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Electronic Structure, Magnetic Ordering and Phonons in Molecules and Solids

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Electronic Structure, Magnetic Ordering and Phonons in Molecules and Solids Electronic structure, magnetic ordering and phonons in molecules and solids Von der Fakultat¨ fur¨ Chemie und Physik der TU Bergakademie Freiberg angenommene Habilitationsschrift zur Erlangung des akademischen Grades doctor rerum naturalium habilitatus Dr. rer. nat. habil. vorgelegt von Dr. rer. nat. Jens Kortus geboren am 1. Oktober 1967 in Potsdam eingereicht am 21. Marz¨ 2003 Gutachter: Prof. Dr. rer. nat. habil. Jochen Monecke, Freiberg Prof. Dr. rer. nat. Ole K. Andersen, Stuttgart Prof. Dr. rer. nat. habil. Gotthard Seifert, Dresden Tag der Verleihung: 9. Dezember 2003 Contents Preface 6 1 Introduction to DFT 7 1.1 Kohn-Sham equation . 8 1.2 The exchange-correlation energy . 9 1.2.1 Local Spin Density Approximation: LSDA . 9 1.2.2 Generalized Gradient Approximations: GGA . 10 1.3 Basis Set Expansion . 10 1.4 NRLMOL implementation . 11 1.4.1 Calculation of vibrational properties . 13 2 Applications 15 2.1 The superconductor MgB . 15 2.1.1 Electronic structure . 16 2.1.2 de Haas-van Alphen effect . 18 2.1.3 Angle resolved photoemission spectroscopy . 19 2.1.4 X-ray spectroscopy . 21 2.1.5 What is special about MgB ? . 21 2.2 Electric field gradients . 22 2.3 Vibrational properties . 25 2.3.1 Host guest interaction in a clathrate . 25 2.3.2 Octanitrocubane . 27 2.3.3 Azidopentazole . 30 2.4 Magnetic ordering . 31 ¤ 2.4.1 Magnetic and vibrational properties of the Fe ¡£¢ O cluster . 31 ¦ 2.4.2 Magnetic moment and anisotropy in Fe ¥ Co clusters . 33 2.5 Molecular magnets . 35 2.5.1 Spin-orbit coupling and magnetic anisotropy energy . 36 2.5.2 Electronic structure of the Fe ¤ magnet . 39 2.5.3 Magnetic anisotropy in single molecule magnets . 42 2.5.4 The V ¡£§ spin system . 46 3 Summary 51 3 CONTENTS Bibliography 53 Appendices 62 A Kortus J., Mazin I.I., Belashchenko K.D., Antropov V.P., and Boyer L.L. Superconductivity of metallic boron in MgB Phys. Rev. Lett. 86, 4656-4659 (2001) 63 B Mazin I.I. and Kortus J. Interpretation of the de Haas - van Alphen experiments in MgB Phys. Rev. B 65, 180510-1/4 (R) (2002) 69 C Kurmaev E.Z., Lyakhovskaya I.I., Kortus J., Moewes A., Miyata N., Deme- ter M., Neumann M., Yanagihara M., Watanabe M., Muranaka T. and Akimitsu J. Electronic structure of MgB : X-ray emission and absorption studies Phys. Rev. B 65 134509-1/4 (2002) 75 D Dietrich M., Kortus J., Cordts W. and Unterricker S. Electric Field Gradients in Wurtzite - Type Semiconductors phys. stat. sol. (b) 207, 13-17 (1998) 81 E Kortus J., Irmer G., Monecke J. and Pederson M.R. Influence of cage structures on the vibrational modes and Raman activity of methane Modelling Simul. Mater. Sci. Eng. 8, 403-411 (2000) 87 F Kortus J., Pederson M. R. and Richardson S. L. Density functional based prediction of the electronic, structural and vibra- tional properties of the energetic molecule: Octanitrocubane Chem. Phys. Lett. 322, 224-230 (2000) 99 G Kortus J., Richardson S. L. and Pederson M. R. First-Principles DFT study of the structural, electronic and vibrational prop- erties of azidopentazole Chem. Phys. Lett. 340, 565-570 (2001) 107 H Kortus J. and Pederson M. R. 4 CONTENTS ¤ Magnetic and vibrational properties of the uniaxial Fe ¡£¢ O cluster Phys. Rev. B 62, 5755-5759 (2000) 115 I Kortus J., Baruah T., Pederson M. R., Khanna S. N. and C. Ashman ¦ Magnetic moment and anisotropy in Fe ¥ Co Clusters Appl. Phys. Lett. 80, 4193-4195 (2002) 123 J Kortus J., Hellberg C. S., Pederson M. R. and Khanna S. N. ¡£§ DFT studies of the molecular nanomagnet Fe ¤ and the V spin system Eur. Phys. J. D 16, 177-180 (2001) 129 K Kortus J., Baruah T., Bernstein N., and Pederson M.R. Magnetic ordering, electronic structure and magnetic anisotropy energy in the high-spin Mn ¡£¨ single molecule magnet Phys. Rev. B 66, 092403-1/4 (2002) 135 L Kortus J., Hellberg C. S. and Pederson M. R. Hamiltonian of the V ¡£§ Spin System from First-Principles Density-Functional Calculations Phys. Rev. Lett. 86, 3400-3403 (2001) 141 5 Preface The present work gives an overview of the authors work in the field of elec- tronic structure calculations. The density functional theory (DFT) is the the- oretical background used for various applications from physics and chemistry. Therefore DFT is also the combining principle which joins the selected publi- cations of the author. The main objective is to show how electronic structure methods can be used for the description and interpretation of experimental re- sults in order to enhance our understanding of physical and chemical properties of materials. DFT is a very powerful method in that respect, as recognized by the Nobel prize in chemistry 1998. This method has been successfully applied to many different classes of ma- terials and properties. In order to place the present work in a proper context a brief introduction in the basics of DFT will be given. The main part of the work discusses the electronic structure of a few interesting examples. Today the electronic structure itself is an experimental property measured by angle resolved photo emission spectroscopy (ARPES), resonant inelastic X-ray scatter- ing or probed directly by the de Haas-van Alphen (dHvA) effect. The recently found superconductor MgB is an example where the electronic structure was the key to our understanding of the surprising properties of this material. The experimental confirmation of the predicted electronic structure from first prin- ciples calculations was very important for the acceptance of earlier theoretical suggestions. The only input parameter to DFT is the atomic composition of the material, no further experimental information is required. Because it is possible to de- termine the forces acting on the atoms within the DFT, one can optimize the atomic postions, predict stable structures and calculate vibrational spectra. IR and Raman spectroscopy are widely used tools for materials characterization and first principles electronic structure calculations deliver very important in- formation for interpreting these spectra. Not only vibrational frequencies but also IR and Raman intensities can be calculated from DFT, as will be discussed in a few examples. Molecular crystals build from magnetic clusters containing a few transition metal ions and organic ligands show fascinating magnetic properties at the nanoscale. DFT allows for the investigation of magnetic ordering and magnetic anisotropy energies. The magnetic anisotropy which results mainly from the spin-orbit coupling determines many of the poperties which make the single molecule magnets interesting. Some recent results of the author in this field will be given. 6 Chapter 1 Introduction to DFT Molecules and solids are complicated many-particle systems which are in prin- ciple described by the corresponding Schrodinger¨ equation. Unfortunately, an- alytic solutions are available only for a few very simple systems [1], and even numerically exact solutions are limited to a small number of atoms or electrons. Therefore one is forced to find accurate approximations in order to describe real materials. The density-functional theory is one of the most successful methods in that respect. Within the scope of this work we will give only a brief introduction to the basic foundations of density functional theory [2, 3]. There are excellent re- views available on that topic [4–8] which will give much more insight in the theoretical basics, accuracy and limitations than this short overview. In principle electrons and atomic nuclei can be treated at the same theoret- ical basis, although the problem can be simplified by decoupling the electronic and nuclear degrees of freedom. In many cases it is very appropriate to assume that electronic and nuclear dynamics take place at different time scales and can therefore be separated. This approach called Born-Oppenheimer approxima- tion considers all nuclei as fixed in space during the electronic relaxation. The stationary Schrodinger¨ equation of a system consisting of © electrons is given by: %$ $<;>= 103254 ! #" '& ( £" " " " ¡769898:8:6 )+*-, (1.1) ¨/. %$<; where ! #" ( ? " )+*-, (1.2) ¨/. @ . ? @ and are the coordinates and atomic numbers of the nuclei. The spin vari- " ables have been omitted for clarity (or the could be interpreted as generalized coordinates which include the spin degrees of freedom too). From here on we will use atomic units (1 Hartree = 27.2116 eV, 1 ACB =0.529177 A).˚ 7 CHAPTER 1. INTRODUCTION TO DFT The most important observation for a non-degenerate ground state of this many-electron system is, that the total$ energy can be expressed as a unique £" functional of the electron density D [2], which is determined solely by the locations @ of the nuclei. ;KJ $PO $ $ $PO $YO 0FEHGIE 0TS 0FVXW #" " £" #" £" #" ¨MLND D LUD LND RQ d (1.3) $ $ $PO;\[ $ $;][ S S £" #" ^ Z where Z #" " £" " "_^ #" D D " " ^ LND D Q Q<Q $YO $ J d d d (1.4) . #" £" 0TS D ¨MLND is the kinetic energy of a non-interacting electron gas with density , $YO is the Hartree0FVXW energy due to the average electrostatic interaction of the £" $ electrons. LUD is the exchange-correlation energy, which is in general an £" unknown functional of the electron density. The potential for an interact- $ ing © -electron system is defined as a general external potential. The knowledge £" of the ground state density D determines the external potential within an ir- relevant constant. The initial proof of the above statement given by Hohenberg and Kohn in 1964 [2] has been extended also to the lowest state of each sym- metry [9], degenerate ground states [7] or spin density functional theory [8]. 1.1 Kohn-Sham equation The ground state density must give the lowest energy by definition. This al- lows to use the variational principle in order to obtain the ground state energy.
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