In DFT calculations, the most computationally intensive part is the solution of the Kohn-Sham equations, which has an O(n^3) computational complexity with respect to the number of electrons (n). However, the eigendecomposition is not the bottleneck in DFT simulations, and diagonalization is not explicitly performed during the iterative solution of the Kohn-Sham equations.
Instead, the most computationally demanding operation in DFT simulations is the repeated multiplication of large matrices, which has an O(n^3) computational complexity. This operation is required to solve the Kohn-Sham equations and obtain the self-consistent electron density and potential. The iterative solution of the Kohn-Sham equations involves the use of numerical methods such as the self-consistent field (SCF) method or the density mixing method, which do not explicitly involve diagonalization.
While the eigendecomposition is not explicitly performed during the iterative solution of the Kohn-Sham equations, it is required in the initial setup of the Kohn-Sham equations to obtain the set of orbitals that are used to construct the density matrix. However, this eigendecomposition is typically performed only once, and the resulting set of orbitals is then used for the iterative solution of the Kohn-Sham equations.
Overall, the computational complexity of DFT simulations is dominated by the repeated multiplication of large matrices, rather than the eigendecomposition. Therefore, optimizing the matrix multiplication operations is essential for improving the performance of DFT simulations.
