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In most DFT codes like QE and VASP, spin-orbit coupling (SOC) can be incorporated by employing fully relativistic pseudopotentials. My question is with regard to the implementation of SOC among different functionals. If one believes that using a fully relativistic PP is equivalent to solving the Dirac equation, are the results simply transferrable between different functionals?

For example, let us say I calculate the total energy from DFT-PBE with SOC off and then turned on. The difference in these two configurations should give me the amount of SOC for my system. Now, if I perform DFT-LDA on the same system, should I expect just the same amount of SOC? My main concern here is because we use different pseudopotentials.

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VASP with the PAW method can't take a fully relativistic effect, which can be taken into account only by solving the Dirac equation. The SOC effect in the PAW method is included with the following perturbed Hamiltonian (zeroth-order-regular approximation):

$$H_{SO}^{\alpha\beta}=-\dfrac{\hbar^2}{(2m_ec)^2} \dfrac{K(r)}{r}\dfrac{dV(r)}{dr}\vec{\sigma}^{\alpha\beta}\cdot\vec{L} \tag{1}$$

Here the angular momentum operators $\vec{L}$ is defined as $\vec{L}=\vec{r}\times \vec{p}$ and $\sigma=(\sigma_x, \sigma_y, \sigma_z)$ are the $(2 \times 2)$ Pauli spin matrices, $V(\vec{r})$ is the spherical part of the effective all-electron (AE) potential within the PAW sphere, and

$$K(\vec{r})=\left(1-\dfrac{V(\vec{r})}{2m_e c^2} \right)^{-2}.$$

Take the bulk Bi2Te3 as an example, I test what you are asking.

  • PBE Potentials:

    TITEL  = PAW_PBE Bi 08Apr2002
    TITEL  = PAW_PBE Te 08Apr2002
    
  • PBE total energy without SOC (eV):

    E0= -.18384597E+02
    
  • PBE total energy with SOC (eV):

    E0= -.19687316E+02
    
  • PBE energy difference (eV):

    -1.302718
    
  • LDA Potentials:

      TITEL  = PAW Bi 03Oct2001
      TITEL  = PAW Te 03Oct2001
    
  • LDA total energy without SOC (eV):

      E0= -.21427747E+02
    
  • LDA total energy with SOC (eV):

      E0= -.22785609E+02
    
  • LDA energy difference (eV):

      -1.3578620
    

There is a difference here.

Hope it helps.

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