4030:
2626:
4025:{\displaystyle {\begin{aligned}\delta E&=\sum _{i=1}^{N}\int {\text{d}}\mathbf {x} _{i}\,{\hat {h}}(\mathbf {x} _{i})\phi _{i}(\mathbf {x} _{i})\delta (\mathbf {x} _{i}-\mathbf {x} _{k})\delta _{ik}\\&+\sum _{i=1}^{N}\sum _{j=1}^{N}\int \mathrm {d} \mathbf {x} _{i}\int {\text{d}}\mathbf {x} _{j}\phi _{j}^{*}(\mathbf {x} _{j}){\frac {1}{|\mathbf {r} _{i}-\mathbf {r} _{j}|}}\phi _{i}(\mathbf {x} _{i})\phi _{j}(\mathbf {x} _{j})\delta (\mathbf {x} _{i}-\mathbf {x} _{k})\delta _{ik}\\&-\sum _{i=1}^{N}\sum _{j=1}^{N}\int {\text{d}}\mathbf {x} _{i}\int {\text{d}}\mathbf {x} _{j}\phi _{j}^{*}(\mathbf {x} _{j}){\frac {1}{|\mathbf {r} _{i}-\mathbf {r} _{j}|}}\phi _{i}(\mathbf {x} _{j})\phi _{j}(\mathbf {x} _{i})\delta (\mathbf {x} _{i}-\mathbf {x} _{k})\delta _{ik}\\&-\sum _{i=1}^{N}\epsilon _{i}\int {\text{d}}\mathbf {x} _{i}\,\phi _{i}(\mathbf {x} _{i})\delta (\mathbf {x} _{i}-\mathbf {x} _{k})\delta _{ik}\\&={\hat {h}}(\mathbf {x} _{k})\phi _{k}(\mathbf {x} _{k})\\&+\sum _{j=1}^{N}\int {\text{d}}\mathbf {x} _{j}\phi _{j}^{*}(\mathbf {x} _{j}){\frac {1}{|\mathbf {r} _{k}-\mathbf {r} _{j}|}}\phi _{k}(\mathbf {x} _{k})\phi _{j}(\mathbf {x} _{j})\\&-\sum _{j=1}^{N}\int {\text{d}}\mathbf {x} _{j}\phi _{j}^{*}(\mathbf {x} _{j}){\frac {1}{|\mathbf {r} _{k}-\mathbf {r} _{j}|}}\phi _{k}(\mathbf {x} _{j})\phi _{j}(\mathbf {x} _{k})\\&-\epsilon _{k}\phi _{k}(\mathbf {x} _{k})=0.\\\end{aligned}}}
1713:
916:
1708:{\textstyle {\begin{aligned}E&=\left\langle \psi ^{HF}|{\hat {H}}^{e}|\psi ^{HF}\right\rangle \\&=\sum _{i=1}^{N}\int {\text{d}}\mathbf {x} _{i}\,\phi _{i}^{*}(\mathbf {x} _{i}){\hat {h}}(\mathbf {x} _{i})\phi _{i}(\mathbf {x} _{i})\\&+{\frac {1}{2}}\sum _{i=1}^{N}\sum _{j=1}^{N}\int \mathrm {d} \mathbf {x} _{i}\int {\text{d}}\mathbf {x} _{j}\phi _{i}^{*}(\mathbf {x} _{i})\phi _{j}^{*}(\mathbf {x} _{j}){\frac {1}{|\mathbf {r} _{i}-\mathbf {r} _{j}|}}\phi _{i}(\mathbf {x} _{i})\phi _{j}(\mathbf {x} _{j})\\&-{\frac {1}{2}}\sum _{i=1}^{N}\sum _{j=1}^{N}\int {\text{d}}\mathbf {x} _{i}\int {\text{d}}\mathbf {x} _{j}\phi _{i}^{*}(\mathbf {x} _{i})\phi _{j}^{*}(\mathbf {x} _{j}){\frac {1}{|\mathbf {r} _{i}-\mathbf {r} _{j}|}}\phi _{i}(\mathbf {x} _{j})\phi _{j}(\mathbf {x} _{i})\end{aligned}}}
4971:
850:, the eigenfunctions of the Fock operator are in turn new orbitals, which can be used to construct a new Fock operator. In this way, the Hartree–Fock orbitals are optimized iteratively until the change in total electronic energy falls below a predefined threshold. In this way, a set of self-consistent one-electron orbitals is calculated. The Hartree–Fock electronic wave function is then the Slater determinant constructed from these orbitals. Following the basic postulates of quantum mechanics, the Hartree–Fock wave function can then be used to compute any desired chemical or physical property within the framework of the Hartree–Fock method and the approximations employed.
4407:
2192:
4966:{\displaystyle {\begin{aligned}{\hat {J}}(\mathbf {x_{k}} )&\equiv \sum _{j=1}^{N}\int \mathrm {d} \mathbf {x} _{j}{\frac {\phi _{j}^{*}(\mathbf {x} _{j})\phi _{j}(\mathbf {x} _{j})}{|\mathbf {r} _{k}-\mathbf {r} _{j}|}}=\sum _{j=1}^{N}\int \mathrm {d} \mathbf {x} _{j}{\frac {\rho (\mathbf {x} _{j})}{|\mathbf {r} _{k}-\mathbf {r} _{j}|}},\\{\hat {K}}(\mathbf {x_{k}} )\phi _{k}(\mathbf {x} _{k})&\equiv \sum _{j=1}^{N}\phi _{j}(\mathbf {x} _{k})\int {\text{d}}\mathbf {x} _{j}{\frac {\phi _{j}^{*}(\mathbf {x} _{j})\phi _{k}(\mathbf {x} _{j})}{|\mathbf {r} _{k}-\mathbf {r} _{j}|}}.\\\end{aligned}}}
1753:
5708:
immediately preceding wave function. A clever dodge, employed by
Hartree, for atomic calculations was to increase the nuclear charge, thus pulling all the electrons closer together. As the system stabilised, this was gradually reduced to the correct charge. In molecular calculations a similar approach is sometimes used by first calculating the wave function for a positive ion and then to use these orbitals as the starting point for the neutral molecule. Modern molecular Hartree–Fock computer programs use a variety of methods to ensure convergence of the Roothaan–Hall equations.
2187:{\displaystyle {\begin{aligned}\psi ^{HF}=\psi (\mathbf {x} _{1},\mathbf {x} _{2},\ldots ,\mathbf {x} _{N})={\frac {1}{\sqrt {N!}}}{\begin{vmatrix}\phi _{1}(\mathbf {x} _{1})&\phi _{2}(\mathbf {x} _{1})&\cdots &\phi _{N}(\mathbf {x} _{1})\\\phi _{1}(\mathbf {x} _{2})&\phi _{2}(\mathbf {x} _{2})&\cdots &\phi _{N}(\mathbf {x} _{2})\\\vdots &\vdots &\ddots &\vdots \\\phi _{1}(\mathbf {x} _{N})&\phi _{2}(\mathbf {x} _{N})&\cdots &\phi _{N}(\mathbf {x} _{N})\end{vmatrix}}.\end{aligned}}}
741:. However, the label "electron correlation" strictly spoken encompasses both the Coulomb correlation and Fermi correlation, and the latter is an effect of electron exchange, which is fully accounted for in the Hartree–Fock method. Stated in this terminology, the method only neglects the Coulomb correlation. However, this is an important flaw, accounting for (among others) Hartree–Fock's inability to capture
762:
796:, where the last two approximations of the Hartree–Fock theory as described above are completely undone. It is only when both limits are attained that the exact solution, up to the Born–Oppenheimer approximation, is obtained.) The Hartree–Fock energy is the minimal energy for a single Slater determinant.
842:
attraction terms. The second are
Coulombic repulsion terms between electrons in a mean-field theory description; a net repulsion energy for each electron in the system, which is calculated by treating all of the other electrons within the molecule as a smooth distribution of negative charge. This is
628:
The
Hartree–Fock method, despite its physically more accurate picture, was little used until the advent of electronic computers in the 1950s due to the much greater computational demands over the early Hartree method and empirical models. Initially, both the Hartree method and the Hartree–Fock method
543:
introduced a procedure, which he called the self-consistent field method, to calculate approximate wave functions and energies for atoms and ions. Hartree sought to do away with empirical parameters and solve the many-body time-independent Schrödinger equation from fundamental physical principles,
354:
The rest of this article will focus on applications in electronic structure theory suitable for molecules with the atom as a special case. The discussion here is only for the restricted
Hartree–Fock method, where the atom or molecule is a closed-shell system with all orbitals (atomic or molecular)
304:
required the final field as computed from the charge distribution to be "self-consistent" with the assumed initial field. Thus, self-consistency was a requirement of the solution. The solutions to the non-linear
Hartree–Fock equations also behave as if each particle is subjected to the mean field
5707:
or damping. With F-mixing, once a single-electron wave function is calculated, it is not used directly. Instead, some combination of that calculated wave function and the previous wave functions for that electron is used, the most common being a simple linear combination of the calculated and
4296:
558:. However, many of Hartree's contemporaries did not understand the physical reasoning behind the Hartree method: it appeared to many people to contain empirical elements, and its connection to the solution of the many-body Schrödinger equation was unclear. However, in 1928
2482:
5633:
837:
between themselves. The Fock operator is an effective one-electron
Hamiltonian operator being the sum of two terms. The first is a sum of kinetic-energy operators for each electron, the internuclear repulsion energy, and a sum of nuclear–electronic
781:
wave function corresponding to the given
Hamiltonian. Because of this, the Hartree–Fock energy is an upper bound to the true ground-state energy of a given molecule. In the context of the Hartree–Fock method, the best possible solution is at the
5293:
5684:. However, in most modern computer programs for molecular Hartree–Fock calculations this procedure is not followed due to the high numerical cost of orthogonalization and the advent of more efficient, often sparse, algorithms for solving the
5752:) modify the Hartree–Fock wave function by multiplying it by a correlation function ("Jastrow" factor), a term which is explicitly a function of multiple electrons that cannot be decomposed into independent single-particle functions.
5900:
4045:
828:
context. The orbitals are optimized by requiring them to minimize the energy of the respective Slater determinant. The resultant variational conditions on the orbitals lead to a new one-electron operator, the
637:. Even so, calculating a solution by hand using the Hartree–Fock equations for a medium-sized atom was laborious; small molecules required computational resources far beyond what was available before 1950.
6088:
6008:
2203:
2614:
4412:
2631:
1758:
921:
629:
were applied exclusively to atoms, where the spherical symmetry of the system allowed one to greatly simplify the problem. These approximate methods were (and are) often used together with the
5759:, which treats both exchange and correlation energies, albeit approximately. Indeed, it is common to use calculations that are a hybrid of the two methods—the popular B3LYP scheme is one such
5776:
525:
of the bare nuclear charge. These early researchers later introduced other potentials containing additional empirical parameters with the hope of better reproducing the experimental data.
521:). The existence of a non-zero quantum defect was attributed to electron–electron repulsion, which clearly does not exist in the isolated hydrogen atom. This repulsion resulted in partial
4399:
4349:
320:
The
Hartree–Fock method finds its typical application in the solution of the Schrödinger equation for atoms, molecules, nanostructures and solids but it has also found widespread use in
190:
5418:
2528:
5400:
5373:
5334:
904:
511:
5100:
5031:
2558:
625:
and was too abstract for contemporary physicists to understand and implement. In 1935, Hartree reformulated the method to be more suitable for the purposes of calculation.
450:. It was observed from atomic spectra that the energy levels of many-electron atoms are well described by applying a modified version of Bohr's formula. By introducing the
824:
The orbitals above only account for the presence of other electrons in an average manner. In the
Hartree–Fock method, the effect of other electrons are accounted for in a
1745:
448:
5004:
665:), the problem is solved numerically. Due to the nonlinearities introduced by the Hartree–Fock approximation, the equations are solved using a nonlinear method such as
585:
in its older formulation, forbidding the presence of two electrons in the same quantum state. However, this was shown to be fundamentally incomplete in its neglect of
336:
theory, calculations may be for a spectrum with many excited energy levels, and consequently, the
Hartree–Fock method for atoms assumes the wave function is a single
5092:
5054:
685:
is inherently assumed. The full molecular wave function is actually a function of the coordinates of each of the nuclei, in addition to those of the electrons.
351:
For both atoms and molecules, the Hartree–Fock solution is the central starting point for most methods that describe the many-electron system more accurately.
5745:
5733:
90:
5703:
can be a problem with this procedure and there are various ways of combatting this instability. One of the most basic and generally applicable is called
273:
spin orbitals. A solution of these equations yields the Hartree–Fock wave function and energy of the system. Hartree–Fock approximation is an instance of
183:
4291:{\displaystyle {\hat {F}}(\mathbf {x} _{k})\phi _{k}(\mathbf {x} _{k})\equiv \left\phi _{k}(\mathbf {x} _{k})=\epsilon _{k}\phi _{k}(\mathbf {x} _{k}),}
5725:
645:
The Hartree–Fock method is typically used to solve the time-independent Schrödinger equation for a multi-electron atom or molecule as described in the
78:
5672:
is performed in order to produce a set of orthogonal basis functions. This can in principle save computational time when the computer is solving the
737:
is implied. Effects arising from deviations from this assumption are neglected. These effects are often collectively used as a definition of the term
112:
317:
algorithm does not always converge. This solution scheme is not the only one possible and is not an essential feature of the Hartree–Fock method.
74:
360:
166:
770:
124:
120:
6536:
6518:
5650:
818:
176:
5732:
of the Fock operator. Others expand the true multi-electron wave function in terms of a linear combination of Slater determinants—such as
5808:
5741:
545:
94:
5668:
are used in practice, most of which are composed of Gaussian functions. In some applications, an orthogonalization method such as the
682:
646:
6254:
4035:
The factor 1/2 in the molecular Hamiltonian drops out before the double integrals due to symmetry and the product rule. We may define
6478:
6459:
6440:
6414:
6378:
6339:
5913:
5716:
Of the five simplifications outlined in the section "Hartree–Fock algorithm", the fifth is typically the most important. Neglect of
633:
to impose the condition that electrons in the same shell have the same radial part and to restrict the variational solution to be a
5657:. Furthermore, it is very common for the "atomic orbitals" in use to actually be composed of a linear combination of one or more
2477:{\displaystyle \delta E=\delta \left\langle \psi ^{HF}|{\hat {H}}^{e}|\psi ^{HF}\right\rangle -\delta \left{\stackrel {!}{=}}\,0,}
6551:
128:
843:
the major simplification inherent in the Hartree–Fock method and is equivalent to the fifth simplification in the above list.
6556:
6546:
5685:
2563:
562:
and J. A. Gaunt independently showed that the Hartree method could be couched on a sounder theoretical basis by applying the
364:
5775:
For a list of software packages known to handle Hartree–Fock calculations, particularly for molecules and solids, see the
5724:
methods, have been devised to include electron correlation to the multi-electron wave function. One of these approaches,
6355:
5851:
337:
384:
in 1926. Douglas Hartree's methods were guided by some earlier, semi-empirical methods of the early 1920s (by E. Fues,
630:
158:
5720:
can lead to large deviations from experimental results. A number of approaches to this weakness, collectively called
48:
6513:
in E. Pavarini, E. Koch, J. van den Brink, and G. Sawatzky: Quantum materials: Experiments and Theory, Jülich 2016,
6541:
5970:
1747:
is the one electron operator including electronic kinetic operators and electron-nucleus Coulombic interaction and
154:
142:
56:
4357:
4307:
6334:(2nd ed.). Baffins Lane, Chichester, West Sussex PO19 1UD, England: John Wiley & Sons Ltd. p. 186.
6205:
6168:
5756:
5737:
792:
582:
325:
135:
105:
82:
5669:
5036:
Although Hartree-Fock equation appears in the form of a eigenvalue problem, the Fock operator itself depends on
2197:
To derive Hartree-Fock equation we minimize the energy functional for N electrons with orthonormal constraints.
6561:
400:
150:
63:
52:
864:
617:. The original Hartree method can then be viewed as an approximation to the Hartree–Fock method by neglecting
5749:
5628:{\displaystyle E_{HF}=2\sum _{i=1}^{N/2}{\hat {h}}_{ii}+\sum _{i=1}^{N/2}\sum _{j=1}^{N/2}+V_{\text{nucl}}.}
799:
The starting point for the Hartree–Fock method is a set of approximate one-electron wave functions known as
381:
297:
247:
208:
6409:
Levine, Ira N. (1991). Quantum Chemistry (4th ed.). Englewood Cliffs, New Jersey: Prentice Hall. p. 402-3.
116:
5856:
5717:
834:
650:
380:
The origin of the Hartree–Fock method dates back to the end of the 1920s, soon after the discovery of the
5832:
5721:
750:
5665:
5644:
2617:
868:
704:
614:
563:
314:
309:
below), and hence the terminology continued. The equations are almost universally solved by means of an
204:
281:
allows interaction terms to be replaced with quadratic terms, obtaining exactly solvable Hamiltonians.
5649:
Typically, in modern Hartree–Fock calculations, the one-electron wave functions are approximated by a
5288:{\displaystyle E_{HF}=\sum _{i=1}^{N}{\hat {h}}_{ii}+\sum _{i=1}^{N}\sum _{j=1}^{N/2}+V_{\text{nucl}}}
2490:
6304:
6263:
6214:
6177:
6140:
6096:
6060:
6016:
5978:
5938:
5827:
5378:
5339:
5300:
773:
states that for a time-independent Hamiltonian operator, any trial wave function will have an energy
738:
719:
457:
as an empirical parameter, the energy levels of a generic atom were well approximated by the formula
385:
33:
6495:
The Wave Mechanics of an Atom with a Non-Coulomb Central Field. Part II. Some Results and Discussion
873:
513:, in the sense that one could reproduce fairly well the observed transitions levels observed in the
460:
6501:, Mathematical Proceedings of the Cambridge Philosophical Society, Volume 24, 111–132, January 1928
5764:
5729:
5700:
5654:
5009:
2533:
98:
41:
6504:
5661:, rather than Slater-type orbitals, in the interests of saving large amounts of computation time.
6349:
6230:
6193:
6112:
6032:
5822:
5689:
5673:
4976:
The exchange operator has no classical analogue and can only be defined as an integral operator.
907:
810:
723:
700:
689:
598:
586:
389:
262:
239:
1721:
817:
or crystalline calculation, the initial approximate one-electron wave functions are typically a
409:
4982:
6514:
6510:
6474:
6455:
6436:
6410:
6374:
6335:
6166:
Fock, V. A. (1930). "Näherungsmethode zur Lösung des quantenmechanischen Mehrkörperproblems".
5909:
5871:
5803:
5798:
5760:
4352:
825:
814:
787:
774:
742:
734:
727:
712:
662:
618:
606:
578:
329:
274:
224:
5966:"SiGe superlattice nanocrystal infrared and Raman spectra: A density functional theory study"
605:
of one-particle orbitals first used by Heisenberg and Dirac in 1926, trivially satisfies the
306:
6312:
6271:
6222:
6185:
6148:
6104:
6068:
6024:
5986:
5946:
5658:
5067:
4302:
839:
597:
A solution to the lack of anti-symmetry in the Hartree method came when it was shown that a
534:
522:
518:
333:
310:
293:
278:
228:
162:
5405:
It should be emphasized that the total energy is not equal to the sum of orbital energies.
5039:
6498:
6132:
6052:
5681:
554:
540:
321:
301:
86:
6006:
Hartree, D. R. (1928). "The Wave Mechanics of an Atom with a Non-Coulomb Central Field".
6308:
6267:
6218:
6181:
6144:
6100:
6064:
6020:
5982:
5942:
5861:
5677:
806:
634:
559:
451:
341:
833:. At the minimum, the occupied orbitals are eigensolutions to the Fock operator via a
6530:
6234:
6197:
6116:
6036:
5950:
830:
658:
574:
359:
systems, where some of the electrons are not paired, can be dealt with by either the
220:
146:
846:
Since the Fock operator depends on the orbitals used to construct the corresponding
5409:
801:
778:
622:
577:
independently pointed out that the Hartree method did not respect the principle of
345:
258:
17:
6292:
677:
The Hartree–Fock method makes five major simplifications to deal with this task:
4036:
847:
602:
6435:(4th ed.). Englewood Cliffs, New Jersey: Prentice Hall. pp. 455–544.
284:
Especially in the older literature, the Hartree–Fock method is also called the
6108:
6050:
Slater, J. C. (1928). "The Self Consistent Field and the Structure of Atoms".
6028:
5866:
708:
396:
356:
6152:
5402:
is the total electrostatic repulsion between all the nuclei in the molecule.
813:(an atom with only one electron, but the appropriate nuclear charge). For a
6494:
5375:
are matrix elements of the Coulomb and exchange operators respectively, and
666:
6316:
6276:
6249:
6072:
761:
786:; i.e., the limit of the Hartree–Fock energy as the basis set approaches
693:
6203:
Fock, V. A. (1930). ""Selfconsistent field" mit Austausch für Natrium".
5908:(Corrected version ed.). Oxford New York: Oxford University Press.
6226:
6189:
243:
5991:
5965:
749:
Relaxation of the last two approximations give rise to many so-called
5902:
Many-body quantum theory in condensed matter physics: an introduction
610:
567:
517:
region (for example, see the empirical discussion and derivation in
726:, an antisymmetrized product of one-electron wave functions (i.e.,
219:) method is a method of approximation for the determination of the
5929:
Froese Fischer, Charlotte (1987). "General Hartree-Fock program".
5755:
An alternative to Hartree–Fock calculations used in some cases is
760:
514:
251:
238:-body wave function of the system can be approximated by a single
570:(trial wave function) as a product of single-particle functions.
5837:
669:, which gives rise to the name "self-consistent field method."
5033:
are called molecular orbital and orbital energy respectively.
6454:. Chichester: John Wiley & Sons, Ltd. pp. 153–189.
6086:
Gaunt, J. A. (1928). "A Theory of Hartree's Atomic Fields".
5787:
2530:, we choose a basis in which the Lagrange multiplier matrix
548:. His first proposed method of solution became known as the
5412:, the total energy according to the Hartree-Fock method is
5777:
list of quantum chemistry and solid state physics software
765:
Algorithmic flowchart illustrating the Hartree–Fock method
6505:
An Introduction to Hartree-Fock Molecular Orbital Theory
2609:{\displaystyle \lambda _{ij}=\epsilon _{i}\delta _{ij}}
609:
property of the exact solution and hence is a suitable
1857:
919:
711:. The finite basis set is assumed to be approximately
696:
operator is assumed to be completely non-relativistic.
6250:"Self-consistent field, with exchange, for beryllium"
5421:
5381:
5342:
5303:
5103:
5070:
5042:
5012:
4985:
4410:
4360:
4310:
4048:
2629:
2566:
2536:
2493:
2206:
1756:
1724:
876:
463:
412:
234:
The Hartree–Fock method often assumes that the exact
809:
calculation, these are typically the orbitals for a
581:of the wave function. The Hartree method used the
6130:Slater, J. C. (1930). "Note on Hartree's Method".
5627:
5394:
5367:
5328:
5287:
5086:
5048:
5025:
4998:
4965:
4393:
4343:
4290:
4024:
2608:
2552:
2522:
2476:
2186:
1739:
1707:
898:
707:, which are usually (but not always) chosen to be
505:
442:
344:and that the energy level is not necessarily the
277:, where neglecting higher-order fluctuations in
5094:can be written in terms of molecular orbitals.
6293:"A Simplification of the Hartree-Fock Method"
184:
8:
4394:{\displaystyle {\hat {K}}(\mathbf {x} _{k})}
4344:{\displaystyle {\hat {J}}(\mathbf {x} _{k})}
699:The variational solution is assumed to be a
5734:multi-configurational self-consistent field
5056:and must be solved by different technique.
657:solutions for one-electron systems such as
621:. Fock's original method relied heavily on
91:Multi-configurational self-consistent field
5899:Bruus, Henrik; Flensberg, Karsten (2014).
777:that is greater than or equal to the true
191:
177:
29:
6275:
5990:
5616:
5597:
5586:
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5561:
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3978:
3955:
3950:
3940:
3927:
3922:
3912:
3900:
3894:
3889:
3879:
3874:
3868:
3862:
3853:
3848:
3838:
3833:
3823:
3818:
3812:
3803:
3792:
3769:
3764:
3754:
3741:
3736:
3726:
3714:
3708:
3703:
3693:
3688:
3682:
3676:
3667:
3662:
3652:
3647:
3637:
3632:
3626:
3617:
3606:
3583:
3578:
3568:
3555:
3550:
3535:
3534:
3515:
3502:
3497:
3487:
3482:
3466:
3461:
3451:
3446:
3440:
3435:
3429:
3420:
3410:
3399:
3376:
3363:
3358:
3348:
3343:
3327:
3322:
3312:
3299:
3294:
3284:
3272:
3266:
3261:
3251:
3246:
3240:
3234:
3225:
3220:
3210:
3205:
3195:
3190:
3184:
3175:
3170:
3164:
3155:
3144:
3134:
3123:
3100:
3087:
3082:
3072:
3067:
3051:
3046:
3036:
3023:
3018:
3008:
2996:
2990:
2985:
2975:
2970:
2964:
2958:
2949:
2944:
2934:
2929:
2919:
2914:
2908:
2899:
2894:
2888:
2879:
2868:
2858:
2847:
2824:
2811:
2806:
2796:
2791:
2775:
2770:
2760:
2747:
2742:
2727:
2726:
2725:
2719:
2714:
2708:
2699:
2688:
2665:
2652:
2647:
2630:
2628:
2597:
2587:
2571:
2565:
2541:
2535:
2511:
2498:
2492:
2467:
2459:
2454:
2452:
2451:
2432:
2414:
2401:
2378:
2368:
2357:
2347:
2336:
2307:
2298:
2292:
2281:
2280:
2274:
2265:
2238:
2225:
2220:
2205:
2160:
2155:
2145:
2125:
2120:
2110:
2095:
2090:
2080:
2041:
2036:
2026:
2006:
2001:
1991:
1976:
1971:
1961:
1944:
1939:
1929:
1909:
1904:
1894:
1879:
1874:
1864:
1852:
1837:
1825:
1820:
1804:
1799:
1789:
1784:
1765:
1757:
1755:
1726:
1725:
1723:
1692:
1687:
1677:
1664:
1659:
1649:
1637:
1631:
1626:
1616:
1611:
1605:
1599:
1590:
1585:
1575:
1570:
1557:
1552:
1542:
1537:
1527:
1522:
1516:
1507:
1502:
1496:
1487:
1476:
1466:
1455:
1441:
1422:
1417:
1407:
1394:
1389:
1379:
1367:
1361:
1356:
1346:
1341:
1335:
1329:
1320:
1315:
1305:
1300:
1287:
1282:
1272:
1267:
1257:
1252:
1246:
1237:
1232:
1226:
1217:
1206:
1196:
1185:
1171:
1152:
1147:
1137:
1124:
1119:
1104:
1103:
1094:
1089:
1079:
1074:
1069:
1063:
1058:
1052:
1043:
1032:
1004:
995:
989:
978:
977:
971:
962:
934:
920:
918:
890:
879:
878:
875:
867:, the expectation value of energy of the
722:is assumed to be describable by a single
497:
476:
462:
434:
425:
411:
5712:Weaknesses, extensions, and alternatives
399:of the atom, the energy of a state with
305:created by all other particles (see the
113:Time-dependent density functional theory
75:Semi-empirical quantum chemistry methods
6511:Mean-Field Theory: Hartree-Fock and BCS
6473:. Mineola, New York: Dover Publishing.
6373:. Mineola, New York: Dover Publishing.
5891:
692:effects are completely neglected. The
328:for a discussion of its application in
134:
104:
62:
40:
32:
6347:
5838:Direct inversion of iterative subspace
292:). In deriving what is now called the
125:Linearized augmented-plane-wave method
121:Orbital-free density functional theory
6452:Essentials of Computational Chemistry
5651:linear combination of atomic orbitals
5639:Linear combination of atomic orbitals
2487:Since the we can choose the basis of
819:linear combination of atomic orbitals
242:(in the case where the particles are
7:
6248:Hartree, D. R.; Hartree, W. (1935).
5653:. These atomic orbitals are called
757:Variational optimization of orbitals
6396:The Theory of Intermolecular Forces
5809:Quantum chemistry computer programs
5742:quadratic configuration interaction
95:Quantum chemistry composite methods
6469:Szabo, A.; Ostlund, N. S. (1996).
6369:Szabo, A.; Ostlund, N. S. (1996).
5746:complete active space SCF (CASSCF)
5726:Møller–Plesset perturbation theory
4627:
4477:
2889:
1227:
296:as an approximate solution of the
79:Møller–Plesset perturbation theory
25:
5763:method. Another option is to use
653:for many-electron systems (there
6507:by C. David Sherrill (June 2000)
4938:
4923:
4901:
4873:
4840:
4817:
4761:
4737:
4733:
4691:
4676:
4654:
4633:
4581:
4566:
4544:
4516:
4483:
4436:
4432:
4378:
4328:
4272:
4231:
4198:
4165:
4132:
4094:
4066:
3999:
3951:
3923:
3890:
3875:
3849:
3819:
3765:
3737:
3704:
3689:
3663:
3633:
3579:
3551:
3498:
3483:
3462:
3436:
3359:
3344:
3323:
3295:
3262:
3247:
3221:
3191:
3171:
3083:
3068:
3047:
3019:
2986:
2971:
2945:
2915:
2895:
2807:
2792:
2771:
2743:
2715:
2523:{\displaystyle \phi _{i}(x_{i})}
2156:
2121:
2091:
2037:
2002:
1972:
1940:
1905:
1875:
1821:
1800:
1785:
1688:
1660:
1627:
1612:
1586:
1553:
1523:
1503:
1418:
1390:
1357:
1342:
1316:
1283:
1253:
1233:
1148:
1120:
1090:
1059:
869:molecular electronic Hamiltonian
6450:Cramer, Christopher J. (2002).
5931:Computer Physics Communications
5750:variational quantum Monte Carlo
5395:{\displaystyle V_{\text{nucl}}}
5368:{\displaystyle {\hat {K}}_{ij}}
5329:{\displaystyle {\hat {J}}_{ij}}
129:Projector augmented wave method
6332:Modelling Molecular Structures
6089:Math. Proc. Camb. Philos. Soc.
6009:Math. Proc. Camb. Philos. Soc.
5964:Abdulsattar, Mudar A. (2012).
5686:generalized eigenvalue problem
5606:
5591:
5566:
5553:
5477:
5350:
5311:
5269:
5254:
5229:
5216:
5148:
4949:
4917:
4911:
4896:
4883:
4868:
4827:
4812:
4771:
4756:
4743:
4728:
4722:
4702:
4670:
4664:
4649:
4592:
4560:
4554:
4539:
4526:
4511:
4442:
4427:
4421:
4388:
4373:
4367:
4338:
4323:
4317:
4282:
4267:
4241:
4226:
4208:
4193:
4187:
4175:
4160:
4154:
4142:
4127:
4121:
4104:
4089:
4076:
4061:
4055:
4009:
3994:
3961:
3946:
3933:
3918:
3901:
3869:
3859:
3844:
3775:
3760:
3747:
3732:
3715:
3683:
3673:
3658:
3589:
3574:
3561:
3546:
3540:
3508:
3478:
3472:
3457:
3369:
3339:
3333:
3318:
3305:
3290:
3273:
3241:
3231:
3216:
3093:
3063:
3057:
3042:
3029:
3014:
2997:
2965:
2955:
2940:
2817:
2787:
2781:
2766:
2753:
2738:
2732:
2674:
2671:
2658:
2640:
2517:
2504:
2299:
2286:
2275:
2247:
2244:
2231:
2213:
2166:
2151:
2131:
2116:
2101:
2086:
2047:
2032:
2012:
1997:
1982:
1967:
1950:
1935:
1915:
1900:
1885:
1870:
1831:
1780:
1731:
1698:
1683:
1670:
1655:
1638:
1606:
1596:
1581:
1563:
1548:
1428:
1413:
1400:
1385:
1368:
1336:
1326:
1311:
1293:
1278:
1158:
1143:
1130:
1115:
1109:
1100:
1085:
996:
983:
972:
943:
927:
899:{\displaystyle {\hat {H}}^{e}}
884:
683:Born–Oppenheimer approximation
647:Born–Oppenheimer approximation
506:{\displaystyle E=-1/(n+d)^{2}}
494:
481:
326:Hartree–Fock–Bogoliubov method
1:
5026:{\displaystyle \epsilon _{k}}
2553:{\displaystyle \lambda _{ij}}
167:Korringa–Kohn–Rostoker method
6537:Electronic structure methods
5951:10.1016/0010-4655(87)90053-1
5852:Vladimir Aleksandrovich Fock
406:is given in atomic units as
376:Early semi-empirical methods
338:configuration state function
286:self-consistent field method
5408:If the atom or molecule is
649:. Since there are no known
631:central field approximation
269:-coupled equations for the
159:Empty lattice approximation
6578:
5728:, treats correlation as a
5642:
1740:{\displaystyle {\hat {h}}}
532:
443:{\displaystyle E=-1/n^{2}}
388:, and himself) set in the
265:, one can derive a set of
143:Nearly free electron model
57:Modern valence bond theory
6398:, Oxford: Clarendon Press
6330:Hinchliffe, Alan (2000).
6109:10.1017/S0305004100015851
6029:10.1017/S0305004100011920
5757:density functional theory
5748:. Still others (such as
5738:configuration interaction
5064:The optimal total energy
4999:{\displaystyle \phi _{k}}
583:Pauli exclusion principle
136:Electronic band structure
106:Density functional theory
83:Configuration interaction
27:Method in quantum physics
6471:Modern Quantum Chemistry
6371:Modern Quantum Chemistry
6354:: CS1 maint: location (
6153:10.1103/PhysRev.35.210.2
4039:to rewrite the equation
854:Mathematical formulation
735:mean-field approximation
663:diatomic hydrogen cation
401:principal quantum number
225:quantum many-body system
151:Muffin-tin approximation
64:Molecular orbital theory
53:Generalized valence bond
6552:Computational chemistry
6431:Levine, Ira N. (1991).
5690:Roothaan–Hall equations
5674:Roothaan–Hall equations
4401:are defined as follows
2560:becomes diagonal, i.e.
155:k·p perturbation theory
6317:10.1103/PhysRev.81.385
6291:Slater, J. C. (1951).
6277:10.1098/rspa.1935.0085
6073:10.1103/PhysRev.32.339
5977:(4): 044306–044306–4.
5659:Gaussian-type orbitals
5629:
5552:
5523:
5469:
5396:
5369:
5330:
5289:
5215:
5186:
5140:
5088:
5087:{\displaystyle E_{HF}}
5050:
5027:
5000:
4967:
4801:
4622:
4472:
4395:
4345:
4292:
4026:
3808:
3622:
3415:
3160:
3139:
2884:
2863:
2704:
2610:
2554:
2524:
2478:
2373:
2352:
2188:
1741:
1709:
1492:
1471:
1222:
1201:
1048:
900:
835:unitary transformation
766:
703:of a finite number of
641:Hartree–Fock algorithm
507:
444:
367:Hartree–Fock methods.
49:Coulson–Fischer theory
6557:Computational physics
6547:Theoretical chemistry
6255:Proc. R. Soc. Lond. A
5645:Basis set (chemistry)
5630:
5524:
5495:
5441:
5397:
5370:
5331:
5290:
5187:
5166:
5120:
5089:
5051:
5049:{\displaystyle \phi }
5028:
5001:
4968:
4781:
4602:
4452:
4396:
4346:
4293:
4027:
3788:
3602:
3395:
3140:
3119:
2864:
2843:
2684:
2611:
2555:
2525:
2479:
2353:
2332:
2189:
1742:
1710:
1472:
1451:
1202:
1181:
1028:
901:
764:
615:variational principle
564:variational principle
508:
445:
361:restricted open-shell
315:fixed-point iteration
205:computational physics
6394:A. J. Stone (1996),
5718:electron correlation
5670:Gram–Schmidt process
5655:Slater-type orbitals
5419:
5379:
5340:
5301:
5101:
5068:
5040:
5010:
4983:
4408:
4358:
4308:
4046:
2627:
2564:
2534:
2491:
2204:
1754:
1722:
917:
874:
790:. (The other is the
739:electron correlation
720:energy eigenfunction
573:In 1930, Slater and
461:
410:
382:Schrödinger equation
298:Schrödinger equation
223:and the energy of a
34:Electronic structure
6309:1951PhRv...81..385S
6268:1935RSPSA.150....9H
6219:1930ZPhy...62..795F
6182:1930ZPhy...61..126F
6145:1930PhRv...35..210S
6101:1928PCPS...24..328G
6065:1928PhRv...32..339S
6021:1928PCPS...24..111H
5983:2012JAP...111d4306A
5943:1987CoPhC..43..355F
5765:modern valence bond
5701:Numerical stability
5696:Numerical stability
4867:
4510:
3843:
3657:
3215:
2939:
2657:
2230:
1580:
1547:
1310:
1277:
1084:
865:Slater–Condon rules
771:variational theorem
99:Quantum Monte Carlo
71:Hartree–Fock method
42:Valence bond theory
18:Hartree-Fock theory
6227:10.1007/BF01330439
6190:10.1007/BF01340294
5823:Roothaan equations
5680:effectively to an
5676:by converting the
5625:
5392:
5365:
5326:
5285:
5084:
5046:
5023:
4996:
4963:
4961:
4853:
4496:
4391:
4341:
4288:
4022:
4020:
3829:
3643:
3201:
2925:
2643:
2606:
2550:
2520:
2474:
2216:
2184:
2182:
2171:
1737:
1705:
1703:
1566:
1533:
1296:
1263:
1070:
908:Slater determinant
896:
811:hydrogen-like atom
784:Hartree–Fock limit
767:
724:Slater determinant
701:linear combination
651:analytic solutions
635:spin eigenfunction
599:Slater determinant
587:quantum statistics
503:
440:
390:old quantum theory
340:with well-defined
263:variational method
261:. By invoking the
240:Slater determinant
117:Thomas–Fermi model
6542:Quantum chemistry
6519:978-3-95806-159-0
6433:Quantum Chemistry
5992:10.1063/1.3686610
5882:
5881:
5872:Reinhart Ahlrichs
5833:Post-Hartree–Fock
5828:Koopmans' theorem
5804:Molecular physics
5799:Quantum chemistry
5771:Software packages
5761:hybrid functional
5722:post-Hartree–Fock
5619:
5594:
5569:
5480:
5389:
5353:
5314:
5282:
5257:
5232:
5151:
4954:
4836:
4725:
4707:
4597:
4424:
4370:
4353:exchange operator
4320:
4190:
4157:
4124:
4058:
3906:
3815:
3720:
3629:
3543:
3432:
3278:
3187:
3167:
3002:
2911:
2735:
2711:
2616:. Performing the
2464:
2289:
1850:
1849:
1734:
1643:
1519:
1499:
1449:
1373:
1249:
1179:
1112:
1055:
986:
887:
826:mean-field theory
815:molecular orbital
775:expectation value
751:post-Hartree–Fock
743:London dispersion
613:for applying the
355:doubly occupied.
330:nuclear structure
275:mean-field theory
246:) or by a single
201:
200:
16:(Redirected from
6569:
6484:
6465:
6446:
6418:
6407:
6401:
6399:
6391:
6385:
6384:
6366:
6360:
6359:
6353:
6345:
6327:
6321:
6320:
6288:
6282:
6281:
6279:
6245:
6239:
6238:
6201:
6163:
6157:
6156:
6127:
6121:
6120:
6083:
6077:
6076:
6047:
6041:
6040:
6003:
5997:
5996:
5994:
5961:
5955:
5954:
5926:
5920:
5919:
5907:
5896:
5857:Clemens Roothaan
5788:
5692:are an example.
5634:
5632:
5631:
5626:
5621:
5620:
5617:
5605:
5604:
5596:
5595:
5587:
5580:
5579:
5571:
5570:
5562:
5551:
5547:
5538:
5522:
5518:
5509:
5491:
5490:
5482:
5481:
5473:
5468:
5464:
5455:
5434:
5433:
5401:
5399:
5398:
5393:
5391:
5390:
5387:
5374:
5372:
5371:
5366:
5364:
5363:
5355:
5354:
5346:
5335:
5333:
5332:
5327:
5325:
5324:
5316:
5315:
5307:
5294:
5292:
5291:
5286:
5284:
5283:
5280:
5268:
5267:
5259:
5258:
5250:
5243:
5242:
5234:
5233:
5225:
5214:
5210:
5201:
5185:
5180:
5162:
5161:
5153:
5152:
5144:
5139:
5134:
5116:
5115:
5093:
5091:
5090:
5085:
5083:
5082:
5055:
5053:
5052:
5047:
5032:
5030:
5029:
5024:
5022:
5021:
5005:
5003:
5002:
4997:
4995:
4994:
4972:
4970:
4969:
4964:
4962:
4955:
4953:
4952:
4947:
4946:
4941:
4932:
4931:
4926:
4920:
4914:
4910:
4909:
4904:
4895:
4894:
4882:
4881:
4876:
4866:
4861:
4851:
4849:
4848:
4843:
4837:
4834:
4826:
4825:
4820:
4811:
4810:
4800:
4795:
4770:
4769:
4764:
4755:
4754:
4742:
4741:
4740:
4727:
4726:
4718:
4708:
4706:
4705:
4700:
4699:
4694:
4685:
4684:
4679:
4673:
4667:
4663:
4662:
4657:
4644:
4642:
4641:
4636:
4630:
4621:
4616:
4598:
4596:
4595:
4590:
4589:
4584:
4575:
4574:
4569:
4563:
4557:
4553:
4552:
4547:
4538:
4537:
4525:
4524:
4519:
4509:
4504:
4494:
4492:
4491:
4486:
4480:
4471:
4466:
4441:
4440:
4439:
4426:
4425:
4417:
4400:
4398:
4397:
4392:
4387:
4386:
4381:
4372:
4371:
4363:
4350:
4348:
4347:
4342:
4337:
4336:
4331:
4322:
4321:
4313:
4303:Coulomb operator
4297:
4295:
4294:
4289:
4281:
4280:
4275:
4266:
4265:
4256:
4255:
4240:
4239:
4234:
4225:
4224:
4215:
4211:
4207:
4206:
4201:
4192:
4191:
4183:
4174:
4173:
4168:
4159:
4158:
4150:
4141:
4140:
4135:
4126:
4125:
4117:
4103:
4102:
4097:
4088:
4087:
4075:
4074:
4069:
4060:
4059:
4051:
4031:
4029:
4028:
4023:
4021:
4008:
4007:
4002:
3993:
3992:
3983:
3982:
3967:
3960:
3959:
3954:
3945:
3944:
3932:
3931:
3926:
3917:
3916:
3907:
3905:
3904:
3899:
3898:
3893:
3884:
3883:
3878:
3872:
3863:
3858:
3857:
3852:
3842:
3837:
3828:
3827:
3822:
3816:
3813:
3807:
3802:
3781:
3774:
3773:
3768:
3759:
3758:
3746:
3745:
3740:
3731:
3730:
3721:
3719:
3718:
3713:
3712:
3707:
3698:
3697:
3692:
3686:
3677:
3672:
3671:
3666:
3656:
3651:
3642:
3641:
3636:
3630:
3627:
3621:
3616:
3595:
3588:
3587:
3582:
3573:
3572:
3560:
3559:
3554:
3545:
3544:
3536:
3527:
3523:
3522:
3507:
3506:
3501:
3492:
3491:
3486:
3471:
3470:
3465:
3456:
3455:
3445:
3444:
3439:
3433:
3430:
3425:
3424:
3414:
3409:
3388:
3384:
3383:
3368:
3367:
3362:
3353:
3352:
3347:
3332:
3331:
3326:
3317:
3316:
3304:
3303:
3298:
3289:
3288:
3279:
3277:
3276:
3271:
3270:
3265:
3256:
3255:
3250:
3244:
3235:
3230:
3229:
3224:
3214:
3209:
3200:
3199:
3194:
3188:
3185:
3180:
3179:
3174:
3168:
3165:
3159:
3154:
3138:
3133:
3112:
3108:
3107:
3092:
3091:
3086:
3077:
3076:
3071:
3056:
3055:
3050:
3041:
3040:
3028:
3027:
3022:
3013:
3012:
3003:
3001:
3000:
2995:
2994:
2989:
2980:
2979:
2974:
2968:
2959:
2954:
2953:
2948:
2938:
2933:
2924:
2923:
2918:
2912:
2909:
2904:
2903:
2898:
2892:
2883:
2878:
2862:
2857:
2836:
2832:
2831:
2816:
2815:
2810:
2801:
2800:
2795:
2780:
2779:
2774:
2765:
2764:
2752:
2751:
2746:
2737:
2736:
2728:
2724:
2723:
2718:
2712:
2709:
2703:
2698:
2670:
2669:
2656:
2651:
2615:
2613:
2612:
2607:
2605:
2604:
2592:
2591:
2579:
2578:
2559:
2557:
2556:
2551:
2549:
2548:
2529:
2527:
2526:
2521:
2516:
2515:
2503:
2502:
2483:
2481:
2480:
2475:
2466:
2465:
2463:
2458:
2453:
2450:
2446:
2445:
2441:
2440:
2439:
2424:
2420:
2419:
2418:
2406:
2405:
2386:
2385:
2372:
2367:
2351:
2346:
2320:
2316:
2315:
2314:
2302:
2297:
2296:
2291:
2290:
2282:
2278:
2273:
2272:
2243:
2242:
2229:
2224:
2193:
2191:
2190:
2185:
2183:
2176:
2175:
2165:
2164:
2159:
2150:
2149:
2130:
2129:
2124:
2115:
2114:
2100:
2099:
2094:
2085:
2084:
2046:
2045:
2040:
2031:
2030:
2011:
2010:
2005:
1996:
1995:
1981:
1980:
1975:
1966:
1965:
1949:
1948:
1943:
1934:
1933:
1914:
1913:
1908:
1899:
1898:
1884:
1883:
1878:
1869:
1868:
1851:
1842:
1838:
1830:
1829:
1824:
1809:
1808:
1803:
1794:
1793:
1788:
1773:
1772:
1746:
1744:
1743:
1738:
1736:
1735:
1727:
1714:
1712:
1711:
1706:
1704:
1697:
1696:
1691:
1682:
1681:
1669:
1668:
1663:
1654:
1653:
1644:
1642:
1641:
1636:
1635:
1630:
1621:
1620:
1615:
1609:
1600:
1595:
1594:
1589:
1579:
1574:
1562:
1561:
1556:
1546:
1541:
1532:
1531:
1526:
1520:
1517:
1512:
1511:
1506:
1500:
1497:
1491:
1486:
1470:
1465:
1450:
1442:
1434:
1427:
1426:
1421:
1412:
1411:
1399:
1398:
1393:
1384:
1383:
1374:
1372:
1371:
1366:
1365:
1360:
1351:
1350:
1345:
1339:
1330:
1325:
1324:
1319:
1309:
1304:
1292:
1291:
1286:
1276:
1271:
1262:
1261:
1256:
1250:
1247:
1242:
1241:
1236:
1230:
1221:
1216:
1200:
1195:
1180:
1172:
1164:
1157:
1156:
1151:
1142:
1141:
1129:
1128:
1123:
1114:
1113:
1105:
1099:
1098:
1093:
1083:
1078:
1068:
1067:
1062:
1056:
1053:
1047:
1042:
1021:
1017:
1013:
1012:
1011:
999:
994:
993:
988:
987:
979:
975:
970:
969:
942:
941:
905:
903:
902:
897:
895:
894:
889:
888:
880:
659:hydrogenic atoms
535:Hartree equation
512:
510:
509:
504:
502:
501:
480:
449:
447:
446:
441:
439:
438:
429:
334:atomic structure
311:iterative method
294:Hartree equation
250:(in the case of
229:stationary state
193:
186:
179:
163:GW approximation
30:
21:
6577:
6576:
6572:
6571:
6570:
6568:
6567:
6566:
6562:1927 in science
6527:
6526:
6524:
6491:
6481:
6468:
6462:
6449:
6443:
6430:
6427:
6422:
6421:
6408:
6404:
6393:
6392:
6388:
6381:
6368:
6367:
6363:
6346:
6342:
6329:
6328:
6324:
6297:Physical Review
6290:
6289:
6285:
6247:
6246:
6242:
6213:(11): 795–805.
6202:
6165:
6164:
6160:
6129:
6128:
6124:
6085:
6084:
6080:
6049:
6048:
6044:
6005:
6004:
6000:
5963:
5962:
5958:
5928:
5927:
5923:
5916:
5905:
5898:
5897:
5893:
5888:
5883:
5785:
5773:
5714:
5698:
5688:, of which the
5682:identity matrix
5647:
5641:
5612:
5584:
5559:
5470:
5422:
5417:
5416:
5382:
5377:
5376:
5343:
5338:
5337:
5304:
5299:
5298:
5275:
5247:
5222:
5141:
5104:
5099:
5098:
5071:
5066:
5065:
5062:
5038:
5037:
5013:
5008:
5007:
4986:
4981:
4980:
4960:
4959:
4936:
4921:
4915:
4899:
4886:
4871:
4852:
4838:
4815:
4802:
4774:
4759:
4746:
4732:
4713:
4712:
4689:
4674:
4668:
4652:
4645:
4631:
4579:
4564:
4558:
4542:
4529:
4514:
4495:
4481:
4445:
4431:
4406:
4405:
4376:
4356:
4355:
4326:
4306:
4305:
4270:
4257:
4247:
4229:
4216:
4196:
4163:
4130:
4114:
4110:
4092:
4079:
4064:
4044:
4043:
4019:
4018:
3997:
3984:
3974:
3965:
3964:
3949:
3936:
3921:
3908:
3888:
3873:
3867:
3847:
3817:
3779:
3778:
3763:
3750:
3735:
3722:
3702:
3687:
3681:
3661:
3631:
3593:
3592:
3577:
3564:
3549:
3525:
3524:
3511:
3496:
3481:
3460:
3447:
3434:
3416:
3386:
3385:
3372:
3357:
3342:
3321:
3308:
3293:
3280:
3260:
3245:
3239:
3219:
3189:
3169:
3110:
3109:
3096:
3081:
3066:
3045:
3032:
3017:
3004:
2984:
2969:
2963:
2943:
2913:
2893:
2834:
2833:
2820:
2805:
2790:
2769:
2756:
2741:
2713:
2677:
2661:
2625:
2624:
2593:
2583:
2567:
2562:
2561:
2537:
2532:
2531:
2507:
2494:
2489:
2488:
2428:
2410:
2397:
2396:
2392:
2391:
2387:
2374:
2331:
2327:
2303:
2279:
2261:
2260:
2256:
2234:
2202:
2201:
2181:
2180:
2170:
2169:
2154:
2141:
2139:
2134:
2119:
2106:
2104:
2089:
2076:
2073:
2072:
2067:
2062:
2057:
2051:
2050:
2035:
2022:
2020:
2015:
2000:
1987:
1985:
1970:
1957:
1954:
1953:
1938:
1925:
1923:
1918:
1903:
1890:
1888:
1873:
1860:
1853:
1819:
1798:
1783:
1761:
1752:
1751:
1720:
1719:
1702:
1701:
1686:
1673:
1658:
1645:
1625:
1610:
1604:
1584:
1551:
1521:
1501:
1432:
1431:
1416:
1403:
1388:
1375:
1355:
1340:
1334:
1314:
1281:
1251:
1231:
1162:
1161:
1146:
1133:
1118:
1088:
1057:
1019:
1018:
1000:
976:
958:
957:
953:
946:
930:
915:
914:
877:
872:
871:
861:
856:
759:
705:basis functions
675:
643:
595:
555:Hartree product
537:
531:
493:
459:
458:
430:
408:
407:
378:
373:
342:quantum numbers
322:nuclear physics
313:, although the
279:order parameter
197:
165:
161:
157:
153:
149:
145:
127:
123:
119:
115:
97:
93:
89:
87:Coupled cluster
85:
81:
77:
73:
55:
51:
28:
23:
22:
15:
12:
11:
5:
6575:
6573:
6565:
6564:
6559:
6554:
6549:
6544:
6539:
6529:
6528:
6522:
6521:
6508:
6502:
6490:
6489:External links
6487:
6486:
6485:
6479:
6466:
6460:
6447:
6441:
6426:
6423:
6420:
6419:
6402:
6386:
6379:
6361:
6340:
6322:
6303:(3): 385–390.
6283:
6240:
6176:(1): 126–148.
6158:
6139:(2): 210–211.
6122:
6095:(2): 328–342.
6078:
6059:(3): 339–348.
6042:
5998:
5971:J. Appl. Phys.
5956:
5937:(3): 355–365.
5921:
5914:
5890:
5889:
5887:
5884:
5880:
5879:
5875:
5874:
5869:
5864:
5862:George G. Hall
5859:
5854:
5842:
5841:
5840:
5835:
5830:
5825:
5813:
5812:
5811:
5806:
5801:
5793:Related fields
5786:
5784:
5781:
5772:
5769:
5713:
5710:
5697:
5694:
5678:overlap matrix
5643:Main article:
5640:
5637:
5636:
5635:
5624:
5615:
5611:
5608:
5603:
5600:
5593:
5590:
5583:
5578:
5575:
5568:
5565:
5558:
5555:
5550:
5546:
5542:
5537:
5534:
5531:
5527:
5521:
5517:
5513:
5508:
5505:
5502:
5498:
5494:
5489:
5486:
5479:
5476:
5467:
5463:
5459:
5454:
5451:
5448:
5444:
5440:
5437:
5432:
5429:
5425:
5385:
5362:
5359:
5352:
5349:
5323:
5320:
5313:
5310:
5296:
5295:
5278:
5274:
5271:
5266:
5263:
5256:
5253:
5246:
5241:
5238:
5231:
5228:
5221:
5218:
5213:
5209:
5205:
5200:
5197:
5194:
5190:
5184:
5179:
5176:
5173:
5169:
5165:
5160:
5157:
5150:
5147:
5138:
5133:
5130:
5127:
5123:
5119:
5114:
5111:
5107:
5081:
5078:
5074:
5061:
5058:
5045:
5020:
5016:
4993:
4989:
4974:
4973:
4958:
4951:
4945:
4940:
4935:
4930:
4925:
4919:
4913:
4908:
4903:
4898:
4893:
4889:
4885:
4880:
4875:
4870:
4865:
4860:
4856:
4847:
4842:
4832:
4829:
4824:
4819:
4814:
4809:
4805:
4799:
4794:
4791:
4788:
4784:
4780:
4777:
4775:
4773:
4768:
4763:
4758:
4753:
4749:
4745:
4739:
4735:
4730:
4724:
4721:
4715:
4714:
4711:
4704:
4698:
4693:
4688:
4683:
4678:
4672:
4666:
4661:
4656:
4651:
4648:
4640:
4635:
4629:
4625:
4620:
4615:
4612:
4609:
4605:
4601:
4594:
4588:
4583:
4578:
4573:
4568:
4562:
4556:
4551:
4546:
4541:
4536:
4532:
4528:
4523:
4518:
4513:
4508:
4503:
4499:
4490:
4485:
4479:
4475:
4470:
4465:
4462:
4459:
4455:
4451:
4448:
4446:
4444:
4438:
4434:
4429:
4423:
4420:
4414:
4413:
4390:
4385:
4380:
4375:
4369:
4366:
4340:
4335:
4330:
4325:
4319:
4316:
4299:
4298:
4287:
4284:
4279:
4274:
4269:
4264:
4260:
4254:
4250:
4246:
4243:
4238:
4233:
4228:
4223:
4219:
4214:
4210:
4205:
4200:
4195:
4189:
4186:
4180:
4177:
4172:
4167:
4162:
4156:
4153:
4147:
4144:
4139:
4134:
4129:
4123:
4120:
4113:
4109:
4106:
4101:
4096:
4091:
4086:
4082:
4078:
4073:
4068:
4063:
4057:
4054:
4033:
4032:
4017:
4014:
4011:
4006:
4001:
3996:
3991:
3987:
3981:
3977:
3973:
3970:
3968:
3966:
3963:
3958:
3953:
3948:
3943:
3939:
3935:
3930:
3925:
3920:
3915:
3911:
3903:
3897:
3892:
3887:
3882:
3877:
3871:
3866:
3861:
3856:
3851:
3846:
3841:
3836:
3832:
3826:
3821:
3811:
3806:
3801:
3798:
3795:
3791:
3787:
3784:
3782:
3780:
3777:
3772:
3767:
3762:
3757:
3753:
3749:
3744:
3739:
3734:
3729:
3725:
3717:
3711:
3706:
3701:
3696:
3691:
3685:
3680:
3675:
3670:
3665:
3660:
3655:
3650:
3646:
3640:
3635:
3625:
3620:
3615:
3612:
3609:
3605:
3601:
3598:
3596:
3594:
3591:
3586:
3581:
3576:
3571:
3567:
3563:
3558:
3553:
3548:
3542:
3539:
3533:
3530:
3528:
3526:
3521:
3518:
3514:
3510:
3505:
3500:
3495:
3490:
3485:
3480:
3477:
3474:
3469:
3464:
3459:
3454:
3450:
3443:
3438:
3428:
3423:
3419:
3413:
3408:
3405:
3402:
3398:
3394:
3391:
3389:
3387:
3382:
3379:
3375:
3371:
3366:
3361:
3356:
3351:
3346:
3341:
3338:
3335:
3330:
3325:
3320:
3315:
3311:
3307:
3302:
3297:
3292:
3287:
3283:
3275:
3269:
3264:
3259:
3254:
3249:
3243:
3238:
3233:
3228:
3223:
3218:
3213:
3208:
3204:
3198:
3193:
3183:
3178:
3173:
3163:
3158:
3153:
3150:
3147:
3143:
3137:
3132:
3129:
3126:
3122:
3118:
3115:
3113:
3111:
3106:
3103:
3099:
3095:
3090:
3085:
3080:
3075:
3070:
3065:
3062:
3059:
3054:
3049:
3044:
3039:
3035:
3031:
3026:
3021:
3016:
3011:
3007:
2999:
2993:
2988:
2983:
2978:
2973:
2967:
2962:
2957:
2952:
2947:
2942:
2937:
2932:
2928:
2922:
2917:
2907:
2902:
2897:
2891:
2887:
2882:
2877:
2874:
2871:
2867:
2861:
2856:
2853:
2850:
2846:
2842:
2839:
2837:
2835:
2830:
2827:
2823:
2819:
2814:
2809:
2804:
2799:
2794:
2789:
2786:
2783:
2778:
2773:
2768:
2763:
2759:
2755:
2750:
2745:
2740:
2734:
2731:
2722:
2717:
2707:
2702:
2697:
2694:
2691:
2687:
2683:
2680:
2678:
2676:
2673:
2668:
2664:
2660:
2655:
2650:
2646:
2642:
2639:
2636:
2633:
2632:
2603:
2600:
2596:
2590:
2586:
2582:
2577:
2574:
2570:
2547:
2544:
2540:
2519:
2514:
2510:
2506:
2501:
2497:
2485:
2484:
2473:
2470:
2462:
2457:
2449:
2444:
2438:
2435:
2431:
2427:
2423:
2417:
2413:
2409:
2404:
2400:
2395:
2390:
2384:
2381:
2377:
2371:
2366:
2363:
2360:
2356:
2350:
2345:
2342:
2339:
2335:
2330:
2326:
2323:
2319:
2313:
2310:
2306:
2301:
2295:
2288:
2285:
2277:
2271:
2268:
2264:
2259:
2255:
2252:
2249:
2246:
2241:
2237:
2233:
2228:
2223:
2219:
2215:
2212:
2209:
2195:
2194:
2179:
2174:
2168:
2163:
2158:
2153:
2148:
2144:
2140:
2138:
2135:
2133:
2128:
2123:
2118:
2113:
2109:
2105:
2103:
2098:
2093:
2088:
2083:
2079:
2075:
2074:
2071:
2068:
2066:
2063:
2061:
2058:
2056:
2053:
2052:
2049:
2044:
2039:
2034:
2029:
2025:
2021:
2019:
2016:
2014:
2009:
2004:
1999:
1994:
1990:
1986:
1984:
1979:
1974:
1969:
1964:
1960:
1956:
1955:
1952:
1947:
1942:
1937:
1932:
1928:
1924:
1922:
1919:
1917:
1912:
1907:
1902:
1897:
1893:
1889:
1887:
1882:
1877:
1872:
1867:
1863:
1859:
1858:
1856:
1848:
1845:
1841:
1836:
1833:
1828:
1823:
1818:
1815:
1812:
1807:
1802:
1797:
1792:
1787:
1782:
1779:
1776:
1771:
1768:
1764:
1760:
1759:
1733:
1730:
1716:
1715:
1700:
1695:
1690:
1685:
1680:
1676:
1672:
1667:
1662:
1657:
1652:
1648:
1640:
1634:
1629:
1624:
1619:
1614:
1608:
1603:
1598:
1593:
1588:
1583:
1578:
1573:
1569:
1565:
1560:
1555:
1550:
1545:
1540:
1536:
1530:
1525:
1515:
1510:
1505:
1495:
1490:
1485:
1482:
1479:
1475:
1469:
1464:
1461:
1458:
1454:
1448:
1445:
1440:
1437:
1435:
1433:
1430:
1425:
1420:
1415:
1410:
1406:
1402:
1397:
1392:
1387:
1382:
1378:
1370:
1364:
1359:
1354:
1349:
1344:
1338:
1333:
1328:
1323:
1318:
1313:
1308:
1303:
1299:
1295:
1290:
1285:
1280:
1275:
1270:
1266:
1260:
1255:
1245:
1240:
1235:
1229:
1225:
1220:
1215:
1212:
1209:
1205:
1199:
1194:
1191:
1188:
1184:
1178:
1175:
1170:
1167:
1165:
1163:
1160:
1155:
1150:
1145:
1140:
1136:
1132:
1127:
1122:
1117:
1111:
1108:
1102:
1097:
1092:
1087:
1082:
1077:
1073:
1066:
1061:
1051:
1046:
1041:
1038:
1035:
1031:
1027:
1024:
1022:
1020:
1016:
1010:
1007:
1003:
998:
992:
985:
982:
974:
968:
965:
961:
956:
952:
949:
947:
945:
940:
937:
933:
929:
926:
923:
922:
893:
886:
883:
860:
857:
855:
852:
807:atomic orbital
758:
755:
747:
746:
731:
716:
697:
686:
674:
673:Approximations
671:
642:
639:
594:
591:
550:Hartree method
533:Main article:
530:
529:Hartree method
527:
500:
496:
492:
489:
486:
483:
479:
475:
472:
469:
466:
452:quantum defect
437:
433:
428:
424:
421:
418:
415:
377:
374:
372:
369:
199:
198:
196:
195:
188:
181:
173:
170:
169:
139:
138:
132:
131:
109:
108:
102:
101:
67:
66:
60:
59:
45:
44:
38:
37:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
6574:
6563:
6560:
6558:
6555:
6553:
6550:
6548:
6545:
6543:
6540:
6538:
6535:
6534:
6532:
6525:
6520:
6516:
6512:
6509:
6506:
6503:
6500:
6499:D. R. Hartree
6496:
6493:
6492:
6488:
6482:
6480:0-486-69186-1
6476:
6472:
6467:
6463:
6461:0-471-48552-7
6457:
6453:
6448:
6444:
6442:0-205-12770-3
6438:
6434:
6429:
6428:
6424:
6416:
6415:0-205-12770-3
6412:
6406:
6403:
6397:
6390:
6387:
6382:
6380:0-486-69186-1
6376:
6372:
6365:
6362:
6357:
6351:
6343:
6341:0-471-48993-X
6337:
6333:
6326:
6323:
6318:
6314:
6310:
6306:
6302:
6298:
6294:
6287:
6284:
6278:
6273:
6269:
6265:
6261:
6257:
6256:
6251:
6244:
6241:
6236:
6232:
6228:
6224:
6220:
6216:
6212:
6209:(in German).
6208:
6207:
6199:
6195:
6191:
6187:
6183:
6179:
6175:
6172:(in German).
6171:
6170:
6162:
6159:
6154:
6150:
6146:
6142:
6138:
6135:
6134:
6126:
6123:
6118:
6114:
6110:
6106:
6102:
6098:
6094:
6091:
6090:
6082:
6079:
6074:
6070:
6066:
6062:
6058:
6055:
6054:
6046:
6043:
6038:
6034:
6030:
6026:
6022:
6018:
6014:
6011:
6010:
6002:
5999:
5993:
5988:
5984:
5980:
5976:
5973:
5972:
5967:
5960:
5957:
5952:
5948:
5944:
5940:
5936:
5932:
5925:
5922:
5917:
5915:9780198566335
5911:
5904:
5903:
5895:
5892:
5885:
5878:
5873:
5870:
5868:
5865:
5863:
5860:
5858:
5855:
5853:
5850:
5849:
5848:
5847:
5843:
5839:
5836:
5834:
5831:
5829:
5826:
5824:
5821:
5820:
5819:
5818:
5814:
5810:
5807:
5805:
5802:
5800:
5797:
5796:
5795:
5794:
5790:
5789:
5782:
5780:
5778:
5770:
5768:
5766:
5762:
5758:
5753:
5751:
5747:
5743:
5739:
5735:
5731:
5727:
5723:
5719:
5711:
5709:
5706:
5702:
5695:
5693:
5691:
5687:
5683:
5679:
5675:
5671:
5667:
5662:
5660:
5656:
5652:
5646:
5638:
5622:
5613:
5609:
5601:
5598:
5588:
5581:
5576:
5573:
5563:
5556:
5548:
5544:
5540:
5535:
5532:
5529:
5525:
5519:
5515:
5511:
5506:
5503:
5500:
5496:
5492:
5487:
5484:
5474:
5465:
5461:
5457:
5452:
5449:
5446:
5442:
5438:
5435:
5430:
5427:
5423:
5415:
5414:
5413:
5411:
5406:
5403:
5383:
5360:
5357:
5347:
5321:
5318:
5308:
5276:
5272:
5264:
5261:
5251:
5244:
5239:
5236:
5226:
5219:
5211:
5207:
5203:
5198:
5195:
5192:
5188:
5182:
5177:
5174:
5171:
5167:
5163:
5158:
5155:
5145:
5136:
5131:
5128:
5125:
5121:
5117:
5112:
5109:
5105:
5097:
5096:
5095:
5079:
5076:
5072:
5059:
5057:
5043:
5034:
5018:
5014:
4991:
4987:
4979:The solution
4977:
4956:
4943:
4933:
4928:
4906:
4891:
4887:
4878:
4863:
4858:
4854:
4845:
4830:
4822:
4807:
4803:
4797:
4792:
4789:
4786:
4782:
4778:
4776:
4766:
4751:
4747:
4719:
4709:
4696:
4686:
4681:
4659:
4646:
4638:
4623:
4618:
4613:
4610:
4607:
4603:
4599:
4586:
4576:
4571:
4549:
4534:
4530:
4521:
4506:
4501:
4497:
4488:
4473:
4468:
4463:
4460:
4457:
4453:
4449:
4447:
4418:
4404:
4403:
4402:
4383:
4364:
4354:
4333:
4314:
4304:
4285:
4277:
4262:
4258:
4252:
4248:
4244:
4236:
4221:
4217:
4212:
4203:
4184:
4178:
4170:
4151:
4145:
4137:
4118:
4111:
4107:
4099:
4084:
4080:
4071:
4052:
4042:
4041:
4040:
4038:
4037:Fock operator
4015:
4012:
4004:
3989:
3985:
3979:
3975:
3971:
3969:
3956:
3941:
3937:
3928:
3913:
3909:
3895:
3885:
3880:
3864:
3854:
3839:
3834:
3830:
3824:
3809:
3804:
3799:
3796:
3793:
3789:
3785:
3783:
3770:
3755:
3751:
3742:
3727:
3723:
3709:
3699:
3694:
3678:
3668:
3653:
3648:
3644:
3638:
3623:
3618:
3613:
3610:
3607:
3603:
3599:
3597:
3584:
3569:
3565:
3556:
3537:
3531:
3529:
3519:
3516:
3512:
3503:
3493:
3488:
3475:
3467:
3452:
3448:
3441:
3426:
3421:
3417:
3411:
3406:
3403:
3400:
3396:
3392:
3390:
3380:
3377:
3373:
3364:
3354:
3349:
3336:
3328:
3313:
3309:
3300:
3285:
3281:
3267:
3257:
3252:
3236:
3226:
3211:
3206:
3202:
3196:
3181:
3176:
3161:
3156:
3151:
3148:
3145:
3141:
3135:
3130:
3127:
3124:
3120:
3116:
3114:
3104:
3101:
3097:
3088:
3078:
3073:
3060:
3052:
3037:
3033:
3024:
3009:
3005:
2991:
2981:
2976:
2960:
2950:
2935:
2930:
2926:
2920:
2905:
2900:
2885:
2880:
2875:
2872:
2869:
2865:
2859:
2854:
2851:
2848:
2844:
2840:
2838:
2828:
2825:
2821:
2812:
2802:
2797:
2784:
2776:
2761:
2757:
2748:
2729:
2720:
2705:
2700:
2695:
2692:
2689:
2685:
2681:
2679:
2666:
2662:
2653:
2648:
2644:
2637:
2634:
2623:
2622:
2621:
2619:
2601:
2598:
2594:
2588:
2584:
2580:
2575:
2572:
2568:
2545:
2542:
2538:
2512:
2508:
2499:
2495:
2471:
2468:
2460:
2455:
2447:
2442:
2436:
2433:
2429:
2425:
2421:
2415:
2411:
2407:
2402:
2398:
2393:
2388:
2382:
2379:
2375:
2369:
2364:
2361:
2358:
2354:
2348:
2343:
2340:
2337:
2333:
2328:
2324:
2321:
2317:
2311:
2308:
2304:
2293:
2283:
2269:
2266:
2262:
2257:
2253:
2250:
2239:
2235:
2226:
2221:
2217:
2210:
2207:
2200:
2199:
2198:
2177:
2172:
2161:
2146:
2142:
2136:
2126:
2111:
2107:
2096:
2081:
2077:
2069:
2064:
2059:
2054:
2042:
2027:
2023:
2017:
2007:
1992:
1988:
1977:
1962:
1958:
1945:
1930:
1926:
1920:
1910:
1895:
1891:
1880:
1865:
1861:
1854:
1846:
1843:
1839:
1834:
1826:
1816:
1813:
1810:
1805:
1795:
1790:
1777:
1774:
1769:
1766:
1762:
1750:
1749:
1748:
1728:
1693:
1678:
1674:
1665:
1650:
1646:
1632:
1622:
1617:
1601:
1591:
1576:
1571:
1567:
1558:
1543:
1538:
1534:
1528:
1513:
1508:
1493:
1488:
1483:
1480:
1477:
1473:
1467:
1462:
1459:
1456:
1452:
1446:
1443:
1438:
1436:
1423:
1408:
1404:
1395:
1380:
1376:
1362:
1352:
1347:
1331:
1321:
1306:
1301:
1297:
1288:
1273:
1268:
1264:
1258:
1243:
1238:
1223:
1218:
1213:
1210:
1207:
1203:
1197:
1192:
1189:
1186:
1182:
1176:
1173:
1168:
1166:
1153:
1138:
1134:
1125:
1106:
1095:
1080:
1075:
1071:
1064:
1049:
1044:
1039:
1036:
1033:
1029:
1025:
1023:
1014:
1008:
1005:
1001:
990:
980:
966:
963:
959:
954:
950:
948:
938:
935:
931:
924:
913:
912:
911:
909:
891:
881:
870:
866:
863:According to
858:
853:
851:
849:
844:
841:
836:
832:
831:Fock operator
827:
822:
820:
816:
812:
808:
804:
803:
802:spin-orbitals
797:
795:
794:
793:full-CI limit
789:
785:
780:
776:
772:
763:
756:
754:
752:
744:
740:
736:
732:
729:
725:
721:
717:
714:
710:
706:
702:
698:
695:
691:
687:
684:
680:
679:
678:
672:
670:
668:
664:
660:
656:
652:
648:
640:
638:
636:
632:
626:
624:
620:
616:
612:
608:
607:antisymmetric
604:
600:
592:
590:
588:
584:
580:
576:
571:
569:
565:
561:
557:
556:
551:
547:
542:
541:D. R. Hartree
536:
528:
526:
524:
520:
519:Moseley's law
516:
498:
490:
487:
484:
477:
473:
470:
467:
464:
456:
453:
435:
431:
426:
422:
419:
416:
413:
405:
402:
398:
393:
391:
387:
386:R. B. Lindsay
383:
375:
371:Brief history
370:
368:
366:
362:
358:
352:
349:
347:
343:
339:
335:
332:theory). In
331:
327:
323:
318:
316:
312:
308:
307:Fock operator
303:
299:
295:
291:
287:
282:
280:
276:
272:
268:
264:
260:
259:spin-orbitals
257:
253:
249:
245:
241:
237:
232:
230:
226:
222:
221:wave function
218:
214:
210:
206:
194:
189:
187:
182:
180:
175:
174:
172:
171:
168:
164:
160:
156:
152:
148:
147:Tight binding
144:
141:
140:
137:
133:
130:
126:
122:
118:
114:
111:
110:
107:
103:
100:
96:
92:
88:
84:
80:
76:
72:
69:
68:
65:
61:
58:
54:
50:
47:
46:
43:
39:
35:
31:
19:
6523:
6470:
6451:
6432:
6405:
6395:
6389:
6370:
6364:
6331:
6325:
6300:
6296:
6286:
6259:
6253:
6243:
6210:
6204:
6173:
6167:
6161:
6136:
6131:
6125:
6092:
6087:
6081:
6056:
6051:
6045:
6012:
6007:
6001:
5974:
5969:
5959:
5934:
5930:
5924:
5901:
5894:
5876:
5845:
5844:
5816:
5815:
5792:
5791:
5774:
5754:
5730:perturbation
5715:
5704:
5699:
5663:
5648:
5410:closed shell
5407:
5404:
5297:
5063:
5060:Total energy
5035:
4978:
4975:
4300:
4034:
2620:, we obtain
2486:
2196:
1717:
862:
845:
823:
800:
798:
791:
788:completeness
783:
779:ground-state
768:
748:
690:relativistic
676:
654:
644:
627:
623:group theory
596:
593:Hartree–Fock
579:antisymmetry
572:
560:J. C. Slater
553:
549:
538:
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394:
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365:unrestricted
353:
350:
346:ground state
319:
289:
285:
283:
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213:Hartree–Fock
212:
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688:Typically,
603:determinant
6531:Categories
6262:(869): 9.
6133:Phys. Rev.
6053:Phys. Rev.
6015:(1): 111.
5886:References
5867:John Pople
5666:basis sets
4301:where the
859:Derivation
709:orthogonal
575:V. A. Fock
397:Bohr model
357:Open-shell
6350:cite book
6235:120921212
6198:125419115
6117:119685329
6037:121520012
5767:methods.
5592:^
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753:methods.
667:iteration
546:ab initio
539:In 1927,
523:screening
471:−
420:−
392:of Bohr.
248:permanent
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6206:Z. Phys.
6169:Z. Phys.
5817:Concepts
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5705:F-mixing
5664:Various
4351:and the
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2258:⟨
1015:⟩
955:⟨
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728:orbitals
713:complete
694:momentum
661:and the
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244:fermions
6425:Sources
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6264:Bibcode
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