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  2. DFT matrix - Wikipedia

    en.wikipedia.org/wiki/DFT_matrix

    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 .

  3. Discrete Fourier transform over a ring - Wikipedia

    en.wikipedia.org/wiki/Discrete_Fourier_transform...

    The Fourier transform can therefore be seen to relate the coefficients and the values of a polynomial: the coefficients are in the time-domain, and the values are in the frequency domain. Here, of course, it is important that the polynomial is evaluated at the n th roots of unity, which are exactly the powers of α {\displaystyle \alpha } .

  4. Cooley–Tukey FFT algorithm - Wikipedia

    en.wikipedia.org/wiki/Cooley–Tukey_FFT_algorithm

    The Cooley–Tukey algorithm, named after J. W. Cooley and John Tukey, is the most common fast Fourier transform (FFT) algorithm. It re-expresses the discrete Fourier transform (DFT) of an arbitrary composite size = in terms of N 1 smaller DFTs of sizes N 2, recursively, to reduce the computation time to O(N log N) for highly composite N (smooth numbers).

  5. Density functional theory - Wikipedia

    en.wikipedia.org/wiki/Density_functional_theory

    The same name is used for quantum DFT, which is the theory to calculate the electronic structure of electrons based on spatially dependent electron density with quantum and relativistic effects. Classical DFT is a popular and useful method to study fluid phase transitions , ordering in complex liquids, physical characteristics of interfaces and ...

  6. Discrete Fourier transform - Wikipedia

    en.wikipedia.org/wiki/Discrete_Fourier_transform

    It has the same sample-values as the original input sequence. The DFT is therefore said to be a frequency domain representation of the original input sequence. If the original sequence spans all the non-zero values of a function, its DTFT is continuous (and periodic), and the DFT provides discrete samples of one cycle.

  7. Linearized augmented-plane-wave method - Wikipedia

    en.wikipedia.org/wiki/Linearized_augmented-plane...

    The linearized augmented-plane-wave method (LAPW) is an implementation of Kohn-Sham density functional theory (DFT) adapted to periodic materials. [1] [2] [3] It typically goes along with the treatment of both valence and core electrons on the same footing in the context of DFT and the treatment of the full potential and charge density without any shape approximation.

  8. Density matrix - Wikipedia

    en.wikipedia.org/wiki/Density_matrix

    The states of the C*-algebra of compact operators K(H) correspond exactly to the density operators, and therefore the pure states of K(H) are exactly the pure states in the sense of quantum mechanics. The C*-algebraic formulation can be seen to include both classical and quantum systems.

  9. Time-dependent density functional theory - Wikipedia

    en.wikipedia.org/wiki/Time-dependent_density...

    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 ...