Homonuclear diatomic molecule (N_2 and O_2)- question about magnetic quantum number

1. Jan 4, 2008

grunf

I have one question about wave functions and molecular orbitals (MO). In the computational quantum chemistry scientists often use Linear Combination Atomic Orbitals (LCAO) to construct MO.

$$\varphi_{i\lambda\alpha}(\vec{r})= \sum\limits_{p=1}^{N}\chi_{p\lambda\alpha}(\vec{r}) C_{i\lambda p},$$
where
$$\chi_{p\lambda\alpha}(\vec{r})=2^{-1/2} \left[\chi_{n_{\lambda p} l_{\lambda p}m_{\lambda\alpha}}(\vec{r}_{A}) +\sigma_{\lambda}\chi_{n_{\lambda p} l_{\lambda p}m_{\lambda\alpha}}(\vec{r}_{B})\right].$$

Here $$\chi_{n_{\lambda p} l_{\lambda p}m_{\lambda\alpha}}(\vec{r}_{J})$$ (J=A,B) are the usual Slater-type functions (STF-s) centered on A and B, respectively. $$\lambda$$ is the symmetry species $$(\textrm{for example } \sigma$$ or $$\pi)$$ and $$\alpha$$ (for example g or u) is the subspecies of symmetry $$\lambda$$.

Note, that in the above sum's there is no dependence $$m_{\lambda\alpha}$$
from p!?

The question is: Does it means that in some MO ($$\sigma$$ or $$\pi$$, for example) all electrons must have the same magnetic quantum number? Why is that?
Is there some symmetry, some rule, some law or some common property for all electrons in the same MO.

In some papers I have found that for LCAO molecule $$N_{2}$$ all electrons have
$$m=0$$ while in the case of molecule $$O_{2}$$ $$m=1$$. Why is that? Is it possible that some electrons have $$m=-1$$?

Regards

Last edited: Jan 4, 2008
2. Jan 6, 2008

muppet

I have absolutely no idea at all about computational chemistry or STFs I'm afraid... (which I think is what may have put a lot of physicists off trying to answer!)
What I can tell you is that magnetic quantum numbers can range over the integers (-L, L) where L is the orbital quantum number, so yes electrons can have a magnetic moment of minus one. See here for a little more info, or google 'magnetic quantum numbers'.

3. Jan 6, 2008

olgranpappy

all? really? All fourteen electrons?

4. Jan 7, 2008

grunf

I was quite unaccurate. In this case I was thinking on the electron in the Highest Ocuppied Molecular Orbital (HOMO). My mistake

5. Jan 7, 2008

grunf

Thanx for the tips. I already know that. I am sure that is the key in the some molecular symmetries, but I am not sure. I am looking for some rule, law... Thanx again