Information carriers and (reading them through) information theory in quantum chemistry

This Perspective discusses the reduction of the electronic wave function via the second-order reduced density matrix to the electron density ρ(r), which is the key ingredient in density functional theory (DFT) as a basic carrier of information. Simplifying further, the 1-normalized density function turns out to contain essentially the same information as ρ(r) and is even of preferred use as an information carrier when discussing the periodic properties along Mendeleev's table where essentially the valence electrons are at stake. The Kullback-Leibler information deficiency turns out to be the most interesting choice to obtain information on the differences in ρ(r) or σ(r) between two systems. To put it otherwise: when looking for the construction of a functional F(AB) = F[ζ(A)(r),ζ(B)(r)] for extracting differences in information from an information carrier ζ(r) (i.e. ρ(r), σ(r)) for two systems A and B the Kullback-Leibler information measure ΔS is a particularly adequate choice. Examples are given, varying from atoms, to molecules and molecular interactions. Quantum similarity of atoms indicates that the shape function based KL information deficiency is the most appropriate tool to retrieve periodicity in the Periodic Table. The dissimilarity of enantiomers for which different information measures are presented at global and local (i.e. molecular and atomic) level leads to an extension of Mezey's holographic density theorem and shows numerical evidence that in a chiral molecule the whole molecule is pervaded by chirality. Finally Kullback-Leibler information profiles are discussed for intra- and intermolecular proton transfer reactions and a simple S(N)2 reaction indicating that the theoretical information profile can be used as a companion to the energy based Hammond postulate to discuss the early or late transition state character of a reaction. All in all this Perspective's answer is positive to the question of whether an even simpler carrier of information than the electron density function ρ(r) can be envisaged: the shape function, integrating to 1 by construction fulfils this role. On the other hand obtaining the information (or information difference) contained in one (or two) systems from ρ(r) or σ(r) can be most efficiently done by using information theory, the Kulback-Leibler information deficiency being at the moment (one of) the most advisable functionals.     

Monotonicity properties of the atomic charge density function

The present knowledge of the monotonicity properties of the spherically averaged electron density ρ(r) and its derivatives, which comes mostly from Roothan-Hartree-Fock calculations, is reviewed and extended to all Hartree-Fock ground-state atoms from hydrogen (Z = 1) to uranium (Z = 92). In looking for electron functions with universal (i.e., valid in the whole periodic table) monotonicity properties, it is found that there exist positive values of α so that the function g_o(r; α) = ρ(r)/r^α is convex, and g₁(r;α) = −ρ′(r)/r^α is not only monotonically decreasing from the origin but also convex. This is, however, not the case for the function g₂(r; α) = ρ′(r)/r^α. Additionally, the conditions which specify values for β such that the function g_n(r; β) = (−1) ′ρ⁽ⁿ⁾(r)/r^β is logarithmically convex are obtained and numerically calculated for n = 0,1 in all neutral atoms below uranium. The last property is used to obtain inequalities of general validity involving three radial expectation values which generalize all the similar ones known to date, as well as other relationships among these quantities and the values of the electron density and its derivatives at the nucleus. © 1996 John Wiley & Sons, Inc.     

Comparative density functional theory and post-Hartree-Fock (CCSD, CASSCF) studies on the electronic structure of halogen nitrites ClONO and BrONO using quantum chemical topology

In this paper, the electronic structures of cis- and trans-ClONO and BrONO are studied at the CCSD∕aug-cc-pVTZ, CASSCF(14,12)/aug-cc-pVTZ, and B3LYP/aug-cc-pVTZ computational levels. For the Cl-O bond, topological analysis of the electron density field, ρ(r), shows the prevalence of the shared-electron type bond (?(2)ρ((3,-1)) < 0). The Br-O bond, however, represents the closed-shell interaction (?(2)ρ((3,-1)) > 0). Topological analysis of the electron localization function, η(r), and electron localizability indicator (ELI-D), (D) (σ)(r), shows that the electronic structure of the central N-O bond is very sensitive to both electron correlation improvements (coupled-cluster single double (CCSD), CASSCF, density functional theory (DFT)) and bond length alteration. Depending on the method used, the N-O bond can be characterized as a "normal" N-O bond with a disynaptic V(N,O) basin (DFT); a protocovalent N-O bond with two monosynaptic, V(N) and V(O), basins (CCSD, CASSCF); or a new type, first discovered for FONO, characterized by a single monosynaptic, V(N) basin (CCSD, DFT). The total basin population oscillates between 0.46-0.96 e (CCSD) and 0.86-1.02 e (CASSCF). The X-O bond is described by the single disynaptic basin, V(X,O), with a basin population between 0.76 and 0.81 e (CCSD) or 0.77 and 0.85 e (CASSCF). Analysis of the localized electron detector distribution for the cis-Cl-O1-N=O2 shows a manifold in the Cl···O2 region, associated with decreased electron density.     

Electron-pair shell density approximation applied to inner and outer densities of atoms

The shell density approximation to the electron-pair radial density of atoms is applied to the inner $D_< (r)$ and outer $D_> (r)$ densities, which are two components of the single-electron density $D(r)$ . The inner and outer densities are found to be expressed by product sums of shell densities and shell distributions or their complements. The expressions clarify physical meaning of the two densities and give examples for constructing two-electron properties from single-electron properties. Examination of the 53 atoms He through Xe shows that the quantum similarity indices between the original and approximate densities, bounded by 0 (complete dissimilarity) and 1 (complete similarity), are never smaller than 0.99998 and 0.99987 for the inner and outer densities, respectively. The local nature of the shell density and the monotonically increasing property of the shell distribution are used to derive simple shellwise lower and upper bounds to $D_< (r)$ and $D_> (r)$ in terms of $D(r)$ and the numbers of shell electrons. Numerical tests of the bounds demonstrate their utility.     

Slater's nonlocal exchange potential and beyond

The local density approximation (LDA) to the exchange potential V_x( r ), namely the ρ^1/3 electron gas form, was already transcended in Slater's 1951 paper. Here, using Dirac's 1930 form for the exchange energy density ? _x( r ), the Slater (Sl) nonlocal exchange potential V( r ) is defined by 2? _x( r )/ρ( r ). In spherical atomic ions, say the Be or Ne‐like series, this form V( r ) already has the correct behavior in both r → 0 and r → ∞ limits when known properties of the exchange energy density ? _x( r ) and the ground‐state electron density ρ( r ) are invoked. As examples, some emphasis will first be given to the use of the so‐called 1/Z expansion in such spherical atomic ions, for which analytic results can be obtained for both ? _x( r ) and ρ( r ) as the atomic number Z becomes large. The usefulness of the 1/Z expansion is directly demonstrated for the U atomic ion with 18 electrons by comparison with the optimized effective potential prediction. A rather general integral equation for the exchange potential is then proposed. Finally, without appeal to large Z, two‐level systems are considered, with specific reference to the Be atom and to the LiH molecule. In all cases treated, the Slater potential V( r ) is a valuable starting point, even though it needs appreciable quantitative corrections reflecting directly atomic shell structure. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2005     

Charge densities for singlet and triplet electron pairs

Analytic properties of charge densities associated with singlet and triplet electron pairs, ρ₀( r ) and ρ₁( r ), are presented. In an N‐electron system with total spin S, distributions ρ₀( r ) and ρ₁( r ) are independent of the spin projection quantum number M (spin rotation invariance), as opposed to the usual spin‐up and spin‐down electron densities, ρ_α( r ) and ρ_β( r ). We derive equations showing that in the case of a wave function which is a spin‐eigenfunction, ρ₀( r ) and ρ₁( r ) are linear combinations of the total charge density ρ( r ) and the uncompensated spin density ρ_s( r )=[ρ_α( r )−ρ_β( r )]/2M. For a wave function which is not an eigenfunction of $\mathcal{S}^{2}$, no such relationship exists. In a related discussion, a definition of the high‐spin solution corresponding to a given spin‐unrestricted Hartree–Fock wave function is proposed, and a notion of effectively unpaired electrons is introduced. The distributions ρ₀( r ) and ρ₁( r ) are shown not to be invariant under spin coupling between isolated systems. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 77: 651–660, 2000     

Si-O bonded interactions in silicate crystals and molecules: a comparison

Bond critical point, local kinetic energy density, G(rc), and local potential energy density, V(rc), properties of the electron density distributions, rho(r), calculated for silicates such as quartz and gas-phase molecules such as disiloxane are similar, indicating that the forces that govern the Si-O bonded interactions in silica are short-ranged and molecular-like. Using the G(rc)/rho(rc) ratio as a measure of bond character, the ratio increases as the Si-O bond length, the local electronic energy density, H(rc)= G(rc) + V(rc), and the coordination number of the Si atom decrease and as the accumulation of the electron density at the bond critical point, rho(rc), and the Laplacian, inverted Delta2 rho(rc), increase. The G(rc)/rho(rc) and H(rc)/rho(rc) ratios categorize the bonded interaction as observed for other second row atom M-O bonds into discrete categories with the covalent character of each of the M-O bonds increasing with the H(rc)/rho(rc) ratio. The character of the bond is examined in terms of the large net atomic charges conferred on the Si atoms comprising disiloxane, stishovite, quartz, and forsterite and the domains of localized electron density along the Si-O bond vectors and on the reflex side of the Si-O-Si angle together with the close similarity of the Si-O bonded interactions observed for a variety of hydroxyacid silicate molecules and a large number of silicate crystals. The bond critical point and local energy density properties of the electron density distribution indicate that the bond is an intermediate interaction between Al-O and P-O bonded interactions rather than being a closed-shell or a shared interaction.     

Exact relations between the electron density and external potential for systems of interacting and noninteracting electrons

The differential virial theorem (DVT) is an explicit relation between the electron density ρ( r ), the external potential, kinetic energy density tensor, and (for interacting electrons) the pair function. The time‐dependent generalization of this relation also involves the paramagnetic current density. We present a detailed unified derivation of all known variants of the DVT starting from a modified equation of motion for the current density. To emphasize the practical significance of the theorem for noninteracting electrons, we cast it in a form best suited for recovering the Kohn–Sham effective potential v_s( r ) from a given electron density. The resulting expression contains only ρ( r ), v_s( r ), kinetic energy density, and a new orbital‐dependent ingredient containing only occupied Kohn–Sham orbitals. Other possible applications of the theorem are also briefly discussed. © 2012 Wiley Periodicals, Inc.     

Molecular density of states from estimated vapor phase heat capacities

Heat capacity data between 298 and 1500K are used to derive a reduced set of apparent vibrational frequencies that can be used for estimation of molecular density of states, ρ(E). Estimates for a number of molecule and radical species, using a reduced set of three frequencies with noninteger degeneracies, are shown to compare favorably to direct count methods, which require specification of the complete frequency set. Use of the reduced set of three frequencies leads to significant improvement in calculations of ρ(E)/Q as compared to similar calculations which use only a single geometric- or arithmetic-mean frequency approximation. Since vapor phase heat capacity data of molecules and radicals can be estimated accurately by a group additivity formalism, this approach provides a method to estimate ρ(E) for use in calculations of pressure effects in unimolecular and chemical activation reaction systems. The accuracy of the ρ(E)/Q distributions obtained from heat capacity data makes this a viable method for those cases where the complete frequency distribution is not known. It is especially valuable for those cases where contributions to ρ(E) from internal rotors or low frequency vibrations such as inversions are not well known. This approach is useful for quantum RRK or inverse Laplace transform calculations of k(E) since no assignment of transition state properties is necessary. The reduced frequency set can also be combined with ΔH_f(298) and S(298) to provide a compact data set to describe thermodynamic properties at any temperature. © 1997 John Wiley & Sons, Inc.     

Exact non-additive kinetic potentials in realistic chemical systems

In methods based on frozen-density embedding theory or subsystem formulation of density functional theory, the non-additive kinetic potential (v(t) (nad)(r)) needs to be approximated. Since v(t) (nad)(r) is defined as a bifunctional, the common strategies rely on approximating v(t) (nad)[ρ(A),ρ(B)](r). In this work, the exact potentials (not bifunctionals) are constructed for chemically relevant pairs of electron densities (ρ(A) and ρ(B)) representing: dissociating molecules, two parts of a molecule linked by a covalent bond, or valence and core electrons. The method used is applicable only for particular case, where ρ(A) is a one-electron or spin-compensated two-electron density, for which the analytic relation between the density and potential exists. The sum ρ(A) + ρ(B) is, however, not limited to such restrictions. Kohn-Sham molecular densities are used for this purpose. The constructed potentials are analyzed to identify the properties which must be taken into account when constructing approximations to the corresponding bifunctional. It is comprehensively shown that the full von Weizsa?cker component is indispensable in order to approximate adequately the non-additive kinetic potential for such pairs of densities.     

Density functional theory for open-shell systems using a local-scaling transformation scheme. II. Euler-Lagrange equation for f(r) versus that for ρ(r)

Following the previous article (Part I), we express the total nonrelativistic energy for spin manifolds of open-shell multielectronic systems, within an orbit θ^N induced by a model wave function (MWF) _Ψ using a single local-scaling transformation (LST) as an exact functional of the single-particle density ρ( r ) or, alternatively, of the LST scalar function f( r ). We derive the corresponding Euler–Lagrange variational equations: one implicit in ρ( r ), which can be solved iteratively through steps involving f( r ), and one explicit in f( r ), derived from the total energy as a functional of f( r ). Both equations fulfill the space and spin symmetries characterizing the system. The problems arising from the specificities of these two highly nonlinear integrodifferential equations are discussed. The optimal charge density ρ( r ) derived from these equations is N- and v-representable and determines the optimal spin density σ( r ) as well. Accurate optimal values of all observables can be derived from this scheme using standard procedures. © 1997 John Wiley & Sons, Inc. Int J Quant Chem 65 : 257–268, 1997     

Bond length and local energy density property connections for non-transition-metal oxide-bonded interactions

For a variety of molecules and earth materials, the theoretical local kinetic energy density, G(r(c)), increases and the local potential energy density, V(r(c)), decreases as the M-O bond lengths (M = first- and second-row metal atoms bonded to O) decrease and the electron density, rho(r(c)), accumulates at the bond critical points, r(c). Despite the claim that the local kinetic energy density per electronic charge, G(r(c))/rho(r(c)), classifies bonded interactions as shared interactions when less than unity and closed-shell when greater, the ratio was found to increase from 0.5 to 2.5 au as the local electronic energy density, H(r(c)) = G(r(c)) + V(r(c)), decreases and becomes progressively more negative. The ratio appears to be a measure of the character of a given M-O bonded interaction, the greater the ratio, the larger the value of rho(r(c)), the smaller the coordination number of the M atom and the more shared the bonded interaction. H(r(c))/rho(r(c)) versus G(r(c))/rho(r(c)) scatter diagrams categorize the M-O bonded interactions into domains with the local electronic energy density per electron charge, H(r(c))/rho(r(c)), tending to decrease as the electronegativity differences for the bonded pairs of atoms decrease. The values of G(r(c)) and V(r(c)), estimated with a gradient-corrected electron gas theory expression and the local virial theorem, are in good agreement with theoretical values, particularly for the bonded interactions involving second-row M atoms. The agreement is poorer for shared C-O and N-O bonded interactions.     

Approximate scaling properties of the density functional theory Tc for atoms

Revealed are scaling properties for T(c)[rho], the kinetic-energy component of the correlation energy density functional for atoms, in terms of the total number of electrons N, the nuclear charge Z, and the total electron density at the nucleus rho(0). T(c) scales well as Nrho(0)/Z(8/3) for both neutral atoms up to Z=18 and the four-electron Be-like cationic species. A model is given that describes these findings, involving a density encoding the cusp information and an effective potential going like r(-4/3).     

Electrochemical and surfaced-enhanced Raman spectroscopic investigation of CO and SCN- adsorbed on Au(core)-Pt(shell) nanoparticles supported on GC electrodes

Core-shell Au-Pt nanoparticles were synthesized by using a seed growth method and characterized by transmission electron microscopy, X-ray diffraction, and UV-vis spectroscopy. Au(core)-Pt(shell)/GC electrodes were prepared by drop-coating the nanoparticles on clean glassy carbon (GC) surfaces, and their electrochemical behavior in 0.5 M H2SO4 revealed that coating of the Au core by the Pt shell is complete. The electrooxidation of carbon monoxide and methanol on the Au(core)-Pt(shell)/GC was also examined, and the results are similar to those obtained on a bulk Pt electrode. High quality surface-enhanced Raman scattering (SERS) spectra of both adsorbed CO and thiocyanate were observed on the Au(core)-Pt(shell)/GC electrodes. The potential-dependent SERS features resemble those obtained on electrochemically roughened bulk Pt or Pt thin films deposited on roughened Au electrodes. For thiocyanate, the C-N stretching frequency increases with the applied potential, yielding two distinctly different dnu(CN)/dE. From -0.8 to -0.2 V, the dnu(CN)/dE is ca. 50 cm(-1)/V, whereas it is 90 cm(-1)/V above 0 V. The bandwidth along with the band intensity increases sharply above 0 V. At the low-frequency region, Pt-NCS stretching mode at 350 cm(-1) was observed at the potentials from -0.8 to 0 V, whereas the Pt-SCN mode at 280 cm(-1) was largely absent until around 0 V and became dominant at more positive potentials. These potential-dependent spectral transitions were attributed to the adsorption orientation switch from N-bound dominant at the negative potential region to S-bound at more positive potentials. The origin of the SERS activity of the particles is briefly discussed. The study demonstrates a new method of obtaining high quality SERS on Pt-group transition metals, with the possibility of tuning SERS activity by varying the core size and the shell thickness.     

Atomic shell structure and electron numbers

The electron localization function (ELF) was calculated for the atoms Li to Sr. The ELF maxima reveal the atomic shell structure for all these atoms. The shells are separated from each other by ELF minima. The integration of the electron density in a shell gives electron numbers. For the valence shell those are in good agreement with the ones expected from the Periodic Table of Elements. © 1996 John Wiley & Sons, Inc.     

Quantification of the trans influence in hypervalent iodine complexes

The trans influence of various X ligands in hypervalent iodine(III) complexes of the type CF(3)[I(X)Cl] has been quantified using the trans I-Cl bond length (d(X)), the electron density ρ(r) at the (3, -1) bond critical point of the trans I-Cl bond, and topological features of the molecular electrostatic potential (MESP). The MESP minimum at the Cl lone pair region (V(min)) is a sensitive measure of the trans influence. The trans influence of X ligands in hypervalent iodine(V) complexes is smaller than that in iodine(III) complexes, while the relative ordering of this influence is the same in both complexes. In CF(3)[I(X)Y] complexes, the mutual trans influence due to the trans disposition of the X and Y ligands is quantified using the energy E(XY) of the isodesmic reaction CF(3)[I(X)Cl] + CF(3)[I(Y)Cl] → CF(3)[I(Cl)Cl] + CF(3)[I(X)Y]. E(XY) is predicted with good accuracy using the trans-influence parameters of X and Y, measured in terms of d(X), ρ(r), or V(min). The bond dissociation energy (E(d)) of X or Y in CF(3)[I(X)Y] is significantly influenced by the trans influence as well as the mutual trans influence. This is confirmed by deriving an empirical equation to predict E(d) using one of the trans-influence parameters (d(X), ρ(r), or V(min)) and the mutual trans-influence parameter E(XY) for a large number of complexes. The quantified values of both the trans influence and the mutual trans-influence parameters may find use in assessing the stability of hypervalent iodine compounds as well as in the design of new stable hypervalent complexes. Knowledge about the I-X bond dissociation energies will be useful for explaining the reactivity of hypervalent iodine complexes and the mechanism of their reactions.     

Structural, textural and adsorption characteristics of nanosilica mechanochemically activated in different media

The structural, textural, and adsorption characteristics of mechanochemically activated (MCA) fumed silica A-300 as dry or water, ethanol, or water/ethanol-wetted powders (0.5 g of a solvent per gram of silica) in a ball mill for 1-6 h were studied in comparison with those of the initial powder. The MCA treatment enhances bulk density (ρ(b)) of the powder (from 0.045 g/cm(3) for the initial silica to 0.4 g/cm(3) for 6 h-MCA-treated water-wetted silica) depending on medium type and MCA time (t(MCA)). Stronger effects are observed for the MCA treatment of water-wetted silica than of dry or ethanol- or water/ethanol-wetted samples. The MCA treatment weakly affects the specific surface area (S(BET)). However, void (pore) size distribution, porosity, particle aggregation and size distribution in aqueous suspension, behavior of interfacial water, properties of poly(vinyl alcohol)/silica composites and adsorption of gelatin depend more strongly on the t(MCA) and ρ(b) values. Some of the observed changes in the characteristics (e.g., gelatin adsorption) depend on the ρ(b) value but are independent of the medium type used on the MCA. Other characteristics are nonlinear functions of both t(MCA) and ρ(b) values.     

Letter: correlation between phosphorylation ratios by matrix-assisted laser desorption/ionization time-of-flight mass spectrometric analysis and enzyme kinetics

To identify the correlation between the phosphorylation ratios by matrix-assisted laser desorption/ionization-time-of-flight mass spectrometry (MALDI-ToF MS) analysis and enzyme kinetics (K(m), V(max), and V(max)/K(m)) is important to understand whether MALDI-TOF MS can be applied for monitoring the properties of peptides that are substrates of protein kinases. The correlation between phosphorylation ratios and enzyme kinetics was examined using peptides for protein kinase C (PKC) and for 60 kDa phosphoprotein, encoded by the cellular sarcoma gene (c-Src). Phosphorylation ratios, analyzed by MALDI-ToF MS, showed higher correlation coefficient (r = > +0.7) for V(max)/K(m) compared with that (r = < -/+0.6) for K(m) or V(max). For ion modes, a higher correlation coefficient between phosphorylation ratios and V(max)/K(m) was identified in the positive mode (r = > +0.7) compared to that in the negative mode (r = < +0.5). These results suggest that MALDI-ToF MS is a useful tool to evaluate V(max)/K(m) of peptides for protein kinases.