1925:
1325:
The PAW transformation allows all-electron observables to be calculated using the pseudo-wavefunction from a pseudopotential calculation, conveniently avoiding having to ever represent the all-electron wavefunction explicitly in memory. This is particularly important for the calculation of properties
216:
tend to have rapid oscillations near ion cores due to the requirement that they be orthogonal to core states; this situation is problematic because it requires many
Fourier components (or in the case of grid-based methods, a very fine mesh) to describe the wavefunctions accurately. The PAW approach
217:
addresses this issue by transforming these rapidly oscillating wavefunctions into smooth wavefunctions which are more computationally convenient, and provides a way to calculate all-electron properties from these smooth wavefunctions. This approach is somewhat reminiscent of a change from the
2140:
1704:
1563:
1303:
701:
1316:
The PAW method is typically combined with the frozen core approximation, in which the core states are assumed to be unaffected by the ion's environment. There are several online repositories of pre-computed atomic PAW data.
1639:
1032:
960:
1457:
384:
519:
1308:
Outside the augmentation regions, the pseudo partial waves are equal to the all-electron partial waves. Inside the spheres, they can be any smooth continuation, such as a linear combination of polynomials or
1920:{\displaystyle {\tilde {A}}={\hat {A}}+\sum _{i,j}|p_{i}\rangle \left(\langle \phi _{i}|{\hat {A}}|\phi _{j}\rangle -\langle {\tilde {\phi }}_{i}|{\hat {A}}|{\tilde {\phi }}_{j}\rangle \right)\langle p_{j}|}
1995:
1395:
884:
1465:
1141:
1189:
561:
424:
295:
1957:
run over all projectors on all atoms. Usually only indices on the same atom are summed over, i.e. off-site contributions are ignored, and this is called the "on-site approximation".
1095:
1696:
1178:
1056:
818:
756:
452:
323:
258:
728:
588:
1987:
1672:
783:
389:
Note that the "all-electron" wavefunction is a Kohn–Sham single particle wavefunction, and should not be confused with the many-body wavefunction. In order to have
1955:
608:
2146:
which can be seen as the basis for implementation of pseudopotentials within PAW, as the nuclear coulomb potential can now be substituted with a smoother one.
176:
619:
1579:
965:
1330:, which strongly depend on the form of the wavefunction near the nucleus. Starting with the definition of the expectation value of an operator:
892:
1404:
331:
460:
76:
169:
2135:{\displaystyle {\hat {B}}-\sum _{i,j}|p_{i}\rangle \langle {\tilde {\phi }}_{i}|{\hat {B}}|{\tilde {\phi }}_{j}\rangle \langle p_{j}|}
64:
98:
1336:
826:
60:
2200:
Dal Corso, Andrea (2010-08-11). "Projector augmented-wave method: Application to relativistic spin-density functional theory".
202:
110:
106:
2486:
162:
152:
2425:
Pickard, Chris J.; Mauri, Francesco (2001). "All-electron magnetic response with pseudopotentials: NMR chemical shifts".
2282:
1558:{\displaystyle a_{i}=\langle {\tilde {\Psi }}|{\mathcal {T}}^{\dagger }{\hat {A}}{\mathcal {T}}|{\tilde {\Psi }}\rangle }
194:
80:
1034:, are typically chosen to be solutions to the Kohn–Sham Schrödinger equation for an isolated atom. The transformation
2501:
1298:{\displaystyle {\mathcal {T}}=1+\sum _{i}\left(|\phi _{i}\rangle -|{\tilde {\phi }}_{i}\rangle \right)\langle p_{i}|}
2491:
1102:
2357:
1960:
In the original paper, Blöchl notes that there is a degree of freedom in this equation for an arbitrary operator
2496:
528:
392:
263:
2383:
2171:
Kresse, G.; Joubert, D. (1999). "From ultrasoft pseudopotentials to the projector augmented-wave method".
144:
1065:
1327:
128:
42:
34:
140:
206:
121:
91:
68:
1677:
1148:
1037:
788:
737:
429:
300:
239:
136:
49:
38:
706:
566:
56:
1963:
1648:
102:
218:
234:
613:
Around each atom, it is useful to expand the pseudo wavefunction into pseudo partial waves:
2446:
2323:
2209:
2180:
761:
19:
2404:
8:
1989:, that is localised inside the spherical augmentation region, to add a term of the form:
1934:
84:
27:
2450:
2327:
2213:
2184:
2462:
2436:
2313:
2157:
593:
222:
2466:
2427:
2339:
2304:
696:{\displaystyle |{\tilde {\Psi }}\rangle =\sum _{i}|{\tilde {\phi }}_{i}\rangle c_{i}}
2454:
2331:
2277:
2217:
2188:
1310:
148:
198:
72:
785:
can be written as an inner product with a set of so-called projector functions,
2458:
2221:
2335:
1634:{\displaystyle {\tilde {A}}={\mathcal {T}}^{\dagger }{\hat {A}}{\mathcal {T}}}
1027:{\displaystyle |\phi _{i}\rangle ={\mathcal {T}}|{\tilde {\phi }}_{i}\rangle }
2480:
2192:
213:
132:
2229:
2343:
2441:
2318:
2361:
955:{\displaystyle \langle p_{i}|{\tilde {\phi }}_{j}\rangle =\delta _{ij}}
2252:
1452:{\displaystyle |\Psi \rangle ={\mathcal {T}}|{\tilde {\Psi }}\rangle }
379:{\displaystyle |\Psi \rangle ={\mathcal {T}}|{\tilde {\Psi }}\rangle }
1674:
is local and well-behaved we can expand this using the definition of
2162:
209:
calculations to be performed with greater computational efficiency.
514:{\displaystyle {\mathcal {T}}=1+\sum _{R}{\hat {\mathcal {T}}}_{R}}
2249:(in form of their Gaussian and Augmented Plane Wave (GAPW) Method)
2156:
Rostgaard, Carsten (2010). "The
Projector Augmented-wave Method".
2257:
197:
electronic structure calculations. It is a generalization of the
1401:
where you can substitute in the pseudo wavefunction as you know
2262:
2240:
2235:
2267:
1390:{\displaystyle a_{i}=\langle \Psi |{\hat {A}}|\Psi \rangle }
879:{\displaystyle c_{i}=\langle p_{i}|{\tilde {\Psi }}\rangle }
2246:
563:
is non-zero only within some spherical augmentation region
2272:
2230:
Software implementing the projector augmented-wave method
2302:
Blöchl, P.E. (1994). "Projector augmented-wave method".
454:
differ only in the regions near the ion cores, we write
1998:
1966:
1937:
1707:
1680:
1651:
1582:
1468:
1407:
1339:
1192:
1151:
1105:
1068:
1040:
968:
895:
829:
791:
764:
740:
709:
622:
596:
569:
531:
463:
432:
395:
334:
303:
266:
242:
2384:"Periodic Table of the Elements for PAW Functions"
2134:
1981:
1949:
1919:
1690:
1666:
1633:
1557:
1451:
1389:
1297:
1172:
1135:
1089:
1050:
1026:
954:
878:
812:
777:
750:
722:
695:
602:
582:
555:
513:
446:
418:
378:
317:
289:
252:
2478:
228:
260:transforms the fictitious pseudo wavefunction
2170:
1136:{\displaystyle |{\tilde {\phi }}_{i}\rangle }
170:
2424:
2114:
2111:
2048:
2045:
1899:
1891:
1828:
1822:
1777:
1769:
1552:
1482:
1446:
1416:
1384:
1353:
1277:
1269:
1239:
1167:
1130:
1084:
1021:
984:
933:
896:
873:
843:
807:
680:
640:
441:
413:
373:
343:
312:
284:
77:Multi-configurational self-consistent field
177:
163:
2440:
2317:
2199:
2161:
2155:
1320:
556:{\displaystyle {\hat {\mathcal {T}}}_{R}}
419:{\displaystyle |{\tilde {\Psi }}\rangle }
290:{\displaystyle |{\tilde {\Psi }}\rangle }
99:Time-dependent density functional theory
61:Semi-empirical quantum chemistry methods
1183:and we can explicitly write it down as
1058:is thus specified by three quantities:
2479:
2301:
111:Linearized augmented-plane-wave method
107:Orbital-free density functional theory
1062:a set of all-electron partial waves
1698:to give the PAW operator transform
203:linear augmented-plane-wave methods
81:Quantum chemistry composite methods
13:
2149:
1683:
1626:
1601:
1543:
1530:
1505:
1488:
1437:
1424:
1413:
1381:
1356:
1195:
1090:{\displaystyle |\phi _{i}\rangle }
1043:
992:
962:. The all-electron partial waves,
864:
743:
711:
631:
571:
537:
495:
466:
438:
404:
364:
351:
340:
309:
275:
245:
65:Møller–Plesset perturbation theory
14:
2513:
2358:"PAW atomic data for ABINIT code"
297:to the all-electron wavefunction
191:projector augmented wave method
115:Projector augmented wave method
2418:
2397:
2376:
2350:
2295:
2128:
2099:
2088:
2081:
2071:
2058:
2031:
2005:
1973:
1913:
1879:
1868:
1861:
1851:
1838:
1808:
1801:
1791:
1755:
1729:
1714:
1691:{\displaystyle {\mathcal {T}}}
1658:
1618:
1589:
1569:from which you can define the
1546:
1536:
1522:
1498:
1491:
1440:
1430:
1409:
1377:
1370:
1360:
1291:
1257:
1246:
1225:
1173:{\displaystyle |p_{i}\rangle }
1153:
1118:
1107:
1099:a set of pseudo partial waves
1070:
1051:{\displaystyle {\mathcal {T}}}
1009:
998:
970:
921:
910:
867:
857:
813:{\displaystyle |p_{i}\rangle }
793:
751:{\displaystyle {\mathcal {T}}}
668:
657:
634:
624:
541:
499:
447:{\displaystyle |\Psi \rangle }
434:
407:
397:
367:
357:
336:
318:{\displaystyle |\Psi \rangle }
305:
278:
268:
253:{\displaystyle {\mathcal {T}}}
1:
2288:
2243:(to calculate NMR properties)
1145:a set of projector functions
229:Transforming the wavefunction
193:(PAW) is a technique used in
153:Korringa–Kohn–Rostoker method
2487:Electronic structure methods
758:is linear, the coefficients
7:
723:{\displaystyle \Omega _{R}}
583:{\displaystyle \Omega _{R}}
145:Empty lattice approximation
10:
2518:
2459:10.1103/PhysRevB.63.245101
2222:10.1103/PhysRevB.82.075116
1982:{\displaystyle {\hat {B}}}
1667:{\displaystyle {\hat {A}}}
129:Nearly free electron model
43:Modern valence bond theory
2336:10.1103/PhysRevB.50.17953
207:density functional theory
122:Electronic band structure
92:Density functional theory
69:Configuration interaction
2502:Condensed matter physics
2193:10.1103/PhysRevB.59.1758
1573:, indicated by a tilde:
137:Muffin-tin approximation
50:Molecular orbital theory
39:Generalized valence bond
2492:Computational chemistry
141:k·p perturbation theory
2136:
1983:
1951:
1921:
1692:
1668:
1635:
1559:
1453:
1391:
1321:Transforming operators
1299:
1174:
1137:
1091:
1052:
1028:
956:
880:
814:
779:
752:
724:
697:
604:
584:
557:
515:
448:
420:
380:
319:
291:
254:
35:Coulson–Fischer theory
2497:Computational physics
2435:(24): 245101–245114.
2137:
1984:
1952:
1922:
1693:
1669:
1636:
1560:
1454:
1392:
1300:
1175:
1138:
1092:
1053:
1029:
957:
881:
815:
780:
778:{\displaystyle c_{i}}
753:
734:Because the operator
725:
698:
605:
585:
558:
516:
449:
421:
381:
320:
292:
255:
235:linear transformation
2364:on 11 September 2015
1996:
1964:
1935:
1705:
1678:
1649:
1580:
1466:
1405:
1337:
1190:
1149:
1103:
1066:
1038:
966:
893:
827:
789:
762:
738:
707:
620:
594:
567:
529:
461:
430:
393:
332:
301:
264:
240:
20:Electronic structure
2451:2001PhRvB..63x5101P
2405:"Atomic PAW Setups"
2328:1994PhRvB..5017953B
2312:(24): 17953–17978.
2214:2010PhRvB..82g5116D
2185:1999PhRvB..59.1758K
1950:{\displaystyle i,j}
219:Schrödinger picture
85:Quantum Monte Carlo
57:Hartree–Fock method
28:Valence bond theory
2132:
2029:
1979:
1947:
1931:where the indices
1917:
1753:
1688:
1664:
1631:
1555:
1449:
1387:
1295:
1218:
1170:
1133:
1087:
1048:
1024:
952:
876:
810:
775:
748:
720:
693:
655:
600:
580:
553:
511:
489:
444:
416:
376:
315:
287:
250:
223:Heisenberg picture
103:Thomas–Fermi model
2428:Physical Review B
2305:Physical Review B
2202:Physical Review B
2173:Physical Review B
2102:
2084:
2061:
2014:
2008:
1976:
1882:
1864:
1841:
1804:
1738:
1732:
1717:
1661:
1621:
1592:
1549:
1525:
1494:
1443:
1373:
1260:
1209:
1121:
1012:
924:
870:
671:
646:
637:
603:{\displaystyle R}
544:
502:
480:
410:
370:
281:
205:, and allows for
187:
186:
2509:
2471:
2470:
2444:
2442:cond-mat/0101257
2422:
2416:
2415:
2413:
2411:
2401:
2395:
2394:
2392:
2390:
2380:
2374:
2373:
2371:
2369:
2360:. Archived from
2354:
2348:
2347:
2321:
2319:cond-mat/0201015
2299:
2278:Quantum ESPRESSO
2225:
2196:
2179:(3): 1758–1775.
2167:
2165:
2141:
2139:
2138:
2133:
2131:
2126:
2125:
2110:
2109:
2104:
2103:
2095:
2091:
2086:
2085:
2077:
2074:
2069:
2068:
2063:
2062:
2054:
2044:
2043:
2034:
2028:
2010:
2009:
2001:
1988:
1986:
1985:
1980:
1978:
1977:
1969:
1956:
1954:
1953:
1948:
1926:
1924:
1923:
1918:
1916:
1911:
1910:
1898:
1894:
1890:
1889:
1884:
1883:
1875:
1871:
1866:
1865:
1857:
1854:
1849:
1848:
1843:
1842:
1834:
1821:
1820:
1811:
1806:
1805:
1797:
1794:
1789:
1788:
1768:
1767:
1758:
1752:
1734:
1733:
1725:
1719:
1718:
1710:
1697:
1695:
1694:
1689:
1687:
1686:
1673:
1671:
1670:
1665:
1663:
1662:
1654:
1645:If the operator
1640:
1638:
1637:
1632:
1630:
1629:
1623:
1622:
1614:
1611:
1610:
1605:
1604:
1594:
1593:
1585:
1564:
1562:
1561:
1556:
1551:
1550:
1542:
1539:
1534:
1533:
1527:
1526:
1518:
1515:
1514:
1509:
1508:
1501:
1496:
1495:
1487:
1478:
1477:
1458:
1456:
1455:
1450:
1445:
1444:
1436:
1433:
1428:
1427:
1412:
1396:
1394:
1393:
1388:
1380:
1375:
1374:
1366:
1363:
1349:
1348:
1311:Bessel functions
1304:
1302:
1301:
1296:
1294:
1289:
1288:
1276:
1272:
1268:
1267:
1262:
1261:
1253:
1249:
1238:
1237:
1228:
1217:
1199:
1198:
1179:
1177:
1176:
1171:
1166:
1165:
1156:
1142:
1140:
1139:
1134:
1129:
1128:
1123:
1122:
1114:
1110:
1096:
1094:
1093:
1088:
1083:
1082:
1073:
1057:
1055:
1054:
1049:
1047:
1046:
1033:
1031:
1030:
1025:
1020:
1019:
1014:
1013:
1005:
1001:
996:
995:
983:
982:
973:
961:
959:
958:
953:
951:
950:
932:
931:
926:
925:
917:
913:
908:
907:
885:
883:
882:
877:
872:
871:
863:
860:
855:
854:
839:
838:
819:
817:
816:
811:
806:
805:
796:
784:
782:
781:
776:
774:
773:
757:
755:
754:
749:
747:
746:
729:
727:
726:
721:
719:
718:
702:
700:
699:
694:
692:
691:
679:
678:
673:
672:
664:
660:
654:
639:
638:
630:
627:
609:
607:
606:
601:
589:
587:
586:
581:
579:
578:
562:
560:
559:
554:
552:
551:
546:
545:
540:
535:
520:
518:
517:
512:
510:
509:
504:
503:
498:
493:
488:
470:
469:
453:
451:
450:
445:
437:
425:
423:
422:
417:
412:
411:
403:
400:
385:
383:
382:
377:
372:
371:
363:
360:
355:
354:
339:
324:
322:
321:
316:
308:
296:
294:
293:
288:
283:
282:
274:
271:
259:
257:
256:
251:
249:
248:
179:
172:
165:
149:GW approximation
16:
15:
2517:
2516:
2512:
2511:
2510:
2508:
2507:
2506:
2477:
2476:
2475:
2474:
2423:
2419:
2409:
2407:
2403:
2402:
2398:
2388:
2386:
2382:
2381:
2377:
2367:
2365:
2356:
2355:
2351:
2300:
2296:
2291:
2232:
2152:
2150:Further reading
2127:
2121:
2117:
2105:
2094:
2093:
2092:
2087:
2076:
2075:
2070:
2064:
2053:
2052:
2051:
2039:
2035:
2030:
2018:
2000:
1999:
1997:
1994:
1993:
1968:
1967:
1965:
1962:
1961:
1936:
1933:
1932:
1912:
1906:
1902:
1885:
1874:
1873:
1872:
1867:
1856:
1855:
1850:
1844:
1833:
1832:
1831:
1816:
1812:
1807:
1796:
1795:
1790:
1784:
1780:
1776:
1772:
1763:
1759:
1754:
1742:
1724:
1723:
1709:
1708:
1706:
1703:
1702:
1682:
1681:
1679:
1676:
1675:
1653:
1652:
1650:
1647:
1646:
1625:
1624:
1613:
1612:
1606:
1600:
1599:
1598:
1584:
1583:
1581:
1578:
1577:
1571:pseudo operator
1541:
1540:
1535:
1529:
1528:
1517:
1516:
1510:
1504:
1503:
1502:
1497:
1486:
1485:
1473:
1469:
1467:
1464:
1463:
1435:
1434:
1429:
1423:
1422:
1408:
1406:
1403:
1402:
1376:
1365:
1364:
1359:
1344:
1340:
1338:
1335:
1334:
1323:
1290:
1284:
1280:
1263:
1252:
1251:
1250:
1245:
1233:
1229:
1224:
1223:
1219:
1213:
1194:
1193:
1191:
1188:
1187:
1161:
1157:
1152:
1150:
1147:
1146:
1124:
1113:
1112:
1111:
1106:
1104:
1101:
1100:
1078:
1074:
1069:
1067:
1064:
1063:
1042:
1041:
1039:
1036:
1035:
1015:
1004:
1003:
1002:
997:
991:
990:
978:
974:
969:
967:
964:
963:
943:
939:
927:
916:
915:
914:
909:
903:
899:
894:
891:
890:
862:
861:
856:
850:
846:
834:
830:
828:
825:
824:
801:
797:
792:
790:
787:
786:
769:
765:
763:
760:
759:
742:
741:
739:
736:
735:
714:
710:
708:
705:
704:
687:
683:
674:
663:
662:
661:
656:
650:
629:
628:
623:
621:
618:
617:
595:
592:
591:
590:enclosing atom
574:
570:
568:
565:
564:
547:
536:
534:
533:
532:
530:
527:
526:
505:
494:
492:
491:
490:
484:
465:
464:
462:
459:
458:
433:
431:
428:
427:
402:
401:
396:
394:
391:
390:
362:
361:
356:
350:
349:
335:
333:
330:
329:
304:
302:
299:
298:
273:
272:
267:
265:
262:
261:
244:
243:
241:
238:
237:
231:
199:pseudopotential
183:
151:
147:
143:
139:
135:
131:
113:
109:
105:
101:
83:
79:
75:
73:Coupled cluster
71:
67:
63:
59:
41:
37:
12:
11:
5:
2515:
2505:
2504:
2499:
2494:
2489:
2473:
2472:
2417:
2396:
2375:
2349:
2293:
2292:
2290:
2287:
2286:
2285:
2280:
2275:
2270:
2265:
2260:
2255:
2250:
2244:
2238:
2231:
2228:
2227:
2226:
2197:
2168:
2151:
2148:
2144:
2143:
2130:
2124:
2120:
2116:
2113:
2108:
2101:
2098:
2090:
2083:
2080:
2073:
2067:
2060:
2057:
2050:
2047:
2042:
2038:
2033:
2027:
2024:
2021:
2017:
2013:
2007:
2004:
1975:
1972:
1946:
1943:
1940:
1929:
1928:
1915:
1909:
1905:
1901:
1897:
1893:
1888:
1881:
1878:
1870:
1863:
1860:
1853:
1847:
1840:
1837:
1830:
1827:
1824:
1819:
1815:
1810:
1803:
1800:
1793:
1787:
1783:
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1771:
1766:
1762:
1757:
1751:
1748:
1745:
1741:
1737:
1731:
1728:
1722:
1716:
1713:
1685:
1660:
1657:
1643:
1642:
1628:
1620:
1617:
1609:
1603:
1597:
1591:
1588:
1567:
1566:
1554:
1548:
1545:
1538:
1532:
1524:
1521:
1513:
1507:
1500:
1493:
1490:
1484:
1481:
1476:
1472:
1448:
1442:
1439:
1432:
1426:
1421:
1418:
1415:
1411:
1399:
1398:
1386:
1383:
1379:
1372:
1369:
1362:
1358:
1355:
1352:
1347:
1343:
1322:
1319:
1306:
1305:
1293:
1287:
1283:
1279:
1275:
1271:
1266:
1259:
1256:
1248:
1244:
1241:
1236:
1232:
1227:
1222:
1216:
1212:
1208:
1205:
1202:
1197:
1181:
1180:
1169:
1164:
1160:
1155:
1143:
1132:
1127:
1120:
1117:
1109:
1097:
1086:
1081:
1077:
1072:
1045:
1023:
1018:
1011:
1008:
1000:
994:
989:
986:
981:
977:
972:
949:
946:
942:
938:
935:
930:
923:
920:
912:
906:
902:
898:
887:
886:
875:
869:
866:
859:
853:
849:
845:
842:
837:
833:
809:
804:
800:
795:
772:
768:
745:
732:
731:
717:
713:
690:
686:
682:
677:
670:
667:
659:
653:
649:
645:
642:
636:
633:
626:
599:
577:
573:
550:
543:
539:
523:
522:
508:
501:
497:
487:
483:
479:
476:
473:
468:
443:
440:
436:
415:
409:
406:
399:
387:
386:
375:
369:
366:
359:
353:
348:
345:
342:
338:
314:
311:
307:
286:
280:
277:
270:
247:
230:
227:
185:
184:
182:
181:
174:
167:
159:
156:
155:
125:
124:
118:
117:
95:
94:
88:
87:
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2019:
2015:
2011:
2002:
1992:
1991:
1990:
1970:
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1944:
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224:
220:
215:
214:wavefunctions
210:
208:
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180:
175:
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168:
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161:
160:
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133:Tight binding
130:
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119:
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2408:. Retrieved
2399:
2387:. Retrieved
2378:
2366:. Retrieved
2362:the original
2352:
2309:
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2297:
2205:
2201:
2176:
2172:
2145:
1959:
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1570:
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211:
190:
188:
114:
2410:14 February
2389:13 February
2368:13 February
2163:0910.1921v2
2481:Categories
2289:References
2467:119510292
2115:⟨
2112:⟩
2100:~
2097:ϕ
2082:^
2059:~
2056:ϕ
2049:⟨
2046:⟩
2016:∑
2012:−
2006:^
1974:^
1900:⟨
1892:⟩
1880:~
1877:ϕ
1862:^
1839:~
1836:ϕ
1829:⟨
1826:−
1823:⟩
1814:ϕ
1802:^
1782:ϕ
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1770:⟩
1740:∑
1730:^
1715:~
1659:^
1619:^
1608:†
1590:~
1553:⟩
1547:~
1544:Ψ
1523:^
1512:†
1492:~
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1483:⟨
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1438:Ψ
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1371:^
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1278:⟨
1270:⟩
1258:~
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874:⟩
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641:⟩
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542:^
500:^
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442:⟩
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414:⟩
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374:⟩
368:~
365:Ψ
344:⟩
341:Ψ
313:⟩
310:Ψ
285:⟩
279:~
276:Ψ
195:ab initio
2273:S/PHI/nX
1326:such as
212:Valence
2447:Bibcode
2344:9976227
2324:Bibcode
2210:Bibcode
2181:Bibcode
703:within
221:to the
22:methods
2465:
2342:
2263:ONETEP
2253:CP-PAW
2241:CASTEP
2236:ABINIT
889:where
525:where
2463:S2CID
2437:arXiv
2314:arXiv
2268:PWPAW
2158:arXiv
2412:2012
2391:2012
2370:2012
2340:PMID
2283:VASP
2258:GPAW
2247:CP2K
426:and
233:The
201:and
189:The
2455:doi
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2218:doi
2189:doi
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