Making four- and two-component relativistic density functional methods fully equivalent based on the idea of “from atoms to molecule”

Abstract

It is shown that four- and two-component relativistic Kohn-Sham methods of density functional theory can be made fully equivalent in all the aspects of simplicity, accuracy, and efficiency. In particular, this has been achieved based solely on physical arguments rather than on mathematical tricks. The central idea can be visualized as "from atoms to molecule," reflecting that the atomic information is employed to "synthesize" the molecular no-pair relativistic Hamiltonian. That is, the molecular relativistic Hamiltonian can, without loss of accuracy, be projected onto the positive energy states of the isolated Dirac atoms with the projector approximated simply by the superposition of the atomic ones. The dimension of the four-component Hamiltonian matrix then becomes the same as that of a two-component one. Another essential ingredient is to formulate quasirelativistic theory on matrix form rather than on operator form. The resultant quasi-four-component, normalized elimination of the small component, and symmetrized elimination of the small component approaches are critically examined by taking the molecules of MH and M(2) (M=At, E117) as examples.

Exact two-component relativistic theory for nuclear magnetic resonance parameters

“…Other forms of F such as symmetrized elimination of the small component 28 are also possible but which will not be considered here. By setting Z 00 =1 4 , the zeroth order equations ͑original NESC͒ of Eqs.…”

“…That is, the matrix representation of the full Dirac operator in a magnetic-field-dependent basis can be block-diagonalized in a single step, just like the previous matrix formulation of the X2C ͑algebraic͒ Hamiltonians in the absence of magnetic fields. [25][26][27][28][29] The resulting X2C Hamiltonian and transformation matrices can then be expanded to obtain the expressions for NMS, SSC, and magnetizabilities.…”

Exact decoupling of the relativistic Fock operator

“…The only difference is that the iterative equations are slightly different and may have different convergence behavior. The U X transformation is indeed obtained within matrix representation via the so-called eXact-2-Component (X2C) approach [47][48][49][50][51][52][53]. However, an important characteristic of the X2C approach is that it invokes a non-iterative construction of the key operator X in U X .…”

“…These authors suggested the non-iterative construction scheme for many-electron calculations. Later, the noniterative construction approach was implemented into the BDF program [51,52] by Peng and Liu for molecular calculations. The method was first called XQR (exact quasirelativistic) or infinite-order quasi-relativistic method.…”

Relativistic calculations of magnetic resonance parameters: background and some recent developments

“…A relatively straightforward approximation is the use of a model potential [51,110,111] for the e-e interaction, for instance, from relativistic calculations on atomic fragments of a molecule. As pointed out by van Wüllen [110], the potential must be calculated from the correct electron charge density and therefore needs to correspond to the four-component density of an atomic fragment.…”