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.     

Electron momentum spectroscopy of the valence orbitals of H2O and D2O: Quantitative comparisons using Hartree—Fock limit and correlated wavefunctions

The large discrepancies found earlier between experimental measurements and calculations based on near Hartree—Fock wavefunctions for the valence orbital electron momentum distributions of H₂O are reinvestigated. New and improved electron momentum spectroscopy measurements for the valence orbitals of H₂O and D₂O, together with existing experimental data, have been placed on a common intensity scale using the binding energy spectra. Investigation of possible vibrational effects by means of new measurements of the momentum distributions of D₂O indicates no detectable differences with the H₂O results, within experimental error. A quantitative comparison of these experimental results with both the shapes and magnitudes of momentum distributions calculated in the PWIA and THFA approximations using new, very precise Hartree—Fock (single-configuration) wavefunctions is made. These wavefunctions, which include considerable polarization and which are effectively converged at the HF limit for total energy, dipole moment and momentum distribution permit establishment of basis set independence. The significant discrepancies between theory and experiment which still remain for the momentum distributions of the 1b₁, 3a₁ and 2a₁ orbitals at the THFA level are largely removed by CI calculations of the full ion—neutral overlap amplitude. These CI wavefunctions for the final ion and neutral ground states, generated from the accurate HF limit basis sets, recover up to 88% of the correlation energy. The present work clearly shows the need for adequate consideration of electron correlation effects in describing the low-momentum parts of the 1b₁, 3a₁ and 2a₁ electron distributions, a region which is of crucial importance in problems related to chemical bonding and reactivity. The high level of quantitative agreement obtained between experiment and calculations using sufficiently sophisticated wavefunctions provides support for the essential validity of the plane wave impulse approximation as used in the interpretation of EMS experiments on small molecules.     

The valence orbitals of NH3 by electron momentum spectroscopy: Quantitative comparisons using Hartree-Fock limit and correlated wavefunctions

The complete valence shell binding energy spectra and valence orbital electron momentum distributions for NH₃ have been measured by high-momentum-resolution electron momentum spectroscopy (EMS). The results are quantitatively compared with theoretical calculations using SCF wavefunctions ranging from DZ quality to a newly developed 126-GTO wavefunction essentially at the Hartree-Fock limit. The 3a₁ and to a lesser extent the 2a₁ valence orbital are not adequately described even at the Hartree-Fock limit with basis set saturation including diffuse functions. The differences between theory and experiment are largely resolved by ion-neutral overlap calculations using CI wavefunctions to incorporate the effects of electron correlation. The 126-G (CI) wavefunctions provide accurate calculation of a wide range of electronic properties of NH₃ and also give good quantitative prediction of the three valence orbital momentum distributions as well as a reasonable prediction of the many-body pole strength distribution observed in the (2a₁)⁻¹ inner valence binding energy spectrum. The present EMS results are compared with recent investigations of wavefunction tails by exterior electron distribution calculations and Penning ionization electron spectroscopy measurements reported by Ohno et al.     

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.     

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.     

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.     

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.     

Symmetric orthogonalisation in momentum space: A numerical study

Summary Symmetric orthogonalisation is favourable to perform in momentum space, as this article will show. We have used a model of a body centered cubic lattice with 1s- and 2s-Slater orbitals centered at each atom site. Computer programs have been written to calculate the eigenvalues of the overlap matrix which play an important role in constructing symmetrically orthogonalised wavefunctions.     

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.     

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.     

He原子基态波函数的实验检验

以作者所在实验室最近完成的He原子基态的电子动全港学实验结果为基础，对三类He原子基态波函数进行了分析与检验．结果表明，电子动量话学是获取电子波西数信息的有用手段．且实验结果与理论计算的联系文＊已报导了本实验室最近完成的He原子基态的电子动量话学实验结果．此实验的条件满足准自由碰撞的要求问，平面波冲量近似是适用的，因而实验测定的（e，Ze）反应的符合计数N可表为问本文自始至终采用原子单位（an），除非另有说明．在（1）中，C是比例常数，只与实验条件有关；PO，PI和马分别是人射电子和两个出射电子的动量，Th是…     

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.     

Variational cellular method for polyatomic molecules: SF6

This is the third paper on the cellular method for polyatomic systems. We show how to deal with nonspherical Coulomb potentials. We also show how to modify the variational expression for the energy eigenvalues so as to obtain a faster convergence in the angular momentum series for the wavefunctions. We apply both techniques to the self-consistent calculation of SF₆. Contrary to what we obtained in CH₄ and SiH₄, the cellular method cannot yield the correct equilibrium interatomic distance in the present case. The calculated ionization potentials are in the correct order but are all shifted by 2–3 eV. This shift is attributed to the wrong expression for exchange correlation.     

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.     

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.

     

Elastic scattering of electrons by H2 in the born approximation

This paper presents accurate computation of the differential cross sections for the elastic and vibrational excitation of ground state H₂ by fast electrons. The electronic matrix elements are evaluated at 20 internuclear separations and proper averaging over vibrational wavefunctions is carried out. The comparison to experiment yields excellent agreement. Different computational procedures appropriate for low and moderate values of momentum transfer are compared.     

The atomic form factor for vanadium: a treatment including correlation andLS-dependent relativistic effects

The atomic form factor for the ground state of vanadium is evaluated using non-relativistic and “relativistic-corrected” configuration interaction wavefunctions in the |LSM _L M _s 〉 representation. Relaxing the constraints imposed by the Hartree-Fock model results in a very small reduction of the atomic factor at small momentum transfer with a negligible change at higher momentum transfer. Better agreement with the relativistic Hartree-Fock atomic form factor at small momentum transfer is obtained when theLS-dependent relativistic effects are included in the Breit-Pauli approximation. The sensitivity of the atomic form factor to small changes in the magnitude of the expansion coefficients of the configurational functions is also discussed.     

Energy shift in (dtμ)e due to the finite size of the muonic molecular ion (dtμ)+

The energy shift due to the presence of the extended (dtμ)₁₁ pseudonucleus (in its first excited state with one unit of angular momentum) in the quasihydrogenlike system (dtμ)₁₁ e _1s is estimated using perturbation theory up to second order with two choices of zeroth order electron wavefunctions. The energy shift is found to be 0.50 meV using pure Coulomb electron wavefunctions and 0.58 meV using electron wavefunctions calculated with a potential modified to take partial account of the finite size of (dtμ)₁₁. In both cases, the perturbation Hamiltonian is expanded in multipoles, retaining terms up to and including octupole terms.     

Vibrational corrections to the charge and momentum densities of H2

Vibrational corrections to the charge and momentum densities of H₂ are calculated using the Wang wavefunction and various expressions for the dependence of the effective nuclear charge on the interatomic distance. Both harmonic oscillator and Morse potentials are employed for the vibrational wavefunctions, and excited as well as ground vibrational states are considered. The corrections to the momentum density, though considerably smaller than the charge density corrections, appear to be somewhat more sensitive qualitatively to the choice of both nuclear and electronic wavefunctions.     

A comparison of theoretical compton profiles for aromatic and non-aromatic molecules

This paper compares Compton profiles and electronic momentum distributions derived from minimal Slater orbital basis set SCF MO wavefunctions for the molecules benzene, fulvene, trimethylene cyclopropane, dimethylene cyclobutene, pyridine and borazine. Except for borazine the total molecular Compton profiles are expected to be experimentally indistinguishable. It is concluded that momentum space properties seem to be more dependent on electronegativity than on aromaticity.