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The effective nuclear charge (often symbolized as \( Z_{\mathrm{eff}} \) or \( Z^\ast) \) is the net positive charge experienced by an electron in a polyelectronic atom. The term "effective" is used because the shielding effect of negatively charged electrons prevents higher orbital electrons from experiencing the full nuclear charge of the nucleus due to the repelling effect of inner-layer electrons. The effective nuclear charge experienced by the electron is also called the core charge. It is possible to determine the strength of the nuclear charge by the oxidation number of the atom. Most of the physical and chemical properties of the elements can be explained on the basis of electronic configuration. Consider the behavior of ionization energies in the periodic table. It is known that the magnitude of ionization potential depends upon the following factors:

(a) Size of atom;

(b) The nuclear charge;

(c) The screening effect of the inner shells, and;

(d) The extent to which the outermost electron penetrates into the charge cloud set up by the inner lying electron.

In an atom with one electron, that electron experiences the full charge of the positive nucleus. In this case, the effective nuclear charge can be calculated by Coulomb's law.

However, in an atom with many electrons, the outer electrons are simultaneously attracted to the positive nucleus and repelled by the negatively charged electrons. The effective nuclear charge on such an electron is given by the following equation:

\( Z_\mathrm{eff} = Z - S \)

where

Z is the number of protons in the nucleus (atomic number), and
S is the shielding constant.

S can be found by the systematic application of various rule sets, the simplest of which is known as "Slater's rules" (named after John C. Slater). Douglas Hartree defined the effective Z of a Hartree–Fock orbital to be:

\( Z_\mathrm{eff} = \frac{\langle r\rangle_{\rm H}}{\langle r\rangle_Z} \)

where

\( \langle r\rangle_{\rm H} \) is the mean radius of the orbital for hydrogen, and
\( \langle r\rangle_Z \) is the mean radius of the orbital for a proton configuration with nuclear charge Z.

Values

Updated effective nuclear charge values were provided by Clementi et al. in 1963 and 1967.[1][2] In their work, screening constants were optimized to produce effective nuclear charge values that agree with SCF calculations. Though useful as a predictive model, the resulting screening constants contain little chemical insight as a qualitative model of atomic structure.

Effective Nuclear Charges
  H   He
Z 1   2
1s 1.000   1.688
  Li Be   B C N O F Ne
Z 3 4   5 6 7 8 9 10
1s 2.691 3.685   4.680 5.673 6.665 7.658 8.650 9.642
2s 1.279 1.912   2.576 3.217 3.847 4.492 5.128 5.758
2p       2.421 3.136 3.834 4.453 5.100 5.758
  Na Mg   Al Si P S Cl Ar
Z 11 12   13 14 15 16 17 18
1s 10.626 11.609 12.591 13.575 14.558 15.541 16.524 17.508
2s 6.571 7.392 8.214 9.020 9.825 10.629 11.430 12.230
2p 6.802 7.826 8.963 9.945 10.961 11.977 12.993 14.008
3s 2.507 3.308 4.117 4.903 5.642 6.367 7.068 7.757
3p 4.066 4.285 4.886 5.482 6.116 6.764
  K Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn Ga Ge As Se Br Kr
Z 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36
1s 18.490 19.473 20.457 21.441 22.426 23.414 24.396 25.381 26.367 27.353 28.339 29.325 30.309 31.294 32.278 33.262 34.247 35.232
2s 13.006 13.776 14.574 15.377 16.181 16.984 17.794 18.599 19.405 20.213 21.020 21.828 22.599 23.365 24.127 24.888 25.643 26.398
2p 15.027 16.041 17.055 18.065 19.073 20.075 21.084 22.089 23.092 24.095 25.097 26.098 27.091 28.082 29.074 30.065 31.056 32.047
3s 8.680 9.602 10.340 11.033 11.709 12.368 13.018 13.676 14.322 14.961 15.594 16.219 16.996 17.790 18.596 19.403 20.219 21.033
3p 7.726 8.658 9.406 10.104 10.785 11.466 12.109 12.778 13.435 14.085 14.731 15.369 16.204 17.014 17.850 18.705 19.571 20.434
4s 3.495 4.398 4.632 4.817 4.981 5.133 5.283 5.434 5.576 5.711 5.842 5.965 7.067 8.044 8.944 9.758 10.553 11.316
3d 7.120 8.141 8.983 9.757 10.528 11.180 11.855 12.530 13.201 13.878 15.093 16.251 17.378 18.477 19.559 20.626
4p   6.222 6.780 7.449 8.287 9.028 9.338
  Rb Sr Y Zr Nb Mo Tc Ru Rh Pd Ag Cd In Sn Sb Te I Xe
Z 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54
1s 36.208 37.191 38.176 39.159 40.142 41.126 42.109 43.092 44.076 45.059 46.042 47.026 48.010 48.992 49.974 50.957 51.939 52.922
2s 27.157 27.902 28.622 29.374 30.125 30.877 31.628 32.380 33.155 33.883 34.634 35.386 36.124 36.859 37.595 38.331 39.067 39.803
2p 33.039 34.030 35.003 35.993 36.982 37.972 38.941 39.951 40.940 41.930 42.919 43.909 44.898 45.885 46.873 47.860 48.847 49.835
3s 21.843 22.664 23.552 24.362 25.172 25.982 26.792 27.601 28.439 29.221 30.031 30.841 31.631 32.420 33.209 33.998 34.787 35.576
3p 21.303 22.168 23.093 23.846 24.616 25.474 26.384 27.221 28.154 29.020 29.809 30.692 31.521 32.353 33.184 34.009 34.841 35.668
4s 12.388 13.444 14.264 14.902 15.283 16.096 17.198 17.656 18.582 18.986 19.865 20.869 21.761 22.658 23.544 24.408 25.297 26.173
3d 21.679 22.726 25.397 25.567 26.247 27.228 28.353 29.359 30.405 31.451 32.540 33.607 34.678 35.742 36.800 37.839 38.901 39.947
4p 10.881 11.932 12.746 13.460 14.084 14.977 15.811 16.435 17.140 17.723 18.562 19.411 20.369 21.265 22.181 23.122 24.030 24.957
5s 4.985 6.071 6.256 6.446 5.921 6.106 7.227 6.485 6.640 (empty) 6.756 8.192 9.512 10.629 11.617 12.538 13.404 14.218
4d 15.958 13.072 11.238 11.392 12.882 12.813 13.442 13.618 14.763 15.877 16.942 17.970 18.974 19.960 20.934 21.893
5p   8.470 9.102 9.995 10.809 11.612 12.425

Comparison with Nuclear charge

Nuclear charge is the electric charge of a nucleus of an atom, equal to the number of protons in the nucleus times the elementary charge. In contrast, the effective nuclear charge is the attractive positive charge of nuclear protons acting on valence electrons, which is always less than the total number of protons present in a nucleus due to the shielding effect[3]
See also

Atomic orbitals
Core charge
d-block contraction (or scandide contraction)
Electronegativity
Lanthanide contraction
Shielding effect
Slater-type orbitals
Valence electrons
Weak charge

References

Clementi, E.; Raimondi, D. L. (1963). "Atomic Screening Constants from SCF Functions". J. Chem. Phys. 38 (11): 2686–2689. Bibcode:1963JChPh..38.2686C. doi:10.1063/1.1733573.
Clementi, E.; Raimondi, D. L.; Reinhardt, W. P. (1967). "Atomic Screening Constants from SCF Functions. II. Atoms with 37 to 86 Electrons". Journal of Chemical Physics. 47: 1300–1307. Bibcode:1967JChPh..47.1300C. doi:10.1063/1.1712084.

Resources

Brown, Theodore; LeMay, H.E.; & Bursten, Bruce (2002). Chemistry: The Central Science (8th revised edition). Upper Saddle River, New Jersey 07458: Prentice-Hall. ISBN 0-13-061142-5.

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