Coexistence in small inert gas clusters

Prediction of 3D RNA Structures from Sequence Using Energy Landscapes of RNA Dimers: Application to RNA Tetraloops

Access to the three-dimensional structure of RNA enables an ability to gain a more profound understanding of its biological mechanisms, as well as the ability to design RNA-targeting drugs, which can take advantage of the unique chemical environment imposed by a folded RNA structure. Due to the dynamic and structurally complex properties of RNA, both experimental and traditional computational methods have difficulty in determining RNA's 3D structure. Herein, we introduce TAPERSS (Theoretical Analyses, Prediction, and Evaluation of RNA Structures from Sequence), a physics-based fragment assembly method for predicting 3D RNA structures from sequence. Using a fragment library created using discrete path sampling calculations of RNA dinucleoside monophosphates, TAPERSS can sample the physics-based energy landscapes of any RNA sequence with relatively low computational complexity. We have benchmarked TAPERSS on 21 RNA tetraloops, using a combinatorial algorithm as a proof-of-concept. We show that TAPERSS was successfully able to predict the apo-state structures of all 21 RNA hairpins, with 16 of those structures also having low predicted energies as well. We demonstrate that TAPERSS performs most accurately on GNRA-like tetraloops with mostly stacked loop-nucleotides, while having limited success with more dynamic UNCG and CUYG tetraloops, most likely due to the influence of the RNA force field used to create the fragment library. Moreover, we show that TAPERSS can successfully predict the majority of the experimental non-apo states, highlighting its potential in anticipating biologically significant yet unobserved states. This holds great promise for future applications in drug design and related studies. With discussed improvements and implementation of more efficient sampling algorithms, we believe TAPERSS may serve as a useful tool for a physics-based conformational sampling of large RNA structures.

100 Years of the Lennard-Jones Potential

It is now 100 years since Lennard-Jones published his first paper introducing the now famous potential that bears his name. It is therefore timely to reflect on the many achievements, as well as the limitations, of this potential in the theory of atomic and molecular interactions, where applications range from descriptions of intermolecular forces to molecules, clusters, and condensed matter.

Combining experiment and energy landscapes to explore anaerobic heme breakdown in multifunctional hemoproteins

To survive, many pathogens extract heme from their host organism and break down the porphyrin scaffold to sequester the Fe2+ ion via a heme oxygenase. Recent studies have revealed that certain pathogens can anaerobically degrade heme. Our own research has shown that one such pathway proceeds via NADH-dependent heme degradation, which has been identified in a family of hemoproteins from a range of bacteria. HemS, from Yersinia enterocolitica, is the main focus of this work, along with HmuS (Yersinia pestis), ChuS (Escherichia coli) and ShuS (Shigella dysenteriae). We combine experiments, Energy Landscape Theory, and a bioinformatic investigation to place these homologues within a wider phylogenetic context. A subset of these hemoproteins are known to bind certain DNA promoter regions, suggesting not only that they can catalytically degrade heme, but that they are also involved in transcriptional modulation responding to heme flux. Many of the bacterial species responsible for these hemoproteins (including those that produce HemS, ChuS and ShuS) are known to specifically target oxygen-depleted regions of the gastrointestinal tract. A deeper understanding of anaerobic heme breakdown processes exploited by these pathogens could therefore prove useful in the development of future strategies for disease prevention.

Energy Landscapes and Heat Capacity Signatures for Monomers and Dimers of Amyloid-Forming Hexapeptides

Amyloid formation is a hallmark of various neurodegenerative disorders. In this contribution, energy landscapes are explored for various hexapeptides that are known to form amyloids. Heat capacity (CV) analysis at low temperature for these hexapeptides reveals that the low energy structures contributing to the first heat capacity feature above a threshold temperature exhibit a variety of backbone conformations for amyloid-forming monomers. The corresponding control sequences do not exhibit such structural polymorphism, as diagnosed via end-to-end distance and a dihedral angle defined for the monomer. A similar heat capacity analysis for dimer conformations obtained using basin-hopping global optimisation shows clear features in end-to-end distance versus dihedral correlation plots, where amyloid-forming sequences exhibit a preference for larger end-to-end distances and larger positive dihedrals. These results hold true for sequences taken from tau, amylin, insulin A chain, a de novo designed peptide, and various control sequences. While there is a little overall correlation between the aggregation propensity and the temperature at which the low-temperature CV feature occurs, further analysis suggests that the amyloid-forming sequences exhibit the key CV feature at a lower temperature compared to control sequences derived from the same protein.

Exploiting Sequence-Dependent Rotamer Information in Global Optimization of Proteins

Rotamers, namely amino acid side chain conformations common to many different peptides, can be compiled into libraries. These rotamer libraries are used in protein modeling, where the limited conformational space occupied by amino acid side chains is exploited. Here, we construct a sequence-dependent rotamer library from simulations of all possible tripeptides, which provides rotameric states dependent on adjacent amino acids. We observe significant sensitivity of rotamer populations to sequence and find that the library is successful in locating side chain conformations present in crystal structures. The library is designed for applications with basin-hopping global optimization, where we use it to propose moves in conformational space. The addition of rotamer moves significantly increases the efficiency of protein structure prediction within this framework, and we determine parameters to optimize efficiency.

Evaluating Geometric Definitions of Stacking for RNA Dinucleoside Monophosphates Using Molecular Mechanics Calculations

RNA modulation via small molecules is a novel approach in pharmacotherapies, where the determination of the structural properties of RNA motifs is considered a promising way to develop drugs capable of targeting RNA structures to control diseases. However, due to the complexity and dynamic nature of RNA molecules, the determination of RNA structures using experimental approaches is not always feasible, and computational models employing force fields can provide important insight. The quality of the force field will determine how well the predictions are compared to experimental observables. Stacking in nucleic acids is one such structural property, originating mainly from London dispersion forces, which are quantum mechanical and are included in molecular mechanics force fields through nonbonded interactions. Geometric descriptions are utilized to decide if two residues are stacked and hence to calculate the stacking free energies for RNA dinucleoside monophosphates (DNMPs) through statistical mechanics for comparison with experimental thermodynamics data. Here, we benchmark four different stacking definitions using molecular dynamics (MD) trajectories for 16 RNA DNMPs produced by two different force fields (RNA-IL and ff99OL3) and show that our stacking definition better correlates with the experimental thermodynamics data. While predictions within an accuracy of 0.2 kcal/mol at 300 K were observed in RNA CC, CU, UC, AG, GA, and GG, stacked states of purine-pyrimidine and pyrimidine-purine DNMPs, respectively, were typically underpredicted and overpredicted. Additionally, population distributions of RNA UU DNMPs were poorly predicted by both force fields, implying a requirement for further force field revisions. We further discuss the differences predicted by each RNA force field. Finally, we show that discrete path sampling (DPS) calculations can provide valuable information and complement the MD simulations. We propose the use of experimental thermodynamics data for RNA DNMPs as benchmarks for testing RNA force fields.

The energy landscapes of bidisperse particle assemblies on a sphere

The interplay between crystalline ordering, curvature, and size dispersity make the packing of bidisperse mixtures of particles on a sphere a varied and complex phenomenon. These structures have functional significance in a broad range of systems, such as cellular organisation in spherical epithelia, catalytic activity in binary colloidosomes, and chemical activity in heterofullerenes. In this contribution, we elucidate the potential energy landscapes for systems of repulsive, bidisperse particles confined to the surface of a sphere. It is commonly asserted that particle size dispersity destroys ordered arrangements, leading to glassy landscapes. Surprisingly, across a range of compositions, we find highly ordered global minima. Moreover, a minority of small particles is able to passivate defects, stabilising bidisperse global minima relative to monodisperse systems. However, our landscape analysis also reveals that bidispersity introduces numerous defective, low-lying states that are expected to cause broken ergodicity in corresponding experimental and computational systems. Probing the global minimum structures further, particle segregation is energetically preferred at intermediate compositions, contrasting with the approximate icosahedral global packing at either end of the composition range. Finally, changing the composition has a dramatic effect on the heat capacity: systems with low-symmetry global minima have melting temperatures an order of magnitude lower than monodisperse or high-symmetry systems. This observation may provide a further example of the principle of maximum symmetry: higher symmetry global minima exhibit a larger energy separation from the minima that define the high-entropy phase-like region of configuration space, raising the transition temperature.

Competition between tubular, planar and cage geometries: a complete picture of structural evolution of Bn (n = 31–50) clusters

Stimulated by the early theoretical prediction of B₈₀ fullerene and the experimental finding of the B₄₀ cage, the structures of medium-sized boron clusters have attracted intensive research interest during the last decade, but a complete picture of their size-dependent structural evolution remains a puzzle.

Nested Basin-Sampling

We report an embarrassingly parallel method for the evaluation of thermodynamic properties over an energy landscape exhibiting broken ergodicity, nested is the likelihood of the observed data D givenbasin-sampling (NBS). We also introduce the No Galilean U-Turn Sampler (NoGUTS), a new sampling scheme based on the No U-Turn Sampler (NUTS) introduced by Hoffman and Gelman (2014) that works with the Galilean Monte Carlo scheme introduced by Betancourt (2012) to aid the efficient generation of new live points. NoGUTS can be thought of as a form of reflective slice sampling with an automatic stopping criterion. We apply this approach to a benchmark atomic cluster of 31 Lennard-Jones atoms, which exhibits a low temperature solid-solid heat capacity peak. The calculated heat capacity is compared with results generated by parallel tempering (PT), basin-sampling parallel tempering (BSPT), and standard nested sampling (NS) simulations. NBS reproduces the full heat capacity curve predicted by PT and BSPT, while the NS calculation with similar computational cost fails to resolve the low-temperature solid-solid phase transition.

Temperature and isomeric effects in nanoclusters

The canonical thermodynamics of clusters is determined quantum mechanically in the general case of multiple minima of the potential energy surface (PES) using the superposition approximation. As an illustration of the consequences of our approach, we study in detail the thermodynamic properties of CuN clusters with N from 2 to 150 as a function of cluster size, temperature, and number of isomers. Thereby, for instance, solid-solid transition temperatures for several cluster sizes are determined. We show that the cluster vibrations have a strong impact on the stability of the clusters and that this can be observed not only at medium and high temperatures but also at low temperatures and even at T = 0 K. Thus, including zero-point corrections can change the relative energetic ordering of different isomers. This isomeric effect results in an oscillatory dependence of the heat capacity on cluster size at moderate and high temperatures. Moreover, for the identification of magic clusters at non-vanishing temperature, the Helmholtz free energy is analyzed as a function of cluster size and temperature within a one-, two-, and three-minima model of the PES. Thereby, we demonstrate that the concept of magic clusters is strongly temperature dependent so that in several cases clusters change from being magic or non-magic at T = 0 K to be the opposite at non-vanishing temperature. We emphasize that all these effects are not specific for copper clusters alone but can also be observed in other metal or semiconductor nanoclusters.

Exploring Energy Landscapes

Recent advances in the potential energy landscapes approach are highlighted, including both theoretical and computational contributions. Treating the high dimensionality of molecular and condensed matter systems of contemporary interest is important for understanding how emergent properties are encoded in the landscape and for calculating these properties while faithfully representing barriers between different morphologies. The pathways characterized in full dimensionality, which are used to construct kinetic transition networks, may prove useful in guiding such calculations. The energy landscape perspective has also produced new procedures for structure prediction and analysis of thermodynamic properties. Basin-hopping global optimization, with alternative acceptance criteria and generalizations to multiple metric spaces, has been used to treat systems ranging from biomolecules to nanoalloy clusters and condensed matter. This review also illustrates how all this methodology, developed in the context of chemical physics, can be transferred to landscapes defined by cost functions associated with machine learning.

Atomic clusters with addressable complexity

A general formulation for constructing addressable atomic clusters is introduced, based on one or more reference structures. By modifying the well depths in a given interatomic potential in favour of nearest-neighbour interactions that are defined in the reference(s), the potential energy landscape can be biased to make a particular permutational isomer the global minimum. The magnitude of the bias changes the resulting potential energy landscape systematically, providing a framework to produce clusters that should self-organise efficiently into the target structure. These features are illustrated for small systems, where all the relevant local minima and transition states can be identified, and for the low-energy regions of the landscape for larger clusters. For a 55-particle cluster, it is possible to design a target structure from a transition state of the original potential and to retain this structure in a doubly addressable landscape. Disconnectivity graphs based on local minima that have no direct connections to a lower minimum provide a helpful way to visualise the larger databases. These minima correspond to the termini of monotonic sequences, which always proceed downhill in terms of potential energy, and we identify them as a class of biminimum. Multiple copies of the target cluster are treated by adding a repulsive term between particles with the same address to maintain distinguishable targets upon aggregation. By tuning the magnitude of this term, it is possible to create assemblies of the target cluster corresponding to a variety of structures, including rings and chains.

Exploring biomolecular energy landscapes

The potential energy landscape perspective provides both a conceptual and a computational framework for predicting, understanding and designing molecular properties. In this Feature Article, we highlight some recent advances that greatly facilitate structure prediction and analysis of global thermodynamics and kinetics in proteins and nucleic acids. The geometry optimisation procedures, on which these calculations are based, can be accelerated significantly using local rigidification of selected degrees of freedom, and through implementations on graphics processing units. Results of progressive local rigidification are first summarised for trpzip1, including a systematic analysis of the heat capacity and rearrangement rates. Benchmarks for all the essential optimisation procedures are then provided for a variety of proteins. Applications are then illustrated from a study of how mutation affects the energy landscape for a coiled-coil protein, and for transitions in helix morphology for a DNA duplex. Both systems exhibit an intrinsically multifunnel landscape, with the potential to act as biomolecular switches.

Energy landscapes for machine learning

Machine learning techniques are being increasingly used as flexible non-linear fitting and prediction tools in the physical sciences. Fitting functions that exhibit multiple solutions as local minima can be analysed in terms of the corresponding machine learning landscape. Methods to explore and visualise molecular potential energy landscapes can be applied to these machine learning landscapes to gain new insight into the solution space involved in training and the nature of the corresponding predictions. In particular, we can define quantities analogous to molecular structure, thermodynamics, and kinetics, and relate these emergent properties to the structure of the underlying landscape. This Perspective aims to describe these analogies with examples from recent applications, and suggest avenues for new interdisciplinary research.

Structure, thermodynamics, and rearrangement mechanisms in gold clusters-insights from the energy landscapes framework

We consider finite-size and temperature effects on the structure of model AuN clusters (30 ≤ N ≤ 147) bound by the Gupta potential. Equilibrium behaviour is examined in the harmonic superposition approximation, and the size-dependent melting temperature is also bracketed using molecular dynamics simulations. We identify structural transitions between distinctly different morphologies, characterised by various defect features. Reentrant behaviour and trends with respect to cluster size and temperature are discussed in detail. For N = 55, 85, and 147 we visualise the topography of the underlying potential energy landscape using disconnectivity graphs, colour-coded by the cluster morphology; and we use discrete path sampling to characterise the rearrangement mechanisms between competing structures separated by high energy barriers (up to 1 eV). The fastest transition pathways generally involve metastable states with multiple fivefold disclinations and/or a high degree of amorphisation, indicative of melting. For N = 55 we find that reoptimising low-lying minima using density functional theory (DFT) alters their energetic ordering and produces a new putative global minimum at the DFT level; however, the equilibrium structure predicted by the Gupta potential at room temperature is consistent with previous experiments.

Energy Landscape and Pathways for Transitions between Watson-Crick and Hoogsteen Base Pairing in DNA

The recent discovery that Hoogsteen (HG) base pairs are widespread in DNA across diverse sequences and positional contexts could have important implications for understanding DNA replication and DNA-protein recognition. While evidence is emerging that the Hoogsteen conformation could be a thermodynamically accessible conformation of the DNA duplex and provide a means to expand its functionality, relatively little is known about the molecular mechanism underlying the Watson-Crick (WC) to HG transition. In this Perspective, we describe pathways and kinetics for this transition at an atomic level of detail, using the energy landscape perspective. We show that competition between the duplex conformations results in a double funnel landscape, which explains some recent experimental observations. The interconversion pathways feature a number of intermediates, with a variable number of WC and HG base pairs. The relatively slow kinetics, with possible deviations from two-state behavior, suggest that this conformational switch is likely to be a challenging target for both simulation and experiment.

Structure and Thermodynamics of Metal Clusters on Atomically Smooth Substrates

We analyze the structure of model Ni_N and Cu_N clusters (N = 55, 147) supported on a variety of atomically smooth van der Waals surfaces. The global minima are mapped in the space of two parameters: (i) the laterally averaged surface stickiness, γ, which controls the macroscopic wetting angle, and (ii) the surface microstructure, which produces more subtle but important templating via epitaxial stresses. We find that adjusting the substrate lattice (even at constant γ) can favor different crystal plane orientations in the cluster, stabilize hexagonal close-packed order, or induce various defects, such as stacking faults, twin boundaries, and five-fold disclinations. Thermodynamic analysis reveals substrate-dependent solid-solid transitions in cluster morphology, with tunable transition temperature and sometimes exhibiting re-entrant behavior. These results shed new light on the extent to which a supporting surface can be used to influence the equilibrium behavior of nanoparticles.

Decoding heat capacity features from the energy landscape

A general scheme is derived to connect transitions in configuration space with features in the heat capacity. A formulation in terms of occupation probabilities for local minima that define the potential energy landscape provides a quantitative description of how contributions arise from competition between different states. The theory does not rely on a structural interpretation for the local minima, so it is equally applicable to molecular energy landscapes and the landscapes defined by abstract functions. Applications are presented for low-temperature solid-solid transitions in atomic clusters, which involve just a few local minima with different morphologies, and for cluster melting, which is driven by the landscape entropy associated with the more numerous high-energy minima. Analyzing these features in terms of the balance between states with increasing and decreasing occupation probabilities provides a direct interpretation of the underlying transitions. This approach enables us to identify a qualitatively different transition that is caused by a single local minimum associated with an exceptionally large catchment volume in configuration space for a machine learning landscape.

Impurity effects on solid-solid transitions in atomic clusters

We use the harmonic superposition approach to examine how a single atom substitution affects low-temperature anomalies in the vibrational heat capacity (C_V) of model nanoclusters. Each anomaly is linked to competing solidlike "phases", where crossover of the corresponding free energies defines a solid-solid transition temperature (T_s). For selected Lennard-Jones clusters we show that T_s and the corresponding C_V peak can be tuned over a wide range by varying the relative atomic size and binding strength of the impurity, but excessive atom-size mismatch can destroy a transition and may produce another. In some tunable cases we find up to two additional C_V peaks emerging below T_s, signalling one- or two-step delocalisation of the impurity within the ground-state geometry. Results for Ni₇₄X and Au₅₄X clusters (X = Au, Ag, Al, Cu, Ni, Pd, Pt, Pb), modelled by the many-body Gupta potential, further corroborate the possibility of tuning, engineering, and suppressing finite-system analogues of a solid-solid transition in nanoalloys.

Energy landscapes for a machine-learning prediction of patient discharge

The energy landscapes framework is applied to a configuration space generated by training the parameters of a neural network. In this study the input data consists of time series for a collection of vital signs monitored for hospital patients, and the outcomes are patient discharge or continued hospitalisation. Using machine learning as a predictive diagnostic tool to identify patterns in large quantities of electronic health record data in real time is a very attractive approach for supporting clinical decisions, which have the potential to improve patient outcomes and reduce waiting times for discharge. Here we report some preliminary analysis to show how machine learning might be applied. In particular, we visualize the fitting landscape in terms of locally optimal neural networks and the connections between them in parameter space. We anticipate that these results, and analogues of thermodynamic properties for molecular systems, may help in the future design of improved predictive tools.

Grand and Semigrand Canonical Basin-Hopping

We introduce grand and semigrand canonical global optimization approaches using basin-hopping with an acceptance criterion based on the local contribution of each potential energy minimum to the (semi)grand potential. The method is tested using local harmonic vibrational densities of states for atomic clusters as a function of temperature and chemical potential. The predicted global minima switch from dissociated states to clusters for larger values of the chemical potential and lower temperatures, in agreement with the predictions of a model fitted to heat capacity data for selected clusters. Semigrand canonical optimization allows us to identify particularly stable compositions in multicomponent nanoalloys as a function of increasing temperature, whereas the grand canonical potential can produce a useful survey of favorable structures as a byproduct of the global optimization search.

Self-assembly of colloidal magnetic particles: energy landscapes and structural transitions

The self-assembly of colloidal magnetic particles is of particular interest for the rich variety of structures it produces and the potential for these systems to be reconfigurable.

Temperature dependent structural variations of OH−(H2O)n, n = 4–7: effects on vibrational and photoelectron spectra

In this work, we identified a large number of structurally distinct isomers of midsized deprotonated water clusters using first-principles methods.

Exploring energy landscapes: from molecular to mesoscopic systems

We review a comprehensive computational framework to survey the potential energy landscape for systems composed of rigid or partially rigid molecules. Illustrative case studies relevant to a wide range of molecular clusters and soft and condensed matter systems are discussed.

Energy landscapes of planar colloidal clusters

A short-ranged pairwise Morse potential is used to model colloidal clusters with planar morphologies. Potential and free energy global minima as well as rearrangement paths, obtained by basin-hopping global optimisation and discrete path sampling, are characterised. The potential and free energy landscapes are visualised using disconnectivity graphs. The short-ranged potential is found to favour close-packed structures, with the potential energy primarily controlled by the number of nearest neighbour contacts. In the case of quasi-degeneracy the free energy global minimum may differ from the potential energy global minimum. This difference is due to symmetry effects, which result in a higher entropy for structures with lower symmetry.

Nanothermodynamics of large iron clusters by means of a flat histogram Monte Carlo method

The thermodynamics of iron clusters of various sizes, from 76 to 2452 atoms, typical of the catalyst particles used for carbon nanotubes growth, has been explored by a flat histogram Monte Carlo (MC) algorithm (called the σ-mapping), developed by Soudan et al. [J. Chem. Phys. 135, 144109 (2011), Paper I]. This method provides the classical density of states, gp(Ep) in the configurational space, in terms of the potential energy of the system, with good and well controlled convergence properties, particularly in the melting phase transition zone which is of interest in this work. To describe the system, an iron potential has been implemented, called "corrected EAM" (cEAM), which approximates the MEAM potential of Lee et al. [Phys. Rev. B 64, 184102 (2001)] with an accuracy better than 3 meV/at, and a five times larger computational speed. The main simplification concerns the angular dependence of the potential, with a small impact on accuracy, while the screening coefficients S(ij) are exactly computed with a fast algorithm. With this potential, ergodic explorations of the clusters can be performed efficiently in a reasonable computing time, at least in the upper half of the solid zone and above. Problems of ergodicity exist in the lower half of the solid zone but routes to overcome them are discussed. The solid-liquid (melting) phase transition temperature T(m) is plotted in terms of the cluster atom number N(at). The standard N(at)(-1/3) linear dependence (Pawlow law) is observed for N(at) >300, allowing an extrapolation up to the bulk metal at 1940 ±50 K. For N(at) <150, a strong divergence is observed compared to the Pawlow law. The melting transition, which begins at the surface, is stated by a Lindemann-Berry index and an atomic density analysis. Several new features are obtained for the thermodynamics of cEAM clusters, compared to the Rydberg pair potential clusters studied in Paper I.

A conformational factorisation approach for estimating the binding free energies of macromolecules

We present a conformational factorization approach. The theory is based on a superposition partition function, decomposed as a sum over contributions from local minima. The factorisation greatly reduces the number of minima that need to be considered, by employing the same local configurations for groups that are sufficiently distant from the binding site. The theory formalises the conditions required to analyse how our definition of the binding site region affects the free energy difference between the apo and holo states. We employ basin-hopping parallel tempering to sample minima that contribute significantly to the partition function, and calculate the binding free energies within the harmonic normal mode approximation. A further significant gain in efficiency is achieved using a recently developed local rigid body framework in both the sampling and the normal mode analysis, which reduces the number of degrees of freedom. We benchmark this approach for human aldose reductase (PDB code 2INE). When varying the size of the rigid region, the free energy difference converges for factorisation of groups at a distance of 14 Å from the binding site, which corresponds to 80% of the protein being locally rigidified.

Where macro meets micro

Reconciling or somehow linking the macroscopic and microscopic approaches to chemical and physical processes has been a challenge unaddressed for many years. One approach, presented here, treats the issue by examining individual phenomena well described by a macro approach that fails when applied to small systems. The key to the approach is determining the approximate system size below which the breakdown of the macro description is observable. The most developed example is the failure of the Gibbs phase rule for sufficiently small atomic clusters. Other examples, such as the onset, at sufficient size, of the insulator-to-metal transition, are discussed, as are some still more challenging phenomena.

Exploring Energy Landscapes: Metrics, Pathways, and Normal-Mode Analysis for Rigid-Body Molecules

We present new methodology for exploring the energy landscapes of molecular systems, using angle-axis variables for the rigid-body rotational coordinates. The key ingredient is a distance measure or metric tensor, which is invariant to global translation and rotation. The metric is used to formulate a generalized nudged elastic band method for calculating pathways, and a full prescription for normal-mode analysis is described. The methodology is tested by mapping the potential energy and free energy landscape of the water octamer, described by the TIP4P potential.