# Is This Calculation of Sz for Molecular Orbitals Correct?

• gazepdapi1
In summary, the determinant can be simplified to psi=1/sqrt(2)psi(b)^2[alpha(1)beta(2) - alpha(2)beta(1)].
gazepdapi1
[SOLVED] molecular orbital wavefunctions

http://img255.imageshack.us/img255/7012/37626561fb6.jpg

starting with

Sz(total) = Sz1 + Sz2
Sz(alpha) = +(1/2)hbar(alpha)
Sz(beta) = -(1/2)hbar(beta)

I first found the determinant and then simplified

psi = 1/sqrt(2)psi(b)^2[alpha(1)beta(2) - alpha(2)beta(1)]

So,

Sz1psi = 1/sqrt(2)psi(b)^2[beta(2)(1/2)h(bar)alpha(1) + alpha(2)(1/2)h(bar)beta(1)]

Sz1psi = 1/sqrt(2)psi(b)^2[-alpha(1)(1/2)h(bar)beta(2) - beta(1)(1/2)h(bar)alpha(2)]

So...
Sztotal = Sz1 + Sz2 = 0

Can someone please verify if this is correct?
I would appreciate it.
thank you

Last edited by a moderator:
I realize it's hard to read without the LaTeX, but no one can check if it's correct or not?

yes it seems to be ok,

learn LaTeX, it is worth it :)

nertil1 said:
http://img255.imageshack.us/img255/7012/37626561fb6.jpg

starting with

Sz(total) = Sz1 + Sz2
Sz(alpha) = +(1/2)hbar(alpha)
Sz(beta) = -(1/2)hbar(beta)

I first found the determinant and then simplified

psi = 1/sqrt(2)psi(b)^2[alpha(1)beta(2) - alpha(2)beta(1)]

So,
$$Sz_1 \psi = 1/\sqrt{2} ~\psi_b^2 (\beta_2 ~ (1/2) \hbar \alpha_1 + \alpha_2 ~(1/2) \hbar \beta_1)$$
I don't understand that result. Shouldn't the second term be negative since it's a beta?

$$Sz_1 \psi = 1/\sqrt{2} ~\psi_b^2 ~(-\alpha_1 ~ (1/2) \hbar \beta_2 - \beta_1~(1/2) \hbar \alpha_2)$$
You means Sz2 I think here. But the second term should be positive since it's an alpha, no?

Last edited by a moderator:
No because when you multiply by the -(1/2)hbar, the beta portion (the second term) becomes a positive, the opposite goes for the alpha, I think, and yes it should be an Sz2.

nertil1 said:
No because when you multiply by the -(1/2)hbar, the beta portion (the second term) becomes a positive, the opposite goes for the alpha, I think, and yes it should be an Sz2.

you are absolutely right, my mistake. For some reason, when i replied to your post, I forgot about the Slater determinant and I was thinking about a symmetric wavefunction. Sorry about that.

Thank you all for your help

## 1. What is a molecular orbital wavefunction?

A molecular orbital wavefunction is a mathematical representation of the probability of finding an electron in a given region of a molecule. It is a combination of atomic orbitals from individual atoms that form the molecule.

## 2. How are molecular orbital wavefunctions calculated?

Molecular orbital wavefunctions are calculated using mathematical equations and quantum mechanics principles. These calculations take into account the positions and energies of the individual atomic orbitals and the interactions between them.

## 3. What is the significance of molecular orbital wavefunctions?

Molecular orbital wavefunctions provide insight into the electronic structure and properties of molecules. They can help predict the reactivity, stability, and other properties of a molecule, which is crucial in fields such as chemistry and materials science.

## 4. How do molecular orbital wavefunctions differ from atomic orbitals?

Molecular orbital wavefunctions describe the behavior of electrons in a molecule as a whole, while atomic orbitals describe the behavior of electrons around individual atoms. Additionally, molecular orbital wavefunctions are delocalized over the entire molecule, while atomic orbitals are localized to a specific atom.

## 5. Can molecular orbital wavefunctions be visualized?

Yes, molecular orbital wavefunctions can be visualized using various computational tools and software. These visualizations can aid in understanding the shape, symmetry, and energy levels of the molecular orbitals within a molecule.

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