Two-Dimensional fully numerical solutions of molecular Schrödinger equations. II. Solution of the Poisson equation and results for singlet states of H2 and HeH+

Two-dimensional fully numerical solutions of the Hartree–Fock problem are reported for the singlet ground states of H^?, He, H₂, and HeH⁺. The H₂ energy at R = 1.4 a.u. is ?1.13362957 a.u.     

Fragmentation processes in phthalimide- and pyridine-2,3-dicarboimidoalkyl-α-diazoketones under resonant electron capture

Resonant electron capture (REC) mass spectra of phthalimide- and pyridine-2,3-dicarboimidoalkyl-α-diazoketones have been investigated. Based on calculations using the Hartree–Fock method and density functional theory with the B3LYP functional the structure of the negative ions (NIs) [M–N₂]^? and [M–N₂–C₃H₃RO]^? as well as the reactions leading to their formation have been proposed.     

Electronic structure and bonding of the pentasulfur hexanitride (S5N6) molecule

An ab initio version of the Hartree–Fock–Slater method is applied to obtain molecular orbitals and eigenvalues for S₅N₆. The electronic structure, bonding, stability, and electronic spectrum are discussed.     

Electronic structure of long cumulene chains

The analysis of the equations of the unrestricted Hartree–Fock (UHF) method for polyenes C_NH_N+2 with even and odd N » 1 is carried out. The equations of the UHF method are shown to be the same in both cases. The comparison of the UHF method with the extended Hartree–Fock (EHF) method applied to large systems is performed. The ground state and π-electron spectra of long cumulene chains C_NH₄ are treated by the EHF Method. The end effects are taken into consideration. It is shown that the EHF method gives a finite value of the first optical transition frequency and, at the same time, zero value of torsion barrier of end CH₂–groups in long cumulene chains (N → ) in contrast to previous calculations of cumulenes by the Huckel method and the restricted Hartree–Fock method.     

Atomic Xα calculations based on EHFS(α) = Eexp

The total energies and one-electron energies for first- and second-row atoms were calculated by using the Hartree–Fock and the Hartree–Fock-Slater Hamiltonian with X_α orbitals, u_i(α_exp); α was parametrized from E^HFS (α_exp) = E^exp. The E^HF (α_exp) total energies are always higher than the Hartree–Fock energies for the atoms. The relation of the calculated ionization potential to the experimental ionization potential depends on the α used to define u_i(α), α_exp, or α_HF.     

Foundations of the relativistic theory of many-electron bound states

Most of the existing calculations of relativistic effects in many-electron atoms or molecules are based on the Dirac–Coulomb Hamiltonian H_DC. However, because the electron–electron interaction mixes positive- and negative-energy states, the operator H_DC has no normalizable eigenfunctions. This fact undermines the quantum-theoretic rationale for the Dirac–Hartree–Fock (DHF ) equations and therefore that of the relativistic configuration-interaction (RCI ) and multiconfiguration Dirac–Fock (MCDF ) methods. An approach to this problem based on quantum electrodynamics is reviewed. It leads to a configuration-space Hamilton H which involves positive-energy projection operators dependent on an external potential U; identification of U with the nuclear potential V_ext corresponds to use of the Furry bound-state interaction picture. It is shown that the RCI method can be reinterpreted as an approximation scheme for finding eigenvalues of a Hamiltonian H, with U identified as the DHF potential; the theoretical interpretation of the MCDF method needs further clarification. It is emphasized that if U differs from V_ext one must consider the effects of virtual-pair creation by the difference potential δU = V_ext ? U; an approximate formula for the level-shift arising from δU is derived. Some ideas for dealing with the technical problems introduced by the projection operators are discussed and relativistic virial theorems are given. Finally, a possible scheme for adapting current MCDF methods to Hamiltonians involving projection operators is described.     

Gradient expansion of the atomic kinetic energy functional

It is demonstrated that the ground-state atomic kinetic energy functional T[?], where ? is the electron density, can be computed to surprising accuracy from the truncated gradient expansion: T[?] = + T₂[?] + T₄[?], with T_o[?] = $\text{3}\text{10}$(3π²)$\text{2}\text{3}$ ∫ ?$\text{5}\text{3}$ d_τ, T₂ [?] = $\text{1}\text{72}$ ∫ (??)²?^?1 d_τ, and T₄ [?] given by the formula of Hodges. Calculations of T₀, T₂ and T₄ are reported for He with ? both the Hartree—Fock and a very accurate density, and for Ne, Ar and Kr with ? the Hartree—Fock density. For Kr, T₀ + T₂ + T₄ is within 0.3% of the exact Hartree—Fock T, with T₂/T₀ = 0.05, T₄/T₂ = 0.17.     

Large-Z and -N dependence of atomic energies from renormalization of the large-dimension limit

By combining Hartree–Fock results for nonrelativistic ground-state energies of N-electron atoms with analytic expressions for the large-dimension limit, we have obtained a simple renormalization procedure. For neutral atoms, this yields energies typically threefold more accurate than the Hartree–Fock approximation. Here, we examine the dependence on Z and N of the renormalized energies E(N, Z) for atoms and cations over the range Z, N = 2 → 290. We find that this gives for large Z = N an expansion of the same form as the Thomas–Fermi statistical model, E → Z^7/2(C₀ + C₁Z^?1/3 + C₂Z^?2/3 + C₃Z^?3/3 + ?), with similar values of the coefficients for the three leading terms. Use of the renormalized large-D limit enables us to derive three further terms. This provides an analogous expansion for the correlation energy of the form δE δZ^4/3(δC₃ + δC₅Z^?2/3 + δC₆Z^?3/3 + ?); comparison with accurate values of δE available for the range Z ? 36 indicates the mean error is only about 10%. Oscillatory terms in E and δE are also evaluated. © 1994 John Wiley & Sons, Inc.     

Conformation of cationic N,N‐di­methyl­glycine in di­methyl­glycinium tri­fluoro­acetate

In the title compound, C₄H₁₀NO₂⁺·C₂F₃O₂^?, the main N—C—COOH skeleton of the protonated amino acid is nearly planar. The C=O/C—N and C=O/O—H bonds are syn and the two methyl groups are gauche to the methyl­ene H atoms. The conformation of the cation in the crystal is compared to that given by ab initio calculations (Hartree–Fock, self‐consistent field molecular‐orbital theory). The tri­fluoro­acetate anion has the typical staggered conformation with usual bond distances and angles. The cation and anion form dimers through a strong O—H?O hydrogen bond which are further interconnected in infinite zigzag chains running parallel to the a axis by N—H?O bonds. Weaker C—H?O interactions involving the methyl groups and the carboxy O atoms of the cation occur between the chains.     

Calculation of ns2 configurations for the Helium isoelectronic series without matching

In the Hartree–Fock equations for the He isoelectronic series, the two-point boundary conditions on one of the differential equations is replaced by initial conditions specified at large distances. [The condition Y′(∞) = 0 replaces the condition Y (0) = 0.] This permits eigenvalues of ns² configurations to be determined as the zeros of a certain function arising from inward integrations, without having to match the solution with a corresponding outward integration. Calculations are performed for n = 1,2,3 for H^? through Be²⁺. Resulting energy values and radial densities are presented. Agreement is found, to the eight significant figures calculated here, with the n = 1 results given by Roothaan and Soukup.     

Ab initio studies of F−(H2O)n and Cl−(H2O)n clusters for n = 1, 2

Ab initio Hartree–Fock calculations are performed on hydrates of the F^? and Cl^? ions using 6-31G, 6-31G**, and 6-21G basis sets. Geometries and binding energies are obtained. An estimate of the correlation energy is provided by an MP2/6-31G (Møller-Plesset second-order perturbation) calculation. Comparisons are made between the Cl^?(SO₂) and the Cl^?(H₂O) complexes.     

Ab initio study of organic mixed valency

A series of six radical cations of the type (D L D)⁺ was investigated at the ab initio unrestricted Hartree–Fock level. One localized and one delocalized conformation were systematically searched by full geometry optimization. At both nuclear arrangements, mostly found as being minima in the symmetry‐restrained Hartree–Fock framework, excitation energies were calculated through the expansion of the wave function on single electronic excitations of the Hartree–Fock fundamental determinant and at the unrestricted Hartree–Fock or at the multiconfigurational self consistent field levels. Few calculations were also performed by taking into account some part of the electronic correlation. Except for N,N,N′,N′‐tetramethyl p‐phenylenediamine, all the studied compounds are localized stable cations, at the symmetry‐restrained Hartree–Fock level. However, the reoptimization of their wave function changes this observation since only three of them seem to conserve a localized stable conformation. Most of the studied systems are characterized by one or two excited electronic states very close to the fundamental one and should thus present an unresolved broadened first absorption band in the near‐infrared region. These features are in agreement with the available experimental data. Strong Hartree–Fock instabilities are found for the delocalized structure and put in relation with the existence of the large nonadiabatic coupling in this conformational region. The solvent influence is discussed in the Onsager dipolar reaction field framework. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 76: 552–573, 2000     

Multiplicity,instability, and SCF convergence problems in Hartree–Fock solutions

We present a study of the instability and convergence of Hartree–Fock (HF) ab initio solutions for the diatomic systems H₂, LiH, CH, C₂, and N₂. In our study, we consider real molecular orbitals (MOs) and analyze the classes of single‐determinant functions associated to Hartree–Fock–Roothaan (HFR) and Hartree–Fock–Pople–Nesbet (HFPN) equations. To determine the multiple HF solutions, we used either an SCF iterative procedure with aufbau and non‐aufbau ordering rules or the algebraic method (AM). Stability conditions were determined using TICS and ASDW stability matrices, derived from the maximum and minimum method of functions (MMF). We examined the relationship between pure SCF convergence criterion with the aufbau ordering rule, and the classification of the HF solution as an extremum point in its respective class of functions. Our results show that (i) in a pure converged SCF calculation, with the aufbau ordering rule, the solutions are not necessarily classified as a minimum of the HF functional with respect to the TICS or ASDW classes of solutions, and (ii) for all studied systems, we obtained local minimum points associated only with the aufbau rule and the solutions of lower energies. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 76: 600–610, 2000     

A comparison of Hartree—Fock,MP2, and DFT results for the HCN dimer and crystal

A number of hydrogen-bond related quantities—geometries, interaction energies, dipole moments, dipole moment derivatives, and harmonic vibrational frequencies—were calculated at the Hartree—Fock, MP2, and different DFT levels for the HCN dimer and the periodic HCN crystal. The crystal calculations were performed with the Hartree—Fock program CRYSTAL92, which routinely allows an a posteriori electron-correlation correction of the Hartree—Fock obtained lattice energy using different correlation-only functionals. Here, we have gone beyond this procedure by also calculating the electron-correlation energy correction during the structure optimization, i.e., after each CRYSTAL92 Hartree—Fock energy evaluation, the a posteriori density functional scheme was applied. In a similar manner, we optimized the crystal structure at the MP2 level, i.e., for each Hartree—Fock CRYSTAL92 energy evaluation, an MP2 correction was performed by summing the MP2 pair contributions from all HCN molecules within a specified cutoff distance. The crystal cell parameters are best reproduced at the Hartree—Fock and the nongradient-corrected HF + LDA and HF + VWN levels. The BSSE-corrected MP2 method and the HF + P91, HF + LDA, and HF + VWN methods give lattice energies in close agreement with the ZPE-corrected experimental lattice energy. The (HCN)₂ dimer properties are best reproduced at the MP2 level, at the gradient-corrected DFT levels, and with the B3LYP and BHHLYP methods. © 1996 John Wiley & Sons, Inc.     

16α,17α‐Epoxy‐20‐oxopregn‐5‐en‐3β‐yl acetate

The title compound, C₂₃H₃₂O₄, has a 3β configuration, with the epoxy O atom at 16α,17α. Rings A and C have slightly distorted chair conformations. Because of the presence of the C5=C6 double bond, ring B assumes an 8β,9α‐half‐chair conformation slightly distorted towards an 8β‐sofa. Ring D has a conformation close to a 14α‐envelope. The acetoxy and acetyl substituents are twisted with respect to the average molecular plane of the steroid. The conformation of the mol­ecule is compared with that given by a quantum chemistry calculation using the RHF–AM1 (RHF = Roothaan Hartree–Fock) Hamiltonian model. Cohesion of the crystal can be attributed to van der Waals interactions and weak intermolecular C—H?O interactions, which link the mol­ecules head‐to‐tail along [101].     

Author index

On the basis of high-quality Hartree—Fock MO calculations, the barrier to internal rotation in Si₂H₆ is predicted to be 2.317 kJ mol^?1. The importance of geometry optimisation in such calculations is illustrated. In view of the Mulliken population indices, it seems unlikely that a simple electrostatic model will ever be able to explain the origin of the barrier.     

Hartree—Fock—Slater functions as basis for one-electron perturbation calculations for molecules: I. Calculation of ground state electric dipole polarizabilities

The analysis of the decoupling of Hartree—Fock—Slater SCF perturbation equations for an external field is undertaken. The points of departure from the corresponding Hartree—Fock perturbation equations are stressed. Both formal and numerical results suggest that the fully uncoupled Hartree—Fock—Slater expression is a less drastic approximation than the same Hartree—Fock one. The uncoupled expression for the ground state electric dipole polarizability is calculated for CO, N₂, ethylene, acethylene and trans-butadiene in the dipole length—dipole length, dipole velocity—dipole length and dipole velocity—dipole velocity alternative formulations with an ab initio Hartree—Fock—Slater SCF basis set. The results compare well with other non-empirical results and the dipole velocity-dipole length results are in remarkably good agreement with experiments.     

3,17‐Dioxoandrost‐4‐en‐4‐yl acetate

The title compound, C₂₁H₂₈O₄, has a 4‐acetoxy substituent positioned on the steroid α face. The six‐membered ring A assumes a conformation intermediate between 1α,2β‐half chair and 1α‐sofa. A long Csp³—Csp³ bond is observed in ring B and reproduced in quantum‐mechanical ab initio calculations of the isolated molecule using a molecular‐orbital Hartree–Fock method. Cohesion of the crystal can be attributed to van der Waals interactions and weak C—H...O hydrogen bonds.     

Dipole moment derivatives of H2O and H2S

The dipole and quadrupole derivatives of H₂O and H₂S are calculated analytically, using the coupled Hartree—Fock method first proposed by Gerratt and Mills. The greater efficiency, of this method allows SCF wave functions very, close to the Hartree—Fock limit to be used. Agreement, with experimental data is good.     

New basis sets for the evaluation of interaction‐induced electric properties in hydrogen‐bonded complexes

Interaction‐induced static electric properties, that is, dipole moment, polarizability, and first hyperpolarizability, of the CO? (HF)_n and N₂? (HF)_n, n = 1–9 hydrogen‐bonded complexes are evaluated within the finite field approach using the Hartree–Fock, density functional theory, Møller–Plesset second‐order perturbation theory, and coupled cluster methods, and the LPol‐n (n = ds, dl, fs, fl) basis sets. To compare the performance of the different methods with respect to the increase of the complex size, we consider as model systems linear chains of the complexes. We analyze the results in terms of the many‐body and cooperative effects. © 2012 Wiley Periodicals, Inc.