By means of second quantization one may construct the ground state of a closed shell atom(He, Ne, etc.) using single-particle creation operators on the vacuum state:
|Ф0>=N/2Пi (bi↑+)*(bi↓+) |0>
Can a closed SUBshell atom be constructed in a simmilar manner?
In the UHF method, from what I understand, if there is one more electron of spin alpha(N+1 electrons) than beta spin(N electrons), the ground state is a doublet but it may be contaminated by a quartet. In the case of Mg-like sequence there are two electrons, with spin alpha and beta, respectively, besides the Ne-like closed shell core. Can their corresponding ground state be contaminated by a triplet excited state as similar to the previous case?
What are the assumptions needed for one to be able to construct the ground-state of a closed SUBshell atom?
What you mean by ground state of a closed subshell? The ground state of an atom is only one, with level number n=1.
About the UHF method you mention, won't you give some details? Not everybody is familiar with it.
Best regards!
Dear Sofia,
Thank you for your response!
In most textbooks of second quantization, in the case of a closed shell atom, or more generally, in the case where the spins of electrons are paired, one can act with a pair of creation operators (for spin up and spin down, respectively) on the so called vacuum state in order to create the ground state of the system.
When there is an uneven number of electron spins in the system, one needs to include in the calculation an unpaired creation operator corresponding to an electron belonging to an open shell. The question arises: how does the single-particle wavefunction of this unpaired electron look compared to the rest of the wavefunctions of the electrons paired in different shells?
The single-particle wavefunction of any electron is a combination between a radial function, an angular function and an element from the spin basis denoting if the spin is up(alpha) or down(beta).
In the Restricted open shell Hartree-Fock method (RHF), the radial part for two electrons belonging to the same shell are considered equal, thus, one uses a single Slater determinant describing all the electron pairs of the system, while the unpaired electron is described by a radial and angular part, along with a combination between a spin up and spin down function. An important thing to note is that the single-particle wavefunction of an electron with spin up from a pair and the single-particle wavefunction of the unpaired electron with spin up are eigenstates of the same spin operator. Same goes for the spin down case. This means that the ground state is an eigenfunction of the S^2 operator.
In the Unrestricted Hartree-Fock method (UHF), one uses completely different single-particle wavefunctions to describe electrons with different spins. This method uses a single Slater determinant containing different single-particle wavefunctions for electrons with different spin, thus it is not an eigenstate of the S^2 operator. This results in the so called „contamination” of the ground state with excited states through the spin space.
From what I read it is said that the UHF method is much more easier to implement but the user must be cautious on where to apply it, while The RHF method is generally available but it is computationally costly to use.
I am pretty new to this domain and to reformulate my question I would say: in the case of an even number of electrons (Be, 4 electrons: 1s2 2s2), with the last two electrons belonging to an open shell (the n=2 shell is open since it can accomodate up to 8 electrons: 2s2 2p6, but it is only filled with two), is it possible to construct the ground state of such an atom by applying a product of pairs of creation operators corresponding to electrons with opposite spins belonging to the same subshell on the vacuum state? Intuitively, I would say yes, since the number of spins is even and if they are distributed equally in each shell. But if one considers Hund’s law, one would fill the 2s subshell with an electron with spin up and then fill the 2p subshell with spin up, so in the end, the ground state of such a system may be contaminated by excited states?!
I came accros this in my adiacent studies but I find it very interesting and I would like to receive some advices on where to find relevant papers in this matter. Up until now I found Roothaan’s papers useful.
Please excuse my late answer and feel free to correct or add any information you think might help!
With kind regards!
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