Line-integral method

Assuming that the diabatic space can be truncated to the same size as the adiabatic space, Eqs. (64) and (65) clearly define the relationship between the two representations, and methods have been developed to obtain the tians-formation matrices directly. These include the line integral method of Baer [53,54] and the block diagonalization method of Pacher et al. [179]. Failure of the truncation assumption, however, leads to possibly important nonremovable derivative couplings remaining in the diabatic basis [55,182]. 

CORRELATION ENERGY FOR ISOELECTRONIC SERIES OF ATOMS BY THE LINE INTEGRAL METHOD... 

The line integral method advsmced by van Leeuwen and Baerends [Phys. Rev. A 51, 170, (1995)] is applied to the calculation of correlation energies of the iso-electronic series terms with nuclear charge Z -f 1 from exact densities Peioct.Zi Pexact,z+i and the total energy of the term with nuclear charge Z. Numerical calculations are performed for He(Z = 1,2,3)-, Li(Z = 3,4,5,6,7,8,9)- and Be(Z = 4,5,6) isoelectronic series. 

A fundamental problem in density-functional theory (DFT) is that of how to calculate the exact total energy of a Coulombic system starting from the mere knowledge of the exact density. Of course, this problem could be trivially solved if one were able to express the exact energy as an explicit functional of the density. However, this would imply that we have at our disposal explicit expressions for both the kinetic and the exchange-correlation functionals. As this is not yet possible, one must rely on round-about methods such as the line-integral method recently advanced by van Leeuwen and Baerends. ... 

In Section II, the basic formalism of the line integral method is described. In particular, we discuss the choice of integration path employed in the present work. In Section III, numerical results are presented for a few terms of the He-, Li- and Be-isoelectronic series. Finally, in Section IV we discuss the numerical results and assess their accuracy with respect to the exact values. The latter are obtained in the context of the Local Scaling Transformation (LST) DFT formalism. 

Table 1. "Exact correlation energies and those calculated by the line integral method for the path pexact,z Pexact,z+i for the He, Li and Be isoelectronic series (in Hartrees).

Correlation Energy for Isoelectronic Series of Atoms by the Line Integral Method... 

Since to date summaiies about the practical implementation of the line integral have been given recently (in [108,282] as also in the chapter by Baer in the present volume), and the method was applied also to a pair of ci s [282], we do not elaborate here on the fonn of the phase associated with one or more ci s, as obtained through this method. 

An alternative method that can be used to characterize the topology of PES is the line integral technique developed by Baer [53,54], which uses properties of the non-adiabatic coupling between states to identify and locate different types of intersections. The method has been applied to study the complex PES topologies in a number of small molecules such as H3 [55,56] and C2H [57]. 

Developments. A variety of process modifications aimed at improving surface finish or weld line integrity have been described. They include gas assisted, co-injection, fusible core, multiple Hve feed, and push—pull injection mol ding (46,47). An important development includes computer-aided design (CAD) methods, wherein a proposed mold design is simulated by a computer and the melt flow through it is analy2ed (48). 

Fiber and Karplus [38] presented an effective set of numerical methods for computing the reaction paths based on this approximation. First the path is discretized—it is expressed as a chain of intermediate configurations of the system rj,. The line integrals of Fq. (19) are then written as... 

In the Self Penalty Walk (SPW) method the whole reaction path is approximated by minimizing the average energy along the path, given as a line integral between the reactant and product geometries (R and P). 

Schiesser, W.E. (1991) The Numerical Method of Lines Integration of Partial... 

The integration method illustrated above becomes somewhat more complicated iftiHb detttifflihator botitiddStepeated line f Actors. Thus, if the denbmi-oator contains a factor such as (x — a), it identical factors would result which could of course be coipbined. To avoid this problem it is assumed (hat l/(x - a)" can bereplaced by... 

The line-integral expression (189), for the exchange-correlation potential of the HF-KS approach in Sect. 2.4, offers an interesting way to reconstruct the exchange potential for a given system, from the known HF solution for this system. Being alternative to schemes discussed in Sect. 3, this method provides an expression for the exchange potential solely in terms of the HF orbitals in the form... 

Fig. 2. Comparison for Pe = 103 between Eq. (137) (broken line) and the results of the integral method (solid line).

In the integration method, an assumed rate equation is integrated and mathematically manipulated to obtain the best straight line plot to fit the experimental data of concentrations against time. 

---