We approach the problem of combining top-ranking association statistics or P-values from a new perspective which leads to a remarkably simple and powerful method. Statistical methods, such as the rank truncated product (RTP), have been developed for combining top-ranking associations, and this general strategy proved to be useful in applications for detecting combined effects of multiple disease components. To increase power, these methods aggregate signals across top ranking single nucleotide polymorphisms (SNPs), while adjusting for their total number assessed in a study. Analytic expressions for combined top statistics or P-values tend to be unwieldy, which complicates interpretation and practical implementation and hinders further developments. Here, we propose the augmented rank truncation (ART) method that retains main characteristics of the RTP but is substantially simpler to implement. ART leads to an efficient form of the adaptive algorithm, an approach where the number of top ranking SNPs is varied to optimize power. We illustrate our methods by strengthening previously reported associations of μ-opioid receptor variants with sensitivity to pain. L studies, potentially providing a higher level of assurance that the studied risk factor is associated with disease. Furthermore, if samples in those studies are taken from populations that are similar with respect to the effect size magnitude, the combined meta-analytic P-value will well approximate the one that would have been obtained by pooling together all raw data and performing a single test  mapping individual SNPs to genes, followed by combining their association P-values into gene-scores, and (2) grouping genes into pathways and combining gene-scores into pathway-scores.P-values can also be combined when the i,iP = 1, …, L) are often based on the sum of iP‘s transformed by some function H. For example, H(iP)=–21n(iP), which are then added up to form a test statistics L degrees of freedom. When a portion of L distinct associations is expected to be spurious, it is advantageous to combine only some of the predictors using a truncated variation of combined P-value methods or emphasize the smallest P-values. For instance, I(iP≤ α), that is equal to zero if iP> α, and one if iP≤ α; 0 < α ≤ 1 is a truncation threshold. The combined P-value, PTPM, is then given by the cumulative distribution function (CDF) of W is random. Similarly inspired methods include the “tail strength measure” Existing tools for combining P-values  . In RTP,P(i) is the ith smallest P-value, i = 1,…,k. RTP leads to an appealing extension, where k can be chosen adaptively, to maximize statistical power  of possible values 1 ≤ k ≤ L. Adaptive extensions for TPM are not as straightforward because the threshold α is a continuous variable, but one can resort to evaluating the distribution over a set of grid points  being a random variable: when P(k+1) happens to be relatively small, the k P-values have to squeeze into a relatively small interval from zero to that value. This induces positive dependency between random sets of k smallest P-values, similar to the clustering effect in random effects models. Although the linear correlation can be eliminated by scaling the largest P-value, P(k), the k values remain dependent, as illustrated in The RTP null distribution is considerably more complicated than that of TPM. Complexity of the RTP distribution is due to dependency between ordered P-values. When k is chosen beforehand and fixed) and for the adaptive variations of RTP and ART (aRTP and ART-A). Finally, we illustrate usefulness of the proposed methods by strengthening an overall, gene-based association via combining previously reported P-values between pain sensitivity and individual SNPs within the μ-opioid receptor.Applications of combining independent P-values remain important in statistical research, and there is clear preference among practitioners for methods that are based on simple and transparent approaches, such as the Fisher or the inverse normal (Stouffer’s) tests . Here, wk< L, as:Even when P-values are independent, previously proposed theoretical forms of the RTP distribution are cumbersome and result in expressions that involve repeated integration . For exak smallest P-values can be shuffled. This step does not change the value of the product, kW (or its logarithm), which is our statistic of interest, but implies that we can treat the joint k-variate distribution as governed by the same pair-wise dependence for every pair of variables. Moreover, variables of that shuffled distribution are identical marginally. The dependency is induced completely through randomness of P(k+1), and conditionally on its value, the {kW|p(k+1)} distribution is given by standard independence results. Then, PRTP is given by the marginal CDF of kW. Based on this conceptual model, we derived the following representation of RTP where a single integral is evaluated in a bounded interval , which allows one to evaluate the RTP distribution as a simple average of standard functions. Specifically, we derive a simple expression for the RTP distribution as the expectation of a function of a uniform (0 to 1) random variable:Theoretical forms of the RTP distribution  may retain order-specific terms. Here, we proceed to a simpler representation by noting that every random realization of k+1,L−k) distribution, kG(·) is CDF of Gamma, and PRTP(k) is the combined RTP P-value. Notably, given the value of the product of k P-values, W = w, we can simultaneously evaluate PRTP(k+1):where Details and the derivation are given in k−1 smallest P-values, given a value of the beta-distributed k-th smallest P-value, leads to a simple statistic which (just as RTP) is a function of the product of the k-th smallest P-values. This statistic and its distribution are not an approximation to kW and the RTP distribution. However, similarly to RTP, the new statistic is designed to capture information contained in the first k smallest P-values. Motivation for our augmented RTP method (ART) follows from the intermediate results of the derivation of the RTP distribution. Note that the distribution of kW conditional on (k + 1) involves a product of k uniforms and a beta random variable. Moreover, the product and the beta variable are independent. On the one hand, we could have proceeded by dividing every P(1:k) value by the observed p(k+1) and employed the gamma CDF to P(k+1) magnitude. On the other hand, we can exploit the independence of the gamma-distributed sum and P(k+1) by transforming P(k+1) to a gamma random variable and adding the result to the sum. Specifically, to construct the new statistic, we propose the following transformation that involves the product k−1W and the variable (k)P. These transformations yield three independent variables, which are next added together and give a gamma-distributed random variable,The conditional independence of k,1),where k – 1) × E {– ln ((k)P)} = (k – 1) (Γ′ (L + 1)/Γ (L + 1) – Γ′ (k)/Γ (k)),λ = (kB(x) is the CDF of Beta distribution evaluated at x. The shape parameter λ is chosen so that the two last terms in Eq. 4 (that are both transformations of (k)P) have the same expectation. The shape parameter here plays a role of a weight given to the last term. This is similar to the Lancaster method for combining P-values, where the gamma shape parameter  has been used to give differential weights to P-values  that ART has either the same or higher power than RTP. Furthermore, the combined P-value, ART, can be easily computed in R using its standard functions. A short example and an R code are given in Under the null hypothesis, as illustrated by Introduction, the number of k P-values to be combined by the RTP method (or ART) is fixed and needs to be pre-specified. The choice of k is somewhat arbitrary; so, a researcher may wish to evaluate ART at several values of k, consequently choosing k that corresponds to the smallest combined P-value. However, this additional step creates another layer of multiple comparisons, which needs to be accounted for. B×L matrix, with every row containing L P-values generated under the null distribution over B simulated experiments. As we discussed earlier in via decorrelation by orthogonal transformation approach, DOT. Let L correlated P-values,  originate from statistics that jointly follow a standard multivariate normal distribution, y ∼ MVN , under H0. For two-sided P-values, the elements of y are squared. Elements of the vector of squared variables, Σ, such that −1Σ = QΛQ, where Q is a square matrix, with ith column containing eigenvector qi of Σ, and Λ is the diagonal matrix of eigenvalues λ1, λ2, …, λL. Next, define an orthogonal matrix H = QΛ−1/2QT and ye = HTy. P-values are decorrelated as 1-Φ−1(ye). Then, the first k smallest decorrelated P-values can be used to calculate various combined statistics. The choice of this particular transformation is motivated by its “invariance to order” property. Briefly, in the equicorrelation case, including the special case of ρ = 0, i.e., independence, a permutation of y should yield the same (possibly permuted) values in the decorrelated vector, ye. Extensive evaluation of the decorrelation approach is presented by us elsewhere  and to the adaptive RTP [where k is varied and the distribution is evaluated by single-layer simulations as in et al. proposed a clever shortcut based on a rectangular matrix, in which each i-th row is a set of L P-values that corresponds to the i-th simulation. The shortcut avoids nested and slow simulations but has a downside of keeping quite large matrix in computer memory.We used simulation experiments to evaluate the type I error rate and power of the proposed methods relative to the previously studied RTP  ≤ iα/L for at least one i. Equivalently, the overall (or the “combined”) Simes P-value can be obtained as min {kp(i)/i}. The Simes test is a useful benchmark, because it is related to the combined P-value methods with truncation, as well as to multiple testing adjustment procedures. At the extreme, the RTP with k = 1 becomes equivalent to L. The Simes P-value is at least as small as Bonferroni-corrected P-value. In addition, there is a connection of the Simes test to the i, such that P(i) ≤ iα/L, and rejects H0 for all P(j), j ≤ i, to control the expectation of FDR.When various combined P-value methods are being compared, it is meaningful to gauge their performance against methods designed for multiple testing adjustments. This is especially relevant with methods that employ truncation due to their emphasis on small P-values. Therefore, we included the k or from 1 to L, if k = L. Both tables confirm that all methods maintain the correct Type I error rate at various α-levels.L statistics had the same underlying effect size (μ = 0.5). From this table, it is evident that our ART has the highest power, closely followed by RTP. In general, the ART P-values tend to be similar to the P-values obtained by the RTP, and we show their similarity graphically in L tests and absence of true nulls. For the results in k (RTP and ART). Nonetheless, in practice, a good choice for k may not be immediately clear, so a small sacrifice in power may be preferable to an arbitrary and possibly poor choice of k. However, when L is large, it can be impractical or unreasonable to vary candidate truncation points all the way up to L. Therefore, the value of k in these tables is the upper bound for sets of candidate truncation points, and that explains the difference in powers. For example, for the combination k = 10, L = 200, the power of ART-A in k = 100, L = 200, the power is 0.37. Adaptive methods are considerably slower than methods where the value of k is fixed. A promising solution, which is currently under development, is removal of candidate truncation points for which the corresponding partial products are highly correlated.k = L, but for illustrative purposes, we also give power values for some choices where k < L. We emphasize that, in our tables, these chosen maximum k values do not have any specific meaning. In general, the maximum value of k for the adaptive methods can be varied up to L, unless L is either very large, or if there is a priori assumption that the real signals are rare. As discussed in k is usually lower than the actual number of real signals; however, the upper bound for k can be set to reflect a priori knowledge about potential maximum number of real signals.We note that power values in L hypotheses were true nulls (μ = 0), while the remaining hypotheses were true signals (μ = 0.5). The results follow the same pattern as in the previous tables, with ART having the highest power.Finally, k and L values. However, decorrelation-based methods become quite powerful, and their power is increasing with k and L. The steady power increase is due to the decorrelation effect on the combined noncentrality that involves the sum μ. More details on the performance of the decorrelation approach are given by us elsewhere  with pain sensitivity. MOR gene, from which the 11×11 LD correlation matrix can be computed. The ijP frequencies are given by the sum of frequencies of those 11-SNP haplotypes that contain both of the minor alleles for SNPs i and j. Similarly, ip allele frequency is the sum of haplotype frequencies that carry the minor allele of the SNP i. The LD correlations within the MOR region spanned by the 11 SNPs ranged from −0.82 to 0.99, with the average absolute value ≈0.55 and the median absolute value ≈0.66. Half of pairwise LD correlations were smaller than −0.23 or larger than 0.82. Our analysis . Similar to what we found via simulation experiments, where correlation is controlled by reshuffling the phenotype values while keeping the original LD structure intact, RTP and aRTP (without the decorrelation step) do not benefit from inclusion of additional SNPs. P-values in the ART column are very similar to those in the RTP column, which reassures our theoretical expectations. In contrast to previously proposed methods that control correlation by resampling , the results in columns marked by “decorr” are substantially lower. In these columns, we used the proposed transformation to independence, which gives much stronger combined P-values. In all “decorr” columns, k = 7 results in the smallest combined P-value, implying that the number of real effects (including proxy associations) is at least seven.In several popular variations of the gene-based approach , future k” and its optimal value. Unfortunately, k cannot be interpreted as an estimate of the number of real signals, for the reason that the k value that yields the minimum P-value depends on both the numbers of real signals, as well as on their strength of association, both of which are unknown. This issue has been investigated in detail in μ-opioid gene that have been shown to affect sensitivity to pain. We find strengthened evidence of overall association within the 11-SNP block. In this application, the LD correlation matrix was reconstructed from the haplotype frequencies, which might be slightly different from the correlation of  values between pairs of SNPs  has been designed with this objective in mind. The RTP and related approaches had been shown to be valuable tools in analysis of genetic associations with disease. In this article, we derive a mathematically simple form of the RTP distribution that leads a to new method, ART and its adaptive version, ART-A, which searches through a number of candidate values of truncation points and finds an optimal number in terms of combined P-value. Two important questions are the meaning of “ of SNPs . Furtherhttps://github.com/dmitri-zaykin/Total_Decor.The URL for software referenced in this article is available at: The datasets generated for this study are available on request to the corresponding author.DZ and OV conceived the study, interpreted the results, and wrote the manuscript. FH performed extensive simulations. All authors read and approved the final manuscript.The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.


A large number of topological materials with a CsCl-type structure in crystallographic databases are explored and their crystal and electronic structures are studied. i.e. simple in structure, ease of synthesis and good stability at room temperature, thus are an excellent choice for designing functional materials. Using high-throughput first-principles calculations, a large number of topological semimetals/metals (TMs) were designed from CsCl-type materials found in crystallographic databases and their crystal and electronic structures have been studied. The CsCl-type TMs in this work show rich topological character, ranging from triple nodal points, type-I nodal lines and critical-type nodal lines, to hybrid nodal lines. The TMs identified show clean topological band structures near the Fermi level, which are suitable for experimental investigations and future applications. This work provides a rich data set of TMs with a CsCl-type structure.CsCl-type materials have many outstanding characteristics,  Such enthusiasm was initially generated by the discovery of topological insulators  , CsCl-type materials have a simple crystal structure that possesses only two atomic positions, known as 1a  and 1b  Wyckoff sites. CsCl-type materials are quite easy to synthesize, and most of them can be directly prepared from melting without any further chemical treatment , which clearly shows a band-crossing point at the M–R path near the Fermi level (at E − EF = 0.08 eV). Excluding spin, this band-crossing point has a triple degeneracy formed by one non-degenerate band and one doubly degenerate band. It should be noted that, with the exception of the bands which form the triple nodal point, there exists no other extraneous band nearby. Such a clean band structure generally favors experimental detection of triple nodal points in YIr. One of the representative signatures of triple-nodal-point TMs is the existence of Fermi-arc surface states . We can clearly observe the Fermi-arc states originating from the triple nodal points. It is worth noting that, since the triple nodal points are nearly situated in the middle of the M–R path, the Fermi arc in YIr spans a large scale of momenta. Evidence of Fermi-arc states in YIr can be readily detected by experiment.YIr is a typical triple-nodal-point TM; its electronic band structure is shown in Fig. 53.3.a), the band crossing forms a one-dimensional nodal line. On the nodal line, each point exhibits linear band crossing. According to the slope of band dispersion, band crossing can be termed as three types, namely type-I, critical-type and type-II, as shown in Fig. 6b). For type-I, the bands have conventional dispersion, where the electron-like and hole-like states are completely separated by the crossing point. For type-II, the conical spectrum is tipped over. As a result, the electron-like and hole-like states coexist at each energy level. For critical type, which is formed by a flat band and a dispersive band . At the Γ–M, M–X and R–M paths, there exists type-I band-crossing points, as indicated by the arrows in Fig. 7a). Because of the protection of the inversion symmetry (P) and the time reversal symmetry (T) in the CaTe system, these crossing points in fact reside on nodal lines centering on the M point. By performing more detailed computations, we find the nodal line is situated in the kz = π plane, under additional protection of the mirror symmetry (Mz). Considering the cubic symmetry of the CaTe lattice, the system possesses in total three equivalent nodal lines crossing each other, as shown in Fig. 7b). CaTe is an excellent type-I nodal-line TM because of the following: (1) it has a clean nodal-line band structure; (2) the nodal lines are quite close to the Fermi level; and (3) the crossing bands have a very large linear energy range (∼2 eV). With the SOC effect, the nodal line is gapped beside a pair of Dirac points leaving at the R–M path (protected by the C4 rotation symmetry), as shown in Figs. 7c) and 7d). In fact, Du et al.  is shown in Fig. 7 al. 2017 reporteda), without the SOC effect, type-I band-crossing points occur at the Γ–X, X–M and R–X paths. Under the same protection mechanism with CaTe, these band-crossing points also belong to three crossing nodal lines. However, unlike CaTe, these nodal lines center on the X point instead of the M point, as shown in Fig. 8b). With the SOC effect, these nodal lines are fully gapped to a sizeable gap (SOC-induced gap) along all the high-symmetry paths involving the X point, as shown in Fig. 8c).TiOs is another type-I nodal-line TM. As shown in Fig. 8a). It can be observed there are two band crossing points at the R–X and X–M paths. Different from CaTe and TiOs, the band-crossing points show a critical-type band-crossing feature, namely, one of the crossing bands is almost flat in CaPd. This band crossing produces a nodal line centering on the X point in the kz = π plane. The three-dimensional energy dispersion of the two crossing bands is plotted in Fig. 9b). We can observe from the critical-type nodal-line signature that the flat band is nearly non-dispersive near the nodal line at every k-path in the kz = π plane. Such a critical-type nodal line was first proposed by Liu, Jin et al. . For each k-path  from the M point in the plane, we can always obtain a type-II band crossing, as shown in Fig. 11b). However, we noticed that the slopes of the two crossing bands between M–C and M–D paths have opposite sign. Thus we select another four k-paths  between M–C and M–D to observe the evolution of the slope. The band structures along these paths are shown in Fig. 11d). It is evident that, the band crossing along these k-paths exhibits a type-II–type-I–type-II transition. Therefore, the nodal line in Fig. 11a) is a hybrid nodal line. In addition, we need to clarify that compared with common hybrid nodal lines, including Ca2As  and 12b), respectively. We can observe clear surface bands on each k-path from the point.We performed a careful investigation of the band structure around the type-II crossing point and found it belongs to a nodal line centering on the 2As . It manifests two band-crossing points near the Fermi level, which are denoted as point A and point B [see Fig. 13a)]. The two nodal points have different degeneracy without counting spin: point A is doubly degenerate and point B is triply degenerate. By performing more detailed computations, we find point A resides on a hybrid nodal line centering on the M point, which is similar with that in YCd. Point B is a triple nodal point at the R–M path and is similar to that in YIr. As a result, a hybrid nodal line and a pair of triple nodal points coexist in YMg, as depicted in Fig. 13b). Benefiting from the clean band structure near the nodal line and the triple nodal point, topological surface states for the nodal line and the triple nodal point can be clearly identified, as shown in Figs. 13c) and 13d), respectively.Several CsCl-type materials possess multiple types of band crossing. YMg is one of these materials The band structure of YMg is shown in Fig. 134.E − EF| < 0.5 eV). Most of the TMs identified show a clean topological band structure where there are no interfering bands around the band crossing. Moreover, these CsCl-type materials can be easily synthesized and are stable at room temperature, which greatly favors the experimental investigation of their topological properties.By using high-throughput computational screening based on first-principles, we have identified 61 TMs in existing CsCl-type materials. The identified TMs show rich topological character, including triple nodal point, type-I nodal line, critical-type nodal line and hybrid nodal line. In their band structures, the band crossing occurs near the Fermi level (|


Callicebus nigrifrons). This species uses two alarm calls, the A-call and the B-call, and arranges them into diverse sequences. The composition of those sequences differs depending on the type and location of the predator  a. A raptor in a non-ground position (flying or perched in the canopy) elicits a sequence of A-calls, (hereafter A+)b. A raptor on the ground elicits a sequence of A-calls followed by B-calls (A+B+)c. A cat in the canopy elicits a sequence a single A-call followed by B-calls (AB+)d. A cat on the ground /the presence of a terrestrial animal/ a monkey descending near the ground elicits a sequence of B-calls only (B+).Schlenker et al.  proposed(2) a. A+ is given to a serious non-ground threatb. a raptor on the ground attacks by flying, hence it is treated as a non-ground threat (A+) that stops being serious at some point because it is not in a hunting position (hence B+)c. a cat in the canopy is a serious non-ground threat (hence A). However, the cat's hunting strategy relies on surprise, and spotting a felid may deter it from attacking  a. A+ is given to a serious threatb. A raptor on the ground is still a dangerous threat (A+) but stops being serious at some point for it is not in a hunting position (hence B+)c. a cat in the canopy is a serious threat (A) that stops being serious once spotted (B+)d. B are used in non-serious alerts.and the location of the threat (“non-ground”) while we argue that A-calls only refer to the level of alarm. Our analysis is therefore more parsimonious, and in line with the animal communication literature that described several animal communication systems as urgency-based (e.g., Rauber and Manser, In Schlenker et al. , A-callsWhen hearing A-calls, titi monkeys mainly look upwards (Cäsar et al., We reckon that decisive field experiments are now necessary to verify the meaning of A-calls. One could, for instance, present titi monkeys that are on the ground with a cat model. According to Schlenker et al. , this woBeyond reflecting on the titi monkeys alarm system, our own analysis proves that the approach introduced by Schlenker et al. ,b and enSC: conception and first draft; MB: further redaction and revision of the manuscript.The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.


Nature Communications 10.1038/s41467-019-10742-8, published online 28 June 2019.Correction to: The original version of this Article contained an error in Fig. 6e, in which the labels ‘IAA’ and ‘Phle’ in the bottom right where inadvertently omitted. This has been corrected in both the PDF and HTML versions of the Article.


The structure of L-cysteine adsorption under acidic conditions can be further verified by the addition of an L-cysteine molecule through light-induced coupling reaction to form cystine. Finally, in-situ two-dimensional Raman scattering spectroscopy confirmed the feasibility and uniformity of the coupling reaction.Under a controlled adsorption environment, L-cysteine molecules can be chemically adsorbed to the dendritic silver (Ag-D) surface by electrochemical methods with different functional groups. It is verified by surface-enhanced Raman spectroscopy that under alkaline conditions (pH = 13.50), the two functional groups of thiol and acid are simultaneously adsorbed on the surface of Ag-D, while NH For example, abnormal concentrations of cysteine in the blood are important indicators of skin damage and Alzheimer’s disease; the thiol groups of two L-cysteine molecules can form a disulfide-bonded Cys–Cys via oxidation3. This dimer is called cystine. Its importance lies in (1) its role of stabilizing the spatial structure of the peptide chain, (2) the fact that more there are disulfide bonds, the more stably the protein adapts to the outside, (3) the stable tertiary structure4. It is commonly present in keratin and collagen in the hair and skin of the human body, providing structural support or retention of toughness, as well as insulin or oxytocin that regulates physiological functions. In addition, L-cysteine molecules can spontaneously self-assemble with metal surfaces through thiol groups, carboxyl groups, or amine groups, modifying metal properties or increasing the possibility of practical application10. The modified metal surface can be an important technical application in chemical sensing or catalysis14. For example, proteins can be more easily and stably adsorbed on the surface to facilitate the provision of a substrate for a nano-process involving biological substances16. This may help further understand the molecular characteristics of the L-cysteine monolayer and appropriately control the adsorption conditions of the protein; it may even be used to capture and detect heavy metals18.L-cysteine is a natural amino acid with a thiol side chain that plays a key role in the organism19. The selection of the substrate, the molecular environment, the adsorption time, and other conditions are also important factors affecting adsorption and thus the more single and stable adsorption of L-cysteine on the metal. There are many studies on the surface-enhancement Raman scattering (SERS) spectra and theoretical calculations of the adsorption behavior or structure of L-cysteine on the surface of silver nanostructures21. Because of the small Raman scattering cross section of Cysteine molecule and low molecular symmetry, however, the SERS signal of this type of biological small molecule is usually very weak22. Only weak enhancement effect occurs: this effect is due to the Herzberg–Teller mechanism (Albrecht’s term B), an enhancement that allows for enhanced non-complete symmetry modes25. Therefore, high concentrations of target molecules are usually required to generate detectable signals21, thus constituting a technical obstacle in the practical application of SERS to biomolecules.Many factors influence the adsorption morphology of L-cysteine on the metal surface, such as temperature, applied potential control, and environmental pH26. Generally, chemisorbed molecules show an effect more pronounced than that of physically adsorbed molecules27. In other words, chemisorbed molecules show the most significant enhancement effects, namely, the highest intensity electromagnetic mechanism (EM) and chemical mechanism (CM) effects28. The chemical amplification is affected by the degree of overlap and probability of the bond electron domain between the target molecule and the SERS substrate. However, in SERS measurements, the target molecule can contain a variety of functional groups that can dissociate and subsequently interact with the active substrate. Cysteine molecules contain three different functional groups of amine groups, carboxyl groups and thiol groups30. Their properties and electrokinetic behavior are largely dependent on the solution pH31. Thus, the formation of anionic, zwitterionic, or cationic species is suitable for studying the competitive interaction of functional groups with metal surfaces32. In addition, cysteine can form a chemical bond on the metal surface through lone pairs on the functional group. This promotes chemical enhancement through the charge-transfer mechanism, enhancing the Raman signal33. However, the cysteine molecule has a small scattering cross section and is not easy to measure, and there are three functional groups that can form a bond with the substrate atom. If the adsorption between the molecule and the substrate is not effectively controlled, the contribution of the chemical amplification effect to Raman scattering makes the signal complex and difficult to judge. In many studies, the adsorption mechanism for this molecular structure is still blurred in SERS analysis. Therefore, this study will examine in detail the adsorption behavior of this molecule on the surface of dendritic silver (Ag-D) on a SERS-active substrate. The cysteine ion state is adjusted by the pH of the solution, and its structure adsorbed on the surface is optimized by a step function. By control of cysteine molecular adsorption behavior, cystine was formed on the surface of SERS substrate by in situ SERS measurement and induced coupling reaction of cysteine molecules.On the surface of the SERS-active substrate, both the physically adsorbed and chemisorbed molecules exhibit an enhanced Raman signal due to the effect of the additive effect34. The dendritic Ag surface showed high Raman enhancement, enabling the detection of analytes from dilute solution by SERS. 4-Nitrothiophenol was used as the probe molecule to examine the analyte-concentrating ability and SERS activity of Ag-D. Quantitative SERS detection using the Ag-D has detection limits in the nanomolar (5 nM) concentration range35. It is a very good substrate for SERS. However, surface silver atoms are reactive and easily oxidized, and preservation and SERS measurement conditions require precise control. In this study, we used L-cysteine as the target molecule, and the Ag-D electrode was a surface-enhanced Raman substrate; that is, the reactivity of the surface of the SERS substrate to the target molecule was investigated. In aqueous solution, the functional groups of L-cysteine ionize with different net charges as the pH changes  and material characterization are described in SI (Supporting Information). Ag-D is characterized by surface plasmon resonance (SPR), easy preparation, and surface modification. As shown in the Scanning electron microscope (SEM) image of Ag-D in Fig. ges Fig. , with pK−1 can be observed. They are for ν(Ag–S), ν(C–S), and ν(COO−) respectively, as shown in Table 38. It can be seen that L-cysteine directly bonded to the silver surface by S−, and that C–S bond has a clear signal enhancement and a relatively low wavenumber compared with the L-cysteine solid powder. It is speculated that in this environment, the sulfur generates a strong interaction  between the negative charge of the deprotonation of thiol group and positive side of the silver surface, and the increase in the COO− signal intensity indicates that the COO− and the silver surface have a strong interaction. It can be seen that this part can also be verified by the ν(Ag–O) shoulder at 247 cm−1. From this result, it is inferred that the adsorption behavior of L-cysteine is that COO− and S− adsorb on the surface of Ag-D, while NH2 is far from the surface, as shown in Fig. −1, and its vibration corresponds to ν(Ag–O), indicating that COO− forms a bond with the silver surface. The characteristic peak is ν(C–S) at 674 cm−1 shifts to 620 cm−1. As the pH value is lower, S gradually changes to the form of SH. No bond formed at this time, but it was still close to the surface of the Ag-D, so that the increase in this signal was observable. Therefore, it is inferred that the adsorption behavior is such that COO− and SH are oriented toward the Ag-D surface, and that NH3+ is relatively far from the surface, as shown in Fig. −1 and ν(COO−) at 1525 cm−1 increased, indicating that COO− and the dendritic silver surface bonded. In addition, there is a tendency for the shoulder peak at 1586 cm−1 to appear next to 1525 cm−1. It is presumed that the signal is enhanced with the vibration ν(NH3+) corresponding to 1127 cm−1, indicating that NH3+ is also close to the surface of Ag-D. Finally, it was observed that there was no characteristic peak of ν(C–S) between 600 and 750 cm−1, indicating that SH was far from the surface. Therefore, this adsorption behavior suggests that both NH3+ and COO− are oriented toward the Ag-D surface, and that SH is far from the surface, as shown in Fig. −1 and for ν(C–N) at 950–1050 cm−1. When the pH is 9.12, there are two structures, as shown in Fig. −1 gradually shifts to 620 cm−1. At this time, the torsion angle of L-cysteine χ1 (N–Cα–Cβ–S) is changed from trans to gauche. However, the ν(C–N) characteristic peak at 1024 cm−1 is also gradually shifted to low frequency to 973 cm−1. This phenomenon is presumed to be due to changes in adsorption behavior and structure, which in turn causes changes in the electron density of C–S or C–N and causes the change in Raman scattering signals.Figure In different acid–base environments, cysteine forms a chemical bond with the silver atoms on the surface of the dendritic silver SERS-active substrate. In addition to the electromagnetic field enhancement considering the SPR effect of dendritic silver, there is a clear chemical contribution. In addition, L-cysteine can be selectively adsorbed onto the surface of the Ag-D electrode by controlling the pH and potential. Thus, L-cysteine can be further used as a finishing layer for developing subsequent, broader applications by reaction with different exposed functional groups with additional added compounds.−1 is stronger than that of ν(C–N) 972 cm−1; only at 680 cm−1, is the ν(C–N) signal intensity consistent with the result for pH = 13.50 in Fig. −1 was gradually stronger than that of ν(C–N) at 1042 cm−1. This means that under the same illumination intensity, prolonged exposure causes L-cysteine to slowly undergo partial structure reorganization. When the laser power is increased to 80% (15.97 mW), it is clear that after 10 seconds of exposure, not only was the characteristic peak of 972 cm−1 significantly enhanced, but also the ν(C–S) of 680 cm−1 had a trend of signal enhancement. When the exposure time reached 30 seconds, the intensity of the two characteristic peaks became stronger. Increasing the light intensity can make L-cysteine result in faster structure variation, further changing the electron distribution of C–S and C–N and resulting in this growth and decline. The SERS spectrum with pH has been judged to have an adsorption behavior at pH of 13.50, as shown as structure A. In the literature39, the NH2, COO− and S− groups in the three structures of L-cysteine are close to the silver surface, and the torsion angle of χ1(N–Cα–Cβ–S) is the most stable state when it is structure B. Therefore, the inference is due to the fact that L-cysteine itself has a rotating dihedral angle. When the laser source supplied additional energy to L-cysteine, driving the original structure A, it was easier to change to a state structure B, which is adsorbed on the Ag-D surface. On the other hand, L-cysteine modified with pH = 5.22 was further exposed to 60% (12.18 mW) of the laser power, as shown in Fig. The effects of laser intensity and exposure time on the adsorption structure of L-cysteine were further investigated Fig. . The SERIt has been confirmed that the adsorption configuration of L-cysteine on the surface of the Ag-D electrode can be successfully controlled by the solution pH environment and the adsorption–desorption potential. Next, the functional groups of the chemisorbed modified layer will be further utilized to form a dimer with a laser-light-induced dimerization reaction with an added L-cysteine molecule. Upon oxidation of the two L-cysteine monomers, a cystine with a disulfide bond is formed. From the experimental results Fig.  it is kn−1 in the SERS spectrum, and no dimerization occurred at this time. The addition of a drop of 0.01 M L-cysteine aqueous solution onto the surface of the substrate will wet the surface and change the original chemical environment, so that the original signal disappears. When the reaction time was 6 seconds, it was observed that ν(S–S) gradually appeared, indicating that the thiol groups of the two L-cysteine monomers began to dimerize to form cystine37. After 150 seconds of continuous reaction, the intensity of the ν(S–S) signal peak had reached near saturation, and the relatively weak characteristic peaks of other signals were obvious; 602, 664, 1330, and 1401 cm−1 correspond to vibrations of ν(C–S), ν(C–S), δ(NC–H), and ν(COO−), respectively.The excitation wavelength was 532 nm, the objective lens was 50 times, and the integral time of each spectrum was 3 seconds/time for real-time reaction measurement. Figure −1 gradually appeared, and the other signal peaks increased with time. When the reaction reaches 90 seconds, the signal peak intensity tends to be saturated, indicating that the cystine almost completely formed at this time. Comparing the reaction times for pH = 1.67 and 5.85, we found that the reaction time of the latter was longer than that of the former. It is speculated that at lower pH values, the presence of L-cysteine is relatively simple. As shown in Fig. Figure −1 still does not appear. This means that under these conditions, cystine cannot be formed by dimerization reaction. However, we observed a slight signal peak; we presume that unreacted L-cysteine monomer molecules randomly located on the surface of the L-cysteine-modified Ag-D substrate. From the experimental results of the pH control, it is known that in this pH environment, the sulfur end of L-cysteine bond with the silver surface, and that there is no exposed thiol group; thus, cystine does not form.Finally, Fig. −1 corresponding to the ν(COO−) vibration of L-cysteine  was mapped by using an optical microscopy image. The SERS spectra of this range were scanned to match the results of L-cysteine modification on Ag-D at pH = 1.67 Fig. . Accordiine Fig. , the intsee Fig. . After tAfter a series of SEM, XRD, and XPS characterization, it was proved that the high-density Ag-D electrode can be prepared by electrochemical deposition method. It is suitable for surface-enhanced Raman substrate, and it can successfully utilize the electrochemical method of adsorption of L-cysteine on it.− and S− with the silver surface, and NH2 movement away from the surface ; or with NH3+ and COO− reacts on the silver surface, while the SH moves away from the surface (acid). The last one is when NH3+, COO−, and SH are all close to the silver surface. Therefore, it is possible to simultaneously adjust the environmental pH value and to control the adsorption and desorption potential, so that L-cysteine has different adsorption structures. However, in the SERS spectrum that changes the laser power and illumination time, L-cysteine is converted from the structure A (trans structure) to the more stable structure B (gauche) because of the applied energy (pH = 13.40).The structure of the adsorbed molecule L-cysteine can be optimized by adjusting the adsorption and desorption of the target molecule on the Ag-D surface multiple times by electrochemical potential. In the SERS spectrum of the effect of pH on the adsorption of L-cysteine, three different adsorption behaviors can be obtained by adjusting the pH. This is the action of COOFinally, the three adsorption states obtained under the control of pH were respectively subjected to oxidation reaction by light-induced L-cysteine. It was confirmed that the pH value at the most acidic and isoelectric point conditions, allows two L-cysteines to undergo oxidation reaction to form a disulfide because of the non-bonded thiol group. However, as the pH is higher, the reaction rate is slower and less likely to proceed. Two-dimensional Raman scattering spectroscopy confirmed the feasibility and uniformity of the coupling reaction. The modified cysteine on the dendritic silver electrode has the possibility of further react in the measurement range. It can be seen that in this study, we can effectively control the adsorption behavior of L-cysteine and that we can used it as a modification to improve the surface selectivity and to conduct more chemical reactions.35. The glassy carbon electrode (12 mm × 5 mm × 1 mm) was polished with of 0.3 μm and 0.05 μm alumina powder, and then ultrasonically shaken with ethanol and deionized water for 15 minutes. Finally, the glassy carbon electrode was rinsed with deionized water. The dendritic silver was directly deposited on the glassy carbon electrode and was easily peeled off. Therefore, before preparing the dendritic silver, nanogold was deposited on the surface of the glassy carbon electrode for 500 seconds, and then dendritic silver was deposited thereon. About 30.0 mL of 1.0 mM tetrachloroauric acid, 0.10 mM cysteine, and 0.50 M aqueous sulfuric acid solution was prepared. A three-electrode electrochemical system (CHI6111) was used. The reference electrode was saturated calomel electrode (SCE), the auxiliary electrode was platinum wire, and the working electrode was glassy carbon. We used the step function for gold electrodeposition for 500 seconds. The parameter settings were 0 to −0.8 V potential range, 0.1 s step time, and 5000 step segments. Next, dendritic silver deposition was performed. The solution was changed to 1.0 mM silver sulfate, 0.10 mM cysteine, and 0.50 M aqueous sulfuric acid solution, and we used about 30.0 mL. Three electrodes were reference electrodes: silver wire, auxiliary electrode (platinum wire), and working electrode (gold-modified glassy carbon electrode). We used the step function for silver electrodeposition for 1000 seconds. Parameter settings were −0.2 to −1.2 V potential range, 0.1 s step time, and 10000 step segment. The deposited dendritic silver electrode was taken out, washed with deionized water, and dried with nitrogen. The dendritic silver electrode should be washed with 0.2 M Na2S2O3 for 10 seconds to remove the surface Ag2SO4 residue to obtain a Raman substrate with no background interference signal. The dendritic silver obtained was identified by SEM, XRD, and XPS.Ag-Ds were prepared through a modification of our previous report3 at approximately 30 mL. We set up a three-electrode system: the reference electrode was silver wire, the auxiliary electrode was platinum wire, and the working electrode was dendritic silver electrode. We used a step function to adsorb L-cysteine for 1500 seconds. Parameter settings were potential range of −0.2 to −1.2 V, step time of 0.1 s, and step segment of 15000. The L-cysteine-modified dendritic silver electrode was taken out, rinsed, dried with N2 gas, and then measured. Samples were irradiated with a laser illumination of 532 (25 mW), focused on a sample of 1 μm diameter with a 50X objective lens , MRI532S, Protrustech Co., Ltd, Taiwan.First we prepared a 100 μM L-cysteine solution and adjusted the pH to be measured with KOH and HNO−1. The objective lens magnification was 50 times. The integration time was as indicated in the figure, and the instantaneous reaction measurement was performed. A drop of 0.01 M L-cysteine aqueous solution was placed onto the surface of the L-cysteine-modified dendritic silver obtained under the conditions of pH = 1.67, 5.85, 13.40. The coupling reaction was induced by a 532 nm light source while the change in the SERS spectrum was monitored during the reaction. The source intensity was 9.42 mW at the sample position.All Raman scattering spectra were obtained at 532 nm excitation wavelength was corrected by the signal peak of polystyrene at 1001.4 cmSupplementary Information


The charge-transfer electronic states appearing at the donor-acceptor interfaces in organic solar cells mediate exciton dissociation, charge generation, and charge recombination. To date, the characterization of their nature has been carried out on the basis of models that only involve the charge-transfer state and the ground state. Here, we demonstrate that it is essential to go beyond such a two-state model and to consider explicitly as well the electronic and vibrational couplings with the local absorbing state on the donor and/or acceptor. We have thus developed a three-state vibronic model that allows us: to provide a reliable description of the optical absorption features related to the charge-transfer states; to underline the erroneous interpretations stemming from the application of the semi-classical two-state model; and to rationalize how the hybridization between the local-excitation state and charge-transfer state can lead to lower non-radiative voltage losses and higher power conversion efficiencies. i.e., the local absorbing state on the donor and/or acceptor, needs to be considered.Previous descriptions of the charge-transfer absorptions in organic solar cells only involve the charge transfer state and the ground state. Here Chen et al. underline that a third state,  In OSCs, the excitons photogenerated in the donor (D) and/or acceptor (A) components dissociate at the D/A interfaces. This process results in either immediate long-range charge separation or formation of interfacial charge-transfer (CT) states; the nongeminate (bimolecular) recombination of initially separated charges also results in the formation of CT states. These interfacial CT states can either separate into free charges or recombine to the electronic ground state via radiative or non-radiative pathways. Thus, with respect to the energy of the local excitonic states on the absorbing species, there exist two significant sources of VOC (energy) losses, related to exciton dissociation and charge recombination that both are mediated by the CT states14. Thus, given the critical role played by these CT states, it is crucial to develop robust methodologies that provide reliable information on their electronic and optical characteristics.The most efficient organic solar cells (OSC) based on blends of conjugated polymers or oligomers (electron donors) and (non-)fullerene derivatives (electron acceptors) currently reach power conversion efficiencies (PCE) up to 14–15% in single-junction devices15. Then, extracting the electronic-structure information from the CT absorption bands has been commonly performed by analyzing the low-energy tail of the measured PDS or FTPS spectra in the framework of the semi-classical two-state Mulliken–Hush model. Importantly, this model exclusively considers the ground (G) electronic state and the full electron-transfer (Coulomb-bound electron–hole pair) state, which we refer to here as the D+A− state17; it completely neglects any potential role played by the strongly absorbing local-excitation (LE) state on the donor and/or acceptor, even though the LE and D+A− states have to be electronically coupled in order for exciton dissociation to take place.In most OSC materials, the CT absorptions are very weak, which means that highly sensitive photo-thermal deflection spectroscopy (PDS) or Fourier-transform photocurrent spectroscopy (FTPS) techniques are required to detect the CT statesVOC losses, a large number of experimental efforts have been devoted to develop D/A active layers in which there is as small as possible an energy offset between the LE and D+A− states25. In these instances, it can be anticipated that a strong coupling occurs between the LE and D+A− states; such a strong coupling has potentially a significant impact on the low-energy absorption of the D/A system and, consequently, on the charge-separation efficiency as well as on the radiative and non-radiative transitions. Therefore, especially in the case of such systems that form the basis of a next generation of efficient OSCs, it is critical to rely on a robust methodology that goes beyond the conventional two-state Mulliken–Hush approach.Recently, in order to reduce +A−, and LE states to investigate the electronic-structure properties in the framework of the perturbation theory27. Based on these early works, the impact of the LE-D+A− electronic coupling on the intensity of the CT absorption in polymer/fullerene systems has been discussed in terms of intensity borrowing, in the context of the weak electronic coupling limit28. Here, we go beyond the perturbation theory and develop a three-state dynamic vibronic model that incorporates the G, D+A−, and LE states in order to describe reliably the nature of the CT absorptions in D/A active layers. Thus, the electronic couplings and electron-vibrational couplings are both treated non-perturbatively and the vibrational modes are treated quantum mechanically. Our objectives in the present work are: (i) to provide a comprehensive description of the nature of the CT absorptions in D/A complexes; (ii) to bring caution to the use of the semi-classical two-state Mulliken–Hush models as they can lead to serious misinterpretations; and (iii) to describe a first application of our methodology to the PBTTT/PCBM and PIPCP/PCBM systems that have recently attracted a great deal of attention34.A number of earlier investigations on D/A small-molecule complexes or D-A isolated molecules have employed a three-state model that includes the G, D17:A(E) is the optical absorption intensity  per donor/acceptor pair; E, the photon energy; +A− state; +A−-state excitation energy  state. The related potential energies are shown in Fig. +A− state, in addition to being coupled to the ground state, is also coupled to the LE state by an electronic coupling The three-state model is obtained by considering in addition to the G and Dθ = 0° or 180° θ = 90° 6037, the It is informative to consider two model cases that are related to the main geometric configurations (edge-on and face-on packing modes) appearing at the D/A interfaces: one where the axis corresponding to the electron transfer from D to A is parallel , since the orientations of the D/A complexes in actual OSC active layers are expected to be combinations of these limit configurations. Note that, in what follows, we denote by “LE”’ the local-excitation absorbing state intrinsic to , say, the donor component, and by “D+A−” the pure charge-transfer state corresponding to a full electron transfer from D to A; the coupled states are denoted by “S1” and “CT”.It is well-established that the details of the donor/acceptor interfacial molecular packings greatly impact the electronic structure of the CT states+A− states: (a) 1.1 eV ; (b) 0.4 eV ; and (c) 0.2 eV . Without any loss of generality , we assume that the vertical excitation energy of the diabatic D+A− state is 1.0 eV (8065 cm−1); although the absolute value of the D+A−-state energy chosen here is small, this choice does not affect the conclusions derived from the three-state approach since, it is the energy offset 42: a high-frequency (HF) vibration mode (ℏωHF = 1200 cm−1), typical of a carbon–carbon bond stretch44, and a low-frequency (LF) vibration mode (ℏωLF = 100 cm−1), which represents rotations between intramolecular fragments as well as intermolecular motions46 ). The values of electronic couplings and electron-vibration couplings are chosen in such a way as to match commonly used literature values48; they are summarized in Table +A−-G electronic coupling (30 cm−1) while the D+A−-LE electronic coupling is varied within the range from 0 to 300 cm−1 in order to easily assess the role of the D+A−-LE electronic coupling in the absorption spectra.We consider three values for the offset, mentclass2pt{minimmentclass2pt{minimmentclass2pt{minim−1) for the configuration where the electron-transfer axis is parallel (θ = 0°) to the transition dipole of the LE state, i.e., 7 , a low-energy absorption shoulder, with an intensity some 102–104 times weaker than that of the pure donor absorption, can be clearly distinguished. When the D+A− state is considered not to be coupled with the LE state  for the CT absorption derived from our vibronic solutions .We start with a discussion of the results in the case where mentclass2pt{minim+A− state and the LE state has a marked effect on both the shape and intensity of the resulting CT absorption band. There are two contributions to the CT absorption band (see Eq. +A− state itself while the second is due to the hybridization between the intrinsic LE and D+A− states (corresponding to an intensity borrowing effect). Depending on the relative orientations between the dipole moment of the D+A−state 27. In the case where θ = 0°, these contributions are constructive. As a result, in comparison with the CT absorption band derived when −1, an increase in −1  results in an increase in absorption intensity by one order of magnitude.As seen from Fig. see Fig. . Importa+A−, and G states would be described by the semi-classical two-state model, which we recall is the model that has been applied to date in order to characterize the CT absorption bands. There are two procedures commonly used to interpret the CT absorption bands:It is instructive at this point to address how the CT absorption bands derived on the basis of an explicit consideration of the couplings among the LE, DμCT) can be obtained, and the electronic couplings 54. Here, taking as example the absorption spectrum simulated via our three-state model in the case where +A− state and the ground state since the absorption intensity (or transition dipole moment) borrowed from the LE state is incorrectly attributed to the D+A− state itself; (b) overestimates the relaxation energy in the D+A− state; and (c) importantly underestimates the D+A−-state energy, (a) The Mulliken–Hush Eqs  and 3)  3) are e via Eq. . This pr11. For example, for the absorption spectrum simulated via our three-state model in the case where −1 (0.70 eV)) is close to the input value in our three-state model (5713 cm−1 (0.71 eV)); on the other hand, the fitted −1 (0.27 eV)) is smaller than our input value (2352 cm−1 (0.29 eV)), which comes from the fact that the quantum effect of high-frequency vibration is neglected in the semi-classical two-state model. When a series of Gaussian functions is used to reproduce the whole CT absorption band , as done in the recent work of Belova et al.40 However, according to the three-state approach, these higher-energy absorptions correspond in fact to hot hybrid (D+A−-LE) vibronic states, which are induced by D+A−-LE coupling. Thus, our results bring to light another explanation for the higher-energy absorptions observed experimentally40.(b) In the OPV community, the usual procedure is to use a Gaussian function to fit the low-energy absorption tails, in order to extract the electronic-structure parameters of the CT statessee Fig. , the fitsee Fig. , the genθ = 180°, the contributions of the D+A− and LE states to the CT absorptions act in a destructive fashion. As a result, the D+A− and LE-state contributions cancel each other, especially in the low-energy tail  case are shown in Fig. +A− state is not coupled with the LE state, the CT absorption for small −1) that a slight shoulder appears in the low-energy region of the absorption spectrum  to  to 32 an33. A low-energy shoulder is clearly observed in the experimental absorption spectra, with an intensity about 100 times weaker than that of the neat polymer absorption , and 3510 cm−1 (0.44 eV) for μCT, 32. Based on this Gaussian fit32, the relaxed D+A−-state energy −1 (0.92 eV) and 5555 cm- 1 (0.69 eV), respectively, and the D+A−-G electronic coupling ca. 573 cm−1 (0.07 eV). In the framework of our three-state model, a very good comparison between the experiment and fit is obtained , 8400 cm−1 (1.04 eV), 5055 cm−1 (0.63 eV), and 450 cm−1 (0.06 eV) ; (b) an underestimation of the relaxed D+A−-state energy by 1000 cm−1 (0.12 eV); and (c) an overestimation of the relaxation energy in the D+A− state by ca. 500 cm−1 (0.06 eV). Note also that the results of a two-state Gaussian fit specifically depend on the chosen procedure, which can sometimes lead to a significant error.We first apply our three-state approach to the PBTTT/PCBM system, which is representative of systems with large see Fig. . In the see Fig.  when \domentclass2pt{minimVOC loss of about 0.52 eV (4193 cm−1)29. The experimental absorption spectrum in the blend is red-shifted by ca. 60 meV (484 cm−1) compared with that of the pure polymer, with no appearance of a low-energy CT absorption shoulder in either photo-thermal deflection spectroscopy (PDS) or external quantum efficiency (EQE) spectra29. The lack of a low-energy CT absorption shoulder in the experimental EQE spectra has also been reported in other, high-efficient systems with low-energy loss56. Now, we exploit the three-state vibronic model to fit the absorption spectrum of the PIPCP/PCBM blend.We now turn to the PIPCP/PCBM system  of the PIPCP backbone when going from the neat polymer film to the blend structure29. However, even when the absorption spectrum of the blend is displaced so that its vibrational structure matches that of the neat polymer absorption, the blend still shows an additional absorption feature at the low-energy edge ; a −1; and a −1 . Interestingly, the D+A−-LE electronic coupling is seen to be much larger than the D+A−-G coupling; this result is in fact consistent with the electronic couplings calculated via long-range corrected density functional theory (see Supplementary Table ECT – eVOC) energy loss of only ca. 0.4 eV (3226 cm−1). While there remain differences between the fitted and experimental spectra, the three-state approach is able to reproduce the main characteristics of the absorption spectrum: (1) the absence of any low-energy absorption shoulder; (2) the presence of vibronic peaks in the region above 1.6 eV (12904 cm−1); and (3) the red-shift of the low-energy absorption edge of the blend compared with that of the pure polymer. The three-state vibronic model locates the D+A− state about 0.2 eV (1613 cm−1) below the polymer LE state, which is about four times larger than the value of 50 meV (403 cm−1) estimated previously from the electroluminescence spectrum29.The experimental high-resolution transmission electron microscopy indicates that PIPCP in the blend has a somewhat higher crystallinity than in the polymer neat film.2 eV 161 cm−1; a 58, we assume the related reorganization energy to be ca. 0.3 eV (2419 cm−1). With −1) and ca. 1 × 1014 s−1. The experimental data of Menke et al. have shown that, in the PIPCP/PCBM blend, exciton dissociation after photoexcitation is very fast 31, which is consistent with a reasonable 0.2 eV (1613  cm−1) driving force and a large D+A−-LE electronic coupling.We note that the rate of exciton dissociation in PIPCP/PCBM can be crudely estimated via Marcus electron-transfer theory where, following earlier investigations59. As reported by Köhler et al. 51, static disorder can lead to a broadening of the CT absorption spectrum. Thus, when neglecting static disorder, the CT-state reorganization energies 59. Thus, our objective in future work will be to account for static disorder and electronic delocalization, although their inclusion is not expected to affect the main conclusions of the present work regarding the impact of the electronic coupling between the D+A− and LE states.Finally, we would like to point out that our present three-state approach is a starting point and does not consider other factors such as static disorder or electronic delocalization due to aggregation effects, which can also impact the CT absorption spectra+A−) and LE states on the characteristics of the CT states at the D/A interfaces.We have developed a three-state vibronic model, with which we can explicitly take into account the electronic and vibronic couplings between the charge-transfer states appearing at the donor/acceptor interfaces and not only the ground state but also the strongly absorbing, local-excitation (LE) state on the donor and/or acceptor. We recall that coupling to only the ground state (two-state model) has been the hallmark of the analysis tools commonly applied to date to characterize the CT states appearing in the active layers of OSC. Here, the application of the three-state vibronic model has allowed us to clarify the impact of hybridization of the pure charge-transfer (D(i)+A− and LE states, a low-energy CT-absorption shoulder is clearly apparent. However, the electronic coupling between the D+A− and LE states makes the shape and intensity of the low-energy shoulder substantially different from those that the semi-classical two-state Mulliken–Hush model would predict. In fact, application of the semi-classical Mulliken–Hush model to analyze the whole CT absorption bands leads to an overestimation of the reorganization energy and electronic coupling with the ground state and an underestimation of the adiabatic energy of the D+A− state. In addition, if a Gaussian fit to the low-energy CT absorption tails is performed, this procedure could provide a good fit to the adiabatic energy of the D+A− state, but also misinterpret the higher-energy absorptions (which correspond to hot hybrid (D+A−-LE) vibronic states induced by the D+A−-LE coupling).In the case of a large energy offset  between the D(ii)+A−-LE coupling gain a more substantial transition dipole moment; these hot CT vibronic states, which are directly optically accessible, have energies that can allow them overcome, entirely or at least in part, the electron-hole Coulomb barrier and to dissociate easily into free charge carriers. This feature could thus result in improved power conversion efficiencies.In comparison with the characteristics of the (two-state derived) conventional CT states, the hot CT vibronic states resulting from D(iii)+A− and LE states, which is representative of the next generation of highly efficient OSCs, the absorption spectra hardly show any low-energy absorption feature. Instead, the D+A−-LE coupling leads to a red-shift of the whole low-energy absorption edge, which is consistent with the experimental data reported for the PIPCP/PCBM blend. In such instances, a three-state model is mandatory in order to extract reliable information on the CT states from the absorption spectra.In the case of a small energy offset  between the DA number of important conclusions can be drawn:Overall, our work provides a robust and comprehensive framework to understand the CT absorption spectra at donor/acceptor interfaces and also provides a pathway for a reliable extraction of the electronic-structure parameters describing the CT states from these spectra. However, more experimental data, such as temperature dependence and polarization dependence of the absorption, would be desirable to achieve these parameters. Our model can be further applied to multiple donor/acceptor complexes in order to account for the effects of electron delocalization and static disorder. Taking into account a three-state vibronic approach or its extensions is necessary to adequately characterize the voltage (or energy) loss mechanisms in efficient OSC devices and to guide the development of improved systems.The total Hamiltonian for a donor/acceptor molecular pair can be written as:r and q are standard notations for the electronic and nuclear coordinates, respectively.Here, +A− diabatic states, respectively, the αth eigenfunction In the basis of the three electronic states η(q) denotes the expansion coefficients. The electronic wavefunctions q = 0) of the G state is selected as the reference point, see Fig. Here, mentclass2pt{minimqi, the eigenenergies The eigenenergies ith vibrational normal mode with energy ℏωi, which is defined by λ of the D+A− and LE states  in Eq. (χk(qi)〉60:The full dynamic solution of the vibronic Hamiltonian given by Eq.  can be o) in Eq.  in termsΨα is the superposition of the |ΦG〉, |ΦLE〉 and ci terms are the expansion coefficients. By using a finite but large enough number of vibrational functions (large k), the eigenenergies and eigenfunctions of the vibronic Hamiltonian can be obtained with any desirable accuracy.Each adiabatic solution f(Eα) stands for the thermal (Boltzmann) population of the vibronic state Eα; 60. In the case of vibronic models including a few effective modes, the calculated vibronic peaks are separated by an energy related to that of the lower-energy vibrational mode. In these instances, it is a standard practice to use a Gaussian smearing of the vibronic peaks. In order to obtain a smooth profile of the simulated absorption spectra, we used a Gaussian broadening with a width (100 cm−1) set according to the energy of the low-frequency vibration (100 cm−1). We note that this procedure has a negligible impact on the overall width and asymmetry of the simulated absorption spectra. The transition dipole moment, μCT, of the CT absorption can be then obtained via the integration of A(E)/E54.Based on the calculated eigenenergies and eigenfunctions of the vibronic Hamiltonian, the optical absorption intensity  per donor/acceptor pair is obtained from:X axis, and the difference between the dipole moments of the G and D+A− states is given by 35, where Without any loss of generality, we assume that the LE state is localized on the donor and that the transition dipole moment Supplementary Information


Nature Commununications; 10.1038/s41467-018-07070-8, published online 7 Nov 2018Correction to: The original version of this Article contained an error in Fig. 2. In panel a, the two legend items “rare” and “common” were inadvertently swapped. This has been corrected in both the PDF and HTML versions of the Article.


We propose a generalization of the Su-Schrieffer-Heeger (SSH) model of the bipartite lattice, consisting of a periodic array of domain walls. The low-energy description is governed by the superposition of localized states at each domain wall, forming an effective mono-atomic chain at a larger scale. When the domain walls are dimerized, topologically protected edge states can appear, just like in the original SSH model. These new edge states are formed exclusively by soliton-like states and therefore, the new topological states are qualitatively different from the regular SSH edge states. They posses a much longer localization length and are more resistant to on-site disorder, in marked contrast to the standard SSH case. Different experimental setups, such as plasmonic nanoparticles and phononic lattices, have been used as practical realizations of one-dimensional models8. This emergence of novel experimental escenarios, in turn, has motivated further theoretical research, often on basic aspects or phenomena beyond the scope of common electronic systems, such as non-linearities10 and Floquet insulators12. The interest on one-dimensional topologically protected modes is not only related to basic understanding, but to the practical implementation in the design of low-loss devices14.The last years have witnessed a growing interest on one-dimensional models of non-trivial topological systems. This has been largely favored by the rapid advance of photonics and nano-photonics as an ideal playground to experimentally corroborate theoretical predictions15 (SSH). It consists on a tight-binding model for the bipartite lattice, and displays soliton-like localized edge states at domain walls (i.e. stacking faults of the bipartite lattice). Recently, this model has been the subject of generalizations in order to observe new phenomena17. Other one-dimensional models, not restricted to the bipartite lattice, have been proposed and realized, showing novel edge states20.Perhaps, the most studied one-dimensional model with a non-trivial topology is the Su-Schrieffer-Heeger model of polyacetylene21. Similarly, it was shown that some edges states in 2D systems with a negligible spin-orbit coupling -for instance some terminations of black phosporus- are indeed topologically protected22. Also, the natural extension of the SSH model to two dimensions can host different topological phases, even with a zero Berry phase23.One of the main purposes behind the focus in simpler models as the SSH is to get insight on more complex systems or materials. For instance, depending on the termination, a non-trivial Zak phase in nanoribbons can arise, implying topologically-protected edge statesN interacting copies of the SSH model, originated from an array of domain walls. If the lattice of domain walls is dimerized -a bipartite lattice of domain walls-, we recover a SSH-like behavior formed by the superposition of localized modes. Naturally, this model has a non-trivial phase featuring edge states. In Sec. 3 the effect of disorder in the model is studied. Unlike the SSH model, the new edge modes are somewhat robust to on-site disorder.In this article we start by reproducing the results of the famous SSH model, Sec. 1, to set the notation and make its generalization easier. In Sec. 2, we introduce a new model, consisting on 15, or more precisely, the tight-binding formulation of the bipartite lattice, see Fig. 24. In the optical context, the edge states from this model have been experimentally observed and theoretically explained in photonic superlattices25.We start with a brief summary of the SSH model of polyacetyleneIn a finite bipartite lattice, the two different orderings , Fig. t and v are the hopping amplitudes and each cell has two equal –but inequivalent– sites a, b (e.g. the sub-lattices). In spite of its simplicity, this system has a very rich physics. In the case of t < v, it has  zero-energy and sub-lattice polarized edge states solutions. In the limit N → ∞ the zero-energy modes have a simple expressionψL is localized at the left edge of the chain, with width ψR is localized at the right edge. α, β are normalization constants. For a finite chain size, N, the interaction between both edge states is small but finite The one-dimensional bipartite lattice, see Fig. entclass1pt{minimai.e. excluding edge states) is The bulk-like states, have energies entclass1pt{minimaN sites, after imposing periodic boundary conditions, i.e. by setting Both bonding schemes of the bipartite lattice, see Fig. ψL, ψR, Eqs , ψa is localized at j = 0 and ψb localized on ε just reverses the positions of the DW centers.We can use the previous solutions  ψR, Eqs  and 3 asM mono-spaced DWs –see Fig. hN, h′N are SSH-like Hamiltonians, but with different topological phase. The periodicity in HPDW is two DWs, or 2N cells, but the spacing between domain walls is just N. This Hamiltonian is quite cumbersome, but if we focus on its low-energy excitations, it can be greatly simplified, by just keeping the superposition of modes ψa, ψb, at each domain wall, see Eqs (A bipartite lattice with  see Eqs  and 6. UN sites is an effective ‘cell’ with two sub-lattices per cell (i.e. both DWs), just like in the standard bipartite lattice, but with just one single hopping t′. Therefore, the effective low-energy excitations are just like those in a monoatomic chain with period N, see Figs mN, (2m + 1)N aremN, we need to multiply Eq. (m + 1)N. The simple interpretation of this b sites, see the lower inset from Fig. Each group of 2d in Eqs  and 6. Wd in Eqs  is alreaα′, β′ are almost independent of N, N the first term in the parenthesis dominates, but in some contexts -like in optics- the most common arrays consist of a limited number of waveguides.The interaction between two localized states is, approximately:locally sub-lattice polarized , but on a larger scale (NM sites) the sub-lattices are mixed. Also, their energy is genuinely finite, forming an s-band, Fig. 26 and the results agree with ours.The low-energy states given by Eqs  or 10),,10), areIn this section we introduce a bipartite lattice of domain walls, starting by its Hamiltonian, and derive its low-energy version, which is a new version of the SSH model, but with smaller hopping strenghts. As the SSH model, this new model has a phase with topologically protected edge states, see Fig. i.e. by setting the spacing from one DW to the next one as N cells to the right and N′ cells to the left. This changes slightly the Hamiltonian of a periodic lattice of DWs, from HPDW, Eq. , and in the middle of it two zero-energy states appear. These states are built from DW states, localized at the edges of the system and are fully sublattice polarized  is the kernel of the Hamiltonian. We can decompose h(k) as a linear combination of the Pauli matrices σx, σy, σz),t′ + v′cos(k), v′sin(k), 0). Figure N, N′, and keeping the original hopping strengths constant. The geometric place of this vector is a circle and we can define a topological invariant related to it, the winding number or Chern number:nc is very simple, it counts how many times the curve k) encircle the origin. The two possible values of nc = {0, 1} defines the phase diagram of the system24, see Fig. N > N′ the system is in the ‘topological phase’, that is nc = 1, conversely N < N′ is the trivial phase with nc = 0. In the remaining case, N = N′, the curve nc = 1 has topologically protected edge states.The analogy with the regular SSH modelian, Eq. , gives19While ian, Eq.  in the n27. The band gap due to off-diagonal disorder is not shown since in the SSH model it is below our numerical precision. For the edges states of the BDW model, it lies below 10−4 for a disorder amplitude of δ = . In smaller SSH chains the off-diagonal disorder is able to close the  gap, but the chains considered in Fig. The resilience of topologically protected states in the SSH model to off-diagonal disorder is well-known, as well as its weakness against disorder on the on-site energiesThe non-uniformity of on-site disorder directly breaks the chiral symmetry, destroying the topological protection. Figure εBDW –which averages to zero for large εBDW. In the Figure, the asymptotic-like band gap is well below the band gap of the bands formed by the DWs states, which in turn is much smaller than the bandgap from the SSH model. Given t = 1, v = 1.5, the characteristic length of the SSH model is t, v– the characteristic length of the SSH model, see Eq. . At moderate disorder . The Blöch states of the SSH lattice become localized with disorder, regardless of whether it is on-site or off-diagonal disorder. The regular Blöch states of the BDW have a very similar behavior -they are Blöch states too-, but they are slighty more localized for any finite value of δ. This is a consequence of the ‘fragmentation’ of the extended states  into smaller groups with smaller bandgaps –see the green region in Fig. IPR of these states is very similar to the regular Blöch states, and for clarity they are not included in Fig. To get a deeper insight on the effect of disorder on both, the SSH and the BDW models, we calculated the inverse participation ratio (IPR)IPR doubling its value for tiny amounts of on-site disorder. When δ is of the order of the interaction between the edges states, they no longer form an bonding anti-bonding pair, i.e., the IPR should be ~0.18 for no disorder), but it shows that a very small on-site disorder can prevent the occurrence of charge fractionalization in the model29.The BDW’s edges states  and local disorder (Anderson) is a necesary step towards the design of future robust low-loss devices. The effect of nonlinearity on the topological robustness of these systems is another subject of interest (under investigation) and will be reported elsewhere.


Scientific Reports 10.1038/s41598-018-33609-2, published online 12 October 2018Correction to: The original version of this Article contained an error in the title of the paper, where the word “graphene” was incorrectly given as “grapheme”. This has now been corrected in the PDF and HTML versions of the Article.


Scientific Reports 10.1038/s41598-018-25119-y, published online 27 April 2018Correction to: A supplementary file, “Example of a dual-task trial for the Feedback group” was omitted from the original version of this Article. This has been corrected in the HTML version of the Article; the PDF version was correct at time of publication.


Longitudinal studies of tumor volume have used certain named mathematical growth models. The Bertalanffy-Pütter differential equation unifies them: It uses five parameters, amongst them two exponents related to tumor metabolism and morphology. Each exponent-pair defines a unique three-parameter model of the Bertalanffy-Pütter type, and the above-mentioned named models correspond to specific exponent-pairs. Amongst these models we seek the best fitting one.The best fitting model curve within the Bertalanffy-Pütter class minimizes the sum of squared errors (SSE). We investigate also near-optimal model curves; their SSE is at most a certain percentage (e.g. 1%) larger than the minimal SSE. Models with near-optimal curves are visualized by the region of their near-optimal exponent pairs. While there is barely a visible difference concerning the goodness of fit between the best fitting and the near-optimal model curves, there are differences in the prognosis, whence the near-optimal models are used to assess the uncertainty of extrapolation.For data about the growth of an untreated tumor we found the best fitting growth model which reduced SSE by about 30% compared to the hitherto best fit. In order to analyze the uncertainty of prognosis, we repeated the search for the optimal and near-optimal exponent-pairs for the initial segments of the data (meaning the subset of the data for the first n days) and compared the prognosis based on these models with the actual data (i.e. the data for the remaining days). The optimal exponent-pairs and the regions of near-optimal exponent-pairs depended on how many data-points were used. Further, the regions of near-optimal exponent-pairs were larger for the first initial segments, where fewer data were used.While for each near optimal exponent-pair its best fitting model curve remained close to the fitted data points, the prognosis using these model curves differed widely for the remaining data, whence e.g. the best fitting model for the first 65 days of growth was not capable to inform about tumor size for the remaining 49 days. For the present data, prognosis appeared to be feasible for a time span of ten days, at most. Historically, the systematic application of mathematical models for tumor growth has begun in the 1960s . In the v(t) in mm3 over time t in days, using five model parameters that are to be determined from fitting the model to size-at-age data: Four parameters are displayed in the equations, namely the non-negative exponent-pair a < b and the constants p and q. A fifth parameter is the initial tumor volume at the start of monitoring, i.e. v (0) = v0 > 0.This equation describes tumor volume a, b) defines a unique model with three free parameters . Thus, for these models the “model muddle” can be reduced by considering them within the context of the larger unifying class (1) of models. Figure a, b) =  defines exponential growth (i.e. v(t) = v0·eq·t-, assuming p = 0, q < 0), and bounded exponential growth (i.e. v(t) = (p/q)·(1-d·eq·t-), assuming p, q, v0 > 0 and defining d from these parameters). The logistic growth model of Verhulst [a, b) = . The Gompertz [a = b = 1; it uses a different differential equation [In this paper, we perceive eq.  as a defng class  of modelVerhulst  is definng class  of modelGompertz  model isequation . These ma = 1, b > 1) and the generalized Bertalanffy model  are represented as classes of models. In the theory of economic growth, the latter model (class) is known as Solow-Swan model [Richards’  model Fa = , b > 1 aan model –22.a = 2/3, as the inflow of energy would be proportional to the surface area , and the exponent b = 1, as the energy needs for sustenance would be proportional to volume (cell count). This model appears to be plausible for the avascular stage of a solid tumor (nutrients enter only through the periphery). However, other authors proposed different biophysical explanations of growth and different exponent-pairs [a) may be above the value 2/3 of [v´/v over time, as these would inform about metabolism [A drawback of this type of phenomenological models is the difficulty in relating the comparatively easy to observe macroscopic data (size-at-age) to actual biological processes. According to von Bertalanffy , 14, thent-pairs , 24. Thue 2/3 of . Furthere 2/3 of , a statie 2/3 of , 27 anale 2/3 of  in the ctabolism .A different modeling approach describes tumor growth at the more detailed tissue scale in terms of partial differential equations related to invasion-proliferation and diffusion-reaction; e.g. Fisher-Kolmogorov equation , 30. ForWe reconsider the findings of . They coAs these findings depended on a few models only, and as there is no generally valid tumor growth model, which ensures a clear understanding and prognosis of tumor growth, the present paper revisits this issue and considers models from a more comprehensive class. The differential eq. (1) defines such a class that encompasses the most popular models (see above). We therefore aim at comparing the models from the model class (1) in terms of their goodness of fit (see methods) to the data of  and we a− 30) may result in very poorly fitting model curves.This approach has the following advantages: First, using a larger class of models with different growth patterns for comparison will provide a high flexibility in data-fitting and results in a better fit of the overall optimal model; for such a model we expect a better prognosis. Second, using a large class of models allows for a better comparability of the results with the literature. For even if a certain model may not be an element of the large class, some other element of the class may approximate it. And third, this approach allows estimating the uncertainty of prognosis. We claim that there may be limits to prognosis that are inherent to the research problem (c.f. prognosis of weather), and that near-optimal models might highlight these limits. For near-optimal models –34 the fa, b, p, q, and v0 of (1) may fail with standard optimization software. We will therefore also explain the optimization tools.This working plan has not yet been implemented in literature, because the optimization of the five parameters We aim at explaining the findings of  and we tWe demonstrate our method to find the best fitting and the near-optimal models for this dataset, only, as the paper aims at a “proof of principle”. While the conclusions about the limitations of prognosis may not apply to other data, the method to obtain such conclusions for concrete data generalizes. As a visual inspection of the data Fig.  would sua = m·0.01, b = a + n·0.01, with integers m ≥ 0, n ≥ 1). If the exponent-pair of the best fitting model was attained on the upper boundary line or on the right boundary line of this grid . Thus, if the exponent-pair with the best fitting model curve was a grid point next to the diagonal, then there remained the possibility that a Gompertz-type model could fit better. However, we expected that the improvement would be small. For, as noted by [a < b.The diagonal tial eq.  needs tonoted by , the GomSSE, which is defined as follows; v(t) is a solution of eq. (1), using certain exponents a < b and parameters p, q, v0, and vi are the data from above: assesseda, b) defines a unique model from class (1) with three free parameters . For each of these models (exponent-pairs) we wished to identify the best fitting model parameters p, q, v0. Thereby, we used the method of least squares; i.e. for each exponent-pair  on the grid we sought parameters  that minimized SSE. This defined the following function on the grid to identify, for each exponent-pair a < b, the minimal SSE that can be achieved:Thus, each exponent-pair  from the grid of Fig. SSEopt attained the minimum over the grid at this exponent-pair. The exponent-pair was defined as near-optimal, if SSEopt was at most a certain percentage  larger than this minimal SSE.In terms of this definition, an exponent-pair  are normally distributed, we tested this assumption using the Anderson-Darling and Cramér-von Mises distribution fit tests [As using it tests . We inveAIC [Alternative criteria for model selection, e.g. the well-known Akaike information criterion AIC –42, focuAIC , for theSSEopt (i.e. optimization) was done using Mathematica® 12.0 software of Wolfram Research. The authors provide a Mathematica file as supporting material. The output was exported to a spreadsheet, Microsoft Excel®. It is provided as a supporting material in the format ) of the table.The computation of a, b) were visited by means of an outer loop running through a = m·0.01 and an inner loop that for each a ran through the values b = a + n·0.01. Given , the optimization of p, q, and v0 was done using a custom-made variant of the method of simulated annealing [p, q, v0 > 0 for optimal parameters. The candidate parameters were altered by multiplying them with positive random numbers close to 1.  These new parameters were accepted as new candidates, if for them SSE became smaller, but in order to escape from suboptimal solutions, with a certain probability new parameter values were also accepted, if for them SSE became larger.  This step was then repeated with the accepted parameter. At the begin of each inner loop, i.e. at a grid point near the diagonal of the form , the optimization of p, q, v0 used 50,000 of these simulated annealing steps, starting with the estimate p = q = 1 and the initial condition v(t1) = v1 (first data point). For the subsequent grid points in the b-direction (inner loop), the previous optimization results were used as starting values and improved in 10,000 annealing steps. Thereby, after each 1000 steps the simulated annealing restarted with the hitherto best parameters. Further, the probability of accepting parameters with a higher SSE was lowered slightly. (This was made dependent on the hitherto obtained optimization results and is known as adaptive cooling: [SSEopt exceeded the minimum SSE by at most 1%) was visually inspected. Where the plot had a frayed appearance, the optimization exercise was repeated with more simulated annealing steps. CPU-time for the computations took about one week.For optimization, the grid-point exponent-pairs  To assv0, p, q (Mathematica command NonlinearModelFit) and further improve SSEopt.The optimal parameters from simulated annealing were finally used as starting values for a nonlinear regression for the model with the optimal exponent-pair, using standard methods to determine SSEopt  = 1.274·105 with the parameters v0, p, and q from the caption of Fig. v0, p, and q as starting values for a standard nonlinear regression for the model defined from the exponent-pair a = 1.62, b = 2.44 resulted in a slight improvement below the displayed accuracy. Figure SSEopt = 1.61·105.  was sought for. Figure  1.62, 2.4 = 1.27461·105. ( obtained3 (computed as the limit of the model curve v(t) for infinite t) remained close to the maximally observed volume  Further, as shown by Fig. 3. Distribution fit tests did not refute the implicit assumption for using the method of least squares, normally distributed fit residuals .The best fitting model curve supported the hypothesis of bounded growth, as its asymptotic volume of 4,034 mmxcessive ) and as To explore the potential for prognosis,  fitted sSSE = 33,700  = 32,177 and identified a still smaller SSEopt  = 32,087 (rounding to integers).Table Figure b, necessitated the use of extremely small parameters, q. The frayed character of the red region reflects the numerical problems of using such exponents and parameters; due to such problems conventional all-purpose optimization software was doomed to fail. For the former data, the optimal exponent-pair was still on the upper boundary of the search grid, whence the optimality of the exponent-pair was not secured.The optimization for the data for 98 and 107 days was particularly time consuming, as 63,377 and 64,150 grid points were searched. For the latter data, Fig. Figure Our results confirm the finding of , that thFigure For a practitioner, who uses the past data to extrapolate into the future, the failure of forecasting may indicate problems for the patient, e.g. a different phase of growth, where the apparent trend of the growth curve changes due to a biological cause (e.g. angiogenesis). It may indicate problems with the data, such as the presence of outliers. Or it may merely indicate that the true nature of the growth curve could not be identified, because its S-shape could not (yet) be discerned from the data.v´/v over time for the best fitting models of the top-1% of the near-optimal exponents. Its reverted U-shape suggests that the tumor size may have approached the carrying capacity, whence further growth would be inhibited by lack of resources, unless other drivers of growth (e.g. angiogenesis) were activated. This information might not have been readily available, if v´/v were estimated from a numeric differentiation of the data (blue line).For the present data the latter reason may apply, as Fig. The analysis of relative growth rates in Fig. Further, the size of the region of near-optimal exponent-pairs is related to the information inherent to the growth data: The larger the region is, the fewer information can be retrieved, as for a larger region the data would be compatible with more (too many) possible shapes of the growth curve. As was shown in Fig. For the data of  the progHowever, even for peculiar data, prognosis is not futile, as for practitioners any discrepancy between observed and forecasted growth may be an important warning signal that the tumor biology may change. The present paper provided methods for a more accurate prognosis.v´/v. However, for the present data also this analysis of the past did depend on how much information about the growth was available at the time when the assessment was done. For, the assessment switched from an initially decreasing relative growth rate, if only seven or eight data points were considered, to an initially increasing relative growth rate, when more data were utilized (Fig. In addition to prognosis, practitioners may use best fitting model curves to assess the character of past growth in terms of the relative growth rate zed Fig. .