Basis Sets for Relativistic Calculations

In proceeding to the relativistic description of molecular systems, one would like to be able to draw on the advances and developments of the non-relativistic case. However, as we shall show, the relativistic formulation as well as the effects that this formulation place demands on the basis sets that are not necessarily satisfied by a simple transfer of the non-relativistic framework. The subject of our presentation here is therefore to describe the special features and requirements for basis sets to be used in relativistic calculations. As this volume will show, there axe numerous approaches to describing relativity for molecular systems. Here we shall relate our discussion to the conceptually simplest of these, the 

4-component method arising from the Dirac equation. While this entails a number of challenges in the practical computational applications, it is simplest in the sense that it entails no further approximations, and thus relates to the raw relativistic wavefunctions. In developing basis sets for other methods, e.g. relativistic pseudopotentials or density functional methods, additional considerations enter beyond these which apply to the Dirac spinor formulation. We believe that these axe more naturally treated in connection with the various alternative approaches, and will not discuss basis sets for these in the present context. 

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While molecules in general are many-nucleus many-electron systems, the special conditions that a basis set for relativistic calculation must fulfill follow to a large extent from the Dirac equation for the hydrogen atom. The solutions of the Schrodinger equation for the non-relativistic hydrogen atom are well known and take the form... 

Above we have presented some of the considerations and practical difficulties that must be taken into account in the derivation of 4-component basis sets for relativistic calculations. The ultimate validation of any basis set will have to take place through applications. Despite the availability of programs for 4-component calculations for more than 10 years, the field is still quite unexplored, compared to the wealth of information and experience that exists with regard to non-relativistic basis sets. Such experience will eventually accumulate also for relativistic work, but the size and cost of most calculations where relativity is of interest, indicates that this will be a slower process than what it has been for non-relativistic basis sets. 

If the atomic wavefunctions Xiii, 9, four-spinor basis Ix ) are introduced (see chapter 5 by Faegri and Dyall on basis sets for relativistic calculations [161] in the first part of this book)... 

X. Faegri, K. DyaU, Basis sets for relativistic calculations, in P. Schwerdtfeger (Ed.), Relativistic Electronic Structure Theory, Part 1, Fundamentals, Elsevier, Netherlands, 2002, pp. 259-290. 

L. Visscher, P. J. C. Aerts, O. Visser, W. C. Nieuwpoort. Kinetic Balance in Contracted Basis Sets for Relativistic Calculations. Int. J. Quantum Chem., Quantum Chem. Symp., 25 (1991) 131-139. 

Visscher, L., Aerts, P.J.C., Visser, O., Nieuwpoort, W.C. Kinetic balance in contracted basis sets for relativistic calculations. Int. J. Quant. Chem. 40, 131-139 (1991)... 

It is not possible to use normal AO basis sets in relativistic calculations The relativistic contraction of the inner shells makes it necessary to design new basis sets to account for this effect. Specially designed basis sets have therefore been constructed using the DKH Flamiltonian. These basis sets are of the atomic natural orbital (ANO) type and are constructed such that semi-core electrons can also be correlated. They have been given the name ANO-RCC (relativistic with core correlation) and cover all atoms of the Periodic Table.36-38 They have been used in most applications presented in this review. ANO-RCC are all-electron basis sets. Deep core orbitals are described by a minimal basis set and are kept frozen in the wave function calculations. The extra cost compared with using effective core potentials (ECPs) is therefore limited. ECPs, however, have been used in some studies, and more details will be given in connection with the specific application. The ANO-RCC basis sets can be downloaded from the home page of the MOLCAS quantum chemistry software (http //www.teokem.lu.se/molcas). 

VIII. Basis Sets for Relativistic Electronic Structure Calculations. 482... 

The ideas and concepts concerning the use of basis sets in relativistic calculations which have been described in the previous subsections allow the Dirac-Fock equations for many-electron systems to be formulated within the algebraic approximation. A discussion of these equations lies outside the scope of the present chapter. 

Up to now the basis sets have appeared mostly as formal entities. However, the success of any computational scheme is crucially dependent on the size and suitability of the basis sets used. In this section, we discuss the various considerations that influence the choice of a good basis set for relativistic four-component calculations. It is obvious that we want basis sets that are generally applicable, give a good approximation of the exact functions with a minimum number of terms in the expansion, and provide for fast and easy calculation of matrix elements. 

When it comes to contracted basis sets, kinetic balance strictly applied to the contracted large component can lead to problems. While it would be possible to apply the kinetic balance relation to derive a small-component basis from a set of large-component contracted basis functions, this procedure has been shown to be unsuitable in practice (Visscher et al. 1991). The best approach for generating contracted basis sets for relativistic four-component calculations has been to start with an uncontracted large-component basis, and to construct a small-component basis from this basis using kinetic balance. This set is then used in an uncontracted DHF calculation for the atom in question, yielding large- and small-component atomic functions that are kinetically... 

There exists a wide selection of exponents for Gaussian basis sets for nonrelativistic calculations, although most of these are for lighter elements which for most purposes do not require a relativistic treatment. For four-component relativistic calculations, nonrelativistic basis sets can be used for lighter atoms, but as the relativistic effects of orbital contraction and spin-orbit splitting increase in importance, these nonrelativistic basis sets become inadequate. In some measure the orbital contraction for inner orbitals is counteracted by the use of a finite nucleus, which tends to push out the inner parts of the spinors. A major concern is the 2/ i/2 space (Matsuoka and Okada 1989) due to the 5-character of the small component at least two extra functions relative to the nonrelativistic basis are needed for the 6/ block to reduce the error in the energy to 0.5 h-... 

T. Koga, H. Tatewaki, and O. Matsuoka, Relativistic Gaussian basis sets for molecular calculations Cs-Hg, J. Chem. Phys. 117, 7813-7814 (2002). 

Koga T, Tatewaki H, Matsuoka O. Relativistic Gaussian basis sets for molecular calculations Tl-Lr. J Chem Phys. 2003 119(2) 1279-80. 

All calculations are scalar relativistic calculations using the Douglas-Kroll Hamiltonian except for the CC calculations for the neutral atoms Ag and Au, where QCISD(T) within the pseudopotential approach was used [99], CCSD(T) results for Ag and Au are from Sadlej and co-workers, and Cu and Cu from our own work, using an uncontracted (21sl9plld6f4g) basis set for Cu [6,102] and a full active orbital space. 

Kello, V. and Sadlej, A.J. (1996) Standardized basis sets for high-level-correlated relativistic calculations of atomic and molecular electric properties in the spin-averaged Douglas-Kroll (nopair) approximation 1. Groups Ib and 11b. Theoretica Chimica Acta, 94, 93-104. 

The electronic structure calculations were carried out using the hybrid density functional method B3LYP [15] as implemented in the GAUSSIAN-94 package [16], in conjunction with the Stevens-Basch-Krauss (SBK) [17] effective core potential (ECP) (a relativistic ECP for Zr atom) and the standard 4-31G, CEP-31 and (8s8p6d/4s4p3d) basis sets for the H, (C, P and N), and Zr atoms, respectively. 

On the other hand, high-level computational methods are limited, for obvious reasons, to very simple systems.122 Calculations are likely to have limited accuracy due to basis set effects, relativistic contributions, and spin orbit corrections, especially in the case of tin hydrides, but these concerns can be addressed. Given the computational economy of density functional theories and the excellent behavior of the hybrid-DFT B3LYP123 already demonstrated for calculations of radical energies,124 we anticipate good progress in the theoretical approach. We hope that this collection serves as a reference for computational work that we are certain will be forthcoming. 

Table 7 Estimates of total relativistic correction, E , and the first-order Breit energy correction,, obtained by combining the atomic or ionic contributions indicated by the second column. They may be compared with the values of the total relativistic correction, Er. and the first-order Breit interaction, Es, obtained directlyfrom matrix Dirac-Elartree-Fock and Elartree-Fock calculations of the molecular structure using BERTEIA [12], Only the results of the 13s7p2d atom-centred basis sets for Er and Eb are quoted. All energies in atomic units.

An application of the variational principle to an unbounded from below Dirac-Coulomb eigenvalue problem, requires imposing upon the trial function certain conditions. Among these the most important are the symmetry properties, the asymptotic behaviour and the relations between the large and the small components of the wavefunction related to the so called kinetic balance [1,2,3]. In practical calculations an exact fulfilment of these conditions may be difficult or even impossible. Therefore a number of minimax principles [4-7] have been formulated in order to allow for some less restricted choice of the trial functions. There exist in the literature many either purely intuitive or derived from computational experience, rules which are commonly used as a guidance in generating basis sets for variational relativistic calculations. 

Relativistic charge-current densities expressed in terms of G-spinor basis sets for stable and economical numerical calculations [2]. 

For systems with heavy atoms we often employ pseudopotential basis sets (frequently relativistic), which reduce the computational burden of large numbers of electrons. Transition metals present problems beyond those of main-group heavy atoms not only can relativistic effects be significant, but electron d- or f-levels, variably perturbed by ligands, make possible several electronic states. DFT calculations, with pseudopotentials, are the standard approach for computations on such compounds. 

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