Quantifying energetic information in density functional theory
Energy and information are two fundamental concepts in physics and chemistry. In density functional theory (DFT), all information pertaining to stability, reactivity, and other properties is encompassed in the ground state electron density. The basic theorems of DFT govern that energy is a universal functional of the density and thus it can be regarded as a special kind of information. In this work, we quantify the energetic information in terms of Shannon entropy and Fisher information for energetic distributions of atoms and molecules. Two identities are unveiled for the energetic density, its gradient and Laplacian to rigorously satisfy. A new partition scheme to decompose atoms in molecules has been proposed using the energetic distribution. We also show that our approach can simultaneously quantify both two-body and many-body interactions. This new framework should provide us with new analytical tools to appreciate electronic properties of molecular systems including stability and reactivity. More importantly, this work establishes the missing link in DFT between energy and information, the two most fundamental quantities in quantum theory.
<p>The relationship among matter, energy, and information is controversial and appears to be philosophical. Nevertheless, quantum mechanics does provide with us meaningful clues about the relationship. 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. So, within the wavefunction theory framework, matter comes in first, and energy and information are secondary but independent. In density functional theory (DFT), however, matter is represented by the external potential, and information by the electron density. The one-toone 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.</p><p>In past few decades, tremendous progresses in DFT have been achieved in looking for better functional approximations for energetic components such as exchange-correlation energy and kinetic energy. [1][2][3][4][5] In recent years, we invested considerable efforts in establishing an information-based framework in DFT with the electron density as the probability distrbution function, which we call the information-theoretic approach (ITA). [6][7][8][9][10] 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.</p><p>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, 𝐸 = ∫ 𝑒(𝐫) 𝑑𝐫.</p><p>(</p><p>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 TS, [1][2][3] Weizsäcker kinetic energy TW, 11 steric energy ES, 12 classical Coulombic repulsion J, [1][2][3] exchange-correlation energy EXC, [1][2][3] Pauli energy, 12 Fermionic quantum energy Eq, 12 etc. 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, (r)  e(r)/E, so that the following normalization condition to unit is met, [6][7][8][9][10] ∫ (𝐫</p><p>With this new probability function, following the literature, [6][7][8][9][10] we can then introduce its corresponding information-theoretic quantities, such as Shannon entropy SS 13</p><p>Fisher information IF, 14</p><p>and the alternative Fisher information I'F, 15 𝐼′ 𝐹 ≡ ∫ 𝑖′ 𝐹 (𝐫) 𝑑𝐫 = − ∫ 𝛻 2 𝜀(𝐫) ln 𝜀 (𝐫) 𝑑𝐫,</p><p>where 𝑠 𝑆 (𝐫) is the Shannon entropy density, 𝑖 𝐹 (𝐫) and 𝑖′ 𝐹 (𝐫) are two forms of Fisher information densities, and 𝛻𝜀(𝐫) and 𝛻 2 𝜀(𝐫) are gradient and Laplacian of the energetic density 𝜀(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. 15 As can be readily proved with the same spirit, 15 the following identity should be also true for the energetic probability function (r),</p><p>With the help of Eq. ( 2), integrating both sides of Eq. ( 6) yields</p><p>Equations ( 6) and (7) provide strict constraints for the energetic density 𝜀(𝐫), its gradient 𝛻𝜀(𝐫), and Laplacian 𝛻 2 𝜀(𝐫) to satisfy. It is the first time that such stringent constraints for any energetic function 𝜀(𝐫) and its associated quantities are reported in the literature.</p><p>Using three energetic components, total kinetic energy TS, Weizsäcker kinetic energy TW, and classical Coulombic repulsion J as examples, we illustrate the radial distribution of Shannon entropy of these energetic components for Ne and Ar atoms. Hartree-Fock densities are employed with tight numerical integration using 1024 nonuniform grids along the radial distance axis from 0 to 10 a.u. 16 Figure 1 shows the results for Ne and Fig. 2 for Ar. As can be seen from the Figures, local behaviors of Shannon entropy of these energetic distribution functions are smooth and well-behaved. They are different from their counterparts when the electron density is employed as the probability function. 15 Since the energetic information as reflected by its local behaviors, gradient and Laplacian included in Shannon entropy and Fisher information is crucial in determining molecular stability and reactivity, 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.</p><p>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, etc. [17][18][19][20] 𝐼 𝐺 ≡ ∫ 𝑖 𝐺 (𝐫) 𝑑𝐫 = ∫ 𝜀(𝐫) ln</p><p>where 𝑖 𝐺 (𝐫) is the information gain density of the energetic distribution 𝜀(𝐫), and 𝜀 0 (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 the following 9,21,22</p><p>and</p><p>We 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. 23 With the electron density (𝐫) replaced by the energy distribution 𝜀(𝐫) as the probability function, as can readily be shown, 23 the same identity must also be true</p><p>with</p><p>Same as Eq. ( 6), Eq. ( 11) provides another stringent constraint for the energetic density, its gradient, and Laplacian to satisfy.</p><p>Shown in Fig. 3 is the radial distribution of the relative Shannon entropy, Eq. ( 8 with the Hartree-Fock density 16 and He as the reference. As the nuclear charge Z increases, Shannon entropy monotonically decreases but 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.</p><p>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 representation 24 for the total or component energy density, e(r), whose probably function was defined in Eq. ( 1) (r),</p><p>where  𝐴 (𝐫) and  𝐴</p><p>where the summation is over all atoms or groups in the molecule. Following Nalewajski and Parr, [25][26][27][28] if we minimize Eq. ( 13) subject to the condition of Eq. ( 14), there results in the Hirshfeld's stock-holder partition,</p><p>Equation ( 15) provides a novel atoms-in-molecules approach to decompose the total or</p><p>For instance, for a molecular system composed of two atoms or fragments, A and B, we have</p><p>and</p><p>The second example of possible applications is to use the energy density e(r), or its magnitude if it is negative, not its probability function 𝜀(𝐫), as the probability function. In such a case, instead of the normalization condition to unit in Eq. ( 2), e(r) is normalized to E in Eq.</p><p>(1). In</p><p>Following what we proposed earlier, [17][18][19][20] because eA and 𝑒 𝐴 0 behave quite similarly and with the introduction of a dimensionless variable x  [𝑒 𝐴 (𝐫) − 𝑒 𝐴 0 (𝐫)]/𝑒 𝐴 (𝐫), Eq. ( 19) becomes</p><p>Using ln 1 1−𝑥 ≈ 𝑥 as the first-order approximation of the Taylor expansion, we obtain</p><p>This result suggests that under the first-order approximation, the relative Shannon entropy (also called information gain, Kullback-Leibler divergence, etc.) of a given energy distribution is simply the contribution of the given component to the total interaction energy. Keep in mind that this energy distribution can be either the total energy itself or any of its components. The result in Eq. ( 21) provides a new partition scheme for the interaction energy, in lieu of that for the total energy in DFT. 12,[29][30][31][32] Also, expanding the Taylor series to second or higher orders in Eq. ( 20) comes arise contributions from many-body interactions, which are found to be important in quantifying such phenomena as cooperativity, [33][34][35][36] frustration, 37 and homochirality. 38,39 On other words, relative Shannon entropy together with other ITA quantities defined in Eqs. ( 3) -( 12) should be able to provide with us novel physiochemical insights previously unavailable in the literature. Applications of this new analytical framework to molecular stability, reactivity and other properties are in progress.</p><p>In summary, in this work, we present a theoretical framework to quantify energetic information in density functional theory. To that end, Shannon entropy, two forms of Fisher information, relative Shannon entropy and two forms of relative Fisher information have been introduced for any energetic distribution function, which can be the total energy itself or any of its components. Two identities have been unveiled for these energetic information quantities. A new partition similar to 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. More importantly, this approach 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 stability and reactivity from this completely new perspective are anticipated and currently in progress.</p>