Information processing system, information processing method, and program

Abstract

One aspect of the present invention provides an information processing system comprising at least one processor. The processor is configured to execute a program for executing the following steps of: an acquisition step in which substance information regarding atoms constituting a substance to be calculated is acquired; a first generation step in which a trial electron density of the substance is generated on the basis of the substance information, and the trial electron density is stipulated by a linear combination of expansion functions localized at the positions of the atoms; an optimization step in which expansion coefficients p for the expansion functions constituting the trial electron density are optimized with the energy of the substance set as an objective function in a state in which the positions of the atoms in the trail electron density are fixed; a calculation step in which a force to act on the atoms at the fixed positions is calculated on the basis of the optimization process and result; and a second generation step in which a trial electron density in which the positions of the atoms are updated is generated on the basis of the calculated force. The calculation step includes: calculating, on the basis of the energy of the substance and the expansion coefficients p obtained through the optimization, Hellmann-Feynman forces acting on the atoms; calculating, on the basis of at least one of the expansion coefficients p and the fixed positions of the atoms, at least one of a gradient of the expansion functions with respect to the change in positions of the atoms and a partial differentiation of the expansion coefficients p after the optimization by the fixed positions of the atoms; and correcting the Hellmann-Feynman forces on the basis of at least one of the gradients of the expansion functions with respect to a change in positions of the atoms and the partial differentiation of the expansion coefficients p after the optimization by the fixed positions of the atoms.

Description

Information processing system, information processing method and program

[0001] The present invention relates to an information processing system, an information processing method, and a program.

[0002] a potential energy calculation unit that calculates the total potential energy based on the nuclear coordinates, the internal coordinates, the bond order, and the coordination state; a first-order differential calculation unit that calculates the force acting on each nuclear coordinate and internal coordinate by first-order differential processing of the potential energy; a simulation unit that calculates the amount of movement over a specified short time period based on the nuclear coordinates and the force acting on the internal degree of freedom that determines the direction of the covalent bond of each atom, and updates the coordinate information, and performs molecular dynamics calculations on the time evolution of the nuclear coordinates and the internal degree of freedom variables while repeating the bond order calculation, potential energy calculation, and differential calculation a specified number of times based on the coordinate information; and an output unit that outputs the simulation results.

[0003] By applying the technology described in Patent Document 1, in a molecular dynamics simulation of a large-scale system dealing with a group of several thousand atoms or more, it is possible to efficiently reproduce the structural changes that accompany the rearrangement of covalent bonds in a substance having a complex composition in which multiple elements are mixed, while faithfully reproducing the structural energy changes that depend on the coordination state of each atom.

[0004] Japanese Patent Application Laid-Open No. 2008-052308

[0005] Incidentally, when performing simulations of materials, the arrangement of atoms constituting the material may be important. Therefore, it is desirable to accurately determine the arrangement of atoms. However, for example, when searching for the optimal arrangement of atoms in molecular dynamics calculations, structural optimization, etc., using density functional theory (especially orbital-free density functional theory: OFDFT), the expansion function itself, which serves as the basis for the electron density function, may change due to translation or the like as the atomic positions are updated. In this case, the conversion of the expansion function may reduce the accuracy of estimating the atomic positions based solely on the Hellmann-Feynman force.

[0006] According to one aspect of the present invention, there is provided an information processing system comprising at least one processor, the processor being configured to execute a program for executing the following steps: in the acquisition step, material information relating to atoms constituting a material to be calculated is acquired; in the first generation step, a trial electron density of the material is generated based on the material information, and the trial electron density is defined by a linear combination of expansion functions localized at each position of the atoms; in the optimization step, the energy of the material is used as an objective function, and the expansion coefficients p of the expansion functions constituting the trial electron density are optimized with the positions of the atoms in the trial electron density fixed; in the calculation step, a force to be applied to the atoms at the fixed positions is calculated based on the process and results of the optimization; and The generating step generates a trial electron density in which the positions of the atoms are updated based on the calculated forces, and the calculating step includes calculating a Hellmann-Feynman force acting on each atom based on the energy of the material and the expansion coefficient p obtained by the optimization; calculating at least one of a gradient of the expansion function with respect to a change in the position of the atom and a partial derivative of the expansion coefficient p after optimization with the fixed atomic positions, based on at least one of the fixed atomic positions and the expansion coefficient p; and correcting the Hellmann-Feynman force based on at least one of the gradient of the expansion function with respect to a change in the position of the atom and the partial derivative of the expansion coefficient p after optimization with the fixed atomic positions.

[0007] According to this configuration, it is possible to suppress a decrease in the accuracy of estimating the positions of atoms, which may occur when the expansion function itself is changed by translation or the like.

[0008] It is a configuration diagram showing an information processing system 1. It is a block diagram showing the hardware configuration of an information processing device 2. It is a block diagram showing the hardware configuration of a user terminal 3. It is a block diagram showing the functional configuration of a processor 23. It is a flowchart showing the flow of information processing according to the present embodiment.

[0009] DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS The present invention will be described below with reference to the accompanying drawings. Various features shown in the following embodiments can be combined with each other.

[0010] Incidentally, a program for realizing the software appearing in one embodiment may be provided as a non-transitory computer-readable recording medium, or may be provided so as to be downloadable from an external server, or may be provided so that the program is started on an external computer and its functions are realized on a client terminal (so-called cloud computing).

[0011] Furthermore, various information processing according to an embodiment may realize input and output corresponding to the input. Here, the form of information referenced in such information processing (hereinafter referred to as reference information) is not limited as long as an output is obtained as a result of the input. The reference information may be, for example, rule-based information such as a database, a lookup table, or a predetermined function (including a determination formula such as a regression formula constructed using a statistical method), a trained model that has previously learned the correlation between input and output, or a generative AI such as a large-scale language model or a visual language model that can output a desired result by inputting a prompt.

[0012] In one embodiment, the term "unit" may include, for example, a combination of hardware resources implemented by a circuit in the broad sense and software information processing that can be specifically realized by these hardware resources. In one embodiment, various information is handled, and this information is represented, for example, by physical values ​​of signal values ​​representing voltage or current, high or low signal values ​​as a binary bit set consisting of 0 or 1, or quantum superposition (so-called quantum bits), and communication and calculations can be performed on the circuit in the broad sense.

[0013] Furthermore, a circuit in a broad sense is a circuit realized by at least an appropriate combination of a circuit, circuitry, a processor, a memory, etc. The processor may be a general-purpose processor or a dedicated circuit. That is, it includes application specific integrated circuits (ASICs), programmable logic devices (e.g., simple programmable logic devices (SPLDs), complex programmable logic devices (CPLDs), and field programmable gate arrays (FPGAs)), etc.

[0014] 1. Hardware Configuration This section describes the hardware configuration of an information processing system 1 according to this embodiment. <Information Processing System 1> Fig. 1 is a configuration diagram showing the information processing system 1. The information processing system 1 includes an information processing device 2 and a user terminal 3. The information processing device 2 and the user terminal 3 are configured to be able to communicate with each other via a telecommunications line. In one embodiment, the information processing system 1 is made up of one or more devices or components. For example, if the information processing system 1 is made up of only the information processing device 2, the information processing system 1 can be the information processing device 2. These components will be described below.

[0015] 2 is a block diagram showing the hardware configuration of the information processing device 2. The information processing device 2 includes a communication unit 21, a storage unit 22, and at least one processor 23, and these components are electrically connected via a communication bus 20 inside the information processing device 2. Each component will be further described.

[0016] <Communication Unit 21> The communication unit 21 is preferably a wired communication means such as USB, IEEE 1394, Thunderbolt (registered trademark), or wired LAN network communication, but may also include wireless LAN network communication, mobile communication such as 3G / LTE / 5G, or BLUETOOTH (registered trademark) communication as needed. In other words, it is more preferable to implement the communication unit 21 as a collection of multiple communication means. In other words, the information processing device 2 may communicate various information from the outside via the communication unit 21 and the network.

[0017] <Storage Unit 22> The storage unit 22 stores various pieces of information defined above. This may be implemented, for example, as a storage device such as a solid state drive (SSD) that stores various programs and the like related to the information processing device 2 executed by the processor 23, or as a memory such as a random access memory (RAM) that stores temporarily required information (arguments, arrays, etc.) related to program calculations. The storage unit 22 stores various programs, variables, etc. related to the information processing device 2 executed by the processor 23.

[0018] <Processor 23> The processor 23 processes and controls the overall operations related to the information processing device 2. The processor 23 is, for example, a central processing unit (CPU) not shown. The processor 23 realizes various functions related to the information processing device 2 by reading out predetermined programs stored in the storage unit 22. In other words, information processing by software stored in the storage unit 22 is specifically realized by the processor 23, which is an example of hardware, and can be executed as each functional unit included in the processor 23. These will be described in more detail in the next section. Note that the processor 23 is not limited to being single, and multiple processors 23 may be provided for each function. A combination of these may also be used.

[0019] <User Terminal 3> Next, the hardware configuration of the user terminal 3 will be described. Fig. 3 is a block diagram showing the hardware configuration of the user terminal 3. The user terminal 3 includes a communication unit 31, a storage unit 32, a processor 33, a display unit 34, and an input unit 35, and these components are electrically connected via a communication bus 30 inside the user terminal 3. The description of the communication unit 31, the storage unit 32, and the processor 33 will be omitted as they are the same as the description of each unit in the information processing device 2.

[0020] <Display Unit 34> The display unit 34 may be included in the housing of the user terminal 3 or may be externally attached. The display unit 34 displays a graphical user interface (GUI) screen that can be operated by the user. This is preferably implemented by selectively using display devices such as a CRT display, a liquid crystal display, an organic EL display, or a plasma display depending on the type of user terminal 3.

[0021] <Input Unit 35> The input unit 35 may be included in the housing of the user terminal 3 or may be externally attached. For example, the input unit 35 may be implemented as a touch panel integrated with the display unit 34. A touch panel allows the user to input tapping, swiping, and the like. Of course, switch buttons, a mouse, a QWERTY keyboard, and the like may be used instead of a touch panel. That is, the input unit 35 accepts operation inputs made by the user. The inputs are transferred as command signals to the processor 33 via the communication bus 30, and the processor 33 can execute predetermined control and calculations as necessary.

[0022] 2. Functional Configuration of the Processor 23 In this section, the functional configuration of the processor 23 of the information processing device 2 according to this embodiment will be described. Fig. 4 is a block diagram showing the functional configuration of the processor 23. The processor 23 includes an acquisition unit 231, a generation unit 232, an optimization unit 233, a calculation unit 234, and an output unit 235.

[0023] <Acquisition Unit 231> The acquisition unit 231 is configured to acquire various pieces of information related to information processing, which will be described later, from the user terminal 3. The acquisition unit 231 is configured to be able to acquire various pieces of information by reading out various pieces of information stored in a storage area, which is at least a part of the memory unit 22, and writing the read out information in a working area, which is at least a part of the memory unit 22. The storage area is, for example, an area of ​​the memory unit 22 that is implemented as a storage device such as an SSD. The working area is, for example, an area that is implemented as a memory, such as a RAM.

[0024] <Generation Unit 232> The generation unit 232 generates various information based on the acquired information. For example, the generation unit 232 generates atomic arrangements, trial electron densities, etc. based on the information acquired by the acquisition unit 231.

[0025] <Optimization Unit 233> The optimization unit 233 is configured to perform optimization of the arrangement of the generated atoms, trial electron density, and the like.

[0026] <Calculation Unit 234> The calculation unit 234 is configured to calculate various pieces of information based on the acquired information and information related to optimization.

[0027] <Output Unit 235> The output unit 235 is configured to be able to output various types of information. The information can be presented to the user via the display unit 34 of the user terminal 3 or another device. In such a case, for example, the output unit 235 controls the display unit 34 of the user terminal 3 to display visual information such as screens, images including still images or videos, icons, messages, etc. The output unit 235 may generate only rendering information for displaying the visual information on the user terminal 3. Note that the output unit 235 may present the output information to the user without going through the user terminal 3 or another device.

[0028] 3. Information Processing Method This chapter describes the flow of information processing executed in the information processing system 1 described above. Note that the information processing may include any exception processing. Exception processing may include interrupting the information processing or omitting each process. Selection or input performed in the information processing may be based on a user operation or may be performed automatically without relying on a user operation.

[0029] The information processing method can be used, for example, to identify the optimal atomic arrangement in any substance. In the following description, the natural unit system is used, but other unit systems, such as the SI unit system or the CGS unit system, can also be used.

[0030] First, the acquisition unit 231 acquires substance information relating to atoms that constitute the substance to be calculated.

[0031] Next, the generating unit 232 generates a trial electron density of the material based on the material information.

[0032] Here, the trial electron density is defined by a linear combination of expansion functions localized at each atom position.

[0033] Next, the optimization unit 233 optimizes the expansion coefficients p of the expansion functions that make up the trial electron density, with the positions of the atoms in the trial electron density fixed, using the energy of the material as an objective function.

[0034] Next, the calculation unit 234 calculates the forces acting on the atoms at the fixed positions based on the process and results of the optimization. Here, the calculation unit 234 includes the following processes: (a) calculating the Hellmann-Feynman forces acting on each atom based on the energy of the material after optimization and the expansion coefficients p after optimization, (b) calculating the partial derivatives of the expansion coefficients p after optimization with respect to the fixed atomic positions based on the fixed atomic positions and the transition of the expansion coefficients p due to the optimization, and (c) correcting the Hellmann-Feynman forces based on the partial derivatives of the atomic positions.

[0035] For example, optimization may be performed by updating the input expansion coefficients p. In this case, the calculation unit 234 may calculate the partial derivatives by converting the updated expansion coefficients p into a format in which the positions of the atoms fixed during optimization and the expansion coefficients p before the update are variables. This configuration can reduce the complexity of partial derivative calculations on a computer. For example, the calculation unit 234 may calculate partial derivatives of the expansion coefficients p after optimization based on the fixed positions of the atoms using automatic differentiation that uses the fixed positions of the atoms and the updated expansion coefficients p as input. This configuration can accurately perform differentiation based on the positions of the fixed atoms.

[0036] Specifically, for example, the calculation unit 234 calculates the correction term F defined by the formula (1) ν （Ｐｕｌａｙ－ｃ） can be used to correct the Hellman-Feynman force.

[0037] In equation (1), τ indicates the position of an atom fixed during optimization, ν is a subscript corresponding to each atom, and b′ is a subscript corresponding to each expansion function representing a certain atom.

[0038] Specifically, for example, the calculation unit 234 may correct the Hellmann-Feynman force using the following two correction terms:

[0039] For the specific content of these formulas, please refer to the contents shown in the reference diagram.

[0040] Furthermore, the calculation unit 234 may include a configuration in which, when the set of expansion functions constitutes a complete system, the Hellmann-Feynman force is not corrected based on the partial derivative of the position of the atom. With this configuration, it is possible to reduce the calculation load on the processor 23 due to the processing for correction when there is a possibility that the contribution of the correction is small.

[0041] Next, the generation unit 232 generates a trial electron density in which the atomic positions are updated based on the calculated force. By repeating this process, the trial electron density is updated and the most stable atomic positions are obtained.

[0042] Thereafter, the output unit 235 outputs the finally generated trial electron density.

[0043] With this configuration, in a complex system in which the expansion functions change with the movement of the atoms, when performing molecular dynamics (MD) calculations or structural optimization based on density functional theory (particularly orbit-free density functional theory: OFDFT), it is possible to take into account the effect of changes in the expansion functions on the forces acting on the atoms, thereby improving the accuracy of MD simulations of atomic arrangements. Furthermore, it becomes possible to more accurately calculate physical quantities derived from atomic vibrations, such as phonon dispersion and infrared spectrum.

[0044] 4. Example of Information Processing Method In this section, an information processing method according to this embodiment will be described based on the information processing methods described in the previous section. Figure 5 is a flowchart showing the flow of information processing according to this embodiment. Note that the information processing method according to this embodiment should not be interpreted as being limited based on the information processing method in the previous section.

[0045] [Step S1] As shown in FIG. 5 , first, in step S1, the acquisition unit 231 executes an acquisition process to acquire information to be used in calculations of material information, etc. The material information may include, for example, information about atomic nuclei constituting a system representing the material to be calculated (e.g., atomic species, composition ratio, etc.) and information about electrons (e.g., total number of electrons, chemical potential, etc.). The material information may be in any format and may be input via a preset input user interface or acquired from a specified model such as a CIF file. The processor 23 may use the acquired material information to set a system describing the material (e.g., a region where atoms and electrons can exist, the number of electrons present in space, chemical potential, spatiotemporal resolution in the region, presence or absence of periodic boundary conditions, etc.).

[0046] The acquiring unit 231 may also acquire information (e.g., an expansion function) used to define the trial electron density through the acquisition process, and a condition for terminating the optimization, which will be described later. The condition may include a convergence condition indicating whether the trial electron density ρ has converged through the optimization, and a termination condition (e.g., the number of iterations) for terminating the optimization if the trial electron density ρ has not converged.

[0047] An expansion function is a function used to describe an object of analysis. A set of expansion functions can, for example, form an orthonormal basis in a certain state space. Note that the inner product of expansion functions may be nonzero. Any expansion function may be set appropriately depending on the properties of the object of analysis. It may be a Bessel function, a spherical harmonic function, a plane wave basis, or a combination thereof. In particular, if the object of analysis may have a peak structure, the expansion function is preferably a localized function having a peak at a representative position. With this configuration, for example, if the object of analysis includes at least one peak, an expansion function optimized for the peak position can be obtained. The localized function is, for example, a function of the same form as the electron localized function. Examples of electron localized functions include Bloch functions, Wannier functions, delta functions, Slater functions, and Lorentz functions. The localized function may also be a function representing a statistical distribution. For example, the localized function may be a function representing a Gaussian distribution or a Poisson distribution. In particular, the expansion function is preferably a Lorentz function. The representative position of an expansion function corresponds to the origin in a state space defined by the input variables of the expansion function. However, the representative position is not limited to this, and may be any point that characterizes the expansion function, such as the mean value, median value, or mode value of the function. Hereinafter, for convenience of explanation, the expansion function is assumed to be a localized function.

[0048] In this embodiment, the desired trial electron density is expressed as a linear combination of a plurality of expansion functions using the following formula:

[0049] In formula (3), ν is a subscript corresponding to each atom, b is a subscript corresponding to each expansion function χ that represents a certain atom, τ is the position of the atom fixed in the optimization described below, ρ is the electron density distribution, E is the energy of the material, and ρ ＧＳ，τ （ＬＤ） is the electron density distribution when the energy E is minimized in the atomic configuration represented by τ, and p ＧＳ denotes the expansion coefficients optimized for fixed atomic positions. More specifically, r is a vector indicating an arbitrary position in space, and τν is a vector indicating the position τ of the νth atom. νｂ is the bth expansion function for expressing the electron density around the νth atom, and p νｂ is the expansion function χ νｂ In this embodiment, the expansion functions are assumed to be local functions. For convenience of explanation, each expansion coefficient p νｂ These are sometimes collectively expressed as the expansion coefficient p. Also, the positions τ of the atoms ν These are sometimes collectively referred to as the trial atomic configuration τ.

[0050] [Step S2] Next, in step S2, the generation unit 232 generates an initial state of a trial atomic configuration τ of the system to be calculated based on the acquired material information, etc. The initial state of the trial atomic configuration τ may be obtained by randomly arranging atoms of each atomic species within the system that describes the material based on the material information, or, in cases where the material information includes the positional relationships of atoms, such as in a cif file, the initial state of the trial atomic configuration τ may be generated based on the positional relationships.

[0051] [Step S3] Next, in step S3, the optimization unit 233 executes an optimization process to calculate the energy E （ＬＤ） With (p, τ) as the objective function, and with the atomic position τ in the trial electron density ρ fixed, the expansion function χ that constitutes the trial electron density ρ νｂ Each expansion coefficient p νｂ For example, the optimization unit 233 optimizes the energy E （ＬＤ） The expansion coefficient p νｂ Explore.

[0052] Material Energy E （ＬＤ） (p, τ) is defined as follows, for example, as a density functional using the electron density distribution expressed by equation (3):

[0053] In this case, the optimized expansion coefficients p that give the ground state (i.e., the trial electron density ρ with the minimum energy, which is the objective function) are ＧＳ It is expressed as (τ).

[0054] However, p ＧＳ is the optimized expansion coefficient p νｂ Using this, the ground state energy can be expressed as

[0055] In addition, the energy E is the smallest E ＧＳ （ＬＤ） The set of expansion coefficients p ＧＳ Any algorithm can be used to search for the electron density function ρ that minimizes the energy at a fixed trial atomic configuration τ, including exhaustive search, local search, iterative improvement, local neighborhood search, cuckoo search, genetic algorithm, and particle swarm optimization. （ＬＤ） ＧＳ The expansion coefficient p ＧＳ is obtained.

[0056] [Step S4] Next, in step S4, the calculation unit 234 executes a calculation process to calculate forces to be applied to atoms at fixed positions (i.e., each atom arranged in the trial atomic arrangement τ) based on the process and results of the optimization in step S3. Here, the forces acting on each atom will be described.

[0057] Force F acting on the νth atom at τ ν (τ) can be expressed as follows based on the relationship between energy and force:

[0058] However, it is generally difficult to precisely calculate such an energy derivative, and in the field of conventional numerical calculations using OFDFT, the trial atomic configuration τ has been optimized by approximating the right-hand side of equation (7) as the Hellmann-Feynman force expressed as follows:

[0059] However, due to its nature, the Hellmann-Feynman force does not incorporate the modulation of the force that accompanies a change in the electron density function ρ as the trial atomic configuration τ changes, so there was a risk that the accuracy of the optimization would decrease if a process of updating the atomic configuration was included.

[0060] Therefore, the present inventors have discovered a new force expression that can be used in information processing to simulate the optimal structure of a material, including atomic arrangement, particularly when the electron density distribution itself is treated as the optimization target, as in OFDFT. This force expression is particularly useful, for example, when the structure of a material (especially electrons) is described as a function that takes non-negative values, such as an electron density distribution, rather than a function that includes phase information, such as a wave function (especially a function that can be inverted). The new force expression will be described below.

[0061] Specifically, in this information processing method, the processor 23 calculates two correction terms F as correction terms for the Hellmann-Feynman force. ν （Ｐｕｌａｙ－ｂ） , F ν （Ｐｕｌａｙ－ｃ） The force in equation (7) is corrected as follows using the formula F, and the corrected force is applied to each atom appearing in the trial atomic configuration τ, thereby updating the atomic positions. Note that, in the following, these two correction terms F ν （Ｐｕｌａｙ－ｂ） , F ν （Ｐｕｌａｙ－ｃ） Although a case where both of these are used will be described, force correction may be performed using only one of them.

[0062] The correction term F on the right side of equation (9) ν （Ｐｕｌａｙ－ｂ） , F ν （Ｐｕｌａｙ－ｃ） are expressed as follows, respectively:

[0063] However, the parameters in the equations (10) and (11) are expressed as follows:

[0064] In the formulas (12) to (14), τ represents the position of an atom fixed during optimization, ρ represents the electron density distribution, E represents the energy of the substance, and ρ ＧＳ，τ （ＬＤ） is the electron density distribution when the energy E is minimum in the atomic configuration represented by τ, χ represents the expansion function, and p ＧＳ denotes the expansion coefficients optimized for the fixed atomic positions, ν is a subscript corresponding to each atom, and b is a subscript corresponding to each expansion function χ that represents an atom.

[0065] From this formulation, two correction terms F ν （Ｐｕｌａｙ－ｂ） , F ν （Ｐｕｌａｙ－ｃ） is the force F shown in equation (7) ν It can be seen that it contains a component related to the first derivative of F ν （Ｐｕｌａｙ－ｂ） indicates the effect of a change in the expansion function χ itself (for example, a change in the representative position) due to a change in the trial atomic configuration τ on the force, and F ν （Ｐｕｌａｙ－ｃ） is the expansion coefficient p ＧＳ Therefore, in order to perform the above correction, the calculation unit 234 calculates the parameters shown in the formulas (12) to (14) by using the trial electron density ρ (i.e., the expansion coefficient p ＧＳ That is, the calculation unit 234 calculates the energy E （ＬＤ） and the expansion coefficient p ＧＳ Based on this, the Hellman-Feynman force F acting on each atom is ＨＦ and calculating the gradient of the expansion function χ with respect to the change in the position τ of the atom and the expansion coefficient p after optimization with the position τ of the fixed atom based on at least one of the position τ of the fixed atom and the expansion coefficient p ＧＳ and calculating at least one of the gradient of the expansion function with respect to the change in the position of the atom and the partial derivative of the expansion coefficient p after optimization with the fixed atomic positions, ＨＦIn particular, the Hellmann-Feynman force F ＨＦ F ν （Ｐｕｌａｙ－ｂ） When correcting using the formula, the calculation unit 234 calculates the gradient of the expansion function with respect to the change in the position of the atom based on the position τ of the fixed atom, and calculates the Hellmann-Feynman force F based on the gradient of the expansion function χ with respect to the change in the position of the atom. ＨＦ According to this configuration, the influence of the change in the expansion function itself that may occur due to the change in the position of the atom on the system can be corrected by the Hellmann-Feynman force F ＨＦ Therefore, atomic arrangement by MD can be performed more accurately and efficiently.

[0066] The functional of the total energy E is expressed, for example, as follows:

[0067] On the right side of the formula (15), the first term represents kinetic energy, the second term represents electron-electron interaction (Hartree energy), the third term represents electron-ion interaction, and the fourth term represents exchange interaction. By using the expression for the energy, g in formula (14) can be calculated as follows: ＧＳ is expressed as follows:

[0068] Among these, all but the functional derivative of the kinetic energy are obtained as parameters. The functional derivative of the kinetic energy may be calculated analytically or approximately. When the functional derivative is calculated approximately, for example, the calculation unit 234 calculates the functional T ｓ [ρ] is a functional of kinetic energy T whose expression is known. ｓ （ｋｎｏｗｎ） The function T that depends explicitly on [ρ] and the expansion coefficient p ｓ Using (p), it can be approximated as follows: ｓ （ｋｎｏｗｎ） The original T ｓ If it can be considered that is a sufficiently good approximation, the second term on the right-hand side can be omitted.

[0069] The differential terms included in the right-hand side of the approximate formula (17) can be calculated in a normal way if their expressions are known explicitly, and even if the differential expressions are not known explicitly, they can be calculated using an existing numerical differential calculation algorithm such as automatic differentiation. ＧＳ Calculate the g ＧＳ Based on F ν （Ｐｕｌａｙ－ｂ） can be calculated.

[0070] Furthermore, F ν （Ｐｕｌａｙ－ｃ） When performing correction based on ∂p ＧＳνｂ (τ) / ∂τ ν As will be described later, in this information processing, the processor 23 applies minute displacements to each atom ν in the directions (x, y, z) to perform optimization processing equivalent to step S3, and calculates the expansion coefficient p ＧＳνｂ (τ) and its numerical differentiation with respect to τ. Alternatively, by constructing some approximation or machine learning model, the expansion coefficients p ＧＳνｂ If the τ dependence of (τ) can be written explicitly, automatic differentiation or numerical differentiation can be used to obtain ∂p ＧＳνｂ (τ) / ∂τ ν may be calculated.

[0071] The calculation unit 234 calculates the Hellmann-Feynman force F by, for example, the above-mentioned method. ＨＦ and the correction term F ν （Ｐｕｌａｙ－ｂ） , F ν （Ｐｕｌａｙ－ｃ） and the force F is calculated based on these.

[0072] In addition, when a predetermined omission condition is satisfied, such as the number of electrons that defines the electron density distribution of the system being constant, the processor 23 ν （Ｐｕｌａｙ－ｃ） Calculations related to the number of electrons in the system, n, may not be performed. Not performing a calculation may include omitting the calculation or not including a process for performing the calculation itself. For example, the processor 23 may ｅ The optimized expansion coefficient p ＧＳIf the following normalization conditions are satisfied in relation to F ν （Ｐｕｌａｙ－ｃ） There is no need to perform calculations related to

[0073] Equation (18) is a formula for the case where the number of electrons in the entire system is constant, n ｅ Here, both sides of equation (18) are fixed at τ ν When differentiating with respect to , the following equation (19) is obtained.

[0074] In addition, the total energy E at the trial atomic configuration τ is calculated using the optimized expansion coefficient p ｇｓ (τ) takes a stationary value. Therefore, if we use Lagrange's undetermined constant λ, the following stationary condition is satisfied:

[0075] From equations (11), (12), (19), and (20), F ν （Ｐｕｌａｙ－ｃ） Therefore, especially when the assumption is made that the number of electrons in the entire system is constant, the processor 23 can calculate F ν （Ｐｕｌａｙ－ｃ） Therefore, correction by the method described above may not be performed.

[0076] Furthermore, when the set of expansion functions χ substantially constitutes a complete system, the calculation unit 234 calculates the Hellmann-Feynman force F based on the partial differential of the position of the atom. ＨＦ It is not necessary to perform the correction. With this configuration, it is possible to reduce the calculation load on the processors 23 and 33 due to the processing for correction when there is a possibility that the contribution of the correction will be small. Whether or not a substantially complete system is configured can be determined, for example, by whether or not the number of expansion functions χ is equal to or greater than a predetermined number, or whether or not the ratio of the number of expansion functions χ to the number of dimensions of the state space to be optimized is equal to or greater than a predetermined value. The greater the number of mutually independent expansion functions χ, the closer the set of expansion functions χ becomes to a complete system.

[0077] [Step S5] Then, in step S5, the processor 23 determines whether the convergence condition set in step S1 is satisfied based on the force F calculated in step S4. Note that, since it is assumed here that structural optimization will be performed using OFDFT, step S5 of the convergence determination condition is included, but when performing molecular dynamics calculations (MD calculations), step S5 of the convergence determination condition may be omitted.

[0078] [Step S6] If it is determined in step S5 that the convergence condition is not satisfied, in step S6, the processor 23 updates the atomic positions (i.e., trial atomic configuration τ) based on the calculated force F (i.e., the corrected Hellmann-Feynman force). The position τ can be updated based on the classical equation of motion. Thereafter, the process returns to step S3, and the optimization unit 233 executes the optimization process based on the updated trial atomic configuration τ. As a result, the processor 23 sequentially updates the trial atomic configuration τ based on the corrected Hellmann-Feynman force, optimizes the electron density distribution with the updated trial atomic configuration τ fixed, and calculates the Hellmann-Feynman force F based on the optimized trial electron density ρ (i.e., the expansion coefficient p). ＨＦ and correction term F ν （Ｐｕｌａｙ－ｂ） , F ν （Ｐｕｌａｙ－ｃ） Calculate.

[0079] [Step S7] On the other hand, if it is determined in step S5 that the convergence condition is satisfied, the output unit 235 outputs information about the optimization result in step S7. The information may include the latest trial atomic configuration τ and the trial electron density ρ corresponding to τ as the optimized atomic configuration and electron density. Thereafter, the information processing system 1 terminates this information processing.

[0080] According to the above-described configuration, in a complex system in which the expansion functions change as the atomic positions move, when performing MD or structural optimization based on density functional theory (particularly OFDFT), it is possible to take into account the effect that the change in the expansion functions has on the forces acting on the atoms, thereby improving the accuracy of the MD simulation of atomic arrangement.

[0081] [Others] The above embodiment can be implemented as follows, for example.

[0082] The calculation unit 234 calculates the correction term F ν （Ｐｕｌａｙ－ｂ） Whether or not the correction term F ν （Ｐｕｌａｙ－ｃ） Using the Hellman-Feynman force F ＨＦ In other words, the calculation unit 234 may calculate partial derivatives of the expansion coefficients p after optimization with respect to the fixed atomic positions based on the positions of the fixed atoms and the transition of the expansion coefficients p due to the optimization, and correct the Hellmann-Feynman forces based on the partial derivatives of the positions of the atoms. Furthermore, the optimization may be performed by updating the input expansion coefficients p. In this case, the calculation unit 234 may calculate the partial derivatives of the expansion coefficients p by converting the updated expansion coefficients p into a format in which the positions of the atoms fixed during optimization and the expansion coefficients p before the update are variables. Furthermore, the calculation unit 234 may calculate the partial derivatives of the expansion coefficients p after optimization with respect to the fixed atomic positions using automatic differentiation with the positions of the fixed atoms and the expansion coefficients p after the update as input. With this configuration, differentiation based on the positions of the fixed atoms can be performed with high accuracy.

[0083] The specific form of the correction term for correcting the Hellmann-Feynman force is arbitrary and is not limited to the above-mentioned expressions (10) and (11). For example, any expression including an element expressed in a first-order or higher differential form of the force may be adopted.

[0084] In the above embodiment, the processor 23 corrects the Hellmann-Feynman force in the simulation based on OFDFT, but this is not limiting. The correction can also be applied to first-principles calculations using DFT calculations that use algorithms other than OFDFT, first-principles calculations other than DFT calculations, and molecular dynamics calculations.

[0085] The information processing device 2 may be an on-premise type or a cloud type. As the information processing device 2 in the cloud type, for example, the above-described functions and processes may be provided in the form of SaaS (Software as a Service) or cloud computing.

[0086] In the above embodiment, the information processing device 2 performs various storage and control operations, but multiple external devices may be used instead of the information processing device 2. That is, various information and programs may be distributed and stored in multiple external devices using blockchain technology or the like.

[0087] The above-described embodiment is not limited to the information processing system 1, and may be an information processing method or an information processing program. The information processing method includes each step of the information processing system 1. The program causes at least one computer to execute each step of the information processing system 1.

[0088] The information processing system 1 and the like may be provided in the following aspects.

[0089] (1) An information processing system, comprising at least one processor, the processor being configured to execute a program for executing each of the following steps: in an acquisition step, material information on atoms constituting a material to be calculated is acquired; in a first generation step, a trial electron density of the material is generated based on the material information, wherein the trial electron density is defined by a linear combination of expansion functions localized at each position of the atoms; in an optimization step, the energy of the material is used as an objective function, and an expansion coefficient p of each of the expansion functions constituting the trial electron density is optimized with the positions of the atoms in the trial electron density fixed; in a calculation step, a force to be applied to the atoms at the fixed positions is calculated based on the process and results of the optimization; and in a second generation step, and generating a trial electron density in which the positions of the atoms are updated based on the calculated forces, wherein the calculating step includes: calculating a Hellmann-Feynman force acting on each of the atoms based on the energy of the material and the expansion coefficient p obtained by the optimization; calculating at least one of a gradient of the expansion function with respect to a change in the position of the atom and a partial derivative of the expansion coefficient p after the optimization with respect to the fixed atomic positions, based on at least one of the fixed atomic positions and the expansion coefficient p; and correcting the Hellmann-Feynman force based on at least one of the gradient of the expansion function with respect to a change in the position of the atom and a partial derivative of the expansion coefficient p after the optimization with respect to the fixed atomic positions.

[0090] According to this configuration, in a complex system in which the expansion functions change as the atomic positions move, when performing molecular dynamics calculations (MD) or structural optimization based on density functional theory (particularly orbital-free density functional theory: OFDFT), it is possible to take into account the influence that the change in the expansion functions has on the forces acting on the atoms, and therefore it is possible to improve the accuracy of the simulation of atomic arrangement by MD.

[0091] (2) In the information processing system described in (1) above, the calculation step includes calculating a gradient of the expansion function with respect to a change in the position of the atom based on the position of the fixed atom, and correcting the Hellmann-Feynman force based on the gradient of the expansion function with respect to the change in the position of the atom.

[0092] With this configuration, the influence of the change in the expansion function itself, which may occur due to the change in the atomic position, on the system can be incorporated into the Hellmann-Feynman force, thereby enabling more accurate and efficient atomic arrangement by MD.

[0093] (3) In the information processing system described in (2) above, in the updating step, the position of the atom is updated based on the corrected Hellmann-Feynman force, and in the calculating step, a gradient of the expansion function with respect to a change in the position of the atom is calculated based on at least two of an expression of the expansion function, the position of the atom before the update, and the position of the atom after the update.

[0094] With this configuration, it is possible to improve the efficiency of calculating the gradient of the expansion function.

[0095] (4) In the information processing system according to any one of (1) to (3) above, in the calculation step, a correction term F defined by the formula (1-1) is calculated. ν （Ｐｕｌａｙ－ｂ） An information processing system that corrects the Hellmann-Feynman force using [Equation 1]. (where, in formulas (1-1) to (1-3), τ represents the position of an atom fixed during optimization, ρ represents the electron density distribution, E represents the energy of the substance, and ρ ＧＳ，τ （ＬＤ） is the electron density distribution when the energy E is minimum in the atomic configuration represented by τ, χ represents the expansion function, and p ＧＳ denotes the expansion coefficients optimized for the fixed atomic positions, ν is a subscript corresponding to each atom, and b is a subscript corresponding to each expansion function χ that represents an atom.

[0096] (5) In the information processing system according to any one of (1) to (4) above, the calculation step includes calculating partial derivatives of the expansion coefficients p after the optimization with respect to the positions of the fixed atoms, based on the positions of the fixed atoms and transitions of the expansion coefficients p due to the optimization, and correcting the Hellmann-Feynman force based on the partial derivatives of the positions of the atoms.

[0097] (6) In the information processing system described in (5) above, the optimization is performed by updating the expansion coefficients p that are input, and the calculation step includes calculating partial derivatives of the expansion coefficients p by converting the expansion coefficients p after the update into a format in which the positions of the atoms fixed during the optimization and the expansion coefficients p before the update are variables.

[0098] This configuration can reduce the complexity of partial differential calculations on a computer.

[0099] (7) In the information processing system described in (6) above, the calculation step calculates partial derivatives of the expansion coefficients p after the optimization with respect to the fixed atomic positions using automatic differentiation with the positions of the fixed atoms and the expansion coefficients p after the update as inputs.

[0100] With this configuration, differentiation based on the positions of the fixed atoms can be performed with high accuracy.

[0101] (8) In the information processing system according to any one of (1) to (7) above, in the calculation step, a correction term F defined by the formula (2-1) is calculated. ν （Ｐｕｌａｙ－ｃ） An information processing system that corrects the Hellmann-Feynman force using [Equation 2]. (where, in formulas (2-1) to (2-3), τ represents the position of an atom fixed during optimization, ρ represents the electron density distribution, E represents the energy of the substance, and ρ ＧＳ，τ （ＬＤ）is the electron density distribution when the energy E is minimum in the atomic configuration represented by τ, χ represents the expansion function, and p ＧＳ indicates the expansion coefficients optimized for the fixed atomic positions, ν and ν' are subscripts corresponding to each atom, and b and b' are subscripts corresponding to each expansion function χ that represents a certain atom.)

[0102] (9) In the information processing system according to any one of (1) to (8) above, the calculation step includes not correcting the Hellmann-Feynman force based on a partial derivative of the position of the atom when the set of expansion functions substantially constitutes a complete system.

[0103] With this configuration, it is possible to reduce the calculation load on the processor due to the processing for correction when there is a possibility that the contribution of the correction will be small.

[0104] (10) An information processing method, comprising the steps of the information processing system described in any one of (1) to (9) above.

[0105] (11) A program that causes at least one computer to execute each step of the information processing system described in any one of (1) to (9) above. Of course, this is not a limitation.

[0106] Finally, while various embodiments of the present invention have been described, these are presented by way of example only and are not intended to limit the scope of the invention. The novel embodiments may be embodied in various other forms, and various omissions, substitutions, and modifications may be made without departing from the spirit of the invention. Such embodiments and modifications are intended to be included within the scope and spirit of the invention, as well as within the scope of the inventions and their equivalents as defined in the accompanying claims.

[0107] 1: Information processing system, 2: Information processing device, 20: Communication bus, 21: Communication unit, 22: Storage unit, 23: Processor, 231: Acquisition unit, 232: Generation unit, 233: Optimization unit, 234: Calculation unit, 235: Output unit, 3: User terminal, 30: Communication bus, 31: Communication unit, 32: Storage unit, 33: Processor, 34: Display unit, 35: Input unit

Claims

1. An information processing system comprising at least one processor, the processor being configured to execute a program that executes each of the following steps: in an acquisition step, material information regarding atoms that constitute a material to be calculated is acquired; in a first generation step, a trial electron density of the material is generated based on the material information, wherein the trial electron density is defined by a linear combination of expansion functions localized at the positions of each of the atoms; in an optimization step, with the energy of the material as an objective function, each expansion coefficient p of the expansion functions that constitute the trial electron density is optimized while the positions of the atoms in the trial electron density are fixed; in a calculation step, forces to be applied to the atoms in the fixed positions are calculated based on the process and results of the optimization; and in a second generation step, a trial electron density in which the positions of the atoms are updated based on the calculated forces is generated, wherein the calculation step includes: calculating a Hellmann-Feynman force acting on each of the atoms based on the energy of the material and the expansion coefficient p obtained by the optimization; calculating, based on at least one of the positions of the fixed atoms and the expansion coefficients p, at least one of a gradient of the expansion function with respect to a change in the positions of the atoms and a partial derivative of the expansion coefficients p after the optimization with respect to the positions of the fixed atoms; and correcting the Hellmann-Feynman force based on at least one of the gradient of the expansion function with respect to a change in the positions of the atoms and a partial derivative of the expansion coefficients p after the optimization with respect to the positions of the fixed atoms.

2. An information processing system according to claim 1, wherein the calculation step includes: calculating a gradient of the expansion function with respect to a change in the position of the atom based on the fixed position of the atom; and correcting the Hellmann-Feynman force based on the gradient of the expansion function with respect to a change in the position of the atom.

3. An information processing system according to claim 2, wherein in the updating step, the position of the atom is updated based on the corrected Hellmann-Feynman force, and in the calculating step, the gradient of the expansion function with respect to a change in the position of the atom is calculated based on at least two of an expression of the expansion function, the position of the atom before the update, and the position of the atom after the update.

4. In the information processing system according to any one of claims 1 to 3, the calculation step includes calculating a correction term F defined by the formula (1-1): ν （Ｐｕｌａｙ－ｂ） and correcting the Hellmann-Feynman force using the (where, in formulas (1-1) to (1-3), τ represents the position of an atom fixed during optimization, ρ represents the electron density distribution, E represents the energy of the substance, and ρ ＧＳ，τ （ＬＤ） is the electron density distribution when the energy E is minimum in the atomic configuration represented by τ, χ represents the expansion function, and p ＧＳ denotes the expansion coefficients optimized for the fixed atomic positions, ν is a subscript corresponding to each atom, and b is a subscript corresponding to each expansion function χ that represents an atom.

5. An information processing system according to any one of claims 1 to 4, wherein the calculation step includes: calculating partial derivatives of the expansion coefficients p after the optimization with respect to the positions of the fixed atoms, based on the positions of the fixed atoms and changes in the expansion coefficients p due to the optimization; and correcting the Hellmann-Feynman force based on the partial derivatives of the positions of the atoms.

6. An information processing system according to claim 5, wherein the optimization is performed by updating the expansion coefficients p that are input, and the calculation step includes calculating partial derivatives of the expansion coefficients p by converting the expansion coefficients p after the update into a form in which the positions of the atoms fixed during the optimization and the expansion coefficients p before the update are variables.

7. An information processing system according to claim 6, wherein the calculation step calculates partial derivatives of the expansion coefficients p after optimization with respect to the fixed atomic positions using automatic differentiation with the fixed atomic positions and the expansion coefficients p after the update as inputs.

8. In the information processing system according to any one of claims 1 to 7, in the calculation step, a correction term F defined by the formula (2-1) is calculated. ν （Ｐｕｌａｙ－ｃ） and correcting the Hellmann-Feynman force using the (where, in formulas (2-1) to (2-3), τ represents the position of an atom fixed during optimization, ρ represents the electron density distribution, E represents the energy of the substance, and ρ ＧＳ，τ （ＬＤ） is the electron density distribution when the energy E is minimum in the atomic configuration represented by τ, χ represents the expansion function, and p ＧＳ indicates the expansion coefficients optimized for the fixed atomic position, ν and ν' are subscripts corresponding to each atom, and b and b' are subscripts corresponding to each expansion function χ that represents a certain atom.) 9. An information processing system according to any one of claims 1 to 8, wherein the calculation step includes not correcting the Hellmann-Feynman force based on partial derivatives of the positions of the atoms when the set of expansion functions substantially constitutes a complete system.

10. An information processing method, comprising the steps of the information processing system according to any one of claims 1 to 9.

11. A program that causes at least one computer to execute each step of the information processing system according to any one of claims 1 to 9.

Images