A method for calculating molecular diameter using minimum circumscribed circle encompassing molecular entity projection

Abstract

The application discloses a method for calculating a molecular diameter by using a minimum circumscribed circle encompassing a spatial projection of a molecular entity, and relates to the field of adsorption separation of compounds, wherein the molecular entity is abstracted into a three-dimensional van der Waals surface / electron density isosurface, and the minimum circumscribed circle capable of encompassing a figure obtained by projecting the three-dimensional van der Waals surface / electron density isosurface after spatial rotation is the quantum chemical diameter of the molecule. Through systematic analysis, the quantum chemical diameter of the molecule calculated in this way has a good correlation (Pearson correlation coefficient) with a measured dynamic diameter of the molecule, and the value is stable.

Description

Technical Field

[0001] This invention relates to the field of adsorption and separation of compounds, and more particularly to a method for calculating molecular diameter using the smallest circumscribed circle that encompasses the projection of the molecular entity. Background Technology

[0002] The size of compound molecules is closely related to their adsorption and diffusion behavior within the separation medium. When developing adsorption separation methods and processes for sieving mixtures using porous materials, it is necessary to select appropriate porous materials based on the kinetic diameters of each component in the mixture. The molecular kinetic diameter can be obtained by fitting the measured second virial coefficient or viscosity data of the gaseous state of the compound to the Lennard-Jones potential; however, the results obtained from fitting the second virial coefficient and viscosity are usually highly biased and not unique in determining the compound. Precise theoretical calculations of molecular size are complex, time-consuming, and labor-intensive, and are only applicable to small molecules. Currently developed methods for estimating molecular diameter using electron density and other methods not only require analysis of the types and distribution of surface functional groups but also visualization analysis of the spatial distribution characteristics of electron density. Furthermore, some of the setup and data analysis processes are quite complex, requiring manual completion, which is time-consuming, labor-intensive, prone to human error, and inefficient.

[0003] The dynamic diameter (size) of a molecule is a key parameter in adsorption separation, catalysis, and porous materials science. In existing techniques, spherical approximation models (such as those using PyWindow or Multiwfn software) suffer from simulation biases in dynamic processes like gas permeation because they neglect the anisotropy of molecular structure. While some studies have attempted to introduce cylindrical models to characterize molecular orientation, the lack of a universal optimization logic limits existing methods to simple molecules, and their determination process heavily relies on subjective human judgment, resulting in high subjectivity and a lack of universality. Furthermore, traditional semi-empirical methods such as the maximum diameter method and the spherical diameter method, although well-defined, generally ignore the anisotropy of molecular structure, leading to significant errors when dealing with non-spherical or complex molecules. While quantum chemical methods based on quantum electron density consider the anisotropy of molecular structure, existing schemes largely depend on subjective human judgment. Summary of the Invention

[0004] Traditional semi-empirical methods, such as the maximum diameter method and the spherical diameter method, while clearly defined, generally ignore the anisotropy of molecular structure and suffer from significant errors when dealing with non-spherical or complex molecules. This invention proposes a method for calculating molecular diameter using the smallest circumscribed circle encompassing the spatial projection of the molecular entity. This invention abstracts the molecular entity into a three-dimensional van der Waals surface / electron density isosurface. The smallest circumscribed circle that includes the projection of this surface after spatial rotation can be used to calculate the quantum chemical diameter of the molecule. When the molecular entity is an electron density isosurface, the diameter of the smallest circumscribed circle is the quantum chemical diameter of the molecule. When the molecular entity is represented by the Cartesian coordinates of atoms within the molecule, the quantum chemical diameter must be calculated by referring to the maximum van der Waals radius of the surface atoms. Systematic analysis shows that the quantum chemical diameter of the molecule calculated in this way has a good correlation with the measured kinetic diameter of the molecule (correlation coefficient). ), and the values ​​are stable.

[0005] The technical means employed in this invention are as follows: A method for calculating molecular diameter using the smallest circumscribed circle encompassing the molecular entity's projection, characterized by using the size of the smallest circumscribed circle including the molecular entity's projection as the measurement means, and including the following steps: Step 1: Based on the original data file of the target molecule, obtain the electron density or atomic coordinate dataset of the target molecule, and construct a three-dimensional spatial point set based on the atomic dataset. and the three-dimensional spatial point set Defined as the initial reference for geometric calculations; Step 2: Using the geometric center of the target molecule as the origin of rotation, utilize Euler angles... A parameterized 3D spatial rotation matrix is ​​constructed; the original 3D point set is rotated around the X and Y axes respectively, and the rotation matrix is ​​sampled and traversed throughout the entire space. The initial point set is then... Transformed into projection point sets under different spatial orientations ; Step 3: If the molecular entity is not represented by atomic coordinates, after each rotation, project the three-dimensional point set onto the XOY plane to obtain a two-dimensional point set. As a result from the observation perspective, the two-dimensional coordinates of the center of the circle are obtained, and then the minimum circumcircle algorithm is used to calculate the minimum circumcircle diameter of the projection point set under this orientation. If the molecular entity is represented in atomic coordinates, then the corresponding minimum circumscribed circle diameter is determined by referring to the van der Waals radius database. Step 4: By comparing the minimum circumcircle diameter of the projection point set obtained during the rotation, the smallest one is selected as the quantum mechanical diameter of the target molecule; by performing spatial search through angular rotation, the minimum circumcircle diameter results of the projection point set under all rotation orientations are compared, and the orientation that makes the projection diameter, i.e. the diameter of the smallest circle surrounding all projection points, reach a global minimum value is selected as the optimal orientation of the target molecule. The global minimum value is defined as the geometric basis parameter for obtaining the quantum chemical diameter of the molecule. Meanwhile, since the axial direction of the molecular entity is consistent with the Z-axis direction after rotation-projection, the original coordinate system can be restored by inverse rotation transformation to obtain the angle and axial direction of rotation required to rotate to the minimum circle position.

[0006] Furthermore, in step 3, When rotating about the X-axis:

[0007] Rotation about the Y-axis:

[0008] in, Represents the first point in the set of origins. The coordinates of the points and These represent the coordinates after rotation around the X-axis and the Y-axis, respectively; Indicates the rotation angle around the X-axis; This indicates the rotation angle around the Y-axis.

[0009] Furthermore, the original three-dimensional point set is rotated around the X-axis and Y-axis respectively, with a rotation step of less than 5°. The state during the rotation process is projected, and the minimum circumcircle radius of the projected two-dimensional point set is calculated, taking the minimum value.

[0010] Furthermore, the inverse rotation transformation to restore the original coordinate system includes the following steps: First, rotate counterclockwise around the Y-axis: ; Then rotate counterclockwise around the X-axis: ; in, This indicates the direction of the molecular axis after rotation-projection. The direction of the molecular axis in the original coordinate system. This is the angle required to rotate the Y-axis to the smallest circumcircle. This is the angle required to rotate the Y-axis to the smallest circumcircle. The molecular axis orientation is deduced from the calculation and recording results in step 3, thus determining the transformation process from the original position to the target position.

[0011] Furthermore, when the target molecule is calculated using quantum mechanical electron cloud density, the minimum circumscribed circle radius is the obtained molecular radius, and twice that radius is the quantum chemical diameter of the molecule.

[0012] Furthermore, when the target molecule is a semi-empirical van der Waals surface, the quantum chemical diameter of the molecule can be obtained by adding the minimum circumscribed circle diameter to the maximum van der Waals radius of the surface atoms. :

[0013] in, Indicates the minimum circumcircle radius. The largest atomic van der Waals radius appearing on the surface.

[0014] Compared with the prior art, the present invention has the following advantages: 1. Comprehensiveness and accuracy of geometric feature extraction: Traditional methods generally ignore the anisotropy of molecules and their overall spatial configuration. This invention extracts the geometric features of molecules in the entire space from their physical geometric essence through a cyclic computational framework of "spatial rotation-planar projection-minimum circumcircle fitting," ensuring that the predicted diameter value can truly reflect the physical occupancy of molecules in the dynamic process, effectively solving the problem of finding the optimal direction in traditional methods.

[0015] 2. Objectivity and numerical reproducibility of results: Existing technologies rely heavily on manual intuition or specific subjective experience benchmarks when determining geometric orientation, leading to inconsistencies in measurements of the same molecule in different studies. The global optimization algorithm introduced in this invention achieves automatic orientation determination, eliminates interference from human factors, ensures the standardization and objectivity of molecular diameter prediction results, and greatly improves data reproducibility under different computational scenarios.

[0016] 3. Stability and robustness of computational accuracy: Through a full-space traversal optimization mechanism, this invention effectively overcomes the bottleneck of traditional local approximation methods easily falling into "local optimal orientation." This invention, through global minimization at the algorithm level, ensures that the calculated diameter is the minimum geometric boundary of the molecule in space, thereby providing higher prediction accuracy. Attached Figure Description

[0017] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0018] Figure 1 This is a schematic diagram of the overall process of the present invention.

[0019] Figure 2 This is a projection analogy diagram of the present invention.

[0020] Figure 3 This is a schematic diagram of the rotation of the present invention.

[0021] Figure 4 The method for determining the diameter of a sphere in this invention ( ) and the maximum molecular diameter method ( The correlation analysis diagram between the obtained results and the dynamic diameter is shown.

[0022] Figure 5 The quantum mechanical diameter of the molecule calculated using electron density data in this invention ( ) and dynamic diameter ( Correlation analysis diagram of ).

[0023] Figure 6 This invention provides the molecular quantum mechanical diameter calculated using a combination of molecular structure and semi-empirical van der Waals radius. ) Results and dynamic diameter ( Correlation analysis diagram of ).

[0024] Figure 7 This is a schematic diagram of the molecular structure at the input position of the NH3 molecule in Embodiment 1 of the present invention.

[0025] Figure 8 This is a schematic diagram of the molecular structure of NH3 molecules when the molecular axis is perpendicular to the xy-screen in Embodiment 1 of the present invention.

[0026] Figure 9 This is a schematic diagram of the molecular structure at the input position of the NH3 molecule in Embodiment 2 of the present invention.

[0027] Figure 10 This is a schematic diagram of the molecular structure of the NH3 molecule in Embodiment 2 of the present invention when the molecular axis is perpendicular to the xy-screen. Detailed Implementation

[0028] To enable those skilled in the art to better understand the present invention, the technical solutions of the present invention will be clearly and completely described below with reference to the accompanying drawings of the embodiments of the present invention. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort should fall within the scope of protection of the present invention.

[0029] It should be noted that the terms "first," "second," etc., in the specification, claims, and accompanying drawings of this invention are used to distinguish similar objects and are not necessarily used to describe a specific order or sequence. It should be understood that such data can be interchanged where appropriate so that the embodiments of the invention described herein can be implemented in orders other than those illustrated or described herein. Furthermore, the terms "comprising" and "having," and any variations thereof, are intended to cover a non-exclusive inclusion; for example, a process, method, system, product, or apparatus that comprises a series of steps or units is not necessarily limited to those steps or units explicitly listed, but may include other steps or units not explicitly listed or inherent to such processes, methods, products, or apparatus.

[0030] like Figure 1 As shown, this invention provides a method for calculating the molecular diameter using the smallest circumcircle encompassing the projection of the molecular entity, comprising the following steps: Step 1: Based on the original data file of the target molecule, obtain the electron density or atomic coordinate dataset of the target molecule, and construct a three-dimensional spatial point set based on the atomic dataset. and the three-dimensional spatial point set Defined as the initial reference for geometric calculations; Step 2: Using the geometric center of the target molecule as the origin of rotation, utilize Euler angles... A parameterized 3D spatial rotation matrix is ​​constructed; the original 3D point set is rotated around the X and Y axes respectively, and the rotation matrix is ​​sampled and traversed throughout the entire space. The initial point set is then... Transformed into projection point sets under different spatial orientations ; When rotating about the X-axis:

[0031] Rotation about the Y-axis:

[0032] in, Represents the first point in the set of origins. The coordinates of the points and These represent the coordinates after rotation around the X-axis and the Y-axis, respectively; Indicates the rotation angle around the X-axis; This indicates the rotation angle around the Y-axis.

[0033] Step 3: If the molecular entity is not represented by atomic coordinates, after each rotation, project the three-dimensional point set onto the XOY plane to obtain a two-dimensional point set. As a result from the observation perspective, the two-dimensional coordinates of the center of the circle are obtained, and then the minimum circumcircle algorithm is used to calculate the minimum circumcircle diameter of the projection point set under this orientation. If the molecular entity is represented in atomic coordinates, then the corresponding minimum circumscribed circle diameter is determined by referring to the van der Waals radius database. Step 4: By comparing the minimum circumcircle diameter of the projection point set obtained during the rotation, the smallest one is selected as the quantum mechanical diameter of the target molecule. By performing a spatial search through angular rotation, the minimum circumcircle diameter of the projection point set under all rotation orientations is compared. The orientation that makes the projection diameter, i.e. the diameter of the smallest circle surrounding all projection points, reach the global minimum value is selected (taking the initial position seen from the observation perspective as the base point (assuming the reference axis is (0,0,1) at this time), then the angle of rotation required to see the smallest circumcircle from the observation perspective is reflected on the axis as (X,Y,1), which is to rotate X degrees around the x-axis and Y degrees around the y-axis) as the optimal orientation of the target molecule. The global minimum value is defined as the geometric basis parameter for obtaining the quantum chemical diameter of the molecule. Meanwhile, since the axial direction of the molecular entity is consistent with the Z-axis direction after rotation-projection, the original coordinate system can be restored by inverse rotation transformation to obtain the angle and axial direction of rotation required to rotate to the minimum circle position.

[0034] In this application, preferably, the original three-dimensional point set is rotated around the X-axis and Y-axis respectively, with a rotation step of less than 5°. The state during the rotation process is projected, and the minimum circumcircle radius of the projected two-dimensional point set is calculated, with the minimum value being taken.

[0035] In a preferred embodiment, the inverse rotation transformation to restore the original coordinate system in this application includes the following steps: First, rotate counterclockwise around the Y-axis: ; Then rotate counterclockwise around the X-axis: ; in, This indicates the direction of the molecular axis after rotation-projection. The direction of the molecular axis in the original coordinate system. This is the angle required to rotate the Y-axis to the smallest circumcircle. This is the angle required to rotate the Y-axis to the smallest circumcircle. The molecular axis orientation is deduced from the calculation and recording results in step 3, thus determining the transformation process from the original position to the target position.

[0036] In this application, when the target molecule is calculated using quantum mechanical electron cloud density, the minimum circumscribed circle radius is the obtained molecular radius, and twice that radius is the molecule's quantum chemical diameter. When the target molecule is a semi-empirical van der Waals surface, the molecule's quantum chemical diameter can be obtained by adding the minimum circumscribed circle diameter to the maximum van der Waals radius of the surface atoms. :

[0037] in, Indicates the minimum circumcircle radius. The largest atomic van der Waals radius appearing on the surface.

[0038] like Figure 2 As shown, the silver-gray surface in the figure represents a molecular entity, which can be an isoelectronic density surface or a van der Waals surface. The molecular model is a water molecule (the large sphere is an oxygen atom, and the two small spheres are hydrogen atoms). The smallest circumcircle is marked by the projected elliptical region. This method of finding the smallest circumcircle uses a similar projection idea.

[0039] like Figure 3 As shown, the optimal minimum circumcircle is found through two-step rotation. First, the rotation around the x-axis is used to find the local optimum. Then, the rotation around the y-axis is used to find the optimal minimum circumcircle. Finally, the rotation angles of the x-axis and y-axis are determined to determine the correct angle of the minimum circumcircle.

[0040] like Figure 4 As shown, the reported diameter of the molecular circumsphere ( ) and maximum diameter ( Molecular structure description and experimental data obtained Correlation analysis of ) The left figure shows the fitting of the diameter of the circumscribed sphere of the molecule to experimental data. The formula for this method is as follows: ; in, It represents the diameter of the circumscribed sphere of the molecular center of mass set (i.e., the diameter of the particle sphere). This represents the van der Waals radius of a surface atom.

[0041] The right figure shows the fitting of the maximum diameter method with experimental data. The formula for this method is as follows: ; in, This represents the straight-line distance between the two farthest atoms in a molecule. and These represent the van der Waals radii of the two atoms, respectively.

[0042] like Figure 5As shown, the calculation results of molecular quantum chemical diameter using the electron density isosurface method of this patent are compared with experimental data. Correlation analysis: The left figure shows the correlation between the results obtained and the experimental data in the range of electron density of 0.002 to 0.00201 au; the right figure shows the correlation between the results obtained and the experimental data in the range of electron density of 0.003 to 0.00205 au.

[0043] like Figure 6 As shown, by applying the method of this patent, the correlation between the molecular quantum chemical radius obtained from processing molecular semi-empirical van der Waals surfaces or molecular isoelectronic density surfaces and experimental results is far superior to the traditional maximum diameter method and molecular circumscribed ball method.

[0044] Example 1: Taking NH3 as an example, calculations were performed based on the electron density dataset. First, electron density points are extracted, focusing on electron densities within a specified range (0.003 to 0.00305 in this case). Next, a coarse search is performed with a step size of 10°, traversing the X and Y axes 1296 times. Each time, the original point set is copied, rotated around both axes, and projected onto the XY plane. The minimum circumcircle diameter of the projected point set is calculated, and the diameter and axis vectors are recorded. Based on this, a search with a step size of 1° is performed to find the final minimum circumcircle. The axis direction vectors are calculated by reverse rotation around the reference axis [0,0,1] around the Y-axis, and then around the X-axis. Finally, the target molecule diameter is determined.

[0045] Table 1. Partial process data for calculating the quantum mechanical diameter using the electron density dataset of NH3.

[0046] The final result shows that the minimum value, or the specified molecular diameter of 3.64 Å, is achieved when the axis is rotated to [0.0, 0.829, 0.5592]. The initial state of the NH3 molecule and its molecular axis... xy - Molecular structure when the plane is perpendicular, such as Figure 7 , 8 As shown.

[0047] Example 2: Taking NH3 as an example, calculations were performed based on molecular structure data. First, the atomic coordinates in the molecule are read, as shown in Table 2.

[0048] Table 2 Cartesian coordinates of NH3 molecules

[0049] The process involves traversing 360° increments of 1° along the X and Y axes. Molecular coordinates are copied and rotated around both axes, projecting atomic coordinates onto the XY plane and calculating the minimum enclosing circle diameter of the projected point set. Once the minimum diameter is obtained, the reference axis [0,0,1] is rotated in the opposite direction around the Y-axis, and then in the opposite direction around the X-axis, based on its corresponding rotation angle. Next, the surface atoms are identified by rotating the molecule backward along the minimum circumscribed circle axis. The surface atoms are then identified; in NH3, the surface atoms are designated as "H". The Bondi van der Waals radius is used for calculation, where the van der Waals radius of "H" is 1.2 Å. This radius is multiplied by 2 and then added to the previously found minimum circumscribed circle diameter.

[0050] The calculation results are shown in Table 3 below. Table 3. Partial process data for calculating the quantum mechanical diameter using NH3 structure data.

[0051] The final result shows that the molecular diameter reaches its minimum value, or the specified molecular diameter of 3.81 Å, when the axis is rotated to [0.866, -0.5, -0.0]. The initial state of the NH3 molecule and its molecular axis... xy - Molecular structure when the plane is perpendicular, such as Figure 9 , 10 As shown.

[0052] The sequence numbers of the above embodiments of the present invention are for descriptive purposes only and do not represent the superiority or inferiority of the embodiments. In the above embodiments of the present invention, the descriptions of each embodiment have their own emphasis; parts not described in detail in a certain embodiment can be referred to in the relevant descriptions of other embodiments. It should be understood that the disclosed technical content in the several embodiments provided in this application can be implemented in other ways.

[0053] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, and not to limit them; although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some or all of the technical features; and these modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the scope of the technical solutions of the embodiments of the present invention.

Claims

1. A method for calculating the diameter of a molecule using the smallest circumcircle encompassing the projection of the molecular entity, characterized in that, Includes the following steps: Step 1: Based on the original data file of the target molecule, obtain the electron density or atomic coordinate dataset of the target molecule, and construct a three-dimensional spatial point set based on the atomic dataset. and the set of three-dimensional spatial points Defined as the initial reference for geometric calculations; Step 2: Using the geometric center of the target molecule as the origin of rotation, utilize Euler angles... A parameterized 3D spatial rotation matrix is ​​constructed; the original 3D point set is rotated around the X and Y axes respectively, and the rotation matrix is ​​sampled and traversed throughout the entire space. The initial point set is then... Transformed into projection point sets under different spatial orientations ; Step 3: If the molecular entity is not represented by atomic coordinates, after each rotation, project the three-dimensional point set onto the XOY plane to obtain a two-dimensional point set. As a result from the observation perspective, the two-dimensional coordinates of the center of the circle are obtained, and then the minimum circumcircle algorithm is used to calculate the minimum circumcircle diameter of the projection point set under this orientation. If the molecular entity is represented in atomic coordinates, then the corresponding minimum circumscribed circle diameter is determined by referring to the van der Waals radius database. Step 4: By comparing the minimum circumcircle diameter of the projection point set obtained during the rotation, the smallest one is selected as the quantum mechanical diameter of the target molecule; by performing spatial search through angular rotation, the minimum circumcircle diameter of the projection point set under all rotation orientations is compared, and the orientation that makes the minimum circumcircle diameter of the projection point set reach the global minimum value is selected as the optimal orientation of the target molecule axis. The global minimum value is defined as the basic geometric parameter for obtaining the quantum chemical diameter of the molecule. Meanwhile, since the axial direction of the molecular entity is consistent with the Z-axis direction after rotation-projection, the original coordinate system can be restored by inverse rotation transformation to obtain the angle of rotation required for the molecule to rotate to the optimal orientation and the optimal orientation of the axis.

2. The method for calculating molecular diameter using the smallest circumcircle encompassing the projection of the molecular entity, as described in claim 1, is characterized in that... In step 3 When rotating about the X-axis: Rotation about the Y-axis: in, Represents the first point in the set of origins. The coordinates of the points and These represent the coordinates after rotation around the X-axis and the Y-axis, respectively; Indicates the rotation angle around the X-axis; This indicates the rotation angle around the Y-axis.

3. The method for calculating the molecular diameter using the smallest circumcircle encompassing the projection of the molecular entity, as described in claim 1, is characterized in that... The original 3D point set is rotated around the X-axis and Y-axis respectively, with a rotation step of less than 5°. The state during the rotation is projected, and the minimum circumcircle radius of the projected 2D point set is calculated. Finally, the minimum value is taken.

4. The method for calculating the molecular diameter using the smallest circumcircle encompassing the projection of the molecular entity, as described in claim 1, is characterized in that... The inverse rotation transformation to restore the original coordinate system includes the following steps: First, rotate counterclockwise around the Y-axis: ； Then rotate counterclockwise around the X-axis: ； in, This indicates the direction of the molecular axis after rotation-projection. The direction of this axis in the original coordinate system. This is the angle required to rotate the Y-axis to the smallest circumcircle. This is the angle required to rotate the Y-axis to the smallest circumcircle. The molecular axis orientation is deduced from the calculation and recording results in step 3, thus determining the transformation process from the original position to the target position.

5. The method for calculating the molecular diameter using the smallest circumcircle encompassing the projection of the molecular entity, as described in claim 1, is characterized in that... When the target molecule is calculated using quantum mechanical electron cloud density, the minimum circumscribed circle radius is the obtained molecular radius, and twice that radius is the quantum chemical diameter of the molecule.

6. The method for calculating the molecular diameter using the smallest circumcircle encompassing the projection of the molecular entity, as described in claim 1, is characterized in that... When the molecular entity is represented by the Cartesian coordinates of the atoms within the molecule, the quantum chemical diameter of the molecule can be obtained by adding the smallest circumcircle diameter to the largest van der Waals radius of the surface atoms. : ； in, Indicates the minimum circumcircle radius. The largest atomic van der Waals radius appearing on the surface.

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