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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
EditorinCharge 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 CoupledCluster Methods (Volume 3) ed. Rodney J. Bartlett
Recent_Advances_in_Density_Functional_Methods_Part_III/9810248253/files/00003___ef8e66279320f81b29864574ab3a36e5.pdfRecent Advances in Computational ChemistryVol. 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 MilanoBicocca, 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
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British Library CataloguinginPublication 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.
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Recent_Advances_in_Density_Functional_Methods_Part_III/9810248253/files/00005___3f4c62516964a2c3460f1e77beba8c3e.pdfV
PREFACE
Density Functional Theory (DFT) aims at the description of the physicochemical 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 manybody 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 6fh10th 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 oneday meeting organized by 14 chemists, wishing to share their experiences in MSX? and DVMHartreeFockSlater 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, MilanoBicocca, 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 TransitionMetal Silonyl Complexes MSiO 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. BonacicKoutecky, 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 ExchangeCorrelation Kernel in TimeDependent Density Functional Theory 67 K. Burke, M. Petersilka and E. K. U. Gross
Calculation of Vertical Ionization Potentials Using a Density Functional TotalEnergy 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. PfisterGuillouzo
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
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10. Local Density Approach to RealSpace MultipleScattering Calculations of InnerShell 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 LowDimensional Iron Systems 205 J. Izquierdo, A. Vega, L. C. Balbds, P. Ordejon, D. SanchezPortal, 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. FirstPrinciples Calculation of Multiplets of TransitionMetal Ions in Crystals Based on Density Functional Method 278 K. Ogasawara, T. Ishii, I. Tanaka and H. Adachi
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19. Correlated Multideterminantal Potential Energy Curves for Diatomic Molecules with One ValenceBond Pair 293 E. SanFabian, L. P. Abia and J. M. PerezJordd
20. The Description of the Photoionization Process by the BSpline 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. SpinDensities in ChargeTransfer Complexes Derived from DFT Calculations Using An OrbitalFree 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
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THEORETICAL STUDY OF THE TRANSITIONMETAL SILONYL COMPLEXES MSIO 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
Email : [email protected]
The spectroscopic properties of MSiO and M(SiO)2 (11 and 12 complexes with M = Cu, Ag, or Au) have been theoretically studied. It has been shown
that both MSiO and M(SiO)2 compounds in their ground state are bent with a metalSi 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 MSi bond energy correlates with the
electron density located at the MSi 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 compounds 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 exchangecorrelation functional seems an adequate approach to study such a system 3. In particular, hybrid approach proposed by Becke 4 manages the advantages of HartreeFock (HF) and DFT models with an improved overall accuracy 5 ' 6 . A number of theoretical studies have shown that the B3LYP variant is particularly effective in the description of the physicochemical properties of the binary complexes 7 _ 9 . Among small transition metal complexes, metalmonocarbonyl, M(CO) has been studied extensively both experimentally 1 0 _ 2 2 and theoretically 2>3>79>2332; whereas the corresponding M(SiO) is not yet sufficiently investigated 33~39. The infrared (IR) and electron spin resonance (ESR) properties of M:CO (M = Cu, Ag, or Au) have been measured in several experimental works 1 2 _ 1 4 ' 1921,36_ These complexes were also studied using several theoretical methods 8,9,28,30,31 _ ^From both experimental and theoretical
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points of view, it has been shown that the Ag:CO complex would be actually a weak van der Waals complex (bound by only 4050 cm  1 ) , dissociating into Ag and CO 2 '2028 . ^From ESR experiment, the Cu and AuCO complexes were found to be linear (2S electronic state) 36, whereas the theoretical investigations proposed a bent structure 2>8>31. Barone using DFT calculations showed that the linear CuCO 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 CuCO 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 metalligand compounds. Furthermore, in the case of AgSiO it has been shown that the vibrational 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 reproduced 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, Sibonded or Obonded. In order to calculate the structural, energetic, and vibrational properties, these compounds have been investigated using the DFT/Hybrid method. Furthermore, the nature of MSi 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 LeeYangParr correlation functional 42 (denoted as B3LYP). We have used the 19valence electron of Stuttgart pseudo potential for the group 11 metals (relativistic (MDF) for Cu 43 and pseudorelativistic (MWB) for Ag and Au 44) and the 6311+G(2d) extended basis set of Pople et al. 45~47 for the other atoms. The topological properties of the bonding have been investigated using the program EXTREME (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 onetoone complex M(SiO)
2.1 Structural and vibrational analysis
onetoone 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 MOSi structures (Figures 1a and 1b) are found to be unbound in all of the cases. The cyclic geometry (Figure 1c) is calculated to be bound only for Cu(SiO). For two other complexes (Ag(SiO) and Au(SiO)), the triangular structure collapses to a bent metalSi bonded structure upon optimization process. Concerning the metalSi bonded structures (Figures 1d and 1e), it has been shown that the bent structure is more stable than the linear one. A vibrational
3
Si
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Table 1. Structural and vibrational parameters of bent MSiO in the 2A' state
Parameters r i (A) T2 La (deg.)
De (kcal/mol)
P, (M) (%) P.(Af) * SiO stret. (cm1) c Exp. SiM stret. MSiO bend.
fsiO (mdyn/A)
CuSiO 1.528 2.328 120.0
13.2 4.24 61 71
1187[133]
229[3] 108[3]
8.5
AgSiO 1.526 2.558 114.5
8.1 4.08 74 74
1192[125] 1163 d 182 [6] 93[5]
8.6
AuSiO 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)  EMsiO, 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 MSiO 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 metalSi bonded structure is more stable than the cyclic one. These results allow one to consider the bent MSiO structure as ground electronic state (2A'). Therefore, in the following we study only the bent MSiO geometry. In Table 1 are reported the spectroscopic properties of the 11 complexes. In the MSiO molecules, the Si0 bond length is slightly larger than that of free SiO. The Si0 lengthening decreases from Cu to AuSiO. For all compounds, the Si0 bond distance remains less than 1.550 A, which is in the range of Si=0 double bonds 49. The MSi bond length varies as CuSiO < AuSiO < AgSiO. The dipole moment is enhanced in MSiO with respect to free SiO. As expected, the Si0 bond lengthening correlates with the SiO frequency shift and the Si0 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.
onetotwo 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 onetotwo complexes, excepted the electron density of Ag which tentatively attributed to Ag(SiO)234,36 ^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 MSiO + 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 MSiO compound. The Si0 bond length in the M(SiO)2 system is found to be slightly smaller than that in the MSiO one, while the lengthening of the MSi bond is more
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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. (cm1) 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 MSiO 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 MSiO 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 MSiO 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,
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Table 3. Topological properties of MSiO and M(SiO)2. a
Compounds
SiO CuSiO AgSiO AuSiO
Cu(SiO) 2 Ag(S iO) 2 Au(S iO) 2
SiO 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
MSi 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 Si0 bond critical point. Whereas, the bonding between the metal atom and the silicon one within both MSiO and M(SiO)2 complexes presents a very small covalent character, because of the very small negative value for the energy density at the MSi 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 MSi 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 MSi bond critical point. For all of the compounds studied in this paper, the correlation coefficient calculated for the binding energy as function of p(MSi) 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 Si0 bcp (p(SiO)) is excellent, because the correlation coefficient value is equal to 0.99.
3 Concluding Remarks
It has been shown that both the onetoone and onetotwo 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
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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 metalsilicon bond path, and between the SiO frequency (or its frequency shift with respect to free SiO) and the electron density at the Si0 bond path. It seems reasonable to consider the electron density located at the metalligand bond critical point as a measure of the bond strength.
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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. Montgomery, K. Raghavachari, M. A. AlLaham, 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. Binkley, D. J. Defrees, J. Baker, J. P. Stewart, M. HeadGordon, C. Gonzalez, and J. A. Pople, Gaussian 94, Revision D.4 (Gaussian Inc. and Pittsburgh 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).
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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).
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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, E39005 Santander, Spain
* Departamento Ciencias de la Tierra y Fisica de la Materia Condensada, Facultad de Ciencias, Universidad de Cantabria, E39005 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 spinhamiltonian parameters) due to a Transition Metal (TM) impurity, M, can be understood to a long extent [1] only on the
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basis of the MXN complex (formed with the N nearest anions) at the equilibrium metalligand 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 [57], the energy of a charge transfer transition [8] or the ZeroPhonon 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 HartreeFock (including CI) or in the Density Functional framework one would expect that a valuable information on the equilibrium impurityligand 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
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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 [1419] 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 impurityligand 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 lF~ 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 lF" distance is left as variable. This assumption is reasonable provided u7R01 where R0 means the MF distance of the perfect host lattice and U1= ReRo
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.
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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 BeckePerdew 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
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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 KohnSham 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
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cluster only some close neighbours to the MXN complex. This conclusion is consistent with the following facts, i) In a BornMayer 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 BornMayer 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 impurityligand 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. PB980190.
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5 References
1. S. Sugano, Y. Tanabe and H. Kamimura, Multiplets of TransitionMetal 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. LopezMoraza, 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)
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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)
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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)
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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
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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
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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
MnF64FeF63CrO/
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
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Fig. 1. Cluster of 21 atoms used in the ADF calculation of AMF3:Mn2+
systems
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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+
WaltherNernatInstitut fur Physikalische und Theoretische Chemie, HumboldtUniversitat zv, Berlin,
Bunsenstrafie 1, D10117 Berlin, Germany FAX: +4930/2093 5573, phone: +4930/2093 5590, Email:
[email protected]. chemie.huberlin. de
"PERMANENT ADDRESS: KONRADZUSE ZENTRUM FUR INFORMATION
STECHNIK BERLIN, TAKUSTR. 7, D14195 BERLIN, EMAIL: REICHA[email protected] tPERMANENT ADDRESS: DEPARTMENT OF BIOTECHNOLOGIES AND
BIOSCIENCES, UNIVERSITY OF MILANOBICOCCA, ITALY, EMAIL: [email protected]
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We present the outline of abinitio molecular dynamics method based on gradient
corrected DF approach using gaussian atomic basis (AIMDGDF) for the adiabatic 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 KohnSham
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 liquidlike 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 iontoneutraltopositive ion) pumpprobe 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 fsmultistate nuclear dynamics involving AIMDGDF
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).
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1 Introduction
It has been recognized that discoveries of novel physical and chemical phenomena of atomic clusters are closely connected with their nonscalable properties 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 CarParrinello molecular dynamics procedure based on the DF method (AI MDEDF) has been originally designed employing the plane wave expansion 5 . The purpose of this contribution is to present an abinitio MD method based on a gradient corrected DF approach using Gaussian atomic basis (AIMDGDF) 6 '7 for the adiabatic ground states in the context of the single and multi state dynamics.
The single ground state AIMDGDF 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 AIMDGDF involving ground states of negative, neutral and cationic clusters is designed to address the femtosecond pumpprobe NeNePo spectroscopy (negative ionto neutralto 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 twophoton ionization 8.
In this paper we will first briefly outline the AIMDGDF which have been developed to study the temperature dependent ground state dynamical and structural properties. The results of the single state AIMDGDF 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 pumpprobe 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 connection with a DF procedure employing a plane waves expansion, is here revisited
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in terms of a quantum chemical approach based on fully self consistent energies and gradients in the framework of the HF a10 or GDF 6 '7 procedures, employing the AO basis sets centered at the nuclei.
The classical trajectories are calculated using the Verlet algorithm according 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 AIMDHF or AIMDDF programs the total molecular energy obtained from iterative HF or iterative KohnSham 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 determination of forces is required for the conservation of the total energy. In order to obtain meaningful information on dynamical properties sufficiently long trajectories 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 AIMDHF and AIMDDF 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 inputoutput 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 AIMDDF with gradient corrections owing to timeconsuming numerical threedimension integration of the functionals for the exchangecorrelation (e.g. those proposed by Becke 12 and Lee, Yang and Parr 13 (BLYP)).
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)
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clusters. The latter aspect is particularly important, since it allows to investigate 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 highaccuracy criteria for calculations of energies and derivatives.
A small split valence basis set with three sfunctions and one pfunction was used in our AIMD calculations 6 '7, 911. Cluster geometries, which were generated along the trajectories at high internal energies were employed as initial coordinates in a gradientbased 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 (BLYP). 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 pentagonal 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 BLYP 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
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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 AIMDBLYP simulations will be analyzed using the following four quantities. We find particularly instructive to analyze the trajectories 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 [7779]).
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 of0.40e 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 singlettriplet 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 adatom on tetrahedral Si site yields a 5coordinated Si). The triplet is the ground state of this species. Upon excitation into a singlet state (which requires 0.20.3eV 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,
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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 alkaliearth 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 nanocluster representation of the polarisable host lattice via the shell model. Relaxation of both electronic and ionic subsystems of the host lattice is performed selfconsistently with the charge density of the QM cluster calculated using the B3LYP density functional. The selfconsistent 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 631G basis set. Cations at the border of the QM cluster have been treated using Hay and Wadt effective core pseudopotentials [84] and one contracted sfunction.
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 adatom: results of Natural Population Analysis (NPA) [85] suggest that about 0.75e is associated with the adatom. 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.
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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 oneelectron spectrum of the system has three oneelectron states due to the bonding peroxide orbitals below the valence band and two oneelectron states due to the antibonding orbitals just above the top of the valence band. The highest occupied oneelectron 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 oneelectron state associated with a surface anion in the forbidden gap found by us was not observed in the periodical planewave 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
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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, semiionic 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.
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A HYBRID FUNCTIONAL FOR THE EXCHANGECORRELATION KERNEL IN
TIMEDEPENDENT DENSITY FUNCTIONAL THEORY
K. BURKE Departments of Chemistry and Physics, Rutgers University, 610 Taylor Road,
Piscataway, NJ 08854, USA Email: [email protected]
M. PETERSILKA AND E. K. U. GROSS Institut fur Theoretische Physik, Universitdt Wurzburg, Am Hubland, 97074
Wiirzburg, Germany Email: [email protected]
A review of the approximations in any timedepedendent density functional calculation of excitation energies is given. The singlepole approximation for the susceptibility is used to understand errors in popular approximations for the exchangecorrelation kernel. A new hybrid of exact exchange and adiabatic local density approximation is proposed and tested on the He and Be atoms.
1 Introduction
Groundstate density functional theory is wellestablished as an inexpensive alternative to traditional ab initio quantum chemical methods.1 Now timedependent density functional theory (TDDFT) is rapidly emerging as an inexpensive 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.611 Already, this method has been implemented in several quantum chemistry packages, such as deMon,12 Turbomole, 13,14 ADF,15 and QCHEM.16 Important calculations include the calculation of excitedstate crossings in formaldehyde,12 excitations with significant doublyexcited character,16 the photospectrum of chlorophyll A,17 and the response of 2D quantum strips.18
How are excitation energies calculated using TDDFT? First, a selfconsistent groundstate KohnSham calculation is performed, using some approximation for the exchangecorrelation energy Xci such as B3LYP19 or PBE.20 This yields a set of KohnSham eigenvalues ej and orbitals i. Even with the exact groundstate energy functional and potential, these eigenvalues are in general not the true excitations of the system, but are closely related. In
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a second step, the central equation of TDDFT response theory is solved, which extracts the true linear response function from its KohnSham counterpart. This equation includes a second unknown functional, the exchangecorrelation kernel /Xc(r, r'; w), which is the Fourier transform of the functional derivative of the timedependent exchangecorrelation 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 requires two distinct density functional approximations: one for the groundstate energy, which implies a corresponding approximation for the exchangecorrelation potential vxc(r) = SExc/5n(r), and a second for the exchangecorrelation kernel. Most calculations now appearing in the chemical literature use the adiabatic local density approximation (ALDA) for / x c Adiabatic implies that the frequencydependence of / x c is ignored, and its u