(Expanding my comments into an answer)

I am afraid you are wasting your time. This is a 3 body problem. There is no closed form analytic solution. Take a look at Quantum Three-Body Problem amongst others if you want to see some of the fun and games that are required here for a good approximate, numerical solution.

More generally this is the usual problem in Chemistry - few if any problems are exactly solvable, even the solution of the Schrödinger equation for the H atom is an approximation if we dig a bit deeper than we usually do. So we make models which while not exact give us some insight to help understand what is going on. Screening is one such model. Admittedly it is a very crude one, one we learn early in our chemical career when we lack the tools to do better, but that doesn't stop it being useful so long as we understand it is a model and don't try to use it where it is not applicable.

Finally about your maths where you say "I took the initial condition" you mean "I used the boundary condition" - initial conditions are for time dependent problems, which this ain't. I also can't justify the boundary conditions you propose. They should be something along the lines of

• When both electrons are far from the nucleus the wavefunction should approach zero
• The solution should be regular - basically acceptable solutions should not "blow up" due to the infinities in the Coulomb operators at $r_{12}=0$

These are probably most easily written by a change of variables to $r_1+r_2$ and $r_1-r_2$, but given this is all ultimately an exercise in futility I can't be bothered to think harder.

(Expanding my comments into an answer)

I am afraid you are wasting your time. This is a 3 body problem. There is no closed form analytic solution. Take a look at Quantum Three-Body Problem amongst others if you want to see some of the fun and games that are required here for a good approximate, numerical solution.

More generally this is the usual problem in Chemistry - few if any problems are exactly solvable, even the solution of the Schrödinger equation for the H atom is an approximation if we dig a bit deeper than we usually do. So we make models which while not exact give us some insight to help understand what is going on. Screening is one such model. Admittedly it is a very crude one, one we learn early in our chemical career when we lack the tools to do better, but that doesn't stop it being useful so long as we understand it is a model and don't try to use it where it is not applicable.

Finally about your maths where you say "I took the initial condition" you mean "I used the boundary condition" - initial conditions are for time dependent problems, which this ain't. I also can't justify the boundary conditions you propose. They should be something along the lines of

• When both electrons are far from the nucleus the wavefunction should approach zero
• The solution should be regular - basically acceptable solutions should not "blow up" due to the infinities in the Coulomb operators at $r_{12}=0$

These are probably most easily written by a change of variables to $r_1+r_2$ and $r_1-r_2$, but given this is all ultimately an exercise in futility I can't be bothered to think harder.

(Expanding my comments into an answer)

I am afraid you are wasting your time. This is a 3 body problem. There is no closed form analytic solution. Take a look at Quantum Three-Body Problem amongst others if you want to see some of the fun and games that are required here for a good approximate, numerical solution.

More generally this is the usual problem in Chemistry - few if any problems are exactly solvable, even the solution of the Schrödinger equation for the H atom is an approximation if we dig a bit deeper than we usually do. So we make models which while not exact give us some insight to help understand what is going on. Screening is one such model. Admittedly it is a very crude one, one we learn early in our chemical career when we lack the tools to do better, but that doesn't stop it being useful so long as we understand it is a model and don't try to use it where it is not applicable.

Finally about your maths where you say I took the initial condition" you mean "I used the boundary condition" - initial conditions are for time dependent problems, which this ain't. I also can't justify the boundary conditions you proposes. They should be something along the lines of

• When both electrons are far from the nucleus the wavefunction should approach zero
• The solution should be regular - basically acceptable solutions should not "blow up" due to the infinities in the Coulomb operators at $r_{12}=0$

These are probably most easily written by a change of variables to $r_1+r_2$ and $r_1-r_2$, but given this is all ultimately an exercise in futility I can't be bothered to think harder.

(Expanding my comments into an answer)

I am afraid you are wasting your time. This is a 3 body problem. There is no closed form analytic solution. Take a look at Quantum Three-Body Problem amongst others if you want to see some of the fun and games that are required here for a good approximate, numerical solution.

More generally this is the usual problem in Chemistry - few if any problems are exactly solvable, even the solution of the Schrödinger equation for the H atom is an approximation if we dig a bit deeper than we usually do. So we make models which while not exact give us some insight to help understand what is going on. Screening is one such model. Admittedly it is a very crude one, one we learn early in our chemical career when we lack the tools to do better, but that doesn't stop it being useful so long as we understand it is a model and don't try to use it where it is not applicable.

Finally about your maths where you say "I took the initial condition" you mean "I used the boundary condition" - initial conditions are for time dependent problems, which this ain't. I also can't justify the boundary conditions you propose. They should be something along the lines of

• When both electrons are far from the nucleus the wavefunction should approach zero
• The solution should be regular - basically acceptable solutions should not "blow up" due to the infinities in the Coulomb operators at $r_{12}=0$

These are probably most easily written by a change of variables to $r_1+r_2$ and $r_1-r_2$, but given this is all ultimately an exercise in futility I can't be bothered to think harder.