering ambiguous acts[END_REF] Let P be a class of preferences over a set A.

Assume that a binary relation "more ambiguity averse" is given, which is a strict partial order. Assume that each ∈ P is related to an ambiguity neutral element of P. Given two acts f, g ∈ A, f is a more ambiguous act than g if the following conditions are satisfied:

(i) if ∈ P is ambiguity neutral, then g ∼ f ;

(ii) for all A , B ∈ P such that A is an ambiguity neutral preference and B is more (less) ambiguity averse than A , we have g B ( B ) f . According to this definition, an act f is more ambiguous than an act g if an ambiguityaverse DM prefers g to f , but an ambiguity-neutral DM is indifferent between the acts.

It should be clear, however, that this order of more ambiguous arises on the back of a specific relation on preferences. In the context of the smooth ambiguity model (i.e, when the preference class partially ordered by a more ambiguity-averse relation is characterized by a more concave function φ), we can order the partial ambiguity situations presented in Table 2 exclusively in terms how much they are affected by ambiguity. In line with Observation 2, we obtain the following results Proposition 2.

(i) Under I n and T n , the higher the n, the more ambiguous is the situation, (ii) Under D n , the lower the n, the more ambiguous is the situation.

In addition, the ambiguous situations encountered in Thought Experiments 1 and 2-for which partial ambiguity remains silent-can now be classified using the more ambiguous relation. In particular, by letting E N denote the two-color ambiguous urn with a total number of balls N in it, we have Proposition 3. For ambiguous urns E N , a larger N is always preferred under ambiguity aversion.

Propositions 2 and 3 thus indicate whether the class of smooth ambiguity deems some acts to be more ambiguous than others. These results are achieved without further restriction on ambiguity and risk attitudes but with a fixed belief (within the families discussed previously).11 They make clear that the counterintuitive Corollary 1, i.e. that full ambiguity may be more desirable than partial ambiguity, is due to the fact that, for example, the situation associated with the set M = 0 100 , 100 100 is more ambiguous than the one associated with the set M = 0 100 , 1 100 , ..., 100 100 . Such a result is in line with Ellsberg's quote above: situations like the one with M = 0 100 , 100 100 present a high degree of ambiguity, as the available information is highly conflicting and leads to expectations that differ widely.

Finally note that, in a recent study, [START_REF] Filiz-Ozbay | Comparing ambiguous urns with different sizes[END_REF] investigate experimentally whether the number of balls in Ellsberg's two-color urn matters for decisions. Specifically, they use binary comparisons between urns E 2 , E 10 , and E 1000 and show that the majority of subjects (around 60%) prefer urns with larger N . In accordance with Proposition 3, they further show that this preference for the larger urn is mainly driven by ambiguity averse subjects.

Concluding remarks

The notion of partial ambiguity has been recently proposed to describe intermediate forms of ambiguous situations, lying between risk and full ambiguity. In this paper, I show that partial ambiguity, which implicitly relies on the quantity of information provided about the possible probability models describing the stochastic phenomenon of interest, may lead to counterintuitive results. Rather, I argue that the quality of the information is ultimately what determines whether an ambiguous situation is deemed more desirable than another. Following this idea, I propose a reinterpretation of partial ambiguity in the light of the more ambiguous relation proposed by [START_REF] Jewitt | Ordering ambiguous acts[END_REF]. I show that, in the cases in which full ambiguity may surprisingly prove more desirable than partial ambiguity, it is just because it is less ambiguous.

C.

  The collection of all acts is denoted by A. It is assumed that the DM has a preference relation over acts, a binary relation that describes how she ranks alternatives. In particular, I write a b if the DM either strictly prefers act a to act b or is indifferent between the two. As usual, ∼ denotes indifference and strict preference. I assmue that the DM is able, because of her ex ante information, to posit a set of potential models M describing the likelihoods of the different states. This set of models is taken as a datum of the decision problem: the DM behaves as if she knows that states are generated by a probability model m that belongs to the collection M . Each probability model m therefore describes a given risk and, as such, represents the intrinsic randomness that states feature. The epistemic uncertainty is about which is the correct model among the collection M . I write M = {m θ }, indicating that each model is indexed by a parameter θ. For example, the Ellsberg's urn E100 has 101 possible compositions, which can be summarized by M = m θ = θ 100 for θ ∈ {0, 1, ..., 100} . 5

  Number of possible models in parenthesis (|M |)

  Figure A.1: Cumulative distribution functions of different bets over the set of possible models describing E 1 , E 10 , E 100 , and E 1000

Table 2 :

 2 Sets of models in the partial ambiguity situations

Note that, technically, this result coincides with Predictions 1 and

[START_REF] Chew | Partial ambiguity[END_REF].

Note that this thought experiment had already appeared in[START_REF] Keynes | A treatise on probability[END_REF].

The number of red (black) balls in the urn necessarily belongs to {0, 1, 2, ..., 100}.

Except in the case of compound risk, in which there is an objective probability distribution on the possible compositions of the urn. In this case, the compound lottery takes the form (...; Li, pi; ...), yielding the lottery Li = (...; xij, pij; ...) with probability pi.

In an abuse of notations, I let the model mp = xpy be fully characterized by the probability p.

Under the assumption of uniform prior µ and after normalizing u(0) = 0.

As noted byJewitt and Mukerji (2017, p. 215), "Fixing belief is natural and necessary given that we are in a framework with subjective belief on states and choice objects are acts, rather than lotteries involving given distributions on outcomes. In this framework one needs such a restriction even when working with expected utility (say, to investigate risk orders on acts) since under different beliefs the same act will induce correspondingly different lotteries over outcomes; an act that is riskier under one belief can be the opposite under another belief."

This research is supported by a grant from the French Agence Nationale de la Recherche (ANR-17-CE03-0008-01 INDUCED) and from the European Research Council (ERC-2013-StG 336703-RISICO). I am grateful to Mohammed Abdellaoui,

Thought Experiment 1 Imagine that you confront two urns containing red and black balls, from one of which a ball will be drawn at random. You can choose the color and the urn on which to bet. With a correct bet, you win $100 (and nothing happens if your bet is incorrect). You have the following information. The first urn, called E100, contains 100 red and black balls, but in a ratio entirely unknown to you. The second urn, called E1, contains only 1 ball that may be either red or black. On which urn do you prefer to bet?

If the answer to this question is E1, then facing the bet with two models M = 0 100 , 100 100 is deemed more desirable than facing the bet with 101 models M = 0 100 , 1 100 , ..., 100 100 . However, there is no reason, in this case, to consider that some (partial) information is available in E1 compared with E100 (the only difference between the two urns being their total number of balls). Instead, if, following the simple stochastic dominance argument exposed above, the answer is E100, then we need to conclude that, in the terminology proposed by [START_REF] Chew | Partial ambiguity[END_REF], a preference is expressed for full ambiguity over partial ambiguity. This counterintuitive result challenges the actual meaning of partial ambiguity.

Second, it is also unclear what is the exact meaning of full ambiguity. To understand why, consider the following experiment.

Thought Experiment 2 Imagine that you confront two urns containing red and black balls, from one of which a ball will be drawn at random. You can choose the color and the urn on which to bet. With a correct bet, you win $100 (and nothing happens if your bet is incorrect). You have the following information. The first urn, E100, contains 100 red and black balls, but in a ratio entirely unknown to you. The second urn, E1000, contains 1000 red and black balls, but also in a ratio entirely unknown to you. On which urn do you prefer to bet?

In this experiment, which of the two urns represents the situation of partial ambiguity, and which urn represents full ambiguity? Following the terminology adopted in [START_REF] Chew | Partial ambiguity[END_REF], E100 is the full ambiguity situation. However, following the line of arguments exposed above, E100-which entails 101 models-may be seen as encompassing partial information compared with E1000, which entails 1001 possible models. Indeed, remark that E100 could be taken as a special case of E1000, in which the DM is told that the 1000 balls are gathered in groups of 10 balls of the same color. Such type of partial information enables the DM to restrict the set of potential models to take in consideration (in terms of the sets modeling the information, we indeed have that I E100 is a subset of I E1000 ). Does it then imply that E100 is now the partial ambiguity situation and E1000 the full ambiguity situation? Pushing the reasoning further, how should be called the situation of an urn containing even more balls (e.g., 10,000)? In the limiting case, wouldn't the hypothetical ambiguous urn with an infinite number of balls (or Schmeidler's (1989) coin 10 ) be 10 Schmeidler (1989) proposed an example in which an individual is given the choice between betting on

A Proofs

Proof of Proposition 2 Directly follows from Observation 1 and the fact that all partial ambiguity situations are equivalent under ambiguity neutrality (i.e., under SEU).

Proof of


Marcio T Do

N Varella
email: mvarella@if.usp.br

Ljiljana Stojanović

Stephan Van Quan Vuong

Thomas A Irle

Mario Niehaus

Barbatti

Márcio T Do

Van Quan Vuong

Stephan Irle

Thomas A Niehaus

Mario Barbatti

How the Size and Density of Charge-Transfer Excitons Depend on Heterojunction's Architecture

des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

I. INTRODUCTION

Efficient charge separation at donor-acceptor (DA) interfaces is critical for the performance of electronic devices. In organic solar cells (OSCs), light absorption typically gives rise to bound excitons, and their dissociation into charge-separated (CS) states can be significantly constrained by radiative and non-radiative recombination processes 1,2 . Chargetransfer (CT) excitons at the interface between donor and acceptor domains attracted considerable attention in recent years [3][4][5][6][7] , since they are believed to mediate the long-range charge separation in OCSs. Despite the outstanding progress in conversion efficiency [8][9][10][11] , the mechanisms for free-carrier production are still under debate. For instance, the generation of free charges from high-energy (hot) CT excitons has been pointed out [12][13][14] , but significant yields have also been reported for light absorption at sub-excitation energies 15,16 . The dissociation mechanisms of interfacial CT excitons, having binding energies ∼10 times higher than ambient thermal energies 1,15 , are also currently in discussion. Several physical effects can impact the exciton dissociation dynamics, such as entropy 6,[17][18][19] , vibronic couplings [19][20][21][22] , energy offset 5,[22][23][24][25][26][27][28][29][30][31] , disorder [32][33][34][35][36] , hybridization between CT and localized states 7,22,[37][38][39] , and also electronic delocalization and polarization 14,34,[40][41][42][43][44][45][46][47][48][49][50] , among others.

A number of recent studies have addressed the microscopic structures of the DA interfaces and their impact on free-carrier-generation [5][6][7]34,38,46,47,[51][52][53][54] . While atomistic simulations can provide invaluable mechanistic and structure-property insights, the importance of going beyond minimal (bimolecular) interface models has been emphasized 34,47,52,54 . Therefore, the computational methods are challenged by the size of the DA heterojunction models, by the need to account for different interface geometries, and also by disorder and finitetemperature effects. The balance between numerical effort and accuracy is a prominent issue, but the analysis of the excited states of complex molecular systems should not be overlooked. The inspection of molecular orbitals is often tedious and sometimes ineffective to assign the CT character of electronic states, e.g., when many electron-hole pairs have significant weights in a given excited state or for systems having multiple chromophores.

In this work, we address the CT excitons of polymer:fullerene interfaces combining the time-dependent long-range corrected density functional tight-binding (TD-LC-DFTB) method [56][57][58] with the fragment-based analysis of the calculated excited-states [59][60][61] . The fragmentation into subsystems has proved a useful tool to assign, quantitatively and un-ambiguously, the CT character of exciton states. Its application to DA heterojunctions is particularly appealing since the interface can be viewed as composed of donor and acceptor fragments, which can be further decomposed into oligomer and fullerene fragments. The methodology is applied to an interface model in which the acceptor domain is composed of PCBM (phenyl-C61-butyric acid methyl ester), and the donor domain by PTBTBTz, a copolymer comprised of thiophene (T), benzothiadiazole (BT), and benzotriazole (BTz) subunits, as shown in Fig. 1a. PTBTBTz is an example of D-A 1 -D-A 2 dual-band polymers 62 , which have been considered for applications in photovoltaics [62][63][64][65] and organic electronics 66,67 .

Those copolymers are characterized by bright excitations from the donor subunit (T in the present case) to the strong and weak acceptor subunits (BT and BTz) -the latter acceptor subunits in the PTBTBTz donor domain should not be confused with the acceptor domain of the interface model.

The PTBTBTz:PCBM interface is an interesting model for our purposes since (i) in addition to cold and hot CT excitons, the dual-band character is expected to give rise to interband CT states; (ii) the relation between interface geometry and CT states can be affected by the two absorption bands arising from electron transfer to two distinct polymer subunits; and (iii) the BT and BTz acceptor subunits are moderately sized compared to those employed in other D-A 1 -D-A 2 polymers 62 , thus helping to manage the computational effort. We explore 17 DA interface models of different sizes and geometries obtained from both geometry optimizations and room-temperature molecular dynamics (MD) simulations.

The excited state analysis employed different fragmentation schemes and exciton properties to draw, as much as possible, general conclusions not restricted to the properties of the PTBTBTz:PCBM system. We explore edge-on and face-on orientations of the DA domains, which are known to correlate to device performance 6,47,51,52 , different closest-approach positions of the acceptor molecules along the donor chains, and also the size of the interface domais. The fragment-based analysis is further applied to physical properties expected to affect the excited state dynamics, such as the degree of charge transfer, delocalization of the CT excitons, and the electron-hole distances.

II. METHODS

The excitation spectra were calculated with the TD-LC-DFTB method, described in detail elsewhere 57,58 , as implemented in the DFTB+ code 69 . From the formal standpoint, the approach is similar to linear-response time-dependent density functional theory (TD-DFT), i.e., one has to solve the generalized eigenvalue equation

  A B B * A *     X Y   = Ω   1 0 0 -1     X Y   , (1)
where the A and B matrices are given by

A iaσ,jbσ = δ ij δ ab δ σσ (ε aσ -ε iσ ) + (iaσ||jbσ ) B iaσ,jbσ = (iaσ||bjσ ) . (2)
In the expressions above, Ω denotes the excitation energies (eigenvalues), ε iσ,aσ are orbital energies obtained from a previous ground-state calculation for spin σ, and (iaσ||jaσ ) are generalized electron repulsion matrix elements, which involve occupied (i, j) and virtual (a, b) orbitals, and account for the Hartree and exchange-correlation interactions. In TD-DFTB the two-electron integrals are computed with a Mulliken approximation to increase the numerical efficiency 55 .

The TD-LC-DFTB method is built on the the LC-DFTB counterpart. The Coulomb interaction is split into short-and long-range components using a Yukawa ansatz 57 , and the DFTB approximation for the short-range contribution is obtained from the Baer, Neuhauser, and Livshits (BNL) exchange-correlation functional 71,72 . A second-order expansion of the Kohn-Sham total energy around a reference density matrix is carried out, ρ = ρ 0 + δρ, where ρ 0 is the sum of BNL atomic density matrices. While we do not discuss the approximations employed to obtain the DFTB ground state energies 70 , we mention the decomposition of the total energy into two contributions, E total = E el + E rep . The electronic part, E el , involves the computation of the Hamiltonian matrix elements which only depend on the reference density. The corresponding diagonal elements are given by the atomic orbital energies, while the off-diagonal elements are precomputed for pairs of elements over a range of internuclear distances, along with the overlap matrix and the repulsive energy component. These precomputed values are tabulated in the Slater-Koster (SK) files 70 . The electronic part further accounts for deviations from the reference density through a charge-charge interaction term.

An SK parametrization with the range-separation parameter ω = 0.3 a -1 0 , referred to as the OB2 set, was recently reported for the H, C, N, and O elements 73 . The present calculations used a re-parametrization of this OB2 set, which includes sulfur and was optimized for five different values of the range-separation parameter, namely ω = 0.1, 0.2, 0.3, 0.4, and 0.5 a 0 -1 . The parameter sets used in this work are deposited in the Supporting Information.

The inclusion of sulfur is essential for applications in organic electronics, and the range of ω values allows for a partial tuning of range separation, which is considered important to describe CT excitations 4 . The complete re-parameterization of the OB2 set will be published elsewhere in the near future.

Our LC-DFTB and TD-LC-DFTB calculations included dispersion interactions via the Slater-Kirkwood model 74 . The ground-state MD simulations were based on LC-DFTB energies and gradients either in the NVE or NVT ensemble. In the latter case, the Andersen thermostat, built in the DFTB+ code, was employed, with a reselection probability of 0.2. We did not account for the dielectric environment since continuum solvation models are currently not available in the DFTB+ package. A similar study of polymer:fullerene (P3HT:PCBM) interfaces 52 pointed out that accounting for the dielectric medium essentially gives rise to a systematic shift of the excitation spectra, not significantly affecting the photophysics. We expect that the lack of a solvation model should not affect the main conclusions of the present work, as they rely only on the relative energies of the excited states.

Finally, the charge separation parameter was set to ω = 0.2 a -1 0 for the PTBTBTz:PCBM interface models. This value was chosen because it provided, among the available ω values, the smallest |IP -(E + -E 0 )| differences (in absolute value), where E + and E 0 are the energies of the cationic and neutral species, and IP is the ionization potential of the latter.

Other supramolecular models of interest for organic electronics have been recently investigated using an alternative version of TD-LC-DFTB 56,[START_REF] Darghouth | [END_REF][82][83] . In addition to a different parametrization, that version also differs from the present one by not including Hartree-Fock exchange in the DFTB zeroth-order Hamiltonian (see Ref. 58). In a nonadiabatic dynamics study of a pentacene:fullerene interface based on that TD-LC-DFTB version, Darghouth and co-workers 83 have proposed a value for ω, 0.07 a -1 0 , significantly smaller than those used in our simulations.

The exciton analysis employed the recently developed 75 interface between the DFTB+ 69 and TheoDORE 76 codes. The latter is a toolbox for the fragment-based analysis of excited states 61 , exploring the one-electron transition density matrix (1-TDM) between the ground and the I-th excited state, γ 0I (r e , r h ), where the electron and hole coordinates are indicated as r e,h . Groups of atoms belonging to the system of interest define molecular fragments.

A charge transfer number, for fragments A and B, can be obtained from the 1TDM by restricting the integrations over the electron and hole coordinates to the A and B fragments, respectively,

Ω AB = A dr h B dr e γ 2 0I (r e , r h ) . (3)
Ω AB gives the probability to find the hole on fragment A, with the electron in fragment B, and it can be evaluated by resorting to a population analysis procedure 59,61 . The interpretation of Ω AB as a matrix defines the electron-hole correlation plots and allows for the definition of several excited-state descriptors 59 . The CT number is obtained from the off-diagonal elements,

CT = 1 Ω A B =A Ω AB , (4)
with Ω = A,B Ω AB , and it ranges from CT = 0 (ideal Frenkel exciton) to CT = 1 (ideal charge-transfer exciton). In case the fragments can be arranged along some direction, such that an ordering like A = 1, B = 2, etc. can be meaningfully assigned, the position descriptor (POS) for the hole (h) and electron (e) orbitals can be computed as,

POS h = Ω -1 A A ( B Ω AB ) POS e = Ω -1 B B ( A Ω AB ) , (5)
with the exciton position given by

POS = 1 2 (POS h + POS e ) . (6)
Ideal Frenkel excitons localized on the donor (LD) and on the acceptor (LA) are represented in Fig. 1b, along with an ideal CT exciton. The CT and POS descriptors for these excited states are also indicated, and the corresponding electron-hole correlation plots are shown in Fig. 1c. The POS descriptor is calculated with the convention D = 1 and A = 2 for the positions of the donor and acceptor, respectively, whenever the system is decomposed into two fragments.

The 1TDM can also be interpreted as the exciton wave function 60 , χ exc (r e , r h ) := γ 0I (r e , r h ), which allows for the definition of the exciton size as the root-mean-square (rms) distance between the hole and the electron,

d 2 exc = χ exc | (r h -r e ) 2 |χ exc χ exc |χ exc . (7)
The rms distance can be approximately evaluated by defining atomic pairs, breaking the integrals into atomic volumes, and assuming the electron-hole distance to be equal to the distance between the nuclei (d M N ) for each atomic pair 60 ,

d exc ≈ 1 Ω M,N Ω M N d 2 M N , (8)
where Ω M N is the Ω matrix, given in eq. 3, for the single-atom fragments M and N .

LA exciton =

   0 ≤ CT ≤ 0.25 1.75 ≤ POS ≤ 2.00 , ( 10
) CT exciton =    0.75 ≤ CT ≤ 1.00 1.35 ≤ POS ≤ 1.65 , (11)
and also assign those not matching the above definitions as mixed-character (MX) excitons.

Although we explore different fragmentation schemes, the interfacial CT excitons are always defined using only two fragments corresponding to the D and A domains.

III. RESULTS

Before we discuss the models for PTBTBTz:PCBM interfaces, we comment on the quality of the TD-LC-DFTB calculations compared to the TD-DFT counterparts. Table I presents the excitation energies and oscillator strengths for five lowest-energy excited states, S 1 to S 5 , of the PTBTBTz monomer (Fig. 1a) and the stacked dimer, PTBTBTz 2 (the latter is shown in the Supporting Information, Fig. S1). The DFT and TD-DFT computations were performed with the Gaussian09 package [START_REF] Frisch | Gaussian 09, revision D.01[END_REF] employing different correlation-exchange functionals, standard values for the range separation parameters, and the 6-31G* basis set.

For PBTBTz and PBTBTz 2 , the TD-LC-DFTB range separation parameter was set to ω = 0.3 a 0 -1 , based on the same criterion that was described in Sec. II. We first compare the excitation spectrum of the PTBTBTz monomer obtained with CAM-B3LYP, LC-ωPBE, ωB97XD, as well as TD-LC-DFTB, at the ground-state geometry optimized with the B3LYP functional. There is a good agreement ( 0.1 eV) between the CAM-B3LYP and ωB97XD excitation energies, while LC-ωPBE overestimates them by ≈ 0.4 eV with respect to the other functionals. The discrepancies between TD-LC-DFTB energies and those obtained with CAM-B3LYP and ωB97XD vary more significantly from state to state, being mostly around ≈ 0.1 eV to ≈ 0.3 eV, although larger for the S 3 state, ≈ 0.5 eV. The agreement between TD-LC-DFTB and ωB97XD is similar at the ground-state geometry optimized with the latter functional, although there is some improvement when we compare the excitation energies computed at the respective optimal geometries ( 0.4 eV discrepancies). For the PTBTBTz 2 dimer, the discrepancies between the TD-ωB97XD and TD-LC-DFTB energies are somewhat lower, in spite of the less favorable agreement for the S 1 state.

The dual-band character of the PTBTBTz polymer is already evident in the monomer.

As discussed elsewhere 62 , the bright S 1 and S 2 states mainly arise from excitations out of the T subunit to the BT and BTz subunits, respectively. The oscillator strengths calculated with the TD-DFT and TD-LC-DFTB methods are consistent, with f ≈ 1 and f ≈ 0.1 for the transitions to the S 1 and S 2 states, respectively. The situation is similar for the dimer, although with larger differences in the magnitudes of the oscillator strengths, especially when TD-LC-DFTB and TD-ωB97XD are compared at their respective optimal geometries.

Particularly at regions where the excited-state spectra are denser, the oscillator strengths, and even the character of the excited states, should be more strongly dependent on the geometry. To further inspect this aspect, we calculated the absorption spectrum of the monomer by performing a Monte Carlo integration of the Wigner distribution function of the vibrational ground state [START_REF] Crespo-Otero | [END_REF] . The simulated spectra, shown in the Supporting Information (Section S-II), were calculated with the Newton-X code 79,80 using the excitation energies, oscillator strengths, and vibrational normal modes obtained with the TD-ωB97XD and TD-LC-DFTB methods. The shapes of the absorptions bands and the overall magnitudes of the cross sections are in good agreement, despite the ≈ 0.2 eV shift between the maxima of the first bands, which could be anticipated from the vertical excitations reported in Table I. indicate the LD, LA, CT, and MX excitons defined in eqs. 9 to 11. The excitation spectra were obtained from stationary structures, as described in the Supporting Information (Sec- The models with n = 3 stacked PTBTBTz dimers and one PCBM with face-on orien-tation, shown in Fig. 4(b), have denser spectra of LD states, as expected. Once more, the excited states were calculated for stationary structures, with no important differences arising from the different positions (p = bt, btz, t) of the acceptor with respect to the donor (see also Table II). The cross sections show an absorption band around 2.9 eV, not present in the models with n = 2 stacking (Fig. 4(a)), in addition to the bands around 2.5 eV and 2.7 eV.

The second band has the strongest optical coupling for the n = 3 models, with bright states having either LD or MX character. Despite the differences in the absorption cross sections, the n = 2, 3 stacking models mostly have high-energy CT states, with only a few of such states lying around and below the brightest band.

The spectra of the n = 2, 3 stacking models with edge-on orientation are shown in Fig. 5 for the PCBM positions p = bt, t. As described in the Supporting Informatiom (Section S-I), we performed MD simulations before obtaining the stationary structures. The edge-on models with p = btz positions sometimes dissociated along the dynamics, so we did not We defined cold CT excitons based on two criteria. For each system, we considered the CT states within a 0. in Table III. The average number of CT states lying below the brightest ones, n low CT , are also presented. In general, the different positions have a mild impact on the number of cold CT excitons, although n cold CT , n cold CT and n low CT tend to be smaller for p = bt and larger for p = t. The significance of these differences should be viewed with caution in view of the small number of geometries considered in the averages. The impact of orientation on the number of cold CT states is much clearer than that of position, confirming the trends suggested by the spectra of the stationary structures.

The excitation spectra of the models comprising two PCBM molecules are presented in Fig. 6. For the face-on orientation, we considered side-by-side arrangements in which the acceptors lie either on top of BT and BTz subunits (p = bt btz) or on top of two T subunits (p = t t), while for the edge-on orientation we only considered the p = t t case to prevent dissociation during the MD. In panel (b) we show the results for the PCBMs arranged in a row at the p = t position. In view of the weak dependence of the calculated spectra on positions, the p = bt case is shown in the Supporting Information (Fig. S5), while p = btz models were not explored. It is clear from Fig. 6 and Tabs. II and III that the 2-PCBM models share the basic features with the single-PCBM counterparts. The spectra of CT states has a significant dependence on the orientation, although not as much on the position of the acceptor molecules. The bright states mostly have LD or MX character, despite the denser spectra of LA states. The edge-on orientation increases the density of cold CT excitons, an effect that is more evident for the p = t position than p = bt, in case the PCBMs arranged in a row (see Fig. 6(b) and the Supporting Information, Fig. S5). Compared to the other interface models with edge-on orientation, the 2-PCBM systems with 2rw arrangement and e orientation have more significant differences between n cold CT and n cold

CT

. The reasons of this discrepancy, which indicates cold CT excitons lying further above the S 1 states, are not clear.

IV. DISCUSSION

Our analysis has shown significantly larger densities of cold CT states in edge-on than in face-on interfaces. Such a density dependence on orientation has also been previously pointed out 6,52 for other interface models with fullerene and non-fullerene acceptors. In the present work, however, we deliver a much in-depth picture of the CT exciton's dependence on the interface's architecture.

The orientation effect on the cold CT states seems to be a general property of DA interfaces 6,47,51,52,54 and can be further investigated by the fragment-based analysis outlined in Sec. II. We averaged the CT numbers, defined in eq. 4, for the different models using the geometries sampled from the MD simulations. We considered the cold CT states lying up to 0.2 eV above the S 1 state, such that the number of data points in each average is given by N × n cold CT , where N is the number of structures and n cold CT is the average number of cold CT states. In addition to increasing the spectral density of low-energy CT states, the edge-on orientation enhances the level of charge separation. In nearly all cases, one finds CT cold < 0.9 and CT cold ≥ 0.95 for the face-on and edge-on models, respectively, regardless of the oligomer stacking as well as the number, position, and arrangement of the PCBM molecules. Although not shown here, a similar trend was found for the cold CT states defined with respect to the lowest-lying CT exciton, and also for the CT states lying below the brightest states. The CT states at the bottom of the excited state spectra are believed to dissociate thermally 5,6 , a process that should be favored by edge-on orientation given the increased interfacial charge transfer and the higher density of cold CT states 6 .

Further insight into the orientation effect on the CT excitons can be gained from the analysis of the POS descriptor, defined in eqs. neighboring fragments, (1, 3 → 4), or uniform delocalization of the hole, (1, 2, 3 → 4). Only hot CT states lying above the third absorption band have prevailing (1 → 4) character.

These high-energy excitons with large electron-hole separations also show the most effective charge transfer, with CT ≈ 1, although in general, one finds POS > 3.0 and CT ≈ 0.9

for the face-on models. The electron-hole correlation plots for representative CT excitons of the (dim-PT) 3 :(PCBM@t) 1 :f structure are shown in Fig. 8(b). In contrast, the edge-on orientation mostly gives rise to POS ≈ 3, suggesting more effective hole delocalization over the donor chains, according to the ideal transitions described above. Only for high-energy CT excitons, around and above the third band, the values of the position descriptors are more spread. In general, the charge transfer is more effective, as previously pointed out, and the hole delocalization is also clear from the correlation plots of the representative CT excitons, shown in Fig. 8(b).

The POS descriptor is expected to correlate with the exciton size (d exc , see eqs. 7 and 8), which would significantly impact the dissociation of cold interfacial excitons 50 . The connection between exciton size and the average positions of the hole and electron (POS, see eq. 6) might be affected by several aspects, such as the degree of (de)localization of the particles within a given fragment, the fragmentation scheme employed to define the POS descriptors, and the detailed structure of the 1-TDM. The approximations underlying eq. 8, particularly the definition of atomic volumes with constant electron-hole distances for the atomic pairs, might also impact the results. While we did not explore more complex fragmentation schemes to investigate the quasi-particle delocalization in detail, d exc is shown in Fig. 9 While it makes physical sense that these transitions produce large excitons, as they maximize the delocalization of the hole, the present analysis provides limited information since smaller CT excitons have similar fragment contributions. Breaking the interface models into more fragments could provide more detailed information, but we do not pursue this goal in the present work. In any case, exciton sizes as large as d exc ≈ 15 Å are found for both orientations, although the electron-hole characteristics are not the same. Apart from the delocalization along the chains, the face-on models produce larger excitons by placing the hole on oligomer chains lying far from the acceptor, whereas larger excitons are produced by the delocalization of the hole in edge-on models.

The delocalization of the hole along the oligomer chains was further explored with models composed of PTBTBTz tetramers, namely (tet-PT) 2 (PCBM@p) 1 :f , and also PT3BTBTz dimers, (dim-PT3) 2 (PCBM@p) 1 :f . The data in Fig. 9(b), obtained for stationary structures, spread considerably, such that exciton sizes around and below 14 Å are found for the face-on models built on (dim-PT) In the case of edge-on orientation (triangles), we mostly obtained delocalized holes for the lower-lying CT excitons produced by (1, 2 → 3) or (1, 2 → 4) transitions, while both the hole and the electron tend to delocalize for the higher-energy CT states. The fully delocalized high-energy CT states are illustrated by the correlation plot in Fig. 10, having d exc ≈ 13.5 Å. As for the face-on orientation, the addition of a PCBM molecule does not significantly alter the sizes of the largest excitons compared to the similar edge-on models with a single PCBM molecule (not shown). This happens because the delocalization of the electron is compensated by the smaller electron-hole distances.

For the models with two PCBM molecules arranged in a row, the dependence of the CT exciton sizes with respect to the POS descriptor and CT number is presented in Fig. 11, along with electron-hole correlation plots for representative states. Once more, we do not distinguish the positions and concentrate on the orientations. The CT states of the face-on systems (circles) fall into three distinct regions. The low-energy excitons are typically formed The delocalization of the interfacial CT excitons is, of course, limited by the system sizes, but some of the trends are compatible with those obtained from tight-binding models for much larger interfaces 6 . In the latter models, the higher-energy CT states have a band-like structure, approaching charge-separated states. On average, the electron-hole distances are smaller for the model in which the hole tends to delocalize along the DA interface. This result is in line with the present ones for the models having two PCBM molecules arranged side by side since the high-energy exciton sizes are not particularly large even when delocalized over all fragments. In case the acceptor molecules are arranged in a row, the delocalization of the high-energy CT excitons is more perpendicular to the interface, producing larger electronhole distances. From the photophysics perspective, the present study points out to similar trends as the previous TD-DFT study on P3HT:PCBM models 52 . Hot CT states, having energies around the absorption bands, would be expected to play a more significant part in interfaces with face-on orientation. Cold CT excitons, lying close to the S 1 state, would be more relevant to those with edge-on orientation. Population transfer to cold excitons is favored by the higher density of those states and also by the cascade of CT excitons lying below the brightest states (see Fig. 5 and6). Once hot CT excitons are populated from the bright states, relaxation to lower CT states would require smaller electronic rearrangement, compared to LD and LA states, such that stronger diabatic couplings could be expected.

Finally, the role of entropic forces arising from the density of electronic states in the dissociation of hot CT excitons has been recently pointed out 6 , but we cannot explore those entropy effects based on the present models.

V. CONCLUSIONS

We have employed the TD-LC-DFTB method and fragment-based 1-TDM analysis to study the CT states of polymer:fullerene interfaces, exploring models of different sizes, as well as different positions and orientations of the acceptor molecules with respect to those of the donor domain. The orientation effect is significant since the edge-on models have denser spectra of cold CT excitons, lying below the absorption bands. This effect was observed for all models addressed in the present study, taking into consideration both the stationary structures and the geometries obtained from MD simulations. The interface orientation is thus expected to impact the excited state dynamics and charge separation mechanisms.

Phases with face-on orientation favor dissociation from hot excitons, lying around the

Fig. 1 (

 1 Fig.1(d) shows the CT and POS descriptors for the 50 lowest-lying singlet excited states of 10 interface models out of those considered in the present study. The systems were decomposed into two fragments, corresponding to the donor (D) and acceptor (A) domains, and the POS descriptors were calculated with the positions of the D and A fragments chosen as 1 and 2, respectively (see eqs. 5 and 6). The data points are somewhat scattered, making the assignment of Frenkel and CT excitons arbitrary to some extent. Nevertheless, most of the points lie close to the corners of a triangle whose vertices correspond to the ideal exciton characters, namely LD (CT = 0.0, POS = 1.0), LA (CT = 0.0, POS = 2.0), and CT (CT = 1.0, POS = 1.5). We, therefore, employ the following practical definitions,

FIG. 1 .

 1 FIG. 1. (a) Structures of PCBM and PTBTBTz monomer, with the T, BT, and BTz subunits indicated for the latter. The sulfur atoms are represented in yellow, oxygen in red, nitrogen in deep blue, carbon in light blue, and hydrogen in white. (b) Ideal excited states: Frenkel exciton localized on the donor domain (LD); Frenkel exciton localized on the acceptor domain (LA); and charge-transfer exciton (CT). In the three panels, the electron and hole locations are indicated by the negative (red) and positive (blue) signs. The values of the CT and POS descriptors for the ideal excitations are also given in the panels, according to eqs. 4 and 6. The donor and acceptor positions are defined as 1 and 2, respectively. (c) Electron-hole correlation plots for the ideal LD, LA, and CT excited states, according to eq. 3. The electron and hole axes are also shown. (d) CT and position (POS) descriptors for the excited states of several interface models, where the lowest-lying 50 excited states were calculated in each case. The model systems are (A) (dim-PT) 2 :(PCBM@bt) 1 :f ; (B) (dim-PT) 2 :(PCNM@btz) 1 :f ; (C) (dim-PT) 2 :(PCBM@t) 1 :f ; (D) (dim-PT) 2 :(F@bt) 1 :e; (E) (dim-PT) 2 :(PCBM@t) 1 :e; (F) (dim-PT) 3 :(PCBM@bt) 1 :f ; (G) (dim-PT) 3 :(PCBM@btz) 1 :f ; (H) (dim-PT) 3 :(PCBM@t) 1 :f ; (I) (dim-PT3) 2 :(PCBM@btz) 1 :f ; (J) (dim-PT3) 2 :(PCBM@t) 1 :f . The shaded areas highlight the practical definitions of LD (green), LA (purple), and CT (red) excitons.

FIG. 2 .

 2 FIG. 2. The main panel shows the vertical excitation spectra for the PTBTBTz:PCBM dimer obtained with the (I) TD-LC-DFTB and (II) TD-ωB97XD/6-31G* methods at their respective ground-state stationary geometries. The LD states are indicated in green, the LA in purple, the CT in red, and MX in gray. The horizontal arrows indicate the low-energy bright transitions. The right panel shows the absorption cross section (in arbitrary units) obtained from the broadening of the vertical spectra with Gaussian line shapes.

FIG. 3 .

 3 FIG. 3. (a) Structures of the interface models addressed in the present study (the remaining structures are shown in the Supporting Information, Figs. S2and S3). In all panels, the donor domain is composed of stacked PTBTBTz dimers, (dim-PT) n , with n = 3 stacked chains in panels (e) to (h), and n = 2 stacked chains in the other panels. A single PCBM molecule forms the acceptor domain in panels (a) to (h). Its relative position, defined by the closest-lying donor subunit, is indicated as (PCBM@p) 1 , where p = t (thiophene) in panels (b), (d), (f), (h), and p = bt (benzothiodiazole) in panels (a), (c), (e) and (g). The orientation of the PCBM subunits is either face-on, (PCBM@p) 1 :f , as in panels (a), (b), (e), (f), or edge-on, (PCBM@p) 1 :e, as in panels (c), (d), (g), (h). In panels (i) to (o), the acceptor domain is formed by two PCBM molecules, either arranged side by side (2sd), as in panels (i), (j), (m), or in a row (2rw), as in panels (k), (l), (n), (o). The positions are denoted as (PCBM@p 1 p 2 ), with p 1 p 2 =t t, panels (j) and (m), and p 1 p 2 b=bt btz, panel (i), where btz indicates the benzotriazole donor subunit.

  FIG. 4. (a) The main panel shows the vertical excitation spectra for the (dim-PT) 2 :(PCBM@bt) 1 :f (I), (dim-PT) 2 :(PCBM@btz) 1 :f (II) and (dim-PT) 2 :(PCBM@t) 1 :f (III) systems. The LD states are indicated in green, LA in purple, CT in red, and MX in gray. The horizontal arrows point to the brightest transitions, and the geometries of the model systems are shown on top of the spectra. The right panel shows the absorption cross section (in arbitrary units) obtained from the broadening of the vertical spectra with Gaussian line shapes. (b) Same as in panel (a) for the (dim-PT) 3 :(PCBM@bt) 1 :f (I), (dim-PT) 3 :(PCBM@btz) 1 :f (II) and (dim-PT) 3 :(PCBM@t) 1 :f (III) systems.

FIG. 5 .

 5 FIG. 5. (a) The main panel shows the vertical excitation spectra for the (dim-PT) 2 :(PCBM@bt) 1 :e (I) and (dim-PT) 2 :(PCBM@t) 1 :e (II) systems. The LD states are indicated in green, LA in purple, CT in red, and MX in gray. The horizontal arrows point to the brightest transitions, and the geometries of the model systems are shown on top of the spectra. The right panel shows the absorption cross section (in arbitrary units) obtained from the broadening of the vertical spectra with Gaussian line shapes. (b) Same as in panel (a) for the (dim-PT) 3 :(PCBM@bt) 1 :e (III), and (dim-PT) 3 :(PCBM@t) 1 :e (IV) systems.

FIG. 6 .

 6 FIG. 6. (a) The main panel shows the vertical excitation spectra for the (dim-PT) 2 :(PCBM@bt btz) 2sd :f (I), (dim-PT) 2 :(PCBM@t t) 2sd :f (II) and (dim-PT) 2 :(PCBM@t t) 2sd :e (III) systems. The LD states are indicated in green, LA in purple, CT in red, and MX in gray. The horizontal arrows point to the brightest transitions, and the geometries of the model systems are shown on top of the spectra. The right panel shows the absorption cross section (in arbitrary units) obtained from the broadening of the vertical spectra with Gaussian line shapes. (b) Same as in panel (a) for the (dim-PT) 2 :(PCBM@t) 2rw :f (IV), and (dim-PT) 2 :(PCBM@t) 2rw :e (V) systems.

5 and 6 .

 6 The fragmentation scheme follows the conventions shown in Figs. 7(b) and 8(b) for the models with n = 2, 3 stacking, respectively.For a decomposition into N fr fragments, the oligomer chains are labeled A = 1 to A = (N fr -1), from the furthest-to the closest-lying one with respect to the PCBM molecule, which is labeled A = N fr . While the positions might not be as meaningful for the edge-on orientation, since the fragments are not aligned, the POS descriptor can still be interpreted as a weighted contribution from the fragments to the excitons. The POS descriptor for the CT states of the n = 2 stacking models are presented in Fig.7(a) as a function of the excitation energy. We only show the data points for the stationary structures to avoid overloading the panel, but the results are representative. The vertical shaded areas indicate the energy range of the bright states, where the brightest ones lie around ≈ 2.55 eV. The CT numbers, given in the color map, were obtained for two fragments, corresponding to the donor and acceptor domains, for consistency with the previous results (this procedure also avoids that a LD state involving charge transfer between the oligomer chains be assigned as a CT exciton).From eqs. 5 and 6, a CT exciton with the hole fully localized on the fragment A = 1 and the electron fully localized on the fragment A = 3 will have POS = 2.00. Accordingly, POS = 2.50 results from hole localization on the A = 2 fragment, while POS = 2.25 from an evenly delocalized hole over the A = 1, 2 fragments. These ideal cases are also indicated by the dashed lines. The cold excitons of the structures with face-on orientation essentially

FIG. 7 .

 7 FIG. 7. (a) Position (POS) descriptor for the CT states as a function of the excitation energy. The color map indicates the corresponding CT numbers. The data points were obtained for the stationary structures of the interface models with n = 2 stacked chains in the donor domain, (dim-PT) 2 :(PCBM@p) 1 :f (circles) with p = bt, btz, t, and (dim-PT) 2 :(PCBM@p) 1 :e (triangles) with p = bt, t. The POS numbers were computed for the three-fragment decomposition shown on the left-hand side of the panel (b), although the CT numbers were obtained from a two-fragment (DA) decomposition scheme. The horizontal dashed lines correspond to the POS values of ideal CT states with the hole localized on fragment 1 (POS = 2.00), localized on fragment 2 (POS = 2.50), or evenly delocalized over fragments 1 and 2 (POS = 2.25). The shaded regions indicate the energy range where the bright states are found (the brightest states of the five structures lie at ≈ 2.55 eV). (b) The structures of the (dim-PT) 2 :(PCBM@t) 1 :o models, with o = f, e orientations, are shown on the left-hand side along with the fragmentation convention. Electron-hole correlation plots for selected CT states are presented on the right-hand side, following the convention given in Fig. 1(c).The CT states are indicated as "high" and "low" in case they lie above or below the brightest state.

FIG. 8 .

 8 FIG. 8. (a) Position (POS) descriptor for the CT states as a function of the excitation energy. The color map indicates the corresponding CT numbers. The data points were obtained for the stationary structures of the interface models with n = 3 stacked chains in the donor domain, (dim-PT) 3 :(PCBM@p) 1 :f (circles) with p = bt, btz, t, and (dim-PT) 3 :(PCBM@p) 1 :e (triangles) with p = bt, t. The POS numbers were computed for the four-fragment decomposition shown on the left-hand side of panel (b), although the CT numbers were obtained from a two-fragment (DA) decomposition scheme. The horizontal dashed lines correspond to the POS values of ideal CT states with the hole localized on fragment 1 (POS = 2.5), localized on fragment 3 (POS = 3.5), and either localized on fragment 2 or evenly delocalized over fragments 1,2 and 3 (POS = 3.0). The shaded regions indicate the energy range where the bright states are found (the brightest states of the five structures lie at ≈ 2.7 eV). (b) The structures of the (dim-PT) 3 :(PCBM@t) 1 :o models, with o = f, s orientations, are shown on the left-hand side along with the fragmentation convention. Electron-hole correlation plots for selected CT states are presented on the right-hand side, following the convention given in Fig. 1(c). The CT states are indicated as "high" and "low" in case they lie above or below the brightest state.

  (a) for the CT states of the models with n = 3 stacking (stationary structures). The exciton size is plotted against the POS descriptor obtained from four fragments as before, and also the two-fragment CT number, indicated by the color map. Different positions are not discriminated, so we focus on the orientations. The face-on models (circles) display a stronger d exc dependence on position since the (de)localization of the hole over different donor chains has a clear effect on the charge separation, with the electron essentially localized on the PCBM molecule. In case the hole is localized on the A = 3 fragment, which is the oligomer chain lying the closest to the acceptor (POS ≈ 3.5), we obtain d exc ≈ 9.5 Å. For the CT states with POS ≈ 2.5, consistent with (1 → 4) transitions, one finds d exc ≈ 14.5 Å. As discussed above, the localization of the hole on the A = 3 and A = 1 fragments typically produces low-and high-energy CT excitons, respectively. A few ideal excitations might be of help to better understand the intermediate values of the POS descriptor. The even delocalization of the hole over two fragments leads to (1, 2 → 4) and (2, 3 → 4) transitions, and hence to POS = 2.75 and POS = 3.25, respectively. As mentioned above, POS = 3.00 can arise from either (2 → 4), (1, 3 → 4) or (1, 2, 3 → 4) ideal excitations. Most of the excited states have POS > 3.00, with the degree of charge transfer generally increasing with the exciton size. As a rule of thumb, face-on models give rise to CT excitons that increase in size and charge separation as a function of the excitation energy, according to the results shown in Figs. 8(a) and 9(a). The edge-on models (triangles) do not show a clear relation between position and excitation energy, as discussed above. The d exc dependence on position is not evident, either, since nearly all points lie within (2.75 < POS < 3.25). This suggests hole delocalization over the donor chains, although with no clear trend, since several ideal transitions have POS descriptors lying in that range. Also as a rule of thumb, the exciton size increases with the degree of charge transfer and the excitation energy, as inferred from Figs. 8(a) and 9(a). Inspection of the electron-hole correlation plots (not shown) indicates that the largest excitons arise from excitations having either dominant (1, 3 → 4) or (1, 2, 3 → 4) characters.

FIG. 9 .

 9 FIG. 9. (a) Exciton size against the position descriptor (POS) and the charge transfer number (CT), given in the color map, for the interface models indicated in the panel. In each case we consider different positions of the PCM molecule: p = bt, btz, t (circles); p = bt, t (triangles). The vertical lines indicate the POS descriptor values for ideal excitations (see Fig. 8(a)). (b) Exciton size against the charge transfer number (CT) and the excitation energy, given in the color map, for the face-on interface models indicated in the panel. The PCBM positions are p = bt, btz (circles); p = bt, t (squares); p = bt, btz, t (triangles) (c) Same as in panel (b), for the edge-on models with positions p = bt, t (circles and triangles).

FIG. 10 .

 10 FIG.10. Exciton size against the position descriptor (POS) and the charge transfer number (CT), given in the color map, for the interface models (dim-PT) 2 (PCBM@p) 2sd :f (circles) and (dim-PT) 2 (PCBM@p) 2sd :e (triangles). In both cases, the data points include the p = bt btz, t t positions. representative electron-hole correlation plots are shown above and below the main panel, with the fragment labels (1 to 4) indicated on the top.

FIG. 11 .

 11 FIG.11. Exciton size against the position descriptor (POS) and the charge transfer number (CT), given in the color map, for the interface models (dim-PT) 2 (PCBM@p) 2rw :f (circles) and (dim-PT) 2 (PCBM@p) 2rw :e (triangles). In both cases, the data points include the p = bt, t positions. Representative electron-hole correlation plots are shown above and below the main panel, with the fragment labels (1 to 4) indicated on the left.

FIG. 12 .

 12 FIG.12. Mechanisms for charge separation. (a) Face-on orientation. Photoexcitation to a LD bright state is followed by non-adibatic population transfer to a hot CT state (rate constant k hot ). Decay to the cold CT states (gray wavy line) is not likely to take place, so charge separation mostly proceeds from the hot CT exciton (rate constant k CS ). (b) Edge-on orientation. Relaxation from the hot CT exciton to the cold CT excitons (rate constant k cold ) is favored by the high density of low-energy CT states. Charge separation mostly proceeds from the cold CT excitons at the bottom of the excitation band.

  83 A. A. H. M. Darghouth, M. E. Casida, X. Zhu, B. Natarajan, H. Su, A. Humeniuk, E. Titov, X. Miao, R. Mitric, arXiv 1803.11121 (2020).

TABLE I .

 I Lowest-lying excited states of the PTBTBTz monomer and the PTBTBTz 2 stacked dimer. For the monomer, we consider the five lowest-lying excited states, S 1 to S 5 , while for dimer, S 1 and the four lowest-lying bright states, S B1 to S B4 . The energies, E (in eV), and the oscillator strengths, f , were obtained with several methods. TD-M1//M2 indicates that the excited states were calculated with the time-dependent method M1 at the ground-state geometry optimized with the method M2. All DFT and TD-DFT calculations were performed with the 6-31G* basis set. As shown in TableII, there are two low-lying absorption bands at ≈ 2.5 eV and ≈ 2.8 eV with oscillator strengths around f ≈ 2 and f ≈ 1, respectively (in case nearly degenerate states contribute to a given band, we refer to the sum of their strengths). For p = bt, t, the bright states have LD character, although MX character for p = btz. For the three systems, a significant density of LA states is found at the bottom of the bands, i.e., lying below the

	System	Method		S 1	S 2	S 3	S 4	S 5
		TD-CAMB3LYP//	E	3.017 3.596 4.308 4.530 4.579
		B3LYP	f	1.022 0.227 0.023 0.015 0.036
		TD-LCwPBE//	E	3.378 4.011 4.740 4.878 4.993
		B3LYP	f	1.219 0.103 0.136 0.095 0.007
		TD-ωB97XD//	E	2.917 3.531 4.382 4.465 4.530
		B3LYP	f	1.121 0.172 0.038 0.009 0.048
	PTBTBTz	TD-LC-DFTB//	E	2.997 3.639 3.842 4.192 4.201
		B3LYP	f	1.254 0.200 0.000 0.119 0.000
		TD-ωB97XD//	E	3.097 3.670 4.538 4.583 4.651
		ωB97XD	f	1.084 0.180 0.043 0.013 0.060
		TD-LC-DFTB//	E	3.195 3.796 3.957 4.290 4.294
		ωB97XD	f	1.187 0.219 0.000 0.038 0.099
		TD-LC-DFTB//	E	3.351 4.001 4.297 4.520 4.549
		LC-DFTB	f	1.394 0.070 0.000 0.000 0.031
		Method		S 1	S B1	S B2	S B3	S B4
		TD-ωB97XD//	E	2.852 3.133 3.544 3.656 4.725
		ωB97XD	f	0.067 1.247 0.441 0.170 0.266
	PTBTBTz 2	TD-LC-DFTB//	E	3.049 3.262 3.472 3.564 3.799
		ωB97XD	f	0.095 1.380 0.177 0.111 0.343
		TD-LC-DFTB//	E	3.204 3.444 4.049 4.559 4.713
		LC-DFTB	f	0.043 2.247 0.169 0.121 0.165

tion S-II), and approximate absorption cross sections are also shown. The face-on models have similar features regardless of the positions (p = bt, t, btz) of the PCBM molecule. first bright state (E 2.55 eV). Only a few CT excitons are found either below the first bright state, close to the second bright state (E ≈ 2.8 eV), and between these two bright states (2.55 eV E 2.75 eV).

  2 , (tet-PT) 2 and (dim-PT3) 2 donors. Nevertheless, larger excitons are produced by the two larger models, as expected, and the trend of size increase with the CT number is once more noticeable. Despite the limited number of data points, the largest excitons (15 Å < d ext < 18 Å) are mostly built on PT3BTBTz dimers. In general,

TABLE II .

 II Lowest-lying absorption bands of PTBTBTz:PCBM models. In each case, we show the major contributions from the vertical excited states of the stationary structures, indicating the energies, in eV, and the oscillator strengths (given in parenthesis). The letter in parenthesis indicates the interface in Fig.3. The two interfaces with (*) are shown in Fig.S3a and b of the Supporting Information.

	System	1 st band	2 nd band	3 rd band
	(dim-PT) 2 (PCMB@bt) 1 :f (a)	2.557 (2.245)	2.778 (1.345)	weak
	(dim-PT) 2 (PCMB@btz) 1 :f (*)	2.534 (0.919)	2.802 (0.808)	weak
		2.547 (1.177)	2.806 (0.609)
	(dim-PT) 2 (PCMB@t) 1 :f (b)	2.612 (2.572)	2.775 (0.919)	weak
	(dim-PT) 2 (PCMB@bt) 1 :e (c)	2.518 (0.183)	2.771 (1.266)	3.142 (0.737)
		2.587 (2.239)
	(dim-PT) 2 (PCMB@t) 1 :e (d)	2.613 (2.406)	2.709 (0.923)	3.071 (0.228)
		2.780 (0.576)		3.143 (0.561)
	(dim-PT) 3 (PCMB@bt) 1 :f (e)	2.465 (1.558)	2.691 (0.591)	2.940 (0.990)
			2.696 (1.231)	2.945 (0.176)
			2.706 (0.423)
	(dim-PT) 3 (PCMB@btz) 1 :f (*)	2.460 (1.512)	2.702 (2.197)	2.934 (1.237)
				2.828 (0.215)
	(dim-PT) 3 (PCMB@t) 1 :f (f)	2.428 (0.149)	2.636 (0.516)	2.937 (1.061)
		2.471 (1.319)	2.710 (2.171)
	(dim-PT) 3 (PCMB@bt) 1 :e (g)	2.464 (1.572)	2.723 (2.453)	2.969 (1.165)
				3.036 (0.130)
	(dim-PT) 3 (PCMB@t) 1 :e (h)	2.490 (0.860)	2.603 (0.149)	2.914 (0.174)
			2.678 (3.244)	2.917 (0.776)
				2.931 (0.120)
	(dim-PT) 2 (PCMB@bt btz) 2sd :f (i)	2.526 (2.040)	2.790 (1.110)	2.880 (0.108)
			2.804 (0.159)	2.887 (0.138)
				2.943 (0.147)
	(dim-PT) 2 (PCMB@t t) 2sd :f (j)	2.531 (2.032)	2.807 (1.036)	2.941 (0.211)
			2.869 (0.269)	3.070 (0.111)
	(dim-PT) 2 (PCMB@t t) 2sd :e (m)	2.632 (0.461)	2.731 (0.822)	weak
		2.636 (0.877)
		2.652 (1.606)
	(dim-PT) 2 (PCMB@bt) 2rw :f (k)	2.571 (2.236)	2.793 (1.008)	2.900 (0.129)
			2.799 (0.114)	2.955 (0.111)
				3.173 (0.592)
	(dim-PT) 2 (PCMB@t) 2rw :f (l)	2.569 (2.035)	2.713 (0.207)	2.900 (0.129)
		2.602 (0.185)	2.787 (1.110)	3.173 (0.600)
	(dim-PT) 2 (PCMB@bt) 2rw :e (n)	2.579 (2.316)	2.754 (0.235)	2.915 (0.388)
			2.794 (0.880)	2.964 (0.231)
				3.172 (0.552)
	(dim-PT) 2 (PCMB@t) 2rw :e (o)	2.628 (2.481)	2.723 (0.422)	2.921 (0.145)
			2.773 (0.861)	3.159 (1.081)

ACKNOWLEDGMENTS M. T. do N. V. acknowledges support from FAPESP (grant 2018/22948-0) and also from the Brazilian National Council for Scientific and Technological Development (CNPq), grant 304571/2018-0. The calculations were partly performed with HPC resources from STI (University of São Paulo), Centro Nacional de Processamento de Alto Desempenho em São Paulo

Contract No. DE-AC05-00OR22725 with the U.S. Department of Energy. manuscript, or allow others to do so, for United States Government purposes. The Department of Energy will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan (http://energy.gov/downloads/doe-public-access-plan).

1

brightest bands. Edge-on phases favor dissociation following relaxation to cold CT states, which is enhanced by the cascade of CT excitons. These photophysical mechanisms, summarized in Fig. 12, are consistent with previous atomistic simulations 52 . The exciton analysis also points out to the formation of low-energy interfacial CT states with increased electronhole separation in the edge-on phases. This effect would also make the charge separation from cold CT states more efficient, along with the higher density of these low-energy states.

The position of the acceptor molecules with respect to the donor is generally less important, although the change in the excitation spectra from the face-on to the edge-on orientation tends to be more clear when the PCBM molecule lies closer to the T subunit of the donor chains. The face-on orientation gives rise to fewer cold CT states, which also have, on average, smaller CT numbers compared to the cold CT states in the models with edge-on orientation. The degree of charge transfer and the CT exciton size tends to increase with the excitation energy. In the edge-on models with a single PCBM molecule, the charge transfer is favored by longer electron-hole distances arising from hole localization on donor chains lying far from the acceptor molecules, while hole delocalization over different donor chains prevails in the edge-on interfaces. The models with two acceptor molecules suggest that the electron remains localized on a single molecule in the cold CT states. The delocalization over two PCBM molecules was observed for higher-energy CT states, although the impact on the exciton size also depends on the orientation. Larger excitons are produced by the delocalization of the electron perpendicularly to the DA interface. In case the delocalization is parallel to the interface, it is compensated by the smaller electron-hole distances, producing moderately sized CT excitons.

SUPPORTING INFORMATION

Details of the MD simulations and geometry optimization. Structures of the PTBTBTz stacked dimer, the PT3BTBtz monomer, and also of the interface models not shown in the main text. Photo-absorption cross sections for the PTBTBTz monomer calculated with the nuclear ensemble method. Vertical excitation spectra of the (dim-PT) 2 :(PCBM@bt) 2rw :f and (dim-PT) 2 :(PCBM@bt) 2rw :e models. Photo-absorption cross sections for geometries randomly chosen from the MD simulations for the (dim-PT) 2 (PCBM@t) 1 :f , (dim-PT) 2 (PCBM@t) 1 :e, (dim-PT) 3 (PCBM@t) 1 :f , and (dim-PT) 3 (PCBM@t) 1 :e models. Geometries of the interface CT , and the average values of the charge-transfer number for the cold CT excitons defined with respect to S 1 , CT cold , are also shown. For each system, the averages are computed for the different relative positions of the PCBM molecules (bt, btz, t) and also without discriminating the positions (all). The number of geometries, N , used to calculate the n CT averages and standard deviations (given in parenthesis) is indicated. The CT cold values are averaged over the CT states, not over the geometries. The letter in parenthesis indicates the interface in Fig. 3. The two interfaces with (*) are shown in Fig. S3 a and b of the Supporting Information.