Hartree-Fock Wave Functions Research Articles

Interaction of the NO 3pπ Rydberg state with Ar: Potential energy surfaces and spectroscopy

We present the experimental and simulated (2+1) REMPI spectrum of the C(2)Π state of the NO-Ar complex, in the vicinity of the 3p Rydberg state of NO. Two Rydberg states of NO are expected in this energy region: the C(2)Π (3pπ) and D(2)Σ(+) (3pσ) states, and we concentrate on the former here. When the C(2)Π (3pπ) state interacts with Ar at nonlinear orientations, the symmetry is lowered to C(s), splitting the degeneracy of the (2)Π state to yield C((2)A") and C((2)A') states. For these two states of NO-Ar, we calculate potential energy surfaces using second order Møller-Plesset perturbation theory, exploiting a procedure to converge the reference Hartree-Fock wavefunction to describe the excited states, the maximum overlap method. The bound rovibrational states obtained from the surfaces are used to simulate the electronic spectrum, which is in excellent agreement with experiment, providing assignments for the observed spectral lines from the calculated rovibrational wavefunctions.

Gauge origin independent calculations of molecular magnetisabilities in relativistic four-component theory

The use of magnetic-field dependent London atomic orbitals, also called gauge including atomic orbitals, is known to be an efficient choice for accurate non-relativistic calculations of magnetisabilities. In this work, the appropriate formulas were extended and implemented in the framework of the four-component relativistic linear response method at the self-consistent field single reference level. Benefits of employing the London atomic orbitals in relativistic calculations are illustrated with Hartree–Fock wave functions on the XF3 (X = N, P, As, Sb, Bi) series of molecules. Significantly better convergence of magnetisabilities with respect to the basis set size is observed compared to calculations employing a common gauge origin. In fact, it is mandatory to use London atomic orbitals unless you want to use ridiculously large basis sets. Relativistic effects on magnetisabilities are found to be quite small (<5%) for this particular set of molecules, but should be investigated on a larger set of molecules. We emphasise the breakdown of the connection between the paramagnetic contribution to magnetisabilities and rotational g tensors in the relativistic domain and discuss its origin. Finally, we visualise the magnetisability density which shows markedly atomic features evocative of Pascal’s rules.

Resonating Hartree–Fock Studies on the Electronic Structures of the Two-Dimensional Hole-Doped Hubbard Model

Electronic structures of the two-dimensional doped Hubbard model are investigated by using the resonating Hartree–Fock method. It is shown that doped holes form polarons in a wide range of doping regions relevant to superconductivity. The momentum distribution of electrons and the spin correlation function indicate that carriers change from polarons to conventional holes as we enhance hole doping. We will visualize quantum fluctuations in these electronic states by analyzing the structures of the Slater determinants generating the resonating Hartree–Fock wave functions.

Competition betweenT=1andT=0pairing inpf-shell nuclei withN=Z

The pairing correlation energy for two-nucleon configurations with the spin-parity and isospin of $J^\pi=0^+$, $T$=1 and $J^\pi=1^+$, $T$=0 are calculated with $T$=1 and $T$=0 pairing interactions, respectively. To this end, we consider the $(1f2p)$ shell model space, including single-particle angular momenta of $l=3$ and $l=1$. It is pointed out that a two-body matrix element of the spin-triplet $T$=0 pairing is weakened substantially for the $1f$ orbits, even though the pairing strength is much larger than that for the spin-singlet $T$=1 pairing interaction. In contrast, the spin-triplet pairing correlations overcome the spin-singlet pairing correlations for the $2p$ configuration, for which the spin-orbit splitting is smaller than that for the $1f$ configurations, if the strength for the T=0 pairing is larger than that for the T=1 pairing by 50% or more. Using the Hartree-Fock wave functions, it is also pointed out that the mismatch of proton and neutron radial wave functions is at most a few % level, even if the Fermi energies are largely different in the proton and neutron mean-field potentials. These results imply that the configuration with $J^\pi=0^+$, $T$=1 is likely in the ground state of odd-odd $pf$ shell nuclei even under the influence of the strong spin-triplet $T$=0 pairing, except at the middle of the $pf$ shell, in which the odd proton and neutron may occupy the $2p$ orbits. These results are consistent with the observed spin-parity $J^{\pi}=0^+$ for all odd-odd $pf$ shell nuclei except for $^{58}_{29}$Cu, which has $J^{\pi}=1^+$.

N-electron Slater determinants from nonunitary canonical transformations of fermion operators

Mean-field methods such as Hartree-Fock (HF) and Hartree-Fock-Bogoliubov (HFB) constitute the building blocks upon which more elaborate many-body theories are based. The HF and HFB wave functions are built out of independent quasiparticles resulting from a unitary linear canonical transformation of the elementary fermion operators. Here, we discuss the possibility of allowing the HF transformation to become nonunitary. The properties of such HF vacua are discussed, as well as the evaluation of matrix elements among such states. We use a simple ansatz to demonstrate that a nonunitary transformation brings additional flexibility that can be exploited in variational approximations to many-fermion wave functions. The action of projection operators on nonunitary-based HF states is also discussed and applied, in a variation-after-projection approach, to the one-dimensional Hubbard model with periodic boundary conditions.

Dielectronic recombination and resonant transfer excitation processes for helium-like krypton

The relativistic configuration interaction method is employed to calculate the dielectronic recombination (DR) cross sections of helium-like krypton via the 1s2lnl′ (n = 2, 3, …, 15) resonances. Then, the resonant transfer excitation (RTE) processes of Kr34+ colliding with H, He, H2, and CHx (x = 0−4) targets are investigated under the impulse approximation. The needed Compton profiles of targets are obtained from the Hartree—Fock wave functions. The RTE cross sections are strongly dependent on DR resonant energies and strengths, and the electron momentum distributions of the target. For H2 and H targets, the ratio of their RTE cross sections changes from 1.85 for the 1s2l2l′ to 1.88 for other resonances, which demonstrates the weak molecular effects on the Compton profiles of H2. For CHx (x = 0−4) targets, the main contribution to the RTE cross section comes from the carbon atom since carbon carries 6 electrons; as the number of hydrogen increases in CHx, the RTE cross section almost increases by the same value, displaying the strong separate atom character for the hydrogen. However, further comparison of the individual orbital contributions of C(2p, 2s, 1s) and CH4(1t2, 2a1, 1a1) to the RTE cross sections shows that the molecular effects induce differences of about 25.1%, 19.9%, and 0.2% between 2p−1t2, 2s−2a1, and 1s−1a1 orbitals, respectively.

Comparison of self-consistent field convergence acceleration techniques

The recently proposed ADIIS and LIST methods for accelerating self-consistent field (SCF) convergence are compared to the previously proposed energy-DIIS (EDIIS) + DIIS technique. We here show mathematically that the ADIIS functional is identical to EDIIS for Hartree-Fock wavefunctions. Convergence failures of EDIIS + DIIS reported in the literature are not reproduced with our codes. We also show that when correctly implemented, the EDIIS + DIIS method is generally better than the LIST methods, at least for the cases previously examined in the literature. We conclude that, among the family of DIIS methods, EDIIS + DIIS remains the method of choice for SCF convergence acceleration.

Lifetimes of excited levels for atomic silicon

The lifetimes of some excited levels for atomic silicon are calculated using the weakest bound electron potential model theory (WBEPMT) and the quantum defect orbital theory. In the WBEPMT framework, we have employed both numerical Coulomb Approximation wave functions and numerical non-relativistic Hartree–Fock wave functions for expectation values of radii. The obtained lifetime results have been compared with theoretical and experimental data in the literature.

The analysis of polarization effects on the interelectronic separations in the atoms and molecules of the G1 test set

AbstractWe have calculated the differences between kinetic, nuclear attraction, and two‐electron energies as well as the position intracules, P(u), for 56 molecules and 17 atoms contained in the G1 test set, using basis sets differing in their degree of polarization. The calculations were performed using an unrestricted Hartree–Fock wave function that was expanded using the 6‐31G, 6‐31G(d,p), 6‐31G(3d,3p), 6‐31G(3df,3pd), 6‐311G, 6‐311G(d,p), 6‐311G(3d,3p), 6‐311G(3df,3pd), 6‐311++G, 6‐311++G(d,p), 6‐311++G(3d,3p), and 6‐311++G(3df,3pd) basis sets. In addition to these Pople basis sets, the Dunning's cc‐pVDZ and cc‐pVTZ bases as well as the non‐polarized portions of these basis sets were used. We observe a consistent contraction of electron pairs (as measured by the distribution in P(u)) as the number of polarization functions is increased beyond zero in molecular systems. This contraction is related to an increase in the electron density in the internuclear bonding regions of molecules as the wave functions are polarized. We note that the increased flexibility in the polarized basis set allows for a more accurate description of the chemical bond and is energetically favorable due to an increased level of attraction to the nuclei (despite the obvious increase in electronic repulsion energy). Because of the increase in electron density in internal regions of the molecule (at the expense of electron density depletions in outer regions), the probability of observing electrons closer together (i.e., at smaller u) increases due to the relationship between P(u) and its Coulomb component, J(u). © 2012 Wiley Periodicals, Inc.

Oscillator strengths for Be I

The electric dipole oscillator strengths for lines between some singlet and triplet levels have been calculated using the weakest bound electron potential model theory and the quantum defect orbital theory for Be I. In the calculations both multiplet and fine structure transitions are studied. We employed both the numerical Coulomb approximation method and numerical non-relativistic Hartree–Fock wavefunctions for expectation values of radii. The necessary energy values have been taken from experimental energy data in the literature. The calculated oscillator strengths have been compared with available theoretical results. A good agreement with the results in the literature has been obtained.

Electron-induced proton knockout from neutron rich nuclei

We study the evolution of the (e,e'p) cross section on nuclei with increasing asymmetry between the number of neutrons and protons. The calculations are done within the framework of the nonrelativistic and relativistic distorted-wave impulse approximation. In the nonrelativistic model phenomenological Woods-Saxon and Hartree-Fock wave functions are used for the proton bound-state wave functions, in the relativistic model the wave functions are solutions of Dirac-Hartree equations. The models are first tested against experimental data on 40 Ca and 48Ca nuclei, and then they are applied to a set of spherical calcium isotopes.

Application of the weak-field asymptotic theory to the analysis of tunneling ionization of linear molecules

The recently developed weak-field asymptotic theory [Phys. Rev. A 84, 053423 (2011)] is applied to the analysis of tunneling ionization of a molecular ion (H${}_{2}^{+}$), several homonuclear (H${}_{2}$, N${}_{2}$, O${}_{2}$) and heteronuclear (CO, HF) diatomic molecules, and a linear triatomic molecule (CO${}_{2}$) in a static electric field. The dependence of the ionization rate on the angle between the molecular axis and the field is determined by a structure factor for the highest occupied molecular orbital. This factor is calculated using a virtually exact discrete variable representation wave function for H${}_{2}^{+}$, very accurate Hartree-Fock wave functions for the diatomics, and a Hartree-Fock quantum chemistry wave function for CO${}_{2}$. The structure factors are expanded in terms of standard functions and the associated structure coefficients, allowing the determination of the ionization rate for any orientation of the molecule with respect to the field, are tabulated. Our results, which are exact in the weak-field limit for H${}_{2}^{+}$ and, in addition, under the Hartree-Fock approximation for the diatomics, are compared with results from the recent literature.

Interatomic Coulombic decay of fixed-in-space neon dimers

The detailed theoretical and experimental analysis of the angular distributions of electrons from interatomic Coulombic decay (ICD) of the Ne dimer in the molecular frame is performed. In the initial state the doubly charged dimer ion has one $2s$ vacancy and one $2p$ vacancy on one atom. After the ICD process the neutral neon atom is ionized and the triply charged molecular ion dissociates into singly and doubly charged atomic ions, ${\text{Ne}}^{2+}(2{p}^{\ensuremath{-}2})+{\text{Ne}}^{+}(2{p}^{\ensuremath{-}1})$. From the coincident measurement of kinetic energy release (KER) of the ions and the ICD electron the decay channel can be identified unambiguously. The most detailed experimental data have been obtained for the singlet dicationic state Ne${}^{2+}(2{p}^{\ensuremath{-}2}){[}^{1}\phantom{\rule{-0.16em}{0ex}}D]$. Different KER energies correspond to different internuclear distances at which the ICD process takes place. In experiment the data have been presented for three regions of KER energies, and the corresponding calculations have been performed for three fixed internuclear distances. In calculations we imply that all the electrons in Ne${}_{2}$ to a good approximation are localized. However, we need to retain the molecular character of the dimer wave functions which opens the possibility for the ICD process. To do it, we calculate at first the Hartree-Fock ground state wave functions of the neutral Ne${}_{2}$ dimer using the standard procedure for homonuclear diatomic molecules corresponding to the ${D}_{\ensuremath{\infty}h}$ symmetry group. For the doubly charged ion Ne${}_{2}{}^{2+}$ with two vacancies on one atom the symmetry is lowered to ${C}_{\ensuremath{\infty}v}$, and we are looking now for the set of one-electron Hartree-Fock wave functions which are localized either on the left or on the right atom as a linear combination of symmetry-adopted wave functions. The theory correctly reproduces the experimental data and predicts the sharp variation of the angular distributions as a function of internuclear distance.

Polarized–unpolarized ground state of small polycyclic aromatic hydrocarbons

AbstractDo polyacenes, circumacenes, periacenes, nanographenes, and graphene nanoribbons show a spin polarized ground state? In this work, we present monodeterminantal (Hartree–Fock (HF) and density functional theory (DFT) types), and multideterminantal calculations (Møller–Plesset and Coupled Cluster), for several families of unsaturated organic molecules (n‐Acenes, n‐Periacenes and n‐Circumacenes). All HF calculations and many DFT show a spin‐polarized (antiferromagnetic) ground state, in agreement with previous calculations. Nevertheless, the multideterminantal calculations, carried out with perturbative and variational wavefunctions, show that the more stable state is obtained starting from the unpolarized HF wavefunction. The trend of the stabilization of wavefunctions (polarized or unpolarized) with respect to exchange and correlation potentials, and to the number of benzene rings, has been analyzed. A study of the spin (〈Ŝ2〉) and the spin density on the carbon atoms has also been carried out. © 2012 Wiley Periodicals, Inc.

Charge Model 5: An Extension of Hirshfeld Population Analysis for the Accurate Description of Molecular Interactions in Gaseous and Condensed Phases

We propose a novel approach to deriving partial atomic charges from population analysis. The new model, called Charge Model 5 (CM5), yields class IV partial atomic charges by mapping from those obtained by Hirshfeld population analysis of density functional electronic charge distributions. The CM5 model utilizes a single set of parameters derived by fitting to reference values of the gas-phase dipole moments of 614 molecular structures. An additional test set (not included in the CM5 parametrization) contained 107 singly charged ions with nonzero dipole moments, calculated from the accurate electronic charge density, with respect to the center of nuclear charges. The CM5 model is applicable to any charged or uncharged molecule composed of any element of the periodic table in the gas phase or in solution. The CM5 model predicts dipole moments for the tested molecules that are more accurate on average than those from the original Hirshfeld method or from many other popular schemes including atomic polar tensor and Löwdin, Mulliken, and natural population analyses. In addition, the CM5 charge model is essentially independent of a basis set. It can be used with larger basis sets, and thereby this model significantly improves on our previous charge models CMx (x = 1-4 or 4M) and other methods that are prone to basis set sensitivity. CM5 partial atomic charges are less conformationally dependent than those derived from electrostatic potentials. The CM5 model does not suffer from ill conditioning for buried atoms in larger molecules, as electrostatic fitting schemes sometimes do. The CM5 model can be used with any level of electronic structure theory (Hartree-Fock, post-Hartree-Fock, and other wave function correlated methods or density functional theory) as long as an accurate electronic charge distribution and a Hirshfeld analysis can be computed for that level of theory.

Reply to the ‘Comment on “Dynamic factors in the reactions between the magic cluster Al13− and HCl/HI”: A wavefunction instability problem.’ by Young-Kyu Han, Phys. Chem. Chem. Phys. 2012, DOI: 10.1039/C2CP23908F

The reaction between a metal cluster Al13− and an inorganic molecule HI involves charge transfer and density functional calculations has deficiency to estimate the activation barrier due to the self-interaction error. Coupled-cluster calculation with single and double excitation (CCSD) based on spin-unrestricted Hartree-Fock wavefunction can improve the calculated barrier and the choice of basis set is important for the calculation. Energy difference can be as large as 4.9 kcal mol−1 for the reaction by using aug-cc-pvdz and 6-31++G** basis sets respectively.

On the nature of electron correlation in C60

The ground state restricted Hartree Fock (RHF) wave function of C(60) is found to be unstable with respect to spin symmetry breaking, and further minimization leads to a significantly spin contaminated unrestricted Hartree Fock (UHF) solution (<S(2)> = 7.5, 9.6 for singlet and triplet, respectively). The nature of the symmetry breaking in C(60) relative to the radicaloid fullerene, C(36), is assessed by energy lowering of the UHF solution, <S(2)>, and the unpaired electron number. We conclude that the high value of each of these measures in C(60) is not attributable to strong correlation behavior as is the case for C(36). Instead, their origin is from the collective effect of relatively weak, global correlations present in the π space of both fullerenes. Second order perturbation (MP2) calculations of the singlet triplet gap are significantly more accurate with RHF orbitals than UHF orbitals, while orbital optimized opposite spin second order correlation (O2) performs even better.

Accurate Bond Energies of Hydrocarbons from Complete Basis Set Extrapolated Multi‐Reference Singles and Doubles Configuration Interaction

Quantum chemistry has become one of the most reliable tools for characterizing the thermochemical underpinnings of reactions, such as bond dissociation energies (BDEs). The accurate prediction of these particular properties (BDEs) are challenging for ab initio methods based on perturbative corrections or coupled cluster expansions of the single-determinant Hartree-Fock wave function: the processes of bond breaking and forming are inherently multi-configurational and require an accurate description of non-dynamical electron correlation. To this end, we present a systematic ab initio approach for computing BDEs that is based on three components: 1) multi-reference single and double excitation configuration interaction (MRSDCI) for the electronic energies; 2) a two-parameter scheme for extrapolating MRSDCI energies to the complete basis set limit; and 3) DFT-B3LYP calculations of minimum-energy structures and vibrational frequencies to account for zero point energy and thermal corrections. We validated our methodology against a set of reliable experimental BDE values of CC and CH bonds of hydrocarbons. The goal of chemical accuracy is achieved, on average, without applying any empirical corrections to the MRSDCI electronic energies. We then use this composite scheme to make predictions of BDEs in a large number of hydrocarbon molecules for which there are no experimental data, so as to provide needed thermochemical estimates for fuel molecules.

The Nature of the Binding of Au, Ag, and Pd to Benzene, Coronene, and Graphene: From Benchmark CCSD(T) Calculations to Plane-Wave DFT Calculations.

The adsorption of Ag, Au, and Pd atoms on benzene, coronene, and graphene has been studied using post Hartree–Fock wave function theory (CCSD(T), MP2) and density functional theory (M06-2X, DFT-D3, PBE, vdW-DF) methods. The CCSD(T) benchmark binding energies for benzene–M (M = Pd, Au, Ag) complexes are 19.7, 4.2, and 2.3 kcal/mol, respectively. We found that the nature of binding of the three metals is different: While silver binds predominantly through dispersion interactions, the binding of palladium has a covalent character, and the binding of gold involves a subtle combination of charge transfer and dispersion interactions as well as relativistic effects. We demonstrate that the CCSD(T) benchmark binding energies for benzene–M complexes can be reproduced in plane-wave density functional theory calculations by including a fraction of the exact exchange and a nonempirical van der Waals correction (EE+vdW). Applying the EE+vdW method, we obtained binding energies for the graphene–M (M = Pd, Au, Ag) complexes of 17.4, 5.6, and 4.3 kcal/mol, respectively. The trends in binding energies found for the benzene–M complexes correspond to those in coronene and graphene complexes. DFT methods that use empirical corrections to account for the effects of vdW interactions significantly overestimate binding energies in some of the studied systems.

Rainbow scattering under axial surface channeling from a KCl(001) surface

Fast He, Ne, Ar, and N atoms with projectile energies from 1 up to 60 keV are scattered under grazing polar angles of incidence from a flat and clean KCl(001) surface. For the scattering along low-index directions (axial surface channeling) we observe pronounced peaks in the angular distributions of scattered projectiles which can be attributed to rainbow scattering. From classical trajectory calculations based on universal and individual pair as well as density functional theory (DFT) potentials, we obtained corresponding rainbow angles for comparison with the experimental data. Fair agreement was found for DFT and individual pair potentials calculated from Hartree-Fock wave functions.