Bertrand Rivet
email: bertrand.rivet@gipsa-lab.grenoble-inp.fr

Hubert Cecotti

Margaux Perrin

Emmanuel Maby

Jérémie Mattout

Adaptive training session for a P300 speller brain-computer interface

Keywords: brain-computer interface, P300 speller, adaptive signal processing, angles between subspace, xDAWN spatial filters, Bayesian Linear Discriminant Analysis

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Introduction

Brain-Computer Interfaces (BCI), which aim at providing a direct communication between the user's brain and a computer, have recently became an existing pluridisciplinary field of research, e.g., [START_REF] Wolpaw | Brain-computer interfaces for communication and control[END_REF]. Indeed the BCI field requires research effort in many aspects: signal processing (e.g., signal enhancement), machine learning (e.g., classification), user interface (e.g., ergonomics) and of course neuroscience for understanding the different neural processings. In fact, most of the recent progress in BCI during the last decade has been achieved by coupling knowledge from different research fields.

While BCI is a pluridisciplinary field of research, the evolution of each field does not progress at the same pace over time. The brain responses that are being detected have been known for decades in neuroscience, e.g., [START_REF] Vidal | Toward direct brain-computer communication[END_REF]. Machine learning has grown a lot in the last two decades. To some extends, some signal processing and classification techniques remain relatively simple but they are efficient, due to the noisy brain signals aspect and Occam's razor principle. It is the case of Linear Discriminant Analysis [START_REF] Fisher | The use of multiple measurements in taxonomic problems[END_REF], which stays very popular in the field. Moreover, in spite of the recent improvements of BCI, they shall face the hard reality of their early and still pending promises, e.g., pragmatic problems for potential daily user. This paper focuses on some practical aspects for non-invasive BCI.

The BCI performances are both related to the performance of the signal processing techniques that are used for assigning some EEG signal to a command and to the possibility of the user to adapt herself/himself to the system over time. Indeed, advanced machine learning, signal processing and classifier techniques have been widely used for improving BCIs [START_REF] Bashashati | A survey of signal processing algorithms in brain-computer interfaces based on electrical brain signals[END_REF][START_REF] Lotte | A review of classification algorithms for EEG-based brain-computer interfaces[END_REF][START_REF] Müller | Machine learning for real-time single-trial EEGanalysis: From brain-computer interfacing to mental state monitoring[END_REF]. In spite of these recent improvements in the BCI community, several obstacles remain to fully transfer laboratory demonstrator BCIs to commercial applications. Whereas tuning the different parameters of a BCI in relation to a specific individual can improve the performance of the system, this procedure is time consuming. For training a classifier, a large database containing labeled EEG signal is often needed. To obtain these data, a training session is required where the subject has to follow a calibration procedure which can be long and discouraging.

Several types of potential BCI users can be distinguished. First, BCIs are usually dedicated to people with severe motor disabilities who are unable to communicate through any other means. For these people, a BCI is the only way to communi-cate and the main challenge is to have a functional BCI. While reducing the time of the training session can be a certain advantage, it is not the main objective. For persons who suffer from severe disabilities, like the locked-in syndrome, having a working BCI can still be challenging. Besides, one may add that for some particular disabilities, it would be more advantageous to consider an invasive solution. Second, BCIs can be used by people with disabilities or elderly people who need a device to facilitate their daily life. In this situation, current challenges in BCI are to get a reliable BCI, which could be efficient daily in real condition. For this group of potential users, the time dedicated to the training session and the expected performance shall be well balanced. Third, BCIs can be used by healthy people as a complement to other device in multi tasks situations. In this type of situation, a BCI should ideally be flawless: the calibration step shall be as short as possible. Indeed, once the excitement of controlling something with the mind has passed, the usability stays an important satisfaction criterion. Some potential BCI users, even disabled, are highly demanding in term of performance and usability. The last decade has shown that BCIs can effectively work, they should now ideally become plug'n'play [START_REF] Cecotti | A self-paced and calibration-less SSVEP based braincomputer interface speller[END_REF]. The capacity to transfer technologies to commercial applications will be critical for the future of BCI.

On a more practical aspect for BCI users, the calibration step can be a drawback. With a lot of available data, it is possible to train classifiers and to improve the reliability of the detection procedure. However, one may wonder when the signal processing steps show their limits and it becomes useless to pursue the training procedure. Therefore, a challenge is to determine the ideal training session duration for the personalization of a BCI. On one hand, a too short training session would not be enough and would involve the creation of a poorly trained classifier. On the other hand a too long training session would frustrate the user, decrease her/his motivation. The calibration step shall not be considered as the final state of the system adaptation but only one first step for providing an efficient system: it allows to switch automatically between the training session and the online spelling use. For instance, classifiers can be trained in an unsupervised way over time in such fashion that the system adaptability stays invisible to the user. In addition, the user can improve his performance by finding appropriate ways to adapt his/her behavior to the system or to some feedbacks [START_REF] Martinez | Fully online multicommand brain-computer interface with visual neurofeedback using SSVEP paradigm[END_REF][START_REF] Mcfarland | Brain-computer interface signal processing at the wadsworth center: mu and sensorimotor beta rhythms[END_REF][START_REF] Moore | Real-world applications for brain-computer interface technology[END_REF].

In this paper, we focus on the problem of the required time that is needed to train a classifier and provide efficient results for the P300-Speller. In other words, how much time a subject has to spend during an experiment to efficiently determine the features of his/hers. In the P300-Speller, the duration of the training session can be evaluated as a number of characters to spell. The duration for spelling a character is indeed determined by initial setting of the speller like the inter-stimuli interval. This research is part of an ongoing effort in the French BCI community to reduce the time dedicated to the calibration step of the P300-Speller.

The paper is organized as follows: each step of the P300 detection, from the spatial filtering methods till the classification, is described in the second section. The experimental protocol is then detailed in the third section. Finally, the results of the offline classification are presented and discussed in the last section.

P300 speller and material

P300-Speller brain computer interface

The BCI problem addressed in this paper concerns the P300speller, which enables people to write a text on a computer [START_REF] Farwell | Talking off the top of your head: toward a mental prosthesis utilizing event-related brain potentials[END_REF]. It is based on the oddball paradigm: the task is to discriminate between epochs containing a P300 potential evoked by a (rare) target stimulus and epochs associated with the (frequent) non-target stimuli. In the classical P300 BCI, a 6 × 6 matrix containing all the available symbols is displayed on a screen (Fig. 1). To select a symbol, the user has to focus her/his attention on the symbol she/he wants to spell. The stimuli are the intensification of the rows and columns in a randomly order. Hence, the intensification of the target symbol (i.e. corresponding row and column) is a rare and unexpected event which elicits a P300 potential (i.e. a positive voltage deflection with a latency of about 300ms). Stimulation is organized in blocks of 12 flashes such that each row/column is intensified once per block in a random order. Blocks of 12 intensifications are repeated N epoch times for each symbol: therefore 2 × N epoch epochs amount the 12 × N epoch possible epochs should contain a P300 potential. To identify these epochs allows to determine the target symbol as the intersection of the detected row and column.

Data and pre-processing

Data were recorded from 20 healthy subjects (age = 26±5.7 years, 13 males and 7 females) [START_REF] Maby | The OpenViBE P300-speller scenario: a thorough online evaluation[END_REF]. Subjects were wearing an EEG cap with 32 active electrodes (actiCap, Brain Products GmbH, Munich). The electrodes for reference and the ground were placed on the nose and on the forehead, respectively. Data from five sessions have been recorded:

• training session : 50 symbols with 10 epochs (i.e. N epoch = 10),

• test session 1: 60 symbols with 3 epochs (i.e. N epoch = 3),

• test session 2: 60 symbols with 5 epochs (i.e. N epoch = 5),

• test session 4: 60 symbols with 10 epochs (i.e. N epoch = 10).

For training session, the inter-stimuli interval (ISI) is equal to 170ms. For each test session, four different values of ISI were tested (110ms, 150ms, 190ms, and 230ms) with 15 symbols for each value. It is worth noting that different ISIs may impact the classification accuracy, however several values of ISIs have been used to test the robustness of the proposed algorithm. For all these sessions, the stimulus onsets are known as well as the corresponding intensified row or column. The users have to copy words, as a consequence the target stimuli onsets are known for all the training and test sessions but this information is only used with the training session. EEG signals were initially sampled at 100Hz. Data preprocessing included a fourth order Butterworth bandpass filter between 1Hz and 20Hz.

Methods: adaptive training session

Spatial filters

Recorded EEG signals not only contain the P300 potentials but also ongoing cerebral activity and eventually muscular/ocular artifacts. A usual step to enhance a particular brain pattern is to use spatial filters: e.g., independent component analysis (Comon and Jutten, 2010;[START_REF] Xu | BCI Competition 2003-Data Set IIb: Enhancing P300 Wave Detection Using ICA-Based Subspace Projections for BCI Applications[END_REF], common spatial pattern [START_REF] Koles | The quantitative extraction and topographic mapping of the abnormal components in the clinical eeg[END_REF][START_REF] Blankertz | Optimizing spatial filters for robust eeg single-trial analysis[END_REF] or more recently xDAWN algorithm [START_REF] Rivet | P300 speller" Brain-Computer Interface: Enhancement of P300 evoked potential by spatial filters[END_REF][START_REF] Rivet | xDAWN algorithm to enhance evoked potentials: Application to brain-computer interface[END_REF]. This latter method (Appendix A) is chosen since it has been shown to provide accurate spatial filters. Let X (i) ∈ R N t ×N s denote the recorded signals related to the i-th symbol, where N t and N s are the number of samples and sensors, respectively. xDAWN algorithm provides spatial filters Û(I) ∈ R N s ×N f , with N f the number of spatial filters and I the set of training symbol used to train these spatial filters. The enhanced signals S (i) ∈ R N t ×N f of the i-th symbol are thus given by S (i) = X (i) Û(I) .

Classification

The enhanced signals are then segmented in order to identify which row/column elicits a P300. Each segment that represents a row/column includes all samples between 0 and 1s after the beginning of the intensification for all sensors: S (i) (k, j) ∈ R N e ×N s is then the enhanced signal of the j-th repetition of the i-th symbol related to the row/column indexed k. N e is the number of samples corresponding to 1s (in this study, N e = 100). Finally, the corresponding feature vector p (i) (k, j) ∈ R (N e N s /4) is obtained from the concatenation of all time-course samples in the enhanced signals S (i) (k, j) which have been decimated by a factor four (i.e. resampled at 25Hz). Among the proposed classifiers that are considered for BCI, Bayesian linear discriminant analysis (BLDA) is chosen since it has been proved to be efficient [START_REF] Hoffmann | An efficient P300-based brain-computer interface for disabled subjects[END_REF]. The discriminant vector d (I) is estimated from the set of pairs {p (i) (k, j) , t (i) (k, j) } i∈I,1≤ j≤N epoch ,1≤k≤12 obtained from the set I of symbols in the training database, where t (i) (k, j) = 0 or 1 is the class index associated with the corresponding feature vector, i.e. 1 (resp. 0) if the related intensification is a target (resp. not a target).

Adaptive training session

The proposed adaptive algorithm aims at estimating the optimal amount of data needed in the training session to learn both the spatial filters U and the classifier d. Starting from one symbol in the training set I, the proposed method increases the number of training symbols until the convergence is reached as detailed bellow. As already pointed out, one can used a cross validation technique to estimate the classification accuracy by splitting the data base into two sets. The first one is used to train the spatial filters and the classifier. The second set is used to test the performance. This splitting is then repeated several times as in the K-fold method. However this procedure needs a lot of data to be accurate. The proposed method avoids this drawback by predicting the accuracy without the need to compute it.

Let us suppose that at the i-th iteration, i symbols compose the training set, i.e.

I i = {1, • • • , i}.
The training database is thus composed of the set of {X ( j) } j∈I i for which the target and non-target onsets are known. This set of training data allows to estimate both

• the spatial filters Û(I i ) thanks to the xDAWN algorithm (Section 3.1),

• the discriminative vector d (I i ) thanks to the BLDA (Section 3.2).

The problem is now to know if adding a new symbol in the training session is needed to improve the classification accuracy of the test session. Since BLDA is a linear classifier, then the performance accuracy directly depends on the values of the components of the discriminative vector. As a consequence, if adding a new data in the training set does not change (too much) these values then one can conclude that the convergence is reached. The similitude between the two discriminative vectors d (I i ) and d (I i-1 ) is based on the principal angle between the two subspaces spanned by these vectors. Indeed, reshape vectors d (•) ∈ R N t N s /4 , which mixes time and space features, into a matrix D (•) ∈ R N t /4×N s , which separates the time features along the rows and space features along the columns, defines a subspace S (•) of dimension N s in R N t /4 (as N s ≤ N t /4). The closeness between d (I i ) and d (I i-1 ) is thus defined as the principal angle between S (I i ) and S (I i-1 ) defined by cos θ

(I i ) = max u∈S (S i ) max v∈S (I i-1 ) u T v (1)
as recalled in Appendix B. Because of the variance in the estimation of the discriminative vector d (•) due to the few number of data in the training session, one can smooth the evolution of θ (I i ) over the iterations i thanks to

θ (i) s = (I + δD T D) -1 θ (i) , (2)
where

θ (i) = [θ (I 1 ) , • • • , θ (I i ) ] T , I ∈ R i×i is the identity matrix, D ∈ R (i-1)×i is the Toeplitz bidiagonal matrix whose first row is [-1, 1, 0, • • • , 0] D =                  -1 1 0 • • • 0 0 0 -1 1 0 • • • 0 . . . . . . . . . . . . . . . . . . 0 0 • • • 0 -1 1                 
and δ > 0 is the smoothing parameter. This latter parameter allows to obtain a trade-off between the fidelity to the original data θ (i) and the smoothness of the regularized angles θ (i) s . Indeed, θ (i) s is the solution of

θ (i) s = arg min θ θ (i) -θ 2 2 + δ Dθ 2 2 ,
where θ (i) -θ 2 2 represents the fidelity to the data and Dθ 2 2 the smoothness of the estimated solution, respectively. The convergence is reached when the last value θ (I i ) s of θ (i) s is larger than a threshold η. Finally, the full algorithm to adaptively select the optimal size of the training set is summarized in Algorithm 1.

Algorithm 1 Adaptive training session.

{Initialization} Choose the threshold η and smoothing parameter δ I 0 = ∅, i = 0, ǫ = TRUE and d (∅) = 0 {Learning loop} while ǫ do Increment the counter: i ← i + 1 Add a symbol in the training set:

I i = I i-1 i
Estimate spatial filters Û(I i ) from all {X ( j) } j∈I i (Section 3.1) Estimate classifier d (I i ) (Section 3.2) Compute angle θ (I i ) between S (I i ) and S (I i-1 ) (1) Smooth angle evolution θ

(I i ) s (2) if θ (I i ) s > η then Stop training: ǫ = FALSE end if end while
The choice of the two parameters η and δ should be related to the following remarks. Choosing a threshold η such that cos η is close to one ensures that the convergence is reached but increases the number of data in the training session. On the contrary a smaller value of cos η will speed up the training session but the classification accuracy could be reduced. Secondly, a too large value of the smoothing parameter δ only allows small variations of the smooth angles θ (I i ) s and as a consequence requires lot of training data to reach the threshold η. On the contrary, a too small value of δ does not smooth enough the variations of θ (I i ) , the convergence criterion could then be verified only due to the noise variance, which means that the training dataset could be not large enough to ensure good classification accuracy.

Results

In this section, the classification accuracy in relation to the number of symbols in the training data set is presented in Section 4.1. Investigations of the influence of the threshold eta and of smoothing parameter δ are then presented in Section 4.2 and Section 4.3, respectively.

In this section, the box plots represent distribution of the displayed quantity for all subjects. The central mark is the median, the edges of the box are the 25th and 75th percentiles, the whiskers extend to the most extreme data points the algorithm considers to be not outliers, and the outliers are plotted individually. For each of the 20 subjects (from bottom left to top right), the continuous line is the classification accuracy (left labels) and the dots are the cosinus of the angle θ (I i ) (right labels).

Number of training symbols

The classification accuracy versus the number of symbols in the training data set is presented on Figure 2. For a fixed number of training symbols, the chronological order during the training session is chosen, i.e., when i symbols are selected, they correspond to the first i symbols in the training session. The spatial filters and the classifier are then tested on the four testing session with 3, 5, 8 or 10 epochs, respectively. Obviously, the classification accuracy increases with the number of symbols in the training set. But one can see that increasing too much the number of training symbol do not increase drastically the classification accuracy. Indeed, with the test session 4 if the objective is to achieve at least 80% (resp. 90%) of classification accuracy, a t-test shows that there is no significative difference with more than 9 (resp. 16) training symbols. Moreover, this is confirmed on the test session 2 and an average objective of 80%, a t-test shows that using more than 15 training symbols does not increase significantly the classification accuracy. This figure highlights the possibility to reduce the number of training symbols without a significant breakdown of the classification accuracy. Clearly, the classification accuracy increases with the number of epochs (from Figure 2(a) to Figure 2(d)).

For the 20 subjects, the classification accuracy and the angle θ (I i ) between d (I i ) and d (I i-1 ) in relation to the number of training symbols is presented on Figure 3. This figure shows the high level of correlation between the classification accuracy and the evolution of the angle θ (I i ) . For most of the subjects, the two curves are very correlated: as a consequence the angles θ (I i ) can be used to predict the number of training symbols needed to reach the convergence of the classification accuracy without using a cross validation approach. This is clearly shown on Figure 4 which displays the distributions of the correlation coefficients between the angles θ (I i ) and the classification accuracy achieved with the four test sessions. Indeed, the median correlation coefficient is comprised between 0.88 and 0.90. For all the correlation coefficients, the p-value is less than 1.6e-6 with a 99.9% confidence interval.

Influence of threshold η

Figure 5 presents the number of selected symbols in the training session by the proposed algorithm (Algo. 1) in relation with the threshold η. The number of selected symbols decreases as the threshold η increases. Moreover, the dispersion of the number of selected symbols is quite small as long as the threshold η is larger than 3 • . There is thus a direct relationship between the threshold η and the number of the selected symbols. As a consequence, one can estimate the number of selected symbols by choosing the threshold η. Finally, this curve presents a knee between η = 5 • and η = 10 • , which appears to be a good trade-off between the duration of the training session and the provided classification accuracy in the test sessions.

Influence of smoothing parameter δ

Figure 6 presents the number of selected symbols in the training session by the proposed method (Algo. 1) in relation with the smoothing parameter δ. The number of selected symbols increases with the smoothing parameter. This growth is relatively limited in average as long as the smoothing parameter δ is less than 10. However, even a limited smoothing of the angles θ (I) (δ less than .3) increases the smallest number of selected symbols from 8 to 12. This has a direct impact of the classification accuracy by improving it in average from 67% to 75%. which exploits all the 50 symbols in the training set. 'Adaptive' is achieved by the proposed method (η = 20 • and δ = 10).

Adaptive training

(N epoch = 3) 2 ≺ 3 ≺ 4 ≺ 6 ≺ 9 ≺ 10 ≺ 19 ≺ 21 ≺ 50 2 (N epoch = 5) 2 ≺ 3 ≺ 5 ≺ 7 ≺ 9 ≺ 15 ≺ 17 ≺ 26 3 (N epoch = 8) 2 ≺ 3 ≺ 4 ≺ 5 ≺ 8 ≺ 11 ≺ 13 ≺ 36 ≺ 49 4 (N epoch = 10) 2 ≺ 4 ≺ 6 ≺ 7 ≺ 9 ≺ 15 ≺ 25 ≺ 43
For the 'Fix' method, the same number of training symbols is used for all subjects which is chosen to be equal to the median of the number of training symbols provided by the 'Adaptive' method. With the chosen parameters η and δ, this fixed number is equal to 8. In this experiment, η is chosen large to have a short training session and as a consequence, δ is also chosen large to avoid a too short training session. It is interesting to note that for almost all the subjects the classification accuracy is similar with the 'Fix' (78%) or 'Adaptive' methods (80%), while the 'reference' achieves 88%. Indeed, the number of selected symbols in the training session is quite equal to 8 for all the subjects. However, the proposed adaptive procedure allows to adjust the training session to the specificity of the subject. For instance, for subject 3, the maximal accuracy is achieved with a few number of training symbols (Fig. 3), the adaptive procedure has reduced the training session to 5 symbols. Moreover, for subject 8 who needs more symbols to reach the convergence (Fig. 3), the adaptive method stops after 10 training symbols. The same remark can be expressed for subject 12. It is worth noting that the subjects, for whom a large number of symbols is selected in the training phase, are generally subjects with the lowest classification accuracy, for instance subjects 8 and 12 (Fig. 3).

Finally, one can show on Table 1 that to add a symbol in the training database until ten symbols increases significantly the CA based on a left-tailed t-test (α = 0.01 and the p-values remain smaller than 0.01). Then, one have to add definitively more symbols in the training database to continue to increase the CA. This last remark shows that the proposed method is well adapted to select automatically the minimum number of necessary symbols in the training database.

Conclusions

The problem of the optimal training session is still challenging to provide user friendly BCIs. However, while almost all the current studies only used a fixed number of symbols to train the BCI elements, the current paper investigates an adaptive method to adjust the number of training data. Indeed, the classification accuracy highly depends on the signal processing methods and on the classification techniques to enhance and 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 to detect the P300 potentials. These steps are of greatest importance to ensure the success of a BCI since a poor accuracy can discourage people to use it. The method proposed in this study to switch automatically between the training session and the online session seems to be very promising. As it is shown in the present work, the angle between two consecutive classifier vectors is suitable to check the convergence of the training session. The proposed method allows to have a good initialization of a P300 speller so that people can start to use the BCI without frustration. Furthermore, the adaptive training session would not only benefit BCI users but also other studies using the P300 detection. Many physiological and psychological studies rely on measurement of the P300 to examine event related potentials, like in combination to behavioral tasks. As the P300 detection has a broad range of uses in clinical research, an adaptive training session would offer a quick evaluation of the P300 characteristics for a particular individual.

Future works will deal with the use of this method in a real context, i.e. a real automatic switch between the training session and the online one. Moreover, the next challenge concerns the improvement of this method and its extension to provide an online training so that the BCI can continuously adjust itself to the environment or to the user's physiology (e.g., tiredness or concentration). subject to u 2 = v 2 = 1, u T u i = 0, and v T v i = 0, ∀ i = {1, • • • , k -1}. Let Q 1 ∈ R m×p and Q 2 ∈ R m×q be two orthonormal basis of S 1 and S 2 , respectively. Assume that p ≥ q. The angles between subspaces can be efficiently obtained by the singular value decomposition of Q T 1 Q 2 :

Y T Q T 1 Q 2 Z = diag(λ 1 , • • • , λ q ).
The principal angles and vectors are then obtained by cos

(θ k ) = λ k , k = {1, • • • , q} and [u 1 , • • • , u p ] = Q 1 Y, [v 1 , • • • , v q ] = Q 2 Z, respectively.
Notice that θ k is such that 0 ≤ cos θ k ≤ 1, cos θ k = 0 means that the two related subspaces are orthogonal and cos θ k = 1 means that there is an intersection of the two subspaces.

Figure 1 :

 1 Figure 1: Classical P300 interface with the third row intensified.

  Test session 4: N epoch = 10.

Figure 2 :

 2 Figure 2: Classification accuracy in relation to the number of symbols used in the training session.

Figure 3 :

 3 Figure 3: Relation between the classification accuracy achieved with the test session 2 (N epoch = 5) and the angle between discriminative vectors d (I i ) and d (I i-1 ) .For each of the 20 subjects (from bottom left to top right), the continuous line is the classification accuracy (left labels) and the dots are the cosinus of the angle θ (I i ) (right labels).

Figure 4 :

 4 Figure 4: Distribution of the correlation coefficients between the angles θ (I i ) and the classification accuracy achieved with the four test session (N epoch =3, 5, 8 or 10.)

Figure 5 :Figure 6 :

 56 Figure 5: Number of selected symbols in the training session vs the threshold η expressed in degree. Smoothing parameter δ is set to 2.

Figure 7

 7 compares the classification accuracy achieved by three different methods of training. 'Ref' is the reference method Test session Increase of classification accuracy 1

Figure 7 :

 7 Figure 7: Comparison of classification accuracies achieved by different training procedures. For the 'adaptive' method, η = 20 • and δ = 10. For 'Fix' and 'Ref' methods, 8 and 50 training symbols are used, respectively. The test session is the second one N epoch = 5.

Table 1 :

 1 Number of necessary symbols in the training database to increase the classification accuracy (CA) of the test session. Column "Increase of classification accuracy" indicates the number of symbols in the training database that are required to increase the CA based on a left-tailed t-test (α = 0.01): n 1 ≺ n 2 means that n 2 is the smallest number such that the first n 2 symbols of the training database are necessary to increase the CA compared to the CA achieved with the first n 1 symbols of the training database.

Acknowledgements

This work has been supported by the French National Research Agency (ANR) through the DEFIS program (project Co-Adapt ANR-09-EMER-002).

Appendix A. xDAWN spatial filters

In this appendix, the principle of the xDAWN algorithm is briefly reprised [START_REF] Rivet | P300 speller" Brain-Computer Interface: Enhancement of P300 evoked potential by spatial filters[END_REF][START_REF] Rivet | xDAWN algorithm to enhance evoked potentials: Application to brain-computer interface[END_REF]. Its goal is to maximize the signal to signal-plus-noise ratio (SSNR). This method relies on two assumptions: i) signal is made of two typical pattern, one evoked by the targets and one evoked by all stimuli (target and non-target ones); ii) responses evoked by the target stimuli could be enhanced by spatial filtering. Let X ∈ R N t ×N s denote the recorded signals, where N t and N s are the number of samples and sensors, respectively. We denote the pattern synchronized with the target and non-target stimuli by

where

where Tr(•) is the trace operator and Σ1 = ÂT 1 D T 1 D 1 Â1 , ΣX = X T X. Note that Â1 is the least mean square estimation of the unknown target evoked response A 1 . Since D 1 A 1 and D 2 A 2 could overlap, Â1 is estimated from

Â1 is thus estimated by Â1 = B T 1 X. To estimate the spatial filters Û1 , one computes the generalized eigenvalue decomposition [START_REF] Golub | Matrix Computation, 3rd Edition[END_REF]

where Λ is a diagonal matrix with entries equal to the generalized eigenvalues λ 1 ≥ • • • ≥ λ N s and U is a full matrix whose columns are the corresponding eigenvectors. Finally, the enhanced signals are given by

where Û1 is the matrix whose N f columns are the eigenvectors associated with the first N f largest eigenvalues λ i .

Appendix B. Angle between subspaces

This appendix summarizes some useful considerations about angles between subspaces [START_REF] Golub | Matrix Computation, 3rd Edition[END_REF]. Let S 1 and S 2 be two subspaces in R m . The principal angles θ k between subspaces S 1 and S 2 are defined by cos(θ k ) = max


Ivan Alata

Michel Broquier

Claude Dedonder

Christophe Jouvet
email: christophe.jouvet@u-psud.fr

Ernesto Marceca

Electronic excited states of protonated aromatic molecules: protonated Fluorene

Keywords: protonated PAHs, photo-fragmentation spectrum, excited state ab initio calculations

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

The excited states of protonated aromatic molecules, in particular those of polycyclic aromatic hydrocarbons (PAHs) have been extensively studied in the condensed phase (1)(2)(3)(4)(5)(6)(7).

However, the optical properties of protonated aromatic molecules isolated in the gas phase have been less studied experimentally, because of difficulties encountered in producing sufficient amount of these reactive species (8)(9)(10). From another point of view, recent developments in quantum calculation methods have improved the description of the excited states of such species (11)(12)(13)(14)(15) and comparison of these calculations with gas phase spectroscopic data is now possible.

The photophysics of protonated polycyclic aromatic hydrocarbons became attractive since it was suggested that these ions might be present in the interstellar medium (14,16,17) being potential carriers for the diffuse interstellar bands (DIBs) that appear in astronomical observations.

We have recently studied the electronic spectra of linear protonated PAHs for which the S 1 -S 0 transitions are all in the visible region and they do not show a monotonic red shift as a function of the molecular size, in contrast to their neutral analogues (18). This behaviour was interpreted using the results of ab-initio calculations as due to the charge transfer character found in the excited states of PAHs, especially those with an even number of aromatic rings.

Previous work was focussed in protonated PAHs composed by different number of condensed six-membered rings. In the present paper, we will investigate the behaviour of protonated fluorene, a PAH analogue having a five-membered ring in the structure, with particular interest in the effect of this moiety on the protonation site and on the charge transfer character of the excited state.

Protonation of fluorene has been used to change the optical properties of conjugated polymers (19). This allows the design and development of new organic emitter and sensor materials tailored toward particular needs. The effect of protonation on the monomer might be useful to understand the optical properties of larger systems.

Finally, many efforts have been done recently to characterise the stability of the different conformers of protonated molecules through their infrared vibrational spectroscopy (20)(21)(22)(23)(24). We will show in the present paper that the recent progress achieved by ab-initio methods to describe an excited state is such that the electronic spectroscopy nowadays represents a quite accurate alternative to assign conformers, as long as the excited states are not too short lived, i.e. some vibrational analysis can be performed.

Methodology

Experimental details

Protonated fluorene is produced in a pulsed high voltage electrical discharge source coupled to a pulsed nozzle (General valve) of 300 µm in diameter. The discharge is produced between two electrodes positioned 3 mm downstream from the nozzle. The gas mixture consists of 50% of Helium and 50% of H 2 seeded with fluorene vapor, obtained by heating the solid at about 120° C. The protonated species are obtained only in presence of H 2 . The typical backing pressure in the source is about 4 bars, while under operation the pressure in the source chamber is 1×10 -5 mbar. The expanding molecular beam containing the ions passes through a 1 mm diameter skimmer located 10 cm downstream from the nozzle orifice, and the collimated ion beam passes into a second differentially pumped chamber (p = 2×10 -6 mbar).

The ions produced in the source are extracted with pulsed voltages (2 kV) into a reflectron time of flight mass spectrometer, perpendicularly to the molecular beam axis. A reasonably good mass resolution (m/∆m > 400) is achieved at the entrance of the reflectron mirror, after the ions have flown a distance of 157 cm along the field-free flight tube. At this point, the protonated species are well separated from the corresponding radical cations (∆m/Z = 1 between them), which are also formed in the discharge.

Photo-fragmentation spectra of the protonated species are recorded by monitoring the intensity of neutral fragments originated during the interaction with a tuneable laser, as a function of its wavelength. This process is carried out inside a field-free 2 cm long interaction region, located in front of the reflectron plates (see the Electronic Supplementary Information for a scheme of the experimental set-up). The fragmentation process is done using a 10 Hz, nanosecond OPO laser (Euroscan) pumped with the third harmonic (355 nm) of a Nd +3 :YAG laser (Quantel YG981C). The OPO laser delivers a tuneable wavelength range from 413 to 670 nm with an output power in the range of 1 to 15 mJ. The laser is mildly focused to intersect the ion bunch in a volume of typically 1 mm 3 . In these conditions the power dependence of the signal evidenced a small saturation, but lower laser energies proved being insufficient to attain the intensity demanded by the experiment.

The detection of neutral fragments is carried out on a MCP detector located directly behind the reflector mirror. The signal from the MCP detector is sent to the digitizing storage oscilloscope interfaced to a PC. Photo-fragmentation of mass selected ions is not the only source of neutrals arriving at the detector. Neutral fragments may also be generated by collisions with the residual gas during the time of flight. To discriminate the neutral fragments produced by collisions from those produced by laser absorption, the parent ions are post-accelerated to a final energy of 5 kV immediately before they enter the interaction region. In this way, the kinetic energy of the neutral fragments produced by laser absorption will be 5 kV, while the fragments produced by collisions in the flight tube will be sensibly slower (2 kV), and hence it becomes easy to differentiate both signals. The interaction region is short enough to assume that the number of neutrals produced by collisions in this region can be neglected.

The signal to noise ratio of the experiment is low and therefore it is necessary to average many scans in order to get a spectrum like that presented in figure 1.

Calculations

Ab initio calculations have been performed with the TURBOMOLE program package (25), making use of the resolution-of-the-identity (RI) approximation for the evaluation of the electron-repulsion integrals (26). The equilibrium geometry of the ground electronic state (S 0 ) has been determined at the MP2 (Moller-Plesset second order perturbation theory) level.

Excitation energies and equilibrium geometry of the lowest excited singlet state (S 1 ) have been determined using a second-order approximate coupled-cluster calculation (RI-CC2). The CC2 method was chosen because it gives reasonable results for medium size closed shell molecules for a moderate computational time (27). Besides it allows excited state geometry optimization more easily than the CASPT2 method. Calculations were performed with the SVP basis set (7s4p1d for carbon and 4s1p for hydrogen), which is quite small but gives reasonable good results for this type of systems (18). Some complementary calculations have been performed with the aug-cc-pVDZ basis set (28). The isomers are noted FC n H + , depending on the carbon atom where the proton is attached to, the numbering of the carbon atoms being shown in the insert of figure 1.

For the two most stable isomers, a complete vibrational analysis has been performed after the excited-state optimization, the vibrational frequencies were calculated in both the ground and the excited states, and a Franck-Condon analysis was performed using the Pgopher software, a program for simulating and fitting rotational, vibrational and electronic spectra [START_REF] Western | PGOPHER, a Program For Simulating Rotational Structure University Of Bristol[END_REF].

Results

Experimental

The experimental spectrum is presented in figure 1: the laser has been scanned to the red and no other bands were detected. Clearly, the spectrum shows two vibrational progressions: a band system A, starting at 479.8 nm (2.58 eV), and a more intense band system B, starting at 447.6 nm (2.77 eV). The vibrational progression found in system B appears well structured, whereas the progression in system A hardly gets out of the experimental noise, although it can be noted a clear onset with a doublet band. The vibronic spectrum is superimposed on a broad background signal, the existence of which will be discussed in the next section.

Using a simple thermodynamic calculation involving tabulated values for the ionization potential [START_REF] Zhang | [END_REF] and proton affinity (31) of fluorene, it was possible to estimate a value of 2.93 eV for the dissociation threshold of protonated fluorene in fluorene + + H (assuming this dissociation channel is major as for other protonated polycyclic aromatic molecules (32,33)), which lie above the position of the observed first vibronic bands, meaning that the recorded photo-fragmentation spectrum results from a resonance enhanced multiphoton process. This is also evidenced by the marked dependence of the signal on the laser power, as mentioned before in the experimental section, changing from being mildly saturated to a total absence of signal for low laser powers. As for linear protonated PAHs, the electronic absorption is strongly red shifted as compared to the absorption of neutral fluorene (296 nm, 4.18 eV) (34,35) the magnitude of the shift being quite similar to the one observed in the case of Naphthalene (18).

The band systems A and B found for protonated fluorene may correspond either to excitations to different electronic states or to the presence of different isomers. We will make use of ab-initio calculations and comparison between recorded and simulated spectra to identify which of the two hypotheses gives the best agreement.

Ab initio calculations

The results of the ab-initio calculations, in particular the ground state relative stabilities and transition energies, are presented in table 1 transition energies of the isomers, the less stable isomers having the lowest transition energies.

This has also been observed in the case of linear PAHs (18) and might be a quite general behaviour, the origin of which must be investigated in more detail. Table 2 shows that the ground state energy difference between the two isomers stays the same, and the transition energies get lower with the size of the basis set. In spite of that, the ordering of the transitions does not change, and the energy gap between the two transitions remains close to the experimental gap between the two band systems (2.77-2.58 = 0.19 eV). In addition, the successive isomerization barriers between isomers FC 2 H + and FC 3 H + and between FC 3 H + and FC 4 H+ have been estimated at the B3-LYP/SVP level of theory.

These barriers are about 1 eV, roughly 3 times higher than in protonated benzene (37)(the potential energy profile is presented in the electronic supplementary material).

Discussion

In all the spectra recorded with our set-up, the structured vibrational progressions are superimposed on a continuous background with a comparable intensity. We ascribe this effect to the presence of protonated molecules formed under very different cooling conditions, the colder ions being responsible for the vibrational band structure while the hot ones contribute to the background. The wide spread in the internal energy of protonated molecules produced by the electric discharge in a supersonic expansion has to be explored further to characterize the ions sampled in the experiment (a qualitative model is presented in the electronic supplementary material).

From the calculations, there are two possibilities for the assignment of the observed two band systems A and B: either they correspond to transitions of two different isomers of protonated fluorene or to two different excited states of a single isomer.

We may first assume that two isomers are responsible for the observed spectrum. A first argument used to identify the isomers in the ion source is based on their calculated ground state energy, which should not be too high as compared to the most stable isomer.

According to this criterion, FC 2 H + and FC 4 H + isomers are then expected to be observed in the experiment since their ground state energies only differ in 0.095 eV, for the other isomers the difference is larger than 0.3 eV. This is in line with previous studies on protonated molecules (18), where only the lowest energy isomers have been observed. The other hypothesis is to assign the two band systems to two different excited states, i.e. S 1 and S 2 , of the most stable isomer. This possibility was explored by performing the S 1 and S 2 excited state optimizations. Table 3 summarizes the calculated adiabatic transition energies, 2.76 eV for S 1 and 3.26 eV for S 2 when the states S 1 and S 2 are optimized independently and when corrections for the difference in the zero point energies are applied.

In addition, the S 1 -S 2 energy gap, which is 0.3 eV for the vertical transition, becomes even larger (0.50 eV) when the excited states are optimized, far more than the observed gap (0.19 eV). The previous criteria are linked to the energetic of the system. We can also use the change in geometry between the ground and the excited states to further test if our assignment is reasonable. This means that the simulated spectra, obtained via the Franck-Condon (FC) factors calculated using the ground and excited state frequencies, should resemble the experimental traces, either if different isomers are present or if different excited states are involved.

Figure 2a shows the simulated spectra obtained for the S 1 -S 0 transition of the FC 2 H + isomer by convolution of the stick spectrum with a Gaussian profile of 10 cm -1 width, at temperatures of 0 and 300 K compared with the most intense observed progression, band system B,. The agreement between the experimental data and the theoretical calculation is quite good, and this strongly supports the assignment of band system B to the most stable FC 2 H + isomer in the first excited state. On the other hand, a poorer agreement is found when the experimental band system B is compared with the simulated spectrum obtained for the S 2 ←S 0 transition of the FC 2 H + isomer (see figure 2b). Among the differences are: i) band intensities decrease with the energy only in the simulated spectrum, and ii) a sequence of hot bands (-20 cm -1 ) appears more markedly in the simulated spectrum when the temperature is increased. These bands correspond to the lowest out of plane vibrations of 76 and 113 cm -1 in S 0 , and 57 and 91 cm -1 in S 2 ), for which the Franck Condon factors are greater in the S 2 ←S 0 transition than in the S 1 ← S 0 transition because the geometry change is smaller between S 0 and S 2 than between S 0 and S 1 . On the whole, the agreement between the observed progression and the simulated spectra is more satisfactory if band system B is assigned to the S 1 state rather than to the S 2 state of the FC 2 H + isomer. Energy (cm -1 )

Figure 2 The simulated spectrum for the FC 4 H + isomer has also been calculated, The main difference with the FC 2 H + isomer being the width of the bands which are broadened by hot bands. Since band system A is weakly observed and hardly resolved, the comparison between the simulated spectra for the FC 2 H + and FC 4 H + isomers is not very demonstrative, although the first doublet, where the first band is a hot band, seems to be best reproduced by the FC 4 H + simulated spectrum (see figure SI-4 in the Electronic Supplementary Information).

Thus, the Franck Condon simulations do not contradict the assignment of the stronger band system B to the FC 2 H + isomer and the weaker band system A to the FC 4 H + isomer, its intensity being weaker because of the 0.09 eV higher ground state energy, but the energy arguments are more convincing than the Franck Condon simulations for the band system A.

It could also be questioned whether band system A could correspond to hot bands However, if we try to simulate the FC 2 H + spectrum with a temperature high enough to have hot bands populated 1500 cm -1 below the origin, then the region of the 0-0 transition is much more congested than what is observed.

The vibrational assignment of the bands in system B can be deduced from the calculated spectra for the S 1 ←S 0 transition of the FC 2 H + isomer and is presented in table 4.

The 2 most actives modes are shown in figure 3. They correspond to in plane symmetric deformations very similar to the modes that have been observed for protonated naphthalene and tetracene (18). Out of plane modes are only weakly active, like mode 66 and mode 63, and are responsible of the hot bands appearing as shoulders on the red side of the main bands.

Mode 43 at 205 cm -1 is also quite similar to the mode observed in the S 1 ←S 0 transition of the neutral molecule (208 cm -1 ) (34) but is more active in the protonated species, which implies the occurrence of a stronger deformation upon excitation (the calculated ground and excited state frequencies are listed in the electronic supplementary information). Energy (cm -1 )

Figure 3 The role of the charge transfer nature of the first excited state has been highlighted previously for protonated linear PAHs (18). The same is true in the case of protonated fluorene, as shown in figure 4, where the HOMO and LUMO orbitals of the FC 2 H + isomer are depicted for comparison. It is observed that the S 1 ←S 0 transition mainly corresponds to a HOMO-LUMO excitation with an electron transfer from the non protonated benzene ring to the protonated one. This charge transfer is probably responsible for the large deformation occurring in the molecule upon excitation, which is more important in the protonated species than in the neutral molecule. The localisation of the electron on the non protonated part of the molecule has been explained in the case of the protonated benzene dimer (38) in terms of the larger ionisation potential of the protonated benzene moiety with respect to that of the neutral homologue, resulting from the presence of a positive charge in one of the rings. As a consequence, the HOMO is more stabilized in the protonated moiety and the first electronic transition basically corresponds to a transition from the neutral part HOMO towards the protonated part LUMO. This also explains why the electronic transitions are red shifted as compared to those in the neutral homologues. The contour in red corresponds to an increase of 0.02 q (more negative charge) and in blue the opposite case. Clearly, the electronic density is increasing on the protonated benzene ring upon excitation.

HOMO LUMO

The Diffuse Interstellar Bands (DIBs) observed in the visible to the near infrared spectral range (39) provide a spectroscopic source of information about the molecules present in the interstellar medium. However, none of them could be assigned to molecular carriers so far, and it has been suggested that protonated PAHs could be possible carriers for the DIBs (16,17). One of the consequences of the charge transfer character of the first excited state is that a relatively large geometry change is produced during the electronic excitation, resulting in an important vibrational activity and fairly low oscillator strengths. On the other hand, the DIBs spectra present no apparent regularity in the line spacing, which means that they cannot be originated in a molecular system with very active Franck Condon progressions. Thus, the first excited state of protonated fluorene is not likely to be responsible for any DIB (a comparison between the observed spectrum and the wavelengths of the DIBs (39) is presented in the supplementary information).

As for other PAHs, protonation leads to a strong red shift of the electronic absorption, which means that the proton affinity of the excited state is much larger than that of the ground state (8,9). The proton affinity of fluorene in the ground state is 8.62 eV (31) eV for the most stable isomer). The effect of protonation on the optical properties of conjugated fluorene-pyridine copolymers (19) is to induce a large red shift of the fluorescence emission from 3.2 eV for non protonated polymers, to 2.6 eV for protonated polymers. In the calculations (19) done to understand this effect, the proton is located on the pyridine moiety since the proton affinity of pyridine, 9.64 eV (31), is higher than the proton affinity of fluorene in its ground state, but lower than the proton affinity of fluorene in its excited state. Besides, the protonation of the pyridine molecule does not induce a shift of the electronic absorption, and the proton affinity of the excited state of pyridine should be similar to the ground state proton affinity. Thus, it is energetically favourable to locate the proton on the fluorene ring in the excited state, which implies that an excited state proton transfer is quite probable. This effect has not been considered previously (19) due the lack of knowledge on the optical properties of protonated molecules and it might be interesting to test this hypothesis by performing time resolved experiments.

Conclusions

Protonation of fluorene leads to a strong red shift of the electronic transition as compared to the neutral molecule. Two vibrational band systems are observed and assigned by comparison with ab-initio calculations to the first excited state transitions of two different isomers. The agreement between the calculations (at the level of theory and basis set used in this paper) and the experiment is extremely good for the transition energies, as it has already been observed for linear protonated PAHs. The large red shift is due to the strong charge transfer character of the excited state as in other protonated PAHs. This shift of the electronic absorption implies a large increase of the excited state proton affinity, which can shade new perspectives on the excited state dynamics of protonated copolymers containing fluorene. Change in the charge density between the ground and the excited states. The contour in red corresponds to an increase of 0.02 q (more negative charge) and in blue the opposite case.

Figure caption

Clearly, the electronic density is increasing on the protonated benzene ring upon excitation.

Figure 1 :

 1 Figure 1: Photo-fragmentation spectrum of protonated fluorene. The numbering of the carbon atoms is represented on the neutral fluorene molecule in the inset.

Figure 2 :

 2 Figure 2: Franck Condon simulation of the vibronic spectrum. a) Comparison of the experimental band system B (middle trace, background subtracted) with the simulated spectrum corresponding to the S 1 ←S 0 transition of the FC 2 H + isomer at 0 K (lower trace) and 300 K (upper trace). b) Comparison between the experimental band system B (middle trace, background subtracted) and the simulated spectrum corresponding to the S 2 ←S 0 transition of the FC 2 H + isomer at 0 K (lower trace) and 300 K (upper trace).

Figure 3 :

 3 Figure 3: Assignment of the vibrational transitions: the upper trace is the experimental spectrum and the lower trace is the calculated one; on the top is shown the assignment of the vibrational transitions and the actives modes responsible for most of the vibrational progression are drawn in the insets.

Figure 4 :

 4 Figure 4: Orbitals involved in the first excited state transition of protonated fluorene. Left: Scheme of the Highest Occupied Molecular Orbital (HOMO) and Lowest Unoccupied Molecular Orbital (LUMO) involved in the S 1 ←S 0 transition for the FC 2 H + isomer. Right: Change in the charge density between the ground and the excited states. The contour in red corresponds to an increase of 0.02 q (more negative charge) and in blue the

Figure 1 :

 1 Figure 1: Photo-fragmentation spectrum of protonated fluorene. The numbering of the carbon atoms is represented on the neutral fluorene molecule in the inset.

Figure 2 :

 2 Figure 2: Franck Condon simulation of the vibronic spectrum. a) Comparison of the experimental band system B (middle trace, background subtracted) with the simulated spectrum corresponding to the S 1 ←S 0 transition of the FC 2 H + isomer at 0 K (lower trace) and 300 K (upper trace). b) Comparison between the experimental band system B (middle trace, background subtracted) and the simulated spectrum corresponding to the S 2 ←S 0 transition of the FC 2 H + isomer at 0 K (lower trace) and 300 K (upper trace).

Figure 3 :

 3 Figure 3: Assignment of the vibrational transitions: the upper trace is the experimental spectrum and the lower trace is the calculated one; on the top is shown the assignment of the vibrational transitions and the actives modes responsible for most of the vibrational progression are drawn in the insets.

Figure 4 :

 4 Figure 4: Orbitals involved in the first excited state transition of protonated fluorene. Left: Scheme of the Highest Occupied Molecular Orbital (HOMO) and Lowest Unoccupied Molecular Orbital (LUMO) involved in the S 1 ←S 0 transition for the FC 2 H + isomer. Right:

  for different isomers of protonated fluorene. The ground state energies of the different isomers are given relative to the most stable FC 2 H + isomer. There are two isomers with similar ground state energies, FC 2 H + and FC 4 H + , while the other isomers have much higher ground state energies, in agreement with calculations(36) of proton affinities of fluorene in the ground state. The stability of isomers FC 2 H + and FC 4 H + can be understood in terms of para and ortho conjugation effect across the planar biphenyl framework relative to carbon C 11 (and C 12 ). Isomers FC 2 H + and FC 4 H + have the highest calculated S 1 ←S 0 transitions energies, which are close to the observed band origins. A rough trend is observed when considering the stability and the S 1 ←S 0

	Neutral Fragment Signal (au)	band system B	C 6 C 7 2.77 eV	2.58 eV C 4 C 11 band system A C 5 C 12 C 1 C 10 C 13 C 8 C 9	C 3 C 2
	420	430	440	450	460	470	480
				Wavelength (nm)

previous

Table 1 :

 1 Ab-initio calculations for the different isomers of protonated fluorene (see the insert of figure1for the numbering of the carbon atoms). The ground state energies are calculated at the MP2 level and referenced to the most stable FC 2 H + isomer. Vertical S 1 ←S 0 and S 2 ←S 0 transition energies as well as adiabatic S 1 ←S 0The optimized geometry of the FC 1 H + , FC 2 H + , FC 1 H + and FC 4 H + isomers in both the ground and the excited states maintains the plane of the carbon atoms as a symmetry plane, while the FC 10 H + and FC 11 H + isomers have a very distorted geometry with a large geometry change between the ground and excited state. The ground and excited state geometries of the two most stable isomers, FC 2 H + and FC 4 H + , are given in the supplementary information.

									S 1 ←S 0	S 1 ←S 0
				Ground	S 1 ←S 0	S 2 ←S 0	S 1 ←S 0	Zero point	Adiabatic	Adiabatic
				state	Vertical	Vertical	Adiabatic	energy	transition	transition
				relative	transition	transition	transition	difference	corrected	corrected
				energy				δZPE	for δZPE	for δZPE
				(eV)	(eV)	(eV)	(eV)	(eV)	(eV)	(nm)
	FC 2 H	+	0.00	3.20	3.5	2.91 (.029)	-0.156	2.76	449.2
	FC 4 H	+	0.09	2.91	3.08	2.74 (.046)	-0.155	2.59	478.7
	FC 10 H	+	0.31	2.91	3.20	2.65 (.196)	-0.157	2.49	497.9
	FC 3 H	+	0.40	2.46	2.76	2.22 (.158)	-0.101	2.12	584.8
	FC 1 H	+	0.49	2.33	2.59	2.07 (.059)	-0.103	1.97	629.4
	FC 11 H	+	0.83	2.81	3.09	2.10 (.001)	-0.093	2.01	616.8

transition energies are calculated at the CC2 level with the SVP basis set. The values in parenthesis are the oscillator strengths of the transitions. The difference between the zero point energies of the ground and the excited states is used to correct the calculated transition energy, in order to compare the calculated value with the experimental measurements.

Since ground state and transition energies of isomers FC 2 H + and FC 4 H + are very close, complementary calculations have been performed with the aug-cc-pVDZ basis set, to see if there is a strong basis set dependence for the ordering of the transitions and the energy gap.

Table 2 :

 2 Comparison of the ground and first excited state energies for the isomers FC 2 H + and FC 4 H + of protonated fluorene using two different basis sets. The energies are given in eV.

				SVP basis set			aug-cc-pVDZ basis set
			Ground state	S 1 ←S 0 Vertical	S 1 ←S 0 Adiabatic	Difference in transition energy	Ground state	S 1 ←S 0 Vertical	S 1 ←S 0 Adiabatic	Difference in transition energy
	FC 2 H	+	0	3.08	2.91		0	3.12	2.84
	FC 4 H	+	0.095	2.91	2.73	0.18	0.093	2.82	2.64	0.20

  The second argument takes into account the calculated adiabatic transition energies, which should be close to the experimental origins of the band systems, when the difference in the zero point energy (δZPE) between S 0 and S 1 is taken into account. The transition for the most stable FC 2 H + isomer is calculated at 2.76 eV, which almost coincides with the band origin of system B, the most intense progression, observed at 2.77 eV. There is also a very good agreement in the case of the FC 4 H + isomer, for which the calculated transition energy is 2.59 eV and the origin of band system A is observed at 2.58 eV. On the other hand, the values of the transition energies calculated for the isomers FC 1 H + , FC 3 H + and FC 11 H + are much lower than the experimental band origins and therefore we consider that these isomers do not contribute to the observed spectrum. Although the transition energy of the H 10 + isomer is closer to the first experimental band origin, its presence is unlikely because its ground state energy is 0.31 eV higher than that of the most stable FC 2 H + isomer.The ground state relative energies and the transitions energies have also been calculated with the aug-cc-pVDZ basis set for the FC 2 H + and FC 4 H + isomers to test the relative stability of the isomers and the ordering of the excited state transition energies. As shown in table 2, the FC 2 H + isomer remains the most stable species and the ground state energy difference between FC 4 H + and FC 2 H + changes only 0.002 eV. The transitions energies of these two isomers are lower with this larger basis set, but the FC 4 H + transition remains to the red of the FC 2 H + transition, and the gap between them (0.2 eV) stays close to the experimental value.

Table 3 :

 3 Excited state transition energies for the FC 2 H + isomer. The values in bold italic have been corrected for the zero point energy difference. All values are given in eV.

					S 1 ←S 0	S 2 ←S 0	S 2 ←S 1	S 2 ←S 1
		S 0	S 1	S 2	Adiabatic	Adiabatic	Vertical	Adiabatic
					transition	transition	energy gap	energy gap
	S 0 optimized geometry	0	3.20	3.50			0.30
	S 1 optimized geometry	0.27	2.64	3.4	2.91 2.76			0.47
	S 2 optimized geometry	0.13	3.08	3.25		3.26 3.38		0.50

Table 4 :

 4 assignment of the vibrational bands in band system B

			42
	66 2	66 4	43
			ν 42
	ν 43
	0	500	1000
	Experimental frequency (cm -1 )	Calculated frequency (cm -1 )	Assignment
	-39	-30	63 1 1
	0	0	0 0 0
	209	206	43 1 0
	397	399 + 411	42 1 0 + 43 2 0
	511	485	66 2 0 43 2 0
	574	577	39 1 0
	602	606 + 616 + 619	43 1 0 42 1 0 + 43 3 0 +38 1 0
	775	783 + 799	39 1 0 43 1 0 + 42 2 0
	807	810 + 822	43 2 0 42 1 0 + 43 4 0
	990	977 + 1004 + 1016	39 1 0 42 1 0 + 43 1 0 42 2 0 + 43 3 0 42 1 0

  and becomes 10.03 eV in the excited state (the energy of the S 1 ←S 0 transition in the neutral is 4.18 eV (34,35) and it is 2.77 eV for the protonated molecule; PA*(S 1 )= 8.62 + (4.18 -2.77) =10.03

Table 1 :

 1 Ab-initio calculations for the different isomers of protonated fluorene (see the insert of figure1for the numbering of the carbon atoms). The ground state energies are calculated at the MP2 level and referenced to the most stable FC 2 H + isomer. Vertical S 1 ←S 0 and S 2 ←S 0 transition energies as well as adiabatic S 1 ←S 0 transition energies are calculated at the CC2 level with the SVP basis set. The values in parenthesis are the oscillator strengths of the transitions. The difference between the zero point energies of the ground and the excited states is used to correct the calculated transition energy, in order to compare the calculated value with the experimental measurements.

							S 1 ←S 0	S 1 ←S 0
		Ground	S 1 ←S 0	S 2 ←S 0	S 1 ←S 0	Zero point	Adiabatic	Adiabatic
		state	Vertical	Vertical	Adiabatic	energy	transition	transition
		relative	transition	transition	transition	difference	corrected	corrected
		energy				δZPE	for δZPE	for δZPE
		(eV)	(eV)	(eV)	(eV)	(eV)	(eV)	(nm)
	FC 2 H +	0.00	3.20	3.50	2.91 (.029)	-0.156	2.76	449.2
	FC 4 H +	0.09	2.91	3.08	2.74 (.046)	-0.155	2.59	478.7
	FC 10 H +	0.31	2.91	3.20	2.65 (.196)	-0.157	2.49	497.9
	FC 3 H +	0.40	2.46	2.76	2.22 (.158)	-0.101	2.12	584.8
	FC 1 H +	0.49	2.33	2.59	2.07 (.059)	-0.103	1.97	629.4
	FC 11 H +	0.83	2.81	3.09	2.10 (.001)	-0.093	2.01	616.8

Table 2 :

 2 Comparison of the ground and first excited state energies for the isomers FC 2 H + and FC 4 H + of protonated fluorene using two different basis sets. The energies are given in eV.

			SVP basis set			aug-cc-pVDZ basis set
		Ground state	S 1 ←S 0 Vertical	S 1 ←S 0 Adiabatic	Difference in transition energy	Ground state	S 1 ←S 0 Vertical	S 1 ←S 0 Adiabatic	Difference in transition energy
	FC 2 H +	0	3.08	2.91		0	3.12	2.84
	FC 4 H + 0.095	2.91	2.73	0.18	0.093	2.82	2.64	0.20

Table 3 :

 3 Excited state transition energies for the FC 2 H + isomer. The values in bold italic have been corrected for the zero point energy difference. All values are given in eV.

					S 1 ←S 0	S 2 ←S 0	S 2 ←S 1	S 2 ←S 1
		S 0	S 1	S 2	Adiabatic	Adiabatic	Vertical	Adiabatic
					transition	transition	energy gap	energy gap
	S 0 optimized geometry	0	3.20	3.50			0.30
	S 1 optimized geometry	0.27	2.64	3.4	2.91 2.76			0.47
	S 2 optimized geometry	0.13	3.08	3.25		3.26 3.38		0.50

Table 4 :

 4 assignment of the vibrational bands in band system B

	Experimental	Calculated	Assignment
	frequency (cm -1 )	frequency (cm -1 )
	-39	-30	63 1 1
	0	0	0 0 0
	209	206	43 1 0
	397	399 + 411	42 1 0 + 43 2 0
	511	485	66 2 0 43 2 0
	574	577	39 1 0
	602	606 + 616 + 619	43 1 0 42 1 0 + 43 3 0 +38 1 0
	775	783 + 799	39 1 0 43 1 0 + 42 2 0
	807	810 + 822	43 2 0 42 1 0 + 43 4 0
	990	977 + 1004 + 1016	39 1 0 42 1 0 + 43 1 0 42 2 0 + 43 3 0 42 1 0

Acknowledgments

The authors thank Prof. O. Dopfer for helpful discussion. This work has been supported by the Université Paris-Sud 11, by the ANR research grant (ANR2010BLANC040501), the PROCOPE 17832NK program and the RTRA "triangle de la physique". I.A. thanks the Atomic Energy Commission of Syria for financial support. The calculations have been performed on the GMPCS cluster of LUMAT. E.M. has been supported by the CONICET/CNRS exchange program.