This document discusses using density functional theory with different basis sets (Gaussian, plane waves, numerical) to calculate exchange coupling constants in transition metal complexes. It compares the accuracy and reliability of these approaches by calculating exchange coupling constants and spin distributions for three test complexes. The plane wave and numerical basis set approaches are found to be accurate alternatives to the more established Gaussian basis functions for calculating these properties, while also allowing for larger system sizes to be studied. Pseudopotentials are also found to not significantly affect the calculation of exchange coupling constants.

1. Exchange coupling in transition-metal complexes via density-functional
theory: Comparison and reliability of different basis set approaches
Eliseo Ruiz,a͒
Antonio Rodríguez-Fortea,b͒
Javier Tercero, and Thomas Cauchy
Department de Química Inorgànica, and Centre Especial de Recerca en Química Teòrica (CeRQT),
Universitat de Barcelona, Diagonal 647, 08028 Barcelona, Spain
Carlo Massobrio
Institut de Physique et de Chimie des Matériaux de Strasbourg (IPCMS) 23, rue du Loess F-67037
Strasbourg, France
͑Received 22 October 2004; accepted 21 June 2005; published online 22 August 2005͒
Theoretical methods based on density-functional theory with Gaussian, plane waves, and numerical
basis sets were employed to evaluate the exchange coupling constants in transition-metal
complexes. In the case of the numerical basis set, the effect of different computational parameters
was tested. We analyzed whether and how the use of pseudopotentials affects the calculation of the
exchange coupling constants. For the three different basis sets, a comparison of the exchange
coupling constants and spin distributions shows that both the plane-wave and the numerical basis set
approaches are accurate and reliable alternatives to the more established Gaussian basis
functions. © 2005 American Institute of Physics. ͓DOI: 10.1063/1.1999631͔
I. INTRODUCTION
Electronic structure methods based on density-functional
theory ͑DFT͒ are widely employed within both the chemistry
and the physics community. Methods originally devised by
physicists for solid-state applications1
became popular
among chemists during the 1990s as a consequence of the
implementation of such methods in widely used commercial
codes for molecular calculations.2
This has contributed to
laying the foundations of a common computational strategy.
However, some notable differences do exist. On the one
hand, physicists have adopted mostly plane waves, due to
their straight implementation to treat periodic systems,3
and
have recently been very much attracted by numerical func-
tions resulting in very fast algorithms.4
On the other hand,
theoretical chemists have implemented the DFT methods
within the Gaussian framework traditionally employed in
conjunction with Hartree-Fock or ab initio multiconfigura-
tion approaches.5
Another fundamental difference is the in-
clusion, by physicists, of some approximations to obtain
faster algorithms.6
One example is found in the calculation
of the Coulomb contributions, which contrasts with the way
that chemists usually compute this term exactly through the
four center integrals.
The study of the magnetic properties of molecular sys-
tems using DFT methods and Gaussian functions as basis set
is by now well established.7,8
By using hybrid functionals,
accurate exchange coupling interactions between the para-
magnetic centers can be obtained, providing local micro-
scopic information of the magnetic properties.9,10
Turning to
periodic systems, the plane-wave approach has been less ex-
ploited, mostly focusing on the qualitative nature of the mag-
netic interactions.11,12
This is due to the lack of a general
procedure to extract the experimental values of the exchange
coupling constants from the magnetic susceptibility data for
periodic systems. It would be highly desirable to extend the
plane-wave approach to the calculation of the exchange cou-
pling constants based on appropriate exchange-correlation
functionals. In this context, we recall that local or nonlocal
functionals usually overestimate the interactions between
paramagnetic centers.13
These considerations also apply to
the fast numerical function-based approaches, essentially un-
exploited for the calculation of exchange couplings of either
isolated or three-dimensional systems.
The main goal of this work is to assess the ability of
computer codes based on plane waves and numerical func-
tions to calculate exchange coupling constants of transition-
metal complexes. Moreover, we intend to check the influence
of the use of pseudopotentials in such calculations. The com-
bination of hybrid functionals and Gaussian basis sets has
proved to be very accurate to evaluate the exchange coupling
constants.8–10,14
However, these calculations are severely
limited by the largest affordable sizes, i.e., around 200 or
300 atoms These sizes are typical of single-molecule
magnets,15
such as the Fe8 and Mn12 complexes.16
This upper
value can largely be increased by resorting to plane-wave or
numerical basis sets ͑1000 atoms͒ for both isolated and pe-
riodic systems,4,17,18
thereby making these approaches valu-
able alternatives to study new systems containing several
transition metals, such as Fe19 or Mn25 complexes.19,20
We
have to keep in mind that accuracy is the foremost require-
ment for the estimate of exchange couplings constants that
are as small as 10–50 cm−1
͑0.001–0.005 eV͒.
II. COMPUTATIONAL DETAILS
The J value for a dinuclear transition-metal complex can
be obtained using the following equation:
a͒
Electronic mail: eliseo.ruiz@qi.ub.es
b͒
Present address: Departament de Química Física i Inorgànica, Universitat
Rovira i Virgili, Marcel.lí Domingo s/n, 43007, Tarragona, Spain.
THE JOURNAL OF CHEMICAL PHYSICS 123, 074102 ͑2005͒
0021-9606/2005/123͑7͒/074102/10/$22.50 © 2005 American Institute of Physics123, 074102-1
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2. EHS − EBS = − ͑2S1S2 + S2͒J, ͑1͒
where S1 and S2 are the total spins of the paramagnetic cen-
ters and S1ϾS2 has been assumed for heterodinuclear com-
plexes using the Heisenberg Hamiltonian:
Hˆ = − JSˆ
1Sˆ
2. ͑2͒
The first energy, EHS, corresponds to the high spin solution, a
triplet state for dinuclear CuII
complexes and the second one,
EBS, to the broken-symmetry solution, a single-determinant
wave function with Sz=0 and opposite spins in both para-
magnetic centers.21
We have found that, when using DFT-
based wave functions, a good estimate of the singlet state
energy for CuII
dinuclear complexes can be obtained directly
from the energy of a broken-symmetry single-determinant
solution.8,9
Polo et al. have shown that this is a consequence
of the inclusion of the nondynamic correlation effects in the
commonly used exchange functionals through the self-
interaction error.22
Hence, if a spin projection is included in
addition, a cancellation ͑or double counting͒ of such corre-
lation effects results.
A detailed description of the computational strategy to
calculate the exchange coupling constants in polynuclear
complexes can be found elsewhere.19,23
The exchange cou-
pling constants are introduced by a phenomenological
Heisenberg Hamiltonian,
Hˆ = − ͚jϾk
JiSˆ
jSˆ
k, ͑3͒
͑where i labels the different kind of exchange coupling con-
stants, while j and k refer to the different paramagnetic cen-
ters͒ to describe the interactions between each pair of para-
magnetic transition-metal atoms present in the polynuclear
complex. At a practical level, for the evaluation of the n
different coupling constants, we need to perform the calcu-
lation of the energy for n+1 different spin distributions.
Thus, we can solve the system of n equations obtained from
the energy differences related to the diagonal terms of the
Hamiltonian matrix.
We employed three different computer codes to perform
the calculations depending on the kind of basis sets. The test
systems are three molecular complexes selected for a pre-
liminary set of calculations in a previous publication.24
The
all-electron calculations for the CuII
complexes with Gauss-
ian functions have been performed by using the GAUSSIAN98
code ͑a.11 version͒.25
We employed a triple-␨ quality all-
electron basis set for copper, manganese, and iron atoms
͑TZP͒26
and a double-␨ all-electron basis set proposed by
Schaefer et al.27
for the other elements. The calculations to
check the effect of the use of pseudopotentials were per-
formed using Stoll-Preuss core potentials ͑SDD͒28
and Los
Alamos large core pseudopotentials ͑Lanl1͒.29
Thus, the
Lanl1+TZP label indicates the use of Lanl1 pseudopotential
together with the valence part of the TZP basis set while the
Lanl1 label is used for the original implementation of Lanl1
with its own basis set.
The plane-wave calculations were performed using the
Car-Parrinello molecular dynamics ͑CPMD͒ code version
3.7.2. All valence electrons were treated explicitly via Fou-
rier expansion in plane waves with an energy cutoff Ecut
equal to 90 Ry at the k=0 point of the supercell. Norm-
conserving pseudopotentials generated following the scheme
by Trouiller and Martins30
are used to account for core-
valence interaction. The simulation box was chosen large
enough so as to minimize the interaction with the periodic
images. We performed the calculations with the hybrid
B3LYP functional,31
recently implemented in this code. Pre-
TABLE I. Calculated exchange coupling constants J ͑cm−1
͒ for copper ac-
etate ͑see Fig. 5͒ using the SIESTA code with PBE functional and different
values of the energy shift, maximum kinetic energy, and basis set quality on
the Cu atom. It is also indicated for comparison the available experimental
data and the all-electron ͑ae͒ Gaussian results using a triple-␨ basis for
copper atoms and double-␨ quality for the other atoms together with the
pseudopotential ͑ps͒ Gaussian results obtained employing the Stuttgart
pseudopotential ͑SDD͒ for the Cu atoms.
Energy shift
͑meV͒
Max. kinetic energy
͑Ry͒ Basis set
J
͑cm−1
͒
200 150 double ␨ −756
200 250 double ␨ −749
100 150 double ␨ −783
50 150 double ␨ −770
50 250 double ␨ −765
50 250 triple-␨ Cu −754
30 250 triple-␨ Cu −747
Gaussian ae PBE −776
Gaussian ps PBE −664
Gaussian ae B3LYP −299
Gaussian ps B3LYP −269
Expt. −297
FIG. 1. Representation of the poly-
nuclear Fe4 complex showing the three
different exchange pathways present
in this structure.
074102-2 Ruiz et al. J. Chem. Phys. 123, 074102 ͑2005͒
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3. viously, we had studied these systems using the BLYP
functional.24,32
The calculations using numerical functions were carried
out using the Spanish Initiative for Electronic Simulations
with Thousands of Atoms ͑SIESTA͒ code version 1.3.33
We
employed the generalized-gradient approximation functional
proposed by Perdew, Burke, and Erzernhof34
͑PBE͒ and the
BLYP functional. Only valence electrons are included in the
calculations as in the case of the plane waves, the cores being
replaced by norm-conserving scalar relativistic pseudopoten-
tials factorized in the Kleinman-Bylander form.35
These
pseudopotentials are generated according to the procedure
proposed by Trouiller and Martins.30
The core radii for the s,
p, and d components for iron and copper atoms are all
2.00 a.u. and we have included partial-core corrections for
both atoms to provide a better description of the core region.
The cutoff radii were 1.15 for oxygen, hydrogen, and nitro-
gen atoms, 1.25 for carbon atoms, and 1.60 for chlorine at-
oms.
We employed different numerical basis sets to analyze
their influence in the calculated exchange coupling constants
obtained via the SIESTA code. In this kind of calculations,
there are two key parameters that control the accuracy.4,17,18
The numerical wave function is zero at a radius larger than
the chosen confinement radius rc, whose value is different for
each atomic orbital. The confinement radius of different or-
bitals is determined by a single parameter, the energy shift
that corresponds to the energy increase of the atomic eigen-
state due to the confinement. The integrals of the self-
consistent terms are calculated with the help of a regular
real-space grid in which the electron density is projected.
The grid spacing is determined by the maximum kinetic en-
ergy of the plane waves that can be represented in that grid.
To the best of our knowledge, no systematic study of the
influence of these parameters on the calculated J values has
been carried out so far. Therefore, one of our goals is to
determine which values have to be employed to achieve the
accuracy needed for the calculation of the exchange coupling
constants.
III. RESULTS AND DISCUSSION
A. Computational parameters in the calculations
with numerical basis sets
In previous papers, we have studied extensively the in-
fluence of the basis sets and functionals on the calculation of
FIG. 2. Dependence of the three calculated exchange coupling constants J
͑cm−1
͒ for the butterfly FeIII
complex using the SIESTA code with PBE func-
tional and different values of the mesh cutoff parameter with an energy shift
of 50 meV using a triple-␨ basis for iron atoms and double-␨ quality basis
with a polarization function for the other atoms.
FIG. 3. Dependence of the three calculated exchange coupling constants J
͑cm−1
͒ for the butterfly FeIII
complex ͑see Fig. 1͒ using the SIESTA code with
PBE functional and different values of the energy shift parameter ͑better
accuracy corresponds to smaller values of this parameter͒ with a mesh cutoff
of 200 Ry employing a triple-␨ basis for iron atoms and double-␨ quality
basis with a polarization function for the other atoms.
074102-3 Exchange coupling in transition-metal complexes J. Chem. Phys. 123, 074102 ͑2005͒
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4. exchange coupling constants using Gaussians as basis
functions.9,21
The plane-wave calculations can be considered
as almost free of parameters because the basis set is con-
trolled just by the value of the energy cutoff employed in the
study.3
However, the numerical calculations as implemented
in the SIESTA code have different parameters that control the
accuracy of the calculation.4,17,18
As far as the magnetic
properties are concerned, copper͑II͒ acetate is one of the
most studied CuII
dinuclear complexes ͑see Fig. 5͒.8
For this
reason, we selected such system to study the influence of the
parameters on the calculations employing numerical basis
sets carried out with SIESTA.
The results shown in Table I for copper͑II͒ acetate show
a small influence of energy shift and the kinetic-energy pa-
rameters on the coupling constant for the different values
used in the calculations. The effect of the quality of the basis
set is quite small in comparison with the changes obtained
when Gaussian functions are used. It is also worth noting
that such results are close to the value obtained with the
GAUSSIAN code with the same functional. As is expected, in
agreement with previous results, all the calculated J values
using the generalized-gradient approximation ͑GGA͒ func-
tionals are too large in comparison with the hybrid function-
als and the experimental data.9,21
However, we also observe
that the use of pseudopotentials for the Cu atoms with
Gaussian functions diminishes in this case the calculated J
value. This effect will be discussed deeper in Sec. III B.
In order to perform a more accurate analysis of the in-
fluence of such parameters, we have selected a more compli-
cated polynuclear system, a butterfly Fe4 complex36
͑see Fig.
1͒ with two central FeIII
cations in the body positions and
two external ones placed in the wing positions. In such com-
plex, each FeIII
cation has five unpaired electrons. In a tetra-
nuclear complex, up to six different Jij exchange interactions
exist ͑J12, J13, J14, J23, J24, and J34͒. Here, as a consequence
of the rhombuslike geometrical arrangement of the FeIII
nu-
clei, four out of six exchange interactions are identical giving
rise to only three different coupling constants: one corre-
sponding to the weak interaction between the central FeIII
cations ͑Jbb͒, another due to the weak interaction between the
two external cations ͑Jww͒, and four identical coupling con-
stants that correspond to the four strong interactions between
central and external cations ͑Jwb͒. The dependence of these
three exchange coupling constants on the energy shift and on
the kinetic-energy parameters is represented in Figs. 2 and 3,
respectively. Experimentally only two J values were consid-
ered to fit the magnetic susceptibility data, Jbb and Jwb, being
−17.8 and −91.0 cm−1
, respectively.36
Only the Jwb value can
be considered accurate because the presence of four strong
wing-body interactions masks the effect of the weak body-
body and wing-wing interactions in the magnetic susceptibil-
ity data.36
Hence, only Jwb can be considered as a good ref-
erence for comparison with the theoretical results ͑see Table
II͒.
The results for the Fe4 complex indicate that 200 Ry and
50 meV are reasonable minimal values of the mesh cutoff
and the energy shift parameters, respectively, to reach accu-
racy around 1 cm−1
. These values will provide a better accu-
racy than those proposed to obtain geometrical structures and
bond energies.18
The analysis of the results also shows a
larger dependence on the mesh cutoff for the weakest ex-
change interactions while, in the case of the energy shift, all
the interactions show a similar sensitivity to changes of this
parameter. The calculated values reproduce correctly the ex-
TABLE II. Calculated exchange coupling constants J ͑cm−1
͒ for the Fe4 complex ͑see Fig. 1͒ using the SIESTA
code with PBE functional and basis sets of different quality ͑energy shift of 50 meV and mesh cutoff of
200 Ry͒. For the Gaussian calculations we have employed the PBE and B3LYP functionals, the double-zeta
basis set of Schaefer et al. for the main group elements and the triple zeta of the same authors for iron atoms
͑TZP͒. All-electron and pseudopotential calculations with different number of core electrons ͑SDD and Lanl1
pseudopotentials with 10 and 18 core electrons, respectively͒ have been employed. The available experimental
values are also provided for comparison36
.
Other elements basis set Fe basis set Jwb ͑cm−1
͒ Jww ͑cm−1
͒ Jbb ͑cm−1
͒
Numerical basis set
PBE DZ TZP −60.5 −5.1 −3.9
PBE DZP TZP −64.4 −5.5 −5.3
PBE DZP TZ −65.3 −5.5 −5.6
PBE DZP DZP −64.1 −5.4 −4.8
Gaussian basis set
PBE ae DZ ae TZP −192.7 −82.0 −51.3
PBE ae DZ ps SDD+TZP −153.9 −39.6 −28.5
PBE ae DZ ps Lanl1+TZP −83.6 −13.8 −9.6
B3LYP ae DZ ae TZP −80.0 −5.8 +8.3
B3LYP ae DZ ps SDD+TZP −69.6 −5.1 +12.5
B3LYP ae DZ ps Lanl1+TZP −60.2 −12.6 +6.7
Expt. −91.0 ¯ −17.8
074102-4 Ruiz et al. J. Chem. Phys. 123, 074102 ͑2005͒
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5. perimental results indicating that in all cases the coupling is
antiferromagnetic.36
Finally, we also analyzed the influence of other param-
eters in the convergence of the calculations with numerical
basis sets, such as the number of cycles for the Pulay ap-
proach, the mixing weight between cycles, and the electronic
temperature. We obtained that the best values for such pa-
rameters are 5 or 6 cycles for the Pulay approach, around
0.20–0.25 for the mixing weight, and 500–600 K for the
electronic temperature. Such values for the electronic tem-
perature provide the best self-consistent-field ͑SCF͒ conver-
gence but it results in too large differences using total ener-
gies or free energies in the calculation of the J value. Hence
we propose to use a value around 300 K that provides a
relative good convergence giving similar J values even if
such value could depend on the system.
B. Influence of the basis sets and pseudopotentials
The dependence of the calculated J values on the quality
of the basis set for the Fe4 complex using the numerical
calculations was also tested and the results are indicated in
Table II. This dependence is quite limited, as observed for
CuII
complexes. The more noticeable changes are provided
by the inclusion of polarization functions in the main group
elements but the calculated J values remain practically un-
changed for different basis sets. However, for the Fe4 com-
plex the comparison of the SIESTA results with those obtained
with all-electron Gaussian calculations with the same GGA
functional shows dramatic differences ͑see Table II͒, while
the results for the copper acetate showed only a slight de-
crease of the J values when pseudopotentials were employed
in the Gaussian calculations ͑see Table I͒.
In order to understand the origin of such discrepancy
when using a GGA functional, we repeated the Gaussian
calculations for the Fe4 complex replacing the core shells by
pseudopotentials. We employed two different sets of pseudo-
potentials with different number of electrons to check gradu-
ally the changes in the calculated J values. The substitution
of the core shells by the pseudopotentials causes an impor-
tant decrease of the J values being those obtained with the
large core pseudopotential close to the SIESTA values. The
decrease of the calculated J values using the GGA function-
als is basically due to the inclusion of pseudopotentials in the
iron atoms. We checked that the use of pseudopotentials for
the other atoms induces very small changes in the J values.
In order to rationalize the influence of the use of pseudopo-
tentials in the electronic structure, we analyzed the orbital
population values of such calculations. Despite the fact that
the atomic-orbital values are relatively similar in all cases,
the substitution of the core shells by pseudopotentials usually
results in an increase of the spin localization of metal d or-
bitals. This effect compensates partially the overestimation
of the spin delocalization produced by the GGA
functionals8,9
giving better results when pseudopotentials are
used in combination with Gaussian functions. In the case of
the hybrid B3LYP functional ͑see Table II͒, however, the
calculated J values do not depend so critically on the use of
pseudopotentials because these functionals provide a more
correct description of the spin delocalization in transition-
metal complexes.
A question that arises from these results what is the rea-
son of the different behavior between the results for the CuII
system where all the basis sets and pseudopotentials produce
similar results while in the Fe4 complex, there is a large
difference between the values obtained with pseudopotentials
and all-electron basis sets. Two possible reasons can be pro-
FIG. 4. Representation of the three dinuclear manganese complexes
͑from up to down͒, complex 1: ͓Mn2͑␮-O͒2͑N-Eth-sal͒2͔
͑N-Eth-sal=N-ethyl-salicyliledeneamine͒,37
complex 2: ͓Mn2͑␮-O͒͑␮
-OAc͒2͑tacn͒2͔2+
͑tacn=1,4,7-trizazacyclononane͒38
and com-
plex 3: ͓Mn2͑OAc͒2͑BpmpH͔͒+
, ͑BpmpH=2,6-bis͓bis͑2-pyridylmethyl͒
aminomethyl͔-4-methyl-phenol͒39
.
074102-5 Exchange coupling in transition-metal complexes J. Chem. Phys. 123, 074102 ͑2005͒
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6. posed, the first one related to the different number of un-
paired electron between the CuII
and FeIII
cations that could
result in important changes in the spin distribution in the
case of the FeIII
complex due to the larger spin density, and
the second possible reason would be the influence of the
charge in the cation that will modify considerably the energy
of the orbitals bearing the unpaired electrons. Hence, in order
to clarify the source of the problem we have selected three
complexes of manganese with different oxidation states37–39
and consequently, different number of unpaired electrons
͑see Fig. 4͒. The results for such systems are collected in
Table III. The analysis of the results indicates that the MnII
complex ͑3͒ shows a similar behavior than that found for the
copper acetate where all the basis set and pseudopotentials
provide similar J values. However, for the 1 and 2 complexes
with MnIV
and MnIII
cations, respectively, there are very im-
portant differences between the calculated values. These re-
sults confirm that the origin for the large difference when
using pseudopotentials in the Gaussian calculations is the
presence of a large charge in the paramagnetic center, while
the amount of spin density does not seem important as it is
proven by the different behavior found for the d5
MnII
and
FeIII
cations both with five unpaired electrons. The calculated
J values obtained the B3LYP functional and all-electron ba-
sis set is in excellent agreement with the experimental data
confirming our previous results. It is worth noting that the
substitution of the core shells by pseudopotentials produces
in all cases a decrease of the antiferromagnetic contribution
due to the larger localization of the spin density. This fact is
clearly reflected in the spin population values of the MnIV
cations in the complex 1, being, respectively, 2.88, 2.92, and
3.52 for the all-electron basis sets, SDD, and Lanl1 pseudo-
potentials. The dramatic changes found especially in the case
of the Lanl1 pseudopotentials for the 1 and 2 complexes due
to the large charge of the cations confirm that a separation of
core and valence orbitals is problematic for the first row
elements as was pointed out previously by other authors.40
C. Comparison between the different approaches
Now we focus on the application of DFT methods based
on Gaussian, plane wave, or numerical basis functions to
calculate the exchange coupling constants in transition-metal
polynuclear complexes. Although Gaussian basis sets have
been extensively employed for this purpose, it is worthwhile
to ascertain whether plane waves and numerical functions are
a good alternative. With this purpose in mind, we have se-
lected three dinuclear CuII
complexes41
as in previous, pre-
liminary study ͑see Fig. 5͒.24
The calculated J values are
collected in Table IV. The values obtained with the numeri-
cal basis and the PBE functional are very similar to those
obtained with the same functional and a Gaussian basis set
for complexes 4 and 6; however, important differences are
noticeable for complex 5. Furthermore, the results using
BLYP functional are in the three complexes very similar
when comparing numerical and Gaussian basis sets, as we
have seen previously. Interestingly, results with plane wave
as basis set with the BLYP functional provide, for the three
complexes, J values closer to the experimental results. Yet,
the discrepancy with experiment is far from being negligible,
due to the well-known shortcomings of the above GGA
functionals.9,22
In what follows, we used a new implementation of the
hybrid B3LYP functional within plane-wave basis sets, with
the intent of providing a new computational framework for
the calculation of the J values. The B3LYP functional was
extensively employed with Gaussian basis sets, but it was
not yet implemented in the SIESTA code. We point out that
TABLE III. Calculated exchange coupling constants J ͑cm−1
͒ for three manganese dinuclear complexes, a
double oxobridged MnIV
complex ͓Mn2͑␮-O͒2͑N-Eth-sal͒2͔ ͑1͒, an oxo- and double acetato-bridged MnIII
complex ͓Mn2͑␮-O͒͑␮-OAc͒2͑tacn͒2͔2+
͑2͒, and a fenoxo- and double acetato-bridged MnII
complex
͓Mn2͑OAc͒2͑BpmpH͔͒+
͑3͒ ͑see Fig. 4͒ using the SIESTA code with PBE functional and basis sets of different
quality ͑energy shift of 50 meV and mesh cutoff of 200 Ry͒. For the Gaussian calculations we have employed
the PBE and B3LYP functionals, the double-zeta basis set of Schaefer et al. for the main group elements and the
triple zeta of the same authors for manganese atoms ͑TZP͒. All-electron and pseudopotential calculations with
different number of core electrons ͑SDD and Lanl1 pseudopotentials with 10 and 18 core electrons, respec-
tively͒ have been employed. The available experimental values are also provided for comparison37–39
.
Other elements basis set Mn basis set Complex 1 Complex 2 Complex 3
Numerical basis set
PBE DZ TZP −376.3 +2.2 −25.1
Gaussian basis set
PBE ae DZ ae TZP −409 −20.7 −26.5
PBE ae DZ ps SDD+TZP −383.1 −12.5 −21.2
PBE ae DZ ps Lanl1 +24.5 +158.4 −5.5
B3LYP ae DZ ae TZP −190.6 +16.6 −13.4
B3LYP ae DZ ps Lanl1 −21.8 +113.2 −5.7
Expt. −300 +18 −9.6
074102-6 Ruiz et al. J. Chem. Phys. 123, 074102 ͑2005͒
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7. this kind of calculation requires very large computational
resources. For instance, a single-point energy calculation for
one of the complexes takes around 30 days on 16 parallel
POWER3 processors. Moreover, a considerable effort in the
optimization of the implementation of exact exchange is nec-
essary in order to reduce the computer time. For all these
reasons, we only calculated two J values corresponding to
the smallest systems selecting two complexes with different
nature in the exchange interaction, that is, one with ferro-
magnetic and the other with antiferromagnetic coupling. The
results are close to those obtained with the Gaussian basis set
showing that this method could be employed to calculate the
exchange coupling constants in periodic systems, for which
the plane-wave codes are more popular and flexible than
those based on Gaussian functions. We note that the J values
obtained with the plane-wave methods are slightly smaller
than those obtained with the same functional and other basis
sets.
The well-known problem of the overestimation of the
calculated J values with GGA functionals and the improved
performances of the hybrid functionals can be rationalized in
connection with a too large delocalization of the spin density
at the ligands, and consequently a smaller spin density at the
metal centers.8,9
In order to further understand this effect,
Table V shows the calculated atomic Mulliken spin popula-
tions for the triplet state of the complex 6 that show ferro-
magnetic coupling. For the plane-wave calculations, we pro-
jected the electron density on the valence orbitals of the basis
set that was employed for the Gaussian calculations and then
performed a Mulliken analysis with the atom-centered basis
set. The spin population values obtained from the plane-
wave calculations indicate a larger localization of the spin
density at the metal centers, thereby accounting for the
smaller J values in comparison with those obtained with the
same functionals and Gaussian functions ͑see Table V͒.
This dependence between the calculated J values and the
localization of the spin density stands out in Fig. 6. In all
cases the spin distributions are relatively similar, but those
corresponding to the plane-wave calculations show a more
localized picture, probably due to the use of pseudopoten-
tials, as we have seen for the Fe4 calculations. This fact can
be clearly identified by the lack of the spin density at the
FIG. 5. Representation of the three dinuclear CuII
complexes ͑from up to
down͒, complex 4: copper acetate molecule, complex 5: ͓Cu2͑␮
-OH͒2͑bipym͒2͔͑NO3͒2 ·4H2O, and complex 6: ͓͑dpt͒Cu͑␮
-Cl͒2Cu͑dpt͔͒Cl2, ͑dpt=dipropylenetriamine͒41
.
TABLE IV. Calculated exchange coupling constants J ͑cm−1
͒ for three CuII
dinuclear complexes, the copper acetate ͑4͒, a hydroxo-bridged CuII
com-
plex ͓Cu2͑␮-OH͒2͑bipym͒2͔͑NO3͒2 ·4H2O ͑5͒, and a chloro-bridged CuII
complex ͓͑dpt͒Cu͑␮-Cl͒2Cu͑dpt͔͒Cl2 ͑6͒ ͑see Fig. 5͒ using three different
approaches described in Sec. II. The experimental data obtained from the
magnetic susceptibility is also provided for comparison41
.
Method Complex 4 Complex 5 Complex 6
Numerical basis set
PBE −747 +208 +83
BLYP −751 +210 +120
Plane-wave basis set
BLYP −518 +95 +61
B3LYP −280 ¯ +38
Gaussian basis set
PBE −776 +117 +95
BLYP −779 +221 +100
B3LYP −299 +113 +56
Expt. −297 +114 +42.9
074102-7 Exchange coupling in transition-metal complexes J. Chem. Phys. 123, 074102 ͑2005͒
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8. terminal chlorine atoms ͑Cl3͒ using the BLYP functional and
plane waves. These results confirm, however, that even when
employing plane waves as basis functions there is a too large
delocalization of the spin density towards the ligands with
the usual GGA functionals when comparing to B3LYP. Thus,
larger localization obtained with the hybrid functional results
in a smaller exchange coupling constant. This behavior ob-
tained for the CuII
complex is the opposite one to the one
found in the manganese complexes. Such difference is prob-
ably due to the predominance of the delocalization mecha-
nism in the copper complexes, hence, larger localization re-
duces the strength of the coupling while in the manganese
complexes the spin polarization is the predominant mecha-
nism because the unpaired electrons are mainly in the non-
bonding t2g orbitals.
Likewise, we have plotted the isodensity surfaces for the
five theoretical approaches in Fig. 7. In all cases the spin
distributions are relatively similar, but those corresponding
to the plane-wave calculations show a more localized pic-
ture. This fact can be clearly ascribed to the lack of the spin
density at the terminal chlorine atoms ͑Cl3͒ using the BLYP
functional and confirms the too large delocalization obtained
with the usual GGA functionals. In all cases, there is a pre-
dominance of the delocalization mechanism being the polar-
TABLE V. Calculated atomic Mulliken spin populations for chloro-bridged CuII
complex ͓͑dpt͒Cu͑␮
-Cl͒2Cu͑dpt͔͒Cl2 ͑6͒ ͑see Fig. 5͒ using three different approaches described in Sec. II. The two bridging
chlorine atoms are indicated with the labels Cl1 and Cl2 while the terminal one is labeled as Cl3.
Method Cu Cl1 Cl2 N Cl3
Numerical basis set
BLYP +0.427 +0.236 −0.003 +0.113 +0.083
Plane-wave basis set
BLYP +0.534 +0.215 −0.005 +0.102 +0.023
B3LYP +0.654 +0.163 −0.007 +0.081 +0.003
Gaussian basis set
BLYP +0.469 +0.196 +0.001 +0.109 +0.102
B3LYP +0.583 +0.176 −0.002 +0.106 +0.022
FIG. 6. Dependence of the calculated exchange coupling constants J ͑cm−1
͒
on the calculated copper Mulliken spin populations for the chloro-bridged
CuII
complex ͓͑dpt͒Cu͑␮-Cl͒2Cu͑dpt͔͒Cl2 using the five different theoretical
approaches indicated in Table V. The labels indicated the functional and the
basis set employed in the calculation.
FIG. 7. Representation of the isodensity surface ͑0.005e−
/bohr3
͒ of the spin
density corresponding to the triplet state of the complex 6, ͓͑dpt͒Cu͑␮
-Cl͒2Cu͑dpt͔͒Cl2, using the BLYP and B3LYP functionals with the three
different basis set approaches.
074102-8 Ruiz et al. J. Chem. Phys. 123, 074102 ͑2005͒
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9. ization effects almost negligible.42
This fact is expected due
to antibonding M-L character of the copper orbitals bearing
the unpaired electrons.
IV. CONCLUSIONS
Computational approaches mostly exploited in solid-
state physics ͑DFT combined with plane waves or numerical
basis sets͒ deserve a precise assessment of their predictive
power when challenging accuracy in the calculation of spe-
cific properties is required. This is the case for the magnetic
exchange couplings of molecular magnets. The same holds
for the more recent frameworks based on numerical func-
tions as basis sets, largely devised for studies of extended
systems. The present article provides information on the re-
liability of these alternative schemes by highlighting their
performances in three benchmark cases.
The results for the studied transition-metal complexes
indicate that the two main parameters in the numerical cal-
culations, the mesh cutoff and the energy shift, must be at
least 200 Ry and 50 meV, respectively, in order to reach
accuracy around 1 cm−1
. The influence of the basis set ap-
pears to be very small in the numerical calculations. The
main limitation of the numerical calculations is the lack of
hybrid functionals that improve the results obtained with the
GGA functionals.
An excellent agreement with the experimental J values is
found when using the hybrid B3LYP functional combined
with all-electron Gaussian basis sets, and this approach
should be recommended in all cases to calculate the ex-
change coupling constants. The use of pseudopotentials in
the Gaussian calculations produces dramatic changes in the J
values for highly charged paramagnetic centers as, for in-
stance, FeIII
, MnIII
, or MnIV
cations, especially in the case of
large core potentials, such as the Lanl1 pseudopotential giv-
ing usually completely wrong values. This fact is probably
due to an inaccurate description of the core-valence effects
of the pseudopotential for highly charged cations. The use of
pseudopotentials in the paramagnetic centers usually pro-
duces a larger localization of the spin density at the metal
centers than that obtained with all-electron calculations. This
localization of the spin density reduces the antiferromagnetic
contribution, hence, the use of pseudopotentials reduces the
antiferromagnetic couplings or enhances the ferromagnetic
values. The effect of the pseudopotentials when using hybrid
functionals is less critical because in such cases the spin
delocalization is smaller than that obtained with GGA func-
tionals but the analysis of our results indicates that their use
must be checked carefully.
The calculated J values using the three different basis
sets, Gaussian, plane wave, and numerical functions, show
an expected overestimation of the J values when GGA func-
tionals such as PBE and BLYP are used, while the B3LYP
method with plane-wave functions gives results very close to
those obtained with Gaussian functions. A detailed analysis
demonstrates that the J values using plane-wave functions
are slightly smaller due to a larger localization of the spin
density at the metal centers probably due to the use of
pseudopotentials. As an interesting consequence, the BLYP
plane-wave results are closer to the values obtained with the
hybrid functional resulting in a better agreement with the
experimental data.
Overall, it appears that the implementation of hybrid
functionals within DFT plane-wave schemes is a valuable
route to obtain accurate exchange couplings. This strategy is
useful for systems requiring an a priori structural optimiza-
tion, commonly easier to achieve within the more flexible
plane-wave scheme.
ACKNOWLEDGMENTS
The research was supported by the Dirección General de
Enseñanza Superior ͑DGES͒ and Comissió Interdepartmen-
tal de Ciència i Tecnologia ͑CIRIT͒ through Grants Nos.
BQU2002-04033-C02-01 and 2001SGR-0044, respectively.
The computing resources were generously made available in
the CIRI and the Centre de Supercomputació de Catalunya
͑CESCA͒ through a grant provided by Fundació Catalana
per a la Recerca ͑FCR͒ and the Universitat de Barcelona.
One of us ͑T.C.͒ thanks the Ministerio de Ciencia y Tec-
nología for a Ph.D. grant. We also acknowledge a generous
allocation of computational resources on the IDRIS and
CINES French national centers.
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