A comparison of finite basis set and finite difference Hartree—Fock calculations for the open-shell (X 2Σ+) molecules BaF and YbF

A comparison is made of the accuracy with which the total electronic energy can be calculated by using the finite basis set approach (the algebraic approximation) and the finite difference method in calculations employing the Hartree—Fock model for the open shell ground (X ²Σ⁺) states of the fluorides BaF and YbF. The convergence of the calculations carried out within the algebraic approximation is monitored by employing systematically constructed basis sets of increasing size. The difference between the finite basis set and finite difference Hartree—Fock energies is 2.6μE _h for BaF and 2.8μE _h for YbF. Dipole moments determined within the algebraic approximation are also compared with the corresponding finite difference expectation values.     

Do Dyson Orbitals resemble canonical Hartree–Fock orbitals?

ABSTRACT

Dyson orbitals are overlaps between states with N and N±1 electrons and provide conceptual links between transition probabilities of electron detachment or attachment, density matrices, total energies and general principles of chemical bonding. Canonical, Hartree–Fock orbitals are compared with Dyson orbitals obtained with electron–propagator calculations that retain all elements of the self–energy matrix, wherein all orbital–relaxation and electron–correlation corrections to Koopmans results reside. For valence ionization energies and electron affinities of representative closed–shell molecules, canonical, Hartree–Fock orbitals usually are excellent approximations to Dyson orbitals, although there are some notable cases where the resemblance is not as strong. Numerical relationships between pole strengths and the Koopmans contributions to Dyson orbitals also are inferred from the data.     

Ab initio SCF investigation of the core and inner valence electron binding and relaxation energies of the CH4, C2 H2 and C2 H6 molecules

Ab initio ΔSCF values of the electron binding and relaxation energies are presented for the four deepest molecular orbitals of methane, acetylene and the two symmetry forms of ethane. Special attention is paid to the inner valence molecular region with regard to the symmetry-constrained Hartree—Fock calculations. The symmetry-constrained effect for the 1t₂ outer valence orbital of methane has also been investigated. The phenomena due to the broken symmetry are discussed on the basis of the configuration-interaction procedure.     

On the orbital dependence of compact,weight-selected configuration interaction and coupled-cluster wave functions

We recently reported that very compact coupled-cluster wave functions may be generated by selecting the most important configurations, by weight, from the full coupled-cluster wave function. Here, we consider how the choice of orbitals may affect these wave functions in the case of the symmetric dissociation of H₂O. We employ unrestricted Hartree–Fock and complete-active-space self-consistent-field orbitals, as well as natural orbitals derived from a coupled-cluster singles and doubles wave function. For a given accuracy, some choices of orbitals can reduce the size of configuration interaction wave functions, but they have little effect on the weight-selected coupled-cluster wave functions.     

Construction of the two-electron contribution to the Fock matrix by numerical integration

A novel method to numerically calculate the Fock matrix is presented. The Coulomb operator is re-expressed as an integral identity, which is discretized. The discretization of the auxiliary t dimension separates the x, y, and z dependencies transforming the two-electron Coulomb integrals of Gaussian-type orbitals (GTO) to a linear sum of products of two-dimensional integrals. The s-type integrals are calculated analytically and integrals of the higher angular-momentum functions are obtained using recursion formulae. The contributions to the two-body Coulomb integrals obtained for each discrete t value can be evaluated independently. The two-body Fock matrix elements can be integrated numerically, using common sets of quadrature points and weights. The aim is to calculate Fock matrices of enough accuracy for electronic structure calculations. Preliminary calculations indicate that it is possible to achieve an overall accuracy of at least 10^?12 E _h using the numerical approach.     

Visualization of deficiencies in approximate molecular wave functions: the local orbital energy function for the matrix Hartree—Fock model

The local orbital energy function is used to assess the quality of approximate Hartree-Fock orbitals obtained by invoking the algebraic approximation and using a finite basis set expansion. Systematic sequences of distributed universal even-tempered basis sets of spherical-harmonic Gaussian-type functions are used to generate orbitals for which the corresponding total Hartree-Fock energy approaches the 1 μE_h level of accuracy. A pilot study of the behaviour of the local energy function is made for the hydrogenic atom described by a sequence of even-tempered Gaussian basis sets. The results of prototype calculations for the Hartree-Fock ground state of the BF molecule at its equilibrium geometry are presented. Sequences of calculations which use atom-centred basis sets are investigated as well as sequences which also include bond centred functions. The effects of the bond centred functions on the local orbital energy function are analysed. The local orbital energy function is seen as a measure of the quality of calculations carried out within the matrix Hartree-Fock approximation which can be employed in cases where the corresponding finite difference Hartree-Fock results are not available.     

Numerical solution of the Sinanoǧlu equation using a multicentre radial-angular grid

ABSTRACT

The pair functions that minimise the correlation energy of second-order many-body perturbation (MP2) theory (the Hylleraas functional) are obtained as solutions to the corresponding Sinano?lu equation by expanding them on a six-dimensional, multicentre, radial-angular grid of two electrons. Cusps in the pair functions at the nuclei are described numerically accurately by the multicentre grid. A cusp in each singlet pair function at the coalescence of the two electrons is taken into account analytically by a correlation factor. With a grid of approximately 10,000 points per atom, the MP2 correlation energies for atoms and polyatomic molecules are obtained usually within 0.1 mE _h of the complete-basis-set results. The correlation factor, auxiliary basis functions, and a judicious choice of integration algorithms are all necessary to stabilise the grid-based MP2 and underlying Hartree–Fock (HF) calculations. The auxiliary basis set, in particular, largely restores the hermiticity and diagonal dominance of the Fock matrix as well as furnishes virtual orbitals used in a resolution-of-the-identity approximation to lower the dimension of some integrals. The results of the grid-based HF and MP2 calculations without a correlation factor are found to suffer from large, nonsystematic errors frequently.     

Calculations of ionization and excitation energies for SiH3 X (X = F,Cl, Br and I) using the discrete variational Xα method

The electronic structure of silyl halide SiH₃ X molecules are investigated using the discrete variational Xα method based on the Hartree—Fock—Slater model. Theoretical ionization and excitation energies are in very good agreement with the experimental results of UPS and UV spectra. The effects of Si 3d orbitals are found to be significant on the bondings and orbital energies.     

A local framework for calculating coupled cluster singles and doubles excitation energies (LoFEx-CCSD)

ABSTRACT

The recently developed Local Framework for calculating Excitation energies (LoFEx) is extended to the coupled cluster singles and doubles (CCSD) model. In the new scheme, a standard CCSD excitation energy calculation is carried out within a reduced excitation orbital space (XOS), which is composed of localised molecular orbitals and natural transition orbitals determined from time-dependent Hartree–Fock theory. The presented algorithm uses a series of reduced second-order approximate coupled cluster singles and doubles (CC2) calculations to optimise the XOS in a black-box manner. This ensures that the requested CCSD excitation energies have been determined to a predefined accuracy compared to a conventional CCSD calculation. We present numerical LoFEx-CCSD results for a set of medium-sized organic molecules, which illustrate the black-box nature of the approach and the computational savings obtained for transitions that are local compared to the size of the molecule. In fact, for such local transitions, the LoFEx-CCSD scheme can be applied to molecular systems where a conventional CCSD implementation is intractable.     

Using the local density approximation and the LYP,BLYP and B3LYP functionals within reference-state one-particle density-matrix theory

For closed-shell systems, the local density approximation (LDA) and the LYP, BLYP and B3LYP functionals are shown to be compatible with reference-state one-particle density-matrix theory, where this recently introduced formalism is based on Brueckner-orbital theory and an energy functional that includes exact exchange and a non-universal correlation-energy functional. The method is demonstrated to reduce to a density functional theory when the exchange-correlation energy-functional has a simplified form, i.e. its integrand contains only the coordinates of two electrons, say r ₁ and r ₂, and it has a Dirac delta function δ(r ₁ - r ₂ as a factor. Since Brueckner and Hartree–Fock orbitals are often very similar, any local exchange functional that works well with Hartree–Fock theory is a reasonable approximation with reference-state one-particle density-matrix theory. The LDA approximation is also a reasonable approximation. However, the Colle–Salvetti correlation-energy functional and the LYP variant are not ideal for the method, since these are universal functionals. Nevertheless, they appear to provide reasonable approximations. The B3LYP functional is derived using a linear combination of two functionals: one is the BLYP functional; the other uses exact exchange and a correlation-energy functional from the LDA.     

The electronic structure of molecules by a many-body approach III. Ionization potentials and one-electron properties of furan and thiophene

The valence ionization potentials (IP's) of furan and thiophene are studied by an ab initio many-body approach which includes the effects of electron correlation and reorganization beyond the Hartree—Fock approximation. For both molecules it is found that the ordering of the IP's as obtained in the Hartree—Fock approximation is correct. The assignment made for furan agrees with the ab initio calculation of Siegbahn, but it does not agree with the ordering proposed by Derrick et al. from their experimental investigations. For thiophene both the ordering of Derrick et al. and the one of Gelius et al. is shown to be incorrect concerning the position of the 1b₁(π) IP. For both molecules the first two IP's are due to the 1a₂(π) and the 2b₁(π) molecular orbitals. For furan four orbitals of σ-type symmetry are placed between the 2b₁ and the 1b₁ π-orbitals, for thiophene there is only one. Several one-electron properties are calculated in the one-particle approximation and compared with experimental and other theoretical data. The localized molecular orbitals are also discussed.     

Electron propagators based on generalised density operators

ABSTRACT

Electron binding energies and Dyson orbitals may be obtained from the poles and residues of the electron propagator. The Dyson quasiparticle equation provides a convenient route to computing this information. Systematic approximations to the latter equation's self-energy, wherein electron correlation and final-state orbital relaxation are described, may be expressed in terms of the elements of the superoperator Hamiltonian matrix. Perturbative methods of electron propagator theory in wide use are based on a reference determinant constructed with canonical, Hartree–Fock orbitals. Generalised matrix elements of the superoperator Hamiltonian that accommodate non-integer occupation numbers associated with general, orthogonal spin orbitals are presented for the first time. Non-Hermitian terms may be systematically eliminated with perturbative corrections to generalised reference density operators. The structure of self-energy approximations that are complete through second, third, fourth or fifth order is presented in terms of superoperator Hamiltonian matrix elements. The present extensions pertain when generalised, zeroth-order density operators expressed in terms of orthonormal spin orbitals are employed.     

Anatomy of molecular properties evaluated with explicitly correlated electronic wave functions

Static electric dipole and quadrupole moments were evaluated at the explicitly correlated second-order Møller–Plesset (MP2-F12) level for BH, CO, H₂O, and HF molecules. The electron correlation contributions to the multipole moments were further decomposed into the direct (unrelaxed) and indirect (orbital response) components; we found that both components are equally important for the conventional (MP2) contribution, whereas the F12 correction to these properties originates primarily from the orbital response effects. Finally, the direct contribution dominates in the perturbative Hartree–Fock basis set incompleteness (CABS singles) correction. Two basis set families were employed: the standard aug-cc-pVXZ series and the cc-pVXZ-F12 series designed specifically for the F12 methods. The aug-cc-pVXZ MP2-F12 multipole moments usually have smaller basis set errors than the cc-pVXZ-F12 counterparts, albeit their differences are small at the triple (X = T) and quadruple (X = Q) zeta level. With the MP2-F12 calculations, the basis set errors of dipole and quadrupole moments can be reduced to ～0.001 a.u., or roughly 0.1%, at the aug-cc-pVDZ and aug-cc-pVTZ levels, respectively.     

Experimental and theoretical investigations into the electronic structure of CF2Br2 by electron momentum spectroscopy

The binding energy spectra and electron momentum density distributions for the valence orbitals of CF₂Br₂ have been obtained by using electron momentum spectroscopy (EMS) at an impact energy of 1200 eV plus binding energy. The measured electron momentum profiles are compared with Hartree–Fock (HF) and density functional theory (DFT) calculations with different-sized basis sets. In general, the DFT-B3LYP calculation using the large basis sets of 6-311++G** and aug-cc-pVTZ fairly describe the experimental results. Moreover, the controversial orderings of the outer valence orbitals have been reassigned. The pole strength of the main ionization transition from the inner valence orbital of 1b₂ is determined.     

Orientation dependence of surface enhanced raman intensities: results from abinitio calculations

Time dependent Hartree—Fock methods have been used to calculate Raman intensifies for H₂ adsorbed onto a model lithium cluster as a function of the orientation of H₂ relative to the cluster surface. The intensity is found to be largest for perpendicular adsorption, dropping to a small value at an intermediate angle, and then rising to a second but smaller maximum for parallel adsorption. The results are interpreted using a model which correlates enhancement to lithium cluster orbital energy shifts induced by the static quadrupole field Of H₂.     

Systematic discrepancy of theoretical predictions of NMR chemical shifts for chlorinated aromatic carbons using the GIAO DFT method

Theoretical calculations of carbon-13 NMR chemical shifts using the gauge including atomic orbitals (GIAO) DFT approach with a moderately large set of basis functions usually yield quite satisfactory results. In the case of chlorinated aromatic carbons, however, abnormally large differences between experimental and calculated values have been noticed. This discrepancy has been proven not to be caused by improper referencing, or the basis set effect, and probably not by neglect of vibrational corrections. One of the possible sources of the chlorine effect could be the impact of relativistic phenomena on electrons moving about the chlorine nucleus. The second, probably more important factor is the influence of electron correlations, ignored in Hartree–Fock SCF and only partially included in DFT calculations. Surprisingly, however, the observed divergence has been significantly larger for DFT than for Hartree–Fock results. In the latter case the observed divergence between theoretical and experimental ¹³C NMR chemical shifts of chlorine-bonded carbons is systematic but rather small (3.4–4.4ppm).