Small component integrals

Four-component methods are computationally expensive since one has to deal with small-component integrals. Therefore, various two-component methods in which small-component degrees of freedom are removed have flourished in the literature. We focus the present discussion on the X2c theory at one-electron level (X2c-le). The X2c-le scheme consists of a one-step block diagonalization of the Dirac Hamiltonian in its matrix representation via a Foldy-Wouthuysen-type matrix unitary transformation ... 

Class I Small machine-trains or individual components integrally connected with the complete machine in its normal operating condition (i.e., drivers up to 20 horsepower). 

TFL is an important sub-discipline of nano tribology. TFL in an ultra-thin clearance exists extensively in micro/nano components, integrated circuit (IC), micro-electromechanical system (MEMS), computer hard disks, etc. The impressive developments of these techniques present a challenge to develop a theory of TFL with an ordered structure at nano scale. In TFL modeling, two factors to be addressed are the microstructure of the fluids and the surface effects due to the very small clearance between two solid walls in relative motion [40]. 

The radial functions Pmi r) and Qn ir) may be obtained by numerical integration [16,17] or by expansion in a basis (for recent reviews see [18,19]). Since the Dirac Hamiltonian is not bound from below, failure to observe correct boundary conditions leads to variational collapse [20,21], where admixture of negative-energy solutions may yield energies much below experimental. To avoid this failure, the basis sets used for expanding the large and small components must maintain kinetic balance [22,23]. 

Note that this changes for high-Z atoms in molecules, where the so-called small component becomes large and thus its contribution to expectation values and integrals is increased. 

In this notation the presence of two upper and two lower components of the four-component Dirac spinor fa is emphasized. For solutions with positive energy and weak potentials, the latter is suppressed by a factor 1 /c2 with respect to the former, and therefore commonly dubbed the small component fa, as opposed to the large component fa. While a Hamiltonian for a many-electron system like an atom or a molecule requires an electron interaction term (in the simplest form we add the Coulomb interaction and obtain the Dirac-Coulomb-Breit Hamiltonian see Chapter 2), we focus here on the one-electron operator and discuss how it may be transformed to two components in order to integrate out the degrees of freedom of the charge-conjugated particle, which we do not want to consider explicitly. 

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