Recent_Advances_in_Density_Functional_Methods_Part_III/9810248253/files/00000___1d6a2b0b22f5348cc2c16a1ecbf95bb9.pdfi
+
Recent Advances in Computational Chemistry - Vol. 1
RECENT ADVANCES IN DENSITY FUNCTIONAL METHODS
03" Part III
'A HOMA 1999 l/i
X ^ {Cc*.^-*.' ( 3 + edited by Vincenzo Barone
Alessandro Bencini Piercarlo Fantucci
World Scientific
Recent_Advances_in_Density_Functional_Methods_Part_III/9810248253/files/00001___332777261848bfd5cb4fdbc4f29b9d7a.pdfRECENT
ADVANCES IN DENSITY FUNCTIONAL METHODS
Part III
Recent_Advances_in_Density_Functional_Methods_Part_III/9810248253/files/00002___cc1b281f50e0b80a5d7dfa2ed982fd5b.pdfRecent
Advances in Computational Chemistry
Editor-in-Charge Delano P. Chong, Department of Chemistry,
University of British Columbia,
Canada
Published Recent Advances in Density Functional Methods, Part I
(Volume 1) ed. D. P. Chong
Recent Advances in Density Functional Methods, Part II (Volume
1) ed. D. P. Chong
Recent Advances in Quantum Monte Carlo Methods (Volume 2) ed. W.
A. Lester
Recent Advances in Coupled-Cluster Methods (Volume 3) ed. Rodney
J. Bartlett
Recent_Advances_in_Density_Functional_Methods_Part_III/9810248253/files/00003___ef8e66279320f81b29864574ab3a36e5.pdfRecent
Advances in Computational Chemistry-Vol. 1
RECENT ADVANCES IN DENSITY FUNCTIONAL METHODS
Part III
edited by Vincenzo Barone
Universita di Napoli Federico II, Italy
Alessandro Bencini Universita di Firenze, Italy
Piercarlo Fantucci Universita di Milano-Bicocca, Italy
U 5 j world Scientific WB New Jersey London* Singapore' New
Jersey'London'Singapore Hong Kong
Recent_Advances_in_Density_Functional_Methods_Part_III/9810248253/files/00004___bb09a27d259155929934c04a8fb4ed4b.pdfPublished
by
World Scientific Publishing Co. Pte. Ltd. P O Box 128, Farrer
Road, Singapore 912805 USA office: Suite IB, 1060 Main Street,
River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden,
London WC2H 9HE
British Library Cataloguing-in-Publication Data A catalogue
record for this book is available from the British Library.
RECENT ADVANCES IN DENSITY FUNCTIONAL METHODS (Part III)
Copyright 2002 by World Scientific Publishing Co. Pte. Ltd. All
rights reserved. This book, or parts thereof, may not be reproduced
in any form or by any means, electronic or mechanical, including
photocopying, recording or any information storage and retrieval
system now known or to be invented, without written permission from
the Publisher.
For photocopying of material in this volume, please pay a
copying fee through the Copyright Clearance Center, Inc., 222
Rosewood Drive, Danvers, MA 01923, USA. In this case permission to
photocopy is not required from the publisher.
ISBN 981-02-4825-3
Printed in Singapore by Uto-Print
Recent_Advances_in_Density_Functional_Methods_Part_III/9810248253/files/00005___3f4c62516964a2c3460f1e77beba8c3e.pdfV
PREFACE
Density Functional Theory (DFT) aims at the description of the
physico-chemical properties of a system using its electron density,
in contrast to traditional quantum chemistry which focuses the
attention on the molecular wavefunction. DFT originated in the
sixties from a fundamental idea by J.C. Slater and received a solid
theoretical background in the papers by P. Hohenberg, W. Kohn and
L.J. Sham.
As a matter of fact, only 20 years later DFT started to enter in
the chemical world and received an immense impulse towards
applications. Since then, the number of applications increased
enormously starting from the calculation of spectroscopic
observables and arriving now to cover almost all the chemical and
physical properties of a system including chemical reactivity. The
key idea of DFT, which makes this theory amenable to a large number
of applications, is to reduce the many-body approach to a formally
single particle formalism through the use of an effective exchange
and correlation (XC) potential. The enormous impact that DFT has
reached in the theoretical community has been recognized in 1998
when W. Kohn shared the Nobel prize in Chemistry with J. Pople.
In the last few years the attention of theoretical chemists was
devoted to development of more accurate model functionals and of
faster computational techniques including also excited electronic
states. The main purpose of the 8th International Conference on the
Applications of Density Functional Theory to Chemistry and Physics,
which was held in Roma (Italy), September 6fh-10th 1999, is to
gather both chemists and physicists to present and discuss state of
the art methodological developments and applications of DFT to
increasingly complex systems. In particular it is expected that
they will share their knowledge and experience in DFT, which should
enable them to face the challenges imposed by the needs in high
level modeling and simulation in their disciplines. The growing use
of DFT in studying organic, inorganic and organometallic molecules,
clusters and solids has provided the basis for the success of this
conference, whose main contributions are collected in this book.
The idea of this series of conferences started from a one-day
meeting organized by 14 chemists, wishing to share their
experiences in MS-X? and DVM-Hartree-Fock-Slater methods, in San
Miniato (Italy) in October 1984. This meeting was followed by an
International Workshop (San Miniato, 1985) from which a series of
conferences to be held every two years, in Venezia (Italy), Aries
(France), Ascona (Switzerland), Como (Italy), Paris (France), and
Wien (Austria) started. The last conferences were held in the
capitals of France and Austria, therefore this meeting was
organized in Roma, the capital of Italy and the mother of the latin
culture. The meeting was attended by 200 participants with a large
number of students
Recent_Advances_in_Density_Functional_Methods_Part_III/9810248253/files/00006___e2f62b7c0ae775a8f6b05bbc319bd43a.pdfVI
and young researchers. New theoretical developments and
achievements in methodology and applications were discussed during
17 invited lectures, 25 oral communications and 2 posters sessions
(91 contributions). The meeting was opened by an exciting lecture
of W. Kohn. The 9th Conference of this series will be organized in
2001 by J. Alonso, J.M. Garcia de la Vega, and M. Moreno in Madrid
(Spain). This book consists of 25 selected and referenced
contributions presented in different formats at the Rome Congress,
which have been ordered putting the name of the first author in
alphabetical order.
The Editors wish to express their warm thanks to all
participants who contributed to the success of this meeting through
their valuable contributions. We also wish to acknowledge our
academic sponsors (Consiglio Nazionale delle Ricerche, Universities
of Basilicata, Firenze, Milano-Bicocca, Napoli), the European
Community (COST D3 Action), and our non academic sponsors (CASPUR,
CILEA.INSTM) for all the help and financial support we received.
Their generosity allowed us to distribute financial helps to
students and to young researchers.
We also gratefully acknowledge the help of the Romacongressi
Society who took care of the practical aspects of the local
organization together with Dr. N. Sanna from CASPUR.
Special thanks are expressed to Dr. Federico Totti and Miss
Elisabetta Berti (University of Firenze) for their assistance in
the organization of the scientific part of the Conference.
Vincenzo Barone, Alessandro Bencini, Piercarlo Fantucci
Recent_Advances_in_Density_Functional_Methods_Part_III/9810248253/files/00007___3ec2a8acb7d1852e4e2f142976872019.pdfVII
CONTENTS
Preface
Theoretical Study of the Transition-Metal Silonyl Complexes
M-SiO and M-(SiO)2: M = Cu, Ag, or Au 1 M. E. Alikhani and D.
Mandelbaum
Local Relaxation for Mn2+ and Fe3+ Impurities in
Fluoroperovskites: Density Functional Study 11 M. T. Barriuso, J.
A. Aramburu and M. Moreno
Theoretical Exploration of Single and Multi State Femtosecond
Nuclear Dynamics of Small Metallic Clusters Using the DF Method 24
V. Bonacic-Koutecky, M. Hartmann, D. Reichardt and P. Fantucci
Applications of Density Functional Theory in Solid State
Chemistry 45 S. T. Bromley, S. A. French, A. A. Sokol, P. V. Sushko
and C. R. A. Catlow
A Hybrid Functional for the Exchange-Correlation Kernel in
Time-Dependent Density Functional Theory 67 K. Burke, M. Petersilka
and E. K. U. Gross
Calculation of Vertical Ionization Potentials Using a Density
Functional Total-Energy Difference Approach 80 G. Cavigliasso and
D. P. Chong
On the Calculation of Ionization Energies within Density
Functional Theory 91 H. Chermette, S. Joanteguy and G.
Pfister-Guillouzo
Modeling Molecular Magnetism Using DFT 106 /. Ciofini, C. A.
Daul and A. Bencini
Conceptual and Computational DFT as a Chemist's Tool 137 P.
Geerlings and F. De Proft
Recent_Advances_in_Density_Functional_Methods_Part_III/9810248253/files/00008___1eb273c51e91750dcd06f26a34882e40.pdfVIII
10. Local Density Approach to Real-Space Multiple-Scattering
Calculations of Inner-Shell Photoabsorption Cross Sections For
Clusters 168 D. L. Foulis, R. F. Pettifer and V. L. Jennings
11. Ab Initio Calculations of Elastic Properties of Solids and
Applications 181 M. Hebbache
12. Structural and Magnetic Properties of Model Spin Probes in
Aqueous Solution: An Application of Recent Developments in Density
Functional Theory and in the Polarizable Continuum Model 189 R.
Improta and V. Barone
13. First Principles Pseudopotential Calculation of the Magnetic
Properties of Low-Dimensional Iron Systems 205 J. Izquierdo, A.
Vega, L. C. Balbds, P. Ordejon, D. Sanchez-Portal, E. Artacho and
J. M. Soler
14. Correlation Energy for Isoelectronic Series of Atoms by the
Line Integral Method 224 V. V. Karasiev, E. V. Ludena and E.
Valderrama
15. Conditions for Cluster Assembled Solids 234 L. M. Molina, J.
A. Alonso, M. J. Lopez, A. Rubio and M. J. Stott
16. Theory for a Single Excited State Differential Virial
Theorem 247 A. Nagy
17. Studies of the Nonadditive Kinetic Energy Functional and the
Coupling between Electronic and Geometrical Structures 257 R. F.
Nalewajski
18. First-Principles Calculation of Multiplets of
Transition-Metal Ions in Crystals Based on Density Functional
Method 278 K. Ogasawara, T. Ishii, I. Tanaka and H. Adachi
Recent_Advances_in_Density_Functional_Methods_Part_III/9810248253/files/00009___76eb9de873e0153243b9b93f69e47fb2.pdfIX
19. Correlated Multideterminantal Potential Energy Curves for
Diatomic Molecules with One Valence-Bond Pair 293 E. San-Fabian, L.
P. Abia and J. M. Perez-Jordd
20. The Description of the Photoionization Process by the
B-Spline Density Functional Method 305 M. Stener and P. Decleva
21. DFT Calculations of Redox Potentials of Disulfide Compounds
Directed Against Retroviral Zinc Fingers 325 /. A. Topol, C.
McGrath, E. N. Chertova and S. K. Burt
22. Prediction of the Structural and Electronic Properties of
Polymeric Systems 339 M. E. Vaschetto, B. A. Retamal, A. P. Monkman
and M. Springborg
23. Chemical Shifts and Coupling Shifts of the Stretch
Vibrations of CO on Cu(100) 360 R. L. C. Wang and H. J. Kreuzer
24. Spin-Densities in Charge-Transfer Complexes Derived from DFT
Calculations Using An Orbital-Free Embedding Scheme for Interacting
Subsystems 371 T. A. Wesotowski and J. Weber
25. Hydroxyl Radical Reactions in Biological Media 387 S. D.
Wetmore, R. J. Boyd, J. Llano, M. J. Lundqvist and L. A.
Eriksson
Index 417
Recent_Advances_in_Density_Functional_Methods_Part_III/9810248253/files/00010___6702bdf2b2b15785b3c743ca9050ab09.pdf1
THEORETICAL STUDY OF THE TRANSITION-METAL SILONYL COMPLEXES
M-SIO A N D M-(SIO)2 : M = C U , AG,
OR AU
M.E. ALIKHANI* AND D. MANDELBAUM Laboratoire de Spectrochimie
Moleculaire (UMR CNRS LADIR),
Universite P. et M. Curie, Boite 49, bdtiment F74, 4 Place
Jussieu Paris Gedex 05, France
E-mail : [email protected]
The spectroscopic properties of M-SiO and M-(SiO)2 (1-1 and 1-2
complexes with M = Cu, Ag, or Au) have been theoretically studied.
It has been shown
that both M-SiO and M-(SiO)2 compounds in their ground state are
bent with a metal-Si bonded structure. The calculated M(ns) spin
density agrees well with
the ESR experimental data. ^From a topological analysis of the
electron density distribution it has been found that the M-Si bond
energy correlates with the
electron density located at the M-Si bond path (bond critical
point). Audentes fortuna juvat
1 Introduction
The interactions involving transition metal atoms play an
essential role in surface and material science. It is of the utmost
important to be enable to perform an accurate theoretical
description of such a system. It is now well established that the
quantum chemical calculation of transition metal com-pounds
requires the explicit treatment of electron correlation and
relativistic effects 1>2. In particular, the relativistic
effects must be taken into account when the chemical compound
involves the third transition metal atoms 2. The density functional
theory (DFT) including the exchange-correlation func-tional seems
an adequate approach to study such a system 3. In particular,
hybrid approach proposed by Becke 4 manages the advantages of
Hartree-Fock (HF) and DFT models with an improved overall accuracy
5 ' 6 . A number of theoretical studies have shown that the B3LYP
variant is particularly effec-tive in the description of the
physico-chemical properties of the binary com-plexes 7 _ 9 . Among
small transition metal complexes, metal-monocarbonyl, M(CO) has
been studied extensively both experimentally 1 0 _ 2 2 and
theoret-ically 2>3>7-9>23-32; whereas the corresponding
M(SiO) is not yet sufficiently investigated 33~39. The infrared
(IR) and electron spin resonance (ESR) prop-erties of M:CO (M = Cu,
Ag, or Au) have been measured in several experi-mental works 1 2 _
1 4 ' 19-21,36_ These complexes were also studied using several
theoretical methods 8,9,28,30,31 _ ^From both experimental and
theoretical
Recent_Advances_in_Density_Functional_Methods_Part_III/9810248253/files/00011___eb4f4ccdbb17111b377e9b44a957b891.pdf2
points of view, it has been shown that the Ag:CO complex would
be actually a weak van der Waals complex (bound by only 40-50 cm -
1 ) , dissociating into Ag and CO 2 '20-28 . ^From ESR experiment,
the Cu- and Au-CO complexes were found to be linear (2S electronic
state) 36, whereas the theoretical in-vestigations proposed a bent
structure 2>8>31. Barone using DFT calculations showed that
the linear Cu-CO corresponds to a transition state 8. In a recent
experimental work, the three fundamental vibrational modes were
identified with the help of isotopic effects and it has been
evidenced that the Cu-CO complex has actually a bent geometry n .
It is worthy noting that the ESR and IR experiments are required to
predict the structure of the metal-ligand compounds. Furthermore,
in the case of Ag-SiO it has been shown that the vi-brational data
obtained from IR experiment 33 and the density of spin located on
the Ag atom measured from ESR 34 '36 are only theoretically well
repro-duced for a bent structure 39, although the ESR experiment
suggested only a linear geometry 36. In the case of the Cu(SiO) and
Au(SiO) complexes, the spin density of unpaired electron is
calculated from the ESR experiment. A linear structure for Cu(SiO)
and a bent structure for Au(SiO) were suggested. To our knowledge,
there is no infrared results on these systems. Therefore, we
thought it interesting to understand if the M(SiO) and M(SiO)2 (M =
Cu, Ag, or Au) are bent or linear, Si-bonded or O-bonded. In order
to calculate the structural, energetic, and vibrational properties,
these compounds have been investigated using the DFT/Hybrid method.
Furthermore, the nature of M-Si bonding has been studied using the
topological theory of atoms in molecules 40.
2 Results and Discussion
All calculations have been performed with the Gaussian 94/DFT
quantum chemical package 41. The DFT calculations have been carried
out with Becke's three parameters hybrid method 4 using the
Lee-Yang-Parr correla-tion functional 42 (denoted as B3LYP). We
have used the 19-valence electron of Stuttgart pseudo potential for
the group 11 metals (relativistic (MDF) for Cu 43 and
pseudo-relativistic (MWB) for Ag and Au 44) and the 6-311+G(2d)
extended basis set of Pople et al. 45~47 for the other atoms. The
topologi-cal properties of the bonding have been investigated using
the program EX-TREME (part of the AIMPAC suite of programs)
developed by Bader et al. 48
Recent_Advances_in_Density_Functional_Methods_Part_III/9810248253/files/00012___c58d90f6779a2bcf07d5de26baf17c21.pdfM
O Si O
(a) M ( b )
O Si
(c) \ / M
O Si M O Si
(d) (e) M
Figure 1. Possible geometries for the one-to-one complex
M(SiO)
2.1 Structural and vibrational analysis
one-to-one complexes: M(SiO) Five geometries have been studied
for all compounds (see Figure 1) at the DFT/B3LYP level of theory.
The linear and bent M-OSi structures (Figures 1-a and 1-b) are
found to be unbound in all of the cases. The cyclic geometry
(Figure 1-c) is calculated to be bound only for Cu(SiO). For two
other complexes (Ag(SiO) and Au(SiO)), the triangular structure
col-lapses to a bent metal-Si bonded structure upon optimization
process. Con-cerning the metal-Si bonded structures (Figures 1-d
and 1-e), it has been shown that the bent structure is more stable
than the linear one. A vibrational
3
Si
Recent_Advances_in_Density_Functional_Methods_Part_III/9810248253/files/00013___d89cc8b5e27769fc6b2067e4ee8b6a8b.pdf4
Table 1. Structural and vibrational parameters of bent M-SiO in
the 2A' state
Parameters r i (A) T2 La (deg.)
De (kcal/mol)
P, (M) (%) P.(Af) * SiO stret. (cm-1) c Exp. SiM stret. MSiO
bend.
fsiO (mdyn/A)
Cu-SiO 1.528 2.328 120.0
13.2 4.24 61 71
1187[133]
229[3] 108[3]
8.5
Ag-SiO 1.526 2.558 114.5
8.1 4.08 74 74
1192[125] 1163 d 182 [6] 93[5]
8.6
Au-SiO 1.524 2.360 127.8
19.2 3.72 56 54
1198[104]
268 [7] 136 [3]
8.6
SiO 1.514
3.23
1246 [56] 1242 e
9.3
a) De = (EM + ESio) - EM-siO, b) Experimental values deduced
from reference 3 e ,
c) In brackets are reported the calculated IR intensities in
km/mole, d) Ref. 3 3 , e) Ref. 5 1 .
analysis of the linear M-SiO compounds, performed with the DFT
approach, indicate that this geometry corresponds actually to a
transition state (two imaginary frequencies with -K symmetry). In
the case of Cu(SiO), the bent metal-Si bonded structure is more
stable than the cyclic one. These results allow one to consider the
bent M-SiO structure as ground electronic state (2A'). Therefore,
in the following we study only the bent M-SiO geometry. In Table 1
are reported the spectroscopic properties of the 1-1 complexes. In
the M-SiO molecules, the Si-0 bond length is slightly larger than
that of free SiO. The Si-0 lengthening decreases from Cu- to
Au-SiO. For all compounds, the Si-0 bond distance remains less than
1.550 A, which is in the range of Si=0 double bonds 49. The M-Si
bond length varies as Cu-SiO < Au-SiO < Ag-SiO. The dipole
moment is enhanced in M-SiO with respect to free SiO. As expected,
the Si-0 bond lengthening correlates with the SiO frequency shift
and the Si-0 force constant, while the dipole moment varies in line
with the IR intensity. It should be noted that there is no
correlation between the SiO frequency shifts and the binding
energies.
The experimental observation of the SiM stretch and MSiO bending
modes
Recent_Advances_in_Density_Functional_Methods_Part_III/9810248253/files/00014___c5deedf5d9abbe5b5e7346c9282e0399.pdfFigure
2. The most stable structures of the M(SiO) and M(SiO)2
compounds
seems very difficult, because of their weak IR intensities.
Finally, we note that the calculated s character of spin densities
of the unpaired electron located on the metal atoms agree well with
the experimental results 36.
one-to-two complexes: M(SiO)2 The spectroscopic parameters of
M(SiO)2 calculated from DFT techniques are reported in Table 2. The
geometrical parameters are displayed in Figure 2.
At our knowledge, there is no experimental data on the
one-to-two complexes, excepted the electron density of Ag which
tentatively attributed to Ag-(SiO)2-34,36 ^ye j^yg founci that the
M-(SiO)2 complex is bound with respect to the M + 2 SiO (labelled
as De(l) in Table 2) and to the M-SiO + SiO (labelled as De{2) in
Table 2) subunits. The geometrical variations in the M-(SiO)2
complex follows the same trend as those in the M-SiO compound. The
Si-0 bond length in the M-(SiO)2 system is found to be slightly
smaller than that in the M-SiO one, while the lengthening of the
M-Si bond is more
Recent_Advances_in_Density_Functional_Methods_Part_III/9810248253/files/00015___9e91d70a9144ad0a6a6a47b81a5b0fc8.pdf6
Table 2. Structural and vibrational parameters of bent M-(SiO)2
in the 2A\ state Parameters ri (A) T2 La (deg.) lb
D e ( l ) (kcal/mol) D=(2) b Ps{M) (%) P ( D ) SiO sym. stret.
(cm-1) SiM sym. stret. MSiO sym. bend. SiMSi bend. MSiO asym. bend.
(O.P.) MSiO sym. bend. (O.P.) SiO asym. stret. SiM asym. stret.
MSiO asym. bend.
fsiO (mdyn/A)
Cu-(SiO) 2 1.522 2.371 168.1 129.1
23.1 9.9 38
5.84
1218 189 130 26 24 41
1203 231 103
8.8
Ag-(S iO) 2 1.522 2.576 168.7 123.9
15.3 7.2
44(50 c) 5.84
1218 161 104 24 16 34
1205 183 94
8.8
Au-(SiC 1.520 2.452 165.2 131.2
32.0 12.8 30
5.41
1222 196 150 27 34 50
1209 232 109
8.9
a) De(l) = (EM + 2 Esio) - EMHSi0h. b) >e(2) = (EM^sio +
ESi0) - S M _ ( s i o ) 2 .
c) Experimental value deduced from reference 3l
pronounced in going from M-SiO to M-(SiO)2- The predicted
frequency shift of the SiO asymmetric stretch for the M-(SiO)2
complex is sufficiently smaller than that for M-SiO allowing this
band to be observed experimentally. The other vibrational
frequencies are found to be very low and relatively weak so that
their experimental observation would be difficult. The dipole
moment is even enhanced in going from M-SiO to M-(SiO)2, leading to
the increase of the IR intensity of the SiO asymmetric stretching
mode. Finally, the predicted s spin density located at the metal
upon complexation allows us to hope his experimental observation
from ESR technique.
Bonding analysis
Table 3 lists the calculated topological properties at the bond
critical points. According to the topological theory of atoms in
molecules 40 the positive values of the electron density Laplacian
at the bond critical point (bcp, where V/>=0) are associated
with closed shell interactions (ionic bonds, hydrogen bonds,
Recent_Advances_in_Density_Functional_Methods_Part_III/9810248253/files/00016___e987fb8fd1a4092be88dc7cbc49ed3db.pdf7
Table 3. Topological properties of M-SiO and M-(SiO)2. a
Compounds
SiO Cu-SiO Ag-SiO Au-SiO
Cu-(SiO) 2 Ag-(S iO) 2 Au-(S iO) 2
Si-O p
0.193 0.187 0.188 0.188
0.189 0.190 0.190
bcp H
-0.099 -0.096 -0.096 -0.097
-0.098 -0.098 -0.098
M-Si P
0.062 0.052 0.079
0.058 0.051 0.070
bcp H
-0.020 -0.015 -0.038
-0.018 -0.014 -0.026
a) From Bader's topological scheme, p and H represent the
electron density and energy density at the bond critical point
(bcp).
and van der Waals molecules), while V2p < 0 indicates shared
interactions (covalent bonds). Another criterion proposed by Cremer
et al. 50 states that the local energy density H at the bond
critical point should be positive for ionic bonds and negative for
partly covalent bond. The results reported in Table 3 show that,
following this criterion, the SiO bond should be considered as a
shared interaction for all compounds, because of the negative
energy density at the Si-0 bond critical point. Whereas, the
bonding between the metal atom and the silicon one within both
M-SiO and M-(SiO)2 complexes presents a very small covalent
character, because of the very small negative value for the energy
density at the M-Si bond critical point. This bonding should be
actually characterized as a weak dative bond, because the very
small value of the electron density at the M-Si bond critical
point. It is very interesting to note that the binding energy
between metal and each of ligand nicely correlates with the
electron density calculated at the M-Si bond critical point. For
all of the compounds studied in this paper, the correlation
coefficient calculated for the binding energy as function of
p(M-Si) is very close to one (0.99). It has been also found that
the correlation between the SiO stretching frequency (vas{SiO)) and
the electron density at the Si-0 bcp (p(Si-O)) is excellent,
because the correlation coefficient value is equal to 0.99.
3 Concluding Remarks
It has been shown that both the one-to-one and one-to-two
complexes studied here are found to be bound, then they could be
experimentally detected from either infrared (IR) or electron spin
resonance (ESR) spectroscopies. A few
Recent_Advances_in_Density_Functional_Methods_Part_III/9810248253/files/00017___44b92b08f7bf69beddcb0aab1ceb8ad3.pdf8
available experimental data are very well reproduced from DFT
calculations. Furthermore, it has been shown that there are
correlation between the binding energy and the electron density
located at the metal-silicon bond path, and between the SiO
frequency (or its frequency shift with respect to free SiO) and the
electron density at the Si-0 bond path. It seems reasonable to
consider the electron density located at the metal-ligand bond
critical point as a measure of the bond strength.
References
1. P. Pyyko, Chem. Rev. 88, 563 (1988). 2. P. Schwerdtfeger and
G. A. Bowmaker, J. Chem. Phys 100, 4487 (1994). 3. S.-C. Chung, S.
Kriiger, G. Pacchioni and N. Rosch, J. Chem. Phys.
102, 3695 (1995). 4. A. D. Becke, J. Chem. Phys. 98, 5648
(1993). 5. A. D. Becke, J. Chem. Phys. 98, 1372 (1993). 6. C. W.
Bauschlicher, Jr. and H. Partridge, J. Chem. Phys. 103, 1788
(1995). 7. V. Barone, Chem. Phys. Lett. 233, 129 (1995). 8. V.
Barone, J. Phys. Chem. 99, 11659 (1995). 9. C. Adamo and F. Lelj,
J. Chem. Phys. 103, 10605 (1995).
10. H. A. Joly and L. Manceron, Chem. Phys. 226, 61 (1998). 11.
B. Tremblay and L. Manceron, Chem. Phys. 242, 235 (1999). 12. J. H.
B. Chenier, C. A. Hampson, J. A. Howard and B. Mile, J. Phys.
Chem. 92, 2745 (1988). 13. J. H. B. Chenier, C. A. Hampson, J.
A. Howard and B. Mile, J. Phys.
Chem. 93, 114 (1989). 14. J. H. B. Chenier, J. A. Howard, H. A.
Joly and M. Tomieto, Can. J.
Chem. 67, 655 (1989). 15. H. Huber, E. P. Kiindig, M. Moskovits
and G. A. Ozin, J. Am. Chem.
Soc. 97, 2097 (1975). 16. L. Hanlan, H. Huber and G. A. Ozin,
Inorg. Chem. 15, 2592 (1976). 17. D. Mcintosh and G. A. Ozin, J.
Am. Chem. Soc. 98, 3167 (1976). 18. D. Mcintosh and G. A. Ozin,
Inorg. Chem. 16, 51 (1977). 19. P. H. Kasai and P. M. Jones, J. Am.
Chem. Soc. 107, 813 (1985). 20. P. H. Kasai and P. M. Jones, J.
Phys. Chem. 89, 1147 (1985). 21. P. H. Kasai and P. M. Jones, J.
Am. Chem. Soc. 107, 6385 (1985). 22. R. J. Van Zee, S. B. H. Bach
and W. Weltner, Jr., J. Phys. Chem. 90,
583 (1986). 23. L. A. Barnes and C. W. Bauschlicher, Jr., J.
Chem. Phys. 91, 314
Recent_Advances_in_Density_Functional_Methods_Part_III/9810248253/files/00018___fb69fdbf1afd815ea16fe6d7c18a3006.pdf9
(1989). 24. G. W. Smith and E. A. Carter, J. Phys. Chem. 95,
2327 (1991). 25. S. M. Mattar and W. Hamilton, J. Mol. Struct.
(Theochem) 226, 147
(1991). 26. G. Jeung, J. Am. Chem. Soc. 114, 3211 (1992). 27. M.
Sodupe, C. W. Bauschlicher, Jr. and T. J. Lee, Chem. Phys.
Lett.
189, 266 (1992). 28. C. M. Marian, Chem. Phys. Lett. 215, 582
(1993). 29. R. Fournier, J. Chem. Phys. 98, 8041 (1993). 30. R.
Fournier, J. Chem. Phys. 99, 1801 (1993). 31. C W. Bauschlicher,
Jr., J. Chem. Phys. 100, 1215 (1994). 32. G. Jeung and S. Haettel,
Int. J. Quant. Chem. 61, 547 (1997). 33. T. Mehner, H. Scnockel, M.
J. Almond and A. J. Downs, J. Chem. Soc,
CHEM. COMMUN. p. 117 (1988). 34. J. H. B. Chenier, J. A. Howard,
H. A. Joly, B. Mile and P. L. Timms, J.
Chem. Soc, CHEM. COMMUN. p. 581 (1990). 35. T. Mehner, R. Koppe
and H. Scnockel, Angew. Chem. Int. Ed. Engl.
31, 638 (1992). 36. A. P. Williams, R. J. Van Zee and W.
Weltner, Jr., J. Am. Chem. Soc.
118, 4498 (1996). 37. G. E. Quelch, R. S. Grev and H. F.
Schaefer III, J. Chem. Soc, CHEM.
COMMUN. p. 1498 (1989). 38. J. S. Tse, J. Chem. Soc, CHEM.
COMMUN. p. 1179 (1990). 39. M. E. Alikhani, J. Chem. Soc, Faraday
Trans. 93, 3305 (1997). 40. R. F. W. Bader, Atoms in Molecules: A
Quantum Theory (Oxford Univ.
Press, Oxford, 1994). 41. M. J. Frisch, G. W. Trucks, H. B.
Schlegel, P. M. W. Gill, B. G. Johnson,
M. A. Robb, J. R. Cheeseman, T. Keith, G. A. Petersson, J. A.
Mont-gomery, K. Raghavachari, M. A. Al-Laham, V. G. Zakrzewski, J.
V. Ortiz, J. B. Foresman, J. Cioslowski, B. B. Stefanov, A.
Nanayakkara, M. Challacombe, C. Y. Peng, P. Y. Ayala, W. Chen, M.
W. Wong, J. L. Andres, E. S. Replogle, R. Gomperts, R. L. Martin,
D. J. Fox, J. S. Bink-ley, D. J. Defrees, J. Baker, J. P. Stewart,
M. Head-Gordon, C. Gonzalez, and J. A. Pople, Gaussian 94, Revision
D.4 (Gaussian Inc. and Pitts-burgh PA, 1995).
42. C. Lee, W. Yang and R. G. Parr, Phys. Rev. B 37, 785 (1988).
43. M. Dolg, U. Wedig, H. Stoll and H. Preuss, J. Chem. Phys. 86,
866
(1987). 44. D. Andrae, U. Haeussermann, M. Dolg, H. Stoll and H.
Preuss, Theor.
Chim. Acta 77, 123 (1990).
Recent_Advances_in_Density_Functional_Methods_Part_III/9810248253/files/00019___52a2dc98531190c7a6604685c0e92377.pdf10
45. R. Krishnan, J. S. Binkley, R. Seeger and J. A. Pople, J.
Chem. Phys. 72, 650 (1980).
46. T. Clark, J. Chandrasekhar, G. W. Spitznagel and P. V. R.
Schleyer, J. Comp. Chem. 4, 294 (1983).
47. M. J. Frisch, J. A. Pople and J. S. Binkley, J. Chem. Phys.
80, 3265 (1984).
48. F. B. Konig, R. Bader and T. Tang, J. Comput. Chem. 13, 317
(1982). 49. Y. Xie and H. F. Schaefer III, J. Chem. Phys. 93, 1196
(1990). 50. D. Cremer and E. Kraka, Angew. Chem. Int. Ed. Engl.
100, 627
(1984). 51. K. P. Huber and G. Herzberg, Molecular Spectra and
Molecular Struc-
ture Constant of Diatomic Molecules (van Nostrand Reinhold, New
York, 1979).
Recent_Advances_in_Density_Functional_Methods_Part_III/9810248253/files/00020___20f4614e84b0d4650c2c7a4d6e843710.pdf11
LOCAL RELAXATION FOR MN2+ AND FE3+ IMPURITIES IN
FLUOROPEROVSKITES: DENSITY FUNCTIONAL STUDY
M.T.BARRIUSO", J.A.ARAMBURU* AND M.MORENO*
* Departamento de Ffsica Moderna, Facultad de Ciencias,
Universidad de Cantabria, E-39005 Santander, Spain
* Departamento Ciencias de la Tierra y Fisica de la Materia
Condensada, Facultad de Ciencias, Universidad de Cantabria, E-39005
Santander, Spain
This work deals with the application of Density Functional (DF)
calculations on clusters of small size for reaching a valuable
information on the equilibrium distance, Re, between an impurity,
M, in an insulating lattice and the nearest anions. In a first step
DF calculations on simple MnF6A8M616+ clusters have been carried
out in order to explain the Re values measured in Mn2+ doped AMF3
fluoroperovskites by means of three independent experimental
techniques. The obtained Re values coincide, within the
experimental uncertainties, with those derived from the analysis of
the experimental isotropic superhyperfine constant, As and the 10Dq
parameter and also with available EXAFS results for KZnF3:Mn2+ and
RbCdF3:Mn2+. This important result is shown to be practically
independent on the quality of the employed basis set. Calculated Re
values for cubic centres in Fe3+ doped fluoroperovskites using
similar clusters are also reported. Although the obtained Re values
are comparable to those derived from the analysis of the
experimental As constant the change undergone by Re on passing from
KMgF3:Fe3+ to CsCdF3:Fe3+ is shown however to be smaller than that
obtained through total energy calculations of FeFeA8M617+ clusters.
This discrepancy is pointed out to arise from the neglect of the
relaxation of second and third neighbours in the present
calculations. Finally the R9 values calculated using a simple MXN
complex in vacuo are compared with experimental figures for Mn2+,
Fe3*, Cr4* and Fe6+ impurities in different insulating lattices.
While the discrepancy between two figures is 25% for Mn2+ it is
only 3% for Fe6+ in oxides. The origin of this fact is briefly
discussed.
1 Introduction
New Physico Chemical properties appear in a given material due
to the presence of impurities. In an insulating lattice the
electronic properties (such as optical transitions or
spin-hamiltonian parameters) due to a Transition Metal (TM)
impurity, M, can be understood to a long extent [1] only on the
Recent_Advances_in_Density_Functional_Methods_Part_III/9810248253/files/00021___4c6400d26031042fd2cd5516e7f5082c.pdf12
basis of the MXN complex (formed with the N nearest anions) at
the equilibrium metal-ligand distance, Re. Such a distance as well
as the vibrational frecuencies related to the impurity depend
however on the interaction between ligand ions and close ions in
the host lattice.
To know the right equilibrium distance, Re, is thus a
prerequisite for gaining a better insight into the properties due
to a TM impurity in an insulating material.
In the realm of impurities such a task is certainly not simple
as standard diffraction methods cannot be used for that purpose.
Moreover the EXAFS technique requires a minimum of impurity
concentration around 1% and the uncertainty in the obtained Rs
values is higher than 1 pm [2,3].
This situation can however be overcome through the analysis of
standard optical and EPR parameters which can be measured for much
lower impurity concentrations. For instance in the case of d5, d7
and d8 ions in halide lattices with octahedral coordination it has
been pointed out that Re can be derived from the experimental
isotropic superhyperfine constant, As. A general view on this
method is given in Ref [4].
In the case of TM impurities with local Oh geometry Re
variations can also be extracted from the experimental 10Dq
parameter [5-7], the energy of a charge transfer transition [8] or
the Zero-Phonon Line (ZPL) energy associated with a 10Dq dependent
transition [9]. By means of the ZPL or the As parameter detected by
the ENDOR technique Re changes down to 0.05pm can be detected
[4,9].
With the advent of powerful theoretical methods based either in
the ab initio Hartree-Fock (including CI) or in the Density
Functional framework one would expect that a valuable information
on the equilibrium impurity-ligand distance can also be obtained
from theoretical calculations on clusters centered around the
impurity. As all calculations of this kind involve some
approximations it is firstly necessary to verify the quality of the
predictions made by a theoretical method on a given cluster. The
set of AMF3 cubic fluoroperovskites doped with Mn2+ is certainly a
good candidate for testing a given method as Re has previously been
determined from the analysis of three different experimental data.
In particular in six of such lattices Re value has been derived [4]
from the analysis of the experimental As constant while in five of
them it has also been obtained from the 10Dq parameter extracted
from optical excitation spectra [6,7]. Moreover for KZnF3:Mn2+ and
RbCdF3:Mn2+ EXAFS measurements have also been carried out [10]. As
shown in Table 1 the Re values derived through the three different
methods are coincident within the experimental uncertainty.
If the set of six fluoroperovskites doped with Mn2+ (shown in
Table 1) is used for controlling the quality of a given method the
test is certainly stringent. In fact for reproducing reasonably the
Re values reached from the analysis of experimental data for the
six systems of the series the calculated
Recent_Advances_in_Density_Functional_Methods_Part_III/9810248253/files/00022___3ddf59c2d8699d612381df25659fb5e0.pdf13
Re value should involve an error less than - 1 % for each member
of the series.
In the present work the Re value for the six systems given in
Tablel is calculated within the Density Functional framework [11]
applied to MnF6A8M616+ clusters (Figure 1). This study is founded
on a recent one on Cr3+ in K3CrF6 lattice [12] showing that
clusters of 21 atoms centered at the Cr3+ ion lead to Re values
very close to experimental ones for fluoride compounds containing
CrF63~ units. According to recent results [13] attention is also
paid to the influence of the basis set quality on the calculated Re
value.
In a second step results about Re for cubic centres formed in
Fe3+ doped fluoroperovskites [14-19] using similar clusters are
reported and compared to those derived from the analysis of the
experimental As constant.
For clarifying the influence on the equilibrium impurity-ligand
distance of lattice ions which are close to the MXN complex the
calculated Re values for some complexes in vacuo are also reported.
The results are compared to experimental figures corresponding to
Mn2+, Fe^.Cr4*, Fe6+.placed inside ionic lattices in order to
explore the influence of the nominal charge of the cation .
2 Theoretical
Following previous results [12] Re values for the series of AMF3
fluoroperovskites doped with a d5 impurity I (l=Mn2+; Fe3+) have
been obtained calculating the energy of a IF6A8M6P+ cluster (p=16
for Mn2+; 17 for Fe3+) as a function of the l-F~ distance, R, for
the 6Ai(t3e2) state. In that cluster the A+ and I ions are kept at
their host lattice position and thus only the l-F" distance is left
as variable. This assumption is reasonable provided u7R01 where R0
means the M-F distance of the perfect host lattice and U1=
Re-Ro-
The calculations shown in this work have been performed using
the Amsterdam Density Functional (ADF) code [20,21]. All of them
are spin not polarized. The influence of the quality of the basis
set on the Re value has been explored taking as a guide the case of
Mn2+ in fluoroperovskites. For this goal two types of basis sets
have been employed. Firstly calculations have been carried out by
means of functions of quality IV (which are implemented in the ADF
code) involving triple zeta basis functions plus a polarization
function. In a second step Re has also been computed using double
zeta functions of quality II for F_and M2+ ions.
Recent_Advances_in_Density_Functional_Methods_Part_III/9810248253/files/00023___5d26dae2d5810d00a66abb4953c780fd.pdf14
For Mn2+ and Fe3+ electrons up to the 3p shell are kept frozen
so as the 1s electrons of F~. The A+ and M2+ ions are treated in a
similar way. For instance in the RbCaF3 host lattice electrons up
to the 3d shell of Rb and the 3p shell of Ca are included in the
core. Some calculations have also been made decreasing the number
of frozen electrons in a given atom.
Calculations have been made in the framework of the Generalized
Gradient Approximation (GGA) by means of the Becke-Perdew
functional [22,23]. It is worth mentioning, however, that in the
case of Cr3+ in fluorides the Re values computed in the Local
Density Approximation (LDA) for a 21 atoms cluster differ only by
about 1% from those obtained through GGA [20]. The effects of the
electrostatic potential due to the rest of the ions not included in
the IF6AeM6p+ cluster have been considered in all the calculations.
When such a potential is not taken into account the computed
distances increase by about 1%. A similar situation is found in the
case of Cr3+in elpasolites [12] where a further analysis on this
point is given.
3 Results and discussion
The equilibrium Mn2+-F" distance, Re, for the six AMF3
fluoroperovskites containing substitutional Mn2+ impurities
obtained using basis functions of quality IV is reported in Tablel.
These results are essentially independent on the number of
electrons included in the core as discussed in Ref [24]. In table 1
the Re values obtained using basis functions of quality II for F"
and M2+ are also reported. It can be seen that the obtained Re
values are practically unaffected by this reduction of quality in
the basis set. This result thus concurs with those of a recent work
[13] showing that DFT results are less sensitive to the quality of
the basis set than in the case of ab initio calculations.
The calculated Re values for the six host lattices compare well
with those derived from EXAFS measurements and the analysis of
experimental 10Dq and As parameters. In fact for each lattice all
the Re values given in Tablel coincide within the experimental
uncertainties. From Tablel it can thus be inferred that the error
involved in the calculated Re would be around 1pm for each one of
the six analysed systems. This result concurs with previous
findings on Cr3+ in halides [12].
From the inspection of tablel it comes out that I u1/R0 I is
always smaller than 4.5%. Let us design by R3e the true equilibrium
Mn2+-M2+ distance between the impurity and the second neighbours in
directions and by R30 the corresponding distance of the perfect
host lattice. Following the theory of elasticity it can be expected
that if I uVR01 < 4.5% then I u3/ R30 I < 0.6% where u3 = R3e
- R30. Therefore the neglect of the
Recent_Advances_in_Density_Functional_Methods_Part_III/9810248253/files/00024___3405e26fc2f2892a3c381a404d73e215.pdf15
relaxation of second neighbours in directions in the present
calculation is reasonably justified a posteriori by this argument.
It is worth recalling here that in the case of CaF2: Mn2+ there is
an inward relaxation of F" ligands which has been measured through
EXAFS [2] and the analysis of the experimental As value [4]. From
both methods it is found u1/R0 = -4%. Moreover the analysis of
EXAFS data indicate that the relaxation of nearest Ca2+ ions is
zero within the experimental uncertainties.
The equilibrium V2+-F_ distance, Re, for three AMF3
fluoroperovskites containing substitutional V2+ impurities has
recently been explored by means of ab initio calculations [25].
The^ obtained values for KMgF3:V2+ (Re =2.074 A) and CsCaF3:V2+ (Re
=2.146 A) are close to the figures obtained for the same host
lattices containing Mn2+. As the ionic radius of Mn2+ is 4% higher
than that of V2+ [26] one would expect however that Re for
KMgF3:V2+ would be smaller than for KMgF3: Mn2+. Further
investigation about this point is necessary.
Although the present calculations account for the experimental
Re values of six AMF3:Mn2+ systems fine details of their electronic
structure cannot be so well reproduced using clusters of 21 atoms
where surface effects are present. For instance in the Kohn-Sham eg
orbital the electronic charge on the M2+ ion is around 5%. In
RbCdF3:Mn2+ such a figure would be smaller than 1% from
experimental hyperfine data on Cd nucleus [19,27]. Along this line
the calculated 10Dq values for AMF3:Mn2+ are displayed in Table 2
and compared to experimental values [6,7]. Although the present
calculations reproduce well, the significant increase undergone by
10Dq on going from KMgF3: Mn2+ to CsCaF3: Mn2+ finer details are
not so well accounted for.
Bearing in mind the good Re values derived for AMF3:Mn2+ using a
simple 21 atoms cluster a parallell study on the local relaxation
for Fe3+ doped fluoroperovskites has also been undertaken. The main
results are shown in Table 3 and compared to those derived from the
analysis of the experimental As constant [4,15]. They firstly
indicate that, as expected, Re values are now closer to 1.91 A
which is the typical figure found in pure compounds containing
FeF63~ units [28]. Moreover the change undergone by Re on passing
from KMgF3:Fe3+ to CsCdF3:Fe3+ derived from total energy
calculations of FeF6A8M617+clusters is shown to be higher than that
obtained through the experimental As values. This discrepancy
likely reflects the neglect of the relaxation of second and third
neighbours in the total energy calculations. In fact such a
relaxation cannot be neglected for CsCdF3:Fe3+ and RbCdF3:Fe3+
(where u1/R0 is about 10%) and, when considered, it would give rise
to a diminution of the Re values with respect to those gathered in
Table3.
The present results outline that experimental Re values of an
impurity in an insulating lattice can be well reproduced by
including in the calculated
Recent_Advances_in_Density_Functional_Methods_Part_III/9810248253/files/00025___1613c539b42687f4cc84c69cf0410a52.pdf16
cluster only some close neighbours to the MXN complex. This
conclusion is consistent with the following facts, i) In a
Born-Mayer description of a complex the stretching force constants
mainly depend on the repulsive part which exhibits a short range
character; ii) Though the presence of a substitutional impurity at
the origin of a lattice induces a strain at the r point such strain
decreases however as |r|~3 if the lattice is tridimensional.
In order to see the influence upon Re of host lattice ions lying
close to the MXN complex some calculations on complexes in vacuo
have also been performed. The obtained Re values for different
complexes are displayed in Table 4. It can be seen that in the case
of MnF64~ in vacuo Re is found to be
o
equal to 2.60 A and thus it is about 25% higher than the typical
Re value measured for Mn2+ in fluoride lattices with octahedral
coordination [26]. Despite this significant discrepancy it can also
be found in Table 4 that the Re values calculated for the complex
in vacuo become closer to experimental ones as far as the nominal
charge of the impurity increases. So the Re obtained for FeF63~ in
vacuo is only 10% higher than experimental ones for Fe3+ in
fluorides while such a difference is reduced to 3% in the case of
Fe6+ in oxides [29]. This situation can be explained considering
the electrostatic energy, Uei, of a complex in vacuo. Writting
Uei=-ae2/R the existence of a stable minimum requires a to be
positive bearing in mind that Born-Mayer interactions are
repulsive. For an octahedral complex if +zMe and -zLe mean the
total charge of the central ion and ligand respectively it is found
a = zL (6zM -10 zL). Therefore if MnF64^ and FeF63~ are considered
as ionic complexes a is equal only to 2 for the former while three
times bigger for the latter.
From the present analysis it can thus be concluded that in the
case of impurities in insulating lattices a reasonable information
about the right impurity-ligand distance can be reached from DF
calculations applied to clusters of small size. This conclusion is
certainly important in order to determine the Re value
corresponding to impurities (like Ag, Zn+, or Pb3+) which are not
usual constituents of known inorganic compounds. Work in this
direction is now in progress.
4 Acknowledgement
This work has partially been supported by the CICYT under
Project No. PB98-0190.
Recent_Advances_in_Density_Functional_Methods_Part_III/9810248253/files/00026___2e65d320db0948c549c6cebdcce320dc.pdf17
5 References
1. S. Sugano, Y. Tanabe and H. Kamimura, Multiplets of
Transition-Metal ions in Crystals (Academic Press, New York,
1970)
2. J. H. Barkyoumb and A. N. Mansour, Phys. Rev. B 46, 8768
(1992) 3. C. Zaldo, C. Prieto, H. Dexpert and P. Fessler, J. Phys.
Condens. Matter 3,
4135(1991) 4. M. Moreno, J. Phys. Chem. Solids 51, 835 (1990);
M. Moreno, M. T. Barriuso
and J. A. Aramburu, Appl. Magn. Reson. 3, 283 (1992) 5. S. J.
Duclos, Y. K. Vohra and A. L. Ruoff, Phys. Rev. B 41, 5372 (1990)
6. F. Rodriguez and M. Moreno, J. Chem. Phys. 84, 692 (1986) 7. C.
Marco de Lucas, F. Rodriguez and M. Moreno, J. Phys.: Condens.
Matter 5,
1437(1993) 8. A. G. Brenosa, M. Moreno, F. Rodriguez and M.
Couzi, Phys. Rev. B 44, 9859
(1991) 9. C. Marco de Lucas, F. Rodriguez and M. Moreno,
Phys.Rev. B 50, 2760 (1994)
10. A. Leble, These d'Etat Universite du Maine, (1982) 11. R. G.
Parr and W. Yang, DensityFunctional Theory of Atoms and
Molecules
(Oxford, New York, 1989) p. 141 12. J. A. Aramburu, M. Moreno,
K. Doclo, C. Daul and M. T. Barriuso, J. Chem.
Phys. 110, 1497 (1999) 13. N. Lopez and F. J. Illas. J. Phys.
Chem.B 102, 1430 (1998) 14. L. Helmholtz, J. Chem. Phys. 32, 302
(1960) 15. J.M. Dance, J. Grannec, A. Tressaud and M. Moreno, Phys.
Stat. So! (b) 173,
579 (1992) 16. R.C. DuVarney, J.R. Niklas and J.M. Spaeth, Phys.
Stat. Sol(b) 103, 329 (1981) 17. R.K. Jeck and J.J. Krebs, Phys.
Rev. B 5, 1677 (1972) 18. T.P.P. Hall, W. Hayes, R.W.H. Stevenson
and J. Wilkens, J. Chem. Phys. 38,
1977(1963) 19. P. Studzinski and J.M. Spaeth, J. Phys. C:
SolidSt.Phys. 19, 6441 (1986) 20. G. te Velde, E. J. Baerends, J.
Comput. Phys. 99, 84 (1992) 21. [21] P. Belanzoni, E. J. Baerends,
S. van Asselt and P.B. Langewen, /. Phys.
Chem. 99, 13094 (1995) 22. A. D. Becke, Phys. Rev. A 38, 3098
(1988) 23. J. P. Perdew, Phys. Rev. B 33, 8822 (1986) 24. M. T.
Barriuso, J. A. Aramburu and M. Moreno, J. Phys.: Condens. Matter
(at
press) 25. S. Lopez-Moraza, L. Seijo and Z. Barandiaran, J.
Chem. Phys. 57, 11974
(1998) 26. D. Babel and A. Tressaud, Solid Inorganic Fluorides,
ed P HagenmuUer
(Academic, New York: 1985)
Recent_Advances_in_Density_Functional_Methods_Part_III/9810248253/files/00027___a5dad2fbbac8d78908c3b9dee979082c.pdf18
27. M. Ziaei, Can. J. Phys 60, 636 (1982) 28. J.L. Fourquet and
H. Duroy, J. Sol. Stat. Chem. 103, 353 (1993) 29. K.Wissing, M.T.
Barriuso, J.A. Aramburu and M. Moreno, J. Chem. Phys. (at
Press)
Recent_Advances_in_Density_Functional_Methods_Part_III/9810248253/files/00028___7166d11f9474752cdf331fb87359bb99.pdf19
Table 1. Theoretical equilibrium distances, Re, obtained through
ADF calculations for Mn2+ doped AMF3 fluoroperovskites. ADF(IV)
means that IV type bases have been used for all the atoms. By
contrary, ADF(II) means that IV type basis has been only used for
manganese while II type bases have been used for the rest of the
atoms. These Re values are compared to the figures obtained from
the analysis of experimental As [7] and 10Dq [14,16] parameters and
also from available EXAFS data [19]. The experimental uncertainties
are also indicated. For comparison purposes the M2+-F" distance of
the perfect host lattice, R0, is also given [29]. All the distances
are in A.
R0 Re Exper.
7 Perovskitas ADF(IV) ADF(II) 10Pq EXAFS As
7.0.1 KMgF3:Mn2+ 1.987 2.057 2.058 2.068(4) 2.07(1)
KZnF3:Mn2+ 2.027 2.086 2.078 2.080(4) 2.08(1) 2.084(6)
RbCdF3:Mn2+ 2.200 2.127 2.133 2.134(4) 2.13(1) 2.124(6)
RbCaF3:Mn2+
CsCdF3:Mn2+
CsCaF3:Mn2+
2.228
2.232
2.262
2.120
2.156
2.158
2.138
2.165
2.165
2.131(4)
2.158(4)
2.142(6)
2.138(6)
2.155(6)
Recent_Advances_in_Density_Functional_Methods_Part_III/9810248253/files/00029___fa5f54cdc0261bf2e1f657b6f73faf0f.pdf20
Table 2. Calculated 10 Dq values (in crrr1) for six
fluoroperovskites. The meaning of ADF(II) and ADF(IV) is the same
than in Table 1. Experimental values from Ref. [7] are included for
comparison.
10Dq calculated 10Dq
Perovskite ADF(IV) ADF(II) experimental
KMgF3:Mn2+ 10831 10662 8432" KZnF3:Mn2+ 9396 9218 8215
RbCdF3:Mn2+ 8670 8275 7264 RbCaF3:Mn2+ 9678 9307 7306 CsCdF3:Mn2+
8476 7662 CsCaF3:Mn2+ 8904 8815 6890
Recent_Advances_in_Density_Functional_Methods_Part_III/9810248253/files/00030___e354794d4ce420f89b5c07e4da138c1f.pdf21
Table 3. Values of the equilibrium Fe3+-F~ distance, Re,
obtained for Fe3+ doped ANF3 (A=K, Rb, Cs; N=Mg, Zn, Cd)
fluoroperovskites from total energy calculations on FeF6A8N617+
clusters. Calculations have been performed using the ADF code, the
local density approximation (LDA) and including the Madelung
potential due to the rest of the lattice. The results are compared
to those derived from the analysis of the experimental isotropic
superhyperfine constant, As.
Host lattice R.(A) (from As)
Re(A) (from total energy)
(Re -Ho) ' H 0 (%)
KMgF3:Fe3+ 1.933 1.933 2.7 KZnF3:Fe3+ 1.937(0.2) 1.966 3
RbCdF3:Fe3+ 1.956(0.2) 1.974 10.5 CsCdF3:Fe3+ 1.957(0.2) 1.986
123
Recent_Advances_in_Density_Functional_Methods_Part_III/9810248253/files/00031___847cf219e0ad05ca97647825952bbde7.pdf22
Table 4. Obtained Re values for different complexes in vacuo by
means of ADF calculations (GGA functional). The values are compared
to those measured for the complex embedded in a crystalline
lattice. The discrepancy between two kinds of values decreases upon
increasing the nominal charge of the cation.
Cation
Mn2+ Fe3+
Cr4*
Complex
MnF64-FeF63-CrO/-
Re
Complex in vacuo
2.60 2.10 1.90
.(A) Complex in a lattice
2.12 1.90 1.70
Fe6+ Fe042- 1.70 1.65
Recent_Advances_in_Density_Functional_Methods_Part_III/9810248253/files/00032___da8060647c91d6983a8a1b0d66072c35.pdf23
-
Fig. 1. Cluster of 21 atoms used in the ADF calculation of
AMF3:Mn2+
systems
Recent_Advances_in_Density_Functional_Methods_Part_III/9810248253/files/00033___40cd1f8791844850bb5176e10420b8ee.pdf24
THEORETICAL EXPLORATION OF SINGLE AND MULTI STATE FEMTOSECOND
NUCLEAR DYNAMICS OF SMALL
METALLIC CLUSTERS USING THE DF METHOD
VLASTA B O N A C I C - K O U T E C K Y , MICHAEL HARTMANN, D E T
L E F R E I C H A R D T * P I E R C A R L O PANTUCCI+
Walther-Nernat-Institut fur Physikalische und Theoretische
Chemie, Humboldt-Universitat zv, Berlin,
Bunsenstrafie 1, D-10117 Berlin, Germany FAX: +4930/2093 5573,
phone: +4930/2093 5590, E-mail:
vbk@kirk. chemie.hu-berlin. de
"PERMANENT ADDRESS: KONRAD-ZUSE ZENTRUM FUR INFORMATION-
STECHNIK BERLIN, TAKUSTR. 7, D-14195 BERLIN, E-MAIL:
[email protected] tPERMANENT ADDRESS: DEPARTMENT OF BIOTECHNOLOGIES
AND
BIOSCIENCES, UNIVERSITY OF MILANO-BICOCCA, ITALY, E-MAIL:
[email protected]
Recent_Advances_in_Density_Functional_Methods_Part_III/9810248253/files/00034___619bb1617da256a10bb2f7186288564d.pdf25
We present the outline of ab-initio molecular dynamics method
based on gradient
corrected DF approach using gaussian atomic basis (AIMD-GDF) for
the adia-batic ground states which is suitable to treat one or more
involved ground elec-
tronic states simultaneously with the motion of the nuclei.
Gradients of the Born-
Oppenheimer ground state energy, obtained by iterative solution
of the Kohn-Sham
equations are used to calculate the forces acting on atoms at
each instantaneous
configuration along trajectories generated by solving classical
equation of motion. First we apply single state MD to investigate
dynamical and structural proper-
ties of Lig" clusters as a function of excess energy. We show
that in spite of large
mobility of alkali metal atoms the isomerization and liquid-like
behaviour occur
at relatively high temperatures. Second, the multistate nuclear
dynamics will be
applied to Ag^" jAg4 / A g J in connection with the NeNePo
(negative ion-to-neutral-to-positive ion) pump-probe femtosecond
spectroscopy. For this purpose the MD on the ground state of AgJ is
necessary for generating the initial conditions. After
photodetachment the dynamics of the ground state of Ag4 is
investigated and for
the probe step the energies of the cation have to be calculated
along the trajectories of the neutral system. The fs-multistate
nuclear dynamics involving AIMD-GDF
ground state of Ag^"/Ag4/Ag4~ allows to introduce temperature
dependence into
initial conditions and to determine time scales of geometrical
relaxation, isomer-
ization and internal vibrational redistribution (IVR).
Recent_Advances_in_Density_Functional_Methods_Part_III/9810248253/files/00035___9431cc629ddb0d91e6e601ffc0dca54a.pdf26
1 Introduction
It has been recognized that discoveries of novel physical and
chemical phe-nomena of atomic clusters are closely connected with
their non-scalable prop-erties in the regime where each atom counts
1. Therefore, the necessity to control the size, shape and
temperature of the clusters confirmed the need for accurate
determination of their structural, optical and dynamical properties
2_4
The gradient corrected DF approach using Gaussian atomic basis
has been widely used for the determination of the structural and
other ground state properties, while the Car-Parrinello molecular
dynamics procedure based on the DF method (AI MDE-DF) has been
originally designed employing the plane wave expansion 5 . The
purpose of this contribution is to present an ab-initio MD method
based on a gradient corrected DF approach using Gaussian atomic
basis (AIMD-GDF) 6 '7 for the adiabatic ground states in the
context of the single and multi state dynamics.
The single ground state AIMD-GDF dynamics is a useful tool to
study the influence of the temperature on the structural properties
of small metallic clusters allowing to find mechanisms of
isomerization processes as well as to formulate precise criteria
for "phase transitions" in finite systems 6 '7.
The multi state AIMD-GDF involving ground states of negative,
neu-tral and cationic clusters is designed to address the
femtosecond pump-probe NeNePo spectroscopy (negative ion-to
neutral-to positive ion), which allows to study, after one photon
detachment, the geometric relaxation and internal vibrational
redistribution (IVR) on the ground state of the neutral species, by
a delayed ionizing pulse via two-photon ionization 8.
In this paper we will first briefly outline the AIMD-GDF which
have been developed to study the temperature dependent ground state
dynamical and structural properties. The results of the single
state AIMD-GDF of Li^ " will be presented as an example of
different temperature behaviour of isomers with distinct structural
properties. The multi state nuclear dynamics will be applied to
Ag^/Ag4/AgJ clusters in connection with the NeNePo fs pump-probe
spectroscopy in order to determine the time scale of the
isomerization process of Ag4 clusters.
2 Computational
The idea of AIMD, originally initiated by Car and Parrinello 5
in connec-tion with a DF procedure employing a plane waves
expansion, is here revisited
Recent_Advances_in_Density_Functional_Methods_Part_III/9810248253/files/00036___2d38b839293ee9249447296c8f765384.pdf27
in terms of a quantum chemical approach based on fully self
consistent ener-gies and gradients in the framework of the HF a-10
or GDF 6 '7 procedures, employing the AO basis sets centered at the
nuclei.
The classical trajectories are calculated using the Verlet
algorithm ac-cording to which the position and velocity of the
nucleus i (rj, Vj) at time step tn = nAt, are obtained
recurrently:
r(n+l) = 2 r(n) _ r ( n - l ) (A*)2 p(, i i i
m i
vr>=v+^(F?+Fr>; The force F\n' acting on nucleus i is
related to the gradients of the total molecular energy computed at
SCF (HF or DF) level. The ground state electronic wavefunction and
energy are computed at each time step (i.e. for each geometric
configuration). In our AIMD-HF or AIMD-DF programs the total
molecular energy obtained from iterative HF or iterative Kohn-Sham
procedure, is the potential energy and its corresponding
derivatives are the forces acting on the nuclei.
The accuracy of calculated energies and gradients must be higher
than usually needed for the geometry optimization, since the
precise determina-tion of forces is required for the conservation
of the total energy. In order to obtain meaningful information on
dynamical properties sufficiently long tra-jectories have to be
calculated. The computational demand is large due to both
requirements. Therefore, different strategies had to be used to
speed up calculations. Since both AIMD-HF and AIMD-DF schemes are
suitable for parallel processing, the efficiency of the programs
has been achieved first by means of a full parallelization of the
most numerically demanding algorithms. Secondly, all input-output
operations have been eliminated by keeping the integrals in memory
and finally, a careful optimization of the sequential parts has
been introduced 7 '10. However, applications are limited to small
systems, particularly in the case of AIMD-DF with gradient
corrections owing to time-consuming numerical three-dimension
integration of the functionals for the exchange-correlation (e.g.
those proposed by Becke 12 and Lee, Yang and Parr 13 (B-LYP)).
3 Results and Discussion of Dynamics of Lig~ Cluster
The AIMD schemes represent suitable tools to study not only
structural features but also the influence of internal energy on
the dynamics of metal
(1)
(2)
Recent_Advances_in_Density_Functional_Methods_Part_III/9810248253/files/00037___1ae3cef8f0fc6f3ec388a078a0ab9068.pdf28
clusters. The latter aspect is particularly important, since it
allows to in-vestigate mechanism of isomerization processes with
increasing temperature 6'7, 9
-
u. For this purpose the trajectories need to be calculated over
a broad
range of different fixed total energies. The total energy of a
cluster is varied by random distortions of its equilibrium geometry
in the case that zero initial atomic velocities are assumed. We
have chosen initial conditions which satisfy the requirement of
zero linear and angular momenta of the clusters. The time step is
equal to 0.5 fs and the lengths of the simulations range from 10 to
40 ps. The conservation of the total cluster energy in the longest
runs was better than 10~2 eV, due to the high-accuracy criteria for
calculations of energies and derivatives.
A small split valence basis set with three s-functions and one
p-function was used in our AIMD calculations 6 '7, 9-11. Cluster
geometries, which were generated along the trajectories at high
internal energies were employed as initial coordinates in a
gradient-based search for stable isomers, which were identified as
local minima by carrying out a full harmonic vibrational analysis.
In addition to the isomers already found by standard optimization
techniques 2-
4 the AIMD procedure gave evidence of existence of other
isomeric forms.
The structures and energy sequence of the isomers of Lin
clusters obtained using a small AO basis were in agreement with the
corresponding results obtained with considerably larger basis sets
14.
The long time average of the kinetic energy (Ek) over the entire
length of a trajectory was used to estimate the temperature T of a
cluster according to T = 2(Ek)/{2>n 6)k, where n is the number
of atoms and k is the Boltzmann constant 6 '7, 9 - n .
In a previous work on structural properties, the centered
antiprism (D.^) has been identified as the most stable isomer of
Lig" at the HF level15 (isomer I). This result was confirmed also
by an electron correlation treatment using large scale CI 2'14 as
well as by the geometry optimization with the gradient corrected
density functional method (B-LYP). However, two other isomeric
forms close in energy have been found by standard geometry
optimization techniques as well as by AIMD schemes: the C21,
structure (isomer II) (which can be obtained by capping the T^ form
of octamer or by bicapping a pentag-onal bipyramid) and the C31,
form (isomer III) (a deformed section of the fee lattice). They
were identified as local minima by harmonic frequency analysis,
both in the framework of the HF and B-LYP procedure. At the HF
level, the energy sequence is D4d < C2v(AE = 0.064eV) <
C3v(AE = O^OeF). The electron correlation effects included in the
density functional treatment give rise to larger energy separation
of the isomers II and III with respect to the isomer I (AE = 0.22
and AE = 0.40 eV) but the energy sequence remains
Recent_Advances_in_Density_Functional_Methods_Part_III/9810248253/files/00038___0ac8de9d889a32e91d9a5cf5b5a6c6d9.pdf29
unchanged. We have chosen to discuss temperature behavior of
individual isomers
of LiJ cluster because they are characterized by distinct type
of structures. Note that the most stable isomer of Lig" (D^)
assumes a shape different from the most stable isomer of NaJ
(C21,). Therefore distinct temperature behavior of Lig" and Nag"
clusters might be expected. Investigation of the temperature
behavior of the neutral Nan cluster has been carried out using Car,
Parrinello method 5 employing DF without gradient corrections and
plane waves 16. Notice that gradient corrections introduced in DF
by Becke 12
for exchange part of the functional, which improved the
determination of binding energy and bond distances considerably,
proved to be also very important for calculation of the energy
sequences of the isomers.
Our simulations were initialized from randomly distorted
geometries of different isomers of Lig", and zero initial
velocities (cf. Ref. 6 ,7 ; 9 - 1 1 ) . The results obtained from
AIMD-B-LYP simulations will be analyzed using the following four
quantities. We find particularly instructive to analyze the
tra-jectories in terms of 'atomic equivalence indeces'
= Si O O Si =, (ii) A peroxy bridge is excited and/or ionised
and a peroxy radical paired with
an E'-centre is formed: = Si O O Si= > = Si O 0\. . .*Sis
(Hi) A hole polaron trapped on an Al impurity also forms a
peroxy radical coordinated to this impurity.
= Si 0 \ . . . Al = +! / 2 0 2 > = Si O 0*....A1= In our
recent work employing a PW91/DNP periodic model based on the
siliceous sodalite as a host material, we have studied these
reactions by calculating the energy minimum defect configurations
and proposed a number of alternative routes leading to the
formation and transformation of the peroxy species [76] (cf. other
theoretical work [77-79]).
Figure 4.3 Configuration and spin density distribution on peroxy
radical defect at Al impurity
One such structure resulting from reaction (iii) is shown in
Fig. 4.3. The possibility of a bridging configuration in which both
oxygen ions are bound to a silicon has also been investigated.
However, we have found only one energy minimum with one bridging
and one terminal oxygen for this defect in the given local
environment. In this defect the electron hole is nearly completely
localised in one of the antibonding TC states of the peroxy,with
the spin density distributed between bridging and terminal oxygens
at 0.23 and 0.72e respectively. The charge of the terminal oxygen
is only -0.09e, and a typical superoxide bond distance of 1.36A has
been obtained. The effective charge on the bridging oxygen of-0.40e
is about half of the effective charge on oxide ions in a perfect
siliceous material i.e. it is close to an idealised 0" ion.
Recent_Advances_in_Density_Functional_Methods_Part_III/9810248253/files/00068___f92eaa8a162429af59d399da87a6efd7.pdf59
Figure 4.4 Oxygen adsorption on silica framework: structural
characteristics, energetics and spin distribution. (Energies in
kcal/mol)
Our calculations have also shown that an excitation of the
peroxy bridge does not proceed by reaction (ii) but is rather
localised on the peroxide species: a Si cannot trap an electron
without further drastic structural transformation (e.g. silicon
inversion through the basal plane of the three oxygens bonded to
it). Fig. 4.4 gives an example how the lowest singlet-triplet
transition would lead to two main configurations. In one, with the
lowest energy of ca. 3 3 kcal/mol, an oxygen adsorption reaction
(i) is reversed with the formation of the physisorbed atomic oxygen
in a triplet state (an O ad-atom on tetrahedral Si site yields a
5-coordinated Si). The triplet is the ground state of this species.
Upon excitation into a singlet state (which requires 0.2-0.3eV
depending on the environment) the defect without crossing a barrier
transforms back into a peroxy bridge. Peroxy Species on the (001)
MgO Surface The question addressed here concerns the positions of
energy levels of adsorbed molecules with respect to some common
level. Clearly this question is directly related to the processes
which involve charge transfer, particularly in chemical reactions
at surfaces. If energy levels of atoms and molecules participating
in the reaction are known, one could, in principle, predict the
direction of the charge transfer and character of the chemical bond
formed. From a physical point of view,
Recent_Advances_in_Density_Functional_Methods_Part_III/9810248253/files/00069___7593b244081acc338d9b2ee38d3274ff.pdf60
knowledge of the relative energy levels of adsorbed species is
also important. It can be used, for example, for the interpretation
of optical absorption and MIES experiments and to provide a better
understanding of the mechanisms of ion desorption induced by
excitation or ionisation of surface species.
We present results of our studies of peroxide species at the
(001) MgO surface and calculate its energy levels with respect to
the vacuum level. The adsorption of atomic oxygen and other oxygen
containing molecules at the surfaces of alkali-earth oxides has
been extensively studied over the last years (e.g. [80,81]), but to
the best of our knowledge the present study is the first attempt to
address the above issue.
mmM
i,eV
2.0-
4.0-
6.0-
.
L
Vacuum level L
ir oi uie ideal ivigu (wi) (o ev
IP of the peroxide at MgO (001) 6.3 eV
Valence Band
Figure 4.5 Geometry of peroxy at the (001) MgO surface: quantum
mechanical cluster used in the calculations (top left) and details
of the optimised structure (bottom left - larger circles are
oxygens, smaller ones - magnesiums); vertical ionisation energy
diagram (right) shows position of the top of the valence band and
peroxy defect with respect to the vacuum level
Recent_Advances_in_Density_Functional_Methods_Part_III/9810248253/files/00070___d09c19f452de102435083649f8cc1031.pdf61
We have employed an embedded cluster approach in which a QM
cluster was embedded into a finite nano-cluster representation of
the polarisable host lattice via the shell model. Relaxation of
both electronic and ionic subsystems of the host lattice is
performed self-consistently with the charge density of the QM
cluster calculated using the B3LYP density functional. The
self-consistent scheme of calculations has been described in more
details elsewhere [82,83]. The QM cluster used in these studies is
shown in Fig. 4.5a. All anions and all fully coordinated cations
have been treated as all electron atoms using the standard 6-31G
basis set. Cations at the border of the QM cluster have been
treated using Hay and Wadt effective core pseudopotentials [84] and
one contracted s-function.
The relaxed geometry and electronic structure of the peroxide
species are in accord with previous studies [80]. Adsorption of the
atomic oxygen leads to the formation of the O22" molecule aligned
approximately along [111] direction. There is a clear indication of
charge transfer from the surface anion to the ad-atom: results of
Natural Population Analysis (NPA) [85] suggest that about 0.75e is
associated with the ad-atom. A similar value of the charge has been
obtained by fitting the electrostatic potential of the QM cluster
at the ions of the host lattice.
The energy level of the surface defect with respect to the
vacuum is given by the ionisation potential of the defect. We
calculate the ionisation potential in two ways: first, we denote it
as IP(0), corresponding to one electron being removed from the
system with electronic relaxation inside the QM cluster only. Thus
the IP is calculated in the conventional cluster model, in which
the effects of polarisation of the host lattice are ignored
completely. Inevitably, this leads to overestimated value. A more
accurate IP, denoted as IP(I), is obtained when response of the
electronic subsystem of the host lattice is included. The latter
manifests itself in the electronic polarisation energy of the host
lattice and the change of the Madelung potential at the atoms of
the QM cluster. The calculated values for EP(0) and IP(I) are 7.0
eV and 6.3 eV respectively. A large difference between the IP(0)
and IP(I) suggests that polarisation of the host lattice in the
processes involving change of the charge state of surface species
should not be neglected. Fig. 4.5 shows energy levels of the
surface with peroxide species on it and the top of the valence band
of defect free surface with respect to the vacuum level (calculated
earlier using the same technique [82,83]). The defect energy level
is only 0.2 eV above the top of the valence band. Such a small
separation of the defect level and the top of the valence band
implies that the valence band states may be mixed with the defect
states and, therefore modelling of this defect on its own, without
valence band states included, may be unreliable.
Recent_Advances_in_Density_Functional_Methods_Part_III/9810248253/files/00071___f9813bb4fdee197d133647c069b21fc5.pdf62
Surface anion nest to peroxide (along [110])
Figure 4.6 Spin density of the system after its ionisation: the
electron hole is mainly localised on the surface anion next to the
peroxy and partly delocalised over all other surface anions and
peroxy
The one-electron spectrum of the system has three one-electron
states due to the bonding peroxide orbitals below the valence band
and two one-electron states due to the anti-bonding orbitals just
above the top of the valence band. The highest occupied
one-electron state, approximately 0.6eV above the top of the
valence band, is associated with the peroxy molecule and a
neighbour surface anion (along [110]). The peroxide molecule is
tilted towards the other surface anion bringing a negative
contribution to the Madelung potential at the surface anion and
destabilises its electronic states. Plots of the spin density (Fig.
4.6) of the ionised system in the plane of the surface and in the
plane perpendicular to the surface demonstrate that most of the
electron hole is associated with the surface anion. The electronic
hole is also partly delocalised over the peroxide species and all
other surface anions. The hole delocalisation correlates with our
previous conclusion about the possibility of mixing of defects
states and valence band states.
All the results discussed above indicate that adsorption of the
atomic oxygen at the MgO surface leads to the formation of a very
extended electronic defect. The energy level of this defect is very
close to the top of the valence band. It is important to note that
the presence of the one-electron state associated with a surface
anion in the forbidden gap found by us was not observed in the
periodical plane-wave DFT calculation of Kantorovich et al [80],
which is possibly related to two differences in our approaches.
First, we used a more accurate hybrid density functional (B3LYP) as
compared to a pure density functional (PW91) employed in [80].
Secondly, periodical calculations predict a larger width of the
valence band. A better representation of the valence band, along
with making use of different density
Recent_Advances_in_Density_Functional_Methods_Part_III/9810248253/files/00072___6819340fa36ff780b9eec572738d96af.pdf63
functionals, is necessary to clarify this inconsistency.
Corresponding calculations will be performed in the near
future.
5 Conclusions
The work reported in this paper has shown that current DFT
techniques may be used to investigate successfully a variety of
problems concerned with the chemistry and physics of complex
materials. The systems studied include ionic, semi-ionic and
molecular materials. The DFT methodology employed perform robustly
in modelling key structural and electronic properties of these
materials.
6 Acknowledgements
We would like to thank EPSRC, Dow Chemical Co, Eastman Kodak
Labs for financial support and MSI for providing software. We also
thank F. Cora, P. E. Sinclair, P. Day, G. Sankar, J. M. Thomas, J.
M. Garces, A. Kuperman, A. A. Shluger, L. N. Kantorovich, D. S.
Shephard and N. A. Ramsahye for useful discussions.
References
1. Hohenberg P. and Kohn W., Phys. Rev. 136 (1964) B864. 2.
Theory of the inhomogeneous electron gas, ed. by Lundqvist S. and
March N.
H. (Plenum Press, 1983). 3. Parr R. G. and Yang W., Density
Functional Theory (Oxford University Press,
1989). 4. McWeeny R., Methods of Molecular Quantum Mechanics
(Academic Press,
1992). 5. Fulde P., Electron Correlation in Molecules and Solids
(Springer Verlag 1996). 6. Catlow C. R. A., Barker C. M., Bell R.
G., Bromley S. T., Coombes D. S., Cora
F., French S., Slater B., Sokol A. A., Whitmore L. and Woodley
S. M., Computer Modelling of Catalysts and Catalysis, in NATO
AS1series, in press.
7. Catlow C. R. A., Ackermann L., Bell R. G., Cora F., Gay D.
H., Nygren M. A., Pereira J. C. G., Sastre G., Slater B. and
Sinclair P. E., Computer Modelling as a Technique in Solid State
Chemistry, Faraday Discussions 106 (1997) 1.
8. Cora F. and Catlow C. R. A., Phys. Stat. Sol. B 111 (2000)
577. 9. Braithwaite J. S., Catlow C. R. A., Gale J. D. and Harding
J. H., Chem. Mater.
11 (1999) 1990. 10. Catlow C. R. A., Coombes D. S. and Pereira
J. C. G., Chem. Mater. 10 (1998)
3249.
Recent_Advances_in_Density_Functional_Methods_Part_III/9810248253/files/00073___b0599c641fa47b936427807f0e54710c.pdf64
11. Pereira J. C. G, Catlow C. R. A. and Price G. D., J. Phys.
Chem. A 103 (1999) 3268, ibid. 3268.
12. Sinclair P. E. and Catlow C. R. A., J. Chem. Soc: Faraday
Trans. 93 (1997) 333.
13. Catlow C. R. A, Ackermann L., Bell R. G, Gay D. H., Holt S,
Lewis D.W., Nygren M. A., Sastre G, Sayle D. C, and Sinclair P. E.,
J. Mol. Catal. A 115 (1997)431.
14. Slater B., Catlow C. R. A., Williams D. E. and Stoneham A. M
, Dissociation of 0 2 on the Reduced (110) Surface, submitted to J.
Chem. Soc. Chem. Commun.
15. Bromley S, T , Sankar G, Catlow C. R. A., Thomas J. M.,
Johnson B., and Maschmeyer T., New Insights into the Structure of
Silica Supported Bimetallic Clusters: a Combined Computational and
Experimental Study, submitted to J. Chem. Soc. Chem. Commun.
16. Kohn W. and Sham L. J., Phys. Rev. 140 (1965) Al 133. 17.
Perdew J. P. and Wang Y., Phys. Rev. B 33 (1986) 8800. 18. Perdew
J.P. and Wang Y., Phys. Rev. B 45 (1992) 13244. 19. Barth U. von
and Hedin L., J. Phys. C 5 (1972) 1629. 20. Pant M. M. and
Rajagopal A. K., Solid State Commun. 10 (1972) 1157. 21. Wang J.
and Boyd R. J., Canad. J. Phys. 72 (1994) 851. 22. Cotton F. A.,
Cowley A. H. and Feng X. J., J. Am. Chem. Soc. 120 (1998)
1795. 23. Noodleman L., J. Chem. Phys. 74 (1981) 5737. 24.
Perdew J. P., Savin A. and Burke K., Phys. Rev. A 51 (1995) 4531.
25. Vosko S.H., Wilk L. and Nusair M., Can. J. Phys. 58 (1980)
1200. 26. Perdew J.P. and Zunger A., Phys. Rev. B 23 (1981) 5048.
27. Perdew J.P. and Wang Y., Phys. Rev. B 40 (1989) 3399. 28. Becke
A.D., Phys. Rev. A 38 (1988) 3098. 29. Lee C , W. Yang, R.G. Parr,
Phys. Rev. B 37 (1988) 785. 30. Becke A.D., J. Chem. Phys. 96
(1992) 2155. 31. Becke A.D., J. Chem. Phys. 97 (1992) 7528. 32.
Becke A. D., J. Chem. Phys. 98 (1993) 5648. 33. Delley B., J. Chem.
Phys. 92 (1990) 508. 34. EOM DSolid User Guide, September 1996 (San
Diego, Molecular Simulations,
1996), Dmof, Cerius2 3.5, User Guide, September 1997 (San Diego,
Molecular Simulations, 1997).
35. Gaussian 98 (Revision A.3, 6, 7) User's Reference (Frisch
M.J. et al., Gaussian Inc., Pittsburgh PA, 1998).
36. Pisani C, Dovesi R and Roetti C, Hartree-Fock ab initio
Treatment of Crystalline Systems, Lecture Notes in Chemistry
(Springer Verlag, Heidelberg, Vol. 48, 1988)
37. CRYSTAL95 User's Manual (Dovesi R., Saunders V. R., Roetti C
, Causa M., Harrison N. M., Orlando R. and Apra E., University of
Torino, Torino, 1996).
Recent_Advances_in_Density_Functional_Methods_Part_III/9810248253/files/00074___9b700c8878ebca8ca9e613944a95c37a.pdf65
38. Causa M. and Zupan A., Int. J. Quantum Chem.: Quantum Chem.
Symposium 28(1994)633.
39. Raja R, Sankar G., Hermans S., Shephard D. S., Bromley S.
T., Thomas J. M, and Johnson B. F. G., J. Chem. Soc: Chem. Commun.
(1999) 1571.
40. Goursot A., Papai I. and Daul C , Int. J. Quant. Chem. 52
(1994) 799. 41. Hirschfeld F. L., Theor. Chim. Acta BAA (1977) 129.
42. Bozarth R. M , Wolff P. A., Davis D. D., Compton V. B. and
Wernick J. H.,
Phys. Rev. 122(1961)1157. 43. Nieuwenhuys G. J., Adv. Phys. 24
(1975) 515. 44. Cable J. W. and David L., Phys. Rev. B 16 (1977)
297. 45. Verbeek B. H., Nieuwenhuys G. J., Mydosh J. A., Van Dijk
C. and Rainford B.
D., Phys. Rev. B 22 (1980) 5426. 46. Piveteau B., Desjonqueres
M.-C, Oles A. M. and Spanjaard D., Phys. Rev. B
53(1996)9251. 47. Reddy B. V., Khanna S. N., Dunlap B. I., Phys.
Rev. Lett. 70 (1993) 3323. 48. Mizuno M., Gariot A. and CavaM., J.
Chem. Soc: Chem. Commun. (1978) 18. 49. Demiralp E., Dasgupta S.
and Goddard W. A., J. Am. Chem. Soc. 117 (1995)
8154. 50. Mallah T , Hollis C , Bott S., Kurmoo M., Day P.,
Allan M. and Friend R. H.,
J. Chem. Soc: Dalton. Trans. (1990) 859. 51. Mori T., Kobayashi
A., Sasaki Y., Kobayashi H., Saito G., Inokuchi H., Bull.
Chem. Soc. Japan. 57 (1984) 627. 52. Demiralp E. and Goddard W.
A., J. Phys. Chem. 98 (1994) 9781. 53. Sato N., Saito G. and
Inokuchi H., Chem. Phys. 76 (1983) 79. 54. Bar G., Magonov S. N.,
Liang W. and Whangbo M. H., Synth. Mets. 72 (1995)
189. 55. Sekiyama A., Susaki T., Fujimori A., Sasaki T., Toyota
N., Kondo T., Saito G.,
Tsunekawa M., Iwasaki T., Muro T., Matsushita T., Suga S., Ishii
H. and Miyahara T., Phys. Rev. B 56 (1997) 9082.
56. Yoshimura M., Shigekawa H., Nejoh H., Saito G., Saito Y. and
Kawazu A., Phys. Rev. B 43 (1991) 13590.
57. Campos C. E., Sandhu P. S., Brooks J. S. and T. Ziman, Phys.
Rev. B 53 (1996) 12725.
58. Shumway J., Chattopadhyay S. and Satpathy S., Phys. Rev. B
53 (1996) 6677. 59. Griscom D.L., J. Ceramic Soc. Japan 99 (1991)
923. 60. Priest V., Cowan D.L., Ysar H. and Ross F.K., Phys. Rev. B
44 (1991) 9877. 61. Nakatsui H. and Nakai H., Chem. Phys. Lett. 197
(1992) 2423. 62. Savariault J.M. and Lehman M.S., J. Am. Chem. Soc.
102 (1980) 1298. 63. Moutiers G., Cassir M. and Devynck J., J.
Electroanalytical Chem. 324 (1992)
175. 64. Mestl G., Rosynek M. P. and Lunsdorf J. H., J. Phys.
Chem. B 101 (1997)
9329. 65. FopplF., AnorgZ. Allg. Chem. 291 (1957) 12.
Recent_Advances_in_Density_Functional_Methods_Part_III/9810248253/files/00075___bd1a2acde12e572861756f4521eb84a2.pdf66
66. Vannerberg N.-G, Prog. Inorg. Chem. 4 (1962) 125. 67. Bremm
T. and Jansen M , Z. Anorg. Allg. Chem. 610 (1992) 64. 68.
Savariault J.-M. and LehmannM. S., J. Am. Chem. Soc. 102 (1980)
1298. 69. Eysel H. H. and Thym S., Anorg Z. Allg. Chem. 411 (1975)
97. 70. Konigstein M., Cora F. and Catlow C. R. A., J. Solid State
Chem. 140 (1998)
103. 71. Konigstein M., Sokol A. A. and Catlow C.R.A., Phys.
Rev. B 60 (1999) 4594. 72. Zhukovskii Yu. F., Sokol A. A., Kotomin
E. A., Catlow C. R. A. and Nieminen
R.M., J. Phys.: Condens. Matter 9 (1997) 3559. 73. Sokol A. A.,
Catlow C.R.A., Garces J. M. and Kuperman A., J. Phys. Chem. B
102 (1998) 10647. 74. Catlow C. R. A, Islam MS. and Zhang X., J.
Phys.: Condens. Matter 10
(1998) L49. 75. West A. R., Solid State Chemistry and its
Applications (John Wiley & Sons,
1989) 42. 76. Sokol A. A., Catlow C. R. A., Garces J. M. and
Kuperman A., In Proceedings
of 12th International Zeolite Conf. Baltimore, Materials
Research Soc. (1999) 457.
77. Edwards A. H. and Fowler W. B., Phys. Rev. B 26 (1982) 6649.
78. Hamann D. R, Phys. Rev. Lett. 81 (1998) 3447. 79. Pacchioni G.
F. and Ierano G, Phys. Rev. B 57 (1998) 818. 80. Kantorovich L. N.
and Gillan M. J., Surf. Sci. 374 (1997) 373. 81. Nygren M. A. and
Pettersson L. G. M , Chem. Phys. Lett, 230 (1994) 456. 82.
Kantorovich L. N., Shluger A. L., Sushko P. V. and Stoneham A. M.,
submitted
to Surf. Sci. (1999). 83. Kantorovich L. N., Shluger A. L.,
Sushko P. V., Gunster J., Stracke P.,
Goodman D. W. and Kempter V., Faraday Discussions, 114 (1999).
84. Hay P. J., Wadt W. R., J. Chem. Phys. 82 (1985) 270 ibid. 284,
299. 85. Reed A. E., Weinstock R B. and Weinhold F., J. Chem. Phys.
83 (1985) 735.
Recent_Advances_in_Density_Functional_Methods_Part_III/9810248253/files/00076___d62903a2f8e8aea24b3818cc06902dce.pdf67
A HYBRID FUNCTIONAL FOR THE EXCHANGE-CORRELATION KERNEL IN
TIME-DEPENDENT DENSITY FUNCTIONAL THEORY
K. BURKE Departments of Chemistry and Physics, Rutgers
University, 610 Taylor Road,
Piscataway, NJ 08854, USA E-mail: [email protected]
M. PETERSILKA AND E. K. U. GROSS Institut fur Theoretische
Physik, Universitdt Wurzburg, Am Hubland, 97074
Wiirzburg, Germany E-mail: [email protected]
A review of the approximations in any time-depedendent density
functional calcu-lation of excitation energies is given. The
single-pole approximation for the suscep-tibility is used to
understand errors in popular approximations for the
exchange-correlation kernel. A new hybrid of exact exchange and
adiabatic local density approximation is proposed and tested on the
He and Be atoms.
1 Introduction
Ground-state density functional theory is well-established as an
inexpensive alternative to traditional ab initio quantum chemical
methods.1 Now time-dependent density functional theory (TDDFT) is
rapidly emerging as an in-expensive accurate method for the
calculation of electronic excitation energies in quantum
chemistry.2'3 Calculation of dynamic response properties using
TDDFT has a long history, since the pioneering work of Zangwill and
Soven.4'5 It is only relatively recently that attention has been
focussed on the direct extraction of excitation energies.6-11
Already, this method has been imple-mented in several quantum
chemistry packages, such as deMon,12 Turbomole, 13,14 ADF,15 and
QCHEM.16 Important calculations include the calculation of
excited-state crossings in formaldehyde,12 excitations with
significant doubly-excited character,16 the photospectrum of
chlorophyll A,17 and the response of 2-D quantum strips.18
How are excitation energies calculated using TDDFT? First, a
self-consistent ground-state Kohn-Sham calculation is performed,
using some ap-proximation for the exchange-correlation energy Xci
such as B3LYP19 or PBE.20 This yields a set of Kohn-Sham
eigenvalues ej and orbitals i. Even with the exact ground-state
energy functional and potential, these eigenvalues are in general
not the true excitations of the system, but are closely related.
In
Recent_Advances_in_Density_Functional_Methods_Part_III/9810248253/files/00077___39bcfaa48a267b38ab3a585f005c178c.pdf68
a second step, the central equation of TDDFT response theory is
solved, which extracts the true linear response function from its
Kohn-Sham counterpart. This equation includes a second unknown
functional, the exchange-correlation kernel /Xc(r, r'; w), which is
the Fourier transform of the functional derivative of the
time-dependent exchange-correlation potential. The poles of the
exact response function are shifted from those of the KS function,
and occur at the true excitations of the system. These steps are
typically repeated for several nuclear positions.
The success of any density functional method, however, depends
on the quality of the approximate functionals employed. The above
calculation re-quires two distinct density functional
approximations: one for the ground-state energy, which implies a
corresponding approximation for the exchange-correlation potential
vxc(r) = SExc/5n(r), and a second for the exchange-correlation
kernel. Most calculations now appearing in the chemical literature
use the adiabatic local density approximation (ALDA) for / x c
Adiabatic im-plies that the frequency-dependence of / x c is
ignored, and its u
