Short Range

Short-range interactions are employed for modeling a wide range of physical phenomena. The Lennard-Jones potential, for example, is extensively used to simulate interactions between uncharged particles. Moreover, the Debye-Hückel potential plays a critical role in modeling charged particles within saline solutions. These potentials, while grounded in physical concepts, are also employed as conceptual models. They are often used to hypothesize about the nature of the studied phenomena, particularly when the exact physical underpinnings are not well-defined. This approach is exemplified by the Lennard-Jones potential, which is employed to represent particles with steric interactions and short-range attractive forces. Additionally, it serves as a foundational model, frequently modified to introduce other effects, showcasing its versatility as a fundamental tool in physical system modeling.

While multi-body short-range potentials are more general (naturally resulting from exact coarse graining theory) and used in some applications, this discussion focuses on two-body (pairwise) potentials. The application of a truncation to these potentials leads to the definition of a cutoff radius r_cut, where U(r) = 0 for all r > r_cut. The choice of r_cut is potential-specific and often depends on the parameters within the model. For instance, with the Lennard-Jones potential, a typical cutoff radius is r_cut = 2.5σ, balancing computational efficiency with physical accuracy.

From a computational standpoint, short-range potentials offer several advantages. In contrast to long-range potentials, where the interaction of a particle with all others must be calculated, short-range interactions simplify computations significantly. Generally, there are three main approaches to computing long-range interactions: N-body approximations, Ewald summation, and the fast multipole method, among others. The first two are implemented in UAMMD and in particular Ewald summation is used for computing electrostatic interactions that will be introduced later. However, a common trait among these methods is their relatively high computational cost, exceeding O(N). In contrast, for short-range interactions, it is possible to construct structures, commonly referred to as neighbor lists, which significantly reduce computational complexity to an O(N) level. These neighbor lists efficiently manage the interactions between particles by only considering those within a predetermined distance. This approach ensures that only the most relevant interactions are computed, thereby saving valuable computational resources and time. This efficiency is particularly advantageous in simulations involving a large number of particles, where every reduction in computational load can lead to significantly faster and more efficient simulations.

In our context, we introduce neighbor lists, neighbor lists are understood as structures that enable access to all neighbors of a given particle. Specifically, for a particle i, these lists include particles j for which the condition distance(i, j) < r_neighbors holds, with r_neighbors ≥ r_cut. This criterion ensures that all the particles within the interaction range defined by the cutoff radius r_cut are included in this list.

Various algorithms exist for constructing neighbor lists in computational simulations. The straightforward approach would involve calculating the distances between all particle pairs and then, for each particle i, assembling a list of neighbors comprising those particles j that lie within a distance less than r_cut. However, this method is computationally expensive, characterized by a O(N^2) complexity, akin to the traditional N-body approach.

To optimize this process, more efficient algorithms have been developed, one of which is the cell list method. This method hinges on subdividing the simulation space into a grid of cells. The number of cells in each dimension α (where α includes x, y, and z dimensions) is determined by dividing the simulation volume length L_α by the cutoff radius r_cut and taking the floor value: n_α = floor(L_α / r_cut). Consequently, the total number of cells is given by N_cells = n_x · n_y · n_z.

Cell list example — Example of a two-dimensional cell list. In this example, the size of each cell is w≈r_cut. When determining the neighboring particles of a certain target particle (the blue particle in the figure), we only check those particles that are located in the same cell as the evaluated particle and in the neighboring cells (within the dashed blue square in the figure). Particles that are not found in these cells (the red particles in the figure) are not considered as candidates, which greatly reduces the compilation time. Of the particles we check, only those that are located at a distance less than r_cut (the green particles in the figure) are considered neighbors, so the remainder particles in the neighbor cells (orange particles in the figure) are not considered when the potential is computed.

In this cell list approach, each particle is assigned to the cell it occupies in space. The neighbor list for a particle i is then constructed by examining particles j not only in the same cell as i but also in adjacent cells, respecting periodic boundary conditions. A particle j is added to the neighbor list of i if the distance distance(i, j) is less than r_cut. This approach drastically reduces the number of distance computations. Assuming a uniform distribution of particles, the average number of particles per cell is N_ppc = N / N_cells. In a 3D space, each cell interacts with its own and the 26 adjacent cells, leading to an average of N_checks = 27 · N_ppc = 27 · N / N_cells distance checks per particle, a significant reduction from the N^2 checks required by the direct pairwise comparison method.

Although the cell list method considerably lowers computational complexity, it does have limitations, particularly in GPU-based implementations. While in CPU implementations, the linked cell approach allows for dynamic updating of cells as particles move, the GPU requires a complete reconstruction of the list at each integration step due to the simultaneous movement of all particles. This need for continual reconstruction introduces additional computational overhead.

To efficiently manage the need for frequent reconstruction of neighbor lists in simulations, the Verlet list approach is utilized. The key principle of Verlet lists is to minimize the continuous rebuilding of these lists. This objective is achieved by creating a neighbor list with an enlarged radius, termed the Verlet radius, r_Verlet, which is larger than the cutoff radius, r_cut. As a result, the neighbor list constructed using r_Verlet remains valid for the interactions within the range of r_cut. In essence, for every particle i, all other particles j that satisfy the condition distance(i, j) < r_cut are included in the neighbor list of i. This inclusion holds as long as no particle travels beyond a distance greater than r_max from its initial position when the list was generated. The distance r_max is defined as follows:

\[r_{\text{max}} = (r_{\text{Verlet}} - r_{\text{cut}}) / 2\]

Verlet radius — Initially, two particles lie outside the neigbhor list of each other, separated by the Verlet distance r_Verlet, which is greater than the cutoff radius r_cut. Over time, if they move closer such that their separation is less than r_cut, they should be included in neighbor list of each other for interaction calculations. The maximum allowable movement without updating the Verlet list is r_max = (r_Verlet - r_cut)/2, ensuring all potential interactions are accounted for within the simulation timeframe.

This particular distance is calculated based on the worst-case scenario, as illustrated in Fig. Here, two particles initially distanced at r_Verlet and therefore not in neighbor lists of each other, start moving towards one another in opposite directions. After some time, their distance diminishes to r_cut, at which point their interaction becomes relevant, necessitating their inclusion in the neighbor lists of each other. The total distance traversed by the particles in this scenario is r_Verlet - r_cut, and consequently, each particle travels a distance given by the equation above. Since we cannot conclusively determine whether this situation will occur, the Verlet list must be reconstructed whenever a particle moves the distance defined by the equation.

In summary, we generate the cell list for a radius larger than the cutoff radius, denoted as r_Verlet (note that we use cell lists, but other algorithms could be used for constructing the neighbor list; the Verlet list is an algorithm to trigger the list reconstruction, not for the construction itself). Each time we update the positions of the particles, we check if they have moved a distance greater than r_max compared to the position when the list was built. If so, we reconstruct the list.

The key parameter in Verlet lists is the Verlet radius, r_Verlet. As we increase the radius, we also increase the number of particles we need to check when evaluating an interaction. Conversely, we can perform more updates before needing to reconstruct the list. Therefore, the efficiency of this approach depends quite a bit on the specific system; its density, diffusion, etc.

Let us now consider how Verlet lists behave for certain types of potentials, specifically examining their behavior for certain coarse-grained models. We will use a protein model as an example, but the principles can be generalized to many other cases. In these models, it is common to employ different types of interactions for particles within the same structure “intra” interactions and those involving interactions between particles from different structures, “inter” interactions, [By a common misuse of language, the terms “intramolecular” and “intermolecular” are often employed, even when not dealing with molecules per se]. This situation is depicted in Fig.

Intramolecular/Intermolecular interactions — Coarse-grained models with three copies of a protein (GFP), each colored differently. Interactions involving particles of the same color (same structure) engage with a different potential than those involving particles of different colors (different structures).

The approach combining cell lists and Verlet lists encounters challenges in handling these types of interactions. The feasible solution involves evaluating each interaction to determine the appropriate potential. This process, particularly on GPUs, faces difficulties due to the SIMD architecture. It causes significant divergence in the code, adversely affecting performance. Furthermore, this approach limits the reusability of interactions because it necessitates the implementation of distinct potentials for each specific scenario. For instance, consider a scenario where the inter interactions are modeled using a Lennard-Jones potential, and the intra interactions employ a WCA-type potential. In such a case, it becomes essential to define a unified potential capable of appropriately applying either the Lennard-Jones or WCA interaction, based on the particle pairs (intra/inter). Ideally, implementing a versatile potential system that allows for selective application to various particle pairs would be more efficient, reducing the need for extensive coding.

Conditional Verlet lists

To address these challenges, we have developed conditioned Verlet lists. This algorithm allows us to generate a set of neighbor lists, ensuring that each pair within these lists meets certain predefined conditions. We will discuss the algorithm in detail in a dedicated section later, but for now, let us describe it in general terms and explain how to use the different lists. This approach enables us to create multiple Verlet lists from a single cell list. Let us consider the previous example: our goal is to generate two lists, one for intra conditions and another for inter conditions (noting that in this case, the conditions are mutually exclusive, although the algorithm is applicable to more general conditions). To determine which structure each particle belongs to, we must set the “modelId” variable for each particle. This is done in the structure section; if two particles have the same “modelId,” they belong to the same structure; if they differ, they belong to different structures. In this instance, we are working with “modelId”, but other identifiers like “resId” or “chainId” could also be used. This allows us to define the use of one potential or another based on the hierarchy in the structure of the interacting particles.

The algorithm operates similarly to the Verlet list approach. In the Verlet list, we construct a cell list with a Verlet radius r_Verlet > r_cut and reconstruct the list when particles move a distance greater than r_max from their position at the time of list construction. The construction of conditioned Verlet lists starts similarly by building the cell list. However, instead of creating a single list, we construct multiple lists depending on whether certain conditions are met. For example, for particle i, we iterate over particles in the same cell and neighboring cells, denoted as j. Then, for each pair (i, j), we iterate over possible conditions, such as intra and inter in our case, and if any condition is satisfied, we add particle j to the neighbor list of particle i that fulfills that condition. At the end of the process, we obtain a Verlet list for each condition. The rest of the procedure remains the same. If a particle moves more than r_max, we repeat the process and reconstruct the lists.

Various sets of conditions are available for selection. As a first example of the input format, we will use the “all” condition, where all particles within the Verlet radius are considered for the list, with no additional conditions applied. This condition generates a result similar to the traditional Verlet list, i.e., it only generates a single neighbor list: the “all” list. In the following example, we demonstrate how to select this list and apply a Lennard-Jones potential and a Debye-Hückel potential:

topology:
  forceField:
    nl:
      type: ["VerletConditionalListSet", "all"]
      parameters:
        cutOffVerletFactor: 1.2
    LennardJones:
      type: ["NonBonded", "LennardJonesType2"]
      labels: ["name_i", "name_j", "epsilon", "sigma"]
      data:
        - ["A", "A", 1.4, 1.5]
      parameters:
        condition: "all"
        cutOffFactor: 2.5
    DebyeHuckel:
      type: ["NonBonded", "DebyeHuckel"]
      parameters:
        condition: "all"
        cutOffFactor: 1.2
        debyeLength: 2.3
        dielectricConstant: 80.0

In our input configuration, it is important to note that the sequence of elements does not affect the outcome. We start by defining a conditioned Verlet list, which we label as “nl,” and select the subtype “all.” This action specifies our choice of the “all” condition. Following this, the “all” condition is consistently applied to the potentials, indicating that both the Lennard-Jones and Debye-Hückel potentials will be applicable to all neighbor pairs within their respective cutoff ranges.

Each potential has its unique way of determining its cutoff radius. For the Lennard-Jones potential, the cutoff is calculated using r_cut = σ_max × cutOffFactor, where σ_max represents the highest sigma value in the potential, with a value of 1.5 in this example. On the other hand, the cutoff for the Debye-Hückel potential is determined as r_cut = λ_D × cutOffFactor, with λ_D being the Debye length.

The selection of the Verlet radius for the “nl” list is based on the following formula:

\[r_{\text{Verlet}} = \max(r_{\text{cut}}) \times \text{cutOffVerletFactor}\]

This formula means that the Verlet radius is set to the maximum value among the cutoff radii of the potentials using the Verlet list, multiplied by a predefined factor specified in the parameters. This setup allows for the possibility of declaring multiple conditioned Verlet lists, each of which can be associated with different potentials.

In molecular dynamics, particularly in coarse-grained models, exclusion lists are a common feature. By using exclusion lists, we enumerate a set of particles j that can never be considered neighbors of particle i, regardless of their distance or any other conditions. This condition, the requirement of not being excluded, necessitates the specification of an exclusion list when selected.

In the following example, we illustrate how to define two sets of conditioned Verlet lists, “vcl1” and “vcl2”. The first set, “vcl1,” is of the type “intra_inter.” This means it generates two Verlet lists: one under the “intra” condition, where particles are assumed to have the same molecular index, and the other under the “inter” condition, where each pair of particles belongs to different models. The second set, “vcl2,” is of the type “nonExclIntra_nonExclInter.” This condition also creates two lists similar to the first (using the same names), but additionally considers the exclusion list indicated in the data section. In other words, the “inter” list will include pairs of particles that belong to different molecules and are not excluded from each other.

In this scenario, besides selecting the condition in the interactions, we must also specify which neighbor list we are referring to. This was not necessary in the previous example, as there was only one neighbor list, which by default was used for all “Short Range” interactions:

topology:
  forceField:
    vcl1:
      type: ["ConditionalVerletListSet", "intra_inter"]
      parameters:
        cutOffVerletFactor: 1.5
    vcl2:
      type: ["ConditionalVerletListSet", "nonExclIntra_nonExclInter"]
      parameters:
        cutOffVerletFactor: 1.5
      labels: ["id", "id_list"]
      data:
        - [0, [1, 2, 3]]
        - [1, [0, 2, 3]]
        - [2, [0, 1, 3]]
        - [3, [0, 1, 2]]
        # ...
    interaction1:
      type: ["NonBonded", ...]
      parameters:
        neighbourList: "vcl1"
        condition: "inter"
      # ...
    interaction2:
      type: ["NonBonded", ...]
      parameters:
        neighbourList: "vcl2"
        condition: "intra"
      # ...
    # ...

The complete list of available conditions for conditioned Verlet list sets, as well as the various lists they generate, can be found in the condtions section. Similarly, a list of the available potentials can also be located there. UAMMD-structured allows users to easily add more conditions and interactions.

The available short-range potentials are listed below: