Real-Space Calculations

One of the most efficient algorithms known for evaluating the Ewald sum is the Particle-mesh Ewald (PME) method of Darden et al. [8, 9]. The use of Ewald s trick of splitting the Coulomb sum into real space and Fourier space parts yields two distinct computational problems. The relative amount of work performed in real space vs Fourier space can be adjusted within certain limits via a free parameter in the method, but one is still left with two distinct calculations. PME performs the real-space calculation in the conventional manner, evaluating the complementary error function within a cutoff... 

This method has already been used successfully for metallic iron. It presents several important advantages. It is a real space calculation for which no translational invariance is required. Calculation up to 500 eV above the edge can be performed because the basis set increases proportionally with the photoelectron energy. The separation between the three contributions (ai, Ojg and ajn) gives also new insight into the physics that is at the origin of XMCD. 

The CCM model allows real-space calculations (formaUy corresponding to the BZ center for the infinite crystal composed of the supercells). From this point of view the cyclic cluster was termed a quasimolecular large unit ceb [289] or a molecular unit ceb [288]. 

However, ffxs is usually a very sparse matrix. For example, in a typical real-space calculation only less than.l % of the elements of H are different from 0. 

This list is not complete, but it should provide a good entry into available software for real-space calculations. It is interesting to note that the real-space field has gone from a few developmental studies to a wide range of relatively mature codes in about 15 years. 

In the Ewald summation method the initial set of charges are surrounded by a Gaussian distribution lated in real space) to which a cancelling change distribution must be added (calculated in reciprocal space). 

The SSW form an ideal expansion set as their shape is determined by the crystal structure. Hence only a few are required. This expansion can be formulated in both real and reciprocal space, which should make the method applicable to non periodic systems. When formulated in real space all the matrix multiplications and inversions become 0(N). This makes the method comparably fast for cells large than the localisation length of the SSW. In addition once the expansion is made, Poisson s equation can be solved exactly, and the integrals over the intersitital region can be calculated exactly. 

D. Van Dyck and W. Coene, The real space method for dynamical electron diffraction calculations in HREM... 

As mentioned in Section 21.4.1, the existence of intermediate phase with slightly stiffer modulus than that of mbber matrix was reported, which was determined by finite element calculation. The author reported that there are two phases around CB—one is almost comparable with bound mbber, 2 nm-thick glassy phase (GH-phase), another is 10 nm-thick uncross-Unked one (SH-phase). The intermediate regions in this smdy were usually observed around CB regions and the Young s modulus of this region was higher as well. Thus, there is a possibility that SH-phase was directly observed in real space for the first time. 

In this work I choose a different constraint function. Instead of working with the charge density in real space, I prefer to work directly with the experimentally measured structure factors, Ft. These structure factors are directly related to the charge density by a Fourier transform, as will be shown in the next section. To constrain the calculated cell charge density to be the same as experiment, a Lagrange multiplier technique is used to minimise the x2 statistic,... 

From the computational point of view the Fourier space approach requires less variables to minimize for, but the speed of calculations is significantly decreased by the evaluation of trigonometric function, which is computationally expensive. However, the minimization in the Fourier space does not lead to the structures shown in Figs. 10-12. They have been obtained only in the real-space minimization. Most probably the landscape of the local minima of F as a function of the Fourier amplitudes A,- is completely different from the landscape of F as a function of the field real space. In other words, the basin of attraction of the local minima representing surfaces of complex topology is much larger in the latter case. As far as the minima corresponding to the simple surfaces are concerned (P, D, G etc.), both methods lead to the same results [21-23,119]. 

The mean-field SCFT neglects the fluctuation effects [131], which are considerably strong in the block copolymer melt near the order-disorder transition [132] (ODT). The fluctuation of the order parameter field can be included in the phase-diagram calculation as the one-loop corrections to the free-energy [37,128,133], or studied within the SCFT by analyzing stability of the ordered phases to anisotropic fluctuations [129]. The real space SCFT can also applied for a confined geometry systems [134], their dynamic development allows to study the phase-ordering kinetics [135]. 

For the calculation of the hydrodynamic thickness we divide the profile artificially into elementary layers, the result being independent of the division chosen provided it is sufficiently fine. The s.a.n.s. data is obtained as a function of Q, the wave vector (4it/A sin(0/2), where X is the neutron wavelength and 0 the scattering angle. The Q resolution corresponds in real space to a fraction of a bond length which is small enough for defining an elementary layer. 

An example of this procedure is shown in Fig. 1. This example shows the build-up of the 2D potential of Ti2S projected along the short c axis, but the principle is the same for creating a 3D potential. The potential is a continuous function in real space and can be described in a map (Fig. 1). On the other hand, the structure factors are discrete points in reciprocal space and can be represented by a list of amplitudes and phases (Table 1). In this Fourier synthesis we have used the structure factors calculated from the refined coordinates of Ti2S °. 

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