Lattice summation

The size-dependence of the intensity of single shake-up lines is dictated by the squares of the coupling amplitudes between the Ih and 2h-lp manifolds, which by definition (22) scale like bielectron integrals. Upon a development based on Bloch functions ((t>n(k)), a LCAO expansion over atomic primitives (y) and lattice summations over cell indices (p), these, in the limit of a stereoregular polymer chain consisting of a large number (Nq) of cells of length ao, take the form (31) ... 

For three-dimensional crystals the lattice summation converges only slowly, and in any brute-force computational scheme one must make sure that, as one sums outward from the "zeroth unit cell" at the center of the crystal, the ions included at any stage should have as close to zero net charge as possible. For some crystals the Madelung constants a have been evaluated (Table 8.4), using component potentials obtained by summing certain infinite series. 

In Eqs. (4), (6) and (11), there are infinite summations in the overlap matrix Sk, the Fock matrix Fk, and the total energy per unit cell E JF. There are also infinite summations in T, V, Jlpq, and Kln. The infinite lattice summations in Sk, Tl and K converge themselves while V , J and the internuclear interactions have to be summed together to get converged results [20-22], In real calculations, cutoffs for the lattice summations have been imposed and multipole expansion techniques have been applied to hasten the convergence [22],... 

For polymers, we can get analytical expressions for the convergence of E and 4k with the lattice summation cutoff N. Let us use E (N) and 4k (N) to denote the approximate MBPT(2) corrections to the total energy per unit cell and energy of the nth band with the reciprocal vector k, respectively. With multipole expansion techniques, we have shown that when N is large enough, E (N) is related to the exact value by [27]... 

Higher-Order MBPT Corrections and Their Convergence with Lattice Summations... 

Table 6 lists the HF and MBPT(2) band gaps with three basis sets for polymethineimine. The structure of the system used in each calculation is the optimized geometry obtained with the same method and basis. The number of unit cells in the lattice summation is the same as that used in geometry optimization, namely 21. From the table, we can see that electron correlation... 

Working out dipole lattice sums, for any wave vector k, became a simpler matter soon after, with computers of the IBM 650 generation, in about 1956-7. Stuart Walmsley (now Reader in Chemistry, University College London) came as my first PhD student after I returned to University College from the University of Sydney in 1956. He had a magic touch with the early computers, and made the first dipole-dipole lattice summations on simple aromatic crystals [220]. Dipole-octupole and octupole-octupole interaction sums were first calculated by Thirunamachandran [221] using the Ferranti Mercury Computer and later extended to quadrupole couplings. These sums are not shape dependent, but were of formidable difficulty at the time. 

The correct on-term programme COR cannot be executed without a corresponding and previous execution of STR to generate the vectors needed in the reciprocal-lattice summations. These vectors are stored in the file VEC/XXX. An example of the additional input used is given in Table 9.3. 

The electrostatic part of the potential is slowly convergent when lattice summations are carried out, and this problem is overcome by the use of the Ewald method, details of which are given in contrasting but complementary approaches by Tosi (1964) and by Jackson and Catlow (1988). A detailed discussion follows in Chapter 4. 

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