In quantum chemistry, a wave function for a single electron is called an orbital or a spin orbital, but what could be a wave function for several electrons? [1] One option is related to the Hartree Product.
Douglas Hartree, a contributor to the field of quantum chemistry.Noninteracting System
According to Attila Szabo and Neil S. Ostlund, it is helpful to consider a noninteracting system, or a system with no interactions between the particles. In physics, an interaction is sometimes represented by potential energy between different particles. The interactions between the particles will be considered later. In this post, I am just considering a system of electrons, so the topic of interest is noninteracting electrons. Szabo and Ostlund start with a Hamiltonian operator for these noninteracting electrons. It is interesting that, at least in quantum chemistry, it is common for the starting point to be a Hamiltonian. In this noninteracting case, the Hamiltonian operator is
$ H \equiv \sum_{n}^{N} h(n) $
in which $h(n)$ “is the operator describing the kinetic energy and potential energy of electron” $n$. So, $n$ is not only an index but also a label for an electron. Here, $N$ is the number of electrons. Also, it is important to note that this mathematical objects are called operators instead of matrices or functions. This simple Hamiltonian operator is the sum of the hamiltonian operator for each noninteracting electron.
The next key step is to define
$h(n) \chi_j(\vec{x}_n) \equiv \varepsilon_j \chi_j(\vec{x}_n) $.
Notice that the subscript of the spin orbital $\chi_j$ is $j$ instead of $n$ because $j$ is a label for a spin orbital whereas $n$ is a label for an electron. Note also that the energy value $\varepsilon_j$ has the same subscript as the spin orbital–not the electron.
Hartree Product
It is also possible to define a so-called Hartree product,
$\Psi_{HP} \equiv \chi_i (\vec{x}_1) \chi_j (\vec{x}_2 ) … \chi_k (\vec{x}_N) $.
With these definitions, it is possible to evaluate the expression $H \Psi_{HP}$:
$H \Psi_{HP} = \sum_{n}^{N} h(n) \chi_i (\vec{x}_1) \chi_j (\vec{x}_2 ) … \chi_k (\vec{x}_N) $.
Since $h(n) \chi_j(\vec{x}_n) \equiv \varepsilon_j \chi_j(\vec{x}_n) $, the previous line becomes
$H \Psi_{HP} = \varepsilon_i \Psi_{HP} + \varepsilon_j \Psi_{HP} + … \varepsilon_k \Psi_{HP}$.
The last steps are the most involved because one has to realize that each $h(n)$ turns into a $\varepsilon$ with a subscript corresponding to the orbital having an argument of $\vec{x}_n$. The sum ranges from $1$ to $N$, and the Hartree Product contains one (and only one) orbital for each of $N$ electrons.
Using the distributive property,
$H \Psi_{HP} = \big( \varepsilon_i + \varepsilon_j + … \varepsilon_k \big) \Psi_{HP}$.
Using a summation for the $\varepsilon$’s, the previous equation is
$H \Psi_{HP} = \big( \sum_z^k \varepsilon_z \big) \Psi_{HP}$.
The index $z$ could be any letter, but $k$ is a necessary upper limit because that is how many orbitals are in the Hartree Product. Note that there are $k$ orbitals instead of $2k$ orbitals as in the previous post which accounted for electron spin.
Using the distributive property, the parentheses may be removed:
$H \Psi_{HP} = \sum_z^k \varepsilon_z \Psi_{HP}$.
In Reference 1, $E \equiv \sum_z^k \varepsilon_z$.
Unless the ordering of the spin orbitals with respect to a subscript is critical–and it might be if the spin orbitals are operators–the order of the spin orbitals in the Hartree Product does not play a role since the spin orbitals are–to my knowledge–functions instead of operators. However, Szabo and Ostlund do require “electron-one being described by the spin orbital $\chi_i$, electron-two being described by the spin orbital $\chi_j$, etc.” Interestingly, the association of the electron with the spin orbital is important, but no mention of the order of the spin orbitals is mentioned. So, at least for the time being, I am going to assume that the order is important, which means that the left-most spin orbital should have the subscript of $I$. This observation leads to the relation between the Hartree product and the antisymmetry principle. This relation is best summarized as follows:
…there is still a basic deficiency in the Hartree product; [it] … specifically distinguishes electron-one as occupying spin orbital $\chi_i$, electron-two as occupying $\chi_j$, etc. The antisymmetry principle does not distinguish between identical electrons and requires that electronic wave functions be antisymmetric (change sign) with respect to the interchange of the space and spin coordinates of any two electrons.
– Szabo and Ostlund
This inconsistency with the Hartree product and the antisymmetry principle is resolved in an after-the-fact way with the so-called Slater determinant, which will be the topic of a subsequent post.
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