Molecular orbital theory question about energy level diagrams

[WMDhamnekar]

Hello,

The sigma $(\sigma)$ molecular orbitals are symmetrical around the bond-axis while pi $(\pi)$ molecular orbitals are not symmetrical. For example, the linear combination of 1s orbitals centered on two nuclei produces two molecular orbitals which are symmetrical around the bond-axis. Such molecular orbitals are of the $\sigma$ type and are designated as $\sigma1s$ and $\sigma^*1s$ [Fig. 4.20(a)].

If internuclear axis is taken to be in the z-direction, it can be seen that a linear combination of $2p_z$ - orbitals of two atoms also produces two sigma molecular orbitals designated as $\sigma 2p_z$ and $\sigma^*2p_z .$ [Fig. 4.20(b)]

Molecular orbitals obtained from $2p_x$ and $2p_y$ orbitals are not symmetrical around the bond axis because of the presence of positive lobes above and negative lobes below the molecular plane. Such molecular orbitals, are labelled as $\pi$ and $\pi^*$ [Fig. 4.20(c)].

A $\pi$ bonding MO has larger electron density above and below the inter -nuclear axis. The $\pi^*$ antibonding MO has a node between the nuclei.

I want explanations about the enclosed areas in the following energy diagram.

In 1s orbital of two atoms, what does +and dot represent?

In $2p_z$ and $2p_x$ orbitals of two atoms, what does +,-, and dot between them represent?

Why 1s orbital doesn't have -(minus) sign along with + and dot symbols?

 

[I like Serena]

In 1s orbital of two atoms, what does +and dot represent?

Let's first introduce:

A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements made on the system can be derived from it. ​

In particular it means that a wave function has a phase, and we can derive the probability that a particle is somewhere in space and time from it.

As an example we have $\ce{H2}$, which has 2 atoms. Each atom has an 1s orbital with 1 electron.

The dot inside an orbital represents an electron; its location is where the electron is on average.
The + inside the orbital represents the phase of the wave function of the electron
The +/- between the orbitals represent whether the orbitals have the same phase, or have opposite phase when they are joined.

In the picture of the $\sigma 1s$ bonding we see only 1 dot in the center, which are actually 2 dots on top of each other.
It's just that the 2 electrons have the same average location in the center.

Wave functions with opposite phase cancel each other out.
That is why the $\sigma^*1s$ bonding shows that the probability for either electron to be in the middle is zero.

In $2p_z$ and $2p_x$ orbitals of two atoms, what does +,-, and dot between them represent?

Same thing. We do see the difference that the 2 electrons have distinct average locations now.

Example is $\ce{O2}$, which has both a $\sigma 2p$ bonding and a $\pi 2p$ bonding.

Why 1s orbital doesn't have -(minus) sign along with + and dot symbols?

The - sign inside the orbital represents a phase of the wave function that is opposite to the phase where a + sign is shown.
Note that in the middle they cancel each other out, which is why a p-orbital looks like 2 lobes.

In the case of the 1s orbital there is no area that has an opposite phase with respect to another area.
That's why the s-orbital is spherical.
The choice of the sign in the orbital is therefore arbitrary. We could also have chosen a - sign instead.

 

[WMDhamnekar]

Let's first introduce:

A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements made on the system can be derived from it. ​

In particular it means that a wave function has a phase, and we can derive the probability that a particle is somewhere in space and time from it.

As an example we have $\ce{H2}$, which has 2 atoms. Each atom has an 1s orbital with 1 electron.

The dot inside an orbital represents an electron; its location is where the electron is on average.
The + inside the orbital represents the phase of the wave function of the electron
The +/- between the orbitals represent whether the orbitals have the same phase, or have opposite phase when they are joined.

In the picture of the $\sigma 1s$ bonding we see only 1 dot in the center, which are actually 2 dots on top of each other.
It's just that the 2 electrons have the same average location in the center.

Wave functions with opposite phase cancel each other out.
That is why the $\sigma^*1s$ bonding shows that the probability for either electron to be in the middle is zero.
Same thing. We do see the difference that the 2 electrons have distinct average locations now.

Example is $\ce{O2}$, which has both a $\sigma 2p$ bonding and a $\pi 2p$ bonding.
The - sign inside the orbital represents a phase of the wave function that is opposite to the phase where a + sign is shown.
Note that in the middle they cancel each other out, which is why a p-orbital looks like 2 lobes.

In the case of the 1s orbital there is no area that has an opposite phase with respect to another area.
That's why the s-orbital is spherical.
The choice of the sign in the orbital is therefore arbitrary. We could also have chosen a - sign instead.

Hello,

Do you mean to say that +/- signs in the orbitals indicates intrinsic spin angular quantum number? Spin angular momentum of the electron — a vector quantity, can have two orientations relative to the chosen axis. These two orientations are distinguished by the spin quantum numbers $m_s$ which can take the values of $+\frac12$ or $–\frac12$. These are called the two spin states of the electron and are normally represented by two arrows, ↑ (spin up) and ↓ (spin down). Two electrons that have different $m_s$ values (one $+\frac12$ and the other $–\frac12$) are said to have opposite spins.

 

[I like Serena]

Hello,

Do you mean to say that +/- signs in the orbitals indicates intrinsic spin angular quantum number? Spin angular momentum of the electron — a vector quantity, can have two orientations relative to the chosen axis. These two orientations are distinguished by the spin quantum numbers $m_s$ which can take the values of $+\frac12$ or $–\frac12$. These are called the two spin states of the electron and are normally represented by two arrows, ↑ (spin up) and ↓ (spin down). Two electrons that have different $m_s$ values (one $+\frac12$ and the other $–\frac12$) are said to have opposite spins.

I did mean that the +/- signs in the orbitals represent the phase of the associated wave probability function.
The shape of the orbital is the representation of that wave probability function after all.
To be honest, I'm not sure if or how it may be tied to electron spin.