Density functional theory/Introduction to functional analysis
Functional (mathematics)Edit
In mathematics, and particularly in functional analysis and the Calculus of variations, a functional is a function from a vector space into its underlying scalar field, or a set of functions of the real numbers. In other words, it is a function that takes a vector as its input argument, and returns a scalar. Commonly the vector space is a space of functions, thus the functional takes a function for its input argument, then it is sometimes considered a function of a function. Its use originates in the calculus of variations where one searches for a function that minimizes a certain functional. A particularly important application in physics is searching for a state of a system that minimizes the energy functional.
Functional detailsEdit
DualityEdit
The mapping
is a function, where is an argument of a function . At the same time, the mapping of a function to the value of the function at a point
is a functional, here is a parameter.
Provided that f is a linear function from a linear vector space to the underlying scalar field, the above linear maps are dual to each other, and in functional analysis both are called linear functionals.
Definite integralEdit
Integrals such as
form a special class of functionals. They map a function f into a real number, provided that H is real-valued. Examples include
- the area underneath the graph of a positive function f
- L^{p} norm of functions
- the arclength of a curve in 2-dimensional Euclidean space
Vector scalar productEdit
Given any vector in a vector space , the scalar product with another vector , denoted or , is a scalar. The set of vectors such that this product is zero is a vector subspace of , called the null space or kernel of .
Local vs non-localEdit
If a functional's value can be computed for small segments of the input curve and then summed to find the total value, a function is called local. Otherwise it is called non-local. For example:
is local while
is non-local. This occurs commonly when integrals occur separately in the numerator and denominator of an equation such as in calculations of center of mass.
Linear functionalsEdit
Linear functionals first appeared in functional analysis, the study of vector spaces of functions. A typical example of a linear functional is integration: the linear transformation defined by the Riemann integral
is a linear functional from the vector space C[a,b] of continuous functions on the interval [a, b] to the real numbers. The linearity of I(f) follows from the standard facts about the integral:
Nonlinear functionalEdit
Functional derivativeEdit
The functional derivative is defined first; Then the functional differential is defined in terms of the functional derivative.
Functional derivativeEdit
Given a manifold M representing (continuous/smooth/with certain boundary conditions/etc.) functions ρ and a functional F defined as
the functional derivative of F[ρ], denoted δF/δρ, is defined by^{[1]}
where is an arbitrary function. is called the variation of ρ.
Functional differentialEdit
The differential (or variation or first variation) of the functional F[ρ] is,^{[2]} ^{[Note 1]}
where δρ(x) = εϕ(x) is the variation of ρ(x).
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This is similar in form to the total differential of a function F(ρ_{1}, ρ_{2}, ..., ρ_{n}),
where ρ_{1}, ρ_{2}, ... , ρ_{n} are independent variables. Comparing the last two equations, the functional derivative δF/δρ(x) has a role similar to that of the partial derivative ∂F/∂ρ_{i} , where the variable of integration x is like a continuous version of the summation index i.^{[3]}
Formal descriptionEdit
The definition of a functional derivative may be made more mathematically precise and formal by defining the space of functions more carefully. For example, when the space of functions is a Banach space, the functional derivative becomes known as the Fréchet derivative, while one uses the Gâteaux derivative on more general locally convex spaces. Note that the well-known Hilbert spaces are special cases of Banach spaces. The more formal treatment allows many theorems from ordinary calculus and analysis to be generalized to corresponding theorems in functional analysis, as well as numerous new theorems to be stated.
PropertiesEdit
Like the derivative of a function, the functional derivative satisfies the following properties, where F[ρ] and G[ρ] are functionals:
- Linear:^{[4]}
- Product rule:^{[5]}
- Chain rules:
- If f is a differentiable function, then
- ^{[6]}
- ^{[7]}
Determining functional derivativesEdit
We give a formula to determine functional derivatives for a common class of functionals that can be written as the integral of a function and its derivatives. This is a generalization of the Euler–Lagrange equation: indeed, the functional derivative was introduced in physics within the derivation of the Lagrange equation of the second kind from the principle of least action in Lagrangian mechanics (18th century). The first three examples below are taken from density functional theory (20th century), the fourth from statistical mechanics (19th century).
FormulaEdit
Given a functional
and a function ϕ(r) that vanishes on the boundary of the region of integration, from a previous section Definition,
The second line is obtained using the total derivative, where ∂f /∂∇ρ is a derivative of a scalar with respect to a vector.^{[Note 2]} The third line was obtained by use of a product rule for divergence. The fourth line was obtained using the divergence theorem and the condition that ϕ=0 on the boundary of the region of integration. Since ϕ is also an arbitrary function, applying the fundamental lemma of calculus of variations to the last line, the functional derivative is
where ρ = ρ(r) and f = f (r, ρ, ∇ρ). This formula is for the case of the functional form given by F[ρ] at the beginning of this section. For other functional forms, the definition of the functional derivative can be used as the starting point for its determination. (See the example Coulomb potential energy functional.)
The above equation for the functional derivative can be generalized to the case that includes higher dimensions and higher order derivatives. The functional would be,
where the vector r ∈ ℝ^{n}, and ∇^{(i)} is a tensor whose n^{i} components are partial derivative operators of order i,
- ^{[Note 3]}
An analogous application of the definition of the functional derivative yields
In the last two equations, the n^{i} components of the tensor are partial derivatives of f with respect to partial derivatives of ρ,
and the tensor scalar product is,
- ^{[Note 4]}
ExamplesEdit
Thomas–Fermi kinetic energy functionalEdit
The Thomas–Fermi model of 1927 used a kinetic energy functional for a noninteracting uniform electron gas in a first attempt of density-functional theory of electronic structure:
Since the integrand of T_{TF}[ρ] does not involve derivatives of ρ(r), the functional derivative of T_{TF}[ρ] is,^{[8]}
Coulomb potential energy functionalEdit
For the electron-nucleus potential, Thomas and Fermi employed the Coulomb potential energy functional
Applying the definition of functional derivative,
So,
For the classical part of the electron-electron interaction, Thomas and Fermi employed the Coulomb potential energy functional
From the definition of the functional derivative,
The first and second terms on the right hand side of the last equation are equal, since r and r′ in the second term can be interchanged without changing the value of the integral. Therefore,
and the functional derivative of the electron-electron coulomb potential energy functional J[ρ] is,^{[9]}
The second functional derivative is
Weizsäcker kinetic energy functionalEdit
In 1935 von Weizsäcker proposed to add a gradient correction to the Thomas-Fermi kinetic energy functional to make it suit better a molecular electron cloud:
where
Using a previously derived formula for the functional derivative,
and the result is,^{[10]}
EntropyEdit
The entropy of a discrete random variable is a functional of the probability mass function.
Thus,
Thus,
ExponentialEdit
Let
Using the delta function as a test function,
Thus,
This is particularly useful in calculating the correlation functions from the partition function in quantum field theory.
Functional derivative of a functionEdit
A function can be written in the form of an integral like a functional. For example,
Since the integrand does not depend on derivatives of ρ, the functional derivative of ρ(r) is,
Application in calculus of variationsEdit
- In the calculus of variations, functionals are usually expressed in terms of an integral of functions, their arguments, and their derivatives. In an integrand L of a functional, if a function f is varied by adding to it another function δf that is arbitrarily small, and the resulting L is expanded in powers of δf, the coefficient of δf in the first order term is called the functional derivative.
For example, consider the functional
where f ′(x) ≡ df/dx. If f is varied by adding to it a function δf, and the resulting integrand L(x, f +δf, f '+δf ′) is expanded in powers of δf, then the change in the value of J to first order in δf can be expressed as follows:^{[11]}^{[Note 5]}
The coefficient of δf(x), denoted as δJ/δf(x), is called the functional derivative of J with respect to f at the point x.^{[3]} For this example functional, the functional derivative is the left hand side of the Euler-Lagrange equation,^{[12]}
Using the delta function as a test functionEdit
In physics, it's common to use the Dirac delta function in place of a generic test function , for yielding the functional derivative at the point (this is a point of the whole functional derivative as a partial derivative is a component of the gradient):
This works in cases when formally can be expanded as a series (or at least up to first order) in . The formula is however not mathematically rigorous, since is usually not even defined.
The definition given in a previous section is based on a relationship that holds for all test functions ϕ, so one might think that it should hold also when ϕ is chosen to be a specific function such as the delta function. However, the latter is not a valid test function.
In the definition, the functional derivative describes how the functional changes as a result of a small change in the entire function . The particular form of the change in is not specified, but it should stretch over the whole interval on which is defined. Employing the particular form of the perturbation given by the delta function has the meaning that is varied only in the point . Except for this point, there is no variation in .
NotesEdit
- ↑ Called differential in (Parr & Yang 1989, p. 246), variation or first variation in (Courant & Hilbert 1953, p. 186), and variation or differential in (Gelfand & Fomin 2000, p. 11, § 3.2).
- ↑ For a three-dimensional cartesian coordinate system,
- ↑ For example, for the case of three dimensions (n = 3) and second order derivatives (i = 2), the tensor ∇^{(2)} has components,
- ↑ For example, for the case n = 3 and i = 2, the tensor scalar product is,
- ↑ According to Giaquinta & Hildebrandt (1996, p. 18), this notation is customary in physical literature.
Edit
Functional equationEdit
The traditional usage also applies when one talks about a functional equation, meaning an equation between functionals: an equation between functionals can be read as an 'equation to solve', with solutions being themselves functions. In such equations there may be several sets of variable unknowns, like when it is said that an additive function is one satisfying the functional equation
- .
Functional derivative and functional integrationEdit
Functional derivatives are used in Lagrangian mechanics. They are derivatives of functionals: i.e. they carry information on how a functional changes, when the function changes by a small amount. See also calculus of variations.
Richard Feynman used functional integrals as the central idea in his sum over the histories formulation of quantum mechanics. This usage implies an integral taken over some function space.
FootnotesEdit
- ↑ (Parr & Yang 1989, p. 246, Eq. A.2).
- ↑ (Parr & Yang 1989, p. 246, Eq. A.1).
- ↑ ^{a} ^{b} (Parr & Yang 1989, p. 246).
- ↑ (Parr & Yang 1989, p. 247, Eq. A.3).
- ↑ (Parr & Yang 1989, p. 247, Eq. A.4).
- ↑ (Greiner & Reinhardt 1996, p. 38, Eq. 7).
- ↑ (Parr & Yang 1989, p. 251, Eq. A.34).
- ↑ (Parr & Yang 1989, p. 247, Eq. A.6).
- ↑ (Parr & Yang 1989, p. 248, Eq. A.11).
- ↑ (Parr & Yang 1989, p. 247, Eq. A.9).
- ↑ (Giaquinta & Hildebrandt 1996, p. 18)
- ↑ (Gelfand & Fomin 2000, p. 28)
ReferencesEdit
- Courant, Richard; Hilbert, David (1953). "Chapter IV. The Calculus of Variations". Methods of Mathematical Physics. Vol. I (First English ed.). New York, New York: Interscience Publishers, Inc. pp. 164–274. ISBN 978-0471504474..
- Frigyik, Béla A.; Srivastava, Santosh; Gupta, Maya R. (January 2008), Introduction to Functional Derivatives, UWEE Tech Report, UWEETR-2008-0001, Seattle, WA: Department of Electrical Engineering at the University of Washington, pp. 7, https://www.ee.washington.edu/techsite/papers/documents/UWEETR-2008-0001.pdf.
- Gelfand, I. M.; Fomin, S. V. (2000) [1963], Calculus of variations, translated and edited by Richard A. Silverman (Revised English ed.), Mineola, N.Y.: Dover Publications, ISBN 978-0486414485, http://store.doverpublications.com/0486414485.html.
- Giaquinta, Mariano; Hildebrandt, Stefan (1996), Calculus of Variations 1. The Lagrangian Formalism, Grundlehren der Mathematischen Wissenschaften, 310 (1st ed.), Berlin: Springer-Verlag, ISBN 3-540-50625-X.
- Greiner, Walter; Reinhardt, Joachim (1996), "Section 2.3 – Functional derivatives", Field quantization, With a foreword by D. A. Bromley, Berlin–Heidelberg–New York: Springer-Verlag, pp. 36–38, ISBN 3-540-59179-6, http://www.physics.byu.edu/faculty/berrondo/wt752/functional%20derivative.pdf.
- Parr, R. G.; Yang, W. (1989). "Appendix A, Functionals". Density-Functional Theory of Atoms and Molecules. New York: Oxford University Press. pp. 246–254. ISBN 978-0195042795.
See alsoEdit
ReferencesEdit
- General
- Citations