A wavefunction modeling method for electronic structure analysis

Abstract

This invention belongs to the subfield of materials science, and in particular to a wavefunction modeling method for electronic structure analysis, the steps of which include: establishing parameter P n The electronic wavefunction model is used to calculate the theoretical structure factor at each diffraction point; the experimental intensity at each diffraction point is obtained through X-ray single-crystal diffraction experiments, and the experimental structure factor is calculated; step three: establish the difference function between the theoretical structure factor and the experimental structure factor; step four: calculate the minimum value of the difference function using the least squares method to obtain the parameter P of the optimal model. n Finally, this invention yields a refined electronic structure for materials analysis; subsequent topology analysis and materials design based on this electronic structure are more accurate.

Description

Technical Field

[0001] This invention relates to the field of materials analysis technology, and in particular to a wavefunction model method for electronic structure analysis. Background Technology

[0002] Materials science is the foundation and precursor of modern science and technology. The level of understanding of material structure and material composition directly determines the research and development capability of new materials.

[0003] The microstructure of materials includes atomic-level structures such as crystal structure, local structure, and defect structure, as well as electronic structure. Among these, electronic structure fundamentally determines the intrinsic properties of materials. Currently, experimental testing techniques for the atomic-level structure of materials are very mature, but experimental testing of electronic structure remains in the exploratory stage. Although electronic structure can be obtained through first-theory calculations, theoretical calculations often rely on numerous assumptions and approximations, leading to discrepancies between the calculated results and actual conditions, making it difficult to guide the design of high-performance materials.

[0004] Therefore, obtaining the experimental electronic structure of materials is a key scientific problem. Solving this problem will help my country's experimental research on material structure to leap from the atomic level to the electronic level, and accelerate the research and development of a number of key functional materials for national defense and civilian use.

[0005] Figure 1 This is an experimental structure diagram using existing X-ray technology. X-rays are incident on the crystal under test and diffract after passing through the crystal. By obtaining high-precision, high-resolution X-ray single-crystal diffraction data (position and intensity information) and refining the electronic structure, it is feasible to deduce the experimental electronic structure of the material. Finally, the experimental electronic structure of the material under static and service conditions can be obtained. The electronic structure can be described using electron density, density matrix, or electron wave function. Summary of the Invention

[0006] This invention proposes a wavefunction modeling method for electronic structure analysis.

[0007] To achieve the above objectives, the present invention adopts the following technical solution: a method for obtaining the electronic structure of a material through an electronic wavefunction model, comprising: Step 1: establishing parameter P n Step 1: Calculate the theoretical structure factor for each diffraction point using the electron wavefunction model; Step 2: Obtain the experimental intensity of each diffraction point through X-ray single-crystal diffraction experiments and calculate the experimental structure factor; Step 3: Establish the difference function between the theoretical and experimental structure factors; Step 4: Calculate the minimum value of the difference function using the least squares method to obtain the parameter P of the optimal model. n .

[0008] The preferred approach is as follows: In step one, a parameterized electronic wavefunction model Ψ is established: φ i (r) represents the i-th basis function (i = 1, 2, 3, ..., m; m is the number of basis functions), and r is the coordinate of the electron; The coefficients of the i-th basis function are parameters that need to be refined; the electron density ρ(r) = Ψ is calculated based on the wave function. 2 Then through Fourier transform Calculate the theoretical structure factor for each diffraction point.

[0009] The preferred scheme is as follows: In step two, the experimental intensity I of each diffraction point is obtained through X-ray single-crystal diffraction experiments. 实验值 The structure factor of the experiment was calculated based on the diffraction point intensities. The energy E(P) is calculated based on the wave function ψ. n )=<ψ|H|ψ>, where H is a known Hamiltonian.

[0010] The preferred approach is as follows: In step three, the difference function between the theoretical and experimental values ​​of the wave function model is defined as:

[0011]

[0012] in N is the experimental structure factor at diffraction point k. r The number of experimental data points is denoted by m, and m is the parameter P. n The number of σ k These are experimental data. The error value, It is the theoretical structure factor calculated from the wave function ψ.

[0013] The preferred solution is: the difference function is: in λ This is the weighting factor.

[0014] The preferred solution is: difference function: Simultaneously satisfying E(P) n Minimum energy Minimum.

[0015] The preferred solution is to calculate the difference function using the least squares method. L(P) n The minimum value of ) yields the optimal model parameters P. n .

[0016] The preferred solution is: Step four calculates using the least squares method. P in n Preset Pn The steps for performing calculations on the values ​​include: A: Randomly selecting two sets of parameter values ​​P 1 and P 2 ,in, Substitution Obtain L(P) respectively 1 ) and L(P 2 That is, first by passing P 1 and P 2 Substitute them separately Calculate Ψ, through E(P) n )=<ψ|H|ψ>Calculate E(P) 1 ) and E(P 2 ) value, then through And λ were calculated Finally passed Calculate L(P) n );

[0017] B: According to the formula A is the step size factor, and P is calculated. 3 ,in, ), that is, n=3, and P 3 Substitution L(P) was calculated using a similar method. 3 )value;

[0018] C: Calculate P using the same formula as in step B. 4 (i.e., n=4), similarly calculate L(P) 4 Repeat n steps until L(P) n )-L(P n -1 If L(P) is less than the preset precision value, then L(P) n P has reached its minimum. n This is the optimal value;

[0019] D:P n Substitution This yields the final experimental electronic wavefunction Ψ, i.e., the refined electronic structure function. The preset precision value is 0.001, 0.0001, or 0.00001. The smaller the preset precision value, the more refined the P... n The more accurate.

[0020] Compared with the prior art, the present invention has the following beneficial effects:

[0021] By establishing parameter P n The electronic wavefunction model Ψ was used to calculate the theoretical structure factor. Experimental structure factor was obtained using X-ray experiments. The theoretical structure factor was calculated using the least squares method. With experimental structure factor The minimum difference between them can be used to deduce the parameter values ​​P of the electron wave function model Ψ. n This allows for a refined electronic structure in the field of materials analysis; thus, subsequent topology analysis and materials design based on this electronic structure can be more accurate. Attached Figure Description

[0022] Figure 1 A schematic diagram of an experimental structure utilizing X-rays in the context of existing technologies;

[0023] Figure 2 This is a flowchart of the wavefunction model method for electronic structure analysis according to the present invention;

[0024] Figure 3 (a) Figure 3 (b) and Figure 3 (c) The electron wave function graphs of LiB3O5 under three conditions: no light, 360nm laser irradiation, and 1064nm laser irradiation. Detailed Implementation

[0025] The technical solutions in the embodiments of the present invention will be clearly and completely described below in conjunction with the drawings in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, rather than all the embodiments.

[0026] Figure 2 The flowchart of the wavefunction model method for electronic structure analysis of this invention is shown below. Figure 2 As shown: This method includes: Step 1: Establishing parameter P n Step 1: Calculate the theoretical structure factor for each diffraction point using the electron wavefunction model; Step 2: Obtain the experimental intensity of each diffraction point through X-ray single-crystal diffraction experiments and calculate the experimental structure factor; Step 3: Establish the difference function between the theoretical and experimental structure factors; Step 4: Calculate the minimum value of the difference function using the least squares method to obtain the parameter P of the optimal model. n .

[0027] Among them, establish P i The parameterized electronic wavefunction model Ψ

[0028]

[0029] φ i (r) represents the i-th basis function (i = 1, 2, 3, ..., m; m is the number of basis functions), and r is the coordinate of the electron; The coefficients of the i-th basis function are parameters that need to be refined; the electron density ρ(r) = |Ψ| is calculated based on the wave function. 2 Then through Fourier transform Calculate the theoretical structure factor for each diffraction point.

[0030] Figure 1 The present invention obtains the experimental structure factor through X-ray single-crystal diffraction experiments, such as Figure 1 As shown: Experimental intensity I obtained at each diffraction point 实验值 The structure factor of the experiment was calculated based on the diffraction point intensities. The energy E(P) = <ψ|H|ψ> is calculated based on the wave function Ψ.

[0031] This embodiment uses theoretical structural factors. and the structural factors of the experiment The difference function between the theoretical and experimental values ​​of the wave function model is set as follows: in N is the experimental structure factor at diffraction point k. r Here, m is the number of experimental data points, σk is the number of parameters P, and σk is the number of experimental data points. The error value, It is the theoretical structure factor calculated from the wave function ψ.

[0032] To achieve more accurate refinement, the difference function is further defined as follows: Where λ is the weighting factor, and the difference function is: Simultaneously satisfying E(P) n Minimum energy, theoretical value of structure factor Experimental values ​​of structural factor Most consistent, that is Also the smallest.

[0033] Then, the difference function is calculated using the least squares method. The minimum value of L(P) is used to obtain the optimal model parameters P. n .

[0034] Specifically, the calculation using the least squares method includes: P in n Value, determined by preset P n The values ​​are used for calculations, and the steps include:

[0035] First, randomly select two sets of parameter values ​​P. 1 and P 2 In this case, the superscripts 1 and 2 of P represent the first and second groups, respectively, and each group of parameters contains m parameters. Substitute Obtain L(P) respectively 1 ) and L(P 2 That is, first by passing P1 and P 2 Substitute them separately Calculate Ψ, through E(P) n ) = <ψ|H|ψ> to calculate E(P) 1 ) and E(P 2 ) value, then through And λ were calculated Finally passed Calculate L(P) n ).

[0036] Then, according to the least squares formula Where A is the step size factor, P is calculated. 3 (i.e., n=3), and at the same time, P 3 It also contains m parameters, that is ), and P 3 Substitution L(P) was calculated using a similar method. 3 )value;

[0037] Then, calculate P using the least squares formula. 4 (i.e., n=4), and calculate L(P) similarly. 4 ), repeat n steps, in this embodiment, until the preset precision value is reached, at which point L(P n P has reached its minimum at this point. n The optimal value is 0.001, 0.0001, or 0.00001. A smaller preset precision value results in a more refined P. n The more accurate.

[0038] Finally, P n Substitution This yields the final experimental electronic wave function Ψ, which is the refined electronic structure function.

[0039] In this embodiment, the difference function is: Simultaneously satisfying E(P) n Minimum energy and theoretical value of structure factor Experimental values ​​of structural factor Most consistent, that is The minimum two conditions.

[0040] The following describes the specific process of obtaining the electronic structure of a material using an electronic wavefunction model, taking LiB3O5 crystal material as an example:

[0041] In the parameterized electronic wavefunction model Ψ,

[0042]

[0043] φi (r) represents the i-th basis function (i = 1, 2, 3, ..., m; m is the number of basis functions), and r is the coordinate of the electron; The coefficients of the i-th basis function are parameters that need to be refined; the electron density ρ(r) = |Ψ| is calculated based on the wave function. 2 Then through Fourier transform Calculate the theoretical structure factor for each diffraction point. Will Substitution Wherein, the basis function φ i Let Ψ be a known function, and substitute the obtained Ψ into ρ(r)=Ψ 2 Calculate ρ(r); substitute ρ(r) into calculate Where κ(h, k, l) represents a set of index parameters consisting of three integers, such as 001, 002, 101, 102, etc.

[0044] In this embodiment, the nonlinear optical crystal material LiB3O5 was used to obtain the experimental intensity of each diffraction point under three different illumination conditions: no illumination, 360nm illumination, and 1064nm illumination.

[0045] Table 1 shows the first five sets of diffraction points κ(h, k, l) of LiB3O5 under three conditions: no light illumination, 360nm laser irradiation, and 1064nm laser irradiation. σ k 、P 1 、P 2 Default value.

[0046]

[0047] Among them, the energy E(P) is calculated based on the wave function Ψ. n )=〈ψ|H|ψ〉.

[0048] A function representing the difference between the theoretical and experimental values ​​of the wave function model:

[0049]

[0050] in N is the experimental structure factor at diffraction point k. r The number of experimental data points is denoted by m, and m is the parameter P. n The number of values, σk is the experimental data. The error value, It is the theoretical structure factor calculated from the wave function ψ.

[0051] Based on LiB3O5 material, this experiment investigated the N... r The number is 6214, σk The error for each experimental data point is provided in the data above, with an m value of 497. The structure factor is calculated based on the above data.

[0052] Next, using functions Where λ is the weighting factor (or Lagrange multiplier), and in this embodiment, λ is set to 0.1.

[0053] Finally, the objective function is calculated using the least squares method. L(P) n Find the minimum value of ) and obtain the optimal model parameters P. n .

[0054] After refinement, the molecular orbital coefficients P of the electron wavefunction of LiB3O5 under three conditions: no light illumination, 360nm laser irradiation, and 1064nm laser irradiation were obtained. n ;

[0055] Step A: Select parameter value P from Table 1 1 and P 2 Substitute into the difference function Calculate L(P) n ), respectively obtain L(P 1 ) and L(P 2 Meanwhile, by substituting P into... Calculate Ψ, through E(P) n )=<ψ|H|ψ〉Calculate the energy E(P) n ),pass And λ calculation

[0056] Table 2 shows the preset P values ​​under three conditions: no illumination, 360nm laser irradiation, and 1064nm laser irradiation. 1 、P 2 The first five sets of data are displayed for analysis and calculation, as shown in Table 2:

[0057] Table 2 shows the calculated values ​​under three conditions: no light, 360nm laser irradiation, and 1064nm laser irradiation. χ 2 (P 1 ), χ 2 (P 2 ), E(P 1 ), E(P 2 ), L(P 1 ) and L(P 2 )value.

[0058]

[0059] Specific calculation process: Table 1 selects the index parameter K(h, k, l) as (0, 0, -13) under no-light conditions for P. 1 Taking the value as an example, select P from Table 1. 1 =0.3345; Substitute into E(P) n In the formula )=〈ψ|H|ψ>, see Table 2: Calculate E(P) 1 ) = -455.959888; then through the difference function And λ = 0.1, calculate Will Substitution The formula calculates L(P) in Table 2. 1 The value is -454.850107; similarly, L(P) is obtained. 1 Value = -454.822211.

[0060] According to the preset P in Table 1 1 、P 2 Value verification L(P) 2 )-L(P 1 The formula -454.822211 - (-454.850107) = 0.027896 is greater than the preset precision value of 0.001, so step B is required: verifying the calculated P. 3 , And by repeating step B above, we obtain P. 4 until L(P) n )-L(P n-1 When the value is less than the preset precision value, the preset precision value in this experiment is 0.001.

[0061] Illustrate the least squares method with examples In this embodiment, A is the step size factor. In this embodiment, the calculation process is set to A = 0.0002, with P in the absence of light. 3 For example, in calculation:

[0062] First, extract L(P) from Table 2. 2 ), L(P 1 ) value, L(P 2 )-L(P 1 The result is 0.027896, which is significantly greater than the preset precision value of 0.001. Therefore, it is necessary to continue using the square method formula to calculate P. 3 The value will be further refined by The formula yields p 3 =p 1 -(0.0002*(0.4968-0.3345) / 0.027896*(-454.850107)=0.8637; For the 360nm and 1064nm cases, the L(P) values ​​in Table 2 are also extracted.1 ), L(P 2 The value is used to perform calculations, and the final result is P in Table 3. 3 Value, F k 理论 (P 3 ).

[0063] Similarly: according to the formula (A is the step size factor, set to 0.02) Calculate P 4 (i.e., n=4), similarly calculate L(P) 4 ), and so on, repeating n times, until L(P n )-L(P n-1 If P is less than 0.001, then P 最终值 As shown in Table 3, the final P 最终值 The value is used to complete the refinement process.

[0064] Table 3 is... Final P 3 、P 最终值 and L(P 2 )-L(P 1 A list of ).

[0065]

[0066]

[0067] Finally, the refined P... 最终值 Substitution The final experimental electronic wavefunction Ψ, i.e., the refined electronic structure function, is obtained, as follows: Figure 3 (a) Figure 3 (b) and Figure 3 (c) shows the electron wave function diagrams of the refined LiB3O5 under three conditions: no light, 360nm laser irradiation, and 1064nm laser irradiation.

[0068] pass Figure 3 (a) Figure 3 (b) and Figure 3 As can be seen from (c) and Table 4, the topological atomic charge, atomic volume and dipole moment of the [B3O5]- group change significantly under laser irradiation. Moreover, the electronic structure of the BO3 structural unit has a larger response amplitude to the external field than that of BO4. This experimentally confirms that the B–O group [B3O5]- is the nonlinear optical functional unit of LBO, and explains the origin of the nonlinear optical effect of LiB3O5 crystal.

[0069] Table 4 shows the charge distribution of Li, B, and O at different positions in LiB3O5.

[0070]

[0071]

[0072] The beneficial effects of this invention are as follows: Taking LiB3O5 as an example, the parameter values ​​P of the refined electron wavefunction model Ψ are obtained according to the analytical method of this invention. 最终值 This allows us to obtain the refined electronic structure of LiB3O5 material. The topological atomic charge, atomic volume, and dipole moment of LiB3O5 material will change significantly, providing a better basis for subsequent topological analysis and material design research based on this electronic structure.

[0073] The above description is only a preferred specific embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any technician familiar with the technical field, within the technical scope disclosed by the present invention, who makes equivalent replacements or changes based on the technical solution and inventive concept of the present invention, should be covered by the scope of protection of the present invention.

Claims

1. A wavefunction modeling method for electronic structure analysis, characterized in that: include: Step 1: Establish parameter P n The electronic wave function model was used to calculate the theoretical structure factor for each diffraction point; Step 2: Obtain the experimental intensity of each diffraction point through X-ray single-crystal diffraction experiments and calculate the experimental structure factor; Step 3: Establish the difference function between the theoretical structure factor and the experimental structure factor; Step 4: Calculate the minimum value of the difference function using the least squares method to obtain the parameter P of the optimal model. n In step one, a parameterized electronic wavefunction model Ψ is established: φ i (r) represents the i-th basis function, i = 1, 2, 3, ..., m; m is the number of basis functions, and r is the coordinate of the electron; The coefficients of the i-th basis function are parameters that need to be refined; the electron density ρ(r) = |Ψ| is calculated based on the wave function. 2 Then through Fourier transform K represents the diffraction point, and the theoretical structure factor for each diffraction point is calculated. In step two, the experimental intensity of each diffraction point is obtained through X-ray single-crystal diffraction experiments, and the structure factor of the experiment is calculated based on the diffraction point intensity; the energy E(P) is calculated based on the wave function. n )=<ψ|H|ψ>, where H is a known Hamiltonian; in step three, the difference function between the theoretical and experimental values ​​of the wave function model is defined as: Where F k 实验 N is the experimental structure factor at diffraction point k. r The number of experimental data points is denoted by m, and m is the parameter P. n The number of σ k These are experimental data. The error value, It is the theoretical structure factor calculated from the wave function ψ.

2. The wavefunction model method for electronic structure analysis according to claim 1, characterized in that, The difference function is: Where λ is the weighting factor.

3. The wavefunction model method for electronic structure analysis according to claim 2, characterized in that, Difference function: Simultaneously satisfying E(P) n Minimum energy Minimum.

4. The wavefunction model method for electronic structure analysis according to claim 2 or 3, characterized in that, The difference function is calculated using the least squares method: L(P) n The minimum value of ) yields the optimal model parameters P. n .

5. The wavefunction model method for electronic structure analysis according to claim 4, characterized in that: Step four involves calculating using the least squares method. P in n Preset P n The steps for performing calculations on the values ​​include: A: Randomly selecting two sets of parameter values ​​P 1 and P 2 ,in, Substitute Obtain L(P) respectively 1 ) and L(P 2 That is, first by passing P 1 and P 2 Substitute them separately Calculate Ψ, through E(P) n )=<ψ|H|ψ>Calculate E(P) 1 ) and E(P 2 ) value, then through And λ were calculated Finally passed Calculate L(P) n ); B: According to the formula A is the step size factor, and P is calculated. 3 , and P 3 Substitution Calculate L(P) 3 )value; C: Calculate P using the same formula as in step B. 4 That is, n=4, calculate L(P) 4 Repeat n steps until L(P) n )-L(P n-1 If L(P) is less than the preset precision value, then L(P) n P has reached its minimum. n This is the optimal value; D:P n Substitution This yields the final experimental electronic wave function Ψ, which is the refined electronic structure function.

6. The wavefunction model method for electronic structure analysis according to claim 5, characterized in that: The preset precision value in step C is 0.001, 0.0001, or 0.00001. The smaller the preset precision value, the more refined the P... n The more accurate.

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