4401:
3741:
4396:{\displaystyle \psi (\mathbf {r} _{1},s_{1},...,\mathbf {r} _{Z},s_{Z})={\frac {1}{\sqrt {Z!}}}det{\begin{bmatrix}\phi _{n_{1}}(\mathbf {r} _{1},s_{1})&\phi _{n_{1}}(\mathbf {r} _{2},s_{2})&...&\phi _{n_{1}}(\mathbf {r} _{Z},s_{Z})\\\phi _{n_{2}}(\mathbf {r} _{1},s_{1})&\phi _{n_{2}}(\mathbf {r} _{2},s_{2})&...&\phi _{n_{2}}(\mathbf {r} _{Z},s_{Z})\\...&...&...&...\\\phi _{n_{Z}}(\mathbf {r} _{1},s_{1})&\phi _{n_{Z}}(\mathbf {r} _{2},s_{2})&...&\phi _{n_{Z}}(\mathbf {r} _{Z},s_{Z})\end{bmatrix}}}
2380:
3626:
1582:
22:
4673:
1275:
1840:
2106:
3321:
1322:
4416:
3139:
From the convergence of the potential we can say that we have a "self consistent" mean field, i.e. a continuous variation from a known potential with known solutions to an averaged mean field potential. In that sense the potential is consistent and not so different from the originally used one as
1884:
At the time of
Hartree the full Pauli exclusion principle was not yet invented, it was only clear the exclusion principle in terms of quantum numbers but it was not clear that the wave function of electrons shall be anti-symmetric. If we start from the assumption that the wave functions of each
1061:
1029:
2375:{\displaystyle V(\mathbf {r} )={\frac {1}{4\pi \epsilon _{0}}}\int {\frac {\rho (\mathbf {r'} )}{|\mathbf {r} -\mathbf {r'} |}}d\mathbf {r'} =-{\frac {e}{4\pi \epsilon _{0}}}\sum _{i\neq j}\int {\frac {|\phi _{n_{j}}(\mathbf {r'} )|^{2}}{|\mathbf {r} -\mathbf {r'} |}}d\mathbf {r'} }
1596:
3621:{\displaystyle \delta \left(\langle \prod _{i}\phi _{n_{i}}(\mathbf {r} _{i},s_{i})|{\hat {H}}|\prod _{i}\phi _{n_{i}}(\mathbf {r} _{i},s_{i})\rangle -\sum _{i}\epsilon _{i}\langle \phi _{n_{i}}(\mathbf {r} _{i},s_{i})|\phi _{n_{i}}(\mathbf {r} _{i},s_{i})\rangle \right)=0}
2688:
3310:
513:
1577:{\displaystyle {\hat {H}}=-{\frac {\hbar ^{2}}{2m}}\sum _{i}{\nabla ^{2}}_{\mathbf {r} _{i}}-\sum _{i}{\frac {Ze^{2}}{4\pi \epsilon _{0}|\mathbf {r} _{i}|}}+{\frac {1}{2}}\sum _{i\neq j}{\frac {e^{2}}{4\pi \epsilon _{0}|\mathbf {r} _{i}-\mathbf {r} _{j}|}}}
4836:
4406:
This determinant guarantees the exchange symmetry (i.e. if the two columns are swapped the determinant change sign) and the Pauli principle if two electronic states are identical there are two identical rows and therefore the determinant is zero.
4668:{\displaystyle \delta \left(\langle \psi (\mathbf {r} _{i},s_{i})|{\hat {H}}|\psi (\mathbf {r} _{i},s_{i})\rangle -\sum _{i}\epsilon _{i}\langle \phi _{n_{i}}(\mathbf {r} _{i},s_{i})|\phi _{n_{i}}(\mathbf {r} _{i},s_{i})\rangle \right)=0}
2095:
2009:
1270:{\displaystyle \Psi (\mathbf {x} _{1},\mathbf {x} _{2},\mathbf {x} _{3},...,\mathbf {x} _{p})=\psi _{\alpha }(\mathbf {x} _{1})\psi _{\beta }(\mathbf {x} _{2})\psi _{\gamma }(\mathbf {x} _{3})...\psi _{\pi }(\mathbf {x} _{p})}
1284:
gives us the wavefunction of a system (many-particle) as a combination of wavefunctions of the individual particles. It is inherently mean-field (assumes the particles are independent) and is the unsymmetrized version of the
884:
1835:{\displaystyle \langle \psi |{\hat {H}}|\psi \rangle =\int \psi ^{*}(\mathbf {r} _{1},s_{1},...,\mathbf {r} _{Z},s_{Z}){\hat {H}}\psi (\mathbf {r} _{1},s_{1},...,\mathbf {r} _{Z},s_{Z})\prod _{i}d\mathbf {r} _{i}}
3091:
2567:
2772:
2578:
1051:
is a solution to the Schrödinger equation by itself, their product should at least approximate a solution. This simple method of combining the wavefunctions of the individual electrons is known as the
351:
3711:
600:
3158:
359:
879:
722:
1876:
which is also unknown and needs to be found together with the eigenfunctions of the problem. We will also neglect all relativistic effects like spin-orbit and spin-spin interactions.
4715:
187:, derived from the field. Self-consistency required that the final field, computed from the solutions, was self-consistent with the initial field, and he thus called his method the
2728:
3099:
Namely, we start from a set of known eigenfunctions (which in this simplified mono-atomic example can be the ones of the hydrogen atom) and starting initially from the potential
801:, is almost always too complex to calculate directly. Hartree's original method was to first calculate the solutions to Schrödinger's equation for individual electrons 1, 2, 3,
638:
3134:
4710:
3656:
3136:
computing at each iteration a new version of the potential from the charge density above and then a new version of the eigen-functions, ideally these iterations converge.
1905:
760:
250:
1870:
1049:
799:
274:
185:
825:
1885:
electron are independent we can assume that the total wave function is the product of the single wave functions and that the total charge density at position
2023:
5008:
Hartree, D. R. (1928). "The Wave
Mechanics of an Atom with a non-Coulomb Central Field. Part III. Term Values and Intensities in Series in Optical Spectra".
1913:
2572:
This is interesting on its own because it can be compared with a single particle problem in a continuous medium where the dielectric constant is given by:
4838:
from which the wave function is built. The orthogonal conditions acts as constraints in the scope of the lagrange multipliers. From this they derived the
1300:, such as electrons, because the resulting wave function is not antisymmetric. An antisymmetric wave function can be mathematically described using the
3096:
This is a non linear system of integro-differential equations, but it is interesting in a computational setting because we can solve them iteratively.
1312:
Let's start from a
Hamiltonian of one atom with Z electrons. The same method with some modifications can be expanded to a monoatomic crystal using the
1024:{\displaystyle \psi _{\alpha }(\mathbf {x} _{1}),\psi _{\beta }(\mathbf {x} _{2}),\psi _{\gamma }(\mathbf {x} _{3}),...,\psi _{\pi }(\mathbf {x} _{p})}
3713:. The orthogonal conditions acts as constraints in the scope of the lagrange multipliers. From here they managed to derive the Hartree equations.
5088:
1313:
5093:
2782:
2391:
766:
105:
39:
2683:{\displaystyle \varepsilon (\mathbf {r} )={\frac {\epsilon _{0}}{1+{\frac {4\pi \epsilon _{0}}{Ze}}|\mathbf {r} |V(\mathbf {r} )}}}
5108:
2733:
4957:
Hartree, D. R. (1928). "The Wave
Mechanics of an Atom with a Non-Coulomb Central Field. Part II. Some Results and Discussion".
86:
5103:
279:
58:
43:
3665:
3305:{\displaystyle \psi (\mathbf {r} _{1},s_{1},...,\mathbf {r} _{Z},s_{Z})=\prod _{i}^{Z}\phi _{n_{i}}(\mathbf {r} _{i},s_{i})}
537:
508:{\displaystyle {\frac {\mathrm {d} ^{2}P(r)}{\mathrm {d} r^{2}}}+\left\{2-{\frac {\ell (\ell +1)}{r^{2}}}\right\}P(r)=0.}
65:
770:
834:
5098:
72:
4831:{\displaystyle \langle \phi _{n_{i}}(\mathbf {r} ,s_{i})|\phi _{n_{j}}(\mathbf {r} ,s_{j})\rangle =\delta _{ij}}
646:
4839:
1293:
54:
119:
32:
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3152:
In 1928 J. C. Slater and J. A. Gaunt independently showed that given the
Hartree product approximation:
608:
151:
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5017:
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4915:
4868:
3659:
212:
208:
135:
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independently then used the Slater determinant instead of the
Hartree product for the wave function
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3634:
2385:
If we now consider the electron i this will also satisfy the time independent Schrödinger equation
155:
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In order to solve the equation of an electron in a spherical potential, Hartree first introduced
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4931:
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79:
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of each electron was the solution of the Schrödinger equation for an electron in a potential
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4974:
4923:
4904:"The Wave Mechanics of an Atom with a Non-Coulomb Central Field. Part I. Theory and Methods"
4876:
218:
1872:
are the spins of the different particles. In general we approximate this potential with a
1848:
1296:. Although it has the advantage of simplicity, the Hartree product is not satisfactory for
1034:
784:
259:
528:
161:
123:
804:
5021:
4970:
4919:
4872:
2090:{\displaystyle \nabla ^{2}V(\mathbf {r} )=-{\frac {\rho (\mathbf {r} )}{\epsilon _{0}}}}
3732:
147:
2004:{\displaystyle \rho (\mathbf {r} )=-e\sum _{i\neq j}|\phi _{n_{j}}(\mathbf {r} )|^{2}}
5082:
4994:
4943:
3728:
5045:
200:
143:
21:
5029:
4978:
4927:
1873:
5037:
4986:
4935:
4903:
4888:
204:
4859:
Lindsay, Robert Bruce (1924). "On the Atomic Models of the Alkali Metals".
4880:
1297:
139:
3141:
1289:
3086:{\displaystyle \left\phi _{n_{i}}=\mathrm {E} _{i}\phi _{n_{i}}}
2562:{\displaystyle \left\phi _{n_{i}}=\mathrm {E} _{i}\phi _{n_{i}}}
5068:
5010:
Mathematical
Proceedings of the Cambridge Philosophical Society
4959:
Mathematical
Proceedings of the Cambridge Philosophical Society
4908:
Mathematical
Proceedings of the Cambridge Philosophical Society
3662:
needed in order to minimize the functional of the mean energy
15:
215:
to show that the solution was a product of a radial function
2767:{\displaystyle \varepsilon (\mathbf {r} )>\epsilon _{0}}
4410:
They then applied the same variational condition as above
203:
to eliminate physical constants. Then he converted the
781:
The wavefunction which describes all of the electrons,
346:{\displaystyle \psi =(1/r)P(r)S_{\ell }(\theta ,\phi )}
3851:
3706:{\displaystyle \langle \psi |{\hat {H}}|\psi \rangle }
3315:
They started from the following variational condition
4718:
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4419:
3744:
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3637:
3324:
3161:
3105:
2785:
2736:
2699:
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2394:
2109:
2026:
2017:
This charge density creates an extra mean potential:
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837:
807:
787:
736:
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362:
282:
262:
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2100:
The solution can be written as the
Coulomb integral
595:{\displaystyle i\,\partial _{t}u+\nabla ^{2}u=V(u)u}
46:. Unsourced material may be challenged and removed.
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2776:Finally, we have the system of Hartree equations
2014:Where we neglected the spin here for simplicity.
5016:(3). Cambridge University Press (CUP): 426–437.
4965:(1). Cambridge University Press (CUP): 111–132.
4712:are a generic orthogonal set of eigen-functions
4914:(1). Cambridge University Press (CUP): 89–110.
874:{\displaystyle \alpha ,\beta ,\gamma ,...,\pi }
118:In 1927, a year after the publication of the
8:
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4547:
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353:. The equation for the radial function was
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717:{\displaystyle V(u)=\pm |x|^{-n}*|u|^{2}}
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106:Learn how and when to remove this message
4851:
2797:
2406:
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881:, which yields individual solutions:
150:together with the electrons formed a
126:formulated what are now known as the
7:
5061:The Calculation of Atomic Structures
3717:Fock and Slater determinant approach
2723:{\displaystyle V(\mathbf {r} )<0}
138:had introduced in his study of many
44:adding citations to reliable sources
5063:. New York: John Wiley & Sons.
4861:Journal of Mathematics and Physics
3056:
2801:
2532:
2410:
2028:
1587:The expectation value is given by
1379:
1314:Born–von Karman boundary condition
1065:
788:
633:{\displaystyle \mathbb {R} ^{d+1}}
562:
546:
393:
368:
14:
3129:{\displaystyle V(\mathbf {r} )=0}
1907:due to all electrons except i is
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1103:
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1073:
1008:
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934:
903:
130:for atoms, using the concept of
20:
1316:and to a crystal with a basis.
767:non-linear Schrödinger equation
519:Hartree equation in mathematics
256:with an angular quantum number
31:needs additional citations for
5089:Partial differential equations
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4785:
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1:
4705:{\displaystyle \phi _{n_{i}}}
3651:{\displaystyle \epsilon _{i}}
5094:Electronic structure methods
5059:Hartree, Douglas R. (1957).
1900:{\displaystyle \mathbf {r} }
755:{\displaystyle 0<n<d}
146:. Hartree assumed that the
5125:
3720:
142:systems in the context of
5030:10.1017/s0305004100015954
4979:10.1017/s0305004100011920
4928:10.1017/s0305004100011919
5109:Computational chemistry
4902:Hartree, D. R. (1928).
3148:Slater–Gaunt derivation
4832:
4706:
4669:
4397:
3707:
3652:
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3253:
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2005:
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1271:
1045:
1025:
875:
821:
795:
756:
718:
634:
596:
509:
347:
270:
246:
245:{\displaystyle P(r)/r}
181:
5104:Theoretical chemistry
4881:10.1002/sapm192434191
4867:(4). Wiley: 191–236.
4833:
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2377:
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2006:
1902:
1867:
1865:{\displaystyle s_{i}}
1837:
1579:
1272:
1046:
1044:{\displaystyle \psi }
1026:
876:
822:
796:
794:{\displaystyle \Psi }
757:
719:
635:
597:
510:
348:
271:
269:{\displaystyle \ell }
247:
213:spherical coordinates
189:self-consistent field
182:
152:spherically symmetric
4716:
4682:
4417:
3742:
3666:
3660:Lagrange multipliers
3635:
3322:
3159:
3103:
2783:
2734:
2697:
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1914:
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1323:
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885:
835:
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785:
734:
647:
609:
538:
523:In mathematics, the
360:
280:
260:
219:
180:{\displaystyle v(r)}
162:
120:Schrödinger equation
40:improve this article
5022:1928PCPS...24..426H
4971:1928PCPS...24..111H
4920:1928PCPS...24...89H
4873:1924PhDT.........3L
4840:Hartree–Fock method
1294:Hartree–Fock method
820:{\displaystyle ...}
769:is in some sense a
156:charge distribution
4828:
4702:
4665:
4536:
4393:
4387:
3723:Slater determinant
3703:
3648:
3618:
3489:
3425:
3345:
3302:
3126:
3083:
2926:
2764:
2720:
2680:
2559:
2372:
2269:
2087:
2001:
1955:
1897:
1880:Hartree derivation
1862:
1832:
1816:
1574:
1504:
1416:
1375:
1302:Slater determinant
1287:Slater determinant
1267:
1041:
1021:
871:
817:
791:
752:
714:
630:
592:
505:
343:
266:
254:spherical harmonic
242:
177:
55:"Hartree equation"
5099:Quantum chemistry
4527:
4479:
3835:
3834:
3689:
3480:
3416:
3408:
3336:
3014:
2911:
2909:
2874:
2819:
2678:
2646:
2483:
2428:
2357:
2254:
2252:
2208:
2150:
2085:
1940:
1807:
1730:
1620:
1572:
1489:
1487:
1474:
1407:
1366:
1364:
1335:
480:
408:
128:Hartree equations
116:
115:
108:
90:
5116:
5073:
5072:
5056:
5050:
5049:
5005:
4999:
4998:
4954:
4948:
4947:
4899:
4893:
4892:
4856:
4837:
4835:
4834:
4829:
4827:
4826:
4805:
4804:
4792:
4784:
4783:
4782:
4781:
4767:
4759:
4758:
4746:
4738:
4737:
4736:
4735:
4711:
4709:
4708:
4703:
4701:
4700:
4699:
4698:
4674:
4672:
4671:
4666:
4658:
4654:
4647:
4646:
4634:
4633:
4628:
4619:
4618:
4617:
4616:
4602:
4594:
4593:
4581:
4580:
4575:
4566:
4565:
4564:
4563:
4546:
4545:
4535:
4517:
4516:
4504:
4503:
4498:
4486:
4481:
4480:
4472:
4469:
4461:
4460:
4448:
4447:
4442:
4402:
4400:
4399:
4394:
4392:
4391:
4381:
4380:
4368:
4367:
4362:
4353:
4352:
4351:
4350:
4320:
4319:
4307:
4306:
4301:
4292:
4291:
4290:
4289:
4270:
4269:
4257:
4256:
4251:
4242:
4241:
4240:
4239:
4172:
4171:
4159:
4158:
4153:
4144:
4143:
4142:
4141:
4111:
4110:
4098:
4097:
4092:
4083:
4082:
4081:
4080:
4061:
4060:
4048:
4047:
4042:
4033:
4032:
4031:
4030:
4009:
4008:
3996:
3995:
3990:
3981:
3980:
3979:
3978:
3948:
3947:
3935:
3934:
3929:
3920:
3919:
3918:
3917:
3898:
3897:
3885:
3884:
3879:
3870:
3869:
3868:
3867:
3836:
3827:
3823:
3815:
3814:
3802:
3801:
3796:
3775:
3774:
3762:
3761:
3756:
3712:
3710:
3709:
3704:
3696:
3691:
3690:
3682:
3679:
3657:
3655:
3654:
3649:
3647:
3646:
3627:
3625:
3624:
3619:
3611:
3607:
3600:
3599:
3587:
3586:
3581:
3572:
3571:
3570:
3569:
3555:
3547:
3546:
3534:
3533:
3528:
3519:
3518:
3517:
3516:
3499:
3498:
3488:
3470:
3469:
3457:
3456:
3451:
3442:
3441:
3440:
3439:
3424:
3415:
3410:
3409:
3401:
3398:
3390:
3389:
3377:
3376:
3371:
3362:
3361:
3360:
3359:
3344:
3311:
3309:
3308:
3303:
3298:
3297:
3285:
3284:
3279:
3270:
3269:
3268:
3267:
3252:
3247:
3232:
3231:
3219:
3218:
3213:
3192:
3191:
3179:
3178:
3173:
3135:
3133:
3132:
3127:
3116:
3092:
3090:
3089:
3084:
3082:
3081:
3080:
3079:
3065:
3064:
3059:
3050:
3049:
3048:
3047:
3033:
3029:
3028:
3027:
3015:
3013:
3012:
3007:
3006:
2994:
2989:
2983:
2982:
2981:
2976:
2967:
2966:
2954:
2953:
2952:
2951:
2937:
2931:
2925:
2910:
2908:
2907:
2906:
2890:
2889:
2880:
2875:
2873:
2872:
2867:
2862:
2857:
2856:
2840:
2839:
2838:
2825:
2820:
2818:
2810:
2809:
2808:
2795:
2773:
2771:
2770:
2765:
2763:
2762:
2747:
2729:
2727:
2726:
2721:
2710:
2689:
2687:
2686:
2681:
2679:
2677:
2673:
2662:
2657:
2652:
2647:
2645:
2637:
2636:
2635:
2619:
2610:
2609:
2600:
2592:
2568:
2566:
2565:
2560:
2558:
2557:
2556:
2555:
2541:
2540:
2535:
2526:
2525:
2524:
2523:
2509:
2505:
2501:
2484:
2482:
2481:
2476:
2471:
2466:
2465:
2449:
2448:
2447:
2434:
2429:
2427:
2419:
2418:
2417:
2404:
2381:
2379:
2378:
2373:
2371:
2370:
2358:
2356:
2355:
2350:
2349:
2337:
2332:
2326:
2325:
2324:
2319:
2310:
2309:
2297:
2296:
2295:
2294:
2280:
2274:
2268:
2253:
2251:
2250:
2249:
2230:
2222:
2221:
2209:
2207:
2206:
2201:
2200:
2188:
2183:
2177:
2173:
2172:
2156:
2151:
2149:
2148:
2147:
2128:
2120:
2096:
2094:
2093:
2088:
2086:
2084:
2083:
2074:
2070:
2058:
2047:
2036:
2035:
2010:
2008:
2007:
2002:
2000:
1999:
1994:
1985:
1977:
1976:
1975:
1974:
1960:
1954:
1927:
1906:
1904:
1903:
1898:
1896:
1871:
1869:
1868:
1863:
1861:
1860:
1841:
1839:
1838:
1833:
1831:
1830:
1825:
1815:
1803:
1802:
1790:
1789:
1784:
1763:
1762:
1750:
1749:
1744:
1732:
1731:
1723:
1717:
1716:
1704:
1703:
1698:
1677:
1676:
1664:
1663:
1658:
1649:
1648:
1627:
1622:
1621:
1613:
1610:
1583:
1581:
1580:
1575:
1573:
1571:
1570:
1565:
1564:
1559:
1550:
1549:
1544:
1538:
1533:
1532:
1516:
1515:
1506:
1503:
1488:
1480:
1475:
1473:
1472:
1467:
1466:
1461:
1455:
1450:
1449:
1433:
1432:
1431:
1418:
1415:
1403:
1402:
1401:
1400:
1395:
1388:
1387:
1386:
1374:
1365:
1363:
1355:
1354:
1345:
1337:
1336:
1328:
1276:
1274:
1273:
1268:
1263:
1262:
1257:
1248:
1247:
1226:
1225:
1220:
1211:
1210:
1198:
1197:
1192:
1183:
1182:
1170:
1169:
1164:
1155:
1154:
1139:
1138:
1133:
1112:
1111:
1106:
1097:
1096:
1091:
1082:
1081:
1076:
1050:
1048:
1047:
1042:
1030:
1028:
1027:
1022:
1017:
1016:
1011:
1002:
1001:
974:
973:
968:
959:
958:
943:
942:
937:
928:
927:
912:
911:
906:
897:
896:
880:
878:
877:
872:
831:, in the states
826:
824:
823:
818:
800:
798:
797:
792:
761:
759:
758:
753:
723:
721:
720:
715:
713:
712:
707:
698:
690:
689:
681:
672:
639:
637:
636:
631:
629:
628:
617:
601:
599:
598:
593:
570:
569:
554:
553:
525:Hartree equation
514:
512:
511:
506:
486:
482:
481:
479:
478:
469:
449:
409:
407:
406:
405:
396:
390:
377:
376:
371:
364:
352:
350:
349:
344:
327:
326:
299:
275:
273:
272:
267:
251:
249:
248:
243:
238:
186:
184:
183:
178:
132:self-consistency
111:
104:
100:
97:
91:
89:
48:
24:
16:
5124:
5123:
5119:
5118:
5117:
5115:
5114:
5113:
5079:
5078:
5077:
5076:
5058:
5057:
5053:
5007:
5006:
5002:
4956:
4955:
4951:
4901:
4900:
4896:
4858:
4857:
4853:
4848:
4815:
4796:
4773:
4768:
4750:
4727:
4722:
4714:
4713:
4690:
4685:
4680:
4679:
4638:
4623:
4608:
4603:
4585:
4570:
4555:
4550:
4537:
4508:
4493:
4452:
4437:
4427:
4423:
4415:
4414:
4386:
4385:
4372:
4357:
4342:
4337:
4335:
4324:
4311:
4296:
4281:
4276:
4274:
4261:
4246:
4231:
4226:
4223:
4222:
4211:
4200:
4189:
4177:
4176:
4163:
4148:
4133:
4128:
4126:
4115:
4102:
4087:
4072:
4067:
4065:
4052:
4037:
4022:
4017:
4014:
4013:
4000:
3985:
3970:
3965:
3963:
3952:
3939:
3924:
3909:
3904:
3902:
3889:
3874:
3859:
3854:
3847:
3806:
3791:
3766:
3751:
3740:
3739:
3725:
3719:
3664:
3663:
3638:
3633:
3632:
3591:
3576:
3561:
3556:
3538:
3523:
3508:
3503:
3490:
3461:
3446:
3431:
3426:
3381:
3366:
3351:
3346:
3332:
3328:
3320:
3319:
3289:
3274:
3259:
3254:
3223:
3208:
3183:
3168:
3157:
3156:
3150:
3101:
3100:
3071:
3066:
3054:
3039:
3034:
3020:
2999:
2984:
2971:
2959:
2943:
2938:
2932:
2898:
2891:
2881:
2848:
2841:
2830:
2826:
2811:
2800:
2796:
2790:
2786:
2781:
2780:
2754:
2732:
2731:
2695:
2694:
2638:
2627:
2620:
2611:
2601:
2577:
2576:
2547:
2542:
2530:
2515:
2510:
2457:
2450:
2439:
2435:
2420:
2409:
2405:
2399:
2395:
2390:
2389:
2363:
2342:
2327:
2314:
2302:
2286:
2281:
2275:
2241:
2234:
2214:
2193:
2178:
2165:
2157:
2139:
2132:
2105:
2104:
2075:
2059:
2027:
2022:
2021:
1989:
1966:
1961:
1912:
1911:
1887:
1886:
1882:
1852:
1847:
1846:
1820:
1794:
1779:
1754:
1739:
1708:
1693:
1668:
1653:
1640:
1595:
1594:
1590:
1554:
1539:
1524:
1517:
1507:
1456:
1441:
1434:
1423:
1419:
1390:
1378:
1376:
1356:
1346:
1321:
1320:
1310:
1282:Hartree product
1252:
1239:
1215:
1202:
1187:
1174:
1159:
1146:
1128:
1101:
1086:
1071:
1060:
1059:
1053:Hartree product
1033:
1032:
1006:
993:
963:
950:
932:
919:
901:
888:
883:
882:
833:
832:
803:
802:
783:
782:
779:
777:Hartree product
732:
731:
702:
676:
645:
644:
612:
607:
606:
561:
545:
536:
535:
529:Douglas Hartree
521:
470:
450:
417:
413:
397:
391:
366:
365:
358:
357:
318:
278:
277:
258:
257:
217:
216:
197:
160:
159:
112:
101:
95:
92:
49:
47:
37:
25:
12:
11:
5:
5122:
5120:
5112:
5111:
5106:
5101:
5096:
5091:
5081:
5080:
5075:
5074:
5051:
5000:
4949:
4894:
4850:
4849:
4847:
4844:
4825:
4822:
4818:
4814:
4811:
4808:
4803:
4799:
4795:
4791:
4787:
4780:
4776:
4771:
4766:
4762:
4757:
4753:
4749:
4745:
4741:
4734:
4730:
4725:
4721:
4697:
4693:
4688:
4678:Where now the
4676:
4675:
4664:
4661:
4657:
4653:
4650:
4645:
4641:
4637:
4632:
4627:
4622:
4615:
4611:
4606:
4601:
4597:
4592:
4588:
4584:
4579:
4574:
4569:
4562:
4558:
4553:
4549:
4544:
4540:
4534:
4530:
4526:
4523:
4520:
4515:
4511:
4507:
4502:
4497:
4492:
4489:
4485:
4478:
4475:
4468:
4464:
4459:
4455:
4451:
4446:
4441:
4436:
4433:
4430:
4426:
4422:
4404:
4403:
4390:
4384:
4379:
4375:
4371:
4366:
4361:
4356:
4349:
4345:
4340:
4336:
4334:
4331:
4328:
4325:
4323:
4318:
4314:
4310:
4305:
4300:
4295:
4288:
4284:
4279:
4275:
4273:
4268:
4264:
4260:
4255:
4250:
4245:
4238:
4234:
4229:
4225:
4224:
4221:
4218:
4215:
4212:
4210:
4207:
4204:
4201:
4199:
4196:
4193:
4190:
4188:
4185:
4182:
4179:
4178:
4175:
4170:
4166:
4162:
4157:
4152:
4147:
4140:
4136:
4131:
4127:
4125:
4122:
4119:
4116:
4114:
4109:
4105:
4101:
4096:
4091:
4086:
4079:
4075:
4070:
4066:
4064:
4059:
4055:
4051:
4046:
4041:
4036:
4029:
4025:
4020:
4016:
4015:
4012:
4007:
4003:
3999:
3994:
3989:
3984:
3977:
3973:
3968:
3964:
3962:
3959:
3956:
3953:
3951:
3946:
3942:
3938:
3933:
3928:
3923:
3916:
3912:
3907:
3903:
3901:
3896:
3892:
3888:
3883:
3878:
3873:
3866:
3862:
3857:
3853:
3852:
3850:
3845:
3842:
3839:
3833:
3830:
3826:
3821:
3818:
3813:
3809:
3805:
3800:
3795:
3790:
3787:
3784:
3781:
3778:
3773:
3769:
3765:
3760:
3755:
3750:
3747:
3721:Main article:
3718:
3715:
3702:
3699:
3695:
3688:
3685:
3678:
3674:
3671:
3645:
3641:
3629:
3628:
3617:
3614:
3610:
3606:
3603:
3598:
3594:
3590:
3585:
3580:
3575:
3568:
3564:
3559:
3554:
3550:
3545:
3541:
3537:
3532:
3527:
3522:
3515:
3511:
3506:
3502:
3497:
3493:
3487:
3483:
3479:
3476:
3473:
3468:
3464:
3460:
3455:
3450:
3445:
3438:
3434:
3429:
3423:
3419:
3414:
3407:
3404:
3397:
3393:
3388:
3384:
3380:
3375:
3370:
3365:
3358:
3354:
3349:
3343:
3339:
3335:
3331:
3327:
3313:
3312:
3301:
3296:
3292:
3288:
3283:
3278:
3273:
3266:
3262:
3257:
3251:
3246:
3242:
3238:
3235:
3230:
3226:
3222:
3217:
3212:
3207:
3204:
3201:
3198:
3195:
3190:
3186:
3182:
3177:
3172:
3167:
3164:
3149:
3146:
3125:
3122:
3119:
3115:
3111:
3108:
3094:
3093:
3078:
3074:
3069:
3063:
3058:
3053:
3046:
3042:
3037:
3032:
3026:
3023:
3018:
3011:
3005:
3002:
2997:
2993:
2988:
2980:
2975:
2970:
2965:
2962:
2957:
2950:
2946:
2941:
2936:
2929:
2924:
2921:
2918:
2914:
2905:
2901:
2897:
2894:
2888:
2884:
2878:
2871:
2866:
2861:
2855:
2851:
2847:
2844:
2837:
2833:
2829:
2823:
2817:
2814:
2807:
2803:
2799:
2793:
2789:
2761:
2757:
2753:
2750:
2746:
2742:
2739:
2719:
2716:
2713:
2709:
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2020:
2019:
2018:
2015:
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1971:
1967:
1962:
1951:
1948:
1945:
1941:
1937:
1934:
1931:
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1879:
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1108:
1098:
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1083:
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1058:
1057:
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1054:
1038:
1031:. Since each
1013:
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990:
987:
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955:
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924:
920:
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889:
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859:
856:
853:
850:
847:
844:
841:
838:
830:
814:
811:
808:
776:
774:
772:
771:limiting case
768:
749:
746:
743:
740:
737:
730:
729:
728:
709:
699:
691:
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493:
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483:
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457:
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445:
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430:
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424:
418:
414:
410:
402:
398:
384:
378:
373:
356:
355:
354:
337:
334:
331:
323:
319:
312:
306:
300:
296:
292:
286:
283:
263:
255:
239:
235:
228:
222:
214:
210:
206:
202:
194:
192:
190:
171:
165:
157:
153:
149:
145:
141:
137:
133:
129:
125:
121:
110:
107:
99:
88:
85:
81:
78:
74:
71:
67:
64:
60:
57: –
56:
52:
51:Find sources:
45:
41:
35:
34:
29:This article
27:
23:
18:
17:
5060:
5054:
5013:
5009:
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4962:
4958:
4952:
4911:
4907:
4897:
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4860:
4854:
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4409:
4405:
3726:
3630:
3314:
3151:
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3098:
3095:
2775:
2692:
2571:
2384:
2099:
2016:
2013:
1883:
1844:
1589:
1586:
1311:
1281:
1279:
1052:
828:
780:
764:
726:
604:
524:
522:
201:atomic units
198:
188:
154:field. The
131:
127:
117:
102:
96:October 2013
93:
83:
76:
69:
62:
50:
38:Please help
33:verification
30:
144:Bohr theory
5083:Categories
4846:References
3631:where the
1874:mean field
1845:Where the
1308:Derivation
66:newspapers
5038:0305-0041
4995:121520012
4987:0305-0041
4944:122077124
4936:0305-0041
4889:0097-1421
4817:δ
4810:⟩
4770:ϕ
4724:ϕ
4720:⟨
4687:ϕ
4652:⟩
4605:ϕ
4552:ϕ
4548:⟨
4539:ϵ
4529:∑
4525:−
4522:⟩
4488:ψ
4477:^
4432:ψ
4429:⟨
4421:δ
4339:ϕ
4278:ϕ
4228:ϕ
4130:ϕ
4069:ϕ
4019:ϕ
3967:ϕ
3906:ϕ
3856:ϕ
3746:ψ
3701:⟩
3698:ψ
3687:^
3673:ψ
3670:⟨
3640:ϵ
3605:⟩
3558:ϕ
3505:ϕ
3501:⟨
3492:ϵ
3482:∑
3478:−
3475:⟩
3428:ϕ
3418:∏
3406:^
3348:ϕ
3338:∏
3334:⟨
3326:δ
3256:ϕ
3241:∏
3163:ψ
3068:ϕ
3036:ϕ
2996:−
2940:ϕ
2928:∫
2920:≠
2913:∑
2900:ϵ
2896:π
2850:ϵ
2846:π
2822:−
2802:∇
2798:ℏ
2792:−
2756:ϵ
2738:ε
2629:ϵ
2625:π
2603:ϵ
2583:ε
2544:ϕ
2512:ϕ
2486:−
2459:ϵ
2455:π
2431:−
2411:∇
2407:ℏ
2401:−
2339:−
2283:ϕ
2271:∫
2263:≠
2256:∑
2243:ϵ
2239:π
2227:−
2190:−
2159:ρ
2153:∫
2141:ϵ
2137:π
2077:ϵ
2061:ρ
2055:−
2029:∇
1963:ϕ
1949:≠
1942:∑
1935:−
1918:ρ
1809:∏
1734:ψ
1728:^
1646:∗
1642:ψ
1638:∫
1632:⟩
1629:ψ
1618:^
1604:ψ
1601:⟨
1552:−
1526:ϵ
1522:π
1498:≠
1491:∑
1443:ϵ
1439:π
1409:∑
1405:−
1380:∇
1368:∑
1348:ℏ
1342:−
1333:^
1245:π
1241:ψ
1208:γ
1204:ψ
1180:β
1176:ψ
1152:α
1148:ψ
1066:Ψ
1039:ψ
999:π
995:ψ
956:γ
952:ψ
925:β
921:ψ
894:α
890:ψ
869:π
851:γ
845:β
839:α
789:Ψ
692:∗
684:−
666:±
563:∇
547:∂
458:ℓ
452:ℓ
446:−
428:−
338:ϕ
332:θ
324:ℓ
284:ψ
276:, namely
264:ℓ
209:Cartesian
205:Laplacian
5046:98842095
3727:In 1930
3658:are the
3025:′
3004:′
2964:′
2368:′
2347:′
2307:′
2219:′
2198:′
2170:′
1298:fermions
191:method.
140:electron
5069:57-5916
5018:Bibcode
4967:Bibcode
4916:Bibcode
4869:Bibcode
1292:in the
195:History
148:nucleus
136:Lindsay
124:Hartree
80:scholar
5067:
5044:
5036:
4993:
4985:
4942:
4934:
4887:
3733:Slater
3142:ansatz
2693:Where
1290:ansatz
640:where
252:and a
82:
75:
68:
61:
53:
5042:S2CID
4991:S2CID
4940:S2CID
1280:This
531:, is
207:from
134:that
87:JSTOR
73:books
5065:LCCN
5034:ISSN
4983:ISSN
4932:ISSN
4885:ISSN
3731:and
3729:Fock
2752:>
2730:and
2715:<
765:The
747:<
741:<
727:and
59:news
5026:doi
4975:doi
4924:doi
4877:doi
605:in
211:to
42:by
5085::
5040:.
5032:.
5024:.
5014:24
5012:.
4989:.
4981:.
4973:.
4963:24
4961:.
4938:.
4930:.
4922:.
4912:24
4910:.
4906:.
4883:.
4875:.
4863:.
4842:.
3144:.
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