Help with a quantum chemistry problem

[chemstudent09]

I'm really struggling on even getting started on this problem. I have read the chapter several times now and I have tried to understand the explanations in the chapter. But I can't even get this problem started. All I ask is if anyone knows how to do this, can you just get me started or point me in the right direction so I can then take it from there?

Thanks for any help.

___________
Prove the theorem <K|H|L> = (N!)^1/2 <K^HP|H|L>

where |K^HP> is the Hartree product corresponding to the determinant |K>, i.e.

|K> = |X_m(X₁)X_n(X₂)...>

and

|K^HP> = X_m(X₁)X_n(X₂)...

Prove this theorem
_______________

H = O₁ + O₂

O₁ = h(1) + h(2) + ... + h(N)

O₂ = r₁₂^-1 + r₁₃^-1 + r₁₄^-1 + ... + r₂₃^-1 + r₂₄^-1 + ... + r_N-1^-1, _N

This problem is from "Modern Quantum Chemistry" by Szabo & Ostlund for anyone who is wondering.

 

[Nino MMP]

A good place to start is by writing out the left-hand side and right-hand side of the equation. Then, you can use the definitions of |K> and |KHP> to expand each side. On the left-hand side, you will use the definition of the Hamiltonian H to expand it. On the right-hand side, you will use the properties of the Hartree product to rewrite it. After that, you can use the orthogonality of the Slater determinants to simplify the equation. Finally, you can use the commutator relation [H,X] = h(X) to simplify any terms involving the Hamiltonian.

 

[Blueshift5]

Hello, I am a quantum chemist and I would be happy to help you with this problem. It is understandable that you are struggling with it, as quantum chemistry can be quite challenging. First, let me assure you that you are on the right track by reading the chapter and trying to understand the explanations. That is the first step towards solving any problem in science.

Now, let's break down the problem and see how we can approach it. The theorem you have been asked to prove is known as the Slater-Condon rules, which relate the matrix elements of a general operator to those of a simpler operator. In this case, we have the operator H, which can be written as the sum of two simpler operators, O1 and O2.

To start, let's look at the simpler operator O1. This is the one-electron operator, which represents the kinetic energy and potential energy of a single electron. The notation h(1), h(2), etc. stands for the Hamiltonian operator for each electron. So, O1 can be written as the sum of all these Hamiltonian operators.

Next, we have the operator O2, which represents the electron-electron interaction. It is a two-electron operator and is written in terms of the inter-electronic distance, r, between two electrons. Note that r12-1 represents the inverse of the distance between the first and second electrons, and so on.

Now, the key to solving this problem is to use the Hartree product, which is a way of representing a determinant in quantum chemistry. In this case, we have a determinant |K> that is represented by the Hartree product |KHP>. We need to use this Hartree product to prove the Slater-Condon rules.

The first step is to expand the determinant |K> in terms of the Hartree product |KHP>. This will involve using the anti-symmetry property of determinants. Once you have expanded |K>, you will see that the term <K|H|L> can be written as a sum of terms involving the operators O1 and O2.

Next, you will need to use the properties of the operators O1 and O2 to simplify the terms in the sum. The key is to use the fact that the operators O1 and O2 commute with each other. This will allow you to rearrange the terms in the sum and eventually prove the desired relation.