Relativistic methods computational details

The calculation of 4f promotion energy requires the combination of an atomic calculation for the 4f shell and a band calculation for the 5d band. The two calculations are based on such drastically different approximations that the combination of the two under one algorithm is a seemingly impossible task. Herbst et al. (1972) overcame this difficulty by using the renormalized atom method first proposed for the d band metals by Watson et al. (1970) and reported in detail by Hodges et al. (1972). We will review here the philosophy of the method, with particular emphasis on the meaning of the various approximations. The computational details are found in the original article. The relativistic version of the calculation has been published recently by Herbst et al. (1976). 

Relativistic effects are significant for the heavier metals. The method of choice is nearly always relativistically derived effective core potentials. Explicit spin-orbit terms can be included in ah initio calculations, but are seldom used because of the amount of computational effort necessary. Relativistic calculations are discussed in greater detail in Chapter 33. 

The transition from (1) and (2) to (5) is reversible each implies the other if the variations 5l> admitted are completely arbitrary. More important from the point of view of approximation methods, Eq. (1) and (2) remain valid when the variations 6 in a trial function are constrained in some systematic way whereas the solution of (5) subject to model or numerical approximations is technically much more difficult to handle. By model approximation we shall mean an approximation to the form of as opposed to numerical approximations which are made at a lower level once a model approximation has been made. That is, we assume that H, the molecular Hamiltonian is fixed (non-relativistic, Born-Oppenheimer approximation which itself is a model in a wider sense) and we make models of the large scale electronic structure by choice of the form of and then compute the detailed charge distributions, energetics etc. within that model. 

Nonrelativistic quantum chemistry has been discussed so far. But transition metal (starting already from the first row) and actinide compounds cannot be studied theoretically without a detailed account of relativity. Thus, the multiconfigurational method needs to be extended to the relativistic regime. Can this be done with enough accuracy for chemical applications without using the four-component Dirac theory Much work has also been done in recent years to develop a reliable and computationally efficient four-component quantum chemistry.25,26 Nowadays it can be combined, for example, with the CC approach for electron correlation. The problem is that an extension to multiconfigurational... 

All calculations were carried out with the software MOLCAS-6.0 [16]. Scalar relativistic effects were included using a DKH Hamiltonian [14,15]. Specially designed basis sets of the atomic natural orbital type were used. These basis sets have been optimized with the scalar DKH Hamiltonian. They were generated using the CASSCF/CASPT2 method. The semi-core electrons (ns, np, n — 3,4, 5) were included in the correlation treatment. More details can be found in Refs. [17-19]. The size of the basis sets is presented in Table 1. All atoms have been computed with basis sets including up to g-type function. For the first row TMs we also studied the effect of adding two h-type functions. 

The use of computational chemistry to address issues relative to process design was discussed in an article. The need for efficient software for massively parallel architectures was described. Methods to predict the electronic structure of molecules are described for the molecular orbital and density functional theory approaches. Two examples of electronic stracture calculations are given. The first shows that one can now make extremely accurate predictions of the thermochemistry of small molecules if one carefully considers all of the details such as zero-point energies, core-valence corrections, and relativistic corrections. The second example shows how more approximate computational methods, still based on high level electronic structure calculations, can be used to address a complex waste processing problem at a nuclear production facility (Dixon and Feller, 1999). 

Up to now all non-empirical computations of barriers to nitrogen inversion (except for ammonia) have been performed within the Hartree-Fock SCF—LCAO—MO theoretical method. Only a brief summary of the problems involved in calculating energy barriers in general and inversion barriers in particular will be given here. A more detailed discussion of the theoretical (correlation and relativistic effects) and computational (basis... 

Because this chapter is a follow-up of previous work in the field it is not necessary to repeat the basics of ab initio methods. This has been done in detail by Basch and Hoz, who also discuss the most important atomic properties of Ge, Sn and Pb. We also recommend the theoretical section in the chapter by Apeloig about organosilicon compounds in this series who gave an excellent overview about the most important aspects of ab initio, semiempirical and force-field methods. The reader will find there an explanation of the most common standard methods which will be mentioned in this review without further explanation. We will focus in the following on those theoretical and computational aspects of methods which are particularly important for heavy-atom molecules that have been advanced in the last decade, i.e. ECPs and DFT. We also briefly discuss relativistic effects. We point out that semiempirical methods" and force field parameters are available for the elements Ge, Sn and Pb. However, the application of the two methods has not gained much popularity and not many papers have been published in the field. Most reports are restricted to special problems. ... 

A further reduction of the computational effort in investigations of electronic structure can be achieved by the restriction of the actual quantum chemical calculations to the valence electron system and the implicit inclusion of the influence of the chemically inert atomic cores by means of suitable parametrized effective (core) potentials (ECPs) and, if necessary, effective core polarization potentials (CPPs). Initiated by the pioneering work of Hellmann and Gombas around 1935, the ECP approach developed into two successful branches, i.e. the model potential (MP) and the pseudopotential (PP) techniques. Whereas the former method attempts to maintain the correct radial nodal structure of the atomic valence orbitals, the latter is formally based on the so-called pseudo-orbital transformation and uses valence orbitals with a simplified radial nodal structure, i.e. pseudovalence orbitals. Besides the computational savings due to the elimination of the core electrons, the main interest in standard ECP techniques results from the fact that they offer an efficient and accurate, albeit approximate, way of including implicitly, i.e. via parametrization of the ECPs, the major relativistic effects in formally nonrelativistic valence-only calculations. A number of reviews on ECPs has been published and the reader is referred to them for details (Bala-subramanian 1998 Bardsley 1974 Chelikowsky and Cohen 1992 Christiansen et... 

A suitable computational approach for the investigation of electronic and geometric structures of transactinide compounds is the fully relativistic Dirac-Slater discrete-variational method (DS-DVM), in a modem version called the density functional theory (DFT) method, which was originally developed in the 1970s (Rosdn and Ellis 1975). It offers a good compromise between accuracy and computational effort. A detailed description can be found in Chapter 4 of this book. 

Aspects of the relativistic theory of quantum electrodynamics are first reviewed in the context of the electronic structure theory of atoms and molecules. The finite basis set parametrization of this theory is then discussed, and the formulation of the Dirac-Hartree-Fock-Breit procedure presented with additional detail provided which is specific to the treatment of atoms or molecules. Issues concerned with the implementation of relativistic mean-field methods are outlined, including the computational strategies adopted in the BERTHA code. Extensions of the formalism are presented to include open-shell cases, and the accommodation of some electron correlation effects within the multi-configurational Dirac-Hartree-Fock approximation. We conclude with a survey of representative applications of the relativistic self-consistent field method to be found in the literature. 

Despite these disadvantages, there is one great advantage of using the nodeless ECP orbitals the primitive functions describing the undulation of the valence orbitals in the vicinity of the nuclei are not needed and the computations are more economic. For more details of the recent developments and applications of the relativistic ECP method, one may refer to the references [14,16,23,32-52]. 

The relativistic calculations on the electronic structure of actinide compounds were reviewed by Pyykko (1987). He also reviewed relativistic quantum chemistry in 1988, whereas the relativistic calculations were limited to small molecules containing one heavy atom only (Pyykko 1988). Calculations on the uranyl and neptunyl ions were introduced in the review article. The general information on the computational chemistry of heavy elements and relativistic calculation techniques appear in the book written by Balasubramanian (1997). There are several first-principle approaches to the electronic structure of actinide compounds. The relativistic effective core potential (ECP) and relativistic density functional methods are widely used for complex systems containing actinide elements. Pepper and Bursten (1991) reviewed relativistic quantum chemistry, while Schreckenbach et al. (1999) reviewed density functional calculations on actinide compounds in which theoretical background and application to actinide compounds were described in detail. The Encyclopedia of computational chemistry also contains examples including lanthanide and actinide elements (Schleyer et al. 1998). The various methods for the computational approach to the chemistry of transuranium elements are briefly described and summarized below. 

In this chapter, we present further computationally efficient methods for relativistic calculations of the electronic structures of molecules and molecular aggregates. While the theory has been developed in detail in the preceding chapters, we now ask the question of how it can be transformed into computationally most feasible methods. 

Current relativistic electronic structure theory is now in a mature and well-developed state. We are in possession of sufficiently detailed knowledge on relativistic approximations and relativistic Hamiltonian operators which will be demonstrated in the course of this book. Once a relativistic Hamiltonian has been chosen, the electronic wave function can be constructed using methods well known from nonrelativistic quantum chemistry, and the calculation of molecular properties can be performed in close analogy to the standard nonrelativistic framework. In addition, the derivation and efficient implementation of quantum chemical methods based on (quasi-)relativistic Hamiltonians have facilitated a very large amount of computational studies in heavy element chemistry over the last two decades. Relativistic effects are now well understood, and many problems in contemporary relativistic quantum chemistry are technical rather than fundamental in nature. 

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