Effective nuclear charge is a very important concept in chemistry, and is the basis for the qualitative explanation of many observed chemical and physical properties, including several periodic trends. Hence it seems to me to be of significant importance that these values are accurately determined (not only for neutral atoms, but for monoatomic ions too). Slater's rules for the calculation of effective nuclear charges, from the 1930s, are in practically any university-level general or inorganic chemistry book, though they are heavily outdated and inaccurate, being mostly of interest due to their simplicity and their historical value. Many books go on to mention updated values for atomic screening constants, calculated by Clementi and Raimondi in 1963 and 1967. This updated table of effective nuclear charges is significantly more interesting, but not used as often probably because all the values for each orbital in each atom must be looked up individually.

However, it's been 50 years since Clementi and Raimondi's first publication on the topic, and I have seen no clear reference to a more recent improved calculation. Surely it is not because the values they found are not without flaws. For example:

  • Screening constants have only been calculated up to $Z = 86$, leaving out a portion of the table, in particular the actinides which have interesting electronic properties
  • No relativistic terms whatsoever were considered in the calculation of the screening constants, which is deleterious especially for atoms beyond about $Z \simeq 70$
  • The calculations are not completely reliable as they are, having not converged well especially some lanthanides, creating odd kinks in screening capacities for electrons in the $4f$ subshell
  • The calculations converged to inaccurate electron distributions in several atoms, such as $[\ce{Kr}]4d^85s^2$ for $\ce{Pd}$ instead of the actual $[\ce{Kr}]4d^{10}$

By now we should be able to do their calculations again with much more accuracy and much more speed, both due to the development of physical and mathematical theory behind the calculations, and due to the stupendous increase in affordable computational capacity. So why has there been no improvement to effective nuclear charge calculations? Has no one bothered? Or have there been several attempts, none of which have gained wide acceptance since then (sort of like what happens with electronegativity)?


3 Answers 3


I would say the most important reason why no development for effective nuclear charges is that the concept of effective nuclear charges has fallen to disuse in the recent years. I can think of two important reasons:

  1. Effective nuclear charges is intended to be used to explain electronic structure of isolated atoms. To this day some atoms can still pose a challenge, but those are the atoms that experience heavy relativistic effects and would need other special treatments than shielding effect along. However, for the most part, chemists are not interested in atoms anymore.

  2. Its theoretical framework is too simple. Chemist need their results to be extremely accurate. A 0.1 eV change can easily be a game changer in an organic reaction. That, unfortunately, cannot be achieved by a model that has only one adjustable constant.

However, the idea still lives on. For example, pseudo-potential or Effective Core Potentials can be considered as more general and modern version of effective nuclear charge concept. Instead of simply changing the charge but keeping the Coulomb potential unchanged, ECP provides the entire effective potential of core electrons. Valence electrons and sometimes even sub-valence electrons has to be excluded from such treatment to be able to produce results with sufficient accuracy. These pseudo-potentials are actively developed and widely used today, can provide chemically relevant accuracy in complex molecular systems and may qualify as "improvement to effective nuclear charge", although they are evolved into different species already.


Analytical Dirac-Hartree-Fock-Slater screening function for atoms (Z=1–92) Phys. Rev. A 36, 467

seems to address the deficiencies mentioned in the question.

Effective nuclear charge is expressed as a function of radius for elements through uranium and relativistic calculations are used.

There is also A new set of relativistic screening constants for the screened hydrogenic model High Energy Density Physics vol. 7 page 169-179.

  • $\begingroup$ It always amazes me how good you are at pulling up sources! I'll look into those articles, hopefully with them I can analyse periodic trends more accurately. $\endgroup$ Mar 17, 2015 at 22:03

As you say, effectively nuclear charges are an interesting tool to provide simple handwaving-type explanations of chemical and physical properties. The reality is more complex, and effective nuclear charges are not a useful tool for heavy elements, where more advanced quantum chemical methods are used. In short: the values of effective nuclear charges we have are good enough for what we want to do with them.

  • $\begingroup$ I can sort of understand that. I'm sure no one is going to rely entirely on effective nuclear charge to put forth an argument (at least not one subtle enough that it is sensitive to details in a quantitative analysis). However, you could say the very same about electronegativity, yet there is a decent amount of research into efforts to redefine it. Why would that get singled out? $\endgroup$ Nov 30, 2013 at 20:04

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