Interatomic interactions in momentum space. Momentum density and kinetic energy in chemical bonding

On the basis of the virial theorem for a uniform scaling process of a polyatomic system, the total energy and its gradient are quantitatively related with the behavior of the electron density in momentum space through the kinetic energy of the system. For attractive and repulsive interactions, the behavior of the momentum density distribution and its effect on the stabilization energy and the interatomic force are examined. Some guiding principles are deduced for their interrelation. The results are used to clarify the role of kinetic energy in chemical bonding. Possible energy partitioning in this approach is also mentioned.     

On the local representation of the electronic momentum operator in atomic systems

The local quantum theory is applied to the study of the momentum operator in atomic systems. Consequently, a quantum-based local momentum expression in terms of the single-electron density is determined. The limiting values of this function correctly obey two fundamental theorems: Kato's cusp condition and the Hoffmann-Ostenhof and Hoffmann-Ostenhof exponential decay. The local momentum also depicts the electron shell structure in atoms as given by its local maxima and inflection points. The integration of the electron density in a shell gives electron populations that are in agreement with the ones expected from the Periodic Table of the elements. The shell structure obtained is in agreement with the higher level of theory computations, which include the Kohn-Sham kinetic energy density. The average of the local kinetic energy associated with the local momentum is the Weizsacker kinetic energy. In conclusion, the local representation of the momentum operator provides relevant information about the electronic properties of the atom at any distance from the nucleus.     

Hartree-Fock energy partitioning in terms of Hirshfeld atoms.

A Hirshfeld decomposition scheme of the Hartree-Fock total molecular energy into atomic energies is presented. The calculations are performed by direct numerical integration and the results are compared for a set of 28 molecules containing different kinds of atoms. The calculated atomic energies show a strong dependency on changes of atomic electron population and hybridization. Linear correlations are found between the energy and the population for H, these being related to the electronegativity of this atom and to the external potential created by the remaining atoms. The proposed energy partitioning scheme appears to be useful for studies such as proton acidity, the anomeric effect and group transferability, and allows atomic virial ratios to be obtained. Finally, the atomic potential energies are found to mimic trends based on exact expressions as well as trends displayed by molecular quantities, thus lending credibility to the partitioning scheme used.     

Theoretical investigations of the momentum densities for molecular hydrogen

A method previously used to solve the Hartree-Fock or the MC SCF molecular equation in momentum space obtained by applying a Dirac transform to the corresponding equations in position space is used to determine numerical wavefunctions directly in p-space. Theoretical radial momentum wavefunctions, total momentum density and density difference (molecule—isolated atoms) maps are compared when going from separated atoms to molecule (taking into account or not configuration interaction effects) and are examined in detail in terms of electronic momentum distributions in chemical bonding. Some comparisons with binary (e,2e) experimental data are performed. An attempt is made to explain the increase of accuracy compared with previous results.     

The prediction of atomic kinetic energies from coordinates of surrounding atoms using kriging machine learning

A novel design of a next-generation force field considers not only the electronic inter-atomic energy but also intra-atomic energy. This strategy promises a faithful mapping between the force field and the quantum mechanics that underpins it. Quantum chemical topology provides an energy partitioning in which atoms have well-defined electronic kinetic energies, and we are interested in capturing how they respond to changes in the positions of surrounding atoms. A machine learning method called kriging successfully creates models from a training set of molecular configurations that can then be used to predict the atomic kinetic energies occurring in previously unseen molecular configurations. We present a proof-of-concept based on four molecules of increasing complexity (methanol, N-methylacetamide, glycine and triglycine). We test how well the atomic kinetic energies can be modelled with respect to training set size, molecule size and elemental composition. For all atoms tested, the mean atomic kinetic energy errors fall below 1.5 kJ mol^?1, and far below this in most cases. This represents errors all under 0.5 % and thus the kinetic energies are well modelled using the kriging method, even when using modest-to-small training set sizes.     

Kinetic energy decomposition scheme based on information theory

We proposed a novel kinetic energy decomposition analysis based on information theory. Since the Hirshfeld partitioning for electron densities can be formulated in terms of Kullback–Leibler information deficiency in information theory, a similar partitioning for kinetic energy densities was newly proposed. The numerical assessments confirm that the current kinetic energy decomposition scheme provides reasonable chemical pictures for ionic and covalent molecules, and can also estimate atomic energies using a correction with viral ratios.     

Analytic correlated wave functions in momentum space and related properties of helium like ions

We obtain analytic correlated wave functions in momentum space as the Fourier transform of correlated wave functions which are able to incorporate almost all of the correlation energy for the ground-state of two-electron atoms. Then we study the atomic momentum-density, the Compton profile and the elastic and inelastic scattering factors for this kind of wave functions. The scattering factors are also compared with the ones provided by a more accurate correlated wave function. All the calculations can be analytically performed, provided the correlated wave function in position space has been determined.     

Atomic shells from the electron localizability in momentum space

The electron localizability indicator in momentum space is proposed as a functional of the same‐spin momentum pair density. This functional yields a discrete distribution of values, which are proportional to the charge needed to form a fixed very small fraction of a same‐spin electron pair. It resolves all atomic shells for the examined atoms (Li–Kr) with reasonable occupation numbers, especially in the valence region. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2006     

Fast density matrix-based partitioning of the energy over the atoms in a molecule consistent with the Hirshfeld-I partitioning of the electron density

For the Hirshfeld-I atom in the molecule (AIM) model, associated single-atom energies and interaction energies at the Hartree-Fock level are efficiently determined in one-electron Hilbert space. In contrast to most other approaches, the energy terms are fully consistent with the partitioning of the underlying one-electron density matrix (1DM). Starting from the Hirshfeld-I AIM model for the electron density, the molecular 1DM is partitioned with a previously introduced double-atom scheme (Vanfleteren et al., J Chem Phys 2010, 132, 164111). Single-atom density matrices are constructed from the atomic and bond contributions of the double-atom scheme. As the Hartree-Fock energy can be expressed solely in terms of the 1DM, the partitioning of the latter over the AIM naturally leads to a corresponding partitioning of the Hartree-Fock energy. When the size of the molecule or the molecular basis set does not grow too large, the method shows considerable computational advantages compared with other approaches that require cumbersome numerical integration of the molecular energy integrals weighted by atomic weight functions.     

Stockholder projector analysis: a Hilbert-space partitioning of the molecular one-electron density matrix with orthogonal projectors

A previously introduced partitioning of the molecular one-electron density matrix over atoms and bonds [D. Vanfleteren et al., J. Chem. Phys. 133, 231103 (2010)] is investigated in detail. Orthogonal projection operators are used to define atomic subspaces, as in Natural Population Analysis. The orthogonal projection operators are constructed with a recursive scheme. These operators are chemically relevant and obey a stockholder principle, familiar from the Hirshfeld-I partitioning of the electron density. The stockholder principle is extended to density matrices, where the orthogonal projectors are considered to be atomic fractions of the summed contributions. All calculations are performed as matrix manipulations in one-electron Hilbert space. Mathematical proofs and numerical evidence concerning this recursive scheme are provided in the present paper. The advantages associated with the use of these stockholder projection operators are examined with respect to covalent bond orders, bond polarization, and transferability.     

Communication: Hilbert-space partitioning of the molecular one-electron density matrix with orthogonal projectors

A double-atom partitioning of the molecular one-electron density matrix is used to describe atoms and bonds. All calculations are performed in Hilbert space. The concept of atomic weight functions (familiar from Hirshfeld analysis of the electron density) is extended to atomic weight matrices. These are constructed to be orthogonal projection operators on atomic subspaces, which has significant advantages in the interpretation of the bond contributions. In close analogy to the iterative Hirshfeld procedure, self-consistency is built in at the level of atomic charges and occupancies. The method is applied to a test set of about 67 molecules, representing various types of chemical binding. A close correlation is observed between the atomic charges and the Hirshfeld-I atomic charges.     

C-Cl bond fission dynamics and angular momentum recoupling in the 235 nm photodissociation of allyl chloride

The photodissociation dynamics of allyl chloride at 235 nm producing atomic Cl((2)P(J);J=1/2,3/2) fragments is investigated using a two-dimensional photofragment velocity ion imaging technique. Detection of the Cl((2)P(1/2)) and Cl((2)P(3/2)) products by [2+1] resonance enhanced multiphoton ionization shows that primary C-Cl bond fission of allyl chloride generates 66.8% Cl((2)P(3/2)) and 33.2% Cl((2)P(1/2)). The Cl((2)P(3/2)) fragments evidenced a bimodal translational energy distribution with a relative weight of low kinetic energy Cl((2)P(3/2))/high kinetic energy Cl((2)P(3/2)) of 0.097/0.903. The minor dissociation channel for C-Cl bond fission, producing low kinetic energy chlorine atoms, formed only chlorine atoms in the Cl((2)P(3/2)) spin-orbit state. The dominant C-Cl bond fission channel, attributed to an electronic predissociation that results in high kinetic energy Cl atoms, produced both Cl((2)P(1/2)) and Cl((2)P(3/2)) atomic fragments. The relative branching for this dissociation channel is Cl((2)P(1/2))/[Cl((2)P(1/2))+Cl((2)P(3/2))]=35.5%. The average fraction of available energy imparted into product recoil for the high kinetic energy products was found to be 59%, in qualitative agreement with that predicted by a rigid radical impulsive model. Both the spin-orbit ground and excited chlorine atom angular distributions were close to isotropic. We compare the observed Cl((2)P(1/2))/[Cl((2)P(1/2))+Cl((2)P(3/2))] ratio produced in the electronic predissociation channel of allyl chloride with a prior study of the chlorine atom spin-orbit states produced from HCl photodissociation, concluding that angular momentum recoupling in the exit channel at long interatomic distance determines the chlorine atom spin-orbit branching.     

Toward a physical understanding of electron-sharing two-center bonds. I. General aspects

In 1916, Lewis and Kossel laid the empirical ground for the electronic theory of valence, whose quantum theoretical foundation was uncovered only slowly. We can now base the classification of the various traditional chemical bond types in a threefold manner on the one- and two-electron terms of the quantum-physical Hamiltonian (kinetic, atomic core attraction, electron repulsion). Bond formation is explained by splitting up the real process into two physical steps: (i) interaction of undeformed atoms and (ii) relaxation of this nonstationary system. We aim at a flexible bond energy partitioning scheme that can avoid cancellation of large terms of opposite sign. The driving force of covalent bonding is a lowering of the quantum kinetic energy density by sharing. The driving force of heteropolar bonding is a lowering of potential energy density by charge rearrangement in the valence shell. Although both mechanisms are quantum mechanical in nature, we can easily visualize them, since they are of one-electron type. They are however tempered by two-electron correlations. The richness of chemistry, owing to the diversity of atomic cores and valence shells, becomes intuitively understandable with the help of effective core pseudopotentials for the valence shells. Common conceptual difficulties in understanding chemical bonds arise from quantum kinematic aspects as well as from paradoxical though classical relaxation phenomena. On this conceptual basis, a dozen different bond types in diatomic molecules will be analyzed in the following article. We can therefore examine common features as well as specific differences of various bonding mechanisms.     

Performance of the density matrix functional theory in the quantum theory of atoms in molecules

The generalization to arbitrary molecular geometries of the energetic partitioning provided by the atomic virial theorem of the quantum theory of atoms in molecules (QTAIM) leads to an exact and chemically intuitive energy partitioning scheme, the interacting quantum atoms (IQA) approach, that depends on the availability of second-order reduced density matrices (2-RDMs). This work explores the performance of this approach in particular and of the QTAIM in general with approximate 2-RDMs obtained from the density matrix functional theory (DMFT), which rests on the natural expansion (natural orbitals and their corresponding occupation numbers) of the first-order reduced density matrix (1-RDM). A number of these functionals have been implemented in the promolden code and used to perform QTAIM and IQA analyses on several representative molecules and model chemical reactions. Total energies, covalent intra- and interbasin exchange-correlation interactions, as well as localization and delocalization indices have been determined with these functionals from 1-RDMs obtained at different levels of theory. Results are compared to the values computed from the exact 2-RDMs, whenever possible.     

Energy partitioning for "fuzzy" atoms

The total energy of a molecule is presented as a sum of one- and two-atomic energy components in terms of "fuzzy" atoms, i.e., such divisions of the three-dimensional physical space into atomic regions in which the regions assigned to the individual atoms have no sharp boundaries but exhibit a continuous transition from one to another. By proper definitions the energy components are on the chemical energy scale. The method is realized by using Becke's integration scheme and weight function permitting very effective numerical integrations.     

Role of angular momentum conservation in unimolecular translational energy release: validity of the orbiting transition state theory

The translational kinetic energy release distribution (KERD) for the halogen loss reaction of the bromobenzene and iodobenzene cations has been reinvestigated on the microsecond time scale. Two necessary conditions of validity of the orbiting transition state theory (OTST) for the calculation of kinetic energy release distributions (KERDs) have been formulated. One of them examines the central ion-induced dipole potential approximation. As a second criterion, an adiabatic parameter is derived. The lower the released translational energy and the total angular momentum, the larger the reduced mass, the rotational constant of the molecular fragment, and the polarizability of the released atom, the more valid is the OTST. Only the low-energy dissociation of the iodobenzene ion (E approximately 0.45 eV, where E is the internal energy above the reaction threshold) is found to fulfill the criteria of validity of the OTST. The constraints that act on the dissociation dynamics have been studied by the maximum entropy method. Calculations of entropy deficiencies (which measure the deviation from a microcanonical distribution) show that the pair of fragments does not sample the whole of the phase space that is compatible with the mere specification of the internal energy. The major constraint that results from conservation of angular momentum is related to a reduction of the dimensionality of the dynamics of the translational motion to a two-dimensional space. A second and minor constraint that affects the KERD leads to a suppression of small translational releases, i.e., accounts for threshold behavior. At high internal energies, the effects of curvature of the reaction path and of angular momentum conservation are intricately intermeddled and it is not possible to specify the share of each effect.     

Core molecular orbital contribution to N2O isomerization as studied using theoretical electron momentum spectroscopy

Core molecular orbital contribution to the electronic structure of N2O isomers has been studied using quantum mechanical density functional theory combined with a plane wave impulse approximation method. Momentum distributions of wave functions for inner shell molecular orbitals of the linear NNO, cyclic and linear NON isomers of N2O are calculated through the (e, 2e) differential cross sections in momentum space. This is possible because this momentum distribution is directly proportional to the modulus squared of the momentum space wave function for the molecular orbital in question. While the momentum distributions of the NNO and cyclic N2O isomers demonstrate strong atomic orbital characteristics in their core space, the outer core molecular orbitals of the linear NON isomer exhibit configuration interactions between them and the valence molecular orbitals. It is suggested that the frozen core approximation breaks down in the prediction of the electronic structure of such an isomer. Core molecular orbital contributions to the electronic structure can alter the order of total energies of the isomers and lead to incorrect conclusions of the stability among the isomers. As a result, full electron calculations should be employed in the study of N2O isomerization.     

Looking at orbitals in the laboratory: The experimental investigation of molecular wavefunctions and binding energies by electron momentum spectroscopy

The experimental technique of electron momentum spectroscopy (EMS ) (i.e., binary (e, 2e) spectroscopy) is discussed together with typical examples of its applications over the past decade in the area of experimental quantum chemistry. Results interpreted within the framework of the plane wave impulse and the target Hartree—Fock approximations provide direct measurements of, spherically averaged, orbital electron momentum distributions. Results for a variety of atoms and small molecules are compared with calculations using a range of Fourier transformed SCF position space wavefunctions of varying sophistication. Measured momentum distributions (MD ) provide a “direct” view of orbitals. In addition to offering a sensitive experimental diagnostic for semiempirical molecular wavefunctions, the MD's provide a chemically significant, additional experimental constraint to the usual variational optimization of wavefunctions. The measured MD's clearly reflect well known characteristics of various chemical and physical properties. It appears that EMS and momentum space chemistry offer the promise of supplementary perspectives and new vistas in quantum chemistry, as suggested by Coulson more than 40 years ago. Binding energy spectra in the inner valence region reveal, in many cases, a major breakdown of the simple MO model for ionization in accord with the predictions of many-body calculations. Results are considered for atomic targets, including H and the noble gases. The measured momentum distribution for H₂ is also compared with results from Compton scattering. Results for H₂ and H are combined to provide a direct experimental assessment of the bond density in H₂, which is compared with calculations. The behavior of the outer valence MD ''s for small row two and row three hydride molecules such as H₂O and H₂S, NH₃, HF, and HCl are consistent with well known differences in chemical and physical behavior such as ligand-donor activity and hydrogen bonding. MD measurements for the outermost valence orbitals of HF, H₂O and NH₃ show significant differences from those calculated using even very high-quality wavefunctions. Measurements of MD's for outer σ_g orbitals of small polyatomic molecules such as CO₂, COS, CS₂, and CF₄ show clear evidence of mixed s and p character. It is apparent that EMS is a sensitive probe of details of electronic structure and electron motion in atoms and molecules.     

Toward a physical understanding of electron-sharing two-center bonds. II. Pseudo-potential based analysis of diatomic molecules

A dozen homo- and heteropolar bonds in a series of diatomic molecules are analyzed by an energy partitioning based on the concepts developed in part I of this series (J Comput Chem 28:411–422, 2007). The different bond types are characterized by various physical contributions to the bond energy, namely classical potential interaction, quantum–mechanical orbital interference, and bond stabilizations by atomic configurational promotion, radial deformation, and angular polarization. The effects of the atomic cores are accounted for by means of pseudo-potential operators. Different atomic cores cause specific bond differences. The various bonding mechanisms can be characterized by several parameters. They describe the quantum–mechanical reduction in the ‘electronic kinetic energy pressure’ due to delocalization (sharing interference) and the increase in the ‘molecular potential energy pull’ due to overlap of atomic electron densities with adjacent atoms in the molecule. In addition, there are one- and two-electron relaxations whose distinctive features depend on the nature of the cores.     

Structure of the electron momentum density of atomic systems

The present paper addresses the controversial problem on the nonmonotonic behavior of the spherically-averaged momentum density γ(p) observed previously for some ground-state atoms based on the Roothaan-Hartree-Fock (RHF) wave functions of Clementi and Roetti. Highly accurate RHF wave functions of Koga et al. are used to study the existence of extrema in the momentum density γ(p) of all the neutral atoms from hydrogen to xenon. Three groups of atoms are clearly identified according to the nonmonotonicity parameter μ, whose value is either equal to, larger, or smaller than unity. Additionally, it is found that the function p^?α γ(p) is (i) monotonically decreasing from the origin for α ≥ 0.75, (ii) convex for α ≥ 1.35, and (iii) logarithmically convex for α ≥ 3.64 for all the neutral atoms with nuclear charges Z = 1–54. Finally, these monotonicity properties are applied to derive simple yet general inequalities which involve three momentum moments 〈p ^t≥. These inequalities not only generalize similar inequalities reported so far but also allow us to correlate some fundamental atomic quantities, such as the electron-electron repulsion energy and the peak height of Compton profile, in a simple manner.