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I've noticed that a lot of computational techniques in physics deal with pseudopotentials in their calculation procedure and especially density functional theory (DFT) projects (open sources such as Quantum Espresso, Abinit, Octopus, etc.) require them with care.

The main algorithm of first-principle calculations consists of diagonalization of (hopefully symmetric, so that diagonalizable) Hamiltonian matrix. Since a length scale of core orbitals with large atomic numbers, Z, is very small, we need a large momentum and $|\vec{k}|$ to describe the behavior of the orbitals. Thus we will meet the giant plane-wave (or sine-wave) basis set and so is Hamiltonian matrix. Consider transition metals. They have both conduction and super-deep-core electrons and they have a variety of $|\vec{k}|$. It is computationally unfavorable to diagonalize the Hamiltonian with million by million dimensions.

I think the pseudopotentials can remedy such tragedy by expressing the influences of the core electrons implicitly. Then we need not use the giant basis set and physicists become happy.

However if the computational costs really matter, why should we use the pseudopotentials for low-Z atoms? I don't know what is going on closed source projects(i.e. VASP?), but as far as Abinit and Quantum Espresso, the DFT calculations always need the pseudopotentials regardless of the kind of the atoms. Can't we run mediocre-cost-computations for light atoms even if there is no pseudopotentials?

Furthermore the quality of the computational results hugely depend on the kind of the pseudopotentials we use (and exchange-correlation functionals...). A lot of scientists have worked to develop more feasible exchange-correlation functionals, from mere LDA and GGA to hybrid functionals. Surely they also have developed more plausible pseudopotentials, but the worry about "poor performance pseudopotential" can be completely overcome if we just use the giant basis set I mentioned above. This makes computational costs skyrocket but why don't we use so-called supercomputers? Any poor property of the pseudopotentials will be erased and hopefully many unsolved problems of theoretical divisions(such as the special properties of transition metal oxides) would be resolved. The size of the required basis set is too large to be dealt with, even, world-level supercomputers?

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Firstly, if you have vast computational resources and accuracy is really important then you'd more likely use a method like quantum Monte Carlo rather than DFT.

Nevertheless, there is no theoretical reason why you can't perform DFT without a pseudopotential (indeed, pseudopotentials were introduced some years after DFT). But, and here is the crucial point: pseudopotentials generally introduce errors that are negligible in comparison to the errors introduced by the exchange-correlation functional (which you have no choice but to use). So you might as well use them given that doing so may speed up your simulations by orders of magnitude and introduce negligible error.

As a side note, your understanding for using pseudopotentials is correct: it allows for using smaller cut-offs since it reduces the width of the Fourier spectrum, plus it demands fewer eigenstates be constructed.

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