Satyajit Das

Kevin J M Martin

Philippe Coussy

Davide Rossi

A Heterogeneous Cluster with Reconfigurable Accelerator for Energy Efficient Near-Sensor Data Analytics

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I. INTRODUCTION

High performance and extreme energy efficiency are strict requirements for many deeply embedded near-sensor processing applications such as wireless sensor networks, end-nodes of the Internet of Things (IoT) and wearables. One of the most traditional approaches to improve energy efficiency of deeply embedded computing systems is achieved exploiting architectural heterogeneity by coupling general-purpose processors with application-or domain-specific accelerators in a single computing fabric [START_REF] Conti | Synthesis-friendly techniques for tightly-coupled integration of hardware accelerators into shared-memory multi-core clusters[END_REF] [START_REF] Taylor | Is dark silicon useful? harnessing the four horsemen of the coming dark silicon apocalypse[END_REF]. On the other hand, most recent ultra-low power designs exploit multiple homogeneous programmable processors operating in near-threshold [START_REF] Rossi | Energy-efficient near-threshold parallel computing: The pulpv2 cluster[END_REF]. Such an approach, which joins parallelism with low-voltage computing, is emerging as an attractive way to join performance scalability with high energy efficiency.

In this paper, we present a heterogeneous architecture which integrates a near-threshold tightly-coupled cluster of processors [START_REF] Rossi | Energy-efficient near-threshold parallel computing: The pulpv2 cluster[END_REF] augmented with the Integrated Programmable Array (IPA) presented in [START_REF] Das | A 142MOPS/mW Integrated Programmable Array Accelerator for Smart Visual processing[END_REF]. This approach joins the programming legacy of instruction processors with the flexible performance and efficiency boost of Coarse Grain Reconfigurable Arrays [START_REF] Sutter | Coarse-grained reconfigurable array architectures[END_REF] (CGRA). A similar approach has been adopted in [START_REF] Duch | Heal-wear: An ultra-low power heterogeneous system for bio-signal analysis[END_REF], which introduced an ultra-low power heterogeneous system featuring a Single Instruction Multiple Data (SIMD) CGRA as reconfigurable accelerator for bio-signal analysis. With respect to this domain-specific architecture, where the computational kernels must be mapped manually on the CGRA, the system proposed in our work is meant for general-purpose near-sensor data analytics, also relying on an automated compilation flow that allows to generate the configuration bitstream for the CGRA starting from a generalpurpose ANSI-C code [START_REF] Das | Efficient mapping of cdfg onto coarse-grained reconfigurable array architectures[END_REF].

We synthesized the architecture in a 28nm FD-SOI technology, and we carried out a quantitative exploration combining physical synthesis results (i.e. frequency, area, power) and benchmarking of a set of signal processing kernels typical of end-nodes IoT applications. Two interesting findings of our exploration show that (1) the performance of the IPA is much less sensitive to memory bandwidth than parallel processor clusters and that (2) the simpler nature of its architecture allows the IPA to run twice as fast as the rest of the system. Exploiting these two features of our architecture, we show that the heterogeneous cluster achieves significant performance and energy improvement for both compute and control intensive benchmarks with respect to the 8 core homogeneous cluster, achieving up to 4.8× speed-up (with a minimum of 1× and an average of 1.79×) and up to 4.4× (with a minimum of 1× and an average of 2.24×) better energy efficiency.

II. BACKGROUND

This section presents the background technology used to design the heterogeneous reconfigurable cluster described in this work.

A. PULP Cluster Architecture

The PULP cluster features 8 32-bit RISC-V cores based on a four pipeline stages micro-architecture optimized for energyefficient operation [START_REF] Gautschi | Near-threshold riscv core with dsp extensions for scalable iot endpoint devices[END_REF] sharing a 64KB multi-banked scratchpad memory through a low-latency interconnect [START_REF] Rahimi | A fully-synthesizable single-cycle interconnection network for shared-l1 processor clusters[END_REF]. The ISA of the cores is extended with instructions targeting energy efficient digital signal processing such as hardware loops, load/store with pre/post increment, SIMD operations. The cores share a 4KB private instruction cache to boost performance and energy efficiency for tightly coupled clusters of processors typically relying on data parallel computational models [START_REF] Loi | The quest for energy-efficient i$ design in ultra-low-power clustered manycores[END_REF]. Off-cluster data transfers are managed by a lightweight multichannel DMA optimized for energy-efficient operation [START_REF] Rossi | Ultra-low-latency lightweight dma for tightly coupled multi-core clusters[END_REF]. Both the (I$) and DMA are connected to an AXI4 cluster bus. A peripheral interconnect is used to communicate with oncluster peripherals such as a timer, a hardware synchronizer and other memory mapped peripherals such as applicationspecific accelerators. To operate at the best operating point for a given workload the cluster can be integrated in an independent voltage and frequency domain, featuring dualclock FIFOs and level shifters at its boundary.

B. IPA Architecture

The IPA consists of an array of 16 PEs communicating through a 2D torus interconnect [START_REF] Das | Efficient mapping of cdfg onto coarse-grained reconfigurable array architectures[END_REF]. Each PE can perform 32bit ALU operations (both arithmetic and logical), 16-bit × 16bit → 32-bit multiplications and control flow operations such as branches. The functional units of each PE features two input operands coming from the neighbouring PEs or the internal register files. The PEs also include an instruction register file which stores the program, a regular register file to store temporary variables and a constant register file to store immediates. To reduce dynamic power consumption in idle mode, each PE contains a tiny Power Management Unit (PMU) which clock gates the PEs when idle [START_REF] Das | A 142MOPS/mW Integrated Programmable Array Accelerator for Smart Visual processing[END_REF]. A parametric number of PEs can be augmented with a load-store unit employing the same request-grant protocol of the PULP lowlatency interconnect [START_REF] Rahimi | A fully-synthesizable single-cycle interconnection network for shared-l1 processor clusters[END_REF], which allows to communicate with a multi-banked shared memory. The configuration of the array is generated automatically by a compilation flow which starts from a ANSI-C code and generates the configuration bitstream for the IPA [START_REF] Das | Efficient mapping of cdfg onto coarse-grained reconfigurable array architectures[END_REF].

III. HETEROGENEOUS CLUSTER ARCHITECTURE

In this work, the PULP cluster is extended with the Integrated Programmable Array accelerator, as shown in Fig. 1. Fig. 2 shows a detailed block diagram of the subsystem embedding the IPA array. The IPA array is configured through a global context memory (GCM), responsible for storing locally the configuration bitstream of the PEs. The GCM is connected through a DMA-capable AXI-4 port to the cluster bus, enabling pre-fetching of IPA contexts from L2 memory. The GCM is considered twice the size of the configuration bitstream of the IPA in the worst case. In this way, it is possible to employ a double-buffering mechanism and load a new bitstream from the L2 to the GCM when the current one is being loaded on the array, completely hiding time for reconfiguration. More details on the structure of the IPA array bitstream can be found in [START_REF] Das | A 142MOPS/mW Integrated Programmable Array Accelerator for Smart Visual processing[END_REF]. A set of memory mapped control registers allow to load a new context to the IPA array, trigger the execution of a kernel and synchronize with the other processors in the cluster.

IPA
As opposed to many CGRA architectures, the IPA can access a multi-banked shared memory through 8 master ports connected to the low-latency interconnect. This eases data sharing with the other processors of the cluster, following the computational model described in [START_REF] Conti | Synthesis-friendly techniques for tightly-coupled integration of hardware accelerators into shared-memory multi-core clusters[END_REF]. The optimal number of port has been chosen to optimize the trade-off between the size of the interconnect and the bandwidth requirements of the IPA. Following the analysis conducted in [START_REF] Das | Efficient mapping of cdfg onto coarse-grained reconfigurable array architectures[END_REF], which shows that the IPA can operate 2× faster than the processors, we have extended the architecture of the cluster in a way that the IPA can work at twice the frequency of rest of the cluster. This approach allows to operate each component in the cluster at the optimal frequency, without paying the overheads of dual-clock FIFOs, requiring a significant amount of logic and synchronization overhead. On the contrary, the hardware support for the dual-frequency mode includes a clock divider to generate the two different edge aligned clocks, and two modules needed to adapt the request-grant protocol of the lowlatency interconnect [START_REF] Rahimi | A fully-synthesizable single-cycle interconnection network for shared-l1 processor clusters[END_REF] to deal with the frequency domain crossing, as shown in Fig. 3.

IV. SOFTWARE INFRASTRUCTURE

To offload jobs to the IPA and synchronize the execution, the cores access the control registers of the IPA, by memory mapped operations. The control registers are composed of a command register and a status register. We designed a simple Application Programming Interface (APIs) to perform the offload and synchronize tasks with the IPA. The main functions are described in Table I. Before execution starts in the IPA accelerator, the cores load the corresponding context and data from the L2 memory to the GCM and L1 memory, programming the system DMA and the IPA DMA, respectively. The context for the IPA consists of instructions and constants for the PEs, generated by the compilation flow proposed in [START_REF] Das | Efficient mapping of cdfg onto coarse-grained reconfigurable array architectures[END_REF]. The functions load data l2totcdm and load context l2togcm contain a set of routines to write data and context into TCDM and GCM respectively. The ipa start execution writes execute command into the command register of the IPA. The completion of the execution is notified by updating the status register. The core is synchronized with the IPA execution by calling the ipa check status function, which checks for the updates in the status register.

V. EXPERIMENTAL RESULTS

In this section we present the implementation results of the heterogeneous PULP cluster. The three possible modes considered in these comparisons are: (a) single-core: running applications in a single core, (b) ipa: running applications in the IPA where the core takes part in offloading only, (c) multi-core: running applications in parallel cores. All the benchmarks are coded in fully portable C, using the OpenMP programming model to express parallelism for PULP. In these benchmarks, matrix multiplication, convolution, FFT, FIR, separable filter, sobel filter are broadly used in near sensor image and multimedia applications. Sampling scheduler of the sensors strongly depends on computation of the greatest common divisor (GCD). Feature extraction in sensor networks widely uses CORDIC.

A. Implementation

The cluster consists of 8 cores featuring 4 kB of shared I$, one IPA with 16 PEs and a GCM of 4KB, while the TCDM is composed of 16 banks of 4 kB each, leading to an overall TCDM size of 64 kB. These architectural parameters were chosen to fit the constraints of the wide range of signal processing benchmarks presented in this paper. The SoC was synthesized with Synopsys Design Compiler 2013.12-SP3 on a STMicroelectronics 28nm UTBB FD-SOI technology library. Since, the achievable frequency of the PEs in the IPA is higher than the RI5KY cores used in the cluster, the IPA is clocked at 100 MHz, while the rest of the cluster runs at 50 MHz (in the SS, 0.6V, -40 • C corner). Synopsys PrimePower 2013.12-SP3 was used for timing and power analysis at the supply voltage of Although, the total area of the IPA with 16 PEs is almost similar to the area of the 8 cores combined, the area occupied by the GCM is much less than the total cache memory, which in turn provides better area efficiency while running applications in IPA.

B. Performance and Energy Efficiency

Table III reports the execution time in nano seconds for different benchmarks running on a single-core, on 8 cores and on the IPA. The IPA execution time includes the time taken for loading the context into the PEs. Comparing to the performance of execution in single-core, the accelerator achieves a maximum of 8× (with a minimum of 2.49× and an average of 5.4×) speed-up. The control intensive kernel like GCD does not exhibit parallelism, hence parallel software execution does improve performance of the homogeneous cluster. On the other hand, the execution on the IPA improves the performance by almost 5×, exploiting also instruction-level parallelism rather than data-level parallelism only. The performance gain in the accelerator for the compute intensive kernels like matrix multiplication, convolution, FIR and separable filters is limited if compared to the performance of parallel-cores. However, the relatively small performance gain compared to the parallel cluster is compensated by the gain in energy consumption (Table V) due to the simpler nature of the compute units of the IPA with respect to full processors, to the smaller number of power-hungry load/store operations (Table VI), and to the fine-grained power management architecture that allows clock gate the inactive PEs during execution (Table V).

Table IV presents the performance improvement of the IPA when moving from iso-frequency to the 2× frequency domain execution in the IPA. This shows that, although there is a reduction of memory bandwidth (see loss due to additional stalls column in Table IV), since the TCDM operates at the same frequency as the rest of the cluster (i.e. half frequency w.r.t. the IPA array), an average of 1.82× speed-up (with maximum of 1.92× and a minimum of 1.73×) can be achieved with this dual-frequency cluster architecture.

The power consumption profiles for the different modes of execution presented in Fig. 4, which shows the percentage of contribution by the several components in the cluster. Fig. 4 (a), (b), (c) represents the power breakdown while executing matrix multiplication in multi-core, single-core and IPA respectively, representative for other compute intensive benchmarks. Similarly, in this figure, (d) and (e) present the profiles for executing GCD, a control intensive benchmark, in single-core and IPA respectively. In Fig. 4 (a), (b), (c), the TCDM contributes to 14.7%, 15% and 7.2% in the multi-core, single-core and IPA configurations, respectively. The reduced memory access in IPA execution helps to achieve better energy efficiency. While executing GCD in single-core and the IPA, the TCDM consumed around 15.9% and 2.5% of the total power in the two analysed configurations, respectively. The simpler nature of the compute units, low burden on the TCDM and data exchange through PEs explains the energy gain of 7× in the IPA execution.

VI. CONCLUSION

In this paper, we present a novel approach towards heterogeneous computing, augmenting ultra-low power reconfigurable accelerator in the PULP multi-core cluster. The experiments integrating IPA in the PULP platform suggests that architectural heterogeneity is a powerful approach to improve energy profile of the computing systems. We have presented three possible executions of the benchmarks in the IPA integrated PULP platform. The heterogeneous cluster achieves achieving up to 4.8× speed-up and up to 4.4× better energy efficiency with respect to an 8-core homogeneous cluster.

Fig. 4 .

 4 Fig. 4. Power consumption breakdown in percentage: Executing Matrix-Multiplication in (a) Multi-Core; (b) Single-Core; (c) IPA. Executing GCD in (d) Single-Core; (e) IPA.

TABLE I .

 I LIST OF APIS FOR CONTROLLING IPA

	Function	Description
	void load data l2totcdm (int DMA CORE ID, int size, unsigned int l2 addr, unsigned int tcdm addr)	Writes data from L2 memory to the TCDM banks through DMA CORE
	void load context l2togcm
	(int DMA IPA ID,	Writes context from L2 memory
	int size, unsigned int l2 addr,	to the GCM through DMA IPA
	unsigned int gcm addr)
	int ipa start execution ()	Initiate IPA execution by writing in the command register
	void ipa check status(in id)	Core synchronization
	void free ipa (int id)	Release IPA

TABLE II

 II 25 • C temperature, in typical process conditions. TableIIpresents the area information of the components in the cluster.

	.	SYNTHESIZED AREA INFORMATION FOR THE PULP
		HETEROGENEOUS CLUSTER
	Components	Area (µm 2 )		% of cluster area
	CORES			160,352	18
	ICACHE			190,089	22
	DMA CORE			41,406	5
		Total		156,323
		IPAC		861
	IPA	GCM INTCNCT	359	18
		PE ARRAY	154,515
		OTHERS	588
	DMA IPA			32,636	4
	GCM			18,704	2
	TCDM			149,638	17
	CLUSTER INTCNCT		63,126	7
	CLUSTER PERIPHERALS		21,610	2
	OTHERS			37,932	4
	Total			871,816	100
	0.6V,

TABLE III

 III

		.	PERFORMANCE EVALUATION IN EXECUTION TIME (NS)
	FOR DIFFERENT CONFIGURATION IN THE HETEROGENEOUS PLATFORM
	Kernels	Data size (KB)	Single-core (ns)	Multi-core (ns)	Speed-up in multi-core (x)	IPA (ns)	Speed-up in IPA (x)
	MatMul	8	3,358,740	435,180	7.72	432,630	7.76
	Conv	8	9,733,380	1,520,840	6.4	1,494,860	6.51
	FFT	1	767,640	142,720	5.38	94,510	8.12
	FIR	0.84	182,500	33,460	5.45	33,410	5.46
	Sep Filter	10	39,870,420	6,404,160	6.23	6,334,700	6.29
	Sobel Filter	10	117,024,880	40,894,260	2.86	28,865,890	4.05
	GCD	0.01	2,951,160	2,951,160	1	61,1300	4.83
	Cordic	0.06	9,000	7,000	1.29	3,610	2.49
	Manh Dist	8	244,640	164,640	1.49	70,300	3.48

TABLE IV

 IV

	.	PERFORMANCE COMPARISON BETWEEN ISO-FREQUENCY
		AND 2× FREQUENCY EXECUTION IN IPA
	Benchmarks	#cycles in iso frequency	#cycles in 2x frequency	Loss due to stalls	overall execution speed-up
	MatMul		39,330	43,263	3,933	1.82
	Convolution	130,896	149,486	18,590	1.75
	FFT		8,182	9,451	1,269	1.73
	FIR		3,122	3,341	219	1.87
	Separable filter	575,882	633,470	57,588	1.82
	Sobel Filter	2,634,172	2,886,589	252,417	1.83
	GCD		58,573	61,130	2,557	1.92
	Cordic		328	361	33	1.82
	ManhDistance	6,391	7,030	639	1.82
	Average					1.82

TABLE V .

 V ENERGY CONSUMPTION EVALUATION IN µ J FOR DIFFERENT CONFIGURATION IN THE HETEROGENEOUS PLATFORM

	Kernels	Single-core	Multi-core	IPA	of
				Energy	Active
					PEs/cycle
	MatMul	1.247	0.313	0.208	58.5
	Convolution	2.876	1.095	0.658	59.2
	FFT	0.292	0.087	0.042	59.7
	FIR	0.08	0.026	0.026	46.1
	Separable filter	16.663	4.611	4.28	55.5
	Sobel Filter	51.491	29.444	12.701	51.2
	GCD	1.151	1.151	0.257	6.25
	Cordic	0.004	0.003	0.001	50
	ManhDistance	0.1	0.095	0.03	48.5

TABLE VI

 VI

	.	COMPARISON BETWEEN TOTAL NUMBER OF MEMORY
		OPERATIONS EXECUTED
	Benchmarks	multi-core	single-core	IPA
	MatMul	66,584	66,561	35,032
	Convolution	135,280	135,114	75,600
	FFT		12,528	11,733	6528
	FIR		5,904	5,893	3,990
	Separable filter	142,840	142,800	95,200
	Sobel Filter	148,240	148,224	120,000
	GCD		64,531	64,531	2
	Cordic		32	28	15
	ManhDistance	2,158	2,049	2,048

ACKNOWLEDGEMENTS

This work was funded by the ERC MultiTherman Project (ERC-AdG-291125), and by the OPRECOMP Project funded from the European Unions Horizon 2020 Research and Innovation Programme under grant agreement No. 732631.


T Miteva

S Kazandjian

N Sisourat

On the computations of decay widths of Fano resonances

In this paper we present an ab initio approach to the computation of decay widths of Fano resonances. The method relies on Fano theory, in which a resonance is described as a bound state embedded in and interacting with a continuum of states. In our approach, we use the Configuration Interaction (CI) method to describe the bound-like and continuum-like parts of the resonance wave function. The aim of this Fano-CI method is to provide decay widths of resonances at a low computational cost such that large systems can be treated. Along with the implementation of the method, we present benchmark calculations of decay widths of Auger and ICD processes in Ne atom, and Ne 2 and NeAr dimers. Our results are in good agreement with the decay widths from other theoretical and experimental works. This makes the Fano-CI approach a promising method for the treatment of Fano resonances.

I. INTRODUCTION

Resonances are metastable states, which decay in time into several subsystems. Among the different types of resonances, a Fano (also called Feshbach) resonance is described as a bound state embedded in one or more continua of an unperturbed Hamiltonian. The coupling due to the full Hamiltonian then turns the bound state into a metastable one, which as time passes breaks into two or more products [START_REF] Fano | [END_REF][START_REF] Moiseyev | Non-Hermitian Quantum Mechanics[END_REF][START_REF] Feshbach | [END_REF]. A clear demonstration of this resonance scattering phenomenon in atomic and molecular physics is the Auger effect [4]: after core ionisation of an atom or a molecule an outer valence electron of this excited ion fills the inner vacancy and the excess energy is used to eject a secondary electron (socalled Auger electron). The Auger effect in atoms has been thoroughly investigated for more than four decades (see [5] and references therein). However, studying the Auger effect in molecules and clusters remains challenging owing to the lack of spherical symmetry and the extra vibrational degrees of freedom.

In this context, Lenz Cederbaum and co-workers have made outstanding contributions to the development of Auger theory, its understanding as well as the computational treatment of this effect in molecules. A major contribution is the elaboration of the so-called Algebraic Diagrammatic Construction (ADC) method [6] (see [7] for a recent review), which allows for a direct and efficient access to the ionisation energies and spectral intensities for small to moderate size molecules. Using this method for doubly ionised molecules combined with a population analysis technique [8,9], they have shown that Auger spectra of molecules can be accurately computed, thus providing a powerful theoretical tool for the routine simulation and interpretation of the spectra of polyatomic systems. Another important aspect in the theoretical description of the Auger process is its lifetime, which is typically in the femtosecond time scale, and thus the process competes with the vibrational motion of light atoms within molecules. The nuclear dynamics must therefore be accounted for in order to provide accurate simulation of Auger processes. To this end, Lenz Cederbaum and co-workers have derived a full time-dependent quantum approach [10,11] which allows for the computation of experimental observables such as Auger electron distribution or kinetic energy of the ionic fragments formed after the Auger decay (the so-called kinetic-energy-release, KER, spectrum). Furthermore, this time-dependent approach is the starting point of other approximate methods that can be used to investigate large systems [12]. resonance phenomena in atomic and molecular systems. They discovered a whole new unsuspected kind of non-radiative decay processes, opening new perspectives in physics, chemistry and recently in biology. It started in 1997 when they revealed that an inner valence ionised molecule can release its excess energy by transferring it to a neighbouring molecule which is then ionised. This process was named Intermolecular Coulombic Decay (ICD) [13].

Since this pioneer work, ICD was shown to be a general decay mechanism for highly excited ions, atoms or molecules embedded in a chemical environment [14]. Following this discovery, Cederbaum and co-workers uncovered other related interatomic and intermolecular decay processes such as Electron Transfer Mediated Decay (ETMD) [15] or Interatomic Coulombic Electron Capture (ICEC) [16]. Beyond their fundamental importance, all these processes have attracted considerable and rising attention for several reasons. First, these processes are direct sources of low energy electrons and radical cations both of which play a central role in radiation chemistry and radiobiology (some examples of their importance in this context can be found in [17][18][19][20]). Furthermore, the processes generally involve the participation of the nearest neighbours and are strongly influenced by the distance and the orientation of the two systems. These properties may be used to probe the close chemical environment of a selected molecule, paving the way for original analytical tools.

An important characteristic of all these resonance phenomena is their energy width, or turning from the energy to the time domain, their lifetime. Accurate calculations of decay widths are a difficult task since one has to deal with both the many-body and the scattering nature of the problem. Various approaches have been developed for the accurate computations of decay widths of rather small systems [21][22][23][24]. More approximate techniques relying on electron population analysis have been devised for larger systems [25]. Several methods specifically designed for the calculations of (both intra-and intermolecular) decay widths of moderate size systems have been developed by Lenz Cederbaum and co-workers.

The first one is the Complex Absorbing Potential (CAP) method. The idea underlying the CAP method is to introduce an absorbing boundary condition in the exterior region of the system by an analytical continuation of the Hamiltonian, which makes the wave function of the scattered electron square integrable, i.e. representable in a L 2 basis. The CAP method has been used in combination with CI [26], ADC [27] and Equation-of-Motion Coupled-Cluster (EOM-CC) [28,29] methods. The second approach relies on the Fano theory of resonances combined with the Stieltjes imaging technique [30,31]. The decaying state is described as a discrete bound state embedded in and interacting with a continuum of states.

The bound state and an approximation of the continuum states are obtained from electronic structure methods. The Stieltjes imaging procedure is finally used to recover the correct normalisation of the continuum states. The simplest approach within this framework is the Wigner-Weisskopf method [30], which is based on the lowest order perturbation theory.

In this approach, the decaying and final electronic states are obtained using the restricted Hartree-Fock solution of the neutral systems. Relaxation and correlation effects are therefore neglected which makes the method only a semi-quantitative one. A more accurate approach relies on the ADC technique [31]. The bound state and an approximation to the continuum states are obtained from ADC calculations which effectively treat the electronic correlation and relaxation effects.

In this work, we describe a method lying between the Wigner-Weisskopf and the Fano-ADC methods. Its aim is to provide a quantitative estimate of decay widths at low computational cost such that large molecular systems can be considered. Our approach relies on the Configuration Interaction (CI) technique in restricted active spaces and we thus called it Fano-CI. In this article, we present in detail the implementation of the method and discuss some important numerical considerations for the efficient calculations of decay widths.

Finally, we benchmark the method against the available data on three well studied examples.

The paper is organised as follows. In the next section we outline the theoretical basis of the Fano-CI method. A derivation of the Fano-CI expressions is presented in Appendix A.

In Sec. III we present benchmark calculations of decay widths of Auger and ICD processes following core ionisation of Ne atom and inner valence ionisation of Ne in Ne 2 and NeAr, respectively. The summary and conclusions are given in Sec. IV.

II. METHODS

For completeness, we briefly outline the Fano theory of resonances following the derivation of Howat et al. [5]. Next, we present and discuss the Fano-CI method. We then sketch the Stieltjes procedure needed to extract a continuous approximation to the decay width from a discrete representation of the continuum. Finally, we discuss the choice of basis set and Hartree-Fock reference in the calculation of Fano-CI decay widths.

Similarly to the Fano-ADC method [31], the Fano-CI approach is devised for the treatment of Feshbach or Fano resonances. In the Fano formalism [START_REF] Fano | [END_REF]5], the wave function at energy E in the vicinity of the resonance is represented as a superposition of a discrete component, Φ, and continuum components, χ β, , corresponding to the N c decay channels

Ψ α,E = a α (E)Φ + Nc β=1 d C β, (E)χ β, , α = 1, . . . N c . (1)
Here the index β runs over the N c decay channels and is the kinetic energy of the outgoing electron. The bound part of the wave function, Φ, is assumed to be an isolated resonance, not interacting with other resonances. It has a mean energy

E Φ = Φ| Ĥ|Φ ( 2
)
where Ĥ is the full electronic Hamiltonian of the system. The continuum states χ β, , in their turn, are assumed to diagonalise the Hamiltonian

χ β, | Ĥ -E|χ β , ≈ δ β,β δ(E β -E β + -)(E β + -E) (3)
and are thus also non-interacting.

Solving the Schrödinger equation ( Ĥ -E)Ψ α,E = 0, one can determine the bound and continuum amplitudes, a α (E) and C β, (E), respectively [START_REF] Fano | [END_REF]5]. From the expression for |a α (E)| 2 [5], one obtains the relationship between partial Γ β and total widths

Γ = Nc β Γ β = 2π Nc β | Φ| Ĥ -E r |χ β, β | 2 (4)
where β is the asymptotic kinetic energy of the emitted electron and E r is the energy of the resonance. In practice, the energy of the resonance is approximated with the expectation value of the Hamiltonian with respect to the bound component of the resonance, i.e. E r ≈ E Φ [31]. This approximation is justified provided that the shift of the resonance, resulting from the interaction with the continuum, is smaller than the error in the calculation of the energy of the bound state [31].

As can be seen from Eq. ( 4), the total decay width is a sum of the partial decay widths to different decay channels. The expression for the partial decay widths given in ( 4) is known as "single channel" approximation, where no interaction between the decay channels is assumed.

The evaluation of expression (4) requires suitable approximations for the bound (Φ)

and continuum (χ β, β ) parts of the resonance. In the following, we discuss the particular approximations made in the Fano-CI method. The discussion is focused on a singly ionised resonance of a closed shell atom or molecule. However, the method can be generalised to n-tuply ionised states, as well as ionisation satellites and excited states of both closed and open shell species.

Let us consider the specific case of an electronic decay process following single ionisation of an N -electron atom or molecule. In this case, the bound part of the resonance state is the initial, singly ionised state of the system, whereas the continuum components comprise all doubly ionised states with an electron in the continuum. Normal Auger decay, ICD and

ETMD are examples of such electronic decay processes. Within the Fano-CI method, the initial singly ionised state is approximated as a one-hole (1h) configuration, i.e. an (N -1)electron Slater determinant, where an electron is removed from orbital

ϕ i | Φ = |Ψ i = c i |Ψ 0 (5)
Here, c i indicates the annihilation operator and |Ψ 0 denotes the Hartree-Fock ground state of the system. The continuum states |χ β, are approximated as discrete square integrable states | χa q , which are linear combinations of two-hole-one-particle (2h1p) configurations

|Ψ akl |χ β, ≈ | χa q = nocc k,l: k,l > i C q akl |Ψ akl (6)
Here, the indices k, l and a stand for occupied and virtual Hartree-Fock orbitals, respectively.

The 2h1p configurations |Ψ akl are characterised with two holes in the space of Hartree-Fock states {ϕ k } of energy k > i , and a particle in ϕ a . Note that in our implementation all 2h1p configurations entering the expansion of a given final state | χa q , have a fixed virtual orbital, ϕ a .

The coefficients C q akl are determined by solving the eigenvalue problem

Ĥ| χa q = E a q | χa q (7)
in the basis of the following doublet spin adapted configurations (or configuration state functions)

|Ψ S 1 akl = 1 √ 2 (c † a c k cl -c † a ck c l )|Ψ 0 (8) |Ψ S 2 akk = c † a c k ck |Ψ 0 (9) |Ψ T akl = 1 √ 6 (c † a c k cl + c † a ck c l + 2c † a c k c l )|Ψ 0 (10)
Here, the superscripts denote the singlet (S 1,2 ) and triplet (T ) spin states of the two holes.

A bar over the creation (c † ) or annihilation (c) operators stands for creation or annihilation of a spin orbital with spin β. A derivation of the expressions for the matrix elements of the Hamiltonian in the basis set of spin adapted configurations is given in Appendix A. The size of the resulting CI matrix is of the order of n 2 occ , where n occ is the number of occupied Hartree-Fock orbitals. In order to obtain all final states | χa q , one needs to solve the eigenvalue problem ( 7) n virt times, where n virt is the number of virtual orbitals. The multiple diagonalisation of a small matrix is a substantial reduction in the computational effort compared to, for example, the effort needed to diagonalise a single ADC(2e) matrix, where the size of the 2h1p block is of the order of n 2 occ n virt . The cost for the substantial reduction in the size of the Hamiltonian matrix in the Fano-CI method is that the coupling between the pseudocontinuum final states | χa q is completely neglected. The 2h1p configurations are however partially coupled through Eq. ( 6).

Finally, the states | χa q are used to compute the coupling matrix elements

γ a q = 2π| Φ| Ĥ -E| χa q | 2 (11)
These coupling matrix elements cannot be directly used in Eq. ( 4) to compute the decay width. However, the total decay width can be recovered from them employing the Stieltjes procedure described below.

C. Stieltjes procedure

The states | χa q computed using the Fano-CI method (Eq. ( 6)) cannot be directly associated with the true continuum wave functions. First of all, these states are normalised to unity rather than energy normalised, and as such they do not satisfy the appropriate scattering boundary conditions. Moreover, owing to the finite basis sets employed in practical calculations, these final states do not satisfy the energy conservation condition for a non-radiative decay process

E β + β = E r (12)
except by a coincidence.

These difficulties can be resolved by applying a mathematical approach known as the Stieltjes imaging technique [START_REF] Hazi | Electron-Molecule and Photon-Molecule Collisions[END_REF][START_REF] Langhoff | Electron-Molecule and Photon-Molecule Collisions[END_REF][START_REF] Müller-Plathe | [END_REF]. It relies on the fact that even though the square integrable (L 2 ) pseudocontinuum states cannot be used to compute the decay width directly, the spectral moments obtained from them are good approximations to the true spectral moments of the decay width.

S(k) = Nc β=1 dEE k | Φ| Ĥ -E r |χ β, β | 2 ≈ q a E k q | Φ| Ĥ -E r | χa q | 2 (13)
The moment theoretical approach is based on the observation that the pseudocontinuum L 2

wave functions approach the behaviour of the true continuum wave functions in the molecular interaction region provided that the basis set is sufficiently large to ensure a high density of pseudocontinuum states around the resonance energy. Usually, a series of calculations with an increasing number of spectral moments S(k) is performed until a consistent result is obtained [31][START_REF] Hazi | Electron-Molecule and Photon-Molecule Collisions[END_REF][START_REF] Langhoff | Electron-Molecule and Photon-Molecule Collisions[END_REF][START_REF] Müller-Plathe | [END_REF].

In practice it is difficult to obtain converged results without requiring a prohibitively large basis set. A question thus arises in the choice of the maximal Stieltjes order to be employed. In principle, higher orders provide more accurate results. However, owing to the finite basis sets, high order moments become inaccurate [START_REF] Müller-Plathe | [END_REF]. In the implementation of Fano-ADC method in Ref. [31], for example, the decay width is taken to be the average over a range of Stieltjes orders, for which it does not vary substantially. In this work, we computed the decay widths as averages over several Stieltjes orders, between 15 and 30 Γ = 1 15

30 n S =15 Γ(n S ) (14)
and we present the results with the respective standard deviation from this average value.

Since the Fano-CI method is devised for large systems, this naturally imposes a limit on the sizes of the basis sets used. The use of small basis sets, however, results in a non-stationary behaviour of the computed decay width as a function of the Stieltjes order. By computing the standard deviation for each averaged decay width (Eq. ( 14)), we can give an estimate of the error resulting from the Stieltjes imaging procedure. As shown below, this error is small even for moderate basis set size.

D. Choice of basis sets and Hartree-Fock reference

The choice of basis set is crucial for the accurate calculation of decay widths as the basis set has to provide a reliable description of both the discrete state and the continuum region of interest. This is usually achieved by employing large Gaussian basis sets (see e.g. [20,31,[35][36][37][38][39]), which are a combination of the standard basis sets and diffuse or compact basis functions centred around the atom or molecule. In the case of ICD due to the low energy of the emitted electrons, the most commonly used basis sets include the standard Gaussian basis sets augmented with a large number of diffuse basis functions specifically designed for the description of Rydberg and continuum states [40]. In contrast, the Auger electrons are much faster and in order to describe them the standard Gaussian basis sets are often uncontracted or augmented with compact basis functions [20,31], which ensures a non-zero density of states in the high energy region of the continuum. In our benchmark calculations of ICD widths we augmented the standard Gaussian basis sets with diffuse functions of the Kaufmann-Baumeister-Jungen (KBJ) type [40] on the atomic centres. In the case of Auger decay, we adopted a different approach: instead of uncontracting the basis set, we augmented it with sets of compact even tempered basis functions.

To construct the bound and continuum parts of the resonance state in the Fano-CI method, one needs to provide Hartree-Fock orbital energies and two-electron integrals as input. In the case of a singly ionised decaying state considered here, the ionisation step is accompanied by orbital relaxation effects, which are not accounted for in the Hartree-Fock calculation on the neutral system. These effects can be included in two ways. First, one can perform a restricted open-shell Hartree-Fock (ROHF) calculation, in which one can force an electron to be removed from the Hartree-Fock orbital of interest. Subsequently, the orbital energies and two-electron integrals generated in the ROHF step can be used in the Fano-CI procedure described in Sec. II B and Appendix A. Even though the expressions in Appendix A are not the rigorous analytical expressions for a ROHF reference function, the obtained decay widths are in good agreement with other theoretical results as we show later (see Sec.

III).

The second way to account for orbital relaxation effects is to employ the so-called core equivalent or (Z + 1) approximation. It relies on the assumption that valence electrons are affected by the ionisation of a core electron in the same way as they would be if a proton is added to the nucleus. This approximation is justified only for core ionised states and is known to give fairly accurate estimates of core electron binding energies [41,42]. Using this approximation, the Hartree-Fock orbital energies and two-electron integrals needed to compute the decay width of a core ionised state of an atom with atomic charge Z are obtained from a closed shell calculation on the singly ionised atom with atomic charge (Z + 1). As we show in the next section, the decay widths obtained using the core equivalent approximation are in good agreement with other works.

III. RESULTS AND DISCUSSION

In this section, we evaluate the accuracy of the Fano-CI method by presenting benchmark calculations of electronic decay widths. Since the Fano-CI method is a general one, it can be applied both to intraatomic processes, such as the Auger decay, and to interatomic processes, such as ICD and ETMD. Therefore, we test the accuracy of the method by computing decay widths for the Auger and ICD processes, and comparing our results with available literature values. We chose the Ne + (1s -1 ) resonance in an isolated Ne atom, and the Ne + (2s -1 ) resonance in Ne 2 and NeAr dimers as our benchmark cases. This choice was incited by the existence of extensive theoretical and experimental studies on the processes following core and inner valence ionisation of Ne [15,26,[43][44][45][46][47][48][49][50][51].

The aim of our benchmark calculations is to compare the accuracy of the Fano-CI method with other methods for the computation of decay widths, in particular, the Fano-ADC(2e) method. Moreover, since the Fano-CI method is primarily intended to describe resonances in large systems for which basis sets of limited size must be employed, a thorough investigation of the dependence of the widths with respect to the basis sets was performed.

A. Auger decay in Ne atom

Let us first consider the decay of the 1s 1 2s 2 2p 6 2 S state of Ne + . The only possible nonradiative relaxation pathway in this case is the KLL Auger decay, and, consequently, the total non-radiative decay width corresponds to the Auger width. The Auger decay leads to the population of the doubly ionised 2s -2 1 S, 2s -1 2p -1 3 P , 2s -1 2p -1 1 P , 2p -2 1 S, 2p -2 1 D states and to the ejection of an Auger electron of kinetic energy around 818 eV [START_REF] Wagner | Faraday Discuss[END_REF]. The Ne + (2s -1 )Ne b Ref. [43] c CAP/CI method, [26] d Ref. [44] e Wigner-Weisskopf theory, Ref. [15] f Electron dynamics calculations, Ref. [46] g Ref. [53] h Experiment, Ref. [54,55] TABLE I: Comparison of the total Auger and ICD widths (meV) computed using the Fano-CI method with theoretical and experimental values for different benchmark systems -Ne 2 , NeAr, Ne. In the case of Ne 2 and NeAr the decay widths are taken at the equilibrium interatomic distances: 3.2 Å and 3.5 Å, respectively. The ICD widths are computed using the cc-pVQZ basis set augmented with 7s, 7p and 7d diffuse basis functions of the KBJ type in the case of Ne 2 and 9s, 9p, 9d functions in the case of NeAr [56,57]. The Auger width of Ne(1s -1 ) is computed using the cc-pVQZ basis set [56] augmented with a set of 7s, 7p, 7d even tempered basis functions.

decay width of the Ne + (1s -1 ) resonance was computed following the procedure described in Sec. II. However, we also included the orbital relaxation effects as described in Sec. II D by performing a restricted open-shell HF calculation on the singly ionised atom (ROHF), where we forced an electron to be removed from the 1s orbital, and a HF calculation on the closedshell Na + ion. The Hartree-Fock solution for the neutral and singly ionised systems was obtained using the GAMESS-US computational package [60], and the cc-pVDZ, cc-pVTZ and cc-pVQZ basis sets [61] augmented with n sets of s, p, d even tempered basis functions,

where n = 1 -7 [START_REF]The exponents of the diffuse s-, p-and d-functions ranged between 100.0 and 1.0[END_REF], thus ensuring sufficient density of states in the high energy part of the continuum.

The Hartree-Fock orbital energies and two-electron integrals were subsequently used to -1 ) computed using the Fano-CI method and the cc-pVDZ (red), cc-pVTZ (green) and cc-pVQZ (blue) basis sets augmented with additional sets of ns, np and nd even tempered basis functions [START_REF]The exponents of the diffuse s-, p-and d-functions ranged between 100.0 and 1.0[END_REF]. The decay widths shown in the lowermost panel were computed using the Hartree-Fock solution for the neutral atom (RHF). The middle panel shows the decay width computed employing the core equivalent approximation, i.e. the solution for Na + . Finally, the total Auger widths presented in the uppermost panel were computed from the Hartree-Fock solution for the singly ionised atom (ROHF).

compute the Auger decay width in the procedure described in Sec. II and Appendix A. The initial state is constructed as an (N -1) Slater determinant, where an electron is removed from the 1s orbital. Then we include the remaining occupied 2s and 2p orbitals in the space of one-particle states, out of which all possible two-hole combinations are formed. Subsequently, we construct and diagonalise the Hamiltonian matrix in the space of the 2h1p configurations as described in Sec. II B.

The results of the three different approaches to the computation of the Auger width are presented in Fig. 1. As the basis set increases, the Auger width of the Ne + (1s -1 ) resonance converges to 152 meV in the RHF case, to 199 meV (Z + 1) case and 298 meV in the ROHF case. The Stieltjes errors are 7 meV, 14 meV and 13 meV, respectively, i.e. smaller than 10%.

The latter two decay widths, computed from the Hartree-Fock solutions for Na + and for the singly ionised Ne atom, are in better agreement with the experimental and theoretical natural linewidths of 220 meV [54,55] and 251 meV [53], respectively (see Table I). Consequently, accounting for the orbital relaxation upon core ionisation, results in a more accurate decay width without increasing the computational overhead.

Fig. 1 also shows the behaviour of the computed decay width as a function of the number of additional sets of even tempered basis functions. A convergence of the Auger width with respect to the basis set augmentation is achieved already for n = 5 for all considered basis sets (cc-pVXZ, X = D, T, Q). In the case of the largest basis set, cc-pVQZ, this result is even more pronounced as the convergence is observed already for 2s, 2p and 2d compact Gaussian functions. Therefore, in the case of Auger decay, in order to obtain a satisfactory result for the decay width, one does not need a very high density of states in the continuum energy region of interest. This makes the Fano-CI method very promising in the treatment of large systems.

B. ICD in Ne 2 and NeAr

Let us now consider the electronic decay processes following inner valence ionisation of Ne in Ne 2 and NeAr. In this case, the intraatomic decay is energetically forbidden. Among the interatomic decay processes, ICD and ETMD are the only energetically allowed relaxation pathways [15,48]. The ETMD channel is open only in the NeAr dimer. Consequently, the total non-radiative decay width of the Ne + (2s -1 ) vacancy corresponds to the ICD width in the case of Ne 2 and to the sum of the ICD and ETMD widths in the case of NeAr. Note that according to Ref. [15], the ETMD contribution to the total decay width at the equi- u state computed at the equilibrium internuclear distance R eq = 3.09 Å [62] using the Fano-CI method and the cc-pVDZ (red), cc-pVTZ (green) and cc-pVQZ (blue) basis sets augmented with a set of ns, np, nd diffuse KBJ functions.The decay widths in the lower panel were computed from the Hartree-Fock solution for the neutral dimer (RHF), whereas for the calculation of the decay widths in the upper panel, we employed the Hartree-Fock solution for the singly ionised dimer (ROHF).

librium interatomic distance is ∼ 10 -4 smaller than ICD. The final states populated in the ICD process are two-site dicationic states -Ne + (2p -1 )Ne + (2p -1 ) and Ne + (2p -1 )Ar + (3p -1 ), whereas the final ETMD states in NeAr are characterised with two holes on the Ar atom:

NeAr 2+ (3p -2 ). The electrons emitted in the ICD process in Ne 2 have energies between 0 and 2 eV [51], whereas the electrons emitted in the decay of the Ne + (2s -1 )Ar resonance, have energies between 3 and 9 eV [15,45].

The Hartree-Fock solutions for both the neutral, and singly ionised dimers were obtained FIG. 3: Total decay width (meV) of the Ne + (2s -1 )Ar 2 Σ + state computed at the equilibrium interatomic distance R eq = 3.5 Å [63] using the Fano-CI method and the cc-pVDZ (red), cc-pVTZ (green) and cc-pVQZ (blue) basis sets augmented with a set of ns, np, nd diffuse KBJ functions.

The decay widths in the lower panel were computed from the Hartree-Fock solution for the neutral dimer (RHF), whereas for the calculation of the decay widths in the upper panel, we employed the Hartree-Fock solution for the singly ionised dimer (ROHF).

using the GAMESS-US computational package [60]. We used the cc-pVDZ, cc-pVTZ and cc-pVQZ [56,57] basis sets on all atoms. The standard atomic basis sets were augmented with sets of diffuse s, p and d functions of the KBJ type [40]. The (Z + 1) approximation is not justified in the case of inner valence ionisation and was therefore not used here.

Then using the HF results, we followed the procedure described in Sec. II. The initial inner valence ionised state Ne + (2s -1 )X (X = Ne, Ar) was constructed as a (N -1) Slater determinant, where an electron is removed from the Ne 2s orbital. In Ne dimer, the 2s The dashed brown line is the R-dependent decay width obtained using the 1/R 6 approximation (15). The atomic data of Refs. [64] and [65] was used for the evaluation of Eq. 15. A vertical black line shows the ground state equilibrium interatomic distance, R eq = 3.09 Å [62].

vacancy is split into two states: a repulsive 2 Σ + g state and a bound 2 Σ + u state, both of which undergo ICD [48]. We computed the decay width for both of these states at the equilibrium distance R eq = 3.09 Å [62]. The one-particle states asymptotically corresponding to the Ne 2p and Ar 3p states were used to construct the final states of the decay processes:

Ne + (2p -1 )Ne + (2p -1 ), Ne + (2p -1 )Ar + (3p -1 ), and NeAr 2+ (3p -2 ). Finally, the total decay widths were computed following the procedure described in Sec. II.

In Figs. 2 and3 we present the total non-radiative decay widths of the Ne + (2s -1 )Ne 2 2 Σ + u and Ne + (2s -1 )Ar 2 Σ + states, respectively, as a function of the basis set. Let us focus on the lower panels, which show the decay widths computed using the Hartree-Fock solution for the neutral dimers. As one can see, the decay widths converge to 10 meV in the case of Ne 2 and 19 meV in the case of NeAr. The Stieltjes errors evaluated for the largest basis sets are, respectively, 1 meV and 2 meV. An important distinction should be made between the two systems. Due to the lower ionisation potential of Ar (15.76 eV, compared to 21.56 eV of Ne [START_REF] Kramida | NIST Atomic Spectra Database (version 5.1)[END_REF]), the energy of the emitted ICD electron in NeAr is higher compared to Ne 2 [15,45,51]. In order to cover the higher energy part of the continuum, one needs a larger Gaussian basis set. Consequently, as can be seen from Figs. 2 and3, the decay width of the Ne + (2s -1 )Ne 2 2 Σ + u state converges already upon the addition of a set of 7s, 7p, 7d diffuse function, whereas in the case of the Ne + (2s -1 )Ar 2 Σ + state, one has to augment the basis sets further.

Let us now consider the upper panels of Figs. 2 and3, where we show the basis set dependence of the total non-radiative decay widths of the Ne + (2s -1 )Ne 2 2 Σ + u and Ne + (2s -1 )Ar 2 Σ + states computed using the HF calculation on the singly ionised dimers. The total nonradiative decay widths converge to 7 meV in the case of Ne 2 and to 16 meV in the case of NeAr. Using a ROHF reference, results in a larger Stieltjes error: 4 meV and 5 meV for Ne 2 and NeAr, respectively, for the largest basis sets. In the case of an interatomic decay process, initiated by inner valence ionisation, the relaxation effects are smaller compared to the core ionisation. Consequently, accounting for the orbital relaxation by computing the decay width using the Hartree-Fock solution for the singly ionised dimers, does not lead to a substantial improvement of the result. Unlike the Auger case, the decay width converges slower, and the Stieltjes error is larger (∼50% in contrast to ∼4% in the case of Auger decay). This indicates that in the case of ICD one needs a high density of states in the low energy continuum, whereas to describe the Auger process, it is sufficient to have a few states in the high energy part of the continuum.

In Table I we present a comparison between the Fano-CI decay widths of the Ne + (2s -1 ) resonance in Ne 2 and NeAr, and the values obtained in other theoretical works. The Fano-CI decay widths are of the same order as those computed employing the Fano-ADC(2e) method. The Fano-ADC(2e) method is expected to be more accurate as it provides a better description of the initial state and fully accounts for the coupling between the final states. Furthermore, the Fano-ADC(2e) decay widths presented in Table I, were computed using large basis sets [43,44]. In the case of Ne 2 , the cc-pVQZ basis set was augmented with a set of 3s, 3p, and 3d Gaussian basis functions on each atom and 6 additional sets of 3s, 3p, and 3d Gaussians were distributed around the dimer [43], in the case of NeAr again the same basis set was used but the number of additional Gaussians was increased to 4s, 4p, 4d [44].

The comparison between the Fano-CI and Fano-ADC(2e) results shows that accurate decay widths can be obtained with the "cheaper" Fano-CI method combined with moderate basis sets, which opens the possibility to investigate decay processes in large systems.

We tested the Fano-CI approach further by computing the ICD width as a function of the distance R between the atoms in the neon dimer. The resulting R-dependent decay widths for both the 2 2 Σ + g and 2 2 Σ + u resonances are shown in Fig. 4. As expected for the interatomic decay process, the decay width decreases monotonically as the distance increases. At large distances, typically R ≥ R eq , the ICD width can be evaluated analytically with the virtual photon transfer mechanism [START_REF] Averbukh | [END_REF][68][69]

Γ(R) = 3h 3π c ω 4 τ -1 rad σ R 6 (15)
where τ rad is the radiative lifetime of the Ne 2s vacancy, σ is the total ionisation cross section at the virtual photon energy hω. In order to evaluate Eq. ( 15), we used the atomic data from references [64,65]. The virtual photon transfer result is also presented in Fig. 4. As can be seen from the figure, the R-dependent decay width computed using the Fano-CI method reproduces well the 1/R 6 dependence. However, similarly to the Fano-ADC(2e) method (see Ref. [43]), there is a discrepancy between the computed value and the one obtained by evaluating (15) at asymptotic distances.

IV. CONCLUSIONS

In conclusion, we have presented a computationally efficient method for calculating decay The Hamiltonian matrix in the basis of 2h1p spin eigenfunctions Eq. ( 8)-( 10) is given in

Ψ S 1 akl | Ĥ -E 0 |Ψ i = 1 √ 2 ai|lk + ai|kl Ψ S 2 akk | Ĥ -E 0 |Ψ i = ai|kk Ψ T akl | Ĥ -E 0 |Ψ i = 3 2 ai|lk -ai|kl
The following file(s) will not be converted:

50 FIG. 1 :

 501 FIG.1: Total Auger decay widths (eV) of Ne + (1s -1 ) computed using the Fano-CI method and the

FIG. 2 :

 2 FIG.2: Total decay width (meV) of the Ne + (2s -1 )Ne 2 2 Σ + u state computed at the equilibrium

FIG. 4 :

 4 FIG.4: Double logarithmic plot of the total non-radiative decay width of the Ne + (2s -1 )Ne 2 2 Σ + g (red circles) and 2 2 Σ + u (blue squares) states as a function of the internuclear distance R. The decay width was computed using the cc-pVQZ basis set augmented with 7s, 7p, 7d diffuse KBJ functions.

  widths of Fano resonances (such as in Auger, ICD and related processes). The method relies on the Fano theory of resonances which describes the resonance state as a discrete bound state embedded in and interacting with a continuum of states. Approximation of the bound and continuum parts are obtained from small CI calculations in restricted active spaces and with moderate basis set size. Benchmark examples show that the method provides accurate decay widths. The present method thus opens new perspectives in the study of Fano resonances in large atomic and molecular systems.

  This project has received funding from the Research Executive Agency (REA) under the European Union's Horizon 2020 research and innovation programme Grant agreement

Fig. 5 . 1 2 2 ka|ka -ka|ak δ kk δ ll - 1 2 2+ -al |al + 3 2 2 - 2 ak|al + ak|la δ kk 2 . C S 1 Tak|ka δ kk δ ll 3 .

 5122222213 Fig.5. From expression Eq. (A1) using the doublet 2h1p spin eigenfunctions (8)-(10), one can derive spin free working expressions for the matrix elements pertaining to the different blocks of the matrix. For the main 2h1p blocks of the Hamiltonian matrix, one obtains

  fig1_ne_auger.eps fig2_nene_icd_sigmau.eps fig3_near_icd.eps fig4_nene_icd_r_qz_7s7p7d.eps

All two-electron integrals are given in the physicists' notation.

In order to construct the final states, one needs to construct and diagonalise the Hamiltonian matrix in the basis of the possible 2h1p configurations for a fixed particle orbital.

In the following, we will briefly outline this procedure as well as the computation of the coupling elements used in the Stieltjes method.

Let us consider the following partitioning of the Hamiltonian

where p denote the Hartree-Fock one-electron energies, V pqrs denote the two-electron integrals in the basis of Hartree-Fock orbitals. The quantities w pq , constituting the non-diagonal one-electron part, are expressed in terms of two-electron integrals

where n r is the occupation number of the rth orbital. Using the above notations, one can derive the matrix element between two 2h1p configurations in a general form (see Eq. A12 in [70])

where a, a denote virtual and k, l, k , l are occupied Hartree-Fock orbitals, respectively; E 0 is the ground state Hartree-Fock energy.

5: Block structure of the Hamiltonian matrix H in the basis set of doublet 2h1p spin eigenfunctions Eq. ( 8)- (10). The main and coupling blocks are denoted as M and C, respectively.