prompt_front: Now, answer the following question based on the text below, only give me the answer and do not output any other words.

Question: What type of distribution do the tail distributions of price returns follow?

Read the following text and answer briefly.

Context 1: Paper Info

Title: Age and market capitalization drive large price variations of cryptocurrencies
Publish Date: 23 Feb 2023
Author List: 

Figure

Figure 3  Illustration of different effects of age and market capitalization on power-law exponents of cryptocurrencies (a) Posterior probability distributions of the linear coefficients associated with the effects of age [p(A)] and (b) the effects of market capitalization [p(C)] on power-law exponents related to large positive returns Panels (c) and (d) show the analogous distributions for the association with power-law exponents related to large negative returns In all panels, the different curves show the distributions for each of the top 20 cryptoassets by market capitalization Cryptocurrencies significantly affected by age or market capitalization are highlighted in boldface, and the numbers between brackets show their positions in the market capitalization rank 
Figure S5 There is more probability mass in the positive tail than in the negative tail of price returns (a) Probability distributions of the lower cut-offs (r min ) obtained by applying the Clauset-Shalizi-Newman method to positive (blue) and negative (red) returns The vertical dashed lines indicate the median values of r min for positive and negative returns (b) Probability distributions of 90th percentiles (r 90 ) estimated from the power-law models adjusted to positive (blue) and negative (red) returns The vertical dashed lines indicate the median values of r 90 for positive and negative returns (c) Probability distributions of the fraction of weeks that r 90 estimated from positive returns (r + 90 ) is larger than r 90 estimated from negative returns (r − 90 ) This fraction is calculated only for weeks in which the power-law hypothesis is not rejected for both tails The percentage of cryptoassets for which r + 90 > r − 90 is shown in the panels The first column of panels depicts the results when considering data from all cryptocurrencies, while the second and third columns present the results for the top 2000 and top 200 cryptocurrencies by market capitalization, respectively 
Figure S7 Robustness of the results of Fig  2(b)-(d) against considering only cryptocurrencies with fraction of rejection f r < 0 1 Panels (a) and (b) show the same distributions of Fig  S4 but after filtering out all time series of cryptocurrencies with fraction of rejections f r ≥ 0 1 As in the case related to sampling issues, we observe that these distributions barely change when considering only cryptocurrencies with f r < 0 1 Indeed, the distributions in this figure are not significantly distinguishable from their counterparts in Fig  S4 (two-sample Kolmogorov-Smirnov test, p > 0 05) 

abstract

Cryptocurrencies are considered the latest innovation in finance with considerable impact across social, technological, and economic dimensions  This new class of financial assets has also motivated a myriad of scientific investigations focused on understanding their statistical properties, such as the distribution of price returns 
However, research so far has only considered Bitcoin or at most a few cryptocurrencies, whilst ignoring that price returns might depend on cryptocurrency age or be influenced by market capitalization  Here, we therefore present a comprehensive investigation of large price variations for more than seven thousand digital currencies and explore whether price returns change with the coming-of-age and growth of the cryptocurrency market 
We find that tail distributions of price returns follow power-law functions over the entire history of the considered cryptocurrency portfolio, with typical exponents implying the absence of characteristic scales for price variations in about half of them  Moreover, these tail distributions are asymmetric as positive returns more often display smaller exponents, indicating that large positive price variations are more likely than negative ones 
Our results further reveal that changes in the tail exponents are very often simultaneously related to cryptocurrency age and market capitalization or only to age, with only a minority of cryptoassets being affected just by market capitalization or neither of the two quantities  Lastly, we find that the trends in power-law exponents usually point to mixed directions, and that large price variations are likely to become less frequent only in about 28% of the cryptocurrencies as they age and grow in market capitalization 
Since the creation of Bitcoin in 2008 , various different cryptoassets have been developed and are now considered to be at the cutting edge of innovation in finance   These digital financial assets are vastly diverse in design characteristics and intended purposes, ranging from peer-to-peer networks with underlying cash-like digital currencies (e g 
Bitcoin) to general-purpose blockchains transacting in commodity-like digital assets (e g  Ethereum), and even to cryptoassets that intend to replicate the price of conventional assets such as the US dollar or gold (e g  Tether and Tether Gold)   With more than nine thousand cryptoassets as of 2022 , the total market value of cryptocurrencies has grown massively to a staggering $2 trillion peak in 2021  
Despite long-standing debates over the intrinsic value and legality of cryptoassets , or perhaps even precisely due to such controversies, it is undeniable that cryptocurrencies are increasingly attracting the attention of academics, investors, and central banks, around the world   Moreover, these digital assets have been at the forefront of sizable financial gains and losses in recent years , they have been recognized as the main drivers of the brand-new phenomena of cryptoart and NFTs , but also as facilitators of illegal activities, such as money laundering and dark trade  
Financial research dedicated Our results are based on daily price time series of 7111 cryptocurrencies that comprise a significant part of all currently available cryptoassets (see Methods for details)  From these price series, we have estimated their logarithmic returns 2/16 Log-return, r )  The black horizontal arrow represents a given position of the expanding time window (at t = 2004 days) used to sample the return series over the entire history of Bitcoin 
This time window expands in weekly steps (seven time series observations), and for each position, we separate the positive (blue) from the negative (red) price returns  The gray line illustrates observations that will be included in future positions of the expanding time window (t > 2004)  (b) Survival functions or the complementary cumulative distributions of positive (blue) and negative (red) price returns within the expanding time window for t = 2004 days and above the lower bound of the power-law regime estimated from the Clauset-Shalizi-Newman method  
The dashed lines show the adjusted power-law functions, p(r) ∼ r −α , with α = 4 5 for positive returns and α = 3 0 for negative returns  (c) Time series of the power-law exponents α t for the positive (blue) and negative (red) return distributions obtained by expanding the time window from the hundredth observation (t = 100) to the latest available price return of Bitcoin 
The circular markers represent the values for the window position at t = 2004 days and the dashed lines indicate the median of the power-law exponents ( α+ = 4 50 for positive returns and α− = 2 99 for negative returns)  (d) Time series of the p-values related to the power-law hypothesis of positive (blue) and negative (red) price returns for every position of the expanding time window 
The dashed line indicates the threshold (p = 0 1) above which the power-law hypothesis cannot be rejected  For Bitcoin, the power-law hypothesis is never rejected for positive returns (fraction of rejection f r = 0) and rejected in only 4% of the expanding time window positions (fraction of rejection f r = 0 04) 
where x t represents the price of a given cryptocurrency at day t  All return time series in our analysis have at least 200 observations (see Supplementary Figure for the length distribution)  Figure (a) illustrates Bitcoin's series of daily returns  To investigate whether and how returns have changed over the aging and growing processes of all cryptocurrencies, we sample all time series of log-returns using a time window that expands in weekly steps (seven time series observations), starting from the hundredth observation to the latest return observation 
In each step, we separate the positive from the negative return values and estimate their power-law behavior using the Clauset-Shalizi-Newman method   Figure (a) further illustrates this procedure, where the vertical dashed line represents a given position of the time window (t = 2004 days), the blue and red lines indicate positive and negative returns, respectively, and the gray lines show the return observations that will be included in the expanding time window in future steps 
Moreover, Fig  (b) shows the corresponding survival functions (or complementary cumulative distributions) for the positive (blue) and negative (red) returns of Bitcoin within the time window highlighted in Fig  (a)  These survival functions correspond to return values above the lower bound of the power-law regime (r min ) and dashed lines in Fig  (b) show the power-law functions adjusted to data, that is,
with α = 4 5 for the positive returns and α = 3 0 for the negative returns in this particular position of the time window (t = 2004 days)  We have further verified the goodness of the power-law fits using the approach proposed by Clauset et al  (see also Preis et al  )  As detailed in the Methods section, this approach consists in generating several synthetic samples under the power-law hypothesis, adjusting these simulated samples, and estimating the fraction of times the Kolmogorov-Smirnov distance between the adjusted power-law and the synthetic samples is larger than the value calculated from the empirical data 
This fraction defines a p-value and allows us to reject or not the power-law hypothesis of the return distributions under a given confidence level  Following Refs  we consider the more conservative 90% confidence level (instead of the more lenient and commonly used 95% confidence level), rejecting the power-law hypothesis when p-value ≤ 0 1 
For the particular examples in Fig  (b), the p-values are respectively 1 00 and 0 17 for the positive and negative returns, and thus we cannot reject the power-law hypotheses  After sampling the entire price return series, we obtain time series for the power-law exponents (α t ) associated with positive and negative returns as well as the corresponding p-values time series for each step t of the expanding time window 
These time series allow us to reconstruct the aging process of the return distributions over the entire history of each cryptoasset and probe possible time-dependent patterns  Figures ) and 1(d) show the power-law exponents and p-values time series for the case of Bitcoin  The power-law hypothesis is never rejected for positive returns and rarely rejected for negative returns (about 4% of times) 
Moreover, the power-law exponents exhibit large fluctuations at the beginning of the time series and become more stable as Bitcoin matures as a financial asset (a similar tendency as reported by Begušić et al  )  The time evolution of these exponents further shows that the asymmetry between positive and negative returns observed in Fig  ) is not an incidental feature of a particular moment in Bitcoin's history 
Indeed, the power-law exponent for positive returns is almost always larger than the exponent for negative returns, implying that large negative price returns have been more likely to occur than their positive counterparts over nearly the entire history of Bitcoin covered by our data  However, while the difference between positive and negative exponents has approached a constant value, both exponents exhibit an increasing trend, indicating that large price variations are becoming less frequent with the coming-of-age of Bitcoin 
The previous analysis motivates us to ask whether the entire cryptocurrency market behaves similarly to Bitcoin and what other common patterns digital currencies tend to follow  To start answering this question, we have considered the p-values series of all cryptocurrencies to verify if the power-law hypothesis holds in general 
Figure (a) shows the percentage of cryptoassets rejecting the power-law hypothesis in at most a given fraction of the weekly positions of the expanding time window ( f r )  Remarkably, the hypothesis that large price movements (positive or negative) follow a power-law distribution is never rejected over the entire history of about 70% of all digital currencies in our dataset 
This analysis also shows that only ≈2% of cryptocurrencies reject the power-law hypothesis in more than half of the positions of the expanding time window ( f r ≥ 0 5)  For instance, considering a 10% threshold as a criterion ( f r ≤ 0 1), we find that about 85% of cryptocurrencies have return distributions adequately modeled by power laws 
Increasing this threshold to a more lenient 20% threshold ( f r ≤ 0 2), we find large price movements to be power-law distributed for about 91% of cryptocurrencies  These results thus provide strong evidence that cryptoassets, fairly generally, present large price movements quite well described by power-law distributions 
Moreover, this conclusion is robust when starting the expanding window with a greater   Large price movements are power-law distributed over the entire history of most cryptocurrencies with median values typically smaller than those found for traditional assets  (a) Percentage of cryptoassets rejecting the power-law hypothesis for large positive (blue) or negative (red) price returns in at most a given fraction of the weekly positions of the expanding time window ( f r ) used to sample the return series 
Remarkably, 68% of all 7111 digital currencies are compatible with the power-law hypothesis over their entire history, and about 91% of them reject the power-law hypothesis in less than 20% of the positions of the expanding time window ( f r ≤ 0 2)  (b) Probability distributions obtained via kernel density estimation of the median values of the power-law exponents along the history of each digital currency 
The blue curve shows the distribution of the median exponents related to positive returns ( α+ ) and the red curve does the same for negative returns ( α− )  The medians of α+ and α− are indicated by vertical dashed lines  Panels (c) and (d) show the distributions of these median exponents when considering the top 2000 and the top 200 cryptocurrencies by market capitalization, respectively 
We observe that the distributions of α+ and α− tend to shift toward larger values when considering the largest cryptoassets  number of return observations (between 100 and 300 days) and filtering out cryptoassets with missing observations (Supplementary Figures )  Still, it is worth noticing the existence of a few cryptoassets (9 of them) with relatively small market capitalization (ranking below the top 1000) for which the power-law hypothesis is always rejected (Supplementary Table ) 
Having verified that large price movements in the cryptocurrency market are generally well-described by powerlaw distributions, we now focus on the power-law exponents that typically characterize each cryptoasset  To do so, we select all exponent estimates over the entire history of each digital asset for which the power-law hypothesis is not rejected and calculate their median values for both the positive ( α+ ) and negative ( α− ) returns 
The dashed lines in Fig  ) show these median values for Bitcoin where α+ = 4 50 and α− = 2 99  It is worth noticing that the variance of large price movements σ 2 is finite only for α > 3, as the integral σ 2 ∼ ∞ r min r 2 p(r)dr diverges outside this interval  Thus, while the typical variance of large positive returns is finite for Bitcoin, negative returns are at the limit of not having a typical scale and are thus susceptible to much larger variations 
Figure shows the probability distribution for the median power-law exponents of all cryptoassets grouped by large positive and negative returns  We note that the distribution of typical power-law exponents associated with large positive returns is shifted to smaller values when compared with the distribution of exponents related to large negative returns 
The medians of these typical exponents are respectively 2 78 and 3 11 for positive and negative returns  This result suggests that the asymmetry in large price movements we have observed for Bitcoin is an overall feature of the cryptocurrency market  By calculating the difference between the typical exponents related to positive and negative large returns (∆α = α+ − α− ) for each digital currency, we find that about 2/3 of cryptocurrencies have α+ < α− (see Supplementary Figure for the probability distribution of ∆α) 
Thus, unlike Bitcoin, most cryptocurrencies have been more susceptible to large positive price variations than negative ones  While this asymmetry in the return distributions indicates that extremely large price variations tend to be positive, it does not necessarily imply positive price variations are more common for any threshold in the return values 
This happens because the fraction of events in each tail is also related to the lower bound of the power-law regime (r min )  However, we have found the distribution of r min to be similar among the positive and negative returns [Supplementary Figure ]  The distribution of high percentile scores (such as the 90th percentile) is also shifted to larger values for positive returns [Supplementary Figure ] 
Moreover, this asymmetry in high percentile scores related to positive and negative returns is systematic along the evolution of the power-law exponents [Supplementary Figure ]  These results thus indicate that there is indeed more probability mass in the positive tails than in the negative ones, a feature that likely reflects the current expansion of the cryptocurrency market as a whole 
The distributions in Fig  ) also show that large price variations do not have a finite variance for a significant part of cryptoassets, that is, α+ ≤ 3 for 62% of cryptocurrencies and α− ≤ 3 for 44% of cryptocurrencies  A significant part of the cryptocurrency market is thus prone to price variations with no typical scale 
Intriguingly, we further note the existence of a minority group of cryptoassets with α+ ≤ 2 (7%) or α− ≤ 2 (3%)  These cryptocurrencies, whose representative members are Counos X (CCXX, rank 216) with α − = 1 96 and α + = 1 84 and Chainbing (CBG, rank 236) with α + = 1 87, are even more susceptible to extreme price variations as one cannot even define the average value µ for large price returns, as the integral µ ∼ ∞ r min rp(r)dr diverges for α ≤ 2  We have also replicated the previous analysis when considering cryptocurrencies in the top 2000 and top 200 rankings of market capitalization (as of July 2022) 
Figures ) and 2(d) show the probability distribution for the median power-law exponents of these two groups  We observe that these distributions are more localized (particularly for the top 200) than the equivalent distributions for all cryptocurrencies  The fraction of cryptocurrencies with no typical scale for large price returns ( α+ ≤ 3 and α− ≤ 3) is significantly lower in these two groups compared to all cryptocurrencies 
In the top 2000 cryptocurrencies, 51% have α+ ≤ 3 and 26% have α− ≤ 3  These fractions are even smaller among the top 200 cryptocurrencies, with only 44% and 15% not presenting a typical scale for large positive and negative price returns, respectively  We further observe a decrease in the fraction of cryptoassets for which the average value for large price returns is not even finite, as only 2% and 1% of top 2000 cryptoassets have α+ ≤ 2 and α− ≤ 2  This reduction is more impressive among the top 200 cryptocurrencies as only the cryptoasset Fei USD (FEI, rank 78) has α+ = 1 97 and none is characterized by α− ≤ 2  The medians of α+ and α− also increase from 2 78 and 3 11 for all cryptocurrencies to 2 98 and 3 35 for the top 2000 and to 3 08 and 3 58 for the top 200 cryptocurrencies 
Conversely, the asymmetry between positive and negative large price returns does not differ much among the three groups, with the condition α+ < α− holding only for a slightly larger fraction of top 2000 (69 1%) and top 200 (70 6%) cryptoassets compared to all cryptocurrencies (66 4%)  Moreover, all these patterns are robust when filtering out time series with sampling issues or when considering only cryptoassets that stay compatible with the power-law hypothesis in more than 90% of the positions of the expanding time window (Supplementary Figures ) 
We also investigate whether the patterns related to the median of the power-law exponents differ among groups of cryptocurrencies with different designs and purposes  To do so, we group digital assets using the 50 most common tags in our dataset (e g  "bnb-chain", "defi", and "collectibles-nfts") and estimate the probability distributions of the median exponents α+ and α− (Supplementary Figures ) 
These results show that design and purpose affect the dynamics of large price variations in the cryptocurrency market as the medians of typical exponents range from 2 4 to 3 7 among the groups  The lowest values occur for cryptocurrencies tagged as "doggone-doggerel" (medians of α+ and α− are 2 38 and 2 83), "memes" (2 41 and 2 87), and "stablecoin" (2 65 and 2 79) 
Digital currencies belonging to the first two tags overlap a lot and have Dogecoin (DOGE, rank 9) and Shiba Inu (SHIB, rank 13) as the most important representatives  Cryptoassets with these tags usually have humorous characteristics (such as an Internet meme) and several have been considered as a form of pump-and-dump scheme , a type of financial fraud in which false statements artificially inflate asset prices so the scheme operators sell their overvalued cryptoassets 
Conversely, cryptoassets tagged as "stablecoin" represent a class of cryptocurrencies designed to have a fixed exchange rate to a reference asset (such as a national currency or precious metal)   While the price of stablecoins tends to stay around the target values, their price series are also marked by sharp variations, which in turn are responsible for their typically small power-law exponents 
This type of cryptoasset has been shown to be prone to failures , such as the recent examples of TerraUSD (UST) and Tron's USDD (USDD) that lost their pegs to the US Dollar producing large variations in their price series  The asymmetry between positive and negative large returns also emerges when grouping the cryptocurrencies using their tags 
All 50 tags have distributions of α+ shifted to smaller values when compared with the distributions of α− , with differences between their medians ranging from −0 74 ("okex-blockdream-ventures-portfolio") to −0 14 ("stablecoin")  Indeed, only four ('stablecoin", "scrypt", "fantom-ecosystem" and "alameda-research-portfolio") out of the fifty groupings have both distributions indistinguishable under a two-sample Kolmogorov-Smirnov test (p-value > 0 05) 
Focusing now on the evolution of the power-law exponents quantified by the time series α t for positive and negative returns, we ask whether these exponents present particular time trends  For Bitcoin [Fig  )], α t seems to increase with time for both positive and negative returns  At the same time, the results of Fig  also suggest that market capitalization affects these power-law exponents 
To verify these possibilities, we assume the power-law exponents (α t ) to be linearly associated with the cryptocurrency's age (y t , measured in years) and the logarithm of market capitalization (log c t )  As detailed in the Methods section, we frame this problem using a hierarchical Bayesian model 
This approach assumes that the linear coefficients associated with the effects of age (A) and market capitalization (C) of each digital currency are drawn from distributions with means µ A and µ C and standard deviations σ A and σ C , which are in turn distributed according to global distributions representing the overall impact of these quantities on the cryptocurrency market 
The Bayesian inference process consists of estimating the posterior probability distributions of the linear coefficients for each cryptocurrency as well as the posterior distributions of µ A , µ C , σ A , and σ C , allowing us to simultaneously probe asset-specific tendencies and overall market characteristics 
Moreover, we restrict this analysis to the 2140 digital currencies having more than 50 observations of market capitalization concomitantly to the time series of the power-law exponents in order to have enough data points for detecting possible trends  When considering the overall market characteristics, we find that the 94% highest density intervals for µ A ([-0 01, 0 06] for positive and [-0 02, 0 03] for negative returns) and µ C ([-0 02, 0 03] for positive and [-0 001, 0 04] for negative returns) include the zero (see Supplementary Figure for their distributions) 
Thus, there is no evidence of a unique overall pattern for the association between the power-law exponents and age or market capitalization followed by a significant part of the cryptocurrency market  Indeed, the 94% highest density intervals for σ A ([0 87, 0 93] for positive and [0 63, 0 70] for negative returns) and σ C ([0 57, 0 61] for positive and [0 49, 0 52] for negative returns) indicate that the cryptocurrency market is highly heterogeneous regarding the evolution of power-law exponents associated with large price variations (see Supplementary Figure for the distributions of σ A and σ C )  Figure illustrates these heterogeneous behaviors by plotting the posterior probability distributions for the linear coefficients associated with the effects of age (A) and market capitalization (C) for the top 20 digital assets, where cryptocurrencies which are significantly affected (that is, the 94% highest density intervals for A or C do not include the zero) by these quantities are highlighted in boldface 
Even this small selection of digital  currencies already presents a myriad of patterns  First, we observe that the power-law exponents of a few top 20 cryptocurrencies are neither correlated with age nor market capitalization  That is the case of Shiba Inu (SHIB, rank 13) and Dai (DAI, rank 11) for both positive and negative returns, UNUS SED LEO (LEO, rank 18) and Polkadot (DOT, rank 12) for the positive returns, and USDCoin (USDC, rank 4) and Solana (SOL, rank 9) for negative returns 
There are also cryptocurrencies with exponents positively or negatively correlated only with market capitalization  Examples include Tether (USDT, rank 3) and Dogecoin (DOGE, rank 10), for which the power-law exponents associated with positive returns increase with market capitalization, and Binance USD (BUSD, rank 6), for which power-law exponents associated with positive and negative returns decrease with market capitalization 
We also observe cryptocurrencies for which age and market capitalization simultaneously affect the power-law exponents  Polygon (MATIC, rank 14) is an example where the power-law exponents associated with positive returns tend to increase with age and decrease with market capitalization  Finally, there are also cryptocurrencies with power-law exponents only associated with age 
That is the case of Bitcoin (BTC, rank 1), Ethereum (ETH, rank 2), and Cardano (ADA, rank 8), for which the power-law exponents related to positive and negative returns increase with age, but also the case of Uniswap (UNI, rank 19), for which the exponents decrease with age  Figure systematically extends the observations made for the top 20 cryptoassets to all 2140 digital currencies for which we have modeled the changes in the power-law exponents as a function of age and market capitalization 
First, we note that only 10% of cryptocurrencies have power-law exponents not significantly affected by age and market capitalization  The vast majority (90%) displays some relationship with these quantities  However, these associations are as varied as the ones we have observed for the top 20 cryptoassets 
About 52% of cryptocurrencies have power-law exponents simultaneously affected by age and market capitalization  In this group, these quantities simultaneously impact the exponents related to positive and negative returns of 34% of cryptoassets, whereas the remainder is affected only in the positive tail (9%) or only in the negative tail (9%) 
Moving back in the hierarchy, we find that the power-law exponents of 32% of cryptocurrencies are affected only by age while a much minor fraction (6%) is affected only by market capitalization  Within the group only affected by age, we observe that the effects are slightly more frequent only on the exponents related to negative returns (12%), compared to cases where effects are restricted only to positive returns (10%) or simultaneously affect both tails (10%) 
Finally, within the minor group only affected by market capitalization, we note that associations more frequently involve only exponents related to negative returns (3%) compared to the other two cases (2% only positive returns and 1% for both positive and negative returns)  Beyond the previous discussion about whether positive or negative returns are simultaneously or individually affected by age and market capitalization, we have also categorized the direction of the trend imposed by these two quantities on the power-law exponents 
Blue rectangles in Fig  represent the fraction of relationships for which increasing age or market capitalization (or both) is associated with a raise in the power-law exponents  About 28% of all cryptocurrencies exhibit this pattern in which large price variations are expected to occur less frequently as they grow and age 
Conversely, the red rectangles in Fig  depict the fraction of relationships for which increasing age or market capitalization (or both) is associated with a reduction in the power-law exponents  This case comprises about 25% of all cryptocurrencies for which large price variations are likely to become more frequent as they grow in market capitalization and age 
Still, the majority of associations represented by green rectangles refer to the case where the effects of age and market capitalization point in different directions (e g  exponents increasing with age while decreasing with market capitalization)  About 36% of cryptocurrencies fit this condition which in turn contributes to consolidating the cumbersome hierarchical structure of patterns displayed by cryptocurrencies regarding the dynamics of large price variations 
This complex picture is not much different when considering only cryptocurrencies in the top 200 market capitalization rank (Supplementary Figure )  However, we do observe an increased prevalence of patterns characterized by exponents that rise with age and market capitalization (37%), suggesting that large price variations are becoming less frequent among the top 200 cryptocurrencies than in the overall market 
)  Each of the previous three levels is further classified regarding whether both positive and negative returns are simultaneously affected or whether the effect involves only positive or only negative returns  Finally, the former levels are classified regarding whether the power-law exponents increase, decrease or have a mixed trend with the predictive variables 
Overall, 36% of the associations are classified as mixed trends (green rectangles), 28% are increasing trends (blue rectangles), and 26% are decreasing trends (red rectangles)  We have studied the distributions of large price variations of a significant part of the digital assets that currently comprise the entirety of the cryptocurrency market 
Unlike previous work, we have estimated these distributions for entire historical price records of each digital currency, and we have identified the patterns under which the return distributions change as cryptoassets age and grow in market capitalization  Similarly to conventional financial assets , our findings show that the return distributions of the vast majority of cryptoassets have tails that are described well by power-law functions along their entire history 
The typical power-law exponents of cryptocurrencies (α ∼ 3) are, however, significantly smaller than those reported for conventional assets (α ∼ 4)   This feature corroborates the widespread belief that cryptoassets are indeed considerably more risky for investments than stocks or other more traditional financial assets 
Indeed, we have found that about half of the cryptocurrencies in our analysis do not have a characteristic scale for price variations, and are thus prone to much higher price variations than those typically observed in stock markets  On the upside, we have also identified an asymmetry in the power-law exponents for positive and negative returns in about 2/3 of all considered cryptocurrencies, such that these exponents are smaller for positive than they are for negative returns 
This means that sizable positive price variations have generally been more likely to occur than equally sizable negative price variations, which in turn may also reflect the recent overall expansion of the cryptocurrency market  Using a hierarchical Bayesian linear model, we have also simultaneously investigated the overall market characteristics and asset-specific tendencies regarding the effects of age and market capitalization on the power-law exponents 
We have found that the cryptocurrency market is highly heterogeneous regarding the trends exhibited by each cryptocurrency; however, only a small fraction of cryptocurrencies (10%) have power-law exponents neither correlated with age nor market capitalization  These associations have been mostly ignored by the current literature and are probably related to the still-early developmental stage of the cryptocurrency market as a whole 
Overall, 36% of cryptocurrencies present trends that do not systematically contribute to increasing or decreasing their power-law exponents as they age and grow in market capitalization  On the other hand, for 26% of cryptocurrencies, aging and growing market capitalization are both associated with a reduction in their power-law exponents, thus contributing to the rise in the frequency of large price variations in their dynamics 
Only about 28% of cryptocurrencies present trends in which the power-law exponents increase with age and market capitalization, favoring thus large price variations to become less likely  These results somehow juxtapose with findings about the increasing informational efficiency of the cryptocurrency market  
In fact, if on the one hand the cryptocurrency market is becoming more informationally efficient, then on the other our findings indicate that there is no clear trend toward decreasing the risks of sizable variations in the prices of most considered cryptoassets  In other words, risk and efficiency thus appear to be moving towards different directions in the cryptocurrency market 
To conclude, we hope that our findings will contribute significantly to the better understanding of the dynamics of large price variations in the cryptocurrency market as a whole, and not just for a small subset of selected digital assets, which is especially relevant due to the diminishing concentration of market capitalization among the top digital currencies, and also because of the considerable impact these new assets may have in our increasingly digital economy 
Our results are based on time series of the daily closing prices (in USD) for all cryptoassets listed on CoinMar-ketCap (coinmarketcap com) as of 25 July 2022 [see Supplementary Figure (a) for a visualization of the increasing number cryptoassets listed on CoinMarketCap since 2013]  These time series were automatically gathered using the cryptoCMD Python package and other information such as the tags associated with each cryptoasset were obtained via the CoinMarketCap API  
In addition, we have also obtained the daily market capitalization time series (in USD) from all cryptoassets which had this information available at the time  Earliest records available from CoinMarketCap date from 29 April 2013 and the latest records used in our analysis correspond to 25 July 2022  Out of 9943 cryptocurrencies, we have restricted our analysis to the 7111 with at least 200 price-return observations 
The median length of these time series is 446 observations [see the distribution of series length in Supplementary Figure   We have estimated the power-law behavior of the return distributions by applying the Clauset-Shalizi-Newman method to the return time series r t   In particular, we have sampled each of these time series using an expanding time window that starts at the hundredth observation and grows in weekly steps (seven data points each step) 
For each position of the expanding time window, we have separated the positive returns from the negative ones and applied the Clauset-Shalizi-Newman method to each set  This approach consists of obtaining the maximum likelihood estimate for the power-law exponent, α = 1 + n/ (∑ n t=1 ln r t /r min ) , where r min is the lower bound of the power-law regime and n is the number of (positive or negative) return observations in the power-law regime for a given position of the expanding time window 
The value r min is estimated from data by minimizing the Kolmogorov-Smirnov statistic between the empirical distribution and the power-law model  The Clauset-Shalizi-Newman method yields an unbiased and consistent estimator , in a sense that as the sample increases indefinitely, the estimated power-law exponent converges in distribution to the actual value 
Moreover, we have used the implementation available on the powerlaw Python package   In addition to obtaining the power-law exponents, we have also verified the adequacy of the power-law hypothesis using the procedure originally proposed by Clauset et al  as adapted by Preis et al    This procedure consists of generating synthetic samples under the power-law hypothesis with the same properties of the empirical data under analysis (that is, same length and parameters α and r min ), adjusting the simulated data with the power-law model via the Clauset-Shalizi-Newman method, and calculating the Kolmogorov-Smirnov statistic (κ syn ) between the distributions obtained from the simulated samples and the adjusted power-law model 
Next, the values of κ syn are compared to the Kolmogorov-Smirnov statistic calculated between empirical data and the power-law model (κ)  Finally, a p-value is defined by calculating the fraction of times for which κ syn > κ  We have used one thousand synthetic samples for each position of the expanding time window and the more conservative 90% confidence level (instead of the more lenient and commonly used 95% confidence level), such that the power-law hypothesis is rejected whenever p-value ≤ 0 1 
We have estimated the effects of age and market capitalization on the power-law exponents associated with positive or negative returns of a given cryptocurrency using the linear model where α t represents the power-law exponent, log c t is the logarithm of the market capitalization, and y t is the age (in years) of the cryptocurrency at t-th observation 
Moreover, K is the intercept of the association, while C and A are linear coefficients quantifying the effects of market capitalization and age, respectively  Finally, N (µ, σ ) stands for the normal distribution with mean µ and standard deviation σ , such that the parameter ε accounts for the unobserved determinants in the dynamics of the power-law exponents 
We have framed this problem using the hierarchical Bayesian approach such that each power-law exponent α t is nested within a cryptocurrency with model parameters considered as random variables normally distributed with parameters that are also random variables  Mathematically, for each cryptocurrency, we have
12/16 where µ K , σ K , µ C , σ C , µ A , and σ A are hyperparameters  These hyperparameters are assumed to be distributed according to distributions that quantify the overall impact of age and market capitalization on the cryptocurrency market as a whole  We have performed this Bayesian regression for exponents related to positive and negative returns separately, and used noninformative prior and hyperprior distributions in order not to bias the posterior estimation  
Specifically, we have considered and ε ∼ U (0, 10 2 ) , where U (a, b) stands for the uniform distribution in the interval [a, b] and Inv−Γ(θ , γ) represents the inverse gamma distribution with shape and scale parameters θ and γ, respectively  For the numerical implementation, we have relied on the PyMC Python package and sampled the posterior distributions via the gradient-based Hamiltonian Monte Carlo no-U-Turn-sampler method 
We have run four parallel chains with 2500 iterations each (1000 burn-in samples) to allow good mixing and estimated the Gelman-Rubin convergence statistic (R-hat) to ensure the convergence of the sampling approach (R-hat was always close to one)  In addition, we have also verified that models describing the power-law exponents as a function of only age (C → 0 in Eq  3) or only market capitalization (A → 0 in Eq  3) yield significantly worse descriptions of our data as quantified by the Widely Applicable Information Criterion (WAIC) and the Pareto Smoothed Importance Sampling Leave-One-Out cross-validation (PSIS-LOO) (see Supplementary Table )  ) is larger than r 90 estimated from negative returns (r − 90 ) 
This fraction is calculated only for weeks in which the power-law hypothesis is not rejected for both tails  The percentage of cryptoassets for which r + 90 > r − 90 is shown in the panels  The first column of panels depicts the results when considering data from all cryptocurrencies, while the second and third columns present the results for the top 2000 and top 200 cryptocurrencies by market capitalization, respectively 
Sampling issues refer to missing data and problems caused by prices of cryptoassets decreasing to zero  We note that these distributions barely change when considering only cryptocurrencies without any sampling issue  Indeed, the distributions in this figure are not significantly distinguishable from their counterparts in Fig  (two-sample Kolmogorov-Smirnov test, p > 0 05) 
)  Each of the previous three levels is further classified regarding whether both positive and negative returns are simultaneously affected or whether the effect involves only positive or only negative returns  Finally, the former levels are classified regarding whether the power-law exponents increase, decrease or have a mixed trend with the predictive variables 
Overall, 35% of the associations are classified as mixed trends (green rectangles), 37% are increasing trends (blue rectangles), and 18% are decreasing trends (red rectangles) 
Context 2: \section{Introduction}\label{sec1}
\setcounter{equation}{0} 

Transport problems with highly forward-peaked scattering are prevalent in a variety of areas, including astrophysics, medical physics, and plasma physics \cite{HGK,aristova,multiphysics} 
For these problems, solutions of the transport equation converge slowly when using conventional methods such as source iteration (SI) \cite{adamslarsen} and the generalized minimal residual method (GMRES) \cite{gmres} 
Moreover, diffusion-based acceleration techniques like diffusion synthetic acceleration (DSA) \cite{alcouffe} and nonlinear diffusion acceleration (NDA) \cite{smithetall} are generally inefficient when tackling these problems, as they only accelerate up to the first moment of the angular flux \cite{JapanFPSA} 
In fact, higher-order moments carry important information in problems with highly forward-peaked scattering and can be used to further accelerate convergence \cite{japanDiss} 

This paper focuses on solution methods for the monoenergetic, steady-state transport equation in homogeneous slab geometry 
Under these conditions, the transport equation is given by
\begin{subequations}\label[pluraleq]{eq1}
\begin{equation}
\label{t1}
\mu\frac{\partial}{\partial x} \psi(x,\mu) + \sigma_t \psi(x,\mu) = \int_{-1}^{1} d\mu' \sigma_s(\mu,\mu') \psi(x,\mu') + Q(x, \mu), \,\,\, x\in [0, X],-1\leq\mu\leq 1  ,\\
\end{equation}
with boundary conditions
\begin{align}
\label{t2}
\psi(0,\mu) &= \psi_L(\mu), \quad \mu > 0,\\
\label{t3}
\psi(X,\mu) &= \psi_R(\mu), \quad \mu < 0 
\end{align}
\end{subequations}
Here, $\psi(x,\mu)$ represents the angular flux at position $x$ and direction $\mu$, $\sigma_t$ is the macroscopic total cross section, $\sigma_s(\mu,\mu')$ is the differential scattering cross section, and $Q$ is an internal source 

New innovations have paved the way to better solve this equation in systems with highly forward-peaked scattering 
For instance, work has been done on modified $P_L$ equations and modified scattering cross section moments to accelerate convergence of anisotropic neutron transport problems \cite{khattab} 
In order to speed up the convergence of radiative transfer in clouds, a quasi-diffusion method has been developed \cite{aristova} 
In addition, the DSA-multigrid method was developed to solve problems in electron transport more efficiently \cite{trucksin} 

One of the most recent convergence methods developed is Fokker-Planck Synthetic Acceleration (FPSA) \cite{JapanFPSA,japanDiss} 
FPSA accelerates up to $N$ moments of the angular flux and has shown significant improvement in the convergence rate for the types of problems described above 
The method returns a speed-up of several orders of magnitude with respect to wall-clock time when compared to DSA  \cite{JapanFPSA} 

In this paper, we introduce a new acceleration technique, called \textit{Nonlinear Fokker-Planck Acceleration} (NFPA) 
This  method  returns  a  modified  Fokker-Planck (FP) equation  that  preserves  the  angular moments of the flux given by the transport  equation 
This preservation of moments is particularly appealing for applications to multiphysics problems \cite{multiphysics}, in which the coupling between the transport physics and the other physics can be done through the (lower-order) FP equation 
To our knowledge, this is the first implementation of a numerical method that returns a Fokker-Planck-like equation that is discretely consistent with the linear Boltzmann equation 

This paper is organized as follows 
\Cref{sec2} starts with a brief description of FPSA 
Then, we derive the NFPA scheme 
In \cref{sec3}, we discuss the discretization schemes used in this work and present numerical results 
These are compared against standard acceleration techniques 
We conclude with a discussion in \cref{sec4} 

\section{Fokker-Planck Acceleration}\label{sec2}
\setcounter{equation}{0} 
In this section we briefly outline the theory behind FPSA, describe NFPA for monoenergetic, steady-state transport problems in slab geometry, and present the numerical methodology behind NFPA 
The theory given here can be easily extended to higher-dimensional problems 
Moreover, extending the method to energy-dependence shall not lead to significant additional theoretical difficulties 

To solve the transport problem given by \cref{eq1} we approximate the in-scattering term in \cref{t1} with a Legendre moment expansion:
\begin{equation}
\label{transport1}
\mu\frac{\partial}{\partial x} \psi(x,\mu) + \sigma_t \psi(x,\mu) = \sum_{l=0}^L \frac{(2l+1)}{2} P_l(\mu) \sigma_{s,l} \phi_l(x) + Q(x, \mu),
\end{equation}
with 
\begin{equation}
\label{transport2}
\phi_l(x) =  \int_{-1}^{1} d\mu P_l(\mu) \psi(x,\mu) 
\end{equation}
Here, $\phi_l$ is the $l^{th}$ Legendre moment of the angular flux, $ \sigma_{s,l}$ is the $l^{th}$ Legendre coefficient of the differential scattering cross section,  and $P_l$ is the $l^{th}$-order Legendre polynomial 
For simplicity, we will drop the notation $(x,\mu)$ in the remainder of this section 

The solution to \cref{transport1} converges asymptotically to the solution of the following Fokker-Planck equation in the forward-peaked limit \cite{pomraning1}:
\begin{equation}
\label{fp1}
\mu\frac{\partial \psi}{\partial x} + \sigma_a \psi = \frac{\sigma_{tr}}{2}\frac{\partial }{\partial \mu} (1-\mu^2) \frac{\partial \psi}{\partial \mu} + Q\,,
\end{equation}
where $\sigma_{tr}= \sigma_{s,0} -\sigma_{s,1}$ is the momentum transfer cross section and $\sigma_a = \sigma_t-\sigma_{s,0}$ is the macroscopic absorption cross section 

Source Iteration \cite{adamslarsen} is generally used to solve \cref{transport1}, which can be rewritten in operator notation:
\begin{equation}
\label{si1}
\mathcal{L} \psi^{m+1} = \mathcal{S} \psi^{m} + Q\,,
\end{equation}
where 
\begin{equation}
\mathcal{L} = \mu \frac{\partial}{\partial x} + \sigma_t,
   \quad
\mathcal{S} = \sum_{l=0}^L \frac{(2l+1)}{2} P_l(\mu) \sigma_{s,l} \int_{-1}^{1}d\mu P_l(\mu) ,
\label{trans1}
\end{equation}
and $m$ is the iteration index 
This equation is solved iteratively until a tolerance criterion is met  The FP approximation shown in \cref{fp1} can be used to accelerate the convergence of \cref{transport1} 

\subsection{FPSA: Fokker-Planck Synthetic Acceleration}\label{FPSA}

In the FPSA scheme \cite{JapanFPSA,japanDiss}, the FP approximation is used as a preconditioner to synthetically accelerate convergence when solving \cref{transport1} (cf  \cite{adamslarsen} for a detailed description of synthetic acceleration) 
When solving \cref{si1}, the angular flux at each iteration $m$ has an error associated with it 
FPSA systematically follows a predict, correct, iterate scheme 
A transport sweep, one iteration in \cref{si1}, is made for a prediction 
The FP approximation is used to correct the error in the prediction, and this iteration is performed until a convergence criterion is met 
The equations used are:
\begin{subequations}
\label{fpsaeq}
\begin{align}
\label{predict}
\mathrm{Predict}&: \mathcal{L} \psi^{m+\frac{1}{2}} = \mathcal{S} \psi^{m} + Q\,,\\
\label{correct}
\mathrm{Correct}&: \psi^{m+1} =  \psi^{m+\frac{1}{2}} + \mathcal{P}^{-1} \mathcal{S} \left( \psi^{m+\frac{1}{2}} -  \psi^{m}\right),
\end{align}
\end{subequations}
where we define $\mathcal{P}$ as
\begin{equation}
\label{FPSAsi1}
\mathcal{P} = \mathcal{A}-\mathcal{F} =\underbrace{\left(\mu\frac{\partial}{\partial x} + \sigma_a\right)}_\mathcal{A} - \underbrace{\left(\frac{\sigma_{tr}}{2}\frac{\partial }{\partial \mu} (1-\mu^2) \frac{\partial }{\partial \mu}\right)}_\mathcal{F},
\end{equation}
In this synthetic acceleration method, the FP approximation is used to correct the error in each iteration of the high-order (HO) equation (\ref{predict})  
Therefore, there is no consistency between the angular moments of the flux in the HO and low-order (LO) equations 

\subsection{NFPA: Nonlinear Fokker-Planck Acceleration}\label{NFPA}

Similar to FPSA, NFPA uses the FP approximation to accelerate the convergence of the solution 
We introduce the additive term $\hat{D}_F$ to \cref{fp1}, obtaining the modified FP equation
\begin{equation}
\label{mfp1}
\mu\frac{\partial \psi}{\partial x} + \sigma_a \psi = \frac{\sigma_{tr}}{2}\frac{\partial }{\partial \mu} (1-\mu^2) \frac{\partial \psi}{\partial \mu} + \hat{D}_F + Q\, 
\end{equation}
The role of $\hat{D}_F$ is to force the transport and modified FP equations to be consistent 
Subtracting \cref{mfp1} from \cref{transport1} and rearranging, we obtain the consistency term
\begin{equation}
\label{dfp}
\hat{D}_F = \sum_{l=0}^L \frac{(2l+1)}{2} P_l \sigma_l \phi_l - \frac{\sigma_{tr}}{2}\frac{\partial}{\partial \mu} (1-\mu^2) \frac{\partial \psi}{\partial \mu} - \sigma_{s,0} \psi\, 
\end{equation}

The NFPA method is given by the following equations:
\begin{subequations}\label[pluraleq]{holocons}
\begin{align}
\label{HO1}
\text{HO}&: \mu\frac{\partial \psi_{HO}}{\partial x} + \sigma_t \psi_{HO} = \sum_{l=0}^L \frac{(2l+1)}{2} P_l \sigma_l \phi_{l, LO} + Q\,,\\
\label{LO11}
\text{LO}&: \mu\frac{\partial \psi_{LO}}{\partial x} + \sigma_a \psi_{LO} = \frac{\sigma_{tr}}{2}\frac{\partial }{\partial \mu} (1-\mu^2) \frac{\partial \psi_{LO}}{\partial \mu} + \hat{D}_F + Q\,,\\
\label{con1}
\text{Consistency term}&: \hat{D}_F = \sum_{l=0}^L \frac{(2l+1)}{2} P_l \sigma_l \phi_{l, HO}^m - \frac{\sigma_{tr}}{2}\frac{\partial }{\partial \mu} (1-\mu^2) \frac{\partial \psi_{HO}}{\partial \mu} - \sigma_{s,0} \psi_{HO}\,,
\end{align}
\end{subequations}
where $\psi_{HO}$ is the angular flux obtained from the HO equation and $\psi_{LO}$ is the angular flux obtained from the LO equation 
The nonlinear HOLO-plus-consistency system given by \cref{holocons} can be solved using any nonlinear solution technique \cite{kelley}  Note that the NFPA scheme returns a FP equation that is consistent with HO transport  
Moreover, this modified FP equation accounts for large-angle scattering which the standard FP equation does not  
The LO equation (\ref{fp1}) can then be integrated into multiphysics models in a similar fashion to standard HOLO schemes \cite{patelFBR}  To solve the HOLO-plus-consistency system above, we use Picard iteration \cite{kelley}:
\begin{subequations}
\begin{align}
\label{H1}
\text{Transport Sweep for HO}&:
\mathcal{L} \psi_{HO}^{k+1} = \mathcal{S} \psi_{LO}^{k} + Q, \\
\label{L1}
\text{Evaluate Consistency Term}&: \hat{D}_F^{k+1} = \left(\mathcal{S} - \mathcal{F} - \sigma_{s,0}\mathcal{I}\right) \psi_{HO}^{k+1}, \\
\label{c1}
\text{Solve LO Equation}&: \psi_{LO}^{k+1} = \mathcal{P}^{-1} \left(\hat{D}_F^{k+1} + Q\right), 
\end{align}
\end{subequations}
where $\mathcal{L}$ and $\mathcal{S}$ are given in \cref{trans1}, $\mathcal{P}$ and $\mathcal{F}$ are given in \cref{FPSAsi1}, $\mathcal{I}$ is the identity operator, and $k$ is the iteration index 
Iteration is done until a convergence criterion is met 

The main advantage of setting up the LO equation in this fashion is that the stiffness matrix for LO needs to be setup and inverted \textit{only once}, just as with FPSA \cite{JapanFPSA, japanDiss}  This has a large impact on the method's performance 
A flowchart of this algorithm is shown in \cref{Nalgorithm} 

\begin{figure}[H]
\centering
\begin{tikzpicture}[node distance = 3cm, auto]
   
    \node [block] (init) {Initial guess of flux moments};
    \node [cloud_HO, right of=init, node distance=4cm] (HOm) {HO};
    \node [cloud_LO, below of=HOm, node distance=2cm] (LOm) {LO};
    \node [HO, below of=init] (transport) {One sweep in transport equation};
    \node [decision, below of=transport,node distance=4cm] (decide) {Flux moments converged?};
    \node [LO, left of=decide, node distance=4cm] (dterm) {Solve for consistency term};
    \node [LO, left of=dterm, node distance=3cm] (MFP) {Solve for FP angular flux};
    \node [LO, above of=MFP, node distance=4cm] (moments) {Convert angular flux to moments};
    \node [block, right of=decide, node distance=4cm] (stop) {Stop};
   
    \path [line] (init) -- (transport);
    \path [line] (transport) -- (decide);
    \path [line] (decide) -- node {no} (dterm);
    \path [line] (dterm) -- (MFP);
    \path [line] (MFP) -- (moments);
    \path [line] (moments) -- (transport);
    \path [line] (decide) -- node {yes}(stop);
\end{tikzpicture}
\caption{NFPA algorithm}
\label{Nalgorithm}
\end{figure}

\section{Numerical Experiments}\label{sec3}

In \cref{sec31} we describe the discretization methods used to implement the algorithms 
In \cref{sec32} we provide numerical results for 2 different choices of source $Q$ and boundary conditions 
For each choice we solve the problem using 3 different scattering kernels, applying 3 different choices of parameters for each kernel 
We provide NFPA numerical results for these 18 cases and compare them against those obtained from FPSA and other standard methods 

All numerical experiments were performed using MATLAB 
Runtime was tracked using the tic-toc functionality \cite{matlab17}, with
only the solver runtime being taken into consideration in the comparisons 
A 2017 MacBook Pro with a 2 8 GHz Quad-Core Intel Core i7 and 16 GB of RAM was used for all simulations 


\subsection{Discretization}\label{sec31}

The Transport and FP equations were discretized using linear discontinuous finite element discretization in space \cite{mpd1}, and discrete ordinates (S$_N$) in angle \cite{landm} 
The Fokker-Planck operator $\mathcal{F}$ was discretized using moment preserving discretization (MPD) \cite{mpd1} 
Details of the derivation of the linear discontinuous finite element discretization can be seen in \cite{japanDiss,martin} 
The finite element discretization for the  Fokker-Planck equation follows the same derivation 

A brief review for the angular discretization used for the FP equation is given below 
First, we use Gauss-Legendre quadrature to discretize the FP equation in angle:
\begin{equation}
\mu_n\frac{\partial \psi_n(x)}{\partial x} + \sigma_a \psi_n(x) - \frac{\sigma_{tr}}{2}\nabla^2_n \psi_n(x) =  Q_n(x),
\end{equation}
for $n=1,  ,N$ 
Here, $\nabla^2_n$ term is the discrete form of the angular Laplacian operator evaluated at angle $n$ 

The MPD scheme is then shown as
\begin{equation}
\nabla^2_n \psi_n = M \psi_n = V^{-1} L V \psi_n,
\end{equation}
where $M$ is the MPD discretized operator defined by
\begin{subequations}
\begin{equation}
V_{i,j} = P_{i-1}(\mu_j)w_j,
\end{equation}
and 
\begin{equation}
L_{i,j} = -i(i-1),
\end{equation}
\end{subequations}
for $i,j=1,   ,N$ 
Here, $P_l(\mu_j)$ are the Legendre polynomials evaluated at each angle $\mu_j$ and $w_j$ are the respective weights 
$M$ is defined as a (N x N) operator for a vector of $N$ angular fluxes $ \psi(x)$, at spatial location $x$  

In summary, if we write the FP equation as
\begin{equation}
\mathcal{H} \frac{\partial \psi}{\partial x}(x) + \sigma_a \psi(x) - \mathcal{F} \psi(x) =  Q(x),
\end{equation}
then $\mathcal{H}$ is Diag$(\mu_n)$ for $n=1,   ,N$, $Q(x)$ is a vector of source terms $Q_n(x)$, and $\mathcal{F}$ is represented by $\frac{\sigma_{tr}}{2}M$ 


\subsection{Numerical Results}\label{sec32}

It is shown that for slowly converging problems, typical convergence methods like $L_\infty$ suffer from false convergence \cite{adamslarsen} 
To work around this issue, the criterion is modified to use information about the current and previous iteration:
\begin{equation}
\label{falseconverge}
\frac{|| \phi^{m}_0(x) - \phi^{m-1}_0(x) ||_2}{1-\frac{|| \phi^{m+1}_0(x) - \phi^{m}_0(x) ||_2}{|| \phi^{m}_0(x) - \phi^{m-1}_0(x) ||_2}} < 10^{-8} 
\end{equation}

Two problems were tested using 200 spatial cells, $X$ = 400, $\sigma_a = 0$, $L$ = 15, and $N$ = 16 
Problem 1 has vacuum boundaries and a homogeneous isotropic source $Q$ for $0 < x < X$ 
Problem 2 has no internal source and an incoming beam at the left boundary  The source and boundary conditions used are shown in \cref{parameters}  
\begin{table}[H]
\begin{center}
\scalebox{0 9}{
\begin{tabular}{c | c | c} \hline 
& Problem 1 & Problem 2 \\ \hline \hline
Q(x) & 0 5 & 0 \\
$\psi_L$ & 0 & $\delta(\mu - \mu_N)$  \\
$\psi_R$ & 0 & 0 \\
\end{tabular}}
\end{center}
\caption{Problem Parameters}
\label{parameters} 
\end{table} 
We consider three scattering kernels in this paper: Screened Rutherford \cite{pomraning1}, Exponential \cite{pomraning2}, and Henyey-Greenstein \cite{HGK} 
Three cases for each kernel were tested 
The results obtained with NFPA are compared with those obtained using GMRES, DSA, and FPSA with the MPD scheme 

\subsubsection{SRK: Screened Rutherford Kernel}

The Screened Rutherford Kernel \cite{pomraning1, JapanFPSA} is a widely used scattering kernel for modeling scattering behavior of electrons \cite{SRK} 
The kernel depends on the parameter $\eta$, such that
\begin{equation}
\sigma^{SRK}_{s,l} = \sigma_s  \int_{-1}^{1} d\mu P_l(\mu) \frac{\eta (\eta+1)}{(1+2\eta-\mu)^2} 
\end{equation}
The SRK has a valid FP limit as $\eta$ approaches 0 \cite{patelFBR}  Three different values of $\eta$ were used to generate the scattering kernels shown in \cref{SRK} 
GMRES, DSA, FPSA, and NFPA all converged to the same solution for problems 1 and 2  \Cref{SRK_plots} shows the solutions for SRK with $\eta = 10^{-7}$ 
\begin{figure}[t]
\begin{center}
  \includegraphics[scale=0 1,angle=0]{SRK jpg}
  \caption{Screened Rutherford Kernels}
  \label{SRK}
\end{center}
\end{figure}
\begin{figure}[H]
    \centering
    \subfloat[Problem 1]{{\includegraphics[width=7cm]{s7_iso jpg} }}
    \qquad
    \subfloat[Problem 2]{{\includegraphics[width=7cm]{s7_beam jpg} }}
    \caption{Results for SRK Problems with $\eta = 10^{-7}$}
    \label{SRK_plots}
\end{figure}

\begin{table}[H]
\begin{center}
\scalebox{0 8}{
\begin{tabular}{c || c || c || c} \hline 
Parameter & Solver & Runtime (s) & Iterations \\ \hline \hline
\multirow{4}{*}{$\eta = 10^{-5}$} & GMRES & 98 8 & 12 \\
& DSA & 2380 & 53585 \\
& FPSA & 1 21 & 26 \\
& NFPA & 1 39 & 26 \\ \hline 
\multirow{4}{*}{$\eta = 10^{-6}$} & GMRES & 208 & 84 \\
& DSA & 3040 & 69156 \\
& FPSA & 0 747 & 16 \\
& NFPA & 0 857 & 16 \\ \hline 
\multirow{4}{*}{$\eta = 10^{-7}$} & GMRES & 174 & 124 \\
& DSA & 3270 & 73940 \\
& FPSA & 0 475 & 10 \\
& NFPA & 0 542 & 10 \\ \hline
\end{tabular}}
\end{center}
\caption{Runtime and Iteration Counts for Problem 1 with SRK}
\label{SRKresults1} 
\end{table}
\begin{table}[H]
\begin{center}
\scalebox{0 8}{
\begin{tabular}{c || c || c || c} \hline 
Parameter & Solver & Runtime (s) & Iterations \\ \hline \hline
\multirow{4}{*}{$\eta = 10^{-5}$} & GMRES & 52 4 & 187 \\
& DSA & 1107 & 25072 \\
& FPSA & 0 953 & 20 \\
& NFPA & 1 14 & 20 \\ \hline 
\multirow{4}{*}{$\eta = 10^{-6}$} & GMRES & 108 & 71 \\
& DSA & 1434 & 32562  \\
& FPSA & 0 730 & 14 \\
& NFPA & 0 857 & 14 \\ \hline 
\multirow{4}{*}{$\eta = 10^{-7}$} & GMRES & 94 1 & 185 \\
& DSA & 1470 & 33246 \\
& FPSA & 0 438 & 8 \\
& NFPA & 0 484 & 8 \\ \hline  
\end{tabular}}
\end{center}
\caption{Runtime and Iteration Counts for Problem 2 with SRK}
\label{SRKresults2} 
\end{table}

The results of all solvers are shown in \cref{SRKresults1,SRKresults2} 
We see that NFPA and FPSA tremendously outperform GMRES and DSA in runtime for all cases 
FPSA is a simpler method than NFPA, requiring less calculations per iteration; therefore, it is expected that it outperforms NFPA in runtime 
We see a reduction in runtime and iterations for FPSA and NFPA as the FP limit is approached, with DSA and GMRES requiring many more iterations by comparison as $\eta$ approaches 0 

An advantage that NFPA offers is that the angular moments of the flux in the LO equation will remain consistent with those of the transport equation even as a problem becomes less forward-peaked 
On the other hand, the moments found using only the FP equation and source iteration lose accuracy 
To illustrate this, Problem 1 was tested using different Screened Rutherford Kernels with increasing $\eta$ parameters 
The percent errors (relative to the transport solution) for the scalar flux obtained with the LO equation and with the standard FP equation at the center of the slab are shown in \cref{momcomp} 
It can be seen that the percent relative errors in the scalar flux of the FP solution is orders of magnitude larger than the error produced using the LO equation 
The same trend can be seen when using the exponential and Henyey-Greenstein kernels  

\begin{figure}[H]
\begin{center}
  \includegraphics[scale=0 15,angle=0]{relerrorlog jpg}
  \caption{Log Scale of $\%$ Relative Error vs $\eta$ for Problem 1 at the Center of the Slab with SRK}
  \label{momcomp}
\end{center}
\end{figure}

\subsubsection{EK: Exponential Kernel}

The exponential kernel \cite{pomraning2, JapanFPSA} is a fictitious kernel made for problems that have a valid Fokker-Planck limit \cite{pomraning1} 
The zero$^{\text{th}}$ moment, $\sigma^{EK}_{s,0}$, is chosen arbitrarily; we define $\sigma^{EK}_{s,0}$ as the same zero$^{\text{th}}$ moment from the SRK 
The $\Delta$ parameter determines the kernel: the first and second moments are given by 
\begin{subequations}
\begin{align}
\sigma^{EK}_{s,1} &= \sigma^{EK}_{s,0} (1-\Delta),\\
\sigma^{EK}_{s,2} &= \sigma^{EK}_{s,0} (1-3\Delta+3\Delta^2),
\end{align}
and the relationship for $l\geq 3$ is
\begin{equation}
\sigma^{EK}_{s,l} = \sigma^{EK}_{s,l-2} - \Delta(2l+1) \sigma^{EK}_{s,l-1} 
\end{equation}
\end{subequations}
As $\Delta$ is reduced, the scattering kernel becomes more forward-peaked 

The EK has a valid FP limit as $\Delta$ approaches 0 \cite{patelFBR} 
Three different values of $\Delta$ were used to generate the scattering kernels shown in \cref{EXP} 
The generated scattering kernels are shown in \cref{EXP} 
GMRES, DSA, FPSA, and NFPA all converged to the same solution for problems 1 and 2 
\Cref{EK_plots} shows the solutions for EK with $\Delta = 10^{-7}$ 
\begin{figure}[t]
\begin{center}
  \includegraphics[scale=0 1,angle=0]{EXP jpg}
  \caption{Exponential Kernels}
  \label{EXP}
\end{center}
\end{figure}
\begin{figure}[H]
    \centering
    \subfloat[Problem 1]{{\includegraphics[width=7cm]{dta7_iso jpg} }}
    \qquad
    \subfloat[Problem 2]{{\includegraphics[width=7cm]{dta7_beam jpg} }}
    \caption{Results for EK Problems with $\Delta = 10^{-7}$}
    \label{EK_plots}
\end{figure}

The runtimes and iterations for GMRES, DSA, FPSA, and NFPA are shown in \cref{Expresults1,Expresults2} 
We see a similar trend with the EK as seen with SRK 
Smaller $\Delta$ values lead to a reduction in runtime and iterations for NFPA and FPSA, which greatly outperform DSA and GMRES in both categories 

\begin{table}[h]
\begin{center}
\scalebox{0 8}{
\begin{tabular}{c || c || c || c} \hline 
Parameter & Solver & Runtime (s) & Iterations \\ \hline \hline
\multirow{4}{*}{$\Delta = 10^{-5}$} & GMRES & 196 & 142 \\
& DSA & 3110 & 70140 \\
& FPSA & 0 514 & 11 \\ 
& NFPA & 0 630 & 11 \\\hline 
\multirow{4}{*}{$\Delta = 10^{-6}$} & GMRES & 156 & 132 \\
& DSA & 3120 & 70758 \\
& FPSA & 0 388 & 7 \\ 
& NFPA & 0 393 & 7 \\ \hline 
\multirow{4}{*}{$\Delta = 10^{-7}$} & GMRES & 81 & 127 \\
& DSA & 3120 & 70851  \\
& FPSA & 0 292 & 6 \\ 
& NFPA & 0 318 & 6 \\ \hline
\end{tabular}}
\end{center}
\caption{Runtime and Iteration Counts for Problem 1 with EK}
\label{Expresults1} 
\end{table}
\begin{table}[h]
\begin{center}
\scalebox{0 8}{
\begin{tabular}{c || c || c || c} \hline 
Parameter & Solver & Runtime (s) & Iterations \\ \hline \hline
\multirow{4}{*}{$\Delta = 10^{-5}$} & GMRES & 110 & 73 \\
& DSA & 1455 & 33033 \\
& FPSA & 0 492 & 10 \\ 
& NFPA & 0 613 & 10 \\ \hline 
\multirow{4}{*}{$\Delta = 10^{-6}$} & GMRES & 82 7 & 79 \\
& DSA & 1470 & 33309 \\
& FPSA & 0 358 & 7 \\ 
& NFPA & 0 431 & 7 \\ \hline 
\multirow{4}{*}{$\Delta = 10^{-7}$} & GMRES & 56 8 & 90 \\
& DSA & 1470 & 33339 \\
& FPSA & 0 273 & 5 \\ 
& NFPA & 0 319 & 5 \\ \hline  
\end{tabular}}
\end{center}
\caption{Runtime and Iteration Counts for Problem 2 with EK}
\label{Expresults2} 
\end{table}

\subsubsection{HGK: Henyey-Greenstein Kernel}

The Henyey-Greenstein Kernel \cite{HGK,JapanFPSA} is most commonly used in light transport in clouds 
It relies on the anisotropy factor $g$, such that
\begin{equation}
\sigma^{HGK}_{s,l} = \sigma_s g^l 
\end{equation}
As $g$ goes from zero to unity, the scattering shifts from isotropic to highly anisotropic 
\begin{figure}[H]
\begin{center}
  \includegraphics[scale=0 1,angle=0]{HGK jpg}
  \caption{Henyey-Greenstein Kernels}
  \label{HGK}
\end{center}
\end{figure}
\begin{figure}[H]
    \centering
    \subfloat[Problem 1]{{\includegraphics[width=7cm]{g099_iso jpg} }}
    \qquad
    \subfloat[Problem 2]{{\includegraphics[width=7cm]{g099_beam jpg} }}
    \caption{Results for HGK Problems with $g = 0 99$}
    \label{HGK_plots}
\end{figure}


The HGK does not have a valid FP limit \cite{patelFBR} 
The three kernels tested are shown in \cref{HGK} 
GMRES, DSA, FPSA, and NFPA all converged to the same solution for problems 1 and 2 
\Cref{HGK_plots} shows the solutions for HGK with $g = 0 99$ 
The results of each solver are shown in \cref{HGKresults1,HGKresults2}  
\begin{table}[h]
\begin{center}
\scalebox{0 8}{
\begin{tabular}{c || c || c || c} \hline 
Parameter & Solver & Runtime (s) & Iterations \\ \hline \hline
\multirow{4}{*}{$g=0 9$} & GMRES & 9 88 & 76 \\
& DSA & 24 5 & 554 \\
& FPSA & 1 50 & 32 \\ 
& NFPA & 1 39 & 27 \\ \hline 
\multirow{4}{*}{$g=0 95$} & GMRES & 12 2 & 131 \\
& DSA & 47 7 & 1083 \\
& FPSA & 1 75 & 38 \\ 
& NFPA & 1 83 & 35 \\ \hline 
\multirow{4}{*}{$g=0 99$} & GMRES & 40 0 & 27 \\
& DSA & 243 & 5530  \\
& FPSA & 3 38 & 74 \\ 
& NFPA & 3 93 & 73 \\ \hline
\end{tabular}}
\end{center}
\caption{Runtime and Iteration Counts for Problem 1 with HGK}
\label{HGKresults1} 
\end{table}
\begin{table}[h]
\begin{center}
\scalebox{0 8}{
\begin{tabular}{c || c || c || c} \hline 
Parameter & Solver & Runtime (s) & Iterations \\ \hline \hline
\multirow{4}{*}{$g=0 9$} & GMRES & 24 3 & 135 \\
& DSA & 14 8 & 336  \\
& FPSA & 1 15 & 23 \\ 
& NFPA & 1 35 & 24 \\ \hline 
\multirow{4}{*}{$g=0 95$} & GMRES & 31 3 & 107 \\
& DSA & 29 7 & 675 \\
& FPSA & 1 56 & 32 \\ 
& NFPA & 1 90 & 33 \\ \hline 
\multirow{4}{*}{$g=0 99$} & GMRES & 41 4 & 126 \\
& DSA & 146 & 3345 \\
& FPSA & 3 31 & 67 \\ 
& NFPA & 3 99 & 67 \\ \hline  
\end{tabular}}
\end{center}
\caption{Runtime and Iteration Counts for Problem 2 with HGK}
\label{HGKresults2} 
\end{table}

Here we see that NFPA and FPSA do not perform as well compared to their results for the SRK and EK 
Contrary to what happened in those cases, both solvers require more time and iterations as the problem becomes more anisotropic 
This is somewhat expected, due to HGK not having a valid Fokker-Planck limit 
However, both NFPA and FPSA continue to greatly outperform GMRES and DSA 
Moreover, NFPA outperforms FPSA in iteration count for problem 1 


\section{Discussion}\label{sec4}

This paper introduced the Nonlinear Fokker-Planck Acceleration technique for steady-state, monoenergetic transport in homogeneous slab geometry 
To our knowledge, this is the first nonlinear HOLO method that accelerates \textit{all $L$ moments} of the angular flux 
Upon convergence, the LO and HO models are consistent; in other words, the (lower-order) modified Fokker-Planck equation \textit{preserves the same angular moments} of the flux obtained with the (higher-order) transport equation 

NFPA was tested on a homogeneous medium with an isotropic internal source with vacuum boundaries, and in a homogeneous medium with no internal source and an incoming beam boundary 
For both problems, three different scattering kernels were used 
The runtime and iterations of NFPA and FPSA were shown to be similar 
They both vastly outperformed DSA and GMRES for all cases by orders of magnitude 
However, NFPA has the feature of preserving the angular moments of the flux in both the HO and LO equations, which offers the advantage of integrating the LO model into multiphysics models  

In the future, we intend to test NFPA capabilities for a variety of multiphysics problems and analyze its performance 
To apply NFPA to more realistic problems, it needs to be extended to include time and energy dependence  
Additionally, the method needs to be adapted to address geometries with higher-order spatial dimensions 
Finally, for the NFPA method to become mathematically ``complete", a full convergence examination using Fourier analysis must be performed 
However, this is beyond the scope of this paper and must be left for future work 

\section*{Acknowledgements}

The authors acknowledge support under award number NRC-HQ-84-15-G-0024 from the Nuclear Regulatory Commission 
The statements, findings, conclusions, and recommendations are those of the authors and do not necessarily reflect the view of the U S  Nuclear Regulatory Commission 

J ~K  Patel would like to thank Dr ~James Warsa for his wonderful transport class at UNM, as well as his synthetic acceleration codes 
The authors would also like to thank Dr ~Anil Prinja for discussions involving Fokker-Planck acceleration 




Context 3: Paper Info

Title: Bistability between π-diradical open-shell and closed-shell states in indeno[1,2-a]fluorene
Publish Date: Unkown
Author List: Shantanu Mishra (from IBM Research Europe -Zurich), Manuel Vilas-Varela (from Department of Organic Chemistry, Center for Research in Biological Chemistry and Molecular Materials (CiQUS), University of Santiago de Compostela), Leonard-Alexander Lieske (from IBM Research Europe -Zurich), Ricardo Ortiz (from Donostia International Physics Center (DIPC)), Igor Rončević (from Department of Chemistry, University of Oxford), Florian Albrecht (from IBM Research Europe -Zurich), Diego Peña (from Department of Organic Chemistry, Center for Research in Biological Chemistry and Molecular Materials (CiQUS), University of Santiago de Compostela), Leo Gross (from IBM Research Europe -Zurich)

Figure

Fig  1 | Non-benzenoid non-alternant polycyclic conjugated hydrocarbons a, Classical nonbenzenoid non-alternant polycyclic conjugated hydrocarbons: pentalene, azulene and heptalene b, Generation of indacenes and indenoindenes through benzinterposition and benzannelation of pentalene, respectively Gray filled rings represent Clar sextets c, Closed-shell Kekulé (left) and openshell non-Kekulé (right) resonance structures of QDMs Note that meta-QDM is a non-Kekulé molecule All indenofluorene isomers, being derived through benzannelation of indacenes, contain a central QDM moiety d, Closed-shell Kekulé (top) and open-shell non-Kekulé (bottom) resonance structures of indenofluorenes Compared to their closed-shell structures, 1 and 5 gain two Clar sextets in the openshell structure, while 2-4 gain only one Clar sextet in the open-shell structure Colored bonds in d highlight the ortho-and para-QDM moieties in the two closed-shell Kekulé structures of 5  e, Scheme of on-surface generation of 5 by voltage pulse-induced dehydrogenation of 6 (C20H14) Structures 7 and 8 represent the two monoradical species (C20H13) 
Fig  2 | Characterization of open-shell indeno[1,2-a]fluorene on bilayer NaCl/Au(111) a, DFTcalculated wave functions of the frontier orbitals of 5OS in the triplet configuration for the spin up (occupied) level (isovalue: 0 002 e -Å -3 ) Blue and red colors represent opposite phases of the wave function b, Corresponding DFT-calculated spin density of 5OS (isovalue: 0 01 e -Å -3) Blue and orange colors represent spin up and spin down densities, respectively c, Probability density of the SOMOs of 5OS (isovalue: 0 001 e -Å -3 ) d, DFT-calculated bond lengths of 5OS e, Constant-height I(V) spectra acquired on a species of 5 assigned as 5OS, along with the corresponding dI/dV(V) spectra Open feedback parameters: V = -2 V, I = 0 17 pA (negative bias side) and V = 2 V, I = 0 17 pA (positive bias side) Acquisition position of the spectra is shown in Supplementary Fig 7 f, Scheme of many-body transitions associated to the measured ionic resonances of 5OS Also shown are STM images of assigned 5OS at biases where the corresponding transitions become accessible Scanning parameters: I = 0 3 pA (V = -1 2V and -1 5 V) and 0 2 pA (V = 1 3 V and 1 6 V)  g, Laplace-filtered AFM image of assigned 5OS STM set point: V = 0 2 V, I = 0 5 pA on bilayer NaCl, Δz = -0 3Å The tip-height offset Δz for each panel is provided with respect to the STM setpoint, and positive (negative) values of Δz denote tip approach (retraction) from the STM setpoint f and g show the same molecule at the same adsorption site, which is next to a trilayer NaCl island The bright and dark features in the trilayer NaCl island in g correspond to Cl -and Na + ions, respectively Scale bars: 10 Å (f) and 5 Å (g) 
Fig  3 | Characterization of closed-shell indeno[1,2-a]fluorene on bilayer NaCl/Au(111) a, DFTcalculated wave functions of the frontier orbitals of closed-shell 5 0 (isovalue: 0 002 e -Å -3 ) The wave functions shown here are calculated for the 5para geometry b, DFT-calculated bond lengths of 5ortho (top) and 5para (bottom) c, Constant-height I(V) spectra acquired on a species of 5 assigned as 5para, along with the corresponding dI/dV(V) spectra Open feedback parameters: V = -2 V, I = 0 15 pA (negative bias side) and V = 2 2 V, I = 0 15 pA (positive bias side) Acquisition position of the spectra is shown in Supplementary Fig  7  d, Scheme of many-body transitions associated to the measured ionic resonances of 5para Also shown are STM images of assigned 5para at biases where the corresponding transitions become accessible Scanning parameters: I = 0 15 pA (V = -1 5 V) and 0 2 pA (V = 1 7 V)  e, Laplace-filtered AFM image of assigned 5para STM set point: V = 0 2 V, I = 0 5 pA on bilayer NaCl, Δz = -0 7 Å  f, Selected bonds labeled for highlighting bond order differences between 5para and 5ortho For the bond pairs a/b, c/d and e/f, the bonds labeled in bold exhibit a higher bond order than their neighboring labeled bonds in 5para g, Laplace-filtered AFM images of 5 on bilayer NaCl/Cu(111) showing switching between 5OS and 5para as the molecule changes its adsorption position The faint protrusion adjacent to 5 is a defect that stabilizes the adsorption of 5  STM set point: V = 0 2 V, I = 0 5 pA on bilayer NaCl, Δz = -0 3Å  STM and STS data in c and d are acquired on the same species, while the AFM data in e is acquired on a different species Scale bars: 10 Å (d) and 5 Å (e,g) 
NMR (300 MHz, CDCl3) δ: 7 51 (m, 2H), 7 40 -7 28 (m, 5H), 7 27 -7 20 (m, 2H), 7 13 (d, J = 7 7 Hz, 1H), 2 07 (s, 3H), 1 77 (s, 3H) ppm  13C NMR-DEPT (75 MHz, CDCl3, 1:1 mixture of atropisomers) δ: 141 2 (C), 141 1 (C), 140 0 (C), 139 4 (2C), 137 5 (C), 137 4 (C), 136 0 (3C), 134 8 (C), 134 5 (C), 134 1 (C), 134 0 (C), 133 7 (C), 133 6 (C), 131 6 (CH), 131 2 (CH), 131 1 (CH), 130 7 (CH), 129 8 (CH), 129 7 (CH), 129 5 (CH), 129 4 (CH), 129 0 (CH), 128 9 (CH), 128 7 (2CH), 128 6 (2CH), 127 2 (CH), 127 1 (CH), 127 0 (CH), 126 9 (CH), 126 7 (CH), 126 6 (CH), 20 6 (CH3), 20 5 (CH3), 17 7 (CH3), 17 5 (CH3) ppm MS (APCI) m/z (%): 327 (M+1, 100) HRMS: C20H16Cl2; calculated: 327 0702, found: 327 0709 
NMR (500 MHz, CDCl3) δ: 7 93 (d, J = 7 6 Hz, 1H), 7 85 (d, J = 7 5 Hz, 1H), 7 78 (d, J = 7 7 Hz, 1H), 7 65 (d, J = 7 4 Hz, 1H), 7 61 (d, J = 7 5 Hz, 1H), 7 59 (d, J = 7 7 Hz, 1H), 7 47 (ddd, J = 8 4, 7 2, 1 1 Hz, 1H), 7 42 (dd, J = 8 1, 7 0 Hz, 1H), 7 35 (m, 2H), 4 22 (s, 3H), 4 02 (s, 3H) ppm  13C NMR-DEPT (125 MHz, CDCl3) δ: 144 1 (C), 143 3 (C), 142



ground_truth: Power-law functions.

prompt_front: Now, answer the following question based on the text below, only give me the answer and do not output any other words.

Question: How do the runtimes and iteration counts of NFPA and FPSA compare to GMRES and DSA in the numerical experiments?

Read the following text and answer briefly.

Context 3: 3 (C), 141 9 (C), 141 8 (C), 141 2 (C), 138 2 (C), 136 5 (C), 127 0 (CH), 126 9 (CH), 126 7 (CH), 126 6 (CH), 125 3 (CH), 125 2 (CH), 123 6 (CH), 122 2 (CH), 119 9 (CH), 118 4 (CH), 37 4 (CH2), 36 3 (CH2) ppm MS (APCI) m/z (%): 254 (M+, 88) HRMS: C20H14; calculated: 254 1090, found: 254 1090 

abstract

Indenofluorenes are non-benzenoid conjugated hydrocarbons that have received great interest owing to their unusual electronic structure and potential applications in nonlinear optics and photovoltaics  Here, we report the generation of unsubstituted indeno[1,2-a]fluorene, the final and yet unreported parent indenofluorene regioisomer, on various surfaces by cleavage of two C-H bonds in 7,12-dihydro indeno[1,2-a]fluorene through voltage pulses applied by the tip of a combined scanning tunneling microscope and atomic force microscope 
On bilayer NaCl on Au(111), indeno[1,2a]fluorene is in the neutral charge state, while it exhibits charge bistability between neutral and anionic states on the lower work function surfaces of bilayer NaCl on Ag(111) and Cu(111)  In the neutral state, indeno[1,2-a]fluorene exhibits either of two ground states: an open-shell π-diradical state, predicted to be a triplet by density functional and multireference many-body perturbation theory calculations, or a closedshell state with a para-quinodimethane moiety in the as-indacene core 
Switching between open-and closed-shell states of a single molecule is observed by changing its adsorption site on NaCl  The inclusion of non-benzenoid carbocyclic rings is a viable route to tune the physicochemical properties of polycyclic conjugated hydrocarbons (PCHs)   Non-benzenoid polycycles may lead to local changes in strain, conjugation, aromaticity, and, relevant to the context of the present work, induce an open-shell ground state of the corresponding PCHs  
Many nonbenzenoid PCHs are also non-alternant, where the presence of odd-membered polycycles breaks the bipartite symmetry of the molecular network   Figure shows classical examples of non-benzenoid non-alternant PCHs, namely, pentalene, azulene and heptalene  Whereas azulene is a stable PCH exhibiting Hückel aromaticity ([4n+2] π-electrons, n = 2), pentalene and heptalene are unstable Hückel antiaromatic compounds with [4n] π-electrons, n = 2 (pentalene) and n = 3 (heptalene) 
Benzinterposition of pentalene generates indacenes, consisting of two isomers s-indacene and as-indacene (Fig  )  Apart from being antiaromatic, indacenes also contain proaromatic quinodimethane (QDM) moieties (Fig  ) , which endows them with potential open-shell character  While the parent s-indacene and asindacene have never been isolated, stable derivatives of s-indacene bearing bulky substituents have been synthesized  
A feasible strategy to isolate congeners of otherwise unstable non-benzenoid non-alternant PCHs is through fusion of benzenoid rings at the ends of the π-system, that is, benzannelation  For example, while the parent pentalene is unstable, the benzannelated congener indeno[2,1-a]indene is stable under ambient conditions (Fig  )  
However, the position of benzannelation is crucial for stability: although indeno[2,1a]indene is stable, its regioisomer indeno[1,2-a]indene (Fig  ) oxidizes under ambient conditions   Similarly, benzannelation of indacenes gives rise to the family of PCHs known as indenofluorenes (Fig  ), which constitute the topic of the present work 
Depending on the benzannelation position and the indacene core, five regioisomers can be constructed, namely, indeno [ Practical interest in indenofluorenes stems from their low frontier orbital gap and excellent electrochemical characteristics that render them as useful components in organic electronic devices  
The potential open-shell character of indenofluorenes has led to several theoretical studies on their use as non-linear optical materials and as candidates for singlet fission in organic photovoltaics   Recent theoretical work has also shown that indenofluorene-based ladder polymers may exhibit fractionalized excitations 
Fundamentally, indenofluorenes represent model systems to study the interplay between aromaticity and magnetism at the molecular scale   Motivated by many of these prospects, the last decade has witnessed intensive synthetic efforts toward the realization of indenofluorenes  Derivatives of 1-4 have been realized in solution , while 1-3 have also been synthesized on surfaces and characterized using scanning tunneling microscopy (STM) and atomic force microscopy (AFM), which provide information on molecular orbital densities , molecular structure and oxidation state  
With regards to the open-shell character of indenofluorenes, 2-4 are theoretically and experimentally interpreted to be closed-shell, while calculations indicate that 1 and 5 should exhibit open-shell ground states   Bulk characterization of mesitylsubstituted 1, including X-ray crystallography, temperature-dependent NMR, and electron spin resonance spectroscopy, provided indications of its open-shell ground state  
Electronic characterization of 1 on Au(111) surface using scanning tunneling spectroscopy (STS) revealed a low electronic gap of 0 4 eV (ref  )  However, no experimental proof of an openshell ground state of 1 on Au(111), such as detection of singly occupied molecular orbitals (SOMOs) or spin excitations and correlations due to unpaired electrons , was shown 
In this work, we report the generation and characterization of unsubstituted 5  Our research is motivated by theoretical calculations that indicate 5 to exhibit the largest diradical character among all indenofluorene isomers   The same calculations also predict that 5 should possess a triplet ground state 
Therefore, 5 would qualify as a Kekulé triplet, of which only a handful of examples exist   However, definitive synthesis of 5 has never been reported so far  Previously, Dressler et al  reported transient isolation of mesityl-substituted 5, where it decomposed both in the solution and in solid state , and only the structural proof of the corresponding dianion was obtained 
On-surface generation of a derivative of 5, starting from truxene as a precursor, was recently reported   STM data on this compound, containing the indeno[1,2-a]fluorene moiety as part of a larger PCH, was interpreted to indicate its open-shell ground state  However, the results did not imply the ground state of unsubstituted 5  Here, we show that on insulating surfaces 5 can exhibit either of two ground states: an open-shell or a closed-shell 
We infer the existence of these two ground states based on high-resolution AFM imaging with bond-order discrimination and STM imaging of molecular orbital densities   AFM imaging reveals molecules with two different geometries  Characteristic bond-order differences in the two geometries concur with the geometry of either an open-or a closed-shell state 
Concurrently, STM images at ionic resonances show molecular orbital densities corresponding to SOMOs for the open-shell geometry, but orbital densities of the highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO) for the closed-shell geometry  Our experimental results are in good agreement with density functional theory (DFT) and multireference perturbation theory calculations 
Finally, we observe switching between open-and closed-shell states of a single molecule by changing its adsorption site on the surface  Synthetic strategy toward indeno[1,2-a]fluorene  The generation of 5 relies on the solution-phase synthesis of the precursor 7,12-dihydro indeno[1,2-a]fluorene (6)  Details on synthesis and characterization of 6 are reported in Supplementary Figs 
  Single molecules of 6 are deposited on coinage metal (Au(111), Ag(111) and Cu(111)) or insulator surfaces  In our work, insulating surfaces correspond to two monolayer-thick (denoted as bilayer) NaCl on coinage metal surfaces  Voltage pulses ranging between 4-6 V are applied by the tip of a combined STM/AFM system, which result in cleavage of one C-H bond at each of the pentagonal apices of 6, thereby leading to the generation of 5 (Fig  ) 
In the main text, we focus on the generation and characterization of 5 on insulating surfaces  Generation and characterization of 5 on coinage metal surfaces is shown in Supplementary Fig    )  Blue and orange colors represent spin up and spin down densities, respectively  c, Probability density of the SOMOs of 5OS (isovalue: 0 001 e -Å -3 ) 
d, DFT-calculated bond lengths of 5OS  e, Constant-height I(V) spectra acquired on a species of 5 assigned as 5OS, along with the corresponding dI/dV(V) spectra  Open feedback parameters: V = -2 V, I = 0 17 pA (negative bias side) and V = 2 V, I = 0 17 pA (positive bias side)  Acquisition position of the spectra is shown in Supplementary Fig    f, Scheme of many-body transitions associated to the measured ionic resonances of 5OS 
Also shown are STM images of assigned 5OS at biases where the corresponding transitions become accessible  Scanning parameters: I = 0 3 pA (V = -1 2 V and -1 5 V) and 0 2 pA (V = 1 3 V and 1 6 V)  g, Laplace-filtered AFM image of assigned 5OS  STM set point: V = 0 2 V, I = 0 5 pA on bilayer NaCl, Δz = -0 3
Å  The tip-height offset Δz for each panel is provided with respect to the STM setpoint, and positive (negative) values of Δz denote tip approach (retraction) from the STM setpoint  f and g show the same molecule at the same adsorption site, which is next to a trilayer NaCl island  The bright and dark features in the trilayer NaCl island in g correspond to Cl -and Na + ions, respectively 
Scale bars: 10 Å (f) and 5 Å (g)  To experimentally explore the electronic structure of 5, we used bilayer NaCl films on coinage metal surfaces to electronically decouple the molecule from the metal surfaces  Before presenting the experimental findings, we summarize the results of our theoretical calculations performed on 5 in the neutral charge state (denoted as 5 0 ) 
We start by performing DFT calculations on 5 0 in the gas phase  Geometry optimization performed at the spin-unrestricted UB3LYP/6-31G level of theory leads to one local minimum, 5OS, the geometry of which corresponds to the open-shell resonance structure of 5 (Fig  , the label OS denotes open-shell) 
The triplet electronic configuration of 5OS is the lowest-energy state, with the openshell singlet configuration 90 meV higher in energy  Geometry optimization performed at the restricted closed-shell RB3LYP/6-31G level reveals two local minima, 5para and 5ortho, the geometries of which (Fig  ) exhibit bond length alternations in line with the presence of a para-or an ortho-QDM moiety, respectively, in the as-indacene core of the closed-shell resonance structures of 5 (Fig  )  
Relative to 5OS in the triplet configuration, 5para and 5ortho are 0 40 and 0 43 eV higher in energy, respectively  Additional DFT results are shown in Supplementary Fig    To gain more accurate insights into the theoretical electronic structure of 5, we performed multireference perturbation theory calculations (Supplementary Fig  ) based on quasi-degenerate second-order n-electron valence state perturbation theory (QD-NEVPT2) 
In so far as the order of the ground and excited states are concerned, the results of QD-NEVPT2 calculations qualitatively match with DFT calculations  For 5OS, the triplet configuration remains the lowest-energy state, with the open-shell singlet configuration 60 meV higher in energy  The energy differences between the open-and closed-shell states are substantially reduced in QD-NEVPT2 calculations, with 5para and 5ortho only 0 11 and 0 21 eV higher in energy, respectively, compared to 5OS in the triplet configuration 
We also performed nucleus-independent chemical shift calculations to probe local aromaticity of 5 in the openand closed-shell states  While 5OS in the triplet configuration exhibits local aromaticity at the terminal benzenoid rings, 5OS in the open-shell singlet configuration, 5para and 5ortho all display antiaromaticity (Supplementary Fig  ) 
The choice of the insulating surface determines the charge state of 5: while 5 adopts neutral charge state on the high work function bilayer NaCl/Au(111) surface (irrespective of its openor closed-shell state, Supplementary Fig  ), 5 exhibits charge bistability between 5 0 and the anionic state 5 -1 on the lower work function bilayer NaCl/Ag(111) and Cu(111) surfaces (Supplementary Figs  ) 
In the main text, we focus on the characterization of 5 on bilayer NaCl/Au(111)  Characterization of charge bistable 5 is reported in Supplementary Figs    We first describe experiments on 5 on bilayer NaCl/Au(111), where 5 exhibits a geometry corresponding to the calculated 5OS geometry, and an open-shell electronic configuration 
We compare the experimental data on this species to calculations on 5OS with a triplet configuration, as theory predicts a triplet ground state for 5OS  For 5OS, the calculated frontier orbitals correspond to the SOMOs ψ1 and ψ2 (Fig  ), whose spin up levels are occupied and the spin down levels are empty 
Figure shows the DFT-calculated bond lengths of 5OS, where the two salient features, namely, the small difference in the bond lengths within each ring and the notably longer bond lengths in the pentagonal rings, agree with the open-shell resonance structure of 5 (Fig  )  Figure shows an AFM image of 5 adsorbed on bilayer NaCl/Au(111) that we assign as 5OS, where the bond-order differences qualitatively correspond to the calculated 5OS geometry (discussed and compared to the closed-shell state below) 
Differential conductance spectra (dI/dV(V), where I and V denote the tunneling current and bias voltage, respectively) acquired on assigned 5OS exhibit two peaks centered at -1 5 V and 1 6 V (Fig  ), which we assign to the positive and negative ion resonances (PIR and NIR), respectively  Figure shows the corresponding STM images acquired at the onset (V = -1 2
V/1 3 V) and the peak (V = -1 5 V/1 6 V) of the ionic resonances



ground_truth: NFPA and FPSA greatly outperform GMRES and DSA.

prompt_front: Now, answer the following question based on the text below, only give me the answer and do not output any other words.

Question: Can individual molecules of indeno[1,2-a]fluorene switch between open-shell and closed-shell states?

Read the following text and answer briefly.

Context 3:  To draw a correspondence between the STM images and the molecular orbital densities, we consider tunneling events as many-body electronic transitions between different charge states of 5OS (Fig  )  Within this framework, the PIR corresponds to transitions between 5 0 and the cationic state 5  
At the onset of the PIR at -1 2 V, an electron can only be detached from the SOMO ψ1 and the corresponding STM image at -1 2 V shows the orbital density of ψ1  Increasing the bias to the peak of the PIR at -1 5 V, it becomes possible to also empty the SOMO ψ2, such that the corresponding STM image shows the superposition of ψ1 and ψ2, that is, |ψ1| 2 + |ψ2| 2 (ref 
)  Similarly, the NIR corresponds to transitions between 5 0 and 5 -1   At the NIR onset of 1 3 V, only electron attachment to ψ2 is energetically possible  At 1 6 V, electron attachment to ψ1 also becomes possible, and the corresponding STM image shows the superposition of ψ1 and ψ2  The observation of the orbital densities of SOMOs, and not the hybridized HOMO and LUMO, proves the open-shell ground state of assigned 5OS 
Measurements of the monoradical species with a doublet ground state are shown in Supplementary Fig    Unexpectedly, another species of 5 was also experimentally observed that exhibited a closedshell ground state  In contrast to 5OS, where the frontier orbitals correspond to the SOMOs ψ1 and ψ2, DFT calculations predict orbitals of different shapes and symmetries for 5para and 5ortho, denoted as α and β and shown in Fig   
For 5ortho, α and β correspond to HOMO and LUMO, respectively  The orbitals are inverted in energy and occupation for 5para, where β is the HOMO and α is the LUMO  Fig  shows an AFM image of 5 that we assign as 5para  We experimentally infer its closed-shell state first by using qualitative bond order discrimination by AFM 
In high-resolution AFM imaging, chemical bonds with higher bond order are imaged brighter (that is, with higher frequency shift Δf) due to stronger repulsive forces, and they appear shorter   In Fig  , we label seven bonds whose bond orders show significant qualitative differences in the calculated 5ortho, 5para (Fig  ) and 5OS (Fig  ) geometries 
In 5para, the bonds b and d exhibit a higher bond order than a and c, respectively  This pattern is reversed for 5ortho, while the bond orders of the bonds a-d are all similar and small for 5OS  Furthermore, in 5para bond f exhibits a higher bond order than e, while in 5ortho and 5OS bonds e and f exhibit similar bond order (because they belong to Clar sextets) 
Finally, the bond labeled g shows a higher bond order in 5para than in 5ortho and 5OS  The AFM image of assigned 5para shown in Fig  indicates higher bond orders of the bonds b, d and f compared to a, c and e, respectively  In addition, the bond g appears almost point-like and with enhanced Δf contrast compared to its neighboring bonds, indicative of a high bond order (see Supplementary Fig  for height-dependent measurements) 
These observations concur with the calculated 5para geometry (Fig  )  Importantly, all these distinguishing bond-order differences are distinctly different in the AFM image of 5OS shown in Fig  , which is consistent with the calculated 5OS geometry (Fig  )  In the AFM images of 5OS (Fig  and Supplementary Fig  ), the bonds a-d at the pentagon apices appear with similar contrast and apparent bond length 
The bonds e and f at one of the terminal benzenoid rings also exhibit similar contrast and apparent bond length, while the central bond g appears longer compared to assigned 5para  Further compelling evidence for the closed-shell state of assigned 5para is obtained by STM and STS  dI/dV(V) spectra acquired on an assigned 5para species exhibit two peaks centered at -1 4 V (PIR) and 1 6 V (NIR) (Fig  ) 
STM images acquired at these biases (Fig  ) show the orbital densities of β (-1 4 V) and α (1 6 V)  First, the observation of α and β as the frontier orbitals of this species, and not the SOMOs, strongly indicates its closed-shell state  Second, consistent with AFM measurements that indicate good correspondence to the calculated 5para geometry, we observe β as the HOMO and α as the LUMO 
For 5ortho, α should be observed as the HOMO and β as the LUMO  We did not observe molecules with the signatures of 5ortho in our experiments  We observed molecules in open-(5OS, Fig  ) and closed-shell (5para, Fig  ) states in similar occurrence after their generation from 6 on the surface  We could also switch individual molecules between open-and closed-shell states as shown in Fig  and Supplementary Fig   
To this end, a change in the adsorption site of a molecule was induced by STM imaging at ionic resonances, which often resulted in movement of the molecule  The example presented in Fig  shows a molecule that was switched from 5para to 5OS and back to 5para  The switching is not directed, that is, we cannot choose which of the two species will be formed when changing the adsorption site, and we observed 5OS and 5para in approximately equal yields upon changing the adsorption site 
The molecule in Fig  is adsorbed on top of a defect that stabilizes its adsorption geometry on bilayer NaCl  At defect-free adsorption sites on bilayer NaCl, that is, without a third layer NaCl island or atomic defects in the vicinity of the molecule, 5 could be stably imaged neither by AFM nor by STM at ionic resonances (Supplementary Fig  ) 
Without changing the adsorption site, the state of 5 (open-or closedshell) never changed, including the experiments on bilayer NaCl/Ag(111) and Cu(111), on which the charge state of 5 could be switched (Supplementary Figs  )  Also on these lower work function surfaces, both open-and closed-shell species were observed for 5 0 and both showed charge bistability between 5 0 (5OS or 5para) and 5 -1 (Supplementary Figs  ) 
The geometrical structure of 5 -1 probed by AFM, and its electronic structure probed by STM imaging at the NIR (corresponding to transitions between 5 -1 and the dianionic state 5 -2 ), are identical within the measurement accuracy for the charged species of both 5OS and 5para  When cycling the charge state of 5 between 5 0 and 5 -1 several times, we always observed the same state (5OS or 5para) when returning to 5 0 , provided the molecule did not move during the charging/discharging process 
Based on our experimental observations we conclude that indeno[1,2-a]fluorene (5), the last unknown indenofluorene isomer, can be stabilized in and switched between an open-shell (5OS) and a closed-shell (5para) state on NaCl  For the former, both DFT and QD-NEVPT2 calculations predict a triplet electronic configuration 
Therefore, 5 can be considered to exhibit the spin-crossover effect, involving magnetic switching between high-spin (5OS) and low-spin (5para) states, coupled with a reversible structural transformation  So far, the spin-crossover effect has mainly only been observed in transition-metal-based coordination compounds with a near-octahedral geometry  
The observation that the switching between open-and closedshell states is related to changes in the adsorption site but is not achieved by charge-state cycling alone, indicates that the NaCl surface and local defects facilitate different electronic configurations of 5 depending on the adsorption site 
Gas-phase QD-NEVPT2 calculations predict that 5OS is the ground state, and the closed-shell 5para and 5ortho states are 0 11 and 0 21 eV higher in energy  The experiments, showing bidirectional switching between 5OS and 5para, indicate that a change in the adsorption site can induce sufficient change in the geometry of 5 (leading to a corresponding change in the ground state electronic configuration) and thus induce switching 
Switching between open-and closed-shell states in 5 does not require the breaking or formation of covalent bonds , but a change of adsorption site on NaCl where the molecule is physisorbed  Our results should have implications for single-molecule devices, capitalizing on the altered electronic and chemical properties of a system in π-diradical open-shell and closed-shell states such as frontier orbital and singlet-triplet gaps, and chemical reactivity 
For possible future applications as a single-molecule switch, it might be possible to also switch between open-and closed-shell states by changing the local electric field, such as by using chargeable adsorbates   Scanning probe microscopy measurements and sample preparation  STM and AFM measurements were performed in a home-built system operating at base pressures below 1×10 -10 mbar and a base temperature of 5 K  Bias voltages are provided with respect to the sample 
All STM, AFM and spectroscopy measurements were performed with carbon monoxide (CO) functionalized tips  AFM measurements were performed in non-contact mode with a qPlus sensor   The sensor was operated in frequency modulation mode with a constant oscillation amplitude of 0 5 Å  STM measurements were performed in constantcurrent mode, AFM measurements were performed in constant-height mode with V = 0 V, and I(V) and Δf(V) spectra were acquired in constant-height mode 
Positive (negative) values of the tip-height offset Δz represent tip approach (retraction) from the STM setpoint  All dI/dV(V) spectra are obtained by numerical differentiation of the corresponding I(V) spectra  STM and AFM images, and spectroscopy curves, were post-processed using Gaussian low-pass filters 
Au(111), Ag(111) and Cu(111) surfaces were cleaned by iterative cycles of sputtering with Ne + ions and annealing up to 800 K  NaCl was thermally evaporated on Au(111), Ag(111) and Cu(111) surfaces held at 323 K, 303 K and 283 K, respectively  This protocol results in the growth of predominantly bilayer (100)-terminated islands, with a minority of trilayer islands 
Sub-monolayer coverage of 6 on surfaces was obtained by flashing an oxidized silicon wafer containing the precursor molecules in front of the cold sample in the microscope  CO molecules for tip functionalization were dosed from the gas phase on the cold sample  Density functional theory calculations  DFT was employed using the PSI4 program package  
All molecules with different charge (neutral and anionic) and electronic (open-and closed-shell) states were independently investigated in the gas phase  The B3LYP exchangecorrelation functional with 6-31G basis set was employed for structural relaxation and singlepoint energy calculations  The convergence criteria were set to 10 −4 eV Å −1 for the total forces and 10 −6 eV for the total energies 
Multireference calculations  Multireference calculations were performed on the DFToptimized geometries using the QD-NEVPT2 level of theory , with three singlet roots and one triplet root included in the state-averaged calculation  A (10,10) active space (that is, 10 electrons in 10 orbitals) was used along with the def2-TZVP basis set  
Increasing either the active space size or expanding the basis set resulted in changes of about 50 meV for relative energies of the singlet and triplet states  These calculations were performed using the ORCA program package   Nucleus-independent chemical shift (NICS) calculations  Isotropic nucleus-independent chemical shift values were evaluated at the centre of each ring using the B3LYP exchangecorrelation functional with def2-TZVP basis set using the Gaussian 16 software package  
Starting materials (reagent grade) were purchased from TCI and Sigma-Aldrich and used without further purification  Reactions were carried out in flame-dried glassware and under an inert atmosphere of purified Ar using Schlenk techniques  Thin-layer chromatography (TLC) was performed on Silica Gel 60 F-254 plates (Merck) 
Column chromatography was performed on silica gel (40-60 µm)  Nuclear magnetic resonance (NMR) spectra were recorded on a Bruker Varian Mercury 300 or Bruker Varian Inova 500 spectrometers  Mass spectrometry (MS) data were recorded in a Bruker Micro-TOF spectrometer  The synthesis of compound 6 was developed following the two-step synthetic route shown in Supplementary Fig  , which is based on the preparation of methylene-bridge polyarenes by means of Pd-catalyzed activation of benzylic C-H bonds  
Supplementary Figure | Synthetic route to obtain compound 6  The complex Pd2(dba)3 (20 mg, 0 02 mmol) was added over a deoxygenated mixture of 1,3-dibromo-2,4-dimethylbenzene (9, 100 mg, 0 38 mmol), boronic acid 10 (178 mg, 1 14 mmol), K2CO3 (314 mg, 2 28 mmol) and XPhos (35 mg, 0 08 mmol) in toluene (1:1, 10 mL), and the resulting mixture was heated at 90 °C for 2 h 
After cooling to room temperature, the solvents were evaporated under reduced pressure  The reaction crude was purified by column chromatography (SiO2; hexane:CH2Cl2 9:1) affording 11 (94 mg, 76%) as a colorless oil  The complex Pd(OAc)2 (7 mg, 0 03 mmol) was added over a deoxygenated mixture of terphenyl 11 (90 mg, 0 27 mmol), K2CO3 (114 mg, 0 83 mmol) and ligand L (26 mg, 0 06 mmol) in NMP (2 mL) 
The resulting mixture was heated at 160 °C for 4 h  After cooling to room temperature, H2O (30 mL) was added, and the mixture was extracted with EtOAc (3x15 mL)  The combined organic extracts were dried over anhydrous Na2SO4, filtered, and evaporated under reduced pressure  The reaction crude was purified by column chromatography (SiO2; hexane:CH2Cl2 9:1) affording compound 6 (8 mg, 11%) as a white solid  in AFM imaging due to their reduced adsorption height compared to the rest of the carbon atoms 
We attribute this observation to the significantly different lattice parameter of Cu(111) (2 57 Å) compared to Au(111) and Ag(111) (2 95 Å and 2 94 Å, respectively) , such that the apical carbon atoms of the pentagonal rings of 5 adsorb on the on-top atomic sites on Au(111) and Ag(111), but not on Cu(111) 
Our speculation is based on a previous study of polymers of 1 on Au(111) by Di Giovannantonio et al  , where both tilted and planar individual units of 1 were observed depending on whether the apical carbon atoms of the pentagonal rings in 1 adsorbed on the on-top or hollow sites of the surface, respectively 
Given the strong molecule-metal interaction, we found no electronic state signatures of 5 on all three metal surfaces  STM set point for AFM images: V = 0  e, Frontier orbital spectrum of 5 -1   In the anionic state, ψ2 becomes doubly occupied and ψ1 is the SOMO  Filled and empty circles denote occupied and empty orbitals, respectively 
For each panel, zero of the energy axis has been aligned to the respective highest-energy occupied orbital 



ground_truth: Yes, individual molecules of indeno[1,2-a]fluorene can switch between open-shell and closed-shell states by changing their adsorption site on the surface.