# Improvement of Performance, Stability and Continuity by Modified Size-Consistent Multipartitioning Quantum Mechanical/Molecular Mechanical Method

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## Abstract

**:**

## 1. Introduction

## 2. Theory

#### 2.1. Size-Consistent Partitioning

_{(M,m)}and thus, it is not feasible to computationally cover all of them. To reduce the number of partitionings to be considered, we introduce a weight function σ that satisfies the following boundary conditions: σ = 0 in the limit of ordered and disordered partitionings as shown in Scheme 1. Here, we define a partitioning as ordered if all the nearest m solvent molecules to the QM solute are QM. Since the QM solvent molecules diffuse away from the solute in a QM/MM partitioning, the QM region is supposed to be disordered in the course of the MD simulation.

#### 2.2. Score Function

^{(n)}mentioned in Section 2.1, we define a score function for a solvent molecule j in the n-th partitioning as Λ

_{j}

^{(n)}. According to the molecular definition, this score function varies as follows:

^{(n)}

_{QM}contains the QM solvent molecules in the n-th partitioning and r

_{j}is the distance between the QM solute and the nearest j-th solvent molecule; λ

_{QM}(r

_{j}) is a score function for a QM solvent, which are parameterized by range parameters s

_{QM}and t

_{QM}(λ

_{QM}(r

_{j}) = 1 if r

_{j}≤ s

_{QM}and λ

_{QM}(r

_{j}) = 0 if t

_{QM}≤ r

_{j}). Likewise, λ

_{MM}(r

_{j}) is a score function for an MM atom using s

_{MM}and t

_{MM}(λ

_{MM}(r

_{j}) = 0 for r

_{j}≤ s

_{MM}and λ

_{MM}(r

_{j}) = 1 for t

_{MM}≤ r

_{j}). In the present study, we employed spline curves for λ

_{QM}so that

_{MM}satisfies

#### 2.3. Fade-in and Fade-out Functions

^{(n)}

_{QM}is defined for the QM solvent molecules in the n-th partitioning as follows:

^{(n)}

_{QM}= 0 if any QM solvent molecules in the n-th partitioning diffuses far away (beyond s

_{QM}) from the QM solute. Otherwise, O

^{(n)}

_{QM}> 0. Next, we define the fade-in function I

^{(n)}for the QM solvent molecules in the n-th partitioning as follows:

^{(n)}

_{QM}= 0 if all the QM solvent molecules in the n-th partitioning are within s

_{QM}from the QM solute. Otherwise, I

^{(n)}

_{QM}> 0.

^{(n)}

_{MM}and the fade-in function I

^{(n)}

_{MM}for the MM solvent molecules in the n-th partitioning, respectively

^{(n)}

_{MM}= 0 if any MM solvent molecules in the n-th partitioning come within s

_{MM}of the QM solute. Otherwise, O

^{(n)}

_{QM}> 0. On the other hand, I

^{(n)}

_{MM}= 0 if all the MM solvent molecules in the n-th partitioning are far away (beyond t

_{MM}) from the QM solute. Otherwise, I

^{(n)}

_{MM}> 0.

#### 2.4. Weight Functions, Effective Energy and Forces

^{(n)}, can be written as:

^{(n)}satisfies the previously mentioned boundary conditions. Then, the effective potential energy V

^{eff}in MD simulations is

## 3. Modification of the Update Protocol

_{QM}from a QM solute molecule, all the partitionings can have simultaneously a weight of zero, which would result in the collapse of the MD simulation. This would also be the case for the solvent molecules that are defined as MM in all the weighted partitionings. Therefore, to achieve the stability of the MD simulations, the partitionings should have variety in the selection of the QM solvent molecules and as many partitionings as possible should have nonzero weight. To assess simulation stability, let us define σ

_{max}as

_{max}ranges from 1/N to 1, where N represents the total number of partitionings. To achieve stable simulations, σ

_{max}should be kept as small as possible.

_{t}(j)

_{t}(j) describes how much a solvent molecule that is the j-th nearest neighbor to the QM solute behaves as a QM molecule at time t [11].

^{(n)}is normalized in Equation (8), a QM profile ranges from 0 to 1 where the values of unity or zero indicate that the solvent molecule corresponds to a perfectly QM or MM model, respectively. Thus, as a solvent molecule approaches the QM solute, the QM profile should become larger and vice versa. Therefore, to have a smooth connection between the QM and MM regions, the QM profiles should gradually and monotonically decrease as the molecular number j increases. Then, the time-averaged QM profile ω(j) is written as

#### 3.1. Partitioning Update Types

_{QM}= 0 or I

_{MM}= 0, at a certain MD step becomes effectively weighted at the next MD step. Likewise, let MM inside entry be a process where a partitioning has nonzero weight because I

_{MM}> 0. Here, we note that inside updates are not always available depending on the range parameters s and t. For example, suppose all partitionings have consistently m QM solvent molecules. Let d

_{j}be the distance between the QM solute and the j-th nearest solvent molecule. Note that I

_{QM}= 0 for s

_{QM}≥ d

_{m}and I

_{MM}= 0 for d

_{m}

_{+1}≥ t

_{MM}. Since d

_{j}fluctuates during the MD simulations, if t

_{MM}> s

_{QM}, there can be a moment in which t

_{MM}> d

_{m+}

_{1}≥ d

_{m}> s

_{QM}. In this case, partitionings satisfying either I

_{QM}= 0 or I

_{MM}= 0 do not exist. If s

_{QM}≥ t

_{MM}, in contrast, there always exists at least one zero-weighted partitioning satisfying either I

_{QM}= 0 or I

_{MM}= 0 regardless of d

_{m}. To make the update available at every MD time step, we assume s

_{QM}≥ t

_{MM}hereafter.

_{QM}= t

_{MM}and s

_{QM}> t

_{MM}. To have stable simulations, ideally, partitionings should have nonzero weights immediately after the update. Suppose that an updated partitioning with σ = 0 has I

_{QM}= 0 for s

_{QM}> d

_{m}. Because of the diffusion of the QM solvent molecules, sooner or later, the updated partitioning by inside entry will have a nonzero weight, when s

_{QM}< d

_{m}. However, if s

_{QM}>> d

_{m}, it would take time for the partitioning to have a nonzero weight; as a result, a small number of partitionings would have large weights, making the simulation unstable. Thus, s

_{QM}should be as small as possible. For the same reason, t

_{MM}should be as large as possible. Therefore, we assume that s

_{QM}= t

_{MM}is the most efficient situation to suppress σ

_{max}and this will be the condition that we will apply for efficient update unless otherwise stated.

_{QM}= 0 happens to be reweighted again. For instance, suppose that a QM solvent molecule diffuses beyond t

_{QM}and accordingly O

_{QM}= 0. Then, the diffused QM solvent molecule may happen to come back within t

_{QM}again, leading to σ > 0 with O

_{QM}> 0. Likewise, let an MM outside entry be a process where σ > 0 when an MM solvent molecule moves beyond s

_{MM}of the QM solute. In contrast to the QM entries, the MM entries are always available regardless of range parameters s and t.

_{MM}= 0. In such case, it is less likely that this partitioning can have a nonzero weight again in the limited simulation time. Thus, partitionings updated should be carefully checked to see if there are better candidates with nonzero weights among other possible partitionings. In contrast, a QM outside entry seems to happen more frequently because the sphere surface at r = t

_{QM}is larger than those at r = s

_{MM}and s

_{QM}and therefore, the frequency of solvent molecules crossing the surface is also higher.

_{max}. For efficiency, when a partitioning becomes disordered (σ = 0 by either O

_{QM}= 0 or O

_{MM}= 0), we make the partitioning partially-ordered by tuning the solvent molecules irrelevant to O = 0. Otherwise, as previously mentioned, the outside-entered partitioning is highly likely to become disordered again.

#### 3.2. Degree of Order

_{QM}= t

_{MM}, the perfectly ordered partitioning whose QM region consists of the nearest m solvent molecules has always zero weight as described in the previous section. Using this property, in a previous study, we employed a minimum update protocol where the partitioning to be updated is always replaced by the perfectly ordered partitioning, namely the QM region consists of the nearest solvent molecules. Note that when the minimum update is once performed, the partitioning update is not available until solvent diffusion occurs to some extent. Otherwise, more than one partitioning can become identical. As a result, idling times can happen between partitioning updates, which can cause an increase of the maximum weight σ

_{max}and destabilize the MD simulation. Thus, the minimum update protocol is inefficient.

_{QM}O

_{QM}or I

_{MM}O

_{MM}is zero in a new partitioning, while the nonzero one is arbitrary. Thus, keeping σ = 0, new partitionings can be disordered to some extent. The degree of order D indicates how ordered a new partitioning is. If D = 1, the fade-in function I = 0 (O = 1) and the scenario is equivalent to that of the minimum update. If D = 0, I = 1 (O = 0), which indicates that the partitioning is already completely disordered. The four types of update protocols in combination with the degree of order are visualized in Scheme 2 and detailed in Appendix A. We assume that the optimal value of D makes the updated partitionings to become effectively weighted after an entry. Although the optimal value of D is not trivial, it should obviously be between D = 0 and D = 1. Thus, we assessed the optimal value of D in the section below.

## 4. Results

#### 4.1. MD Performance

#### 4.2. Simulation Stability

_{max}sampled along the MD simulations as shown in Figure 2a. As expected, the maximum value of the partitioning weights obviously depends on the number of partitionings; a larger number of partitionings leads to a smaller σ

_{max}value.

_{max}for a SCMP simulation with 60 partitionings using various degrees of order (D = 0.50, 0.75, 0.90 and 0.99) as shown in Figure 2b. Notably, a larger value of D seems to keep σ

_{max}smaller, although there is not a distinct difference for D = 0.90 and 0.99. Thus, we conclude that in what regards simulation stability, the optimal value of D seems to be between 0.90 and 0.99. On the contrary, we emphasize that all the simulations conducted in the present study lasted for 100 ps and never collapsed even with an unfavorable condition such as D = 0.50. This is in contrast to previous studies where the MD simulations sometimes crashed. We think that stabilization is achieved because of the new partitioning update protocol. While we used only two protocols (QM and MM inside entries) in previous studies, we made use of four types of the partitioning update protocol in the preset study (see Appendix A).

#### 4.3. Spatial Continuity

## 5. Computational Details

_{2}O molecules in a cubic box with a side length of 39.48 Å. The range parameters were set as s

_{MM}= 3.5 Å and t

_{MM}= s

_{QM}= 6.0 Å and t

_{QM}= 8.8 Å.

## 6. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

**QM inside entry**: Among the solvent molecules within s

_{QM}of the QM solute, set the furthest solvent molecule QM. Let k solvent molecules be in the range from s

_{MM}to t

_{MM}(s

_{QM}) from the QM center. The nearest D × k solvent molecules are defined as QM, while the rest of k solvent molecules are defined as either QM or MM molecules in a new partitioning.

**MM inside entry**: Among the solvent molecules more than t

_{MM}from the QM solute, set the nearest solvent molecule MM, while the solvent within t

_{MM}of the QM solute should be QM. Let k solvent molecules be in the range from t

_{MM}(s

_{QM}) to t

_{QM}from the QM center. The furthest (1 – D) × k solvent molecules are defined as MM, while the rest of k solvent molecules are defined as either QM or MM molecules in a new partitioning.

**QM outside entry**: Among the solvent molecules more than t

_{QM}from the QM solute, the nearest one should be QM. Suppose the QM region contains m QM solvent molecules and the m-th solvent is at a distance larger than r

_{m}from the QM center. Let k solvent molecules ranging from r

_{m}to t

_{QM}from the QM center. The furthest D × k solvents are defined as MM. The rest of the k solvents are defined as either QM or MM. Likewise, let be l solvent molecules ranging from r

_{m}to t

_{QM}from the QM center. The nearest D × l solvent molecules are defined as QM. The rest of the l solvent molecules are defined as either QM or MM.

**MM outside entry**: Among the solvent molecules within s

_{MM}of the QM solute, the furthest one should be MM. Let the QM region contain m QM solvent molecules and r

_{m}be the distance between the m-th solvent molecule and the QM center. Suppose k solvent molecules exist in the range from r

_{m}to t

_{QM}. The furthest D × k solvent molecules are defined as MM. The rest of the k solvent molecules are defined as either QM or MM. Likewise, let l solvent molecules exist in the range from r

_{m}to t

_{QM}from the QM center. The nearest D × l solvent molecules are defined as QM. The rest of the l solvent molecules are defined as either QM or MM.

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Sample Availability: Not available. |

**Scheme 1.**Concept of weight function. The horizontal axis represents a conceptual index related to the degree of disorder of the QM region.

**Scheme 2.**Four possible patterns of partitioning update. Filled and open circles represent QM and MM solvent, respectively. The open squares stand for either QM or MM solvent.

**Figure 1.**MD performances (wall-time per one MD step) of the SCMP simulations relative to a conventional QM/MM simulation that has the same size of the QM region. Filled circles and the solid line represent the results in the present study and open circles and the dashed line represent the results in a previous study [14]. The benchmark simulations were conducted for a system composed by 2048 water molecules, where the QM region consisted of one solute water and 32 solvent water molecules.

**Figure 2.**Distribution of the maximum value, σ

_{max}(t), sampled over 100 ps MD simulations where respective partitionings contain 1 QM solute and 32 QM solvent molecules. (

**a**) The black, red, green, blue and purple lines represent the simulations with 40, 60, 80, 100 and 200 partitionings, respectively. All the simulations employed D = 0.75; (

**b**) The black, red, green and blue lines represent the results from the simulations with D = 0.50, 0.75, 0.90 and 0.99, respectively; all the simulations were performed with 60 partitionings.

**Figure 3.**(

**a**) Time-averaged QM profiles, ω, and (

**b**) their standard deviations obtained from 100 ps SCMP simulations. The horizontal axis represents the solvent number from the QM solute. Black, red, green and blue lines represent the results obtained by the SCMP method with number of partitionings equal to 40, 60, 100 and 200, respectively.

**Figure 4.**Standard deviations of instantaneous QM profiles, ω

_{t}, obtained from 100 ps SCMP simulations with 60 partitionings containing one QM solute water and 32 QM solvent water molecules. The horizontal axes represent the solvent number from the QM solute. Black, red, green and blue lines represent the results obtained by SCMP with degrees of order D = 0.50, 0.75, 0.90 and 0.99, respectively.

**Figure 5.**RDFs obtained from 100 ps SCMP simulations with 60 partitionings. Black, red, green and blue lines represent the results obtained by SCMP with degrees of order D = 0.50, 0.75, 0.90 and 0.99, respectively. Green line represents the result obtained by the previous SCMP code. Dashed line stands for the full DFTB3/3OB result.

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**MDPI and ACS Style**

Watanabe, H.C.
Improvement of Performance, Stability and Continuity by Modified Size-Consistent Multipartitioning Quantum Mechanical/Molecular Mechanical Method. *Molecules* **2018**, *23*, 1882.
https://doi.org/10.3390/molecules23081882

**AMA Style**

Watanabe HC.
Improvement of Performance, Stability and Continuity by Modified Size-Consistent Multipartitioning Quantum Mechanical/Molecular Mechanical Method. *Molecules*. 2018; 23(8):1882.
https://doi.org/10.3390/molecules23081882

**Chicago/Turabian Style**

Watanabe, Hiroshi C.
2018. "Improvement of Performance, Stability and Continuity by Modified Size-Consistent Multipartitioning Quantum Mechanical/Molecular Mechanical Method" *Molecules* 23, no. 8: 1882.
https://doi.org/10.3390/molecules23081882