2419:
1906:
2414:{\displaystyle {\begin{aligned}E_{\text{MP2}}&=2\sum _{i,j,a,b}{\frac {\langle \varphi _{i}\varphi _{j}|{\hat {\tilde {v}}}|\varphi _{a}\varphi _{b}\rangle \langle \varphi _{a}\varphi _{b}|{\hat {\tilde {v}}}|\varphi _{i}\varphi _{j}\rangle }{\varepsilon _{i}+\varepsilon _{j}-\varepsilon _{a}-\varepsilon _{b}}}-\sum _{i,j,a,b}{\frac {\langle \varphi _{i}\varphi _{j}|{\hat {\tilde {v}}}|\varphi _{a}\varphi _{b}\rangle \langle \varphi _{a}\varphi _{b}|{\hat {\tilde {v}}}|\varphi _{j}\varphi _{i}\rangle }{\varepsilon _{i}+\varepsilon _{j}-\varepsilon _{a}-\varepsilon _{b}}}\\\end{aligned}}}
2956:≠ 0. The solution of the zeroth-order MP equation is the sum of orbital energies. The zeroth plus first-order correction yields the Hartree–Fock energy. As with the original formulation, the first non-vanishing perturbation correction beyond the Hartree–Fock treatment is the second-order energy. To reiterate, the second- and higher-order corrections are the same in both formulations.
861:
2995:(ROHF). There are many ROHF based MP2-like methods because of arbitrariness in the ROHF wavefunction(for example HCPT, ROMP, RMP (also called ROHF-MBPT2), OPT1 and OPT2, ZAPT, IOPT, etc.). Some of the ROHF based MP2-like theories suffer from spin-contamination in their perturbed density and energies beyond second-order.
1751:
2681:
2964:
Second (MP2), third (MP3), and fourth (MP4) order Møller–Plesset calculations are standard levels used in calculating small systems and are implemented in many computational chemistry codes. Higher level MP calculations, generally only MP5, are possible in some codes. However, they are rarely used
1246:
2524:
Equivalent expressions are obtained by a slightly different partitioning of the
Hamiltonian, which results in a different division of energy terms over zeroth- and first-order contributions, while for second- and higher-order energy corrections the two partitionings give identical results. The
2968:
Systematic studies of MP perturbation theory have shown that it is not necessarily a convergent theory at high orders. Convergence can be slow, rapid, oscillatory, regular, highly erratic or simply non-existent, depending on the precise chemical system or basis set. The density matrix for the
2944:
634:
1412:
1552:
1055:
3245:
Raghavachari, Krishnan.; Pople, John A.; Replogle, Eric S.; Head-Gordon, Martin (1990). "Fifth order
Moeller-Plesset perturbation theory: comparison of existing correlation methods and implementation of new methods correct to fifth order".
2516:. The solution of the zeroth-order MP equation (which by definition is the Hartree–Fock equation) gives the Hartree–Fock energy. The first non-vanishing perturbation correction beyond the Hartree–Fock treatment is the second-order energy.
2531:
2791:
2977:. The eigenvalues of the response density matrix (which are the occupation numbers of the MP2 natural orbitals) can therefore be greater than 2 or negative. Unphysical numbers are a sign of a divergent perturbation expansion.
1878:
1887:
In order to obtain the MP2 formula for a closed-shell molecule, the second order RS-PT formula is written in a basis of doubly excited Slater determinants. (Singly excited Slater determinants do not contribute because of the
1070:
503:
3884:
Roos, Bjrn O; Andersson, Kerstin; Flscher, Markus P; Malmqvist, Per-ke; Serrano-Andrs, Luis; Pierloot, Kristin; Merchn, Manuela (1996). "Multiconfigurational
Perturbation Theory: Applications in Electronic Spectroscopy".
593:
2802:
856:{\displaystyle \left({\hat {H}}_{0}+\lambda V\right)\left(\sum _{i=0}^{m}\lambda ^{i}\Psi ^{(i)}\right)=\left(\sum _{i=0}^{m}\lambda ^{i}E^{(i)}\right)\left(\sum _{i=0}^{m}\lambda ^{i}\Psi ^{(i)}\right).}
388:
3527:
Hubač, Ivan; Čársky, Petr (1980). "Correlation energy of open-shell systems. Application of the many-body
Rayleigh-Schrödinger perturbation theory in the restricted Roothaan-Hartree-Fock formalism".
3082:
This article contains several minor, albeit annoying problems in the mathematics as published. For a concise derivation of MP perturbation theory to n order, see any good quantum mechanics textbook.
2496:
are the corresponding orbital energies. Clearly, through second-order in the correlation potential, the total electronic energy is given by the
Hartree–Fock energy plus second-order MP correction:
1911:
1265:
3343:
Wiberg, Kenneth B.; Hadad, Christopher M.; Lepage, Teresa J.; Breneman, Curt M.; Frisch, Michael J. (1992). "Analysis of the effect of electron correlation on charge density distributions".
1421:
is the number of electrons in the molecule under consideration (a factor of 2 in the energy arises from the fact that each orbital is occupied by a pair of electrons with opposite spin),
3632:
Lauderdale, Walter J.; Stanton, John F.; Gauss, Jürgen; Watts, John D.; Bartlett, Rodney J. (1991). "Many-body perturbation theory with a restricted open-shell
Hartree—Fock reference".
1746:{\displaystyle {\hat {F}}\Phi _{0}-\langle \Phi _{0}|{\hat {F}}|\Phi _{0}\rangle \Phi _{0}=0\implies {\hat {H}}_{0}\Phi _{0}=\langle \Phi _{0}|{\hat {H}}|\Phi _{0}\rangle \Phi _{0},}
937:
409:, which thus serves as the unperturbed operator. The perturbation is the correlation potential. In RS-PT the perturbed wave function and perturbed energy are expressed as a
284:
1490:
3006:(CASPT2), and Multi-Configuration Quasi-Degenerate Perturbation Theory (MCQDPT), have been developed. MCSCF based methods are not without perturbation series divergences.
891:
626:
1783:
1544:
1448:
313:
2676:{\displaystyle \langle \Phi _{0}|({\hat {H}}-{\hat {F}})|\Phi _{0}\rangle \neq 0\qquad \Longleftrightarrow \qquad E_{\text{HF}}\neq 2\sum _{i=1}^{N/2}\varepsilon _{i}.}
3030:
2696:
3959:
Granovsky, A. A. (2011). "Extended multi-configuration quasi-degenerate perturbation theory: The new approach to multi-state multi-reference perturbation theory".
3273:
Leininger, Matthew L.; Allen, Wesley D.; Schaeferd, Henry F.; Sherrill, C. David (2000). "Is Moller–Plesset perturbation theory a convergent ab initio method?".
2525:
formulation is commonly used by chemists, who are now large users of these methods. This difference is due to the fact, well known in
Hartree–Fock theory, that
179:
3597:
Knowles, Peter J.; Andrews, Jamie S.; Amos, Roger D.; Handy, Nicholas C.; Pople, John A. (1991). "Restricted Møller—Plesset theory for open-shell molecules".
1799:
2980:
Additionally, various important molecular properties calculated at MP3 and MP4 level are no better than their MP2 counterparts, even for small molecules.
1241:{\displaystyle {\hat {V}}\equiv {\hat {H}}-{\hat {H}}_{0}={\hat {H}}-\left({\hat {F}}+\langle \Phi _{0}|({\hat {H}}-{\hat {F}})|\Phi _{0}\rangle \right),}
419:
2987:
reference functions (since UHF states are not in general eigenvectors of the Fock operator). However, the resulting energies often suffer from severe
509:
3752:
Kozlowski, P. M.; Davidson, Ernest R. (1994). "Construction of open shell perturbation theory invariant with respect to orbital degeneracy".
924:
The MP-energy corrections are obtained from
Rayleigh–Schrödinger (RS) perturbation theory with the unperturbed Hamiltonian defined as the
3822:
Murray, Christopher; Davidson, Ernest R. (1992). "Different forms of perturbation theory for the calculation of the correlation energy".
2999:
2939:{\displaystyle E_{\text{MP0}}=2\sum _{i=1}^{N/2}\varepsilon _{i},\qquad E_{\text{MP1}}=E_{\text{HF}}-2\sum _{i=1}^{N/2}\varepsilon _{i}.}
79:
3002:(MCSCF) methods use several determinants and can be used for the unperturbed operator, although not uniquely, so many methods, such as
172:
3787:
Murray, Christopher W.; Handy, Nicholas C. (1992). "Comparison and assessment of different forms of open shell perturbation theory".
3181:
3143:
3003:
321:
101:
3562:
Amos, Roger D.; Andrews, Jamie S.; Handy, Nicholas C.; Knowles, Peter J. (1991). "Open-shell Møller—Plesset perturbation theory".
3218:
Krishnan, Raghavachari; Pople, John A. (1978). "Approximate fourth-order perturbation theory of the electron correlation energy".
3308:
Handy, Nicholas C.; Schaefer, Henry F. (1984). "On the evaluation of analytic energy derivatives for correlated wave functions".
3020:
908:
242:
222:
63:
4174:
2992:
113:
109:
4150:
4131:
4112:
4093:
4074:
4011:
3902:
3429:
165:
155:
1407:{\displaystyle {\hat {F}}\Phi _{0}\equiv \sum _{k=1}^{N}{\hat {f}}(k)\Phi _{0}=2\sum _{i=1}^{N/2}\varepsilon _{i}\Phi _{0}.}
3370:
Gordon, Mark S.; Schmidt, Michael W.; Chaban, Galina M.; Glaesemann, Kurt R.; Stevens, Walter J.; Gonzalez, Carlos (1999).
2998:
These methods, Hartree–Fock, unrestricted
Hartree–Fock and restricted Hartree–Fock use a single determinant wave function.
3490:"On the energy invariance of open-shell perturbation theory with respect to unitary transformations of molecular orbitals"
206:
83:
225:(RS-PT), usually to second (MP2), third (MP3) or fourth (MP4) order. Its main idea was published as early as 1934 by
4169:
247:
The MP perturbation theory is a special case of RS perturbation theory. In RS theory one considers an unperturbed
4027:
Glaesemann, Kurt R.; Gordon, Mark S.; Nakano, Haruyuki (1999). "A study of FeCO+ with correlated wavefunctions".
2949:
Obviously, with this alternative formulation, the Møller–Plesset theorem does not hold in the literal sense that
117:
4179:
3703:
2984:
3445:
Glaesemann, Kurt R.; Schmidt, Michael W. (2010). "On the
Ordering of Orbital Energies in High-Spin ROHF†".
3107:
Head-Gordon, Martin; Pople, John A.; Frisch, Michael J. (1988). "MP2 energy evaluation by direct methods".
1050:{\displaystyle {\hat {H}}_{0}\equiv {\hat {F}}+\langle \Phi _{0}|({\hat {H}}-{\hat {F}})|\Phi _{0}\rangle }
397:
is an arbitrary real parameter that controls the size of the perturbation. In MP theory the zeroth-order
3920:"Quasidegenerate perturbation theory with multiconfigurational self-consistent-field reference functions"
147:
2991:, leading to large errors. A possible better alternative is to use one of the MP2-like methods based on
1900:-electron matrix elements with Slater determinants in bra and ket and integrating out spin, it becomes
131:
45:
37:
143:
124:
94:
71:
253:
1457:
139:
52:
41:
869:
605:
1893:
214:
210:
59:
3201:
3163:
1759:
1520:
1424:
599:
289:
3667:
Murray, Christopher; Davidson, Ernest R. (1991). "Perturbation theory for open shell systems".
1451:
218:
105:
3372:"A natural orbital diagnostic for multiconfigurational character in correlated wave functions"
2786:{\displaystyle {\hat {H}}_{0}\equiv {\hat {F}},\qquad {\hat {V}}\equiv {\hat {H}}-{\hat {F}}.}
2690:
equal to the sum of occupied-orbital energies). In the alternative partitioning, one defines
3025:
203:
3704:"An open-shell restricted Hartree—Fock perturbation theory based on symmetric spin orbitals"
4036:
3968:
3931:
3849:
Fletcher, Graham D; Gordon, Mark S; Bell, Robert S (2002). "Gradient of the ZAPT2 energy".
3796:
3761:
3715:
3676:
3641:
3606:
3571:
3536:
3501:
3454:
3383:
3317:
3282:
3116:
3065:
3015:
248:
22:
8:
3050:
87:
30:
4040:
3972:
3935:
3800:
3765:
3719:
3680:
3645:
3610:
3575:
3540:
3505:
3458:
3387:
3321:
3286:
3120:
3069:
226:
3866:
3399:
3144:"Variational configuration interaction methods and comparison with perturbation theory"
2988:
1252:
3618:
3583:
3371:
4146:
4127:
4108:
4089:
4070:
4007:
3984:
3898:
3773:
3727:
3688:
3653:
3470:
3425:
3128:
1889:
1873:{\displaystyle E_{\text{MP1}}\equiv \langle \Phi _{0}|{\hat {V}}|\Phi _{0}\rangle =0}
230:
200:
3870:
4044:
3976:
3939:
3890:
3858:
3831:
3804:
3769:
3723:
3684:
3649:
3614:
3579:
3544:
3509:
3462:
3391:
3352:
3325:
3290:
3255:
3227:
3193:
3155:
3124:
3073:
151:
3403:
3417:
498:{\displaystyle \Psi =\lim _{m\to \infty }\sum _{i=0}^{m}\lambda ^{i}\Psi ^{(i)},}
75:
3894:
3862:
4163:
3548:
928:
406:
402:
398:
135:
3988:
3474:
3077:
410:
3835:
3231:
3197:
3159:
588:{\displaystyle E=\lim _{m\to \infty }\sum _{i=0}^{m}\lambda ^{i}E^{(i)}.}
3356:
3259:
1501:
is the orbital energy belonging to the doubly occupied spatial orbital
3980:
3466:
4048:
3944:
3919:
3808:
3513:
3395:
3329:
3294:
2983:
For open shell molecules, MPn-theory can directly be applied only to
3731:
3489:
3244:
4126:. Englewood Cliffs, New jersey: Prentice Hall. pp. 511–515.
3851:
Theoretical Chemistry Accounts: Theory, Computation, and Modeling
3488:
Crawford, T. Daniel; Schaefer, Henry F.; Lee, Timothy J. (1996).
3416:
2969:
first-order and higher MP2 wavefunction is of the type known as
4145:. Mineola, New York: Dover Publications, Inc. pp. 350–353.
3272:
3051:"Note on an Approximation Treatment for Many-Electron Systems"
1884:
Hence, the first meaningful correction appears at MP2 energy.
4002:
Davidson, Ernest R.; Jarzecki, A. A. (1999). K. Hirao (ed.).
3883:
3369:
2959:
383:{\displaystyle {\hat {H}}={\hat {H}}_{0}+\lambda {\hat {V}}.}
4069:. Chichester: John Wiley & Sons, Ltd. pp. 207–211.
3631:
236:
3180:
Pople, John A.; Binkley, J. Stephen; Seeger, Rolf (1976).
3031:
List of quantum chemistry and solid state physics software
1789:, the Hartree-Fock energy. Similarly, it can be seen that
3342:
1756:
i.e. the zeroth-order energy is the expectation value of
3596:
3561:
3182:"Theoretical models incorporating electron correlation"
598:
Substitution of these series into the time-independent
4026:
3106:
2460:
are virtual (or unoccupied) orbitals. The quantities
4086:
Exploring Chemistry with Electronic Structure Methods
2805:
2699:
2534:
1909:
1802:
1762:
1555:
1523:
1460:
1427:
1268:
1073:
940:
872:
637:
608:
512:
422:
324:
292:
256:
4140:
4083:
3487:
3141:
4088:. Pittsburgh, PA: Gaussian Inc. pp. 267–271.
3848:
3179:
2938:
2785:
2675:
2413:
1872:
1777:
1745:
1538:
1484:
1442:
1406:
1240:
1049:
885:
855:
620:
587:
497:
382:
307:
278:
3751:
3444:
286:, to which a small (often external) perturbation
4161:
4001:
520:
430:
3142:Pople, J. A.; Seeger, R.; Krishnan, R. (1977).
1259:is the lowest eigenstate of the Fock operator:
3821:
3666:
3049:Møller, Christian; Plesset, Milton S. (1934).
914:
3701:
3307:
3217:
3048:
173:
2601:
2535:
2350:
2278:
2275:
2203:
2112:
2040:
2037:
1965:
1861:
1816:
1727:
1682:
1626:
1581:
1227:
1161:
1044:
978:
3786:
3702:Lee, Timothy J.; Jayatilaka, Dylan (1993).
3526:
3000:Multi-configurational self-consistent field
80:Multi-configurational self-consistent field
3824:International Journal of Quantum Chemistry
3220:International Journal of Quantum Chemistry
3186:International Journal of Quantum Chemistry
3148:International Journal of Quantum Chemistry
2960:Use of Møller–Plesset perturbation methods
2519:
1649:
1645:
180:
166:
4084:Foresman, James B.; Æleen Frisch (1996).
4004:Recent Advances in Multireference Methods
3958:
3943:
3004:complete active space perturbation theory
221:effects by means of Rayleigh–Schrödinger
237:Rayleigh–Schrödinger perturbation theory
102:Time-dependent density functional theory
64:Semi-empirical quantum chemistry methods
4141:Szabo, Attila; Neil S. Ostlund (1996).
3021:Perturbation theory (quantum mechanics)
1492:is the one-electron Fock operator, and
919:
243:Perturbation theory (quantum mechanics)
4162:
4121:
4064:
3917:
3420:; Poul Jorgensen; Jeppe Olsen (2000).
897:th-order perturbation equation, where
114:Linearized augmented-plane-wave method
110:Orbital-free density functional theory
4102:
4067:Essentials of Computational Chemistry
3422:Molecular Electronic Structure Theory
16:Method in ab initio Quantum Chemistry
4006:. World Scientific. pp. 31–63.
3094:
2973:, which differs from the more usual
2442:are canonical occupied orbitals and
1060:and the perturbation defined as the
4107:. Harlow: Longman. pp. 83–85.
4029:Physical Chemistry Chemical Physics
3447:The Journal of Physical Chemistry A
84:Quantum chemistry composite methods
13:
4058:
2993:restricted open-shell Hartree–Fock
2592:
2539:
1852:
1820:
1731:
1718:
1686:
1670:
1630:
1617:
1585:
1569:
1392:
1337:
1282:
1218:
1165:
1035:
982:
830:
713:
615:
530:
477:
440:
423:
193:Møller–Plesset perturbation theory
68:Møller–Plesset perturbation theory
14:
4191:
3345:The Journal of Physical Chemistry
3248:The Journal of Physical Chemistry
3042:
4065:Cramer, Christopher J. (2002).
4020:
3995:
3952:
3924:The Journal of Chemical Physics
3911:
3877:
3842:
3815:
3789:The Journal of Chemical Physics
3780:
3745:
3695:
3660:
3625:
3590:
3555:
3520:
3494:The Journal of Chemical Physics
3481:
3438:
3410:
3363:
3336:
3310:The Journal of Chemical Physics
2864:
2796:Clearly, in this partitioning,
2737:
2614:
2610:
118:Projector augmented wave method
3301:
3266:
3238:
3211:
3173:
3135:
3100:
3087:
2774:
2759:
2744:
2728:
2707:
2611:
2587:
2583:
2577:
2562:
2553:
2549:
2326:
2319:
2314:
2302:
2251:
2244:
2239:
2227:
2088:
2081:
2076:
2064:
2013:
2006:
2001:
1989:
1847:
1840:
1830:
1769:
1713:
1706:
1696:
1657:
1646:
1612:
1605:
1595:
1562:
1530:
1513:Since the Slater determinant Φ
1479:
1473:
1467:
1434:
1333:
1327:
1321:
1275:
1213:
1209:
1203:
1188:
1179:
1175:
1152:
1132:
1111:
1095:
1080:
1030:
1026:
1020:
1005:
996:
992:
969:
948:
840:
834:
783:
777:
723:
717:
650:
612:
577:
571:
527:
487:
481:
437:
371:
347:
331:
299:
279:{\displaystyle {\hat {H}}_{0}}
264:
1:
3889:. Vol. 93. p. 219.
3619:10.1016/S0009-2614(91)85118-G
3584:10.1016/S0009-2614(91)85057-4
3036:
1892:). After application of the
1485:{\displaystyle {\hat {f}}(k)}
156:Korringa–Kohn–Rostoker method
3887:Advances in Chemical Physics
3774:10.1016/0009-2614(94)00763-2
3728:10.1016/0009-2614(93)85024-I
3689:10.1016/0009-2614(91)80281-2
3654:10.1016/0009-2614(91)90478-R
3129:10.1016/0009-2614(88)85250-3
2686:(The Hartree–Fock energy is
886:{\displaystyle \lambda ^{k}}
621:{\displaystyle m\to \infty }
7:
3009:
915:Møller–Plesset perturbation
148:Empty lattice approximation
10:
4196:
1896:for the simplification of
1778:{\displaystyle {\hat {H}}}
1546:, it follows readily that
1539:{\displaystyle {\hat {F}}}
1443:{\displaystyle {\hat {H}}}
308:{\displaystyle {\hat {V}}}
240:
132:Nearly free electron model
46:Modern valence bond theory
4175:Post-Hartree–Fock methods
4103:Leach, Andrew R. (1996).
3918:Nakano, Haruyuki (1993).
3895:10.1002/9780470141526.ch5
3863:10.1007/s00214-001-0304-z
2985:unrestricted Hartree–Fock
2975:expectation value density
893:in this equation gives a
125:Electronic band structure
95:Density functional theory
72:Configuration interaction
4143:Modern Quantum Chemistry
3754:Chemical Physics Letters
3710:(Submitted manuscript).
3708:Chemical Physics Letters
3669:Chemical Physics Letters
3634:Chemical Physics Letters
3599:Chemical Physics Letters
3564:Chemical Physics Letters
3549:10.1103/PhysRevA.22.2392
3109:Chemical Physics Letters
866:Equating the factors of
602:gives a new equation as
140:Muffin-tin approximation
53:Molecular orbital theory
42:Generalized valence bond
4170:Computational chemistry
4122:Levine, Ira N. (1991).
2965:because of their cost.
2520:Alternative formulation
211:computational chemistry
144:k·p perturbation theory
3093:See all volumes under
3078:10.1103/PhysRev.46.618
2940:
2922:
2850:
2787:
2677:
2659:
2415:
1874:
1779:
1747:
1540:
1486:
1452:electronic Hamiltonian
1444:
1408:
1380:
1314:
1242:
1051:
887:
857:
818:
761:
701:
622:
589:
555:
499:
465:
384:
309:
280:
38:Coulson–Fischer theory
4180:Theoretical chemistry
3836:10.1002/qua.560430604
3232:10.1002/qua.560140109
3198:10.1002/qua.560100802
3160:10.1002/qua.560120820
2941:
2894:
2822:
2788:
2678:
2631:
2416:
1875:
1780:
1748:
1541:
1487:
1445:
1409:
1352:
1294:
1251:where the normalized
1243:
1062:correlation potential
1052:
888:
858:
798:
741:
681:
623:
590:
535:
500:
445:
385:
310:
281:
213:. It improves on the
3016:Electron correlation
2803:
2697:
2532:
1907:
1800:
1760:
1553:
1521:
1517:is an eigenstate of
1458:
1425:
1266:
1071:
938:
920:Original formulation
870:
635:
606:
600:Schrödinger equation
510:
420:
322:
290:
254:
249:Hamiltonian operator
219:electron correlation
199:) is one of several
23:Electronic structure
4105:Molecular Modelling
4041:1999PCCP....1..967G
3973:2011JChPh.134u4113G
3936:1993JChPh..99.7983N
3801:1992JChPh..97.6509M
3766:1994CPL...226..440K
3720:1993CPL...201....1L
3681:1991CPL...187..451M
3646:1991CPL...187...21L
3611:1991CPL...186..130K
3576:1991CPL...185..256A
3541:1980PhRvA..22.2392H
3506:1996JChPh.105.1060C
3459:2010JPCA..114.8772G
3388:1999JChPh.110.4199G
3357:10.1021/j100181a030
3322:1984JChPh..81.5031H
3287:2000JChPh.112.9213L
3260:10.1021/j100377a033
3121:1988CPL...153..503H
3070:1934PhRv...46..618M
1894:Slater–Condon rules
1791:in this formulation
909:perturbation theory
223:perturbation theory
88:Quantum Monte Carlo
60:Hartree–Fock method
31:Valence bond theory
2989:spin contamination
2936:
2783:
2673:
2411:
2409:
2199:
1961:
1870:
1775:
1743:
1536:
1482:
1440:
1404:
1253:Slater determinant
1238:
1047:
911:for more details.
883:
853:
618:
585:
534:
495:
444:
380:
305:
276:
106:Thomas–Fermi model
4152:978-0-486-69186-2
4133:978-0-205-12770-2
4124:Quantum Chemistry
4114:978-0-582-23933-3
4095:978-0-9636769-4-8
4076:978-0-471-48552-0
4013:978-981-02-3777-6
3981:10.1063/1.3596699
3930:(10): 7983–7992.
3904:978-0-470-14152-6
3529:Physical Review A
3467:10.1021/jp101758y
3453:(33): 8772–8777.
3431:978-0-471-96755-2
3281:(21): 9213–9222.
3254:(14): 5579–5586.
3026:Post-Hartree–Fock
2885:
2872:
2813:
2777:
2762:
2747:
2731:
2710:
2622:
2580:
2565:
2405:
2322:
2317:
2247:
2242:
2172:
2167:
2084:
2079:
2009:
2004:
1934:
1921:
1890:Brillouin theorem
1843:
1810:
1785:with respect to Φ
1772:
1709:
1660:
1608:
1565:
1533:
1470:
1437:
1324:
1278:
1206:
1191:
1155:
1135:
1114:
1098:
1083:
1023:
1008:
972:
951:
653:
519:
429:
374:
350:
334:
302:
267:
231:Milton S. Plesset
217:method by adding
207:ab initio methods
204:post-Hartree–Fock
201:quantum chemistry
190:
189:
4187:
4156:
4137:
4118:
4099:
4080:
4053:
4052:
4049:10.1039/a808518h
4024:
4018:
4017:
3999:
3993:
3992:
3956:
3950:
3949:
3947:
3945:10.1063/1.465674
3915:
3909:
3908:
3881:
3875:
3874:
3846:
3840:
3839:
3819:
3813:
3812:
3809:10.1063/1.463680
3784:
3778:
3777:
3760:(5–6): 440–446.
3749:
3743:
3742:
3740:
3739:
3730:. Archived from
3699:
3693:
3692:
3664:
3658:
3657:
3629:
3623:
3622:
3605:(2–3): 130–136.
3594:
3588:
3587:
3570:(3–4): 256–264.
3559:
3553:
3552:
3535:(6): 2392–2399.
3524:
3518:
3517:
3514:10.1063/1.471951
3485:
3479:
3478:
3442:
3436:
3435:
3418:Helgaker, Trygve
3414:
3408:
3407:
3396:10.1063/1.478301
3382:(9): 4199–4207.
3367:
3361:
3360:
3340:
3334:
3333:
3330:10.1063/1.447489
3305:
3299:
3298:
3295:10.1063/1.481764
3270:
3264:
3263:
3242:
3236:
3235:
3215:
3209:
3208:
3206:
3200:. Archived from
3177:
3171:
3170:
3168:
3162:. Archived from
3154:(S11): 149–163.
3139:
3133:
3132:
3104:
3098:
3095:#Further reading
3091:
3085:
3084:
3055:
3046:
2971:response density
2945:
2943:
2942:
2937:
2932:
2931:
2921:
2917:
2908:
2887:
2886:
2883:
2874:
2873:
2870:
2860:
2859:
2849:
2845:
2836:
2815:
2814:
2811:
2792:
2790:
2789:
2784:
2779:
2778:
2770:
2764:
2763:
2755:
2749:
2748:
2740:
2733:
2732:
2724:
2718:
2717:
2712:
2711:
2703:
2682:
2680:
2679:
2674:
2669:
2668:
2658:
2654:
2645:
2624:
2623:
2620:
2600:
2599:
2590:
2582:
2581:
2573:
2567:
2566:
2558:
2552:
2547:
2546:
2515:
2420:
2418:
2417:
2412:
2410:
2406:
2404:
2403:
2402:
2390:
2389:
2377:
2376:
2364:
2363:
2353:
2349:
2348:
2339:
2338:
2329:
2324:
2323:
2318:
2310:
2308:
2305:
2300:
2299:
2290:
2289:
2274:
2273:
2264:
2263:
2254:
2249:
2248:
2243:
2235:
2233:
2230:
2225:
2224:
2215:
2214:
2201:
2198:
2168:
2166:
2165:
2164:
2152:
2151:
2139:
2138:
2126:
2125:
2115:
2111:
2110:
2101:
2100:
2091:
2086:
2085:
2080:
2072:
2070:
2067:
2062:
2061:
2052:
2051:
2036:
2035:
2026:
2025:
2016:
2011:
2010:
2005:
1997:
1995:
1992:
1987:
1986:
1977:
1976:
1963:
1960:
1923:
1922:
1919:
1879:
1877:
1876:
1871:
1860:
1859:
1850:
1845:
1844:
1836:
1833:
1828:
1827:
1812:
1811:
1808:
1784:
1782:
1781:
1776:
1774:
1773:
1765:
1752:
1750:
1749:
1744:
1739:
1738:
1726:
1725:
1716:
1711:
1710:
1702:
1699:
1694:
1693:
1678:
1677:
1668:
1667:
1662:
1661:
1653:
1638:
1637:
1625:
1624:
1615:
1610:
1609:
1601:
1598:
1593:
1592:
1577:
1576:
1567:
1566:
1558:
1545:
1543:
1542:
1537:
1535:
1534:
1526:
1491:
1489:
1488:
1483:
1472:
1471:
1463:
1449:
1447:
1446:
1441:
1439:
1438:
1430:
1413:
1411:
1410:
1405:
1400:
1399:
1390:
1389:
1379:
1375:
1366:
1345:
1344:
1326:
1325:
1317:
1313:
1308:
1290:
1289:
1280:
1279:
1271:
1247:
1245:
1244:
1239:
1234:
1230:
1226:
1225:
1216:
1208:
1207:
1199:
1193:
1192:
1184:
1178:
1173:
1172:
1157:
1156:
1148:
1137:
1136:
1128:
1122:
1121:
1116:
1115:
1107:
1100:
1099:
1091:
1085:
1084:
1076:
1056:
1054:
1053:
1048:
1043:
1042:
1033:
1025:
1024:
1016:
1010:
1009:
1001:
995:
990:
989:
974:
973:
965:
959:
958:
953:
952:
944:
906:
902:= 0, 1, 2, ...,
892:
890:
889:
884:
882:
881:
862:
860:
859:
854:
849:
845:
844:
843:
828:
827:
817:
812:
792:
788:
787:
786:
771:
770:
760:
755:
732:
728:
727:
726:
711:
710:
700:
695:
675:
671:
661:
660:
655:
654:
646:
627:
625:
624:
619:
594:
592:
591:
586:
581:
580:
565:
564:
554:
549:
533:
504:
502:
501:
496:
491:
490:
475:
474:
464:
459:
443:
389:
387:
386:
381:
376:
375:
367:
358:
357:
352:
351:
343:
336:
335:
327:
314:
312:
311:
306:
304:
303:
295:
285:
283:
282:
277:
275:
274:
269:
268:
260:
227:Christian Møller
209:in the field of
182:
175:
168:
152:GW approximation
19:
18:
4195:
4194:
4190:
4189:
4188:
4186:
4185:
4184:
4160:
4159:
4153:
4134:
4115:
4096:
4077:
4061:
4059:Further reading
4056:
4025:
4021:
4014:
4000:
3996:
3957:
3953:
3916:
3912:
3905:
3882:
3878:
3847:
3843:
3820:
3816:
3785:
3781:
3750:
3746:
3737:
3735:
3700:
3696:
3665:
3661:
3630:
3626:
3595:
3591:
3560:
3556:
3525:
3521:
3486:
3482:
3443:
3439:
3432:
3415:
3411:
3368:
3364:
3341:
3337:
3306:
3302:
3271:
3267:
3243:
3239:
3216:
3212:
3204:
3178:
3174:
3166:
3140:
3136:
3105:
3101:
3092:
3088:
3053:
3047:
3043:
3039:
3012:
2962:
2955:
2927:
2923:
2913:
2909:
2898:
2882:
2878:
2869:
2865:
2855:
2851:
2841:
2837:
2826:
2810:
2806:
2804:
2801:
2800:
2769:
2768:
2754:
2753:
2739:
2738:
2723:
2722:
2713:
2702:
2701:
2700:
2698:
2695:
2694:
2664:
2660:
2650:
2646:
2635:
2619:
2615:
2595:
2591:
2586:
2572:
2571:
2557:
2556:
2548:
2542:
2538:
2533:
2530:
2529:
2522:
2514:
2507:
2497:
2495:
2486:
2477:
2468:
2459:
2450:
2441:
2432:
2408:
2407:
2398:
2394:
2385:
2381:
2372:
2368:
2359:
2355:
2354:
2344:
2340:
2334:
2330:
2325:
2309:
2307:
2306:
2301:
2295:
2291:
2285:
2281:
2269:
2265:
2259:
2255:
2250:
2234:
2232:
2231:
2226:
2220:
2216:
2210:
2206:
2202:
2200:
2176:
2160:
2156:
2147:
2143:
2134:
2130:
2121:
2117:
2116:
2106:
2102:
2096:
2092:
2087:
2071:
2069:
2068:
2063:
2057:
2053:
2047:
2043:
2031:
2027:
2021:
2017:
2012:
1996:
1994:
1993:
1988:
1982:
1978:
1972:
1968:
1964:
1962:
1938:
1924:
1918:
1914:
1910:
1908:
1905:
1904:
1855:
1851:
1846:
1835:
1834:
1829:
1823:
1819:
1807:
1803:
1801:
1798:
1797:
1793:the MP1 energy
1788:
1764:
1763:
1761:
1758:
1757:
1734:
1730:
1721:
1717:
1712:
1701:
1700:
1695:
1689:
1685:
1673:
1669:
1663:
1652:
1651:
1650:
1633:
1629:
1620:
1616:
1611:
1600:
1599:
1594:
1588:
1584:
1572:
1568:
1557:
1556:
1554:
1551:
1550:
1525:
1524:
1522:
1519:
1518:
1516:
1509:
1500:
1462:
1461:
1459:
1456:
1455:
1429:
1428:
1426:
1423:
1422:
1395:
1391:
1385:
1381:
1371:
1367:
1356:
1340:
1336:
1316:
1315:
1309:
1298:
1285:
1281:
1270:
1269:
1267:
1264:
1263:
1258:
1221:
1217:
1212:
1198:
1197:
1183:
1182:
1174:
1168:
1164:
1147:
1146:
1145:
1141:
1127:
1126:
1117:
1106:
1105:
1104:
1090:
1089:
1075:
1074:
1072:
1069:
1068:
1038:
1034:
1029:
1015:
1014:
1000:
999:
991:
985:
981:
964:
963:
954:
943:
942:
941:
939:
936:
935:
922:
917:
898:
877:
873:
871:
868:
867:
833:
829:
823:
819:
813:
802:
797:
793:
776:
772:
766:
762:
756:
745:
740:
736:
716:
712:
706:
702:
696:
685:
680:
676:
656:
645:
644:
643:
642:
638:
636:
633:
632:
607:
604:
603:
570:
566:
560:
556:
550:
539:
523:
511:
508:
507:
480:
476:
470:
466:
460:
449:
433:
421:
418:
417:
366:
365:
353:
342:
341:
340:
326:
325:
323:
320:
319:
294:
293:
291:
288:
287:
270:
259:
258:
257:
255:
252:
251:
245:
239:
186:
154:
150:
146:
142:
138:
134:
116:
112:
108:
104:
86:
82:
78:
76:Coupled cluster
74:
70:
66:
62:
44:
40:
17:
12:
11:
5:
4193:
4183:
4182:
4177:
4172:
4158:
4157:
4151:
4138:
4132:
4119:
4113:
4100:
4094:
4081:
4075:
4060:
4057:
4055:
4054:
4035:(6): 967–975.
4019:
4012:
3994:
3967:(21): 214113.
3951:
3910:
3903:
3876:
3841:
3814:
3779:
3744:
3694:
3675:(5): 451–454.
3659:
3640:(1–2): 21–28.
3624:
3589:
3554:
3519:
3480:
3437:
3430:
3409:
3362:
3335:
3300:
3265:
3237:
3210:
3207:on 2012-10-20.
3172:
3169:on 2013-01-05.
3134:
3115:(6): 503–506.
3099:
3086:
3064:(7): 618–622.
3040:
3038:
3035:
3034:
3033:
3028:
3023:
3018:
3011:
3008:
2961:
2958:
2953:
2947:
2946:
2935:
2930:
2926:
2920:
2916:
2912:
2907:
2904:
2901:
2897:
2893:
2890:
2881:
2877:
2868:
2863:
2858:
2854:
2848:
2844:
2840:
2835:
2832:
2829:
2825:
2821:
2818:
2809:
2794:
2793:
2782:
2776:
2773:
2767:
2761:
2758:
2752:
2746:
2743:
2736:
2730:
2727:
2721:
2716:
2709:
2706:
2684:
2683:
2672:
2667:
2663:
2657:
2653:
2649:
2644:
2641:
2638:
2634:
2630:
2627:
2618:
2613:
2609:
2606:
2603:
2598:
2594:
2589:
2585:
2579:
2576:
2570:
2564:
2561:
2555:
2551:
2545:
2541:
2537:
2521:
2518:
2512:
2505:
2491:
2482:
2473:
2464:
2455:
2446:
2437:
2428:
2422:
2421:
2401:
2397:
2393:
2388:
2384:
2380:
2375:
2371:
2367:
2362:
2358:
2352:
2347:
2343:
2337:
2333:
2328:
2321:
2316:
2313:
2304:
2298:
2294:
2288:
2284:
2280:
2277:
2272:
2268:
2262:
2258:
2253:
2246:
2241:
2238:
2229:
2223:
2219:
2213:
2209:
2205:
2197:
2194:
2191:
2188:
2185:
2182:
2179:
2175:
2171:
2163:
2159:
2155:
2150:
2146:
2142:
2137:
2133:
2129:
2124:
2120:
2114:
2109:
2105:
2099:
2095:
2090:
2083:
2078:
2075:
2066:
2060:
2056:
2050:
2046:
2042:
2039:
2034:
2030:
2024:
2020:
2015:
2008:
2003:
2000:
1991:
1985:
1981:
1975:
1971:
1967:
1959:
1956:
1953:
1950:
1947:
1944:
1941:
1937:
1933:
1930:
1927:
1925:
1917:
1913:
1912:
1882:
1881:
1869:
1866:
1863:
1858:
1854:
1849:
1842:
1839:
1832:
1826:
1822:
1818:
1815:
1806:
1786:
1771:
1768:
1754:
1753:
1742:
1737:
1733:
1729:
1724:
1720:
1715:
1708:
1705:
1698:
1692:
1688:
1684:
1681:
1676:
1672:
1666:
1659:
1656:
1648:
1644:
1641:
1636:
1632:
1628:
1623:
1619:
1614:
1607:
1604:
1597:
1591:
1587:
1583:
1580:
1575:
1571:
1564:
1561:
1532:
1529:
1514:
1505:
1496:
1481:
1478:
1475:
1469:
1466:
1436:
1433:
1415:
1414:
1403:
1398:
1394:
1388:
1384:
1378:
1374:
1370:
1365:
1362:
1359:
1355:
1351:
1348:
1343:
1339:
1335:
1332:
1329:
1323:
1320:
1312:
1307:
1304:
1301:
1297:
1293:
1288:
1284:
1277:
1274:
1256:
1249:
1248:
1237:
1233:
1229:
1224:
1220:
1215:
1211:
1205:
1202:
1196:
1190:
1187:
1181:
1177:
1171:
1167:
1163:
1160:
1154:
1151:
1144:
1140:
1134:
1131:
1125:
1120:
1113:
1110:
1103:
1097:
1094:
1088:
1082:
1079:
1058:
1057:
1046:
1041:
1037:
1032:
1028:
1022:
1019:
1013:
1007:
1004:
998:
994:
988:
984:
980:
977:
971:
968:
962:
957:
950:
947:
921:
918:
916:
913:
880:
876:
864:
863:
852:
848:
842:
839:
836:
832:
826:
822:
816:
811:
808:
805:
801:
796:
791:
785:
782:
779:
775:
769:
765:
759:
754:
751:
748:
744:
739:
735:
731:
725:
722:
719:
715:
709:
705:
699:
694:
691:
688:
684:
679:
674:
670:
667:
664:
659:
652:
649:
641:
617:
614:
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596:
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579:
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526:
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515:
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489:
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483:
479:
473:
469:
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432:
428:
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390:
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373:
370:
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361:
356:
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346:
339:
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330:
301:
298:
273:
266:
263:
238:
235:
188:
187:
185:
184:
177:
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162:
159:
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121:
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3962:
3961:J. Chem. Phys
3955:
3946:
3941:
3937:
3933:
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3921:
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3892:
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3767:
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3734:on 2018-11-04
3733:
3729:
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3717:
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3713:
3709:
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1431:
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416:
415:
414:
412:
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407:Fock operator
404:
403:eigenfunction
400:
399:wave function
396:
377:
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359:
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318:
317:
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3747:
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411:power series
401:is an exact
394:
392:
246:
215:Hartree–Fock
196:
192:
191:
67:
3795:(9): 6509.
3500:(3): 1060.
4164:Categories
3830:(6): 755.
3738:2018-11-04
3351:(2): 671.
3205:(abstract)
3167:(abstract)
3037:References
315:is added:
241:See also:
3857:(2): 57.
3424:. Wiley.
3058:Phys. Rev
2925:ε
2896:∑
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2605:≠
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3871:95857722
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3010:See also
4037:Bibcode
3969:Bibcode
3932:Bibcode
3797:Bibcode
3762:Bibcode
3716:Bibcode
3677:Bibcode
3642:Bibcode
3607:Bibcode
3572:Bibcode
3537:Bibcode
3502:Bibcode
3455:Bibcode
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3066:Bibcode
926:shifted
405:of the
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413:in λ:
393:Here,
3867:S2CID
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1417:Here
4147:ISBN
4128:ISBN
4109:ISBN
4090:ISBN
4071:ISBN
4008:ISBN
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2451:and
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229:and
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1603:F
1596:|
1590:0
1574:0
1560:F
1528:F
1515:0
1507:i
1503:φ
1498:i
1494:ε
1480:)
1477:k
1474:(
1465:f
1432:H
1419:N
1402:.
1397:0
1387:i
1377:2
1373:/
1369:N
1364:1
1361:=
1358:i
1350:2
1347:=
1342:0
1334:)
1331:k
1328:(
1319:f
1311:N
1306:1
1303:=
1300:k
1287:0
1273:F
1257:0
1255:Φ
1236:,
1232:)
1223:0
1214:|
1210:)
1201:F
1186:H
1180:(
1176:|
1170:0
1159:+
1150:F
1143:(
1130:H
1124:=
1119:0
1109:H
1093:H
1078:V
1040:0
1031:|
1027:)
1018:F
1003:H
997:(
993:|
987:0
976:+
967:F
956:0
946:H
904:m
900:k
895:k
879:k
851:.
847:)
841:)
838:i
835:(
825:i
815:m
810:0
807:=
804:i
795:(
790:)
784:)
781:i
778:(
774:E
768:i
758:m
753:0
750:=
747:i
738:(
734:=
730:)
724:)
721:i
718:(
708:i
698:m
693:0
690:=
687:i
678:(
673:)
669:V
663:+
658:0
648:H
640:(
610:m
583:.
578:)
575:i
572:(
568:E
562:i
552:m
547:0
544:=
541:i
525:m
517:=
514:E
493:,
488:)
485:i
482:(
472:i
462:m
457:0
454:=
451:i
435:m
427:=
395:λ
378:.
369:V
360:+
355:0
345:H
338:=
329:H
297:V
272:0
262:H
195:(
181:e
174:t
167:v
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