*(Fig.1) No solution → just "choose" trial functions ! = useless*

Schrodinger equation of two-electron helium contains interelectronic Coulomb energy. So it has No solution of helium.

They just choose **artifical** trial function as "*imaginary*" solution.

"Choosing" convenient hypothetical solution out of **infinite** choices means Schrödinger equation has **no** ability to predict multi-electron atoms.

And it's **impossible** to try *infinite* kinds of trial wavefunctions and find the one giving the lowest energy in them.

*(Fig.2) Insert hydrogen solution e ^{-r/a} into Schrödinger equation ↓*

Only one-electron Hydrogen atom has its solution in Schrödinger equation.

Kinetic energy term just **cancel**s out Coulomb energy term to *satisfy* Schrödinger equation in all electron's position r.

Inserting one of hydrogen solutions (ex. e^{-r/a} ) into Schrödinger equation, you find its *identical* equation **holds** true in **all** position r.

*(Fig.3) its kinetic energy terms must cancel all Coulomb terms.*

Why multi-electron atoms such as Helium **don't** have any exact solutions in their Schrödinger equations ?

If Helium has a solution (= ψ_{He} ), when you insert it into Schrödinger equations, its *kinetic* energy terms need to **cancel** all Couloumb energy terms to get constant total energy E.

But there is a **serious** problem in Schrödinger wavefunction.

Schrödinger wavefunction always spreads **all** space. Its helium approximately consists of two hydrogen wavefunctions, so each electron of helium can move independently everywhere over spreading wavefunction !

It means these three different Coulomb terms must be **independently** canceled by its corresponding kinetic energy term → three "*fake*" hydrogens !?

*(Fig.4) ↓ Helium atom can be divided into three unreal atoms ?*

If Helium atom has its solution, it means Schrödinger equation *holds* true in **all** its electrons' positions (= r_{1}, r_{2} ).

In Schrödinger equation helium, three Coulomb energy terms can be changed independently.

So each Coulomb energy term must be cancelled out **independently** by its corresponding kinetic energy term.

Coulomb term ① must be canceled by its kinetic term ① (← hydrogen ? )

Coulomb term ② must be canceled by its kinetic term ② (← hydrogen ? )

Coulomb term ③ must be canceled by its kinetic term ③ (← **unreal** atom ! )

This means Helium atom can be divided into three *independent* atoms which have **NO** relation to each other, if helium has solution !

Divided (= independent ) energy term **cannot** express the property specific to the *whole* atom or molecule.

*(Fig.5) ①,② = hydrogen-like ? ③ = unreal electron-electron atom !*

If Helium has its solution in Schrödinger equation, three different Coulomb terms have to be cancelled out independently by their kinetic energy terms.

It means, if a helium has solution, a helium atom can be **divided** into *three* independent atoms.

①,② are hydrogen-like atoms. ③ is a electron-electron atom which **doesn't** exist.

Furthermore, each of these three strange independent atoms **doen't** contain other atoms' information, which **cannot** describe the **whole** atom or molecule containing all informations !

For example, average electron-electron energies are different in different atoms or molecules due to their different nuclei.

But the above independent unreal atom (= ③ contains only electron-electron information ) cannot express it !

*(Fig.6) ↓ Helium approximate quantum method is just fake !*

In Helium atom, the first thing you do is choose trial function as "**fake**" Helium solution, and insert it into Schrödinger equation.

After putting Schrödinger equation between two tial function (= ψ_{He} ), you integrate it instead of solving it, because you **cannot** solve it.

The total energy E by this method is **not** true Helium energy.

You just repeat this ( choose different trial functions → insert → integral ) until you can get convenient fake energy.

You have to try *infinite* different trial functions as Helium fake solution and compare them to find the solution which gives the **lowest** ground state energy E.

But of course, it's **impossible** to try infinite kinds of trial function to find the lowest Helium energy E. So Schrödinger equation is **useless** in Helium.

2017/6/2 updated. Feel free to link to this site.