THE JOURNAL OF CHEMICAL PHYSICS 127, 174110 共2007兲
Core ionization potentials from self-interaction corrected Kohn-Sham
orbital energies
Guangde Tu
Institute of Chemical-Physical Processes, Area of Research CNR, via Moruzzi 1, 56124 Pisa, Italy and
Laboratory of Theoretical Chemistry, Royal Institute of Technology, SE-106 91 Stockholm, Sweden
Vincenzo Carravetta
Institute of Chemical-Physical Processes, Area of Research CNR, via Moruzzi 1, 56124 Pisa, Italy
Olav Vahtras and Hans Ågrena兲
Laboratory of Theoretical Chemistry, Royal Institute of Technology, SE-106 91 Stockholm, Sweden
共Received 29 May 2007; accepted 6 August 2007; published online 7 November 2007兲
We propose a simple self-interaction correction to Kohn-Sham orbital energies in order to apply
ground state Kohn-Sham density functional theory to accurate predictions of core electron binding
energies and chemical shifts. The proposition is explored through a series of calculations of organic
compounds of different sizes and types. Comparison is made versus experiment and the
“⌬Kohn-Sham” method employing separate state optimizations of the ground and core hole states,
with the use of the B3LYP functional and different basis sets. A parameter ␣ is introduced for a best
fitting of computed and experimental ionization potentials. It is found that internal parametrizations
in terms of basis set expansions can be well controlled. With a unique ␣ = 0.72 and basis set larger
than 6-31G, the core ionization energies 共IPs兲 of the self-interaction corrected Kohn-Sham
calculations fit quite well to the experimental values. Hence, self-interaction corrected Kohn-Sham
calculations seem to provide a promising tool for core IPs that combines accuracy and efficiency.
© 2007 American Institute of Physics. 关DOI: 10.1063/1.2777141兴
I. INTRODUCTION
Removing one electron from the inner shells of a molecule leads to ionized states having energies of hundreds of
eV above the ground state energy level. Such states, the socalled core-ionized states, have lifetimes of the order of femtoseconds and can be considered as quasistationary states
embedded in the electronic continuum. Core-ionized and
core-excited states have been and still are the subject of a
large number of investigations based on the use of electron
or x-ray beams as excitation agents. In fact, the chemical
shift of core ionization potentials has constituted one of the
most intriguing aspects of photoelectron spectroscopy and
has formed the basis for the notion of ESCA, Electron Spectroscopy for Chemical Analysis.1,2
Today, synchrotron radiation spectroscopies are increasingly applied to study core excitation of molecules either in
gas phase or adsorbed on surfaces, thanks to the recent technological improvements in the quality of x-ray sources and
of photoelectron detectors, which exalt the main characteristics of chemical selectivity and orientational selectivity. By
high resolution x-ray photoemission spectroscopy 共XPS兲, it
is possible, nowadays, to determine electronic structure, vibrational properties, and geometries of both adsorbed and
isolated molecules, and even of different conformers of a
molecule,3 with high accuracy.
The experimental investigations may supply information
a兲
Author to whom correspondence should be addressed. Electronic mail:
[email protected]
0021-9606/2007/127共17兲/174110/11/$23.00
on molecular structure and properties more efficiently when
they are combined with computational modeling of the core
ionization/excitation processes. A limited number of quantum chemical methods commonly employed for the description of the lowest excited states of a molecule have been
adapted, by appropriate constraints on the wave function
variational space, to the study of such “superexcited” states.
The application of the most advanced computational methods presently available, either based on wave function or
propagator techniques, is, however, quite difficult or simply
impossible in practice due to intrinsic and computational
problems and are limited to systems with a small number of
electrons. For large molecular systems the presence of a
complex vibrational structure in the XPS spectra makes the
basic knowledge of vertical ionization potentials obtained by
quantum chemical calculations very useful for the interpretation of the spectra in terms of electronic structure. The
calculations must be able to reasonably predict the chemical
shift of the core ionization potential 共IP兲 for the chemically
nonequivalent atoms of the same kind that, in a large molecule as a biomolecule, may be present in high numbers.
Most of such calculations have been performed in the independent particle approximation using the ⌬SCF method, in
which two separate Hartree-Fock 共HF兲 calculations performed by the self-consistent field 共SCF兲 approach are run
for the ground state and for the core-hole state corresponding
to a specific ionization site. The relatively great success of
the ⌬SCF approach in predicting core IPs derives essentially
from its capability to correctly describe the electron relaxation occurring in the molecule after ionization that screens
127, 174110-1
© 2007 American Institute of Physics
Downloaded 23 Jan 2008 to 130.237.79.166. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/jcp/copyright.jsp
174110-2
J. Chem. Phys. 127, 174110 共2007兲
Tu et al.
the core hole. This phenomenon is the most relevant one
from an energetic point of view and contributes to the total
energy of the ionic state by tens of eV at the K edge of the
light second row atoms. In fact, electron correlation effects,
not considered in the independent particle approximation,
contribute to the core IPs with a term that is one order of
magnitude smaller than the relaxation energy and, for light
atoms, relativistic effect are even smaller.4 The error of the
⌬SCF approach in predicting core IPs may be estimated below 1 – 2 eV, and the chemical shifts between IPs of identical
elements at different chemical positions are generally more
accurate than that, as a consequence, of effective cancelations of the correlation energies.
Despite the relative simplicity of SCF calculations that
can be easily run nowadays by efficient direct algorithms
with good scaling of computational time versus the number
of electrons, the ⌬SCF approach requires a number of separate iterative calculations equal to the number of nonequivalent ionization sites, that, as already observed, may be a quite
large number in an extended molecular system. First order
predictions of the core IPs of a molecule could be obtained
by making use of Koopman’s theorem, which allows estimating the ionization thresholds of the different electronic shells
as the negative of the orbital energies of the SCF ground
state. This is certainly an efficient way from a computational
point of view, requiring only a single SCF calculation, but
the “frozen” IPs obtained by Koopman’s theorem are affected by the large error due to neglecting the relaxation
energy and may predict a wrong order of the chemical shifts.
This can be considered in the light of the perhaps most popular and useful quantum chemical methods, including correlation effects, for computing molecular properties, namely,
density functional theory 共DFT兲 as implemented by the
Kohn-Sham 共KS兲 approach. DFT requires the iterative solution of differential equations similar to that of the SCF
method and which are solved in a similar way by projection
on a basis set. Despite that DFT is essentially a ground state
theory for the total energy of the system, the similarity of the
KS and SCF equations suggests to attach a physical meaning
to the KS orbitals and orbital energies, rather than that they
are just considered as mathematical quantities in the solution
of the DFT problem. It was early recognized, in fact, that
Koopman’s theorem is not valid for the KS orbital energies
and that the experience shows that while the 共negative兲 orbital energies of the valence shell may be, in practice, somehow close to the experimental IPs,5 the deviation for the core
orbitals is usually quite large. The pioneering work of Perdew and Zunger6 demonstrated that the origin of this large
deviation for core shells can be traced to an error due to the
spurious self-interaction for a singly occupied orbital that is
much enhanced for strongly localized orbitals such as core
orbitals and that therefore undermines the results of the
DFT-KS model in different practical applications. It was also
shown that by correcting for the self-interaction error 共SIE兲
in some appropriate way, the corrected KS orbital energies
will account not only for the electron correlation effects, as it
could be hoped from the character of DFT, but also for the
electron relaxation effects.
The main purpose of the present paper is to investigate
the possibility of correcting the KS orbital energies of the
core shells of a molecule by an appropriate self-interaction
correction 共SIC兲 that may allow an estimation of core IPs
with good accuracy from DFT calculations for the ground
state of a molecule. This approach, requiring a single calculation to reproduce the XPS spectra at different K edges,
would be particularly efficient for applications to large molecular systems. In Sec. II we briefly present our proposal for
a self-interaction correction of the core KS orbital energies
that has been tested on a large set of molecules of different
dimensions and chemical structures. The results are presented and discussed in Sec. III.
II. METHOD
A. DFT and KS orbital energies
DFT is nowadays one of the most useful quantum
chemical tools for the study of molecular electronic structures. Compared to ab initio quantum chemical methods such
as HF, configuration interaction, and coupled cluster, DFT
may combine the high accuracy of correlated methods to the
low computational cost of independent particle methods.
Thus, DFT is the most competitive tool for applications to
large molecules. In the last decades, DFT has been successfully applied in several different fields: simulations of chemical reactions, prediction of molecular properties, and general
descriptions of electronic molecular structure. However, the
method fails, for instance, in some applications to chemical
reactivity: the dissociation processes of order-electron tworesidue systems are not properly described, the reaction barriers of some reactions are too low, and the intermolecular
interactions are not properly described for some charge transfer systems. These problems are mainly caused6 by the selfinteraction error in the DFT methods 共SIE兲, which is the
spurious residue interaction of an electron with itself, that is
present in standard DFT-KS implementations. In a molecular
dissociation, the fragments may easily have singly occupied
orbitals, which may vary in localization. The SIE is very
dependent on the orbital localization, thus, SIE affects by a
different extension initial and final products of the reaction.
For what concerns the subject of the present investigation on
the ionization of core orbitals, the localization of the orbital
does not change substantially in the initial and final states,
but the occupancy of the orbital is suddenly changed by the
removal of one electron.
Perdew and Zunger proposed6 a self-interaction correction 共PZ-SIC兲 for the SIE in the total energy obtained by
using the local density approximation functional. By such a
procedure, both the total electronic energy and the orbital
energies are sensibly improved. However, the total electronic
energy depends on the orbital rotation between occupied orbitals in this PZ-SIC method. Thus, the usual self-consistent
field technique implemented in DFT-KS is not applicable for
PZ-SIC DFT. A simple approximation is offered by the
post-KS PZ-SIC DFT method, which consists in applying
PZ-SIC as a first order perturbative correction after a conventional DFT-KS calculation.
Fermi and Amaldi proposed an averaged density SIC
共ADSIC兲 DFT method,7,8 which uses ␳occ / Nocc to replace the
Downloaded 23 Jan 2008 to 130.237.79.166. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/jcp/copyright.jsp
174110-3
J. Chem. Phys. 127, 174110 共2007兲
Core ionization
single orbital density in the PZ-SIC expression. By this approximation, the total energy of PZ-SIC DFT becomes invariant for orbital rotations between occupied orbitals. ADSIC provides good results for delocalized states and large
molecules, while its performance is poorer for small molecules and localized states.
In the last two years, a self-consistent SIC DFT method
using optimized effective potentials has been developed.9,10
This method corrects the asymptotic behavior of dissociation
and provides very good potential energy curves. More recently, a self-consistent PZ-SIC DFT method was implemented using the quasi-Newton Broyden-Fletcher-GoldfarbShanno method.11 While it also provides excellent potential
energy curves, the quasi-Newton method is not particularly
efficient and its low convergence rate prevents its application
to the simulation of large systems.
All the mentioned SIC methods provide accurate potential energy curves, while the orbital energies remain affected
by large errors when compared to experimental IPs. Compared to HF, both DFT and SIC DFT predict worse values for
the IPs. It would thus be a great advantage to figure out a
simple SIC procedure that can provide IP values comparable
to those offered by ⌬SCF calculations or even closer to the
experimental values.
B. Self-energy corrections of core orbital energies
In the HF method, the nonlocal exchange potential is
expressed for the orbital i as
Vi␴共r1兲 = 兺
j
冕
␾ j␴共r2兲␾l␴共r2兲 ␾ j␴共r1兲
,
dr2
r12
␾i␴共r1兲
共1兲
共2兲
where ␳ is the total electron density of the system.
With this choice of the potential field, DFT becomes
more applicable to large systems than HF and by choosing a
proper density functional; DFT can also provide better results than HF. For a single electron system, the correct density functional gives a null two-electron contribution to the
total energy, and Coulomb plus exchange and correlation
terms cancel
J关␳兴 + Vxc关␳兴 = 0.
共4兲
i␴
where the sum runs over singly occupied spin orbitals.
Taking the functional derivative of the PZ-SIC DFT energy with respect to the electronic density of the i␴ spin
orbital, we obtain a SIC potential,
VPZ = VDFT −
冉冕
dr1
冊
␳i␴共r1兲
+ Vxc关␳i␴,0兴 .
r1
共5兲
And the orbital energy SIC expressed by
冉
␧SIC = − 2J关␳i␴兴 +
冕
冊
d␶Vxc关␳i␴,0兴␳i␴ .
共6兲
For a one-electron system, the spurious SIE contribution to
the total energy is exactly compensated by ␧SIC. However, in
a many-electron system, the total DFT exchange potential
cannot be written as the sum of the exchange potential of a
single electron and the exchange potential of the remaining
electrons. The difference is a potential term which represents
the exchange contribution to the electron coupling,
Vcoupling = Vxc关␳兴 − 共Vxc关␳ − ␳i␴兴 + Vxc关␳i␴,0兴兲,
共3兲
However, any approximated density functional will, instead,
lead to a spurious self-interaction contribution to the molecular energy when applied to an open shell system. As mentioned before, many SIC methods have been proposed to
solve the SIE problem. The earliest and basic one proposed
by Perdew and Zunger, which leads to better potential energy
curves but give little improvement for the orbital energies, is
based on the simple correction,
共7兲
that should be considered in a more correct SIC for the spin
orbital i␴. We propose to include the contribution of the
coupling term to SIC in a pragmatic way
DFT
ViSIC
−
␴ =V
再冕
dr1
␳i␴共r1兲
+ Vxc关␳i␴,0兴
r1
冎
+ ␣共Vxc关␳ − ␳i␴兴 + Vxc关␳i␴,0兴 − Vxc关␳兴兲 ,
where ␴ is the spin coordinate. Therefore, for different orbitals in the system, the exchange potential is different. However, in the DFT-KS method, a unique exchange 共and correlation兲 potential field is used for any orbital i,
Vi共r1兲 = Vxc关␳共r1兲兴,
EPZ = EDFT − 兺 共J关␳i␴兴 + Exc关␳i␴,0兴兲,
共8兲
where ␣ is an empirical parameter, that, as discussed in the
following section, will be optimized by a best fitting of the
estimated core IPs to the experimental values for a large set
of molecules. The SIC to a core orbital energy is then expressed as
␧iSIC
␴ =
冕
d␶ViSIC
␴ ␳ i␴ .
共9兲
III. RESULTS
Calculations of the ground state of a set of molecules of
different dimensions and electronic structures have been performed by the DFT-KS approach followed by the evaluation
of ␧iSIC
␴ to estimate core IPs at the K edges of C, N, and O
using a locally modified version of DALTON.12 The Boys localization procedure was applied to the delocalized core orbitals of chemically equivalent atoms. In a DFT procedure, if
we run a calculation of a molecule with chemically equivalent atoms, the core orbitals will be delocalized between
equivalent atoms. However, this is not the case in a real
system since core orbitals are rarely affected by their chemical environments. This means that orbital localization is required in SIC DFT. Here we applied the Boys localization
procedure only in cases that there are chemically equivalent
atoms in a molecule. For a molecule without chemically
equivalent atoms, the core orbital energies of different atoms
Downloaded 23 Jan 2008 to 130.237.79.166. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/jcp/copyright.jsp
174110-4
Tu et al.
J. Chem. Phys. 127, 174110 共2007兲
FIG. 1. Structure of 2-mercaptobenzoxazole 共MBO兲.
are usually not equivalent and the orbitals are not mixed.
This means that it is not necessary to apply the Boys localization to a molecule without chemically equivalent atoms in
it. Our SIC DFT method is related to the first order correction to the conventional DFT, and the orbitals from conventional DFT are not optimized further. Therefore the requirement of orbital localization in our method is not related to
the orbital rotation dependence of the self-correction scheme.
All the molecular geometries, except that of the peptide
共alanina-glutammic acid兲 which is represented by the optimized geometry of the adsorbate on a TiO2 surface,14 were
optimized by DFT calculations with the B3LYP functional
and 6-311G共d , p兲 basis set, using the GAUSSIAN code.
In Figs. 1–3, the geometrical structures of the largest
molecules 2-mercaptobenzoxazole 共MBO兲, glycine, and depeptide 共alanina-glutammic acid兲 are displayed. Figure 4
shows an example of optimization of the empirical parameter
␣ for the core IPs 共C, N, O K edge兲 of formamide, formic
acid, methanol, and methyl format. We show the difference
between experimental ionization potentials and minus the
self-interaction corrected orbital energies for a range of values for the ␣ parameter. The ␣ value corresponding to the
lines crossing is the best estimation of ␣ for a certain density
functional and for a specific orbital. In Fig. 4, the best value
of ␣ is slightly different for different core orbitals. However,
for the sake of convenience and clearness, we prefer to use a
unique ␣ in the calculation. We find that the differences be-
FIG. 2. Structure of glycine.
FIG. 3. Structure of dipeptide 共alanina-glutammic acid兲.
tween experimental and theoretical IPs are all very close to
zero when ␣ = 0.72; such value, optimized for the B3LYP
functional, was then used in the SIC DFT calculation for all
the C, N, O 1s orbitals.
Figures 5–7 show the deviation of the computed IPs
from the available experimental values for the considered
molecules, using the hybrid B3LYP exchange correlation
functional and different basis sets. A full comparison of
theory, at different levels of approximation, and experiment
is also presented in Tables I–III, where we listed the IPs from
the conventional DFT and the SIC DFT calculations. In this
paper, we use the approximation that the core IP is simply
minus the core orbital energy. In the Kohn-Sham DFT
scheme, where Koopman’s theorem is not valid, this approximation is quite unreasonable, as clearly shown in Tables
I–III. Compared to the experimental values, the IPs from the
conventional DFT calculation bear large errors. Furthermore,
such errors are element dependent, i.e., the errors are nearly
the same for IPs at the same K edge in different molecules
but are different for IPs of different K edges in the same
molecule. For all the molecules we studied, the mean absolute errors in the theoretical prediction, by the conventional
DFT approach, are around 18 eV for the O K edge, about
16 eV for the N K edge, and about 14 eV for the C K edge.
In the SIC-DFT approach, this element dependence disappears. SIC-DFT is seen to properly correct the main errors of
conventional DFT and produces reasonable IPs. However,
using a unique ␣ for all the core orbitals, the SIC term rarely
reflects the chemical environment. The IPs at a specific element K edge in different molecules, which is the “chemical
shifts,” show small differences with respect to different
chemical environments, of the order of 1 eV. Experimentally, the following IP sequence is observed for the set of
molecules C2H2, C2H4, C2H6 at the C K edge: IP共C2H2兲
⬎ IP共C2H4兲 ⬎ IP共C2H6兲. This particular IP sequence is already predicted by the conventional DFT procedure and SIC
DFT does not introduce any variation. In most cases, we can
observe that the chemical shift is reproduced by the conven-
Downloaded 23 Jan 2008 to 130.237.79.166. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/jcp/copyright.jsp
174110-5
Core ionization
J. Chem. Phys. 127, 174110 共2007兲
FIG. 4. Difference between orbital energies and experimental ionization potentials as a function of the parameter ␣ in Eq. 共8兲. Results include the IPs for the
1s shell of C, N, and O for 共a兲 CH3CN, 共b兲 CH4, 共c兲 CO, and 共d兲 CH3COOH.
tional DFT, while only SIC DFT will correct the elementdependent errors and will provide reasonable absolute values
of the core IPs.
Figures 5–7 show the deviation of the computed IPs
from the available experimental values for the considered
molecules using different basis sets. The basis sets employed
were
6-31G = 共10s,4p兲 → 关3s,2p兴,
6-311Gdp = 共11s,5p兲 → 关4s,3p兴 + 关1p,1ddiffuse兴,
6-311G2d2p = 共11s,5p兲 → 关4s,3p兴 + 关2p,2ddiffuse兴.
In these figures, the smaller basis set 6-31G results in larger
deviations, where the mean absolute error is 0.96 eV. With a
larger basis set, 6-311G共d , p兲, the deviations become smaller
and the mean absolute error is reduced to 0.57 eV. However,
the even larger basis set 6-311+ + G共2d , 2p兲 does not further
improve the performance, the mean absolute deviation remaining practically equal 共0.59 eV兲. These results indicate
that a proper SIC correction can only be obtained by a basis
set of better than 6-31G quality.
A problem exists for fluorine ionization, which leads to a
large negative error. However, the use of a unique ␣ value, as
we have discussed, is an approximation adopted only for
sake of convenience and clearness. While the best estimated
␣ is nearly the same for C, N, O 1s orbitals in different
molecules, for fluorine, which has a very large electron affinity, the best value of ␣ may differ significantly from that
of C, N, O.
The deviation between the SIC orbital energy and experiment is quite system dependent; however, in a system
with many possible substituents, the SIC seems quite independent of the type of substituting group, thus in the series of
benzene substituents with progressively increasing electronegativity, CH3, NH2, OH, F, the variation is only within
1.4 eV for the C K edge with a 6-311G共d , p兲 basis set.
The variation between different nonequivalent elements
is also of concern as they reflect the chemical shift. The
MBO molecule with five nonequivalent carbon atoms is a
Downloaded 23 Jan 2008 to 130.237.79.166. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/jcp/copyright.jsp
174110-6
J. Chem. Phys. 127, 174110 共2007兲
Tu et al.
FIG. 5. Deviations for IPs between theoretical calculation and experiment for C, N, O K edges for different molecules. Calculations are done by SIC DFT
using B3LYP functional with 6-31G basis set.
good example, from this point of view. Here we notice a
variation well within 1 eV of the errors for the MBO and
glycine molecules, see Figs. 5–7. We thus observe that by
adopting a proper, but unique, ␣, SIC-DFT may produce
good quality IPs in most cases where a comparison between
theory and experiment is feasible. It therefore sounds reasonable to apply this method for theoretical predictions of unknown vertical IPs and/or to compare a computed spectrum
to a poorly resolved experimental photoelectron spectrum.
To illustrate this point, we applied the approach to predict the
core IPs of dipeptide AE 共alanina-glutammic acid兲. The results are listed in Table IV and are there compared to the IPs
from ⌬DFT calculations. It may be observed that the deviations between the results of the SIC-DFT calculations and
those of the much more time consuming KS-DFT calculations are at most 1 eV and around 0.5 eV in the average.
Such average deviation is below the experimental resolution
that could be achieved in collecting the XPS spectra of
dipeptide adsorbed on a TiO2 surface.14 This is a confirmation that SIC-DFT can be used as an efficient and sufficiently
accurate method for the interpretation of XPS spectra of
large biomolecules, where the presence of complex unresolved vibrational structure requires a theoretical prediction
of the vertical IPs for an assignment of the main spectral
features in terms of electronic structure.
IV. DISCUSSION
The development of quantum chemistry methodologies
has produced ever more sophisticated tools to compute photoelectron binding energies and pole strengths. Combined
with photon energy dependent oscillator strength computa-
Downloaded 23 Jan 2008 to 130.237.79.166. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/jcp/copyright.jsp
174110-7
Core ionization
J. Chem. Phys. 127, 174110 共2007兲
FIG. 6. Deviations for IPs between theoretical calculation and experiment for C, N, O K edges for different molecules. Calculations are done by SIC DFT
using B3LYP functional with 6-311G共d , p兲 basis set.
tions, these methodologies often provide unequivocal assignments of small symmetric molecules. This goes especially
for the outer valence region where the quasiparticle approximation holds, but sometimes also—using electron correlated
techniques—for the more complex inner valence regions
with hole mixing and breakdown effects.15,16 As for many
other properties and spectra, the most problematic aspect is
the poor particle scaling for techniques beyond the selfconsistent field 共SCF兲 or the ⌬SCF approximation levels,
making larger systems inaccessible for a treatment including
dynamical electron correlation. Density functional theory is
natural to consider in this context since it indeed accounts for
dynamical correlation and possesses good particle scaling.
However, its use has been hampered by its less inherent theoretical rigor and by the lack of a direct interpretation of the
orbital energies corresponding to Koopman’s theorem in
Hartree-Fock theory. The Kohn-Sham orbital formulation of
DFT together with transition potential techniques seems,
however, to provide a way out; Chong17,18 used this combination for direct calculations of ionization potentials and
chemical shifts in the core region with high accuracy 共the
so-called unrestricted generalized transition-state 共␮GTS
method兲. The ionization potential is then directly identified
as the energy of the Kohn-Sham orbital optimized for a transition state with a fractional occupation number. The original
study was followed by several works for both core electron
binding or excitation energies in Refs. 19–22.
The transition potential method is a rigorous approximation of ⌬SCF energies and forms a viable alternative also to
the ⌬Kohn-Sham method in the DFT case. An extensive
comparison of the two methods was carried out in Ref. 22. It
is clear from that work that while at a sub-eV scale there is a
Downloaded 23 Jan 2008 to 130.237.79.166. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/jcp/copyright.jsp
174110-8
Tu et al.
J. Chem. Phys. 127, 174110 共2007兲
FIG. 7. Deviations for IPs between theoretical calculation and experiment for C, N, O K edges for different molecules. Calculations are done by SIC DFT
using B3LYP functional with 6-311+ + G共2d , 2p兲 basis set.
significant functional dependence of Kohn-Sham theory for
core IP calculations, the overall precision improves that of
⌬SCF. ⌬KS does not seem to improve much on the already
quite successful ␮GTS method, however, the ␮GTS
method17,18 shows a larger functional dependence than
⌬KS.22
It is clear that the SIC corrected core IPs are of lower
precision than ⌬Kohn-Sham values. While the latter show
IPs with less than 1 eV error with respect to experiment 共for
certain compounds even below 0.5 eV兲, the SIC corrections
are clearly above this limit. It should be considered that this
work presents a first application of the new SIC DFT
method, where the use of a unique ␣ value is just an approximation. However, the main quantity to qualify is the chemical shift, which is the variation of errors for the same element
over the different compounds. This is of course considerably
smaller than the absolute IP error, but needs to go well below
1 eV to make the calculation useful for experimental assignments. It is clear that to obtain this level of precision we
should reach for optimum ␣ and a suitable functional and
basis sets beyond the 6-31G level. It is in that context relevant to remind that since DFT theory is semiempirical in
nature, accuracy and efficiency go hand in hand. The error
from the SIC corrected orbital energies should be weighed
against the computational cost of producing them. Ground
state DFT scales very well and the new SIC calculation,
being a first order correction to the conventional DFT, scales
nearly the same as the ground state calculation.
A qualification of the results in terms of experimental
precision also calls for a consideration of relativistic effects
associated with the removal of one electron from the core
orbital. This energy is to a good approximation an atomic,
Downloaded 23 Jan 2008 to 130.237.79.166. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/jcp/copyright.jsp
174110-9
J. Chem. Phys. 127, 174110 共2007兲
Core ionization
TABLE I. Comparison of experimental K-shell IPs and KS-DFT orbital
energies ␧ including the self-interaction correction ⌬␧SIC in Eq. 共9兲. All the
calculations are done using B3LYP functional with 6-311+ + G共2d , 2p兲 basis
set. In the SIC part, the parameter ␣ is set to be 0.72. All results are in eV.
Molecule
Atom
−␧
−⌬␧SIC −共␧ + ⌬␧SIC兲
IPa
⌬
TABLE II. Comparison of experimental K-shell IPs and KS-DFT orbital
energies ␧ including the self-interaction correction ⌬␧SIC in Eq. 共9兲. All the
calculations are done using B3LYP functional with 6-311G共d , p兲 basis set.
In the SIC part, the parameter ␣ is set to be 0.72. All results are in eV.
Molecule
Atom
−␧
−⌬␧SIC −共␧ + ⌬␧SIC兲
IPa
⌬
CO
O
C
523.36
280.02
18.26
14.64
541.62
294.66
542.10 −0.48
295.90 −1.24
CO
O
C
523.14
279.82
18.25
14.64
541.39
294.46
542.10 −0.71
295.90 −1.44
H 2O
O
520.54
18.43
538.97
539.93 −0.96
H 2O
O
519.69
18.43
538.12
539.93 −1.81
CH4
C
276.13
14.81
290.94
290.83
CH4
C
276.15
14.83
290.98
290.83
CH3CN
CH3CN*
C*H3CN
CH3C*N
389.43
278.19
277.91
16.53
14.79
14.52
405.96
292.98
292.43
405.60 0.36
292.98 0.00
292.44 −0.01
CH3CN
CH3CN*
C*H3CN
CH3C*N
389.24
278.14
277.76
16.53
14.84
14.54
405.77
292.98
292.30
405.60 0.17
292.98 0.00
292.44 −0.14
521.86
520.22
280.62
277.16
18.39
18.36
14.85
14.81
540.25
538.58
295.47
291.97
540.09
538.36
295.38
291.55
0.16
0.22
0.09
0.42
CH3COOH CH3COO*H
CH3CO*OH
CH3C*OOH
C*H3COOH
521.47
519.84
280.38
277.07
18.38
18.35
14.85
14.83
539.85
538.19
295.23
291.90
540.09 −0.24
538.36 −0.17
295.38 −0.15
291.55 0.45
521.97
520.43
389.15
280.67
278.04
18.39
18.36
16.72
14.84
14.87
540.36
538.79
405.87
295.51
292.91
540.20
538.40
405.40
295.20
292.30
0.16
0.39
0.47
0.31
0.61
Glycine
COO*H
CO*OH
N
C*OOH
C *H
521.60
520.08
388.76
280.43
277.85
18.39
18.35
16.73
14.84
14.88
539.99
538.43
405.49
295.27
292.73
540.20 −0.21
538.40 0.03
405.40 0.09
295.20 0.07
292.30 0.43
MBO
S
O
N
C *S
C *O
C *N
C*CN
C*CO
2416.39
522.62
391.62
280.83
279.06
278.75
277.65
277.56
30.46
18.40
16.69
14.91
14.69
14.67
14.57
14.41
2446.85
541.02
408.31
295.74
293.75
293.42
292.22
291.97
540.58 0.44
407.01 1.30
295.71 0.03
293.91 −0.16
293.01 0.41
291.45 0.77
291.67 0.30
CH3COOH CH3COO*H
CH3CO*OH
CH3C*OOH
C*H3COOH
0.11
0.15
Glycine
COO*H
CO*OH
N
C*OOH
C *H
MBO
S
O
N
C *S
C *O
C *N
C*CN
C*CO
2416.46
522.77
391.66
280.87
279.08
278.77
277.67
277.59
30.46
18.40
16.68
14.91
14.67
14.65
14.53
14.38
2446.92
541.17
408.34
295.78
293.75
293.42
292.20
291.97
C6H5CH3
C *H 3
C*CH3
276.59
277.00
14.80
14.59
291.39
291.59
290.10
290.90
1.29
0.69
C6H5CH3
C *H 3
C*CH3
276.57
276.93
14.81
14.61
291.38
291.54
290.10
290.90
1.28
0.64
C6H5NH2
N
C*NH2
389.64
277.86
16.71
14.81
406.35
292.67
405.30
291.20
1.05
1.47
C6H5NH2
N
C*NH2
389.39
277.73
16.72
14.82
406.11
292.55
405.30
291.20
0.81
1.35
C6H5OH
O
C*OH
520.86
278.38
18.45
14.83
539.31
293.21
538.90
292.00
0.41
1.21
C6H5OH
O
C*OH
520.47
278.24
18.45
14.84
538.92
293.08
538.90
292.00
0.02
1.08
C 6H 5F
F
C *F
671.81
279.21
20.07
14.85
691.88
294.06
693.30 −1.42
292.90 1.16
C 6H 5F
F
C *F
671.41
279.11
20.06
14.87
691.47
293.98
693.30 −1.83
292.90 1.08
C 2H 2
C
276.94
14.64
291.58
291.20
0.38
C 2H 2
C
276.76
14.63
291.39
291.20
0.19
C 2H 4
C
276.70
14.75
291.45
290.70
0.75
C 2H 4
C
276.60
14.76
291.36
290.70
0.66
C 2H 6
C
276.29
14.84
291.13
290.60
0.53
C 2H 6
C
276.30
14.86
291.16
290.60
0.56
540.58 0.59
407.01 1.33
295.71 0.07
293.91 −0.16
293.01 0.41
291.45 0.75
291.67 0.30
a
a
transferable, quantity and can accordingly be obtained from
calculations on the respective atoms using large uncontracted
basis sets and including an electron-correlated approach 共for
instance, the modified coupled pair functional approach in
Ref. 23兲. For chemical shifts it is the differential energy
which is relevant; this quantity was obtained in modified
coupled pair functional calculations23 as first order perturbation theory estimates of the mass-velocity and Darwin terms
as 0.2, 0.3, 0.4, and 0.7 eV for C, N, O, and F, respectively.
These values actually agree well with the early estimates by
Pekeris24 共0.10, 0.21, 0.39, and 0.68 eV, respectively兲 using
data of two-electron ions—indicating thereby the small contribution from the valence shell—while they roughly double
the values derived semiempirically by Chong.17,18 These values have subsequently been used in relativistic corrections of
core IPs.
Depending on the required precision, zero-point vibrational energy corrections 共ZPVECs兲 are also relevant to consider for core ionization, since XPS bands, in general, possess vibrational progressions. These progressions are most
often not resolved experimentally and still often indicate
dominance of the 0-0 transition. It is relevant to approximate
Taken from Ref. 13.
Taken from Ref. 13.
Downloaded 23 Jan 2008 to 130.237.79.166. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/jcp/copyright.jsp
174110-10
J. Chem. Phys. 127, 174110 共2007兲
Tu et al.
TABLE III. Comparison of experimental K-shell IPs and KS-DFT orbital
energies ␧ including the self-interaction correction ⌬␧SIC in Eq. 共9兲. All the
calculations are done using B3LYP functional with 6-3G basis set. In the
SIC part, the parameter ␣ is set to be 0.72. All results are in eV.
Molecule
Atom
−␧
−⌬␧SIC −共␧ + ⌬␧SIC兲
IPa
⌬
CO
O
C
524.10
280.75
18.38
14.77
542.48
295.52
542.10 0.38
295.90 −0.44
H 2O
O
520.55
18.60
539.15
539.93 −0.78
CH4
C
276.33
14.94
290.27
290.83 −0.56
CH3CN
CH3CN*
C*H3CN
CH3C*N
390.17
278.36
278.32
16.70
13.11
12.94
406.87
291.47
291.26
405.60 1.27
292.98 −1.51
292.44 −1.18
522.29
520.91
281.09
277.36
18.52
18.51
14.88
14.92
540.81
539.42
295.97
292.28
540.09
538.36
295.38
291.55
0.72
1.06
0.59
0.73
522.46
521.19
389.38
281.20
278.24
18.52
18.51
16.87
14.87
14.95
540.98
539.70
406.25
296.07
293.19
540.20
538.40
405.40
295.20
292.30
0.78
1.30
0.85
0.87
0.89
S
O
N
C *S
C *O
C *N
C*CN
C*CO
2416.82
523.32
392.18
281.23
279.44
279.19
278.03
277.93
30.54
18.52
16.82
14.91
14.74
14.75
14.77
14.67
2447.36
541.84
409.00
296.14
294.18
293.94
292.80
292.60
540.58
407.01
295.71
293.91
293.01
291.45
291.67
1.26
1.99
0.43
0.27
0.93
1.35
0.93
C6H5CH3
C *H 3
C*CH3
276.75
277.18
14.93
14.79
291.68
291.97
290.10
290.90
1.58
1.07
C6H5NH2
N
C*NH2
389.90
278.04
16.88
14.92
406.78
292.96
405.30
291.20
1.48
1.76
C6H5OH
O
C*OH
521.12
278.56
18.60
14.91
539.72
293.47
538.90
292.00
0.82
1.47
C 6H 5F
F
C *F
671.90
279.40
20.16
14.91
692.06
294.31
693.30 −1.24
292.90 1.41
C 2H 2
C
277.08
14.83
291.91
291.20
0.71
C 2H 4
C
276.86
14.90
291.76
290.70
1.06
C 2H 6
C
276.50
14.96
291.46
290.60
0.86
CH3COOH CH3COO*H
CH3CO*OH
CH3C*OOH
C*H3COOH
Glycine
MBO
COO*H
CO*OH
N
C*OOH
C *H
a
Taken from Ref. 13.
the ZPVEC by assuming a multimode decoupling of harmonic oscillators. There are two situations then to consider,
one in which the lower and upper potential curves for a certain mode have the same interatomic distance. In that case
the ZPVEC is given by the difference of the ZPVEs of the
upper and lower curves, which, in general, is a small number,
in the order of hundredths of eV. The second situation occurs
if the upper core hole potential curve is displaced from that
of the ground state 共when the vertical distance falls outside
the turning points of the zero-level vibrations兲. In that case it
is the ZPVE of the lower potential that should be subtracted
TABLE IV. Ala-Glu 1s orbital energies from B3LYP calculation with
6-311G共d , p兲 basis set; ␣ = 0.72. All the results are in eV.
Atom
−␧
−⌬␧SIC
−共␧ + ⌬␧SIC兲
⌬DFT
⌬
O共3兲
O共2兲
O共4兲
O共1兲
N共6兲
N共7兲
N共5兲
C共14兲
C共15兲
C共10兲
C共11兲
C共8兲
C共13兲
C共12兲
C共9兲
521.65
520.05
518.92
518.86
390.32
390.06
388.63
280.62
279.41
279.14
278.19
277.86
277.36
276.93
276.89
18.40
18.36
18.37
18.36
16.67
16.67
16.74
14.88
14.85
14.81
14.89
14.90
14.84
14.84
14.81
540.05
538.41
537.29
537.22
406.99
406.73
405.37
295.50
294.26
293.95
293.08
292.76
292.20
291.77
291.70
539.99
537.80
536.38
536.18
406.15
405.85
404.89
295.19
293.90
293.44
290.09
291.91
291.54
290.94
291.20
0.06
0.61
0.91
1.04
0.84
0.88
0.48
0.31
0.36
0.51
0.99
0.85
0.66
0.83
0.50
from the computed vertical IP. This gives a larger correction,
in the order of tenth of eV. The corrections from each decoupled mode should so be summed up. For polyatomic molecules, the second case should only be considered for those
modes excited during core ionization, i.e., those modes
which are locally attached to the site of ionization, while the
first situation prevails for the “inactive” modes. So although
the total ZPVEs become larger for a larger molecule, the
appropriate fraction of the total ZPVEs used for the actual
ZPVEC thus becomes smaller. The identification of the appropriate fraction of the ZVPE calls for a complete normal
mode and gradient analysis. It is quite clear that these errors
are considerably smaller than the size of the errors in the IPs
obtained from the SIC corrected orbital energies.
The main gain in using Kohn-Sham theory to analyze
XPS data may not necessarily be associated with the precision for small or large organic species, but could rather be
the applicability to a broader spectrum of species. This is
illustrated, for instance, in Ref. 22 by the calculations of
large metal-adsorbate clusters, for which the ⌬KS method
works just as well as for the small organic molecules. These
systems, like many other systems containing metallic elements, are notoriously problematic for ⌬SCF methods based
on traditional Hartree-Fock. Metallic and organometallic
compounds are thus suitable candidates for studies with SIC
orbital corrections, something we hope to accomplish in the
near future.
V. SUMMARY AND CONCLUSION
In order to explore the possibility to predict core electron
ionization potentials based on density functional theory, we
have tested the applicability of a self-energy correction to the
Kohn-Sham orbital energies. We focused in this work on
core ionization potentials as these, in opposition to valence
levels, are very poorly described by the canonical KohnSham orbital energies. Among several possibilities we propose a new SIC DFT method that adds a further correction of
the potential coupling to the Perdew-Zunger SIC DFT. We
introduce an empirical parameter ␣ that weighs the strength
Downloaded 23 Jan 2008 to 130.237.79.166. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/jcp/copyright.jsp
174110-11
of the coupling term with respect to the one-electron potential. For different orbitals, ␣ should be different in principle.
However, by testing on a large set of molecules and at different K edges, we find that, in practice, there is only a slight
variation of the best ␣ value for the core 1s orbitals of C, N,
O. The proposition is evaluated by calculations of C, N, O,
and F element core ionization energies for a number of different organic compounds. Tests on basis set dependence,
made assuming the optimized ␣ parameter, indicate a
1 – 2 eV absolute error compared with experimental core
binding energies and a chemical shift error of the order of
1 eV or less. The present work focuses on orbital energies as
they appear in photoelectron spectroscopy, but the analysis
will be applicable to other members of the family of x-ray
spectroscopies, for instance, x-ray absorption and emission
spectroscopy.
ACKNOWLEDGMENTS
This work has been supported by the European Research
and Training Network “Understanding Nanomaterials from a
Quantum Perspective” 共NANOQUANT兲, Contract No.
MRTN-CT-2003-506842.
1
J. Chem. Phys. 127, 174110 共2007兲
Core ionization
K. Siegbahn, C. Nordling, A. Fahlman et al., ESCA-Atomic, Molecular
and Solid State Structure Studied by Means of Electron Spectroscopy
共North-Holland, Amsterdam, 1966兲.
2
K. Siegbahn, C. Nordling, G. Johansson et al., ESCA Applied to Free
Molecules 共North-Holland, Amsterdam, 1969兲.
3
O. Plekan, V. Feyer, R. Richter, M. Coreno, M. de Simone, K. C. Prince,
and V. Carravetta 共unpublished兲.
4
V. Carravetta, H. Årgen, and A. Cesar, Chem. Phys. Lett. 148, 210
共1988兲.
5
O. Plashkevych, H. Ågren, L. Karlsson, and L. G. M. Pettersson, J. Electron Spectrosc. Relat. Phenom. 106, 51 共2000兲.
6
J. P. Perdew and A. Zunger, Phys. Rev. B 23, 5048 共1981兲.
7
E. Suraud, C. Legrand, and P. G. Reinhard, J. Phys. B 35, 1115 共2002兲.
8
H. Chermette, I. Ciofini, and C. Adamo, Chem. Phys. Lett. 380, 12
共2003兲.
9
J. A. Nichols, J. Garza, R. Vargas, and D. A. Dixon, J. Chem. Phys. 114,
639 共2001兲.
10
S. Patchkovskii and T. Ziegler, J. Chem. Phys. 116, 7806 共2002兲.
11
O. A. Vydrov and G. E. Scuseria, J. Chem. Phys. 121, 8187 共2004兲.
12
DALTON, a molecular electronic structure program, Release 2.0, 2005, see
http://www.kjemi.uio.no/software/dalton/dalton.html
13
W. L. Jolly, K. D. Bomben, and C. J. Eyermann, Ann. Phys. 共N.Y.兲 31,
433 共1984兲.
14
G. Polzonetti, C. Battocchio, G. Lucci, M. Dettin, R. Gambaretto, C. Di
Bello, and V. Carravetta, Mater. Sci. Eng., C 26, 929 共2006兲.
15
L. S. Cederbaum, W. Domcke, J. Schirmer, and W. von Niessen, Adv.
Chem. Phys. 65, 115 共1986兲.
16
S. J. Desjardins, A. D. O. Bagawan, Z. F. Liu, K. H. Tan, Y. Wang, and
E. R. Davidson, J. Chem. Phys. 103, 6385 共1995兲.
17
D. P. Chong, Chem. Phys. Lett. 232, 486 共1995兲.
18
D. P. Chong, J. Chem. Phys. 103, 1842 共1995兲.
19
C. Bureau, D. P. Chong, K. Endo, J. Delhalle, G. Lecayon, and A. Le
Moel, Nucl. Instrum. Methods Phys. Res. B 269, 1 共1997兲.
20
C. H. Hu and D. P. Chong, Chem. Phys. Lett. 262, 729 共1996兲.
21
M. Stener, A. Lisini, and P. Decleva, Chem. Phys. 191, 141 共1995兲.
22
L. Triguero, O. Plashkevych, H. Ågren, and L. G. M. Pettersson, J. Electron Spectrosc. Relat. Phenom. 104, 195 共1999兲.
23
D. P. Chong and S. R. Langhoff, J. Chem. Phys. 84, 5606 共1986兲.
24
C. L. Pekeris, Phys. Rev. 112, 1649 共1958兲.
Downloaded 23 Jan 2008 to 130.237.79.166. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/jcp/copyright.jsp