Hartree-Fock Limit Research Articles

Average electron radii in many-electron atoms.

In many-electron atoms, the average electron radius r represents the mean distance of a single electron from the nucleus when all the interelectronic interactions are averaged. If the electron-electron interaction is explicitly considered, the average radius r splits into two different radii, inner radius r(<) and outer radius r(>). For the 102 atoms He through Lr in their ground states, the radii r(<) and r(>) are systematically examined at the Hartree-Fock limit level. The effect of electron correlations on r(<) and r(>) is also discussed for the He atom and its isoelectronic ions.

Coulomb hole in N2, CO and O2 deduced from X-ray scattering cross-sections

Accurate total (elastic and inelastic) X-ray scattering cross-sections of N2, CO and O2 were measured by the use of the energy dispersive method up to a momentum transfer of q = 12 au. The radial electron pair distribution function P(r 12) was extracted from the cross-sections. The Coulomb hole, defined as the difference between the exact P(r 12) and the corresponding function evaluated at the Hartree–Fock limit, has been derived from experimental data for the first time. Comparison of multi-reference configuration interaction (MRCI) and averaged quadratic coupled cluster (MR-AQCC) calculations indicates substantial shortcomings of MRCI due to the lack of size extensivity. The overall agreement with experiment is good but some differences between the theoretical and experimental results remain.

Hartree-Fock dynamical electron-correlation effects in C60 after laser excitation.

In the static Hartree-Fock limit, the on-site electron-electron interaction has no effect on the electronic properties of C60. But upon the laser excitation, the dynamical correlation effect appears. The time-dependent Hartree-Fock simulation through the Hubbard model shows that such an effect originates from the charge fluctuation, but an increase in the on-site electron-electron interaction suppresses the charge fluctuation and reduces the absorbed energy and the bond structure distortion.

Analytic theory of ground-state properties of a three-dimensional electron gas with arbitrary spin polarization

We present an analytic theory of the spin-resolved pair distribution functions $g_{\sigma\sigma'}(r)$ and the ground-state energy of an electron gas with an arbitrary degree of spin polarization. We first use the Hohenberg-Kohn variational principle and the von Weizs\"{a}cker-Herring ideal kinetic energy functional to derive a zero-energy scattering Schr\"{o}dinger equation for $\sqrt{g_{\sigma\sigma'}(r)}$. The solution of this equation is implemented within a Fermi-hypernetted-chain approximation which embodies the Hartree-Fock limit and is shown to satisfy an important set of sum rules. We present numerical results for the ground-state energy at selected values of the spin polarization and for $g_{\sigma\sigma'}(r)$ in both a paramagnetic and a fully spin-polarized electron gas, in comparison with the available data from Quantum Monte Carlo studies over a wide range of electron density.

Rationalization of the optical rotatory power of chiral molecules into atomic terms: a study of N 2 H 4

We applied a strategy to assign the individual contributions that atoms make to the optical rotation angle and, more generally, to the molecular chirality. The method resolves the optical rotatory power tensor into atomic contributions employing the formalism of the acceleration gauge for the electric dipole and the torque formalism for the magnetic dipolar moment. The gross atomic isotropic contributions have been evaluated for nitrogen and hydrogen in hydrazine, employing Gaussian basis sets of very good quality, in order to achieve the Hartree–Fock limit.

Sum rules related to third-order properties: a numerical check

Abstract Although electrodynamics is formally invariant in a gauge transformation, the values of some physical quantities, e.g., magnetic properties, depend on the approximation employed to calculate them. The conditions for gauge independence of third-rank tensor properties that describe the response of a molecule in the presence of three perturbations, that is, external electric and magnetic field, and intramolecular nuclear magnetic dipoles, are discussed. The relationships for invariance of the physical properties to a gauge translation are exactly the same as the constraints for charge conservation. They are expressed in terms of second-rank response properties, namely electric polarizabilities and electric shielding at the nuclei. An extended numerical test has been carried out to determine the Hartree–Fock limit for a series of quantities entering the gauge-invariance sum rules.

Accurate electric multipole moment, static polarizability and hyperpolarizability derivatives for N2

We report accurate values of the electric moments, static polarizabilities, hyperpolarizabilities and their respective derivatives for N2. Our values have been extracted from finite-field Møller–Pleset perturbation theory and coupled cluster calculations performed with carefully designed basis sets. A large [15s12p9d7f] basis set consisting of 290 CGTF is expected to provide reference self-consistent-field values of near-Hartree–Fock quality for all properties. The Hartree–Fock limit for the mean hyperpolarizability is estimated at γ̄=715±4e4a04Eh−3 at the experimental bond length Re=2.074 32a0. Accurate estimates of the electron correlation effects were obtained with a [10s7p6d4f] basis set. Our best values are Θ=−1.1258ea02 for the quadrupole and Φ=−6.75ea04 for the hexadecapole moment, ᾱ=11.7709 and Δα=4.6074e2a02Eh−1 for the mean and the anisotropy of the dipole polarizability, C̄=41.63e2a04Eh−1 for the mean quadrupole polarizability and γ̄=927e4a04Eh−3 for the dipole hyperpolarizability. The latter value is quite close to Shelton’s experimental estimate of 917±5e4a04Eh−3 [D. P. Shelton, Phys. Rev. A 42, 2578 (1990)]. The R dependence of all properties has been calculated with a [7s5p4d2f] basis set. At the CCSD(T) level of theory the dipole polarizability varies around Re as ᾱ(R)/e2a02Eh−1=11.8483+6.1758(R−Re)+0.9191(R−Re)2−0.8212(R−Re)3−0.0006(R−Re)4, Δα(R)/e2a02Eh−1=4.6032+7.0301(R−Re)+1.9340(R−Re)2−0.5708(R−Re)3+0.1949(R−Re)4. For the Cartesian components and the mean of γαβγδ, (dγzzzz/dR)e=1398, (dγxxxx/dR)e=867, (dγxxzz/dR)e=317, and (dγ̄/dR)e=994e4a03Eh−3. For the quadrupole polarizability Cαβ,γδ, we report (dCzz,zz/dR)e=19.20, (dCxz,xz/dR)e=16.55, (dCxx,xx/dR)e=10.20, and (dC̄/dR)e=23.31e2a03Eh−1. At the MP2 level of theory the components of the dipole–octopole polarizability (Eα,βγδ) and the mean dipole–dipole–octopole hyperpolarizability B̄ we have obtained (dEz,zzz/dR)e=36.71, (dEx,xxx/dR)e=−12.94e2a03Eh−1, and (dB̄/dR)e=−108e3a03Eh−2. In comparison with some other 14-electron systems, N2 appears to be less (hyper)polarizable than most, as near the Hartree–Fock limit we observe ᾱ(N2)&amp;lt;ᾱ(CO)&amp;lt;ᾱ(HCN)&amp;lt;ᾱ(BF)&amp;lt;ᾱ(HCCH) and γ̄(N2)&amp;lt;γ̄(CO)&amp;lt;γ̄(HCN)&amp;lt;γ̄(HCCH)&amp;lt;γ̄(BF).

Electron Momentum Spectroscopy of the Frontier Orbitals ofChlorodifluoromethane

We report on the first measurement of the electron momentumdistributions of the three outermost valence orbitals forchlorodifluoromethane (CHF2Cl) by binary (e,2e) electronmomentum spectroscopy. The experimental data are compared withHartree-Fock and density functional theory (DFT) calculationsemploying 6-31G, 6-311++G* * and AUG-cc-pVQZ basis sets. Forthe summed momentum distribution of 8a' + 5a'' + 7a' orbitals, theDFT/AUG-cc-pVQZ calculation gives the best fit. A very large anddiffuse basis set, AUG-cc-pVQZ, is employed in the calculations toapproach the Hartree-Fock limit of the basis set, but theimprovement of the calculation quality is little in comparison withthat calculated with the 6-311++G* * basis set. This indicatesthat the 6-311++G* * basis set is nearly saturated for thecalculations of these three orbitals of CHF2Cl, and it isunnecessary to employ a larger basis set in the calculations.

Nuclear mass corrections for atoms and ions

To take the nuclear motion contribution into account, we traditionally evaluate the normal mass ε nm and specific mass (mass polarization) ε sm corrections separately and add them to the total electronic energy of atoms and ions calculated in the fixed-nucleus approximation. We point out that the sum of these mass corrections ε mass= ε nm+ ε sm is immediately and rigorously obtained from the electron-pair moments in momentum space. The Hartree–Fock limit values of the mass correction sum ε mass are reported for the 102 neutral atoms He through Lr, singly charged 53 cations Li + through Cs +, and 43 stable anions H − through I − in their experimental ground states.

Orbital electron densities of ethane: Comparison of electron momentum spectroscopy measurements with near Hartree–Fock limit and density functional theory calculations

Electron density distributions in momentum space of the valence orbitals of ethane (C2H6) are measured by electron momentum spectroscopy (EMS) in a noncoplanar symmetric geometry. The impact energy was 1200 eV plus binding energy and energy resolution of the EMS spectrometer was 0.95 eV. The measured experimental momentum distributions of the valence orbitals are compared with Hartree–Fock and density functional theory (DFT) calculations. The shapes of the experimental momentum distributions are generally quite well described by both the Hartree–Fock and DFT calculations when large and diffuse basis sets are used. A strong “turn up” of the experimental cross section is observed for the HOMO 1eg orbital in the low momentum region, compared with the theoretical calculations. The pole strengths for the main ionization peaks in the inner-valence region are estimated.

On the oscillator strength sums S(±1) of atoms and ions

Using rigorous relations discovered recently, the Hartree–Fock limit values of the dipole oscillator strength sums S(−1) and S(+1) are obtained for the 102 neutral atoms He through Lr, singly charged 53 cations Li+ through Cs+, and 43 stable anions H− through I− in their experimental ground states. The present results for the ionic species are the first compilation of the sums. For heavy neutral atoms, the literature S(+1) values are found to be insufficiently accurate.

Valence orbital electron momentum distributions for dimethylsulfide: comparison of EMS measurements with near Hartree–Fock limit and density functional theory calculations

The binding energy spectra and orbital electron momentum density distributions of the complete valence shell of (CH 3) 2S have been measured by electron momentum spectroscopy at a total energy of 1200 eV+binding energy. Binding energies for the three inner valence (6a 1, 3b 2 and 5a 1) orbitals which have not been well characterized by photoelectron spectroscopy have been assigned in the present work. The extensive structure observed in the high energy region from 30 to 55 eV is attributed to satellite peaks from the inner valence orbitals. Measured electron momentum profiles for each of the valence orbitals are compared with calculations using the target Hartree–Fock approximation and SCF wavefunctions with a range of basis sets up to the large, saturated and diffuse aug-cc-pV5Z basis set. The observed momentum profiles are also compared with DFT momentum profiles calculated using the target Kohn–Sham approximation with the B3LYP and B3PW91 gradient corrected functionals. Individual experimental and calculated momentum profiles are presented for the three outermost orbitals (3b 1, 8a 1 and 5b 2) while the summed experimental data and calculations are presented for the four closely-spaced 1a 2, 4b 2, 7a 1 and 2b 1 higher binding energy orbitals in the outer valence region. Experimental and theoretical momentum profiles for the three inner valence orbitals are also presented. In general, both the Hartree–Fock and DFT calculations with high-level basis sets give good descriptions of the experimental electron momentum profiles.

Adapted Gaussian basis sets for atoms from Li through Xe generated with the generator coordinate Hartree-Fock method

The generator coordinate Hartree-Fock method is used to generate adapted Gaussian basis sets for the atoms from Li (Z=3) through Xe (Z=54). In this method the Griffin-Hill-Wheeler-Hartree-Fock equations are integrated through the integral discretization technique. The wave functions generated in this work are compared with the widely used Roothaan-Hartree-Fock wave functions of Clementi and Roetti (1974), and with other basis sets reported in the literature. For all atoms studied, the errors in our total energy values relatively to the numerical Hartree-Fock limits are always less than 7.426 mhartree.

Electron densities for the outer valence orbitals of pyridine: comparison of EMS measurements with near Hartree–Fock limit and density functional theory calculations

Momentum distributions of the outer valence orbitals of the aromatic molecule pyridine (C 5H 5N), measured by electron momentum spectroscopy (EMS), are compared with Hartree–Fock (HF) and density functional theory (DFT) calculations. The experiment was performed using a multichannel EMS spectrometer at a total energy of 1200 eV plus the binding energy and in symmetric non-coplanar kinematics. Binding energy spectra have been measured in the energy range of 4–45 eV at the azimuthal angles φ=0° and φ=8°, and in the range of 4–24 eV over a range of azimuthal angles from 0° to ±30°. Theoretical momentum profiles are calculated in the plane wave impulse approximation using the target HF approximation, and also with DFT using the target Kohn–Sham approximation with the B3LYP and B3PW91 gradient corrected functionals. Basis sets range from STO-3G to aug-cc-pVQZ. The shapes of the experimental momentum profiles, and thus the frontier orbital electron behaviour, are generally quite well reproduced by both the HF and the DFT calculations of the delocalised (canonical) molecular orbitals when using the larger diffuse basis sets. It is also found that the calculated localised molecular orbitals (i.e. valence bond hybrid orbitals involving resonance forms) completely fail to describe the frontier orbital electron behaviour observed in the EMS experiments for pyridine.

Vibrational spectra and scaled quantum-mechanical molecular force fields

A number of questions connected with the use of scaled quantum-mechanical force fields in interpreting molecular vibrational spectra are considered briefly.The Pulay method of scaling (congruent transformation of the force constant matrix) is noted to be applicable in the case where the relative accuracies of the determination of diagonal and off-diagonal quantum-mechanical force constants are approximately equal. This requirement is satisfied for a quantum-mechanical force field determined close to the Hartree–Fock limit. Then it is possible to carry out its correction, preserving to a maximum the features inherent to the molecule under investigation.Several examples of predicting the vibrational spectra of isotopomers, rotational isomers, and related compounds are given. Unlike using force fields obtained by the traditional method of solving the inverse vibrational problem and the additive scheme for transferring the force constants, scaled quantum-mechanical force fields make it possible to perform well-grounded theoretical vibrational analyses of all the related compounds of a molecule.

Overlap repulsion with L�wdin's pairing theorem

Using Löwdin's pairing theorem one can obtain an explicit expression for the overlap repulsion of two “unperturbed” closed-shell molecules at the single determinant level. After some regrouping, the interaction energy can be presented as a sum of terms of different physical meaning, each of which is expressed explicitly in terms of the (paired) molecular orbitals of the interacting molecules. The interaction energy can be decomposed into the following terms: the “naive” electrostatic interaction corresponding to undisturbed charge distributions of the two molecules; the Hartree–Fock exchange calculated neglecting the orbital overlap; “finite basis” terms originating from the deviation of the individual molecular orbitals (MOs) from the Hartree–Fock limit; overlap effects modifying the intramolecular energies as well as the electrostatic and exchange interactions; “true” overlap effects containing intermolecular overlap integrals and intermolecular charge distributions in the one- or two-electron integrals. © 2001 John Wiley & Sons, Inc. Int J Quant Chem 82: 53–59, 2001

Electron–electron coalescence and counterbalance functions for atoms

For many-electron atoms, spherically averaged electron–electron coalescence h0(R) and counterbalance d0(u) functions are studied which, respectively, represent the probability densities that any electron pairs with zero interelectronic distance are located at a radius R from the nucleus and that any electron pairs with zero center-of-mass radius have a relative distance u. For the exact and Hartree–Fock (HF) wave functions, cusp relations h0′(0)/h0(0)=−4Z and d0′(0)/d0(0)=−2Z are derived theoretically, where the prime denotes the first derivative and Z is nuclear charge. At the Hartree–Fock limit level, both functions h0(R) and d0(u) are found to be monotonically decreasing with a single maximum at R=0 or u=0 for all the 102 atoms He through Lr. The long-range asymptotic behavior of the coalescence and counterbalance functions is governed in general by the orbital energy of the highest occupied atomic orbital.

Orbital imaging for the valence shell of sulphur dioxide: comparison of EMS measurements with near Hartree–Fock limit and density functional theory

The momentum distributions of the valence orbitals of sulphur dioxide have been studied by electron momentum spectroscopy (EMS), Hartree–Fock (HF) and density functional theory (DFT) calculations. The experiments were performed using an energy dispersive multichannel EMS spectrometer at an impact energy of 1200 eV plus the binding energy and in symmetric non-coplanar kinematics. The valence-shell binding energy spectra have been measured for all valence orbitals over the energy range 8–60 eV. Calculated synthetic binding energy spectra derived from many-body Green’s function and DFT calculations are compared with experiment. In the inner valence region, strong splitting of the 3b 2 and 5a 1 ionization is observed due to final state electron correlation effects. The measured momentum profiles of the valence orbitals are compared with HF calculations at the level of the target HF approximation, and with DFT calculations using B3LYP functionals at the level of the target Kohn–Sham approximation, with basis sets ranging from minimum (STO-3G) to large (AUG-cc-pV5Z(-h)). Generally, the shapes of the experimental momentum profiles are well reproduced by HF and DFT calculations using large diffuse basis sets. However, small discrepancies still exist between all theoretical treatments and experiment in the low momentum region for the (4b 2+7a 1+2b 1) and 6a 1 orbitals.

The basis set convergence of the Hartree-Fock energy for H 3 + , Li 2 and N 2

By using completely optimized basis functions it is shown that the convergence of the Hartree–Fock energy for the H3+, Li2 and N2 molecules is significantly better described by exponential behavior than by inverse power dependence. This is the case both with respect to the number of basis functions of a given type and with respect to the highest angular momentum function included. The Hartree–Fock limit for H3+ is estimated to be −1.300372125 hartree.

Static hyperpolarizability of the water dimer and the interaction hyperpolarizability of two water molecules

We report an extensive investigation of the electric dipole moment (μα), static polarizability (ααβ), and hyperpolarizability (βαβγ and γαβγδ) of the water dimer. Calculations were performed at both rigid and relaxed monomer geometries. At the rigid monomer geometry (RIMG), a very large [9s6p6d4f/6s5p3d2 f] basis set consisting of 370 Gaussian-type functions is thought to provide self-consistent field (SCF) values very close to the Hartree–Fock limit for all properties: total dipole moment μ=1.0706ea0, mean and anisotropy of the dipole polarizability ᾱ=16.98 and Δα=2.69e2a02Eh−1, first hyperpolarizability (in the direction of the dipole moment vector) β̄=−2.9e3a03Eh−2, and mean second dipole hyperpolarizability γ̄=1906e4a04Eh−3. Very large electron correlation effects are observed for the hyperpolarizability. At the CCSD(T) level, coupled-cluster theory with single, double and perturbatively linked triple excitations, our best values are μ=1.0204ea0, ᾱ=19.54, and Δα=3.06e2a02Eh−1, β̄=−6.5e3a03Eh−2, and γ̄=3669e4a04Eh−3. ᾱ((H2O)2) and γ̄((H2O)2) are not drastically different than twice the size of ᾱ(H2O) and γ̄(H2O). Our efforts to estimate the interaction properties of two water molecules in the dimer lead to the conclusion that ᾱinter and γ̄inter are rather small. Further calculations at a relaxed monomer geometry (REMG) corroborate this conclusion. We have fully explored basis set effects at all levels of theory and for all properties. We rely on a sequence of small-sized but sufficiently flexible basis sets in order to propose reliable computational strategies for the extension of electric property calculations to large water clusters.