Summation index

Note that contributions from the secondary sector of the eigenvectors, Uf, do not appear in the residues, for the summation index, r, pertains to spin-orbitals only. 

Using the distributive property of the matrix product, and since each summation index is independent, we get... 

The summation index n has the same meaning as in Eq. (31), i.e., it enumerates the components of the interaction between the nuclear spin I and the remainder of the system (which thus contains both the electron spin and the thermal bath), expressed as spherical tensors. are components of the hyperfine Hamiltonian, in angular frequency units, expressed in the interaction representation (18,19), with the electron Zeeman and the ZFS in the zeroth order Hamiltonian Hq. The operator H (t) is evaluated as ... 

Here the summation index i covers both traps and recombination centers. We suppose initially that there is only one trap and that the rest of the impurities or defects act as recombination centers. A trap, of course, is by definition more likely to release its charge to the appropriate band than to hold it until annihilation by subsequent capture of the opposite charge. The reverse is true for recombination centers. We can often lump the sum effect of all the recombination centers into a single lifetime t ... 

Thus, the first-order correction to the zero-order (unperturbed) wave function is obtained by substituting equation (A.104) into equation (A.103) and changing the summation index ... 

Note that the summation index k is a dummy variable. (By this, we mean that the value of the sum is the same no matter what letter we use for the summation index.) Equation (2.11) is worth remembering. 

It is worth emphasizing that coefficients (20.29) and (20.30) do not depend on orbital quantum numbers. These numbers define only the parity of summation index k, following from the conditions of the nonvanishing of radial integrals in (20.27) and (20.28). For the direct term, parameter k acquires even values, whereas for the exchange part the parity of k equals the parity of sum h + h-If one or two subshells are almost filled, then the following equalities are valid ... 

All potentially molar-mass-dependent quantities have been labelled with a summation index k in Eq. (30), which, in this form, also holds for dilute solutions of mixtures of chemically different species or copolymers with heterogeneity of both chemical composition and degree of polymerization. 

In a regime of not knowing what to do the TVOC indicator was bom. The TVOC s mass/m3 was suggested as an indicator of exposure to a mixture of VOCs and of its potency to cause effects reviewed by IVtelhave (2000b, 2001). From the beginning it was underlined that this TVOC indicator is intended as an ad hoc tool meant for screening and for sensory irritation (i.e., non-adverse secondary causalities as described above). The indicator is based on the assumption that additivity can be used to calculate a summation index which indicates low risk of effects when below one. 

Here the summation index k runs over all common spin orbitals. 

The summation index k runs over common spin orbitals. The second Coulomb-type integral (k(l)i(2) M so(l,2) k(l)j(2)) vanishes because is linear in... 

Frozen-core orbitals are doubly occupied and a spin integration has been performed for the core electrons. The summation index k therefore runs over spatial orbitals only. Employing the frozen-core approximation considerably shortens the summation procedure in single excitation cases. Double excitation cases are left unaltered at this level of approximation. The computational effort can be substantially reduced further if one manages to get rid of all explicit two-electron terms. 

The summation index k now runs over all spatial molecular orbitals. 

At T = 20 °C, the coefficient connecting the summation index n with the imaginary eigenfrequency is... 

Or, after dividing by dr dc dt and adopting Einsteins summation index notation ... 

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