Method for measuring coulomb interaction energy of s electron cloud and other electron clouds

Abstract

The invention discloses a method for measuring coulombic interaction energy of s electron cloud and other electron clouds, and relates to the technical field of electron cloud coulombic interaction energy calculation optimizing.After spatial symmetry transformation, the distance between a hydrogen atom 1s electron cloud microcell and a second electron cloud microcell is replaced with an equivalent projection distance, and the coulombic interaction energy of the s electron cloud and other electron clouds is calculated. The equivalent projection distance is the distance between the hydrogen atom 1s electron cloud microcell and the rotated second electron cloud microcell, and the projection distance is a measurable numerical value on a connecting line of the electron cloud microcells, so that only four parameters, namely a radial parameter and an angle parameter, of the hydrogen atom 1s electron cloud microcell and the second electron cloud microcell need to be integrated, and then the hydrogen atom 1s electron cloud microcell and the second electron cloud microcell can be obtained. The coulomb interaction energy can be solved by using quadruple integration, which is reduced by one time compared with the previous quintuple integration, and the calculation efficiency of the coulomb interaction energy between the s electron cloud and other electron clouds can be greatly improved.

Description

Technical Field

[0001] This invention relates to the field of electron cloud Coulomb interaction energy calculation and optimization technology, and particularly to a method, system, device and medium for measuring the Coulomb interaction energy of an s electron cloud with other electron clouds. Background Technology

[0002] Calculating the Coulomb interaction energy between electron clouds in hydrogen atoms and multi-electron systems is a core problem in quantum chemistry and computational physics. Traditional methods calculate the Coulomb interaction energy by directly solving a six-dimensional spatial integral (a quintuple integral involving the coordinates of two electrons). For example, for the interaction energy between the 1s electron cloud and other electron clouds in a hydrogen atom, it is necessary to calculate the interaction energy of r... 12 Integrating multiple variables such as r1, θ1, r2, and θ2 results in extremely high computational complexity, especially when dealing with high-energy electron clouds (such as 2p and 3d orbitals), where the analytical and numerical computation time for the integral terms increases significantly.

[0003] To address the problem of high computational complexity, Monte Carlo integration or numerical approximation methods are currently commonly used to reduce the computational load. However, these two methods still have the same integration multiples as traditional computational methods, namely five-fold integrals. This results in low computational efficiency for the Coulomb interaction energy between hydrogen atoms and different electron clouds, making it difficult to apply to high-precision quantum chemical and physical computations. Summary of the Invention

[0004] This invention provides a method for measuring the Coulomb interaction energy between s electron clouds and other electron clouds. This method can solve the problem in the prior art that the calculation efficiency of the Coulomb interaction energy between hydrogen atoms and different electron clouds is low, making it difficult to apply to high-precision quantum chemical and physical calculations.

[0005] This invention provides a method for measuring the Coulomb interaction energy between an s-electron cloud and other electron clouds, comprising the following steps: Taking the 1s electron cloud of a hydrogen atom as the first electron cloud, and based on the spherical symmetry of the 1s electron cloud of a hydrogen atom, the coordinate system of the second electron cloud is rotated clockwise around the horizontal symmetry axis of the 1s electron cloud of a hydrogen atom, so that the micro-region of the second electron cloud is projected onto the electron line connecting the center of the first electron cloud and the center of the second electron cloud; wherein, the second electron cloud is any electron cloud other than the first electron cloud. The Coulomb interaction energy between the 1s electron cloud and the second electron cloud microregion of the hydrogen atom before rotation is equivalently replaced with the Coulomb interaction energy between the 1s electron cloud and the second electron cloud microregion of the hydrogen atom after rotation, so as to replace the electron spacing between the 1s electron cloud and the second electron cloud microregion of the hydrogen atom with an equivalent distance; where the electron spacing is the spacing between the 1s electron cloud and the second electron cloud microregion of the hydrogen atom before rotation. The equivalent distance is the measured distance on the electron connection line between the 1s electron cloud and the second electron cloud micro-region of the hydrogen atom after rotation; the equivalent distance can be projected onto the horizontal symmetry axis of the 1s electron cloud of the hydrogen atom to form a measured projected distance. By using the measured projection distance, as well as the radius and angle parameters between the 1s electron cloud and the second electron cloud micro-region of the hydrogen atom, the Coulomb interaction energy between the 1s electron cloud and the second electron cloud of the hydrogen atom is obtained.

[0006] Preferably, the step of projecting the second electron cloud micro-region onto the electron connection line between the center of the first electron cloud and the center of the second electron cloud includes: Based on the spatial spherical symmetry of the 1s electron cloud of the hydrogen atom, the micro-region of the second electron cloud is rotated by an angle ω around the center point A of the 1s electron cloud of the hydrogen atom. The micro-region C on the second electron cloud falls on the line AB connecting the center A of the first electron cloud and the center B of the second electron cloud. The point C on the electron connection line AB is the micro-region C. ' ; Where the distance between AB is R, And point C ' The coordinates satisfy the following formula: ; Where: r2 represents the radial component of the coordinates of the second electron cloud micro-region; θ2 represents the angular component of the coordinates of the second electron cloud micro-region.

[0007] Preferably, replacing the electron spacing between the 1s electron cloud of a hydrogen atom and the micro-region of the second electron cloud with an equivalent distance includes: The Coulomb interaction energy between the 1s electron cloud and the micro-region C of the second electron cloud of a hydrogen atom before rotation, and the micro-region C of the 1s electron cloud and the second electron cloud of a hydrogen atom after rotation. ' If the Coulomb interaction energies between them are equal, then the electron spacing r between the 1s electron cloud and the second electron cloud micro-region of the hydrogen atom will be equal. 12 Replace with equivalent distance r ' 12 , is represented as: ; ; Where: r1 represents the radial component of the coordinates of the first electron cloud micro-region; θ1 represents the angular component of the coordinates of the first electron cloud micro-region.

[0008] Preferably, the acquisition of the Coulomb interaction energy between the 1s electron cloud and the second electron cloud of the hydrogen atom includes: The 1s electron cloud and the second electron cloud micro-region C of a hydrogen atom ' The Coulomb interaction energy I(r²θ²) between them is expressed as: ; ; The Coulomb interaction energy Q between the 1s electron cloud and the second electron cloud of a hydrogen atom is expressed as: ; in: The wavefunction representing the 1s electron cloud of a hydrogen atom; This represents the charge within the volume element dτ1; The wave function representing the second electron cloud; This represents the charge within the volume element dτ².

[0009] Preferably, the wavefunction of the 1s electron cloud of the hydrogen atom is expressed as: ; The charge within the volume element dτ1 is represented as: ; The wave function of the second electron cloud is expressed as: ; The charge within the dτ2 volume element is represented as: ; in: φ 1 represents the angle between the projection of the electron cloud micro-region 1 onto the xy plane and the x-axis in the spherical coordinate system; φ 2 represents the angle between the projection of the 2nd electron cloud micro-region onto the xy plane and the x-axis in the spherical coordinate system.

[0010] This invention also provides a system for measuring the Coulomb interaction energy between an s-electron cloud and other electron clouds, comprising: The simulation module is used to take the 1s electron cloud of the hydrogen atom as the first electron cloud. Based on the spherical symmetry of the 1s electron cloud of the hydrogen atom, the coordinate system of the second electron cloud is rotated clockwise around the horizontal symmetry axis of the 1s electron cloud of the hydrogen atom, so that the micro-region of the second electron cloud is projected onto the electron line connecting the center of the first electron cloud and the center of the second electron cloud; wherein, the second electron cloud is an electron cloud other than the first electron cloud. The substitution module is used to equivalently replace the Coulomb interaction energy between the 1s electron cloud and the second electron cloud micro-region of the hydrogen atom before rotation with the Coulomb interaction energy between the 1s electron cloud and the second electron cloud micro-region of the hydrogen atom after rotation, so as to replace the electron spacing between the 1s electron cloud and the second electron cloud micro-region of the hydrogen atom with an equivalent distance; wherein, the electron spacing is the spacing between the 1s electron cloud and the second electron cloud micro-region of the hydrogen atom before rotation; The equivalent distance is the measured distance on the electron connection line between the 1s electron cloud and the second electron cloud micro-region of the hydrogen atom after rotation; the equivalent distance can be projected onto the horizontal symmetry axis of the 1s electron cloud of the hydrogen atom to form a measured projected distance. The calculation module is used to obtain the Coulomb interaction energy between the 1s electron cloud and the second electron cloud of a hydrogen atom by using the measured projection distance and the radius and angle parameters between the micro-regions of the 1s electron cloud and the second electron cloud of a hydrogen atom.

[0011] This invention also provides an electronic device, including a memory and a processor; The memory is used to store computer programs; When the processor executes the computer program stored in the memory, it implements the steps of the method for measuring the Coulomb interaction energy of an s-electron cloud and other electron clouds as described above.

[0012] This invention also provides a computer-readable storage medium for storing a computer program, which, when executed by a processor, implements the steps of a method for measuring the Coulomb interaction energy of an s-electron cloud with other electron clouds as described above.

[0013] This invention provides a method for measuring the Coulomb interaction energy between an s-electron cloud and other electron clouds. Compared with the prior art, its advantages are as follows: This invention replaces the electron distance between the micro-regions of the 1s and second electron clouds of a hydrogen atom with the measured equivalent projected distance. In other words, the original calculation of the Coulomb interaction energy involved integrating five parameters—radius, angle, and electron distance—from the micro-regions of the 1s and second electron clouds. This invention, through equivalent substitution, replaces the electron distance between the micro-regions of the 1s and second electron clouds with a measured equivalent projected distance. The equivalent projected distance, which is the measurable projected distance on the electron line connecting the micro-regions of the 1s electron cloud and the second electron cloud of the hydrogen atom after rotation, is a measurable value on the electron line. Therefore, only the radius and angle parameters of the micro-regions of the 1s electron cloud and the second electron cloud of the hydrogen atom need to be integrated. The Coulomb interaction energy can be obtained by using a quadruple integral, which reduces the number of steps compared to the previous five-fold integral, and can greatly improve the calculation efficiency of the Coulomb interaction energy between the hydrogen atom and different electron clouds. Attached Figure Description

[0014] Figure 1 A schematic diagram of the overall process for measuring the Coulomb interaction energy between an s-electron cloud and other electron clouds, provided in an embodiment of the present invention; Figure 2 A schematic diagram of the spatial distribution of electron clouds for a method of measuring the Coulomb interaction energy between an s-electron cloud and other electron clouds provided in an embodiment of the present invention; Figure 3 The diagram illustrates the relationship between the Coulomb interaction energy of H1s electron cloud and H1s electron cloud as a function of their distance (ε) in a method for measuring the Coulomb interaction energy of s electron cloud and other electron clouds provided in an embodiment of the present invention. The inset in the upper right corner shows the relative error of the Coulomb interaction energy before and after the transformation as a function of ε. Figure 4 The H1s-H1s and H1s-H2p values ​​are provided in an embodiment of the present invention for measuring the Coulomb interaction energy between an s-electron cloud and other electron clouds. z H1s- H3dz 2 A schematic diagram showing how the relative error of the Coulomb interaction energy changes with ε before and after the transformation. Detailed Implementation

[0015] To make the above-mentioned objects, features, and advantages of the present invention more apparent and understandable, specific embodiments of the present invention will be described in detail below with reference to the accompanying drawings. Many specific details are set forth in the following description to provide a thorough understanding of the present invention. However, the present invention can be practiced in many other ways different from those described herein, and those skilled in the art can make similar modifications without departing from the spirit of the present invention. Therefore, the present invention is not limited to the specific embodiments disclosed below.

[0016] See Figure 1 This invention provides a method for measuring the Coulomb interaction energy between an s-electron cloud and other electron clouds, specifically: like Figure 2 The diagram shows the spatial distribution of electron clouds, including point charges a and b. The gray area in the diagram represents electron cloud 1, i.e., electron cloud 1s, and also includes electron cloud 2.

[0017] Depend on Figure 2 It can be seen that the wave function of electron cloud 1, with point A as the origin (center), is: .

[0018] Then the charge within the volume element dτ1 is: .

[0019] With point B as the origin (center), the 1s wavefunction of electron cloud 2 is: .

[0020] The charge within the volume element dτ² is: .

[0021] in: φ 1 represents the angle between the projection of the electron cloud micro-region 1 onto the xy plane and the x-axis in the spherical coordinate system; φ 2 represents the angle between the projection of the 2nd electron cloud micro-region onto the xy plane and the x-axis in the spherical coordinate system.

[0022] The Coulomb interaction energy between electron clouds 1 and 2 can be obtained as follows: .

[0023] Where: r 12 R represents the distance between the two electron cloud micro-regions; R represents the distance between the centers of the two electron clouds; r1 represents the radial component of the coordinates of the first electron cloud micro-region; θ1 represents the angular component of the coordinates of the first electron cloud micro-region; r2 represents the radial component of the coordinates of the second electron cloud micro-region; θ2 represents the angular component of the coordinates of the second electron cloud micro-region; φ represents the angle between the plane containing the first electron cloud micro-region and the plane containing the second electron cloud micro-region.

[0024] Because r 12 If the integral is a function of R, r1, θ1, r2, θ2, and φ, then the integral of the above equation is a five-fold integral, which takes a relatively long time to calculate the Coulomb interaction energy between the point charge and the electron cloud. To improve computational efficiency and reduce the number of integrals, considering that the 1s electron cloud is spherical and symmetric about the c-axis, the micro-region of electron cloud 2 is rotated clockwise by ω, so that point C falls on the line AB. ' At this point, let AC ' =ζ;Assume C and C ' If the location is a unit point charge, then it is at C and C. ' The Coulomb interaction energy between the point charge and the electron cloud 1 (gray shaded area) is equal. The Coulomb interaction energy between C and the electron cloud 1 is expressed as C0. ' The Coulomb interaction energy substitution between a point charge and the electron cloud of number 1 is expressed by the following formulas: .

[0025] .

[0026] .

[0027] .

[0028] .

[0029] .

[0030] .

[0031] .

[0032] Unit point charge of electron cloud 2 (C 'The Coulomb interaction energy between the point (where r is located) and electron cloud 1 is defined as I(r²θ²), and expressed as: .

[0033] .

[0034] The Coulomb interaction energy between electron clouds 1 and 2 can be expressed as: .

[0035] As can be seen from the above formula, using the method of equivalent substitution, the formula for calculating the Coulomb interaction energy Q between electron clouds 1 and 2 (1s electron clouds) is a quadruple integral. Compared with the formula before the substitution, the number of integrals has decreased by one. The integral calculation results before and after the equivalent substitution are as follows: Figure 3 As shown; Figure 3 This represents the relationship between the Coulomb interaction energy between the H1s electron clouds and the H1s electron cloud as a function of their distance (ε). Figure 3 The inset in the upper right corner shows how the relative error of the Coulomb interaction energy changes with ε before and after the transformation.

[0036] If electron cloud 2 is replaced with 2p and 3d electron clouds, the integral calculation results before and after the equivalent substitution are also equivalent; next, this invention will calculate the Coulomb interaction energy between 1s electron cloud and 2p and 3d electron clouds to prove the accuracy of the equivalent substitution described in this invention; specifically including: (1) 1s and 2p z Coulomb interaction energy between electron clouds.

[0037] 2p of hydrogen atom z The expression for the electron cloud function is: .

[0038] The Coulomb interaction energy before the equivalent substitution is: .

[0039] The Coulomb interaction energy after equivalent substitution is: .

[0040] Its I(r2θ2) is expressed as: .

[0041] (2) 1s and 3dz 2 Coulomb interaction energy between electron clouds.

[0042] 3dz of hydrogen atom 2 The electron cloud wavefunction is: .

[0043] The Coulomb interaction energy before the equivalent substitution is: .

[0044] The Coulomb interaction energy after equivalent substitution is: .

[0045] like Figure 4 As shown, H1s-H1s and H1s-H2p are... z H1s- H3dz 2 A schematic diagram illustrating how the relative error of the Coulomb interaction energy changes with ε before and after the transformation.

[0046] This invention reduces the integral multiplicity through equivalent substitution, thereby significantly improving the computational efficiency of the Coulomb interaction energy between the 1s electron cloud of a hydrogen atom and other electron clouds. This improvement makes the originally complex and time-consuming calculation process more efficient, helping to accelerate the speed of quantum chemical calculations. This invention is not only applicable to the calculation of interactions between the 1s electron clouds of hydrogen atoms, but also to the calculation of interactions between other types of electron clouds. The widespread application of the method proposed in this invention will help promote the development of the field of quantum chemical calculations and provide a more powerful tool for understanding the electronic structure and properties of atoms. This invention verifies through numerical calculations that the equivalence of the calculation results is maintained before and after the equivalent substitution, ensuring the accuracy of the calculation results.

[0047] The embodiments described above are merely illustrative of several implementations of the present invention, and while the descriptions are relatively specific and detailed, they should not be construed as limiting the scope of the invention patent. It should be noted that those skilled in the art can make various modifications and improvements without departing from the concept of the present invention, and these all fall within the protection scope of the present invention. Therefore, the protection scope of this invention patent should be determined by the appended claims.

Claims

1. A method for measuring the Coulomb interaction energy between an s-electron cloud and other electron clouds, characterized in that, Includes the following steps: Taking the 1s electron cloud of a hydrogen atom as the first electron cloud, and based on the spherical symmetry of the 1s electron cloud of a hydrogen atom, the coordinate system of the second electron cloud is rotated clockwise around the horizontal symmetry axis of the 1s electron cloud of a hydrogen atom, so that the micro-region of the second electron cloud is projected onto the electron line connecting the center of the first electron cloud and the center of the second electron cloud; wherein, the second electron cloud is any electron cloud other than the first electron cloud. The Coulomb interaction energy between the 1s electron cloud and the second electron cloud microregion of the hydrogen atom before rotation is equivalently replaced with the Coulomb interaction energy between the 1s electron cloud and the second electron cloud microregion of the hydrogen atom after rotation, so as to replace the electron spacing between the 1s electron cloud and the second electron cloud microregion of the hydrogen atom with an equivalent distance; where the electron spacing is the spacing between the 1s electron cloud and the second electron cloud microregion of the hydrogen atom before rotation. The equivalent distance is the measured distance on the electron connection line between the 1s electron cloud and the second electron cloud micro-region of the hydrogen atom after rotation; the equivalent distance can be projected onto the horizontal symmetry axis of the 1s electron cloud of the hydrogen atom to form a measured projected distance. By using the measured projection distance, as well as the radius and angle parameters between the 1s electron cloud and the second electron cloud micro-region of the hydrogen atom, the Coulomb interaction energy between the 1s electron cloud and the second electron cloud of the hydrogen atom is obtained.

2. The method for measuring the Coulomb interaction energy between an s-electron cloud and other electron clouds according to claim 1, characterized in that, The projection of the second electron cloud micro-region onto the electronic connection line between the center of the first electron cloud and the center of the second electron cloud includes: Based on the spatial spherical symmetry of the 1s electron cloud of the hydrogen atom, the micro-region of the second electron cloud is rotated by an angle ω around the center point A of the 1s electron cloud of the hydrogen atom. The micro-region C on the second electron cloud falls on the line AB connecting the center A of the first electron cloud and the center B of the second electron cloud. The point C on the electron connection line AB is the micro-region C. ' ; Where the distance between AB is R, And point C ' The coordinates satisfy the following formula: ； Where: r2 represents the radial component of the coordinates of the second electron cloud micro-region; θ2 represents the angular component of the coordinates of the second electron cloud micro-region.

3. The method for measuring the Coulomb interaction energy between an s-electron cloud and other electron clouds according to claim 2, characterized in that, The replacement of the electron spacing between the 1s electron cloud and the second electron cloud micro-region of a hydrogen atom with an equivalent distance includes: The Coulomb interaction energy between the 1s electron cloud and the micro-region C of the second electron cloud of a hydrogen atom before rotation, and the micro-region C of the 1s electron cloud and the second electron cloud of a hydrogen atom after rotation. ' If the Coulomb interaction energies between them are equal, then the electron spacing r between the 1s electron cloud and the second electron cloud micro-region of the hydrogen atom will be equal. 12 Replace with equivalent distance r ' 12 , is represented as: ； ； Where: r1 represents the radial component of the coordinates of the first electron cloud micro-region; θ1 represents the angular component of the coordinates of the first electron cloud micro-region.

4. The method for measuring the Coulomb interaction energy between an s-electron cloud and other electron clouds according to claim 3, characterized in that, The acquisition of the Coulomb interaction energy between the 1s electron cloud and the second electron cloud of the hydrogen atom includes: The 1s electron cloud and the second electron cloud micro-region C of a hydrogen atom ' The Coulomb interaction energy I(r²θ²) between them is expressed as: ； ； The Coulomb interaction energy Q between the 1s electron cloud and the second electron cloud of a hydrogen atom is expressed as: ； in: The wavefunction representing the 1s electron cloud of a hydrogen atom; This represents the charge within the volume element dτ1; The wave function representing the second electron cloud; This represents the charge within the volume element dτ².

5. The method for measuring the Coulomb interaction energy between an s-electron cloud and other electron clouds according to claim 4, characterized in that, The wavefunction of the 1s electron cloud of the hydrogen atom is expressed as: ； The charge within the volume element dτ1 is represented as: ； The wave function of the second electron cloud is expressed as: ； The charge within the dτ2 volume element is represented as: ； in: φ 1 represents the angle between the projection of the electron cloud micro-region 1 onto the xy plane and the x-axis in the spherical coordinate system; φ 2 represents the angle between the projection of the 2nd electron cloud micro-region onto the xy plane and the x-axis in the spherical coordinate system.

6. A system for measuring the Coulomb interaction energy between an s-electron cloud and other electron clouds, characterized in that, include: The simulation module is used to take the 1s electron cloud of the hydrogen atom as the first electron cloud. Based on the spherical symmetry of the 1s electron cloud of the hydrogen atom, the coordinate system of the second electron cloud is rotated clockwise around the horizontal symmetry axis of the 1s electron cloud of the hydrogen atom, so that the micro-region of the second electron cloud is projected onto the electron line connecting the center of the first electron cloud and the center of the second electron cloud; wherein, the second electron cloud is an electron cloud other than the first electron cloud. The substitution module is used to equivalently replace the Coulomb interaction energy between the 1s electron cloud and the second electron cloud micro-region of the hydrogen atom before rotation with the Coulomb interaction energy between the 1s electron cloud and the second electron cloud micro-region of the hydrogen atom after rotation, so as to replace the electron spacing between the 1s electron cloud and the second electron cloud micro-region of the hydrogen atom with an equivalent distance; wherein, the electron spacing is the spacing between the 1s electron cloud and the second electron cloud micro-region of the hydrogen atom before rotation; The equivalent distance is the measured distance on the electron connection line between the 1s electron cloud and the second electron cloud micro-region of the hydrogen atom after rotation; the equivalent distance can be projected onto the horizontal symmetry axis of the 1s electron cloud of the hydrogen atom to form a measured projected distance. The calculation module is used to obtain the Coulomb interaction energy between the 1s electron cloud and the second electron cloud of a hydrogen atom by using the measured projection distance and the radius and angle parameters between the micro-regions of the 1s electron cloud and the second electron cloud of a hydrogen atom.

7. An electronic device, characterized in that, include: Memory and processor; The memory is used to store computer programs; When the processor executes the computer program stored in the memory, it implements the steps of the method for measuring the Coulomb interaction energy of an s-electron cloud with other electron clouds as described in any one of claims 1 to 5.

8. A computer-readable storage medium, characterized in that, Used to store a computer program, which, when executed by a processor, implements the steps of a method for measuring the Coulomb interaction energy of an s-electron cloud with other electron clouds as described in any one of claims 1 to 5.

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