Atomic orbital self-energy and electronegativity
In this work, atomic calculations were performed within the local-density and generalized-gradient approximations of exchange and correlation density functionals within density-functional theory to provide accurate periodic trends of first ionization energies and electron affinities of the atomic series from hydrogen to xenon. Electronegativities were determined directly from Mulliken’s formula and were shown to be equivalently calculated rather by using Slater-Janak’s transition state or by calculating the electrostatic self-energies of the orbitals involved in the transition to ions. Finally, comparisons were made with other theoretical and experimental results, including Mulliken-Jaffé’s electronegativity scale.
Electronegativity is the measure of an atom, molecule or solid substance to attract electrons to itself. The first connection of the electronegativity () concept with quantum mechanics within density-functional theory (DFT) is assigned to Parr et al. Parr et al. (1978), where , where is the chemical potential in DFT Fiolhais et al. (2002a). Slater claimed that the total energy () of an atom was a continuous function of its orbital occupations Slater (1974). In a more general way one can describe the electronegativity of ions as
for an ion with atomic number and a variable number of electrons. If , then Eq. 1 is valid for the neutral atom. By writing as a parabolic function of Hinze et al. (1963), and choosing coefficients from atomic first ionization energy (I) and electron affinity (A), we reach the following:
which is precisely Mulliken’s electronegativity formula Mulliken (1934) with which I will work in this manuscript, with and .
Usually, as a first approximation to the calculation of excitation energies, we simply take the difference between the ground-state Kohn-Sham eigenvalues. However, the Kohn-Sham eigenvalues (and wave functions) do not have any physical interpretation, excepting those related to the highest occupied state, which is regarded as the negative of the ionization energy of the system
where and refer to atomic states which, for convenience, we label highest occupied atomic orbital (HOAO) and lowest unoccupied atomic orbital (LUAO). This means one has to be able to calculate excited states to calculate electronegativities. However, DFT does not account for calculating excited states, as is widely known and described in many references elsewhereFiolhais et al. (2002a); Perdew and Levy (1983); Sham and Schlüter (1983); Mori-Sánchez et al. (2008); Cohen et al. (2008). Therefore, the calculation of gap within DFT defined at a first approximation by (not considering spin-orbit effects) is not accurately determined within DFT, as the exchange-correlation energy is only roughly approximated Fiolhais et al. (2002a). In this work, theoretical results for periodic properties of ionization energy, electron affinity and electronegativity are presented using a method of fractional occupations of the orbitals involved in the transition from neutral atoms to ions (0.5 electron, for negative/positive ions) using different exchange-correlation functionals within density-functional theory. The numerical fact of the (piecewise) linearity of the eigenvalue as a function of its occupation (at least for the range tried, of 0.5 electron, for which it proved valid from H to Xe) is used to calculate self-energy of the orbital(s) involved in the transition, for each element, and those same properties were determined and compared with the direct ion calculation. Details are described in next section.
Ii Methods of calculation
Electronegativity calculations by means of fractional occupations is not novelty, and was performed successfully several times Manoli and Whitehead (1984); Robles and Bartolotti (1984); Bartolotti et al. (1980); Sen et al. (1981). To rewrite the electronegativity expression in terms of half-occupations, let us consider a system - let’s say, an atom - in its ground state to have a fully occupied valence orbital and unoccupied conduction orbital. If we write as the explicit dependence of the eigenvalue with the occupations at HOAO () and LUAO (), then we can label the formerly LUAO of the atom in its ground state, , that has just received half-electron. Similarly, will be the formerly occupied valence state of the ground state system, , that has been half-ionized. Assuming that the gap depends only on and orbitals occupations ( and ), we can expand the energy in Taylor series around the ground state until second order:
with second derivatives with respect to total energies being regarded as the hardness tensor Chermette (1998). Now, Janak proved that the variation of total energy in DFT, with respect to the orbital occupation is equal to the eigenvalue of that orbital Janak (1978),
in which is the total energy and is the occupation of the Kohn-Sham state with eigenvalue . Differentiating Eq. 6 a second time with respect to the orbital occupation, we obtain the relation , in which is regarded as being the (electrostatic) self-energy function for the state Ferreira et al. (2008, 2013, 2011) (v. Appendix for details on this derivation). Writing
with , we rewrite Eq. 5 as
where is the ground state energy. Using Janak’s theorem and differentiating Eq. 8 in relation to and we obtain
So, we can calculate for half-ionized HOAO state, , and the half-occupied orbital (formerly LUAO orbital), :
It is now possible to write and in terms of half-occupations for an atom with electrons as
Now, we can rewrite Eq. 4, always considering the linearity of with occupation Slater and Wood (1971); ?, a necessary condition to be satisfied by exchange-correlation functionalsPerdew et al. (1982):
where I labelled and from Eq. 10 as and , respectively. and can be calculated from the slope of the vs trend line.
It is essential bearing in mind that there can be four situations for and , which can be [a] and are orbitals of the same spin and same quantum numbers and (e.g. Si, B, C); [b] and are orbitals of different spins but same quantum numbers and (e.g. N, P, As); [c] and are orbitals of the same spin, but different quantum numbers and (e.g. Tc,Nb, Fe); and [d], and are orbitals of different spins and different quantum numbers and (e.g. Ar, Ti, Kr).
All the calculations were ”all-electron” and performed with the ATOM program, contained in SIESTA package Soler et al. (2002). The preferred exchange-correlation functional used was the GGA by Perdew, Burke and Ernzerhof (PB) Perdew et al. (1996). I encountered numerical convergence problems when trying to calculate electron affinities for some elements due to the prediction of unbound statesManoli and Whitehead (1984), specially when adding half-electrons to d-states. Here, differently from previously described by Robles and Bartolotti Robles and Bartolotti (1984), where they calculate indirectly through , I simply changed the exchange-correlation functional to the BLYP functional Becke (1988); ?, or the LDA by Perdew and Wang (PW) Perdew and Wang (1992), and calculated directly. In the cases it also failed, as a common practice I then extrapolated the fitted trade line to versus until Fiolhais et al. (2002b). Although ionization energies were easily calculated for all elements using the PBE functional, for the final electronegativities table, I preferred to choose the best fit result regardless the exchange-correlation functional. I will comment this in more details in Results section.
Iii Results and discussion
Ionization energies are presented in Tab. 1. Also indicated in Tab. 1 are the exchange-correlation functionals used for each element. It is worth mentioning that any single functional used (BLYP, PB and PW) provides very accurate results – e.g. an average percent error of less than 1.6% if only PB is used – but I decided to filter the best results out of the three functionals used for each element. Catches the eye the impressive agreement with experimental results, with an average error of just 0.9% (v. Fig. 1). Specially for heavier elements – and for 3d transition metals as well – relativistic corrections (excluding spin-orbit) improved the resultsRinaldo et al. (2008); Engel et al. (2001).
|Element||XC||nl||I||I (exp)WEB ()||(%)||(meV)|
|Element||XC||nl||A||A (exp)WEB ()||(%)||(meV)||Contrib. to EN (%)|
The sign convention for electron affinity was adopted as a positive energy value that indicates the spontaneous attachment of an electron to a single atomic orbital. Although first ionization energies are calculated with good accuracy using transition statesFerreira et al. (2011), electron affinity still poses challenges from theoretical point of view, specially for numerical calculations within DFTAndersen (2004); Sen and Jørgensen (1987); Lee et al. (2010); Galbraith and Schaefer (1996), which traditionally found restrictions of applicability to this type of system due to large self-interaction errors that causes outermost occupied states to become unbound, specially in small ionsShore et al. (1977); Lee et al. (2010); Galbraith and Schaefer (1996), thus making anions unstable upon electrons detachments as consequence of Koopman’s theorem Rienstra-Kiracofe et al. (2002). In any case, I followed the tradition of, in principle, ignoring DFT drawbacks to calculate negative ions, and the electron affinity results are presented in Table 2. This time we can notice the much larger percent errors along the list in column 7, with the average percent error in calculations of 11.4% in relation to experimental results. Some elements presented errors as large as 42% (Boron), or even 77% (Calcium). However, as the main objective in this work is to provide results for electronegativities, column 9 in Tab. 2 collects the contributions, to the electronegativity, of each individual electron affinity calculation. We can note that, for elements with the greatest electron affinity percent errors, like Ca, Mg and B, those electron affinity contribute little to the final electronegativities results (0.09%, 3.75% and 1,90%, respectively).
It is also important to mention the absolute errors in electron-volts. Typically, the LDA (spin-polarized) functional provides electron affinity results with absolute errors of the order of 0.3 to 1 eV (v. Ref. Rienstra-Kiracofe et al. (2002) and references therein). Comparing to present work, the average absolute error if merging the best results of each exchange-correlation functional is of only 0.07 eV, a considerable improvement. The average contribution of the electron affinity to the electronegativity is around 8.2%, which indeed shows that the electron affinity value is not so important in final electronegativity value in Mulliken’s formulation. Figures 2, 3 and 4 compile the results of Tab. 2. Interestingly, in Fig. 4 we can notice an increase in percent contribution of to from left to right in periodic table, with peaks in F, Cl, Br and I. To summarize, sorting the best results among the exchange-correlation functionals used revealed the best strategy for DFT-based calculations, specially for electron affinity results, that can only be improved upon better exchange-correlation functionals, in which correlations play an important role Fiolhais et al. (2002b).
But the point here is not to provide the most accurate , and calculations, but to demonstrate that the transition state technique is still powerful in face of the modern exchange-correlation functionals, and that there is a connection of it with self-energies that could facilitate the numerical calculation and predict the correct trends for those atomic properties. Final electronegativities are grouped in Tab. 3, together with some other DFT electronegativity scales like older results of Robles and BartolottiRobles and Bartolotti (1984); Bartolotti et al. (1980) obtained by the and spin-polarized techniques, the Mulliken-Jaffé EN scale and the experimental results from direct applying (v. Tabs. 1 and 2).
|Symbol||XalphaBartolotti et al. (1980)||Xalpha spinRobles and Bartolotti (1984)||Mulliken-JafféPutz et al. (2005)||Experimentala||Present work|
Figure 5 summarizes the comparisons with calculations and experiments. The agreement with experimental results represented a significant improvement, with an average percent error of only 1.2%, compared to 16.3% and 22.3% of and spin-polarized , respectively. Also, there is a clear improvement in describing periodic trends from K to Zn, including the 3d transition metals, and from Rb to Cd, including 4d transition metals.
The calculation of atomic orbital electrostatic self-energy is straightforward since the linearity condition of is valid to the extent of electron, for the PW, PB and BLYP exchange-correlation functionals, for all elements from H (Z=1) to Xe (Z=54). Table 4 contains the results of self-energy calculations of a number of elements, obtained using Eq. 7. Electronegativity for some elements like Tc, Mn, Mg, Ne and Cd can be calculated with accurate results just by the first ionization energy, because of lack of numerical convergence and proven negative ion instability Andersen (2004). Apart from little numerical differences when considering the linear fit in different regions of the curvePerdew et al. (1982) (and as Eq. 10 is an approximation to second order), self-energy-derived electronegativities results present excellent agreement with Tab. 3.
|Element||a||a||I||in I (%)||A||in A (%)||EN b||in EN (%)c|
In summary, I calculated first ionization energies, electron affinities and electronegativities for elements ranging from H to Xe using transition state techniques within density-functional theory using different exchange-correlation functionals. This method showed extremely accurate, with an average percent error of only 1.2% compared to experimental results. Numerical convergence was easily achieved for all calculations and most calculations as well, and alternatives were used to overcome convergence problems. A direct connection of electronegativity and atomic orbital self-energy was tested and used to endorse preliminary results, which showed excellent agreement with experimental ones, even without considering spin-orbit effects.
I believe this method can be employed with other computational techniques that include spin-orbit effects, like Weizmann theory Martin and de Oliveira (1999).
Calculating the second derivative of (Eq. 6),
Calculating the partial derivative of with relation to ,
But, how to calculate ? Let and , we reach
But, from Hellmann-Feynman theorem, , and then
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