Normalized Elimination of the Small Component NESC

Some of the other developments in the area have been derived by Dyall et al. [47,48] and improved recently by Filatov et al. [49-51]. The method is known as the normalized elimination of the small component (NESC) method. One has to also mention the recent work of Liu et al. (see the review [52]) and Ilias [15]. 

In order to analyze this question, the best point to start with is the so-called modified Dirac equation [547,718]. The modified Dirac equation is the basis of the so-called normalized elimination of the small component (NESC) worked out by Dyall [608,719-721]. Here, the small component ip of the 4-spinor tp is replaced by a pseudo-large component

[Pg.531]

X2C ( eXact 2-Component ) is an umbrella acronym [56] for a variety of methods that arrive at an exactly decoupled two-component Hamiltonian, with X2C referring to one-step approaches [65]. Related methods to arrive at formally exact two-component relativistic operators are, for example, infinite-order methods by Barysz and coworkers (BSS = Barysz Sadlej Snijders, lOTC = infinite-order two-component) [66-69] and normalized elimination of the small component (NESC) methods [70-77]. We discuss here an X2C approach as it has been implemented in a full two-component form with spin-orbit (SO) coupling and transformation of electric property operators to account for picture-change (PC) corrections [14],... 

One-component calculations or two-component calculations including also spin-orbit coupling effects provide a firm basis for the calculations of higher-order relativistic corrections by means of perturbation theory. Several quasi-relativistic approximations have been proposed. The most successful approaches are the Douglas-Kroll-Hess method (DKH) [1-7], the relativistic direct perturbation theory (DPT) [8-24], the zeroth-order regular approximation (ZORA) [25-48], and the normalized elimination of small components methods (NESC) [49-53]. Related quasi-relativistic schemes based on the elimination of the small components (RESC) and other similar nonsingular quasi-relativistic Hamiltonians have also been proposed [54-61]. 

The modified Dirac equation can now be viewed from two different perspectives. The first perspective is the fact that the approximate kinetic balance condition of Eq. (5.137) has been exploited. The normalized elimination procedure then results in energy eigenvalues which deviate only in the order c from the correct Dirac eigenvalues, whereas the standard un-normalized elimination techniques are only correct up to the order c. In addition, the NESC method is free from the singularities which plague the un-normalized methods and can be simplified systematically by a sequence of approximations to reduce the computational cost [562,720,721]. From a second perspective, Eq. (14.1) defines an ansatz for the small component, which, as such, is not approximate. Hence, Eq. (14.4) can be considered an exact starting point for numerical approaches that aim at an efficient and accurate solution of the four-component SCF equations (without carrying out the elimination steps). We will discuss this second option in more detail in the next section. 

Scalar relativistic effects may be included at the levels of DKH2, DKH3, normalized elimination of small components (NESC) (Dyall 2002), and relativistic elimination of small components (RESC) (Nakajima and Hirao 1999) methods. 

---