A crystal tight-binding parameter prediction method with symmetry of rotation invariance and degenerate orbit

Abstract

This application relates to the field of materials computation and tight-binding models, and discloses a method for predicting crystal tight-binding parameters with rotational invariance and degenerate orbital symmetry. The method includes: obtaining the crystal material's lattice matrix, atom types and coordinates, orbital types, and DFT band structure; constructing a crystal graph structure; obtaining atomic local environment representations via graph neural network message passing; obtaining orbital features through a degenerate orbital weight-sharing encoder; constructing a lattice-anchored local reference frame based on lattice basis vectors; projecting inter-atomic direction vectors to obtain intrinsic direction cosines; and combining this with Slater-Koster geometric factors to predict orbital coupling parameters and generate tight-binding parameters. This application achieves overall crystal rotational symmetry while maintaining orbital degeneracy.

Description

Technical Field

[0001] This application relates to the field of materials calculation and tight-binding models, specifically to a method for predicting crystal tight-binding parameters that are rotationally invariant and degenerate orbital symmetric. Background Technology

[0002] In crystal electronic structure modeling, tight-binding (TB) parameters directly determine orbital energy levels, interatomic transition strengths, and band dispersion relations, serving as crucial intermediate quantities connecting atomic structure and electronic transport properties. Traditional tight-binding models typically rely on manually selecting orbital basis sets, empirically setting coupling strengths, or fitting parameters based on first-principles calculations. While computationally efficient and with clear physical meaning, the parameter construction process often lacks automatic adaptation capabilities in complex crystal configurations, defect structures, or strained systems. In recent years, graph neural networks have been used to learn local chemical environments from atomic structures and further predict tight-binding parameters, thereby reducing the need for manual parameter tuning.

[0003] However, existing graph neural networks typically still establish orbital orientations in a global Cartesian coordinate system when predicting orbital coupling parameters. For tight-binding models described in the Slater-Koster form, orbital coupling strength depends not only on interatomic spacing but also on the projection relationship between interatomic orientations and the orbital basis. If global coordinate orientation components are used directly, the same crystal may receive different directional inputs after global rotation, causing the transition parameters output by the network to change with coordinate orientation, thus affecting the consistency of bandgap predictions. Existing methods relying on rotational data augmentation or result correction struggle to guarantee strict rotational invariance at the model structure level. On the other hand, physically equivalent or approximately degenerate orbitals often exist in highly symmetric crystals. These orbitals should maintain consistent or constrained feature representations under the same local environment, but existing models often set independent encoders or independent parameter branches for different orbitals. Random initialization and independent training can easily introduce non-physical differences, causing inconsistent mappings of orbitals that should be equivalent, leading to spurious energy level splitting or destruction of degeneracy relationships. Therefore, in tight-binding parameter prediction models, it is necessary to simultaneously consider the consistency of directional features under crystal rotation and the preservation of symmetry of equivalent orbitals during the feature encoding stage. Summary of the Invention

[0004] The purpose of this application is to provide a rotation-invariant and degenerate orbital symmetry-based method for predicting crystal tight-binding parameters, addressing the problems of existing graph neural networks' susceptibility to global coordinate system influences in predicting tight-binding parameters, inconsistent prediction results after overall crystal rotation, and the tendency for independent encoding of equivalent orbits to disrupt orbital degenerate symmetry. To achieve the above objective, this application provides the following solution:

[0005] This application provides a method for predicting the tight-binding parameters of a crystal, including: acquiring crystal structure data, wherein the crystal structure data includes lattice matrix, atom type and coordinates, orbital type, and band structure calculated by density functional theory (DFT);

[0006] Crystal graph structure data is constructed based on the crystal structure data. The crystal graph structure data includes atomic node features, edge features, and orbital index information. It is input into a graph neural network, and the atomic local environment representation of each atom is obtained through message passing.

[0007] Based on the atomic local environment representation and orbit type, orbit features that satisfy orbit symmetry are obtained through an orbit encoder that shares degenerate orbit weights;

[0008] A lattice-anchored local reference system is constructed based on the lattice basis vectors in the lattice matrix. The interatomic direction vectors are projected onto the lattice-anchored local reference system to obtain the intrinsic direction cosines.

[0009] Based on the orbital features, edge features, and intrinsic direction cosines, a graph neural network is used to predict rotationally invariant orbital coupling parameters, and tight-binding parameters are generated from the orbital coupling parameters.

[0010] In some embodiments, the orbital index information of the crystal diagram data adopts a preset orbital mapping table, which includes a Slater-Koster orbital index, an orbital encoding index, and an orbital type index corresponding to the atom type and local orbital number.

[0011] In some embodiments, the edge features are jointly determined by distance encoding and bond type embedding. To enable the graph neural network to continuously represent local interactions under different bond lengths, the interatomic distance d is... ij Radial basis function expansion yields the distance code:

[0012]

[0013] In the formula, ϕ k (d ij ) represents the distance code corresponding to the k-th radial basis function, μ k γ represents the center of the k-th radial base, and γ represents the radial base width parameter.

[0014] The distance encoding and key type embedding are then input into the edge feature encoding function to obtain the edge embedding features:

[0015]

[0016] In the formula, ϕ(d ij ) represents a distance code consisting of multiple radial basis functions, bij Indicates key type embedding, e ij This represents edge embedding features.

[0017] In some embodiments, the graph neural network performs message passing based on node features and edge features. For a graph edge pointing from node j to node i, the edge message is calculated based on the node representation and edge embedding:

[0018]

[0019] Then, the neighbor edge messages are aggregated, and the node representation is updated:

[0020]

[0021] In the formula, h i and h j Let N(i) represent the node representations of node i and node j respectively, and let N(i) represent the set of neighboring nodes of node i. ' i The updated node representation shows that, through the above process, atomic nodes can fuse neighborhood structure information, edge feature information, and local environment information, providing atomic environment characterization for subsequent orbit coupling prediction.

[0022] In some embodiments, the degenerate orbit weight-sharing orbit encoder sets a shared coding branch according to the orbit symmetry relationship, and uses the same multi-layer perceptron (MLP) encoder for feature extraction of physically degenerate orbits. The MLP includes a linear transform layer, a SiLU activation function layer, and a layer normalization layer. For orbit α on atom i, its orbital features are represented as:

[0023]

[0024] In the formula, o iα h represents the orbital characteristic corresponding to orbital α of atom i. i The local atomic environment of atom i is represented by s. α The orbital type embedding of orbital α is represented; if orbital α and orbital β belong to the same degenerate orbital group, then the following condition is satisfied: Encoder ord (α) = Encoder ord (β).

[0025] In some embodiments, the lattice-anchored local reference system is dynamically constructed from lattice basis vectors. Let the first lattice basis vector of the lattice matrix be a, and the second lattice basis vector be b. Then, the lattice-anchored local reference system consists of three locally orthogonal axes:

[0026]

[0027]

[0028]

[0029] In the formula, u is a local axis along the direction of the first lattice basis vector, v is a local axis located in the lattice plane and orthogonal to u, and w is a local axis perpendicular to the lattice plane.

[0030] In some embodiments, the interatomic direction vectors are projected onto the lattice-anchored local reference frame to obtain intrinsic direction cosines, which are expressed as:

[0031]

[0032]

[0033]

[0034] In the formula, l ' ij m ' ij n ' ij Used to replace the direction components in the global coordinate system.

[0035] In some embodiments, the intrinsic direction cosine remains unchanged before and after the overall crystal rotation. Let the overall rotation matrix be R, then the following is satisfied:

[0036]

[0037]

[0038]

[0039] In the formula, Rr ij R represents the interatomic interaction vector after rotation, and Ru, Rv, and Rw represent the three local axes of the lattice-anchored local reference frame after rotation.

[0040] In some embodiments, the orbital coupling parameters are jointly determined by orbital representation, atomic local environment representation, edge features and intrinsic direction cosine, and form interatomic tight-binding transition terms through Slater-Koster geometric factors;

[0041] The Slater-Koster geometric factor between orbits α and β is determined based on the intrinsic direction cosine vector:

[0042]

[0043] Predicting the σ, π, and δ type orbital coupling strength between orbits α and β using graph neural networks:

[0044]

[0045] The final interatomic tight-binding transition term is expressed as:

[0046]

[0047] In the formula, G αβ Let f represent the Slater-Koster geometric factor function. SK Let t represent the orbital coupling prediction function in a graph neural network. αβ ij This represents the tight-binding transition term between orbital α of atom i and orbital β of atom j.

[0048] Compared with the prior art, this application has at least the following beneficial effects: by obtaining intrinsic direction cosines by anchoring the local reference system to the lattice, the orbital orientation features can be avoided from directly depending on the global Cartesian coordinate system, thus improving the consistency of tight-binding parameter prediction under overall crystal rotation; by sharing the weights of the degenerate orbital encoder, the encoding consistency of equivalent or degenerate orbitals in high-symmetry crystals can be maintained, reducing the risk of non-physical orbital splitting; combining the above-mentioned rotation-invariant orientation features and orbital symmetry encoding mechanism with a graph neural network model helps to improve the generalization ability, physical consistency, and reliability of electronic structure prediction of tight-binding parameters. Attached Figure Description

[0049] Figure 1 This is a schematic flowchart of a method for predicting the tight-binding parameters of a crystal that is rotationally invariant and degenerate orbit symmetric, provided in an embodiment of this application.

[0050] Figure 2 This is a schematic diagram of the degenerate track weight sharing track encoder process provided in the embodiments of this application.

[0051] Figure 3 This is a schematic diagram of the orbital projection band structure of a MoS2 crystal unit cell provided in an embodiment of this application.

[0052] Figure 4 This is a schematic diagram of the construction and rotation invariance of the lattice-aligned local reference frame provided in the embodiments of this application.

[0053] Figure 5 This is a schematic diagram of the band structure comparison and tight-binding parameter matrix of MoS2 crystal provided in the embodiments of this application. Detailed Implementation

[0054] The technical solution of this application will be further described below with reference to the accompanying drawings. It should be understood that the following embodiments are only used to illustrate the technical solution of this application and are not intended to limit the scope of protection of this application. All equivalent substitutions, conventional modifications or combinations made based on the technical concept of this application should be included within the scope of protection of this application.

[0055] This embodiment provides a method for predicting the tight-binding parameters of a crystal that is rotationally invariant and degenerate orbital symmetric. Specifically, MoS2 crystal material is used as an example for illustration, but this application is not limited to MoS2 crystal material and can also be applied to other crystal materials with periodic structures and orbital coupling relationships.

[0056] Combination Figure 1 As shown, the method described in this embodiment includes: acquiring crystal structure data, constructing crystal graph structure data based on the crystal structure data, wherein the crystal graph structure data includes atomic node features, edge features, and orbital index information, and obtaining an atomic local environment representation through message passing via a graph neural network;

[0057] Based on the atomic local environment representation and orbit type, symmetric orbit features are obtained through an orbit encoder with degenerate orbit weight sharing;

[0058] A lattice-anchored local reference system is constructed based on the lattice basis vectors. The interatomic direction vectors are projected onto the lattice-anchored local reference system to obtain the intrinsic direction cosines.

[0059] The orbit coupling parameters are predicted based on the orbit features, edge features, and intrinsic direction cosines, and tight-binding parameters are generated from the orbit coupling parameters.

[0060] In this embodiment, the orbital index information adopts a preset orbital mapping table. The orbital mapping table includes a Slater-Koster orbital index, an orbital encoding index, and an orbital type index corresponding to the atom type and local orbital number. Through the orbital mapping table, different local orbitals on different atoms can be uniformly mapped to the corresponding positions in the tight-bound Hamiltonian matrix, and an index basis is provided for subsequent orbital encoder selection, orbital coupling parameter prediction, and Slater-Koster geometric factor calculation.

[0061] The edge features are jointly determined by distance encoding and bond type embedding. To enable the graph neural network to continuously represent local interactions under different bond lengths, the interatomic distance d is... ij Radial basis function expansion yields the distance code:

[0062]

[0063] In the formula, ϕ k (d ij) represents the distance code corresponding to the k-th radial basis function, μ k γ represents the center of the k-th radial base, and γ represents the radial base width parameter.

[0064] The distance encoding and key type embedding are then input into the edge feature encoding function to obtain the edge embedding features:

[0065]

[0066] In the formula, ϕ(d ij ) represents a distance code consisting of multiple radial basis functions, b ij Indicates key type embedding, e ij This represents edge embedding features.

[0067] The graph neural network performs message passing on node and edge features. For a graph edge pointing from node j to node i, the edge message is calculated based on the node representation and edge embedding:

[0068]

[0069] Then, the neighbor edge messages are aggregated, and the node representation is updated:

[0070]

[0071] In the formula, h i and h j Let N(i) represent the node representations of node i and node j respectively, and let N(i) represent the set of neighboring nodes of node i. ' i The updated node representation shows that, through the above process, atomic nodes can fuse neighborhood structure information, edge feature information, and local environment information, providing atomic environment characterization for subsequent orbit coupling prediction.

[0072] Based on the atomic local environment representation and orbit type, orbital features satisfying orbital symmetry are obtained through an orbital encoder using degenerate orbital weight sharing. For example... Figure 2 As shown, the track encoder identifies and groups tracks based on node information and track type information; for independent tracks, an independent track encoder is used; for physically degenerate tracks or equivalent tracks, the same encoder is used; for other track groups, the corresponding encoder is set according to the track category.

[0073] The orbit encoder employs a multilayer perceptron, which includes a linear transform layer, a SiLU activation function layer, and a layer normalization layer. For orbit α on atom i, its orbital characteristics are represented as:

[0074]

[0075] In the formula, oiα h represents the orbital characteristic corresponding to orbital α of atom i. i The local atomic environment of atom i is represented by s. α The orbital type embedding for orbital α;

[0076] If orbits α and β belong to the same degenerate orbit group, then they use the same orbit encoder, i.e.: Encoder ord (α) = Encoder ord (β) By using weight sharing, the feature extraction rules of physically equivalent orbits can be kept consistent in the same or similar local environment, avoiding non-physical differences caused by random initialization and independent training of different orbit encoders.

[0077] Specifically, by calculating the orbital projection band structure of a MoS2 crystal unit cell, the contribution relationship between the d orbitals of Mo atoms and the p orbitals of S atoms in the band structure can be determined, and the orbitals involved in the prediction of tight-binding parameters can be grouped accordingly. For example... Figure 3 As shown, in MoS2 crystal, the energy band near the conduction band bottom mainly originates from the d(z) of Mo atoms. 2 The energy band near the valence band apex mainly originates from the d(xy) and d(xy) orbitals of Mo atoms. 2 -y 2 The orbitals contribute to the electron band structure of MoS2 crystals. Furthermore, in the lower valence band region and part of the conduction band region, there is a certain degree of hybridization between the p orbitals of S atoms and the d orbitals of Mo atoms. Therefore, both the d orbitals of Mo atoms and the p orbitals of S atoms have a significant impact on the electronic band structure of MoS2 crystals and should be selected as the orbital basis sets in the tight-binding parameter prediction process.

[0078] Furthermore, based on the orbital symmetry relationship of the MoS2 crystal unit cell, the above orbitals are grouped. In this embodiment, the d(xy) orbitals and d(xy) orbitals of the Mo atom are grouped. 2 -y 2 In a highly symmetric crystal environment, the d(xz) and d(yz) orbitals of Mo atoms have equivalent or approximately degenerate relationships and can be grouped into the same equivalent orbital group. The d(xz) and d(yz) orbitals of Mo atoms have the same orbital symmetry properties and can also be grouped into the same equivalent orbital group. The p(x) and p(y) orbitals of S atoms are equivalent in the in-plane direction and can be grouped into the same equivalent orbital group. For the d(z) orbitals of Mo atoms... 2 The p(z) orbitals of the S atom and the p(z) orbitals of the S atom, due to their different orbital orientation characteristics from the above-mentioned in-plane orbitals, can be encoded as independent orbitals or independent orbital groups, respectively.

[0079] Therefore, the Mo-(d) orbitals and S-(p) orbitals with degeneracy or equivalence in the MoS2 crystal can be uniformly constrained in the orbital encoding stage, making the orbital coupling parameters predicted later more consistent with the physical laws of crystal symmetry and band structure.

[0080] Construct a lattice-anchored local reference frame based on lattice basis vectors. For example... Figure 4 As shown, the lattice-anchored local reference frame is dynamically constructed from lattice basis vectors and rotates synchronously with the overall crystal rotation. Let the first lattice basis vector of the lattice matrix be *a*, and the second lattice basis vector be *b*, then the first local axis *u* is defined as:

[0081]

[0082] The projection components along the first local axis in the second lattice basis vector are removed and normalized to obtain the second local axis v:

[0083]

[0084] The third local axis w is obtained by the cross product of the first and second local axes:

[0085]

[0086] In the formula, u represents a local axis along the direction of the first lattice basis vector, v represents a local axis located in the lattice plane and orthogonal to u, and w represents a local axis perpendicular to the lattice plane. Since u, v, and w are all dynamically constructed from the lattice basis vectors, the local reference frame is bound to the crystal's own orientation.

[0087] By projecting the interatomic direction vectors onto the lattice-anchored local reference frame, intrinsic direction cosines are obtained. For the interatomic interaction vector r... ij Its direction cosine in a lattice-anchored local reference frame is expressed as:

[0088]

[0089]

[0090]

[0091] In the formula, l ' ij m ' ij n ' ij These represent the projection components of the interatomic direction vector onto the local axes u, v, and w, respectively, used to replace the direction components in the global Cartesian coordinate system. When the crystal undergoes a rigid rotation, assuming the rotation matrix is ​​R, the interatomic interaction vector and the lattice-anchored local reference frame rotate synchronously:

[0092]

[0093]

[0094]

[0095] In the formula, Rr ij Let Ru, Rv, and Rw represent the three local axes of the lattice-anchored local reference system after rotation, respectively. The intrinsic direction cosine remains unchanged before and after the overall crystal rotation. Therefore, using the intrinsic direction cosine as the orbital coupling direction input can avoid the directional features from directly depending on the global Cartesian coordinate system, thus improving the rotational consistency of the tight-binding parameter predictions at the model structure level.

[0096] The Slater-Koster geometric factor is calculated based on the intrinsic direction cosine. For orbital α of atom i and orbital β of atom j, their corresponding Slater-Koster geometric factors are determined according to the intrinsic direction cosine:

[0097]

[0098] Predicting the σ, π, and δ type orbital coupling strength between orbits α and β using graph neural networks:

[0099]

[0100] The final interatomic tight-binding transition term is expressed as:

[0101]

[0102] In the formula, G αβ Let f represent the Slater-Koster geometric factor function. SK Let t represent the orbital coupling prediction function in a graph neural network. αβ ij This represents the tight-binding transition term between orbital α of atom i and orbital β of atom j.

[0103] The application effect of this embodiment is described in detail below. Table 1 shows the errors of onsite, hopping, and total TB parameters under different rotation angles of the MoS2 crystal structure. The onsite parameter error represents the difference in potential energy parameters of the same orbital of the same atom before and after rotation; the hopping parameter error represents the difference in transition parameters between different atomic orbitals before and after rotation; and the total TB parameter error represents the average difference in the complete tight-binding parameter matrix. As can be seen from Table 1, after the overall rotation of the MoS2 crystal, the predicted onsite parameter, hopping parameter, and total tight-binding parameter errors are all within the numerical calculation tolerance range, indicating that the method described in this embodiment can maintain the consistency of the predicted tight-binding parameter results before and after the overall rotation of the crystal.

[0104] Table 1

[0105] 0° 0 0 0 Reference Structure 60° <![CDATA[1.10×10 −5 ]]> <![CDATA[2.74×10 −5 ]]> <![CDATA[2.72×10 −5 ]]> Keep the parameters consistent 90° <![CDATA[1.36×10 −7 ]]> <![CDATA[9.68×10 −6 ]]> <![CDATA[9.57×10 −6 ]]> Keep the parameters consistent 120° <![CDATA[3.61×10 −5 ]]> <![CDATA[2.01×10 −5 ]]> <![CDATA[2.03×10 −5 ]]> Keep the parameters consistent 180° <![CDATA[1.82×10 −7 ]]> <![CDATA[9.29×10 −6 ]]> <![CDATA[9.18×10 −6 ]]> Keep the parameters consistent

[0106] To further verify the stability of the overall rotation of the MoS2 crystal and its preservation of degenerate orbital symmetry by the method described in this embodiment, tight-binding parameters were predicted for both the original MoS2 crystal structure and the MoS2 crystal structure after overall rotation. The predicted tight-binding parameters were then used to reconstruct the electronic band structure. For example... Figure 5 As shown, Figure 5 (a) presents a comparison of the DFT reference band, the TB predicted band corresponding to the original structure, and the TB predicted band corresponding to the rotated structure. The original TB band and the rotated TB band basically coincide on the high symmetry path, and both are in good agreement with the DFT reference band, indicating that the overall rotation of the crystal did not cause any non-physical changes to the predicted band. Figure 5 (b) Further, a heatmap of the corresponding tight-binding parameter matrix is ​​presented, where the diagonal elements represent onsite parameters, and the off-diagonal elements represent hopping parameters between different orbitals or coupling parameters between orbitals of the same atom. The heatmap shows that the tight-binding parameter matrices of the original structure and the rotated structure maintain consistency in numerical distribution and main coupling regions.

[0107] The above results demonstrate that this embodiment obtains intrinsic direction cosines by anchoring a local reference frame to the lattice, ensuring that the Slater-Koster geometric factor and orbital coupling parameters are independent of external global coordinate orientation. This guarantees the consistency of the onsite parameters, hopping parameters, and the reconstructed band structure of the MoS2 crystal before and after overall rotation. Simultaneously, the use of a shared orbital encoder for feature extraction of equivalent or degenerate orbitals helps maintain the symmetry constraint relationship of equivalent orbital groups in Mo-(d) and S-(p) orbitals, reducing the risk of non-physical level splitting or orbital degeneracy violation. Therefore, this verification result shows that the method of this application can simultaneously satisfy the rotational invariance of tight-binding parameter prediction, orbital degeneracy symmetry, and accuracy of electronic band reconstruction.

[0108] Using the above method, this embodiment can automatically predict tight-binding parameters with rotational invariance and orbital degeneracy symmetry based on MoS2 crystal structure data. On one hand, this embodiment uses an orbit encoder with shared degenerate orbit weights, enabling physically degenerate orbits to be extracted using the same encoder, thereby maintaining the characteristic consistency of degenerate orbits in high-symmetry crystals and reducing the risk of non-physical energy level splitting. On the other hand, this embodiment constructs a lattice-anchored local reference frame based on lattice basis vectors, replacing the global coordinate direction components with the intrinsic direction cosines under this local reference frame, ensuring that the Slater-Koster geometric factor remains consistent before and after overall crystal rotation. Therefore, this application can improve the physical consistency, rotational stability, and reliability of crystal tight-binding parameter prediction.

[0109] The technical features in the above embodiments are not isolated from each other. They can be combined, replaced, or adjusted according to actual needs, provided that no logical conflict or technical contradiction occurs. To avoid making the specification too complicated, this document does not describe all possible combinations of the technical features one by one. However, as long as the relevant combination can achieve the technical purpose of this application, it should be considered as being included in the technical scope disclosed in this specification.

[0110] This document illustrates the technical principles, implementation methods, and beneficial effects of this application through specific embodiments. The above embodiments are only used to help understand the technical solution and core concept of this application and do not constitute a limitation on the scope of protection of this application. For those skilled in the art, without departing from the technical concept of this application, adaptive adjustments, equivalent substitutions, or improvements can be made to the specific implementation methods, parameter settings, model structures, or application scope according to actual application scenarios. All changes made based on the technical concept of this application should fall within the scope of protection of this application. Therefore, the content of this specification should not be construed as a limitation of this application.

Claims

1. A method for predicting the tight-binding parameters of a crystal that is rotationally invariant and degenerate orbitally symmetric, characterized in that, The method includes: Acquire crystal structure data, which includes lattice matrix, atom type and coordinates, orbital type, and DFT band structure; Crystal graph data is constructed based on the crystal structure data. The crystal graph data includes atomic node features, edge features, and orbital index information. It is input into a graph neural network, and the atomic local environment representation of each atom is obtained through message passing. Based on the atomic local environment representation and orbit type, orbit features that satisfy orbit symmetry are obtained through an orbit encoder with degenerate orbit weight sharing; A lattice-anchored local reference system is constructed based on the lattice basis vectors in the lattice matrix. The interatomic direction vectors are projected onto the lattice-anchored local reference system to obtain the intrinsic direction cosines. Based on the orbital features, edge features, and intrinsic direction cosines, a graph neural network is used to predict rotationally invariant orbital coupling parameters, and tight-binding parameters are generated from the orbital coupling parameters.

2. The method according to claim 1, characterized in that, The orbital index information adopts a preset orbital mapping table, which includes the Slater-Koster orbital index, orbital encoding index, and orbital type index corresponding to the atom type and local orbital number.

3. The method according to claim 1, characterized in that, The edge features are jointly determined by distance encoding and key type embedding, and the interatomic distance d ij Radial basis function expansion is performed to obtain the distance encoding: In the formula, φ k (d ij ) represents the distance coding corresponding to the kth radial basis function, μ k represents the kth radial basis center, and γ represents the radial basis width parameter. The distance encoding and key type are embedded into the input edge feature encoding function to obtain the edge embedding features: In the formula, ϕ(d ij ) represents a distance code consisting of multiple radial basis functions, b ij Indicates key type embedding, e ij This represents edge embedding features.

4. The method according to claim 1, characterized in that, The graph neural network performs message passing based on node features and edge features; For a graph edge pointing from node j to node i, the edge message is computed based on the node representation and edge embedding: Then, the neighbor edge messages are aggregated, and the node representation is updated: In the formula, h i and h j Let N(i) represent the node representations of node i and node j respectively, and let N(i) represent the set of neighboring nodes of node i. ' i f represents the updated node representation. msg The message function, f upd This represents the node update function.

5. The method according to claim 1, characterized in that, The degenerate orbit weight-sharing orbit encoder sets a shared coding branch based on orbit symmetry. The same multilayer perceptron encoder is used for feature extraction of physically degenerate orbits. The multilayer perceptron includes a linear transform layer, a SiLU activation function layer, and a layer normalization layer. For orbit α on atom i, its orbital features are represented as: In the formula, o iα h represents the orbital characteristic corresponding to orbital α of atom i. i The local atomic environment of atom i is represented by s. α The orbital type embedding of orbital α is defined such that if orbital α and orbital β belong to the same degenerate orbital group, then the following condition is satisfied: Encoder ord (α) =Encoder ord (β).

6. The method according to claim 1, characterized in that, The lattice-anchored local reference system is dynamically constructed from lattice basis vectors. Let the first lattice basis vector of the lattice matrix be a, and the second lattice basis vector be b. Then the lattice-aligned local reference system is constructed as follows: In the formula, u is a local axis along the direction of the first lattice basis vector, v is a local axis located in the lattice plane and orthogonal to u, and w is a local axis perpendicular to the lattice plane.

7. The method according to claim 6, characterized in that, Projecting the interatomic direction vectors onto the lattice-anchored local reference frame, we obtain the intrinsic direction cosines: In the formula, l ' ij m ' ij n ' ij Used to replace the direction components in the global coordinate system.

8. The method according to claim 7, characterized in that, The intrinsic direction cosine remains unchanged before and after the overall crystal rotation. Let the overall crystal rotation matrix be R, then the following is satisfied: In the formula, Rr ij R represents the interatomic interaction vector after rotation, and Ru, Rv, and Rw represent the three local axes of the lattice-anchored local reference frame after rotation.

9. The method according to claim 1, characterized in that, The orbital coupling parameters are jointly determined by orbital representation, atomic local environment representation, edge features and intrinsic direction cosine, and form interatomic tight-binding transition terms through Slater-Koster geometric factors; The Slater-Koster geometric factor between orbits α and β is determined based on the intrinsic direction cosine vector: Predicting the σ, π, and δ type orbital coupling strength between orbits α and β using graph neural networks: The final interatomic tight-binding transition term is expressed as: In the formula, G αβ Let f represent the Slater-Koster geometric factor function. SK Let t represent the orbital coupling prediction function in a graph neural network. αβ ij This represents the tight-binding transition term between orbital α of atom i and orbital β of atom j.

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