# A novel approach to generate datasets with XAI ground truth to evaluate
image models

Miquel Miró-Nicolau<EMAIL_ADDRESS>Antoni Jaume-i-Capó
<EMAIL_ADDRESS>Gabriel Moyà-Alcover<EMAIL_ADDRESS>UGiVIA Research
Group, University of the Balearic Islands, Dpt. of Mathematics and Computer
Science, 07122 Palma (Spain)  Laboratory for Artificial Intelligence
Applications (LAIA@UIB), University of the Balearic Islands, Dpt. of
Mathematics and Computer Science, 07122 Palma (Spain)

###### Abstract

With the increased usage of artificial intelligence (AI), it is imperative to
understand how these models work internally. These needs have led to the
development of a new field called eXplainable artificial intelligence (XAI).
This field consists of on a set of techniques that allows us to theoretically
determine the cause of the AI decisions. One main issue of XAI is how to
verify the works on this field, taking into consideration the lack of ground
truth (GT). In this study, we propose a new method to generate datasets with
GT. We conducted a set of experiments that compared our GT with real model
explanations and obtained excellent results confirming that our proposed
method is correct.

###### keywords:

Fidelity , Explainable Artificial Intelligence (XAI) , Objective evaluation

###### PACS:

0000 , 1111

###### MSC:

0000 , 1111

††journal: Knowledge-Based Systems

##  1 Introduction

Deep neural networks have become a de facto standard to solve complex task as
computer vision [8] or natural language processing [21], obtaining astonishing
results in most fields. These good results were reached through the
combination of larger data sets and increase model complexity. However, this
complexity also provoked a set of caveats. One of them is the difficulty to
understand the causes of their results. This opacity of the deep learning
models is known as the black-box problem. This limitation is addressed by
eXplainable artificial intelligence (XAI), which aims to open this models and
avoid the black-box problem. As stated by Barredo Arrieta _et al._ [4] “XAI
proposes creating a suite of ML techniques that 1) produce more explainable
models while maintaining a high level of learning performance (e.g.,
prediction accuracy), and 2) enable humans to understand, appropriately trust,
and effectively manage the emerging generation of artificially intelligent
partners.”

XAI techniques have been widely used in multiple fields, as can be seen in the
work of Murdoch _et al._ [13]. In particular, and due to the ethical and
regulatory concerns, XAI had been largely used in sensitive contexts as
medicine, as can be seen in the recent published reviews of the state-of-art
([12], [20]). These studies assess that most XAI related work is not
validated. In this same direction, Miller [11] assessed that most of the work
about explainability relies on the authors’ intuition, and an essential point
is to have metrics that describe the overall explainability with the aim to
compare different models regarding their level of explainability. This problem
is not only for the different XAI methods, as discussed by Adebayo _et al._
[1], but also for adjacent element of the field as fidelity metrics [19]. All
these limitations arise from the fact that, in a real scenario, it is not
possible to have a ground truth for the explanations, because, a priori,
multiple explanation can be correct to obtain good results, however, only one
of the possible explanations is the one used by the backbone AI model.

Recently, a set of studies emerged that proposed to generate artificial
datasets with ground truth for the explanations ([5], [3], [6], and [10]).
These datasets allowed to validate the different XAI techniques, without
adding new sources of problems or having to take unproven assumptions. Cortez
and Embrechts [5] proposed a new XAI method and generated a synthetic dataset
with a ground truth for the explanation as a benchmark. The dataset consisted
of 1000 different samples of four random values. This dataset also contained a
synthetic label that was built using a weighted sum of the four values. An AI
model was trained to regress this synthetic label from the original data, and
for this reason a well-trained model, weights each feature of the dataset with
the weights of the sum that generates the label. These weights represented the
explanation ground truth. Nonetheless, their proposal had two limitations: it
was only applied to regression problems and only used with tabular data. Arras
_et al._ [3] proposed a methodology for generating a synthetic attribution
dataset of images. The proposed methodology was based on visual question
answering (VQA) models. These models are specific types of AI models that can
answer questions regarding the content of an image. Arras _et al._ [3]
proposed asking questions depending only on one object from the image. They
considered an explanation correct if only the object of interest in the
question was highlighted. Following this methodology, they compared 11
different XAI methods. Arras _et al._ [3] proposal was capable to handle
images data. Nevertheless, only in a very specific problem, the VQA models.
They also did not take into account the importance value of the explanations;
instead, they only considered the position of the most important elements.
Guidotti [6] acknowledge the black-box models trained on datasets with GT for
explanation become transparent models due to the availability of the real
explanation. He proposed to establish a set of generators specifically
designed for this type of dataset. In the case of images, these resulting
datasets can be effectively utilized for classification tasks. However, it is
important to note that this approach focuses solely on a single pattern. As a
result, the evaluation of explanations is limited to their spatial location.
Mamalakis _et al._ [10] named this type of datasets synthetic attribution
benchmark and formalized them. They defined them as the conjunction of three
elements: synthetic input $X$, function $F$ and synthetic output $Y$, the
latter is the result of applying $F$ to $X$. In addition, the methodology they
proposed was used for classification problems, similarly to Guidotti [6]
proposal, but with multiple pattern for an explanation. The $X$ of this
dataset is a set of binary images of circles and squares. $Y$ was one of two
classes: the first one indicated more area of circles than squares, and the
second indicated the contrary. With this information, they compared 12
methods. All 12 methods indicate the positive or negative of each pixel in a
given class. Mamalakis _et al._ [10] proposal can be used with multiple
pattern classes, but it only verifies whether the features have a negative or
positive impact on the classification. Similar to Arras _et al._ [3] and
Guidotti [6] did not consider the value of the explanation.

In this paper, we proposed a new methodology based on the previous works to
build synthetic attribution benchmark datasets with a defined GT for image
tasks. Our proposal overcome the limitations of the other proposals on the
existing state-of-art: not taking into account the value of the explanation
and being unable to use it for most typical image-related tasks. With our
methodology, we use the exact explanation value as a ground truth, and we can
use it indistinctly in regression and classification contexts. This dataset
going to allow the community to verify the different XAI techniques used and
future developments.

The rest of this paper is organized as follows. In the next section, we
describe our methodology for building synthetic attributions benchmarks
dataset. In Section 3, we specify the experimental environment, and we
describe the datasets, measures, metrics, and predictive models used for
experimentation. Section 4 discuss the results and comparison experiments
obtained after applying the two datasets generated using the proposed method
to multiple common XAI methods. Finally, in Section 5 we present the
conclusions of the study.

##  2 Method

We based our proposal on the work of Cortez and Embrechts [5]. As explained in
the previous section, these authors generated a synthetic attribution
benchmark (SAB) dataset for tabular data. The proposed methodology enhances
their method and applies it to images. Similar to the original work, we used
it for regression, but also for classification tasks. This was accomplished by
changing how the label was generated. Instead of tabular features, we proposed
to use numerical information found on the images, such as the number of times
a particular pattern was found on an image.

To define our methodology, we use the formalization of SAB proposed by
Mamalakis _et al._. We defined three different elements: the input data
$\mathcal{X}$, attribution function $F$, and output data $\mathcal{Y}$. This
was conducted as follows:

Input data: In this step, we defined the input data, which is a collection of
$n$ images: $\mathcal{X}=\\{I_{1},I_{2},...I_{n}\\}$. In each image, we can
find multiple occurrences of different patterns $p\in\mathcal{C}$, where
$\mathcal{C}$ is the set of all possible patterns that can be identified in an
image. The definition of $p$ is strongly related to the nature of the
experiment. In figure 1, for example, we have six different $p$, $C=\\{$
circles, crosses, uniform figures, blue texture figures, uniform circles,
uniform crosses, blue texture circles, and blue texture crosses $\\}$.
However, depending on the problem definition only a subset of $C$ can be
considered.

Attribution function. We define a function $F:\mathcal{X}\to\mathbb{R}$. This
function takes an image $I$ and uses the $p$ present on it,
$F(I)=\sum_{i=1}^{m}w_{i}\cdot g(p_{i},I)$, where $m$ is the total number of
different $p$ considered, $g(p_{i},I)$ is a function that extracts some
information of the pattern $p_{i}$ of the image $I$, for example the number of
times that this pattern appears on the image, and $w_{i}$ is the ground truth
for the importance of pattern $p_{i}$ for image $I$.

Figure 1: Image containing different patterns. The total amount of patterns
presented in the image is $|C|=4$. Nonetheless, depending on the problem, only
a subset of this $C$ is considered, e.g., if the problem is based on texture
analysis, we have two different $p$, the uniform elements and the ones
containing the blue texture.

The attribution function can be either a continuous or a discrete function.
The selection of which kind of function used depends on which is the task to
evaluate (regression or classification).

Output:. We generated labels, or output data, by combining the function and
input data. When $\mathcal{X}$ and $F$ are defined, the output data is
$\mathcal{Y}=\\{y:\forall I\in\mathcal{X},y\Rightarrow F(I)\\}$.

All this methodology is summarized in the algorithm 1 and figure 2.

In the following section, we describe the experimental setup used to test
whether this methodology allowed us to analyse the quality of a given XAI
method.

Algorithm 1 Proposed algorithm to generate synthetic attribution benchmark
datasets

$I\leftarrow$ Random Image Generator

$Y\leftarrow f(I)$

$\hat{f}\leftarrow$ Initial model

$\hat{Y}\leftarrow\hat{f}(I)$

while $Y\not\approx\hat{Y}$ do

$\hat{f}\leftarrow$ train($\hat{f},I,Y$)

$\hat{Y}\leftarrow\hat{f}(I)$

end while

Figure 2: Summary of the proposed methodology to generate Synthetic
Attribution Benchmark datasets for XAI methods.

##  3 Experimental setup

The end-goal of the experimental setup was to verify whether the ground truth,
generated by the proposed methodology corresponded to the actual explanations.
To determine whether this was the case, we conducted multiple experiments. To
do these experiments, we needed an XAI method as a proxy to obtain
explanations. We need to trust that the results of the XAI method were
faithful to the learned relations of the predictive model.

###  3.1 Datasets

Following the proposed methodology, we generated two different datasets: AIXI-
Shape (Artificial Intelligence eXplainable Insights) and AIXI-Color. These
datasets consist of 50000 images for training and 2000 for validation. All
images had a size of 128 pixels per 128 pixels. We used this image size to
simplify the training process. Both the datasets and scripts used to generate
them are available at https://github.com/miquelmn/aixi/releases/tag/1.0.0.

|
---|---

(a) (b)

Figure 3: Example of an image from the dataset (a) AIXI-Shape and (b) AIXI-
Color.

Each dataset allowed us to assess whether the proposed methodology was
accurate for patterns of different natures (colours and shapes).

####  3.1.1 AIXI-Shape

The images from AIXI-Shape were built using a combination of three different
patterns: circles, squares, and crosses, as shown in figure 3(a). We used
these forms for the different types of visual patterns. We used these simple
forms because they are easily recognizable.

These shapes were combined to form each image in the dataset. All the patterns
had random sizes, positions, and appearances. Nonetheless, we define a set of
restrictions, with the goal to simplify the posterior usage of this dataset:

  * 1.

One image contained a maximum of two objects of the same pattern.

  * 2.

We did not allow overlapping shapes.

  * 3.

All patterns had the same intensity.

With the proposed methodology, we must define the function $g(p_{i},I)$. As
already explained in the previous section, this function obtains numerical
data from the visual patterns of the images. Here, $g(p_{i},I)=|p_{i}\in I|$.
The position of these, $p_{i}$ in addition to the $w_{i}$ given by the
function used, allows the generation of a GT explanation for each pixel.

####  3.1.2 AIXI-Color

AIXI-Color, in contrast to AIXI-Shape, is a dataset containing only one type
of shape, circles, but of different sizes and values. These circles have three
possible intensity values, as shown in Figure 3(b). This randomly defined
intensity value is the pattern used for the attribution function. Similarly,
the positions of the circles were randomized. The random position has only one
constraint: the circles must not intersect. This constrain is to avoid
possible the existence of completely occluded pattern that could produce miss-
prediction and erroneous explanations.

We used the same $g(p_{i},I)$ of the previous dataset. In addition to the
weights of the functions, these features enabled, as already explained for the
AIXI-Shape dataset, the generation of ground truth as a saliency map.

###  3.2 Functions

Following the proposed methodology, we must also define the attribution
function $F$. All functions, as discussed in the previous section, must comply
with a set of requirements. We define three different functions: $ssin$,
$suum$ and $class$.

####  3.2.1 SSIN

First proposed by Cortez and Embrechts [5] the $ssin$ function (see equation
1), is defined as the sum of three factors based on the $sin$ function. Each
of these three factors has an associated weight, $w_{i}$. These weights were
defined by the authors as theoretical relative importance of ($0.55$, $0.27$,
$0.18$), e.g., $0.55$ = $0.5/(0.5+0.25+0.125)$. This function has one main
restriction: the output of the associated function, $g(p_{i},I)$, must be in
the range [0, 1]. This range was defined because the maximum value of $sin$
was obtained using $\pi/2$. Therefore, the maximum value for the $ssin$
function was obtained when all factors had the maximum value of $1$,
essentially when $x_{1}$, $x_{2}$ and $x_{3}$ were equal to $\pi/2$.

$ssin(x)=1/2\cdot sin(\frac{\pi}{2}x_{1})+1/4\cdot
sin(\frac{\pi}{2}x_{2})+1/6\cdot sin(\frac{\pi}{2}x_{3}).$ (1)

We adapted the original function to our methodology and notation as follows:

$ssin(x)=w_{1}\cdot sin(\frac{\pi}{2}g(p_{1},x))+w_{2}\cdot
sin(\frac{\pi}{2}g(p_{2},x))+w_{3}\cdot sin(\frac{\pi}{2}g(p_{3},x)),$ (2)

where $g(p_{i},x))$ indicates some information of the image $x$ regarding the
patten $p_{i}$, and $w_{i}$ is the weight for each factor, where $w_{1}=0.55$,
$w_{2}=0.27$, $w_{3}=0.18$

This function, as stated by Cortez and Embrechts [5], aims to mimic an
additive response of independent nonlinear inputs. The result was a continuous
value between zero and one. For this reason, we considered this to be a
regression task.

####  3.2.2 SUUM

We simplify the $ssin$ function to verify whether our methodology can also
work with a linear function. We call the resulting function $suum$:

$suum(x)=w_{1}\cdot g(p_{1},x)+w_{2}\cdot g(p_{2},x)+w_{3}\cdot g(p_{3},x),$
(3)

similar to the $ssin$ function, the weights had the values $w_{1}=0.55$,
$w_{2}=0.27$, $w_{3}=0.18$. The theoretical function outputs are in the range
of $(-\infty,+\infty)$. We normalized the information output of $g(p_{1},x)$
to be in the range $[0,1]$; for this reason, the range of $suum(x)$ was also
reduced to $[0,1]$. Similar to the $ssin$ function, we consider it a
regression problem.

####  3.2.3 Class

We define a new function that allows us to simulate a classification problem.
To accomplish this, we define a discrete function $class$. This function is
defined as follows:

$class(x)=w_{1}\cdot g(p_{1},x)-w_{2}\cdot g(p_{2},x)\geq 0,$ (4)

where $w_{1}$ and $w_{2}$ are 1 and 0.5, respectively, $g(p_{i},x)$ indicates
numerical information regarding the visual pattens. We can see, that the
function only considered two pattens the rest, if they existed, were ignored.
Because the function has a threshold operation, the only possible values are
$0$ or $1$. Furthermore, we considered the function of having semantic
meaning. For example, if the $g(p_{i},x)$ function indicates the number of
occurrences of pattern $p_{i}$, then the function $class(x)$ is $1$ when there
are more objects of the first pattern than of the second. The fact that the
output of the function was binary, led us to consider any model that estimates
the function as a classification model.

###  3.3 Predictive model

As explained in previous sections, with this experimental setup, we aimed to
verify the quality of the ground truth generated with the proposed
methodology. To measure this quality, we must have something to compare it
with. For this reason, we required two different elements: a predictive model
and an XAI method. We used both for comparison with the proposed GT.

We define a predictive model as a function $f(x)$, where $x$ is a 2D image and
$f(x)$ is a prediction of the content of $x$. Explanations were extracted from
the predictive model. We used two types of predictive model. First, we used as
the model the functions defined in section 3.2. Second, we used a deep
learning model trained to approximate the same functions.

We first used directly the function as a predictive model to avoid any
possible errors in the prediction. We used the explanations obtained from this
model as baseline punctuation, which allowed us to identify the limitations of
the successive model. Because it was not a real AI model, we could only use
agnostic XAI methods. An agnostic method was defined in [15] as the method
that treat the original model as a black box.

We also used a deep learning model to simulate a real scenario. Because deep
learning models are universal function estimators [7], these types of models
can learn any $F$ that we had previously defined. If the model correctly
estimated this function, necessarily also had learned the weight associated to
each pattern. For this reason, we consider the $w_{i}$ of each pattern the
correct explanation for the result. We used a convolutional neural network
(CNN), a special neural network for images-related tasks. The model was
reintroduced by Krizhevsky _et al._ [8]. The authors proposed a model with two
different parts: a convolutional part, responsible for pattern recognition,
and a classification part, responsible for combining these recognized patterns
into semantic information. The convolutional part was built with three main
elements: convolutional layers, rectified linear unit (ReLU), and max pooling
operations. The classification component was a multilayer perceptron (MLP)
with a softmax operation as the activation function. All these building blocks
were discussed in the original work. These model apply to classification and
regression tasks.

We developed a compact CNN architecture in response to the straightforward
nature of the images within the dataset. We defined this architecture
empirically to ensure that the value of the validation metrics was at least on
the best decile. We needed to obtain good results to ensure that the AI model
learned the desired $F$ function. To construct the feature extractor portion
of our model, we utilized two MLP layers. The proposed architecture is
illustrated in Figure 4.

Figure 4: Diagram of the proposed predictive model.

In both classification and regression tasks, we employed the same
architecture, with the only distinction being the addition of a $sigmoid$
activation function for the classification tasks.

In all tasks, we obtained accuracy values for the validation set higher than
$0.93$.

###  3.4 XAI method

Once we had defined the predictive models, we also needed to select the XAI
method. We used an XAI method to obtain explanations and compare them to our
ground truth. Because our goal was to verify the quality of the ground truth
and not vice versa, we needed to ensure that the obtained explanations were
accurate. We defined a set of requirements to select the best XAI method
according to them:

  * 1.

Verified. The XAI method needed to be largely tested and verified to have the
minimum number of unknown errors possible. Because it is unlikely to identify
a perfect method, the important part of this requirement is to know the
possible errors and avoid them.

  * 2.

End to end. The method must be able to analyse the entire AI model. This
restriction was set because our proposal generated exact explanations.

  * 3.

Fidelity. Fidelity, as explained in the previous section, is the similarity
between the explanation obtained and the actual casual relation learned by the
model. Because this experimental setup aimed to verify whether the generated
ground truth was correct, we needed to extract the cause of the models with
high accuracy to compare it to the ground truth. For this reason, the most
important requirement was that the method had a high fidelity.

If a method meets all three criteria, we consider that the explanation
obtained is correct. Local Interpretable Model-agnostic Explanations (LIME)
was the XAI solution that best matched these constraints.

LIME was first introduced by Ribeiro _et al._ [15]. The authors proposed
building a local environment around a particular input using occlusions. This
local environment of the solution space, allows learning simpler methods,
interpretable by design, as a proxy for the underlying, more complex model to
be explained.

We verified whether the LIME were complied with the requirements that we
defined. First, this method had been widely used, tested, and verified, as
demonstrated by over 5000 citations 111Last time checked on 12-01-2022 on
scopus. Therefore, we considered that a major part of the limitation of the
method was already discovered: low performance in the presence of out-of-
distribution samples [14] and the sensibility to adversarial attacks [18].
Both problems show the lack of robustness of LIME. Robustness, as explained by
Alvarez-Melis _et al._ [2], is the expectation that if the data is slightly
modified therefore the explanation of this modified data should be similar to
the original explanation. Visani _et al._ [22] identified the origin of this
lack of robustness to “the randomness introduced in the sampling step”. All of
these problems can be avoided with a well-curated datasets, as the ones we
proposed, and small modification to the original LIME implementation. In
particular, the fact that at most exist $2^{6}$ possible samples, due to the
maximum number of possible patterns in the images, allowed us to remove the
random sampling from the algorithm. Even more, we allowed to select a subset
of these possible $2^{6}$ samples, via a parameter (sample size).

Second, LIME [15] is an agnostic method, considering the entire model to
obtain its explanations. Finally, as already discussed the sample size defines
a trade-off between fidelity and interpretability (or robustness), however and
due to the changes we made to the original method, that we already explained,
we can increase the overall fidelity of the method without losing robustness.
We tested whether this assumption was correct in the experimental setup,
analysing how the fidelity increases with a larger sample size. Once we tested
it, we maximized the fidelity of the method to ensure that the obtained
results provided an exact explanation of the data.

###  3.5 Performance measures

The results of the explanation method were measured and compared with the
ground truth of the datasets. We can find in the literature multiple measures
to compare two saliency maps. We based the selection of the measures in the
work of Riche _et al._ [16], in which the authors review and analyse the
state-of-art. Based on their findings, we can deduce that in order to
accurately compare two saliency maps, we must treat them as probability
distributions. This allowed us to use two well-established measures:

  * 1.

Earth Mover Distance ($\mathit{EMD}$). First proposed by Rubner _et al._ [17].
$\mathit{EMD}$ can be defined as the minimum amount of work required to
transform one distribution into the other. See equation 5.

$\displaystyle\mathit{EMD}={}$
$\displaystyle(\min_{f_{ij}}\sum_{i,j}f_{ij}\cdot
d_{ij})+|\sum_{i}\mathit{SM}_{i}-\sum_{j}\hat{\mathit{SM}_{j}}|\max_{i,j}d_{i,j}$
(5) $\displaystyle s.t.f_{i,j}\geq
0,\sum_{j}f_{i,j}\leq\mathit{SM}_{i},\sum_{i}f_{i,j}\leq\hat{\mathit{SM}_{j}}$
$\displaystyle\sum_{i,j}f_{i,j}=\min(\sum_{i}\mathit{SM}_{i}-\sum_{j}\hat{\mathit{SM}_{j}})$

where each $f_{ij}$ represents the amount transported from the $i_{th}$ supply
to the $j_{th}$ demand. $d_{ij}$ is the ground distance between the sample $i$
and sample $j$ in the distribution. $\mathit{SM}$ is the probability
distribution of the ground truths and $\hat{\mathit{SM}}$ the probability
distribution of the predicted saliency map.

  * 2.

Kullback–Leibler divergence ($\mathit{KL}_{div}$). First proposed by Kullback
and Leibler [9]. The $\mathit{KL}_{div}$ is a measure of the information lost
from one distribution to the other. See equation 6.

$\displaystyle\mathit{KL}_{div}=\sum^{X}_{x=1}\mathit{SM}(x)\cdot\mathit{log}(\frac{\mathit{SM(x)}}{\hat{\mathit{SM(x)}+\epsilon}}+\epsilon),$
(6)

where $\mathit{SM}$ is the probability distribution of the ground truths and
$\hat{\mathit{SM}}$ the probability distribution of the predicted saliency
map.

Both measures have the best result equal to $0$, however while $\mathit{EMD}$
have a lower boundary with value $1$ as the worst possible results,
$\mathit{KL}_{div}$ do not have any boundary. This fact is important for any
further analysis.

###  3.6 Experiments

Three experiments were conducted to study the proposed methodology. The first
experiment tests a basic assumption of the rest of experiments, that the
fidelity of LIME [15] can be increased through the size of the sample. In the
second experiment, we compared the proposed ground truth with the results of
LIME [15], configured to obtain the maximum fidelity possible. In this
experiment, we used the function itself instead of an AI model. Finally, in
the third experiment, we did the same as in the second experiment but with an
AI model. The last two experiments had six sub-experiments each, all of these
sub-experiments share that they were built with a combination of one dataset
and a function.

These experiments aimed to assess whether the GT generated by our methodology
was in fact the correct ground truth.

In all experiments, we used LIME as the XAI method, to obtain explanations. We
compared these explanations with the GT generated by our method. We used both
metrics defined in Section 3.5. However, as already discussed, the
$\mathit{KL}_{div}$ measure does not have a reference value to be able to
further analyse the results. We used as a baseline value the results obtained
with the smallest possible sample size calculated in the first experiment.

In the second experiment, we used a function $F$ as the predictor to avoid
unexpected errors provoked by the AI model. Because we did not train any
model, we could be sure that the prediction was perfect, and for this reason,
we know that all errors were produced either by the XAI method or the proposed
methodology. We considered the results obtained from these experiments as
baseline results. However, because the $class(x)$ function is a binary and the
result value only varies if there is a big enough perturbation, we expect that
the explanations from the AI model surpass in fidelity this function. This
counter-intuitive behaviour can be even exaggerated if the sigmoid activation
function of the last layer is removed.

In the third experiment, we used an AI model. We use the model explained in
Section 3.3. This experiment aimed to simulate a real scenario in a simplified
manner. Similar to the second experiment, we extracted the explanations using
LIME [15]. The major caveat of using an AI model was the possibility of adding
new errors without relation to our methodology.

These two last experiments were divided into six sub-experiment. Each of these
six sub-experiments was defined by the dataset and function used, as presented
in sections 3.1 and 3.2:

  * 1.

Sub-experiments $\bm{ssin}$. These experiments aimed to test the proposed
methodology for the regression of non-linear functions. We defined two sub-
experiments with different datasets that aimed to test this. We used, in both
cases, the $ssin$ function.

We used in the first sub-experiment the AIXI-Shape dataset and in the second
the AIXI-Color. In both cases, the $g(p_{i},I)$ used to calculate the $ssin$
was $g(p_{i},I)=|p_{i}\in I|$, indicating how many objects of the respective
pattern were found on the image.

  * 2.

Sub-experiments $\bm{suum}$. These experiments were defined to assess whether
our methodology was useful for the regression of linear functions. We used the
same two datasets from the previous experiments. We used the $suum$ function
and again $g(p_{i},I)=|p_{i}\in I|$.

  * 3.

Sub-experiments $\bm{class}$. These experiments were defined to assess whether
our methodology was useful for classification tasks. We used the $class$
function, defined in Section 3.2, and similarly to the previous experiments,
both AIXI-Shape and AIXI-Color dataset and $g(p_{i},I)=|p_{i}\in I|$.

##  4 Results and discussion

In this section, we discuss and analyse the results of the experiments defined
in Section 3.6.

###  4.1 Experiment 1

Figure 5 depicts the $\mathit{EMD}$ and $\mathit{KL}_{div}$ results to compare
our proposed ground truth and the saliency map results from LIME [15] with
different sampling size.

We can observe, for both metrics, that, clearly, with an increased sample size
we obtain results more similar to our ground truth. These results are
consistent with the original work from Ribeiro _et al._ [15] that expects an
increase in the size of the sample generates explanations with higher
fidelity. In addition, and because we removed the stochasticity of the
sampling process, as already explained in the previous section, the danger of
a lack of robustness with an increased sampling is removed. We realized all
the existing perturbations, $2^{6}$, and we obtained its corresponding
explanations.

(a)

(b)

Figure 5: Results of the first experiment.

The results of this experiment confirm that with our modified LIME [15]
proposal, we can obtain high fidelity values while maintaining the robustness
of the method. These two facts allowed us to realize further experimentation,
knowing that the results of LIME [15] (when the sampling process uses all
possible samples) are a good comparison element to know whether our ground
truth is correct or not. In addition, we used the results of LIME with the
minimum samples obtained in this experiment as the baseline results to compare
the results of the rest of experiments. This baseline values can be seen in
table 1.

Function | Dataset | $\mathit{KL}_{div}$ | $\mathit{EMD}$
---|---|---|---
$SSIN$ | AIXI-Shape | 3.8873 | 0.8722
AIXI-Color | 1.2284 | 0.5631
$SUUM$ | AIXI-Shape | 3.8926 | 0.8722
AIXI-Color | 1.2286 | 0.5631
$CLASS$ | AIXI-Shape | 4.4907 | 0.3579
AIXI-Color | 2.2768 | 0.2880
Table 1: Results obtained for experiment 1 with the minimum sample size. These
values can be used as a baseline method.

###  4.2 Experiment 2

Table 3 shows the $\mathit{KL}_{div}$ and $\mathit{EMD}$ results obtained
within the second experiment.

We observe that the $\mathit{EMD}$, are nearly approaching 0, indicating a
high similarity between our ground truth and the outcomes generated by LIME
[15]. Similarly, we can see that $\mathit{KL}_{div}$ have low values.
Evaluating this latter metric is somewhat intricate due to the requirement for
a baseline value. We used the results obtained in the first experiment as the
baseline (see table 1), and we can see that from the LIME [15] with less
fidelity to the one used in this experiment the distance is reduced by two
points.

We can also see, in both metrics results, that the worst results obtained were
with the $class$ function. As already explained, we expected this behaviour to
the limitation of the occluding process in a scenario where it can exist
perturbation that do not modify in any degree the output value. We analysed
the non-agreggated results, and we detected that the images with worse results
were the ones that even occluding all the objects did not provoke change on
the function output. Nonetheless, even the results of the classification tasks
had nearly perfect measures.

Function | Dataset | $\mathit{KL}_{div}$ | $\mathit{EMD}$
---|---|---|---
$SSIN$ | AIXI-Shape | 1.4895 | 0.0618
AIXI-Color | 0.1993 | 0.0414
$SUUM$ | AIXI-Shape | 1.2537 | 0.0469
AIXI-Color | 0.1954 | 0.0375
$CLASS$ | AIXI-Shape | 2.3396 | 0.0394
AIXI-Color | 1.4000 | 0.0878
Table 2: Results obtained for experiment 2. The values indicated the result
obtained with a sample size of 128.

Based on the previous analysis, we can state that the proposed methodology
generates the ground truth very similar to the explanations obtained from a
high fidelity and highly robust LIME [15]. Even so, there were sub-experiments
with worse results, we expected that behaviour, and we assessed that it was
provoked by the limitation of the method used. For this reason, we conclude
that, in this controlled environment we generated a correct ground truth for
the explanations. In the following experiment, we analyse whether these
results are consistent in a more realistic scenario.

###  4.3 Experiment 3

Table 3 lists the results obtained in Experiment 3. Similar to the previous
experiment, this table summarizes the $\mathit{KL}_{div}$ and $\mathit{EMD}$
results from all sub-experiments. This experiment used an AI model as the
predictive model that estimates the respective $F:\mathcal{X}\to\mathbb{R}$
function.

We can see that the results, in the regression tasks, are worse than the
previous experiment. We expect this behaviour to the possible inclusion of
errors from the neural network. However, we can see that, in the
classification tasks, the results are slightly better than the previous
experiment. We already explained that we also expected this behaviour to the
sensitivity of this method to small changes, in comparison to the function
inability to be affected by small changes.

Function | Dataset | $\mathit{KL}_{div}$ | $\mathit{EMD}$
---|---|---|---
$SSIN$ | AIXI-Shape | 1.6722 | 0.0728
AIXI-Color | 0.3029 | 0.0784
$SUUM$ | AIXI-Shape | 1.6708 | 0.0734
AIXI-Color | 0.3175 | 0.0854
$CLASS$ | AIXI-Shape | 1.5751 | 0.0679
AIXI-Color | 1.2971 | 0.0856
Table 3: Results obtained for experiment 3. The values indicated the result
obtained with a sample size of 128.

Although the results have a small diverge from the previous experiment, we can
see that are still almost perfect results. This fact is more easily concluded
from the $\mathit{EMD}$ results. All sub-experiments had results near to 0.
Furthermore, the $\mathit{KL}_{div}$ had also small values in comparison to
the baseline from the first experiment.

Based on the previous analysis, we can state that our methodology is
consistent in a more realistic environment. The fact that the results of this
third experiment were almost perfect, and for this reason also similar to
those of the second experiment, indicated that the AI model approximates
correctly the function $F$. The above demonstrates that our assumptions hold:
when we used the proposed methodology, the GT generated was the actual cause
for the predictions for the AI models.

##  5 Conclusion

In this study, we propose a novel methodology to generate a Synthetic
attribution benchmark (SAB) dataset of images. Our methodology allowed us to
define the precise importance for each pixel of an image and can be used to
verify most of XAI techniques, methods, and metrics equally, proposed in the
literature. This ground truth (GT) allowed us to ignore the ad hoc solutions
proposed in the state-of-art.

We verified through three different experiments, both in a controlled
environment, the correctness of this approach. We obtained explanations using
LIME [15] to verify whether our GT was correct. We used this algorithm because
its results achieved high fidelity, as demonstrated in the first experiment.
Our GT was very similar to the LIME [15] results, when this method was forced
to be both robust and faithful to the real cause of the prediction. Therefore,
we can state that we generated the correct ground truth.

We included as results the implemented code for the generation of the datasets
(see https://github.com/miquelmn/aixi/releases/tag/1.0.0). As an additional
contribution we constructed and opened to the scientific community two
synthetic datasets, AIXI-Color and AIXI-Value, with the positions of the
objects and their classes as a ground truth. For the sake of scientific
progress, it would be beneficial if authors published their raw data, code,
and the image datasets.

##  6 Declaration of competing interest

The authors declare that they have no known competing financial interests or
personal relationships that could have influenced the work reported in this
study.

##  7 Funding

Project PID2019-104829RA-I00 “EXPLainable Artificial INtelligence systems for
health and well-beING (EXPLAINING)” funded by
MCIN/AEI/10.13039/501100011033. Miquel Miró-Nicolau benefited from the
fellowship FPI_035_2020 from Govern de les Illes Balears.

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# Intermolecular Coulombic decay by concerted transfer of energy from
photoreceptors to a reaction center

Saroj Barik  Department of Physics, Indian Institute of Technology Madras,
Chennai, India  Nihar Ranjan Behera  Department of Physics, Indian Institute
of Technology Madras, Chennai, India  Saurav Dutta  Department of Physics,
Indian Institute of Technology Madras, Chennai, India  Y. Sajeev
<EMAIL_ADDRESS>Theoretical Chemistry Section, Bhabha Atomic Research
Centre, Mumbai, India  G. Aravind<EMAIL_ADDRESS>Department of Physics,
Indian Institute of Technology Madras, Chennai, India

###### Abstract

Molecular mechanisms that enable concerted transfer of energy from several
photoacceptors to a distinct reaction center are most desirable for the
utilization of light-energy. Here we show that intermolecular Coulombic decay,
a channel which enables non-local disposal of energy in photoexcited
molecules, offers an avenue for such a novel energy-transfer mechanism. On
irradiation of pyridine-argon gas mixture at 266 nm and at low laser
intensities, we observed a surprisingly dominant formation of argon cations.
Our measurements on the laser-power dependence of the yield of the Ar cations
reveal that intermolecular Coulombic interactions concertedly localize the
excitation energy of several photoexcited pyridines at the argon reaction
center and ionize it. The density of the reaction center offers an efficient
handle to optimize this concerted energy-transfer. This mechanism paves the
way for a new $\pi$-molecular light-harvesting system, and can also contribute
to biomolecular stability against photodamage.

##  I Introduction

Intermolecular Coulombic decay (ICD) is the most efficient and ubiquitous
relaxation pathway by which excited molecules in a molecular environment relax
by ionizing their neighbours [1, 2, 3, 4, 5]. Following the theoretical
prediction of ICD [1], a number of experiments revealed this relaxation
channel in several molecular environments [4, 6, 7, 5, 8, 9, 10, 11, 12, 13,
14, 15, 16, 17, 18]. Due to the strong dependence of the ICD rates on the
intermolecular interactions, early reports on its measurements were
essentially on rare gas clusters. While ICD has also been reported in many
natural molecular systems, including liquid water [19], $\pi$-stacked aromatic
molecules [20], and hydrogen-bonded dimers, they all are mainly concerned with
the electronic relaxation of an inner-valence ionized state created by
ionizing radiation or electron impact. A recently reported experiment in which
$\pi$-molecules collectively relax via a highly efficient multi-center ICD,
following their $\pi-\pi^{*}$ photoexcitation at ambient UV intensity,
demonstrates that this mechanism is also of great importance in general
photochemistry [31]. The efficiency of this multi-center mechanism is revealed
by the fact that the ICD operates through an associative covalent interaction
even when the $\pi$-molecules are unbound in their ground state. It is,
therefore, apparent that this multi-center ICD mechanism is a fundamental
reaction channel to explore for the nonlocal utilization of excitation energy
in $\pi$-molecular systems.

The nonlocal and simultaneous utilization of excitation energy, by its
transfer from several photoexcited $\pi$-molecules to a distinct non-
photoabsorbing reaction center, is not known but is most desirable in many
fields of science. DNA and light-harvesting (LH) systems are two examples
where many $\pi$ molecules can undergo simultaneous photoexcitation. DNA, made
up of $\pi$-electron nucleobases, is one of the macromolecules susceptible to
UV induced direct damage [22]. When multiple nucleobase moieties of DNA absorb
UV radiation, a deposition in excess of 3 eV on each of the bases can trigger
a variety of photochemical reactions in the DNA leading to mutagenesis and
carcinogenesis [23, 24, 25, 7]. Ultrafast and nonlocal disposal of this excess
electronic energy accumulated on adjacent bases is the key to prevent its
photodamage. Therefore, the possibility that excited nucleobases can use their
excitation energy to ionize the DNA counter-ion and thereby relax collectively
warrants a discussion. Similarly, given the fact that many receptors are
simultaneously photoexcited in a $\pi$-molecular LH system, the availability
of nonlocal molecular mechanisms that help to control the rapid and concerted
flow of electronic energy from several receptors to a reaction center can be
considered a major challenge for the development of future LH systems.

As a relaxation channel facilitating nonlocal transfer of electronic energy
from photoexcited molecules, the ICD mechanism is an obvious solution to the
two challenges discussed above. But, although ICD mechanisms involving energy
transfer between photoexcited molecules and, between a photoexcited molecule
and its neighbour, have been reported, there is a void in our knowledge on ICD
mechanisms that involve concerted and efficient transfer of electronic energy
from several photoexcited molecules to a distinct non-photoabsorbing reaction
center. Therefore, by keeping these two challenges in focus, to study
concerted transfer of electronic energy within an unbound environment
consisting of photoreceptors and reaction centers, we explore here an ICD that
ensues photoabsorption in an unbound system of pyridine monomers diluted in
argon (Ar) gas.

##  II Results and Discussion

The pyridine-argon molecular environment was irradiated with 266 nm photon
beam. Even at low laser intensities between $10^{4}-10^{6}$ Wcm-2, we observed
a surprisingly dominant formation of argon cations, as shown in Figure 1. All
the cations that were observed in the undiluted pyridine experiments [31] were
also observed. Velocity map image (VMI) of the electrons emitted in the
present experiment, i.e., on irradiation of the pyridine-argon environment, is
shown in Figure 5. The VMI shows isotropic emission of electrons with kinetic
energies ranging between 0-0.6 eV.

Figure 1: Time-of-flight mass spectrum of cations and the laser-power
dependence for Ar+ yield: The mass spectrum in the top panel shows the
dominant formation of argon cations and other cations from pyridine upon
irradiation of pyridine-Ar gas mixture. For this measurement, pyridine at its
vapour pressure was diluted with 1 atm. of Ar gas. The laser wavelength
employed was 266 nm at an intensity of 2.4 $\times$ 106 Wcm-2. Adjacent-
averaging of the raw data points was performed using Origin 7.5 software for
this plot. The laser power dependence of the Ar+ yield is linear as shown in
the bottom panel. a.u., arbitrary units.  Figure 2: VMI of ICD electrons: VMI
showing isotropic emission of ICD electrons from Ar atoms and pyridine
molecules, where the imaging is not done in coincidence with any particular
cation. au, atomic units.

We now consider the formation of Ar+ cations in our experiments. The single-
photon energy (4.661 eV) is way below the ionization threshold of Ar (15.75
eV). Thus, four-photon energy is required for ionizing the Ar atoms. The
laser-intensities employed here are low for the occurrence of multi-photon
ionization (MPI) in argon. There are no resonant transitions in argon at 266
nm which could enhance the MPI. Further, since four photons are required for
the MPI of argon, the plot for log (Ar+ yield) vs log (laser-pulse intensity)
should have a value of 4 for its slope. However, in the present experiments,
the value for this slope is nearly 1, as shown in Figure 1. The observed
isotropic emission of electrons from argon atoms is also not expected in a MPI
process. Thus, the linear variation of the yield of Ar+ cations with the laser
power and the isotropic emission of electrons unambiguously rule out the
photoionization of argon atoms in our experiments. Therefore, considering the
fact that pyridine undergoes $\pi-\pi^{*}$ resonant excitation at 266 nm, the
observation of dominant Ar+ cations in Ar-pyridine gas mixture, at ambient
intensity of this light, is a key to explore the mechanism of concerted
transfer of electronic energy from multiple photoacceptors to a reaction
center.

Three important experimental observations shed light on the mechanism behind
the observed formation of Ar+ cations. First, Ar+ cations are formed under
experimental conditions where pyridine monomers are dissolved in Ar gas.
However, under the same laser-beam conditions, Ar+ cations are not formed when
Ar gas alone is used. Second, as mentioned above, despite the requirement of
four-photon energy, the laser-power dependence for the yield of Ar+ cations is
linear. These two features affirm a resonant single-photon absorption in
pyridine molecules followed by a non-local transfer of electronic energy from
$\pi^{*}$-excited pyridine molecules to Ar atoms. Since the non-local transfer
of electronic energy depends only on the electron-electron correlations, the
overall laser-power dependence for the yield of Ar+ cations will be that of
the preceding step viz. single-photon absorption [2, 10]. Hence, this linear
laser-power dependence for the yield of Ar+ cations affirms ICD ionization of
the Ar atoms. Lastly, the observation of isotropic emission of electrons is
characteristic of ICD ionization. In this experiment, the ICD ionization is a
Coulombic decay involving concerted transfer of electronic energy from four
photoexcited pyridine molecules to an Ar atom.

We now discuss the Coulombic decay in photoexcited pyridine molecules that
leads to the formation of Ar+ cations under our experimental conditions. The
$\pi$-electron molecules exhibit long-range associative covalent interactions
in their $\pi^{*}$ excited state. Therefore, under our experimental
conditions, the dominant ICD channels are expected to originate through the
dimers of excited state pyridines, which are formed by this interaction. The
two-photon energy in such dimers is barely above the ionization potential of
pyridine and hence the intramolecular vibrational energy redistribution (IVR)
would render the two-photon energy off-resonant for ionization in pyridine
dimers [31]. Hence, such a dimer of excited pyridines can primarily relax
through an ICD channel by its direct interactions with an excited monomer or
with another dimer of excited pyridines. This direct ICD channel is very
efficient because it is driven by covalent interactions between the
$\pi^{*}$-excited states of the pyridine units [31]. The insertion of argon
atoms into this photoexcited pyridine system opens a new competitive ICD
channel involving the argon atom and two dimers of excited pyridines. Since
the relative yield of the argon cation is observed to be significant and
competing with the direct ICD products involving only photoexcited pyridine
units, this channel must also be driven by efficient covalent interactions
between the participating units, i.e., two dimers of excited pyridines and an
argon atom. To understand the extent of the covalent interaction between the
argon atom and the two dimers, we calculated the electron density distribution
of the molecular units that comprise them (see the Methods section for
computational details).

On the basis of our experimental observation and quantum chemical
calculations, a molecular mechanism for the ionization of an argon atom via
ICD due to its interaction with two dimer units of excited-state pyridines is
shown in Figure 3. When an argon atom diffuses into the noninteracting
interior space of two dimers of photoexcited pyridines, the intermolecular
covalent interaction between the dimers is activated by the Ar atom. As a
result, the ICD is activated and the molecular system relaxes by ionizing the
Ar atom. A covalent mixing of 1$\pi$ orbitals of pyridines with the 3$p_{z}$
atomic orbital of the argon atom is responsible for the concerted relaxation
of energy via the ICD channel. This covalent mixing of orbitals is seen even
when Ar and excited pyridines are separated by more than 10$\AA$ .

Figure 3:  Molecular mechanism of photoexcited pyridine molecules ionizing an
argon neighbor via ICD: a, Argon atom is shown in green while pyridine is
shown in red. The haze represents the valence electron density, and the blue
outer haze represents the photoexcited electron density of pyridine.
Intermolecular covalent interactions between two non-interacting dimers of
photoexcited pyridines are activated by the diffusion of an Ar atom into the
noninteracting inter dimer space. Consequently, the ICD becomes operative, and
ionizes the Ar atom. b-c, The overlap of the wavefunction due to diffusion of
the Ar atom into the non-interacting inner dimer space is shown by comparison
of the total electron density surfaces with and without the argon atom, where
the surfaces are plotted for a very small iso-density value of
$1\text{x}10^{-6}$. d, The intermolecular orbital formed from the covalent
mixing of 1$\pi$ orbitals of pyridines with the 3$p_{z}$ orbital of the argon
atom is shown, where a $4\AA$ of pyridine-argon separation is used for better
visualization of the covalent mixing.

As mentioned earlier, a doubly-excited pyridine dimer could associate with a
photoexcited dimer or a monomer and undergo ICD. The electronic energy
transfer from the doubly excited dimers to argon atoms should compete with the
above mentioned ICD that occurs between photoexcited pyridine units. At higher
laser intensities, when the number density of photoexcited dimers increases,
the ICD between the photoexcited pyridine units would increasingly dominate
over the ionization of Ar. At lower number densities of photoexcited dimers,
the separations between them being larger, the probability for ionizing Ar
atoms en route to their association increases. Hence, the ratio of the yield
of Ar+ cation to that of the total yield of all the other cations originating
from photoexcited pyridines should increase with decreasing laser intensities,
and this is observed in our experiments, as shown in Figure 4. Further, the
relative yield of argon cations were observed to also increase with increasing
partial pressure of the argon gas, as shown in Figure 4. These two
observations unequivocally assert the concerted ICD mechanism between several
photoreceptors and a distinct reaction center. Increasing relative yield of
Ar+ cations implies efficient energy transfer. Hence, it is seen that both the
laser intensity and the partial pressure of the argon reaction center offer an
efficient handle to optimize this concerted energy transfer.

Figure 4: Relative yield of Ar+: The ratio of the Ar+ yield to the total yield
of all the other cations originating from photoexcited pyridine monomers is
displayed. In the upper panel the laser intensity is varied and the argon
pressure is fixed at 0.5 atm. In the bottom panel, the argon pressure is
varied for a fixed laser intensity of $1.49\times 10^{6}$ Wcm-2. For the
details on the partial pressure of pyridine monomers, see the Methods section.
These two measurements show that it is possible to optimize the concerted
energy transfer from the photoexcited pyridine to argon by reducing the
intensity of the laser or by increasing the partial pressure of argon.

A few important remarks on the generality of this concerted ICD mechanism is
in order. The ionization of the argon atom serves here as a model for the
reaction that would occur due to the concerted energy-transfer from several
excited pyridine centers to a distinct reaction center. Any reaction occurring
at the reaction center due to the availability of electronic energy, by
concerted localization of excitation energy in general, but not limited to
such hole-particle separation, can be initiated through this mechanism.
Moreover, since the mechanism is initiated due to the overlap of
wavefunctions, this concerted transfer of energy to the reaction center may
originate not only from a collection of $\pi-\pi^{*}$ excited states, but also
from a set of $n-\pi^{*}$ states or molecular Rydberg states.

To conclude, our measurements on the laser-power dependence for the yield of
Ar cations, the isotropic ejection of slow electrons, and the laser-power
dependence of the relative yield of Ar cations, unequivocally reveal concerted
and nonlocal energy transfer from four photoexcited pyridine units to an Ar
atom. This novel ICD channel, wherein $\pi^{*}$-excited molecules ionize their
atomic environment via concerted and nonlocal energy transfer, has wide
implications in several fields. For instance, an efficient molecular mechanism
that transfers energy from multiple photoacceptors to a distinct reaction
center characterizes all chemical systems of energy synthesis in Nature,
including light harvesting (LH) systems. But unlike conventional LH systems,
where the energy transfer from a photoacceptor to its reaction center is
limited to resonant condition set by the charge separated state of the
reaction center, a reaction is possible at the reaction center via ICD without
the limitation of a resonance condition. Furthermore, our finding that
photoexcited $\pi$-molecules of a system flare of their excitation energy by
ionizing a distinct neighbouring atom raises a fundamental biological
question: Has a similar concerted nonlocal energy transfer mechanism evolved
along with the chemical evolution of $\pi$-chromophore-rich macromolecules,
such as DNA, to avoid the apparent consequences of multiple photoabsorption by
their chromophores? The fact that life on Earth has evolved under the
conditions of intense UV light validates the existence of such a protective
bio-molecular mechanism. Clearly, the advent of ICD as a fundamental molecular
mechanism deep into many frontiers of science is evidenced by our observation
that $\pi^{*}$-excited molecules ionize a distinct reaction center.

## METHODS

### Experimental methods

Gas-phase pyridine was diluted in argon gas before letting it into the
interaction chamber through a solenoidal pulsed valve. The gas line was
evacuated to 3.6 $\times$ 10-2 mbar using a dry pump before letting pyridine
into the gas line. For the measurements on the relative yield of Ar+, as
discussed in Figure 4 of the manuscript, pyridine vapour was let into the
evacuated gas line through a needle valve to obtain a gas line pressure of
about $1.1\times 10^{-1}$ mbar. Pure Ar gas of various partial pressures
ranging from 0.25 to 1.2 atm was added to the same gas line. The gas mixture
was then pulsed into the interaction chamber at 10 Hz. Measurements were made
with a chamber pressure of $1\times 10^{-5}$ mbar, while operating at the
above mentioned repetition rate. Argon ionization was observed even at
$5\times 10^{-6}$ mbar pressure in the interaction chamber. The base pressure
was maintained around $8\times 10^{-8}$ mbar. For the measurements on the
pressure dependence of the relative yield of Ar+, the laser intensity was
maintained at $1.49\times 10^{6}$ Wcm-2.

The aperture of the nozzle was of 0.8 mm diameter and there was no formation
of pyridine clusters at the above mentioned operating conditions. Nearly 10 cm
downstream the nozzle exit, the gas pulse was collided with an unfocussed 266
nm Nd: YAG laser beam, midway between the first two electrodes of a time-of-
flight mass spectrometer (TOFMS) [31]. A pulsed high voltage was applied to
the repeller plate of the TOFMS to drive the ions towards the detector. The
repeller voltage pulse was applied after a delay of about one microsecond post
the laser pulse. The data acquisition was triggered by the TOF extraction
pulse and hence the MCP dark counts were completely timed out. For electron
measurements, the pyridine-Ar mixture was pulsed between the first two
electrodes of a velocity-map imaging spectrometer [26] and the gas pulse was
collided with 266 nm laser beam. The VMI spectrometer was carefully calibrated
by performing photodetachment of anions with known electron affinities.

When the partial pressure of pyridine behind the nozzle is 10 mbar, it
corresponds to a number density of $2.4\times 10^{17}cm^{-3}$. The interaction
region is 125 nozzle diameters away from the nozzle. The density at the
central line of the interaction region is thus about $2.4\times
10^{13}cm^{-3}$. Laser beam intersects the neutral beam at right angles. The
number of photons in a 7 mJ (avg energy of the range of pulse energies
employed) pulse of 266 nm beam = 9.3 $\times 10^{15}$. The pyridine number
density would vary as we move away from central beam, so considering an
overlap cylinder of diameter 5 mm and length 5mm. Overlap volume is thus =
0.0981 cm-3. The total number of molecules in this volume is $2.355\times
10^{12}$. The photoabsorption cross-section at 266 nm, $\sigma=19.3\times
10^{-19}cm^{2}.$ $\sigma\times$ target density $\times$ photon flux =
$4.3\times 10^{11}cm^{-1}$ per pulse. The number of photoexcited pyridine
molecules in the overlap volume = $4.3\times 10^{11}cm^{-1}\times 0.5cm$ per
pulse = $2.15\times 10^{11}$ per pulse. Hence, in the interaction volume
$V=0.0981cm^{-3}$, the number of photoexcited pyridine is $2.15\times 10^{11}$
per pulse and the number of Ar atoms (partial pressure = 0.5 atm) present in
the same volume $\approx$ $1\times 10^{14}$. We detect about three to four Ar+
cations per laser shot. ICD, being a long-range non-collisional process [4],
does not come under the realm of hard-sphere collision model. Calculation of
ICD rates in the present case should take into account the attractive
interactions between the dimers along with the initial and final states of the
dimers.

### Computational methods

Computation of the electronic structure of a molecular system consisting of
four excited pyridine moieties and one argon atom using currently available
bound state-based quantum chemical codes is a challenge. This is because the
electronic state of the system containing four excited pyridines is above the
ionization threshold when intermolecular interactions are active. Available
quantum chemical methods are inadequate to account for such a state
interacting with the ionization continuum. To overcome this difficulty, we
used a real-valued continuum remover potential based stabilization method
[27]. The real-valued continuum remover potential, applied at the non-
interacting asymptote of the molecular potential, removes the discretized
continuum descriptions of the fourfold excited state and transforms it into a
bound state of the modified Hamiltonian.

In order to show the extent of covalent interaction in the excited state, the
electron density in the stabilized excited state was calculated at the
excited-state Hartree-Fock level [28]. This calculation was performed using an
in-house modified Gamess-US quantum chemical package [29, 27]. We imposed the
symmetry of the orbitals as a constraint on the construction of the excited
state density at each iteration step of the excited-state Hartree-Fock
calculation. Since the intermolecular covalent interaction between $\pi^{*}$
orbitals of pyridines is maximum when they cofacially approach, such geometric
structures were chosen to measure the extent of inter-molecular covalent
interaction. The molecular geometry of pyridine reported in Ref. 30 is used
for the pyridine moieties. An atom centered 6-311 split valence Gaussian basis
set, which is further augmented with three diffused even tempered Gaussians
for the non-hydrogen atoms, is used in the calculation.

#### Acknowledgements

This work was supported by the Department of Science and Technology through
project no: EMR/2016/005247 (G.A.).

#### Author contribution

S.B., G.A. and Y.S. conceived the concerted ICD. S.B. and G.A. planned the
experiment and carried out the measurements with the support of N.R.B. and
S.D. Y.S. planned the quantum-chemistry calculations. The results were
discussed among the authors. Y.S. and G.A. wrote the manuscript.

#### Additional Information

Electronic supplementary information is available online for:

  1. 1.

Kinetic energy spectrum of the emitted electrons.

  2. 2.

Additional details on the experiment.

  3. 3.

Absence of electron-impact ionization of argon in the present experiments.

#### Corresponding authors

Correspondence and request for materials should be addressed to G.A
<EMAIL_ADDRESS>or Y.S. (sajeevy@barc.gov.in).

##  III Supplementary Material

## Kinetic energy spectrum of the emitted electrons.

Figure 5: Velocity Map Imaging (VMI) of electrons: The top panel displays the
VMI of the electrons emitted and the bottom panel displays the kinetic energy
of the released electrons. au, atomic units. a.u., arbitrary units.

Figure 5 shows the velocity map image of the ICD electrons with low kinetic
energies. In most of the ICD experiments reported so far, excitation of one
sub-unit of a dimer is the first step. Then an ultra-fast energy transfer from
the excited sub-unit to the other sub-unit of the dimer ionizes the latter. In
such cases, the maximum kinetic energy of the ejected electrons will be the
difference between the excitation energy in the dimer and the ionization
energy. However, in the present experiment, since the initial excitation is in
unbound molecular monomers, we expect slow ICD electrons, as discussed below.
On photoexcitation of pyridine monomers, Intramolecular vibrational
redistribution (IVR) leads to transfer of the excess electronic energy into
the nuclear modes. When ICD occurs en route to the association of these
excited monomers, only a small fraction of energy in the nuclear modes becomes
available for ICD ionization leading to slow electrons. This is because ICD
occurs faster than the relaxation of energy from the nuclear modes back to the
electronic mode. Since IVR is the characteristic of the photoabsorbing
$\pi$-molecules, the range of kinetic energy for the ICD electrons should not
differ much for the ICD ionization of pyridine molecules [31] and Ar atoms.
Further, the observed isotropic emission of the ICD electrons is
characteristic of ICD ionization.

## Additional details on the experiment

Electron measurements: The VMI electrodes were plated with gold, whose
workfunction is above the photon energy (4.661 eV).

Ion measurements: The photon energy is above the workfunction of the stainless
steel TOFMS electrodes ($\approx$ 4.4 eV). In order to avoid the acceleration
of the photoelectrons, if any, produced from the surfaces of the TOFMS
electrodes, we had applied the TOFMS pulsed electric-field with a finite time-
delay after the laser-pulse. As mentioned in the Methods section, this delay
was about 1 microsecond, which is sufficient to get rid of the photoelectrons
($\approx$0.26 eV) from the volume contained by the TOFMS electrodes. We did
not apply an electric-field exclusively to sweep photoelectrons produced, if
any. We had also performed SIMION simulations, which show that even if
photoelectrons were produced from the TOF electrode surfaces and are
accelerated, their path would result in the formation of Ar+ much away from
the central volume of the first two TOF plates. In such a situation we see
that the field is unable to manoeuvre the ions to the detector, which is
located 1 meter away from the interaction region. TOF electric field is
optimized to collect ions produced at the overlap of laser-gas beam region.
Ions produced at random locations will also result in a very broad TOF peak,
which is also not seen for any of our ions. Thus, with great care we have
ensured the absence of any role of photoelectrons.

Thus Ar+ were not formed on irradiation of pure-Ar gas alone, as shown in the
Figure 6 (blue points). We observe Ar+ only when pyridine-Ar gas mixture is
irradiated and most importantly, the Ar+ yield depends on the density of the
pyridine gas, as seen by comparing the black and red data points in the Figure
6. The latter clearly asserts that the ionization of Ar is due to concerted
energy-transfer from photoexcited pyridine molecules. Note that the yield in
Figure 6 decreases with increasing time-delay. This is due to the degrading
collection efficiency as the initial position of the ions deviate from the
volume centered between the first two TOFMS electrodes.

Figure 6: The yield of Ar+ cations as a function of the delay between the
laser pulse and the TOFMS electric-field pulse. The laser energy was
maintained the same for these data points.

## Absence of electron-impact ionization of Argon in the present experiments

We describe below our experiments to that rule out electron-impact ionization
of argon in our experiments. Figure 7 shows the schematic of our time-of-
flight spectrometer (TOF) along with the voltages applied. The distance
between adjacent electrodes = 2 cm.

Figure 7:  Electrode 1: A pulsed voltage $V_{1}$ is applied over a DC base
voltage $V_{0}$. Electrode 2: A pulsed voltage $V_{2}$ is applied w.r.t. to
the ground. The ions and electrons formed between the first two electrodes
experience a DC electric field of 50$V_{0}$ V/m for $t_{0}$ microseconds,
before the application of the high voltage pulses that drive the cations
towards the detector.

  * For a DC potential difference of $V_{0}$ between the first two electrodes, the acceleration for an electron is

$a=\cfrac{Eq}{m_{e}}=\cfrac{50V_{0}*q}{m_{e}}=8.8V_{0}\times
10^{12}~{}m/s^{2}$

When $V_{0}=0.5V$:

  * (a) The time taken by an electron produced at the center of the interaction region, with zero KE, to hit the first TOF electrode is

$t=\sqrt{\cfrac{2\times 0.01m}{4.4\times 10^{12}m/s^{2}}}=67~{}ns$

  * (b) An electron produced with 0.2 eV and with an initial velocity directed away from the first TOF electrode will travel 8 mm before it turns around. The total time for it to hit the first TOF electrode is

$t=\left(\cfrac{1}{a}\times\sqrt{\cfrac{2\times
0.2eV}{m_{e}}}\right)+\sqrt{\cfrac{2\times 18mm}{a}}\approx 150ns$

  * Therefore for $V_{0}=0.5V$ and a delay of $1~{}\mu s$ between the laser-hit and the high-voltage extraction, there won’t be any electrons present in the interaction region when the high-voltage pulse arrives!

Figure 8:  The yield of Ar+ cations as a function of $V_{0}$, the DC voltage
that sweeps the electrons.

  * Figure 8 shows that the yield of Ar+ is nearly the same with and without the application of the electron sweeping voltage $V_{0}$. For different $V_{0}$ values, the initial positions and energies of the Ar+ cations prior to the application of the high-voltage pulse differ, resulting in slight variation in the efficiency of their collection at the detector, which is situated one meter away from the interaction point. This reflects in the minor variation in the Ar+ yield as $V_{0}$ changes. Thus, there is no contribution of ICD electrons in ionizing the Argon atoms.

  * Earth’s magnetic field: Since the above experiments reveal the absence of any electron during the arrival of the high-voltage pulse, the Earth’s magnetic field does not also have a role here.

Additional experiment

Figure 9: Mass spectra obtained by applying $V_{1}=402V$ and $V_{2}=392V$.
Laser and gas-beam conditions are the same. No electron sweeping potential is
applied.

We also performed an additional experiment, wherein $V_{0}=0$ but the
potential difference between the first two electrodes was maintained at 10
Volts! $V_{1}=402V$ and $V_{2}=392V$. Thus the maximum energy gained by any
electron is 10 eV, which is insufficient to ionize Argon (I.P. = 15.6 eV).
Figure 9 shows that the mass spectra thus obtained is the same as the mass
spectra obtained in our original experiments (Figure 1 of the main
manuscript). Thus, there is no electron-impact ionization of Argon in our
experiments.

  * (a) In fact, the kinetic energy of the ICD electrons from pyridine peaks at 80 meV, corresponding to a velocity of 1.7 $\times 10^{5}$ m/s. Hence the ICD electrons will move away from the interaction region (2 cm wide) in 0.1 $\mu s$. Therefore these electrons will not be present when the high-voltage pulse arrives 1 $\mu s$ post laser-hit, as done in our original experiments.

  * (b) Moreover, collisional ionization of Ar by the ICD electrons from pyridine, being a secondary process, should result in a very low yield of Ar+ relative to the pyridine cations. On the contrary, as shown in Fig 4 of the main manuscript, with decreasing laser intensity, the relative yield [Ar+] / [pyridine cations] tends to 1!

  * Points (a), (b), the results of the weak DC-electric field experiment and the additional experiment described above, together with the observed variation of the relative yield [Ar+] / [pyridine cations] with the laser intensity, unambiguously assert the concerted ICD ionization of Ar in our experiments.

Ion-TOF spectra with no gas and with only Argon gas

We also present the ion-TOF spectra for (a) no gas (background) and (b) only
argon gas. There is no Ar+ signal seen, when only pure argon gas is irradiated
with 266nm. These two spectra were taken even without a delay in the
extraction pulse w.r.t to the laser pulse. These spectra also assert that
there is no formation of Ar+ due to impact with accelerated photoelectrons.
These data were taken with the laser intensity and data collection time same
as that for Fig 1 of the manuscript and hence the yields could be compared
with each other. Only a very weak background ion signal is seen.

Figure 10: Ion TOF spectra for the background and pure Argon gas are
presented. These data were taken with the laser intensity and data collection
time same as that for Fig 1 of the manuscript.

Electron spectra with no gas and with only Argon gas

The spectra Fig. 11 and Fig. 12 are the electron spectra obtained for (a) no
gas and (b) argon only, respectively. The data were taken for 10000 laser
shots with the same laser intensity as employed in Fig 2 of the manuscript. A
structureless weak background electron signal is seen for both the spectra
with equal counts! This unambigoussly rules out Ar+ signal when only Argon is
irradiated with 266nm.

Figure 11: Electron VMI with no gas. The data is taken for 10000 laser shots
with intensity same as employed for Fig 2. of our manuscript. This weak
structureless spectra has count equal to 85. Figure 12: Electron VMI with only
argon gas. The data is taken for 10000 laser shots with intensity same as
employed for Fig 2. of our manuscript. This weak structureless spectra has
count equal to 84.

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