Restricted hartree–fock method instability

The explicit forms of the spin density matrix variations which are responsible for the external instability in the restricted Hartree–Fock (RHF ) method are found. The RHF open shell instability matrix which guarantees the distorted wave function to belong to the same space as the initial one is derived for arbitrary spin.     

Coupled hartree–fock calculation of static and shielding factors for open shell atomic systems

In this paper a calculation of the static quadrupole polarizabilities and the shielding factors of the atoms and ions of the 2p open shell systems is presented. A fully coupled Hartree–Fock method using the Roothaan formalism is adopted. Calculations have been done for the ground as well as for some valence excited states. An antishielding effect is observed for all the atoms and ions under study.     

Numerical method for the open-shell restricted hartree–fock density-matrix direct calculation

A new computation procedure for direct calculation of the density matrix in the LCAO version of the restricted Hartree–Fock–Roothaan open-shell theory is analyzed. It is proved that the procedure is quadratically convergent and stable to the round-off errors independently of the Fock operator spectrum. The dependence of the limit matrix of the initial matrix is examined.     

New algorithm for high-order time-dependent hartree–fock theory for nonlinear optical properties

A new formalism and algorithm is developed for solving the general-order time-dependent Hartree–Fock (TDHF) problem. It is shown that for any order a generalization of the TDHF equations can be derived where all lower-order solutions constitute a constant term. This makes it very easy to obtain high-order solutions. As the space required for the mapping of density matrices to Fock matrices in a problem of a giver order is largely reduced, we can perform the most time-consuming steps within the core memory of the machine and easily manipulate vector products via optimum routines. The second hyperpolarizability γ is obtained from the secondorder TDHF solution via a 2n rule. The formalism also allows for expressing all terms in the equation diagrammatically, which provides additional physical insight and a more systematic evaluation of terms. To illustrate the method, TDHF results are presented for trans-butadiene and carbon monoxide for several optical processes, including correlation corrections to their static hyperpolarizabilities obtained via coupled cluster (CCSD) and many-body perturbation theory. The hybrid TDHF/CCSD method provides excellent agreement with the DC–SHG experiments (χ||⁽²⁾ = 11.4 x 10^?32 esu/mol compared to 12.9 ± 11.4 × 10^?32 esu/mol and χ||⁽³⁾ = 149 compared to 144 ± 4 × 10^?39 esu/mol).     

An alternative computational strategy for direct space hartree–fock calculations on extended model chains

Based on previous analyses of slowly convergent exchange lattice sums entering the configuration space restricted Hartree–Fock–Roothaan scheme for chain systems, an alternative computational strategy is developed. Within the present formalism, the traditionally used finite Fourier transform of k-dependent LCAO density matrices are by-passed and an advantageous computational organization is obtained.     

Bond length alternation in cyclic polyenes. II. Unrestricted hartree–fock method

The problem of bond length alternation in cyclic polyene models as described by the Pariser–Parr–Pople π-electron Hamiltonian, together with an empirical quasi harmonic σ-core potential is investigated using the unrestricted Hartree–Fock wave function employing different spatial orbitals for different spins. It is shown that in contrast to the restricted Hartree–Fock method, which favors bond alternation in large cyclic polyenes, the unrestricted Hartree–Fock method stabilizes the symmetric structures with equidistant internuclear separation. An assessment of the amount of correlation error recovered by the unrestricted Hartree–Fock procedure is examined and the qualitatively different behavior of the cyclic polyene models when described by restricted and unrestricted Hartree–Fock wave functions is discussed from this viewpoint.     

Predictive chemical kinetics: Density functional and hartree–fock calculations on free-radial reaction transition states

Density functional theory (DFT ) using gradient-corrected «nonlocal» functionals is used to calculated the thermochemistry and barrier heights for several types of peroxyl radical isomerizations currently being studied by kineticists. The calculations are generally in good agreement with experimental data, where such data are available. An important exception is that the O—H bond strengths in hydroperoxides are all predicted to be too weak by about 7 kcal/mol. The calculated reaction barriers are a few kcal/mol lower than the experimental estimates, but comparable in accuracy to the much more computationally expensive second-order Møller–Plesset (MP 2) predictions. Various theoretical methods converge on a 42 ± 2 kcal/mole barrier for CH₃OO → H₂CO + OH. The DFT calculations can be used to predict reaction barriers in cases where no reliable experimental data are available. The effects of the choice of basis set and correlation functional are explored. Improvements needed to make these calculations most valuable to the chemical kineticist are discussed. © 1994 John Wiley & Sons, Inc.     

First order pair functions for the beryllium isoelectronic sequence

The first order pair equations have been solved for the four electron sequence Li^– to Ne⁶⁺. A variational perturbation procedure is blended with numerical methods to generate a method for computing pair functions which is both accurate and efficient with respect to time of computation.The variation in the pair correlation energies as a function of atomic number is discussed and the asymptotic behaviour ofE ²(l) asl ^–4 andl ^–6 for singlet and triplet states respectively is corroborated. An estimate is made for the electron affinity of lithium.

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Zusammenfassung Die Paargleichungen 1. Ordnung wurden für die Reihe der Ionen mit 4 Elektronen von Li^– bis Ne⁶⁺ gelöst. Ein Variations-Störungsverfahren wird mit numerischen Methoden verknüpft, um eine Methode zur Berechnung von Paarfunktionen zu erhalten, die genau ist und möglichst geringe Rechenzeiten benötigt. Die Variation der Paarkorrelationsenergien als Funktion der Atomnummer wird diskutiert und das asymptotische Verhalten vonE ²(l) wiel ^–4 undl ^–6 für Singulet- und Tripletzustände wird bestätigt. Für Lithium wird die Elektronenaffinität abgeschätzt.

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Résumé Résolution des équations de paires au premier ordre pour la série à 4 électrons: Li^– à Ne⁶⁺. Un procédé de perturbation variationnel est lié à des méthodes numériques pour engendrer une méthode de calcul des fonctions de paires simultanément précise et économique. La variation des énergies de corrélation de paires en fonction du nombre atomique est discutée ainsi que le comportement asymptotique deE ²(l) par rapport àl ^–4 etl ^–6 pour les états singlet et triplet respectivement.

     

Calculations on the helium isoelectronic sequence using Coulomb Green's function variables

Calculations on the ground states of the helium isoelectronic series are carried out using variational wavefunctions of the form ψ (x,y), in which x and y are the combinations r₁ + r₂ ± r₁₂ occurring in the Coulomb Green's function. The results for helium are the most accurate to date using a two-variable wavefunction accounting for 71.5% of the correlation energy.     

Scaling in the S and P states of the helium isoelectronic sequence

Mean values of r ₁ ^r and r _r ¹² for the ground and several excited states of the helium isoelectronic sequence are used to demonstrate that a simple scaling which superimposes the distribution function f(r ₁₂) as a function of the atomic number leads to a similar result for the electron density distribution D(r₁). On the basis of a screening interpretation of the scaling parameter , it is concluded that screening is greater in the singlet than the triplet state of a particular configuration, that screening is greater in the P states than the corresponding S states, and that the screening approaches the limiting value of 1 for the highly excited states. The perturbation expansions of Scherr and Knight are used to determine the limiting value of when Z and the relationship between the scaling parameter and the scale factor, chosen so that a trial wave function satisfies the virial theorem, is discussed. A brief discussion of the scaling of the Coulomb hole is presented.     

Sur la theorie quantique des equations de hartree–fock dependant du temps. IV. Expression matricielle du tenseur d'hyperpolarisabilité

Following a paper drawing up a matrix form of the time-dependent Hartree–Fock equations which allow the calculation of the nth harmonic generation, we establish the matrix form of the hyperpolarizability tensor. In case of linear polarization, we show that, if the ground-state orbitals are real, so are the perturbed ones. This involves an easy expression of the hyperpolarizability tensor.     

The Coulomb hole in the 23S state of the helium isoelectronic sequence

The pair distribution functions evaluated for the 2³S state of the helium isoelectronic sequence from the Hart and Herzberg correlated wave functions and those corresponding to the Hartree-Fock approximation are used to determine the shape of the corresponding Coulomb holes. As a consequence of a discontinuity in the Hartree-Fock solution between He and Li⁺, the Coulomb hole has a different shape for He than for Li⁺ and the other isoelectronic ions.     

Fermi and Coulomb correlations in the 21 S state of the helium isoelectronic sequence

The Fermi and Coulomb holes of the 2¹ S state of the helium isoelectronic sequence are investigated. Several interesting differences between the results obtained and those which might be expected on the basis of the corresponding 2³ S state are pointed out and discussed.     

Bounds to atomic properties: Electric polarizabilities of lithium isoelectronic sequence ions

Using the optimal wave functions obtained by means of the superposition of correlated configurations method, elaborated previously, static and dynamic polarizabilities and rigorous lower bounds to them have been calculated for the ground states of three-electron atoms in the lithium isoelectronic sequence (Li I –C IV ). The results can be treated with considerable confidence, especially for higher members of the sequence. Furthermore, the calculated rigorous lower bounds for static polarizabilities enable us to rule out several theoretical and experimental values obtained previously. The results obtained by us are then employed to evaluate the leading unknown terms in the Z^?1-type expansion for the polarizability, extending thereby our treatment to much higher members of the isoelectronic sequence.     

Kramers' restricted hartree—fock method for polyatomic molecules using ab initio relativistic effective core potentials with spin—orbit operators

A two-component Kramers' restricted Hartree–Fock method (KRHF) has been developed for the polyatomic molecules with closed shell configurations. The present KRHF program utilizes the relativistic effective core potentials with spin–orbit operators at the Hartree–Fock (HF) level and produces molecular spinors obeying the double group symmetry. The KRHF program enables the variational calculation of spin–orbit interactions at the HF level. KRHF calculations have been performed for the HX, X₂, XY(X, Y = I, Br), and CH₃I molecules. It is demonstrated that the orbital energies from KRHF calculations are useful for the interpretation of spin-orbit splittings in photoelectron spectra. In all molecules studied, bond lengths are only slightly expanded, harmonic vibrational frequencies are reduced, and bond energies are significantly decreased by the spin–orbit interactions.     

Improved Hylleraas calculations for ground state energies of lithium ISO–electronic sequence

The ²S ground state of lithium iso–electronic sequence is calculated by the use of Hylleraas-type wave functions. A 92 term one-spin wave function was used for lithium atom calculations. The energy obtained was ?7.478031 a.u. as compared with the previous best value of ?7.478025 a.u. calculated by Larsson. In addition, improved energies for Z = 4 to 8 were calculated by the use of 60 term wave functions. This work thus provides the lowest ab initio ground state energies for lithium sequence to date.     

Correlation energies for He isoelectronic sequence with Z=2-116 from four-component relativistic configuration interactions

The relativistic correlation energies (CEs) for the He isoelectronic sequence from 2He to 116Uuh were investigated using configuration-interaction (CI) calculations. We used a large universal-type Gaussian basis set, which gives accurate Dirac-Fock total energies for the ions under consideration. In contrast to nonrelativistic CEs, the relativistic CEs decrease monotonically with increasing nuclear charge, but the p-, d-, and f-partial CEs have a hump like the relativistic Hylleraas CI.     

Accurate nonrelativistic energies for 2Pe states of the Li isoelectronic series

Highly accurate upper bounds for several ²P^e states of the Li isoelectronic series obtained by extensive Hylleraas-Cl calculations are given. The best value for the 2²P^e state (1s2p²) of Li is −5.2137392₀ au. The evaluation of the occurring integrals is given explicitly. Additionally, we present some expectation values and isotope energies of the Li isoelectronic series. © 1997 John Wiley & Sons, Inc.