Effective Core Potentials Applications

Thble3.6 Bond length Rc (A), vibrational constant coe (cm-1) and binding energy De (eV) of Eka-Au hydride (111)H without (with) counterpoise correction of the basis-set superposition error. All-electron (AE) values based on die Dirac-Coulomb-Hamiltonian (Seth and Schw-erdtfeger 2000) are compared with valence-only results obtained with energy-consistent (EC) (Dolg etal. 2001) and shape-consistent (SC) (Han and Hirao 2000) pseudopotentials (PP). The numbers 19 and 34 in parentheses denote the number of valence electrons for the Eka-Au PP. 

In many cases, less costly treatments of the SO interaction may be successful as well. Scalar-relativistic ECPs are employed at the independent particle level to generate a set of (real) orbitals which are used for the integral transformation. The SO term of the ECP is included in the calculations at the correlated level, i.e. electron correlation and spin-orbit effects are treated on an equal footing. Double group symmetry may again be applied to reduce the computational effort (Chang and Pitzer 

The extension of the applicability of quantum Monte Carlo (QMC) calculations to systems with heavy elements depends critically on the availability and accuracy of large-core PPs, possibly augmented by CPPs. Besides the usual problems of QMC, 

Institut fur Theoretische Physik, J. W. Goethe Universitat Frankfurt 

Most frequently, however, a purely density-dependent version of RDFT is used. In this context we have examined the role of relativistic corrections to the exchange-correlation (xc) energy functional. In view of the limited accuracy of the relativistic local density approximation (RLDA) (Das etal. 1980 Engel etal. 1995a Ramana et 

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Pettersson,L.G.M. and Stromberg,A. (1983), A study of the value of the interaction integrals in effective core potential applications , Chem.Phys. 

Finally, some spectroscopic applications for pseudopotentials within SOCI methods are presented in section 3. We focus our attention on applications related to relativistic averaged and spin-orbit pseudopotentials (other effective core potentials applications are presented in chapters 6 and 7 in this book). Due to the large number of theoretical studies carried out so far, we have chosen to illustrate the different SOCI methods and discuss a few results, rather than to present an extensive review of the whole set of pseudopotential spectroscopic applications which would be less informative. Concerning the works not reported here, we refer to the exhaustive and up-to-date bibliography on relativistic molecular studies by Pyykko [21-24]. The choice of an application is made on the basis of its ability to illustrate the performances on both the pseudopotential and the SOCI methods. One has to keep in mind that it is not easy to compare objectively different pseudopotentials in use since this would require the same conditions in calculations (core definition, atomic basis set, SOCI method). The applications are separated into gas phase (section 3.1) and embedded (section 3.2) molecular applications. Even if the main purpose of this chapter is to deal with applications to molecular spectroscopy, it is of great interest to underline the importance of the spin-orbit coupling on the ground state reactivity of open-shell systems. A case study is presented in section 3.1.4. 

The transition metals have been a very active area for effective core potential applications. Several monographs and reviews are available. Many ot the issues discussed for main group elements are also pertinent to computational TM chemistry, in particular, core size. For transition metals, most chemists would agree the valence orbitals are the nd, (n -I- l)s, and (n -I- l)p atomic orbitals. However, most ECP researchers have derived schemes in which the outer core orbitals are not replaced by the potential. -7 Hay and Wadt have derived semi- and full-core ECP schemes for the d-block metals, ... 

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). 

Pettersson,L.G.M., Wahlgren,U. and Gropen,0. (1983), Effective core potential calculations using frozen orbitals. Applications to transition metals Chem.Phys. 80, 7... 

Another application of the GVB-CI method is the determination of the chemisorption geometries of oxygen and aluminum on the GaAs (110) surface by Barton et al./187/ The 28 core electrons of Ga and As are replaced by model effective core potentials./184/... 

Two methods are mainly responsible for the breakthrough in the application of quantum chemical methods to heavy atom molecules. One method consists of pseudopotentials, which are also called effective core potentials (ECPs). Although ECPs have been known for a long time, their application was not widespread in the theoretical community which focused more on all-electron methods. Two reviews which appeared in 1996 showed that well-defined ECPs with standard valence basis sets give results whose accuracy is hardly hampered by the replacement of the core electrons with parameterized mathematical functions" . ECPs not only significantly reduce the computer time of the calculations compared with all-electron methods, they also make it possible to treat relativistic effects in an approximate way which turned out to be sufficiently accurate for most chemical studies. Thus, ECPs are a very powerful and effective method to handle both theoretical problems which are posed by heavy atoms, i.e. the large number of electrons and relativistic effects. 

Effective core potentials can be developed from an exact application of perturbation theory. For this reason it is sometimes controversial to call such methods "semi-empirical". In electronic structure theory the term "semi-empirical" has been pretty much reserved for modifications introduced into the electronic Hamiltonian itself. 

The AIMP method as a common strategy for effective core potential calculations in molecules and for embedded cluster calculations, has been detailed and reviewed [17]. In this paper, we will pay special attention to its applications in the field of structure and spectroscopy of crystal defects created by actinide element impurities, where relativistic effects are a determinant factor, electron correlation and host embedding effects are also key elements, and not only the ground state but also large manifolds of hundreds of excited states are involved in the chemical and physical processes of interest. 

Relativistic and electron correlation effects play an important role in the electronic structure of molecules containing heavy elements (main group elements, transition metals, lanthanide and actinide complexes). It is therefore mandatory to account for them in quantum mechanical methods used in theoretical chemistry, when investigating for instance the properties of heavy atoms and molecules in their excited electronic states. In this chapter we introduce the present state-of-the-art ab initio spin-orbit configuration interaction methods for relativistic electronic structure calculations. These include the various types of relativistic effective core potentials in the scalar relativistic approximation, and several methods to treat electron correlation effects and spin-orbit coupling. We discuss a selection of recent applications on the spectroscopy of gas-phase molecules and on embedded molecules in a crystal enviromnent to outline the degree of maturity of quantum chemistry methods. This also illustrates the necessity for a strong interplay between theory and experiment. 

Pseudopotentials have been the subject of considerable attention in the last two decades, and they have been developed by a number of different groups. They are also the most widely used effective core potentials in chemical applications either for the study of chemical reactions or spectroscopy. A large variety of pseudopotentials are now available, and all the coupling schemes at the SCF step have been implemented four-component, two-component, and scalar relativistic along with spin-orbit pseudopotentials. However, it is well known that four-component calculations can (in the worst cases) be 64 times more expensive than in the non relativistic case. In addition, the small component of the Dirac wave function has little density in the valence region, and pseudopoten-... 

A large number of spectroscopic applications using spin-orbit Cl methods are nowadays performed using effective core potentials, especially when heavy atoms are involved. However, by nature all SOCI methods can in principle be used either in an all-electron or in an ECP scheme, provided the code can compute the appropriate integrals (see section 2.1.1). Although we are dealing in... 

We review the Douglas-Kroll-Hess (DKH) approach to relativistic density functional calculations for molecular systems, also in comparison with other two-component approaches and four-component relativistic quantum chemistry methods. The scalar relativistic variant of the DKH method of solving the Dirac-Kohn-Sham problem is an efficient procedure for treating compounds of heavy elements including such complex systems as transition metal clusters, adsorption complexes, and solvated actinide compounds. This method allows routine ad-electron density functional calculations on heavy-element compounds and provides a reliable alternative to the popular approximate strategy based on relativistic effective core potentials. We discuss recent method development aimed at an efficient treatment of spin-orbit interaction in the DKH approach as well as calculations of g tensors. Comparison with results of four-component methods for small molecules reveals that, for many application problems, a two-component treatment of spin-orbit interaction can be competitive with these more precise procedures. 

Despite this ubiquitous presence of relativity, the vast majority of quantum chemical calculations involving heavy elements account for these effects only indirectly via effective core potentials (ECP) [8]. Replacing the cores of heavy atoms by a suitable potential, optionally augmented by a core polarization potential [8], allows straight-forward application of standard nonrelativistic quantum chemical methods to heavy element compounds. Restriction of a calculation to electrons of valence and sub-valence shells leads to an efficient procedure which also permits the application of more demanding electron correlation methods. On the other hand, rigorous relativistic methods based on the four-component Dirac equation require a substantial computational effort, limiting their application in conjunction with a reliable treatment of electron correlation to small molecules [9]. 

Lee YS, Ermler WC, Pitzer KS (1977) Ab initio effective core potentials including relativistic effects. I. Formalism and applications to the Xe and Au atoms. J Chem Phys 67(12) 5861-5876... 

Pseudopotentials describe the interaction of a valence electron with the core of the atoms. They are known in the literature under various names, such as model potentials, effective core potentials,. Model potentials are generally parametrized from atomic spectroscopic data whereas effective core potentials and pseudopotentials are most often derived from ab initio calculations. There is a huge literature on the subject and several review articles. " The recent paper by Krauss and Stevens is recommended for an overall survey of the subject with applications and comparisons with all-electron calculations. The recent review paper of Pelissier et al is devoted to transition elements. In the following we shall only review the main characteristics of the determination of atomic pseudopotentials by the ab initio simulation techniques of Section II.B. 

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