[Enthalpy]

For butadiene, the lowest energy is ++++, the second lowest ++--, second highest +--+, and highest +-+-.
http://www.chem.ucla.edu/harding/butadieneorbitals.html
http://www.chem.ucalgary.ca/courses/351/Carey5th/Ch10/ch10-6-2.html
which corresponds to the general idea that electrons have a lower kinetic energy when the sign or phase changes slowly over distance - in other words, the energy usually increases with the wave vector, similar to E=(h/2pi)²k²/2m (but not always, especially at the top of a crystal's energy band).

Spin has only indirect relations with that. We're speaking about electron pairs in ground state, that is with ++++ and ++-- occupied in butadiene. From time to time, I've read attempts to explain chemical bonds by magnetic moment attraction, and this is flawed. Spin permits two electron to occupy the same favourable orbital, but the magnetic energy is far too weak to explain the bond.

[Irlanur]

so what has this to do with the previous discussion?

[orgopete]

For butadiene, the lowest energy is ++++, the second lowest ++--, second highest +--+, and highest +-+-.
http://www.chem.ucla.edu/harding/butadieneorbitals.html

Spin has only indirect relations with that. We're speaking about electron pairs in ground state, that is with ++++ and ++-- occupied in butadiene. From time to time, I've read attempts to explain chemical bonds by magnetic moment attraction, and this is flawed. Spin permits two electron to occupy the same favourable orbital, but the magnetic energy is far too weak to explain the bond.

I was really trying to explain how the poster may have been confused. I'm not sure this is helping. Electrons are given four designations, shell, subshell, magnetic, and spin. In order for a bond to form, the spins must be anti-parallel. If one is drawing the lowest energy form of butadiene, the all pink energy state, the pi-electron spins must all be anti-parallel, or I presume so. You may see this written with up and down arrows, but you also see it written with + and - signs in the orbitals. This is different than the psi designation, though I think it is not unrelated.

If this is in no way related to the poster's question, then I'm sorry for creating any confusion. Sometimes, this does come up. I believe the psi states correspond to the different possible bonding and spin states for the electrons.

[Enthalpy]

It is possible - only you can decide - that you mix up the spin with the sign of the orbitals.
The + and - tell the sign of the molecular orbital near the individual atoms. Each of these orbitals contains two electrons of opposed spin, so these signs are NOT the spin orientation.

In the lowest molecular orbital of butadiene, ++++ means that the pi orbital has identical signs at all four carbons, for instance plus above the molecule and minus below, if the carbon chain is horizontal. Within this ++++ orbital, you find two electrons, one up and the other down. These signs do NOT describe the spin.

Same for the other occupied orbital at rest, ++--, which contains one electron pair with up and down orientation, both being spread identically among the four carbons.

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In addition to the excited ethylene example, where two atoms keep together despite some electrons are not paired, you have molecules with some unpaired electrons in the ground state. One example is the normal oxygen molecule 0₂ (=triplet)
http://en.wikipedia.org/wiki/Triplet_oxygen
which has two more electrons than the binding molecular orbitals can contain, so these electrons go to antibinding ones, one electron to each orbital because both orbitals have the same energy and electrons repel an other.

So normal oxygen at rest is a case where
- some electrons must populate antibonding orbitals
- two electrons are not paired