Complex Potential Energy Surface Research Articles

Computational Investigations of the Detailed Mechanism of Reverse Intersystem Crossing in Inverted Singlet-Triplet Gap Molecules.

Inverted singlet-triplet gap (INVEST) materials have promising photophysical properties for optoelectronic applications due to an inversion of their lowest singlet (S1) and triplet (T1) excited states. This results in an exothermic reverse intersystem crossing (rISC) process that potentially enhances triplet harvesting, compared to thermally activated delayed fluorescence (TADF) emitters with endothermic rISCs. However, the processes and phenomena that facilitate conversion between excited states for INVEST materials are underexplored. We investigate the complex potential energy surfaces (PESs) of the excited states of three heavily studied azaphenalene INVEST compounds, namely, cyclazine, pentazine, and heptazine using two state-of-the-art computational methodologies, namely, RMS-CASPT2 and SCS-ADC(2) methods. Our findings suggest that ISC and rISC processes take place directly between the S1 and T1 electronic states in all three compounds through a minimum-energy crossing point (MECP) with an activation energy barrier between 0.11 to 0.58 eV above the S1 state for ISC and between 0.06 and 0.36 eV above the T1 state for rISC. We predict that higher-lying triplet states are not populated, since the crossing point structures to these states are not energetically accessible. Furthermore, the conical intersection (CI) between the ground and S1 states is high in energy for all compounds (between 0.4 to 2.0 eV) which makes nonradiative decay back to the ground state a relatively slow process. We demonstrate that the spin-orbit coupling (SOC) driving the S1-T1 conversion is enhanced by vibronic coupling with higher-lying singlet and triplet states possessing vibrational modes of proper symmetry. We also rationalize that the experimentally observed anti-Kasha emission of cyclazine is due to the energetically inaccessible CI between the bright S2 and the dark S1 states, hindering internal conversion. Finally, we show that SCS-ADC(2) is able to qualitatively reproduce excited state features, but consistently overpredict relative energies of excited state structural minima compared to RMS-CASPT2. The identification of these excited state features elaborates design rules for new INVEST emitters with improved emission quantum yields.

Identifying the charge states of carbon vacancies in 4H-SiC by ab initio metadynamics

4H Silicon carbide (4H-SiC) is widely recognized as a highly promising material for high-voltage and high-power electronic applications due to its exceptional properties. The performance of devices based on 4H-SiC is often weakened by the presence of carbon-related point defects, particularly carbon vacancies (VC). The defects of VC introduce deep-level traps (e.g., Z1/2 and EH6/7) that deteriorate device functionality. Experimental and theoretical studies on VC have led to some conflicting results about the charge states of VC, especially for the charge state ordering of EH6/7. We now employ ab initio metadynamics (META) to systematically investigate configuration space including the direction and magnitude of bond distortion and identify the most stable structures of VC. Eventually, the charge states of VC in 4H-SiC are identified. The Z1 (EH6) and Z2 (EH7) indicate transitions from acceptor (donor) levels of VC, located on the h and k sublattice sites, respectively. Z1 and Z2 demonstrate negative-U ordering, characterized by U values of −0.16 and −0.37 eV, respectively. Conversely, EH6 and EH7 display positive-U ordering, with U values of 0.16 and 0.08 eV, respectively. The current results provide insights into the properties of VC in 4H-SiC, highlighting the effectiveness of META in the exploration of complex potential energy surfaces associated with point defects in solids.

Mechanism of Asymmetric Homologation of Alkenylboronic Acids with CF3-Diazomethane via Borotropic Rearrangement.

Density functional theory calculations have been performed to investigate the mechanism for the BINOL-catalyzed asymmetric homologation of alkenylboronic acids with CF3-diazomethane. The reaction proceeds via a chiral BINOL ester of the alkenylboronic acid substrate. The calculations reveal a complex scenario for the formation of the chiral BINOL-alkenylboronate species, which is the key intermediate in the catalytic process. The aliphatic alcohol additive plays an important role in the reaction. This study provides a rationalization of the stereoinduction step of the reaction, and the enantioselectivity is mainly attributed to the steric repulsion between the CF3 group of the diazomethane reagent and the γ-substituent of the BINOL catalyst. The complex potential energy surface obtained by the calculations is analyzed by means of microkinetic simulations.

Global Neural Network Potential with Explicit Many-Body Functions for Improved Descriptions of Complex Potential Energy Surface.

The high dimensional machine learning potential (MLP) that has developed rapidly in the past decade represents a giant step forward in large-scale atomic simulation for complex systems. The long-range interaction and the poor description of chemical reactions are typical problems of high dimensional MLP, which are mainly caused by the poor structure discrimination of the atom-centered ML model. Herein, we propose a low-cost neural-network-based MLP architecture for fitting global potential energy surface data, namely, G-MBNN, that can offer improved energy and force resolution on a complex potential energy surface. In G-MBNN, a set of many-body energy terms based on the local atomic environment are explicitly included in computing the total energy─the total energy of the system is written as the sum of atomic energy and many-body energy contributions. These extra many-body energy terms are computationally low-cost and, importantly, can provide easy access to delicate energy terms in complex systems such as very short repulsion, long-range attractions, and sensitive angular-dependent covalent interactions. We implement G-MBNN in the LASP code and demonstrate the improved accuracy of the new framework in representative systems, including ternary-element energy materials LiCoOx, TiO2 with defects, and a series of organic reactions.

Fragment-based models for dissociation of strong acids in water: Electrostatic embedding minimizes the dependence on the fragmentation schemes.

Fragmentation methods such as MIM (Molecules-in-Molecules) provide a route to accurately model large systems and have been successful in predicting their structures, energies, and spectroscopic properties. However, their use is often limited to systems at equilibrium due to the inherent complications in the choice of fragments in systems away from equilibrium. Furthermore, the presence of charges resulting from any heterolytic bond breaking may increase the fragmentation error. We have previously suggested EE-MIM (Electrostatically Embedded Molecules-In-Molecules) as a method to mitigate the errors resulting from the missing long-range interactions in molecular clusters in equilibrium. Here, we show that the same method can be applied to improve the performance of MIM to solve the longstanding problem of dependency of the fragmentation energy error on the choice of the fragmentation scheme. We chose four widely used acid dissociation reactions (HCl, HClO4, HNO3, and H2SO4) as test cases due to their importance in chemical processes and complex reaction potential energy surfaces. Electrostatic embedding improves the performance at both one and two-layer MIM as shown by lower EE-MIM1 and EE-MIM2 errors. The EE-MIM errors are also demonstrated to be less dependent on the choice of the fragmentation scheme by analyzing the variation in fragmentation energy at the points with more than one possible fragmentation scheme (points where the fragmentation scheme changes). EE-MIM2 with M06-2X as the low-level resulted in a variation of less than 1 kcal/mol for all the cases and 1 kJ/mol for all but three cases, rendering our method fragmentation scheme-independent for acid dissociation processes.

Semiempirical Methods for Molecular Systems in Strong Magnetic Fields.

A general scheme is presented to extend semiempirical methods to include the effects of arbitrary strength magnetic fields, while maintaining computational efficiency. The approach utilizes three main modifications; a London atomic orbital (LAO) basis set is introduced, field-dependent kinetic energy corrections are added to the model Hamiltonian, and spin-Zeeman interaction energy terms are included. The approach is applied to the widely available density-functional tight-binding method GFN1-xTB. Considering the basis set requirements for the kinetic energy corrections in a magnetic field leads to two variants: a single-basis approach GFN1-xTB-M0 and a dual-basis approach GFN1-xTB-M1. The LAO basis in the latter includes the appropriate nodal structure for an accurate representation of the kinetic energy corrections. The variants are assessed by benchmarking magnetizabilities and nuclear magnetic resonance shielding constants calculated using weak magnetic fields. Remarkably, the GFN1-xTB-M1 approach also exhibits excellent performance for strong fields, || ≤ 0.2B0 (B0 = 2.3505 × 105 T), recovering exotic features such as the para- to dia-magnetic transition in the BH molecule and the preferred electronic configuration, molecular conformation, and orientation of benzene. At stronger field strengths, || > 0.2B0, a degradation in the quality of the results is observed. The utility of GFN1-xTB-M1 is demonstrated by performing conformer searches in a range of field strengths for the cyclooctatetraene molecule, with GFN1-xTB-M1 capturing the transition from tub to planar conformations at high field, consistent with much more computationally demanding current-density functional theory calculations. Magnetically induced currents are also shown to be well described for the benzene and infinitene molecules, the latter demonstrating the flexibility and computational efficiency of the approach. The GFN1-xTB-M1 approach is a useful tool for the study of structure, conformation, and dynamics of large systems in magnetic fields at the semiempirical level as well as for preoptimization of molecular structure in ab initio calculations, enabling more efficient exploration of complex potential energy surfaces and reactivity in the presence of external fields.

Machine learning meets quantum mechanics in catalysis

Over the past decade many researchers have applied machine learning algorithms with computational chemistry and materials science tools to explore properties of catalysts. There is a rapid increase in publications demonstrating the use of machine learning for rational catalyst design. In our perspective, targeted tools for rational catalyst design will continue to make significant contributions. However, the community should focus on developing high-throughput simulation tools that utilize molecular dynamics capabilities for thorough exploration of the complex potential energy surfaces that exist, particularly in heterogeneous catalysis. Catalyst-specific databases should be developed to contain enough data to represent the complex multi-dimensional space that defines structure-function relationships. Machine learning tools will continue to impact rational catalyst design; however, we believe that more sophisticated pattern recognition algorithms would yield better understanding of structure-function relationships for heterogeneous catalysis.

Theoretical kinetic investigation of the multichannel mechanism of O(3P) atmospheric oxidation reaction of but-3-enal

Several levels of theory such as Møller-Plesset MP2, G3, and CBS-QB3, have been used in order to investigate the complex and multichannel potential energy surface of the reaction of but-3-enal with the triplet oxygen atom. The results show that the O-addition channel is dominant. The different possible pathways of oxygen atom attack are thoroughly studied to better understand and explain the reaction mechanism. Regarding the oxidation of but-3-enal by triplet oxygen O(3P), it is shown that the major thermodynamic product is H3CC(O)CH2C(O)H (P3) being the most stable for the whole reaction. However, the most favored product kinetically is H2CC(OH)CH2C(O)H (P2). For the H-abstraction second possible pathway, the most favored product both kinetically and thermodynamically is found to be P8. The activation energy and calculated rate constants are consistent with the proposed addition mechanism.

Machine-learning driven global optimization of surface adsorbate geometries

The adsorption energies of molecular adsorbates on catalyst surfaces are key descriptors in computational catalysis research. For the relatively large reaction intermediates frequently encountered, e.g., in syngas conversion, a multitude of possible binding motifs leads to complex potential energy surfaces (PES), however. This implies that finding the optimal structure is a difficult global optimization problem, which leads to significant uncertainty about the stability of many intermediates. To tackle this issue, we present a global optimization protocol for surface adsorbate geometries which trains a surrogate machine learning potential on-the-fly. The approach is applicable to arbitrary surface models and adsorbates and minimizes both human intervention and the number of required DFT calculations by iteratively updating the training set with configurations explored by the algorithm. We demonstrate the efficiency of this approach for a diverse set of adsorbates on the Rh(111) and (211) surfaces.

Robosample: A software using constrained molecular dynamics coupled with Gibbs sampling to investigate macromolecules.

An ongoing challenge for the simulation of macromolecules is their complex potential energy surface which is a result of high dimensionality and chemical complexity. The surface contains high energy barriers usually difficult to overcome by existing simulation methods. We present here Robosample, a software combining the advantages of robot mechanics for fast constrained simulations and of Gibbs sampling for achieving ergodicity. While energy calculation is the main computational bottleneck for macromolecule simulations, a big part of the work can be short-circuited by increasing the trajectory integration timestep. Usually this is achieved by modifying the system's mass matrix. The way this is implemented in Robosample is by using constraints in the framework of reduced coordinates. A good example is torsional dynamics, which transforms the system in Bond-Angle-Torsion coordinates and then ignores the motion of the bonds and the angles. Robosample sampling method is Hamiltonian Monte Carlo coupled with Gibbs sampling (GCHMC), with random variables divided among molecular degrees of freedom. With GCHMC, certain molecule segments are considered rigid and sub-sampled, while focusing on soft degrees of freedom as Gibbs blocks. Ergodicity is achieved by interleaving fully flexible Cartesian dynamics Monte Carlo proposals. The validity of the method was first proven on small systems correctly reproducing alanine dipeptide free energy landscape and showed to increase sampling efficiency of macrocycle containing small molecules which are notoriously difficult to simulate. The software was subsequently extended to include spherical and cylindrical robotic joints and proved to be efficient on medium sized biological molecules, such as glycans. Now we have reached a stage where we can successfully sample big macromolecules such as antibodies on graphical processing units.

PESPIP: Software to fit complex molecular and many-body potential energy surfaces with permutationally invariant polynomials.

We wish to describe a potential energy surface by using a basis of permutationally invariant polynomials whose coefficients will be determined by numerical regression so as to smoothly fit a dataset of electronic energies as well as, perhaps, gradients. The polynomials will be powers of transformed internuclear distances, usually either Morse variables, exp(-ri,j/λ), where λ is a constant range hyperparameter, or reciprocals of the distances, 1/ri,j. The question we address is how to create the most efficient basis, including (a) which polynomials to keep or discard, (b) how many polynomials will be needed, (c) how to make sure the polynomials correctly reproduce the zero interaction at a large distance, (d) how to ensure special symmetries, and (e) how to calculate gradients efficiently. This article discusses how these questions can be answered by using a set of programs to choose and manipulate the polynomials as well as to write efficient Fortran programs for the calculation of energies and gradients. A user-friendly interface for access to monomial symmetrization approach results is also described. The software for these programs is now publicly available.

Growth Pattern of Large Morse Clusters with Medium-Range Potentials.

Due to the extremely complex potential energy surfaces of large Morse clusters with medium-range potentials (i.e., ρ = 6 and 10), global optimization studies in the literature are limited to a cluster size (N) of ≤240. Starting from completely random structures, we successfully systematically studied Morse clusters with up to 700 atoms using our unbiased fuzzy global optimization (FGO) method. While all of the putative global minima reported previously have been efficiently obtained, new global minima with lower energies are identified for N values of 176, 258, 485, 561, 817, and 923 with ρ = 6 and for N values of 151, 202, 226, and 229 with ρ = 10. A detailed growth pattern and magic clusters are obtained. For the first time, we find that a central vacancy is present in Morse clusters containing 542, 543, 548, and 922 atoms with ρ = 6. FGO has achieved high performance in large clusters with different interatomic interaction ranges, thus showing great application potential in the global structure optimization of general clusters.

Automation of chemical kinetics: Status and challenges

Driven by synergic advancements in high performance computing and theory, the capability to estimate rate constants from first principles has evolved considerably recently. When this knowledge is coupled with a procedure to determine a list of all reactions relevant to describe the evolution of a reacting system, it becomes possible to envision a methodology to predict theoretically the reaction kinetics. However, if a thorough examination of all possible reaction channels is desired, the number of reactions for which a rate constant estimate is needed can become quite large. This determines the need for rate constant estimation automation. In the present work, the status of this rapidly evolving field is reviewed, with emphasis on recent advancements and present challenges. Thermochemistry is the field where automation is most advanced. Entropies, heat capacities, and enthalpies can be determined efficiently with accuracy comparable to experiments for most chemical species containing a limited number of atoms, while machine learning can be used to improve the computational predictions for large chemical species using reduced computational resources. Several approaches have been proposed to automatically investigate the reactivity over complex potential energy surfaces, while rate constants for elementary steps can be determined accurately for several reaction classes, such as abstraction, addition, beta-scission, and isomerization. Kinetic mechanisms can be automatically generated using methodologies that differ for level of complexity and required physical insight. Among the challenges that are still to be met are the estimation of rate constants for intrinsically multireference reaction classes, such as barrierless processes, the containment of the number of reactions to screen in mechanism development, and the integration of the existing automated software. It is suggested that the synergy between experiment and theory should evolve towards a stage where experiments are focused on the estimation of parameters where theoretical tools are least predictive, and vice versa.

Experimental study of the removal of excited state phosphorus atoms by H2O and H2: implications for the formation of PO in stellar winds

ABSTRACTThe reactions of the low-lying metastable states of atomic phosphorus, P(2D) and P(2P), with H2O and H2 were studied by the pulsed laser photolysis at 248 nm of PCl3, combined with laser-induced fluorescence detection of P(2D), P(2P), and PO. Rate coefficients between 291 and 740 K were measured, along with a yield for the production of PO from P(2D or 2P) + H2O of (35 ± 15) %. H2 reacts with both excited P states relatively efficiently; physical (i.e. collisional) quenching, rather than chemical reaction to produced PH + H, is shown to be the more likely pathway. A comprehensive phosphorus chemistry network is then developed using a combination of electronic structure theory calculations and a Master Equation treatment of reactions taking place over complex potential energy surfaces. The resulting model shows that at the high temperatures within two stellar radii of a MIRA variable AGB star in oxygen-rich conditions, collisional excitation of ground-state P(4S) to P(2D), followed by reaction with H2O, is a significant pathway for producing PO (in addition to the reaction between P(4S) and OH). The model also demonstrates that the PN fractional abundance in a steady (non-pulsating) outflow is underpredicted by about 2 orders of magnitude. However, under shocked conditions where sufficient thermal dissociation of N2 occurs at temperatures above 4000 K, the resulting N atoms convert a substantial fraction of PO into PN.

Projected Complex Absorbing Potential Multireference Configuration Interaction Approach for Shape and Feshbach Resonances.

Anion resonances are formed as metastable intermediates in low-energy electron-induced reactions. Due to the finite lifetimes of resonances, applying standard Hermitian formalism for their characterization presents a vexing problem for computational chemists. Numerous modifications to conventional quantum chemical methods have enabled satisfactory characterization of resonances, but specific issues remain, especially in describing two-particle one-hole (2p-1h) resonances. An accurate description of these resonances and their coupling with single-particle resonances requires a multireference approach. We propose a projected complex absorbing potential (CAP) implementation within the multireference configuration interaction (MRCI) framework to characterize single-particle and 2p-1h resonances. As a first application, we use the projected-CAP-MRCI approach to characterize and benchmark the 2Πg shape resonance in N2-. We test its performance as a function of the size of the subspace and other parameters, and we compute the complex potential energy surface of the 2Πg shape resonance to show that a smooth curve is obtained. One key benefit of MRCI is that it can describe Feshbach resonances (most common examples of 2p-1h resonances) at the same footing as shape resonances. Therefore, it is uniquely positioned to describe mixing between the different channels. To test these additional capabilities, we compute Feshbach resonances in H2O- and anions of dicyanoethylene isomers. We find that CAP-MRCI can efficiently capture the mixing between the Feshbach and shape resonances in dicyanoethylene isomers, which has significant consequences for their lifetimes.

The complex vibrational spectrum of proline explained through the adiabatically switched semiclassical initial value representation.

Proline, a 17-atom amino acid with a closed-ring side chain, has a complex potential energy surface characterized by several minima. Its IR experimental spectrum, reported in the literature, is of difficult and controversial assignment. In particular, the experimental signal at 3559 cm-1 associated with the OH stretch is interesting because it is inconsistent with the global minimum, trans-proline conformer. This suggests the possibility that multiple conformers may contribute to the IR spectrum. The same conclusion is obtained by investigating the splitting of the CO stretch at 1766 and 1789 cm-1 and other, more complex spectroscopic features involving CH stretches and COH/CNH bendings. In this work, we perform full-dimensional, on-the-fly adiabatically switched semiclassical initial value representation simulations employing the ab initio dft-d3-B3LYP level of theory with aug-cc-pVDZ basis set. We reconstruct the experimental spectrum of proline in its main features by studying the vibrational features of trans-proline and cis1-proline and provide a new assignment for the OH stretch of trans-proline.

Assessment of the van der Waals, Hubbard U parameter and spin-orbit coupling corrections on the 2D/3D structures from metal gold congeners clusters.

The coinage-metal clusters possess a natural complexity in their theoretical treatment that may be accompanied by inherent shortcomings in the methodological approach. Herein, we performed a scalar-relativistic density functional theory study, considering Perdew, Burke, and Ernzerhof (PBE) with (empirical and semi empirical) van der Waals (vdW), spin-orbit coupling (SOC), +U (Hubbard term), and their combinations, to treat the , , and clusters in different structural motifs. The energetic scenario is given by the confirmation of the 3D lowest energy configurations for and within all approaches, while for there is a 2D/3D competition, depending on the applied correction. The 2D geometry is 0.43 eV more stable with plain PBE than the 3D one, the SOC, +U, and/or vdW inclusion decreases the overestimated stability of the planar configurations, where the most surprising result is found by the D3 and D3BJ vdW corrections, for which the 3D configuration is 0.29 and 0.11 eV, respectively, more stable than the 2D geometry (with even higher values when SOC and/or +U are added). The D3 dispersion correction represents 7.9% (4.4%) of the total binding energy for the 3D (2D) configuration, (not) being enough to change the hybridization and the position of the occupied -states. Our predictions are in agreement with experimental results and in line with the best results obtained for bulk systems, as well as with hybrid functionals within D3 corrections. The properties description undergoes small corrections with the different approaches, where general trends are maintained, that is, the average bond length is smaller (larger) for lower (higher)-coordinated structures, since a same number of electrons are shared by a smaller (larger) number of bonds, consequently, the bonds are stronger (weaker) and shorter (longer) and the hybridization index is larger (smaller). Thus, Au has a distinct behavior in relation to its lighter congeners, with a complex potential energy surface, where in addition to the relevant relativistic effects, correlation and dispersion effects must also be considered.

Structures and structural evolution of MN (M = Pt, Ag, Au, N=2-20) from combined revised particle swarm optimization and density function theory

ABSTRACT Structure is the key to understanding the properties of metal nanoclusters. Finding a globally optimal structure on an extremely complex potential energy surface is a great challenge for theoretical calculations. In this article, we combine the Revised Particle Swarm Optimization algorithm (RPSO) with Density Functional Theory (DFT), called RPSO-DFT, to explore the potential energy surfaces of platinum, silver, and gold nanoclusters within 20 atoms. Four previously unreported platinum nanocluster structures and two silver nanocluster structures were found. The discovered platinum cluster structures overturn the previous definition of the magic number of platinum clusters. The gold nanocluster calculated by RPSO-DFT is the same as the previous reports, but this algorithm can find the global optimal solution of Au20 cluster with fewer iterations. The test results in this work show that the RPSO-DFT method is accurate and efficient. In addition, RPSO-DFT helped to explore the significance of the difference between the lowest energy value and the second lowest energy value, as well as the variation of the average coordination number with the cluster size based on the new lowest energy structure of small metal clusters.

Photoinduced Desorption Dynamics of CO from Pd(111): A Neural Network Approach.

Modeling the ultrafast photoinduced dynamics and reactivity of adsorbates on metals requires including the effect of the laser-excited electrons and, in many cases, also the effect of the highly excited surface lattice. Although the recent ab initio molecular dynamics with electronic friction and thermostats, (Te,Tl)-AIMDEF [AlducinM.;Phys. Rev. Lett.2019, 123, 246802]31922860, enables such complex modeling, its computational cost may limit its applicability. Here, we use the new embedded atom neural network (EANN) method [ZhangY.;J. Phys. Chem. Lett.2019, 10, 496231397157] to develop an accurate and extremely complex potential energy surface (PES) that allows us a detailed and reliable description of the photoinduced desorption of CO from the Pd(111) surface with a coverage of 0.75 monolayer. Molecular dynamics simulations performed on this EANN-PES reproduce the (Te,Tl)-AIMDEF results with a remarkable level of accuracy. This demonstrates the outstanding performance of the obtained EANN-PES that is able to reproduce available density functional theory (DFT) data for an extensive range of surface temperatures (90–1000 K); a large number of degrees of freedom, those corresponding to six CO adsorbates and 24 moving surface atoms; and the varying CO coverage caused by the abundant desorption events.

Exploring energy landscapes at the DFTB quantum level using the threshold algorithm: the case of the anionic metal cluster Au$$_{20}^{-}$$

We report the combination of the threshold algorithm with the Density Functional-based Tight Binding method allowing for the exploration of complex potential energy surfaces and the evaluation of probability flows between their regions, at the quantum level. This original scheme is used to explore the energy landscape of an anionic 20-atom gold cluster, Au $$_{20}^{-}$$ . On the basis of the relevant structures, 19 structural groups are highlighted, all of them being variations about the pyramidal shape: (1) distorted pyramids, (2) pyramids in which the atom of one of the facets has been removed, leaving a hole, and placed at different positions on the cluster and (3) pyramids on which an atom located at a vertex has been removed and placed on an edge or on a facet. Upper limits of the energies required to connect the basins of the 19 groups on the potential energy surface are evaluated. Moreover, the attractive basins are identified on the basis of the analysis of the probability flows on the landscape. The comparison of the disconnectivity tree with the results of the flux analysis provides a consistent representation of the Au $$_{20}^{-}$$ basins’ proximity. Finally, we show how the new scheme allowed for the identification of counter-intuitive transition pathways.