Hartree Fock Symmetric Energy Expression

In summary, the Hartree Fock energy expression is symmetric because the repulsive energy between different electrons is averaged, leading to indistinguishable contributions from different electrons. This symmetry is not what makes the method variational, but rather the use of a Slater determinant as the trial wave function to produce the lowest energy. While there are other methods with non-symmetric energy expressions, they can still be variational if designed correctly.
  • #1
arsc
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Hello.

I just wonder why the energy expression of Hartree Fock method is symmetric. I tried to find out the reason on the Internet but I could only find that: since the Hartree Fock energy expression is symmetric, it is variational.

In Hartree Fock method, the repulsive energy between different electrons is averaged, so does that mean the amount of contribution of different electrons to the repulsive energy is indistinguishable which leads to the symmetric energy expression?

Thank you for your reply
 
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  • #2
What do you mean by "symmetric"? The HF energy equation is
[tex]E=\langle\Phi|H|\Phi\rangle[/tex]
with Phi being a Slater determinant. It is simply the expectation value of the trial wave function (if you evaluate this expression in terms of matrix elements, you end up with the usual sum over h_ii + 2<ij|ij>-<ii|jj> for a closed-shell determinant). HF is not variational because of the form of its energy expression, but because it is defined to be the method giving the wave function of this form (=determinant) which produces the lowest energy.

While there are plenty of many body methods with non-symmetric energy expressions of the form
[tex]E=\langle\Phi|H|\Psi\rangle[/tex]
(e.g., standard coupled cluster), and this form indeed prevents a variational solution of the equation (it is obviously not bounded for variations of Psi and fixed Phi), it is perfectly possible to make theories with symmetric expectation values which are *not* variational.
 

What is the Hartree Fock Symmetric Energy Expression?

The Hartree Fock Symmetric Energy Expression is a mathematical formula used in computational chemistry to approximate the energy of a quantum system, specifically the electronic energy of a multi-electron system. It is based on the Hartree Fock method, which is a self-consistent field method used to calculate the electronic structure of atoms and molecules.

Why is the Hartree Fock Symmetric Energy Expression important?

The Hartree Fock Symmetric Energy Expression is important because it allows scientists to calculate the electronic energy of a multi-electron system, which is crucial in understanding the behavior and properties of atoms and molecules. It also serves as the basis for more advanced computational methods used in theoretical chemistry.

How is the Hartree Fock Symmetric Energy Expression derived?

The Hartree Fock Symmetric Energy Expression is derived by approximating the total energy of a quantum system as a sum of one-electron energies, also known as orbitals. These orbitals are then optimized to minimize the total energy of the system, resulting in the Hartree Fock energy expression.

What are the limitations of the Hartree Fock Symmetric Energy Expression?

Although the Hartree Fock Symmetric Energy Expression is a powerful tool in computational chemistry, it has certain limitations. It does not take into account electron correlation, which is the interaction between electrons in a multi-electron system. This can lead to discrepancies between the calculated energy and the actual energy of a system.

Are there any variations of the Hartree Fock Symmetric Energy Expression?

Yes, there are variations of the Hartree Fock Symmetric Energy Expression, such as the Restricted Hartree Fock (RHF) and the Unrestricted Hartree Fock (UHF) methods. These variations take into account different types of electron spin and allow for a more accurate calculation of the electronic energy of a system.

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