Four-component relativistic

Esgri, K. (2004) Even tempered basis sets for four-component relativistic quantum chemistry. Chemical Physics, 311,... 

Anton, J., Ericke, B. and Schwerdtfeger, P. (2005) Non-collinear and collinear four-component relativistic molecular density functional calculations. Chemical Physics, 311, 97-103. 

Wang, F. and Liu, W. (2005) Benchmark four-component relativistic density functional calculations on Cu2, Ag2, and Au2. Chemical Physics, 311, 63-69. 

Abe, M., Mori, S., Nakajima, T. and Hirao, K. (2005) Electronic structures of PtCu, PtAg, and PtAu molecules a Dirac four-component relativistic study. Chemical Physics, 311, 129-137. 

Thierfelder, C., Schwerdtfeger, P. and Saue, T. (2007) Cu and Au Nuclear Quadrupole Moments from Four-Component Relativistic Density Functional Calculations using Exact Long-Range Exchange. Physical Review A, 76, 034502-1-034502-4. 

Belpassi, L., Tarantelli, F., Sgamellotti, A., Gdtz, A.W. and Visscher, L. (2007) An indirect approach to the determination of the nuclear quadrupole moment by four-component relativistic DFT in molecular calculations. Chemical Physics Letters, 442, 233-237. 

Electronic Structures of PtCu, PtAg, and PtAu Molecules A Dirac Four-Component Relativistic Study. 

Four-component relativistic molecular calculations are based directly on the Dirac equation. They include both scalar relativistic effects and spin-orbit... 

In this chapter, we therefore consider whether it is possible to eliminate spin-orbit coupling from four-component relativistic calculations. This is a situation quite different from that of more approximate relativistic methods where a considerable effort is required for the inclusion of spin-orbit coupling. We have previously shown that it is indeed possible to eliminate spin-orbit coupling from the calculation of spectroscopic constants [12,13]. In this chapter, we consider the extension of the previous result to the calculation of second-order electric and magnetic properties, i.e., linear response functions. Although the central question of this article may seem somewhat technical, it will be seen that its consideration throws considerable light on the fundamental interactions in molecular systems. We will even claim that four-component relativistic theory is the optimal framework for the understanding of such interactions since they are inherently relativistic. 

The resulting property operators at the four-component relativistic level are listed in Table 1. From the property operators associated with uniform electric and magnetic fields one may directly read off the relativistic operators of electric dipole moment p,= —r, and magnetic dipole moment m = — c(r,o X a,), respectively [36]. 

From the above discussion it becomes clear that in order to eliminate the spin-orbit interaction in four-component relativistic calculations of magnetic properties one must delete the quaternion imaginary parts from the regular Fock matrix and not from other quantities appearing in the response function (35). It is also possible to delete all spin interactions from magnetic properties, but this requires the use of the Sternheim approximation [57,73], that is calculating the diamagnetic contribution as an expectation value. 

We then turn to the question of how to eliminate the spin-orbit interaction in four-component relativistic calculations. This allows the assessment of spin-orbit effects on molecular properties within the framework of a single theory. In a previous publication [13], we have shown how the spin-orbit interaction can be eliminated in four-component relativistic calculations of spectroscopic properties by deleting the quaternion imaginary parts of matrix representations of the quaternion modified Dirac equation. We show in this chapter how the application of the same procedure to second-order electric properties takes out spin-forbidden transitions in the spectrum of the mercury atom. Second-order magnetic properties require more care since the straightforward application of the above procedure will extinguish all spin interactions. After careful analysis on how to proceed we... 

Instead of a two-component equation as in the non-relativistic case, for fully relativistic calculations one has to solve a four-component equation. Conceptually, fully relativistic calculations are no more complicated than non-relativistic calculations, hut they are computationally demanding, in particular, for correlated molecular relativistic calculations. Unless taken care of at the outset, spurious solutions can occur in variational four-component relativistic calculations. In practice, this problem is handled by employing kinetically balanced basis sets. The kinetic balance relation is... 

The ability to use precisely the same basis set parameters in the relativistic and non-relativistic calculations means that the basis set truncation error in either calculation cancels, to an excellent approximation, when we calculate the relativistic energy correction by taking the difference. The cancellation is not exact, because the relativistic calculation contains additional symmetry-types in the small component basis set, but the small-component overlap density of molecular spinors involving basis functions whose origin of coordinates are located at different centres is so small as to be negligible. The non-relativistic molecular structure calculation is, for all practical purposes, a precise counterpoise correction to the four-component relativistic molecular... 

In general, the computational requirements for full four-component relativistic calculations on molecules are so severe that cheaper alternatives must be explored. 

On the other hand, in the pioneering DHF and post-DHF program package MOLFDIR [3] and the well-developed four-component relativistic program package DIRAC [4], the molecular four-component spinors are expanded into decoupled... 

Norman and Jensen27 have implemented a method for obtaining second order response functions within the four component (relativistic) time-dependent Hartree-Fock scheme. Results are presented for the first order hyperpolarizabilities for second harmonic generation, />(—2o o),o ) for CsAg and CsAu. A comparison of the results with those of non-relativistic calculations implies that the nonrelativistic results are over-estimated by 18% and 66% respectively. In this method transitions that are weakly-allowed relativistically can lead to divergences in the frequency-dependent response, which would be removed if the finite lifetimes of the excited states could be taken into account. 

Relativistic effects remarkably influence the electronic structure and the chemical bonding of heavy atoms [15]. In order to calculate the relativistic effects a four-component relativistic formulation by solving the Dirac equation is essential [16]. 

The standard representation of the four-component relativistic Dirac equa-tion for a single particle in an electrostatic potential V(r) and a vector potential (r) reads... 

Visscher, L., Aerts, P. J. C. and Visser, O. (1991a) General contraction in four-component relativistic Hartree-Fock calculations. In Wilson et al. (1991), pp. 197-205. 

Among several other additional operators, which occur in the complete exact treatment of the parity-nonconserving weak interaction within a four-component relativistic approach, the following nuclear spin-dependent operator [123] is the most important one ... 

This requirement does not fix the Dirac matrices uniquely, and thus the whole Dirac theory and all systematic approximations to it could equally well be formulated in terms of general four-dimensional quaternions, which are independent of a special representation and rely only on the algebraic properties of the Clifford algebra [8-10]. Such an implementation of the Dirac theory is known to speed up diagonalisation procedures significantly, and has successfully been employed in modem four-component relativistic program packages like Dirac... 

Historically, approaches to treat the electronic structure relativistically have split into two camps one is the four-component relativistic approach and another is the two-component one. Focusing on our recent studies, in this section, we will introduce these two types of relativistic approaches. The reader is referred to the detailed reviews for our recent relativistic works [118-120]. 

Despite recent implementations of an efficient algorithm for the four-component relativistic approach, the DC(B) equation with the four-component spinors composed of the large (upper) and small (lower) components stiU demands severe computational efforts to solve, and its applications to molecules are currently limited to small- to medium-sized systems. As an alternative approach, several two-component quasi-relativistic approximations have been proposed and applied to chemically interesting systems containing heavy elements, instead of explicitly solving the four-component relativistic equation. 

In consequence, the several numerical results including the present results show that the third-order DK transformation to both one-electron and two-electron Hamiltonians gives excellent agreement with the four-component relativistic approach. The first-order DK correction to the two-electron interaction is shown to be satisfactory for both atomic and molecular systems. 

Scalar relativistic effects (e.g. mass-velocity and Darwin-type effects) can be incorporated into a calculation in two ways. One of these is simply to employ effective core potentials (ECPs), since the core potentials are obtained from calculations that include scalar relativistic terms [50]. This may not be adequate for the heavier elements. Scalar relativity can be variationally treated by the Douglas-Kroll (DK) [51] method, in which the full four-component relativistic ansatz is reduced to a single component equation. In gamess, the DK method is available through third order and may be used with any available type of wavefunction. 

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