Effective nuclear charge
Encyclopedia
The effective nuclear charge is the net positive charge experienced by an electron
Electron
The electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...

 in a multi-electron atom
Atom
The atom is a basic unit of matter that consists of a dense central nucleus surrounded by a cloud of negatively charged electrons. The atomic nucleus contains a mix of positively charged protons and electrically neutral neutrons...

. The term "effective" is used because the shielding effect
Shielding effect
The shielding effect describes the decrease in attraction between an electron and the nucleus in any atom with more than one electron shell. It is also referred to as the screening effect or atomic shielding.-Cause:...

 of negatively charged electrons prevents higher orbital electrons from experiencing the full nuclear charge by the repelling effect of inner-layer electrons. The effective nuclear charge experienced by the outer shell electron is also called the core charge
Core charge
Core charge is the effective nuclear charge experienced by an outer shell electron. In other words core charge is an expression of the attractive force experienced by the valence electrons to the core of an atom which takes into account the shielding effect of core electrons...

. It is possible to determine the strength of the nuclear charge by looking at the oxidation number of the atom.

Calculating the effective nuclear charge

In an atom with one electron, that electron experiences the full charge of the positive nucleus
Atomic nucleus
The nucleus is the very dense region consisting of protons and neutrons at the center of an atom. It was discovered in 1911, as a result of Ernest Rutherford's interpretation of the famous 1909 Rutherford experiment performed by Hans Geiger and Ernest Marsden, under the direction of Rutherford. The...

. In this case, the effective nuclear charge can be calculated from Coulomb's law
Coulomb's law
Coulomb's law or Coulomb's inverse-square law, is a law of physics describing the electrostatic interaction between electrically charged particles. It was first published in 1785 by French physicist Charles Augustin de Coulomb and was essential to the development of the theory of electromagnetism...

.

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:

where
Z is the number of protons in the nucleus (atomic number
Atomic number
In chemistry and physics, the atomic number is the number of protons found in the nucleus of an atom and therefore identical to the charge number of the nucleus. It is conventionally represented by the symbol Z. The atomic number uniquely identifies a chemical element...

), and
S is the average number of electrons between the nucleus and the electron in question (the number of nonvalence electrons).


S can be found by the systematic application of various rule sets, the simplest of which is known as "Slater's rules
Slater's rules
In quantum chemistry, Slater's rules provide numerical values for the effective nuclear charge concept. In a many-electron atom, each electron is said to experience less than the actual nuclear charge owning to shielding or screening by the other electrons...

" (named after John C. Slater
John C. Slater
John Clarke Slater was a noted American physicist who made major contributions to the theory of the electronic structure of atoms, molecules and solids. This work is of ongoing importance in chemistry, as well as in many areas of physics. He also made major contributions to microwave electronics....

). Douglas Hartree
Douglas Hartree
Douglas Rayner Hartree PhD, FRS was an English mathematician and physicist most famous for the development of numerical analysis and its application to the Hartree-Fock equations of atomic physics and the construction of the meccano differential analyser.-Early life:Douglas Hartree was born in...

 defined the effective Z of a Hartree-Fock
Hartree-Fock
In computational physics and chemistry, the Hartree–Fock method is an approximate method for the determination of the ground-state wave function and ground-state energy of a quantum many-body system....

 orbital to be:
where
H is the mean radius of the orbital for hydrogen, and
Z is the mean radius of the orbital for an electron configuration with nuclear charge Z.


Note: Zeff is also often written Z*.

Example

Consider a sodium
Sodium
Sodium is a chemical element with the symbol Na and atomic number 11. It is a soft, silvery-white, highly reactive metal and is a member of the alkali metals; its only stable isotope is 23Na. It is an abundant element that exists in numerous minerals, most commonly as sodium chloride...

 cation, a fluorine
Fluorine
Fluorine is the chemical element with atomic number 9, represented by the symbol F. It is the lightest element of the halogen column of the periodic table and has a single stable isotope, fluorine-19. At standard pressure and temperature, fluorine is a pale yellow gas composed of diatomic...

 anion, and a neutral neon
Neon
Neon is the chemical element that has the symbol Ne and an atomic number of 10. Although a very common element in the universe, it is rare on Earth. A colorless, inert noble gas under standard conditions, neon gives a distinct reddish-orange glow when used in either low-voltage neon glow lamps or...

 atom. Each has 10 electrons, and the number of nonvalence electrons is 2 (10 total electrons - 8 valence) but the effective nuclear charge varies because each has a different atomic number:


So, the sodium cation has the largest effective nuclear charge, and thus the smallest atomic radius
Atomic radius
The atomic radius of a chemical element is a measure of the size of its atoms, usually the mean or typical distance from the nucleus to the boundary of the surrounding cloud of electrons...

.

Values

Updated values of screening constants were provided by Clementi et al.
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 26.047
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

Resources

  • Brown, Theodore; LeMay, H.E.; & Bursten, Bruce (2002). Chemistry: The Central Science (8th revised edition). Upper Saddle River, NJ 07458: Prentice-Hall. ISBN 0-61155-61141-5.
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