Still confused about (anti)symmetrization of wavefunctions

[HAYAO]

I am currently confused about anti-symmetrization of wavefunctions. In a thread "Still confused about super position and mixed state", kith responded that anti-symmetrization was not done thus the none of the bras and kets shows any properties of a possible state of a molecule I have mentioned in the thread OP.

1) So then how can it be done? The wavefunctions used in the problem has no information about what kind of equation it is composed of. So we have no way of knowing how to anti-symmetrize the wavefunctions.

2) Couldn't the wavefunctions used in the thread be considered as wavefunction that have already been antisymmetrized? Otherwise, quite some quantum chemical theories (such as spin-orbit coupling, excitonic interactions, ligand field, etc.) would fails because most textbooks and papers regarding these theory typically don't consider antisymmetrization or at least don't mention it.

3) Practically, how is anti-symmetrization done in general? It seems to me that anti-symmetrization is a tedious work considering how many electrons are included in a molecule. Both Antisymmetrizer and Slater determinants seems to be tons of work. Are there any way to come around this tremendous amount of work?

 

[vanhees71]

Have a look at this

https://en.wikipedia.org/wiki/Slater_determinant

This explains the basics about antisymmetrization in the context of wave mechanics.

 

[HAYAO]

Have a look at this

https://en.wikipedia.org/wiki/Slater_determinant

This explains the basics about antisymmetrization in the context of wave mechanics.

Yes, and I am well aware of this.

I am, however, not aware of the actual calculation when handling relatively larger molecules. For example, mathematically, no analytical solution is available for Slater determinants for matrix over 4x4 unless in special cases (no formula is available if my memory is right). But typical organic molecules surely have electrons over 4.

The specific case I have presented above is a general representation of excitonic interaction. So each of the wavefunctions presented in the thread shouldn't be able to be anti-symmetrized due to lack of information. Despite this, kith has responded that none of the wavefunction can actually represent a molecule. That means it is absolutely impossible to generalize any quantum chemical theory. But the fact is, there are tons of famous theory that uses wavefunction that is not defined specifically (thus antisymmetrization is not possible). If kith is right, then all these theory cannot be true. I feel that there is some contradiction or confusion here that I need clarification of.

 

[vanhees71]

Ok, can you point me to a concrete example, where non-antisymmetrized wave functions for many-fermion systems are successfully used? For atomic and molecular structure, for sure the Pauli principle is mandatory. How else can one, e.g., understand the periodic table of elements?

 

[HAYAO]

Ok, can you point me to a concrete example, where non-antisymmetrized wave functions for many-fermion systems are successfully used? For atomic and molecular structure, for sure the Pauli principle is mandatory. How else can one, e.g., understand the periodic table of elements?

For example, page 2491 of http://www.itc.univie.ac.at/~wichard/JCP120_2490.pdf from Journal of Chemical Physics uses wavefunction $\left | S_{i_{A}} \right \rangle$ and $\left | S_{j_{B}} \right \rangle$ for singlet excited state of molecule A and B. These two wavefunction is clearly a general representation of unknown molecules A and B, and thus there is no way to antisymmetrize this. We lack the information of the actual wavefunction to antisymmetrize. I always aｌthought that these wavefunctions are used under the premise that they are already antisymmetrized but according to kith's post, he argues otherwise.

Of course, I am well aware of the Pauli principle and its importance. That is why I was extremely confused when kith pointed out that none of the wavefunction used in the thread I've mentioned in OP represent any of the state of the molecules mentioned. Because if what kith said is true, then the paper I mentioned above is already wrong starting from the premise.