_{1}

^{*}

It is shown that the process of conventional functional differentiation does not apply to functionals whose domain (and possibly range) is subject to the condition of integral normalization, as is the case with respect to a domain defined by wave functions or densities, in which there exists no neighborhood about a given element in the domain defined by arbitrary variations that also lie in the domain. This is remedied through the generalization of the domain of a functional to include distributions in the form of , where is the Dirac delta function and is a real number. This allows the determination of the rate of change of a functional with respect to changes of the independent variable determined at each point of the domain, with no reference needed to the values of the functional at different functions in its domain. One feature of the formalism is the determination of rates of change of general expectation values (that may not necessarily be functionals of the density) with respect to the wave functions or the densities determined by the wave functions forming the expectation value. It is also shown that ignoring the conditions of conventional functional differentiation can lead to false proofs, illustrated through a flaw in the proof that all densities defined on a lattice are -representable. In a companion paper, the mathematical integrity of a number of long-standing concepts in density functional theory are studied in terms of the formalism developed here.

Functionals and functional derivatives [

Functional or not, the rate of change defines the change induced in the dependent variable with respect to a change at a single point in the domain of its independent variable and in this interpretation can be applied to a single pair,

Indeed, the concept of the derivative as rate of change is well known from the calculus: The derivative of a function, say

of the ratio of the difference in the function corresponding to two different points in the domain, to the dif- ference between the two corresponding points in the domain, as the latter difference is allowed to vanish. In this case, the rate of change is synonymous with that of derivative. This exhibits the requirement that every point in the domain of

A characteristic feature of the derivative is the connection through the definition in (0.1) of the value of the function at

Analogously to a function, a functional is defined as a set of ordered pairs,

As for ordinary functions, the collection of first entries,

Again in analogy with ordinary differentiation, the functional derivative,

ing procedure [

where

relevant integrals must be well-defined.)

The need for arbitrariness of the test function,

Appendix.

This formalism is immediately applicable to functionals of the form,

written as explicit functions of the coordinates, the independent variable and its derivative. For reference purposes and to distinguish such functionals from more general structures that enter the discussion in this paper,

we will refer to them as functionals of form. In the integral above,

that, in general, can become arbitrarily large.

Beginning with the next section, we examine the analytic properties of derivatives of functionals as compared to those of a function.

In view of (0.4), the change in the functional under a continuous change in the independent variable, takes the form

implying that

is a linear superposition of the changes at each point,

the whole range of

respect to the change in the independent variable at each point in its domain. This rate of change multiplied by a change at each point,

This is analogous to the change in a function,

The change in the function at

The analogous feature in the case of functionals rests on the existence of a small neighborhood of functions about a value (a function) in the domain of the functional in which the functional is uniquely and continuously differentiable through (0.4). In that case, there exists a neighborhood defined by small but arbitrary changes in the independent variable that are elements of the domain of the functional, so that the procedure in (0.4) is well-defined. As in the case of a function, the change in the functional induced by change in the independent variable takes the form, (first term in a Taylor series-like expansion of the functional),

where

that

In the study of interacting N-particle systems, a density,

The normalization is also a formal requirement of the definition of the density in terms of the wave function describing the configurations of the system. For the case of electrons, our interest in this work, the system of

particles is described by a wave function,

antisymmetric with respect to interchange of any coordinate pair (and spins, for electrons). In terms of a wave function, the density takes the form,

Under the general condition of unit normalization for the wave function,

the normalization condition, Equation (0.9), follows. We now examine the determination of functional de- rivatives of functionals over the domain of densities, i.e., whose elements are subject to the condition of in- tegral normalization.

As an illustrative example, consider the linear functional, of Equation (0.3), defined over the set of densities normalized to

Assuming the existence of an arbitrary function,

leads to the identity, (see also Appendix),

Unfortunately, in the present case, the derivation of the last equality is invalid.

When the domain of the functional is constrained to integral normalization, the test function,

or the changed function,

defined. Crucially, it is no longer possible to obtain the value of a functional at a function,

its value at

The difficulties associated with functional derivatives with respect to the density under the condition of integral normalization of the differentiating quantity have already been noted in the literature [

Although the condition in (0.15) is characteristic of domains restricted to integral normalization, the rate of change of functionals defined over such domains can still be rigorously defined.

For reasons mentioned below, it is instructive to consider the identity functional,

A particular interpretation of the last expression is useful: We view the quantity,

Now, using the definition of the functional derivative, Equation (0.4), and for general domains, unconstrained by normalization, we obtain,

from which follows the result,

We note that the identical result is obtained through the procedure,

Before turning to the justification for this definition of the functional derivative, and its associated rate of change, we note some important properties: The rate of change of the identity functional is independent of the particular function in the domain of the functional and is thus an inherent property of the functional. Most

importantly, the rate of change (the ratio,

We can now write the general expression,

for all cases, irrespective of normalization requirements. Appendix establishes this general result along intuitive grounds.

It remains to establish that the definition of Equation (0.19) indeed corresponds to the derivative of the iden- tity functional.

In formal mathematics, the Dirac delta function is an example of a distribution (or generalized function) that lies outside the realm of ordinary functions [

This is accomplished by extending the domain,

the union of the domain of the functional and distributions proportional to the Dirac delta function. A schematic representation of this generalization is given in figure 1.

Let one of the axes in a three-dimensional Cartesian coordinate system represent the functions in

The generalized identity functional can be expressed in the language of a function mapped onto itself,

where the integral is evaluated through the recursion relation for delta functions [

along with the property,

that is generally non-integral, even if the integral over

the domain of the identity functional in the sense of a functional that maps a function in its domain onto itself.

The derivative of the identity functional at any point in the plane defined by

form,

where

(0.27)

where

The general validity of this definition follows because the derivative of any function with respect to itself is not a function, but a distribution. As such it lies outside the domain of functions (that may be constrained by normalization). By writing

Furthermore, the definition is applicable to all functionals of form because the derivative in these cases can be expressed as the ordinary derivative of a differentiable function times the functional derivative of the identity. In fact, it is applicable to isolated single pairs of the form (independent variable, dependent variable) that may not form part of a functional (collection of pairs). The significance of these results is highlighted below.

The following is a well-known property of functional differentiation. Let a functional of form,

that the quantity,

which justifies the general result in (0.27).

Although the formalism just completed provides a justification of the derivative given by (0.27), this result is already freely used in the literature [

A simple example illustrates the point. Given the functional,

This is an example of the rule of parametric differentiation that leads to the rate of change of any functional of form (that is written explicitly in terms of the independent variable): Differentiate the function,

The rate of change of a functional with respect to the change of the independent variable at one point in space is an inherent property of a single point in the functional (a single ordered pair, (independent variable, dependent variable)), and in every case can be defined without reference to the value of the functional, or the independent variable, at different ordered pairs. As shown below, this allows the definition of the rate of change of quantities

dependent on functions even if the dependence extends only to a single pair,

not be part a functional.

However, a Taylor-series like representation of the functional at at points close to a given independent variable can only be defined through the integral in (0.8) if the domain of the functional admits all possible infinitesimal neighborhoods about each point in the domain that differ arbitrarily from that point and throughout which the functional is continuously differentiable. In that case, the rate of change can be used to determine the value of the dependent variable at points (functions) near the function where the rate of change has been defined.

At the same time, the lack of a Taylor series representation is of no consequence in cases where the functional consists of a single pair of independent variable and its associated dependent variable, where the concept of a Taylor series becomes moot. As pointed out in the following discussion, far from being a limitation, this feature is consistent with the analytic properties of wave functions as well as with the manner in which the study of nature proceeds in terms of non-interacting systems (see following sections and comment in the Discussion section).

We consider the case in which the domain of a functional is required to satisfy a set of additional conditions such as that of normalization to an integer, as in the case of the wave functions of a many-particle Hilbert space and the corresponding densities obtained from them, Equations (0.9), (0.10) and (0.11).

Now, there exists no neighborhood about a particular density such that arbitrary variations of the form,

In this case, however, the concept of functional derivative as a rate of change remains valid and applicable where it retains all the properties attending to the rate of change such as assessments of the value of a quantity at a particular point with respect to its minimum based on the value of the rate of change at that point. The remainder of the paper is devoted to the exploitation of this feature and the derivation of formal results resting on it.

Two more advantages emerge. First, the rate of change of an expression such as an expectation value of an operator with respect to a wave function (defining the expectation value), or the density (defined by the wave function) can be obtained irrespective of whether or not the expectation value is a functional of the wave function or the density.

The second advantage is concerned with expressions, possibly functionals of wave functions or the density, that do not exhibit explicitly the independent variable, but are non-the-less dependent on it. Such expressions are not defined as mere functions of the independent variable, functionals of form, but rather by means of a procedure based on the independent variable (see Section 7.2). We shall refer to such functionals as functionals of process and in the following, we develop the general formalism for their derivatives (rates of change).

We generalize the concept of independent and dependent variable to apply to any ordered pair of the form,

function or a density, and

on

In all that follows, we seek to determine the rate of change of functionals (or generally expectation values that are not necessarily functionals over wave functions or densities) with respect to changes at one point of the

independent variable. We identify cases in which

function of its independent variable,

differentiation derived from the identity in (0.29). In each case, we express the dependent quantity (functional or not) in terms of the independent variable and proceed to obtain its rate of change with respect to that variable at a given, fixed independent variable.

The first demonstration of parametric differentiation (functional differentiation through (0.29)), is to determine

the derivative,

potential,

where

where

Analogously, the evaluation of

leading to the series,

where,

(0.35)

is the single-particle Green function in the presence of the potential. The Green function can also be written in the form of a resolvent,

In the limit,

It is worth emphasizing that the last expression for the derivative (the rate of change of

with respect to a change in the potential at

Consistent with the result established above, the functional derivative in (0.37) is confined to a single Hilbert

space, the Hilbert space defined by the potential,

Consider a generally complex, many-particle wave function,

ordinate space, subject to the normalization condition,

and the expectation value,

for an operator,

For a fixed

pairs,

over all multidimensional functions that obey the normalization condition in (0.38) and for which the integral in (0.39) is well defined.

We seek to determine the functional derivative,

multidimensional point,

where

runs into an insurmountable barrier.

Even though the functional is one of form, exhibiting explicitly the independent variable, the requirement of

arbitrariness may cause the quantity,

to satisfy the definition of a many-particle wave function.

The functional derivative interpreted as a rate of change can now be obtained, however, through a ge- neralization of the concept of parametric derivative to multi-dimensional space. In this procedure, using the

definition of the multidimensional Dirac function and its properties under integration, and where

an infinitesimal, (a number) we have,

so that

The feature that allows differentiation of expectation values is clear: The quantity,

The derivation of the time-dependent Schrödinger equation proceeds from the consideration of the action in- tegral,

evaluated at the state (wave function),

dependent Hamiltonian,

we have (in operator form),

Setting the derivative in (0.45) equal to zero yields the TDSE,

to be solved under some initial condition. This is none other than the well-known condition determining the TDSE as the vanishing of the derivative of the action at its stationary point. In this case, the TDSE is derived through the vanishing of the rate of change of the action with respect to changes in the wave function at points in multidimensional space.

We recall that in a many-particle system described by a wave function,

probability of finding a particle at

Consequently, a density function satisfies the conditions of positive semi-definiteness,

As a direct result of the normalization condition, conventional functional differentiation, requiring the value of a functional,

We consider the set,

wave functions (in the following we suppress spin),

relation in (0.47). Cioslowski [

minants, each constructed form

The set of all possible expectation values of many-particle operators with respect to the elements of

Although rigorous, Cioslowski’s formal procedure for the derivative is computationally out of reach. Here, we develop an equally rigorous alternative technique for differentiating the elements of

We recall the basic requirement of parametric differentiation of one function by another: The symbol,

Therefore, as stated above, the expression,

dependence of the function

derivative of the functional identity, (0.18), to the expression providing an exact representation of

The following point is possibly both self-evident as well as subtle: There is nothing in the form of a function (its dependence on coordinates) that betrays its functional dependence on a particular density. A given function, (e.g.,

Now, in the case in which a function is to be differentiated with respect to a given density, there exists an immediate and exact mapping of its dependence on coordinates to an analytic, differentiable form that explicitly exhibits the density. It is an expansion in the equidensity basis [

For the sake of completeness, in this subsection, we introduce the spin variable [

An exact mapping of a function,

dinger equation), onto an analytic (differentiable) expression can be had in terms of the expansion of the func-

tion in terms of the equidensity basis. Namely, for each orbital,

nant contributes to the formation of a density,

with the sum ranging over some collection of orbitals, we write,

with

being the elements of an orthonormal and complete basis [

with

where

In accordance with the closing remarks in the previous section, an orbital may contribute to a number of different densities (in principle an infinite number) ant thus possesses unique parametric derivatives with respect to each of these densities.

The particular choice of

The coefficients,

and are generally complex numbers. These coefficients change according to the density used to construct the equidensity basis. That density, on the other hand, is uniquely chosen as the density to which a given orbital may be contributing through (0.49), with no connection existing between the coefficients at one expansion (some density) to those at another. (It may be tempting to think of the equidensity basis as a functional of the density concluding that the coefficient for a given orbital at one density may be connected through functional dif- ferentiation to coefficients of the same orbital of another density. Recall, however, that the very concept of con- ventional functional differentiation is disallowed over the domain of densities, and the only differentiation available is that of parametric differentiation confined to the space of given density. Furthermore, any general function of coordinates, one that is not necessarily a solution of a Sxhrödinger equation and hence not a functional of a density can be expanded in terms of the equidensity basis for any density, each expansion defined in terms of coefficients that have no functional connection to those at another density. It follows that the question of the derivatives of the coefficients is mathematically moot.)

Finally, the parametric derivative of an expectation value with respect to a density formed by a wave function that leads to that density can be determined through the parametric derivative of the expansion of the wave function in the equidensity basis for that density. No connection of that derivative to that at another density that may contain the same orbital exists, or can be sought in terms of functional derivatives of the coefficients of the expansions.

Explicit examples of such derivatives of the Coulomb potential energy determined with respect to Slater determinants leading to a density are given in other work [

Cioslowski [

Suppressing spin, a brief summary of the procedure is given below, with details to be found in the original papers [

Any antisymmetric N-particle wave function can be written as a linear superpositions of mutually ortho- normal Slater determinants (in Einstein summation notation),

where

ments (orbitals),

coordinate space. The orbitals,

squared and integrated over all coordinates but one yields the density through (0.56).

This property follows from the form of the orbitals (see [

where the functions,

where in the second line we have used the expansion of a determinant along its first row,

minor of

whose elements are given by the integrals,

The non-linear nature of the transformation,

The existence and convergence of

Cioslowski [

by the expression,

where,

In the last two expressions, a vector-matrix notation is used, so that

We now examine the properties of the orbitals defining

of the matrix

Clearly, the set,

orbitals,

Cioslowski also shows that near its converged limit,

Because of the explicit presence of the density at each iteration step, the matrix,

where the tensor product,

Expressions for the parametric derivative of

The orbitals contributing to

the corresponding values of

To summarize: Each density,

dual orbitals used to construct the antisymmetric, N-particle wave functions,

metric derivatives of constrained search functionals, or general expectation values of operators, with respect to these wave functions can be obtained by means of the parametric derivatives of the orbitals under the integral signs defining the expectation value. They, in turn, can be obtained by the rigorous means of expanding in the equidensity basis and differentiating the expansion.

Ensembles, or mixed states, are described by a density matrix [

with,

density,

The expression in (0.67) describes a system that is known to occupy a state,

The functional derivative of

In the last expression, each differentiation is performed at the density defined by the individual states,

In a well-known paper [

Kohn’s proof relies on the following theorem:

If

the trivial conditions

The proof hinges on the existence of a small neighborhood around a given density throughout which derivatives with respect to the density can be uniquely obtained in a continuously differentiable manner.

The existence of such a neighborhood guarantees the existence of uniquely defined Frèchet derivatives. As already stated above, however, no such neighborhood exists for the case of domains defined in terms of func- tions that satisfy the definition of a density.

The flaw in the proof is evident in the expression of the theorem. The function,

The central result of this paper is the use of the Dirac delta function, rather than an arbitrary test function defined over an extended domain in coordinate space, as in the conventional formulation, in the determination of functional derivatives of quantities whose determination depends on a function of coordinates. In this procedure, the formalism leads to the rate of change of a dependent variable with respect to the change at one point of the independent variable (a function) in all cases, even when the function is constrained to integral normalization, thus blocking conventional formulations of the functional derivative.

In the case of general functionals, whose domains are free of conditions of normalizations, conventional functional derivatives coincide with parametric derivatives and Equation (0.6) remains valid. In the presence of externally imposed conditions, such as fixed normalization, for example, conventional functional differentiation becomes inapplicable whereas rates of change can still be readily evaluated through parametric differentiation.

This feature is particularly useful when a functional is known to exhibit an extremum (e.g., a minimum). Then, the rate of change at the minimum vanishes, while that at any other point is non-zero (signifying that the value of the functional is higher than that at its minimum, for example).

The paper provides a rigorous, exact procedure for the functional differentiation of expectation values of operators in quantum mechanics with respect to wave functions or the densities obtained from the wave func- tions entering the determination of the expectation value in question. Emphasis is placed on derivatives with respect to the density, especially since wave functions generally are not written explicitly in terms of the density. In this case, the dependence of the wave function on coordinates can be mapped exactly onto that of the density by means of an expansion in an orthonormal and complete basis (the equidensity basis at the density of the dis- cussion) [

Finally, a comment on the absence of a Taylor-series-like expression that connects expectation values at one density to those at another, possibly nearby density. There are both mathematical as well as fundamental, phy- sical reasons for this absence.

First, as mentioned in the text, a density defines a unique functional as the set of all antisymmetric wave functions that lead to the density. There is, however, no one-to-one correspondence between the elements of the

sets defined by two different densities, and no functional of the form,

the dependent variable is a single wave function, can be established. In this case, the very concept of Taylor series connection between different wave function corresponding to different densities becomes moot.

Second, recall that a density may correspond to the ground state of a many-particle system and thus belong to the Hilbert space determined by the potential acting on, and number of particles of, the system. This Hilbert space is separate and disjoint from that of any other system that is independent of (non-interacting with) the one in question. Consequently, there exists no connection between the spaces provided by the properties (quantum states or functional derivatives) of either system separately.

Discussions with colleagues at LLNL and ORNL provided motivation for this work. I am also grateful to the referees whose comments significantly improved the paper. Work supported by the US DOE under Contract DE-AC52-07NA27344 with LLNS, LLC.

The procedure of conventional differentiation, outlined in Section 2, relies on the existence of a small neigh- borhood about each point,

the domain,

reason for this condition and the consequences of its violation.

Consider the simple example of the linear functional,

tion of functional derivative, we obtain the relations,

Provided that

as follows from the well-known theorem on equality of integrants when two functions integrated agains any arbitrary function yield identical results, i.e., if

for any (sufficiently well-defined) function

For the sake of completeness, this theorem is proven below. At this point we expose its content, beginning with a counterexample:

Suppose that

last equality does not follow. For example,

clusion that 1 = 2.

We now prove the theorem:

If for any two continuous and differentiable functions the equality,

holds for all arbitrarily chosen, that is unconstrained,

identity).

The theorem implies that the equality holds for any and all specific choices of

Choose as

implying,

for every (arbitrary) point

It can be readily shown that if the function

tives can be determined only modulo an arbitrary constant. This has immediate consequences in the deter- mination of functional derivatives: Contrary to the demands of Frèchet differentiation, it leads to arbitrariness in the derivative defined conventionally with respect to an arbitrary test function [

In the case of the identity function,

It is useful to inquire as to the analogous relation for the case of functionals.

We seek to determine the rate of change,

examine first the somewhat simpler case of functions defined on a lattice of distinct points,

Given a change,

some point,

where

except when

mains valid regardless of the manner in which the change in the function at one of its points was introduced, either arbitrarily or as the difference at a single point between the function

since each term in the sum vanishes except when the indices coincide where it equals unity.

In the continuum case, we set,

and once again we demand that,

since the change in the function at all points other than

Therefore, a function can be changed in terms of a distribution (the Dirac delta function), so that,

an expression that is intended to make clear that

The Dirac delta function is an example of a distribution, or generalized function [