Kronecker’s delta function

Here 3 i) is the Kronecker s delta function which has value of 1 for i = 0 and 0 for i 7 0. Note that growing chains at the linear catalyst do not carry branches hence the second index is always zero. Now, we will derive the equations for the pseudodistributions in a stepwise maimer. Reactions not affecting the number of branches per chain, that is, those describing propagation, transfer to monomer. 

Here, the angular brackets denote an ensemble average, which is the same as time average from the ergodic hypothesis. 8ap and 8,- are Kronecker delta, and 8(f — t ) is Dirac s delta function. 

Here 8, is Kronecker s delta i, j= 1,2,3 correspond to Cartesian (the crystalline) axes, 8(t) is the Dirac delta-function. The random variable H (f) must also obey Isserlis s theorem [21]. By introducing [9] the dipole vector... 

The Kronecker delta functions, 5 and 6,p, resulting from Eq. [21], cannot be simplified to 1 or 0 because the indices p and q may refer to either occupied or virtual orbitals. The important point here, however, is that the commutator has reduced the number of general-index second-quantized operators by one. Therefore, each nested commutator from the Hausdorff expansion of H and T serves to eliminate one of the electronic Hamiltonian s general-index annihilation or creation operators in favor of a simple delta function. Since f contains at most four such operators (in its two-electron component), all creation or annihilation operators arising from f will be eliminated beginning with the quadruply nested commutator in the Hausdorff expansion. All higher order terms will contain commutators of only the cluster operators, T, and are therefore zero. Hence, Eq. [52] truncates itself naturally after the first five terms on the right-hand side. ° This convenient property results entirely from the two-electron property of the Hamiltonian and from the fact that the cluster opera-... 

Three of the five terms in the final rearrangement contain operator strings of reduced length, and the first term contains only Kronecker delta functions. Note also that all the operator strings on the right-hand side of the final equality are normal-ordered by Merzbacher s definition. If we now evaluate the quan-... 

The Kronecker symbol is then must be replaced by (1/N) (s - s ) since one of the approximation of delta function is... 

Here 3 is a Kronecker delta function with value 1 if bead s is of type / and 0 otherwise. The trace Tr is limited to the integration over the coordinates of one chain... 

The components of the hyperfine tensor A in Eq. (12) comprise an anisotropic (dipolar) and an isotropic part as written in frequency units in Eq. (13). The term dy denotes the Kronecker delta function h in Eq. (13) is Planck s constant. The distance r and location (r ,ry) with respect to the center of spin density in the point-dipole approximation, as well as the isotropic coupling, determine the ENDOR response in the case of unit spin density. If a spin density p is distributed over various spin centres with p =, the components of the dipolar part of the tensor... 

This equation relates the nth component of displacement to the near-field (first term on right), the far-field P-wave, and the far-field S-wave radiation for a unit force in the p-direction. a, and jS are the P- and S-wave velocities, S is the Kronecker delta function, and y are direction cosines relating the direction of ground motion, n, to the direction of the unit force, p. The far-field terms are proportional to the applied force, and the near-field term is proportional to its integral. Single-force solutions may be used to smdy the seismic excitation due to an impact, mass ejection (Kanamori et al. 1984), or landslides (e.g., Kawakatsu 1989). Other types of seismic excitation are described by combinations of dipole forces, which are obtained by taking spatial derivatives of the single-force solutions, as described below. 

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