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The DFT is used in the Fourier analysis of many practical applications. [2] In digital signal processing, the function is any quantity or signal that varies over time, such as the pressure of a sound wave, a radio signal, or daily temperature readings, sampled over a finite time interval (often defined by a window function [3]).
Density functional theory (DFT) is a computational quantum mechanical modelling method used in physics, chemistry and materials science to investigate the electronic structure (or nuclear structure) (principally the ground state) of many-body systems, in particular atoms, molecules, and the condensed phases.
Both transforms are invertible. The inverse DTFT reconstructs the original sampled data sequence, while the inverse DFT produces a periodic summation of the original sequence. The Fast Fourier Transform (FFT) is an algorithm for computing one cycle of the DFT, and its inverse produces one cycle of the inverse DFT.
Quantum chemistry computer programs are used in computational chemistry to implement the methods of quantum chemistry. Most include the Hartree–Fock (HF) and some post-Hartree–Fock methods. They may also include density functional theory (DFT), molecular mechanics or semi-empirical quantum chemistry methods.
The DFT is (or can be, through appropriate selection of scaling) a unitary transform, i.e., one that preserves energy. The appropriate choice of scaling to achieve unitarity is 1 / N {\displaystyle 1/{\sqrt {N}}} , so that the energy in the physical domain will be the same as the energy in the Fourier domain, i.e., to satisfy Parseval's theorem .
By far the most commonly used FFT is the Cooley–Tukey algorithm. This is a divide-and-conquer algorithm that recursively breaks down a DFT of any composite size = into smaller DFTs of size , along with () multiplications by complex roots of unity traditionally called twiddle factors (after Gentleman and Sande, 1966).
Over the complex numbers, it is often customary to normalize the formulas for the DFT and inverse DFT by using the scalar factor in both formulas, rather than in the formula for the DFT and in the formula for the inverse DFT. With this normalization, the DFT matrix is then unitary.
TDDFT is an extension of density-functional theory (DFT), and the conceptual and computational foundations are analogous – to show that the (time-dependent) wave function is equivalent to the (time-dependent) electronic density, and then to derive the effective potential of a fictitious non-interacting system which returns the same density as ...