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Gaussian Basis Sets for Lanthanide and Actinide Elements: Strategies for Their Development and Use Kirk A Peterson1 and Kennith G. Dyall2 1 Department
8.1 Introduction
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of Chemistry, Washington State University, Pullman, WA 99164-4630, USA Electronic mail:
[email protected] 2 Dirac Solutions, 10527 NW Lost Park Drive, Portland, OR 97229, USA Electronic mail:
[email protected]
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Molecules containing f-block elements, particularly the actinides, are currently of very strong interest due to their role in the nuclear fuel cycle and waste remediation efforts. In particular, those containing actinide elements demonstrate a rich chemistry due to the existence of numerous stable oxidation states of the metal together with partially filled 6d and 5f shells, especially amongst the early members of the block. For experimental spectroscopic studies this can lead to complicated, congested spectra due to the high density of states, while the theoretical work is very challenging due to the large number of electrons, strong relativistic effects, and often multireference character due to nearly degenerate 5f, 6d, 7s, and 7p orbitals. In contrast to the actinides, lanthanides are generally found in just their +3 oxidation states, but they can still exhibit some of the same challenges as the actinides due to partially filled 4f shells. Fundamental studies on small molecular systems involving f-block elements have been an active area for both experiment and theory, since the results and insights arising from these studies can often be used for understanding the chemistry of f-block elements in complex molecular environments.
Computational Methods in Lanthanide and Actinide Chemistry, First Edition. Edited by Michael Dolg. © 2015 John Wiley & Sons, Ltd. Published 2015 by John Wiley & Sons, Ltd.
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8.2 Basis Set Design 8.2.1
General Considerations
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In carrying out ab initio quantum chemistry calculations it is now well known that the choice of basis set can be nearly as important as the electronic structure method that is used. For the main group elements, families of accurate Gaussian basis sets have been available for some time, which in many cases can be used to estimate the complete basis set (CBS) limit for density functional and wavefunction-based approaches. Basis sets for the transition metals have also received considerable attention, and families of accurate Gaussian basis sets suitable for extrapolation are available for these elements as well. The lanthanides and actinides have received less attention, partly because the challenges of developing basis sets are greater than for the transition metals. This chapter aims to outline the challenges of the f block for the development of basis sets and present considerations and strategies for developing well-balanced, accurate basis sets. The existing basis sets for the lanthanide and actinide elements are enumerated and evaluated, and some results presented for illustration.
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The design of a basis set depends on many factors: which Hamiltonian is to be used, which orbitals need to be represented, how correlation is to be treated (wavefunction or DFT), which properties need to be represented, whether contraction is used and by what contraction method, what accuracy level is required, what nuclear model is used. The process of developing a basis set starts with generating a suitable set of primitive functions. This process usually involves generating a primitive set that represents the atomic occupied orbitals and perhaps the most important low-lying virtual orbitals; adding primitives for polarization of the atom in the molecular environment, which is necessary for both DFT and wave function correlation methods; adding primitives for correlation of various orbitals or shells; and adding primitives for properties. Of crucial importance in designing a basis set is the issue of linear dependence, because this affects the numerical stability of the atomic or molecular calculations that use the basis sets. Linear dependence in the primitive set can be controlled by the use of even-tempered or well-tempered basis sets, which minimize the linear dependence by construction. However, such basis sets tend to be larger than energy-optimized basis sets, where linear dependence problems can become significant as the basis set size increases. Contraction of the basis set can alleviate or remove entirely any problems with linear dependence. Of course this depends on how the contraction is done and how the basis set is to be used. If a general contraction is used for all basis functions, as is common for ANO sets, there are no linear dependence problems, and even the addition of a few primitives for extra flexibility does not usually cause problems. Segmented contractions in which the inner core is contracted and the rest of the basis is largely left uncontracted could suffer from linear dependence if the primitive basis has linear dependence, whereas segmented contractions in which all or most shells are represented by a contraction are less likely to suffer from linear dependence. When primitives for correlation are added on top of an SCF contraction, as in the correlation-consistent style of basis set, the correlating set usually includes higher angular momentum functions that are not represented in the SCF set, and functions of lower
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8.2.2 Basis Sets for the f Block
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angular momentum that may already be represented in the SCF set. Care must be taken with the latter, to ensure that the functions added for correlation or valence flexibility do not create a set that is linearly dependent on the SCF occupied functions, particularly if multiple shells are correlated. A further issue is the relation between the correlating functions and the SCF functions, particularly when primitives are used for correlation. For efficiency in sets like the correlation-consistent sets, the primitive functions added to an SCF orbital set for correlation in the occupied symmetries are normally taken from the SCF set, because this requires less work in the integral code. However, it is usually the case that the optimal exponents for the correlating functions do not match very well with those of the SCF primitive functions: the ratio between the correlating exponents is often larger than that in the SCF basis set, except perhaps for the outer few SCF primitives, which tend to be more widely spaced. Some compromise must usually be made: for example, to choose the SCF set to match the optimal correlating functions, as has been done for the correlation consistent sets, or to choose the correlating functions from the SCF set that maximize the correlation contribution, as done in the Dyall basis sets. A third possibility, also used in the Dyall basis sets where the valence correlating functions overlap with the outer core orbitals is to replace the linearly dependent SCF functions with the valence correlating functions, and then reoptimize the SCF set with the correlating functions frozen. Such a strategy is useful if the basis is to be used uncontracted, because the tail of the outer core functions is not as important as the correlation of the valence shell. Other approaches include optimizing the SCF and the correlating exponents together, or replacing them with an even-tempered set.
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When applying the principles of basis set design, the characteristics of the elements for which the basis sets are developed must be taken into account. The f series start out like the d-block transition metals, with the 6d orbital occupied for several of the early actinides, and the 5d occupied for La and Ce. These early elements even have some low-lying states in which the d orbital is multiply occupied. Further along the series, the f orbital is the dominant occupied open-shell orbital and the d is unoccupied, except in the middle of the block. The outer s orbital is doubly occupied in all of these elements. The chemistry of the lanthanides is largely (but by no means solely) that of the +3 oxidation state, whereas higher oxidation states are of importance in the early part of the actinide series, and the +3 oxidation state becomes dominant later in the series. For the development of basis sets, the radial behavior of the orbitals is important, because it determines the range of exponents of the Gaussian functions that are used, and to what extent the exponent sets for different shells overlap, particularly those for electron correlation. Because the f shell is fairly compact, any basis set must cover a radial range that extends from that of the f shell to that of the outer valence s and d shells. The radial behavior is elaborated below. Relativistic effects are also critical, particularly for the actinides, where both direct and indirect effects significantly change the radial behavior of the orbitals compared to those of the lanthanides. The relativistic effects are not negligible in the lanthanides, though, because they contribute a good fraction of the lanthanide contraction. The spin-orbit
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splitting is important for the 6p of the actinides and for the core shells of the same principal quantum number as the f shell. The accessibility of the orbitals (or spinors) for bonding, and hence whether they are considered part of the valence, the outer core or the inner core, must be assessed on the basis of both the energetics and the radial extent of the orbitals. How the orbitals are partitioned is also relevant to the development of basis sets, which must describe not only the atomic behavior but the bonding behavior as well. For the energy, the eigenvalue is usually sufficient, because it gives an approximate measure of the ionization potential. For the radial extent, mean properties, although useful, are insufficient to portray the entire radial behavior, because the behavior at large distances depends on the angular momentum due the power of r in the radial functions (rl ). Thus, two orbitals of different angular momentum with the same mean radius or radial maximum will have different radial extents: the higher angular momentum function will extend further than the lower. Here we use three measures of the radial extent: the root-mean-square radius, r(rms), the position of the radial maximum, r(max), and the radius that contains 95% of the density, r(95%). The last of these is a measure of how far out the charge distribution extends. Together with the spinor eigenvalues, these quantities are plotted for the lanthanides and actinides in Figures 8.1–8.4, based on Dirac-Hartree-Fock (DHF) calculations on the (degeneracy-weighted) average energy of the fn−1 d1 s2 configuration. For the lanthanides, the rms radii and the 95% density radius are in shell order, i.e., 4d < 4f
