Diatomic interactions in momentum space. The effect of polarization and floating functions

The method of momentum density for interatomic interactions is used to investigate the pictures and roles of the polarization and floating functions in momentum (p-) space. Referring to the previous results from the minimal LCAO (Finkelstein-Horowitz) momentum density, we quantitatively discuss the effect of these functions for the bonding process in the ground state of H ₂ ⁺ system. The essence of the polarization and floating effects is found to be a modulation of the oscillation in the two-center part of the momentum density. The polarization function introduces a term with a phase and the floating function enlarges the period of the oscillation. An increased migration of the density from the one-center to the two-center part is also important. As a result, both the functions contribute to emphasize the contraction and expansion of momentum density observed previously. However, the floating function disturbs the density distribution in high momentum region, reflecting the destruction of cusps in position (r-) space. We point out an error in the pioneer work of Duncanson.     

Quantum information entropies of multiple quantum well systems in fractional Schrödinger equations

In this work, we study the position and momentum information entropies of multiple quantum well systems in fractional Schrödinger equations, which, to the best of our knowledge, have not so far been studied. Through a confining potential, their shape and number of wells (NOW) can be controlled by using a few tuning parameters; we present some interesting quantum effects that only appear in the fractional Schrödinger equation systems. One of the parameters denoted by the L_d can affect the position and momentum probability densities if the system is fractional (1 < α < 2). We find that the position (momentum) probability density tends to be more severely localized (delocalized) in more fractional systems (ie, in smaller values of α). Affecting the L_d on the position and momentum probability densities is a quantum effect that only appears in the fractional Schrödinger equations. Finally, we show that the Beckner Bialynicki-Birula-Mycieslki (BBM) inequality in the fractional Schrödinger equation is still satisfied by changing the confining potential amplitude V_conf, the NOW, the fractional parameter α, and the confining potential parameter L_d .     

On the basis‐set dependence of local and integrated electron density properties: Application of a new computer program for quantum‐chemical density analysis

A new computer program for post‐processing analysis of quantum‐chemical electron densities is described. The code can work with Slater‐ and Gaussian‐type basis functions of arbitrary angular momentum. It has been applied to explore the basis‐set dependence of the electron density and its Laplacian in terms of local and integrated topological properties. Our analysis, including Gaussian/Slater basis sets up to sextuple/quadruple‐zeta order, shows that these properties considerably depend on the choice of type and number of primitives utilized in the wavefunction expansion. Basis sets with high angular momentum (l = 5 or l = 6) are necessary to achieve convergence for local properties of the density and the Laplacian. In agreement with previous studies, atomic charges defined within Bader's Quantum Theory of Atoms in Molecules appear to be much more basis‐set dependent than the Hirshfeld's stockholder charges. The former ones converge only at the quadruple‐zeta/higher level with Gaussian/Slater functions. © 2008 Wiley Periodicals, Inc. J Comput Chem, 2009     

Application of the unitary group approach to evaluate spin density for configuration interaction calculations in a basis of S2 eigenfunctions

We present an implementation of the spin‐dependent unitary group approach to calculate spin densities for configuration interaction calculations in a basis of spin symmetry‐adapted functions. Using S² eigenfunctions helps to reduce the size of configuration space and is beneficial in studies of the systems where selection of states of specific spin symmetry is crucial. To achieve this, we combine the method to calculate U(n) generator matrix elements developed by Downward and Robb (Theor. Chim. Acta 1977, 46, 129) with the approach of Battle and Gould to calculate U(2n) generator matrix elements (Chem. Phys. Lett. 1993, 201, 284). We also compare and contrast the spin density formulated in terms of the spin‐independent unitary generators arising from the group theory formalism and equivalent formulation of the spin density representation in terms of the one‐ and two‐electron charge densities.     

Influence of electronic correlation in monoelectronic density in p-space

The radial molecular monoelectronic density and their orbital contributions have been calculated in the momentum space. For these purposes, densities for the ground state of several atoms and molecules, using a cc-pVTZ basis set at HF level, as well as some post-HF and DFT methods are computed. The difference between the radial monoelectronic density computed with each method and that using the HF wave function is used as a tool to study the influence of the electronic correlation in the momentum space. Densities obtained with post HF calculations show a similar behavior around p = 1.0 and 2.0, that are different from the DFT results. Radial momentum densities (p-densities) are more influenced by the electronic correlation than the exchange part of the DFT methods. CISD p-density is more affected than DFT p-density when the intermolecular distance increases. An analysis of the powers of moments calculated with different methods has been carried out. Contribution to the Serafin Fraga Memorial Issue.     

Expansion coefficients and moments of electron momentum densities for singly charged ions

Numerical Hartree–Fock calculations of the first three coefficients of the MacLaurin expansion and the leading coefficient of the large-p asymptotic expansion of the electron momentum densities Π(p) are reported for 54 singly charged atomic cations from He⁺ (atomic number Z = 2) to Cs⁺ (Z = 55) and 43 anions from H⁻ (Z = 1) to I⁻ (Z = 53) in their experimental ground states. We also report all the finite moments <p ^k > (−2≤k≤+4) of the momentum densities Π(p) for the above-mentioned 97 ionic species. The results are compared with the previous ones for neutral atoms [Koga and Thakkar (1996) J Phys B 29: 2973], and the dependence of the expansion coefficients and moments on nuclear charge is discussed among isoelectronic species. Received: 20 November 1998 / Accepted: 15 January 1999 / Published online: 7 June 1999     

Basis set quality. II. Information theoretic appraisal of various s- orbitals

The expectation values 〈r^k〉 (?2 ? k ? 4, k = 10), values of the charge density ρ(r) at selected points, and coefficients in the MacLaurin expansion of ρ(r) are used to test the quality of 71 orbital basis sets used for the atomic helium Hartree–Fock problem. These tests in position space are complementary to the momentum space tests previously carried out [Int. J. Quantum Chem. 21 , 419 (1982)]. Information theoretic measures with respect to either or both position and momentum space properties are subsequently defined and the orbitals are ranked accordingly. These measures indicate that, for a given orbital, momentum space properties are more poorly predicted than position space ones. Moreover an improvement in the virial ratio does not necessarily lead to a more balanced orbital with respect to position and momentum space properties. Basis sets containing Slater-type orbitals are again found to be rather accurate. The exponentially damped rational function is confirmed to be the outstanding two-parameter unconventional orbital.     

Transferable integrals in a deformation-density approach to crystal orbital calculations. I

In the usual ab initio method of calculating molecular orbitals, the number of integrals to be evaluated increases as M⁴, where M is the number of basis functions. In this paper, an alternative method is discussed, where the computation time increases much less violently with the number of basis functions. Matrix elements of the deformation potential are evaluated by Fourier transform methods, while matrix elements of the neutral-atom potential are evaluated by means of transferable integrals. The transferable integrals (moments of the neutral-atom potentials) can be evaluated once and for all and incorporated as input data in computer programs. In an appendix to the paper, a general expansion theorem is discussed. This theorem allows an arbitrary spherically symmetric function to be expanded about another center.     

Interesting properties of Thomas–Fermi kinetic and Parr electron–electron‐repulsion DFT energy functional generated compact one‐electron density approximation for ground‐state electronic energy of molecular systems

The reduction of the electronic Schrodinger equation or its calculating algorithm from 4N‐dimensions to a (nonlinear, approximate) density functional of three spatial dimension one‐electron density for an N‐electron system, which is tractable in the practice, is a long desired goal in electronic structure calculation. If the Thomas‐Fermi kinetic energy (～∫ρ^5/3d r ₁) and Parr electron–electron repulsion energy (～∫ρ^4/3d r ₁) main‐term functionals are accepted, and they should, the later described, compact one‐electron density approximation for calculating ground state electronic energy from the 2nd Hohenberg–Kohn theorem is also noticeable, because it is a certain consequence of the aforementioned two basic functionals. Its two parameters have been fitted to neutral and ionic atoms, which are transferable to molecules when one uses it for estimating ground‐state electronic energy. The convergence is proportional to the number of nuclei (M) needing low disc space usage and numerical integration. Its properties are discussed and compared with known ab initio methods, and for energy differences (here atomic ionization potentials) it is comparable or sometimes gives better result than those. It does not reach the chemical accuracy for total electronic energy, but beside its amusing simplicity, it is interesting in theoretical point of view, and can serve as generator function for more accurate one‐electron density models. © 2008 Wiley Periodicals, Inc. J Comput Chem 2009     

Analysis of tertiary phosphanes, arsanes, and stibanes as bridging ligands in dinuclear Group 9 complexes

The unusual bridging and semi‐bridging binding mode of tertiary phosphanes, arsanes, and stibanes in dinuclear low‐valent Group 9 complexes have been studied by density functional methods and bonding analyses. The influence of various parameters (bridging and terminal ligands, metal atoms) on the structural preferences and bonding of dinuclear complexes of the general composition [A¹ M¹(μ‐CH₂)₂(μ‐EX₃)M² A²] (M¹, M²=Co, Rh, Ir; A¹, A²=F, Cl, Br, I, κ²‐acac; E=P, As, Sb, X=H, F, CH₃) has been analyzed. A number of factors have been identified that favor bridging or semi‐bridging modes for the phosphane ligands and their homologues. A more symmetrical position of the bridging ligand EX₃ is promoted by more polar E? X bonding, but by less electronegative (softer) terminal anionic ligands. Among the Group 9 metal elements Co, Rh, and Ir, the computations clearly show that the 4d element rhodium exhibits the largest preference for a {M¹(μ‐EX₃)M²} bridge, in agreement with experimental observation. Iridium complexes should be valid targets, whereas cobalt does not seem to support well a symmetric bridging mode. Analyses of the Electron Localization Function (ELF) indicate a competition between a delocalized three‐center bridge bond and direct metal–metal bonding.     

Density functionals in the presence of magnetic field

In this article, density functionals for Coulomb systems subjected to electric and magnetic fields are developed. The density functionals depend on the particle density ρ and paramagnetic current density j^p. This approach is motivated by an adapted version of the Vignale and Rasolt formulation of current density functional theory, which establishes a one‐to‐one correspondence between the nondegenerate ground‐state and the particle and paramagnetic current density. Definition of N‐representable density pairs (ρ,j^p) is given and it is proven that the set of v‐representable densities constitutes a proper subset of the set of N‐representable densities. For a Levy–Lieb‐type functional Q(ρ,j^p), it is demonstrated that (i) it is a proper extension of the universal Hohenberg–Kohn functional to N‐representable densities, (ii) there exists a wavefunction ψ₀ such that , where H₀ is the Hamiltonian without external potential terms, and (iii) it is not convex. Furthermore, a convex and universal functional F(ρ,j^p) is studied and proven to be equal the convex envelope of Q(ρ,j^p). For both Q and F, we give upper and lower bounds. © 2014 Wiley Periodicals, Inc.     

A quantitative analysis of light‐driven charge transfer processes using voronoi partitioning of time dependent DFT‐derived electron densities

An analytical method is presented that provides quantitative insight into light‐driven electron density rearrangement using the output of standard time‐dependent density functional theory (TD‐DFT) computations on molecular compounds. Using final and initial electron densities for photochemical processes, the subtraction of summed electron density in each atom‐centered Voronoi polyhedron yields the electronic charge difference, Q ^VECD. This subtractive method can also be used with Bader, Mulliken and Hirshfeld charges. A validation study shows Q ^VECD to have the most consistent performance across basis sets and good conservation of charge between electronic states. Besides vertical transitions, relaxation processes can be investigated as well. Significant electron transfer is computed for isomerization on the excited state energy surface of azobenzene. A number of linear anilinepyridinium donor‐bridge‐acceptor chromophores was examined using Q ^VECD to unravel the influence of its pi‐conjugated bridge on charge separation. Finally, the usefulness of the presented method as a tool in optimizing charge transfer is shown for a homologous series of organometallic pigments. The presented work allows facile calculation of a novel, relevant quantity describing charge transfer processes at the atomic level. © 2017 The Authors Journal of Computational Chemistry Published by Wiley Periodicals, Inc.     

Diatomic interactions in momentum space. The 1sσg and 2pσ u states of H 2 + system

The method of momentum electron density for interatomic interactions has been applied to the two lowest σ states of the H ₂ ⁺ system. For attractive (1sσ_g) and repulsive (2pσ _u ) interactions, the behaviour of momentum density and its effect on the stabilization energy of the system are examined quantitatively. The concept of contraction and expansion of the momentum density is shown to form an important guiding principle in this approach. The origin of covalent bonding is discussed based on the energy partitioning proposed previously.     

Maximum-entropy analysis of momentum densities in diatomic molecules

The one-particle density in momentum space γ(p) is studied for diatomic molecules by using the maximum-entropy technique. The knowledge of one or more momentum expectation values <p"> provides approximations on the density γ(p) for any value of the momentum, which are convergent when increasing the number of known moments. Other unknown expectation values are estimated in terms of the constructed maximum-entropy densities. A numerical study of the quality of the approximations is carried out by means of experimental and theoretical data for the momentum expectation values involved. Experimental errors are also taken into account to have an idea of the sensibility of the results to the information from which they are obtained. © 1997 John Wiley & Sons, Inc.     

Analytical Hartree-Fock electron densities for singly charged cations and anions

The Hartree-Fock (HF) electron density has an important property that it is identical to the unknown exact density to the first order in the perturbation theory. We generate the spherically averaged HF electron density ρ(r) by using the numerical HF method for the singly charged 53 cations from Li⁺ to Cs⁺ and 43 anions from H⁻ to I⁻ in their ground state. The resultant density is then accurately fitted into an analytical function F(r), which is expressed by a linear combination of basis functions r ⁿⁱ exp(−ζ _i r). The present analytical approximation F(r) has the following properties: (1) F(r) is nonnegative, (2) F(r) is normalized, (3) F(r) reproduces the HF moments <r ^k > (k=−2 to +6), (4) F(0) is equal to ρ(0), (5) F ^′(0) satisfies the cusp condition and (6) F(r) has the correct exponential decay in the long-range asymptotic region. The present results together with our previous ones for neutral atoms provide a compilation of accurate analytical approximations of the HF electron densities for all the neutral and singly charged atoms with the number of electrons N≤54. Received: 11 July 1997 / Accepted: 27 August 1997     

Electron momentum density and band structure calculations of α- and β-GeTe

We have measured isotropic experimental Compton profile of α-GeTe by employing high energy (662 keV) γ-radiation from a ¹³⁷Cs isotope. To compare our experiment, we have also computed energy bands, density of states, electron momentum densities and Compton profiles of α- and β-phases of GeTe using the linear combination of atomic orbitals method. The electron momentum density is found to play a major role in understanding the topology of bands in the vicinity of the Fermi level. It is seen that the density functional theory (DFT) with generalised gradient approximation is relatively in better agreement with the experiment than the local density approximation and hybrid Hartree–Fock/DFT.     

Metallobiliverdin Radicals—DFT Studies

Several aspects of the molecular and electronic structure of biliverdin derivatives have been studied using density functional theory (DFT). The calculations have been performed for complexes of trianion (BvO₂)^3? and dianion [BvO(OH)]^2?, derived from two tautomeric forms of biliverdin, BvO₂H₃ and [BvO(OH)]H₂, with redox innocent metal ions: lithium(I ), zinc(II ), and gallium(III ). One‐electron‐oxidized and ‐reduced forms of each complex (cation and anion radicals) have been also considered. The molecular structures of all species investigated are characterized by a helical arrangement of tetrapyrrolic ligands with the metal ion lying in the plane formed by the two central pyrrole rings. The spin density distribution in four types of metallobiliverdin radicals—[(BvO₂^.)Mⁿ⁺]^n‐2, [{BvO(OH)^.}Mⁿ⁺]^n‐1 (cation radicals), [(BvO2^.)Mⁿ⁺]^n‐4, [{BvO(OH)^.}Mⁿ⁺]^n‐3 (anion radicals)—has been investigated. In general, the absolute values of spin density on meso carbon atoms were larger than for the β‐carbon atoms. Sign alteration of spin density has been found for meso positions, and also for the β‐carbon atoms of at least two pyrrole rings. The calculated spin density maps accounted for the essential NMR spectroscopic features of iron biliverdin derivatives, including the considerable isotropic shifts detected for the meso resonances and shift alteration at the meso and β‐positions.     

Theoretical Study on the High Energy Density Compound Hexanitrohexaazatricyclotetradecanedifuroxan

Density functional theory （DFT） has been employed to study the molecular geometries, electronic structures, infrared （IR） spectra, and thermodynamic properties of the high energy density compound hexanitrohexaazatricyclotetradecanedifuroxan （HHTTD） at the B3LYP/6-31G^＊＊ level of theory. The calculated results show that there are four conformational isomers （α, β, γ and δ） for HHTTD, and the relative stabilities of four conformers were assessed based on the calculated total energies and the energy-gaps between the frontier molecular orbitals. The computed harmonic vibrational frequencies are in reasonable agreement with the available experimental data. Thermodynamic properties derived from the IR spectra on the basis of statistical thermodynamic principles are linearly correlated with the temperature. Detonation performances were evaluated by using the Kamlet-Jacobs equations based on the calculated densities and heats of formation. It was found that four HHTTD isomers with the predicted densities of ca. 2 g·cm^-3, detonation velocities near 10 km·s^-1, and detonation pressures over 45 GPa, may be novel potential candidates of high energy density materials （HEDM）. These results may provide basic information for the molecular design of HEDM.     

Electron pair density information measures in atomic systems

Shannon entropies of the pair density, conditional entropies, and mutual information are studied in position and in momentum space for ground state neutral atoms and selected excited states at the Hartree‐Fock level. We show that the mutual information, a measure of correlation, is larger in position space than in momentum space. This result also holds for a mutual information defined in terms of the exchange density; however, these quantities display much more structure than the corresponding ones based on the pair densities. The interpretation of this behavior is that exchange effects are smaller in momentum space than in position space in these systems. © 2010 Wiley Periodicals, Inc. Int J Quantum Chem, 2011     

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