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Article Cite This: J. Phys. Chem. A XXXX, XXX, XXX−XXX

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Toward Electron Correlation and Electronic Properties from the Perspective of Information Functional Theory Mojtaba Alipour* and Zeinab Badooei Department of Chemistry, College of Sciences, Shiraz University, Shiraz 71946-84795, Iran

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S Supporting Information *

ABSTRACT: Over the last years, immense efforts have been made to apply the quantities of information theory to the electronic structure and properties of various systems. In this context, one can make use of one or many of the information theoretic quantities together to describe the total energy, its components, and other electronic properties. Such an idea is feasible through an approach so-called information functional theory, which in turn constitutes the cornerstone of the present investigation. More specifically, in this work several information theoretic quantities like Fisher information, Shannon entropy, Onicescu information energy, and Ghosh−Berkowitz−Parr entropy with the two representations of electron density and shape function are considered for reliable prediction of atomic and molecular correlation energies as well as several electronic properties such as atomization energies, electron affinities, and ionization potentials. It is shown that with more or less different accountabilities of the information theoretic quantities they can be introduced as useful descriptors for estimation of electron correlation energies for a large variety of systems including neutral atoms, cations, isoelectronic series, and molecules. This is also indeed the case for the electronic properties under study. Considering different notions of the information theoretic quantities with various scaling properties and varied physiochemical meanings about the electron density distribution, we find that instead of simulating all data using one of these quantities individually taking all of them together provides a better view for the description of correlation effects and electronic properties of systems.

1. INTRODUCTION When studying the cohesive energy of metals, Wigner and Seitz introduced the concept of electron correlation.1 However, it was Löwdin who defined the correlation energy as the energy correction needed to make the Hartree−Fock (HF) energy exact.2 The origin of this correction can be traced to the HF approximation where the instantaneous electron−electron repulsion is replaced by an averaged electron−electron interaction. Accordingly, the electron correlation energy Ec can be defined as the difference between exact energy and energy of the HF approximation Ec = ⟨Ψ|Ĥ |Ψ⟩ − ⟨ΨHF|Ĥ |ΨHF⟩

correlation energy can be obtained from eq 1 if both the HF and the exact nonrelativistic energies of systems are known. However, given the fact that the latter energy is not available in most cases, there has been a tremendous interest in the literature to present different approximate schemes to estimate the correlation energies of atomic and molecular systems. As some representative efforts in this context, several empirical equations and scaling relations have recently been proposed to predict electron correlation energy in different systems.12−15 In this work, however, we take a different approach to address the issue where the quantities from information theory16−19 will be used for the prediction of atomic and molecular correlation energies as well as some electronic properties. Information theory is a branch of applied mathematics and computer science that deals with the quantification of information, which is often a probability distribution function, for a system. In recent years, much efforts have been invested to appreciate and understand physical and chemical processes in the framework of information theory; see, for instance, refs 20−52. Some related studies have also been performed in the context of electron correlation energies.53−62 Considering atomic and molecular species as information systems with the electron density as the probability distribution function, there

(1)

where Ψ and ΨHF denote respectively the exact and HF ground state wave functions of a system described by the nonrelativistic Hamiltonian Ĥ . A proper treatment of correlation effects for a quantum system poses one of the most challenging problems in theoretical chemistry. Electron correlation affects not only the electronic energy but also many other characteristics such as bond dissociation, electron delocalization, chemical shifts, spin, optical properties, and so on.3−11 Among others, imperative roles of electron correlation in condensed matter and physics, superconductivity, and semiconductor materials should also be noted. Accurate values of the electron correlation energies are useful for the calibration of nonrelativistic electronic structure theories and parametrization of correlation density functionals. The © XXXX American Chemical Society

Received: June 14, 2018 Revised: July 13, 2018

A

DOI: 10.1021/acs.jpca.8b05703 J. Phys. Chem. A XXXX, XXX, XXX−XXX

Article

The Journal of Physical Chemistry A

Thomas−Fermi kinetic energy density given by tTF(r;ρ) = ck[ρ ( 3 r)] 5/3 with ck = 10 (3π 2)2/3.72−75 There are some remarkable points regarding to the quantities defined in eqs 2−5). From the viewpoint of information theory, Fisher information including the both electron density and its gradient, eq 2, is a gauge of the sharpness or localization of the electron density distribution while Shannon entropy with the electron density as the only variable, eq 3, measures the spatial delocalization of the electronic density.76 Aiming to define a finer measure of dispersion distribution than Shannon entropy, Onicescu defined information energy of order n as the integral of the nth power of the electron density, eq 4. Notice that the Onicescu information energy does not have the dimension of energy in the physical sense and consequently it is a purely statistical concept. Finally, the GBP entropy defined in eq 5 originates from the reformulation of the ground state DFT into a local thermodynamics through the phase space distribution function. The information theoretic quantities defined in eqs 2−5) include the electron density as the probability distribution function. However, there is another distribution function, namely the shape function σ(r),77−80 upon which the information functionals can also be redefined. The two variables σ(r)and ρ(r) are related through the following relation

are various quantities in information theory to evaluate their properties. Such quantities as Fisher information,16 Shannon entropy,17 Onicescu information energy,18 and Ghosh− Berkowitz−Parr entropy19 should be named. As will be discussed in what follows, these key quantities are density functionals with different scaling properties. On the one hand, according to density functional theory (DFT), the electron density includes all the information necessary to determine all properties in the ground state and consequently any property of a system can be expressed as a function of the electron density.63−66 On the other hand, the information theoretic quantities as density functionals are also expected to be able to describe various properties as well. Putting these perceptions together, one can make use of one or many of these quantities together to describe the total energy, its components, and other electronic properties of systems. Effort in the line of this research field is called information functional theory (IFT).67 We notice that although to accomplish this goal there is a pathway through the traditional DFT as a well-documented approach in the literature which in turn depends on various approximations for the exchange-correlation functional, as an alternative route differing from the original idea in DFT the IFT quantities are also of interest.68−71 Using information theoretic quantities within the framework of IFT for prediction of atomic and molecular correlation energies as well as several electronic properties constitutes the subject of the present contribution. In particular, we would like to dissect whether there exist information theoretic functionals with superior performance compared to others in applicability and accountability for exploring the electron correlation effects. Furthermore, we will also analyze the combination forms of the information theoretic quantities to generate a sufficiently accurate description of some representative electronic properties such as atomization energies, electron affinities, and ionization potentials through IFT.

ρ(r) = Nσ(r)

with the normalization condition∫ σ(r) dr = 1. Accordingly, the information theoretic quantities can be rewritten based on the shape function as follows 2

IF[σ ] =

EOn [σ ] =

SS[ρ] = − EOn [ρ] =

∫

∫ [ρ(r)]n dr

(n ≥ 2)

ÄÅ ÉÑ ÅÅ t(r; ρ) ÑÑÑ 3 Å ÑÑ dr kρ(r)ÅÅÅc + ln ÅÅÇ t TF(r; ρ) ÑÑÑÖ 2

(7) (8)

1 n−1

(9)

∫

∫ [σ(r)]n dr

ÄÅ ÉÑ ÅÅ t(r; σ ) ÑÑÑ 3 Å Å ÑÑ dr kσ(r)ÅÅc + ln ÅÅÇ 2 t TF(r; σ ) ÑÑÑÖ

(10)

Using the relation between σ(r) and ρ(r), eq 6, it can be shown that the information theoretic quantities under study in the two representations of electron density and shape function are related through eqs 11−14),

(2)

∫ ρ(r) ln ρ(r) dr

1 n−1

SGBP[ρ] =

dr

dr

∫ σ(r) ln σ(r) dr

SGBP[σ ] =

2

∫ |∇ρρ((rr))|

∫ |∇σσ((rr))|

SS[σ ] = −

2. THEORETICAL AND METHODOLOGICAL ASPECTS 2.1. Information Theoretic Quantities. The central information theoretic quantities of concern in this work are Fisher information, IF,16 Shannon entropy, SS,17 Onicescu information energy of order n, EnO,18 and Ghosh−Berkowitz− Parr (GBP) entropy, SGBP.19 These functionals are respectively defined as follows: IF[ρ] =

(6)

(3)

IF[σ ] =

IF[ρ] N

(11)

(4)

SS[σ ] =

SS[ρ] + ln N N

(12)

(5)

where ρ(r) is the ground state electron density of an N-electron system which satisfies the normalization condition ∫ ρ(r) dr = N and ∇ρ(r) is the density gradient. In eq 4, the cases of n = 2 and n = 3 are often used, though other options are also possible.67−71 4π c 5 Moreover, in eq 5, k is the Boltzmann constant, c = 3 + ln 3 k , t(r;ρ) is the kinetic energy density which is related to the total kinetic energy TS via ∫ t(r;ρ) dr = TS, and tTF(r;ρ) is the

EOn[σ ] =

EOn[ρ] Nn

(13)

SGBP[σ ] =

SGBP[ρ] N

(14)

Regarding eq 14 we mention in passing that since the scaling properties for the kinetic energy density t(r;ρ) included in eq 5 are unknown, it is unclear whether there exists any explicit relationship between SGBP[ρ] and SGBP[σ]. However, it has numerically81 and analytically82 been proven that the two B


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The Journal of Physical Chemistry A representations of GBP entropy based on σ(r) and ρ(r) can be related via eq 14. 2.2. IFT Framework. According to the related theoretical proofs in the field,83−85 one or more of the information theoretic quantities should suffice in determining the total electron density and therefore all the ground state properties. As a matter of fact, because of the intercorrelation among different information theoretic quantities, when one quantity can be employed to describe a system other quantities should also be able to do the same job. However, although the analytical justifications are robust and valid, their practical implementations and numerical verifications in this context seem to be difficult, at least at first glance. To address this issue, the IFT framework is of concern herein.67 Pragmatically, we rely on a strategy where electronic energy any consequently any electronic property P can be expressed in terms of information theoretic quantities Q as follows P[ρ] = P[Q ]

P[ρ] =

Here the summations run over information theoretic quantities Qj and cj and cne are the expansion coefficients to be determined by the least-squares fitting procedure. We note in passing that similar approaches have previously been proposed to expand energy functional using homogeneous density functionals in density or coordinate scaling as well as in terms of moments; see, for instance, refs 86−91. Nonetheless, unlike earlier efforts the central density functionals of concern herein are the intercorrelated information theoretic quantities having different scaling properties and varied physiochemical propensities.

3. COMPUTATIONAL DETAILS As benchmark sets under study, we first considered the neutral atoms and their single charged cations with N = 2 to 18. Next, different isoelectronic series with the number of electrons from 2 up to 18, making 315 atomic species in total, have also been considered as other test sets. As reference correlation energies for these systems, the widely used data reported in refs 92 and 93 were employed. On the molecular species, we have taken another set of 56 molecules along with the reference correlation energies from the work of O’Neill and Gill whose applicability and reliability have earlier been advocated in the related efforts.94 Regarding the electronic properties, we have considered atomization energies, electron affinities, and ionization potentials for a benchmark set of molecules with the recommended geometries and reference data taken from ref 95. It has earlier been found that the numerical values of information theoretic quantities and specially their trend for reproducing different properties are qualitatively consistent, where no substantial differences and visible dependence on the choice of level of theory (functionals and basis sets forms) have been observed.71,96 Thus, for all the atomic and molecular systems investigated in this work, we have employed the widely used exchange-correlation functional B3LYP,97,98 and the standard Pople basis set, 6-311++G(d,p), in our calculations. All the geometry optimizations and frequency calculations of molecules to ensure the absence of imaginary frequencies as well as the subsequent computations on atomic and molecular species have been performed using the Gaussian09 suite of programs.99 The required files for computation of the information theoretic quantities and related analyses have been derived from the Gaussian package. Then, all the runs to obtain the values of Fisher information, Shannon entropy, GBP entropy, and Onicescu information energy of the orders of 2 and 3 were implemented in multifunctional wave function analyzer program developed by Lu and Chen.100 The numerical values of the information theoretic quantities have been derived from the integration of the corresponding formulas with employing 75 points for the radial part and 434 points for the angular part. As metrics for gauging the performance of the performed fittings we have used four statistical measures as follows: mean signed deviation (MSD), mean absolute deviation (MAD), maximum absolute deviation (MaxAD), and root-mean-square deviation (RMSD). With definition of Δi as the deviation of a predicted correlation energy/electronic property from the reference value and n as the given data, these descriptors are 1 n 1 n respectively defined as MSD = n ∑i Δi , MAD = n ∑i |Δi |,

(15)

∫ ρυ dr

(16)

where F[ρ] is the universal energy density functional and υ is the external potential. Comparing eqs 15 and 16 and considering the intrinsic correspondence between density and external potential it seems that when using information theoretic quantities to predict electronic energies and other electronic properties based on eq 15, the second term in the right side of eq 16 as an explicit density functional needs to be considered (note that the latter term is also accessible experimentally through X-ray crystallography). This in turn leads to an alternative strategy to express P as follows: P[ρ] = P[Q , Vne]

(17)

Here the nucleus−electron attraction energy Vne as a component representing the second term in the right side of eq 16 has also been included. We mention in passing that although the presence of this latter term may not change the obtained results significantly, its role in our calculations has also been considered for the sake of completeness. Altogether, the two strategies, eqs 15 and 17, are of concern for evaluating the applicability of information theoretic quantities to predict electron correlation energies and other electronic properties. However, there are many options to express the relations defined in eqs 15 and 17 explicitly. Herein, following earlier suggestions in the field67,71 the form of linear combination of information theoretic quantities has been considered, though other cases are also possible which their applicability may be examined elsewhere. Accordingly, the working expressions employed in this work take the following forms: P[ρ] =

∑ cjQ j j

(19)

j

When the number of the information theoretic quantities is large enough, P[ρ] can sufficiently be accessed byP[Q]. In other words, the Q quantities as density functionals might form a complete set in such a way that they can provide an opportunity to accurately describe the properties of systems. With this in mind, should more information theoretic quantities be employed, more improved accuracy would be achieved. On the other hand, the conventional presentation of the total energy in DFT has the below form63−66 E [ρ ] = F [ ρ ] +

∑ cjQ j + cneVne

MaxAD = max |Δi|, and RMSD =

(18) C

1/2

( 1n ∑in Δi2)

.


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Table 1. Computed Values of Shannon Entropy, Fisher Information, GBP Entropy, and Onicescu Information Energies of Second and Third Orders Based on Electron Density for the Neutral Atoms He−Ara atom

Ec

SS[ρ]

IF[ρ]

SGBP[ρ]

EO2[ρ]

EO3[ρ]

Vne

He Li Be B C N O F Ne Na Mg Al Si P S Cl Ar

−0.042 −0.045 −0.094 −0.125 −0.156 −0.199 −0.258 −0.325 −0.390 −0.396 −0.438 −0.470 −0.505 −0.540 −0.605 −0.666 −0.722

4.10 7.71 8.85 8.83 7.85 6.14 3.91 1.11 −2.18 −0.86 −1.24 −1.84 −3.25 −5.22 −7.61 −10.36 −13.53

22.90 57.39 109.10 175.48 255.96 349.61 460.35 585.14 723.32 881.74 1058.09 1251.47 1460.74 1686.22 1928.83 2187.77 2463.43

14.42 22.21 28.83 36.07 43.08 49.95 56.37 62.72 69.02 76.88 83.65 90.57 97.34 104.01 110.22 116.41 122.47

0.76 3.13 8.39 17.53 31.83 52.63 81.54 120.13 170.21 233.76 312.78 408.80 523.26 658.18 815.46 996.24 1203.46

0.39 6.15 42.67 181.11 579.81 1535.32 3556.63 7431.03 14325.15 26115.04 45397.81 75391.40 120402.00 185999.14 279178.85 408606.69 584565.95

−6.74 −17.15 −33.68 −56.98 −88.23 −128.42 −178.13 −238.67 −311.03 −389.71 −479.12 −578.71 −689.63 −812.45 −947.14 −1093.97 −1254.32

R2 b

0.924

0.953

0.989

0.856

0.660

a

Also given in the table are the reference values of correlation energies as well as nucleus-electron attraction energies. Units are in atomic units. b Regression coefficients for the relationship between the reference correlation energies and information theoretic quantities.

Table 2. Computed Values of Shannon Entropy, Fisher Information, GBP Entropy, and Onicescu Information Energies of Second and Third Orders Based on Electron Density for the Cations Li+−K+ a cation

Ec

SS[ρ]

IF[ρ]

SGBP[ρ]

EO2[ρ]

EO3[ρ]

Vne

+

−0.044 −0.047 −0.111 −0.139 −0.167 −0.194 −0.261 −0.325 −0.389 −0.400 −0.452 −0.486 −0.522 −0.556 −0.622 −0.683 −0.739

1.14 4.14 4.98 4.55 3.26 1.44 −1.02 −3.88 −7.35 −6.37 −6.83 −7.66 −9.19 −11.14 −13.53 −16.27 −19.42

57.80 108.36 178.09 259.98 359.80 468.81 596.76 729.84 882.75 1058.10 1252.61 1462.34 1689.40 1931.55 2191.25 2465.87 2757.03

14.28 22.26 28.85 35.99 42.45 49.33 55.71 62.41 68.61 76.73 83.54 90.36 96.71 103.36 109.62 116.06 122.15

3.07 8.23 17.69 32.03 52.98 81.81 120.46 170.24 233.52 312.40 409.00 523.46 658.39 815.62 996.36 1203.48 1441.02

5.98 41.52 185.95 591.85 1563.69 3605.17 7515.64 14457.49 26058.15 45294.30 75523.80 120601.71 186322.02 279587.96 409172.50 585274.06 821330.02

−16.11 −31.87 −54.63 −84.94 −123.80 −172.57 −231.79 −302.93 −386.44 −474.97 −574.64 −684.60 −806.12 −940.01 −1085.69 −1245.01 −1418.66

Li Be+ B+ C+ N+ O+ F+ Ne+ Na+ Mg+ Al+ Si+ P+ S+ Cl+ Ar+ K+ R2 b

0.952

0.966

0.990

0.881

0.688

a

Also given in the table are the reference values of correlation energies as well as nucleus-electron attraction energies. Units are in atomic units. b Regression coefficients for the relationship between the reference correlation energies and information theoretic quantities.

4. RESULTS AND DISCUSSION We first deal with the corresponding results for the atomic and cationic species. The computed values of the information theoretic quantities under study, Shannon entropy, Fisher information, GBP entropy, and Onicescu information energy of the orders of 2 and 3 as well as the nucleus-electron attraction energies for the neutral atoms from He to Ar and single charged cations from Li+ to K+ have been reported in Tables 1 and 2, respectively. Also shown in the tables are the reference values of electron correlation energies for these species. The regression coefficients R2 for the correlation between each of the

information theoretic quantities and correlation energies have also been provided in the last row of the tables. We can observe from this data that the GBP entropy, Fisher information, and Shannon entropy with the regression coefficients better than about 0.95 for the both sets have the best relationships with the reference correlation energies. The next respectable linear correlations come from the second order Onicescu information energy with the regression coefficients no smaller than 0.85 on the two sets, followed finally by the decreased correlations arising from the Onicescu information energy of the order 3. Superior performance of the former set of information theoretic D


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Table 3. Predicted Values of Correlation Energies Based on the Two Strategies in Equations 18 and 19, Fit-A and Fit-B, Respectively, for the Neutral Atoms and the Singly Charged Cationsa atom

Fit-A

Fit-B

Ec

cation

Fit-A

Fit-B

Ec

He Li Be B C N O F Ne Na Mg Al Si P S Cl Ar

−0.046 −0.053 −0.077 −0.115 −0.161 −0.212 −0.265 −0.321 −0.381 −0.402 −0.432 −0.465 −0.506 −0.552 −0.602 −0.659 −0.725

−0.043 −0.057 −0.080 −0.116 −0.160 −0.210 −0.260 −0.318 −0.384 −0.406 −0.434 −0.465 −0.505 −0.552 −0.602 −0.659 −0.726

−0.042 −0.045 −0.094 −0.125 −0.156 −0.199 −0.258 −0.325 −0.390 −0.396 −0.438 −0.470 −0.505 −0.540 −0.605 −0.666 −0.722

Li+ Be+ B+ C+ N+ O+ F+ Ne+ Na+ Mg+ Al+ Si+ P+ S+ Cl+ Ar+ K+

−0.043 −0.064 −0.089 −0.129 −0.168 −0.216 −0.263 −0.322 −0.382 −0.410 −0.441 −0.479 −0.520 −0.570 −0.622 −0.680 −0.739

−0.042 −0.067 −0.091 −0.130 −0.166 −0.213 −0.260 −0.320 −0.380 −0.413 −0.446 −0.482 −0.521 −0.568 −0.618 −0.677 −0.742

−0.044 −0.047 −0.111 −0.139 −0.167 −0.194 −0.261 −0.325 −0.389 −0.400 −0.452 −0.486 −0.522 −0.556 −0.622 −0.683 −0.739

a

Electron density has been employed as distribution function for information theoretic quantities. For the sake of comparison, the corresponding values of reference correlation energies are also given. Units are in atomic units.

Table 4. Obtained Values of the Coefficients in the Least Squares Fitting Processes Based on the Two Strategies in Equations 18 and 19, Fit-A and Fit-B, Respectively, for the Neutral Atoms and the Singly Charged Cationsa atom Fit-A SS[ρ] IF[ρ] SGBP[ρ] EO2[ρ] EO3[ρ] Vne constant R2 MSD MAD maxAD RMSD

−2

cation Fit-B

−2

Fit-B

1.0 × 10 5.1 × 10−4 −9.5 × 10−3 −7.0 × 10−4 2.1 × 10−7

−5.9 × 10−3

1.0 × 10 8.8 × 10−4 −8.6 × 10−3 1.2 × 10−4 −1.0 × 10−7 1.4 × 10−3 2.8 × 10−2

5.3 × 10−2

7.4 × 10−3 6.2 × 10−4 −7.9 × 10−3 6.2 × 10−5 −6.4 × 10−8 9.9 × 10−4 4.2 × 10−2

0.999 0.000 0.007 0.017 0.008

0.999 0.000 0.006 0.014 0.008

0.998 0.000 0.008 0.022 0.011

0.998 0.000 0.008 0.020 0.010

1.0 × 10 3.4 × 10−5 −5.7 × 10−3 1.2 × 10−4 −1.9 × 10−7

−2

Fit-A

a

Electron density has been employed as distribution function for information theoretic quantities. Also given in the table are the regression coefficients and the corresponding statistical descriptors (in atomic unit) for the performed fittings.

quantities, GBP entropy, Fisher information, and Shannon entropy, can be deduced from the corresponding definitions. For instance, we find from the definition of Fisher information in eq 2 that this information theoretic quantity differs from the Weizsäcker kinetic energy TW, a major portion of the total noninteracting kinetic energy of DFT energy expression, by a factor of only 1/8 (IF = 8TW).101 This may reflect the good performance of Fisher information to estimate the energetic component of electron correlation. On the other hand, the satisfactory correlations obtained from Shannon entropy provide in turn another verification to what has earlier been justified theoretically on the applicability of Shannon entropy as a reliable measure to adequately describe an electronic system.84 Putting all these findings together, it seems that given the weaknesses and strengths of the information theoretic quantities most of them disclose reasonable relationships with electron correlation energies.

One question that may arise herein is that why can different forms of information theoretic quantities correlate simultaneously with electron correlation energies? This is, however, understandable. As previously mentioned, the analytical expressions, physiochemical origins, and scaling properties of the information theoretic quantities as density functionals are significantly different. Nevertheless, they are not independent of each other. Indeed, all the information theoretic quantities under study are functions of the electron density and consequently it is not unusual that these quantities are interrelated, a relationship which has theoretically been verified and numerically illustrated elsewhere.102,103 The presence of correlations among different information theoretic quantities with various scaling properties and distinct physiochemical propensities, on the one hand, and their relationships with electron correlation energies, on the other hand, unveil that the correlation effects may be considered as a multifacets problem. As a piece of supporting evidence to this E


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Table 5. Regression Coefficients and Statistical Descriptors (in kcal/mol) on the Performed Least Squares Fittings Based on Equations 18 and 19, Fit-A and Fit-B, Respectively, for the Isoelectronic Series Under Study Fit-A

Fit-B

N

data points

R2

MSD

MAD

maxAD

RMSD

R2

MSD

MAD

maxAD

RMSD

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

19 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11

0.9996 0.9996 0.9999 0.9999 0.9999 0.9999 0.9997 0.9992 0.9993 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999

−0.0006 0.0035 0.0017 0.0016 0.0015 0.0067 0.0039 0.0020 −0.0012 −0.0027 0.0052 0.0019 0.0051 0.0082 0.0033 −0.0006 −0.0002

0.0125 0.0195 0.1481 0.0627 0.0439 0.0491 0.0821 0.1207 0.0841 0.0242 0.0160 0.0148 0.0221 0.0954 0.0205 0.0036 0.0050

0.0307 0.0393 0.3088 0.1416 0.1253 0.1443 0.1764 0.3254 0.2592 0.0628 0.0314 0.0314 0.0502 0.2838 0.0941 0.0103 0.0251

0.0147 0.0229 0.1710 0.0731 0.0540 0.0595 0.0993 0.1478 0.1076 0.0297 0.0188 0.0169 0.0260 0.1233 0.0309 0.0046 0.0096

0.9998 0.9996 0.9999 0.9999 0.9999 0.9999 0.9998 0.9994 0.9997 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999

0.0008 0.0000 0.0010 0.0005 0.0075 0.0063 0.0031 0.0001 0.0021 0.0019 −0.0002 0.0012 0.0013 0.0057 0.0020 −0.0130 −0.0037

0.0097 0.0185 0.1459 0.0621 0.0326 0.0245 0.0789 0.1079 0.0608 0.0238 0.0119 0.0142 0.0195 0.0925 0.0167 0.0155 0.0044

0.0301 0.0406 0.3070 0.1416 0.1139 0.1004 0.1842 0.3267 0.1883 0.0553 0.0251 0.0274 0.0504 0.2871 0.0251 0.1569 0.0251

0.0122 0.0221 0.1694 0.0732 0.0443 0.0338 0.0917 0.1316 0.0725 0.0287 0.0136 0.0164 0.0228 0.1180 0.0187 0.0454 0.0086

point, we observed for instance that the electron correlation energies are correlated reasonably with both Fisher information and Shannon entropy, where the role of the sharpness or concentration of the density distribution (from Fisher information) and spatial delocalization of the electronic density (from Shannon entropy) in correlation energies is highlighted. This is in turn reminiscent of electron correlation effects arising from the inability of the HF wave function to model interelectronic cusps and dispersion interactions (dynamic regime) and the near-degeneracy of HF occupied and virtual orbitals (nondynamic regime). Note that from the perspective of electron density the former regime only produces local variations in the density of electrons with respect to the HF view while the latter induces global changes.11 As earlier noted different information theoretic quantities have diverse scaling properties and consequently measure different propensities of electron density. Therefore, using none of them alone should not be adequate to accurately describe an electronic system. Accordingly, one can make use of several information theoretic quantities simultaneously in the framework of IFT.67 To this end, we have considered the two strategies based on eqs 18 and 19. At first, all the information theoretic quantities have been taken into account concurrently in a linear combination fashion, eq 18, denominated as Fit-A. In another tack, called hereafter as Fit-B, besides the information theoretic quantities the numerical values of Vne have also been included in the fittings based on eq 19 to account for the second term in the right side of Eq. 16. The predicted correlation energies from the two strategies for both neutral and cationic species under study are reported in Table 3, where for the straightforward comparisons the reference correlation energies of these species have also been given. In addition, the numerical values of the expansions coefficients, regression coefficients, and statistical descriptors to gauging the performance of the both fittings are gathered in Table 4. It can be seen from the data in the tables that, as compared to the previous fittings where only one quantity was used, the improved correlations with higher regression coefficients and better fitted results are achieved when using linear combinations of more than one information theoretic quantity. Furthermore, the predicted correlation

energies from the two strategies (Table 3) and the performed statistical analyses on the fittings (Table 4) point out in turn the role of including Vne as the input in the fittings, where slightly better results were obtained from the Fit-B strategy with respect to those obtained from Fit-A. As a more general viewpoint, this is also indeed the case when considering all the data from the both neutral and cationic sets together (the corresponding results have been collected in Tables S1 and S2 in the Supporting Information). These findings corroborate the idea of employing multiple quantities whiting IFT in correlating with energetic components and electronic properties to achieve improved accuracy, a point that will be used in what follows on the prediction of the properties under study. We have also performed similar analyses using the information theoretic quantities with the shape function as distribution function, eqs 7−10). The corresponding results with the same sequences discussed earlier for the electron density based quantities (Tables 1-4), are detailed in Tables S3− S6 (Supporting Information). Our results show that for the both sets of species under study, the Fisher information and second order Onicescu information energy have the best performance among others. Plus, about the comparison between Fit-A and Fit-B strategies we find almost the same trend and conclusions as in previous cases. However, perusing the results it is turned out that the correlations obtained using the shape function-based information theoretic quantities are not as strong as those we found from the conventional counterparts. These findings are consistent with our earlier results on the using information theoretic quantities for quantitative description of steric effects.51 The accountability of information theoretic quantities for estimating the electron correlation energies is put into broader perspective, where their applicability for prediction of the correlation energies in different isoelectronic series has also been examined. To this end, 17 isoelectronic series constructed from N = 2 up to N = 18 (315 atomic species) have been considered. To save space, the computed values of the information theoretic quantities and nucleus-electron attraction energies as well as reference correlation energies for the species in each isoelectronic series and the detailed results obtained from the F


Article


Table 6. Computed Values of Shannon Entropy, Fisher Information, GBP Entropy, and Onicescu Information Energies of Second and Third Orders Based on Electron Density for the Benchmarked Set of Moleculesa molecule

Ec

SS[ρ]

IF[ρ]

SGBP[ρ]

EO2[ρ]

EO3[ρ]

Vne

H2 LiH BeH Li2 CH CH2 (3B1) CH2 (1A1) NH CH3 NH2 OH CH4 NH3 OH2 FH LiF CN HCCH HCN CO N2 HCO NO SiH2 (1A1) SiH2 (3B1) H2CCH2 H2CO O2 SiH3 PH2 SiH4 PH3 SH2 ClH H3CCH3 H3COH H2NNH2 HOOH F2 CO2 Na2 SiO SC SO ClO ClF CH3Cl CH3SH HOCl Si2 NaCl P2 S2 SO2 Cl2 Si2H6

−0.041 −0.083 −0.094 −0.124 −0.199 −0.213 −0.239 −0.244 −0.259 −0.292 −0.314 −0.299 −0.340 −0.371 −0.389 −0.441 −0.500 −0.480 −0.515 −0.535 −0.549 −0.559 −0.604 −0.567 −0.542 −0.518 −0.586 −0.660 −0.577 −0.616 −0.606 −0.652 −0.683 −0.707 −0.561 −0.629 −0.641 −0.711 −0.757 −0.876 −0.819 −0.879 −0.867 −0.974 −1.009 −1.063 −0.968 −0.946 −1.045 −1.084 −1.101 −1.205 −1.291 −1.334 −1.380 −1.183

6.51 9.50 11.18 14.20 10.18 12.39 12.41 8.39 14.53 10.52 5.98 16.70 12.57 7.94 2.98 4.82 10.74 16.82 12.76 8.42 8.85 11.05 7.46 1.65 1.94 21.86 13.24 5.38 4.27 −0.26 6.61 2.16 −2.78 −8.02 26.89 18.25 18.79 12.62 0.81 9.25 −2.68 −2.74 −2.50 −6.30 −8.45 −10.98 2.22 7.38 −6.13 −8.46 −14.30 −13.12 −17.48 −5.40 −22.36 6.66

9.09 61.53 111.76 114.51 258.69 256.68 260.96 354.70 259.22 352.35 460.62 261.68 352.40 460.06 583.60 636.48 594.01 501.27 594.99 704.67 688.97 705.40 800.43 1467.62 1465.52 507.98 708.22 912.15 1469.25 1691.86 1473.02 1694.60 1933.05 2189.13 513.97 713.02 695.90 920.45 1165.71 1151.93 1763.05 1912.72 2177.64 2381.92 2642.18 2767.58 2442.75 2186.05 2643.42 2919.01 3067.17 3367.62 3853.08 2832.48 4372.05 2936.53

13.90 28.69 35.87 43.03 49.22 55.86 55.34 55.79 61.93 62.38 62.68 68.05 68.57 68.95 69.10 83.69 90.64 96.34 96.66 97.04 96.90 103.74 104.01 109.83 110.23 109.16 109.90 110.63 116.38 116.40 122.52 122.53 122.53 122.57 122.06 122.94 123.02 125.04 124.22 151.54 152.64 151.72 150.92 164.64 171.24 177.86 176.60 176.56 177.65 192.60 191.52 205.26 218.33 219.18 231.36 230.89

0.17 3.20 8.36 6.28 31.87 31.67 31.90 52.75 31.77 52.67 81.55 31.86 52.62 81.53 120.13 123.05 84.35 63.38 84.36 113.45 105.44 113.22 134.12 523.26 522.89 63.47 113.21 163.13 522.93 658.14 522.95 658.10 815.40 996.24 63.53 113.18 105.00 162.97 240.42 194.58 467.57 604.69 847.29 896.89 1077.76 1116.38 1027.93 847.07 1077.75 1046.42 1229.74 1316.27 1630.83 978.25 1992.48 1045.77

0.01 6.13 41.44 12.33 575.79 562.47 570.80 1528.59 559.90 1514.02 3535.27 557.72 1497.63 3508.34 7386.75 7385.47 2092.42 1124.23 2082.23 4105.11 3046.61 4091.32 5061.84 120256.28 120026.23 1120.34 4083.08 7084.12 119971.18 185762.11 119902.47 185624.27 278797.61 408320.27 1115.40 4066.03 2995.21 7087.08 14863.83 7606.28 52240.71 123820.53 279535.08 282605.35 412099.21 416060.65 408910.90 279367.57 412068.44 240678.80 434353.10 371718.07 558126.07 286103.49 817130.80 239828.62

−3.64 −20.45 −37.97 −38.05 −95.02 −103.00 −102.61 −136.49 −111.36 −145.93 −188.21 −120.21 −156.07 −199.25 −250.71 −276.35 −254.51 −228.91 −265.93 −311.31 −303.69 −320.65 −355.36 −712.10 −711.93 −248.63 −331.73 −411.61 −724.63 −837.79 −737.30 −851.37 −975.73 −1109.63 −268.84 −352.48 −344.95 −412.28 −537.58 −560.93 −821.34 −947.22 −1101.55 −1216.23 −1363.23 −1431.62 −1289.46 −1153.68 −1375.68 −1471.95 −1569.95 −1750.87 −2037.49 −1518.66 −2336.46 −1568.59

R2 b

0.506

0.874

0.983

0.762

0.637

a

Also given in the table are the reference values of correlation energies as well as nucleus-electron attraction energies. Units are in atomic units. b Regression coefficients for the relationship between the reference correlation energies and information theoretic quantities. G


Article


Table 7. Computed Values of Shannon Entropy, Fisher Information, GBP Entropy, and Onicescu Information Energies of Second and Third Orders Based on Shape Function for the Benchmarked Set of Moleculesa molecule

Ec

SS[σ]

IF[σ]

SGBP[σ]

EO2[σ]

EO3[σ]

Vne

H2 LiH BeH Li2 CH CH2 (3B1) CH2 (1A1) NH CH3 NH2 OH CH4 NH3 OH2 FH LiF CN HCCH HCN CO N2 HCO NO SiH2 (1A1) SiH2 (3B1) H2CCH2 H2CO O2 SiH3 PH2 SiH4 PH3 SH2 ClH H3CCH3 H3COH H2NNH2 HOOH F2 CO2 Na2 SiO SC SO ClO ClF CH3Cl CH3SH HOCl Si2 NaCl P2 S2 SO2 Cl2 Si2H6

−0.041 −0.083 −0.094 −0.124 −0.199 −0.213 −0.239 −0.244 −0.259 −0.292 −0.314 −0.299 −0.340 −0.371 −0.389 −0.441 −0.500 −0.480 −0.515 −0.535 −0.549 −0.559 −0.604 −0.567 −0.542 −0.518 −0.586 −0.660 −0.577 −0.616 −0.606 −0.652 −0.683 −0.707 −0.561 −0.629 −0.641 −0.711 −0.757 −0.876 −0.819 −0.879 −0.867 −0.974 −1.009 −1.063 −0.968 −0.946 −1.045 −1.084 −1.101 −1.205 −1.291 −1.334 −1.380 −1.183

3.95 3.76 3.84 4.16 3.40 3.63 3.63 3.13 3.81 3.37 2.86 3.97 3.56 3.10 2.60 2.89 3.39 3.84 3.55 3.24 3.27 3.44 3.21 2.88 2.89 4.14 3.60 3.11 3.08 2.82 3.26 3.01 2.74 2.44 4.38 3.90 3.93 3.59 2.94 3.51 2.97 2.97 2.98 2.92 2.88 2.84 3.34 3.54 3.02 3.03 2.82 2.96 2.92 3.30 2.87 3.72

4.55 15.38 22.35 19.09 36.96 32.09 32.62 44.34 28.80 39.15 51.18 26.17 35.24 46.01 58.36 53.04 45.69 35.80 42.50 50.33 49.21 47.03 53.36 91.73 91.59 31.75 44.26 57.01 86.43 99.52 81.83 94.14 107.39 121.62 28.55 39.61 38.66 51.14 64.76 52.36 80.14 86.94 98.98 99.25 105.69 106.45 93.95 84.08 101.67 104.25 109.54 112.25 120.41 88.51 128.59 86.37

6.95 7.17 7.17 7.17 7.03 6.98 6.92 6.97 6.88 6.93 6.96 6.81 6.86 6.90 6.91 6.97 6.97 6.88 6.90 6.93 6.92 6.92 6.93 6.86 6.89 6.82 6.87 6.91 6.85 6.85 6.81 6.81 6.81 6.81 6.78 6.83 6.83 6.95 6.90 6.89 6.94 6.90 6.86 6.86 6.85 6.84 6.79 6.79 6.83 6.88 6.84 6.84 6.82 6.85 6.80 6.79

0.04 0.20 0.33 0.17 0.65 0.49 0.50 0.82 0.39 0.65 1.01 0.32 0.53 0.82 1.20 0.85 0.50 0.32 0.43 0.58 0.54 0.50 0.60 2.04 2.04 0.25 0.44 0.64 1.81 2.28 1.61 2.03 2.52 3.07 0.20 0.35 0.32 0.50 0.74 0.40 0.97 1.25 1.75 1.56 1.72 1.65 1.52 1.25 1.59 1.33 1.57 1.46 1.59 0.96 1.72 0.90

0.00 0.10 0.33 0.06 1.68 1.10 1.11 2.99 0.77 2.08 4.85 0.56 1.50 3.51 7.39 4.27 0.95 0.41 0.76 1.50 1.11 1.21 1.50 29.36 29.30 0.27 1.00 1.73 24.42 37.81 20.56 31.83 47.80 70.01 0.19 0.70 0.51 1.22 2.55 0.71 4.91 11.63 26.25 20.44 26.37 23.67 23.27 15.89 23.44 10.96 19.79 13.77 17.03 8.73 20.79 6.10

−3.64 −20.45 −37.97 −38.05 −95.02 −103.00 −102.61 −136.49 −111.36 −145.93 −188.21 −120.21 −156.07 −199.25 −250.71 −276.35 −254.51 −228.91 −265.93 −311.31 −303.69 −320.65 −355.36 −712.10 −711.93 −248.63 −331.73 −411.61 −724.63 −837.79 −737.30 −851.37 −975.73 −1109.63 −268.84 −352.48 −344.95 −412.28 −537.58 −560.93 −821.34 −947.22 −1101.55 −1216.23 −1363.23 −1431.62 −1289.46 −1153.68 −1375.68 −1471.95 −1569.95 −1750.87 −2037.49 −1518.66 −2336.46 −1568.59

0.390

0.252

R2 b

0.191

0.687

0.158

a

Also given in the table are the reference values of correlation energies as well as nucleus-electron attraction energies. Units are in atomic units. b Regression coefficients for the relationship between the reference correlation energies and information theoretic quantities. H


Article


Table 8. Predicted Values of Correlation Energies Based on the Two Strategies in Equations 18 and 19, Fit-A and Fit-B, Respectively, Using Electron Density and Shape Function As Distribution Functions in Information Theoretic Quantities for the Benchmarked Set of Moleculesa electron density

shape function

electron density

shape function

molecule

Ec

Fit-A

Fit-B

Fit-A

Fit-B

molecule

Ec

Fit-A

Fit-B

Fit-A

Fit-B

H2 LiH BeH Li2 CH CH2 (3B1) CH2 (1A1) NH CH3 NH2 OH CH4 NH3 OH2 FH LiF CN HCCH HCN CO N2 HCO NO SiH2 (1A1) SiH2 (3B1) H2CCH2 H2CO O2 SiH3 PH2

−0.041 −0.083 −0.094 −0.124 −0.199 −0.213 −0.239 −0.244 −0.259 −0.292 −0.314 −0.299 −0.340 −0.371 −0.389 −0.441 −0.500 −0.480 −0.515 −0.535 −0.549 −0.559 −0.604 −0.567 −0.542 −0.518 −0.586 −0.660 −0.577 −0.616

−0.008 −0.086 −0.118 −0.143 −0.211 −0.240 −0.236 −0.266 −0.265 −0.295 −0.326 −0.290 −0.322 −0.354 −0.385 −0.473 −0.480 −0.476 −0.505 −0.535 −0.533 −0.560 −0.584 −0.567 −0.567 −0.522 −0.584 −0.638 −0.590 −0.609

−0.008 −0.086 −0.119 −0.144 −0.212 −0.240 −0.236 −0.266 −0.265 −0.295 −0.326 −0.289 −0.321 −0.353 −0.384 −0.471 −0.480 −0.476 −0.505 −0.535 −0.533 −0.559 −0.584 −0.566 −0.566 −0.522 −0.584 −0.638 −0.589 −0.608

−0.084 −0.046 −0.083 −0.142 −0.207 −0.260 −0.296 −0.263 −0.328 −0.309 −0.270 −0.369 −0.361 −0.339 −0.312 −0.417 −0.516 −0.512 −0.547 −0.570 −0.581 −0.572 −0.613 −0.540 −0.520 −0.522 −0.592 −0.665 −0.576 −0.588

−0.093 −0.050 −0.083 −0.139 −0.207 −0.259 −0.296 −0.263 −0.327 −0.309 −0.271 −0.369 −0.361 −0.340 −0.315 −0.417 −0.512 −0.509 −0.543 −0.567 −0.579 −0.569 −0.610 −0.537 −0.517 −0.519 −0.590 −0.662 −0.574 −0.586

SiH4 PH3 SH2 ClH H3CCH3 H3COH H2NNH2 HOOH F2 CO2 Na2 SiO SC SO ClO ClF CH3Cl CH3SH HOCl Si2 NaCl P2 S2 SO2 Cl2 Si2H6

−0.606 −0.652 −0.683 −0.707 −0.561 −0.629 −0.641 −0.711 −0.757 −0.876 −0.819 −0.879 −0.867 −0.974 −1.009 −1.063 −0.968 −0.946 −1.045 −1.084 −1.101 −1.205 −1.291 −1.334 −1.380 −1.183

−0.613 −0.631 −0.659 −0.701 −0.570 −0.633 −0.632 −0.675 −0.747 −0.883 −0.897 −0.871 −0.844 −0.957 −1.017 −1.073 −0.981 −0.940 −1.043 −1.078 −1.159 −1.167 −1.267 −1.304 −1.393 −1.217

−0.612 −0.630 −0.660 −0.705 −0.569 −0.632 −0.631 −0.675 −0.745 −0.882 −0.895 −0.870 −0.846 −0.958 −1.022 −1.076 −0.985 −0.941 −1.047 −1.077 −1.162 −1.167 −1.271 −1.305 −1.378 −1.216

−0.628 −0.645 −0.668 −0.720 −0.522 −0.593 −0.596 −0.635 −0.749 −0.763 −0.867 −0.868 −0.872 −0.969 −1.029 −1.078 −0.965 −0.916 −1.037 −1.144 −1.163 −1.245 −1.343 −1.122 −1.445 −1.140

−0.627 −0.644 −0.667 −0.720 −0.519 −0.591 −0.594 −0.630 −0.748 −0.763 −0.866 −0.868 −0.873 −0.973 −1.035 −1.085 −0.971 −0.921 −1.043 −1.150 −1.172 −1.257 −1.360 −1.135 −1.385 −1.152

a

For the sake of comparison, the corresponding values of reference correlation energies are also given. Units are in atomic units.

18 and 19 for the linear combinations of information theoretic quantities we found the reasonable linear correlations between the predicted values from the fittings and reference correlation energies with R2 values equal to about 0.95. Atomic species aside, similar analyses have also been carried out on molecular systems. The numerical values of information theoretic quantities within the two representations of electron density and shape function along with the reference correlation energies as well as the values of the nucleus-electron attraction energies are gathered in Tables 6 and 7. We have also provided the values of regression coefficients for the correlation between each information theoretic quantity and reference correlation energies in the last row of the tables. A first glance to this data is sufficient to conclude that better correlations are derived from the information theoretic quantities within the electron density representation as compared to the shape function based counterparts. It can also be seen from the results in the tables that the GBP entropy and Fisher information show stronger correlations with respect to others. More importantly, note that these two latter information theoretic quantities have earlier revealed good correlations for the atomic species as well. In contrast, although besides GBP entropy and Fisher information the Shannon entropy provides also reasonable correlations on the atomic species, this is not indeed the case for the molecular systems where no meaningful correlations are observed for Shannon entropy with the both representations of electron

two strategies of linear combinations based on eqs 18 and 19 have been tabulated in the Supporting Information (Tables S7− S57). In order to having a comprehensive view on the obtained results, the corresponding regression coefficients and statistical descriptors on the performed fittings have been summarized in Table 5. We find from the related numerical data in the Supporting Information that depending on the isoelectronic series of concern, a wide range of performance is observed for each of the information theoretic quantities. For the series with low and medium N, Shannon entropy, Fisher information, and GBP entropy outperform others while with an increase in the number of electrons up to N = 18 the role of the second and third orders of Onicescu information energy is also more pronounced. Moreover, scrutinizing the results of all series we find that not only using some of the information theoretic quantities alone provides reasonable correlations with the regression coefficients larger than about 0.9 but also the improved accuracy is obtained when going from the single variable fittings to the multiple linear regression analyses. In fact, zooming in the predicted correlation energies reported in the Supporting Information along with the corresponding statistical measures given in Table 5 the validity and applicability of linear combinations of information theoretic quantities to describe the electron correlation energies for a variety of species are highlighted. Finally, from a more general point of view, considering all the isoelectronic series together (315 atomic species in total) and using the two strategies in eqs I


Article


Table 9. Obtained Values of the Coefficients in the Least Squares Fitting Processes Based on the Two Strategies in Equations 18 and 19, Fit-A and Fit-B, Respectively, Using Electron Density and Shape Function As Distribution Functions in Information Theoretic Quantities for the Benchmarked Set of Moleculesa electron density SS IF SGBP EO2 EO3 Vne constant R2 MSD MAD maxAD RMSD

shape function

Fit-A

Fit-B

Fit-A

Fit-B

9.8 × 10−3 5.5 × 10−5 −7.5 × 10−3 3.7 × 10−4 −5.6 × 10−7

−5.5 × 10−3 −1.9 × 10−2 5.7 × 10−1 7.6 × 10−1 −9.0 × 10−3

3.2 × 10−2

9.2 × 10−3 1.0 × 10−5 −7.2 × 10−3 4.8 × 10−4 −6.5 × 10−7 1.1 × 10−5 3.2 × 10−2

−3.9

4.7 × 10−3 −1.8 × 10−2 5.7 × 10−1 7.5 × 10−1 −8.8 × 10−3 3.6 × 10−5 −4.0

0.996 0.000 0.016 0.078 0.021

0.996 0.000 0.016 0.076 0.021

0.980 0.000 0.034 0.212 0.048

0.980 0.000 0.033 0.199 0.047

Also given in the table are the regression coefficients and the corresponding statistical descriptors (in atomic unit) for the performed fittings.

a

Table 10. Computed Values of Shannon Entropy, Fisher Information, GBP Entropy, and Onicescu Information Energies of Second and Third Orders Based on Electron Density as Well as the Nucleus−Electron Attraction Energies (in Atomic Units) for the Benchmarked Set of Moleculesa molecule

AE

EA

IP

SS[ρ]

IF[ρ]

SGBP[ρ]

EO2[ρ]

EO3[ρ]

Vne

CO C2H4 H2O2 H2S C3H4 CH2CO CO2 CS FCN SiO C2H4O CH2CHF H2CS C4H2 CH2F2 CH3CH2OH CH3SiH3 ClF F2O NaCl P2 CH3CCCH3 CH2CHCHO CH2CF2 CH3NO2 SO2 SiF2 S2 C4H4O ClF3

11.22 24.38 11.62 7.94 29.45 23.03 16.85 7.37 13.19 8.25 28.18 24.81 14.07 30.05 18.99 35.06 27.23 2.68 4.04 4.32 4.99 43.32 35.76 25.38 26.01 10.97 12.93 4.41 42.99 5.43

−1.51 −1.87 −0.92 −0.49 −1.82 −0.51 −0.65 −0.09 −0.66 0.03 −0.86 −0.88 0.28 −0.64 −0.58 −0.53 −0.53 0.44 −0.31 0.65 0.48 −0.68 −0.46 −1.03 −0.37 0.81 0.10 1.53 −0.74 1.20

14.01 10.68 11.7 10.5 9.86 9.64 13.78 11.34 13.65 11.61 10.57 10.63 9.38 10.3 13.27 10.64 11.6 12.77 13.26 9.8 10.62 9.79 10.1 10.7 11.29 12.5 11.08 9.55 8.9 13.05

8.42 21.92 10.08 −2.88 26.37 18.18 9.26 −2.52 9.38 −2.80 22.81 7.24 2.19 26.88 9.93 28.35 16.59 −11.15 3.04 −14.33 −13.13 37.46 28.53 15.03 21.16 −5.67 −6.27 −17.64 32.27 −12.32

704.68 508.05 915.39 1933.06 751.40 952.27 1151.98 2177.59 1168.83 1912.58 958.56 1082.19 2180.43 992.06 1411.50 964.89 1725.21 2767.02 1619.98 3067.12 3367.59 1004.65 1206.00 1655.64 1502.21 2831.70 2616.41 3852.17 1445.96 3925.17

97.04 109.17 123.86 122.51 149.96 150.99 151.55 150.91 151.72 151.70 163.51 164.22 163.65 178.64 178.10 176.91 176.58 177.79 179.15 191.51 205.25 204.38 204.97 219.23 218.81 219.09 220.14 218.42 245.17 301.18

113.45 63.45 163.11 815.42 94.89 144.80 194.58 847.29 204.25 604.71 144.82 183.41 847.06 126.52 271.84 144.90 554.64 1116.40 322.03 1229.75 1316.27 126.58 176.48 303.37 247.09 978.30 763.30 1630.85 207.88 1356.88

4105.11 1120.24 7053.46 278784.57 1676.68 4643.28 7606.46 279534.00 9472.55 123817.53 4640.32 8510.21 279449.40 2245.82 15341.13 4626.23 120461.10 416033.03 18407.90 434352.38 371715.91 2236.94 5202.80 15901.71 9114.34 286070.88 135057.02 558067.96 5750.70 431427.31

−311.31 −248.27 −432.39 −975.89 −397.83 −474.25 −560.70 −1101.55 −565.81 −947.22 −510.93 −554.22 −1127.88 −510.50 −723.49 −526.73 −911.40 −1431.62 −801.16 −1569.95 −1750.87 −559.25 −654.19 −891.68 −825.62 −1518.43 −1370.17 −2037.68 −858.52 −2199.93

a

Also given in the table are the reference values of the properties under study (atomization energy (AE), electron affinity (EA), and ionization potential (IP)) in eV for these molecules.

density and shape function. This is in agreement with earlier

alone to describe molecular systems have been pointed out.67,84,85 On the other hand, the utility of the linear combinations of information theoretic quantities to estimate molecular correla-

numerical findings and theoretical justifications in the field where the problems associated with the use of Shannon entropy J


Article

The Journal of Physical Chemistry A tion energies have also been examined. The corresponding results are collected in Tables 8 and 9. Same as what we have earlier observed for the atomic species, we find from the results in the tables that for the both representations of electron density and shape function when the information theoretic quantities are combined together as linear combinations much better correlations with the regression coefficients of about 0.99 are obtained. In addition, in most cases a reliable predicted value of correlation energy is obtained when using the two strategies of linear combinations. All in all, it can be concluded that considering the information theoretic quantities simultaneously is a useful approach for estimation of the electron correlation energies for both atomic and molecular systems. Finally, following the applicability of information theoretic quantities for predicting electron correlation energy as an important component of electronic energies, describing the electronic properties through the perspective of IFT is also of interest. As representative examples of the electronic properties, we have considered atomization energies, electron affinities, and ionization potentials for a test set of 30 different molecules with a variety of elements. The list of molecules alongside the corresponding reference values of the electronic properties of concern are gathered in Table 10. Also provided in the table are the numerical values of the information theoretic quantities including Shannon entropy, Fisher information, GBP entropy, and Onicescu information energy of second and third orders as well as the nucleus−electron attraction energies for the molecules under study. Before estimating the properties from the mentioned strategies for linear combinations of information theoretic quantities, some remarkable points regarding the obtained numerical data and their using for the purpose are in order. As shown by the results in Table 10, for each molecule the information theoretic quantities have substantially different values. Such large differences reflect in turn that the information theoretic quantities work as quantitative measures of different facets of the molecules. On the other hand, concerning each information theoretic quantity individually we observe a wide range of values for the molecules under study where the values of Fisher information, GBP entropy, and second and third orders of Onicescu information energy are always positive whereas for the Shannon entropy we see also sign changes for some systems. Accordingly, what we would like to highlight the most here is that any effort trying to use only one of the information theoretic quantities for describing electronic properties may lead to an unsuccessful practice. Instead, putting all these meaningful variations and fluctuations as useful clues together it is expected that combining the information theoretic quantities together can provide a more balanced view toward predicting the electronic properties. Therefore, we decided to use the multiple linear regression analyses with all the information theoretic quantities and nucleus-electron attraction energies for estimation of the electronic properties under study. Shown in Figure 1 are the variations trend of the fitted values of the considered properties using eq 19, Fit-B, alongside the reference data for all the studied molecules. We can see from the figures that the fitted electronic properties based on simultaneous considering the information theoretic quantities provide a correct trend against the corresponding reference data. Similar findings have recently been observed when studying other chemical transformation and molecular properties through the IFT framework.71,104 Our numerical results are also quantitatively consistent with the reference data, where such agreements are more pronounced for atomization energies and electron affinities.

Figure 1. Variations trend of the predicted atomization energies (top), electron affinities (middle), and ionization potentials (bottom) against the corresponding reference values for all the molecules in the benchmarked set.

Altogether, the results of our study and earlier related efforts unveil a strong evidence in this sense that strategies of combining information theoretic quantities within the framework of IFT not only describe correlation effects satisfactorily but also suggest an alternative route toward estimating electronic properties as well. Nonetheless, as is the same as any other proposed approach, the employed strategies in this work have in turn pros and cons. As a matter of fact, notwithstanding the promising findings from the considered schemes herein, the relative deviations on both correlation energies and electronic properties are also significant in some cases. This indicates that, alongside the performing statistical analyses, one should also be cautious when trying to describe such characteristics from the information functional theory framework.

5. CLOSING COMMENTS Can we make use of the information functional theory approach to reliable prediction of atomic and molecular correlation energies as well as electronic properties? To answer this question, during this work, several information theoretic quantities like Fisher information, Shannon entropy, Onicescu information energy, and Ghosh−Berkowitz−Parr entropy with the two representations of electron density and shape function were considered. We have shown that with more or less different accountabilities of such quantities they can be introduced as K


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reliable descriptors to estimate electron correlation energies for a large variety of systems including neutral atoms, singly charged positive ions, isoelectronic series, and molecules. However, the obtained relationships from information theoretic quantities are often significantly different from each other, as can be expected from their different views of the electron density distribution and scaling properties as well as varied physiochemical meanings. Therefore, instead of simulating all data using any one of the information theoretic quantities individually, employing them in conjunction with each other was examined. We found that taking all the information theoretic quantities together at the same time provides a more balanced description of the system, which in turn not only leads to the improved correlations against reference data for both atomic and molecular correlation energies but also unveils the multifacet nature of the electron correlation problem. Following good performance of the quantities within information functional theory for electron correlation energies, their utility for describing electronic properties was also investigated. Taking atomization energies, electron affinities, and ionization potentials for a set of molecules as illustrative examples of electronic properties, we observed reasonable correlations between the results obtained from the information theoretic quantities fitted together and reference values of the considered properties, where the validity of multiple linear regressions of the quantities for estimation of electronic properties are showcased. Hopefully, this work can provide with us a different view to the electron correlation problem and electronic properties from the perspective of information functional theory. Sparked by the findings of this study and other efforts in the field, extensions of the theoretical basis of the information functional theory toward designing novel information theoretic quantities with broader applicability for a wide range of properties and other related problems are possible, and this arena of research deserves more attention in future work.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpca.8b05703. Tables of predicted values of correlation energies, coefficients, and computed values of Shannon entropy, Fisher information, GBP entropy, and Onicescu information energies, (PDF)

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AUTHOR INFORMATION

Corresponding Author

*(M.A.) E-mail: [email protected]. Telephone: +98 71 36137160. Fax: +98 71 36460788. ORCID

Mojtaba Alipour: 0000-0003-3037-0232 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors acknowledge Shiraz University for providing computational resources for this project. REFERENCES

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