The treatment of larger molecules that contain a few dozen electrons is computationally tractable by approximate methods such as density functional theory ( DFT ).

Density functional theory ( DFT ) methods are often considered to be ab initio methods for determining the molecular electronic structure, even though many of the most common functionals use parameters derived from empirical data, or from more complex calculations.

Rader's algorithm, exploiting the existence of a generator for the multiplicative group modulo prime, expresses a DFT of prime size as a cyclic convolution of ( composite ) size, which can then be computed by a pair of ordinary FFTs via the convolution theorem ( although Winograd uses other convolution methods ).

DFT is among the most popular and versatile methods available in condensed-matter physics, computational physics, and computational chemistry.

The development of new DFT methods designed to overcome this problem, by alterations to the functional or by the inclusion of additive terms, is a current research topic.

* Built-in DFT methods.

Almost all quantum-chemical methods ( DFT, MP2 etc.

Q-Chem can perform a number of general quantum chemistry calculations, such as Hartree-Fock, density functional theory ( DFT ) including time-dependent DFT ( TDDFT ), Møller – Plesset perturbation theory ( MP2 ), coupled cluster ( CC ), equation-of-motion coupled-cluster ( EOM-CC ), configuration interaction ( CI ), and other advanced electronic structure methods.

* EFP method ( including library of effective fragments, EFP interface with CC / EOM, DFT / TDDFT, and other methods )

By DFT and TDDFT methods one can obtain IR, Raman and UV spectra.

DFT affects and depends on the methods used for test development, test application, and diagnostics.

A key enabling factor for these applications is the fact that the DFT can be computed efficiently in practice using a fast Fourier transform ( FFT ) algorithm.

The transform is sometimes denoted by the symbol, as in or or < ref group =" note "> As a linear transformation on a finite-dimensional vector space, the DFT expression can also be written in terms of a DFT matrix ; when scaled appropriately it becomes a unitary matrix and the X < sub > k </ sub > can thus be viewed as coefficients of x in an orthonormal basis .</ ref >

) This orthogonality condition can be used to derive the formula for the IDFT from the definition of the DFT, and is equivalent to the unitarity property below.

An important simplification occurs when the sequences are of finite length, N. In terms of the DFT and inverse DFT, it can be written as follows:

Another way of looking at the DFT is to note that in the above discussion, the DFT can be expressed as a Vandermonde matrix:

In a real vector space, a unitary transformation can be thought of as simply a rigid rotation of the coordinate system, and all of the properties of a rigid rotation can be found in the unitary DFT.

A useful property of the DFT is that the inverse DFT can be easily expressed in terms of the ( forward ) DFT, via several well-known " tricks ".

First, we can compute the inverse DFT by reversing the inputs:

The conjugation trick can also be used to define a new transform, closely related to the DFT, that is involutary — that is, which is its own inverse.

An FFT is a way to compute the same result more quickly: computing a DFT of N points in the naive way, using the definition, takes O ( N < sup > 2 </ sup >) arithmetical operations, while an FFT can compute the same result in only O ( N log N ) operations.

Since the inverse DFT is the same as the DFT, but with the opposite sign in the exponent and a 1 / N factor, any FFT algorithm can easily be adapted for it.

Also, because the Cooley – Tukey algorithm breaks the DFT into smaller DFTs, it can be combined arbitrarily with any other algorithm for the DFT, such as those described below.

For with coprime and, one can use the Prime-Factor ( Good-Thomas ) algorithm ( PFA ), based on the Chinese Remainder Theorem, to factorize the DFT similarly to Cooley – Tukey but without the twiddle factors.

Indeed, Winograd showed that the DFT can be computed with only irrational multiplications, leading to a proven achievable lower bound on the number of multiplications for power-of-two sizes ; unfortunately, this comes at the cost of many more additions, a tradeoff no longer favorable on modern processors with hardware multipliers.

The inverse DFT cannot reproduce the entire time domain, unless the input happens to be periodic.

The only requirements of these conventions are that the DFT and IDFT have opposite-sign exponents and that the product of their normalization factors be 1 / N.

DFT has been very popular for calculations in solid-state physics since the 1970s.

While these are exact statements in the formalism of DFT, the use of approximate exchange-correlation potentials makes the calculated energies approximate and often the orbital energies are very different from the corresponding ionization energies ( even by several eVs!

( In the presence of round-off error, many FFT algorithms are also much more accurate than evaluating the DFT definition directly, as discussed below.

However, DFT was not considered accurate enough for calculations in quantum chemistry until the 1990s, when the approximations used in the theory were greatly refined to better model the exchange and correlation interactions.

The density functional theory ( DFT ) has been widely used since the 1970s for band structure calculations of variety of solids.

Therefore it is often said that the DFT is a transform for Fourier analysis of finite-domain discrete-time functions.

The input to the DFT is a finite sequence of real or complex numbers ( with more abstract generalizations discussed below ), making the DFT ideal for processing information stored in computers.

Its similarities to the original transform, S ( f ), and its relative computational ease are often the motivation for computing a DFT.

Formally, there is a clear distinction: " DFT " refers to a mathematical transformation or function, regardless of how it is computed, whereas " FFT " refers to a specific family of algorithms for computing DFTs.

The terminology is further blurred by the ( now rare ) synonym finite Fourier transform for the DFT, which apparently predates the term " fast Fourier transform " ( Cooley et al., 1969 ) but has the same initialism.

A normalization of for both the DFT and IDFT makes the transforms unitary, which has some theoretical advantages.

Equivalently, the DFT is often thought of as a matched filter: when looking for a frequency of + 1, one correlates the incoming signal with a frequency of − 1.

In other words, for any N > 0, an N-dimensional complex vector has a DFT and an IDFT which are in turn N-dimensional complex vectors.

If the expression that defines the DFT is evaluated for all integers k instead of just for, then the resulting infinite sequence is a periodic extension of the DFT, periodic with period N.

where the coefficients X < sub > k </ sub > are given by the DFT of x < sub > n </ sub > above, satisfies the interpolation property for.

Consider the unitary form defined above for the DFT of length N, where