Clementi and Raimondi

Clementi and Raimondi refined these results by performing atomic HF-LCAO calculations, treating the orbital exponents as variational parameters. A selection of their results for H through Ne is given in Table 9.3. 

Atomic Screening Constants from SCF Functions E. Clementi and D. L. Raimondi The Journal of Chemical Physics 38 (1963) 2686-2689 

The self-consistent field function for atoms with 2 to 36 electrons are computed with a minimum basis set of Slater-type orbitals. The orbital exponents of the atomic orbitals are optimized so as to ensure the energy minimum. The analysis of the optimized orbital exponents allows us to obtain simple and accurate rules for the 1 s, 2s, 3s, 4s, 2p, 3p, 4p and 3d electronic screening constants. These rules are compared with those proposed by Slater and reveal the need for the screening due to the outside electrons. The analysis of the screening constants (and orbital exponents) is extended to the excited states of the ground state configuration and the positive ions. 

Notice that the 2s and 2p orbitals have a slightly different exponent. Such basis sets, where we simply use the same number of atomic orbitals that we would use in everyday descriptive chemistry, are referred to as minimal basis sets. 

The next step on the road to quality is to expand the size of the atomic orbital basis set, and I hinted in Chapters 3 and 4 how we might go about this. To start with, we double the number of basis functions and then optimize their exponents by systematically repeating atomic HF-LCAO calculation. This takes account of the so-called inner and outer regions of the wavefunction, and Clementi puts it nicely. 

Simple Basis Set for Molecular Wavefunctions Containing First- and Second-Row Atoms E. Clementi 

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Energy-optimized, single-Slater values for the electron subshells of isolated atoms have been calculated by Clementi and Raimondi (1963). For the electron density functions, such values are to be multiplied by a factor of 2. Values for a number of common atoms are listed in Table 3.4, together with averages over electron shells, which are suitable as starting points in a least-squares refinement in which the exponents are subsequently adjusted by variation of k. A full list of the single values of Clementi and Raimondi can be found in appendix F. 

Source Clementi and Raimondi (1963). For double zeta functions, see Clementi (1965). 

Orbital exponents for the calculation (Table I) have been taken from Clementi and Raimondi (1). Their dependence on charge has been assumed to be that given by Slaters formulas for orbital exponents (18). 

To remove the difficulties and inaccuracies in the simplified Slater treatment of shielding, Clementi and Raimondi have obtained effective nuclear charges from... 

The shielding rules of Clementi and Raimondi explicitly account for penetration of outer orbital electrons. They are thus more realistic than Slater s rules, at the expense, however, of more complex computation with a larger number of parameters. If accuracy greater than Uiat afforded by Slater s rules is necessary, it would appear tliat direct application Of the effective nuclear charges from the SCF wave functions is not only simple but also accurate. Such values arc listed in Table 22. Witli the accurate values of Table 22 available, the chief justification of "rules , whetlier Slater s or those of Clementi and Raimondi, is the insiglit tliey provide inlo tlie phenomenon of shielding. 

Compare the values of Z thus obtained with those of Clementi and Raimondi. 

Other prescriptions for the exponents, have been advanced over the years. Clementi and Raimondi (8) proposed in 1963 that the best exponents should be based on the criterion that the atomic energy should be minimized. Clementi, too, (9,10) and others (11) have investigated the use of more than one Slater function to obtain a better representation of the radial wave functions for many-electron atoms. 

There have been attempts to obtain better values for effective nuclear charges, in particular those of Clementi and Raimondi and of Froese and Fischer. As yet, these have not been used for the construction of electronegativity scales for all the elements. 

Slater versus Clementi and Raimondi values of Zeff... 

Clementi and Raimondi [8] have published a refined list of rules for the shielding constant, which extends to the 4p level. Their rules include contributions to shielding due to the presence of electrons in shells outside the orbital under consideration. Such contributions are not large, and, up to the 3d level, there is reasonably good agreement between these two sets of mles. 

To complete the construction of the STO-fcG basis sets, we must specify the Slater exponents to be used for the orbitals. One solution would be to use exponents optimized in separate atomic calculations for each STO-AG set. The resulting exponents are similar for all k and also close to the exponents of Clementi and Raimondi (5.67 for Is, 1.61 for 2s and 1.57 for 2p in the carbon atom), optimized for single STOs in atoms [2], In practice, for the first-row atoms, the atomic exponents of Clementi and Raimondi are used for the K shell. For the L shell, however, a set of standard molecular exponents (obtained from calculations on small molecules) is used - recognizing that, in a molecular environment, the optimum exponents for the valence shells differ somewhat from their atomic values. For the carbon atom, an exponent of 1.72 is used for the 2s and 2p orbitals. For the hydrogen Is orbital, the exponent is 1.24. Thus, the hydrogen exponent differs substantially from its atomic value, reflecting a significant contraction of the atomic electron distribution in a molecular environment. 

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