Rate coefficients can fluctuate in statically and dynamically disordered kinetics. Here, we relate the rate coefficient for an irreversibly decaying population to the Fisher information. From this relationship we define kinetic versions of statistical-length squared and divergence that measure cumulative fluctuations in the rate coefficient. We show the difference between these kinetic quantities measures the amount of disorder, and is zero when the rate coefficient is temporally and spatially unique.

We introduce a new method for sampling a general multidimensional distribution function Px using a quasiregular grid (QRG) of points xi (i = 1, …, N). This grid is constructed by minimizing a pairwise functional, ∑u(xi, xj) → min, with the short-range pair pseudopotential u(xi, xj), defined locally according to the underlying distribution P(x). While QRGs can be useful in many diverse areas of science, in this paper, we apply them to construct Gaussian basis sets in the context of solving the vibrational Schrödinger equation. Using some 2D and 3D model systems, we demonstrate that the resulting optimized Gaussian basis sets have properties superior to other choices explored previously in the literature.

Dynamical disorder motivates fluctuating rate coefficients in phenomenological, mass-action rate equations. The reaction order in these rate equations is the fixed exponent controlling the dependence of the rate on the number of species. Here we clarify the relationship between these notions of (dis)order in irreversible decay, n A → B, n = 1, 2, 3, . . ., by extending a theoretical measure of fluctuations in the rate coefficient. The measure, J n − L 2 n ≥ 0, is the magnitude of the inequality between J n , the time-integrated square of the rate coefficient multiplied by the time interval of interest, and L 2 n , the square of the time-integrated rate coefficient. Applying the inequality to empirical models for non-exponential relaxation, we demonstrate that it quantifies the cumulative deviation in a rate coefficient from a constant, and so the degree of dynamical disorder. The equality is a bound satisfied by traditional kinetics where a single rate constant is sufficient. For these models, we show how increasing the reaction order can increase or decrease dynamical disorder and how, in either case, the inequality J n − L 2 n ≥ 0 can indicate the ability to deduce the reaction order in dynamically disordered kinetics.

We revisit the collocation method of Manzhos and Carrington [224110J. Chem. Phys.2016145 in which a distributed localized (e.g., Gaussian) basis is used to set up a generalized eigenvalue problem to compute the eigenenergies and eigenfunctions of a molecular vibrational Hamiltonian. Although the resulting linear algebra problem involves full matrices, the method provides a number of important advantages, namely, (i) it is very simple both conceptually and numerically, (ii) it can be formulated using any set of internal molecular coordinates, (iii) it is flexible with respect to the choice of the basis, (iv) no integrals need to be computed, and (v) it has the potential to significantly reduce the basis size through optimizing the placement and the shapes of the basis functions. In the present paper, we explore the latter aspect of the method using the recently introduced, and here further improved, quasi-regular grids (QRGs). By computing the eigenenergies of the four-atom molecule of formaldehyde, we demonstrate that a QRG-based distributed Gaussian basis is superior to the previously used choices.

We revisit the collocation method of Manzhos and Carrington (J. Chem. Phys. 145, 224110, 2016) in which a distributed localized (e.g., Gaussian) basis is used to set up a generalized eigenvalue problem to compute the eigenenergies and eigenfunctions of a molecular vibrational Hamiltonian. Although the resulting linear algebra problem involves full matrices, the method provides a number of important advantages. Namely: (i) it is very simple both conceptually and numerically, (ii) it can be formulated using any set of internal molecular coordinates, (iii) it is flexible with respect to the choice of the basis, and (iv) it has the potential to significantly reduce the basis size through optimizing the placement and the shapes of the basis functions.In the present paper we explore the latter aspect of the method using the recently introduced, and here further improved, quasi-regular grids (QRGs). By computing the eigenenergies of the four-atom molecule of formaldehyde, we demonstrate that a QRG-based distributed Gaussian basis is superior to the previously used choices.