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.(adsbygoogle = window.adsbygoogle || []).push({});

[tex]\varphi_{i\lambda\alpha}(\vec{r})=

\sum\limits_{p=1}^{N}\chi_{p\lambda\alpha}(\vec{r})

C_{i\lambda p},[/tex]

where

[tex]\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].[/tex]

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

Note, that in the above sum's there is no dependence [tex]m_{\lambda\alpha}[/tex]

from p!?

The question is: Does it means that in some MO ([tex]\sigma[/tex] or [tex]\pi[/tex], 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 [tex]N_{2}[/tex] all electrons have

[tex]m=0[/tex] while in the case of molecule [tex]O_{2}[/tex] [tex]m=1[/tex]. Why is that? Is it possible that some electrons have [tex]m=-1[/tex]?

If somebody have any sugestions about this question, I will be very appreciate.

Regards

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

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