JP7589546B2 - How to find the route of spread - Google Patents

Description

The present invention relates to a method for searching for diffusion paths.

In order to further improve the performance of various materials, there is active research into new materials and elements that can replace some of the elements that make up a substance.

As mentioned above, when searching for new materials or replacement elements, it is preferable to accurately understand the routes that atoms that affect the desired reaction, function, etc. take to move and diffuse within the crystals of the target substance.

However, elements are often densely packed inside a crystal, and there appear to be only small gaps between the atoms that make up the crystal. For this reason, if the size of an atom whose diffusion path is to be investigated is estimated using the atomic radius, ionic radius, or van der Waals radius, it is difficult to identify the diffusion path of that atom by comparing the size of multiple gaps within the crystal.

In view of the problems inherent in the above-mentioned conventional techniques, one aspect of the present invention aims to provide a method for searching for diffusion paths that can efficiently search for diffusion paths of atoms within a crystal.

In order to solve the above problem, according to one aspect of the present invention,
The computer
an initial structure setting step of setting the positions of a plurality of atoms contained in a crystal used in the diffusion path search;
a calculation step of performing a molecular dynamics calculation using the plurality of atoms whose positions have been set in the initial structure setting step;
an analysis step of performing an independent component analysis on the coordinate data of the plurality of atoms obtained in the calculation step to determine directions in which the plurality of atoms tend to move;
a structural model creation step of creating a plurality of structural models in which the plurality of atoms are positioned in directions in which they are likely to move, based on the results of the analysis step;
The present invention provides a diffusion path searching method for searching for an atomic diffusion path from the plurality of structural models created in the structural model creating step.

According to one aspect of the present invention, a method for searching for diffusion paths can be provided that can efficiently search for diffusion paths of atoms within a crystal.

FIG. 2 is a hardware configuration diagram of a simulation device according to an embodiment of the present invention. 1 is a block diagram showing functions of a simulation device according to an embodiment of the present invention; FIG. 2 is a schematic diagram showing the initial structure of LiMn ₂ O ₄ for which the diffusion path was explored in Example 1. FIG. 2 is a schematic diagram showing the diffusion path of Li atoms in LiMn ₂ O ₄ obtained in Example 1.

The following describes the embodiments of the present invention, but the present invention is not limited to the following embodiments, and various modifications and substitutions can be made to the following embodiments without departing from the scope of the present invention.

The method for searching for a diffusion path according to the present embodiment may include the following steps.
An initial structure setting process for setting the positions of multiple atoms contained in the crystal to be used in the diffusion path search.
A calculation process in which molecular dynamics calculations are performed using multiple atoms whose positions have been set in the initial structure setting process.
An analysis process in which independent component analysis is performed on the coordinate data of multiple atoms obtained in the calculation process to determine the direction in which multiple atoms are likely to move.

A structural model creation process in which multiple structural models are created in which multiple atoms are positioned in directions that are easy for them to move, based on the results of the analysis process.
A diffusion path search process for searching for atomic diffusion paths from the multiple structural models created in the structural model creation process.

The inventors of the present invention have conducted extensive research into an efficient method for searching for atomic diffusion paths within a crystal. When a crystal structure is viewed at the atomic level, the atoms are densely packed. However, when the atoms move in a concerted manner, transitional spaces are formed within the crystal, which often become diffusion paths. By converting the fluctuations of a group of atoms into multiple characteristic movements and analyzing these movements, it becomes possible to discover specific phenomena and find diffusion paths. Furthermore, a multi-component analysis technique is an effective method for the above analysis.

The method for searching for diffusion paths in this embodiment uses independent component analysis, a method of multi-component analysis, to analyze the movements of multiple atoms in a crystal from the results of molecular dynamics. This allows the direction in which atoms are likely to move to be derived, and by searching for diffusion paths from a structural model in which atoms are arranged in the direction in which they are likely to move, it has been discovered that diffusion paths can be derived efficiently, leading to the completion of the present invention.

Each step is explained below.
(Initial structure setting process)
In the initial structure setting step, the positions of a plurality of atoms contained in the crystal used in the diffusion path search can be set, i.e., the initial coordinates of the atoms in the crystal used in the calculation step can be set.

In the initial structure setting step, the specific method for setting the positions of the multiple atoms contained in the crystal used to search for the diffusion path is not particularly limited. For example, the atomic arrangement of each atom can be set based on the crystal structure of the crystal, which is determined experimentally or disclosed in a literature, etc., to form the initial structure.

The initial structure set in the initial structure setting step may or may not include a diffusing atom, which is an atom for searching for a diffusion path. In other words, it may be only a skeleton structure excluding the diffusing atom. In the diffusion path searching method of the present embodiment, a diffusion path is searched for from the size of a space formed by the movement of atoms other than the diffusing atom, so that even if calculation and search are performed for a crystal in a state in which the diffusing atom is not included, there is no significant difference in the result.
(Calculation process)
In the calculation step, a molecular dynamics calculation can be performed using a plurality of atoms whose positions have been set in the initial structure setting step.

Molecular dynamics calculations are a computer simulation method for the physical movement of atoms, and can determine the time evolution of atomic positions by numerically solving Newton's equations of motion. Therefore, by carrying out the calculation process, it is possible to determine the time series changes in the coordinates of multiple atoms set in the initial structure setting process.

In molecular dynamics calculations, information about atoms and interatomic interactions is expressed as a function form for describing potential energy and its parameter set (force field).

The type of force field used in the molecular dynamics calculation in the calculation process is not particularly limited, and various force fields can be used. For example, EAM, MEAM, etc. can be used for metal/alloy systems, Buckingham, BKS, Clay-FF, CVFF_aug, etc. can be used for inorganic compound systems, Tersoff, etc. can be used for semiconductor systems, and PCFF, Compass, MMFF, OPLS-AA, AMBER, CHARMM, UFF, etc. can be used for organic compound systems. In addition, a force field selected from existing force fields such as X-Pol, AMBER polarization force field, CHARMM polarization force field, etc., which are polarization force fields, and ReaxFF, which is a reaction force field, or a force field created by yourself as necessary can be used.

In some cases, the charge of the target atom is not specified in existing force fields. In such cases, it is possible to use the RESP (Restrained ElectroStatic Potential) charge or the AM1-BCC (Bond Charge Correction) charge.

The program (software) used for the molecular dynamics calculation is not particularly limited, but for example, a program selected from existing programs such as LAMMPS, DL_POLY, Gromacs (Groningen Machine for Chemical Simulations), AMBER, CHARMM, NAMD, etc., or a program created by the user can be used.

The environment in which molecular dynamics calculations are performed can be, for example, in a vacuum, or under periodic boundary conditions if a solvent is included.

The numerical integration method for solving Newton's equation of motion when performing molecular dynamics calculations is not particularly limited, but a method selected from, for example, the Verlet method, the velocity Verlet method, the Leap-frog method, the predictor-modifier method, etc. can be used.

The time span for performing molecular dynamics calculations is not particularly limited, but it is preferable to select a time span that allows the directions in which the multiple atoms that make up the crystal tend to move to be understood and that can reduce calculation costs. The time span for performing molecular dynamics calculations can be, for example, 0.5 fs or more and 2 fs or less.

The temperature control method is not particularly limited, but may be selected from the following: velocity scaling method, Nose-Hoover heat bath method, Nose-Hoover chain method, Berendsen heat bath method, Andersen heat bath method, Langevin dynamics method, etc.

The method of controlling pressure under periodic boundary conditions is not particularly limited, but a method selected from, for example, the Berendsen method, the Parinello-Rahman method, etc. can be used.

The cutoff method can be used to calculate long-range interactions such as electrostatic interactions and van der Waals interactions. In particular, the Particle-Mesh Ewald method and multipole expansion method can be used to calculate electrostatic interactions under periodic boundary conditions.

The molecular dynamics calculation in the calculation process can be performed using, for example, a normal computer system equipped with a CPU (Central Processing Unit), RAM (Random Access Memory), various storage media such as a hard disk, an output device such as a display, an input device such as a keyboard, various peripheral devices, etc. Examples of computer systems include network servers, workstations, and personal computers.

Specifically, for example, the above-mentioned molecular dynamics calculation program can be stored in a storage medium or the like, and the program can be executed by a CPU while reading in an initial structure and conditions stored in a storage medium such as a RAM or input from an input device such as a keyboard.
(Analysis process)
In the analysis step, an independent component analysis (ICA) is performed on the coordinate data of the multiple atoms obtained in the calculation step, and the direction in which the multiple atoms tend to move can be found.

Independent component analysis, like principal component analysis, is a method of multi-component analysis. Principal component analysis focuses only on the second-order correlation of components and seeks a transformation that makes them uncorrelated. In contrast, independent component analysis seeks a transformation that separates components into independent components based on higher-order statistics or temporal correlation. In other words, independent component analysis takes the position that the observed components arise from the superposition of multiple independent sources.

Independent component analysis is a statistical method for estimating these sources using only observed data, and can also be used to extract hidden features in observed data.

This section provides an overview of independent component analysis.

Assuming that there are n statistically independent sources s(t), as expressed in the following equation (1), n in equation (1) is n ≥ 2, and the average of each component is 0. Also, T is the transposed matrix.

また、観測データｘ（ｔ）は、以下の式（２）で表されるとする。ただし、式（２）中のｍは、ｍ≧２であり、各成分の平均は０とする。

The observed data x(t) is expressed by the following formula (2): where m in formula (2) is m≧2, and the average of each component is 0.

ここで、上記ｓ（ｔ）とｘ（ｔ）との間に、以下の式（３）で表される線形の関係を仮定する。

Here, it is assumed that there is a linear relationship between s(t) and x(t) as expressed by the following equation (3).

そうすると、独立成分分析は、発生源ｓ（ｔ）と、ｍ×ｎの実数行列Ａの知識を持たずに観測データｘ（ｔ）をｎ個の独立な成分に分離することになる。解法の手順は何らかの方法で、実数行列Ａを求めてから、発生源ｓ（ｔ）を求めることになる。

In this way, independent component analysis separates the observed data x(t) into n independent components without knowledge of the source s(t) and the m x n real matrix A. The solution procedure involves first finding the real matrix A in some way, and then finding the source s(t).

Next, we explain the solution method for independent component analysis.

Independent component analysis converts observed data into independent components, but if they are independent of each other, they are also uncorrelated. In independent component analysis, preprocessing such as centering, whitening, and dimensionality reduction is performed to reduce complexity, and principal component analysis is often used as a preprocessing step. Therefore, we will first explain principal component analysis.
(A) Principal Component Analysis In principal component analysis, a covariance matrix C expressed by the following formula (4) is calculated for m-dimensional time series data to be analyzed, where T indicates the observation time.

さらに、以下の式（５）から、共分散行列Ｃの固有値問題を解いて、固有値行列Λ、固有値ベクトル行列Ｕを求める。

Furthermore, an eigenvalue problem of the covariance matrix C is solved from the following equation (5) to obtain an eigenvalue matrix Λ and an eigenvalue vector matrix U.

固有値ベクトル行列Ｕの列ベクトルｕ_ｉは、無相関な方向を表す。

A column vector u _i of the eigenvalue vector matrix U represents an uncorrelated direction.

The observed data x(t) can be expressed as the following equation (6).

また、Ｉを単位行列とすると、以下の式（７）の関係が成り立つ。

Furthermore, if I is a unit matrix, the relationship of the following equation (7) holds.

つまり、単位行列Ｉのi行j列の成分は、以下の式（８）となる。

That is, the component of row i and column j of the unit matrix I is expressed by the following formula (8).

ただし、δ_ijは、以下の式（９）で表されるクロネッカーのデルタである。

Here, δ _ij is the Kronecker delta expressed by the following equation (9).

このため、観測した時系列データは、以下の式（１０）で表される。

Therefore, the observed time series data is expressed by the following equation (10).

従って、ｉ番目の固有値を持つ固有ベクトルｕ_ｉ方向に射影することで、固有ベクトル方向の成分ａ_ｉを取り出すことができる。すなわち、観測データｘ（ｔ）を固有ベクトルの方向に分離できることになる。
（Ｂ）独立成分分析の解法について
独立成分分析の解法は、確率分布の独立性に基づく分離法と時間構造に基づく分離法（ｔＩＣＡ）があるが、ここでは後者の方法をとる。

Therefore, by projecting in the direction of the eigenvector u _i having the i-th eigenvalue, the component a _i in the eigenvector direction can be extracted. In other words, the observed data x(t) can be separated in the direction of the eigenvector.
(B) Solution Method for Independent Component Analysis There are two solution methods for independent component analysis: a separation method based on the independence of probability distributions and a separation method based on time structure (tICA). Here, the latter method is used.

In the method for searching for diffusion paths in this embodiment, the observed data x(t) uses the trajectory obtained from the molecular dynamics calculations obtained in the calculation process described above. For this reason, the observed data x(t) is assumed to be ergodic. In addition, the conditions dictate that the source s(t) is independent. Therefore, the real matrix A is invariant with respect to the time lag τ, and the observed data x(t+τ) and the source s(t+τ) for the time lag τ have the following relationship.

よって、時間遅れの共分散行列は、以下の式（１２）で表される。

Therefore, the time lag covariance matrix is expressed by the following equation (12).

ただし、式（１２）内のＲ_ｓｉ（τ）は、ｓ_ｉ（ｔ）の自己相関関数である。２つの共分散行列を用いて、以下の式（１３）の一般化固有値問題を解いて固有値行列Ｋと、固有値ベクトル行列Ｆを求める。

Here, R _si (τ) in equation (12) is the autocorrelation function of s _i (t). Using the two covariance matrices, the generalized eigenvalue problem of equation (13) below is solved to obtain an eigenvalue matrix K and an eigenvalue vector matrix F.

式（１３）内のＤは、通常非対称のため、ＫやＦの要素は複素数となる。よって、複素数を避けるためにはＤを対称化しておけばよい。

Since D in the formula (13) is usually asymmetric, the elements of K and F are complex numbers. Therefore, in order to avoid complex numbers, D should be made symmetric.

Although the eigenvectors u _i in the principal component analysis are orthogonal, the eigenvectors f _i obtained by the tICA are not orthogonal. However, the following relationship in equation (14) is satisfied.

このため、以下の式（１５）の様に規定することで、非直交基底として利用できる。

Therefore, by defining it as in the following equation (15), it can be used as a non-orthogonal basis.

したがって、時系列データは以下の式（１６）のように展開できる。

Therefore, the time series data can be expanded as shown in the following equation (16).

このことから、ｇ_ｉが運動の方向を表す独立成分（ｓ_ｉ）であることがわかる。だたし、独立成分へトラジェクトリｘ（ｔ）を射影する場合は、ｇ_ｉではなく、ｆ_ｉを使用することになる。また、固有値は運動の時間スケールを特徴づけており、固有値が大きいほど独立成分方向の運動の時間スケールが遅いことを表している。よって、Ｉ番目の固有ベクトルに射影したトラジェクトリａ_ｉ（ｔ）を解析することでその動的振る舞いがわかる。

From this, it can be seen that g _i is an independent component (s _i ) that represents the direction of movement. However, when projecting the trajectory x(t) onto the independent component, f _i is used instead of g _i . In addition, the eigenvalue characterizes the time scale of the movement, and the larger the eigenvalue, the slower the time scale of the movement in the independent component direction. Therefore, by analyzing the trajectory a _i (t) projected onto the I-th eigenvector, its dynamic behavior can be understood.

As for the time delay τ, ideally, the result does not depend on τ, but in many cases, different results may be obtained depending on τ. For this reason, it is preferable to select a τ that has a similar value even if the value of τ is changed.
(Structural model creation process)
In the structural model creation step, a plurality of models of a structure in which a plurality of atoms are positioned in a direction in which they are likely to move can be created based on the results of the analysis step.

Specifically, for example, the coordinates of multiple atoms set in the initial structure setting process can be calculated in the analysis process, and multiple structural models can be created in which the positions of each atom are shifted in the direction in which it is easiest to move, by changing the distance of movement, etc.

The method for searching for a diffusion path according to the present embodiment aims to search for a diffusion path of a diffusing atom. For this reason, in the structural model creation step, it is preferable to create a structural model in which at least some of the atoms constituting the crystal, which may be a diffusion path, have a large amplitude. Therefore, it is preferable to create a structural model in which the atoms are moved in the direction of an eigenvector having a large eigenvalue among the results obtained in the analysis step.
(Diffusion Path Search Process)
In the diffusion path searching step, the diffusion path of the atom for which the diffusion path is to be searched, i.e., the diffusion path of the diffusing atom, can be searched for from the multiple structural models created in the structural creation step. Specifically, for example, the multiple structural models created are compared, and for atoms constituting the crystal other than the diffusing atom, a portion where the interatomic distance is larger than the atomic radius of the diffusing atom can be recognized as the diffusion path.

According to the method for searching for diffusion paths in this embodiment described above, the direction in which atoms are likely to move is derived using independent component analysis based on the results of molecular dynamics calculations, and the diffusion path is searched for using a structural model that has been moved in that direction. This makes it possible to efficiently search for diffusion paths that would be difficult to find from a static crystal structure obtained in an experiment in which atoms are in their average positions.

In addition, as described above, a diffusion path is searched for using a structural model that reflects atomic fluctuations based on the results of molecular dynamics calculations, making it possible to search for a diffusion path in a structure that is closer to reality.
[Simulation device]
The simulation device of this embodiment is a simulation device for efficiently searching for diffusion paths of atoms in a crystal. Therefore, it can be called a diffusion path searching device, and can include the following members.

An initial structure setting section that sets the positions of multiple atoms contained in the crystal used to search for diffusion paths.

A calculation section that performs molecular dynamics calculations using multiple atoms whose positions are set in the initial structure setting section.

The analysis section performs independent component analysis on the coordinate data of multiple atoms calculated by the calculation section to determine the direction in which multiple atoms are likely to move.

The structural model creation section uses the results of the analysis section to create multiple structural models in which multiple atoms are positioned in directions that are easy for them to move.

The diffusion path search section searches for atomic diffusion paths from multiple structural models created by the structural model creation section.

As shown in the hardware configuration diagram in FIG. 1, the simulation device 10 of this embodiment is, for example, configured as an information processing device (computer), and can be physically configured as a computer system including a CPU (Central Processing Unit: processor) 11, which is an arithmetic processing unit, a RAM (Random Access Memory) 12 and a ROM (Read Only Memory) 13, which are main storage devices, an auxiliary storage device 14, an input/output interface 15, and a display device 16, which is an output device. These are connected to each other by a bus 17. The auxiliary storage device 14 and the display device 16 may be provided externally.

The CPU 11 controls the overall operation of the simulation device 10 and performs various information processing. The CPU 11 can execute programs (simulation programs) stored in the ROM 13 or the auxiliary storage device 14, such as those described below, to perform molecular dynamics calculations, independent component analysis, and exploration of diffusion paths.

RAM 12 is used as a work area for CPU 11 and may include non-volatile RAM for storing key control parameters and information.

ROM 13 can store programs (simulation programs) etc.

The auxiliary storage device 14 is a storage device such as an SSD (Solid State Drive) or an HDD (Hard Disk Drive), and can store various data, files, etc. necessary for the operation of the simulation device.

The input/output interface 15 includes both a user interface such as a touch panel, keyboard, display screen, and operation buttons, and a communication interface that imports information from an external data recording server, etc., and outputs analysis information to other electronic devices.

The display device 16 is a monitor display or the like. An analysis screen is displayed on the display device 16, and the screen is updated in response to input/output operations via the input/output interface 15.

The functions of the simulation device 10 shown in FIG. 1 can be realized by loading a program (simulation program) from a main storage device 14 such as RAM 32 or ROM 33, and executing it with the CPU 11, thereby reading and writing data in the RAM 12, etc., and operating the input/output interface 15 and the display device 16.

Figure 2 shows a functional block diagram of the simulation device 10 of this embodiment.

As shown in FIG. 2, the simulation device 10 can have a reception unit 21, a processing unit 22, and an output unit 23. These units are realized by software and hardware working together in an information processing device such as a personal computer equipped with a CPU, a storage device, various interfaces, etc., of the simulation device 10, by the CPU executing a simulation method or program that is stored in advance, for example, as described below.

The configuration of each part will be explained below.
(A) Reception Unit The reception unit 21 receives input of commands and data from a user related to processing executed by the processing device 22. Examples of the reception unit 21 include a keyboard or mouse operated by a user to input commands, a communication device for inputting data via a network, and a reading device for inputting data from various storage media such as a CD-ROM or DVD-ROM.
(B) Processing Device The processing device 22 can have an initial structure setting unit 221 , a calculation unit 222 , an analysis unit 223 , a structural model creation unit 224 , and a diffusion path search unit 225 .
(B-1) Initial Structure Setting Unit In the initial structure setting unit 221, the positions of a plurality of atoms contained in the crystal used in the diffusion path search can be set.

The data on the positions of a plurality of atoms set by the initial structure setting unit may be data stored in a database or the like, or may be calculated values calculated from experimental results.
(B-2) Calculation section The calculation section can perform molecular dynamics calculations using multiple atoms whose positions are set in the initial structure setting section. The molecular dynamics calculations have already been explained, so the explanation will be omitted here.
(B-3) Analysis Unit The analysis unit 223 performs independent component analysis on the coordinate data of the multiple atoms obtained by the calculation unit 222, and can determine the direction in which the multiple atoms tend to move. Since the independent component analysis has already been described, a description thereof will be omitted here.
(B-4) Structural Model Creation Unit The structural model creation unit 224 can create a plurality of structural models in which a plurality of atoms are positioned in a direction in which they tend to move, based on the analysis results from the analysis unit 223 .
(B-5) Diffusion Path Searching Unit The diffusion path searching unit 225 can search for the diffusion path of atoms from the multiple structural models created in the structural model creation step. Specifically, for example, the multiple structural models created in the structural model creation unit can be compared, and a portion where the interatomic distance is larger than the atomic radius of the diffusing atom for atoms constituting the crystal other than the diffusing atom can be identified as the diffusion path.
(C) Output Unit The output unit 23 may have a display or the like. The search results obtained by the diffusion path search unit 225 may be output to the output unit 23. The contents of the simulation results to be output are not particularly limited, but for example, the search path found as shown in FIG. 4 may be output to the output unit 23 as an image and displayed.

According to the simulation device of this embodiment described above, the direction in which atoms are likely to move is derived using independent component analysis based on the results of molecular dynamics calculations, and a diffusion path is searched for using a structural model that has been moved in that direction. This makes it possible to efficiently search for diffusion paths that would be difficult to find from a static crystal structure obtained in an experiment in which atoms are in their average positions.

In addition, as described above, a diffusion path is searched for using a structural model that reflects atomic fluctuations based on the results of molecular dynamics calculations, making it possible to search for a diffusion path in a structure that is closer to reality.
[program]
Next, the program of this embodiment will be described.

The program of this embodiment is a program for efficiently searching for atomic diffusion paths within a crystal, and can cause a computer to function as each of the following parts:

An initial structure setting section that sets the positions of multiple atoms contained in the crystal used to search for diffusion paths.

A calculation section that performs molecular dynamics calculations using multiple atoms whose positions are set in the initial structure setting section.

The analysis section performs independent component analysis on the coordinate data of multiple atoms calculated by the calculation section to determine the direction in which multiple atoms are likely to move.

The structural model creation section uses the results of the analysis section to create multiple structural models in which multiple atoms are positioned in directions that are easy for them to move.

The diffusion path search section searches for atomic diffusion paths from multiple structural models created by the structural model creation section.

The program of this embodiment can be stored in various storage media, such as the RAM or ROM of the simulation device described above, or in the main or auxiliary storage devices. Then, by loading the program and executing it with the CPU, data can be read and written in the RAM, etc., and the input/output interface and display device can be operated and executed. For this reason, the matters already described for the simulation device will not be described.

The program of the present embodiment described above may be stored on a computer connected to a network such as the Internet and provided by downloading it via the network. The program of the present embodiment may also be configured to be provided and distributed via a network such as the Internet.

The program of this embodiment may be distributed in a state stored on a recording medium such as an optical disk, such as a CD-ROM, or a semiconductor memory.

According to the program of this embodiment described above, the direction in which atoms are likely to move is determined using independent component analysis based on the results of molecular dynamics calculations, and a diffusion path is searched for using a structural model that has been moved in that direction. This makes it possible to efficiently search for diffusion paths that would be difficult to find from a static crystal structure obtained in an experiment in which atoms are in their average positions.

In addition, as described above, the diffusion path is searched for using a structural model that reflects atomic fluctuations based on the results of molecular dynamics calculations, making it possible to search for diffusion paths in a structure that is closer to reality.

The present invention will be described in more detail below with reference to examples, although the present invention is not limited to the following examples.
[Example 1]
The diffusion path of Li atoms in LiMn ₂ O ₄ was explored by the following procedure.
(Initial structure setting process)
An initial structure of LiMn ₂ O ₄ was set. Specifically, an initial structure 30 of LiMn ₂ O ₄ was set in which lithium atoms 31, manganese atoms 32, and oxygen atoms 33 were arranged in a cell as shown in Fig. 3. In Fig. 3, atoms with the same hatching indicate the same type of atoms.
(Calculation process)
Next, molecular dynamics calculations were used to determine the time series changes in the coordinates of the lithium atom 31, manganese atom 32, and oxygen atom 33, the positions of which were set in the initial structure setting step.

The molecular dynamics calculations were performed using the LAMMPS software, and the force field developed by Ishizawa et al. at Nagoya University. The coordinates of each atom were then input to set the environment in the crystal.

The velocity Verlet method was used to calculate the velocity when performing molecular dynamics calculations, with a time span of 1 fs. The Nose-Hoover chain method was used to control the temperature, with a set temperature of 300 K.

The Particle-Mesh Ewald method was used to calculate long-range interactions.

Molecular dynamics calculations were carried out for 10 nanoseconds under the above conditions.
(Analysis process)
Independent component analysis (tICA) was performed on the changes in the coordinates of multiple atoms obtained in the calculation process to determine the directions in which multiple atoms are likely to move.

When the results obtained in the calculation process were subjected to the analysis process, they were corrected using the following formula (17) so that the average value became 0.

なお、上記式中のｘ（ｔ）は、ｔＩＣＡで使用したトラジェクトリ、ｘ´（ｔ）は計算工程で得られたトラジェクトリ、＜ｘ´＞はその時間平均を表す。

In the above formula, x(t) represents the trajectory used in tICA, x'(t) represents the trajectory obtained in the calculation process, and <x'> represents the time average.

A self-written Python code was used to calculate tICA, and the generalized eigenvalue problem was solved using the Python numerical calculation library Numpy. The time delay τ was set to 50 ps.
(Structural model creation process)
From the results of the analysis process, multiple structural models were created in which the coordinates of multiple atoms set in the initial structure setting process were changed in the direction in which each atom would move easily, as calculated in the analysis process, and the movement distance, etc. were changed.
(Diffusion Path Search Process)
The diffusion paths of the diffusing atoms were explored from multiple structures created during the structure creation process.

In general, slow motion has a large amplitude, i.e., it moves slowly and widely. For this reason, in the analysis step, the trajectory obtained in the calculation step was used to calculate projected trajectories projected onto 10 eigenvectors with large eigenvalues. Then, in the diffusion path search step, it was confirmed whether a space that could become a diffusion path could be obtained among them. Specifically, in the structural model creation step, atoms were moved in the eigenvector direction, and in the diffusion path search step, it was confirmed whether a continuous space was formed between the lithium atom 31A located in the center of Figure 3 and the other four lithium atoms 31 in the vicinity of the lithium atom 31A.

The diffusion paths of the diffusing atoms obtained in the diffusion path exploration process are shown in Figure 4.

As shown in FIG. 4, diffusion paths 411 to 414 were found as diffusion paths for lithium atoms. These diffusion paths are formed to connect lithium atom 31A (see FIG. 3) located in the center of the cell to lithium atoms located around lithium atom 31A.

The diffusion paths 411 to 414 are consistent with the diffusion paths of lithium atoms in LiMn ₂ O ₄ that have been reported so far, and it was confirmed that the method of searching for diffusion paths used in this example is in line with the actual phenomenon.

Claims (1)

The computer
an initial structure setting step of setting the positions of a plurality of atoms contained in a crystal used in the diffusion path search;
a calculation step of performing a molecular dynamics calculation using the plurality of atoms whose positions have been set in the initial structure setting step;
an analysis step of performing an independent component analysis on the coordinate data of the plurality of atoms obtained in the calculation step to determine directions in which the plurality of atoms tend to move;
a structural model creation step of creating a plurality of structural models in which the plurality of atoms are positioned in directions in which they are likely to move, based on the results of the analysis step;
a diffusion path searching step of searching for an atomic diffusion path from the plurality of structural models created in the structural model creating step;

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