Basis Sets for the f Block

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 -h3 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 that of the lanthanides. The relativistic effects are not neghgible in the lanthanides, though, because they contribute a good fraction of the lanthanide contraction. The spin-orbit 

Splitting is important for the 6p of the actinides and for the core shells of the same principal quantum number as the f shell. 

For the lanthanides, the rms radii and the 95% density radius are in shell order, i.e., 4d 4f 5s 5p 5d 6s. The radial maximum of the 4f is inside that of the 4d, but otherwise the shell order is observed. What is perhaps not obvious from these plots is that the radial maximum of the 6s is outside the 95% density radius of the 5p shell, whereas the radial maximum of the 5d is inside the 95% density radius of the 5p shell. If the radial maximum is taken as some measure of where the midpoint of a bond would be, this indicates that bonding with the 6s does not incur much repulsion of the ligand orbitals by the outer core (5s and 5p) of the lanthanide, whereas there would be somewhat more repulsion from bonding with the 5d. In any case, the 5d is substantially inside the 6s on all measures of radial extent. 

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