A general method to construct symmetric orbital (SO) is proposed. For all point groups, the corresponding SOs can be uniquely formulated. Due to the closure of SOs with regard to the multiplication operation, the dire...A general method to construct symmetric orbital (SO) is proposed. For all point groups, the corresponding SOs can be uniquely formulated. Due to the closure of SOs with regard to the multiplication operation, the direct products of SOs form an N th rank SO tensor (SOT) group. Moreover, the matrix elements between SOTs can be automatically divided into physical factors and geometric factors. Consequently, the traditional coupling coefficients can be discarded and thus the present method significantly reduces the computation efforts as compared with the irreducible tensor method.展开更多
The explicit representations for tensorial Fourier expansion of 3_D crystal orientation distribution functions (CODFs) are established. In comparison with that the coefficients in the mth term of the Fourier expansion...The explicit representations for tensorial Fourier expansion of 3_D crystal orientation distribution functions (CODFs) are established. In comparison with that the coefficients in the mth term of the Fourier expansion of a 3_D ODF make up just a single irreducible mth_order tensor, the coefficients in the mth term of the Fourier expansion of a 3_D CODF constitute generally so many as 2m+1 irreducible mth_order tensors. Therefore, the restricted forms of tensorial Fourier expansions of 3_D CODFs imposed by various micro_ and macro_scopic symmetries are further established, and it is shown that in most cases of symmetry the restricted forms of tensorial Fourier expansions of 3_D CODFs contain remarkably reduced numbers of mth_order irreducible tensors than the number 2m+1 . These results are based on the restricted forms of irreducible tensors imposed by various point_group symmetries, which are also thoroughly investigated in the present part in both 2_ and 3_D spaces.展开更多
We give a new definition of geometric multiplicity of eigenvalues of tensors, and based on this, we study the geometric and algebraic multiplicity of irreducible tensors＇ eigenvalues. We get the result that the eigen...We give a new definition of geometric multiplicity of eigenvalues of tensors, and based on this, we study the geometric and algebraic multiplicity of irreducible tensors＇ eigenvalues. We get the result that the eigenvalues with modulus p have the same geometric multiplicity. We also prove that two- dimensional nonnegative tensors＇ geometric multiplicity of eigenvalues is equal to algebraic multiplicity of eigenvalues.展开更多
In this two_part paper, a thorough investigation is made on Fourier expansions with irreducible tensorial coefficients for orientation distribution functions (ODFs) and crystal orientation distribution functions (CODF...In this two_part paper, a thorough investigation is made on Fourier expansions with irreducible tensorial coefficients for orientation distribution functions (ODFs) and crystal orientation distribution functions (CODFs), which are scalar functions defined on the unit sphere and the rotation group, respectively. Recently it has been becoming clearer and clearer that concepts of ODF and CODF play a dominant role in various micromechanically_based approaches to mechanical and physical properties of heterogeneous materials. The theory of group representations shows that a square integrable ODF can be expanded as an absolutely convergent Fourier series of spherical harmonics and these spherical harmonics can further be expressed in terms of irreducible tensors. The fundamental importance of such irreducible tensorial coefficients is that they characterize the macroscopic or overall effect of the orientation distribution of the size, shape, phase, position of the material constitutions and defects. In Part (Ⅰ), the investigation about the irreducible tensorial Fourier expansions of ODFs defined on the N_dimensional (N_D) unit sphere is carried out. Attention is particularly paid to constructing simple expressions for 2_ and 3_D irreducible tensors of any orders in accordance with the convenience of arriving at their restricted forms imposed by various point_group (the synonym of subgroup of the full orthogonal group) symmetries. In the continued work -Part (Ⅱ), the explicit expression for the irreducible tensorial expansions of CODFs is established. The restricted forms of irreducible tensors and irreducible tensorial Fourier expansions of ODFs and CODFs imposed by various point_group symmetries are derived.展开更多
An irreducible tensor method based on M. O. theory for solving many-body Schrodinger equation of valence-electron system of metal complexes has been proposed. Electronic structures of several typical octahedral transi...An irreducible tensor method based on M. O. theory for solving many-body Schrodinger equation of valence-electron system of metal complexes has been proposed. Electronic structures of several typical octahedral transition metal complexes have been studied by means of this method. Calculated excitation energies agree with observed ones within several kcm-1.展开更多
基金Supported by the National Natural Science Foundation of China(11601473)the Foundation of Science and Technology Department of Guizhou Province(20191161,20181079)the Natural Science Programs of Education Department of Guizhou Province(2015420)
文摘A general method to construct symmetric orbital (SO) is proposed. For all point groups, the corresponding SOs can be uniquely formulated. Due to the closure of SOs with regard to the multiplication operation, the direct products of SOs form an N th rank SO tensor (SOT) group. Moreover, the matrix elements between SOTs can be automatically divided into physical factors and geometric factors. Consequently, the traditional coupling coefficients can be discarded and thus the present method significantly reduces the computation efforts as compared with the irreducible tensor method.
文摘The explicit representations for tensorial Fourier expansion of 3_D crystal orientation distribution functions (CODFs) are established. In comparison with that the coefficients in the mth term of the Fourier expansion of a 3_D ODF make up just a single irreducible mth_order tensor, the coefficients in the mth term of the Fourier expansion of a 3_D CODF constitute generally so many as 2m+1 irreducible mth_order tensors. Therefore, the restricted forms of tensorial Fourier expansions of 3_D CODFs imposed by various micro_ and macro_scopic symmetries are further established, and it is shown that in most cases of symmetry the restricted forms of tensorial Fourier expansions of 3_D CODFs contain remarkably reduced numbers of mth_order irreducible tensors than the number 2m+1 . These results are based on the restricted forms of irreducible tensors imposed by various point_group symmetries, which are also thoroughly investigated in the present part in both 2_ and 3_D spaces.
基金Acknowledgements The authors are grateful to Mr. Xi He and Mr. Zhongming Chen for their helpful discussion. And the authors would like to thank the reviewers for their suggestions to improve the presentation of the paper. This work was supported in part by the National Natural Science Foundation of China （Grant No. 11271206） and the Natural Science Foundation of Tianjin （Grant No. 12JCYBJC31200）.
文摘We give a new definition of geometric multiplicity of eigenvalues of tensors, and based on this, we study the geometric and algebraic multiplicity of irreducible tensors＇ eigenvalues. We get the result that the eigenvalues with modulus p have the same geometric multiplicity. We also prove that two- dimensional nonnegative tensors＇ geometric multiplicity of eigenvalues is equal to algebraic multiplicity of eigenvalues.
文摘In this two_part paper, a thorough investigation is made on Fourier expansions with irreducible tensorial coefficients for orientation distribution functions (ODFs) and crystal orientation distribution functions (CODFs), which are scalar functions defined on the unit sphere and the rotation group, respectively. Recently it has been becoming clearer and clearer that concepts of ODF and CODF play a dominant role in various micromechanically_based approaches to mechanical and physical properties of heterogeneous materials. The theory of group representations shows that a square integrable ODF can be expanded as an absolutely convergent Fourier series of spherical harmonics and these spherical harmonics can further be expressed in terms of irreducible tensors. The fundamental importance of such irreducible tensorial coefficients is that they characterize the macroscopic or overall effect of the orientation distribution of the size, shape, phase, position of the material constitutions and defects. In Part (Ⅰ), the investigation about the irreducible tensorial Fourier expansions of ODFs defined on the N_dimensional (N_D) unit sphere is carried out. Attention is particularly paid to constructing simple expressions for 2_ and 3_D irreducible tensors of any orders in accordance with the convenience of arriving at their restricted forms imposed by various point_group (the synonym of subgroup of the full orthogonal group) symmetries. In the continued work -Part (Ⅱ), the explicit expression for the irreducible tensorial expansions of CODFs is established. The restricted forms of irreducible tensors and irreducible tensorial Fourier expansions of ODFs and CODFs imposed by various point_group symmetries are derived.
基金Project supported by the National Natural Science Foundation of China.
文摘An irreducible tensor method based on M. O. theory for solving many-body Schrodinger equation of valence-electron system of metal complexes has been proposed. Electronic structures of several typical octahedral transition metal complexes have been studied by means of this method. Calculated excitation energies agree with observed ones within several kcm-1.
