Truth of quantum mechanical variational methods.

What is variation method ?

Single electron is actually quantized particle, NOT as clouds.

It is said that Schrodinger equation of quantum mechanics can give very exact energy values of various atoms.
But we cannot solve the equations other than simple hydrogen-like atoms.
So we need to use some approximation.
In these approimations, variational methods can give most exact values about the energies of atoms.
(Fig.1) Quantum mecanical helium becomes chaotic and unstable.

But as you know, the quantum mechanical atoms can NEVER tell us true states of electron's motion.
A single electron is actually quantized particle with definite mass and charge -e according to experiments.
So it is inconsistent with the experimental results to insist one electron is spreading into all space as clouds.
Furthermore, if two electrons of stable helium has NO angular momentum (= 1s ), this helium model would be extremely unstable.
It means that quantum mechanical peturbation or variational methods cannot express real states of atoms at all.

Perturbation method of helium atom.

(Eq.1) Hamiltonian of helium atom.

Helium-like atom has two nagative electrons ( 1 , 2 ), and one positive nucleus (= +Ze ).
( In neutral helium atom, this Z is "2". )
Two first terms of Hamiltonian in Eq.1 represent kinetic energies of electron 1 and 2.
And the last term of Eq.1 is repulsive Coulomb interaction between two electrons.
( r₁₂ is the distance between two electrons. )
(Eq. 2) "1s" wavefunction of hydrogen-like atoms.

When we try to solve helium's Schrodinger equation, we use 1s wavefunction of hydrogen-like atom.
( Z₁ = 1 is hydrogen atom, and Z₁ = 2 is helium ion. )
Using two wavefunctions of Eq.2, we express neutral helium wavefunction, as follows,
(Eq. 3)

Substituting Eq.1 and Eq.3 into the energy (= W ) equation of
(Eq. 4)

we obtain
(Eq. 5)

When Z = 2, the total energy W of helium becomes -74.833 eV, which is different from the experimental value of -79.0051 eV.
Changing the atomic number Z, we get ground state energies of various helium-like atoms (ions), as follows,

(Table 1) Results in perturbation of helium-like atoms' energies.

Atoms	Experiment (eV)	new Bohr (eV)	Perturbation (eV)	Bohr Error	Pert Error
He	-79.0051	-79.0037	-74.8330	+0.001	+4.172
Li+	-198.093	-198.984	-193.885	-0.89	+4.21
Be2+	-371.615	-373.470	-367.362	-1.85	+4.25
B3+	-599.60	-602.320	-595.262	-2.72	+4.34

In Table 1, "perturbation" means quantum mechanical approximation and "new Bohr" means computed Bohr's orbits which are perpendicular to each other as shown in Top page or Fig.2.
(Fig.2) New Bohr model Helium.

New Bohr's helium model can give just exact ground state energy ( except only for small relativistic effect ).
And of course, Bohr's orbits can also give concrete electron's motion, and agree with de Broglie wave's nature and stability of helium.
So we can apply this model to 1s inner states of various other atoms.
As you notice, quantum mechanical pertubation methods show almost same error (= about +4 eV ), even when the atomic number is bigger. This is unnatural.

Variation method of helium atom.

(Eq. 6) Variation functions.

Next we try quantum mechanical variational methods of helium-like atoms.
Here we consider charge Z₁ as variational parameter and change it.
Substituting Eq.6 into Eq.4,
(Eq. 4)

We obtain,
(Eq. 7)

In Eq.7, "Z" is atomic number included in Hamiltonian of Eq.1.
So this Z is fixed value, which is different from variational parameter Z₁.
We vary Z₁ to minimize the total energy as follows,
(Eq. 8)

For example, in neutral helium ( Z = 2 ), when Z₁ = Z -5/16 = 2-5/16 = +1.6875, the total energy becomes the lowest value.
Substituting this Z₁ = Z - 5/16 into Eq.7, the total energy W becomes,
(Eq.9)

For example, substituting Z = 2 and E_1s = -13.606 into Eq.9, the total energy (E) becomes -77.49 eV, which is closer to the experimental value (-79.005 eV) than the perturbation of Table 1.
But this is the limit of this method about two-electron atom.

(Table 2) Results of variational methods.

Atoms	Experiment (eV)	new Bohr (eV)	Variation (eV)	Bohr Error	Vari Error
He	-79.0051	-79.0037	-77.4904	+0.001	+1.515
Li+	-198.093	-198.984	-196.552	-0.89	+1.55
Be2+	-371.615	-373.470	-370.019	-1.85	+1.60
B3+	-599.60	-602.320	-597.919	-2.72	+1.68

As shown in Table 2, both new Bohr model and quantum mechanical variational method give good results.
But as you notice, variational methods cannot get the exact values especially in neutral helium and lithium ion.
And it is very unnatural that the error ( about +1.5 eV ) doesn't change much, even when the atomic number is bigger.
These results demonstrate that the quantum mechanical variational method does NOT express the true configuration of neutral helium.
( Of course, originally, quantum mechanical model has no reality. )
As the atomic number becomes bigger, true errors ( based on the repulsive interaction between two electrons ) are hidden under the influence by the big positive nucleus.
(Eq.10) Quantum mechanical helium.

To get closer to the experimental values, we heve to use more than a thousand variational terms like Eq.10.
And to express repulsive force between electrons, these terms need to include variable ( r₁₂ ), which is the distance between electrons.
So these wavefunctions can NOT show real helium state at all.
The existence of r₁₂ means the probability at which we find electron 1 in some place is changing with respect to the position of electron 2.
So Eq.10 is NOT constant probability density wave.


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Reference: http://www7b.biglobe.ne.jp/~kcy05t/variational.html