Najat El Mekkaoui

Bérangère Legendre

Does pension information impact savings?

Keywords: Pension statements, savings for retirement, financial literacy, quantile regressions. JEL Classification D83, J18, D14, C31

Many pension reforms in OECD countries included pension statements with the objective of improving individuals' financial security in retirement. Our objective is to assess the effectiveness of the pension information policy implemented in France and to investigate whether the pension statement results in better informed workers, who then increase their retirement savings.

Using regression discontinuity designs combined with quantile regressions, we assess whether the changes in retirement savings and holding of assets are due to the pension information system and then quantify the impact. We conclude that a pension estimate sent to workers encourages the wealthiest to increase their retirement savings while it does not influence the savings of individuals with a low level of wealth.

1-Introduction

The combination of a relatively low birth rate and increasing life expectancy is leading to marked aging in Europe. Thus, in the ten years from 2008 to 2018, the share of people over 65 increased by 4.2 percentage points in the Netherlands and 3.8 in Portugal. In these countries, people over 65 years of age represented 18.8% and 21.5% of the population in 2018, respectively. In France, the share of people over 65 increased over the same period by 3.3 percentage points to represent 19.7% of the population. In Italy and Germany, the increase was less pronounced, with the share of those over 65 increasing by 2.4 and 1.3 percentage points respectively. They now represent 22.6 and 21.4% of the population. To cope with aging, European governments carried out in-depth pension system reforms, thus safeguarding pay-as-you-go systems. At the end of the 1990s, Italy and Sweden adopted notional defined contribution schemes (NDC) within a payas-you-go (PAYG) system. France only recently began a process of systemic reform leading to a point-based system. The current health crisis temporarily interrupted the reform process. This type of system tightly links pension entitlements to pension contributions and reinforces the responsibilities of households in maintaining their standard of living in retirement [START_REF] Benartzi | Behavioral economics and the retirement savings crisis[END_REF][START_REF] Poterba | Retirement security in an aging population[END_REF]. Part of the risk is indeed transferred from the pension system to the insured [START_REF] Debets | Pension Communication in the Netherlands and Other Countries[END_REF]. who are often forced to delay their retirement age in order to obtain a pension equivalent to the one they would have had before the reforms. The effect of these reforms on household financial decisions are therefore particularly complex (Beshears et al., 2006;[START_REF] Fornero | Information and financial literacy for socially sustainable NDC pension schemes[END_REF]. The transfer of responsibility necessarily involves asking households to make decisions that are rational, if not optimal. They must therefore be able to make relevant intertemporal trade-offs (Adam and Rau, 2011), and to formulate hypotheses not only about their own future (employment, family, health) but also about the economic situation (interest rates, inflation, growth). Their choice of retirement has a direct impact on the amount of pension they will receive, and even if a reference retirement age can prevent people retiring too early and receiving too small of a pension, such a legal retirement age is no longer necessary as it does not balance the system [START_REF] Bozio | Is a reference age necessary un a points pension system?[END_REF]. At the same time, households have to anticipate their retirement income and whether their resources will be adequate. The literature on optimal savings decisions demonstrates how ill-equipped, or disparate, households are with respect to the need to prepare for retirement (Prast and von Soest, 2016).

Given the lack of financial literacy and the complexity of issues related to retirement preparation, providing information to households can be a lever to change behavior, help households make long-term decisions and thus prepare for retirement. The abundant literature on financial literacy shows how a better understanding of financial mechanisms and environment is crucial to efficiently prepare for retirement, especially in a context of pension reform [START_REF] Lusardi | Financial literacy and the need for financial education: evidence and implications[END_REF][START_REF] Von Neumann | An empirical study on how financial literacy contributes to preparation for retirement[END_REF]. Many pension reforms in OECD countries have included pension statements with the goal of improving individuals' financial security in retirement. Our objective is to assess the effectiveness of the pension information policy implemented in France and to investigate the pension statement impact in the retirement savings behavior.

Households lack basic financial education and that they do not pay attention to preparing for retirement (Lusardi and Mitchell, 2011;[START_REF] Lusardi | Financial literacy and the need for financial education: evidence and implications[END_REF], which can lead to a sharp drop in living standards if they do not take into account the impact of pension system reforms. The situation appears even more worrying for households with the lowest standard of living, whose financial education seems to be the least developed, leading to lower savings (Lusardi and Mitchell, 2011b), but also to less efficient investment choices [START_REF] Stango | Fuzzy Math and Red Ink: When the Opportunity Cost of Consumption is Not What It Seems[END_REF], or even to retiring too early to be able to have an adequate standard of living. Moreover, [START_REF] Bozio | Is a reference age necessary un a points pension system?[END_REF] show that in the context of the potential introduction of a point-based retirement system in France, informing individuals when they reach a target replacement rate may be a good idea to guarantee that each individual obtains a replacement rate deemed to be sufficient.

Two types of information can be distinguished: on the one hand, general information, explaining to individuals how the pension system works, or the impact of reforms on pension levels at the age of retirement, and on the other hand, more specific and personalized information informing the insured about what they are entitled to and the amount of pension they can expect. The latter type of information can provide precise figures enabling individuals to have a clear picture of their standard of living after they have left the labor market. This is this system that has been introduced in France.

Providing information to households about public and occupational pensions is a way for governments to improve understanding [START_REF] Joo | A retirement investment and savings decision model: influencing factors and outcomes[END_REF][START_REF] Lusardi | Household Saving Behavior: the role of financial literacy, information, and financial education programs[END_REF][START_REF] Lusardi | Financial literacy and the need for financial education: evidence and implications[END_REF][START_REF] Van Rooij | Financial Literacy, Retirement Planning and Household Wealth[END_REF] as well as increase support for pension reforms. The role of the policy maker is therefore to improve the quality of information provided, but also to present it in a clear and uncomplicated way. The institutional environment can facilitate making good use of information at hand within an appropriate decision making environment [START_REF] Altman | Behavourial Economics Perspectives: Implications for Policy and Financial Literacy[END_REF]. Better information appears to be a major issue, as citizens tend to overestimate their pension benefits [START_REF] Dolls | Do retirement savings increase in response to information about retirement and expected pensions[END_REF] and underestimate the cost of public pension systems (Boeri et al.;2001[START_REF] Boeri | Pension reforms and the opinion of European citizens[END_REF].

We focus on the impact of the second type of information on individual savings. We evaluate the French public policy of providing retirement information to individuals about their retirement savings. We examine outcomes after the pension information system has been introduced in France. To do so, we use the SHARE survey, which includes individuals belonging to households where at least one of the members has reached the age of 50. Individuals aged 55 and over since 2007 are the focus of the French retirement information policy. We exploit the conditions of a quasi-natural experiment owing to the progressive implementation of retirement information in France starting in 2007. In 2011, the system was fully implemented and we identify the few cohorts that received pension information -individuals born in 1955 and before -and those who had not yet received it (born in 1956 or after). We therefore use a strict discontinuity regression since only the year of birth determines the individuals to whom the information was sent.

The present article contributes to the literature by providing a public policy assessment that can accompany more structural reforms of pension systems. Clear, transparent, and well-understood information is essential for agents to make informed and rational decisions in the changing context of pension systems.

We concluded that the pension statement sent to workers influences only the wealthiest. Savings for longterm investment is affected by pension information only in the highest quartile. The lowest quartile does not have any long-term assets, and pension information therefore has no impact. When considering financial assets, once again, the pension statement affects the amount held by the wealthiest. Having received the pension estimate increases this amount by 39 to 46% (depending on econometric specifications) on average among those in the highest quartile. We found no significant impact for those with the lowest levels of wealth. This conclusion is in line with part of the existing literature which indicates that households' cognitive resources are limited (Kahneman and Tversky, 2000;[START_REF] Beshears | How are preferences revealed?[END_REF], even if they are well informed. In this case, personalized assistance to make effective use of information could be necessary. Our findings are quite alarming: This first evaluation of pension information in France suggests that the policy to encourage the most vulnerable households to save has failed to reach the target population. To start with, it will reinforce inequalities between the wealthiest individuals and those who are probably the least well-prepared for retirement. Second, the results demonstrate that providing information might not be sufficient: offering an adequate institutional framework for pension information is not enough.

The present paper is organized as follows: in Section 2, we present the literature on financial literacy and retirement-related savings to explain how the expected link between information and savings behavior is established. Section 3 briefly presents the case studied in this paper: the French pension information system. Sections 4 and 5 describe the database used, the Survey of Health, Ageing and Retirement in Europe, and the methodology. The results are discussed in Section 6. We conclude in the last section.

2-Information, literacy and expected retirement-related savings behavior

The literature on financial literacy developed rapidly over the last decade and contributed to our understanding of why households in many developed countries are not well prepared economically for retirement. [START_REF] Lusardi | Financial literacy and the need for financial education: evidence and implications[END_REF], Lusardi and Mitchell (2011) explain that some workers close to retirement did not even think about retirement, which is quite concerning, and tend to overestimate their pension benefits (Dolls et al., 2019;Boeri et al.;2001[START_REF] Boeri | Pension reforms and the opinion of European citizens[END_REF].

The theoretical corpus on financial literacy must be extended to examine why individual rationality might be bounded, thus preventing individuals from saving for their retirement. Only such an extended conceptual framework can lead to a clear understanding of why the role of pension information seems important and why it has been provided in Europe and in the United States over the past 25 years [START_REF] Altman | Behavourial Economics Perspectives: Implications for Policy and Financial Literacy[END_REF].

Although the homo oeconomicus is supposed to be perfectly rational, [START_REF] Simon | Administrative Behavior. A Study of Decision-Making in Administrative Organization[END_REF] explained very early on that this rationality is bounded. First, individual preferences might be temporally inconsistent because of a strong preference for the present (Strotz, 1956): although households seem to be aware of the necessity to save for retirement, they do not. They agree that retirement-related saving is necessary as they would like to maintain their standard of living after they withdraw from the labor market, but they prefer to consume. [START_REF] Akerlof | Procrastination and obedience[END_REF] attributes such a behavior to a lack of imagination: individuals are subject to myopia. According to [START_REF] Laibson | Hyperbolic discount functions, Undersavings and Savings Policy[END_REF], impatience could also explain it [START_REF] Arrondel | Mesurer les préférences individuelles pour le présent[END_REF]. This inconsistency appears because present costs are salient in comparison with future costs. Secondly, complexity in savings choices also reveals the possibly limited rationality of individuals: Beshears and his co-authors (2008) explained that decisions are biased since people tend to avoid more complicated choices.

Financial literacy represents the ability to use information and to make appropriate financial decisions, including retirement-related savings decisions. Individuals with low levels of financial literacy do not have this ability and make temporally inconsistent decisions, suffer from cognitive bias, or do not even think about retirement planning. Several studies [START_REF] Lusardi | Financial literacy and the need for financial education: evidence and implications[END_REF]Lusardi and Mitchell, 2007a, 2007b, 2011a, 2011c[START_REF] Van Rooij | Financial Literacy, Retirement Planning and Household Wealth[END_REF] report a lack of financial literacy and economic awareness preventing individuals from planning for retirement. According to Lusardi and Mitchell (2011c) financially informed individuals are not only more likely to plan their retirement, but they also invest more efficiently (Calvert, Campbell, and Sodini, 2006). Better-educated people have a better propensity to plan, are better able to control their spending and consequently accumulate wealth more efficiently [START_REF] Americks | Wealth accumulation and the propensity to plan[END_REF]. They are able to make long-term financial plans and secure their retirement income. The literature shows that in many countries financial literacy is positively and highly correlated with the propensity to plan [START_REF] Lusardi | Financial literacy and the need for financial education: evidence and implications[END_REF]Lusardi andMitchell, 2011b, 2011c;[START_REF] Bucher-Koenen | Financial literacy and retirement planning in Germany[END_REF][START_REF] Almenberg | Financial literacy and retirement planning in Sweden[END_REF], which ensures that households plan their retirement efficiently. Conversely, people lacking basic financial literacy or numeracy end up close to retirement with a low level of assets [START_REF] Lusardi | Financial literacy and the need for financial education: evidence and implications[END_REF]Lusardi and Mitchell, 2007a) and tend to transact in a high-cost manner [START_REF] Stango | Fuzzy Math and Red Ink: When the Opportunity Cost of Consumption is Not What It Seems[END_REF]. This lack of basic knowledge leads Lusardi and Mitchell's (2011b) to quite worrying conclusions. According to them, many respondents to the US National Financial Capability Survey are then exposed to poor financial decisions. Low-income earners and less educated people are the most vulnerable. Von Rooij et al.

(2012) highlighted little improvement in economic awareness and financial literacy between 2005 and 2010 despite several policy initiatives to increase financial knowledge in the Netherlands. In general, financial literacy has at least a positive impact on the probability of accumulating retirement-related wealth [START_REF] Fornero | Financial literacy and pension plan participation in Italy[END_REF]Klapper and Panos, 2011). In this context, improving financial literacy should be of primary concern for policy makers (Gale and Levine, 2011) and information can be seen as a means. If the lack of financial literacy can be explained in the Simon theoretical framework of bounded rationality, public policy has a role to play in improving the decision-making environment. Providing better quality pension information in a straightforward fashion will improve financial decision making.

Making long-term financial decisions being very complex it can be difficult even when individuals are educated. This calls into question the neoclassical theory which postulates that a rational individual always makes the best possible decisions given the information at their disposal. If individuals are not always able to make the best possible decisions (ie: intertemporal calculations), consequently they make bad choices (CFA Institute, 2009), then the role of the policy maker is reaffirmed. In this context, public information (for instance through pension statements sent to the home) might at least assist in retirement and investment planning. Retirement information could improve the retirement knowledge of consumers and help them to adopt more appropriate behavior [START_REF] Joo | Factors associated with seeking and using professional retirement-planning help[END_REF]. In the US, the social security statement greatly improved the knowledge of workers who did not contact the Social Security Administration (SSA) [START_REF] Mastrobuoni | The Role of Information for Retirement Behavior: Evidence Based on the Stepwise Introduction of the Social Security Statement[END_REF]. [START_REF] Sass | Does the social security "Statement" add value?[END_REF] confirmed that the US social security statement providing estimates of an individual's benefit adds value for workers by improving their pension knowledge. In Germany, Dolls et al. arrived to the same conclusion : the pension letters have a significant and positive impact on the private private retirement account.

Targeted programs addressing differences in household preferences and savings needs as well as financial and economic educational backgrounds could be effective ways to improve retirement planning among different socio-economic groups [START_REF] Lusardi | Financial Literacy Among the Young: Evidence and Implications for Consumer Policy[END_REF]. Searching for information is indeed costly for individuals. Financially distressed older individuals are less likely to seek pension information.

They are more vulnerable as they tend to overestimate their future pensions. According to Kim and Kim (2010), they face a decrease in their standard of living. Private pension funding requires long-term financial plans and pension information. From the literature, we may deduce that government pension information initiatives in many countries are headed in the right direction. By conveying information, they may increase public understanding, the functioning of private pension systems and foster household retirement savings. Free, customized information sent to the home allows the less educated and financially distressed consumers (as described by Kim and Kim, 2010) to be targeted. However, [START_REF] Prast | Is information overrated? Evidence from the pension domain[END_REF] were quite critical about the effectiveness of measures to deliver pension information and underscore that this information is not sufficient to encourage individuals to make appropriate choices. It might be useful to provide clear information, but assistance to interpret it could be necessary if individuals are not able to make effective use of it.

For France, since 2010, there has been a survey called the "Retirement Motivations Survey" to interview retirees shortly after their retirement rights have been settled. This survey assesses the level of information and measures the degree of knowledge which shaped the retirement decision. It is conducted and coordinated by the Directorate of Research, Studies, Evaluation and Statistics (DREES) and the Social Security Department (DSS), the National Insurance Plan for workers in private sector and the Conseil Orientation des Retraites (COR). The survey shows that the insured read the information but it is not always understood. Indeed, the first evaluation showed that knowledge of insurance periods, an essential parameter in the decision to retire, remains imperfect. Comparing the self-reported data with the administrative files of the plans, around 30% report not knowing the insurance period(s). The same applies to the length of insurance required to receive a full pension: around 40% of the respondents indicated that they did not know this (Conseil d'Orientation des retraites, 2016). 2014) carried out a meta-analysis of the relationship between financial literacy, financial education and financial behavior in 168 papers covering 201 studies. They showed that financial education interventions have very little effect on financial behavior with weak effects in low-income samples. They point out that only 0.1% of the variance in financial behavior is due to financial literacy. In this context, following the behavioral economics of Kahneman and Tversky (2000), changing the decision-making environment by providing information might not be enough to enhance the quality of financial decision making.

Fernandes et al. (

3-The French pension information system

The French law to provide information on pensions was introduced with the 2003 pension reform. This law establishes the right of individuals to pension information. Implemented in August 2003, the law states:

Each person has the right to obtain, in conditions specified by decree, a statement of their individual situation regarding all the rights that he/she has acquired in legally compulsory pension plans. The French public interest group Info Retraite provides detailed information on income to those over 35 (see Table 7).

For those close to retirement age, they provide exhaustive information about an individual's pension situation at 55 in a document known as the Indicative Global Estimate (IGE) (in French, Estimation Indicative Globale, or EIG). It provides individuals with a detailed estimate of their future pension benefits as well as the earliest date at which they can expect to retire on a full pension (see Table 7). According to Info Retraite, the reason for providing these statements to all households is to ensure they have the necessary information to make adequate decisions about their retirement.

The implementation of the pension statements in France provides a natural experiment that can be used to test the effects of information provision on individual financial decisions.

The transition period for implementation of the pension information program began in 2007 and ended in 2010. Until 2011, implementation of the pension information system was progressive. At the end of the implementation period, the birth year 1955 constituted a cut-point in receiving the pension statement: individuals born between 1948 and 1955, still in work, had received their pension statement. Conversely, people born after 1955 had not received it. Data from the fourth wave of the Survey of Health, Ageing and Retirement in Europe (SHARE) appear particularly relevant to study this natural experiment.

4-Data and statistics

We used detailed data from the Survey of Health, Ageing and Retirement in Europe (SHARE). This survey takes place in a global context of demographic aging and provides rich economic and social information about individuals in households where at least one member reached the age of 50. This survey appears particularly well suited as we focus on the end of one's working life, and how actively people close to retirement behave, and more specifically save when they receive objective information about their own expected pension level. SHARE is a multidisciplinary and cross-national panel database which includes socio-economic variables, but also detailed information on wealth and family for each respondent. The survey is conducted at regular intervals since 2004. It now comprises seven waves. The survey was launched in 11 countries in 2004.

Today, 27 countries participate, including France. We chose the French part of the survey and used the fourth wave due to the year in which the data was gathered, 2011.

We focused on French data and used the imputed table to obtain aggregated and imputed wealth variables.

This module has the advantage of not being affected by missing data thanks to the imputation procedure implemented in the survey [START_REF] Christelis | Imputation of missing data in waves 1 and 2 of SHARE[END_REF].

We needed to ascertain whether individuals were still in active work when they were supposed to receive their pension statement. We know from French legislation that the 1949 cohort received their estimates in 2007, 1950 and 1951 cohorts in 2008, 1952 and 1953 cohorts in 2009 and 1954 and 1955 cohorts in 2010. As the survey provides the retirement year, we excluded those who declared being retired on the dates above. Starting in 2011, the legal retirement age began to increase, gradually reaching 62 years in 2017.

However, at the time of the survey, individuals aged 60 could still retire, thus explaining the very low number of 60-year-old observations still in active work. This is why we selected cohorts born from 1951 onward inclusive.

We selected a sample of observations whose ages were close to the discontinuity to observe precisely whether the change in legislation had an impact on the generations affected compared with the generations who happened to be older but not affected by the reform. We restricted the bandwith to observations that lie in close proximity to the cut-point and retained cohorts born between 1951 and 1960, particularly because we did not want to include individuals in our sample with very different savings behavior due to their age.

We chose a local analysis of the discontinuity in two outcome variables at the cut-point described above.

The first is the per capita household savings for long-term investment 1 , which includes individual retirement accounts, contractual savings and life insurance. The second is per capita household gross financial assets, i.e., the sum of bank accounts, stocks, bonds, and mutual funds, plus savings for long-term investment.

Tables 1 and 2 provide descriptive statistics for both outcome variables. First, we note that individuals in the lowest quartile of wealth distribution do not have any savings for long-term investments. For 612 observations, this variable is equal to 0. As is common with wealth variables, the median is far lower than the average value for both variables, indicating that some households have very high levels of wealth. In our sample, 953 individuals received a pension estimate, while 792 did not. Savings for long-term investment was 59% higher for the informed cohorts, and their total financial assets were almost 50% higher (See Table 3). These figures are strongly correlated with age, which must be controlled through further econometric analysis, developed in the next section.

5-Methodology

Estimation Procedure

We estimate the causal effect of the pension information program on the savings behavior of individuals in households with one member aged 50 or older. Determining a causal link through the methodology used appears to be crucial in a field of literature where many studies emphasize only a correlation link between financial education and financial decisions [START_REF] Lusardi | Financial literacy and the need for financial education: evidence and implications[END_REF]Lusardi and Mitchell, 2011b).

Implementation of the program created a discontinuity at age 56 in 2011. In this year, only cohorts from 1949 to 1955 had received a pension estimate. In 2011, being 56 or older implies having received information through the mail, therefore having been exposed to the treatment of pension information. This situation is consequently an application of the regression discontinuity design [START_REF] Hahn | Identification and Estimation of Treatment Effects with a Regression Discontinuity Design[END_REF][START_REF] Imbens | Regression Discontinuity Designs: A Guide to Practice[END_REF][START_REF] Lee | Regression Discontinuity Designs in Economics[END_REF]). Specifically, the example of a sharp regression discontinuity (RD) method had to be considered, because depending on birth year, individuals were affected by the pension information reform, or were not. The gradual and transitory implementation of the reform required us to choose a local regression and a reduced cohort window in order to not increase the bias of the estimates. Our aim is to compare observations just below and just above the threshold age of 56 to determine if pension information already had an impact on holding behavior in 2011. The RD method exploits a discontinuity in the treatment assignment to identify a treatment effect. In this case, the known discontinuity is due to the quasi-experimental situation implied by the progressive implementation of the economic policy. In sharp RD design, treatment assignment depends in a deterministic way on variable Z with a known discontinuity at point 𝑍 0 . The assignment of individuals to the treatment pension estimate mail is entirely age dependent. All people born between 1949 and 1955 are considered treated, while those born after 1955 are not. Consequently, two key assumptions were verified: the treatment assignment cannot be manipulated by the individual, and spill-over effects of the treatment are not possible.

Let 𝑝 𝑖 denote the indicator for assignment to the pension information treatment, the rule is then:

𝑝 𝑖 = { 1 𝑖𝑓 𝑍 𝑖 ≤ 𝑍 0 = 1955 0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 (1)
Where 𝑍 𝑖 is the birth year of individual i and 𝑍 0 the threshold, fixed at 19552 .

This empirical approach exploits the discontinuity in available information at age 56 in 2011. We were interested in seeing if receiving the pension estimate had a significant impact on different wealth variables denoted 𝑌. We analyzed both savings for long-term investment, and gross financial assets (in log). Let 𝑌 1 represent the potential outcome if the individual receives the treatment, i.e., the pension information, and 𝑌 0 the potential outcome if they do not receive it. The objective is to estimate the average treatment effect at the threshold 𝑍 0 . This average treatment effect (ATE) can be expressed as:

𝐴𝑇𝐸 = 𝐸[𝑌 1 -𝑌 0 |𝑍 = 𝑍 0 ]
When the support of 𝑍 is continuous, non-parametric and semi-parametric procedures for estimation are appropriate [START_REF] Hahn | Identification and Estimation of Treatment Effects with a Regression Discontinuity Design[END_REF][START_REF] Porter | Estimation in the regression discontinuity model[END_REF]. However, when the support of 𝑍 is discrete, taking 𝐽 distinct values, [START_REF] Lee | Regression Discontinuity Inference with Specification Error[END_REF] showed that parametric methods are preferred.

Identification of the ATE can be achieved by estimating the following regression function:

𝐸[𝑌|𝑍 = 𝑧 𝑗 ] = 𝛽 0 𝑝 𝑗 + ℎ(𝑧 𝑗 ) (2)
Where ℎ(. ) is a continuous function capturing the cohort effect on the outcome variable,

𝑝 𝑗 = 1[𝑧 𝑗 ≥ 0].
The assignment variable 𝑍, here the birth year, is normalized so that the discontinuity point is represented by 𝑧 𝑗 = 0. Therefore, 𝑧 𝑗 = 1955 -𝑏𝑖𝑟𝑡ℎ 𝑦𝑒𝑎𝑟. The link between birth year, age and the assignment variable is summarized in Table 4.

The key identification assumption is the continuity of ℎ(. ). With specification (2), and under this assumption, the treatment effect 𝛽 0 is obtained by estimating the discontinuity in the empirical regression at the point where treatment switches from 0 to 1, in our case at age 56, when the birth year is 1955.

Introducing covariates (𝑋), equation ( 2) can also be expressed as:

𝑌 𝑖𝑗 = 𝛽 0 𝑝 𝑗 + ℎ(𝑧 𝑗 ) + 𝛿𝑋 𝑖 + 𝜀 𝑖𝑗 (3)
In ( 3), 𝑌 𝑖𝑗 is the wealth variable for the i individual, born in year j, i.e., the value of the assignment variable 𝑍. The hypothesis that ℎ(. ) is smooth implies that, controlling for other characteristics, receipt of pension information (i.e., the treatment) is the only source of discontinuity in the wealth variable at age 56. It is common practice to regress 𝑌 𝑖𝑗 on ℎ(. ) assuming it is a low order polynomial function. If the polynomial function assumed is correct, conventional least squares inference is appropriate [START_REF] Lee | Regression Discontinuity Inference with Specification Error[END_REF]. Three different forms are assumed for ℎ(. ): a linear form, a spline linear and a quadratic function.

After suggesting a classic ordinary least squares estimate for equation [2], given the distribution of our data, we propose quantile regressions [START_REF] Koenker | Regression Quantiles[END_REF] to obtain full characterization of the conditional distribution of the dependent variable. Table 1 shows that indeed a large proportion of the sample does save for the long term. We can deduce from these statistics that distribution effects are particularly important with wealth variables. In accordance with the abundant literature on financial literacy, we were concerned that only part of the population can understand and use pension information.

For these reasons, we estimated the treatment effect for the following quantiles: the 25 th percentile, the median and the 75 th percentile.

Two subsamples were then created to target cohorts closer in age to assess the robustness of our estimates. We did not include any age variables as the function h(.) does. Different forms for h(.) being assumed, we revealed a possibly non-linear effect of age on wealth. As suggested in von Rooij et al. ( 2012), a polynomial for the logarithm of per capita disposable household income with a linear, quadratic and cubic term was included, enabling the capture of a potential non-linear effect of income on wealth accumulation.

Finally, we introduced an inheritance variable. Inherited wealth is indeed a source of inequalities in the long run [START_REF] Piketty | Handbook of Income Distribution[END_REF]. In France, individuals inherit at the age of 52 on average, precisely within the range of ages that we studied. The binary variable of inheritance reveals if the household ever received an inheritance of 5000 euros or more. Inherited wealth might be correlated with the social and economic background of each family. For this reason, we also introduce a crossed variable of inheritance and income to control for the risk of multicollinearity.

Graphical Analysis

In case of discreteness of the assignment variable, [START_REF] Lee | Regression Discontinuity Designs in Economics[END_REF] recommend computing and plotting the mean of the outcome variable for each value of the discrete assignment variable. Figure 1 displays an initial approach to the discontinuity analysis and summarizes the mean per capita household wealth and values of the assignment variable 𝑧 𝑗 . A negative value for 𝑧 𝑗 indicates that individuals did not receive their pension estimate. Conversely, if 𝑧 𝑗 ≥ 0, individuals were treated.

The vertical red line represents the discontinuity, the point where the treatment switches from 0 to 1. It seems difficult to draw conclusions from a graphical analysis only. However, we note that mean values of financial assets and savings for long-term investments appear to grow with age (i.e., with 𝑧 𝑗 ), with a possible visual jump at point 𝑧 𝑗 = 0. In addition, RD designs require all other covariates to be continuous at the point of discontinuity. Figure 2 and Figure 3 report the mean variables according to age (specifically, according to the assignment variable). We see that these variables appear to be fairly continuous within the age range that we are interested in. The graphical analysis gives us confidence about the relevance of the methodology.

6-Results

Treatment effects for the total sample are reported in Tables 5 and6 3 . The second column lists the average treatment effect on the total sample. The following columns give the treatment effect on the median and the 75 th percentile respectively4 . Results are reported by functional form assumed for ℎ(. ). We did not detect any impact of pension estimate provision on mean savings for long-term investment nor mean financial assets held. However, quantile regressions reveal a differentiated impact of pension information on wealth distribution. While there is no impact on the lowest percentiles, the wealth stock of the richest seems to be affected. The impact is much stronger on wealth held for long-term investment.

Receiving a pension statement has a significant impact on the 75 th percentile: we see an impact coefficient between 0.62 and 0.69 according to different functional forms of h(.) (see Table 5). The impact displayed with the use of the discontinuity sample is stronger (see Table 10), showing that we do not capture an age effect only after controlling for age.

An effect on median individual long-term wealth is even detected when using the discontinuity samples.

Once again, estimating financial wealth with OLS and a regression discontinuity does not show any significant impact, whatever the form assumed for h(.) (see Table 6). However, allowing for a possibly differentiated impact by percentile shows that the wealthiest behavior is affected by the receipt of pension information. A robust impact on financial wealth is detected for the 75th percentile. We note an elasticity comprised between 0.39 and 0.46 depending on the form of h(.) for the highest quartile of the distribution.

Part of the population reacts to pension information by increasing private savings. As the impact is stronger for long-term savings, we assume that some people adjust their level of wealth with appropriate assets for retirement as a priority. Having an increasing retirement horizon also implies more long-term uncertainty.

However, our results also show that only those who already have the highest levels of wealth seem to be affected. This population is also the most aware of the long-term issues and challenges for the pension system. The significant impact of the education variable, capturing financial literacy in our regressions, confirms that knowledge is the key variable determining long-term wealth accumulated [START_REF] Americks | Wealth accumulation and the propensity to plan[END_REF]Lusardi andMitchell, 2011b, 2011c;[START_REF] Bucher-Koenen | Financial literacy and retirement planning in Germany[END_REF][START_REF] Almenberg | Financial literacy and retirement planning in Sweden[END_REF]Arrondel et al., 2013).

The impact of the provision of pension information being the greatest for the wealthiest households, might lead to greater inequalities. Such a policy encourages the richest households to accumulate more for retirement, but not households with income in the lower percentiles. In other words, pension information does not appear to motivate those who most need to adapt to the decrease in their public pension. Thus, information provision does not represent effective policy. The policy fails to achieve its main purpose.

There are many reasons for this:

-Households with the lowest levels of wealth are not able to increase their savings, even if they have clear information on the amount of pension they will be entitled to.

-Those at the bottom of the wealth distribution, are also those who can expect a high replacement rate in France: as a consequence, they cannot be surprised by the pension evaluation received, and can therefore choose not to change their savings behavior.

-Individuals might not have a sufficient level of financial literacy and not make effective use of the information received. In this case, increasing the efficiency of this policy regime fundamentally requires increasing the capacity to make effective use of the information provided. Currently, those who already have a high level of wealth appear to be able to translate the information provided into a financial decision.

Conclusion

In this research study, we find that receipt of a pension information statement in France is associated with a significant increase in wealth accumulation for high-wealth individuals. This finding is an important contribution to the pension literature as we provide an evaluation of public policies that can lead to more structural reforms of pension systems. Taking advantage of the progressive implementation of economic policy which created the conditions for a quasi-experiment, and thanks to availability of data, we were able to evaluate whether changes in wealth are due to the introduction of the pension information, and to shed light the relationship between a particular aspect of financial awareness: pension information and savings behavior.

We show that pension information has a significant positive impact on accumulation, but only on the highest part of the wealth distribution. With an elasticity for information between 0.62 and 0.67 for the 75th percentile, the impact on long-term savings is much stronger than on financial assets. We estimate that the elasticity of financial wealth to pension estimate is comprised between 0.39 and 0.46 for households within the 75th percentile of wealth.

We emphasize that pension statements reinforce the long-term savings of those who already hold the highest levels of wealth. Households with the lowest levels of wealth are not affected by the policy. Current measures to convey information to households therefore tend to reinforce inequalities between those who have already saved privately and were also probably the most well-informed and the most financially educated, and those with the lowest levels of wealth. The latter are the least informed and the least financially educated. The policy fails to reach them. We also fear that this population, even if it is aware of the necessity to increase retirement-related wealth, will not be able to do this. Consequently, the pension information system, such as in France following the 2003 reform, had perverse effects. Our result confirms that providing information might not be sufficient to influence behavior in in the manner intended by the policy. Making effective use of information also probably requires personalized assistance. For this reason, we conclude that provision of information cannot guarantee the effectiveness of public policy.

French Earnings record

The 'blue envelope' contains a number of compulsory elements:

• A page for each plan to which the insured person has contributed with the start and end dates of each plan; the elements of remuneration to be taken into account for the calculation of the pension; the length of insurance or the number of points acquired (depending on the plans concerned); information on periods or events not related to a given year and which may affect the retirement age or the amount of the pension (e.g. children, national service, etc.).

• A page summarizing the pension rights of the person concerned under each plan;

• A leaflet presenting the organization of the French pension system. From the age of 55, the insured receive the same information as the individual statement above plus an evaluation of the estimated pension amount at different ages. This pension estimate is provided to each insured person in the year in which they turn 55 and they will receive it every 5 years until they retire.

•

  We included individuals in the age bracket 51-60 in 2011 to have enough observations in the study sample. The treatment group consequently included individuals born between 1951 and 1955. Individuals born from 1956 to 1960 constituted the control group.

  The first subsample, referred to as discontinuity sample 1 (DS1) includes cohorts from 1952 to 1958. Cohorts from 1956 to 1958 constitute the control group. The second subsample, referred to as DS2, includes cohorts from 1953 to 1957. Once again, those born in 1956 and 1957 are the control cohorts. This process could enable the comparison of individuals with even more similar accumulation behavior, and reduce the potential bias in the estimate. However, in selecting these cohorts, we also excluded observations and constrained the size of our sample. Discontinuity sample 1 includes 1302 observations while DS2 includes only 951 observations.5.2 VariablesTo investigate the impact of pension information on different wealth variables, we included control variables such as educational attainment for the individual and their partner (years of education and partner's years of education), household composition (marital status and the number of children in the household), and per capita household disposable income to investigate the impact of pension information on different wealth variables. Dummy variables indicate whether the respondent is self-employed, retired or a homeowner. An additional dummy enables capturing of health effects if the individual declared being limited in their daily activities.

Figure 3

 3 Figure 3 Evolution of covariates according to the assignment variable (2)

Table 7

 7 Summary of information available in the French pension statement ('Blue Envelope')

  The amount of the pension is estimated at different ages: At the full retirement age, i.e. at the maximum rate of 50%, taking into account the length of insurance in all plans. This is the age at which the insured person is entitled to a pension without a discount; o At the legal age for obtaining the full rate; o At the age reached in the year of the estimate, if the insured person has exceeded the legal age for obtaining the full rate.

	o At the legal retirement age under the general plan (with a discount or reduction,
	if applicable);
	o

This estimate is based on information known by the various plan on 31 December of the previous year.

Table 8

 8

			Estimates of savings for long-term investment (log)
			OLS			q50			q75
		(1)	(2)	(3)	(1)	(2)	(3)	(1)	(2)	(3)
	Pension estimate received 0/1	0.333	0.436	0.410	0.613	0.875	0.940	0.620**	0.686**	0.628**
		(0.45)	(0.56)	(0.55)	(0.98)	(1.31)	(1.47)	(2.15)	(2.21)	(2.12)
	Normalized assignment variable z	-0.0927	-0.181	-0.0993	-0.133	-0.268	-0.176	-0.108** -0.200** -0.104*
		(-0.76)	(-0.91)	(-0.82)	(-1.13)	(-1.41)	(-1.46)	(-1.99)	(-2.26)	(-1.87)
	Years of education (log) 1.521*** 1.528*** 1.531*** 1.075**	1.107**	1.132** 1.630*** 1.659*** 1.689***
		(4.36)	(4.37)	(4.40)	(2.28)	(2.28)	(2.36)	(7.52)	(7.32)	(7.63)
	Partner's years of education (log)	0.181	0.187	0.190	0.100	0.0866	0.0539	-0.0181	0.0224	0.0126

Although the variable is denoted "savings for long-term investment" in the survey, we used stock variables. This variable indicates assets held over long-term horizons.

Individuals born in 1956 received a pension estimate by mail in 2011, however the amount reported was for 2010.

See Tables8 and 9for the complete results of the estimates.

We did not include results for the 25th percentile as they were not significant.


Ernesto García-Alfonso

Maykel Márquez-Mijares

Jesus Rubayo-Soneira

Nadine Halberstadt

Kenneth C Janda

Craig C Martens

Jesús Rubayo-Soneira

Photofragmentation dynamics study

Keywords: 82, 37 Np Dissociation molecular, 33, 80, Gp Photodissociation of moleculas -87, 18, hg Dynamics intermolecular interactions

published or not. The documents may come   L'archive ouverte pluridisciplinaire

Introduction

The noble gas halogen van der Waals (vdW) molecules, where weakly bound rare gas atoms are interacting with a strong chemical bond like Br 2 , are considered ideal systems to study the molecular energy transfer mechanisms and how molecular properties influence intramolecular dynamics. Due to the weakness of the vdW interactions, it is expected that the remaining constituents in these kinds of complexes retain their chemical properties so the energy transfer mechanisms can be easily identified and studied at the state-to-state level. From the experimental point of view, the spectroscopy of these molecules is available as well as real-time dynamics on femtosecond to picosecond time scales. There are a variety of theoretical models which may be formulated and tested with the experimental results [START_REF] Borrell-Grueiro | Fragmentation dynamics of NO-NO dimer: A quasiclassical dynamics study[END_REF][START_REF] Rubayo-Soneira | Vibrational predissociation of I2-Ne. A quasiclassical dynamical study hem[END_REF][START_REF] García-Vela | Quasiclassical dynamics of the I2-Ne2 vibrational predissociation: A comparison with experiment[END_REF][START_REF] Roncero | Photodissociation of NeBr2(B) below and above the dissociation limit of Br2(B)[END_REF][START_REF] Cabrera | Time-dependent pump-probe spectra of NeBr2[END_REF][START_REF] González-Martínez | Quasi-classical trajectories study of Ne 79 Br2(B) vibrational predissociation[END_REF][START_REF] García-Vela | Quantum dynamics of Ne-Br2 vibrational predissociation: The role of continuum resonances and doorway states[END_REF][START_REF] Cline | Structure and Dynamics of Weakly Bound Molecular Complexes[END_REF][START_REF] González-Martínez | Vibrational predissociation of van der Waals complexes: Quasi-classical results with Gaussian-weighted trajectories[END_REF][START_REF] Reed | Cartesian coupled coherence states simulations: NenBr2 dissociation as a test case[END_REF][START_REF] Borrell-Grueiro | Vibrational predissociation dynamics of the nitric oxide dimer[END_REF][START_REF] Prosmiti | Vibrational predissociation of NeBr2(X, v=1) using an ab initio potential energy surface[END_REF][START_REF] Miguel | Time evolution of reactants, intermediates, and products in the vibrational predissociation of Br2 • • • Ne: A theoretical study[END_REF][START_REF] Stephenson | Quantum calculations on the vibrational predissociation of NeBr2: Evidence for continuum resonances[END_REF][START_REF] Bastida | Application of Trajectory Surface Hopping to vibrational predissociation[END_REF][START_REF] Bastida | Vibrational Predissociation Dynamics of I2 • • • Ne2: A Trajectory Surface Hopping Study[END_REF][START_REF] Fernandez Alberti | Intramolecular vibrational redistribution and fragmentation dynamics of I2 • • • Ne|n clusters[END_REF][START_REF] Bastida | Hybrid quantum/classical simulation and kinetic study of the vibrational predissociation of Cl2 • • • Nen (n = 2, 3)[END_REF][START_REF] Rohrbacher | The dynamics of noble gas-halogen molecules and clusters[END_REF] The quasiclassical trajectory method is one of the more popular approaches to study the molecular dynamics of these systems. This formulation solves the classical equa-tions of motion using initial conditions generated from quantum probabilities distributions. It exhibits substantially lower computational costs compared with exact quantum methods. The quasiclassical approach allows the construction of intuitive models for molecular processes. In some cases, the full dimensionality of the system can be included.

The ArBr 2 complex has been the subject of previous experimental and theoretical work [START_REF] Prosmiti | Ab initio ground state potential energy surfaces for Rg-Br2, (Rg=He,Ne,Ar) complexes[END_REF][START_REF] Cabrera | Time and frequency resolved dynamics of ArBr2[END_REF][START_REF] Acosta-Matos | Trajectory-based modeling of the quantum dynamics of vibrational predissociation: Application to the Ar. . .Br2(ν = 24) complex[END_REF][START_REF] Pio | Product state resolved excitation spectroscopy of He,Ne, and Ar -Br2 linear isomers: Experiment and theory[END_REF]. Dissociation rates have been determined for v = 16 -25 of the B electronic state. The results showed that, while the rate generally increases with vibrational quantum number, the trend is not monotonic, suggesting that ArBr 2 undergoes vibrational predissociation (VP) in the sparse intramolecular vibrational distribution (IVR) regime. This was confirmed by a theoretical analysis of the fragmentation dynamics [START_REF] Cabrera | Time and frequency resolved dynamics of ArBr2[END_REF], which was able to qualitatively reproduce the experimental results. So far, no more global studies have been done on this system, so it is of interest to imple-ment the QCTM scheme to investigate other criteria of the processes exhibited by this complex.

The trajectory surface hopping method (TSH) [START_REF] Tully | Molecular dynamics with electronic transitions[END_REF] has demonstrated its validity and efficiency for a variety of molecular dynamics problems [START_REF] Miguel | Time evolution of reactants, intermediates, and products in the vibrational predissociation of Br2 • • • Ne: A theoretical study[END_REF][START_REF] Bastida | Application of Trajectory Surface Hopping to vibrational predissociation[END_REF][START_REF] Bastida | Vibrational Predissociation Dynamics of I2 • • • Ne2: A Trajectory Surface Hopping Study[END_REF][START_REF] Fernandez Alberti | Intramolecular vibrational redistribution and fragmentation dynamics of I2 • • • Ne|n clusters[END_REF][START_REF] García-Alfonso | Study of the Vibrational Predissociation of the NeBr2 Complex by Computational Simulation Using the Trajectory Surface Hopping Method[END_REF][START_REF] Rodríguez-Fernández | Trajectory surface hopping study of the photodissociation dynamics of methyl radical from the 3s and 3pz Rydberg states[END_REF][START_REF] Rodríguez-Fernández | Quasi-Classical study of the photodissociation dynamics of the methyl radical[END_REF][START_REF] Martens | Communication: Fully coherent quantum state hopping[END_REF][START_REF] Bastida | Competition between electronic and vi-brational predissociation in Ar-I2 (B): A molecular dynamics with quantum transitions study[END_REF].

For the study of ArBr 2 we have used a diabatic representation in which the potential energy surfaces are formed by the interaction of the Ar atom with the vibrational levels (v = 16, 17, . . . , [START_REF] García-Alfonso | Study of the Vibrational Predissociation of the NeBr2 Complex by Computational Simulation Using the Trajectory Surface Hopping Method[END_REF] of Br 2 . To understand the evolution of the system we have implemented a kinetic mechanism recently used for NeBr 2 vibrational predissociation [START_REF] García-Alfonso | Study of the Vibrational Predissociation of the NeBr2 Complex by Computational Simulation Using the Trajectory Surface Hopping Method[END_REF] and inspired by earlier work [START_REF] Miguel | Time evolution of reactants, intermediates, and products in the vibrational predissociation of Br2 • • • Ne: A theoretical study[END_REF][START_REF] Fernandez Alberti | Intramolecular vibrational redistribution and fragmentation dynamics of I2 • • • Ne|n clusters[END_REF][START_REF] Bastida | Hybrid quantum/classical simulation and kinetic study of the vibrational predissociation of Cl2 • • • Nen (n = 2, 3)[END_REF]. In this work, the transfer of up to five vibrational quanta is included in the kinetic mechanism.

Theory and Methodology

The photodissociation process of interest in this work is described as:

ArBr2(X,v = 0)+ ω → [ArBr2(B,v = 16-25)] V P → Ar+Br2(B,v")
Two different approaches are used in our simulations: quasiclassical trajectory and trajectory surface hopping in the diabatic representation. Our results have been compared to existing experimental and wave packet results.

We employ Jacobi coordinates (r, R, θ) which are widely used for this kind of molecular geometry. The coordinate r is the bond length of Br 2 , R is the intermolecular distance from the center of mass of the dihalogen to the Ar atom. The angle θ is defined by the r and R vectors. These definitions are illustrated in Figure 1. In our calculations, we take the total angular momentum to be zero ( J = j + l = 0), with j being the angular momentum of Br 2 (associated with r) and l the orbital angular momentum (associated with R), which is a justified constraint for studying photodissociation events occurring at low temperature [START_REF] González-Martínez | Quasi-classical trajectories study of Ne 79 Br2(B) vibrational predissociation[END_REF]. The classical Hamilton function corresponding to these three degree of freedom model of the complex is written as

H(r, R, θ) = H Br2 + 1 2µ ArBr2 p 2 R + p 2 θ R 2 +V ArBr2 (r, R, θ), (1)
where H Br2 is the diatomic Hamiltonian

H Br2 = 1 2µ Br2 p 2 r + p 2 θ r 2 + V Br2 (r), (2)
In the equations above, p r , p R , and p θ are the momenta associated with coordinates r, R, and θ, respectively; µ Br2 and µ ArBr2 stand for the reduced masses for r (Br 2 ) and R (ArBr Both Br-Br (V Br2 (r)) and Ar-Br 2 (V ArBr2 (r, R, θ)) interaction are represented as a Morse potential:

V AB (x) = D(e -2α(x-x) -2e -α(x-x) ), (3)
where x = R is the equilibrium position. Van der Waals interaction is constructed by the pairwise sum of two Morse potential (atom-atom interactions)

V ArBr2 (r, R, θ) = V ArBr (R 1 ) + V ArBr (R 2 ), (4)
with

R 2 1,2 = R 2 + (r/2) 2 ± rR cos(θ),
The interaction potential Br-Br was taken from Ref. [START_REF] Buchachenko | Interaction Potentials and Fragmentation Dynamics of the Ne-Br2, Complex in the Ground and Electronically Excited States[END_REF] which it was obtained adjusting a Morse potential to RKR potential reported in Ref. [30]. It reproduces very well our region of interest, from v = 16 to v = 25. The potential parameters are collected in Table 1.

Quasiclassical trajectory method

The basis of the quasiclassical methodology is to solve the classical equations of Hamilton once the initial state of the system is determined. In particular, our variant of the QCTM is based on the idea that the best trajectory ensemble representation of the initial state is provided by the corresponding quantum distribution function.

We follow here a procedure similar to that used in Ref. [START_REF] González-Martínez | Quasi-classical trajectories study of Ne 79 Br2(B) vibrational predissociation[END_REF] for NeBr 2 vibrational predissociation. We recall here its essential features, adapted to the present study.

We have chosen a sampling of both the initial configurations and the conjugated momenta. After running many trajectories, statistical analysis is used to compute observables from the details of the ensemble of trajectories. Initial conditions representing a microcanonical distribution for the system are obtained by solving the timeindependent Schrödinger equation for the quasibound state in the vibrational level v of the B electronic state, sampling the coordinates following the resulting probability density and deducing the momenta to obtain the bound state energy. The Hamiltonian operator corresponding to the B state is

H(r, R, θ) |Ω B v (r, R, θ) = E |Ω B v (r, R, θ) . (5)
Here, H is the quantum Hamiltonian operator for the system corresponding to J = 0 [Eq. ( 1

)]. A diabatic repre- sentation |Ω B v (r, R, θ) = |ψ B v (r) |Φ B v (R, θ) is employed.
The relevant diatomic wavefunctions |ψ B v (r) and energies E B,v Br2 are obtained from

[H B Br2 -E B,v Br2 ] |ψ B v (r) = 0 (6)
and taking into account p θ = 0.

Besides, |Φ B v (R, θ) is calculated by solving Eq.7 using a Truhlar-Numerov method. We fixed θ = π/2 which the system is found more bound. It corresponds to a "T" shape configuration. We have also taken r = R for the Br-Br and j = 0. The initial rotational quantum number for the Br 2 molecule can be neglected (j = 0), as we will see in the results, because the anisotropy effect is not very strong in this system. Vibrational energies for Br-Br and ArBr 2 are collected in Table 2. Once the initial configurations are determined to statistically reproduce the quantum distributions, the conjugated momenta p r and p R are obtained from energy conservation as in Ref. [START_REF] González-Martínez | Quasi-classical trajectories study of Ne 79 Br2(B) vibrational predissociation[END_REF].

1 2µ ArBr2 -2 ∂ 2 ∂R 2 + ĵ2 R 2 |Φ B v (R, θ) + ψ B v |V ArBr2 (r, R, θ)|ψ B v -(E B -E B,v Br2 ) |Φ B v (R, θ) = 0 (7)

Hamilton's equations, given by

H J=0 = H(r, R, θ) dr dt = ∂ ∂p r [H J=0 ] = p r µ Br2 (8)
dR dt = ∂ ∂p R [H J=0 ] = p R µ ArBr2 (9)
dθ dt = - ∂ ∂p θ [H J=0 ] = p θ 1 µ Br2 r 2 + 1 µ ArBr2 R 2 (10) dp r dt = - ∂ ∂r [H J=0 ] = p 2 θ µ Br2 r 3 - ∂ ∂r [V Br2 (r) + V ArBr2 (r, R, θ)] (11) dp R dt = - ∂ ∂R [H J=0 ] = p 2 θ µ ArBr2 R 3 - ∂ ∂R [V ArBr2 (r, R, θ)] (12) dp θ dt = - ∂ ∂θ [H J=0 ] = - ∂ ∂θ [V ArBr2 (r, R, θ)] (13)
are numerically integrated by using a fourth-order Adams-Moulton algorithm initiated by a fourth-order Runge-Kutta integrator. The time step is ∆t = 0.1 fs. Total energy conservation of ∆E < 10 -8 cm -1 is achieved. Propagation is terminated if any of the following conditions are satisfied: t > 600 ps or R > 10 Å, the former corresponding to 6 • 10 6 maximum integration cycles and the latter assumed to correspond to a dissociated trajectory since the vdW interaction becomes negligible at that distance. The total number of trajectories in the ensemble is N = 5 • 10 3 for each v = 16 -25.

To obtain quantum numbers from classical statistics, the so-called standard binning procedure is used [START_REF] González-Martínez | Quasi-classical trajectories study of Ne 79 Br2(B) vibrational predissociation[END_REF].

-diagonal matrix el- ements | ψ B 20 (r)|VArBr 2 (r, R, θ)|ψ B v (r)
| between v=20 and different vibrational levels as a function of the Ar to Br2 distance R, defining the coupling responsible for VP of v=20. Note that the absolute value of the coupling is represented in order to stress the importance of the repulsive region of the potential, the coupling itself changes sign near R=3.87 Å. Lower plot: Diagonal ψ B v (r)|VArBr 2 (r, R, θ)|ψ B v (r) matrix elements defining the v th surface for TSH classical coordinates R and θ; All curves are calculated for the "T" shape configuration (θ = 90 • )

TSH Method in the Diabatic Representation

The TSH approach for treating vibrational predissociation in this diatomic-rare gas atom system follows the same idea as earlier work [START_REF] Bastida | Application of Trajectory Surface Hopping to vibrational predissociation[END_REF][START_REF] Bastida | Vibrational Predissociation Dynamics of I2 • • • Ne2: A Trajectory Surface Hopping Study[END_REF][START_REF] Fernandez Alberti | Intramolecular vibrational redistribution and fragmentation dynamics of I2 • • • Ne|n clusters[END_REF][START_REF] Bastida | Hybrid quantum/classical simulation and kinetic study of the vibrational predissociation of Cl2 • • • Nen (n = 2, 3)[END_REF] and it is described in detail in Ref. [START_REF] García-Alfonso | Study of the Vibrational Predissociation of the NeBr2 Complex by Computational Simulation Using the Trajectory Surface Hopping Method[END_REF]. We recall here its essential features.

The diatomic vibrational coordinate (r) is treated as a quantum coordinate while the other degrees of freedom R and θ are treated as classical coordinates.

The Hamiltonian for the system can be written as:

H(r, R, θ) = H q (r) + H cl (R) + H int (r, R, θ) (14)
where the quantum Hamiltonian describes Br 2 vibration

H q (r) = - 2 2µ Br2 ∂ 2 ∂r 2 + V Br2 (r) ( 15
)
the classical Hamiltonian describes the intermolecular degrees of freedom

H cl (R) = p 2 R 2µ ArBr2 + p 2 θ 2µ ArBr2 R 2 (16)
and the coupling between the quantum and classical degrees of freedom is

H int (r, R, θ) = p 2 θ 2µ Br2 r 2 + V ArBr2 (r, R, θ). ( 17
)
Defining |v ≡ |ψ B v (r) to simplify notation, the Hamiltonian matrix elements can be written as

v |H(r, R, θ)|v = v |V ArBr2 (r, R, θ)|v + p 2 θ 2µ Br2 v | 1 r 2 |v + + p 2 θ 2µ ArBr2 R 2 + E B,v Br2 + p 2 R 2µ ArBr2 δ vv ,
where δ vv is the Kronecker's delta and E B,v Br2 is the vibrational energy for Br 2 obtained in Eq. ( 6).

The classical equations of motion on the v th surface are then the same ones as Eqs. ( 9), [START_REF] Reed | Cartesian coupled coherence states simulations: NenBr2 dissociation as a test case[END_REF], [START_REF] Prosmiti | Vibrational predissociation of NeBr2(X, v=1) using an ab initio potential energy surface[END_REF], and ( 13

i ∂|Ψ (t) ∂t = [H q (r) + H int (r, R, θ)] |Ψ (t) . ( 18
)
The evolving state |Ψ (t) is expanded as

|Ψ (t) = v c v (t)e -iE B,v Br 2 t/ |v (19)
The c v are propagated using the same numerical integrator algorithm as above.

The time-dependent population of each vibrational level is determined as follows:

ρ vv (t) = c v (t)c * v (t), (20)
where ρ vv (t) is the density matrix.

The transition probabilities, following the Tully's procedure [START_REF] Tully | Molecular dynamics with electronic transitions[END_REF], from the current state v to all other states v = v during the time interval ∆t are computed using the surface hopping probabilities Eq.( 21).

The initial conditions for the classical trajectories are selected randomly as in the QCTM method, for a total energy corresponding to the zero-point of the complex ArBr 2 , for a particular vibrational state of the Br 2 molecule.

When state jumps occur, classical momenta are adjusted in order to conserve the total energy [START_REF] Bastida | Application of Trajectory Surface Hopping to vibrational predissociation[END_REF][START_REF] Crespo-Otero | Recent advances and perspectives on nonadiabatic mixed quantum-classical dynamics[END_REF]. In our case, quantum jumps (vibrational transitions) occur between diabatic surfaces, defined for each vibrational level and coupled by the ArBr 2 potential. The classical linear momentum vector p R and p θ are adjusted after a surface hop to compensate the difference in potential energy, the analogous way as in Ref. [START_REF] García-Alfonso | Study of the Vibrational Predissociation of the NeBr2 Complex by Computational Simulation Using the Trajectory Surface Hopping Method[END_REF].

When a hop to a higher vibrational level is attempted, it can be frustrated, i.e. the energy conservation equation does not have a real solution.

We then applied the fewest-switches with time uncertainty (FSTU) method of Ref. [START_REF] Jasper | Fewestswitches with time uncertainty: A modified trajectory surface-hopping algorithm with better accuracy for classically forbidden electronic transitions[END_REF], which can be interpreted as the ability of a quantum system to borrow energy ∆E for a time ∆t given by the uncertainty principle. This feature allows the trajectory R(t) that experiences a frustrated hop to tunnel and hop successfully at some nearby geometry along R(t) within the time interval ∆t.

If FSTU does not avoid the frustrating hop, we then apply the "∇V " prescription Ref. [START_REF] Jasper | Improved treatment of momentum at classically forbidden electronic transitions in trajectory surface hopping calculations[END_REF]. That allows us change or no the sign of the lineal momentum. We have to take into account the influence of the target PES over the particle. If both the nuclear momentum and the force have the same direction, the particle do not feel the force and therefore no affects its movement. In another case the direction is affected.

In addition, when a hop occurs, we reset the wave function employing the "instantaneous decoherence" (ID) approach [START_REF] Crespo-Otero | Recent advances and perspectives on nonadiabatic mixed quantum-classical dynamics[END_REF] -

c v = 0 ∀ v = v -c v = 1

Kinetic Mechanism

In order to gain insight into the energy relaxation paths followed by the system, we fit the Br 2 (v -i) • • • Ar and Br 2 (v -i) populations resulting from the TSH simulation to a kinetic mechanism introduced by Miguel [START_REF] Miguel | Time evolution of reactants, intermediates, and products in the vibrational predissociation of Br2 • • • Ne: A theoretical study[END_REF] (see supplementary material). This mechanism is based on three types of elementary steps: vibrational predissociation (VP), where the transfer of one vibrational quantum from the diatomic to the Van der Waals degrees of freedom dissociates the complex; intramolecular vibrational relaxation, where one vibrational quantum is transfered but no dissociation occurs; evaporative cooling (EC) when enough energy has been collected in the intermolecular modes and the Ar atom dissociates without transfer of a new vibrational quantum.

In the case of Ar• • • Br 2 , the transfer of one vibrational quantum of Br 2 to the vdW modes is not sufficient to dissociate the complex. Hence it is an IVR step which leads to an intermediate complex, denoted by Br 2 (v -1) • • • Ar ivr , with excited intermolecular modes. The second quantum transfered can go either to the dissociative mode (VP step, if ∆v=-2 is open), leading to Br 2 (v -2) • • • Ar vp which quickly dissociates to Br 2 (v-2) + Ar, or add up to the van der Waals mode excitation (IVR), leading to Br 2 (v-2) • • • Ar ivr . The same processes can be iterated for the loss of up to five quanta. When enough energy has been collected in the intermolecular modes, the complex can dissociate at the same Br 2 vibrational level via evaporative cooling (EC).

The Br 2 (v -i) • • • Ar populations detected in the TSH simulation are thus the sum of two contributions: a shortlived one coming from the VP process, Br 2 (v -i) • • • Ar vp , and a longer-lived one coming from the IVR process, Br 2 (vi) • • • Ar ivr . Depending on the nature of the last step, i.e. the step leading to dissociation, the mechanism is labelled as IVR-VP or IVR-EC.

Results

Vibrational predissociation lifetimes

We have computed the lifetime for QCTM taken two different exponential decay laws, Eqs. ( 22) and [START_REF] Pio | Product state resolved excitation spectroscopy of He,Ne, and Ar -Br2 linear isomers: Experiment and theory[END_REF], where N N DT stands for non-dissociated trajectories and N T the total number of trajectories [START_REF] González-Martínez | Quasi-classical trajectories study of Ne 79 Br2(B) vibrational predissociation[END_REF].

N N DT (t) N T ∼ (α 1 e -α2t -α 2 e -α1t ) α 1 > α 2 (22) N N DT (t) N T ∼ α 1 e -α2t (23)
The parameter α 2 is related with the lifetime (τ ) by the equation τ = α -1 2 , while α -1 1 is associated, mainly with the IVR processes among other factors influence [START_REF] González-Martínez | Quasi-classical trajectories study of Ne 79 Br2(B) vibrational predissociation[END_REF]. The parameter α 1 takes into account the initial slow decay due to the use of classical dynamics [START_REF] González-Martínez | Quasi-classical trajectories study of Ne 79 Br2(B) vibrational predissociation[END_REF]. Figure (3) displays the QCTM dissociation times for v = 23 using Eq. ( 22) or [START_REF] Pio | Product state resolved excitation spectroscopy of He,Ne, and Ar -Br2 linear isomers: Experiment and theory[END_REF] . For t ≤ 2 ps the system is still bound because dissociating atoms have not yet reached the critical distance used as a dissociation criterion. This results in a non-exponential behavior. For this reason, lifetimes in the single exponential fit are larger than in the two exponential fit.

In the case of the TSH method, we use a multiexponential law [START_REF] Bastida | Vibrational Predissociation Dynamics of I2 • • • Ne2: A Trajectory Surface Hopping Study[END_REF] to fit the obtained populations (see supplementary material) : where N p (t) is the vibrational population of the intermediate system at t and N p,∞ the asymptotic value of the fragments vibrational population. In this case, the lifetime is given by (see supplementary material): The resulting lifetimes for each initial vibrational level v are plotted in Figure 4, compared to the experimental product appearance times and to the wave packet results of Ref. [START_REF] Cabrera | Time and frequency resolved dynamics of ArBr2[END_REF]. For the TSH results, a range of values is represented, corresponding to the appearance time of Br 2 in each of the ∆v exit channels, as well as the overall appearance time of Br 2 fragments (see Table 1 in supplementary material).

N p (t) = N p,∞ (1 -e -bt ) c (24)
g v→v = 2∆t Im c * v (t)c v (t)e [i(E B,v Br 2 -E B,v Br 2 )t/ ] |cv(t)| 2 v |Hint(r, R, θ)|v . (21
τ = - c b 1 0 ln(1 -z)z c-1 dz (
From v=16 to 18 there is a good agreement between the TSH and experimental results for the dissociation lifetimes. The most important difference among the lifetimes corresponds to the range v = 19-22, as was also observed in Ref. [START_REF] Cabrera | Time and frequency resolved dynamics of ArBr2[END_REF] between wave packet simulations and experiment. This is associated with the closure of the ∆v=-2 channel. It occurs for v = 19 in the TSH method, and for v = 20 in the wave packet simulation, due to slight differences in the potential used. In both cases the closure is followed by a sudden decrease of the lifetime, although it is preceded by an erratic behavior in the wave packet simulations due to resonances in the sparse IVR regime (discussed below). These characteristics are not reproduced by the QCTM simulations since vibrational energy transfer is not quantified in this method and ∆v = -2 only closes for v = 24. They exhibit a smooth decay of the dissociation lifetime with v, albeit with a small local maximum for v = 20. The single exponential fit of Eq. ( 22) for the QCTM results gives lifetimes closer to the results of the TSH and Wave Packet method (v = 22 -25). This seems to indicate that the appearance time of the fragments is better fitted by depreciating the initial slow decay inherent to the use of classical dynamics.

On the other hand, the wave packet results obtained with the same interaction potentials as the QCTM results should be considered as the reference. However, in the ∆v=-2 range (v ≤ 19) where dissociation requires the transfer of only two vibrational quanta, the dynamics is typical of IVR in the sparse regime, where the initially excited ("bright") v level is coupled to one or very few "dark" levels in the v-1 ensemble. As stated in Refs. [START_REF] Cabrera | Time and frequency resolved dynamics of ArBr2[END_REF][START_REF] Halberstadt | Intramolecular vibrational relaxation in a triatomic Van der Waals molecule: Ar• • • Cl2[END_REF], the rates are then very sensitive to the energy differences between the "bright" and "dark" levels. This effect is highly nonlinear and is strongly affected by the details of the potential (slight changes in the potential parameters strongly affect the dark level energies), the total angular momentum, as well as second order effects such as the (small) mixing of the initial quasibound state of a given v level with the corresponding v ± 1 quasibound states [START_REF] Halberstadt | Intramolecular vibrational relaxation in a triatomic Van der Waals molecule: Ar• • • Cl2[END_REF]. In particular, the lifetime for v=19 strongly depends on the total angular momentum J, as shown in Ref. [START_REF] Cabrera | Time and frequency resolved dynamics of ArBr2[END_REF], whereas they vary much less with J for v=21. The simulations in Fig. 4 are performed for J = 0 whereas a large number of rotational levels are populated in the experiment, and the resonance effect is therefore smoothed out. This resonance effect is not reproduced in classical dynamics, hence the reason why classical results seem to be closer to experimental ones in the region of the closure of ∆v=-2 is probably fortuitous.

For higher vibrational levels v > 19 where ∆v <-2, the density of dark states is higher and IVR is in the intermediate regime where the lifetimes no longer oscillate. The TSH lifetimes show a sudden decrease at the closure of (∆v = -2), followed by a slower decrease with v. It does not show any resonant behavior. This is not really surprising since the intermolecular motion is described classically. However, given that the transfer of vibrational energy is quantized, the phase space accessible in v-2 is very much reduced for v=19 and this could favor quasi-periodic trajectories, hence a longer lifetime as in wavepacket results.

Fragments vibrational state distribution

In Figure [START_REF] Cabrera | Time-dependent pump-probe spectra of NeBr2[END_REF] we show the final vibrational state population of the Br 2 fragments as a function of initial v using the QCTM and TSH methods, and compare with the results of wave packet simulations and of experiment from Ref. [START_REF] Cabrera | Time and frequency resolved dynamics of ArBr2[END_REF].

In the TSH case, v = 19 is closed for two vibrational quanta lost. The probabilities for hopping (vibrational transitions) are governed by the coupling terms v |H int (r, R, θ)|v Eq.( 21). As illustrated in upper Fig- ure (2 ) for v=20, the strongest couplings correspond to ∆v=±1, which justifies considering only ∆v=-1 steps in the kinetic mechanism discussed below.

For v ≥ 22 the results for all methods used in this work are in rather good agreement with those previously reported. As can be seen in that figure, both methods reproduce reasonably well the experimental data. Nevertheless, the TSH method has the advantage in comparison to QCTM that it does not need an approximate definition of the final state quantum numbers. In addition, the TSH method allows describing from a kinetic point of view the path followed the dissociation.

Kinetic mechanism

The TSH populations have been fitted to the kinetic mechanism presented in the supplementary material in terms of the following elementary steps: IVR (intramolecular vibrational redistribution or IVR transfer of one vibrational quantum without dissociation), VP (transfer of one vibrational quantum inducing dissociation), or EC (evaporative cooling or dissociation of a warm intermediate complex without any vibrational quantum transfer).

Figure [START_REF] González-Martínez | Quasi-classical trajectories study of Ne 79 Br2(B) vibrational predissociation[END_REF] shows the proportion of the paths ending with an EC step (IVR-EC) or a VP step (IVR-VP). As can be seen in that Figure, panel (a), the ∆v=-2 channel is dominant among IVR-EC processes for v = 16 -18 before closing, and the IVR-EC process does not occur for v > 18 vibrational levels. Panel (b) shows that except for v ≤ 19 the last step is always IVR-VP (contrary to what occurs upon transfer of the first vibrational quanta, the transfer of the last vibrational quantum does not lead to intramolecular vibrational redistribution but directly to dissociation).

For 16 ≤ v ≤ 18 IVR-VP competes with IVR-EC. However, IVR-EC is mostly for ∆v=-2 and disappears for v ≥ 19, whereas IVR-VP corresponds to ∆v=-3, gradually shifting to -4 and then -5. IVR-EC (v-2) can thus be attributed to the reduced phase space in (∆v = -2) when this channel is open, which is a traduction of the sparse IVR regime of the wave packet study. IVR-VP with more than 2 vibrational quanta involved gradually wins over IVR-EC up to closure of the ∆v=-2 channel. Among the IVR-VP channels, ∆v=-3 (two ∆v=-1 IVR steps followed by ∆v=-1 IVR-VP) dominates, with a gradual increase of ∆v=-4.

Rotational distribution of the Br 2 fragment

In Figure [START_REF] García-Vela | Quantum dynamics of Ne-Br2 vibrational predissociation: The role of continuum resonances and doorway states[END_REF] we compare the average rotational energy E rot and the maximum angular momentum (j max ) for Br 2 using both methods employed in this work. In panel (a) we can observe that for TSH the system acquires lower average rotational energy than for QCTM (for two vibrational quanta energy lost). For three vibrational quanta lost the behavior is reverse. This can be attributed to the continuous vibrational energy transfer in QCTM compared to its quantification in TSH. For ∆v=-2, all the TSH trajectories have received exactly the same amount of energy from Br 2 vibration, E B,v Br2 -E B,v-2

Br2

, whereas most of the QCTM trajectories have received more than that since ∆v=-2 was barely open in the range considered (v=16-18) and -∆v can be up to 2.5. For ∆v=-3, the results tend to show that most of the QCTM trajectories receive less than that E B,v

Br2 -E B,v-3

Br2

before dissociating. In panel (b) the maximum angular momentum (j max ) for Br 2 is higher using QCTM. This is again due to the continuous character of vibrational energy transfer in this method (the transfer of 2.49 quanta is collected in the ∆v=-2 bin for instance). The difference is more important when the system loses two vibrational quanta of energy. Another difference between the methods originates from the PES taken into account. In the QCTM only one surface is involved while for TSH the PES are slightly different for each vibrational level of the diatom v (see the low panel in Fig. 2).

A very convincing approach to improve final state distribution using QCTM is to implement Gaussian binning (GB), where trajectories are weighted following a Gaussian distribution rather than the standard binning used in this work. As recalled in Ref. 9, QCTM is very successful in reproducing final ro-vibrational state distributions except in the sparse IVR regime, in the vicinity of the closure of a vibrational channel. GB is shown to amend this shortcoming of the QCTM, at the expense of increasing by about one order of magnitude the number of trajectories required for convergence. It would be very instructive to perform a systematic comparison for ArBr 2 . However, even in the GB version, QCTM is not able to describe the kinetics mechanism in terms of VP, IVR, and EC steps, since it is based on the kinetics of the vibrational quantum transfer which can only be directly identified in the TSH method.

In order to estimate the importance of the potential anisotropy we have calculated the maximum theoretical value of j max which is determined by energy conservation:

E v + E vdW v v ≥ E v + E vdW v + 2 j(j + 1) 2µ Br2 v | 1 r 2 |v (26)
In Figure [START_REF] Cline | Structure and Dynamics of Weakly Bound Molecular Complexes[END_REF] we compare the theoretical j max by using Eq.26 with the results collected in Figure (7 b). The resulting maximum TSH values are close but not equal to the theoretical values. This shows that the potential anisotropy is important in the dissociation but not very strong.

Conclusions

We have investigated the vibrational predissociation process for the ArBr 2 system employing QCTM and TSH methods, and obtained good results in comparison with previous work. In our simulations, we studied a range of vibrational levels from v = 16 to v = 25. We computed the lifetime, exit channel, average rotational energy, and maximum angular momentum (j max ) of Br 2 .

We have shown that TSH in the diabatic representation provides a natural way of identifying the different paths leading to dissociation, thanks to the quantification of vibrational transfer in this mixed quantumclassical method. The results presented in Figure 7 should be convenient for experimental verification of the theoretical analysis.

Fig. 1 .

 1 Fig. 1. Jacobi coordinates for Ar. . .Br2

8 Fig. 2 .

 82 Fig. 2. Van der Waals interactions and couplings for Ar-Br2 vibrational predissociation. Upper plot: Off-diagonal matrix elements | ψ B 20 (r)|VArBr 2 (r, R, θ)|ψ B v (r) | between v=20 and different vibrational levels as a function of the Ar to Br2 distance R, defining the coupling responsible for VP of v=20. Note that the absolute value of the coupling is represented in order to stress the importance of the repulsive region of the potential, the coupling itself changes sign near R=3.87 Å. Lower plot: Diagonal ψ B v (r)|VArBr 2 (r, R, θ)|ψ B v (r) matrix elements defining the v th surface for TSH classical coordinates R and θ; All curves are calculated for the "T" shape configuration (θ = 90 • )

Fig. 3 .

 3 Fig.3. QCTM population as a function of time for v = 23 and its fit using single [Eq.[START_REF] Acosta-Matos | Trajectory-based modeling of the quantum dynamics of vibrational predissociation: Application to the Ar. . .Br2(ν = 24) complex[END_REF] or double [Eq. (23)] exponential law.

Fig. 4 .

 4 Fig. 4. Lifetimes for the TSH, QCTM, wave packets and experimental results, two last from Ref. [21]. The figure is divided in two parts to highlight the different time scales above and below 55 ps.

Fig. 5 .

 5 Fig. 5. Final vibrational state population (in %) of Br2 as a function of initial vibrational level for the QCTM and the TSH method, compared to wavepacket simulation and experimental results from Ref. [21]. a) ∆v=-2 (Br2(v-2) products); b) ∆v=-3; c) ∆v=-4; d) ∆v=-5.

5 Fig. 6 .

 56 Fig. 6. Path followed (statistically) for each vibrational level. a) IVR-EC and b) IVR-VP process. Both panels correspond to v -2, . . . , 5.

2 Fig. 7 .

 27 Fig. 7. QCTM and TSH results for a) average rotational energy Erot cm -1 and b) the maximum angular momentum (jmax) reached in our simulations for Br2 ∆v = -2, -3.

Fig. 8 .

 8 Fig. 8. Comparison of jmax between theoretical value obtained from energy conservation and the values obtained in our TSH simulation.

Table 1 .

 1 Morse potential parameters

	Interaction D (cm -1 ) α ( Å-1 )	R ( Å)
	Br-Br	3788	2.045	2.667
	Ar-Br	114	1.80	4.10

Table 2 .

 2 v are the vibrational levels for isolated Br-Br. Ev are the energies for those vibrational levels and E vdW

	v	represents

  ), replacing H J=0 with v|H J=0 |v and 1 r 2 with v| 1 r 2 |v [16, 25]. The PES are generated by v|H J=0 |v and the couplings by v |H J=0 |v with v = v, see Figure (2 ). The state vector |Ψ (t) describing the vibration of Br 2 as a function of time is obtained by solving the timedependent Schrödinger equation at the current value of the classical coordinates

5 Acknowledgements -CCM Acknowledges stimulating discussions at the UCI Liquid Theory Lunch (LTL) and the Telluride Science Research Center (TSRC), and support by the US National Science Foundation under grant CHE-1764209. -The authors would also like to thank Advanced Computational Team at Instituto Superior de Tecnologías y Ciencias Aplicadas (InSTEC) for the support provided during the realization of this work. -EGA, MMM, and JRS would like to thank PNAP (Proyecto No Asociado a Programa -Institucional) from InSTEC-UH. -EGA thanks the Université Fédérale Toulouse Midi-Pyrénées for financial support through the "Chaires d'Attractivité 2014" Programme IMDYNHE.

Authors contributions

All the authors were involved in the preparation of the manuscript and contributed equally to this work. All the authors have read and approved the final manuscript.