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Improved adsorption energetics within density-functional theory using revised
Perdew-Burke-Ernzerhof functionals
Hammer, Bjørk; Hansen, Lars Bruno; Nørskov, Jens Kehlet
Published in:
Physical Review B
Link to article, DOI:
10.1103/PhysRevB.59.7413
Publication date:
1999
Document Version
Publisher's PDF, also known as Version of record
Link back to DTU Orbit
Citation (APA):
Hammer, B., Hansen, L. B., & Nørskov, J. K. (1999). Improved adsorption energetics within density-functional
theory using revised Perdew-Burke-Ernzerhof functionals. Physical Review B, 59(11), 7413-7421.
https://doi.org/10.1103/PhysRevB.59.7413
Improved adsorption energetics within density-functional theory using revised
Perdew-Burke-Ernzerhof functionals
B. Hammer
Institute of Physics, Aalborg University, Pontoppidanstræde 103, DK-9220 Aalborg O” st, Denmark
L. B. Hansen and J. K. No”rskov
Center for Atomic-scale Materials Physics, Department of Physics, Technical University of Denmark, DK-2800 Lyngby, Denmark
~Received 23 September 1998!
A simple formulation of a generalized gradient approximation for the exchange and correlation energy of
electrons has been proposed by Perdew, Burke, and Ernzerhof ~PBE!@Phys. Rev. Lett. 77, 3865 ~1996!#.
Subsequently Zhang and Yang @Phys. Rev. Lett. 80, 890 ~1998!# have shown that a slight revision of the PBE
functional systematically improves the atomization energies for a large database of small molecules. In the
present work, we show that the Zhang and Yang functional ~revPBE! also improves the chemisorption ener-
getics of atoms and molecules on transition-metal surfaces. Our test systems comprise atomic and molecular
adsorption of oxygen, CO, and NO on Ni~100!,Ni~111!,Rh~100!,Pd~100!, and Pd~111! surfaces. As the
revPBE functional may locally violate the Lieb-Oxford criterion, we further develop an alternative revision of
the PBE functional, RPBE, which gives the same improvement of the chemisorption energies as the revPBE
functional at the same time as it fulfills the Lieb-Oxford criterion locally. @S0163-1829~99!02711-3#
I. INTRODUCTION
Density-functional theory is widely accepted as a frame-
work for the study of the electronic ground-state properties
of molecules and solids. It has long been realized that the
molecular bond energies and the cohesive energies of the
solids are overestimated when the electronic exchange and
correlation effects are described in the local-density approxi-
mation ~LDA!.
1
However, the development of nonlocal ex-
change and correlation functionals has demonstrated that the
bond energies of molecules, the cohesive energies of solids,
and the energy barriers for molecular reactions can be greatly
improved within density-functional theory.
2–12
In the present paper we investigate the behavior of the
chemisorption energy of different atomic and molecular ad-
sorbates on some late transition-metal surfaces using differ-
ent functionals for the exchange and correlation energy. We
limit ourselves to the study of functionals derived in the
generalized gradient approximation ~GGA!. The functionals
we use include the Perdew-Wang-91 functional
2
~PW91!, the
Perdew-Burke-Ernzerhof ~PBE! functional,
13
and the revised
PBE functional with one parameter,
k
, changed from 0.804
to 1.245 ~revPBE! as proposed by Zhang and Yang.
14
These
three functionals have been reported to give similar values
for the molecular bond energies of about 20 small molecules,
with some evidence that the revPBE functional is the most
accurate. Our calculations show, however, that the function-
als give rather different chemisorption energies. Judging
from a comparison to experimental chemisorption energies,
the revPBE is found to be superior in the description of the
energetics of atomic and molecular bonding to surfaces. The
construction of the revPBE functional involves a softening of
one of the criteria used in the construction of the PBE func-
tional. Specifically, with the PBE functional, the Lieb-
Oxford criterion
15,16
is obeyed locally and hence also glo-
bally by construction,
17
while with the revPBE functional it
is only found empirically to be obeyed globally.
14
To avoid
the uncertainty with respect to the fulfilment of the Lieb-
Oxford criterion associated with the use of the revPBE, we
develop an alternative functional ~RPBE!, which gives prac-
tically identical chemisorption energies. It does not involve
fitting of parameters and it fulfills the Lieb-Oxford criterion
by construction.
The paper is organized as follows: First chemisorption
energies calculated with the PW91, PBE, and revPBE func-
tionals are presented. Next an analysis of the spatial and
gradient-resolved exchange-correlation contributions to the
chemisorption energies for the three different functionals is
given. This leads to the suggestion of the RPBE functional,
which is finally tested.
II. CALCULATIONAL DETAILS
The density-functional theory calculations are done for
adsorbates in p(232) structures on the fcc~100! and
fcc~111! faces of Ni, Rh, and Pd surfaces. The surfaces are
modeled by slabs of four and three layers thickness for the
two facets ~100! and ~111!, respectively @see the Appendix,
issue ~iii!#. The slabs are repeated periodically in three di-
mensions, leaving at least 10 Å of vacuum between the
slabs.
18
The ionic cores are described by ultrasoft
pseudopotentials
19
developed within the PW91 approxima-
tion for the exchange and correlation @see the Appendix, is-
sue ~ii!#. The nonlinear core correction
20
is employed, using
the core density beyond a cutoff, r
c
(r
c
C
5 0.6,r
c
N
5 0.6,r
c
O
5 0.7,r
c
Ni
51.2,r
c
Rh
51.0,r
c
Pd
51.1 bohr! and a second-order
polynomial continuation for smaller r ~for the description of
oxygen we make one improvement over the above—see is-
sue ~i! of the Appendix#. Within the slabs, the PW91 lattice
parameters of 3.52, 3.83, and 3.99 Å for Ni, Rh, and Pd are
used throughout the work except when stated otherwise. For
Ni, the calculations are done spin polarized. For each slab
PHYSICAL REVIEW B 15 MARCH 1999-IVOLUME 59, NUMBER 11
PRB 59
0163-1829/99/59~11!/7413~9!/$15.00 7413 ©1999 The American Physical Society
system, the adsorption of atoms and molecules is done on
only one of the two slab surfaces exposed and the electro-
static potential is adjusted accordingly.
21
The Kohn-Sham one-electron valence states are expanded
in a basis of plane waves with kinetic energies below 25 Ry
at k-point sampling meshes of 64 and 54 k points within the
first Brillouin zone for the fcc~100! and fcc~111!, respec-
tively. The self-consistent densities are determined by subse-
quent iterative diagonalization of the Kohn-Sham Hamil-
tonian, Fermi-population of the Kohn-Sham states (k
B
T
5 0.1 eV), and Pulay mixing of the resulting electronic
density.
22
All total energies have been extrapolated to k
B
T
5 0 eV.
The chemisorbed atoms and molecules have been placed
in atop, bridge, and hollow @for fcc~111!: fcc-hollow# sites
and have been relaxed to find the optimum chemisorption
energy, while the surface ions have been kept fixed at the
truncated bulk positions @see the Appendix, issue ~iv!#. For
each adsorption system, only results for the most stable ad-
sorption site are reported. The chemisorption energy is cal-
culated as the energy difference:
E
chem
5E
AM
2E
A
2E
M
5
(
i5AM,A,M
p
i
E
i
,
p
AM
51, p
A
5 p
M
521, ~1!
where E
AM
is the total energy of the system of adsorbate A
on metal surface M, and E
A
and E
M
are the total energies of
the isolated adsorbate and metal surface, respectively. In the
case of dissociative adsorption of a molecule, AB, the
chemisorption energy is calculated according to
E
chem
5E
AM
1E
BM
2E
AB
22E
M
. ~2!
The isolated atoms and molecules are treated in very large
supercells of dimensions 10.003 10.253 10.50 Å
3
with
G-point sampling of the Brillouin zone. The odd shape of the
supercells guarantees that nonspherical densities result where
required.
23
For self-consistently determined chemisorption energies,
the exchange-correlation potential entering the Kohn-Sham
Hamiltonian is the functional derivative of the exchange-
correlation energy functional evaluated at the ground-state
density. The variational principle of density functional
theory
1,8
, guarantees, however, that the density and the po-
tential input to the Kohn-Sham Hamiltonian may be varied
independently while only giving rise to errors in the total
energies that are second order in the variations of the density
and potential from their ground-state values. We shall there-
fore also report on non-self-consistently determined chemi-
sorption energies, where the electron densities n
a
and ionic
coordinates resulting from self-consistent, relaxed calcula-
tions with one choice of exchange-correlation potential and
energy functional,
m
XC2
a
and E
XC2
a
, are input to different
types of exchange-correlation energy functionals, E
XC2
b
.
Such non-self-consistent chemisorption energies are calcu-
lated according to
E
chem,
b
5
(
i5AM,A,M
p
i
~
E
i
2E
i,XC2
a
@
n
a
#
1E
i,XC2
b
@
n
a
#
!
.
~3!
For
a
and
b
we will use four different functionals from the
literature: LDA,
25
PW91,
2
PBE,
13
and revPBE,
14
and the
new functional, RPBE, described below. For the four nonlo-
cal GGA functionals, PW91, PBE, revPBE, and RPBE, the
exchange-correlation potential is constructed following
White and Bird.
24
The accuracy of the physical and numerical approxima-
tions made with the present calculational setup is discussed
in the Appendix.
III. RESULTS AND DISCUSSION
We start by considering one chemisorption system, CO
adsorption in the fcc hollow on Pd~111!, in some detail.
Table I summarizes the structural and energetic results for
the system as computed fully self-consistently within the
various exchange-correlation approximations considered.
The distance Z of the center of mass of the chemisorbed CO
from the outermost surface layer, and the bond length b of
the chemisorbed CO, have apparently only little dependence
on the choice of nonlocal exchange-correlation functional. Z
is, however, somewhat smaller when using local exchange-
correlation ~LDA!, which is consistent with the LDA also
favoring a smaller lattice constant, a.
3–7
Despite the small
change in structural properties, the chemisorption energy
E
chem
of the CO depends strongly on the choice of
exchange-correlation functional. With the various GGA
functionals considered, E
chem
varies by as much as 0.4 eV
~approximately 20%! and when comparing the result of the
LDA calculation with the set of GGA calculations, the dif-
ference is of the order of 1 eV.
The large differences in chemisorption energies found in
the fully self-consistent calculations remain even when the
exchange-correlation energies are evaluated only non-self-
consistently, according to Eq. ~3!. This is clear from the up-
per half of Table II, which shows such non-self-consistent
chemisorption energies E
chem,
b
as the regular entries and the
self-consistent ones as the highlighted entries. Each column
of the table gives chemisorption energies with one choice of
b
, i.e., one E
XC2
b
, cf. Eq. ~3!. Comparing down a column,
it is seen that the self-consistently and non-self-consistently
determined chemisorption energies differ somewhat less than
the E
chem
for different E
XC2
b
functionals. This is a conse-
quence of the variational principle as discussed above. The
only noteworthy errors in the non-self-consistent E
chem
arise
when an LDA density is input to a GGA E
XC
functional or
TABLE I. The lattice constant a for bulk Pd and some proper-
ties for CO chemisorbed in the fcc-hollow site on Pd~111! calcu-
lated self-consistently with different exchange-correlation function-
als. Z is the separation of the CO center of mass from the outermost
Pd~111! layer, b is the CO bond length upon adsorption, and E
chem
is the chemisorption energy.
LDA PW91 PBE revPBE RPBE
a Å 3.89 3.99 3.99 4.01 4.02
Z Å 1.85 1.93 1.93 1.95 1.95
b Å 1.19 1.19 1.19 1.19 1.19
E
chem
eV -2.74 -2.07 -1.94 -1.64 -1.65
7414 PRB 59
B. HAMMER, L. B. HANSEN, AND J. K. NO” RSKOV
vice versa, i.e., when E
chem,GGA
@
n
LDA
#
or E
chem,LDA
@
n
GGA
#
are evaluated. At first glance, this suggests that the densities
resulting from self-consistent LDA and GGA calculations
differ somewhat. That this is probably not the case can be
deduced from the lower half of Table II. Here are given the
non-self-consistent chemisorption energies that have all been
evaluated on the basis of self-consistent calculations done
using the PW91 value for the Pd lattice constant a
PW91
.
The non-self-consistent chemisorption energies at the
PW91 volume given in Table II are seen for each
b
to be
practically independent of the exchange-correlation func-
tional used in the underlying self-consistent calculations.
This is the case even when LDA densities are used as input
for a non-self-consistent GGA chemisorption energy, and
shows that the densities are indeed very similar in the LDA
and the GGA’s. It is thus the usage of a volume different
from the equilibrium one, which causes most of the error in
a non-self-consistent calculation. The volumes resulting from
the GGA’s presently considered are, however, very similar
~cf. Table I! and working with any of the self-consistent
GGA volumes will therefore result in small errors in non-
self-consistent GGA calculations. In the remainder of the
paper, we therefore choose to report only on chemisorption
energies for the various exchange-correlation functionals cal-
culated non-self-consistently on the basis of self-consistent
PW91 calculations at the PW91 volume. For CO/Pd~111!
these values have been highlighted in the lower half of Table
II and comparing with the self-consistent values highlighted
in the upper half of the table, we judge that the usage of
non-self-consistent chemisorption energies evaluated at con-
stant volume can be made without a notable loss of accuracy.
In Table III calculated chemisorption energies are pre-
sented for a range of different adsorption systems, using the
self-consistent PW91 density and volume. In order to allow
for an assessment of the overall quality of the calculated
chemisorption energies we include in the table—where
available—the experimental initial chemisorption energies
from the microcalorimetric ~MC! measurements reviewed by
Brown, Kose, and King
26,27
@see the Appendix, issue ~v!#.
We consider these values to represent a most reliable set of
experimental chemisorption energies and we therefore com-
pile the root-mean-square ~rms! deviations
s
O
,
s
CO
, and
s
NO
for the calculated chemisorption energies with respect
to the measured ones for O, CO, and NO chemisorption,
TABLE II. Non-self-consistent chemisorption energies, Eq. ~3!,
for CO/Pd~111! as a function of the input density n
a
and the Pd
lattice constant a ~see Table I!. In the upper half of the table, the
lattice constant and hence the volume is chosen to be a
a
, i.e.,
consistent with the density, while in the lower half of the table, the
volume is kept fixed at the PW91 value. The highlighted entries in
the upper half are the self-consistent chemisorption energies, while
the highlighted entries in the lower half are the approximated values
to be used in the remainder of the paper. The two sets of highlighted
chemisorption energies are very similar.
E
chem,
b
LDA PW91 PBE revPBE RPBE
Self-consistent volume
n
LDA
a
LDA
-2.74 -1.80 -1.69 -1.30 -1.28
n
PW91
a
PW91
-2.95 -2.07 -1.96 -1.59 -1.56
n
PBE
a
PBE
-2.94 -2.06 -1.94 -1.57 -1.55
n
re
v
PBE
a
re
v
PBE
-2.98 -2.12 -2.00 -1.64 -1.62
n
RPBE
a
RPBE
-3.00 -2.15 -2.03 -1.67 -1.65
PW91-volume
n
LDA
a
PW91
-3.01 -2.05 -1.93 -1.55 -1.52
n
PW91
a
PW91
-2.95 -2.07 -1.96 -1.59 -1.56
n
PBE
a
PW91
-2.94 -2.06 -1.94 -1.57 -1.55
n
re
v
PBE
a
PW91
-2.92 -2.06 -1.94 -1.58 -1.55
n
RPBE
a
PW91
-2.92 -2.06 -1.94 -1.58 -1.55
TABLE III. Calculated chemisorption energies as a function of
the exchange-correlation energy functional
b
compared to mea-
sured chemisorption energies. All numbers are in eV per adsorbate.
The rms deviations for the calculated chemisorption energies for O,
CO, and NO,
s
O
,
s
CO
, and
s
NO
, and for all three adsorbates,
s
tot
, have been compiled against the experimental numbers from
Refs. 26 and 27 only. The
s
values in parenthesis arise when the
CO/Rh~100! data is neglected. The E
chem
exp
for O have been evaluated
as
1
2
$
E
chem
exp
(O
2
)2E
b
exp
(O
2
)
%
with the value 5.11 eV ~117.96
kcal/mol
51
! for the O
2
bond energy E
b
exp
(O
2
).
E
chem,
b
E
chem
exp
LDA PW91 PBE revPBE RPBE
O~fcc!/Ni~111! -6.68 -5.38 -5.27 -4.83 -4.77 -4.84
a
O~hol!/Ni~100! -6.97 -5.66 -5.55 -5.10 -5.03 -5.41
a
O~hol!/Rh~100! -6.64 -5.34 -5.23 -4.77 -4.71 -4.56
a
O~fcc!/Pd~111! -5.34 -4.08 -3.98 -3.54 -3.49
O~hol!/Pd~100! -5.39 -4.14 -4.04 -3.59 -3.53
s
O
1.84 0.57 0.47 0.22 0.24
CO~fcc!/Ni~111! -2.85 -1.99 -1.88 -1.52 -1.49 -1.35
a
CO~hol!/Ni~100! -3.05 -2.11 -2.00 -1.62 -1.58 -1.26
a
CO~brd!/Rh~100! -3.02 -2.28 -2.16 -1.84 -1.81 -1.19
a
CO~fcc!/Pd~111! -2.95 -2.07 -1.96 -1.59 -1.56 ~-1.47!
b
CO~brd!/Pd~100! -2.77 -1.98 -1.87 -1.53 -1.50 -1.69
a
s
CO
1.58 0.78 0.67 0.39 0.37
~1.49!~0.64!~0.54!~0.25!~0.23!
NO~hol!
*
/Ni~100! -6.31 -4.52 -4.41 -3.79 -3.68 -3.99
a
NO~brd!/Rh~100! -3.73 -2.76 -2.67 -2.31 -2.28
NO~fcc!/Pd~111! -3.27 -2.20 -2.12 -1.72 -1.67 ~-1.86!
c
NO~hol!/Pd~100! -3.19 -2.12 -2.04 -1.63 -1.58 -1.61
d
s
NO
1.98 0.52 0.43 0.14 0.22
s
tot
1.76 0.66 0.56 0.30 0.30
~1.76!~0.58!~0.48!~0.21!~0.23!
*
Dissociative adsorption.
a
MC experiments reviewed by Brown, Kose, and King, Ref. 26.
b
TPD experiment by Conrad et al., Ref. 49.
c
TPD experiment by Ramsier et al., Ref. 50
d
MC experiment by Yeo, Vattuone, and King, Ref. 27.
PRB 59
7415IMPROVED ADSORPTION ENERGETICS WITHIN . . .
respectively, and
s
tot
for all adsorption systems considered.
It appears clearly from the table that the LDA functional
gives adsorbates overbound by more than 1.5 eV. The PW91
and PBE functionals, on the other hand, give more moderate
chemisorption energies, which, however, are still too large
numerically by about 0.6 eV per adsorbate. Finally, the
revPBE functional and the RPBE functional ~to be discussed
below! prove rather accurate in the description of the chemi-
sorption energies for the present set of adsorbate-metal sys-
tems. For these functionals, the typical discrepancy between
the theoretical and experimental chemisorption energies is of
the order 0.30 eV. We note that regardless of which
exchange-correlation functional is considered, in particular,
the CO/Rh~100! system gives rise the the large rms devia-
tions. This raises some question about quality of this one
experimental value of 1.19 eV for the initital chemisorption
energy. The value further compares poorly with the tempera-
ture programmed desorption ~TPD! and isothermal desorp-
tion experiments in Ref. 28, which are reported to give val-
ues for the saturation chemisorption energy around 1.42 eV.
Values for
s
CO
and
s
tot
evaluated without the CO/Rh~100!
system are included in the table in parentheses. It is seen that
neglecting the CO/Rh~100! system the
s
tot
would become
approximately 0.5 eV for the PW91 and PBE functionals and
less than 0.25 eV for the revPBE and RPBE functionals. We
note that the uncertainty of the measurements of the chemi-
sorption energies is of the order 0.2 eV and 0.1 eV for atomic
and molecular chemisorption, respectively.
26
The finding of large overbinding of adsorbates to
transition-metal surfaces in the LDA and less so in the GGA,
e.g., represented by the PW91, was realized early on
8–12
and
many groups are now routinely using the GGA functionals
for such surface studies.
29–40
Besides the evidence for the
improvement of the chemisorption energetics with the
revPBE presented in this work, there is at present one ex-
ample where the revPBE has been used for a chemisorption
study—N
2
adsorption and dissociation on Fe~111!~Ref.
41!—in which case similar improvements were found. Sev-
eral suggestions for the reason for the reduction in the
overbinding have been put forward, the simplest of which is
the following:
42
~i! GGA functionals favor reduced density
gradients, s}
u
¹
r
u
/
r
4/3
; ~ii! the volume of space with large s
values scales with the free surface area, where ‘‘surface’’
may represent both a solid surface and the surface of mol-
ecules and atoms; ~iii! a system of a molecule chemisorbed
on a solid surface exposes less surface ~and therefore has less
volume with large s values! than do the reference systems of
a clean solid surface and a gas phase molecule; ~iv! conse-
quently, the adsorption system is destabilized over the refer-
ence system—i.e., the chemisorption will become less attrac-
tive and the chemisorption energy will become less
negative—when using a GGA. The loss of surface and hence
loss of regions with large reduced density gradients in a
chemisorption event is illustrated schematically in Fig. 1.
Some evidence for this simple picture of the effect of the
GGA functionals on a chemisorption system can be gained
from Fig. 2. In Fig. 2~a! is plotted for CO/Pd~100!, the
gradient-dependent contributions in the GGA to the chemi-
sorption energy:
De
chem,GGA
~
r
!
5
(
i5AM,A,M
p
i
n
i
~
r
!
@
e
i,GGA
~
r
!
2
e
i,LDA
~
r
!
#
,
~4!
where n
i
(r)
e
i,XC
(r) is the XC energy density ~evaluated lo-
cally, XC5LDA, or nonlocally, XC5GGA! at r for each of
the systems, i5AM,A,M. A plane perpendicular to the
Pd~100! surface, through two Pd atoms in the surface and
through a bridge-bonded CO molecule at these Pd atoms is
chosen. It is apparent that, in this plane, the change in the
chemisorption energy comes from the region of contact be-
tween the molecule of the surface, i.e., from the region of
‘‘loss of surface area’’ caused by the adsorption. One should,
FIG. 1. Outside a solid surface, e.g., Pd, and all round a mol-
ecule, e.g. CO, the electron density falls off. These regions of ‘‘sur-
face,’’ here shown shaded, have thus large reduced density gradi-
ents, s}
u
¹
r
u
/
r
4/3
. Once a chemisorption bond between the solid
surface and the molecule is formed less surface results. The loss of
surface appears in the region between the solid surface and the
chemisorbed molecule.
FIG. 2. ~a! The spatially resolved contribution to the chemisorp-
tion energy from nonlocal exchange-correlation, De
chem,PW91
(r) for
CO adsorption on Pd~100!. The contours are separated by 0.19
eV/Å
3
. ~b! The same quantity, but layer integrated, De
chem,GGA
(z)
plotted for the different GGA’s considered.
7416 PRB 59
B. HAMMER, L. B. HANSEN, AND J. K. NO” RSKOV
