Dirac equation for many particles system

[ranytawfik]

Can Dirac equation be used for many particles (fermions) system (i.e. a nucleus with many electrons)? And in this case how do you incorporate the anti-symmetry nature of the wavefunctions? Obviously Slater determined will complicate the equation to a point where it’s almost impossible to solve, and I’m not sure if we can instead use exchange interaction integral.

Thank you.

 

[meopemuk]

Hi ranytawfik,

welcome to the Forums!

I think there is a long-standing confusion about the nature of Dirac equation. Many textbooks regard this equation as a relativistic analog of the Schroedinger equation for particles with spin. This is seemingly justified by the fact that solving Dirac equation with the Coulomb potential one can get very accurate hydrogen spectrum. However, I think this agreement is accidental. Dirac equation is written for quantum fields (operator functions), which have nothing in common with electron wave functions (probability amplitudes). So, there is no direct connection between Dirac and Schroedinger equations.


If you want to have a relativistic Schroedinger equation for electrons you will be served better by the Breit equation described, for example, in section 83 of Berestetskii, Lifgarbagez, Pitaevskii "Quantum electrodynamics". This equation is written for two spinning particles, but it can be easily generalized for N electrons. You are right that one needs to take into account the anti-symmetry of the wave function. So, one can expect a lot of complications (like using linear combinations of Slater determinants) when solving this equation for many-electron systems.

Eugene.

 

[ytuab]

The relativistic calculations of the multi-electron atoms and molecules are so comlicated for me to understand well.

Only the Dirac equation of the hydrogen-like atoms can be precisely solved. (Also in the Schroedinger equation.)
But this solution contains "many accidental coincidences". (See this thread)

There seems to be several methods using the approximations of the relativistic forms such as the Dirac-Hartree-Fock and the relativistic density-functional calculations.)

But these calculations are more difficult than non-relativistic, so a commonly used approach is to do an all-electron atomic Dirac-Fock calculation on each type of atom in the molecule and use the result to derive a relativistic effective core potential(RECP) for that atom.(Since the smallest parts of the relativistic effects are neglected in deriving RECPs.)

The valence electrons are treated nonrelativistically, and the core electrons are represented by adding the operator \Sigma U_{\alpha} to the Fock operator F, where U_{\alpha} is the relativistic ECP for atom \alpha. Sorry, I don't know more about these complicated things.

In the hydrogen atom, the energy levels of 2S1/2 and 2P1/2 states are comletely the same due to the solution of the Dirac equation. And the very small energy difference is the Lamb shift which can be gotten only by QED.
But the Dirac equation of the hydrogen atom contains the Coulomb potential which is known well as the nonrelativistic term. So, before considering the QED, if this small energy difference exists, is it inconsistent?

 

[ranytawfik]

Thanks Eugene for the welcome and the reply. Equal thanks to ytuab.

Eugene,

I was trying to avoid Breit equation due to several reasons:

1. The exhaustive computations due to its 4^N spinors. The only solution to this issue is to use pseudo-potential approach and just freeze the core electrons. I haven’t seen any papers, however, using this technique with Breit equation.

2. I don’t know how the anti-symmetric nature can be embedded in the 4^N matrix and I haven’t seen any derivation for exchange interaction integrals that can be used instead. Have you seen any such integrals used with Breit equation?

I, however, do agree with you that Breit equation is probably the way to go if I want to avoid QFT.

Ytuab, I mostly work with excited states so none of the variational approaches (Hartree-Fock, etc) would really work with me. I might use this approach just to solve for the core electrons and use this solution to construct the pseudo-potential and continue with Breit equation.

Rany

 

[Meir Achuz]

The Breit equation has diseases associated with it. A 'saddle point variation' can be used to solve it. Putting "Saddle point" "Breit equation" into google should find some papers.

 

[meopemuk]

I was trying to avoid Breit equation due to several reasons:

1. The exhaustive computations due to its 4^N spinors. The only solution to this issue is to use pseudo-potential approach and just freeze the core electrons. I haven’t seen any papers, however, using this technique with Breit equation.

2. I don’t know how the anti-symmetric nature can be embedded in the 4^N matrix and I haven’t seen any derivation for exchange interaction integrals that can be used instead. Have you seen any such integrals used with Breit equation?

As I understand, in the Breit formalism electrons are described by 2-component wave functions. 4-component wave functions are characteristic for the Dirac formalism. Or I am missing something?

Eugene.

 

[Meir Achuz]

The Breit equation has four compnents for each particle.

 

[meopemuk]

The Breit equation has four compnents for each particle.

I am not sure we are talking about the same Breit equation. I had in mind the Breit Hamiltonian described in sections 83-84 of Berestetskii, Lifgarbagez, Pitaevskii, "Quantum Electrodynamics" (I have a Russian copy of this book, but I am sure there exists an English translation as well). This Hamiltonian is using 2x2 Pauli matrices \sigma that act on two spin components (spin up and down) of each particle.

Eugene.

 

[Meir Achuz]

I am referring to the equation derived by G. Breit, Phys. Rev. 34 (1929) 553.