The restriction of spin contamination in unrestricted Hartree Fock wave functions

A method is developed to study in detail the contaminating spin components in UHF calculations for free radicals. Applications are made to those cases previously reported in which normal parametrisation of PPP UHF calculations lead to values of S ² which increase on annihilation of the contaminating quartet component.

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Zusammenfassung Es wird eine Methode entwickelt, um detaillierte Untersuchungen der bindenden Spinkomponenten in UHF-Rechnungen für freie Radikale zu ermöglichen. Es werden Anwendungen für solche Fälle, über die schon früher einmal berichtet wurde, durchgeführt, bei denen die normale Parametrisierung von PPP-UHF-Rechnungen zu Werten von S ² führte, welche die Vernichtung der bindenden Quartettkomponente anwachsen lassen.

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Résumé Développement d'une méthode pour étudier en détail les composantes de spin étrangères dans les calculs UHF des radicaux libres. On l'applique aux cas précédemment considérés dans lesquels des calculs UHF avec paramétrisation PPP conduisent à des valeurs de S² qui croissent lorsque l'on annule la composante quartet.

     

Analytical Hartree–Fock wave functions for the atoms Cs to Lr

 Analytical approximations to Hartree–Fock wave functions are constructed using Slater-type functions for the ground states of all 49 neutral atoms from Cs (Z=55) to Lr (Z=103). The current compilation is more extensive and more accurate than previous ones. The wave functions are available upon request from the authors or from the Web page http://www.unb.ca/chem/ajit/download.htm on the Internet. Received: 6 December 1999 / Accepted: 29 February 2000 / Published online: 12 May 2000     

Hartree–Fock High-Precision Analytical Functions of Open-Shell Atoms

The algorithm of high-precision optimization of basis functions suggested previously for calculating the analytical Hartree–Fock orbitals of closed-shell atoms is generalized to open-shell systems described by the Roothaan method (1960). Expressions for the first (free gradient) and second (Hesse matrix) derivatives of the system's energy with respect to the nonlinear parameters (orbital exponents) of the basis functions are derived in terms of density matrices for the filled and open shells. An algorithm is proposed for high-precision optimization of the nonlinear parameters using these equations based on Murtagh–Sargent and Newton minimization procedures. To illustrate the application of this algorithm, we give optimization of the basis sets of Slater type functions for atoms from the second row, as well as for Al, Si, P, K, Sc, and Fe atoms. The analytical Hartree–Fock orbitals giving nearly Hartree–Fock energies are calculated with a high degree of accuracy.     

The calculation of electron localization and delocalization indices at the Hartree–Fock, density functional and post-Hartree–Fock levels of theory

 Localization, λ(A), and delocalization indices, δ(A,B), as defined in the atoms in molecules theory, are a convenient tool for the analysis of molecular electronic structure from an electron-pair perspective. These indices can be calculated at any level of theory, provided that first- and second-order electron densities are available. In particular, calculations at the Hartree–Fock (HF) and configuration interaction (CI) levels have been previously reported for many molecules. However, λ(A) and δ(A,B) cannot be calculated exactly in the framework of Kohn–Sham (KS) density functional theory (DFT), where the electron-pair density is not defined. As a practical workaround, one can derive a HF-like electron-pair density from the KS orbitals and calculate approximate localization and delocalization indices at the DFT level. Recently, several calculations using this approach have been reported. Here we present HF, CI and approximate DFT calculations of λ(A) and δ(A,B) values for a number of molecules. Furthermore, we also perform approximate CI calculations using the HF formalism to obtain the electron-pair density. In general, the approximate DFT and CI results are closer to the HF results than to the CI ones. Indeed, the approximate calculations take into account Coulomb electron correlation effects on the first-order electron density but not on the electron-pair density. In summary, approximate DFT and CI localization and delocalization indices are easy to calculate and can be useful in the analysis of molecular electronic structure; however, one should take into account that this approximation increases systematically the delocalization between covalently bonded atoms, with respect to the exact CI results. Received: 13 February 2002 / Accepted: 24 April 2002 / Published online: 18 June 2002     

Hartree Fock instabilities of sulfur-nitrogen ring systems: S2N2

The Hartree-Fock instablities of S₂N₂ are reported and compared with those of S₃N and S₄N. These unsaturated sulfur nitrogen planar rings are π electron rich and although the symmetry adapted HF solutions are singlet stable at the experimental bond lengths they become unstable with only a very modest increase in bond length. The broken symmetry solutions for S₂N₃, S₃N, and S₄N are of planar C_2v type with one of the nitrogens stripped of its π electrons, producing a π hole.     

Convergence properties of Hartree–Fock SCF molecular calculations

Hartree-Fock equations are viewed as nonlinear algebraic equations that can be solved iteratively. Provided we assume the existence of a solution, valuable properties of convergence may be assessed. The close connection between convergence of the SCF procedure and stability properties of the solution is shown from a nonapproximate standpoint. The convergence features of level-shifting convergence-forcing techniques are analyzed. The connection between this nonlinear algebraic approach and the related gap equation is displayed and the example of the restricted Hartree-Fock hydrogen molecule is discussed.     

On the calculation of bonding energies by the Hartree Fock Slater method

A transition state method has been proposed for the calculation of bonding energies and bond distances within the Hartree Fock Slater Method. Calculations on a number of diatomic molecules and a few transition metal complexes show better agreement with experiment than corresponding Hartree Fock results. The proposed transition state method gives a direct connection between bond orders and bonding energies.     

Electronic and optical properties of functionalized carbon chains with the localized Hartree–Fock and conventional Kohn–Sham methods

The electronic and optical properties of extended functionalized carbyne chains, polyynes and cumulenes, are investigated with the localized Hartree–Fock method, with conventional Kohn–Sham methods, and with the Hartree–Fock method. It is found that even for very long polyynes the carbon–carbon bond lengths within a polyyne alternate while for long cumulenes no carbon–carbon bond length alternation occurs. Polyynes exhibit a finite HOMO–LUMO gap even if they become very long while cumulenes are found to become metallic in the limit of long chain lengths. The geometry and the electro-optical properties of polyynes cannot be influenced significantly by simple sp-σ-bonded end groups. The optically active ¹Σ_u⁺←X¹Σ_g⁺ electronic transition in polyynes is investigated by time-dependent density-functional theory (TDDFT). The known systematic underestimation of excitation energies in large chain-like systems by TDDFT methods is also found for the systems considered here. Deficiencies in the commonly used exchange-correlation kernels are identified as the main source of this shortcoming of TDDFT methods. Unphysical Coulomb self-interactions present in conventional Kohn–Sham potentials seem to not contribute significantly to the problem.     

Combined Open Shell Hartree–Fock Theory of Atomic–Molecular and Nuclear Systems

In this study, the combined Hartree–Fock (HF) and Hartree–Fock–Roothaan equations are derived for multideterminantal single configuration states with any number of open shells of atoms, molecules and nuclei. It is shown that the postulated orbital-dependent energy and Fock operators are invariant to the unitary transformation of orbitals. This new methodology is based entirely on the spin-restricted HF theory. As an application of combined open shell theory of atomic–molecular and nuclear systems presented in this paper, we have solved Hartree–Fock–Roothaan equations for the ground state of electronic configuration C(1s ²2s ²2p ²) using Slater type orbitals as a basis.     

Coupled Hartree Fock calculation of static dipole polarizabilities and shielding factors of open shell systems

Static dipole polarizabilities and shielding factors for the 2p open shell atomic systems are presented using coupled Hartree Fock theory in the framework of the Roothaan formalism. Calculations have been performed for the ground as well as for some valence excited states. A variational approach has been adopted for the determination of the first-order perturbed functions. The results are compared with those obtained from the correlated calculations and other techniques. The shielding factor values are in excellent agreement with the theoretical N/Z ratio.     

Long-range molecular interaction coefficients computed from frost-model wave functions

The average long-range interaction energy between two molecules can be written as an inverse asymptotic series in the intermolecular separation distanceR. Using Frost-model wave functions, the dispersion coefficients of the first three (R ^–6,R ^–8,R ^–10 terms in the series are obtained. Coefficients of three- and four-body non-additive interaction energies are also calculated and the form of the dispersion interaction when retardation effects are included is examined.     

New derivation of the Waller–Hartree–Fock spatial wave function

A new derivation is given for the Waller–Hartree–Fock double-determinantal spatial wave function. One starts from the single-determinant wave function in which a orbitals are doubly occupied, and decomposes it into a sum of products of spatial and spin functions. The spatial product of the first genealogical spin eigenfunction is a double-determinantal function. The derivation is based on the simple form of U_1?(P) when the representation matrix is obtained from the genealogical spin eigenfunction.     

Ab initio Hartree—Fock calculations of model polyacetylene chains using a Christoffersen basis set

Ab initio crystal orbital calculations using a Christoffersen basis set have been performed on the relative stabilities of model polyacetylene chains. Results indicate the all-trans isomer as the most stable. Among the cis structures, the cis-transoid backbone is very slightly preferred to the trans-cisoid backbone.     

The performance of the Hartree–Fock–Wigner correlation model for light diatomic molecules

Results of the Hartree–Fock–Wigner correla- tion model for diatomic molecules with light atoms (H₂, LiH, Li₂, F₂, He₂, Ne₂) using two different atomic parametrizations and one molecular parametrization of the correlation kernel are presented and interpreted in terms of Wigner intracules as well as differences thereof. The molecular parametrization yields encouraging results for simple systems exhibiting covalent or ionic bonding. However, similar to the purely atomic parametrizations severe overestimations of the attractive interaction in van der Waals systems is observed. It is argued that the remaining shortcommings partly result from the restriction of the currently used correlation kernel to be symmetric in relative position and relative momentum.     

Ab initio restricted Hartree—Fock and configuration interaction calculations for Li3

Calculations of the energies for various geometries of Li₃, involving linear and perpendicular approaches of a Li atom to a Li₂ molecule, are described. The basis set was of double zeta plus polarization contracted gaussian orbitals, and both a single configuration restricted Hartree—Fock function, and a five configuration function were used. The perpendicular approach was found to give the lowest energy, and the most stable form of Li₃ is estimated to be about 10 kcal/mole lower in energy than Li + Li₂.     

Application of second-order SCF perturbation theory to the solution of time-dependent Hartree—Fock equations

The self-consistent field based theory of frequency-independent hyperpolarizabilities is adapted to the treatment of the corresponding frequency-dependent quantities. Explicit expressions are given for the double and null frequency polarizations of a molecule in a long-established oscillating electric field.     

Long-range and short-range Coulomb correlation effects as simulated by Hartree–Fock,local density approximation,and generalized gradient approximation exchange functionals

 Exchange functionals used in density functional theory (DFT) are generally considered to simulate long-range electron correlation effects. It is shown that these effects can be traced back to the self-interaction error (SIE) of approximate exchange functionals. An analysis of the SIE with the help of the exchange hole reveals that both short-range (dynamic) and long-range (nondynamic) electron correlation effects are simulated by DFT exchange where the local density approximation (LDA) accounts for stronger effects than the generalized gradient expansion (GGA). This is a result of the fact that the GGA exchange hole describes the exact exchange hole close to the reference electron more accurately than the LDA hole does. The LDA hole is more diffuse, thus leading to an underestimation of exchange and stronger SIE effects, where the magnitude of the SIE energy is primarily due to the contribution of the core orbitals. The GGA exchange hole is more compact, which leads to an exaggeration of exchange in the bond and the nonbonding region and negative SIE contributions. Partitioning of the SIE into intra-/interelectronic and individual orbital contributions makes it possible to test the performance of a given exchange functional in different regions of the molecule. It is shown that Hartree–Fock exchange always covers some long-range effects via interelectronic exchange while self-interaction-corrected DFT is lacking these effects. Received: 25 May 2002 / Accepted: 7 October 2002 / Published online: 21 January 2003 Correspondence to: E. Kraka e-mail: kraka@theoc.gn.se Acknowledgements. This work was supported financially by the Swedish Natural Science Research Council (NFR). Calculations were done on the supercomputers of Nationellt Superdatorcentrum (NSC), Link?ping, Sweden. The authors thank the NSC for a generous allotment of computer time.