Fourier analysis of the Compton profile: Atoms and molecules

Recent work has shown that the one-dimensional projection of the electron momentum density, the Compton profile, can be usefully interpreted as a position space quantity. This has led to an examination of B(r), the Fourier transform of the momentum density. A number of theoretical results relating to this new observable are given. The wave-mechanical representation with (natural) orbitals is employed, and this forms the basis for the subsequent analysis of B(r). The relationship of B(r) to overlap integrals and more generally to other electron density functions is considered. Atomic wavefunctions for krypton are used to illustrate the potential of this new approach to the analysis of momentum density data. General expressions are derived for atoms and molecules, and the radial and angular dependence of B(r) for various orbitals is displayed. The possibility of extracting accurate bond lengths from B(r) is assessed, and an example is given using some recent theoretical data for the fluorine molecule.     

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.     

Valence electronic structure of CH3Br and CH3I: Electron momentum distributions and separation energies

Bromomethane (CH₃Br) and iodomethane (CH₃I) have been studied by binary (e,2e) coincidence spectroscopy at 1200 eV using non-coplanar symmetric kinematics. Separation energy spectra have been determined in the energy range up to 47 eV at azimuthal angles of 0° and 8° for CH₃Br and 0° and 6° for CH₃I. The separation energy spectra and the electron momentum distributions measured for each of the valence orbitals are compared with theoretical predictions employing SCF wavefunctions and outer valence type and extended 2 ph-TDA Green function calculations. Electron density and momentum density maps have been calculated for all the valence orbitals using the SCF wavefunctions, and they are used to explain trends and contrasts in the electronic structure and bonding properties of these halomethanes in both position and momentum space.     

Computed static potentials for AHn molecules: a scattering-orintated form

A symmetry-adapted angular momentum factorization is reported for the density functions of non-linear molecules of the AH_n type with Z_A > 1, and for the static potentials generated by their ground state electronic configurations at equilibrium geometry. Results from one-centre expanded (OCE) wavefunctions are compared with those given by multicentre expansions and the rates of convergence for the electronic and nuclear terms are examined. The usefulness of such a factorization for dealing with electron-molecule scattering is discussed and its suitability in yielding molecule—molecule interactions pointed out for cases where the electron gas model is used to compute such potential surfaces.     

Orbital momentum distributions and binding energies for the complete valence shell of molecular chlorine by electron momentum spectroscopy

The complete valence shall binding energy spectrum (10–50 eV) of Cl₂ has been determined using electron momentum (binary (e,2e)) spectroscopy. The inner valence region, corresponding to 4σ_u and 4σ_g ionization, has been measured for the first time and shows extensive splitting of the ionization strength due to electron correlation effects. These measurements are compared with the results of many-body calculations using Green function and CI methods employing unpolarised as well as polarised wavefunctions. Momentum distributions, measured in both the outer and inner valence regions, are compared with calculations using a range of unpolarised and polarised wavefunctions. Computed orbital density maps in momentum and position space for oriented Cl₂ molecules are discussed in comparison with the measured and calculated spherically averaged momentum distributions.     

Outer valence orbital momentum distributions of carbon dioxide by binary (e,2e) spectroscopy

The outer valence orbital momentum distributions of CO₂ have been reinvestigated using a high momentum resolution (0.1 a_o^?1 fwhm) binary (e,2e) spectrometer operated at 1200 eV impact energy under the non-coplanar symmetric scattering condition. Generally good agreement of the measured momentum distributions with theoretical momentum distributions calculated using literature SCF double-zeta quality wavefunctions has been obtained for the 1π_g, (1π_u + 3σ_u) and 4σ_g orbitals. Although there is a reasonable agreement of the measured momentum distributions with earlier low momentum resolution (0.4 a_o^?1 fwhm) non-coplanar measurements at 400 eV impact energy reported by Cook and Brion, given the large differences in the momentum resolutions much more definitive results are obtained in the present study. In particular, the significantly higher momentum resolution clearly shows the mixed s-p character of the 4σ_g orbital. The present study also gives a much better agreement with theory in the case of the 4σ_g momentum distribution. For each orbital the calculated and where possible the experimentally determined spherically averaged momentum distributions are compared and contrasted with their respective two-dimensional momentum and position density maps. These together with three-dimensional surface plots at selected constant density values of the four outermost orbitals are used to provide a detailed comparison of momentum-space bonding and orbital properties with their more familiar position-space counterparts in the CO₂ triatomic molecule. The calculated momentum-space density contour maps of the core orbitals exhibit rather large density oscillations and the feasibility of future experiments is discussed.     

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.     

Valence electronic structure of CH3F and CH3Cl: electron momentum distributions and separation energies

Fluoromethane (CH₃F) has been studied by binary (e,2e) coincidence spectroscopy at 1200 eV using non-coplanar symmetric kinematics. Separation energy spectra have been determined in the energy range up to 50 eV at azimuthal angles of 0° and 9°. The separation energy spectra and electron momentum distributions measured for the valence orbitals of CH₃F and CH₃Cl are compared with the results of calculations employing SCF wavefunctions and outer valence as well as extended 2ph—TDA Green function methods. Electron density and momentum density maps have been generated for all valence orbitals of both molecules using the SCF wavefunctions and are used to explain differences in the bonding properties of the halomethanes investigated here.     

The self-interaction correction to the local spin density model: Effect on atomic momentum space properties

Perdew and Zunger showed that the exact energy density functional for the ground state is strictly self-interaction-free. However, the local spin density (LSD) approximation lacks this self-interaction correction (SIC). Perdew and Zunger studied the effect on coordinate-space properties of incorporating the SIC. In the present study we examine the effect of SIC on momentum space properties, viz. the electron momentum distribution, ⊂p″⊃ values, electron momentum densities and the Compton profiles for atoms He to Ar. A remarkable improvement is seen in all the momentum space properties of the SIC LSD model over the LSD model when compared to their near Hartree-Fock counterparts.     

Binary (e,2e) spectroscopic study of carbonyl sulfide and the momentum-space chemistry of CO2, CS2 and OCS

The valence-shell binding energy spectra (8–44 eV) and molecular orbital momentum distributions of OCS have been studied by non-coplanar symmetric binary (e,2e) spectroscopy. Existing theoretical binding energy spectra calculated using the many-body 2ph-TDA Green's function (GF) method and using the symmetry-adapted cluster (SAC) on method are compared with the experiment. Intense many-body structure in the measured and calculated binding energy spectra indicates the general breakdown of the independent particle ionization picture. Experimental momentum distributions are compared with those calculated using ab initio SCF wavefunctions of minimal basis set quality and of near Hartree—Fock quality. Excellent agreement between the experimental momentum distributions and those calculated by the near Hartree—Fock wavefunction is obtained for the three innermost valence orbitals: 8σ, 7σ and 6σ. The correct order of the close lying outer-valence 2π and 9σ orbitals is unambiguously identified from the shapes of the measured momentum distributions. Momentum and position contour density maps computed from theoretical wavefunctions of near Hartree—Fock quality are used to interpret the shapes and atomic characters of the observed momentum distributions. The momentum densities of the outermost-valence antibonding π orbitals and of the outermost-valence bonding σ orbitals of the linear triatomic group: CO₂, CS₂ and OCS are compared respectively with each other. The associated chemical trends are discussed within the existing framework of momentum-space chemical principles.     

On the use of NDO approximate wavefunctions in the evaluation of momentum density and radial momentum density distributions in polyatomic molecules

The use of single determinantal approximate molecular wavefunctions of the LCAO MO NDO type for the calculation of the momentum densityρ(p) and the radial momentum density distributionJ(p) is discussed. In each case, these expressions should be orientationally invariant and the momentum density should be normalized. Combining these two requirements, it is shown that only two approximations are physically significant:

1. NDO wavefunctions are used andρ(p) andI(p) are approximated respectively up to an INDO and a CNDO level;
2. Overlap integrals are explicitly taken into account when solving the Roothaan SCF equations or deorthogonalized NDO functions are employed, together with the unapproximated expressions.

     

Molecular orbital momentum distributions and binding energies for nitric oxide

Binding energy spectra of the valence electrons of the open shell molecule NO have been obtained up to 55 eV at azimuthal angles of 0° and 7° using binary (e, 2e) spectroscopy at an impact energy of 1200 eV. The momentum distribution has been obtained for the least tightly bound (unpaired) electron, removal of which leads to formation of the X ¹Σ⁺ ground state of NO⁺. Momentum distributions have also been measured at 21.0 and 40.5 eV. The measured momentum distributions are compared with several literature wavefunctions of varying complexity. They are found to be in excellent agreement with those calculated using the natural spin orbital wavefunctions of Kouba and Ohrn.     

A momentum-space view of the chemical bond. I. The first-row Homonuclear diatomics

The electronic probability distribution in momentum space or electron momentum density (EMD) is studied in detail for the first-row homonuclear diatomics. The total density difference (molecule minus constituting atoms)is analyzed in terms of the separate orbital contributions. The nodal structure shown by the orbital EMD is characteristic for the various types of orbital (σ,σ*,=,=*), and is affected, by the amount of s-p hybridization. Directional and isotropic Compton profiles are used to study the bond-oscillation and bond-directional principles. The bond- directional principle does not hold for pe bonding. Spherically averaged EMD differences (SA Δ EMDs) are related to the changes in kinetic energy (ΔT) upon bond formation. The SA ΔEMDs and ΔT are rationalized by considering the different ranges of internuclear distance that are optimal for 2s-2s, 2p_o-2p_o and 2p_o-2p_o interaction. This leads to a reassessment of the role of the various orbitals in bonding complementing the picture based on orbital Hellmann- Feynman forces.     

Classification of reaction pathways via momentum–space and quantum molecular similarity measures

For four rearrangement reactions, we evaluate (i) values of the moments of momentum 〈pⁿ〉(−2⩽n⩽+1) for reactants, products and transition states (ii) position–space similarity indices between reactants, products and transition states using wavefunctions generated from both Hartree–Fock and density-functional theory (B3LYP). Both the momentum–space expectation values and the similarity measures can be used to distinguish Hammond and anti-Hammond behaviour. In position–space, two new parameters, δ and φ, are introduced to quantify the position and character of the transition state relative to the reactants and products.     

Effects of electron correlation and relativistic effects in X-ray diffraction

Atomic scattering factors for the first row transition metal atoms cobalt and nickel have been evaluated using non-relativistic and ‘relativistic-corrected’ configuration interaction wavefunctions in the |αLSM _LM_S〉 representation. Relaxing the constraints imposed by the Hartree-Fock model results in a very small reduction of the atomic scattering factor at small momentum transfer with a negligible change at higher momentum transfer. Better agreement with the relativistic Hartree-Fock atomic scattering factors at small momentum transfer is obtained when theLS-dependent relativistic effects are included in the Breit-Pauli approximation.     

Electron correlation effects in position and momentum space: the atoms Li through Ar

Electron correlation effects on the electronic structure of atoms were investigated by means of a variety of position and momentum space related properties such as radial one-electron densities and radial electron momentum densities, Compton profiles and radial electron pair distributions. The results were obtained from MR-SDCI wavefunctions utilizing very large basis sets and are discussed in a comparative manner, analysing characteristic features and trends.     

Xα local spin density energy calculations

Possible applications of density functional theory are examined with the intent of obtaining good-quality thermochemical information, such as atomization energies and enthalpies of formation. The local spin density (LSD) approximation is used. Exchange and correlation are treated in the Xα approach, where α is a variable parameter. The tentative hypothesis is that the Xα(LSD) method basically meets the required demands of accuracy, i.e. that the energy should be correct, if α is properly selected. A simple recipe for obtaining α is suggested by the idea that non-local exchangecorrelation contributions should change in a more or less regular fashion when atoms join to form a molecule, i.e. in a fashion largely dictated by the number and type of bonds that involve the individual atoms. Applications presented for organic and inorganic systems using 6–31G^** wavefunctions reveal a reasonable agreement, with a standard deviation of 4.4 kcal mol⁻¹, between calculated and experimental standard gas-phase enthalpies of formation.     

Bond-alternating Hückel-Möbius and related,twisted linear,cyclic and helical systems—their molecular orbitals,energies and phase correlation upon dissociation

Cyclic group formalism and screw symmetry operation are used to clarify and generalize the definition of Hückel and Möbius systems. It is shown that the Möbius ring system has half-integral pseudo-angular momentum similar to that of spin space, and that applications of Möbius electronics to chemical reactions have been based on truncated single-circle Möbius rings which have unique beginning and end (Sect. 2). This concept is illustrated by application to the [1, 7] antarafacial hydrogen shift (Appendix A and Figs. A1–A3). Definition of a Hüickel versus Möbius ring system for in-plane and out-of-plane π, δ and φ orbitals as well as the appropriate relative angle of twists are given (Sect. 2 and Table 1). Using the concept of the compatibility of the twist (screw) angle and rotation around a ring, we also derive the proper phase coherence and energy correlation between a parent cyclic (Hückel or Möbius) molecule and its dissociated linear fragments (Sect. 4). The concept of parentage in diabatic fragmentation is discussed. Forfinite, open, helical chain molecules, an exact periodic boundary condition based on the compatibility of twist angle and number of turns in a helical ring parent molecule is applied to derive their analytic wave functions (Sect. 5 and Table 2). Forbond-alternating “linear” and cyclic Hückel and Möbius systems we also derive the explicit LCAO-MO wavefunctions, energies, their degeneracies and their exact corresponding quantum members for even and odd atom systems at highest bonding and lowest antibonding levels (Sect. 3, Figs. 1–3). The corresponding wavefunctions and energies for uniform-bond systems are given for comparison and for completeness (Sect. 3).     

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.     

Electron correlation and the compton profile of atomic neon

The radial momentum distribution I_o(p) and the Compton profile J_o(q) are determined for atomic neon from several restrictid Hartree-Fock (RHF) wavefunctions and two configuration interaction (CI) wavefunctions. The CI functions are the well correlated (full“second-order”) function of Viers, Schaeffer and Harris, and the Ahlrichs-Hinze multi-configuration Hartree-Fock (MCHF) function which includes only L-shell correlation. It is found for this completely closed shell system that the effects of electron correlation are quite small. This contrasts with the results for systems such as Be(²S) and B(²P) where the semi-internal and internal correlation effects were responsible for significant discrepancies between the RHF and CI results. These results indicate that a wavefunction which carefully includes the semi-internal, orbital polarization, and internal correlations beyond the RHF wavefunction (i.e., a “first-order” or “charge-density” function), should account for the principal correlation effects on the Compton profiles and momentum distributions.