PRL 118, 026401 (2017)

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PHYSICAL REVIEW LETTERS

Absolute Binding Energies of Core Levels in Solids from First Principles
Taisuke Ozaki and Chi-Cheng Lee
Institute for Solid State Physics, The University of Tokyo, Kashiwa 277-8581, Japan(Received 19 July 2016; published 9 January 2017)
A general method is presented to calculate absolute binding energies of core levels in metals andinsulators, based on a penalty functional and an exact Coulomb cutoff method in the framework of density
functional theory. The spurious interaction of core holes between supercells is avoided by the exact
Coulomb cutoff method, while the variational penalty functional enables us to treat multiple splittings due
to chemical shift, spin-orbit coupling, and exchange interaction on equal footing, both of which are notaccessible by previous methods. It is demonstrated that the absolute binding energies of core levels for both
metals and insulators are calculated by the proposed method in a mean absolute (relative) error of 0.4 eV(0.16%) for eight cases compared to experimental values measured with x-ray photoemission spectroscopy
within a generalized gradient approximation to the exchange-correlation functional.
DOI: 10.1103/PhysRevLett.118.026401

Since the pioneering work of Siegbahn and co-workers[1,2], x-ray photoelectron spectroscopy (XPS) has become
one of the most important and widely used techniques instudying chemical composition and electronic states in thevicinity of the surface of materials [3]. Modern advances
combined with synchrotron radiation further extend its
usefulness to enable a wide variety of analyses such as corelevel vibrational fine structure [4], magnetic circulardichroism [5], spin-resolved XPS [6], and photoelectron
holography [7]. The basic physics behind the still advancing XPS measurements dates back to the first interpretation
for the photoelectric effect by Einstein [8]. An incident
x-ray photon excites a core electron in a bulk, and theexcited electron with a kinetic energy is emitted from thesurface to vacuum. The binding energy of the core level inthe bulk can be obtained by measuring the kinetic energy[1,2]. Theoretically, the calculation of the binding energy
involving evaluation of the total energies for the initial andfinal states is still a challenging issue especially forinsulators, since after the emission of the photoelectron
the system is not periodic anymore and ionized due to thecreation of the core hole. The violation of the periodicity
hampers the use of conventional electronic structure methods under a periodic boundary condition, and the Coulomb
potential of the ionized bulk cannot be treated under anassumption of the periodicity due to the Coulombic
divergence. One way to avoid the Coulombic divergenceis to neutralize the final state with a core hole by adding an
excess electron into conduction bands [9–13] or to approximate the bulk by a cluster model [14]. However, the charge
compensation may not occur in insulators because of theshort escape time of the photoelectron (∼10−16 sec) [15],while the treatment might be justified for metals. Even if we
employ the charge compensation scheme, the screened corehole pseudopotential which has been widely used inpseudopotential methods allows us to calculate only thechemical shift of binding energies, but not the absolute
0031-9007=17=118(2)=026401(5)values [9]. In spite of the long history of XPS and its
importance in materials science, a general method has notbeen developed so far to calculate the absolute binding
energies for both insulators and metals, including multiple
splittings due to chemical shift, spin-orbit coupling, andexchange interaction, on equal footing [16]. In this Letter
we propose a general method to calculate absolute binding
energies of core levels in metals and insulators, allowing
treatment of all the issues mentioned above and directcomparison to experimental results, in a single framework
within the density functional theory (DFT) [19,20].
Let us start by defining the absolute binding energy Eb ofcore electrons in bulks measured by a XPS experiment,
based on the energy conservation. The energy of the initialstate is given by the sum of the total energy Ei ðNÞ of theground state of N electrons and an energy hν of a
monochromatic x-ray photon. On the other hand, theenergy of the final state is contributed by the total energy
Ef ðN − 1Þ of the excited state of N − 1 electrons with acore hole and the kinetic energy K spec of photoelectron
placed at the vacuum level V spec , as shown in Fig. 1.
Therefore, the energy conservation in the XPS measurement is expressed byEi ðNÞ þ hν ¼ Ef ðN − 1Þ þ V spec þ K spec :ð1ÞNoting that the chemical potential of the sample is alignedwith that of the spectrometer μ by Ohmic contact, and thatthe vacuum level of the spectrometer V spec is given by
V spec ¼ μ þ φspec using the work function of the spectrometer φspec, Eq. (1) reads ashν − K spec − φspec ¼ Ef ðN − 1Þ − Ei ðNÞ þ μ:

ð2ÞðbulkÞ

The left-hand side of Eq. (2) is the binding energy Ebmeasured by the XPS experiment [21]. The right-hand sideof Eq. (2) provides a useful expression to calculate the026401-1© 2017 American Physical Society

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surface integration is performed over the Fermi surface. By
inserting the above equation into Eq. (3), we obtain thefollowing formula:ðmetalÞ

Eb

ϕ

ϕ
ν

FIG. 1. Schematic energy diagram for a sample and a spectrometer in the XPS measurement.
ðbulkÞ

absolute binding energy Eb
for bulks regardless of theband gap of materials. Instead of using the experimental
chemical potential μ, it is possible to rewrite the totalenergies with the intrinsic chemical potential μ0 , where
intrinsic means a state that is free from the control ofchemical potential, by noting that what the shift of thechemical potential μ ¼ μ0 þ Δμ does is only the constantshift of potential Δμ. Then, we rewrite them as Ei ðNÞ ¼
ð0Þ
ð0Þ
Ei ðNÞ þ NΔμ and Ef ðN − 1Þ ¼ Ef ðN − 1Þ þ ðN − 1ÞΔμ
ð0Þ
ð0Þ
using the intrinsic total energies Ei ðNÞ and Ef ðN − 1Þby assuming the common chemical potential μ for both theinitial and final states due to a very large N. Inserting theseequations into Eq. (2) yieldsðbulkÞ

Eb

ð0Þ

ð0Þ

¼ Ef ðN − 1Þ − Ei ðNÞ þ μ0 :ð3ÞEquation (3) is an important consequence, since only
quantities that can be calculated from first principles are
involved. Thereby, we use Eq. (3) to calculate the absoluteðbulkÞ
binding energy Eb . Note that the chemical potential μ0
is calculated by assuming the Fermi distribution at a finite
electronic temperature. It should be emphasized that Eq. (3)is valid even for semiconductors and insulators. In an
arbitrary gapped system, the common chemical potential μis pinned at either the top of the valence band or the bottomof the conduction band, or located in between them. For all
the possible cases, exactly the same discussion above isvalid. Thus, we conclude that Eq. (3) is a general formula toðbulkÞ
calculate the absolute binding energy Ebof solids.
Especially for metals, Eq. (3) can be further reorganized bynoting a rigorous relation derived with the Janak theorem
R
ð0Þ
ð0Þ
ð0Þ
[23]: Ef ðN − 1Þ − Ef ðNÞ ¼ dn∂Ef =∂n ¼ −μ0 , where
n is an occupation number of a one-particle eigenstate on
the Fermi surface, dn ¼ −ds=S is defined with the area of
the Fermi surface S and an infinitesimal area ds, and the

ð0Þ

ð0Þ

¼ Ef ðNÞ − Ei ðNÞ;

ð4Þwhich allows us to employ the total energy of theð0Þ
neutralized final state Ef ðNÞ instead of that of the ionizedstate. For metals, Eqs. (3) and (4) should result in an
equivalent binding energy in principle; however, the convergence is different from each other as a function of thesystem size, as shown later on.
ð0Þ
Since Ei ðNÞ and μ0 in Eq. (3) can be calculated by aconventional approach with the periodic boundary condition, we now turn to discuss a method of calculatingð0Þ
Ef ðN − 1Þ in Eq. (3) based on the total energy calculation
including many-body effects. Core electrons for which acore hole is created are explicitly included in the calculations to treat multiple splittings due to chemical shift,spin-orbit coupling, and exchange interaction between coreand spin-polarized valence electrons, and to take account of
many-body screening effects. The creation of the core holecan be realized by expressing the total energy of the final
state by the sum of a conventional total energy EDFT within
DFT and a penalty functional Epen asð0Þ

Ef ðN − 1Þ ¼ EDFT þ Epen ;ð5Þwith the definition of Epen :Z
X ðkÞ ðkÞ1ðkÞ
dk3 f μ hψ μ jP̂jψ μ i;
ð6ÞEpen ¼
VB B
μR
where B dk3 is the integration over the first Brillouin
ðkÞ

ðkÞ

M
P̂ ≡ jRΦM
J iΔhRΦJ j;

ð7Þ

zone whose volume is V B , f μ the Fermi function, and ψ μ
the Kohn-Sham wave function of a two-component spinor.
In Eq. (6) the projector P̂ is defined with an angulareigenfunction Φ of the Dirac equation under a sphericalpotential and a radial eigenfunction R obtained by anatomic DFT calculation for the Dirac equation aswith for J ¼ l þ 12 and M ¼ m þ 12:
jΦM
J i¼





l þ m þ 1 1=2 m
l − m 1=2 mþ1
jY l αi þ
jY l βi;2l þ 12l þ 1

ð8Þ

and for J ¼ l − 12 and M ¼ m − 12:
jΦM
J i¼





l − m þ 1 1=2 m−1l þ m 1=2 m
jY l αi −jY l βi;2l þ 12l þ 1ð9Þ

where Y is a spherical harmonic function and α and β are spinbasis functions. The variational treatment of Eq. (5) withrespect to ψ leads to the following Kohn-Sham equation:

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PRL 118, 026401 (2017)
ðkÞ

ðkÞ

ðkÞ

ðT̂ þ veff þ P̂Þjψ μ i ¼ εμ jψ μ i;

ðPÞ

ð10Þ

ðNPÞ

V H ðrÞ ¼ V H ðrÞ þ V H ðrÞ;

ð11Þ

ðPÞ

where V H ðrÞ is the periodic Hartree potential calculatedusing the periodic part ρi ðrÞ via a conventional method using
a fast Fourier transform for the Poisson equation. On the otherðNPÞ
hand, the nonperiodic Hartree potential V H ðrÞ is calculatedusing ΔρðrÞ and an exact Coulomb cutoff method byXðNPÞ~
V H ðrÞ ¼
ð12Þ
ΔρðGÞ~
vðGÞeiG·r ;G~where ΔρðGÞ
is the discrete Fourier transform of ΔρðrÞ andv~ ðGÞ is given by ð4π=G2 Þ½1 − cosðGRc Þ, which is theFourier transform of a cutoff Coulomb potential with thecutoff radius of Rc [24]. If ΔρðrÞ is localized within a sphereof a radius R, as shown in Fig. 2(b), the extent of the Coulombinteraction is 2R at most in the sphere, which leads to
Rc ¼ 2R. In addition, a condition 4R < L should be satisfied
to avoid the spurious interaction between the core holes. In
practice, we set Rc ¼ 12 L, and investigate the convergence ofthe binding energy as a function of L. With the treatment thecore hole is electrostatically isolated from the other periodic
images of the core hole even under the periodic boundary
condition.
We implemented the method in a DFT software package
OPENMX [25], which is based on norm-conserving relativistic pseudopotentials [26,27] and pseudoatomic basis

3

Cubic BN (N−1s) by Eq. (3)Bulk NH3 (N−1s) by Eq. (3)Diamond (C−1s) by Eq. (3)Si (Si−2p) by Eq. (3)Bulk NH3 (N−1s) by Eq. (4)2Relative to the most converged value (eV)where T̂ is the kinetic operator and veff the conventional
Kohn-Sham effective potential originated from EDFT . If alarge number (100 Ry was used in this study) is assigned
for Δ in Eq. (7), the targeted core state ΦM
J specified by
the quantum numbers J and M is penalized through the
projector P̂ in Eq. (10), and becomes unoccupied, resulting
in the creation of a core hole for the targeted state. Since thecreation of the core hole is self-consistently performed, thescreening effects by both core and valence electrons, spinorbit coupling, and exchange interaction are naturally taken
into account in a single framework. It is also straightforward
to reduce the projector P̂ to the nonrelativistic treatment.
After the creation of the core hole, the final state has one
less electron, leading to charging of the system. In the periodic
boundary condition, a charged system cannot be treatedin general because of the Coulombic divergence. The
neutralization of the final state may occur in a metal, andtheoretically such a neutralization can be justified as shown by
Eq. (4). However, it is unlikely that such a charge compensation takes place in an insulator during the escape time of thephotoelectron (∼10−16 sec) [15]. To overcome the difficulty,
we propose a general method of treating the charged statebased on an exact Coulomb cutoff method [24]. It isconsidered that the created core hole is isolated in the sample,
resulting in violation of the periodicity of the system. The
isolation of the core hole can be treated by dividing thecharge density ρf ðrÞ for the final state into a periodic partρi ðrÞ and a nonperiodic part ΔρðrÞ½≡ρf ðrÞ − ρi ðrÞ, which,when integrated over the unit cell, is exactly −1, where ρi ðrÞ isthe charge density for the initial state without the core hole.
Then, as shown in Fig. 2(a), the Hartree potential V H ðrÞ inthe final state is given by
(a)

(b)

(a)

1
0
0

5

10

15

20

25

Graphene (C−1s) by Eq. (3)TiN (N−1s) by Eq. (3)TiC (C−1s) by Eq. (3)Graphene (C−1s) by Eq. (4)TiN (N−1s) by Eq. (4)TiC (C−1s) by Eq. (4)

3
2

30

(b)

1
0
0

10

20

30

40

Intercore hole distance (Å)

FIG. 2. (a) Treatment of the Hartree potential in a system with acore hole under the periodic boundary condition. (b) Configuration to calculate the nonperiodic part of the Hartree potentialðNPÞV H by the exact Coulomb cutoff method for Δρ.

FIG. 3. Calculated binding energies, relative to the most converged value, of (a) gapped systems and (b) a semimetal (graphene)
and metals as a function of intercore hole distance. The referencebinding energies in (a) and (b) were calculated by Eqs. (3) and (4),respectively, for the largest unit cell for each system.

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(a)

(b)

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FIG. 4. (a) Difference charge density Δρ insilicon, induced by the creation of a core hole inthe 2p states, where the unit cell contains 1000
atoms and the intercore hole distance is¯
27.15 Å. (b) Radial distribution of 4πr2 Δρ,¯ is a spherically averaged Δρ. (c) Rawhere ΔρðNPÞdial distribution of V̄ H which is a sphericallyðNPÞ
averaged V H .

(c)functions [28]. A generalized gradient approximation [29]to the exchange-correlation functional and an electronic
temperature of 300 K were used. The details of theimplementation are given in the Supplemental Material[30]. All the molecular and crystal structures used in the
study were taken from experimental ones. Figures 3(a) and3(b) show relative binding energies of core levels in gappedsystems and metals including a semimetal (graphene),
respectively, as a function of intercore hole distance. For
the gapped systems the convergent results are obtained at
the intercore hole distance of ∼15, 20, and 27 Å for cubicboron nitride (diamond), bulk NH3 , and silicon, respectively. This implies that the difference charge ΔρðrÞ
induced by the creation of the core hole is localized withina sphere with a radius of R ¼ L=4, e.g., ∼7 Å for silicon. In
fact, the localization of ΔρðrÞ in silicon can be confirmedby the distribution in real space and the radial distributionof a spherically averaged Δρ, as shown in Figs. 4(a)and 4(b). The deficiency of the electron around 0.3 Åcorresponding to the core hole in the 2p states is compensated by an increase of electron density around 1 Å, whichis the screening on the same silicon atom for the core hole.
As a result of the short-range screening, the nonperiodic
ðNPÞHartree potential V H ðrÞ deviates largely from −1=r, as
TABLE I.

Calculated binding energy of a core level in bulks.

Material

State

Gapped system
c-BN
N-1s
Bulk NH3
N-1s
Diamond
C-1s
Si
Si-2p1=2Si
Si-2p3=2
Semimetal or metal
Graphene
C-1s
TiN
N-1s
TiC
C-1s
a

Ref. [40].
Ref. [41].
Ref. [42] (graphite).
dRef. [43].
bcCalculation (eV) Experiment (eV)398.87398.92286.50100.1399.40

398.1a399.0b285.6c99.8a
99.2a

284.23396.43281.43284.4c397.1d281.5ashown in Fig. 4(c). In Fig. 3(a) it is also shown that thebinding energy of the bulk NH3 calculated with Eq. (4)converges at a value which is larger than that with Eq. (3)by 1.2 eV, implying that Eq. (4) cannot be applied to
gapped systems. On the other hand, for the metallic cases
we see that Eq. (4) provides a much faster convergence thanEq. (3), and both Eqs. (3) and (4) seem to give a practicallyequivalent binding energy, while the results calculated with
Eq. (3) for TiN and TiC do not reach the sufficientconvergence due to computational limitation [39].
Therefore, Eq. (4) is considered to be the choice for thepractical calculation of a metallic system because of thefaster convergence. By compiling the size of the unit cellachieving the convergence into the number of atoms in theunit cell, the use of a supercell including ∼500 and 64
atoms for gapped and metallic systems in three dimensions
might be a practical guideline for achieving a sufficientconvergence by using Eqs. (3) and (4), respectively. The
calculated values of binding energies are well comparedwith the experimental absolute values as shown in Table I
for both the gapped and metallic systems, and the mean
absolute (relative) error is found to be 0.4 eV (0.16%) forthe eight cases. We see that the splitting due to spin-orbit
coupling in the silicon 2p states is well reproduced. In
addition, binding energies of a core level for gaseous
molecules are shown in the Supplemental Material [30],where the mean absolute (relative) error is found to be
0.5 eV (0.22%) for the 23 cases.
In summary, we proposed a general method to calculate
absolute binding energies of core levels in metals andinsulators in the framework of DFT. The method is basedon a penalty functional and an exact Coulomb cutoff
method. The former allows us to calculate multiple splittings due to chemical shift, spin-orbit coupling, andexchange interaction, while the latter enables us to treata charged system with a core hole under the periodic
boundary condition. It was also shown that especially formetals Eq. (4) involving the neutralized final state isequivalent to Eq. (3) involving the ionized final state,
and that Eq. (4) is computationally more efficient thanEq. (3). The remarkable agreement with the absolutebinding energies measured in XPS demonstrates the026401-4

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validity of the proposed method for a variety of materials.
For a better description of a case where an exchangeinteraction plays a dominant role in the splitting, a good
approximation to the exchange-correlation functional
should be adopted and our method provides a natural
way to examine the resulting total energies, while a
possible error by the pseudopotentials and the dependencyof chemical potential on surface structures should also be
addressed in future work. Considering the importance of
the XPS measurement in materials researches, the proposed
method is anticipated to play an indispensable role inquantitatively analyzing absolute binding energies of corelevels in solids.
We would like to thank Y. Yamada-Takamura and J.
Yoshinobu for helpful discussions on the XPS measurements. This work was supported by Priority Issue (creationof new functional devices and high-performance materials
to support next-generation industries) to be tackled byusing Post ‘K’ Computer, MEXT, Japan.

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