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Representation Theory and Operational Calculus for SU(2) and SO(3)
Published in Gregory S. Chirikjian, Alexander B. Kyatkin, Engineering Applications of Noncommutative Harmonic Analysis, 2021
Gregory S. Chirikjian, Alexander B. Kyatkin
Particles with non-integer internal angular momentum (called spin) are described by spinor-valued functions. A spinor-valued function transforms under the action of SU(2) (corresponding to the rotation matrix A) in the space of ψi (i is an index describing internal degrees of freedom). The transformation law for spinor functions is (A^ψi)(x)=∑jA′ijψj(A−1⋅x)
Hyperfine and P, T odd properties in BiO: comparison between coupled-cluster method and multi-reference perturbation method based on a Dirac Hamiltonian
Published in Molecular Physics, 2023
Minori Abe, Takashi Tsutsui, Hideto Kanamori, Masahiko Hada
A significant difference in A// between DHF and CASCI in Table 4 is attributed to the effect of minor configurations having different open-shell orbitals. Because only the large component of the s spinor and the small component of the p spinor have large electron densities at the nucleus, HFCC can be large when open-shell orbitals consist of Bi s or p spinors. Additionally, this effect is more significant for the s spinor than for the p spinor. The open-shell orbital in the main configuration (DHF, 67.8%) was ϕ4, which mainly consisted of O p. In the second largest configuration, with a 10.6% weight, the open-shell orbital was ϕ3, consisting of Bi p. In other sub-configurations, ϕ3 was often an open-shell orbital, as shown in Table 3. Consequently, the A// value at CASCI increases relative to that at DHF. Table 5 shows the matrix elements of HFCC for the CAS orbitals. The matrix elements for ϕ4, the SOMO of DHF, were generally smaller than the other matrix elements, and its diagonal component (1025 MHz) corresponded to A// of DHF. In contrast, the matrix elements for ϕ2 and ϕ3 were relatively large because they consisted of Bi s or p orbitals. In ϕ2, the main component was O p (61%), but Bi s was also mixed in (16%). Thus, the matrix elements were larger than those for ϕ3, which mainly consisted of Bi p (89%).
Master–slave synchronization in a 4D dissipative nonlinear fermionic system
Published in International Journal of Control, 2022
Gursey Model is the only possible 4-D conformally invariant pure fermionic model with a nonlinear self-coupled spinor term. The conformal invariant Gursey spinor wave equation is described by the conformal invariant Lagrangian where the fermion field ψ has scale dimensional and g is positive dimensionless coupling constant (Gursey, 1956). The conformal invariant spinor wave equation that follows the above Lagrangian is The Euclidian configuration of Heisenberg ansatz (Heisenberg, 1954) where C is an arbitrary spinor constant; and are real functions of in the Euclidean space-time, three dimesion of space and one dimension of time and is 4D Dirac matrices .
Theory of chemical bonds in metalloenzymes XXIV electronic and spin structures of FeMoco and Fe-S clusters by classical and quantum computing
Published in Molecular Physics, 2020
Koichi Miyagawa, Mitsuo Shoji, Hiroshi Isobe, Shusuke Yamanaka, Takashi Kawakami, Mitsutaka Okumura, Kizashi Yamaguchi
Double-exchange computational scheme by semi-classical approximation is derived on the basis of Equations (12)–(14) [31–33]. Two sites system composed of S1 and S2 spin vectors is examined. The states where odd electron with spin s is localised to one site are taken. The odd electron can take either alpha (α) or beta (β) spin state in each delocalisation state, there are four states: , , , and . Using the intraatomic effective exchange integral J in the same spin site, the Hamiltonian matrix is represented as where b (not the bond order in chemical index) is transfer integral (t) and . The rotation of odd electron through transfer is for the spinor nature of spin defined by Equation (15). The eigenvalues of this Hamiltonian are