Summation limit

The summation limits over parameters K and K in (19.14) are defined by the conditions of non-zero values of the Clebsch-Gordan and 67-coefficients. The spin-spin operator H (the second sum in (1.22)) may be... 

The summation limits over k are found from the condition that the largest relative uncertainty of approximation does not exceed the given quantity, and the number of terms in expansion (21.23) are minimal. 

In this limit and only in this limit in which each of the materials A, m and B can be considered a dilute gas, does the Hamaker pairwise-summation limit agree rigorously with the modern theory. Compare [Eqs. (L2.123) and (L2.124)]. 

The notation 27 is well established in the literature. It is distinguished from a summation sign by its bold font and lack of summation limits. 

Since the summation limits are always less than or equal to the summation limits for a direc,t product basis representation, the number of operations is rednc.ed by the ESB. A more complete description of the ad antages of the sequential transformation can be fomid in the work of Carrington and coworkers. [18]... 

If the upper summation limit is reached, it means that the series has failed to converge, so say so and exit. / if ( i == maxone ) write(ERR0R QUTPHT.UHIT,200) 200 format ( i > 50 in fmch ) STOP ... 

Here the lower summation limit A Dmin = ( o -ex)/5 = 6 - i o is the activation barrier of formation of the complex N20 ( i +) without changing vibrational quantum number V (see Fig. 6-7) and ex is the kinetic energy of the relative O-N2 motion. The Landau-Zener factors don t change a lot with Ad, whereas the Frank-Condon factors abruptly decrease at higher differences A d in vibrational quantum numbers during transition. Summation (6-21) gives the formation probability of N20 ( i +) ... 

In Appendix M available at booksite.elsevier.com/978-0-444-59436-5, p. el09, we derived the following expressions for the mean value of the Hanultonian using the normalized determinant (without a constant additive term for the nuclear repulsion energy SMO means summation over the spinorbitals i = 1,..N in the RHF method, and the MO summation limit means summation over the orbitals i = I,..., N/2) ... 

Figure 5.5 The lattice energy plotted as a function of summation limit for anthracene, urea, glycine, and succinic acid.

In equation (35) no use of the summation convention was made to show details of the summation limits, and the operator... 

The sum on lok has an additional symmetry requirement that lu + h j + hk) be even. The summation limits for the rest of the sums can be inferred from Eqs.4.12 and/or 4.13. Note that the form of Eq. 4.13 was used for the bra side Eq. 4.12 was used for the ket side of the above integral, so that we could easily recognize where one of the open sums of the orbitals terminates. Thanks to the similarities of the ADTs for both ETO s, the remaining 26 integrals fall into the same pattern and can be worked out rather straightforwardly. 

The lattice summation is very time consuming. For that reason the summation limit has usually been chosen to 8A or even less, resulting in a rather large termination error, often 100 per mille. Kitaigorodsky 36 shown that a summation limit as large as 15A is... 

Three summation limits were used, 6, 8 and 10 A, each with a... 

If K should be the same for repulsive and attractive terms, preferred values are about 0.23, 0.20 and 0.17 A" for summation limits of 6, 8 and 10 A. Proper choice of K is much more critical for a small summation limit, as would be expected. For the Ar case, a good compromise would be a summation limit of 8 A and K = 0.20 A . ... 

The effect of convergence acceleration on the repulsive term is not large, but significant for a summation limit of 6 A, and negligible for 8 and 10 A. A K as large as 0.30 A is unacceptable when summation in reciprocal space is left out. 

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