Zhiyang Shen

Kristiaan Kerstens
email: k.kerstens@ieseg.fr.

Vivian Valdmanis
email: vivian.valdmanis@wmich.ed

Songkai Wang
email: songkai0915@163.com

Evaluation and optimization of hospital system in Chinese provinces: does mortality matter?

Keywords: Data Envelopment Analysis, Mortality, Public hospital efficiency, Undesirable outputs. JEL classification: I10, I12

During the production process of goods and services, sometimes undesirable outputs are difficult to avoid. However, this aspect is often ignored. Hospitals produce patient care, but undesirable outputs do arise. The novelty of this paper is to introduce the mortality as an undesirable output into the derivation of the public hospital efficiency measure. Similar to the production of economic goods and pollution where the latter increase along with the former, our description of mortality in hospital is considered as weakly disposable.

), we evaluate the public hospital efficiency with and without incorporating mortality under four scenarios. We apply this model to measure public hospital efficiency in Chinese provinces. The results indicate that no matter whether one considers undesirable outputs within the objective functions, it has a significant impact on benchmarking once the mortality is included to define the production technology.

Introduction

Over the last 30 years, China's public healthcare system has experienced a number of reforms. The government presented the "Opinions on Deepening the Reform of the Health System," in March 2009, thereby launching the third round of large-scale reform of the hospital system. This round of reforms emphasizes the healthcare equity, mainly focusing on two aspects: improving hospital insurance coverage and strengthening the service capacity of primary hospital institutions [START_REF] Zhou | Supply-induced demand for medical services under price regulation: Evidence from hospital expansion in China[END_REF]. The Chinese government has promised to provide essential hospital services equally to all citizens by 2020 (see [START_REF] Yip | 10 years of health-care reform in China: progress and gaps in Universal Health Coverage[END_REF]. Public hospitals have gradually carried out a series of reforms to improve their performance [START_REF] Meng | What can we learn from China's health system reform[END_REF]. The "Sixth National Health Service Statistical Survey

Report" shows that the coverage rate of basic hospital insurance in 31 provinces has reached 96.8% (National Health Commission). At the same time, the aging population and life expectancy of Chinese society are increasing. According to the seventh census data, the number of persons aged 60 and up has risen to 260 million in China (National Bureau of Statistics). On January 10, 2022, the "14 th Five-Year" Public Service Plan issued by the National Development and Reform Commission and other departments shows that China's average life expectancy in 2025 will reach 78.3 years. With the improvement of the coverage rate of the healthcare security system, the demand for public hospital services increased and it is coupled with increasing expenses. This undoubtedly places higher requirements on the quantity and quality of public hospital supplies in China.

However, along with the growing demand for healthcare services, medical expenditure continues to increase, hence public hospitals face great pressure on resources under this plan [START_REF] Lopreite | The effects of ageing population on health expenditure and economic growth in China: A Bayesian-VAR approach[END_REF][START_REF] Gao | Effect of healthcare system reforms on public hospitals' revenue structures: Evidence from Beijing, China[END_REF][START_REF] Jiang | Population aging and its effects on the gap of urban public health insurance in China[END_REF][START_REF] Zhou | Supply-induced demand for medical services under price regulation: Evidence from hospital expansion in China[END_REF].

Before the third round of healthcare system reforms implemented in 2009, the efficiency of public hospitals in China has been considered poor in terms of quality, access, and costs (Gu & Tang, 1995). Further improvement of public hospitals' efficiency can improve the ability to respond to public health emergencies and effectively alleviate the pressure of the people's growing healthcare needs. Governments, hospitals, and researchers in China are focusing on how to efficiently monitor and improve the performance of public hospitals. In the post-pandemic period, China's overall economic growth slowed down, and health resources' growth has been limited (He et al., 2021).

Facing the challenges of an aging population and the pandemic, improving the utilization efficiency of resources devoted to public hospitals is crucial.

To measure the allocation of resources in China's public hospitals we choose public hospitals' input-output data in various provinces from 2011 to 2019. To clarify the effects of including undesirable outputs into the production technology, we develop four different models. By evaluating the performance of public hospitals under four different scenarios, the characteristics and problems of public hospitals in China are revealed. Targeted suggestions and countermeasures are put forward to provide a scientific basis for formulating development plans, coordinating hospital resource allocation, and rational decision-making. Even though the data are pre-pandemic, recommendations can be made regarding the public hospital system in China to respond to future health-related surges in demand as well as to the continuing growth of the elderly population and their increased needs for healthcare.

The remainder of this research is structured as follows. We compile and review significant studies on healthcare efficiency in Section 2. In Section 3, we introduce data and models, including data description and model setting. Section 4 provides results of efficiency performance, including a discussion of differences across scenarios and regions.

In Section 5, we present conclusions and give suggestions for policy-makers based on empirical results.

Literature review: Hospital efficiency and mortality

The research on efficiency has a long history [START_REF] Bjørnholt | The politics of performance measurement: 'Evaluation use as mediator for politics[END_REF][START_REF] Pollitt | The logics of performance management[END_REF].

Past research has been divided into two categories: parametric methods, such as stochastic frontier analysis (SFA), and nonparametric methods, for instance, data envelopment analysis (DEA). [START_REF] Antunes | Bank efficiency estimation in China: DEA-RENNA approach[END_REF] There are some limitations of the SFA method when measuring hospitals' efficiency.

First, the predefined production function may not be valid for all decision-making units; second, SFA relies on a random error term to set the probability distribution, and a single region can easily impact the frontier production function [START_REF] Lin | Exploring energy efficiency in China's iron and steel industry: A stochastic frontier approach[END_REF]. Also, SFA presupposes a minimum cost function which may incur misspecification especially for hospitals that may not follow the economic norm of cost minimization/profit maximization.

Unlike the SFA approach, using the DEA method, it is not necessary to specify the functional form of the production technology. It is appropriate for boundary production functions with multiple inputs and outputs. Furthermore, the DEA method has no special requirements on the dimension of each index, and inputs and outputs could be measured in their natural units.

Given the reasons mentioned above, DEA is more widely used in computing hospital efficiency [START_REF] Bai | Understanding the trend of total factor carbon productivity in the world: Insights from convergence analysis[END_REF]. Under the framework of dynamic network DEA, See et al. Public hospitals are the main healthcare institutions providing patient care services in China and the main target of the government's healthcare system reform [START_REF] Fang | Effects of healthcare reform on health resource allocation and service utilization in 1110 Chinese county hospitals: data from 2006 to 2012[END_REF]. Therefore, studying whether Chinese public hospitals can effectively provide adequate hospital services is crucial. There has also been extensive literature on this issue in the past two decades (e.g., [START_REF] Chen | Has the medical reform improved the cost efficiency of Chinese hospitals[END_REF][START_REF] Audibert | Health insurance reform and efficiency of township hospitals in rural China: An analysis from survey data[END_REF], especially since China's public hospital system has also undergone many reforms. Every reform has impacted China's public hospital system by increasing insurance coverage leading to more demand for hospital services and by an equitable distribution of resource across hospitals affecting supply.

Driven by the successive reforms of the public hospital system, the efficiency of China's public hospitals has also undergone significant periodic changes. To solve the problem of hospital supports for the rural people, China's government proposed the Rural Cooperative Medical System (RCMS) in the 1950s. The rural cooperative hospital system had covered over 90% of the rural population by the mid-1970s [START_REF] Feng | Awindow-DEA based efficiency evaluation of the public hospital sector in Greece during the 5-year economic crisis[END_REF]. Then, the government amended the basic social and hospital insurance system for city workers in 1998. All local employers and employees must split health insurance premiums under the plan. By 2007, the penetration rate of the scheme had reached about 65% [START_REF] Dong | Medical insurance system evolution in China[END_REF]. In 2009, China implemented its third and most recent reform. This reform proposes establishing and improving the hospital security system to equalize essential public health services. Today, China's hospital insurance system has basically achieved full coverage, covering more than 95% of the population (National Bureau of Statistics).

Taking the last healthcare system reform as a cut-off point, the effects of the third round reform in 2009 have been evaluated in many studies. [START_REF] Jiang | The impact of healthcare reform on the efficiency of public county hospitals in China[END_REF] use DEA to determine the efficiency of 1105 hospitals in 31 Chinese provinces. The results show that from 2008 to 2012, the scale of hospital services and the number of services increased suddenly, but the service performance was not very good and even slightly declined. Chen et al. ( 2020) use a method based on the additivity index and cumulative directional distance function (DDF) to evaluate the public hospitals' regional efficiency from 2011 to 2018. According to their findings, the total factor productivity (TFP) of public hospitals is growing at an annual average rate of 1.38%; driven primarily by technical efficiency (TE). Nevertheless, regional disparities in public hospital performance are rising. Chen et al.

(2020) use the three-stage DEA method to evaluate the operational efficiency of public hospitals in 31 provinces in China over the period 2011-2018. They find that the public hospitals' average efficiency scores have increased from 0.92 to 0.98. Furthermore, they find the performance of public hospitals mainly depends on the operating condition, with 11 district hospitals at the efficiency frontier for the entire period. Although there is an increase in average efficiency, regional differences in public hospital performance persist.

While the Chinese economy is generally centralized, there is an imbalance of regional development, resulting in varying degrees of efficiency of Chinese public hospitals across different provinces or regions. This regional difference has also been found in previous studies. [START_REF] Wang | Bootstrapping data envelopment analysis of efficiency and productivity of county public hospitals in Eastern, Central, and Western China after the public hospital reform[END_REF] It has recently been widely recognized that it is critical to integrate environmental aspects into productive efficiency and productivity measurement. [START_REF] Ancev | Environmentally adjusted productivity and efficiency: A review of concepts, methods and empirical work[END_REF] provide a review of the literature on environmentally adjusted productivity measurement since the 1990s and critically discuss the several concepts and methods developed and applied in various contexts. They highlight, among others, that the weak disposability assumption is often used to model pollution-generating technologies in a non-parametric way, which means that the evaluated units can decrease the negative externalities by reducing production activity [START_REF] Kuosmanen | Weak disposability in nonparametric production analysis with undesirable outputs[END_REF]. Another multi-equation approach relying on the costly disposability assumption (known as the by-production model) is also appealing for economists. The by-production model open the black box of production by modelling the polluting technology as the intersection of an intended-output technology and a residual-generating technology, which can be modeled without explicitly introducing the mass balance condition or avoid violating it [START_REF] Dakpo | Modelling pollution-generating technologies in performance benchmarking: Recent developments, limits and future prospects in the nonparametric framework[END_REF][START_REF] Dakpo | Modelling Environmental Adjustments of Production Technologies: A Literature Review[END_REF]. Obviously, there are also methods that do not belong to these cited categories.

For instance, [START_REF] Arman | Eco-innovation analysis: A data envelopment analysis methodology[END_REF] propose a new approach to find the common set of weights in DEA to examine eco-innovation in the presence of undesirable factors. In the healthcare sector, outputs such as the number of deaths can likewise be viewed as negative externalities. To evaluate hospital efficiency more comprehensively, some studies have begun to consider these undesirable outputs. [START_REF] Hu | Analysis of hospital technical efficiency in China: Effect of health insurance reform[END_REF] adopt the nonparametric method of DEA to deal with the scenario of multiple outputs with undesirable outputs in the healthcare department. Their results suggest that the efficiency of Chinese hospitals is moderate. [START_REF] Clement | Is more better? An analysis of hospital outcomes and efficiency with a DEA model of output congestion[END_REF] measure hospital efficiency using the DEA method, adding undesirable outputs such as patient mortality based on traditional output indicators. Bilsel and Davutyan (2014) use risk-adjusted mortality as an undesirable output when analyzing hospital efficiency in Turkey.

Given the above review of hospital efficiency studies, we contribute to the literature in three aspects: First, by using more recent inputs and outputs data of public hospitals from 2011 to 2019, the hospital system efficiency of each province in China is measured based on the [START_REF] Kuosmanen | Weak disposability in nonparametric production analysis with undesirable outputs[END_REF] model, we can expand on earlier studies. Second, we incorporate the undesirable output gauged as the number of deaths into the DEA model to improve the quality of the evaluation. Third, we specify four different models and use these models to investigate whether the results are impacted by the appearance of undesirable outputs.

Methodology

Production Technologies

In evaluation, each hospital is an evaluated unit. The optimal production frontier is constructed by linear programming (LP), and the inefficiency of each evaluated unit is measured through a directional distance function (DDF). [START_REF] Chambers | Benefit and distance functions[END_REF] introduced the DDF, which is frequently used to calculate the distance between the production frontier (or the best practice) and evaluated decision-making unit (DMU), namely provincial hospital. The DDF can measure the degree of inefficiency of the inputs and outputs separately or simultaneously from the perspective of the inputs to produce outputs. Note that in this study, we are more concerned about the impact of mortality on the evaluation results after it was introduced into the efficiency analysis model as an undesirable output. Therefore, we use the output orientation, to facilitate the comparison of inefficient performance under four different scenarios.

While most previous studies measuring hospital efficiency only consider desirable outputs such as the number of operations and discharged patients, we deviate from this earlier approach along with some other studies (see above) by incorporating undesirable outputs such as the number of deaths (as a proxy for quality) to represent a more realistic measurement of hospital efficiency. In particular, to the best of our knowledge we are the first to introduce mortality within the [START_REF] Kuosmanen | Weak disposability in nonparametric production analysis with undesirable outputs[END_REF] model applied to evaluate hospital efficiency producing jointly good and bad outputs. We next define the methodological steps we use to derive our efficiency measures. Equivalently, these technologies can be represented by their input sets defined as follows:

( )

( ) { } + + ∈ = ∈ ∈ ℝ ℝ P 1 J 1 L , y y x x, y T (3) ( ) ( ) { } + + + ∈ ∈ = ∈ ∈ ℝ ℝ ℝ P Q 2 J 2 L , , ,
,

y z y z x x, y z T (4)
In addition, the production technology mentioned above also needs to satisfy some basic economic assumptions, and these general axioms are necessary to make the production technology T ( 1 T and 2 T ) theoretically sound (Fare et al., 1994). We

denote the axioms (T ) to include the following:

+ + + + + + + ∈ ∈ = = × × ∈ ∈ ∈ × × - ≥ - ℝ ℝ ℝ ℝ ℝ ℝ J P Q N J P Q ( .1)(0,0,0)
, and if (0, , ) , then 0, 0.

( .2) is a closed subset of .

( .3) For each input R , is bounded. These axioms can be explained as follows. Inaction is possible and there is no free lunch.

Technology is closed and bounded. There is strong disposal of inputs and outputs: we can always waste more inputs for given outputs, and we can always produce less for given inputs. Finally, technology is assumed to be convex: this presupposes that technology is time divisible.

Further, to deal with the relationship between hospitals' desirable and undesirable outputs, we need to add the weak disposability hypothesis [START_REF] Shephard | Theory of Cost and Production Functions[END_REF] and the nulljointness hypothesis (Färe & Grosskopf, 2003) in 2 T . The weak disposability hypothesis holds that the desirable outputs and the undesirable outputs must increase or decrease in the same proportion, and the undesirable output cannot be reduced without reducing the desirable output, and the desirable output cannot be increased without increasing the undesirable output. The null-jointness hypothesis means that when the undesirable output is zero, then the desirable output must also be zero. These weak disposability assumptions and null-jointness assumptions are formally defined as follows:

θ θ θ ∈ ≤ ≤ ∈ 2 2 If ( ,

Efficiency Measures

Efficiency measures provide an equivalent representation of production technologies and focus on positioning observations relative to the boundary of the production possibility set. In this contribution, we use the DDF to measure the hospitals' efficiency. [START_REF] Kuosmanen | Weak disposability in nonparametric production analysis with undesirable outputs[END_REF] propose an approach based on the weak disposability assumption while satisfying both variable returns to scale and convexity assumptions. Following [START_REF] Kuosmanen | Weak disposability in nonparametric production analysis with undesirable outputs[END_REF],

we define an output-oriented DDF for four different scenarios. The scenario variations are listed in Table 1. T . The efficiency is measured considering only the production of desirable outputs. Using the [START_REF] Chambers | Benefit and distance functions[END_REF] approach, the DDF framework we employ here can be defined as follows:

( ) ( ) { 1 , ' 1 ' 1 1 ;0, : , 1,..., , 1,..., 1 y y K p p p k k k y k K j j k k k k K k k D x, y g Max y g y y g p P x x j J T θ λ θ θ λ θ λ λ + = = = = ∈ + × ∈ ≥ + = ≤ = =    ℝ } 0, 1,..., k k K λ ≥ = ( 8
)
where θ is the inefficiency value, which represents the potential improvement space of the evaluated unit in the output orientation. The vector ( 0 , ) y g is a nonnegative directional vector of good outputs, usually defined by the outputs corresponding to the DMUs: this yields a proportional interpretation for the DDF. When θ is greater than zero, then we interpret this as the possible increase in the production of outputs holding inputs constant. The activity variables λ k denote the reference set variables. If λ k is greater than zero, then it means that the DMUk is referenced as the benchmark determining the projection hypersurface for the optimal production plan.

In Scenario 2, we include the undesirable outputs, defined as the production technology 2 T . In this scenario, we incorporate both desirable and undesirable outputs data into the DDF framework. However, when measuring inefficiency values, we focus only on the potential improvement in the desirable outputs. That is, we look for the maximum potential when a DMU dedicates all resources to expanding the desirable output (without considering reducing the undesirable output). Therefore, the DDF in this scenario can be defined as:

( ) ( ) { 2 , , ' 1 ' 1 , , ;0, ,0 : , 1,..., ( ) , 1,..., y y K p p p k k k y k K j j k k k k k D x y z g Max R y g T y y g p P x x j J θ λ σ θ θ λ θ λ σ + = = = ∈ + × ∈ ≥ + = + ≤ =   } ' 1 1 , 1,..., ( ) 1 0, 0, 1,..., K q q k k k k K k k k k k z z q Q k K λ λ σ λ σ = = = = + = ≥ ≥ =   (9)
where θ is the inefficiency value of the desirable outputs. The vector , (0, 0 ) y g is a nonnegative directional vector of good outputs. When θ is greater than zero, then the desirable output can be increased by this proportion holding under inputs fixed. The activity variables λ k and σ k are the reference set variables . If λ k and σ k are greater than zero, then it means that the DMUk is referenced as the benchmark for the optimal production plan.

Scenario 3 is similar to Scenario 2 in that it is also focuses on production technology 2 T but now it is considering the undesirable output. The difference is that in scenario 3,

we focus only on the potential for improvement in the undesirable outputs. That is, the maximum potential when a DMU dedicates all resources to reducing the undesirable outputs without considering expanding desirable outputs. Therefore, the DDF in this scenario can be defined as follows:

( ) ( ) { 2 , , ' 1 ' 1
, , ;0,0, : , 1,..., ( ) , 1,...,

ϕ λ σ ϕ ϕ λ λ σ + = = = ∈ -× ∈ ≥ = + ≤ =   z z K p p k k k k K j j k k k k k D x y z g Max R z g T y y p P x x j J } ' 1 1 , 1,..., ( ) 1 0, 0, 1,..., λ ϕ λ σ λ σ = = = - = + = ≥ ≥ =   K q q q k k k z k K k k k k k z z g q Q k K (10)
where ϕ is the inefficiency value of the undesirable outputs. The vector , (0,0 ) z g is a nonnegative direction vector of bad outputs. When ϕ is greater than zero, then the undesirable outputs can be decreased by this proportion holding under inputs fixed.

Finally, in Scenario 4, when measuring the inefficiency value, we consider the objectives in scenario 2 and scenario 3 at same time under the production technology 2 T .

That is, the maximum improvement potential when a DMU uses all resources to expand the desirable outputs and to reduce the undesirable outputs. Therefore, the DDF in this scenario can be defined as:

( ) ( ) { 2 , , , ' 1 ' 1 : , ( ) , 1,..., , 1,...,
, , ; 0, ,

y z K j j k k k k k K p p p k k k y k y z R y g z g T x x j J y y g p P D x y z g g Max θ ϕ λ σ ϕ θ ϕ λ σ λ θ + = = ∈ + × -× ∈ + ≤ = ≥ + = =   ' 1 1 , 1,..., ( ) 1 0, 0, 1,..., K q q q k k k z k K k k k k k z z g q Q k λ ϕ λ σ λ σ = = = - = + = ≥ ≥ =   } K (11)
where θ and ϕ are the inefficiency values of desirable outputs and undesirable outputs, respectively. The vector ) y z g g is a nonnegative directional vector of good and bad outputs. When θ and ϕ is greater than zero, then it is possible to increase the desirable output by θ , while reducing the undesirable output of the ϕ .

, ( 0 ,

Data and Results

4.1Data: Descriptive Statistics

To evaluate the improvement of efficiency performance of public hospitals in various provinces after the latest round of medical reform, we use input and output data of public hospitals in 31 provinces in China from 2011 to 2019 to compute their efficiency. Each province is a DMU, and each year is a period to analyze the efficiency performance of each province in each year.

Due to the differences in the functions of different types of hospitals, there are certain differences in hospital services, staff and patients, and production technologies among hospitals. Therefore, we focus on using two inputs and two outputs.

Efficiency Results

Taking the average efficiency of public hospitals in 31 provinces from 2011 to 2019 by year, we can obtain the time trend change of the national average level shown in Figure 1.

Note that in Scenario 4, we compute both the potential growth of the desirable outputs and the potential reduction of the undesired output, but we report these two inefficiencies separately.

Figure 1 Inefficiency level change curve

As shown in Figure 1, we can find that all inefficiency scores decreased constantly over time. This finding demonstrates that the efficiency performance of Chinese public hospitals is improving. This is in line with the findings of previous studies (see [START_REF] Li | Determinants and differences of township hospital efficiency among Chinese provinces[END_REF][START_REF] Chu | The efficiency and its determinants for China's medical care system: Some policy implications for Northeast Asia[END_REF]. Specifically, scenario 1 has a much higher inefficiency value than the other scenarios when undesirable outputs are not taken into account. This may be due to poor technical modeling. The absence of a key indicator (undesirable output) leads to an overestimation of the improvement potential of public hospitals. In Scenario 2, 3 and 4, we get results considering the undesirable output. Scenario 2 only measures the inefficiency level of the desirable output, Scenario 3 only measures the inefficiency level of the undesirable output, and Scenario 4 measures the inefficiency level of the desirable and undesirable outputs simultaneously. By comparison, it can be found that the values of inefficiency in Scenario 2 and Scenario 3 is higher than that in scenario 4, which is reasonable. As resources are limited, when public hospitals use all resources to expand the desirable outputs or reduce the undesirable outputs, their improvement potential is greater than that when they improve both at the same time. It is worth noting, however, that the overall improvement potential of public hospitals is greater when both desirable and undesirable outputs are improved.

Comparing the historical inefficiency values of desirable and undesirable outputs we can find that the efficiency of Chinese public hospitals is better in terms of desirable output, with an average inefficiency level of less than 2%. Undesirable output performance is relatively poor, with an average inefficiency level of less than 4%. However, the average inefficiency level of desirable output has not changed much from the time trend, and the degree of improvement is small. The average inefficiency level of the undesirable output has been reduced from the initial 10% to 4%, a relatively significant improvement.

Figure 2 shows the kernel densities of each combination of two models: in total, this figure displays 10 subfigures with two densities. Visual inspection of these density figures

shows that the density difference between the models is large in most cases. [START_REF] Li | Nonparametric testing of closeness between two unknown distribution functions[END_REF] first proposed a non-parametric test to evaluate the differences between densities. The null hypothesis is that both densities are identical. The alternative hypothesis is that both densities differ. Subsequently, [START_REF] Simar | On testing equality of distributions of technical efficiency scores[END_REF] further modified this algorithm.

We use the latter revised method to calculate test statistics and significance levels. The test statistic and the p-value results are shown in Table 3.

The results in Table 3 show After removing the 20 efficient provinces or regions, we further compare and analyze the remaining 11 inefficient provinces or regions under the Scenarios 2, 3 and 4 models.

The inefficiency levels of desirable and undesirable outputs are shown in Figure 2 and 40%. In other provinces, the inefficiency level remains close to 5%. Therefore, the desirable output level of public hospitals in Inner Mongolia has the task for improving efficiency to become more similar to other areas lending evidence that regional differences exist. It is worth noting that, among all the provinces whose inefficiency level of undesirable output is not zero, Inner Mongolia and Liaoning have the worst performance, exceeding 50% and 45% respectively. Except for Province, which has a relatively low level of inefficiency (1%), the other eight provinces have a higher level of inefficiency.

Regional Difference Analysis

Furthermore, we divide the 31 provinces into eastern, central and western regions as shown in Table A1 in the appendix. Then, we get the two average levels of inefficiency in Scenario 4 for public hospitals by region. The comparison of these values with the national average is displayed in Figure 4.

Figure 5 Inefficiency level of different regions Obviously, both the eastern and western areas have lower levels of inefficiency for both desirable and undesirable outputs than the national average. The central region far exceeds the national average and has the highest degree of inefficiency and the worst performance of hospital efficiency among the three regions. This conclusion is consistent with previous studies (see [START_REF] Wang | Bootstrapping data envelopment analysis of efficiency and productivity of county public hospitals in Eastern, Central, and Western China after the public hospital reform[END_REF][START_REF] Hu | Analysis of hospital technical efficiency in China: Effect of health insurance reform[END_REF][START_REF] Ng | The productive efficiency of Chinese hospitals[END_REF][START_REF] Zhang | Using data envelopment analysis approach to estimate the health production efficiencies in China[END_REF].

Similarly, we examine the distribution of inefficiencies in different regions. Figure 5 shows the kernel densities of inefficiencies of desirable and undesirable outputs in three regions. This figure displays two subfigures with three densities. The distribution of inefficiencies is basically different in different regions. Also, we do the further examination,

Conclusions, policy recommendations and limitations

As public hospitals are the primary source of healthcare services for the Chinese people, it is critical to investigate their effectiveness in order to improve social welfare. Based on the input and output statistics of 31 Chinese provinces from 2011 to 2019, we measure the changes in the efficiency of public hospitals. To verify the importance of undesirable output in measuring the efficiency of public hospitals, we set four different scenarios and compare the inefficiency values in each scenario. The main conclusions contained in the results are as follows:

First, the overall efficiency level of the Chinese public hospital system has been improving over the past nine years. Second, the inefficiency values for each province show that after accounting for undesirable outputs, more than half of the provinces perform optimally. In terms of the production of the desirable outputs, Inner Mongolia's performance is poor. In terms of the production of the undesirable output, Inner Mongolia and Liaoning's performance are rather poor. We interpret these results that public hospitals operating in Inner Mongolia need to improve both efficiency and quality (as measured by a decrease in mortality rates). Liaoning's public hospitals should focus on improving quality in order to optimize performance. Finally, we compare the average inefficiency levels across regions and found that the inefficiency level in the central region is much higher than in the eastern and western regions.

As we have shown, evaluating hospitals under the four scenarios allows for a more thorough examination of the public hospital system involvement in advancing hospital equalization and optimizing overall hospital efficiency in China. As a result of our findings, the following policy recommendations are made.

First, allocate resources reasonably and control the scale of hospitals. From our findings, it can be found that public hospitals in various provinces differ not only in size, but also in efficiency performance. Regional differences are obvious. Faced with this situation, the government, as the policy-maker in the development of public hospitals, should consider its leading role, allocate resources rationally, and promote the equitable development of public hospitals. When allocating resources, it is necessary to consider the differences geographical and hospital size differences as much as possible. For large hospitals facing diminishing returns to scale expansion should be reduced, and reallocation of redundant human and material resources to avoid wasting resources. For hospitals that are too small, more resources can be appropriately invested to expand their scale of operation and improve their service capabilities. It can also promote the reorganization or merger among various hospital institutions and promote the sharing of high-quality resources while reducing operating costs.

Second, enhance the hospital assessment management via the introduction of an accountability mechanism for asset performance management. Production processes, including the production of undesirable outputs, have costs. The increase in mortality does not only reduce hospital efficiency, but also means a waste of resources. This type of production without regard to consequences is not conducive to sustainable development.

One policy suggestion would be to impose constraints on public hospitals with inadequate Under all these recommendations, proposals presented are geared toward meeting the dual objective of increasing the desirable outputs of patient care while decreasing the undesirable outputs of in-hospital mortality. By meeting these two objectives, the goals of Chinese hospital reform can be better met. Given the experience of the pandemic, it is necessary that governments and hospitals respond accordingly, including maximizing efficiency without an increase in undesirable outputs. The policy recommendations outlined above can further this endeavor to ensure appropriate inputs to meet future health care crises.

Furthermore, focusing on the time interdependence of production decisions and the adjustment path of the decision-making units over time can be an essential orientation for the future research. Variable factors differ in nature from fixed factors. Identifying the types of inputs can suggest more comprehensive remedies to address inefficiencies (see, e.g., [START_REF] Silva | Dynamic efficiency and productivity measurement[END_REF]. Dynamic DEA can measure the overall, technical and allocative efficiency of a multi-period production technology [START_REF] Fare | Intertemporal production frontiers: With dynamic DEA[END_REF]Shephard & Färe, 1980;[START_REF] Esmaeilzadeh | A super-efficiency model for measuring aggregative efficiency of multi-period production systems[END_REF]. This is a suitable challenge for future research.

  propose a new DEA method to analyze a panel dataset covering 39 commercial banks in China from 2010 to 2018 to estimate efficiency. Liddle & Sadorsky (2021) apply SFA to evaluate the energy efficiency of 81 OECD and non-OECD countries from 2000 to 2013. Xie et al. (2018) use SFA approach to calculate the energy efficiency score and potential energy-saving of Chinese provincial transportation sector from 2007 to 2016. Chen et al. (2016) use a Bayesian SFA model considering heterogeneity to examine the cost efficiency of Chinese hospitals at provincial level over 2002-2011.

  (2021) use a non-convex frontier method to evaluate the technical efficiency (TE) of hospital pharmacies between specialty and non-specialty hospital categories. Jahantigh & Ostovare (2020) use an outcome-based DEA model, based on four input factors and eight output factors, to evaluate the effectiveness of Tehran's 40 medical science university hospitals. Sultan & Crispim (2018) use an input-oriented DEA model to investigate the TE of 11 public hospitals in Palestine from 2010 to 2015. Flokou et al. (2017) apply the nonparametric method of DEA to compute the efficiency of the public hospital system in Greece during the 5-year economic crisis (2009-2013). A survey of DEA studies in hospital/healthcare can be found in Nepomuceno _et al (2022), O'Neill et al. (2008), Tiemann, Schreyögg and Busse (2012), and Kohl et al. (2019).

  use a bootstrapped DEA method to assess the productivity and TE changes of Chinese eastern, western and central county-level public hospitals after the 2012 public hospital reform. It is found that the central region's efficiency scores were consistently lower than those in the western and eastern areas. Hu et al. (2012) study the Chinese hospitals' efficiency index during 2002-2008 by using DEA tools. They find that the performance of hospital efficiency index in different provinces ranged from 0.396 to 1.00. Based on the data from 2004 to 2008, Ng (2011) use DEA to study the efficiency changes of hospitals in China. The empirical results show that the overall efficiency of Chinese hospitals improved throughout this period and find that hospitals in the relatively advanced Pearl River Delta region outperform the western regions. Using the DEA method, Zhang et al. (2007) investigate regional healthcare efficiency based on provincial hospital data in 1982, 1990, and 2000, and find large efficiency discrepancies in eastern, central, and western China.

  First, we need to define basic symbols and production technology. Given a J-, and a Q- dimensional undesirable output vector Q z + ∈ℝ . Considering two different production processes 1 T and 2 T , the former's input x only produces the desirable output y , and the latter's input x can produce both the desirable output y and the undesirable output z . Then the production technology or possibility set 1

(

  that only three groups of models fail the test (Scenario-2 & Scenario-4-D; Scenario-3 & Scenario-4-D; and Scenario-3 & Scenario-4-U), and all the other groups significantly reject the null hypothesis at the 5% significance level. in most cases, the distribution of the two models' results is significantly different. This is also consistent with the results of the kernel density figures. Note: Scenario -4-D and Scenario-4-U respectively means the inefficiency of desirable and undesirable outputs.

Figure 2

 2 Figure 2 Kernel density estimates of each two models

Figure 3

 3 Figure 3 below, respectively.

Figure 3 Figure 4

 34 Figure 3 Inefficiency level of desirable output

  asset operation efficiency management. By clarifying the operational positioning of various public hospitals, establishing scale standards for hospitals at all levels, and rationally calculating the input and output between hospitals of different levels, affiliations, and types, the problem can be steadily solved. It can also avoid the waste and inefficiency caused by the misuse of resources. External evaluation methods such as thirdparty evaluation can be introduced if necessary.Third, improve the level of technology, promote technological innovation, and enhance the diagnosis and treatment capacity of complex severe diseases. Improve efficiency performance by reducing mortality. Public hospital can strengthen information management, scientifically manage the use and replacement of new and old equipment, appropriately introduce new technologies and new equipment, and improve the awareness of technological innovation. The clinical pathway implementation process and effect evaluation can also be carried out regularly. The quality monitoring of critical links can be strengthened, and the pathway implementation plan can be continuously improved.For province-level hospitals with low efficiency, technological progress can be promoted through hospital alliances and urban-rural integration and other ways to promote the transfer of high-quality hospital resources to inefficient hospitals. At the same time, hospital managers should also actively encourage hospital staff to innovate diagnosis and treatment techniques and processes. This can be accomplished with more precise data collection regarding the cause of patient mortality. With greater precision, hospital managers and other stakeholders can ascertain hospital related quality and not attribute all mortality to hospital quality, especially if patients who do die were beyond medical intervention.

Table 1

 1 Comparison of four scenarios

	Scenario	Production technology	Objective	Model
	1	Without mortality	Expanding desirable outputs	(8)
	2	With mortality	Expanding desirable outputs	(9)
	3	With mortality	Reducing undesirable outputs	(10)
	4	With mortality	Expanding desirable outputs and reducing undesirable outputs	(11)
	Scenario 1 does not consider the undesirable output, that is, the production
	technology is solely 1

  SD), and the minimum, and the maximum of the data.

	Table 2 Inputs and outputs: Descriptive Statistics
	Variable	Obs	Mean	SD	Min	Max
	Staff1(Physicans)	279 99869.81 65701.84	4043	315311
	Staff2(Nurses)	279 106180	72061.69	1732	356330
	Staff3(Pharmacists)	279 13667.3 8998.969	428	43374
	Staff4(Others)	279 40946.47 25040.35	3133	107581
	Beds	279 226683.6 148158.3	8352	640147
	Surgical operation	279 1565033 1249240	23312	8372426
	Inpatients	279 6920406 4808206	145468	2.02E+07
	Outpatients	279 6894355 4790242	144039	2.01E+07
	Emergency deaths	279 110708.4 135460.7 2275.613	761389.6
	Deaths among hospital discharges	279 24956.45 20036.45 95.78593	90708.35
	Number of deaths in observation room	279 1298.847 1385.697	9.8872	9871.747
						The inputs are
	employees and beds, among which employees include licensed physicians, registered
	nurses, pharmacists, and other hospital personnel. Outputs are desirable outputs and
	undesirable outputs. Desirable outputs include the number of surgeries, hospital
	admissions, and discharges. Undesirable outputs are different types of mortality,
	including emergency room deaths, hospital discharge deaths, and observation room

deaths. All the data comes from China Health Statistical Yearbooks, China Statistical Yearbooks, and China Traditional Chinese Medicine Statistical Yearbooks. Descriptive statistics for these variables are shown in Table 2 below: this table contains mean, standard deviation (

Table 3

 3 Statistical Values of Li-test

		Scenario1	Scenario2	Scenario3	Scenario4-D
	Scenario2	37.265 (0.000)
	Scenario3	30.223 (0.000)	0.326 (0.022)
	Scenario4-D	36.474 (0.000)	-0.068 (0.811)	0.281 (0.052)
	Scenario4-U	30.224 (0.000)	0.331 (0.030)	-0.156 (0.990)	0.284 (0.047)

  Tibet, Shanxi, Guizhou, and Henan are still inefficient in Scenario 4. It is worth noting that among all the provinces where the desirable output inefficiency level is not zero, Inner Mongolia has the highest inefficiency level, reaching

	50%
	45%
	40%
	35%
	30%
	25%
	20%
	15%
	10%
	5%
	0%
	Scenario-2	Scenario-4 Desirable outputs
	70%
	60%
	50%
	40%
	30%
	20%
	10%
	0%
	Scenario-3	Scenario-4 Undesirable outputs

  Undesirable outputs revised Li-test. The test statistic and the p-value results are shown in Table 4. The results in Table4show that only the group (east and west) fail the test in both scenarios, and all the other groups significantly reject the null hypothesis at the 5% significance level. Figure 5 Kernel density estimates of inefficiencies in different regionsTable 4 Statistical Values of Li-test in Regions

	0.00% 2.00% 4.00% 6.00% 8.00% 10.00% 12.00% MIDDLE Scenario-4 Desirable CHINA EAST MIDDLE Scenario-4 Desirable outputs EAST MIDDLE Undesirable 0.573 (0.007) MIDDLE WEST -0.039 (0.733) 0.608 (0.006) WEST	WEST EAST 1.061 (0.002) -0.064 (0.702)	MIDDLE 0.721 (0.008)

regional public hospitals in china based on the three-stage dea. International Journal of Environmental Research and Public Health, 17(24),

Appendix1


Julien Plante
email: julien.plante@obspm.fr

Damien Gratadour

Louis Bondonneau

Cédric Viou

A Novel FRB Detection Pipeline For NenuFAR

NenuFAR is a very large low-frequency radiotelescope, designed to observe the sky at frequencies ranging from 10 to 85 MHz. One of the main science case is the study of pulsars and Fast Radio Bursts (FRBs), and this program already has detected hundreds of these objects and events. However, the current detection pipeline is not able to perform a 24/7 survey of the sky. This is not an issue for the observation of recurring events such as pulsars emissions, but limits the number of FRB detections, and discovery opportunities. Here, we propose a novel real-time FRB detection system for NenuFAR, including a custom hardware platform and a software solution, designed to detect transient events in real-time and trigger signal storage on event detection to reduce memory footprint. This experiment on NenuFAR has also been designed as a pathfinder for more efficient transient events detection on SKA, with scalability as one of the core specifications. In this paper, we present the design of the experiment, detail the underlying hardware and software technologies and discuss initial results from a benchmarking campaign.

Introduction

Multiple Fast Radio Burst (FRB) detection pipelines are available nowadays [START_REF] Petroff | [END_REF]. Because of the low observation frequency of the NenuFAR telescope (10-85MHz) [START_REF] Zarka | & other members of the NenuFAR-France team[END_REF], the dispersive delays of FRBs are up to a thousand times greater than for usual radiotelescopes (with typical observations centered around 1GHz). This makes existing pipelines perform poorly on NenuFAR data (increased memory footprint and latency, reduced Dispersion Measure (DM) search capabilities).

To address this limitation, we propose a pipeline tailored for low-frequency, realtime FRB detection.

Experiment Description

The pipeline currently under development is summarized on Figure 1, while more indepth explanations are given in this section.

NenuFAR data is multicasted to experiments using UDP packets. To maximize the bandwidth and minimize memory footprint, we use DPDK + GPUDirect. This cutting-edge technology introduced by Nvidia enables direct transfer from the Network Interface Controller (NIC) to the GPU, without the need for a bounce buffer on CPU. 1 Incoming data is distributed on GPUs by frames of a few seconds in a round-robin fashion. Using multiple GPUs increases the time available for one iteration, making more complex operations possible.

Radio Frequency Interference (RFI) mitigation is performed after a simple preprocessing (time integration + power law), to identify and remove signal parts contaminated by undesired radio signals such as human emissions. This stage is not yet implemented in our pipeline.

Dedispersion is then applied to compensate the dispersion effect caused by the Interstellar Medium (ISM). We based our work on the Dispersion Measure Transform (DMT), with a custom strategy to address the detection of highly dispersed (> 10 s) bursts. Using piecewise integration, we are able to compute the same DMT by iterating on shorter time frames, while being independent of the underlying DMT algorithm. Current implementation uses the Bruteforce DMT algorithm (Barsdell et al. 2012).

The final stage detects energy peaks in the DMT. We propose an image processing approach to take advantage of highly optimized GPU libraries for image processing, such as Nvidia Performance Primitives (NPP). The current implementation is very limited, simply based on a global maximum and threshold, but we are working on more robust algorithms.

FRB candidates will trigger an external ring buffer of several seconds-long based on memory-only storage. That will allow raw waveform data retrieval and their recording on HDD to bring only signals of interest to the final user. This pipeline will be powered by the COSMIC framework [START_REF] Ferreira | Adaptive Optics Systems VII[END_REF]) for real-time GPU computing, to provide stable performance, flexibility and a comprehensive user interface.

Initial Results

We compared the current performance of our pipeline to that of Heimdall, another GPUaccelerated FRB detection pipeline, and studied the new data acquisition approach.

DPDK + GPUDirect Benchmark

A benchmark has been realized to assess the performance of the DPDK + GPUDirect data acquisition method. Figure 2 show the results of a 1-minute benchmark of the latency, at 1.5 Gbit/s (the maximum bandwidth of NenuFAR for this experiment). These results were obtained with two Mellanox Connectx-5 interconnected with a 100 GbE fibre. The latency is measured as the delay from the first send to the full reception in GPU memory of 100kB chunks of data. -mi =9.97µs -max=169.75µs -ptp=159.78µs gpudirect: -μ=13.66µs -σ=1.52µs -min=9.71µs -max=267.37µs -ptp=257.66µs Latency over 1 minute at 1.5 Gbit/s

The mean latency in both NIC to CPU and NIC to GPU modes is about 13 µs, which will allow for a quick response of the detection pipeline. Some outliers are present, and will have to be dealt with to use this technology in a strong real-time environment. We believe this can be solved by tuning the system configuration (CPU shielding, real-time kernel, ...).

Moreover, we found that this result is strongly dependent on the PCIe topology of the experiment. With the NIC on the same PCIe bridge than the GPU, bandwidths up to 80 Gbit/s were measured. However, with a different topology, bandwidths down to 2 Gbit/s were noticed, with much higher latencies (hundreds of µs).

Computations Benchmark

To compare with Heimdall, we used NenuFAR-like data: 200 kHz sampling frequency (with time integration over 16 samples), 2 21 samples per frame, 256 DM. The benchmark was run on one Nvidia Quadro GV100 (no multi-GPU for fair comparison of the implementations). Both Heimdall and our pipeline are already able to detect FRBs in real-time (2 21 samples at 200 kHz is about 10.5s) using high-end CPU and GPU, and still have a great margin for their processing time compared to a single frame time. This gives us room for thorough RFI mitigation, or opens the door to using lower-end hardware.

Our current implementation of dedispersion is 40% slower than Heimdall's, but enables the detection of FRBs with a dispersive delay longer than the time frame duration. It also allows easy pipelining across multiple GPUs, thus increasing the available processing time with each added GPU. Still, our implementation is very naive, and speedups are expected with further optimizations.

Our implementation of pulse detection is more than 100x faster than Heimdall's, likely thanks to our 2D approach to pulse detection in the DMT space. Though this approach is promising, the current implementation is not robust to RFIs, and needs fine-tuning of a threshold.

Conclusion

We presented initial results concerning a new FRB detection pipeline tailored for use with NenuFAR. This system will make possible to scale down the memory requirement for FRB study, by detecting FRBs on-the-fly, and saving only the FRB-related signal on disk.

Using DPDK + GPUDirect, we are able to provide a low-latency, high-bandwidth, and rather standard data acquisition method. While this technology is currently implemented using Nvidia hardware, we believe this could be ported to other manufacturer's hardware. This data acquisition method is more than enough to fill the requirements in this application, but enables high scalability.

A new approach to dedispersion emerged from this work, especially useful for low-radio observation, by removing the need for overlap and making possible to detect pulses longer than the time frame studied. This approach easily enables multiple-GPUs parallelism, increasing the available time for processing one time frame.

This progress is promising for the deployment of an online robust FRB detection pipeline for the low-frequency radiotelescope NenuFAR, with a first prototype integration planned for the first half of 2022.

Perspectives

The current priority is to complete the pipeline by linking the DPDK + GPUDirect to the rest of the pipeline. Further important steps are to implement a state-of-the-art RFI mitigation tuned for NenuFAR, and more robust pulse detection algorithms to provide an end-to-end prototype ready for integration at Nançay.

Next iterations will also focus on optimizing the DMT implementation, either by optimizing memory accesses in the current Bruteforce DMT, or by using the Fast DMT (Zackay & Ofek 2017), an algorithm with reduced complexity for the DMT computation. The outliers in latency for the data acquisition method are also to be looked into.

  Figure 1. Pipeline architecture

  Figure 2.Latency over 1 minute at 1.5 Gbit/s


Thomas Guillon

Sébastien Bonhommeau

José Sanchez Costa

Antoine Zwick

Jean-François Létard

Philippe Demont

Gábor Molnár

Azzedine Bousseksou

On the dielectric properties of the spin crossover complex [Fe(bpp) 2 ] [BF 4 ] 2

The thermal variation of the dielectric constant in the spin crossover complex [Fe(bpp) 2 ] [BF 4 ] 2 displays a 16 K large hysteresis loop centered around 178 K corresponding to the spin transition temperatures. Unexpectedly, the dielectric constant has a higher value in the low-spin state. DFT calculations were carried out to determine the microscopic electronic polarizabilities, but these latter could not be correlated with the macroscopic permittivity changes pointing the importance of the effect of the counter-anions which must be necessarily taken into account in future DFT calculations.

Introduction

Among bistable molecular materials, spin crossover complexes of 3d 4 -3d 7 transition metal ions have been investigated by many research groups [START_REF]Spin Crossover in Transition Metal Compounds I -III[END_REF]. These compounds hold some promise to be used in electronics based memories because the conversion between the high-spin (HS) and the low-spin (LS) states can be detected via a capacitance measurement [2]. This becomes possible because the dielectric constant in the two spin states differs sufficiently. Since optical addressing is a convenient way for information storage, spin crossover complexes undergoing a LIESST (Light Induced Excited Spin State Trapping) effect have received many attention recently [3]. The spin crossover compound [Fe(bpp) 2 ] [BF 4 ] 2 , for which bpp stands for 2,6-bis(pyrazol-3-yl)pyridine, is a particularly interesting example because in this material the light induced metastable HS state decays at relatively high temperatures (ca. 110 K) [4,5].

In this paper, we report the dielectric properties of the complex [Fe(bpp) 2 ] [BF 4 ] 2 and correlate the macroscopically measured dielectric permittivity with the microscopic (electronic) polarizability calculated by the single molecule DFT (Density Functional Theory) methodology on the cation [Fe(bpp) 2 ] 2+ in the LS and HS states. DFT calculations on this compound have been previously carried out in order to find its free energy [6]. Moreover, DFT methods are also widely used to evaluate and assign the molecular vibrational frequencies [7]. Here, we focus on the determination of electronic polarizabilities.

The dynamic dielectric spectroscopy method was used to measure the thermal variation of the complex dielectric permittivity (ε* = ε′ -iε′′) in the 10 2 -10 7 Hz frequency range, successively in the heating and cooling modes by means of a Novocontrol BDS 4000 broadband spectrometer. The frequency sweeps were carried out isothermally. The powder samples were enclosed in a Teflon sample holder between two stainless steel electrodes. Raman spectra were acquired at room temperature and at 80 K using a Dilor XY triplemate micro-spectrograph coupled to a Princeton Instruments CCD detector. The 632.8 nm line of a He-Ne laser was used as the excitation source. DFT calculations have been carried out on single molecules in vacuo with Gaussian03 package [START_REF] Frisch | GAUSSIAN 03, Revision B.05[END_REF]. The Becke's three-parameter hybrid functional B3LYP [START_REF] Becke | [END_REF] was used with the 3-21G and 6-31G* basis sets. The input structures of the cation [Fe(bpp) 2 ] 2+ for the optimization of the LS and HS geometries were taken from X-ray crystallographic data (Fig. 1) [10]. The optimized structures were calculated without any constraint. From these structures, polarizabilities and intramolecular vibrational frequencies were determined.

Results and discussion

Spin crossover complexes of the family [Fe(bpp) 2 ] [X] y • nH 2 O (where X = NCS -, NCSe -, BF 4 -with y = 2 or X = Fe(CN) 5 (NO) 2-with y = 1) were extensively studied by Sugiyarto et al. [10] who reported X-ray crystallographic data for several dehydrated or hydrated compounds (n = 1 for NCS -, n = 3 for BF 4 -and n = 0 for the other compounds). Unfortunately, single crystal X-ray structures of the title spin crossover compound could not be obtained. However, the crystallographic structure of the cation [Fe(bpp) 2 ] 2+ has been determined in several HS or LS compounds with different counter-anions. These X-ray data were used as the input of the DFT calculations and also to check the reliability of the calculated structures of this cation. Tables 1a, b and c provide a comparison between X-ray crystallographic data and calculated bond distances and angles of the cation [Fe(bpp) 2 ] 2+ . Calculated values of bond distances and angles in the ligand (Table 1b andc) reproduce the measured ones with good accuracy. Calculated values of bond angles involving the central metal atom are in good agreement with the experimental data as well. The metal-ligand bond distances are known to be deeply affected by the spin state change. Indeed, the Fe-N bonds in the HS state are approximately 0.2 Å longer when compared to the LS state both in the experiments and in the calculations (Table 1a). However, the B3LYP/6-31G(d) method slightly overestimates Fe-N bond length values (in both spin states). This tendency was observed earlier for many spin crossover complexes [7]. With this DFT method, the optimized geometry in the LS state presents a D 2d symmetry with a good precision. The optimized geometry in the HS state has a slightly lower symmetry (C 2v ). From these optimized geometries, static electronic polarizability tensors were calculated as the analytic second derivative of the energy with respect to the electric field in order to determine the spin- [10] and (6) [Fe(bpp) 2 ] [Fe(CN) 5 (NO)] form 2 [10].

] [NCS] 2 • H 2 O [10], (2) [Fe(bpp) 2 ] [NCSe] 2 [10], (3) [Fe(bpp) 2 ] [Fe(CN) 5 (NO)] [10], (4) [Fe(bpp) 2 ] [BF 4 ] 2 • 3H 2 O [10], (5) [Fe(bpp) 2 ] [Fe(CN) 5 (NO)] form 1

HS state

X-ray (1) X-ray (2) X-ray LS state X-ray (4) X-ray (5) X-ray (6) B3LYP/6-31G* Fe-N 1A Figure 2 presents Raman spectra associated with the pure HS state (200 K), the pure LS state (140 K) and a mixed HS and LS state on the ascending branch of the hysteresis (180 K). Several marker bands can be chosen to follow the spin transition. For example, peaks at 231 cm -1 and 1224 cm -1 are characteristic of the LS state while the peak at 1577 cm -1 only appears in the HS state. However, the most relevant marker bands are located at ca. 670 cm -1 and 1030 cm -1 and exhibit a well-resolved frequency shift from 667 cm -1 to 678 cm -1 and from 1019 cm -1 to 1034 cm -1 when going from the HS to the LS state. An efficient way to assign these modes to atomic motions consists of calculating molecular vibrational frequencies using the DFT methodology. In the calculated spectra two main frequency ranges stand out.

] [NCS] 2 • H 2 O [10], (2) [Fe(bpp) 2 ] [NCSe] 2 [10], (3) [Fe(bpp) 2 ] [Fe(CN) 5 (NO)] [10].
HS state X-ray (1) X-ray (2) X-ray The 3220-3670 cm -1 frequency range corresponds to C-H and N-H stretches whereas the second range spans up to 1660 cm -1 . In this latter frequency range, the major changes in vibrational frequencies occur below 700 cm -1 and most of the modes associated with the Fe displacement appear below 500 cm -1 . Above 500 cm -1 , the calculations gave two vibrational modes exhibiting frequency shifts from 679 cm -1 to 689 cm -1 and from 1033 cm -1 to 1051 cm -1 when going from the HS to the LS state. These latter modes can be assigned to nearly symmetric deformations of the pyridine rings coupled to a symmetric stretching of the N 5A -Fe-N 5B bonds. This coupling revealed by the DFT calculations explains the somewhat surprising large shift of these two modes observed in the experimental spectra.

The thermal bistability of the [Fe(bpp) 2 ] [BF 4 ] 2 spin crossover complex appears clearly through magnetic susceptibility measurements (Fig. 3a). Cooling the sample leads to a sharp decrease in HS fraction around 180 K. The pure LS state is reached at ca. 140 K. From this state, the sample heating yields a rise in the HS fraction around 174 K. The pure HS state is then attained at ca. 194 K. Thermal variations of [10], (5) [Fe(bpp) 2 ] [Fe(CN) 5 (NO)] form 1 [10] and (6) [Fe(bpp) 2 ] [Fe(CN) 5 (NO)] form 2 [10].

LS state

X-ray (4) X-ray (5) X-ray (6) B3LYP/6-31G* the dielectric constant at 100 kHz frequency for [Fe(bpp) 2 ] [BF 4 ] 2 (Fig. 3b) reveal the same transition temperatures (T up = 186 K, T down = 170 K). The dielectric constant in the HS state is smaller than that of the LS state ∆ε′ = ε′ HS -ε′ LS = -0.11. The change in dielectric constant of SCO materials results from structural modifications accompanying the spin state change. Within the investigated frequency range, the dielectric constant and therefore the total polarizability as well are nearly independent of the frequency of the external applied electric field (Fig. 4). No relaxation process, associated with an orientational polarization of permanent dipole moments, is observed either in the absorption curve ε′′ vs. the frequency. This absence of spontaneous polarization in the [Fe(bpp) 2 ] [BF 4 ] 2 complex rules out any effect of the dipolar polarization on the observed transition and the polarizability is mainly composed of electronic and ionic contributions thereby. As the HS molecules are more voluminous than the LS ones, their electronic polarizability and hence the relative permittivity should decrease when going from the HS to the LS state. Furthermore, since the HS state has a more ionic nature, the ionic polarizability must also decrease to some extent with the HS → LS spin state change. Indeed, for most of the investigated SCO compounds a decrease of the macroscopically measurable ε′ was observed during the HS to LS transition [2]. Therefore the negative value of ∆ε′ HL observed in the compound [Fe(bpp) 2 ] [BF 4 ] 2 can be considered as atypical. This behavior was observed earlier only for the spin crossover complex Fe[5-NO 2 -sal-N [START_REF]Spin Crossover in Transition Metal Compounds I -III[END_REF]4,7,10)] and assigned to structural modifications through DFT calculations [11]. In fact, the local symmetry distortion modifies significantly the local field, inducing a large change in the dipole moment detected through dielectric spectroscopy. On the contrary, in the case of the [Fe(bpp) 2 ] [BF 4 ] 2 molecule, we were unable to correlate the microscopic electronic polarizability obtained by DFT calculations with the experimentally observed behavior of the dielectric constant. In fact, the calculated positive value of the mean electronic polarizability change HS LS (∆ 7.01 a.u.) α α α = -= has an opposite sign compared to the variation of the dielectric constant

N 1A -C 3A 1.328 (8) 1.353 C 3A -C 4A 1.471 (9) 1.462 N 5A -C 4A 1.352 (8) 1.354 N 5A -C 9A 1.341 (8) 1.354 C 8A -C 9A 1.459 (9) 1.462 N 3A -C 8A 1.342 (8) 1.353 N 1B -C 3B 1.340 (8) 1.353 C 3B -C 4B 1.454 (10) 1.462 N 5B -C 4B 1.369 (9) 1.354 N 5B -C 9B 1.347 (8) 1.354 C 8B -C 9B 1.
HS LS (∆ 0.11) ε ε ε = - =- ¢ ¢ ¢
. In summary, this work revealed a clear correlation between the dielectric and magnetic properties of the spin crossover complex [Fe(bpp) 2 ] (BF 4 ) 2 . Interestingly, we have found that the (quasi-static) dielectric constant has a higher value in the LS state. The experimental investigation was completed by DFT calculations on the [Fe(bpp) 2 ] 2+ cation. The comparison of these calculations with the experimental results revealed that the molecular geometry and vibrational frequencies of the cation are determined mainly by the spin state and are less influenced by the nature of the actual counter anion. On the other hand, the calculations suggest that lattice effects (such as the position or orientation of the anions, H-bonds, etc.) must have a strong influence on the macroscopic electrical properties. For this reason, alternative methods like introduction of a Madelung field in a molecular DFT calculation [12] or periodic DFT calculations should be used to get a better description of this system. Especially, periodic DFT calculations provide interesting challenge for future research.

Fig. 1 (

 1 Fig. 1 (online colour at: www.pss-a.com) Optimized structure of the cation [Fe(bpp) 2 ] 2+ in the HS state calculated with B3LYP/6-31G*.

Fig. 2 a

 2 Fig. 2 a) Experimental Raman spectra of [Fe(bpp) 2 ] [BF 4 ] 2 recorded in the 100 -750 cm -1 frequency range at 140, 180 and 200 K. b) Experimental Raman spectra of the [Fe(bpp) 2 ] [BF 4 ] 2 recorded in the 750 -1650 cm -1 frequency range at 140, 180 and 200 K.

Fig. 3 (

 3 Fig. 3 (online colour at: www.pss-a.com) a) Thermal variation of the proportion of high-spin molecules ( HS γ ) in [Fe(bpp) 2 ] [BF 4 ] 2 obtained through magnetic susceptibility measurements. b) Thermal hysteresis of the dielectric constant associated with the spin transition of [Fe(bpp) 2 ] [BF 4 ] 2 .

Fig. 4 (

 4 Fig. 4 (online colour at: www.pss-a.com) Frequency dependence of the dielectric constant of the spin crossover compound [Fe(bpp) 2 ] [BF 4 ] 2 at 186 K (circles) and 170 K (squares).

Table 1a

 1a Experimental Fe -N distances (Å) in some complexes of the [Fe(bpp) 2 ] [X] y • nH 2 O family and calculated values for the [Fe(bpp) 2 ] 2+ ion in the HS and LS states. X-ray crystallographic data are given for (1) [Fe(bpp) 2

Table 1b

 1b Selected experimental bond distances (Å) and bond angles (degrees) in some HS complexes of the [Fe(bpp) 2 ] [X] y • nH 2 O family and calculated values for the [Fe(bpp) 2 ] 2+ ion in the HS state. X-ray crystallographic data are given for (1) [Fe(bpp) 2

Table 1c

 1c Selected experimental bond distances (Å) and bond angles (degrees) in some LS complexes of the [Fe(bpp) 2 ][X] y • nH 2 O family and calculated values for the [Fe(bpp) 2 ] 2+ ion in the LS state. X-ray crystallographic data are given for (4) [Fe(bpp) 2 ] [BF 4 ] 2 • 3H 2 O

Acknowledgements DFT calculations have been performed thanks to the facilities provided by CALMIP (Calculs en Midi-Pyrénées, France), CINES (Centre Informatique National de l'Enseignement Supérieur, France) and IDRIS (Institut du Développement et des Ressources en Informatique Scientifique).