93:
exchange-correlation approximation employed. The LUMO energy shows little correlation with the electron affinity with typical approximations. The error in the DFT counterpart of
Koopmans' theorem is a result of the approximation employed for the exchange correlation energy functional so that, unlike in HF theory, there is the possibility of improved results with the development of better approximations.
308:
was stated that
Koopmans theorem can only be applied for removing the unpaired electron. Later, the validity of Koopmansā theorem for ROHF was revisited and several procedures for obtaining meaningful orbital energies were reported. The spin up (alpha) and spin down (beta) orbital energies do not necessarily have to be the same.
506:
Unlike the approximate status of
Koopmans' theorem in Hartree Fock theory (because of the neglect of orbital relaxation), in the exact KS mapping the theorem is exact, including the effect of orbital relaxation. A sketchy proof of this exact relation goes in three stages. First, for any finite system
101:
While
Koopmans' theorem was originally stated for calculating ionization energies from restricted (closed-shell) HartreeāFock wavefunctions, the term has since taken on a more generalized meaning as a way of using orbital energies to calculate energy changes due to changes in the number of electrons
307:
Koopmans' theorem is also applicable to open-shell systems, however, orbital energies (eigenvalues of
Roothaan equations) should be corrected, as was shown in the 1970s. Despite this early work, application of Koopmans theorem to open-shell systems continued to cause confusion, e.g., it
110:
Koopmansā theorem applies to the removal of an electron from any occupied molecular orbital to form a positive ion. Removal of the electron from different occupied molecular orbitals leads to the ion in different electronic states. The lowest of these states is the ground state and this often, but
667:. Next, as a corollary (since the physically interacting system has the same density as the KS system), both must have the same ionization energy. Finally, since the KS potential is zero at infinity, the ionization energy of the KS system is, by definition, the negative of its HOMO energy, i.e.,
710:
A tuning procedure is able to "impose" Koopmans' theorem on DFT approximations, thereby improving many of its related predictions in actual applications. In approximate DFTs one can estimate to high degree of accuracy the deviance from
Koopmans' theorem using the concept of energy curvature. It
1048:
electron system. When correlated wavefunctions are used, Dyson orbitals include correlation and orbital relaxation effects. Dyson orbitals contain all information about the initial and final states of the system needed to compute experimentally observable quantities, such as total and
291:
Calculations of electron affinities using this statement of
Koopmans' theorem have been criticized on the grounds that virtual (unoccupied) orbitals do not have well-founded physical interpretations, and that their orbital energies are very sensitive to the choice of basis set used in the
92:
energies, although both the derivation and the precise statement differ from that of
Koopmans' theorem. Ionization energies calculated from DFT orbital energies are usually poorer than those of Koopmans' theorem, with errors much larger than two electron volts possible depending on the
706:
While these are exact statements in the formalism of DFT, the use of approximate exchange-correlation potentials makes the calculated energies approximate and often the orbital energies are very different from the corresponding ionization energies (even by several eV!).
241:. The corresponding ionization energies are 539.7, 32.2, 18.5, 14.7 and 12.6 eV. As explained above, the deviations are due to the effects of orbital relaxation as well as differences in electron correlation energy between the molecular and the various ionized states.
776:
The concept of molecular orbitals and a
Koopmans-like picture of ionization or electron attachment processes can be extended to correlated many-body wavefunctions by introducing Dyson orbitals. Dyson orbitals are defined as the generalized overlap between an
502:
electron system but then the exact KS potential jumps by a constant known as the "derivative discontinuity". It can be argued that the vertical electron affinity is equal exactly to the negative of the sum of the LUMO energy and the derivative discontinuity.
997:
665:
2168:
766:
49:
approximation). Ionization energies calculated this way are in qualitative agreement with experiment ā the first ionization energy of small molecules is often calculated with an error of less than
563:
855:
1179:
Hamel, Sebastien; Duffy, Patrick; Casida, Mark E.; Salahub, Dennis R. (2002). "KohnāSham orbitals and orbital energies: fictitious constructs but good approximations all the same".
2545:
1856:
Perdew, John P.; Parr, Robert G.; Levy, Mel; Balduz, Jr., Jose L. (1982). "Density-Functional Theory for
Fractional Particle Number: Derivative Discontinuities of the Energy".
568:
701:
467:
389:
496:
418:
206:), and so on. In this case the order of the ion electronic states corresponds to the order of the orbital energies. Excited-state ionization energies can be measured by
1046:
77:
calculations suggest that in many cases, but not all, the energetic corrections due to relaxation effects nearly cancel the corrections due to electron correlation.
847:
821:
1996:
Almbladh, C. -O.; von Barth, U. (1985). "Exact results for the charge and spin densities, exchange-correlation potentials, and density-functional eigenvalues".
1020:
795:
525:
438:
362:
342:
2320:
Stein, Tamar; Autschbach, Jochen; Govind, Niranjan; Kronik, Leeor; Baer, Roi (2012). "Curvature and frontier orbital energies in density functional theory".
2373:
Andrejkovics, I; Nagy, Ć (1998). "Excitation energies in density functional theory: Comparison of several methods for the H2O, N2, CO and C2H4 molecules".
1608:
Plakhutin, B. N.; Gorelik, E. V.; Breslavskaya, N. N. (2006). "Koopmans' theorem in the ROHF method: Canonical form for the Hartree-Fock Hamiltonian".
2226:
Stein, T.; Eisenberg, H.; Kronik, L.; Baer, R. (2010). "Fundamental gaps of finite systems from the eigenvalues of a generalized Kohn-Sham method".
1704:
Plakhutin, Boris N.; Davidson, Ernest R. (2009). "Koopmans' Theorem in the Restricted Open-Shell HartreeāFock Method. 1. A Variational Approachā ".
1493:
Sauer, Joachim; Jung, Christoph (1975). "Konsequenzen des Koopmansschen Theorems in den Restricted Hartree Fock Methoden fĆ¼r open-shell-Systeme".
2080:
Levy, Mel; Perdew, John P; Sahni, Viraht (1984). "Exact differential equation for the density and ionization energy of a many-particle system".
207:
1592:
1477:
1444:
1383:
1266:
1241:
1216:
299:
Comparisons with experiment and higher-quality calculations show that electron affinities predicted in this manner are generally quite poor.
248:
in contrast, the order of orbital energies is not identical to the order of ionization energies. Near-HartreeāFock calculations with a large
391:. More generally, this relation is true even when the KS system describes a zero-temperature ensemble with non-integer number of electrons
2408:
Perdew, John P; Levy, Mel (1983). "Physical Content of the Exact Kohn-Sham Orbital Energies: Band Gaps and Derivative Discontinuities".
2552:
714:
260:
bonding orbital. In this case the deviation is attributed primarily to the difference in correlation energy between the two orbitals.
1656:"Koopmans's theorem in the restricted open-shell HartreeāFock method. II. The second canonical set for orbitals and orbital energies"
1114:
276:) of the respective systems. However, Koopmans' original paper makes no claim with regard to the significance of eigenvalues of the
2585:
66:, referring to the validity of representing the entire many-body wavefunction using the HartreeāFock wavefunction, i.e. a single
1065:
Koopmans, Tjalling (1934). "Ćber die Zuordnung von Wellenfunktionen und Eigenwerten zu den einzelnen Elektronen eines Atoms".
2590:
296:
the molecule of interest will appear, and care must be taken not to use these orbitals for estimating electron affinities.
1320:
Zhang, Gang; Musgrave, Charles B. (2007). "Comparison of DFT Methods for Molecular Orbital Eigenvalue Calculations".
2563:
1141:
Politzer, Peter; Abu-Awwad, Fakher (1998). "A comparative analysis of HartreeāFock and KohnāSham orbital energies".
324:) very similar in spirit to that of Hartree-Fock theory. The theorem equates the first (vertical) ionization energy
2580:
256:
bonding orbital is the HOMO. However the lowest ionization energy corresponds to removal of an electron from the 3Ļ
2119:
317:
81:
1961:
Kleinman, Leonard (1997). "Reply to "Comment on 'Significance of the highest occupied Kohn-Sham eigenvalue' "".
992:{\displaystyle \phi ^{d}(1)={\sqrt {N}}\int \Psi _{I}^{N}(1,\dots ,n)\Psi _{F}^{N-1}(2,\dots ,n)\,d2\dots dn\;.}
30:
of a molecular system is equal to the negative of the orbital energy of the highest occupied molecular orbital (
1100:
530:
111:
not always, arises from removal of the electron from the HOMO. The other states are excited electronic states.
1683:
42:
23:
1891:
Perdew, John P.; Levy, Mel (1997). "Comment on "Significance of the highest occupied KohnāSham eigenvalue"".
292:
calculation. As the basis set becomes more complete; more and more "molecular" orbitals that are not really
2189:
Baer, R.; Livshits, E.; Salzner, U. (2010). ""Tuned" Range-separated hybrids in density functional theory".
1747:
Glaesemann, Kurt R.; Schmidt, Michael W. (2010). "On the Ordering of Orbital Energies in High-Spin ROHFā ".
660:{\textstyle n(\mathbf {r} )\to \exp \left(-2{\sqrt {{\frac {2m_{\rm {e}}}{\hbar }}I}}|\mathbf {r} |\right)}
284:. Nevertheless, it is straightforward to generalize the original statement of Koopmans' to calculate the
249:
58:
wavefunction. The two main sources of error are orbital relaxation, which refers to the changes in the
670:
2417:
2382:
2245:
2134:
2089:
2052:
2005:
1970:
1935:
1900:
1865:
1809:
1756:
1713:
1667:
1617:
1541:
1408:
1329:
1294:
1074:
1002:
Hartree-Fock canonical orbitals are Dyson orbitals computed for the Hartree-Fock wavefunction of the
70:
composed of orbitals that are the eigenfunctions of the corresponding self-consistent Fock operator.
63:
54:. Therefore, the validity of Koopmans' theorem is intimately tied to the accuracy of the underlying
443:
367:
1469:
45:
if it is assumed that the orbitals of the ion are identical to those of the neutral molecule (the
2355:
2329:
2281:"Excitation Gaps of Finite-Sized Systems from Optimally Tuned Range-Separated Hybrid Functionals"
2269:
2235:
1833:
1799:
1651:
1510:
1353:
1158:
151:
67:
472:
394:
2523:
2515:
2474:
2466:
2347:
2302:
2261:
2216:
2160:
2021:
1825:
1772:
1729:
1633:
1588:
1584:
1557:
1473:
1440:
1379:
1345:
1262:
1237:
1212:
1120:
1110:
285:
269:
85:
35:
27:
1790:
Tsuchimochi, Takashi; Scuseria, Gustavo E. (2010). "Communication: ROHF theory made simple".
55:
2505:
2456:
2425:
2390:
2339:
2292:
2253:
2206:
2198:
2150:
2142:
2097:
2060:
2013:
1978:
1943:
1908:
1873:
1817:
1764:
1721:
1675:
1625:
1576:
1549:
1502:
1461:
1416:
1337:
1302:
1188:
1150:
1082:
1025:
2202:
1462:
826:
800:
2421:
2386:
2249:
2138:
2093:
2056:
2009:
1974:
1939:
1904:
1869:
1813:
1760:
1717:
1671:
1621:
1545:
1412:
1333:
1298:
1078:
1926:
Kleinman, Leonard (1997). "Significance of the highest occupied Kohn-Sham eigenvalue".
1005:
780:
510:
423:
347:
327:
2394:
2065:
2040:
1192:
1086:
217:
O, the near-HartreeāFock orbital energies (with sign changed) of these orbitals are 1a
2574:
1577:
1529:
59:
51:
2359:
2273:
1837:
1514:
1162:
2257:
1357:
238:
62:
and HartreeāFock orbitals when changing the number of electrons in the system, and
2429:
1877:
277:
1982:
1947:
1912:
268:
It is sometimes claimed that Koopmans' theorem also allows the calculation of
2519:
2470:
2101:
2017:
1561:
1306:
1124:
89:
74:
2527:
2478:
2351:
2306:
2265:
2220:
2164:
1829:
1776:
1733:
1637:
1349:
320:(KS-DFT) admits its own version of Koopmans' theorem (sometimes called the
2025:
1257:
Hehre, Warren J.; Radom, Leo; Schleyer, Paul v.R.; Pople, John A. (1986).
1154:
1104:
2211:
2155:
1506:
2510:
2493:
2461:
2444:
2343:
2297:
2280:
2146:
1821:
1768:
1725:
1679:
1629:
1420:
1341:
1285:
Janak, J. F (1978). "Proof that āEāni=Īµin density-functional theory".
1553:
498:
electrons the infinitesimal excess charge enters the KS LUMO of the
1655:
2494:"Dyson-orbital concepts for description of electrons in molecules"
2334:
2240:
1804:
1399:
Duke, Brian J.; O'Leary, Brian (1995). "Non-Koopmans' Molecules".
761:{\textstyle {\frac {\partial E}{\partial n_{i}}}=\varepsilon _{i}}
73:
Empirical comparisons with experimental values and higher-quality
1143:
Theoretical Chemistry Accounts: Theory, Computation, and Modeling
1528:
Sauer, J.; Jung, Ch.; JaffƩ, H. H.; Singerman, J. (1978-07-01).
281:
273:
31:
2279:
Kornik, L.; Stein, T.; Refaely-Abramson, S.; Baer, R. (2012).
2041:"Relationship of KohnāSham eigenvalues to excitation energies"
364:
electrons to the negative of the corresponding KS HOMO energy
1049:
differential photoionization/phtodetachment cross sections.
272:
as the energy of the lowest unoccupied molecular orbitals (
849:
electron wave function of an electron-attached system):
41:
Koopmans' theorem is exact in the context of restricted
158:
HOMO corresponds to the ionization energy to form the H
1181:
Journal of Electron Spectroscopy and Related Phenomena
717:
571:
1028:
1008:
858:
829:
803:
783:
673:
533:
513:
475:
446:
426:
397:
370:
350:
330:
1022:-electron system and Koopmans approximation of the
1040:
1014:
991:
841:
815:
789:
760:
695:
659:
557:
519:
490:
461:
432:
412:
383:
356:
336:
114:For example, the electronic configuration of the H
823:electron wave function of the ionized system (or
711:provides excitation energies to zeroth-order and
565:asymptotic form of the density, which decays as
1232:Michl, Josef; BonaÄiÄ-KouteckĆ½, Vlasta (1990).
1207:Szabo, A.; Ostlund, N. S. (1982). "Chapter 3".
2039:Savin, A; Umrigar, C.J; Gonze, Xavier (1998).
80:A similar theorem (Janak's theorem) exists in
1579:Electron correlations in molecules and solids
154:. From Koopmansā theorem the energy of the 1b
8:
1378:(4th ed.). Prentice-Hall. p. 475.
1234:Electronic Aspects of Organic Photochemistry
1851:
1849:
1847:
772:Orbital picture within many-body formalisms
2285:Journal of Chemical Theory and Computation
1136:
1134:
985:
186:refers to the ion in the excited state (1a
2509:
2460:
2333:
2322:The Journal of Physical Chemistry Letters
2296:
2239:
2210:
2154:
2113:
2111:
2064:
1803:
1369:
1367:
1027:
1007:
969:
936:
931:
900:
895:
881:
863:
857:
828:
802:
797:-electron molecular wavefunction and the
782:
752:
736:
718:
716:
678:
672:
647:
642:
637:
619:
618:
608:
606:
578:
570:
558:{\displaystyle |\mathbf {r} |\to \infty }
544:
539:
534:
532:
512:
474:
445:
425:
396:
375:
369:
349:
329:
182:). The energy of the second-highest MO 3a
1530:"Orbital energies in open shell systems"
312:Counterpart in density functional theory
2546:"Lecture on Koopmans' Theorem Chem 531"
2445:"From orbitals to observables and back"
1464:Introduction to Computational Chemistry
1174:
1172:
1057:
627:
2203:10.1146/annurev.physchem.012809.103321
280:other than that corresponding to the
38:, who published this result in 1934.
7:
1280:
1278:
2191:Annual Review of Physical Chemistry
1749:The Journal of Physical Chemistry A
1706:The Journal of Physical Chemistry A
1322:The Journal of Physical Chemistry A
1109:. Dover Publications. p. 128.
106:Ground-state and excited-state ions
84:(DFT) for relating the exact first
1435:Szabo, A.; Ostlund, N. S. (1982).
1259:Ab initio molecular orbital theory
928:
892:
729:
721:
620:
552:
14:
696:{\displaystyle \epsilon _{H}=-I}
643:
579:
540:
2498:The Journal of Chemical Physics
2449:The Journal of Chemical Physics
2171:from the original on 2021-01-14
2127:The Journal of Chemical Physics
1792:The Journal of Chemical Physics
1686:from the original on 2021-01-14
1660:The Journal of Chemical Physics
1610:The Journal of Chemical Physics
1534:The Journal of Chemical Physics
34:). This theorem is named after
2443:Krylov, Anna I. (2020-08-24).
2258:10.1103/PhysRevLett.105.266802
2118:Salzner, U.; Baer, R. (2009).
1654:; Plakhutin, Boris N. (2010).
966:
948:
924:
906:
875:
869:
648:
638:
586:
583:
575:
549:
545:
535:
453:
1:
2566:. The Nobel Foundation. 1975.
2395:10.1016/S0009-2614(98)01075-6
2066:10.1016/S0009-2614(98)00316-9
1401:Journal of Chemical Education
1193:10.1016/S0368-2048(02)00032-4
1087:10.1016/S0031-8914(34)90011-2
462:{\displaystyle \delta N\to 0}
384:{\displaystyle \epsilon _{H}}
162:O ion in its ground state (1a
88:and electron affinity to the
150:are orbital labels based on
22:states that in closed-shell
16:Theorem in quantum mechanics
2492:Ortiz, J. V. (2020-08-18).
2430:10.1103/PhysRevLett.51.1884
2120:"Koopmans' springs to life"
1878:10.1103/PhysRevLett.49.1691
1103:; Ostlund, Neil S. (1996).
2607:
491:{\displaystyle N+\delta N}
413:{\displaystyle N-\delta N}
208:photoelectron spectroscopy
86:vertical ionization energy
2564:"Koopmans' autobiography"
1983:10.1103/PhysRevB.56.16029
1948:10.1103/PhysRevB.56.12042
1913:10.1103/PhysRevB.56.16021
318:density functional theory
82:density functional theory
2375:Chemical Physics Letters
2102:10.1103/PhysRevA.30.2745
2045:Chemical Physics Letters
2018:10.1103/PhysRevB.31.3231
1437:Modern Quantum Chemistry
1307:10.1103/PhysRevB.18.7165
1209:Modern Quantum Chemistry
1106:Modern quantum chemistry
2586:Computational chemistry
2410:Physical Review Letters
2228:Physical Review Letters
1858:Physical Review Letters
264:For electron affinities
138:), where the symbols a
1460:Jensen, Frank (1990).
1374:Levine, I. N. (1991).
1042:
1041:{\displaystyle N\pm 1}
1016:
993:
843:
817:
791:
762:
697:
661:
559:
521:
492:
463:
434:
414:
385:
358:
338:
303:For open-shell systems
2591:Theoretical chemistry
1583:. Springer. pp.
1575:Fulde, Peter (1995).
1261:. Wiley. p. 24.
1236:. Wiley. p. 35.
1155:10.1007/s002140050307
1043:
1017:
994:
844:
818:
792:
763:
698:
662:
560:
522:
493:
464:
435:
415:
386:
359:
339:
322:DFT-Koopmans' theorem
1026:
1006:
856:
827:
801:
781:
715:
671:
569:
531:
511:
473:
444:
424:
395:
368:
348:
328:
252:indicate that the 1Ļ
64:electron correlation
2422:1983PhRvL..51.1884P
2387:1998CPL...296..489A
2250:2010PhRvL.105z6802S
2139:2009JChPh.131w1101S
2094:1984PhRvA..30.2745L
2057:1998CPL...288..391S
2010:1985PhRvB..31.3231A
1975:1997PhRvB..5616029K
1940:1997PhRvB..5612042K
1905:1997PhRvB..5616021P
1899:(24): 16021ā16028.
1870:1982PhRvL..49.1691P
1814:2010JChPh.133n1102T
1761:2010JPCA..114.8772G
1718:2009JPCA..11312386P
1712:(45): 12386ā12395.
1672:2010JChPh.132r4110D
1652:Davidson, Ernest R.
1622:2006JChPh.125t4110P
1546:1978JChPh..69..495S
1413:1995JChEd..72..501D
1334:2007JPCA..111.1554Z
1299:1978PhRvB..18.7165J
1079:1934Phy.....1..104K
947:
905:
842:{\displaystyle N+1}
816:{\displaystyle N-1}
469:. When considering
270:electron affinities
43:HartreeāFock theory
24:HartreeāFock theory
1507:10.1007/BF01135884
1468:. Wiley. pp.
1433:See, for example,
1205:See, for example,
1038:
1012:
989:
927:
891:
839:
813:
787:
758:
693:
657:
555:
517:
488:
459:
430:
410:
381:
354:
334:
152:molecular symmetry
68:Slater determinant
2581:Quantum chemistry
2511:10.1063/5.0016472
2462:10.1063/5.0018597
2344:10.1021/jz3015937
2298:10.1021/ct2009363
2147:10.1063/1.3269030
2082:Physical Review A
1998:Physical Review B
1963:Physical Review B
1928:Physical Review B
1893:Physical Review B
1864:(23): 1691ā1694.
1822:10.1063/1.3503173
1769:10.1021/jp101758y
1755:(33): 8772ā8777.
1726:10.1021/jp9002593
1680:10.1063/1.3418615
1630:10.1063/1.2393223
1594:978-3-540-59364-5
1495:Theor. Chim. Acta
1479:978-0-471-98425-2
1446:978-0-02-949710-4
1421:10.1021/ed072p501
1385:978-0-7923-1421-9
1376:Quantum Chemistry
1342:10.1021/jp061633o
1293:(12): 7165ā7168.
1287:Physical Review B
1268:978-0-471-81241-8
1243:978-0-471-89626-5
1218:978-0-02-949710-4
1015:{\displaystyle N}
886:
790:{\displaystyle N}
743:
635:
630:
520:{\displaystyle I}
433:{\displaystyle N}
357:{\displaystyle N}
337:{\displaystyle I}
286:electron affinity
118:O molecule is (1a
36:Tjalling Koopmans
28:ionization energy
20:Koopmans' theorem
2598:
2567:
2559:
2557:
2551:. Archived from
2550:
2532:
2531:
2513:
2489:
2483:
2482:
2464:
2440:
2434:
2433:
2405:
2399:
2398:
2370:
2364:
2363:
2337:
2317:
2311:
2310:
2300:
2277:
2243:
2224:
2214:
2186:
2180:
2179:
2177:
2176:
2158:
2133:(23): 231101ā4.
2124:
2115:
2106:
2105:
2077:
2071:
2070:
2068:
2036:
2030:
2029:
2004:(6): 3231ā3244.
1993:
1987:
1986:
1958:
1952:
1951:
1923:
1917:
1916:
1888:
1882:
1881:
1853:
1842:
1841:
1807:
1787:
1781:
1780:
1744:
1738:
1737:
1701:
1695:
1694:
1692:
1691:
1648:
1642:
1641:
1605:
1599:
1598:
1582:
1572:
1566:
1565:
1554:10.1063/1.436380
1525:
1519:
1518:
1490:
1484:
1483:
1467:
1457:
1451:
1450:
1431:
1425:
1424:
1396:
1390:
1389:
1371:
1362:
1361:
1328:(8): 1554ā1561.
1317:
1311:
1310:
1282:
1273:
1272:
1254:
1248:
1247:
1229:
1223:
1222:
1203:
1197:
1196:
1187:(2ā3): 345ā363.
1176:
1167:
1166:
1138:
1129:
1128:
1097:
1091:
1090:
1073:(1ā6): 104ā113.
1062:
1047:
1045:
1044:
1039:
1021:
1019:
1018:
1013:
998:
996:
995:
990:
946:
935:
904:
899:
887:
882:
868:
867:
848:
846:
845:
840:
822:
820:
819:
814:
796:
794:
793:
788:
767:
765:
764:
759:
757:
756:
744:
742:
741:
740:
727:
719:
702:
700:
699:
694:
683:
682:
666:
664:
663:
658:
656:
652:
651:
646:
641:
636:
631:
626:
625:
624:
623:
609:
607:
582:
564:
562:
561:
556:
548:
543:
538:
526:
524:
523:
518:
497:
495:
494:
489:
468:
466:
465:
460:
439:
437:
436:
431:
419:
417:
416:
411:
390:
388:
387:
382:
380:
379:
363:
361:
360:
355:
344:of a system of
343:
341:
340:
335:
26:(HF), the first
2606:
2605:
2601:
2600:
2599:
2597:
2596:
2595:
2571:
2570:
2562:
2555:
2548:
2543:
2540:
2535:
2491:
2490:
2486:
2442:
2441:
2437:
2407:
2406:
2402:
2372:
2371:
2367:
2319:
2318:
2314:
2278:
2225:
2188:
2187:
2183:
2174:
2172:
2122:
2117:
2116:
2109:
2079:
2078:
2074:
2038:
2037:
2033:
1995:
1994:
1990:
1960:
1959:
1955:
1925:
1924:
1920:
1890:
1889:
1885:
1855:
1854:
1845:
1789:
1788:
1784:
1746:
1745:
1741:
1703:
1702:
1698:
1689:
1687:
1650:
1649:
1645:
1607:
1606:
1602:
1595:
1574:
1573:
1569:
1527:
1526:
1522:
1492:
1491:
1487:
1480:
1459:
1458:
1454:
1447:
1439:. p. 127.
1434:
1432:
1428:
1398:
1397:
1393:
1386:
1373:
1372:
1365:
1319:
1318:
1314:
1284:
1283:
1276:
1269:
1256:
1255:
1251:
1244:
1231:
1230:
1226:
1219:
1206:
1204:
1200:
1178:
1177:
1170:
1140:
1139:
1132:
1117:
1099:
1098:
1094:
1064:
1063:
1059:
1055:
1024:
1023:
1004:
1003:
859:
854:
853:
825:
824:
799:
798:
779:
778:
774:
748:
732:
728:
720:
713:
712:
674:
669:
668:
614:
610:
599:
595:
567:
566:
529:
528:
527:determines the
509:
508:
471:
470:
442:
441:
422:
421:
393:
392:
371:
366:
365:
346:
345:
326:
325:
316:KohnāSham (KS)
314:
305:
288:in this sense.
266:
259:
255:
247:
236:
232:
228:
224:
220:
216:
205:
201:
197:
193:
189:
185:
181:
177:
173:
169:
165:
161:
157:
149:
145:
141:
137:
133:
129:
125:
121:
117:
108:
99:
97:Generalizations
17:
12:
11:
5:
2604:
2602:
2594:
2593:
2588:
2583:
2573:
2572:
2569:
2568:
2560:
2558:on 2005-09-30.
2544:Bowman, Joel.
2539:
2538:External links
2536:
2534:
2533:
2484:
2435:
2400:
2365:
2328:(24): 3740ā4.
2312:
2291:(5): 1515ā31.
2234:(26): 266802.
2181:
2107:
2072:
2031:
1988:
1953:
1918:
1883:
1843:
1798:(14): 141102.
1782:
1739:
1696:
1666:(18): 184110.
1643:
1616:(20): 204110.
1600:
1593:
1567:
1540:(1): 495ā496.
1520:
1501:(2): 129ā141.
1485:
1478:
1452:
1445:
1426:
1391:
1384:
1363:
1312:
1274:
1267:
1249:
1242:
1224:
1217:
1198:
1168:
1130:
1115:
1092:
1056:
1054:
1051:
1037:
1034:
1031:
1011:
1000:
999:
988:
984:
981:
978:
975:
972:
968:
965:
962:
959:
956:
953:
950:
945:
942:
939:
934:
930:
926:
923:
920:
917:
914:
911:
908:
903:
898:
894:
890:
885:
880:
877:
874:
871:
866:
862:
838:
835:
832:
812:
809:
806:
786:
773:
770:
755:
751:
747:
739:
735:
731:
726:
723:
692:
689:
686:
681:
677:
655:
650:
645:
640:
634:
629:
622:
617:
613:
605:
602:
598:
594:
591:
588:
585:
581:
577:
574:
554:
551:
547:
542:
537:
516:
487:
484:
481:
478:
458:
455:
452:
449:
429:
409:
406:
403:
400:
378:
374:
353:
333:
313:
310:
304:
301:
265:
262:
257:
253:
245:
234:
230:
226:
222:
218:
214:
203:
199:
195:
191:
187:
183:
179:
175:
171:
167:
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159:
155:
147:
143:
139:
135:
131:
127:
123:
119:
115:
107:
104:
98:
95:
52:electron volts
47:frozen orbital
15:
13:
10:
9:
6:
4:
3:
2:
2603:
2592:
2589:
2587:
2584:
2582:
2579:
2578:
2576:
2565:
2561:
2554:
2547:
2542:
2541:
2537:
2529:
2525:
2521:
2517:
2512:
2507:
2504:(7): 070902.
2503:
2499:
2495:
2488:
2485:
2480:
2476:
2472:
2468:
2463:
2458:
2455:(8): 080901.
2454:
2450:
2446:
2439:
2436:
2431:
2427:
2423:
2419:
2415:
2411:
2404:
2401:
2396:
2392:
2388:
2384:
2380:
2376:
2369:
2366:
2361:
2357:
2353:
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2341:
2336:
2331:
2327:
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2313:
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2304:
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2286:
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2275:
2271:
2267:
2263:
2259:
2255:
2251:
2247:
2242:
2237:
2233:
2229:
2222:
2218:
2213:
2208:
2204:
2200:
2196:
2192:
2185:
2182:
2170:
2166:
2162:
2157:
2152:
2148:
2144:
2140:
2136:
2132:
2128:
2121:
2114:
2112:
2108:
2103:
2099:
2095:
2091:
2087:
2083:
2076:
2073:
2067:
2062:
2058:
2054:
2050:
2046:
2042:
2035:
2032:
2027:
2023:
2019:
2015:
2011:
2007:
2003:
1999:
1992:
1989:
1984:
1980:
1976:
1972:
1969:(24): 16029.
1968:
1964:
1957:
1954:
1949:
1945:
1941:
1937:
1934:(19): 12042.
1933:
1929:
1922:
1919:
1914:
1910:
1906:
1902:
1898:
1894:
1887:
1884:
1879:
1875:
1871:
1867:
1863:
1859:
1852:
1850:
1848:
1844:
1839:
1835:
1831:
1827:
1823:
1819:
1815:
1811:
1806:
1801:
1797:
1793:
1786:
1783:
1778:
1774:
1770:
1766:
1762:
1758:
1754:
1750:
1743:
1740:
1735:
1731:
1727:
1723:
1719:
1715:
1711:
1707:
1700:
1697:
1685:
1681:
1677:
1673:
1669:
1665:
1661:
1657:
1653:
1647:
1644:
1639:
1635:
1631:
1627:
1623:
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1615:
1611:
1604:
1601:
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1590:
1586:
1581:
1580:
1571:
1568:
1563:
1559:
1555:
1551:
1547:
1543:
1539:
1535:
1531:
1524:
1521:
1516:
1512:
1508:
1504:
1500:
1496:
1489:
1486:
1481:
1475:
1471:
1466:
1465:
1456:
1453:
1448:
1442:
1438:
1430:
1427:
1422:
1418:
1414:
1410:
1406:
1402:
1395:
1392:
1387:
1381:
1377:
1370:
1368:
1364:
1359:
1355:
1351:
1347:
1343:
1339:
1335:
1331:
1327:
1323:
1316:
1313:
1308:
1304:
1300:
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1292:
1288:
1281:
1279:
1275:
1270:
1264:
1260:
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1239:
1235:
1228:
1225:
1220:
1214:
1210:
1202:
1199:
1194:
1190:
1186:
1182:
1175:
1173:
1169:
1164:
1160:
1156:
1152:
1148:
1144:
1137:
1135:
1131:
1126:
1122:
1118:
1116:0-486-69186-1
1112:
1108:
1107:
1102:
1101:Szabo, Attila
1096:
1093:
1088:
1084:
1080:
1076:
1072:
1068:
1061:
1058:
1052:
1050:
1035:
1032:
1029:
1009:
986:
982:
979:
976:
973:
970:
963:
960:
957:
954:
951:
943:
940:
937:
932:
921:
918:
915:
912:
909:
901:
896:
888:
883:
878:
872:
864:
860:
852:
851:
850:
836:
833:
830:
810:
807:
804:
784:
771:
769:
753:
749:
745:
737:
733:
724:
708:
704:
690:
687:
684:
679:
675:
653:
632:
615:
611:
603:
600:
596:
592:
589:
572:
514:
504:
501:
485:
482:
479:
476:
456:
450:
447:
427:
407:
404:
401:
398:
376:
372:
351:
331:
323:
319:
311:
309:
302:
300:
297:
295:
289:
287:
283:
279:
278:Fock operator
275:
271:
263:
261:
251:
242:
240:
211:
209:
153:
112:
105:
103:
102:in a system.
96:
94:
91:
90:HOMO and LUMO
87:
83:
78:
76:
71:
69:
65:
61:
60:Fock operator
57:
53:
48:
44:
39:
37:
33:
29:
25:
21:
2553:the original
2501:
2497:
2487:
2452:
2448:
2438:
2416:(20): 1884.
2413:
2409:
2403:
2381:(5ā6): 489.
2378:
2374:
2368:
2325:
2321:
2315:
2288:
2284:
2231:
2227:
2194:
2190:
2184:
2173:. Retrieved
2130:
2126:
2085:
2081:
2075:
2051:(2ā4): 391.
2048:
2044:
2034:
2001:
1997:
1991:
1966:
1962:
1956:
1931:
1927:
1921:
1896:
1892:
1886:
1861:
1857:
1795:
1791:
1785:
1752:
1748:
1742:
1709:
1705:
1699:
1688:. Retrieved
1663:
1659:
1646:
1613:
1609:
1603:
1578:
1570:
1537:
1533:
1523:
1498:
1494:
1488:
1463:
1455:
1436:
1429:
1404:
1400:
1394:
1375:
1325:
1321:
1315:
1290:
1286:
1258:
1252:
1233:
1227:
1208:
1201:
1184:
1180:
1149:(2): 83ā87.
1146:
1142:
1105:
1095:
1070:
1066:
1060:
1001:
775:
709:
705:
505:
499:
420:for integer
321:
315:
306:
298:
293:
290:
267:
243:
212:
113:
109:
100:
79:
72:
56:HartreeāFock
46:
40:
19:
18:
2212:11693/22326
2156:11693/11792
2088:(5): 2745.
233:15.9 and 1b
2575:Categories
2197:: 85ā109.
2175:2019-09-24
1690:2018-04-29
1407:(6): 501.
1053:References
2520:0021-9606
2471:0021-9606
2335:1208.1496
2241:1006.5420
1805:1008.1607
1562:0021-9606
1033:±
977:…
958:…
941:−
929:Ψ
916:…
893:Ψ
889:∫
861:ϕ
808:−
750:ε
730:∂
722:∂
688:−
676:ϵ
628:ℏ
601:−
593:
587:→
553:∞
550:→
483:δ
454:→
448:δ
405:δ
402:−
373:ϵ
250:basis set
221:559.5, 2a
75:ab initio
50:two
2528:32828082
2479:32872858
2360:22495102
2352:26291104
2307:26593646
2274:42592180
2266:21231698
2221:20055678
2169:Archived
2165:20025305
1838:31648260
1830:20949979
1777:20443582
1734:19459641
1684:Archived
1638:17144693
1515:92462090
1350:17279730
1163:96583645
1125:34357385
229:19.5, 3a
2418:Bibcode
2383:Bibcode
2246:Bibcode
2135:Bibcode
2090:Bibcode
2053:Bibcode
2026:9936207
2006:Bibcode
1971:Bibcode
1936:Bibcode
1901:Bibcode
1866:Bibcode
1810:Bibcode
1757:Bibcode
1714:Bibcode
1668:Bibcode
1618:Bibcode
1542:Bibcode
1409:Bibcode
1358:1516019
1330:Bibcode
1295:Bibcode
1075:Bibcode
1067:Physica
225:36.7 1b
2526:
2518:
2477:
2469:
2358:
2350:
2305:
2272:
2264:
2219:
2163:
2024:
1836:
1828:
1775:
1732:
1636:
1591:
1560:
1513:
1476:
1443:
1382:
1356:
1348:
1265:
1240:
1215:
1161:
1123:
1113:
2556:(PDF)
2549:(PDF)
2356:S2CID
2330:arXiv
2270:S2CID
2236:arXiv
2123:(PDF)
1834:S2CID
1800:arXiv
1587:ā26.
1511:S2CID
1472:ā65.
1354:S2CID
1159:S2CID
244:For N
237:13.8
213:For H
202:) (1b
198:) (3a
194:) (1b
190:) (2a
178:) (1b
174:) (3a
170:) (1b
166:) (2a
146:and b
134:) (1b
130:) (3a
126:) (1b
122:) (2a
2524:PMID
2516:ISSN
2475:PMID
2467:ISSN
2348:PMID
2303:PMID
2262:PMID
2217:PMID
2161:PMID
2022:PMID
1826:PMID
1773:PMID
1730:PMID
1634:PMID
1589:ISBN
1558:ISSN
1474:ISBN
1441:ISBN
1380:ISBN
1346:PMID
1263:ISBN
1238:ISBN
1213:ISBN
1121:OCLC
1111:ISBN
440:and
282:HOMO
274:LUMO
32:HOMO
2506:doi
2502:153
2457:doi
2453:153
2426:doi
2391:doi
2379:296
2340:doi
2293:doi
2254:doi
2232:105
2207:hdl
2199:doi
2151:hdl
2143:doi
2131:131
2098:doi
2061:doi
2049:288
2014:doi
1979:doi
1944:doi
1909:doi
1874:doi
1818:doi
1796:133
1765:doi
1753:114
1722:doi
1710:113
1676:doi
1664:132
1626:doi
1614:125
1550:doi
1503:doi
1417:doi
1338:doi
1326:111
1303:doi
1189:doi
1185:123
1151:doi
1083:doi
590:exp
142:, b
2577::
2522:.
2514:.
2500:.
2496:.
2473:.
2465:.
2451:.
2447:.
2424:.
2414:51
2412:.
2389:.
2377:.
2354:.
2346:.
2338:.
2324:.
2301:.
2287:.
2283:.
2268:.
2260:.
2252:.
2244:.
2230:.
2215:.
2205:.
2195:61
2193:.
2167:.
2159:.
2149:.
2141:.
2129:.
2125:.
2110:^
2096:.
2086:30
2084:.
2059:.
2047:.
2043:.
2020:.
2012:.
2002:31
2000:.
1977:.
1967:56
1965:.
1942:.
1932:56
1930:.
1907:.
1897:56
1895:.
1872:.
1862:49
1860:.
1846:^
1832:.
1824:.
1816:.
1808:.
1794:.
1771:.
1763:.
1751:.
1728:.
1720:.
1708:.
1682:.
1674:.
1662:.
1658:.
1632:.
1624:.
1612:.
1585:25
1556:.
1548:.
1538:69
1536:.
1532:.
1509:.
1499:40
1497:.
1470:64
1415:.
1405:72
1403:.
1366:^
1352:.
1344:.
1336:.
1324:.
1301:.
1291:18
1289:.
1277:^
1211:.
1183:.
1171:^
1157:.
1147:99
1145:.
1133:^
1119:.
1081:.
1069:.
768:.
703:.
294:on
239:eV
210:.
2530:.
2508::
2481:.
2459::
2432:.
2428::
2420::
2397:.
2393::
2385::
2362:.
2342::
2332::
2326:3
2309:.
2295::
2289:8
2276:.
2256::
2248::
2238::
2223:.
2209::
2201::
2178:.
2153::
2145::
2137::
2104:.
2100::
2092::
2069:.
2063::
2055::
2028:.
2016::
2008::
1985:.
1981::
1973::
1950:.
1946::
1938::
1915:.
1911::
1903::
1880:.
1876::
1868::
1840:.
1820::
1812::
1802::
1779:.
1767::
1759::
1736:.
1724::
1716::
1693:.
1678::
1670::
1640:.
1628::
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1195:.
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1071:1
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1030:N
1010:N
987:.
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902:N
897:I
884:N
879:=
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873:1
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834:+
831:N
811:1
805:N
785:N
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746:=
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144:2
140:1
136:1
132:1
128:2
124:1
120:1
116:2
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