Structure of phosphorus clusters using simulated annealing—P2 to P8

The Quest for Stable Borozene Core in Main-Group Capped Inverse Sandwich Complexes, [(HE)2B6H6]2- (E=B, Al, Ga, In, and Tl)

The ubiquitous chemistry of benzene led us to explore ways to stabilise analogous borozene, by capping them with appropriate groups. The mismatch in overlap of ring-cap fragment molecular orbitals in [(HB)2B6H6]2- is overcome by replacing the two BH caps with higher congeners of boron. We calculated the relative energies of all the polyhedral structural candidates for [(HE)2B6H6]2- (E=Al-Tl) and found hexagonal bipyramid (HBP) to be more stable with Al-H caps. A global minimum search also gives HBP as the most stable structure for [Al2B6H8]2-. The capped B6H6 ring in [(HAl)2B6H6]2- has aromaticity comparable to that of benzene.

Origins of Severe Structural Changes during Alloying-Dealloying Reactions in Black Phosphorus

Li-alloying reactions facilitate the incorporation of a large number of Li atoms into the crystalline structures of electrodes, such as black phosphorus (BP). However, the reactions inevitably induce multistep phase transitions characterized by drastic atomic rearrangements and lattice collapse. Despite many theoretical and experimental studies on alloying mechanisms, long-term debates persist regarding the structures of the intermediate phases, the accurate pathways of phase transitions, the formation of specific configurations, and alloying/dealloying reversibility. Here, through a combination of operando electron diffraction measurements and ab initio simulations at the atomic and electronic scales, we identify key factors that govern the severe structural changes during alloying-dealloying reactions in BP. P-P bonds of three-bond P atoms are continuously broken during lithiation, generating two-bond P atoms with a high ability to accept inserted electrons and Li ions. Consequently, the pristine layered structure in BP is transformed to P7 cages in Li3P7, which then evolve to chain configurations in LiP and finally to isolated P atoms in Li3P. Specifically, the preferential formation of the P7 cage results from its lowest binding energy with three Li ions compared to other cage isomers. Furthermore, only LiP can be reversibly transformed to the crystalline structure of Li3P7 during charge, but it is thermodynamically favorable for Li3P7 and Li3P intermediates to be delithiated to amorphous structures. Our findings offer unique insights into the alloying mechanisms and deepen the fundamental understanding of alloying anode systems.

PGA: A new particle swarm optimization algorithm based on genetic operators for the global optimization of clusters

We have developed a global optimization program named PGA based on particle swarm optimization algorithm coupled with genetic operators for the structures of atomic clusters. The effectiveness and efficiency of the PGA program can be demonstrated by efficiently obtaining the tetrahedral Au20 and double-ring tubular B20, and identifying the ground stateZrSi 17 - 20 - $$ {\mathrm{ZrSi}}_{17\hbox{--} 20}^{-} $$ clusters through the comparison between the simulated and the experimental photoelectron spectra (PESs). Then, the PGA was applied to search for the global minimum structures ofMg n - $$ {\mathrm{Mg}}_n^{-} $$ (n = 3-30) clusters, new structures have been found for sizes n = 6, 7, 12, 14, and medium-sized 21-30 were first determined. The high consistency between the simulated spectra and the experimental ones once again demonstrates the efficiency of the PGA program. Based on the ground-state structures of theseMg n - $$ {\mathrm{Mg}}_n^{-} $$ (n = 3-30) clusters, their structural evolution and electronic properties were subsequently explored. The performance on Au20, B20,ZrSi 17 - 20 - $$ {\mathrm{ZrSi}}_{17\hbox{--} 20}^{-} $$ , andMg n - $$ {\mathrm{Mg}}_n^{-} $$ (n = 3-30) clusters indicates the promising potential of the PGA program for exploring the global minima of other clusters. The code is available for free upon request.

Probing charge redistribution at the interface of self-assembled cyclo-P5 pentamers on Ag(111)

Phosphorus pentamers (cyclo-P5) are unstable in nature but can be synthesized at the Ag(111) surface. Unlike monolayer black phosphorous, little is known about their electronic properties when in contact with metal electrodes, although this is crucial for future applications. Here, we characterize the atomic structure of cyclo-P5 assembled on Ag(111) using atomic force microscopy with functionalized tips and density functional theory. Combining force and tunneling spectroscopy, we find that a strong charge transfer induces an inward dipole moment at the cyclo-P5/Ag interface as well as the formation of an interface state. We probe the image potential states by field-effect resonant tunneling and quantify the increase of the local change of work function of 0.46 eV at the cyclo-P5 assembly. Our experimental approach suggest that the cyclo-P5/Ag interface has the characteristic ingredients of a p-type semiconductor-metal Schottky junction with potential applications in field-effect transistors, diodes, or solar cells.

Structure search method for atomic clusters based on the dividing rectangles algorithm

The Dividing Rectangles (DIRECT) algorithm is a deterministic optimization method to explore optimal solutions by repeatedly dividing a given hyperrectangle search space into subhyperrectangles. Herein, we propose a structure search method for atomic clusters based on the DIRECT algorithm in combination with a gradient-based local optimizer to enable an efficient structure search in high-dimensional search spaces. We use the Z-matrix representation for defining the hyperrectangle search space, in which the bond lengths, bond angles, and dihedral angles specify a cluster structure. To evaluate its performance, we applied the proposed method to the Lennard-Jones clusters and two kinds of real atomic clusters with many metastable structures, i.e., phosphorus and sulfur clusters, and compared the results with those of conventional methods. The proposed method exhibits a higher efficiency than random search and a comparable efficiency to basin hopping.

Renaissance of elemental phosphorus materials: properties, synthesis, and applications in sustainable energy and environment

The polymorphism of phosphorus-based materials has garnered much research interest, and the variable chemical bonding structures give rise to a variety of micro and nanostructures. Among the different types of materials containing phosphorus, elemental phosphorus materials (EPMs) constitute the foundation for the synthesis of related compounds. EPMs are experiencing a renaissance in the post-graphene era, thanks to recent advancements in the scaling-down of black phosphorus, amorphous red phosphorus, violet phosphorus, and fibrous phosphorus and consequently, diverse classes of low-dimensional sheets, ribbons, and dots of EPMs with intriguing properties have been produced. The nanostructured EPMs featuring tunable bandgaps, moderate carrier mobility, and excellent optical absorption have shown great potential in energy conversion, energy storage, and environmental remediation. It is thus important to have a good understanding of the differences and interrelationships among diverse EPMs, their intrinsic physical and chemical properties, the synthesis of specific structures, and the selection of suitable nanostructures of EPMs for particular applications. In this comprehensive review, we aim to provide an in-depth analysis and discussion of the fundamental physicochemical properties, synthesis, and applications of EPMs in the areas of energy conversion, energy storage, and environmental remediation. Our evaluations are based on recent literature on well-established phosphorus allotropes and theoretical predictions of new EPMs. The objective of this review is to enhance our comprehension of the characteristics of EPMs, keep abreast of recent advances, and provide guidance for future research of EPMs in the fields of chemistry and materials science.

Phosphorus nanoclusters and insight into the formation of phosphorus allotropes

Elemental phosphorus has a striking variety of allotropes, which we analyze by looking at stable phosphorus clusters. We determine the ground-state structures of P_n clusters in a wide range of compositions (n = 2-50) using density functional calculations and global optimization techniques. We explain why the high-energy white phosphorus is so easily formed, compared to the much more stable allotropes - the tetrahedral P₄ cluster is so much more stable than nearby compositions that only by increasing the size to P₁₀ one can get a more stable non-P₄-based structure. Starting from 17 atoms, phosphorus clusters have a single-stranded structure, consisting of a set of well-resolved structural units connected by P₂ linking fragments. The investigation of relative stability has revealed even-odd alternations and structural magic numbers. The former are caused by the higher stability of clusters with even numbers of atoms due to closed electronic shells. The structural magic numbers are associated with the presence of particular stable structural units and lead to enhanced stability of P_18+12k (k = 0, 1, 2) clusters. We also compare the energies of the obtained ground-state structures with clusters of different phosphorus allotropes. Clusters of fibrous phosphorus are energetically the closest to the ground states, white phosphorus clusters are found to be less stable, and the least stable allotrope at the nanocluster scale is black phosphorene.

Cluster Fragments in Amorphous Phosphorus and their Evolution under Pressure

Amorphous phosphorus (a-P) has long attracted interest because of its complex atomic structure, and more recently as an anode material for batteries. However, accurately describing and understanding a-P at the atomistic level remains a challenge. Here, it is shown that large-scale molecular-dynamics simulations, enabled by a machine-learning (ML)-based interatomic potential for phosphorus, can give new insights into the atomic structure of a-P and how this structure changes under pressure. The structural model so obtained contains abundant five-membered rings, as well as more complex seven- and eight-atom clusters. Changes in the simulated first sharp diffraction peak during compression and decompression indicate a hysteresis in the recovery of medium-range order. An analysis of cluster fragments, large rings, and voids suggests that moderate pressure (up to about 5 GPa) does not break the connectivity of clusters, but higher pressure does. The work provides a starting point for further computational studies of the structure and properties of a-P, and more generally it exemplifies how ML-driven modeling can accelerate the understanding of disordered functional materials.

Red Phosphorus: An Up-and-Coming Photocatalyst on the Horizon for Sustainable Energy Development and Environmental Remediation

Photocatalysis is a perennial solution that promises to resolve deep-rooted challenges related to environmental pollution and energy deficit through harvesting the inexhaustible and renewable solar energy. To date, a cornucopia of photocatalytic materials has been investigated with the research wave presently steered by the development of novel, affordable, and effective metal-free semiconductors with fascinating physicochemical and semiconducting characteristics. Coincidentally, the recently emerged red phosphorus (RP) semiconductor finds itself fitting perfectly into this category ascribed to its earth abundant, low-cost, and metal-free nature. More notably, the renowned red allotrope of the phosphorus family is spectacularly bestowed with strengthened optical absorption features, propitious electronic band configuration, and ease of functionalization and modification as well as high stability. Comprehensively detailing RP's roles and implications in photocatalysis, this review article will first include information on different RP allotropes and their chemical structures, followed by the meticulous scrutiny of their physicochemical and semiconducting properties such as electronic band structure, optical absorption features, and charge carrier dynamics. Besides that, state-of-the-art synthesis strategies for developing various RP allotropes and RP-based photocatalytic systems will also be outlined. In addition, modification or functionalization of RP with other semiconductors for promoting effective photocatalytic applications will be discussed to assess its versatility and feasibility as a high-performing photocatalytic system. Lastly, the challenges facing RP photocatalysts and future research directions will be included to propel the feasible development of RP-based systems with considerably augmented photocatalytic efficiency. This review article aspires to facilitate the rational development of multifunctional RP-based photocatalytic systems by widening the cognizance of rational engineering as well as to fine-tune the electronic, optical, and charge carrier properties of RP.

Mechanism for the Reaction of White Phosphorus with Cp2Cr2(CO)6 Leading Ultimately to the Triple-Decker Sandwich Cp2Cr2(μ-η5,η5-P5): A Theoretical Study

The experimentally known reaction of Cp₂Cr₂(CO)₆ with white phosphorus (P₄) to give CpCr(CO)₂(η³-P₃), Cp₂Cr₂(CO)₄(μ-η,²η²-P₂), and the triple-decker sandwich Cp₂Cr₂(μ-η,⁵η⁵-P₅) is of interest since the P₄ reactant having a tetrahedral cluster of four phosphorus atoms is converted to products having P₂, P₃, and P₅ ligands. The mechanism of this obviously complicated reaction can be dissected into three stages using a coupled cluster theoretical method that has been benchmarked with the P₂, Mn(CO)₅, and CpCr(CO)₃ dimerization processes. The first stage of the Cp₂Cr₂(CO)₆/P₄ reaction mechanism generates the unsaturated singlet intermediate Cp₂Cr₂(CO)₅ that combines with the P₄ reactant. Decarbonylation of the resulting Cp₂Cr₂(CO)₅(P₄) complex provides a singlet tetracarbonyl readily fragmenting into the stable triphosphacyclopropenyl complex CpCr(CO)₂(η³-P₃) and the chromium phosphide CpCr(CO)₂(P). The isomeric triplet tetracarbonyl Cp₂Cr₂(CO)₄(P₄), readily fragments into CpCr(CO)₂(η²-P₂), which can generate the stable diphosphaacetylene complex Cp₂Cr₂(CO)₄(η,²η²-P₂) as well as the pentamer [CpCr(CO)₂]₅(P₁₀). Combination of the coordinately unsaturated CpCr(CO)(η³-P₃) with CpCr(CO)₂(η²-P₂) can lead to a ring expansion. This generates the P₅ pentagonal ligand in a Cp₂Cr₂(CO)₃(P₅) precursor to the experimentally observed carbonyl-free triple-decker sandwich Cp₂Cr₂(μ-η,⁵η⁵-P₅) after three successive decarbonylations.

Detecting Dissociation Dynamics of Phosphorus Molecular Ions by Atom Probe Tomography

Dissociation processes involving phosphorus cations were investigated during laser-assisted atom probe tomography of crystalline indium phosphide (InP). This technique not only allows the formation of medium-sized phosphorus cations by means of femtosecond laser pulses under ultrahigh vacuum and high electric field conditions but also allows one to study the time-resolved dissociation dynamics. Data reveal the formation of cations up to P₂₃²⁺ and their subsequent dissociation into two smaller P_k⁺ cations (k > 2). The use of a time- and position-sensitive detector combined with numerical calculations provided information related to the molecule orientation, decay time, and kinetic energy release during dissociation phenomena. Results suggest that the dissociation processes are most likely due to the emission of P_k²⁺ cations in excited states and their subsequent decay in low field regions during their flight toward the detector. This study provides operative guidelines to obtain information on dissociation processes using a tomographic atom probe as a reaction microscope and indicates the current capabilities and limitations of such an approach.

Ozone generation mechanism over phosphorus and the impact of H2O with density functional modeling

The generation of P–O was a continuous process but was not conducive to generate O₃ and a free O atom.

Direct observation of thermal disorder and decomposition of black phosphorus

Theoretical research has been devoted to reveal the properties of black phosphorus as a two-dimensional nanomaterial, but little attention has been paid for the experimental characterization. In this study, the thermal disorder and decomposition of black phosphorus were examined using in situ heating transmission electron microscopy experiments. We observed that the breaking of crystallographic symmetry begins at 380 °C under vacuum condition, followed by the phosphorus evaporates after long-term heating at 400 °C. This decomposition process can be initiated by the surficial vacancy and proceeds toward both interlayer ([010]) and intralayer ([001]) directions. The results on the thermal behavior of black phosphorus provide useful guidance for thin film deposition and fabrication processes with black phosphorus.

Computational Assessment of an Elusive Aromatic N3P3 Molecule

We computationally proved that the planar aromatic hexagonal isomer N₃P₃ with the alteration of N and P is the second most stable structure for the N₃P₃ stoichiometry. We found that the aromatic isomer has high barriers for transition into the global minimum structure or into the three isolated NP molecules, making this structure kinetically stable. We showed that the sandwich N₃P₃CrN₃P₃ molecule corresponds to a minimum on the potential energy surface; thus, the aromatic N₃P₃ molecule has a potential to be a new ligand in chemistry.

Development of a Transferable Reactive Force Field of P/H Systems: Application to the Chemical and Mechanical Properties of Phosphorene

We developed ReaxFF parameters for phosphorus and hydrogen to give a good description of the chemical and mechanical properties of pristine and defected black phosphorene. ReaxFF for P/H is transferable to a wide range of phosphorus- and hydrogen-containing systems including bulk black phosphorus, blue phosphorene, edge-hydrogenated phosphorene, phosphorus clusters, and phosphorus hydride molecules. The potential parameters were obtained by conducting global optimization with respect to a set of reference data generated by extensive ab initio calculations. We extended ReaxFF by adding a 60° correction term, which significantly improved the description of phosphorus clusters. Emphasis was placed on the mechanical response of black phosphorene with different types of defects. Compared to the nonreactive SW potential ( Jiang , J.-W. Nanotechnology 2015 , 26 , 315706 ), ReaxFF for P/H systems provides a significant improvement in describing the mechanical properties of the pristine and defected black phosphorene, as well as the thermal stability of phosphorene nanotubes. A counterintuitive phenomenon is observed that single vacancies weaken the black phosphorene more than double vacancies with higher formation energy. Our results also showed that the mechanical response of black phosphorene is more sensitive to defects in the zigzag direction than that in the armchair direction. In addition, we developed a preliminary set of ReaxFF parameters for P/H/O/C to demonstrate that the ReaxFF parameters developed in this work could be generalized to oxidized phosphorene and P-containing 2D van der Waals heterostructures. That is, the proposed ReaxFF parameters for P/H systems establish a solid foundation for modeling of a wide range of P-containing materials.

Density functional study of structure and dynamics in liquid antimony and Sbn clusters

Density functional/molecular dynamics simulations have been performed on liquid antimony (588 atoms and six temperatures between 600 K and 1300 K) and on neutral Sb clusters with up to 14 atoms. We study structural patterns (coordination numbers, bond angles, and ring patterns, structure factors, pair distribution functions) and dynamical properties (vibration frequencies, diffusion constants, power spectra, dynamical structure factors, viscosity) and compare with available experimental results and with the results of our previous simulations on Bi. Three short covalent bonds characteristic of pnictogens are common in the clusters, and higher temperatures lead in the liquid to broader bond angle distributions, larger total cavity volumes, and weaker correlations between neighboring bond lengths. There are clear similarities between the properties of Sb and Bi aggregates.

The chemistry of cationic polyphosphorus cages--syntheses, structure and reactivity

The aim of this review is to provide a comprehensive view of the chemistry of cationic polyphosphorus cages.

The aim of this review is to provide a comprehensive view of the chemistry of cationic polyphosphorus cages. The synthetic protocols established for their preparation, which are all based on the functionalization of P₄, and their intriguing follow-up chemistry are highlighted. In addition, this review intends to foster the interest of the inorganic, organic, catalytic and material oriented chemical communities in the versatile field of polyphosphorus cage compounds. In the long term, this is envisioned to contribute to the development of new synthetic procedures for the functionalization of P₄ and its transformation into (organo-)phosphorus compounds and materials of added value.

A density functional study of Rh13

A density functional study was performed for the Rh13 cluster using the linear combination of Gaussian-type orbitals density functional theory (LCGTO-DFT) approach. The calculations employed both the local density approximation (LDA) as well as the generalized gradient approximation (GGA) in combination with a quasi-relativistic effective core potential (QECP). Initial structures for the geometry optimization were taken along Born–Oppenheimer molecular dynamics (BOMD) trajectories. The BOMD trajectories were performed at different temperatures and considered different potential energy surfaces (PES). As a result, several hundred isomers of the Rh13 cluster in different spin multiplicities were optimized with the aim to determine the lowest energy structures. All geometry optimizations were performed without any symmetry restriction. A vibrational analysis was performed to characterize these isomers. Structural parameters, relative stability energy, harmonic frequencies, binding energy, and most relevant Kohn–Sham (KS) molecular orbitals are reported. The obtained results are compared with available data from the literature. This study predicts a low symmetry biplanarlike structure as the ground-state structure of Rh13 with 11 unpaired electrons. This isomer was first noticed by inspection of first-principle Born–Oppenheimer molecular dynamics (BOMD) simulations between 300 and 600 K. This represents the most extensive theoretical study on the ground-state structure of the Rh13 cluster and underlines the importance of BOMD simulations to fully explore the PES landscapes of complicated systems.

Finding stable minima using a nudged-elastic-band-based optimization scheme

Optimization is essential in many scientific and economical areas, but it is often too complex to be tackled by simple straightforward calculations or by trial and error. Two well-known methods to find low-lying minima in such complex systems are simulated annealing and the genetic algorithm. In these methods artificial fluctuations control the probability of the system to overcome a local minimum having a certain depth. Here we present a complementary scheme that is based on the nudged-elastic-band method ordinarily used to find saddle points and we apply the scheme to find the most stable isomers of the phosphorus P(4), P(8) molecules and the corresponding molecules of As(n), Sb(n), and Bi(n) (n = 4,8) in the framework of the density functional theory. In the case of n = 8 we have found stable and metastable configurations, some of which are new and have similar energies. As a by-product we obtained an upper bound for the energy barriers between these configurations.

Pressure-dependent structures of amorphous red phosphorus and the origin of the first sharp diffraction peaks

Characterizing the nature of medium-range order (MRO) in liquids and disordered solids is important for understanding their structure and transport properties. However, accurately portraying MRO, as manifested by the first sharp diffraction peak (FSDP) in neutron and X-ray scattering measurements, has remained elusive for more than 80 years. Here, using X-ray diffraction of amorphous red phosphorus compressed to 6.30 GPa, supplemented with micro-Raman scattering studies, we build three-dimensional structural models consistent with the diffraction data. We discover that the pressure dependence of the FSDP intensity and line position can be quantitatively accounted for by a characteristic void distribution function, defined in terms of average void size, void spacing and void density. This work provides a template to unambiguously interpret atomic and void-space MRO across a broad range of technologically promising network-forming materials.

Simulating the phosphorus fluid-liquid phase transition up to the critical point

We report a Car-Parrinello molecular dynamics study of the temperature dependence of the fluid-liquid phase transition in phosphorous, involving the transformation of a molecular fluid phase into a network-like phase. We employed density-functional theory (DFT) with a gradient-corrected functional (B-LYP) to describe the electronic structure and interatomic interactions and performed simulations in a constant pressure ensemble. We spanned a temperature interval ranging from 2500 to 3500 K. With increasing temperature, we found that the structural conversion from the molecular P₄ fluid into the network liquid occurs at decreasing pressures, consistent with experimental observations. At lower temperatures the transition is characterized by a sudden increase of density in the sample. The magnitude of the density change decreases with increasing temperature and vanishes at 3500 K. In the temperature range 3100-3500 K we found signals of near- and super-criticality. We identified local structural changes that serve as seeds triggering the overall structural transition.

Phosphorus: first principle simulation of a liquid-liquid phase transition

We report a Car-Parrinello molecular dynamics study of the liquid-liquid phase transition in phosphorus. We employed a gradient corrected density functional (B-LYP) to describe the electronic structure and performed simulations at constant pressure. Upon increasing pressure we observed, along the 1500 K isotherm, a structural transition converting the molecular P4 liquid into an atomic liquid with a network structure. Our calculations suggest this transition to be first order with a discontinuous density increase accompanied by an insulator into metal transition. The transition pressure is significantly higher than obtained by employing the less accurate local density functional (LDA) [Morishita, Phys. Rev. Lett. 87, 105701 (2001)], which matches the experimental value for the pressure. We argue why the LDA result should be considered fortuitous. The change of the calculated structure factor upon the transition shows the same trend as experimentally observed. Analysis of the structural changes during the phase transition revealed that a chain of linked and opened up ("butterfly") P4 molecules may serve as a seed triggering the transition from the molecular to the network phase.

Parity Alternation of Ground-State Pn- and Pn+ (n = 3−15) Phosphorus Clusters

The ground-state structures of neutral, cationic, and anionic phosphorus clusters P(n), P(n)(+), and P(n)(-) (n = 3-15) have been calculated using the B3LYP/6-311+G* density functional method. The P(n)(+) and P(n)(-) (n = 3-15) clusters with odd n were found to be more stable than those with even n, and we provide a satisfactory explanation for such trends based on concepts of energy difference, ionization potential, electron affinity, and incremental binding energy. The result of odd/even alternations is in good accord with the relative intensities of cationic and anionic phosphorus clusters observed in mass spectrometric studies.

Pseudo-Jahn-Teller origin of geometry and pseudorotations in second row tetra-atomic clusters X4 (X=Na,Mg,Al,Si,P,S)

Experimentally determined or ab initio calculated molecular geometries carry no information about their origin. Employing the Jahn-Teller (JT) vibronic coupling effects as the only source of instability and consequent distortions of high-symmetry molecular configurations, we have worked out a procedure that allows us to trace the origin of particular geometries and determine the detailed electronic mechanism of their formation. This procedure is illustrated by considering a series of X(4) clusters with X=Na, Mg, Al, Si, P, and S. It shows explicitly why Na(4), Si(4), and Al(4) have a rhombic geometry in the ground state, while Mg(4) and P(4) are tetrahedral, whereas S(4) is a trapezium. Even when the minimum-energy geometries are the same (as in the case of rhombic Na(4), Si(4), and Al(4)), the electronic mechanism of their formation is quite different. In particular, in Na(4) and Si(4) the rhombic minima are produced by a strong pseudo JT coupling between two excited states in the square-planar configuration (different in the two cases) that stabilizes one of them and makes it the ground state by rhombic distortions. The rhombic configuration of Al(4) is due to the pseudo JT effect in its ground-state square-planar configuration, and the trapezium in S(4) is formed by two pseudo JT couplings essentially involving excited states. In several cases this analysis shows also the tunneling paths between equivalent configurations.