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http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/helium.html#c1

Thank you!

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- Thread starter MonsieurWise
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- #1

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http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/helium.html#c1

Thank you!

- #2

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Thank you very much, and I'm very sorry if I seems too demanding...but I have only 1 day left before the Science Fair...

Thank you again.

- #3

alxm

Science Advisor

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Spin-orbit coupling probably.

- #4

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- #5

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Oh! Thanks!

Is there a good explanation for this that i can find on the internet?

Is there a good explanation for this that i can find on the internet?

- #6

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"Intermediate Quantum Mechanics" by Hans Bethe and Roman Jackiw. Most

books that discuss the Hartree-Fock method will have discussions on

two-electron atoms.

- #7

alxm

Science Advisor

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Wheeler (1986):

Bob S, I'd have to disagree with the statement it can "only be calculated approximately". It cannot be calculated*analytically* by any *known* method. But it *can* be calculated to arbitrary precision.

There probably does exist an exact mathematical solution, at least for the nonrelativistic equation. They did find one for the classical three-body problem. It's unlikely though, that the mathematically exact solution is something which is itself easier to calculate than a numerical method.

The nonrelativistic ground state energy is -2.903724 a.u. Relativistic: -2.903855 (a difference by ~0.3 J/mol.. not much by any standard)

Just to compare some methods, since I have the numbers handy:

Hartree-Fock: -2.86

Thomas-Fermi model: -2.19

1st order perturbation theory: -2.75

The simplest, most exact method was the one Hylleraas used as early as 1929; a direct variational-principle approach, which is completely exact (nonrelativistic) for a given basis.

A single parameter (analytically solvable): -2.85

3 parameters: -2.847 (Hylleraas 1929)

6 parameters: -2.902 (Hylleraas 1929)

14 parameters: -2.90370 (Chandrasekhar 1955)

10257 parameters: -2.90372437703411959831115924519440444 (Schwartz, 2002)

Which is absurd, really, since relativistic effects come into play. Also, Helium is essentially a bit of a special case; Hylleraas method doesn't scale well to systems of more than two electrons. In other cases you'd probably use full-CI.

"But", Bohr protested, "nobody will believe me unless I can explain every atom and every molecule."; Rutherford was quick to reply, "Bohr, you explain hydrogen and you explain helium and everybody will believe the rest."

Bob S, I'd have to disagree with the statement it can "only be calculated approximately". It cannot be calculated

There probably does exist an exact mathematical solution, at least for the nonrelativistic equation. They did find one for the classical three-body problem. It's unlikely though, that the mathematically exact solution is something which is itself easier to calculate than a numerical method.

The nonrelativistic ground state energy is -2.903724 a.u. Relativistic: -2.903855 (a difference by ~0.3 J/mol.. not much by any standard)

Just to compare some methods, since I have the numbers handy:

Hartree-Fock: -2.86

Thomas-Fermi model: -2.19

1st order perturbation theory: -2.75

The simplest, most exact method was the one Hylleraas used as early as 1929; a direct variational-principle approach, which is completely exact (nonrelativistic) for a given basis.

A single parameter (analytically solvable): -2.85

3 parameters: -2.847 (Hylleraas 1929)

6 parameters: -2.902 (Hylleraas 1929)

14 parameters: -2.90370 (Chandrasekhar 1955)

10257 parameters: -2.90372437703411959831115924519440444 (Schwartz, 2002)

Which is absurd, really, since relativistic effects come into play. Also, Helium is essentially a bit of a special case; Hylleraas method doesn't scale well to systems of more than two electrons. In other cases you'd probably use full-CI.

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- #8

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- #9

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If anyone is interested, and has access to Physical Review journals, search for articles by G. W. F. Drake. He did a lot of work (mostly in the 1970's and 1980's ?) doing precision calculations of helium energy levels. For example:

http://prola.aps.org/abstract/PRL/v59/i14/p1549_1" [Broken]

"New variational techniques for the 1snd states of helium"

http://prola.aps.org/abstract/PRL/v59/i14/p1549_1" [Broken]

"New variational techniques for the 1snd states of helium"

New variational techniques are described which yield a factor of 1000 improvement in accuracy for the energies of the 1snd^{1}D and^{3}D states of helium up to n=8. Convergence to better than ±10 kHz is obtained, making possible high-precision comparisons with experiment for fine structure and singlet-triplet splittings. The comparisons are sensitive to QED, relativistic recoil, and second-order mass-polarization corrections.

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- #10

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Thanks you guys a lot ^^

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