Quantum chemistry of solids LCAO treatment of crystals and nanostructures

Springer Series in solid-state sciences 153 Springer Series in solid-state sciences Series Editors: M Cardona P Fulde K von Klitzing R Merlin H.-J Queisser H Stăormer The Springer Series in Solid-State Sciences consists of fundamental scientif ic books prepared by leading researchers in the f ield They strive to communicate, in a systematic and comprehensive way, the basic principles as well as new developments in theoretical and experimental solid-state physics Please view available titles in Springer Series in Solid-State Sciences on series homepage http://www.springer.com/series/682 Robert A Evarestov Quantum Chemistry of Solids LCAO Treatment of Crystals and Nanostructures Second Edition With 128 Figures 123 Dr Robert A Evarestov St Petersburg State University Chemistry Department Stary Peterghof University Petersburg Russia Series Editors: Professor Dr., Dres h c Manuel Cardona Professor Dr., Dres h c Peter Fulde∗ Professor Dr., Dres h c Klaus von Klitzing Professor Dr., Dres h c Hans-Joachim Queisser Max-Planck-Institut făur Festkăorperforschung, Heisenbergstrasse 1, 70569 Stuttgart, Germany Max-Planck-Institut fă ur Physik komplexer Systeme, Năothnitzer Strasse 38 01187 Dresden, Germany Professor Dr Roberto Merlin Department of Physics, University of Michigan 450 Church Street, Ann Arbor, MI 48109-1040, USA Professor Dr Horst Stăormer Dept Phys and Dept Appl Physics, Columbia University, New York, NY 10027 and Bell Labs., Lucent Technologies, Murray Hill, NJ 07974, USA Springer Series in Solid-State Sciences ISSN 0171-1873 ISBN 978-3-642-30355-5 ISBN 978-3-642-30356-2 (eBook) DOI 10.1007/978-3-642-30356-2 Springer Heidelberg New York Dordrecht London Library of Congress Control Number: 2012954246 © Springer-Verlag Berlin Heidelberg 2007, 2012 This work is subject to copyright All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer Permissions for use may be obtained through RightsLink at the Copyright Clearance Center Violations are liable to prosecution under the respective Copyright Law The use of general descriptive names, registered names, trademarks, service marks, etc in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made The publisher makes no warranty, express or implied, with respect to the material contained herein Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com) This book is dedicated to my teacher and friend Professor Marija I Petrashen • Preface to the Second Edition The first edition of this monograph was published in 2007 and appeared to be useful for solid-state scientists in different countries This is confirmed by numerous references to this monograph in recently published scientific papers It gave me great pleasure to know that a second enlarged edition of my book was planned by Springer-Verlag In the second edition of the book fresh applications of the LCAO method to solids have been added In particular, two new chapters are included in the Part II of the book Chapter 12 deals with the recent LCAO calculations of the bulk and surface properties of crystalline uranium nitrides and illustrates the efficiency of scalarrelativistic LCAO method for solids containing heavy atoms Chapter 13 deals with the symmetry properties and the recent applications of the LCAO method to inorganic nanotubes based on BN, TiO2 and SrTiO3 compounds The efficiency of first-principles LCAO calculations for predicting the structure and stability of single- and double-wall nanotubes is demonstrated New material is also added to Chap devoted to LCAO calculations of perfectcrystal properties ABO3 -type oxides such as barium titanate BaTiO3 are attractive for various technological applications in modern electronics, nonlinear optics, and catalysis We demonstrate that the use of hybrid exchange correlation functional allows reproducing the equilibrium volumes and structural, electronic, dielectric, and vibrational properties of paraelectric cubic and three ferroelectric (tetragonal, rhombohedral, and orthorhombic) BaTiO3 phases in good agreement with the existing experimental data It is also shown that the use of the first-principles LCAO approach allows the calculation of BaTiO3 thermodynamic properties, which provides the valuable information on the low temperature behavior that is not easy to obtain by experimental techniques The efficiency of the LCAO method in the quantum-mechanics–molecular dynamics approach to the interpretation of x-ray absorption is illustrated using perovskite as an example A new section is devoted to recent LCAO calculations of electronic, vibrational, and magnetic properties of tungstates MeWO4 (Me: Zn,Ni) vii viii Preface to the Second Edition The list of references is extended to include papers, published in 2007–2011 and devoted to the application of the LCAO method to the first-principles calculations of crystals and nanostructures This second edition of the book would not be possible without the help of Prof M Cardona, who encouraged me of writing the first edition and gave me useful advice I am grateful to Prof C Pisani and members of the Torino group of Theoretical Chemistry, Prof R Dovesi, Prof C Roetti, for many years of fruitful cooperation Very sadly my colleagues and friends Prof C Pisani and Prof C Roetti passed away recently I will never forget their role in my professional life I am grateful to all my colleagues who took part in our joint research (Prof V Smirnov, Prof K Jug, Prof T Bredow, Prof J Maier, Prof E Kotomin, Prof Ju Zhukovskii, Prof J Choisnet, Prof G Borstel, Prof F Illas, Prof A Dobrotvorsky, Prof A Kuzmin, Prof J Purans, Dr V Lovchikov, Dr V Veryazov, Prof I Tupitsyn, Dr A Panin, Dr A Bandura, Dr D Usvyat, Dr D Gryaznov, Dr V Alexandrov, Dr D Bocharov, Dr A kalinko, and E Blokhin) or sent me fresh results of their research (Prof C Pisani, Prof R Dovesi, Prof C Roetti, Prof P Deak, Prof P Fulde, Prof G Stoll, Prof M Schăutz, Prof A Schluger, Prof L Kantorovich, Prof C Minot, Prof G Scuseria, Prof R Dronskowski, Prof A Titov, and Dr B Aradi) I would like to express my thanks to the members of the Quantum Chemistry Department of St Petersburg State University, Dr A Panin and Dr A Bandura, for their help in preparing the second edition of the book I am especially indebted to Dr C Ascheron of Springer-Verlag for the encouragement and cooperation in the preparation of this edition It goes without saying that I am alone responsible for any shortcomings which remain St Petersburg, Russia April 2012 Robert A Evarestov Preface to the First Edition Nobel Prize Winner Prof Roald Hoffmann introducing a recently published book by Dronskowski [1] on computational chemistry of solid-state materials wrote that one is unlikely to understand new materials with novel properties if one is wearing purely chemical or physical blinkers He prefers a coupled approach—a chemical understanding of bonding merged with a deep physical description The quantum chemistry of solids can be considered as a realization of such a coupled approach It is traditional for quantum theory of molecular systems (molecular quantum chemistry) to describe the properties of a many-atom system on the grounds of interatomic interactions applying the linear combination of atomic orbitals’ (LCAO) approximation in the electronic-structure calculations The basis of the theory of the electronic structure of solids is the periodicity of the crystalline potential and Bloch-type one-electron states, in the majority of cases approximated by a linear combination of plane waves (LCPW) In a quantum chemistry of solids the LCAO approach is extended to periodic systems and modified in such a way that the periodicity of the potential is correctly taken into account, but the language traditional for chemistry is used when the interatomic interaction is analyzed to explain the properties of the crystalline solids At first, the quantum chemistry of solids was considered simply as the energy-band theory [2] or the theory of the chemical bond in tetrahedral semiconductors [3] From the beginning of the 1970s the use of powerful computer codes has become a common practice in molecular quantum chemistry to predict many properties of molecules in the first-principles LCAO calculations In the condensed-matter studies the accurate description of the system at an atomic scale was much less advanced [4] During the last 10 years this gap between the molecular quantum chemistry and the theory of the crystalline electronic structure has become smaller The concepts of standard solid-state theory are now compatible with an atomic-scale description of crystals There are now a number of general-purpose computer codes allowing prediction from the first-principles LCAO calculations of the properties of crystals These codes are listed in Appendix C Nowadays, the quantum chemistry of solids can be considered as the original field of solid-state theory that uses the methods ix x Preface to the First Edition of molecular quantum chemistry and molecular models to describe the different properties of solid materials including surface and point-defect modeling In this book we have made an attempt to describe the basic theory and practical methods of modern quantum chemistry of solids This book would not have appeared without the help of Prof M Cardona who supported the idea of its writing and gave me useful advice I am grateful to Prof C Pisani and members of the Torino group of Theoretical Chemistry, Prof R Dovesi, Prof C Roetti, for many years of fruitful cooperation Being a physicist-theoretician by education, I would never have correctly estimated the role of quantum chemistry approaches to the solids without this cooperation I am grateful to all my colleagues who took part in our common research (Prof V Smirnov, Prof K Jug, Prof T Bredow, Prof J Maier, Prof E Kotomin, Prof Ju Zhukovskii, Prof J Choisnet, Prof G Borstel, Prof F Illas, Dr A Dobrotvorsky, Dr V Lovchikov, Dr V Veryazov, Dr I Tupitsyn, Dr A Panin, Dr A Bandura, Dr D Usvyat, Dr D Gryaznov, and V Alexandrov) or sent me the recent results of their research (Prof C Pisani, Prof R Dovesi, Prof C Roetti, Prof P Deak, Prof P Fulde, Prof G Stoll, Prof M Schăutz, Prof A Schluger, Prof L Kantorovich, Prof C Minot, Prof G Scuseria, Prof R Dronskowski, and Prof A Titov) I am grateful to Prof I Abarenkov, head of the Prof M.I Petrashen named seminar for helpful discussions and friendly support I would like to express my thanks to the members of the Quantum Chemistry Department of St Petersburg State University, Dr A Panin and Dr A Bandura, for help in preparing the manuscript—without their help this book would not be here I am especially indebted to Dr C Ascheron, Mrs A Lahee, and 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phases, 447 phonon frequencies, 452 rhombohedral calculations, 454, 458 free energy, 458 heat capacity, 465 thermal Gruneisen parameter, 464 thermal expansion coefficient, 462, 463 BaZrO3 , 444 calculations, 444, 446 Boundless crystal, 10 Bravais lattice, 12 base-centered, 13 body-centered, 13 cubic, 15 face-centered, 13 hexagonal, 14 monoclinic, 13 orthorhombic, 14 parameters, 13 2D (plane), 543 point symmetry, 12 primitive, 13 tetragonal, 14 triclinic, 13, 14 type, 13 Brillouin zone special points, 109, 134 cubic crystals, 135 modified Monkhorst–Pack, 138 Monkhorst–Pack, 137 supercell generation, 135 symmetry line, 58 point, 58 Bulk crystals band structure MgO, TiO2 , 409 bulk moduli, elastic constants SrTiO3 ; BaTiO3 ; PbTiO3 , 436 bulk modulus Cu2 O, 433 TiO2 phases, 436 cohesive energy corundum-like oxides, 431 TiO2 , 430 R.A Evarestov, Quantum Chemistry of Solids, Springer Series in Solid-State Sciences 153, DOI 10.1007/978-3-642-30356-2, © Springer-Verlag Berlin Heidelberg 2012 729 730 Compton profile Cu2 O, 416 cyclic cluster model Al2 O3 , 513 density of states projected density of states, 412 SrTiO3 , SrZrO3 , 413 electron-density maps Cu2 O, 415 electron-momentum density Cu2 O, 416 magnetic structure LaMnO3 , 419, 421 ScMnO3 , 424 phonon symmetry TiO2 , 440 polymorphs SrZrO3 , 436 structure parameters TiO2 , 430 SnO2 (110), 578 supercell model SrTiOO3 , 529 total energy cohesive energy, 428 vibrational frequences calculation, 438 TiO2 , 442 Chemical bonding in crystals bulk crystal TiO,TiO2 ,Ti2 O3 , 373, 375, 378 crystallographic valence crystalline oxides, 366 cyclic cluster crystalline oxides, 364, 365 Cu4 O3 ; Pb2 O3 ; Pb3 O4 , 369 La2 CuO4 ,La2 NiO4 , 372 YBa2 Cu3 O7 , 371 density matrix LCAO, 358 natural orbitals, 360, 362 electron correlation molecular-crystalline approach, 363 Ti2 O3 , 380 localized orbitals local electronic structure properties, 395 MgO, 400 SrZrO3 , 396, 400 local properties of electronic structure atomic population, 359 bond order, covalency, 359 Index total atomic valency, 359 population analysis projection technique, 401 projection technique Si, SiC, GaAs, MgO, TiO2 , 403 SrTiO3 , SrZrO3 , 405, 406 Wannier functions for valence bands crystalline oxides, 390 Wannier-type atomic functions local properties of electronic structure, 382 PbTiO3 , 385 SrTiO3 , BaTiO3 , 384 Computer programs for LCAO calculations, 697 Conventional unit cell, 16 Crystal structure cubic antifluorite, 33 cesium chloride, 33 diamond, 30 fluorite, 32 perovskite, 34 rocksalt, 31 zincblende, 33 database, 28 description, 27 hexagonal graphite, 42 ScMnO3 , 45 wurtzite ZnS, 44 orthorhombic LaMnO3 , 39 rhombohedral ˛Al2 O3 , corundum, 45 tetragonal anatase, 37 La2 CuO4 , 38 rutile, 35 YBaCuO, 41 type, 7, 27 Crystallographic point groups international notation, 16 Schăonflies notation, 16 Crystal system cubic, 13 hexagonal, 13 monoclinic, 13 orthorhombic, 13 rhombohedral, 13 syngony, 12, 13 tetragonal, 13 triclinic, 13 Index Density-functional theory (crystals) exchange-correlation functionals, 289 LDAU method, 299 SIC-LSDA method, 296 LCAO approximation, 273 HSE03 hybrid functional, 285 HSE screened Coulomb hybrid functional, 283 Kohn–Sham equations, 275 linear-scaling method, 277, 280 screened Coulomb hybrid functional, 286 Density-functional theory (molecules) generalized-gradient approximation GGA exchange-correlation functionals, 264 PBE functional, 265 Hohenberg–Kohn theorems, 253 Kohn–Sham equations, 255, 258 local density approximation LDA exchange-correlation functionals, 260 local spin-density approximation LSDA, 261 orbital-dependent exchange-correlation functionals, 267 hybrid functionals, 268, 271 Thomas-Fermi model, 252 Density matrix crystal, 140 approximate, 149 LCAO approximation, 146 properties, 141, 143 cyclic cluster properties, 144 molecule, 113 Double point group, 605 double-valued representations, 605 single-valued representations, 605 Effective core potentials nonrelativistic energy-consistent, 326 generation, 327 Hay–Wadt potentials, 330 norm-conserving, 329 pseudo wavefunctions, 324 separable embedding potentials, 332 valence basis-sets, 329 relativistic, 342 basis sets, 346 generalized, 344 periodic systems, 349 Stevens–Basch–Krauss potentials, 331 Stuttgart–Dresden potentials, 331 731 Electron correlation crystals embedded cluster, 182 Laplace-MP2, 188, 192 local electron-correlation methods, 177, 179 local MP2 method, 195, 199 method of increments, 179, 183, 187 molecules CI-method, 162 CISDT-method, 164 coupled cluster-method, 166 local electron-correlation methods, 170, 175 MCSCF-method, 165 perturbation theory, 167 spin correlation, 159 Equivalent points of space, Group, diperiodic (layer), 542 plane, 543 point, space, 7, subperiodic, 542 translation, Hartree–Fock method, 112 Coulomb operator, 112 cyclic cluster, 241 exchange operator, 113 LCAO approximation crystals, 128, 129 cyclic cluster, 123 molecules, 121 local exchange approximation, 114 restricted for open shells ROHF, 120 restricted RHF, 118 self-consistent calculation, 114 symmetry of Fockian, 115 UHF crystals, 130 unrestricted UHF, 120 Induced representations of point group, 68 correlation table, 69 of space group composite, 78 k-basis, 75 q-basis, 73 732 simple, 77 tables, 80 Irreducible representations, 7, of space group full representation, 60 small(allowed) representation, 61 of translation group Bloch functions, 53 Brillouin zone, 53 k vector, 52 Layer group Brillouin zone, 547 element, 543 irreducible representations, 546 setting, 544 site-symmetry, 546 table, 544 LCAO and PW basis Wannier-type atomic functions MgO, 388 Line groups, 631 factorization, 636 families, 637 Localized orbitals crystalline, 89, 91, 92 for valence bands, 98 molecular, 71 generation, 71 Local properties of electronic structure molecules atomic covalence, 148 Wiberg index, 148 Molecular cluster, 11 Monoatomic crystal, 12 nanotubes, 640 double wall, 648 BN calculations, 664 SrTiO3 , 684 symmetry, 684 TiO2 calculations, 666 layer folding, 631, 642 multiwall, 649 single wall, 641 BN, 654 calculations, 656 SrTiO3 , 683, 686–688 calculations, 686–688 Index symmetry, 683 TiO2 , 654, 658, 672–674 calculations, 658 rectangular, 672–674 SrTiO3 symmetry, 681 symmetry implementation, 650, 651 Orbits of points, 7, in a crystal, 10 crystallographic, 23 in a molecule, Point defects in solids classification, 490 cyclic-cluster model F center in Al2 O3 , 513, 516 F C center in Al2 O3 , 515 formation energy, 499 models cyclically-embedded cluster, 505 cyclic cluster, 492 molecular cluster, 490 supercell, 492 molecular-cluster model realization, 503 perturbed-cluster model MgO, 507 supercell model binary oxides, 508 charged defects, 502 convergence, 505 F center, 513, 523–526 in Al2 O3 , 513 in SrTiO3 , 523–526 Fe-impurity in SrTiO3 , 528, 531, 532 interstitial oxygen in MgO, 509 MgO crystal, 496, 500 oxygen vacancy, 511, 521 in Al2 O3 , 511 in SrTiO3 , 521 realization, 498, 501 SrTiO3 , BaTiO3 , 520 symmetry, 494–496 TiO2 , 497 vanadium-doped TiO2 , 518 Primitive unit cell, 12 Projective representations characters, 66 factor system, 64 of point group, 63 Index Quantum Mechanics - Molecular Dynamics approach, 466, 467 EXAFS, 467 ReO3 , 473 SrTiO3 , 469, 471 Reference unit cell, 12 Relativistic theory molecules Dirac–Hartree–Fock method, 338, 339 Dirac–Kohn–Sham method, 341 Rod groups axial point groups, 632 correspondence to space groups, 632 families, 632 as line groups, 637 Scalar-relativistic calculations, 604 Semiempirical LCAO methods, 208 crystals CNDO, 226 cyclic cluster, 229, 233, 235 extended Hăuckel, 210 Mulliken-Răudenberg, 218 cyclic cluster CNDO, 237 INDO, 238 MSINDO, 239 molecules CNDO, 219, 221 extended Hăuckel, 209, 211 INDO, 222 MNDO, 222 MSINDO, 223 Mulliken-Răudenberg, 215, 217 PM3, 223 Single wall nanotubes hexagonal lattice, 643 rectangular lattices, 647 square lattice, 645 Site-symmetry group, 9, 23 oriented, 26 Solid solutions supercell model Lac Sr1 c MnO3 , 535, 538 Space group, 18 1D, 18 2D, 18 3D, 18 designations, 22 elements, 18 nonsymmorphic, 19 733 plane, 18 symmorphic, 19 table, 19 Supercell for centered lattices, 17 transformation, 17 symmetrical, 132, 691 Surfaces of crystals (001) cubic perovskites, 556 MgO, 554 (110) TiO2 , 554 atomic charges LaMnO3 (001), 599 bare surface (001) SrTiO3 , SrZrO3 , 589, 591 BSSE SnO2 (110), 579 cyclic cluster TiO2 (110), 566 density of states SrTiO3 , SrZrO3 (001), 594 models, 541 molecular cluster, 541 periodic slab (supercell), 542 semi-infinite crystal, 541 single slab, 542 molecular cluster TiO2 (110), 566, 567 slab SnO2 (110), 578 SnO2 (110) atomic charges, 581 surface energy LaMnO3 (001), 598, 600 MgO (001), 558, 559 slab model, 557 SnO2 (110), 579 SrTiO3 , SrZrO3 (001), 593 TiO2 (110), 559 surface F center SrTiO3 (001), 596 water adsorption SnO2 (100), 587 SnO2 (110), 583, 585 TiO2 (110), 567, 571, 572, 574, 575 Surfaces types type-1 MgO, 548, 550 type-2 TiO2 (110), 551 type-3 SrTiO3 , LaMnO3 , 551, 553 734 Symmetry elements, Symmetry operations, Symmetry properties, 49 of crystalline orbitals, 50 of the Hamiltonian, 49 of molecular orbitals, 50 time-reversal transformation, 51 Translation improper or fractional, 18 proper, 18 Tungstates, 475 NiWO4 , 480 calculations, 481 magnetic ordering, 482 ZnWO4 , 475 calculations, 476 UF6 molecule, 603 fluorine orbitals symmetry, 606 relativistic calculations, 606 scalar-relativistic calculations, 608, 610, 611 uranium atom states, 605 UN crystal, 614 calculations, 615–617 Mulliken charges, 618 oxygen impurity, 626 (001) surface, 621 atomic displacements, 623 O atom adsorption, 624 Index oxygen adsorption, 627, 628 (001) surface atomic charges, 623 (001) surface calculations, 622 (001) surface energies, 623 UN2 crystal, 615 calculations, 619 U2 N3 -crystal, 615 calculations, 603, 615 UO2 crystal, 612 calculations, 613, 614 energy bands, 613 Wannier function, 89, 93 local MP2 method, 200 generation, 93, 95, 97 comparison of methods, 99 variational method, 101, 103, 105 surface MgO (001), 562 slab model, 561 TiO2 (110), 563 symmetry local MP2 method, 205 Wannier type atomic orbitals, 99 Wavevector little group, 61 point-symmetry group, 57 star, 60 Wyckoff positions, notation, 23 parameter-dependent, 23 parameter-free, 23 ... Prof R Dovesi, Prof C Roetti, Prof P Deak, Prof P Fulde, Prof G Stoll, Prof M Schăutz, Prof A Schluger, Prof L Kantorovich, Prof C Minot, Prof G Scuseria, Prof R Dronskowski, Prof A Titov, and. .. research (Prof V Smirnov, Prof K Jug, Prof T Bredow, Prof J Maier, Prof E Kotomin, Prof Ju Zhukovskii, Prof J Choisnet, Prof G Borstel, Prof F Illas, Prof A Dobrotvorsky, Prof A Kuzmin, Prof J Purans,... help of Prof M Cardona who supported the idea of its writing and gave me useful advice I am grateful to Prof C Pisani and members of the Torino group of Theoretical Chemistry, Prof R Dovesi, Prof