Use of basis set in DFT (Density Functional Theory)
This answer only deals with the most common variety of Density Functional Theory, namely, KohnSham DFT. This is what most people mean by "DFT", but, as noted in the comments, things such as orbitalfree DFT exist.
KohnSham DFT was created to solve a historical problem of DFT: The electronic kinetic energy term (as used in the ThomasFermi model) was not accurate enough. Kohn and Sham proposed to make use of what is typically called a "fictitious, noninteracting reference", that is, a wavefunction whose main purpose would be to yield a density to be used in all other (nucleuselectron attraction, Coulomb, exchange, and correlation energy/potential) and whose secondary function is to provide the kinetic energy, which is accurate from wavefunction theory. This wavefunction works just like a Slater determinant and is called the KohnSham determinant. See this question for pointers on where to read more: Equivalent of Szabo and Ostlund book for DFT.
As often in DFT, practicality showed a way that was justified later. When implementing this idea, one learns that one can basically use a HF code and replace the exchange term by the exchangecorrelation (XC) potential (which is projected back on to the AO basis set). The potential needs to evaluated numerically on a grid, a procedure that yields the XC energy for almost all choices of density functionals. Thus one obtains the energy and the KS operator (the equivalent of the Fock operator) and can perform a SCF procedure in a given basis set. For the mathematical details, see e.g. JA Pople, PMW Gill, BG Johnson Chem Phys Lett 199, 557 (1992).
To answer your questions in this context:
1) The initial density depends on the guess, which is a whole different can of worms, but the same is true for HF calculations. (Procedures based on tabulated atomic densities exist, but the initial guess business is a bit of a dark art, and I don't think this is what you meant by your question.) The basis set and its MOlike coefficients (defining the KS determinant) are used on every iteration to yield the density.
2) Naturally, yes. The MOcoefficients of basis sets change on geometry change for a given system, just like in HF. The procedure and workable algorithms are very similar to HF (because it all is).
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Mitradip Das
Updated on November 21, 2020Comments

Mitradip Das almost 3 years
Basis sets are used to guess the electronic wave functions for Hartree Fock or similar methods, which are quite legitimate since these methods deal with the wave function of each and every electron.
Density functional theory, on other hand, uses the electron density at every point of space for optimization and the calculation of properties. This has led to two doubts which I want to clarify:
Is the basis set used to estimate the initial electron density of the system? If so, how the basis functions of an STO or GTF basis set change in order to exhibit the electron density? If not, what exactly is the application of basis set in DFT calculation?
In HF SCF, the coefficients of basis functions (for basis sets with GTF) along with geometry changes with change in nuclear geometry while optimising the structure. Does optimisation with DFT follow a similar procedure?
P.S.: Please do not suggest me this question, since this is not the answer I am searching for.

Ian Bush almost 6 yearsIn a rush, but have a look at en.wikipedia.org/wiki/Orbitalfree_density_functional_theory

awvwgk almost 6 yearsYou may want to have a look at the paper of Kohn and Sham from 1965. It states: We derive two alternative sets of equations [...] which are analogous, respectively, to the conventional Hartree and Hartree–Fock equations, and, although they also include correlation effects, they are no more difficult to solve. Both HF and KSDFT are meanfield methods using quite similar equations.

Admin almost 6 yearsThe modern, mostly used DFT is KohnSham DFT, which has more in common with HF than with DFT...