Convergence with

The comparison of curve 1 and 2 in Fig. 3 yields, that the convergence with respect to the number of projections k is not strongly influenced by noise because of the properties of the reconstruction algorithm. Nevertheless, the noise increases the asymptotic value of o(n)/0 ... 

This gives a good estimate provided At is small enough that the method is truly convergent with order p. This process can also be repeated in the same way Romberg s method was used for quadrature. 

In practice a DFT calculation involves an effort similar to that required for an HF calculation. Furthermore, DFT methods are one-dimensional just as HF methods are increasing the size of the basis set allows a better and better description of the KS orbitals. Since the DFT energy depends directly on the electron density, it is expected that it has basis set requirements similar to those for HF methods, i.e. close to converged with a TZ(2df) type basis. 

The pseudopotential density-functional technique is used to calculate total energies, forces on atoms and stress tensors as described in Ref. 13 and implemented in the computer code CASTEP. CASTEP uses a plane-wave basis set to expand wave-functions and a preconditioned conjugate gradient scheme to solve the density-functional theory (DFT) equations iteratively. Brillouin zone integration is carried out via the special points scheme by Monkhorst and Pack. The nonlocal pseudopotentials in Kleynman-Bylander form were optimized in order to achieve the best convergence with respect to the basis set size. 5... 

If the basic set xpk is chosen complete, the virial theorem will be automatically fulfilled and no scaling is necessary. In such a case, the wave function under consideration may certainly be expressed in the form of Eq. III. 18, but, if the basis is chosen without particular reference to the physical conditions of the problem, the series of determinants may be extremely slowly convergent with a corresponding difficulty in interpreting the results. It therefore seems tempting to ask whether there exists any basic set of spin orbitals. which leads to a most "rapid convergency in the expansion, Eq. III. 18, of the wave function for a specific state (Slater 1951). 

With the aid of the above operator inequalities we are able to produce the necessary a priori estimates and justify the convergence with the rate 0(1/r ) for the scheme in hand. Observe that for p = 2 operator (16) coincides with operator (15). 

Remark 2 Uniform convergence with the rate 0 h + r) of the forward difference scheme with cr = 1 can be established by means of the maximum principle and the reader is invited to carry out the necessary manipulations on his/her own. 

In practical implementations Newton s method converges with any prescribed accuracy e only if... 

Convergence of Newton s method. We are now in a position to find out the conditions under which Newton s method converges. With this aim, the differences... 

Whence the convergence with the rate 0(r + /ip) immediately follows. 

Stability of lOS. The main goal of stability consideration is to establish that the uniform convergence with the rate 0 t + /ip) follows from a summarized approximation obtained. This can be done using the maximum principle and a priori estimates in the grid norm of the space C for a solution of problem (21)-(23) expressing the stability of the scheme concerned with respect to the initial data, the right-hand side and boundary conditions. 

In the second case we obtain a linear equation related to y and then use the elimination method for solving it. The uniform convergence with the rate 0 t + h ) takes place under the extra restrictions concerning the boundedness of the derivatives d k /du, d k /dx du, d k /dx. ... 

Apparently, the cohesive energy of these clusters shows a very slow convergence with the size of the molecule. This should not be surprising, since the number of unsaturated valences "dangling bonds" per carbon atom is one in 1,1/2 in 2 and 1/3 in 3. 

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