A new method for electronic structure refinement with heavy atom relativistic effects

Abstract

The application belongs to the field of material subdivision, and particularly relates to a new method for refining electronic structure containing relativistic effect of heavy atoms, wherein the heavy atoms refer to atoms with atomic mass number greater than 64, and the relativistic effect of the heavy atoms is relatively obvious; a wave function Psi of parameterized electronic relativistic effect correction is established, a minimum difference between theoretical structure factors and experimental structure factors is calculated through a least square method, and a parameter value P in the electronic wave function model Psi is inversely deduced n Thus, the refined electronic structure in the field of material analysis is obtained; in this way, subsequent topological analysis and material design according to the electronic structure are more accurate.

Description

Technical Field

[0001] This invention relates to the field of materials analysis technology, and in particular to a novel method for refining electronic structures containing relativistic effects of heavy atoms. 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] The present invention proposes that for heavy atoms (atoms with an atomic mass number greater than 64), the relativistic effects of heavy atoms are more pronounced, and the relativistic effects must be considered in order to obtain an accurate electronic structure.

[0007] To achieve the above objectives, this invention provides a novel method for refining electronic structures containing relativistic effects of heavy atoms, characterized by comprising: Step 1: Establishing an electronic structure containing parameter P n wave function ψ: φ i (r) represents the i-th basis function (i = 1, 2, 3, ..., m; m is the number of basis functions), and r represents the coordinates 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. 2and then Fourier transforming The theoretical structure factor of each diffraction point is calculated The energy is calculated by E(P n ) = <ψ|H|ψ> where H is the known Hamiltonian, and if the relativistic effect is not considered, H is H 非相对论 : where is the kinetic energy of the electron, is the potential energy of the electron around the nucleus, Z A is the nuclear charge number, is the electron Coulomb interaction, p X is the electron density, r, r' are the coordinates of the electron; if the relativistic effect is considered, the Hamiltonian considering the relativistic effect should be used where c is the speed of light; p is the momentum operator; m is the electron rest mass; V is the electron-nucleus Coulomb interaction potential; σ is the Pauli spin matrix, all of which are known functions;

[0008] Step two: the experimental intensity I 实验值 of each diffraction point is obtained by X-ray single crystal diffraction experiment, and the experimental structure factor F

[0009] Step three: the difference function of the theoretical structure factor and the experimental structure factor is established;

[0010] Step four: the minimum value in the difference function is calculated by the least square method to obtain the parameter P n of the optimal model.

[0011] wherein, the preferred scheme is: the H 相对论 is a 4x4 matrix, and when applied, H 相对论 is diagonalized first, and then only the spin-independent quantity is taken into account to calculate E(P n ) = <ψ|H|ψ>.

[0012] wherein, the preferred scheme is: the difference function of the modified wave function considering the relativistic effect is defined as: wherein, is the experimental structure factor of the diffraction point k, is the theoretical structure factor calculated from the wave function ψ.

[0013] wherein, the preferred scheme is: the difference function is: wherein λ is a weight factor.

[0014] wherein, the preferred scheme is: the difference function: while satisfying the minimum energy E(P n ), the minimum.

[0015] The preferred solution is to calculate the difference function L(P ) by using the least square method, and then obtain the optimal model parameter P n . n .

[0016] The preferred solution is to calculate the difference function L(P ) by using the least square method, and then obtain the optimal model parameter P n . n . Substitute P 1 and P 2 into L(P 1 ) and L(P 2 ) respectively to obtain L(P n ) and L(P 1 ) respectively, substitute P 2 and P 3 into Ψ respectively to obtain Ψ, and then calculate E(P 3 ) and E(P 3 ) by E(P 4 ) = <ψ|H|ψ> and λ, and then calculate and λ. B: Calculate P 4 by the formula P n = P n-1 - A (L(P n ) - L(P n )) / (L'(P n ) - L'(P n )), where A is the step factor, and then calculate L(P n ) by substituting P n into L(P i ). and then obtain the final experimental electronic wave function Ψ by substituting P 2 into Ψ, i.e., the refined electronic structure function. The preset precision value in step C is 0.001 or 0.0001 or 0.00001. The smaller the preset precision value, the more accurate the refined P n .

[0017] ​​The present application has the advantages that: heavy atoms refer to atoms with atomic mass number > 64, the relativistic effect of heavy atoms is relatively obvious, the wave function Ψ of the parameterized electronic relativistic effect correction is established, the minimum difference between the theoretical structure factor and the experimental structure factor is calculated by the least square method , and the parameter value P of the electronic wave function model Ψ is inversely deduced , so that the refined electronic structure in the material analysis field is obtained. n In this way, the subsequent topological analysis and material design based on the electronic structure are more accurate. BRIEF DESCRIPTION OF DRAWINGS

[0018] Figure 1 The existing technology utilizes the experimental structure diagram of X-rays;

[0019] Figure 2 The flowchart of the new method for refining the electronic structure with the relativistic effect of heavy atoms according to the present application.

[0020] Figure 3a The wave function diagram of the crystal material AgGaS2 without being refined by the relativistic effect of heavy atoms according to the present application.

[0021] Figure 3b The wave function diagram of the crystal material AgGaS2 refined by the relativistic effect of heavy atoms according to the present application. DETAILED DESCRIPTION

[0022] The technical solutions in the embodiments of the present application will be clearly and completely described below with reference to the drawings in the embodiments of the present application. Obviously, the described embodiments are only part of the embodiments of the present application, rather than all the embodiments.

[0023] Figure 2 The flowchart of the method for obtaining the electronic structure of a material by an electronic wave function model according to the present application, as shown in the figure, is a new method for refining the electronic structure with the relativistic effect of heavy atoms, and the steps are as follows: Figure 2 Step one: establish the wave function ψ with the parameter P n : φ i (r) is 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; is the coefficient of the i-th basis function, which is the parameter to be refined; the electronic density ρ(r) = Ψ 2 is calculated according to the wave function, and then the theoretical structure factor F of each diffraction point is calculated by Fourier transform The energy E(P n ) = <ψ|H|ψ> is calculated. Wherein H is the known Hamiltonian, if the relativistic effect is not considered, H is H 非相对论 : wherein For electron kinetic energy, Z represents the potential energy of an electron orbiting the nucleus. A Nuclear charge number For electron-coulomb interaction, ρ X Let r be the electron density, and r' be the electron coordinates.

[0024] If relativistic effects are considered, then the Hamiltonian H = H, which takes relativistic effects into account, should be used. Where c is the speed of light; p is the momentum operator; m is the electron rest mass; V is the electron-nuclear Coulomb interaction potential energy; and σ is the Pauli spin matrix, all of which are known functions.

[0025] Step 2: Obtain the experimental intensity I at each diffraction point through X-ray single-crystal diffraction experiments. 实验值 Calculate the experimental structure factor

[0026] Step 3: Establish the difference function between the theoretical structure factor and the experimental structure factor;

[0027] Step 4: Calculate the minimum value of the difference function using the least squares method to obtain the parameters P of the optimal model. n .

[0028] The preferred embodiment is: the H 相对论 It is a 4×4 matrix. When using it, H must first be... 相对论 Diagonalize, then substitute only the spin-independent quantities into E(P). n )=<ψ|H|ψ> to perform the calculation.

[0029] The preferred scheme is as follows: the difference function of the wave function for relativistic effect correction is defined as: in, Let k be the experimental structure factor at diffraction point k. It is the theoretical structure factor calculated from the wave function ψ.

[0030] The preferred solution is: the difference function is: Where λ is the weighting factor.

[0031] The preferred solution is: difference function: Simultaneously satisfying E(P) n Minimum energy and Minimum.

[0032] The preferred solution is to calculate the difference function using the least squares method. Find the minimum value and obtain the optimal model parameters P. n The preferred solution is: Step four uses the least squares method and utilizes... Preset P inn The calculation is performed on the values, and the steps include: A: Randomly select two sets of parameter values ​​P 1 and P 2 ,in, Substitution Obtain L(P) respectively 1 ) and L(P 2 ), by 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, through And λ were calculated B: According to the formula A is the step size factor, and P is calculated. 3 , and P 3 Substitution Calculate L(P) 3 C: Calculate P using the same formula as step B. 4 (i.e., 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 The optimal value is D:P. n Substitution This yields the final experimental electronic wavefunction Ψ, i.e., the refined electronic structure function. 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.

[0033] The following describes the specific process of the novel method for refining the electronic structure with relativistic effects of heavy atoms, using the well-known infrared nonlinear optical crystal material AgGaS2 as an example. As shown in Figure 3, Ga and Ag in AgGaS2 crystal are both heavy metal elements, and relativistic effects should be considered. The experimental electronic wavefunction is tested and refined in two cases: with and without considering relativistic effects.

[0034] The refinement process is the same for non-relativistic and relativistic systems, except that the formula for calculating energy E(P) is used... n The Hamiltonian used in )=<ψ|H|ψ> is different; non-relativistic physics uses the Hamiltonian. The method used to consider relativistic effects is...

[0035] Without considering the relativity:

[0036] Substitute P n into where the base function φ i (r) is a known function, and obtain Ψ by substituting ρ(r) = Ψ 2 into Calculate

[0037] Substitute Ψ into E(P n ) = 〈ψ|H|ψ> to calculate the energy E(P 1 ) and E(P 2 ), according to the function of the difference between the theoretical value and the experimental value of the wave function model: where F (k) is the experimental structure factor of diffraction point k, and F 1 (k) is the theoretical structure factor calculated from the wave function ψ, and χ(P 2 ) and χ(P n ) are calculated.

[0038] Next, use the function where λ is the weight factor (or Lagrange multiplier). In this embodiment, λ is set to 1.

[0039] Finally, use the least squares method to calculate the minimum value of the objective function L(P n ) to obtain the optimal model parameters P 1 . After refinement, the coefficients of the molecular orbitals of the electronic wave function of AgGaS2 can be obtained

[0040] Table 1 shows the and of the first 10 sets of indices κ(h, k, l) of AgGaS2 and the pre-set first 10 sets of P 2 and P 1 .

[0041]

[0042]

[0043] Table 1 shows the and of the first 10 sets of indices κ(h, k, l) of AgGaS2 and the pre-set first 10 sets of P 2 and P 1 .

[0044] 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, and calculate the energy E(P) using E(P) = <ψ|H|ψ>. And λ calculation As described in Table 2

[0045] Similarly, χ(P) can be calculated. 1 = 0.6324.

[0046] Through E(P) n )=ψ|H 非相对论 |ψ>, calculate E(P) 1 E(P) = -3.2261 2 ) = -4.1379; according to λ takes the value 1, L(P) 1 )=E(P 1 )+1*χ(P 1 )=0.5964-3.2261=-2.6297, similarly L(P 2 )=E(P 2 )+1*χ(P 2 = 0.6324 - 4.1379 = -3.5055. Since |L(P 2 )-L(P 1 The formula |=|-3.5055-(-2.6297)|=0.8758, which is clearly greater than the preset precision value of 0.001. Therefore, it is necessary to continue using the square method formula to calculate P. 3 It is worth further refinement.

[0047] Table 2 shows the AgGaS2 values ​​without considering relativity. χ(P 1 ), χ(P 2 ), E(P 1 ), E(P 2 ), L(P 1 ) and L(P 1 )value

[0048]

[0049] Then according to That is, if A = 0.01, then... Taking the index parameter K(0,0,2) as an example,

[0050] Similarly: according to the formula 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 the value is less than 0.001, then P is obtained. 最终 As shown in Table 3, this completes the refinement process.

[0051] Table 3 shows the P values ​​of AgGaS2 without considering relativistic squares. 3 P 最终 A list.

[0052]

[0053] Considering the case of relativity:

[0054] Similarly, using Table 1 P 1 ,P 2 The same applies to the non-relativistic case.

[0055] Table 4 shows the calculations considering relativity. χ(P 1 ), χ(P 2 ), E(P 1 ), E(P 2 ), L(P 1 ) and L(P 1 The first 10 sets of data for the value.

[0056]

[0057] Similarly, χ(P) can be calculated. 1 ) = 0.6324, through E(P nn )=<ψ|H 相对论 |ψ>, calculate E(P) 1 E(P) = -4.3125 2 ) = -5.2684; according to λ takes the value 1, L(P) 1 )=E(P 1 )+1*χ(P 1 L(P) = 0.5964 - 4.3125 = -3.716. Similarly, L(P) = 0.5964 - 4.3125 = -3.716. 2 )=E(P 2 )+1*χ(P 2 = 0.6324 - 5.2684 = -4.636. Since |L(P 2 )-L(P 1The formula |=|-4.636-(-3.716)|=0.92, which is clearly greater than the preset precision value of 0.001. Therefore, it is necessary to continue using the square method formula to calculate P. 3 It is worth further refinement.

[0058] Table 4 shows the calculated values ​​considering relativity. χ(P 1 ), χ(P 2 ), E(P 1 ), E(P 2 ), L(P 1 ) and L(P 1 )value.

[0059]

[0060]

[0061] according to

[0062] have Taking the first indicator (0,0,2) as an example,

[0063]

[0064] Similarly: according to the formula 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 the value is less than 0.001, then P is obtained. 最终 As shown in Table 5, this completes the refinement process.

[0065] Table 5 shows the P values ​​of AgGaS2 calculated considering relativistic effects. 3 P 最终 list

[0066]

[0067] Through Table 6, Figure 3a as well as Figure 3b It is not difficult to see from this: Among them, Figure 3a The data around the atoms and chemical bonds represent atomic charge and chemical bond charge, respectively. Ignoring relativistic effects, the experimental charges of Ag1, Ga2, and S3 atoms are 0.2694, 2.3567, and -0.6260, respectively; considering relativistic effects, Figure 3bThe data around the atoms and chemical bonds in the figure represent the atomic charge and the bond charge, respectively. The experimental atomic charges of Ag1, Ga2 and S3 are 0.4664, 2.6226 and -1.0890, respectively, which are close to the formal charges of Ag+, Ga3+ and S2-, respectively, indicating that the experimental electron wave function obtained after considering the relativistic effect is more reasonable.

[0068] Table 6 is a comparison chart of the atomic valence state of AgGaS2 considering the relativistic effect and not considering the relativistic effect

[0069] Atomic valence state Formal charge Non-relativistic Relativistic Ag1 1.0 0.2694 0.4664 Ga2 3.0 2.3567 2.6226 S3 -2.0 -0.6260 -1.0890

[0070] The above description is only the preferred specific implementation of the present application, but the protection scope of the present application is not limited to this. Any person skilled in the art can make equivalent replacements or changes according to the technical solution and the inventive concept of the present application within the technical range disclosed by the present application, which should be covered in the protection scope of the present application.

Claims

1. A new method for the refinement of electronic structure containing heavy atom relativistic effects, characterized in that: Comprising: Step one: Establishing the parameter P n wave function ψ: φi(r) is 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. Through E(P) n The energy is calculated as <ψ|H|ψ>, where H is the known Hamiltonian; if relativistic effects are neglected, then H is H 非相对论 : in For electron kinetic energy, Z represents the potential energy of an electron orbiting the nucleus. A Nuclear charge number For electron-coulomb interaction, ρ X Let r be the electron density, and r' be the electron coordinates. If the relativistic effect is considered, the Hamiltonian H considering the relativistic effect should be used, i.e. where c is the speed of light; p is the momentum operator; m is the electron rest mass; V is the electron-nucleus Coulomb interaction potential energy; σ is the Pauli spin matrix, which are all known functions; Step two: Obtain the experimental intensity I of each diffraction point by X-ray single crystal diffraction experiment 实验值 , and calculate the experimental structure factor Step three: Establish the difference function of the theoretical structure factor and the experimental structure factor n .

2. The new method for refining the electronic structure with heavy-atom relativistic effects according to claim 1, characterized in that: The H 相对论 is a 4x4 matrix, which, when applied, is first diagonalized, and then only the spin-independent quantities are taken to calculate E(P 相对论 ) = <ψ|H|ψ>. n ) = <ψ|H|ψ>.

3. The new method of heavy-atom relativistic effect-inclusive electronic structure refinement according to claim 1, characterized in that: In step three, the difference function of the modified wave function defining the relativistic effect is defined as: where, is the experimental structure factor of the diffraction point k, is the theoretical structure factor calculated from the wave function ψ.

4. The new method of heavy atom relativistic effect-included electronic structure refinement according to claim 2, characterized in that, The difference function between the theoretical structure factor defined in step three and the experimental structure factor is: where λ is a weighting factor.

5. The new method of heavy-atom relativistic effect-inclusive electronic structure refinement according to claim 4, wherein, Difference function: satisfy both E(P n ) energy minimization and minimization.

6. The new method for refining the electronic structure with heavy-atom relativistic effects according to claim 5, characterized in that, The difference function is calculated using the least squares method The optimal model parameters P obtained when taking the minimum value n .

7. The new method for refining the electronic structure with heavy-atom relativistic effects according to claim 6, characterized in that: Step four is calculated by least square method, using preset P in n , the step includes: A: randomly select two sets of parameter values P 1 and P 2 , wherein, substitute to obtain L(P 1 ) and L(P 2 ), respectively, by substituting P 1 and P 2 into to calculate Ψ, calculate E(P n ) and E(P 2 ) values by E(P 1 ) = <ψ|H|ψ>, and calculate and λ B: according to the formula A is the step factor, calculate P 3 , , and substitute P 3 into to calculate L(P 3 ) value; C: also according to the formula of step B, calculate P 4 , that is, n = 4, calculate L(P 4 ), cycle n steps, until L(P n )-L(P n-1 ) is less than the preset precision value, at this time L(P n ) has reached the minimum, P n is the best value; D: substitute P n into , that is, obtain the final experimental electronic wave function Ψ, that is, the refined electronic structure function.

8. The new method for refining the electronic structure with heavy-atom relativistic effects according to claim 7, characterized in that: The preset precision value in the C step is 0.001 or 0.0001 or 0.00001, and the smaller the preset precision value, the more accurate the refined P n ​

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