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I have a question about exchange integral in quantum chemistry.

Usually, we learn that the sign of exchange integral is positive, so a triplet state has lower energy than singlet state.

Is it possible that the exchange integral has a negative sign? If possible, in what case the exchange integral becomes negative?

Thank you!

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If the spatial part of the orbitals are allowed to overlap the singlet may end up below the triplet state. This happens in antiferromagnets, such as the famous cuprates.

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$\textbf{Note :}$ This is a preliminary vague answer to OP's question. Will soon be expanded with reasonable rigour and details.

Exchange integral can be shown to be positive always (Hint : Use fourier representation of coulomb potential). For determining whether a singlet or triplet state has lower energy, one need also to worry about pairing energy (energy gain you can have by placing both electrons in same spatial orbital). Think why most diatomic molecules except few like oxygen molecule have electronic spin singlet states as ground states.

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