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The problem with Schrödinger's equation is that it isn't exactly solvable for multi-electron species. In the past, atomic orbitals were used to construct a solution for molecules (LCAO). My question is, how were the multi electron atomic orbitals (the ones used for the LCAO) discovered? . Were the atomic orbitals solutions for a one electron system? In other words, were the atomic orbitals calculated the orbital for a $\ce{He+}$, $\ce{Li^2+}$, $\ce{Be^3+}$, etc? What were some approximations that are used to obtain the solution for multi-electron species?

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Already in year 1928 Hartree proposed an approximate method of solving the Schrödinger equation for a many-electron atom that became known as the Hartree method (1, 2, 3). The Hartree method is based on representing the many-electron wave function as a product of one-electron ones with the subsequent application of the variational principle. Two years later, Slater (4) and Fock (5) independently corrected the Hartree method for that did not respect the antisymmetry of a many-electron wave function by representing the wave function by a Slater determinant, rather that a simple product of orbitals. The resulting physically more accurate method became know as the Hartree-Fock method.


In the Hartree–Fock method one solve a set of one-electron equations, called the Hartree–Fock equations, of the form $$ \newcommand{\op}{\hat} \newcommand{\core}{^{\mathrm{core}}} \op{F} \psi_i(\vec{q}_{1}) = \varepsilon_i \psi_i(\vec{q}_{1}) \, , $$ where $\op{F} = \op{H}\core + \sum\nolimits_{j=1}^{n} \big(\op{J}_{j} - \op{K}_{j} \big)$ is the Fock operator and $\psi_i$ are the spin-orbitals of an $n$-electron system. It is customary to suppose that spin orbitals come in pairs: for each pair of electrons two spin orbitals corresponding to two different pure spin states are constructed out of the same spatial orbital, $$ \psi_{2i-1}(\vec{q}_{1}) = \phi_{i}(\vec{r}_{1}) \alpha(m_{s1}) \, , \quad \psi_{2i}(\vec{q}_{1}) = \phi_{i}(\vec{r}_{1}) \beta(m_{s1}) \, . $$ Substitution of such spin orbitals into the Hartree-Fock equations results in a similarly looking system of equations for the corresponding spatial orbitals, $$ \op{F} \phi_{i}(\vec{r}_{1}) = \epsilon_{i} \phi_{i}(\vec{r}_{1}) \, , \quad i = 1, 2, \dotsc, n/2 \, , $$ although, the expression for the Fock operator is different $\op{F} = \op{H}\core + \sum_{j=1}^{n/2} (2 \op{J}_{j} - \op{K}_{j})$. Note here, that for the case of the original Hartree method there was no exchange terms $\op{K}_{j}$.


So, essentially, atomic orbitals of many-electron atoms were obtained by solving the Hartree and Hartree-Fock equations "by hands". Of course, the spherical symmetry of atomic systems greatly simplified the problem, as it can be seen already in the Hartree's original work (1), plus some additional simplification, such as the central field approximation, were often used. But still, in general, a numerical integration was required and it was usually done on some calculating machines (6). Below is the photo (courtesy of AIP) of Douglas Hartree (left) and Arthur Porter (right) viewing one such machine, the meccano differential analyzer.

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And the results of numerical integrations were simply tabulated as the values of the radial part of an orbital for different values of distance from the nucleus (7).

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1) Hartree, D. R. The Wave Mechanics of an Atom with a Non-Coulomb Central Field. Part I. Theory and Methods. Math. Proc. Cambridge Philos. Soc. 1928, 24 (1), 89–110. DOI: 10.1017/S0305004100011919.

2) Hartree, D. R. The Wave Mechanics of an Atom with a Non-Coulomb Central Field. Part II. Some Results and Discussion. Math. Proc. Cambridge Philos. Soc. 1928, 24 (1), 111–132. DOI: 10.1017/S0305004100011920.

3) Hartree, D. R. The Wave Mechanics of an Atom with a non-Coulomb Central Field. Part III. Term Values and Intensities in Series in Optical Spectra. Mathematical Proceedings of the Cambridge Philosophical Society, 1928, 24 (3), 426–437. DOI: 10.1017/S0305004100015954.

4) Slater, J. C. The Theory of Complex Spectra. Phys. Rev. 1929, 34, 1293. DOI: 10.1103/PhysRev.34.1293.

5) Fock V. Näherungsmethode zur Lösung des quantenmechanischen Mehrkörperproblems. Z. Physik. 1930, 61 (1), 126–148. DOI: 10.1007/BF01340294.

6) Hartree, D. R. The Differential Analyser. Nature. 1935, 135, 940-943. DOI: 10.1038/135940a0.

7) Hartree, D. R. and Hartree, W. Results of Calculations of Atomic Wave Functions. III. Results for Be, Ca and Hg. Proc. R. Soc. London, Ser. A. 1935, 149 (867), 210-231. DOI: 10.1098/rspa.1935.0058.

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