Quantifying energetic information in density functional theory

The relationship among matter, energy, and information^1,2 is controversial and appears to be philosophical. Nevertheless, quantum mechanics does provide meaningful clues about the relationship for us. In wavefunction theory, for example, the time-independent Schrödinger equation indicates that matter can be represented by the Hamiltonian operator and information by the total wavefunction. Once the Hamiltonian of a system is given, both energy and information can, in principle, be determined adequately accurately. Hence, within the wavefunction theory framework, matter comes in first, and energy and information are secondary but independent. In density functional theory (DFT), however, matter by the external potential and information by the electron density. The one-to-one correspondence between the external potential and the ground-state electron density suggests that both matter and information are primary, and energy can be regarded as a special kind of information because it is a universal functional of the electron density.

In past few decades, tremendous progress in DFT has been achieved in looking for better functional approximations for energetic components, such as exchange–correlation energy and kinetic energy.^3–7 In recent years, we have invested considerable effort in establishing an information-based framework in DFT with the electron density as the probability distribution function, which we call the information-theoretic approach (ITA).^8–13 There is an apparent missing link, however, between the two efforts. That is, how to establish energy as a special kind of information within the DFT framework has never been seriously addressed in the literature. This is precisely what we plan to do in this work.

Without losing generality, let us consider a total or component energetic quantity E, which is related to its local energy density, e(r), by the following equation:

Continuity and well-behavedness of the energy density are assumed, which are often the case for electronic systems, such as atoms and molecules. This quantity can be either the total energy of a system or its components, including total kinetic energy T_S,^3–5 Weizsäcker kinetic energy T_W,¹⁴ steric energy E_S,¹⁵ classical Coulombic repulsion J,^3–5 exchange–correlation energy E_XC,^3–5 Pauli energy,¹⁵ electrostatic energy E_e,¹⁵ and Fermionic quantum energy E_q.¹⁵ To express the above energetic quantity as a special kind of information, the key is to introduce its local energy density as a continuous probability distribution function, ε(r) ≡ e(r)/E, so that the following normalization condition to unit is met:^8–12 

To demonstrate the feasibility of the above treatment, illustrative examples are shown in Fig. 1 using the total kinetic energy T_S and electrostatic energy density E_e of the neon atom with the Hartree–Fock density.¹⁶ Figures 1(a) and 1(b) show their normalized energy density distributions in Eq. (2) for T_S and E_e, respectively, and Figs. 1(c) and 1(d) show their corresponding radial distributions. From these curves, we can see that these functions are smooth, positive, and well-behaved, so, same as the electron density, they can be regarded as probability functions. Note that the local distribution of E_e is negative everywhere, but as shown in Figs. 1(b) and 1(d), after the normalization in Eq. (2), it becomes a well-defined probability distribution function. We also tested other energetic quantities, and similar conclusions can be obtained. The only exception is the total kinetic energy T_S, which cannot be uniquely defined as we know.^3–5 To avoid its local distribution changing signs caused by the arbitrary Laplacian term, the Lagrangian form of the total kinetic energy must be adopted.

With this new probability function, following the literature,^8–13 we can, then, introduce its corresponding information-theoretic quantities, such as Shannon entropy S_S,¹⁷ 

Fisher information I_F,¹⁸ 

and the alternative Fisher information I′_F,¹⁹ 

where S_S(r) is the Shannon entropy density, i_F(r) and $i′F(r)$ are the two forms of Fisher information densities, and ∇ɛ(r) and ∇²ɛ(r) are the gradient and Laplacian of the energetic density ɛ(r), respectively. Earlier, we have proved that the following identity is valid for atoms and molecules when the electron density is employed as the probability function.¹⁹ As can be readily proved with the same spirit,¹⁹ the following identity should also be true for the energetic probability function ε(r):

With the help of Eq. (2), integrating both sides of Eq. (6) yields

Equations (6) and (7) provide strict constraints for the energetic density ɛ(r), its gradient ∇ɛ(r), and Laplacian ∇²ɛ(r) to satisfy. It is the first time that such stringent constraints for any energetic function ɛ(r) and its associated quantities are reported in the literature.

Using three energetic components: the total kinetic energy T_S, Weizsäcker kinetic energy T_W, and classical Coulombic repulsion J as examples, we illustrate the radial distribution of the Shannon entropy of these energetic components for Ne and Ar atoms. Hartree–Fock densities are employed with the tight numerical integration using 1024 nonuniform grids along the radial distance axis from 0 to 10 a.u.¹⁶ Figure 2 shows the results for Ar. As can be seen from Fig. 2, local behaviors of the Shannon entropy of these energetic distribution functions are smooth and well-behaved. A close examination of these curves, especially when compared with the radial distribution of the electron density, shows that (i) the same as the electron density, distinct atomic shell structures can be unambiguously seen in them; (ii) they are different from their counterparts when the electron density is employed as the probability function,¹⁹ suggesting that they could provide different information about the properties of the systems; and (iii), more importantly, their distributions are more than apparent and visible in the valence-shell region than the electron density. This region is known to be the most important area where chemical processes and transformations take place, indicating that these new quantities should be able to provide a lot of new insights about chemical reactivity. Since the energetic information as reflected by its local behaviors, gradient, and Laplacian included in the Shannon entropy and Fisher information is crucial in determining molecular stability and reactivity, as supported by the profiles shown in Fig. 2, we are certain that these newly introduced energetic information quantities and relationships will prove useful and important in the future in developing a density-based reactivity theory in DFT.

The second set of ITA quantities is based on a reference probability function, which can be from either isolated systems or any other states (e.g., excited states or transition states) with the only condition that it should be normalized to unit. The first ITA quantity of this sort is the relative Shannon entropy, also called information gain, Kullback–Leibler divergence, or information divergence,^20–23 

where i_G(r) is the information gain density of the energetic distribution ɛ(r) and ɛ₀(r) is the energetic distribution function of the reference state, which also satisfies Eq. (2). For the two forms of Fisher information, Eqs. (4) and (5), its relative version is given as follows:^10,12,24,25

and

We have recently proved that there exists an identity for three ITA densities in Eqs. (8)–(10) when the electron density is employed as the probability function.²⁶ With the electron density ρ(r) replaced by the energy distribution ɛ(r) as the probability function, as can readily be shown,²⁶ the same identity must also be true,

with

Same as Eq. (6), Eq. (11) provides another stringent constraint for the energetic density, its gradient, and Laplacian to satisfy.

Figure 3 shows the radial distribution of the relative Shannon entropy, Eq. (8), of energetic components: total kinetic energy T_S, Weizsäcker kinetic energy T_W, and classical Coulombic repulsion J, for Ne and Ar neutral atoms with He as the reference. As can be seen from Fig. 3, their local behaviors are drastically different from those of the Shannon entropy illustrated in Fig. 2. Table I lists the numerical result of Shannon entropy and the relative Shannon entropy of the three energetic components: total kinetic energy, Weizsäcker kinetic energy, and classical Coulombic repulsion, for neutral atoms from He to Xe with the Hartree–Fock density¹⁶ and He as the reference. As the nuclear charge Z increases, the Shannon entropy monotonically decreases, but the relative Shannon entropy monotonically increases. In all cases, these quantities are shown to be well-behaved and have finite values, and no divergence phenomenon is observed. These quantities are anticipated to provide us with new analytical tools to appreciate electronic properties of molecular systems.

As an illustrative example of potential applications of the above energetic information, let us consider the relative Shannon entropy, Eq. (8), under the atoms-in-molecules representation²⁷ for the total or component energy density, e(r), whose probability function is defined in Eq. (1), ε(r),

where ɛ_A(r) and $εA0(r)$ (≡$eA0(r)EA0$⁠) are real and reference probability functions of the local energy density for atom or group A in a molecule, respectively, satisfying the following normalization condition:

where the summation is over all atoms or groups in the molecule. Following Nalewajski and Parr,^28–31 if we minimize Eq. (13) subject to the condition of Eq. (14), it results in Hirshfeld’s stock-holder partition,

Equation (15) provides a novel atoms-in-molecules approach to decompose the total or component energy density into atomic contributions. With the help that $εA0(r)≡eA0(r)EA0$ and $ε(r)≡e(r)E$⁠, Eq. (15) becomes

For instance, for a molecular system composed of two atoms or fragments, A and B, we have

and

The second example of possible applications is to use the energy density e(r) itself in Eq. (1), not its probability function ɛ(r) in Eq. (2), as the probability function in Eqs. (3)–(10). In such a case, instead of the normalization condition to unit in Eq. (2), e(r) is normalized to E in Eq. (1). One thing to keep in mind, though, is that Eqs. (3) and (5) require e(r) to be positive, but this constraint is lifted for all other indices, especially for the relative information quantities in Eqs. (8)–(10). In Table II, we exhibit the numerical result of both the Shannon entropy and the relative Shannon entropy of three such energy components: total kinetic, Weizsäcker kinetic, and classical Coulombic repulsion energies, which are all positive, for neutral atoms from He to Xe with the Hartree–Fock density and He as the reference. The same trend as what we observed in Table I is found in Table II. Although numerical values tend to be larger, no divergence or ill-behavior in their trends is observed, suggesting that this representation of energetic information is also a viable option. It is anticipated that it can also provide valuable insights to appreciate molecular properties. For instance, the relative Shannon entropy under the atoms-in-molecules representation reads

Following what we proposed earlier,^19–21 because e_A and $eA0$ behave quite similarly and with the introduction of a dimensionless variable x ≡ $[eA(r)−eA0(r)]/eA(r)$⁠, Eq. (19) becomes

Using ln $11−x≈x$ as the first-order approximation of the Taylor expansion, we obtain

This result suggests that under the first-order approximation, the relative Shannon entropy (also called information gain, Kullback–Leibler divergence, etc.)^19,26 of a given energetic distribution is simply the contribution of the given component to the total interaction energy. Keep in mind that this energetic distribution can be either the total energy itself or any of its components. In addition, be aware that above formulations are applicable to any energetic distributions and there is no constraint that it be positive. The result in Eq. (21) provides a new partition scheme for the interaction energy, in lieu of two partition schemes of the total energy in DFT already available in the literature.^15,32–35 In addition, expanding the Taylor series to second or higher orders in Eq. (20) yields contributions from nonlinear terms accounting for many-body interactions, which are found to be important in quantifying phenomena such as cooperativity,^36–39 frustration,⁴⁰ and homochirality.^41,42 In other words, the relative Shannon entropy together with other ITA quantities defined in Eqs. (3)–(12) should be able to provide novel physiochemical insights previously unavailable in the literature. Applications of this new analytical framework to molecular stability, reactivity, and other properties are in progress, whose results will be reported elsewhere.

In summary, in this work, we present a theoretical framework to quantify energetic information in density functional theory. To that end, the Shannon entropy, two forms of Fisher information, relative Shannon entropy, and two forms of relative Fisher information have been introduced for an energetic distribution function, which can be either the total energy itself or its components. Two identities have been unveiled for these energetic information quantities. A new partition similar to the Hirshfeld stock-holder scheme has been disclosed using energetic distributions. We also showed that this approach can simultaneously quantify both two-body and many-body interactions. To the best knowledge of the present author, these results are reported for the first time in the literature. More importantly, this work establishes the missing link in density functional theory between energy and information, the two most fundamental quantities in quantum theory. Its applications to appreciate molecular properties, including structure, stability, and reactivity, for chemical processes and transformations from this completely new perspective are anticipated and currently in progress.