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I asked a question previously about "why" it is the case the expanding the size of pi-conjugated systems decreases the required energy to excite an electron from a HOMO to a LUMO band: http://chemistry.stackexchange.com/questions/8910/why-does-the-energy-gap-for-transitions-shrink-with-the-size-of-the-pi-coWhy does the energy gap for π - π* transitions shrink with the size of the pi-conjugated system?. The user "Philipp" (and others) very patiently and kindly explained the reason for this by pointing out the expansion in the number of possible bonding and anti-bonding configurations for a polymethine-like pi-conjugated system.

However, there's something I'm missing that has to do with what is probably my lack of knowledge about chemistry. If we look at this image: http://i.stack.imgur.com/Ti3cx.png (provided by "Philipp") why is that that we have an energy gap between $\Psi_2$ and $\Psi_3$ smaller than the energy gap for a simpler, smaller pi-conjugated system where the HOMO corresponds to a single pair of p-orbitals in a bonding conformation and the LUMO corresponds to the same single pair of p-orbitals in an anti-bonding conformation (i.e. this system: http://i.stack.imgur.com/9cscv.png)?

My guess that this would be because the bonding interactions that exist in the larger pi-conjugated system mitigate the penalty for having the one anti-bonding interaction?

Update --- Could we justify the above guess using something like an overlap integral (if we know the shape of the two bonded p-orbitals forming a pi-bond)? If so, this would be quite intuitive.

I asked a question previously about "why" it is the case the expanding the size of pi-conjugated systems decreases the required energy to excite an electron from a HOMO to a LUMO band: http://chemistry.stackexchange.com/questions/8910/why-does-the-energy-gap-for-transitions-shrink-with-the-size-of-the-pi-co. The user "Philipp" (and others) very patiently and kindly explained the reason for this by pointing out the expansion in the number of possible bonding and anti-bonding configurations for a polymethine-like pi-conjugated system.

However, there's something I'm missing that has to do with what is probably my lack of knowledge about chemistry. If we look at this image: http://i.stack.imgur.com/Ti3cx.png (provided by "Philipp") why is that that we have an energy gap between $\Psi_2$ and $\Psi_3$ smaller than the energy gap for a simpler, smaller pi-conjugated system where the HOMO corresponds to a single pair of p-orbitals in a bonding conformation and the LUMO corresponds to the same single pair of p-orbitals in an anti-bonding conformation (i.e. this system: http://i.stack.imgur.com/9cscv.png)?

My guess that this would be because the bonding interactions that exist in the larger pi-conjugated system mitigate the penalty for having the one anti-bonding interaction?

Update --- Could we justify the above guess using something like an overlap integral (if we know the shape of the two bonded p-orbitals forming a pi-bond)? If so, this would be quite intuitive.

I asked a question previously about "why" it is the case the expanding the size of pi-conjugated systems decreases the required energy to excite an electron from a HOMO to a LUMO band: Why does the energy gap for π - π* transitions shrink with the size of the pi-conjugated system?. The user "Philipp" (and others) very patiently and kindly explained the reason for this by pointing out the expansion in the number of possible bonding and anti-bonding configurations for a polymethine-like pi-conjugated system.

However, there's something I'm missing that has to do with what is probably my lack of knowledge about chemistry. If we look at this image: http://i.stack.imgur.com/Ti3cx.png (provided by "Philipp") why is that that we have an energy gap between $\Psi_2$ and $\Psi_3$ smaller than the energy gap for a simpler, smaller pi-conjugated system where the HOMO corresponds to a single pair of p-orbitals in a bonding conformation and the LUMO corresponds to the same single pair of p-orbitals in an anti-bonding conformation (i.e. this system: http://i.stack.imgur.com/9cscv.png)?

My guess that this would be because the bonding interactions that exist in the larger pi-conjugated system mitigate the penalty for having the one anti-bonding interaction?

Update --- Could we justify the above guess using something like an overlap integral (if we know the shape of the two bonded p-orbitals forming a pi-bond)? If so, this would be quite intuitive.

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I asked a question previously about "why" it is the case the expanding the size of pi-conjugated systems decreases the required energy to excite an electron from a HOMO to a LUMO band: http://chemistry.stackexchange.com/questions/8910/why-does-the-energy-gap-for-transitions-shrink-with-the-size-of-the-pi-co. The user "Philipp" (and others) very patiently and kindly explained the reason for this by pointing out the expansion in the number of possible bonding and anti-bonding configurations for a polymethine-like pi-conjugated system.

However, there's something I'm missing that has to do with what is probably my lack of knowledge about chemistry. If we look at this image: http://i.stack.imgur.com/Ti3cx.png (provided by "Philipp") why is that that we have an energy gap between $\Psi_2$ and $\Psi_3$ smaller than the energy gap for a simpler, smaller pi-conjugated system where the HOMO corresponds to a single pair of p-orbitals in a bonding conformation and the LUMO corresponds to the same single pair of p-orbitals in an anti-bonding conformation (i.e. this system: http://i.stack.imgur.com/9cscv.png)?

My guess that this would be because the bonding interactions that exist in the larger pi-conjugated system mitigate the penalty for having the one anti-bonding interaction?

Update --- Could we justify the above guess using something like an overlap integral (if we know the shape of the two bonded p-orbitals forming a pi-bond)? If so, this would be quite intuitive.

I asked a question previously about "why" it is the case the expanding the size of pi-conjugated systems decreases the required energy to excite an electron from a HOMO to a LUMO band: http://chemistry.stackexchange.com/questions/8910/why-does-the-energy-gap-for-transitions-shrink-with-the-size-of-the-pi-co. The user "Philipp" (and others) very patiently and kindly explained the reason for this by pointing out the expansion in the number of possible bonding and anti-bonding configurations for a polymethine-like pi-conjugated system.

However, there's something I'm missing that has to do with what is probably my lack of knowledge about chemistry. If we look at this image: http://i.stack.imgur.com/Ti3cx.png (provided by "Philipp") why is that that we have an energy gap between $\Psi_2$ and $\Psi_3$ smaller than the energy gap for a simpler, smaller pi-conjugated system where the HOMO corresponds to a single pair of p-orbitals in a bonding conformation and the LUMO corresponds to the same single pair of p-orbitals in an anti-bonding conformation (i.e. this system: http://i.stack.imgur.com/9cscv.png)?

My guess that this would be because the bonding interactions that exist in the larger pi-conjugated system mitigate the penalty for having the one anti-bonding interaction?

I asked a question previously about "why" it is the case the expanding the size of pi-conjugated systems decreases the required energy to excite an electron from a HOMO to a LUMO band: http://chemistry.stackexchange.com/questions/8910/why-does-the-energy-gap-for-transitions-shrink-with-the-size-of-the-pi-co. The user "Philipp" (and others) very patiently and kindly explained the reason for this by pointing out the expansion in the number of possible bonding and anti-bonding configurations for a polymethine-like pi-conjugated system.

However, there's something I'm missing that has to do with what is probably my lack of knowledge about chemistry. If we look at this image: http://i.stack.imgur.com/Ti3cx.png (provided by "Philipp") why is that that we have an energy gap between $\Psi_2$ and $\Psi_3$ smaller than the energy gap for a simpler, smaller pi-conjugated system where the HOMO corresponds to a single pair of p-orbitals in a bonding conformation and the LUMO corresponds to the same single pair of p-orbitals in an anti-bonding conformation (i.e. this system: http://i.stack.imgur.com/9cscv.png)?

My guess that this would be because the bonding interactions that exist in the larger pi-conjugated system mitigate the penalty for having the one anti-bonding interaction?

Update --- Could we justify the above guess using something like an overlap integral (if we know the shape of the two bonded p-orbitals forming a pi-bond)? If so, this would be quite intuitive.

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