Consistency is the last refuge of the unimaginative.

Oscar Wilde

$\left(\right.>separators="">x$ | The mean value of variable $x$ | § |

$\varnothing $ | Empty set | § |

$\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\u02c6$ | Denotes a unit vector; $\widehat{x}=x\u2215\left|x\right|$ | § |

$\left|\phantom{\rule{0.3em}{0ex}}\right|$ | A vector norm; $\left|x\right|=\sqrt{{\sum}_{i=1}^{\mathit{dim}\left(x\right)}{x}_{i}^{2}}$; also, cardinality of a set | § |

$\ominus $ | Denotes symmetric difference of two sets | § |

$\otimes $ | Denotes vector direct product, a.k.a. dyadic; $a\otimes {b}^{T}=c$, ${c}_{i,j}={a}_{i}{b}_{j}$ | § |

$\left(\phantom{\rule{0.3em}{0ex}},\phantom{\rule{0.3em}{0ex}}\right)$ | A scalar product of two vectors, a.k.a. dot product; $\left(a,b\right)={a}^{T}\cdot b$ | § |

$\left\{\phantom{\rule{0.3em}{0ex}}\right\}$ | A set of objects; ${\left\{{x}_{i}\right\}}_{1}^{n}=\left\{{x}_{1},{x}_{2},{x}_{3},\dots ,{x}_{n}\right\}$ | § |

$0$ | A matrix or a vector filled with zeros | § |

$A$ | One of the two superstates in two-state kinetic model | § |

$A$ | Approximation to the inverse Hessian matrix, ${H}^{-1}$ | § |

$\mathit{Adj}\left[i\right]$ | Set of all the nodes adjacent to node $i$ | § |

$\mathit{AdjIn}\left[i\right]$ | Set of all the nodes connected to node $i$ via incoming edges | § |

$\mathit{AdjOut}\left[i\right]$ | Set of all the nodes connected to node $i$ via outgoing edges | § |

$B$ | One of the two superstates in two-state kinetic model | § |

${\mathit{BLJ}}_{n}$ | Binary Lennard-Jones liquid with $n$ atoms in a periodic cell | § |

$C$ | Square matrix, columns of which are eigenvectors | § |

${C}_{N}$ | A chain graph with $N$ nodes | § |

$D$ | Endpoint separation | § |

${\Delta}_{i}\left(j\right)$ | $i$th displacement in magnitude of an atom between structures $j-1$ and $j$ | § |

$E$ | A set of edges | § |

${\mathcal{\mathcal{E}}}_{\alpha}^{G}$ | Probability of escape from $G$ starting from $\alpha $ in a single step | § |

${E}_{-}$ | Energy barrier corresponding to the reverse reaction | § |

${E}_{+}$ | Energy barrier corresponding to the forward reaction | § |

$F$ | Set of all minima connected to the final endpoint | § |

$\mathcal{\mathcal{F}}$ | Frequency distribution function | § |

${G}_{N}$ | An arbitrary graph with $N$ nodes | § |

$H\left(x\right)$ | Hessian matrix evaluated at point $x$ | § |

$I$ | A set containing all the minima that do not belong to $A\cup B$ | § |

${K}_{N}$ | A complete graph with $N$ nodes | § |

$L$ | Lagrangian function | § |

${\mathit{LJ}}_{n}$ | $n$-atom Lennard-Jones cluster | § |

$\mho \left(\right)separators="">x$ | The set of all possible values of the control variable $x$ | § |

$N$ | Number of atoms; number of nodes in a graph | § |

${N}_{c}$ | Cooperativity index | § |

${N}_{f}$ | Number of frames or points sampled along a path | § |

${N}_{i}$ | Number of images in a band | § |

${N}_{p}$ | Participation index | § |

$\stackrel{\u0303}{N}$ | Participation index evaluated using the endpoints alone | § |

${O}_{k}$ | Displacement overlap evaluated for $k$ atoms using displacements ${d}_{i}\left(j\right)$ | § |

$\mathcal{\mathcal{O}}\left(\phantom{\rule{0.3em}{0ex}}\right)$ | $f\left(n\right)=\mathcal{\mathcal{O}}\left(g\left(n\right)\right)$ means $0\le f\left(n\right)\le \mathit{cg}\left(n\right)$ holds for some constants $c>0$ | § |

$P$ | Transition probability matrix | § |

${P}_{i}^{\mathit{eq}}$ | Equilibrium occupation probability of state $i$ | § |

${P}_{i}\left(t\right)$ | Occupation probability of state $i$ at time $t$ | § |

${P}_{j,i}$ | Probability of transition from state $i$ to state $j$ | § |

${\mathcal{\mathcal{P}}}_{\xi}$ | Pathway probability | § |

${R}_{N}$ | A random graph with $N$ nodes | § |

${R}_{\alpha}$ | $3N$-dimensional rotation matrix about axis $\alpha $, $\alpha \in \left\{x,y,z\right\}$ | § |

$\mathbb{R}$ | The set of all real numbers | § |

$S$ | Set of all minima connected to the starting endpoint | § |

${\Sigma}_{\alpha}^{G}$ | Total probability of escape from $G$ if started at node $\alpha $ | § |

${\mathcal{\mathcal{S}}}_{\alpha ,\beta}^{G}$ | Sum of weights of all pathways connecting $\alpha $ and $\beta $ and confined to $G$ | § |

$T$ | Temperature | § |

${\Theta}_{k}$ | Displacement overlap evaluated for $k$ atoms using displacements ${\Delta}_{i}\left(j\right)$ | § |

${\mathcal{\mathcal{T}}}_{i}^{G}$ | Mean escape time from graph $G$ if started at node $i$ | § |

$U$ | Set of all minima that do not belong to $S\cup F$ | § |

$\Upsilon \left(x,\mathit{\epsilon}\right)$ | A set of feasible points contained in the neighbourhood $\mathit{\epsilon}$ of $x$ | § |

$V$ | Potential energy functional; also, a set of graph nodes | § |

$\text{}\mathcal{\mathcal{V}}\text{}$ | $3N$-dimensional vector of velocities^{∗} | § |

$\stackrel{\u0303}{V}$ | Spring potential | § |

$W\left(a,b\right)$ | Weight of the shortest path $\xi =a\leftarrow b$; $W\left(a,b\right)=-\mathrm{ln}\left({\mathcal{\mathcal{W}}}_{\xi}\right)$ | § |

$\mathbb{W}$ | The set of whole numbers; $\mathbb{W}=\left\{0,1,2,\dots \phantom{\rule{0.3em}{0ex}}\right\}$ | § |

${\mathcal{\mathcal{W}}}_{\xi}$ | Product of branching probabilities associated with path $\xi $ | § |

$X$ | $3N$-dimensional vector representing a point in configuration space | § |

$\Xi $ | Pathway ensemble | § |

$a$ | A state that belongs to a superstate $A$ | § |

$\alpha $ | Pathway nonlinearity index | § |

$b$ | A state that belongs to a superstate $B$ | § |

$\beta $ | Energy barrier asymmetry index | § |

$c$ | Eigenvector | § |

$\mathrm{det}M$ | A determinant [33] (a scalar-valued function) of matrix $M$ | § |

${d}_{i}$ | Integrated path length for atom $i$; also, degree of node $i$ | § |

${d}_{i}\left(j\right)$ | Displacement of atom $i$ between structures $j-1$ and $j$ | § |

${e}_{j,i}$ | Directed edge that describes a transition from node $i$ to node $j$ | § |

$\mathit{\epsilon}$ | A parameter in LJ potential (the depth of the potential energy well) | § |

$\mathit{\epsilon}$ | Small positive parameter | § |

$\eta $ | Number of atomic degrees of freedom | § |

$f\left(x\right)$ | Objective function of a vector argument $x$ | § |

$g$ | $3N$-dimensional gradient vector of the true potential | § |

$\gamma $ | Kurtosis of a distribution evaluated using moments about the mean | § |

${\gamma}^{\prime}$ | Kurtosis of a distribution evaluated using moments about the origin | § |

$\stackrel{\u0303}{g}\parallel $ | Spring gradient vector component parallel to the path | § |

$\stackrel{\u0303}{g}\perp $ | Spring gradient vector component perpendicular to the path | § |

$\stackrel{\u0303}{g}$ | $3N$-dimensional gradient vector of the spring potential | § |

$g\parallel $ | True gradient vector component parallel to the path | § |

$g\perp $ | True gradient vector component perpendicular to the path | § |

$i,j,k$ | Indices; range and meaning may vary depending on the context | § |

${k}_{B}$ | Boltzmann’s constant | § |

${k}_{j,i}$ | Rate constant for transitions from state $i$ to state $j$ | § |

${k}_{\mathit{spr}}$ | Spring force constant | § |

$l\left(\xi \right)$ | Length of path $\xi $ | § |

$\lambda $ | Eigenvalue | § |

$m$ | Atomic mass | § |

${m}_{n}$ | $n$th moment of a distribution function about the mean | § |

${m}_{n}^{\prime}$ | $n$th moment of a distribution function about the origin | § |

$n$ | Time parameter of a discrete-time stochastic process | § |

$o\left(\phantom{\rule{0.3em}{0ex}}\right)$ | $f\left(n\right)=o\left(g\left(n\right)\right)$ means $0\le f\left(n\right)\le \mathit{cg}\left(n\right)$ holds for all constants $c>0$^{∗} ^{∗}Otherwise known as an upper bound that is not asymptotically tight. | § |

$p$ | Search direction vector | § |

$\pi $ | Path length asymmetry index | § |

${r}_{i}\left(j\right)$ | Three-dimensional Cartesian coordinates vector of atom $i$ for structure $j$ | § |

$s$ | Integrated path length | § |

$\sigma $ | A parameter in the LJ potential (${2}^{1\u22156}\sigma $ is the pair equilibrium separation) | § |

$\text{}\tau \text{}$ | $3N$-dimensional tangent vector | § |

${\tau}_{i}$ | Mean waiting time in state $i$ before escape | § |

$t$ | Time | § |

$\mathit{\delta t}$ | Time integration step | § |

${\phantom{\rule{0.3em}{0ex}}}^{T}$ | A matrix or vector transpose | § |

$\varpi $ | Step size | § |

${v}_{i}$ | $i$th graph node | § |

$w\left(u,v\right)$ | Weight of the undirected edge connecting nodes $u$ and $v$ | § |

${x}_{i}$ | $i$th component of vector $x$ | § |

$\xi $ | A pathway | § |

$x,y,z$ | Vectors; dimensionality and meaning may vary depending on the context | § |