Numerical integration grid

These calculations employ a grid of points in space in order to perform the numerical integration. Grids are specified as a number of radial shells around each atom, each of which contains a set number of integration points. For example, in the (75,302) grid, 75 radial shells each contain 302 points, resulting in a total of 22,650 integration points. 

The self-consistent field procedure in Kohn-Sham DFT is very similar to that of the conventional Hartree-Fock method [269]. The main difference is that the functional Exc[p] and matrix elements of Vxc(r) have to be evaluated in Kohn-Sham DFT numerically, whereas the Hartree-Fock method is entirely analytic. Efficient formulas for computing matrix elements of Vxc(r) in finite basis sets have been developed [270, 271], along with accurate numerical integration grids [272-277] and techniques for real-space grid integration [278,279]. 

On the other hand, the Coulomb and XC term in O Eq. 18.9 may be combined into an induced potential u d- The potential is known in all points of the numerical integration grid and it allows the calculation of the first-order change in the density dn r) (van Gisbergen et al. 1995). Also, this value is used to calculate the polarizability ... 

This equation can be solved numerically on a grid to determine Vei(ri). The same grid is Iheii used to numerically integrate the four-centre, two-electron integral. Equation (3.60), billows ... 

Some convergence problems are due to numerical accuracy problems. Many programs use reduced accuracy integrals at the beginning of the calculation to save CPU time. However, this can cause some convergence problems for difficult systems. A course DFT integration grid can also lead to accuracy problems, as can an incremental Fock matrix formation procedure. 

G. W. Trucks and M. J. Frisch, Rotational Invariance Properties of Pruned Grids for Numerical Integration, in preparation (1996). 

The coefficients defining the fitted density are obtained via equation (7-27). To avoid the difficulties of dealing with two-electron integrals in a Slater-type basis, is evaluated in this context by a numerical integration on a grid as... 

Here, Rc is a cutoff distance which is made necessary to prevent the potential to become too steep in certain points of the grid and generate instabilities in the numerical integration, and is generally between 0.2 and 0.3A. 

In the variational calculations, the expansions of the kinetic energy factors and the pseudo-potential 1/ [1] are taken to fourth order, and, as mentioned above, the potential energy V is expanded through sixth order. In the numerical integration of the inversion Schrodinger equation a grid of 1000 points is used. The basis set [1] is truncated so that... 

These have hyperbolic form, and in a perfect crystal can be solved analytically. In a distorted crystal they must be solved by numerical integration. Anthier et al. 30 were the first to devise an algorithm for solution over a grid of points. If, in Figure 8.12, M is the point at which we require the wave amplitude and P and Q are neighbouring points at which the amplitudes are known,... 

Figure 8.12 Grid filling the inverted Borrmann fan over which numerical integration of Takagi s equations can be accomplished...

Discretizing the population balance in K+1 grid points results in K ordinary differential equations as the population density n(LQ,t) is determined by the algebraic relation Equation 4. The equation for dn(LQ,t)/dt is therefore not required. In the overal model of the crys lllzer, the population density must be integrated in the calculation of c(t) and In the calculation of m (t) in the nucleatlon rate. As the population balance is discretized these integrals have to be replaced by numerical integration schemes. 

The process inputs are defined as the heat input, the product flow rate and the fines flow rate. The steady state operating point is Pj =120 kW, Q =.215 1/s and Q =.8 1/s. The process outputs are defined as the thlrd moment m (t), the (mass based) mean crystal size L Q(tK relative volume of crystals vr (t) in the size range (r.-lO m. In determining the responses of the nonlinear model the method of lines is chosen to transform the partial differential equation in a set of (nonlinear) ordinary differential equations. The time responses are then obtained by using a standard numerical integration technique for sets of coupled ordinary differential equations. It was found that discretization of the population balance with 1001 grid points in the size range 0. to 5 10 m results in very accurate solutions of the crystallizer model. 

If one wavefunction at one set of nuclear coordinates were sought by numerical integration using only two points in each coordinate, a grid of xo ... 

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