QUANTUM ESPRESSO: A modular and open-source software project for quantum simulations of materials

Journal of Physics Condensed Matter

Volumn 21, Issue 39, 2009, Pages

  Giannozzi, Paolo ^a,b   Baroni, Stefano ^a,c   Bonini, Nicola ^d   Calandra, Matteo ^e   Car, Roberto ^f   Cavazzoni, Carlo ^g,h   Ceresoli, Davide ^d   Chiarotti, Guido L ⁱ   Cococcioni, Matteo ^j   Dabo, Ismaila ^k   Dal Corso, Andrea ^a,c   De Gironcoli, Stefano ^a,c   Fabris, Stefano ^a,c   Fratesi, Guido ^l   Gebauer, Ralph ^a,m   Gerstmann, Uwe ⁿ   Gougoussis, Christos ^e   Kokalj, Anton ^a,o   Lazzeri, Michele ^e   Martin Samos, Layla ^a   Marzari, Nicola ^d   Mauri, Francesco ^e   Mazzarello, Riccardo ^p   Paolini, Stefano ^c,i   Pasquarello, Alfredo ^q,r   Paulatto, Lorenzo ^a,c   Sbraccia, Carlo ^a,t   Scandolo, Sandro ^a,m   Sclauzero, Gabriele ^a,c   Seitsonen, Ari P ^e   Smogunov, Alexander ^m   Umari, Paolo ^a   Wentzcovitch, Renata M ^j,s   more..

^a DEMOCRITOS NATIONAL SIMULATION CENTER   (Italy)

^b UNIVERSITY OF UDINE   (Italy)

^c SISSA   (Italy)

^d Massachusetts Institute of Technology   (United States)

^e UNIVERSITÉ PIERRE ET MARIE CURIE   (France)

^f PRINCETON UNIVERSITY   (United States)

^g CINECA   (Italy)

^h CNR Consiglio Nazionale delle Ricerche   (Italy)

ⁱ AIBS Lab Srl   (Italy)

^j University of Minnesota   (United States)

^k UNIVERSITÉ PARIS EST   (France)

^l UNIVERSITY OF MILANO BICOCCA   (Italy)

^m ABDUS SALAM INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS   (Italy)

ⁿ UNIVERSITY OF PADERBORN   (Germany)

^o JO EF STEFAN INSTITUTE   (Slovenia)

^p ETH ZURICH   (Switzerland)

^q EPFL   (Switzerland)

^r INSTITUT ROMAND DE RECHERCHE NUMÉRIQUE EN PHYSIQUE DES MATÉRIAUX IRRMA   (Switzerland)

^s UNIVERSITY OF MINNESOTA   (United States)

^t Constellation Energy Commodities Group ^*  (United Kingdom)

more..

Author keywords

[No Author keywords available]

Indexed keywords

COMPUTER CODES; ELECTRONIC STRUCTURE CALCULATIONS; GNU GENERAL PUBLIC LICENSE; MATERIALS MODELING; OPEN SOURCE PACKAGE; OPEN SOURCE PROJECTS; OPEN-SOURCE SOFTWARES; PLANE WAVE; PROJECTOR-AUGMENTED WAVES; PSEUDOPOTENTIALS; QUANTUM SIMULATIONS; USER FRIENDLINESS;

DENSITY FUNCTIONAL THEORY; ELECTRONIC STRUCTURE; PARALLEL ARCHITECTURES; SOFTWARE ENGINEERING; TIME VARYING SYSTEMS; WAVE FUNCTIONS;

RESEARCH;

EID: 70349568754     PISSN: 09538984     EISSN: 1361648X     Source Type: Journal    
DOI: 10.1088/0953-8984/21/39/395502     Document Type: Article

References (193)

1. Marzari N 2006 MRS Bull. 31 681
2. Parr R G and Yang W 1989 Density Functional Theory of Atoms and Molecules (Oxford: Oxford University Press)
3. Dreizler R M and Gross E K U 1990 Density Functional Theory (Berlin: Springer)
4. Martin R M 2004 Electronic Structure: Basic Theory and Practical Methods (Cambridge: Cambridge University Press)
5. Baerends E J et al 1995 ADF2008.01, SCM, Theoretical Chemistry (Vrije Universiteit, Amsterdam, The Netherlands) http://www.scm.com
6. Dovesi R, Civalleri B, Orlando R, Roetti C and Saunders V R 2005 Reviews in Computational Chemistry vol 21 ed K B Lipkowitz, R Larter and T R Cundari (New York: Wiley) chapter 1
7. Karlström G et al 2003 Comput. Mater. Sci. 28 222
8. Ahlrichs R, Furche F, Hättig C, Klopper W M, Sierka M and Weigend F 1988 TURBOMOLE http://www.turbomole-gmbh.com
9. Frisch M J et al 2004 Gaussian 03, Revision C.02 (Wallingford, CT: Gaussian, Inc.) http://www.gaussian.com
10. Materials Studio http://accelrys.com/products/materials-studio
11. Jaguar: rapid ab initio electronic structure package, Schrödinger, LLC http://www.schrodinger.com/
12. Spartan, Wavefunction, Inc. http://www.wavefun.com/products/spartan.html
13. Kresse G and Furthmuller J 1996 Comput. Mater. Sci. 6 15-50 http://cmps.mpi.univie.ac.at/vasp
14. http://www.castep.org, http://accelrys.com/products/materials-studio/ modules/CASTEP.html
15. The CPMD consortium page, coordinated by M Parrinello and W Andreoni, Copyright IBM Corp 1990-2008, Copyright MPI für Festkörperforschung Stuttgart 1997-2001 http://www.cpmd.org
16. Bylaska E J et al 2008 Nwchem, a computational chemistry package for parallel computers Version 5.1 (Pacific Northwest National Laboratory, Richland, Washington 99352-0999, USA) http://www.emsl.pnl.gov/docs/nwchem/nwchem.html
17. Kendall R A et al 2000 High performance computational chemistry: an overview of nwchem a distributed parallel application Comput. Phys. Commun. 128 260-83
18. Schmidt M W et al 1993 General atomic and molecular electronic structure system J. Comput. Chem. 14 1347-63
19. Gordon M S and Schmidt M W 2005 Advances in electronic structure theory: GAMESS a decade later Theory and Applications of Computational Chemistry, the First Forty Years ed C E Dykstra, G Frenking, K S Kim and G E Scuseria (Amsterdam: Elsevier) chapter 41 pp1167-89 http://www.msg.chem.iastate.edu/ GAMESS/GAMESS.html
20. http://www.gnu.org/licenses/
21. Gonze X et al 2002 First-principles computation of material properties: the abinit software project Comput. Mater. Sci. 25 478 http://www.abinit.org
22. VandeVondele J, Krack M, Mohamed F, Parrinello M, Chassaing T and Hutter J 2005 Comput. Phys. Commun. 167 103 http://cp2k.berlios.de
23. Bahn S R and Jacobsen K W 2002 An object-oriented scripting interface to a legacy electronic-structure code Comput. Sci. Eng. 4 56 http://www.fysik.dtu. dk/campos
24. Mortensen J J, Hansen L B and Jacobsen K W 2005 Phys. Rev. B 71 035109 https://wiki.fysik.dtu.dk/gpaw
25. Daniel Crawford T et al 2007 J. Comput. Chem. 28 1610-6 http://www.psicode.org
26. http://www.wannier.org/
27. http://qe-forge.org
28. Anderson E et al 1999 LAPACK Users' Guide 3rd edn (Philadelphia, PA: Society for Industrial and Applied Mathematics)
29. Frigo M and Johnson S G 2005 The design and implementation of FFTW3 Proc. IEEE 93 216-31 (special issue on 'Program Generation, Optimization, and Platform Adaptation')
30. Message Passing Interface Forum 1994 Int. J. Supercomput. Appl. 8 159-416
31. http://en.wikipedia.org/wiki/SourceForge
32. Pickett W E 1989 Comput. Phys. Rep. 9 115
33. Dabo I, Kozinsky B, Singh-Miller N E and Marzari N 2008 Phys. Rev. B 77 115139
34. Kleinman L and Bylander D M 1982 Phys. Rev. Lett. 48 1425
35. Hamann D R, Schlüter M and Chiang C 1979 Phys. Rev. Lett. 43 1494
36. Vanderbilt D 1990 Phys. Rev. B 41 7892
37. Blöchl P E 1994 Phys. Rev. B 50 17953
38. Perdew J P, Chevary J A, Vosko S H, Jackson K A, Singh D J and Fiolhais C 1992 Phys. Rev. B 46 6671
39. Tao J, Perdew J P, Staroverov V N and Scuseria G E 2003 Phys. Rev. Lett. 91 146401
40. Perdew J P, Ernzerhof M and Burke K 1996 J. Chem. Phys. 105 9982
41. Becke A D 1993 J. Chem. Phys. 98 1372
42. Stephens P J, Devlin F J, Chabalowski C F and Frisch M J 1994 J. Phys. Chem. 98 11623
43. Kohn W and Sham L J 1965 Phys. Rev. 140 A1133
44. Hellmann H 1937 Einführung in die Quantenchemie (Vienna: Franz Deuticke)
45. Feynman R P 1939 Phys. Rev. 56 340
46. Nielsen O H and Martin R M 1985 Phys. Rev. B 32 3780
47. Pyykkö P 1988 Chem. Rev. 88 563
48. Oda T, Pasquarello A and Car R 1998 Phys. Rev. Lett. 80 3622
49. Gebauer R and Baroni S 2000 Phys. Rev. B 61 R6459
50. Car R and Parrinello M 1985 Phys. Rev. Lett. 55 2471
51. Wentzcovitch R M and Martins J L 1991 Solid State Commun. 78 831
52. Parrinello M and Rahman A 1980 Phys. Rev. Lett. 45 1196-9
53. Wentzcovitch R M 1991 Phys. Rev. B 44 2358-61
54. Baroni S, Giannozzi P and Testa A 1987 Phys. Rev. Lett. 58 1861
55. Baroni S, de Gironcoli S, Dal Corso A and Giannozzi P 2001 Rev. Mod. Phys. 73 515
56. Gonze X 1995 Phys. Rev. A 52 1096
57. Henkelman G and Jónsson H 1999 J. Chem. Phys. 111 7010
58. E W, Ren W and Vanden-Eijnden E 2002 Phys. Rev. B 66 052301
59. Caspersen K J and Carter E A 2005 Proc. Natl Acad. Sci. USA 102 6738-43
60. Choi H J and Ihm J 1999 Phys. Rev. B 59 2267
61. Marzari N and Vanderbilt D 1997 Phys. Rev. B 56 12847
62. Souza I, Marzari N and Vanderbilt D 2001 Phys. Rev. B 65 035109
63. Pickard C J and Mauri F 2001 Phys. Rev. B 63 245101
64. Pickard C J and Mauri F 2003 Phys. Rev. Lett. 91 196401
65. Taillefumier M, Cabaret D, Flank A M and Mauri F 2002 Phys. Rev. B 66 195107
66. Scandolo S, Giannozzi P, Cavazzoni C, de Gironcoli S, Pasquarello A and Baroni S 2005 Z. Kristallogr. 220 574
67. Tersoff J and Hamann D R 1985 Phys. Rev. B 31 805-13
68. Becke A D and Edgecombe K E 1990 J. Chem. Phys. 92 5397-403
69. Szabo A and Ostlund N 1996 Modern Quantum Chemistry (New York: Dover)
70. Baldereschi A, Baroni S and Resta R 1988 Phys. Rev. Lett. 61 734
71. The HDF Group http://www.hdfgroup.org
72. http://www.unidata.ucar.edu/software/netcdf
73. http://www.quantum-simulation.org
74. Bussi G http://www.s3.infm.it/iotk
75. Fuchs M and Scheffler M 1999 Comput. Phys. Commun. 119 67
76. http://www.physics.rutgers.edu/∼dhv/uspp/
77. OPIUM-pseudopotential generation project http://opium.sourceforge.net
78. Dal Corso A 1996 A pseudopotential plane waves program (pwscf) and some case studies Lecture Notes in Chemistry vol 67 ed C Pisani (Berlin: Springer)
79. Kresse G and Joubert D 1999 Phys. Rev. B 59 1758-75
80. Laasonen K, Pasquarello A, Car R, Lee C and Vanderbilt D 1993 Phys. Rev. B 47 10142
81. Giannozzi P, de Angelis F and Car R 2004 J. Chem. Phys. 120 5903
82. White J A and Bird D M 1994 Phys. Rev. B 50 4954
83. Dal Corso A and Mosca Conte A 2005 Phys. Rev. B 71 115106
84. Mosca Conte A 2007 Quantum mechanical modeling of nano magnetism: new tools based on density-functional theory with case applications to solids, surfaces, wires, and molecules PhD Thesis (SISSA/ISAS, Trieste, Italy)
85. Anisimov V I, Zaanen J and Andersen O K 1991 Phys. Rev. B 44 943
86. Cococcioni M and de Gironcoli S 2005 Phys. Rev. B 71 035105
87. Kwee H, Zhang S and Krakauer H 2008 Phys. Rev. Lett. 100 126404
88. Johnson D D 1988 Phys. Rev. B 38 12807
89. Chadi D J and Cohen M L 1973 Phys. Rev. B 8 5747
90. Monkhorst H J and Pack J D 1976 Phys. Rev. B 13 5188
91. Methfessel M and Paxton A 1989 Phys. Rev. B 40 3616
92. Marzari N, Vanderbilt D, de Vita A and M C Payne 1999 Phys. Rev. Lett. 82 3296
93. Blöchl P E, Jepsen O and Andersen O K 1994 Phys. Rev. B 49 16223
94. David Mermin N 1965 Phys. Rev. 137 A1441-3
95. Fletcher R 1987 Practical Methods of Optimization (New York: Wiley)
96. Billeter S R, Turner A J and Thiel W 2000 Phys. Chem. Chem. Phys. 2 2177
97. Billeter S R, Curioni A and Andreoni W 2003 Comput. Mater. Sci. 27 437
98. Micheletti C, Laio A and Parrinello M 2004 Phys. Rev. Lett. 92 170601
99. Verlet L 1967 Phys. Rev. 159 98
100. Allen M P and Tildesley D J 1986 Computer Simulations of Liquids (Oxford: Clarendon)
101. Bernasconi M, Chiarotti G L, Focher P, Scandolo S, Tosatti E and Parrinello M 1995 J. Phys. Chem. Solids 56 501-5
102. Umari P and Pasquarello A 2002 Phys. Rev. Lett. 89 157602
103. Souza I, ̃iguez J and Vanderbilt D 2002 Phys. Rev. Lett. 89 117602
104. Kunc K and Resta R 1983 Phys. Rev. Lett. 51 686
105. Tobik J and Dal Corso A 2004 J. Chem. Phys. 120 9934
106. Scherlis D A, Fattebert J-L, Gygi F, Cococcioni M and Marzari N 2006 J. Chem. Phys. 124 074103
107. Dabo I, Cances E, Li Y and Marzari N 2009 First-principles simulation of electrochemical systems at fixed applied voltage: vibrational stark effect for co on platinum electrodes arXiv:0901.0096v2
108. Cavazzoni C and Chiarotti G L 1999 Comput. Phys. Commun. 123 56
109. Pasquarello A, Laasonen K, Car R, Lee C and Vanderbilt D 1992 Phys. Rev. Lett. 69 1982
110. d'Avezac M, Calandra M and Mauri F 2005 Phys. Rev. B 71 205210
111. Sit H-L, Cococcioni M and Marzari N 2007 J. Electroanal. Chem. 607 107-12
112. Sit H-L, Cococcioni M and Marzari N 2006 Phys. Rev. Lett. 97 028303
113. Marzari N, Vanderbilt D and Payne M C 1997 Phys. Rev. Lett. 79 1337
114. Tassone F, Mauri F and Car R 1994 Phys. Rev. B 50 10561
115. Tobias D J, Martyna G J and Klein M L 1993 J. Phys. Chem. 97 12959
116. Martyna G J, Klein M L and Tuckerman M 1992 J. Chem. Phys. 97 2635-43
117. Payne M C, Teter M P, Allen D C, Arias T A and Joannopoulos J D 1992 Rev. Mod. Phys. 64 1045
118. Marx D and Hutter J 2000 Modern Methods and Algorithms of Quantum Chemistry (FZ Jülich: John von Neumann Institute for Computing) pp301-449
119. Dubois V, Umari P and Pasquarello A 2004 Chem. Phys. Lett. 390 193
120. Giustino F and Pasquarello A 2005 Phys. Rev. Lett. 95 187402
121. Umari P and Pasquarello A 2005 Diamond Relat. Mater. 14 1255
122. Umari P and Pasquarello A 2005 Phys. Rev. Lett. 95 137401
123. Giacomazzi L, Umari P and Pasquarello A 2005 Phys. Rev. Lett. 95 075505
124. Umari P and Pasquarello A 2007 Phys. Rev. Lett. 98 176402
125. Giannozzi P and Baroni S 1994 J. Chem. Phys. 100 8537
126. Gonze X 1997 Phys. Rev. B 55 10337
127. Gonze X and Lee C 1997 Phys. Rev. B 55 10355
128. Dal Corso A, Pasquarello A and Baldereschi A 1997 Phys. Rev. B 56 R11369
129. Dal Corso A 2001 Phys. Rev. B 64 235118
130. Favot F and Dal Corso A 1999 Phys. Rev. B 60 11427
131. Dal Corso A and de Gironcoli S 2000 Phys. Rev. B 62 273
132. Dal Corso A 2009 submitted
133. Dal Corso A 2007 Phys. Rev. B 76 054308
134. Giustino F, Cohen M L and Louie S G 2007 Phys. Rev. B 76 165108
135. Lazzeri M and de Gironcoli S 2002 Phys. Rev. B 65 245402
136. Lazzeri M and Mauri F 2003 Phys. Rev. Lett. 90 036401
137. Lazzeri M and Mauri F 2003 Phys. Rev. B 68 161101(R)
138. Koelling D D and Harmon B N 1977 J. Phys. C: Solid State Phys. 10 3107-14
139. MacDonald A H and Vosko S H 1979 J. Phys. C: Solid State Phys. 12 2977-90
140. Rajagopal A K and Callaway J 1973 Phys. Rev. B 7 1912-9
141. Troullier N and Martins J L 1991 Phys. Rev. B 43 1993-2006
142. Rappe A M, Rabe K M, Kaxiras E and Joannopoulos J D 1990 Phys. Rev. B 41 1227-30
143. Kresse G and Hafner J 1994 J. Phys.: Condens. Matter 6 8245
144. Smogunov A, Dal Corso A and Tosatti E 2004 Phys. Rev. B 70 045417
145. Dal Corso A, Smogunov A and Tosatti E 2006 Phys. Rev. B 74 045429
146. Profeta M, Mauri F and Pickard C J 2003 J. Am. Chem. Soc. 125 541
147. van de Walle C G and Blöchl P E 1993 Phys. Rev. B 47 4244
148. Mauri F and Louie S G 1996 Phys. Rev. Lett. 76 4246
149. Mauri F, Pfrommer B and Louie S G 1996 Phys. Rev. Lett. 77 5300
150. Pickard C J and Mauri F 2002 Phys. Rev. Lett. 88 086403
151. Thonhauser T, Ceresoli D, Mostofi A, Marzari N, Resta R and Vanderbilt D 2009 arXiv:0709.4429v1
152. Gougoussis C, Calandra M, Seitsonen A P and Mauri F 2009 Phys. Rev. B 80 075102
153. Gougoussis C, Calandra M, Seitsonen A, Brouder Ch, Shukla A and Mauri F 2009 Phys. Rev. B 79 045118
154. Mostofi A A, Yates J R, Lee Y-S, Souza I, Vanderbilt D and Marzari N 2008 Wannier90: a tool for obtaining maximally-localised wannier functions Comput. Phys. Commun. 178 685
155. Silvestrelli P L, Marzari N, Vanderbilt D and Parrinello M 1998 Solid State Commun. 107 7
156. Calzolari A, Souza I, Marzari N and Buongiorno Nardelli M 2004 Phys. Rev. B 69 035108
157. Lee Y-S, Buongiorno Nardelli M and Marzari N 2005 Phys. Rev. Lett. 95 076804
158. Yates J R, Wang X, Vanderbilt D and Souza I 2007 Phys. Rev. B 75 195121
159. Kokalj A 1999 Xcrysden-a new program for displaying crystalline structures and electron densities J. Mol. Graph. Modelling. 17 176-9 http://www.xcrysden.org/
160. Humphrey W, Dalke A and Schulten K 1996 VMD-visual molecular dynamics J. Mol. Graph. Modelling. 14 33-8
161. http://www.tcm.phy.cam.ac.uk/∼mdt26/casino2.html
162. Martin-Samos L and Bussi G 2009 Sax: an open source package for electronic-structure and optical-properties calculations in the GW approximation Comput. Phys. Commun. 180 1416 http://sax-project.org
163. Korotin D, Kozhevnikov A V, Skornyakov S L, Leonov I, Binggeli N, Anisimov V I and Trimarchi G 2008 Eur. Phys. J. B 65 91-8 http://dmft.qe-forge. org
164. http://dp-code.org
165. http://www.bethe-salpeter.org
166. Kokalj A 2003 Computer graphics and graphical user interfaces as tools in simulations of matter at the atomic scale Comput. Mater. Sci. 28 155 http://www-k3.ijs.si/kokalj/guib/
167. Giannozzi P and Cavazzoni C 2009 Large-scale computing with Quantum-Espresso Nuovo Cimento C 32 at pressdoi:10.1393/ncc/i2009-10368-9
168. Hutter J and Curioni A 2005 Car-Parrinello molecular dynamics on massively parallel computers ChemPhysChem 6 1788
169. Blackford L S et al 1996 Scalapack: a portable linear algebra library for distributed memory computers-design issues and performance Proc. of the 1996 ACM/IEEE Conf. on Supercomput. (Washington, DC: IEEE Computer Society) p5
170. Gygi F et al 2008 IBM J. Res. Dev. 52 137
171. 2007 Using OpenMP: Portable Shared Memory Parallel Programming (Cambridge, MA: MIT Press)
172. Runge E and Gross E K U 1984 Phys. Rev. Lett. 52 997-1000
173. Marques M A L, Ullrich C L, Nogueira F, Rubio A, Burke K and Gross E K U (ed) 2006 Time-Dependent Density Functional Theory (Lecture notes in Physics vol 706) (Berlin: Springer) doi:10.1007/3-540-35426-3-17
174. Onida G, Reining L and Rubio A 2002 Rev. Mod. Phys. 74 601-59
175. Rocca D, Gebauer R, Saad Y and Baroni S 2008 J. Chem. Phys. 128 154105
176. Umari P, Stenuit G and Baroni S 2009 Phys. Rev. B 79 201104(R)
177. Nguyen H-V and de Gironcoli S 2009 Phys. Rev. B 79 205114
178. Nguyen H-V and de Gironcoli S 2009 Phys. Rev. B 79 115105
179. Nguyen H-V 2008 PhD Thesis (SISSA) http://www.sissa.it/cm/thesis/2008/ VietHuyNguyen-PhDthesis.pdf
180. Marini A, Hogan C, Grüning M and Varsano D 2009 Comput. Phys. Commun. 180 1392 http://www.yambo-code.org
181. Calzolari A, Souza I, Marzari N and Buongiorno Nardelli M 2004 Phys. Rev. B 69 035108 http://www.wannier-transport.org
182. Isaev E qha: Calculation of Thermodynamic Properties using the Quasi-Harmonic Approximation http://qha.qe-forge.org
183. Kokalj A pwtk: a Tcl scripting interface to PWscf http://pwtk.qe-forge. org/
184. da Silveira P, da Silva C R S and Wentzcovitch R M 2008 Comput. Phys. Commun. 178 186
185. da Silva C R S, da Silveira P R C, Karki B B, Wentzcovitch R M, Jensen P A, Bollig E F, Pierce M, Erlebacher G and Yuen D A 2007 Phys. Earth Planet. Inter. 163 321 (Special Issue: Computational Challenges)
186. http://dasilveira.cems.umn.edu:8080/gridsphere/gridsphere, http://www.vlab.msi.umn.edu
187. Wentzcovitch R M, Karki B B, Cococcioni M and de Gironcoli S 2004 Phys. Rev. Lett. 92 018501
188. Raczkowski D, Canning A and Wang L W 2001 Phys. Rev. B 64 R121101
189. Arias T A, Payne M C and Joannopoulos J D 1992 Phys. Rev. B 45 1538
190. Qteish A 1995 Phys. Rev. B 52 14497-504
191. Alden Mead C 1992 Rev. Mod. Phys. 64 51
192. di Meo R, Dal Corso A, Giannozzi P and Cozzini S 2009 Calculation of phonon dispersions on the grid using Quantum ESPRESSO Proc. COST School (Trieste) at press
193. Gygi F and Baldereschi A 1986 Phys. Rev. B 34 4405