Abstract
We have previously proposed a simple empirical equation to reproduce the literature values of the ionization energies of one-electron to four-electron atomic ions with very good agreement. However, we used a potential energy approach in our equation, which has no theoretical basis. This paper discusses an alternative kinetic energy expression for one to five electrons with simple corrections for relativistic and Lamb shift effects and for two- to-five electron ions additional effects including electron relaxation and residual interactions. For calculated values of one-electron (hydrogen-like) and two electron (helium like) atomic ions, the difference with the literature values is typically 0.001% or less. Agreement with the literature values for three-, four-, and five-electron ions is 99% or better. First electron affinities calculated by our expression also agree fairly well with generally recommended values. These results show that there is strong evidence that our methodology can be developed to reliably predict, with fairly good accuracy, ionization energies of multielectron atomic ions that have not been measured.
1. Introduction
A knowledge of ionization energies is essential for understanding the chemistry of the elements and other fundamental concepts, such as lattice energies of inorganic solids.
With the development of quantum theory, the two-particle problem can be solved exactly and the kinetic energy of the electron in a hydrogen atom can be calculated using the Schrödinger equation. Since the Schrödinger equation does not take account of relativistic effects, Dirac [1] produced an equation which included a relativistic correction for the electron energy levels. However, Lamb and Retherford [2–4] in a series of experiments showed that there is a small shift in the energy levels of the hydrogen atom not accounted for by the Dirac equation. This energy shift is now commonly called the Lamb shift. Theoretical atomic energy levels were calculated from a nonrelativistic model, and then relativistic and quantum electrodynamic effects were accounted for by treating them as perturbation corrections.
General availability of powerful computers allowed highly complicated theoretical calculations of the energy levels and ionization energies of hydrogen [5] and helium-like ions [6] to be performed. These sophisticated equations for one- and two-electron atomic ions, which need complex computer routines to compute, include corrections for the variation of mass with velocity, reduced mass, mass polarization, and Lamb shift, and for two-electron ions, corrections for interactions between the two electrons. Values computed by these equations are quoted to cm−1 (wave number) level or small fractions of a cm−1. Updated theoretical ionization energies and Lamb shifts for one-electron [7] and two-electron [8] atomic ions have been published more recently and are generally accepted as very accurate. These recent results differ very little from the earlier computed values of Garcia and Mack [5] or Midtdal and Aashamar [6].
We have previously devised a simple empirical equation to reproduce literature values for the ionization energies of one-electron [9] and two-electron [10] atomic ions with very good agreement. More recently, we extended calculations to three- and four-electron isoelectronic ions [11], which reproduced literature values with good agreement. However, this equation is based on a potential energy approach (when the potential energy is maximum, the kinetic energy approaches zero, that is, zero velocity, and relativistic correction for the variation of mass with velocity is not required) and has no theoretical basis. In this paper we use a kinetic energy expression with a relativistic correction, which has theoretical justification to calculate the ionization energies of one- to five-electron atomic ions. Besides applying a relativistic and a Lamb shift correction, we have also made a few hypotheses on factors that influence the size of the ionization energy. To maintain our aim of simplicity and ease of computation, the equations only contain fundamental constants or values derived from fundamental constants and simple numbers.
2. Sources of Data
The extensive publications by Moore [12–15] contain detailed tables of atomic energy levels in reciprocal wavelengths or wave numbers (cm−1), ionization reciprocal wavelengths (in cm−1), and values converted from wave numbers to electron volts (eV) for atoms and atomic ions with estimated experimental errors and references to original work. This remains the most extensive survey of ionization energies. Recently, the CRC Handbook of Chemistry and Physics [16] presented extensive authoritative data of ionization energies with reference to Moore and other up-to-date works. Andersen et al. [17] provided an extensive compilation of first electron affinities (binding energies of single negative ions) based on both theory and experiment. The majority of compilations of ionization energy data are now available on the National Institute of Standards and Technology web site (http://www.nist.gov/srd/). These compilations include values of ionization energies that are accurately measured as well as crude estimates.
On closer examination of many of the original papers and compilations, we have found that, in general, the first few ionization energies are most reliably measured, experimental errors increase across the higher members of a series, and some results have errors of over one eV. For example, the ionization energy of (five-electron boron series) is . Hence, we consider that when comparisons are made, agreement with the first few members of a series is more relevant than with higher members of a series. There is no reason to doubt the reliability of the values published in the CRC Handbook of Chemistry and Physics of all fundamental constants and ionization energies. Values of all fundamental constants used for calculation in this work and ionization energies for comparison purposes are taken directly from it. Table 1 shows the symbols, elementary constants and conversion factors used in this work.
Ionization energies are presented in electron volts (eV) in the Handbook of Chemistry and Physics and eV is a more commonly used unit, so, as with our previous work, ionization energies are presented here in eV and rounded to four decimal places (please note that 0.0001 eV is equivalent to 0.8 cm−1.) We consider that this is sufficiently precise, since some literature values are quoted only to one or two decimal places or less because the levels of uncertainty/experimental error for some ionization energies can be of the order of an eV or more. For comparison purposes, values that are originally presented in cm−1 are converted to eV with the conversion factor k in Table 1. Results are presented up to atomic number 20 for the following two reasons. Firstly, uncertainties of the ionization energies of those elements with atomic number 21 or more (for an isoelectronic sequence) are usually much higher than the differences between the calculated values produced by our expression and literature values for elements with atomic number below 21. Secondly, our results are compared with ionization energies computed by Garcia and Mack [5] and Midtdal and Aashamar [6] who presented their results up to atomic number 20.
3. Contributions to the Electron Energy
The kinetic energy of an electron in an isolated atom or atomic ion is not simply because it is moving at high speed. When there is more than one electron in the system, it is logical to assume that the attractive energy between the nucleus and remaining electron(s) changes because of the change in screening experienced by the remaining electron(s) before and after ionization. The energy released by the electron relaxation or transition needs to be taken into account as well as the Lamb shift correction when calculating the ionization energy. There may be residual electron interactions because of the finite size of the electron. Hence, we assume that the ionization energy of an electron can be considered to contain the following components in a multielectron system: where is the ionization energy, is the kinetic energy term, is the Lamb shift term, is the relaxation/transition energy term, and represents residual electron interaction energy term. is the principal quantum number and is the reduced mass [18] (because the nucleus and electron being ionized, both revolve around the centre of mass). represent multiplied by , respectively, because ionization occurs in energy level , where is the principal quantum number. Values of ionization energy calculated by our expression are converted to eV by dividing them with the elementary electron charge (given in Table 1).
4. Kinetic Energy and the Relativistic Correction
The maximum velocity of the electron moving in a Bohr orbit [19] can be calculated from the relationship: where is the electron rest mass, is the velocity of the electron, stand for the charges of the electron and nucleus, is the permittivity of a vacuum, and , the Bohr radius, is . The velocity of the electron in the hydrogen atom can be calculated from (2) and is equal to m/sec. The velocity of the electron in successive atoms of the one-electron series increase by times, where is the atomic/proton number or . When there is more than one electron in the system, the velocity of the electron increases by , where is the screening constant for that electron. A common interpretation of relativity theory [20] shows the total energy of a particle of mass and velocity is and the kinetic energy of the particle is then In reality, only the first three terms (in (4)) are significant as shown in the expression, and all other terms are minute and can be ignored. In an equilibrium situation, the energy of the electron is exactly half potential and half kinetic. Assuming that an electron is ionized in an equilibrium condition, only half the relativistic correction should be included, that is, the relativistic correction for a one-electron atom is then For a hydrogen-like ion, there is no relaxation energy and no residual electron interactions, and the ionization energy is then represented by Columns 2, 3, 5, and 6 of Table 2 list the nonrelativistic kinetic energy, the relativistic correction calculated by (5), the Lamb shift, and the ionization energy as calculated by (6), respectively. However, as shown below, values calculated by (6) for one-electron atoms are too large when compared to generally accepted values.
We consider that, as theory of relativity points out, the mass of a moving particle is given by the expression , where is the rest mass of particle. Expansion of this expression gives Therefore, it follows that .
This expression for total kinetic energy can be simplified to: and the relativistic correction for a one electron atom at the equilibrium position is then The ionization energy of a one-electron atom is then Columns 4 and 7 of Table 2 list the relativistic corrections calculated by (9) and the ionization calculated by (10), respectively. As described below, when expression (9) is used, the ionization energies calculated have excellent agreement with values calculated by complex computer routines and other generally accepted literature values.
5. The Lamb Shift
The Lamb Shift is usually considered as a quantum electrodynamic effect (see e.g., Garcia and Mack [5] or Johnson and Soff [7]) and is computed by complex formulas.
We have made a number of assumptions (without theoretical justification) that the Lamb shift is a secondary relativistic effect. We assume that the Lamb shift is caused by a combination of effects related to mass and size and the whole system in motion. The energy reduction is approximated to the following expression: where is the electron mass, is the proton mass, where the factor for the reduced mass correction for hydrogen is . is the fine structure constant and is a crude approximation of the square root of the ratio of nuclear to atomic size for hydrogen. is the mass number of the atom. S is the screening constant and in a one-electron system S is zero. The size of the nucleus increases [21] roughly in proportion to . As shown in Table 3, the Lamb shift values calculated using this simple equation agree well with those computed by Johnson and Soff [7] for the one-electron system. For the two-electron system, the values calculated by (11) are higher than those computed by Midtdal and Aashamar [6], but they compare well with those computed by Drake [8], which are more recent and probably more accurate.
6. Electron Relaxation Energy
After an electron is ionized, one or more of the remaining electron(s) is/are attracted more closely to the nucleus. The attractive energy between the nucleus and remaining electron(s) changes because of the change in screening experienced by the remaining electron(s) before and after ionization. This relaxation or transition energy is proportional to where is the screening constant for remaining electron(s) after ionization and is the screening constant for the remaining electron(s) before ionization.
In the helium system, there are two electrons and both occupy the orbital. Since each electron occupies half of the space and each is repelled by only one other electron (here we have assumed that the two electrons act as in a two-particle problem and are equivalent), the screening constant is a half (0.5). After ionization, there is only one electron in the system and so there is zero screening.
In the lithium series, the electron that is ionized occupies a higher orbital and is shielded by two electrons, and the screening increases by 1 to 1.5. The extra screening experienced by the two inner electrons in the orbital increases to 0.625, which is an increase by or 0.125 rather than 0.5 because the third electron occupies a different orbital and in a different electron shell (i.e., 0.5 of 0.5 of 0.5). After ionization, only two electrons are left in the system and the screening reduces to just 0.5.
In the beryllium series, the electron that is ionized occupies the orbital. Since there are four electrons and each moving in an elliptical orbit, each may at any time interact differently with the other three electrons. We suppose that the screening of the fourth electron increases by a half to 2 and the other electron in the orbital experiences a screening of between 1.5 and 2 and is 1.75. After ionization of the fourth electron, the screening experienced by the third electron drops back to 1.5.
For the boron system, the outermost electron occupies a new orbital and the screening increases by 1 to 3. The screening of the fourth electron increases by 0.25 to 2.25 and after ionization the screening reduces back to 2. There is no reason to expect screening constants to be complicated numbers, and the screening constants used in the calculations are listed in Table 4.
7. Residual Electron Interaction
The reduced mass calculation implicitly assumes that the electron and nucleus are point charges, but the nucleus and the electron have a finite size [22]. We assume that when there is more than one electron in the atom/ion, the finite size of the electron generates two opposite and competing kinds of residual electron-electron interactions. The first type is residual electron-electron repulsion because the electrons are not point charges (have a finite size) and are slightly closer to each other. This increases the electron-electron repulsion and reduces the energy required to ionize the electron. However, this reduction diminishes very rapidly with increasing number of protons as the electrons become much more tightly bound to the nucleus.
This residual electron-electron repulsion is directly proportional to the number of electron-electron interactions and an inverse function of proton-electron interactions. The number of electron-electron interactions QI, before ionization, is 1, 3, and 6 when there are two, three, or four electrons and so on. The reduction in energy resulting from this interaction is approximated to , the electron radius, is is the Bohr radius , and is the number of electrons after ionization.
We assume that the second type of residual interaction, which is opposite to the first type and a competing type of electron-electron interaction, occurs when there are three or more electrons in the system before ionization. In a three-electron system, two electrons occupy the orbital and one occupies the 2s orbital. The finite size of the two electrons causes them to repel each other slightly more and tend to move further from each other. This asymmetric distribution of electrons causes the electron being screened slightly less from each of the electrons. The result is that it is attracted to the nucleus a bit more than expected and hence increases amount of energy required to remove it from the atom/ion. The small increase in energy is a function of the number of electron-electron interactions, there are because the more electron-electron interactions, the further they are apart and the bigger will be the reduction in screening of the nucleus from the ionizing electron. This effect also increases as the number of protons increases (i.e., attraction between the nucleus and outermost electron(s) increases), and for a system with three or more electrons it is approximately is the number of electron interactions after ionization and the total residual electron interaction energy change is
8. Ionization Energy of One-Electron Atoms
There is only one electron, n is 1, and the expression for calculating the ionization energy is Column 2 of Table 2 shows the nonrelativistic kinetic energy , for the purpose of comparison, column 3 shows the relativistic correction calculated by (5), column 4 shows the relativistic correction calculated by (9), column 5 shows the Lamb shift correction, column 6 shows the ionization energy calculated by expression (6), and column 7 shows the ionization energy calculated by expression (16) (which is equivalent to (10)).
The one-electron ionization energies calculated by (16) are shown in column 2 of Table 5. Ionization energies computed by Johnson and Soff [7] and Garcia and Mack [5] and values published in the CRC Handbook are shown in columns 3, 4, and 5, respectively. It is very clear that (9) is a much more appropriate expression than (5) to calculate the relativistic correction. Ionization energies calculated by expression (16) show exceptional agreement with values computed by complex formulas and with the latest accepted values. Compared to ionization energies published in the CRC Handbook, values calculated by (16) agree to 99.999% or differ by less than 0.001% (or <10 parts per million) in 18 out of the 20 values as shown in column 6 of Table 5, the biggest absolute difference is 0.0860 eV from a calculated value of 5469.95 eV.
9. Ionization Energy of Two-Electron Atoms
The ionization energy of a two-electron system is , where:
Since, for simplicity, we have not applied a relativistic correction to , we have made a crude approximation of reducing the relativistic correction in by 5% to 0.45 (rather than 0.5) and where is , so it is Columns 2, 3, 4, 5, and 6 of Table 6 show separately and the calculated ionization energies of two-electron atoms, respectively. Table 7 is a comparison of the ionization energies calculated in this paper and those computed by Drake [8], Midtdal, and Aashamar [6] and values published in the CRC Handbook. As shown in the table, there is very good agreement with the other computed results. When compared to values published in the CRC Handbook, all values except one (atomic number 8) agree to better than 99.99% (or difference of less than 0.01% as shown in column 6), and the largest absolute difference is 0.2161 eV from a calculated value of 2437.8641 eV.
10. Ionization Energy of Three, Four, and Five Electron Atoms
The ionization energy of a three- to-five electron system is where The screening constants used are shown in Table 4, and , the number of electron-electron interactions before ionization, is 3, 6, and 10, respectively, for three-, four-, and five-electron systems. Columns 2, 3, and 4 of Table 8 list the calculated ionization energies, literature values, and the percentage differences of three-electron isoelectronic systems. The calculated ionization energies of four- and five-electron isoelectronic sequences are shown in columns 2 and 5; the corresponding literature values in columns 3 and 6 and their differences are shown in columns 4 and 7, respectively, in Table 9. The agreement with literature values is 99% or better in all cases (99.9% or better for a third of all values).
11. First Electron Affinities
First electron affinities calculated by our expression and recommended values are listed in Table 10. Our calculated value of the first electron affinity of helium differs from the generally accepted value by less than 0.5% which is an excellent agreement. There is also fairly good agreement between our other calculated values and recommended values.
12. Discussion
Johnson and Soff [7] used an approximate empirical formula as part of the set of expressions to calculate the finite nuclear size correction. Drake [8] reported that uncertainties of calculated Lamb shifts in some cases can be up to 10%. Our expression (19) is fairly simple and can be calculated quite easily as compared to the complex equations, which required computer routines to compute.
We have only used the size of the (one-electron) hydrogen atom in the Lamb shift equation for one to five electrons. It is fairly remarkable that in the majority of cases agreement between our calculated values and the values computed by Johnson and Soff [7] and Drake [8] is 90% or better. The size of hydride ion (negative hydrogen ion) is different from the hydrogen atom, and obviously the sizes of the negative helium, lithium, and beryllium ions are as well. We have not applied any corrections to negative ions for the two- to five-electron systems, because the size of the hydride ion [23] is contentious and there is no accurate measurement of the size of the other negative ions. There are no reliable Lamb shift calculations for three- to five-electron systems, so a comparison with our results is not possible. Even though our assumptions and approximations are simple and straight forward, our results show excellent agreement with ionization energies published in the CRC Handbook as well as with Johnson and Soff [7] and Garcia and Mack [5] for one-electron and Midtdal and Aashamar [6] and Drake [8] for two-electron ions.
We have found that, as shown by up-to-date publications [24], Moore is still quoted as a major source of experimental information. For some isoelectronic sequences, only the first five or six members are reliably measured and there may be many members somewhere along the sequence without any reliably measured ionization energies, and in these cases values are obtained by semiempirical interpolation or extrapolation formulas [25]. Recent papers [26, 27] also quoted ionization energies of some members of many isoelectronic sequences estimated by Lotz [28] in 1967. We have recently demonstrated [29, 30] that ionization energies and ionization pathways of some elements or sequences are fairly complicated. Hence, there is scope to update the measurements and calculations of ionization energies of isoelectronic sequences.
Although some of our hypotheses may be controversial and there is no proof that our interpretation of the relativistic correction for mass change is correct, there is no doubt that they form part of expressions that produce very precise values of the ionization energies of one- to five-electron isoelectronic sequences. This is good evidence that both expression (19) or our alternative potential energy approach can be developed to predict, with good accuracy, ionization energies of multi-electron atomic ions that have not been reliably measured.