Topography of molecular scalar fields. II. An appraisal of the hierarchy principle for electron momentum densities

The previously observed hierarchy principle for nondegenerate critical points (CPs) of the electron momentum density (EMD) of molecules [Kulkarni, Gadre, and Pathak, Phys. Rev. A. 45, 4399 (1992)] is verified at a reliable level of theory. Application of Morse inequalities and the Poincare-Hopf relation to EMD leads to some rigorous results viz (i) for total number of CPs, NCP=3,7,11,15, ...there must be either a (3,+3) or a (3,-1) CP at the center of symmetry, (ii) for N(CP)=1,5,9,13, ...there must be either a (3,-3) or a (3,+1) CP at the center of symmetry. A single directional maximum on every ray, starting from p=0 has been observed for all the test molecules and is suggested as a working topographical principle in p space. This working principle is shown to satisfy the sufficiency condition for the hierarchy principle.     

Theoretical investigations into the blue-shifting hydrogen bond in benzene complexes.

The benzene...X complexes (X=benzene, antracene, ovalene) were optimised at the MP2/6-31G** level with the C2v symmetry of the complex and planarity of the proton acceptor being preserved. The resulting stabilisation energies amount to 1.2, 2.3 and 2.9 kcal mol(-1), and the C-H bond of the proton donor is contracted by 0.0035, 0.0052 and 0.0055 A, respectively. The contraction is connected with a blue-shift of the C-H stretch vibration frequency. A two-dimensional anharmonic vibration treatment based on a MP2/6-31G** potential energy surface yields the following blue shifts for the complexes studied: 28, 42 and 43 cm(-1). The dominant attraction in the complexes is London dispersion, while the attractive contribution from electrostatic quadrupole-quadrupole interactions is considerably smaller.     

Theoretical studies on the molecular electron densities and electrostatic potentials in azacubanes

Energetics and the charge distributions in azacubanes (C₈NH_8–) have been obtained using the ab initio Hartree–Fock, second-order Mø øller–Plesset perturbation theory and hybrid density functional methods. For diazacubane to hexaazacubane the lowest-energy conformers have nitrogen atoms occupying the face opposite corners of a cube. The topography of the molecular electrostatic potential and the electron density of azacubane conformers have been investigated. The electrostatic potential studies have shown that successive substitution of nitrogen instead of CH groups of cubane engenders smaller and more localized electron-rich regions around the nitrogens of a cube. Further the bond ellipticity and the electron density at the bond critical point of the X–N bonds (X=C or N) in a cubanoid increase from azacubane to octaazacubane. The heats of formation of azacubanes calculated by the isodesmic reaction approach using different levels of theory correlate well with the electron density at the bond critical point of X–N (X=C or N) bonds in a cubanoid.     

Compton profiles for ejected electrons: Determination of momentum densities

Electron Compton scattering effects usually lead to the determination of momentum densities in atoms and molecules by cross-section measurements on scattered electrons. Similar information is also shown to be directly available from the study of ejected electrons. The proposed treatment results from a simple alteration of the impulse assumptions and appears to give a satisfactory representation of ejected profiles for medium and high energy electron scattering. Preliminary investigations are discussed here for hydrogen and helium atoms.     

New bounds for the atomic charge and momentum densities at the origin

The “Stieltjes moment problem” technique together with the positivity and monotonic decreasing properties of the electronic density of an atom is used to find new and more accurate lower bounds for the charge density at the nucleus and the momentum density at the origin, in terms of radial and momentum expectation values, respectively. Bounds depending on two and three expectation values are given explicitly and a Hartree-Fock study of their quality is carried out. Also, the behavior of the new bounds at largeZ's is discussed. The Stieltjes technique allows to find lower bounds of better accuracy by including expectation values of higher order.     

Experimental and calculated momentum densities for the valence orbitals of carbon tetrafluoride

Carbon tetraflouoride has been investigated by binary (e,2e) spectroscopy at 1200 eV impact energy. Binding energy spectra (10–60 eV) at azimuthal angles of 0° and 8° are reported and are found to be in quantitative agreement with a previous Green's function calculated spectrum. Momentum distributions corresponding to individual orbitals are also reported and compared with theoretical momentum distributions evaluated using double-zeta quality SCF wavefunctions. Excellent agreement between experimental and theories is found for the strongly bonding 3t₂ orbital and the antibonding 4a₁ orbital but agreement is less good for the outermost non-bonding orbitals. Intense structure due to molecular density (bond) oscillation is observed experimentally in the region above 1.0 a_o^?1 in the case of the non-bonding 4t₂ orbital. It is also notable that the measured 4a₁ momentum distribution exhibits an extremely well-defined “p” character with clear separation between the s and p components. Contour maps of the position-space and momentum-space orbital densities in the F-C-F plane of the molecule are used to provide a qualitative interpretation of the features observed in the momentum distribution. In order to further extend momentum-space chemical concepts to three-dimensional systems, constant density surface plots are also used to give a more comprehensive view of the density functions of the CF₈ molecule.     

Atoms-in-molecules in momentum space: a Hirshfeld partitioning of electron momentum densities

A direct application of the Hirshfeld atomic partitioning (HAP) scheme is implemented for molecular electron momentum densities (EMDs). The momentum density contributions of individual atoms in diverse molecular systems are analyzed along with their topographical features and the kinetic energies of the atomic partitions. The proposed p-space HAP-based charge scheme does seem to possess the desirable attributes expected of any atoms in molecules partitioning. In addition to this, the main strength of the p-space HAP is the exact knowledge of the kinetic energy functional and the inherent ease in computing the kinetic energy. The charges derived from HAP in momentum space are found to match chemical intuition and the generally known chemical characteristics such as electronegativity, etc.     

Density matrices from position and momentum densities

Summary One-electron density matrices, which are representable in single-centers-orbital basis sets, have been investigated with respect to their reconstruction from densities. The maximum allowed dimension for reconstruction from a combination of position & momentum density dependent properties is only slightly bigger than the dimension in the case of position (or momentum) densities only. Since for a given one-particle basis of dimensionM, the number of one-matrix elements which can be determined is also of orderM only, while the total number of one-matrix elements is of orderM ², it is in general necessary to introduce severe constraints and restrictions. The accuracy demands on the data and algorithms increase exponentially for linearly increasing size of basis set.     

Interelectronic angle densities of atoms in momentum space

For the 102 atoms from He to Lr in their ground states, the Hartree–Fock interelectronic angle densities,¯A(¯₁₂), in momentum space are reported, where ¯₁₂ is the angle between the momentum vectorsp₁ and p₂ of two electrons. In the first three atoms, He–Be, ¯A(¯₁₂) is found to be uniform independent of ¯₁₂, while in the remaining 99 atoms,¯A(¯₁₂) is larger for a large ¯₁₂ than for a small ¯₁₂. Accordingly, the average interelectronic angles in momentum space are 90° precisely for the three atoms and greater than 90° for the 99 atoms.     

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.     

Theoretical investigations on 1,2-ethanediol: The problem of intramolecular hydrogen bonds

The conformational space of 1,2-ethanediol is studied on the basis of ab initio and semiempirical calculations. All possible conformers are treated. The relative energies of the conformers are systematically studied using various basis sets up to 6–311 + G(3df, 3pd) in order to perform calculations as accurate as possible within a reasonable amount of computer time. Electron correlation is included using Møller-Plesset perturbation theory. We propose two methods to evaluate the basis set superposition error associated with the intramolecular hydrogen bond appearing in some of the conformers. The results of semiempirical calculations are compared with these ab initio calculations. © 1996 by John Wiley & Sons, Inc.     

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     

Imaging momentum orbital densities of conformationally versatile molecules: a benchmark theoretical study of the molecular and electronic structures of dimethoxymethane

The main purpose of the present work is to predict from benchmark many-body quantum mechanical calculations the results of experimental studies of the valence electronic structure of dimethoxymethane employing electron momentum spectroscopy, and to establish once and for all the guidelines that should systematically be followed in order to reliably interpret the results of such experiments on conformationally versatile molecules. In a first step, accurate calculations of the energy differences between stationary points on the potential energy surface of this molecule are performed using Hartree-Fock (HF) theory and post-HF treatments of improving quality (MP2, MP3, CCSD, CCSD(T), along with basis sets of increasing size. This study focuses on the four conformers of this molecule, namely the trans-trans (TT), trans-gauche (TG), gauche-gauche (G+G+), and gauche-gauche (G+G-) structures, belonging to the C2v, C1, C2, and Cs symmetry point groups, respectively. A focal point analysis supplemented by suited extrapolations to the limit of asymptotically complete basis sets is carried out to determine how the conformational energy differences at 0 K approach the full CI limit. In a second step, statistical thermodynamics accounting for hindered rotations is used to calculate Gibbs free energy corrections to the above energy differences, and to evaluate the abundance of each conformer in the gas phase. It is found that, at room temperature, the G+G+ species accounts for 96% of the conformational mixture characterizing dimethoxymethane. In a third step, the valence one-electron and shake-up ionization spectrum of dimethoxymethane is analyzed according to calculations on the G+G+ conformer alone by means of one-particle Green's function [1p-GF] theory along with the benchmark third-order algebraic diagrammatic construction [ADC(3)] scheme. A complete breakdown of the orbital picture of ionization is noted at electron binding energies above 22 eV. A comparison with available (e,2e) ionization spectra enables us to identify specific fingerprints of through-space orbital interactions associated with the anomeric effect. At last, based on our 1p-GF/ADC(3) assignment of spectral bands, accurate and spherically averaged (e,2e) electron momentum distributions at an electron impact energy of 1200 eV are computed from the related Dyson orbitals. Very significant discrepancies are observed with momentum distributions obtained for several outer-valence levels using standard Kohn-Sham orbitals.     

Rovibrational momentum densities of diatomic molecules in the rigid rotor-harmonic oscillator approximation

The rigid rotor-harmonic oscillator model in the momentum representation is used to examine the rovibrational momentum density of a diatomic molecule. The effects on this function of rotational and vibrational excitation, thermal averaging, and nuclear spin statistics are exhibited in calculations on N₂ and thehydrogen halides, and a brief comparison with electron momentum densities is made.     

Theoretical investigations on the solvation process

A model to facilitate the computation of the most stable conformer of associated M · H₂O (M being a polar molecule) which depends upon the electrostatic interaction energy between the two associated molecules is proposed and tested. SCF electrostatic potentials for the M molecule and a suitable point charge distribution for H₂O were employed in the model computations. Energies predicted by the model are found to be in good agreement with those resulting from an ab initio minimal STO basis SCF treatment of some conformations of the H₂O dimer.

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Zusammenfassung Ein Modell zur Durchführung der Berechnung des stabilsten Konformeren eines Assoziationskomplexes M · H₂O, wobei M ein polares Molekül ist, wird vorgeschlagen und untersucht. Es basiert auf der elektrostatischen Wechselwirkung zwischen beiden Partnern, und zwar wird für das Molekül M der elektrostatische Anteil seines SCF-Potentials und für H₂O eine angemessene Punktladungsverteilung zugrunde gelegt. Die resultierenden Energien sind in guter Übereinstimmung mit denen, die sich bei einer ab initio Rechnung mit minimaler STO Basis ergeben.

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Work performed with the financial support of the Consiglio Nazionale delle Ricerche, through its Laboratorio di Chimica Quantistica ed Energetica Molecolare.     

Theoretical investigations on the solvation process

The conformation energies for monohydrated associates M · H₂O, where M stands for oxirane, aziridine, oxaziridine and cyclopropene, have been obtained by using an electrostatic method, tested in a preceding paper, which relies on SCF molecular potentials calculated exactly. Stable associates have been found in the heterogroup region as well as near the bent bonds (single or double).A discussion is made on the errors in calculating thermodynamic properties for such associates in gas phase by using a priori calculations. As a numerical example the free energy change in the association process is compared for two monohydrated associates of aziridine.

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Zusammenfassung Die Konformationsenergien für Anlagerungsverbindung M · H₂O, wobei M für Oxiran, Aziridin, Oxaziridin und Cyclopropen steht, werden mit einer elektrostatischen Methode berechnet. Dieses Verfahren wurde in einer vorhergehenden Veröffentlichung getestet und beruht auf exakt berechneten SCF-Molekülpotentialen. Stabile Anlagerungsverbindungen wurden für Konformationen gefunden, bei denen das Wassermolekül in der Nähe der Heterogruppe oder der gezogenen Einfach- oder Doppelbindung liegt. Die Fehler bei der Berechnung thermodynamischer Eigenschaften für derartige Anlagerungsverbindungen in der Gasphase werden abgeschätzt. Als ein numerisches Beispiel wird die Differenz der freien Energie bei der Anlagerung für zwei Anlagerungsverbindungen von Aziridin und Molekül Wasser verglichen.

     

Theoretical investigations on the solvation process

STO double-zeta SCF wavefunctions for various configurations of the H₅O ₂ ⁺ associate have been computed. Results are discussed and compared with other authors' similar calculations on gaussian bases.An electrostatic picture of the monosolvation of H₃O⁺ is proposed as a fairly satisfactory one.