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Effective nuclear charge
Effective nuclear charge
Effective nuclear charge
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In atomic physics, the effective nuclear charge of an electron in a multi-electron atom or ion is the number of elementary charges () an electron experiences by the nucleus. It is denoted by Zeff. The term "effective" is used because the shielding effect of negatively charged electrons prevent higher energy electrons from experiencing the full nuclear charge of the nucleus due to the repelling effect of inner layer. The effective nuclear charge experienced by an 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:

  1. The size of an atom
  2. The nuclear charge; oxidation number
  3. The screening effect of the inner shells
  4. The extent to which the outermost electron penetrates into the charge cloud set up by the inner lying electron

In the periodic table, effective nuclear charge decreases down a group and increases left to right across a period.

Description

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The effective atomic number Zeff, (sometimes referred to as the effective nuclear charge) of an electron in a multi-electron atom is the number of protons that this electron effectively 'sees' due to screening by inner-shell electrons. It is a measure of the electrostatic interaction between the negatively charged electrons and positively charged protons in the atom. One can view the electrons in an atom as being 'stacked' by energy outside the nucleus; the lowest energy electrons (such as the 1s and 2s electrons) occupy the space closest to the nucleus, and electrons of higher energy are located further from the nucleus.

The binding energy of an electron, or the energy needed to remove the electron from the atom, is a function of the electrostatic interaction between the negatively charged electrons and the positively charged nucleus. For instance, in iron (atomic number 26), the nucleus contains 26 protons. The electrons that are closest to the nucleus will 'see' nearly all of them. However, electrons further away are screened from the nucleus by other electrons in between, and feel less electrostatic interaction as a result. The 1s electron of iron (the closest one to the nucleus) sees an effective atomic number (number of protons) of 25. The reason why it is not 26 is that some of the electrons in the atom end up repelling the others, giving a net lower electrostatic interaction with the nucleus. One way of envisioning this effect is to imagine the 1s electron sitting on one side of the 26 protons in the nucleus, with another electron sitting on the other side; each electron will feel less than the attractive force of 26 protons because the other electron contributes a repelling force. The 4s electrons in iron, which are furthest from the nucleus, feel an effective atomic number of only 5.43 because of the 25 electrons in between it and the nucleus screening the charge.

Effective atomic numbers are useful not only in understanding why electrons further from the nucleus are so much more weakly bound than those closer to the nucleus, but also because they can tell us when to use simplified methods of calculating other properties and interactions. For instance, lithium, atomic number 3, has two electrons in the 1s shell and one in the 2s shell. Because the two 1s electrons screen the protons to give an effective atomic number for the 2s electron close to 1, we can treat this 2s valence electron with a hydrogenic model.

Mathematically, the effective atomic number Zeff can be calculated using methods known as "self-consistent field" calculations, but in simplified situations is just taken as the atomic number minus the number of electrons between the nucleus and the electron being considered.

Calculations

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See also: Shielding effect

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.[1]

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

  • is the number of protons in the nucleus (atomic number) and
  • is the shielding constant.

S can be found by the systematic application of various rule sets.

Slater's rules

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Main article: Slater's rules

The simplest method for determining the shielding constant for a given electron is the use of "Slater's rules", devised by John C. Slater, and published in 1930.[2] These algebraic rules are significantly simpler than finding shielding constants using ab initio calculation.

Hartree–Fock method

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See also: Hartree–Fock method

A more theoretically justified method is to calculate the shielding constant using the Hartree-Fock method. Douglas Hartree defined the effective Z of a Hartree–Fock orbital to be: where

  • is the mean radius of the orbital for hydrogen, and
  • is the mean radius of the orbital for a proton configuration with nuclear charge Z.

Values

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Updated effective nuclear charge values were provided by Clementi et al. in 1963 and 1967.[3][4] 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

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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.[5]

See also

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References

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Further reading

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Revisions and contributorsEdit on WikipediaRead on Wikipedia

Effective nuclear charge

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from Grokipedia
The effective nuclear charge is the net positive charge experienced by a in a multi-electron atom, for the partial screening of the nucleus by inner electrons. This value, denoted as ZeffZ_{\text{eff}}, is less than the full nuclear charge ZZ (the ) because electrons closer to the nucleus repel outer electrons, reducing the attractive force felt by valence electrons. The is fundamental to understanding atomic structure and in chemistry. The arises from electron-electron repulsions, where form a "cloud" that diminishes the nuclear pull on outer electrons. Quantitatively, Zeff=Z−σZ_{\text{eff}} = Z - \sigma, where σ\sigma is the shielding constant representing the average screening by all other electrons. is more effective from inner shells, while electrons in the same shell provide partial screening. To approximate σ\sigma, , developed by physicist John C. Slater, provide an empirical method by grouping electrons into shells: (1s), (2s,2p), (3s,3p), (3d), (4s,4p), etc. Under these rules, for a given : electrons in groups with n' ≤ n-2 contribute 1.00 to σ\sigma; those in the (n-1) group contribute 0.85 (for s and p electrons); electrons in the same group contribute 0.35 each (or 0.30 for the 1s group). For d or f electrons, all other electrons in the same group and all electrons in lower groups (including the n-1 group) contribute 1.00 each. These rules yield reasonable estimates but are approximations, as actual shielding varies with orbital type and quantum mechanical effects. The effective nuclear charge governs key periodic properties: it increases across a period (left to right) as ZZ rises while shielding remains similar, causing atomic radii to decrease and ionization energies to increase. For instance, in the third period, ZeffZ_{\text{eff}} for valence electrons rises from approximately 2.2 in sodium (Z=11) to approximately 6.8 in argon (Z=18). Down a group, added inner shells enhance shielding, so ZeffZ_{\text{eff}} changes little despite increasing ZZ, leading to larger atomic sizes. It also influences electron affinity, electronegativity, and spectral properties of atoms and ions.

Basic Concepts

Definition

In multi-electron atoms, the effective nuclear charge, denoted as ZeffZ_{\text{eff}}, represents the net positive charge experienced by a , accounting for the attractive pull of the nucleus partially offset by the repulsive interactions from inner electrons. This concept arises because the full nuclear charge ZZ, equal to the , is not fully felt by outer electrons due to electron-electron repulsions that reduce the net attraction. Formally, ZeffZ_{\text{eff}} is expressed as Zeff=Z−σZ_{\text{eff}} = Z - \sigma, where σ\sigma is the shielding constant that quantifies the screening effect of inner electrons. In contrast, for single-electron atoms like , there is no shielding, so Zeff=[Z](/page/Z)Z_{\text{eff}} = [Z](/page/Z), and the experiences the full nuclear charge without interference from other electrons. This distinction highlights how multi-electron systems deviate from simple hydrogen-like models, necessitating the effective nuclear charge to describe the modified electrostatic environment.

Physical Significance

The effective nuclear charge, ZeffZ_{\text{eff}}, plays a central role in governing the radial distribution of electrons and their energy levels in multi-electron atoms. In multi-electron atoms, Zeff<ZZ_{\text{eff}} < Z due to electron shielding by inner electrons. The effective nuclear charge experienced by electrons varies depending on the orbital type: for orbitals with the same principal quantum number nn but different azimuthal quantum numbers ll, those with higher ll (e.g., p orbitals compared to s) experience lower ZeffZ_{\text{eff}} due to reduced penetration towards the nucleus, resulting in higher orbital energies, while s orbitals penetrate closer to the nucleus, experiencing a higher ZeffZ_{\text{eff}} and thus lower energy levels. As ZeffZ_{\text{eff}} increases, electrons experience a stronger net attraction to the nucleus, resulting in wavefunctions that are more compact and shifted toward smaller radial distances from the nucleus. This tighter binding elevates the energy levels relative to those in isolated atoms, making electron removal more energetically costly and influencing the overall electronic structure. A higher ZeffZ_{\text{eff}} intensifies the electron-nucleus attraction, pulling valence electrons closer to the nucleus and thereby reducing atomic and ionic radii while enhancing stability. This effect is particularly evident in atoms where the nuclear charge grows without a proportional increase in shielding, leading to smaller, more stable configurations that resist deformation or ionization. Such dynamics underpin the compactness of inner-shell electrons and the progressive tightening of outer orbitals across the periodic table. In quantum mechanical models of multi-electron systems, ZeffZ_{\text{eff}} serves as an approximation to the mean-field potential, where electron-electron interactions are averaged into an effective central potential experienced by each electron. This concept, foundational to methods like the Hartree approximation, simplifies the many-body problem by treating electrons as moving independently in this averaged field, capturing essential features of atomic orbitals without full correlation effects. A clear illustration of ZeffZ_{\text{eff}}'s influence appears in isoelectronic ions, such as Na+^+, Mg2+^{2+}, and Al3+^{3+}, all sharing the same electron configuration ([Ne]) but differing nuclear charges. As the atomic number ZZ increases from 11 to 13, ZeffZ_{\text{eff}} rises due to the greater nuclear pull unmitigated by additional electrons, resulting in progressively smaller ionic radii: 102 pm for Na+^+, 72 pm for Mg2+^{2+}, and 53.5 pm for Al3+^{3+} (Shannon ionic radii, coordination number 6). This trend demonstrates how increasing ZeffZ_{\text{eff}} contracts the electron cloud, enhancing nuclear dominance.

Shielding and Screening

Shielding Effect

The in multi-electron atoms arises from the electrostatic repulsion between , whereby inner form a diffuse electron cloud that partially screens the positive charge of the nucleus from outer electrons. This screening reduces the net attractive force experienced by valence electrons, leading to a diminished effective nuclear charge. The extent of this shielding is quantified by the shielding constant σ, which represents the average number of electrons effectively opposing the nuclear attraction for a given outer electron. Although the inner electron cloud provides substantial screening, the shielding is incomplete for outer electrons because the electron density distribution is non-uniform, with regions of higher probability closer to the nucleus where the full nuclear charge is more directly felt. This non-uniformity means that outer electrons do not experience complete neutralization of the nuclear charge, resulting in a partial penetration of the nuclear field. Core electrons, being closer to the nucleus, valence electrons almost completely, such that σ is approximately equal to the number of , effectively isolating the valence shell from much of the nuclear attraction. In contrast, valence electrons one another poorly due to their similar radial distribution, with each contributing only a fractional amount to σ, typically much less than 1. This differential ing explains why remain tightly bound while valence electrons are more loosely held. From a quantum mechanical perspective, the shielding effect emerges from the variational principle applied to the wavefunctions of multi-electron atoms, where optimizing the trial wavefunction minimizes the energy and incorporates electron correlation through mutual repulsions. This correlation adjusts the effective nuclear charge to account for the screening, as seen in approximations where the Hamiltonian is simplified by treating repulsions as an average field.

Penetration Effect

The penetration effect describes the phenomenon where valence electrons in outer orbitals have a non-zero probability that extends into the regions occupied by inner , allowing them to overlap with the nuclear charge and thereby experience incomplete shielding. This overlap reduces the shielding constant σ, leading to a higher effective nuclear charge Z_eff for the penetrating compared to what would occur with perfect screening. The extent of penetration depends on the orbital's angular momentum quantum number l, with s-orbitals (l=0) exhibiting the greatest penetration, followed by (l=1), d (l=2), and f (l=3) orbitals within the same principal quantum shell n. Consequently, s-electrons penetrate closer to the nucleus and sense a stronger nuclear attraction, resulting in higher Z_eff values for s-orbitals relative to other subshells. This variation in Z_eff affects the orbital energies, with energies increasing as l increases for a given n; for example, s orbitals have lower energy than p orbitals in the same shell due to their greater penetration, which allows them to experience a higher effective nuclear charge and greater stabilization. For instance, this contributes to the , where the poor shielding by 4f electrons is amplified by the penetration of 6s valence electrons, causing a steeper decrease in atomic radii across the series. Mathematically, the reduction in σ due to penetration is often quantified through integrals of the radial wavefunctions, such as the overlap between the valence orbital's radial probability |R_{nl}(r)|^2 r^2 and the core regions near the nucleus, which directly influences the computed Z_eff = Z - σ. A representative example occurs in alkali metals, where the ns valence electrons penetrate the core, experiencing a Z_eff greater than +1. In sodium (Na, [Ne] 3s^1), the 3s electron has a Z_eff of approximately 2.20, which elevates its first to 496 kJ/mol—higher than anticipated if shielding were complete and Z_eff were solely +1—explaining the relatively modest decrease in ionization energies down the group despite increasing atomic size.

Calculation Methods

Slater's Rules

Slater's rules were developed by John C. Slater in 1930 as an empirical approximation to estimate the effective nuclear charge ZeffZ_{\text{eff}} experienced by an in a multi-electron atom based on its . These rules simplify the calculation of the shielding constant σ\sigma by grouping electrons into shells and assigning fixed numerical contributions to shielding, derived from early atomic structure computations. The effective nuclear charge is then given by Zeff=Z−σZ_{\text{eff}} = Z - \sigma, where ZZ is the . This method facilitates quick estimates without solving the full , making it useful for understanding qualitative trends in atomic properties. The electrons are grouped according to their nn and subshell type as follows: [1s], [2s, 2p], [3s, 3p], [3d], [4s, 4p], [4d], [4f], and similarly for higher nn. To calculate σ\sigma for a specific electron, consider its position in one of these groups relative to the others in the configuration. For an electron in an ns or np group:
  • Electrons in groups to the right (higher effective nn) contribute 0 to σ\sigma.
  • Each of the other electrons in the same group contributes 0.35.
  • Each electron in the immediately preceding group (n-1) contributes 0.85.
  • Each electron in all groups further inward (n ≥ 2 lower) contributes 1.00.
For an electron in an nd or nf group, all electrons to the left (lower effective nn) contribute 1.00 each, each other electron in the same nd or nf group contributes 1.00, while those to the right contribute 0. Special cases apply for the innermost shells: for a 1s electron, any other 1s electron contributes 0.30; for 2s or 2p electrons, the [1s] electrons contribute 0.85 each instead of 1.00. The following table illustrates the standard electron grouping used in Slater's rules:
nGroups
1[1s]
2[2s, 2p]
3[3s, 3p], [3d]
4[4s, 4p], [4d], [4f]
5[5s, 5p], [5d], [5g]
To apply the rules, write the in this grouped form and sum the contributions for the of interest. For example, in sodium (Z = 11, configuration [1s]^2 [2s,2p]^8 [3s]^1), for the 3s : other electrons in [3s] = 0 × 0.35 = 0; [2s,2p] = 8 × 0.85 = 6.80; [1s] = 2 × 1.00 = 2.00; thus σ=8.80\sigma = 8.80 and Zeff=11−8.80=2.20Z_{\text{eff}} = 11 - 8.80 = 2.20. Similar calculations for atoms up to (Z = 36, configuration [1s]^2 [2s,2p]^8 [3s,3p]^8 [3d]^{10} [4s,4p]^8) yield Zeff≈8.25Z_{\text{eff}} \approx 8.25 for a 4p , illustrating increasing ZeffZ_{\text{eff}} across periods. Although effective for main-group elements, Slater's rules have limitations, including overestimation of ZeffZ_{\text{eff}} for valence electrons in transition metals due to the assumption of complete ing (1.00) by inner d electrons, which actually shield less effectively. The method also neglects relativistic effects, reducing accuracy for heavier atoms beyond . Overall, the approximation achieves reasonable results for valence electrons, with errors typically on the order of 10-20% compared to more precise methods.

Hartree–Fock Method

The method provides an approximation to the many-electron by treating s as independent particles moving in an average mean-field potential created by the nucleus and other s. This approach assumes the total wavefunction is a of single-particle orbitals, ensuring antisymmetry and incorporating exchange effects, which allows for the computation of atomic orbitals and energies from first principles. The effective nuclear charge ZeffZ_{\text{eff}} emerges naturally from the HF orbitals, as it quantifies the net attraction experienced by an after accounting for shielding by the electron cloud, and can be extracted from the orbital eigenvalues or the asymptotic behavior of the . The procedure involves iteratively solving the HF equations in a self-consistent manner. Starting with an initial guess for the orbitals, the electron density ρ(r)\rho(\mathbf{r}) is constructed, leading to the effective potential Veff(r)=−Zr+∫ρ(r′)∣r−r′∣dr′V_{\text{eff}}(\mathbf{r}) = -\frac{Z}{r} + \int \frac{\rho(\mathbf{r}')}{|\mathbf{r} - \mathbf{r}'|} d\mathbf{r}', where the first term is the nuclear attraction and the second is the classical Coulomb repulsion from the mean electron density (with exchange contributions added via a nonlocal operator). The single-particle HF equations are then solved for updated orbitals f^ψi(r)=ϵiψi(r)\hat{f} \psi_i(\mathbf{r}) = \epsilon_i \psi_i(\mathbf{r}), where f^\hat{f} is the Fock operator incorporating kinetic energy, nuclear potential, and the mean-field terms; this process repeats until convergence. For large rr, Veff(r)≈−ZeffrV_{\text{eff}}(r) \approx -\frac{Z_{\text{eff}}}{r}, so ZeffZ_{\text{eff}} is obtained as the limit Zeff=−lim⁡r→∞rVeff(r)Z_{\text{eff}} = -\lim_{r \to \infty} r V_{\text{eff}}(r), reflecting the screened charge seen by valence electrons. Alternatively, for inner orbitals, ZeffZ_{\text{eff}} can be approximated from the orbital energy via ϵ≈−13.6Zeff2n2\epsilon \approx - \frac{13.6 Z_{\text{eff}}^2}{n^2} eV, adjusted for relativistic effects in heavier atoms. Compared to empirical methods like , the HF approach accounts explicitly for quantum exchange-correlation effects in the mean field, yielding more accurate ZeffZ_{\text{eff}} values without adjustable parameters; for example, in , HF orbital energies give Zeff≈5.2Z_{\text{eff}} \approx 5.2 for 2p electrons, closer to spectroscopic estimates than Slater's approximation of 5.85. This results in errors typically below 5% for ionization potentials and atomic radii in light elements (Z < 20), establishing a rigorous foundation for understanding shielding and penetration. In practice, HF calculations for atoms distinguish between restricted (closed-shell, all electrons paired) and unrestricted (open-shell, allowing spin polarization) formulations to handle different electron configurations. Numerical implementation often employs basis set expansions, such as the minimal STO-3G set, which approximates Slater-type orbitals with three Gaussian functions for efficient linear combination of atomic orbitals (LCAO) solutions, enabling computation of ZeffZ_{\text{eff}} for systems up to moderate size. These methods form the cornerstone of modern quantum chemistry software for atomic structure analysis. The HF method originated with Douglas Hartree's 1928 development of the self-consistent field using product wavefunctions for atoms like sodium, followed by Vladimir Fock's 1930 incorporation of antisymmetrization via to include exchange, transforming it into a fully quantum mechanical framework that underpins subsequent computational advances.

Other Computational Approaches

Configuration interaction (CI) methods account for electron correlation by expanding the wave function as a linear combination of multiple from different electron configurations, yielding more precise effective nuclear charges than single-determinant approaches, especially in heavy atoms where correlation significantly influences shielding. These techniques improve upon Hartree-Fock results by capturing dynamic correlation effects that reduce the overestimation of nuclear attraction in core regions. Density functional theory (DFT) provides an alternative framework for computing effective nuclear charges by mapping the interacting electron system onto a non-interacting one via the Kohn-Sham equations, where the exchange-correlation energy is approximated using functionals like the local density approximation (LDA) or generalized gradient approximation (GGA). The effective nuclear charge is then extracted from the Kohn-Sham orbitals or associated eigenvalues, offering computational efficiency for larger systems while incorporating approximate correlation effects through the functional choice. For high-Z atoms, relativistic effects must be included, as the Dirac-Hartree-Fock (DHF) method incorporates the Dirac equation to account for spin-orbit coupling and other relativistic phenomena that modify electron shielding and increase the effective nuclear charge experienced by valence electrons. This approach reveals periodic trends where relativistic contraction enhances Z_eff more pronouncedly across heavier elements. In transition metals, where d-orbitals lead to strong electron correlation and near-degeneracies invalidating single-reference descriptions, multi-reference methods such as multi-reference configuration interaction (MRCI) are employed to construct wave functions from multiple active configurations, yielding reliable effective nuclear charges for these systems. These methods address the limitations of Hartree-Fock by properly describing the multi-configurational character of ground and excited states. Comparisons of accuracy show that DFT functionals typically reproduce valence effective nuclear charges with errors of 1-2% relative to benchmark coupled-cluster or CI results for main-group elements, making it a practical choice for routine calculations in software like Gaussian and ORCA. More advanced post-Hartree-Fock methods like achieve higher precision for heavy atoms and transition metals but at greater computational cost.

Influence on Atomic Properties

The effective nuclear charge, ZeffZ_{\text{eff}}, plays a central role in determining atomic and ionic radii by influencing the attraction between the nucleus and valence electrons. A higher ZeffZ_{\text{eff}} pulls valence electrons closer to the nucleus, contracting the electron cloud and resulting in smaller atomic radii. This effect is particularly evident across a period in the periodic table, where the nuclear charge increases while shielding remains relatively constant, leading to a progressive increase in ZeffZ_{\text{eff}} and a corresponding decrease in atomic radii. For ionic radii, the trend is amplified in cations, where removal of electrons reduces shielding, effectively increasing ZeffZ_{\text{eff}} experienced by the remaining electrons and further contracting the ion size compared to the neutral atom. Ionization energy (IE), the energy required to remove a valence electron, is directly related to ZeffZ_{\text{eff}}, as a stronger effective nuclear attraction binds electrons more tightly. An approximate formula for the first IE in multi-electron atoms modifies the hydrogen atom energy expression to IE≈13.6Zeff2n2\text{IE} \approx 13.6 \frac{Z_{\text{eff}}^2}{n^2} eV, where nn is the principal quantum number; thus, higher ZeffZ_{\text{eff}} significantly raises IE by quadratically increasing the binding energy. This relationship explains why IE generally increases across a period as ZeffZ_{\text{eff}} rises, making electron removal more difficult. Electron affinity (EA), the energy change upon adding an electron to a neutral atom, follows a similar trend influenced by ZeffZ_{\text{eff}}. Higher ZeffZ_{\text{eff}} enhances the attraction for the incoming electron, leading to more negative (exothermic) EA values and greater stability for the resulting anion. Across a period, the increasing ZeffZ_{\text{eff}} strengthens this nuclear pull despite added electron-electron repulsion, promoting higher EA, though exceptions occur in elements like nitrogen due to half-filled orbital stability. Electronegativity, as defined on the Pauling scale, measures an atom's tendency to attract electrons in a bond and correlates positively with ZeffZ_{\text{eff}}, since greater effective nuclear attraction facilitates electron withdrawal from bonding partners. This correlation underlies the periodic increase in electronegativity across periods, where rising ZeffZ_{\text{eff}} enhances electron-pulling power without a proportional increase in atomic size. The Allred-Rochow scale explicitly incorporates ZeffZ_{\text{eff}} in its formulation, χAR=3590×Zeffrcov2+0.744\chi^{\text{AR}} = \frac{3590 \times Z_{\text{eff}}}{r_{\text{cov}}^2} + 0.744 (where rcovr_{\text{cov}} is covalent radius), aligning closely with Pauling values and reinforcing the link. To illustrate these influences, consider approximate ZeffZ_{\text{eff}} values for valence electrons: in Group 1 (alkali metals) like sodium (Zeff≈2.2Z_{\text{eff}} \approx 2.2), the low effective charge results in larger atomic radii (~180 pm), low IE (~496 kJ/mol), near-zero EA, and low electronegativity (0.9), promoting easy electron loss and metallic reactivity. In contrast, Group 17 (halogens) like chlorine (Zeff≈6.1Z_{\text{eff}} \approx 6.1) exhibit smaller radii (~100 pm), high IE (1251 kJ/mol), highly negative EA (−349-349 kJ/mol), and high electronegativity (3.0), favoring electron gain and forming stable anions. These differences, derived from Slater's rules, predict the contrasting reactivities: alkali metals as strong reducing agents and halogens as oxidizing agents. The effective nuclear charge (ZeffZ_\text{eff}) experienced by valence electrons increases progressively across a period in the periodic table. As the atomic number (ZZ) rises, the nuclear charge grows, but the shielding constant (σ\sigma) increases more slowly because additional electrons enter the same principal shell, providing limited shielding to one another. This results in ZeffZ_\text{eff} rising by approximately +1 for each added proton, drawing valence electrons closer to the nucleus and causing atomic radii to contract. Moving down a group, ZeffZ_\text{eff} for valence electrons remains nearly constant. Although the nuclear charge increases with higher ZZ, the addition of new electron shells enhances shielding by inner electrons, largely offsetting the greater attraction from the nucleus. For instance, in the alkali metals (group 1), the valence ns1ns^1 electrons experience a similar ZeffZ_\text{eff} from to cesium, contributing to the relatively consistent reactivity observed in this group. In the d-block transition series, ZeffZ_\text{eff} increases more steadily across each row due to the poor shielding provided by electrons in (n−1)d(n-1)d subshells. These d-electrons, occupying orbitals that overlap less effectively with valence orbitals, fail to screen the nucleus adequately, leading to a stronger pull on valence electrons and smaller-than-expected atomic sizes compared to main-group elements. This effect is even more pronounced in the f-block lanthanide and actinide series, where 4f and 5f electrons offer minimal shielding, causing ZeffZ_\text{eff} to rise sharply and resulting in the lanthanide and actinide contractions—unexpectedly small decreases in ionic radii across these series that influence the sizes of subsequent elements. One notable exception to these trends occurs in heavier p-block elements, manifesting as the inert pair effect. Here, the valence ns2ns^2 electrons are reluctant to participate in bonding because they penetrate closer to the nucleus, experiencing a higher effective nuclear charge due to inadequate shielding from intervening d- and f-electrons, which stabilizes lower oxidation states. For heavy elements (high ZZ), relativistic effects further modify these patterns by accelerating inner electrons to near-light speeds, contracting s- and p-orbitals and enhancing their shielding efficiency. This indirectly boosts ZeffZ_\text{eff} for valence electrons in some cases, deviating from non-relativistic predictions and altering trends like atomic radii and ionization energies in superheavy elements. Graphical representations of ZeffZ_\text{eff} versus ZZ for valence electrons typically illustrate a stepwise increase across periods, with plateaus down groups, steeper gradients in d- and f-blocks, and subtle deviations at high ZZ due to relativistic influences. These plots underscore how variations in shielding and penetration drive the periodic table's structural and chemical patterns.

Tabulated Values

Representative Effective Nuclear Charges

Effective nuclear charges for valence electrons in selected neutral atoms from lithium to zinc are tabulated below, comparing approximate values from Slater's rules with more accurate Hartree–Fock (HF) computations from Clementi and Raimondi (1963). Slater's rules provide a simple empirical estimate, while HF values derive from self-consistent field calculations of atomic wavefunctions. These examples highlight typical magnitudes, with Slater's often underestimating for transition metals due to its simplified shielding assumptions.
ElementOrbitalSlater's ZeffHF Zeff (Clementi-Raimondi)
Li2s1.31.28
Be2s1.951.91
B2p2.62.42
C2p3.253.14
N2p3.93.85
O2p4.554.45
F2p5.25.07
Ne2p5.855.66
Na3s2.22.51
Mg3s2.853.15
Al3p3.54.07
Si3p4.154.21
P3p4.84.79
S3p5.455.38
Cl3p6.16.12
Ar3p6.756.76
K4s2.22.30
Ca4s2.853.50
Sc4s3.054.61
Zn4s4.355.58
In an isoelectronic series with the neon electron configuration, the effective nuclear charge experienced by valence 2p electrons decreases as the nuclear charge diminishes while the electron count remains fixed at 10. Representative values using are approximately 5.9 for neutral Ne, 4.9 for F-, and 3.9 for O2-. For inner 1s electrons in Ne, Zeff approaches the full nuclear charge of 10 due to minimal shielding. Orbital variations within the same atom arise from differences in radial distribution and penetration. For carbon, the 2s orbital experiences a higher Zeff of 3.22 compared to 3.14 for the 2p orbital, as s electrons penetrate closer to the nucleus and are less shielded. The Clementi-Raimondi HF values typically increase by about 0.8 to 1.0 per successive element in a period, accounting for the added proton outweighed partially by increased shielding from the new . Modern computations using (DFT) with relativistic corrections extend these estimates to heavier elements; for instance, Dirac-Fock calculations for yield a 4p1/2 Zeff of 11.98, reflecting enhanced penetration and relativistic contraction in valence shells.

Comparison with Nuclear Charge

The nuclear charge, denoted as ZZ, represents the total number of protons in the and is fully experienced by in hydrogen-like atoms or ions with no other present. In multi-electron atoms, the effective nuclear charge ZeffZ_\text{eff} is the net positive charge sensed by an electron after accounting for shielding by other , resulting in Zeff<ZZ_\text{eff} < Z for most orbitals. For the innermost 1s in He-like ions, such as He0^{0} or heavier analogs, Zeff≈ZZ_\text{eff} \approx Z because the single accompanying 1s electron provides negligible shielding near the nucleus. This reduction is particularly pronounced for valence electrons, where the ratio Zeff/Z<1Z_\text{eff}/Z < 1, indicating substantial screening by . For example, in sodium (Z = 11), the 3s experiences Zeff≈2.2Z_{\text{eff}} \approx 2.2, corresponding to approximately 80% screening of the nuclear charge by the inner 10 electrons. The effective nuclear charge exhibits radial dependence, Zeff(r)Z_{\text{eff}}(r), which arises from the varying degree of shielding at different distances from the nucleus. In –Fock calculations, Zeff(r)=−rV(r)Z_{\text{eff}}(r) = -r V(r), where V(r)V(r) is the total potential; near the nucleus (small rr), Zeff(r)→ZZ_{\text{eff}}(r) \to Z due to dominant nuclear attraction, while at large rr, it asymptotically approaches the valence ZeffZ_{\text{eff}} as outer electrons are screened by the entire core. This monotonic decrease from Z to the valence value is evident in self-consistent field potentials for atoms like or . These differences have key implications for : transitions involving , such as K-shell emission edges, probe regions near the nucleus and thus reflect the full nuclear charge with minimal shielding, whereas optical spectra from excitations sense the reduced valence ZeffZ_{\text{eff}}. For instance, in aluminum (Z = 13), L-shell electrons in spectra experience an effective charge close to Z due to poor shielding near the nucleus.

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