Improved generator coordinate Hartree–Fock method for molecular systems: application to H2, Li2 and LiH

The improved generator coordinate Hartree–Fock (GCHF) method is extended to molecular systems. The Griffin–Hill–Wheeler–HF equations were solved by an integral discretization technique. The method is then implemented with the use of the GAMESS program and applied to the H₂, Li₂, and LiH molecules. For these molecules, sequences of basis sets of atom-centred Gaussian-type functions are employed to explore the accuracy achieved with our approach. For all systems studied, our ground-state HF total energies are better than those obtained with basis sets generated with the original GCHF method for molecules and larger even-tempered basis sets. For H₂, Li₂, and LiH, the differences between our best energies and the corresponding numerical HF results are about 2 × 10⁻², 1, and 4 × 10⁻¹ μhartree, respectively. The dipole, quadrupole, and octupole moments at the center of mass and electric field, the electric field gradient, the electrostatic potential, and the electron density at the nuclei were evaluated and compared with results reported in the literature. Received: 4 May 1999 / Accepted: 22 July 1999 / Published online: 2 November 1999     

Roothaan–Hartree–Fock wave functions for cations and anions in Slater-type basis sets with doubly even tempered exponents

Summary Roothaan-Hartree-Fock wave functions in Slater-type basis sets are reported for the cations Li⁺-Cs⁺ and anions H^–-I^– using the double even tempering (DET) method of selecting orbital exponents. The DET total energies do not differ from the corresponding numerical Hartree-Fock values by more than 0.2 millihartrees for the cations and anions. The present results together with the previous ones for neutral atoms [Theor Chim Acta 88:273 (1994)] provide a compilation of DET wave functions of near Hartree-Fock quality for all the neutral and singly-charged atoms with the number of electronsN54.     

Relativistic Gaussian basis sets for the elements K – Uuo

A series of energy-minimized relativistic Gaussian basis sets for the elements with atomic numbers 19–118 is presented. The basis sets have been derived at the self-consistent field level as weighted average energies of the respective electronic configurations. A spherical Gaussian charge distribution has been used to model the nucleus. The basis sets are constructed as interleaving dual family sets with shared exponents within each family. The quality of the basis sets is better than double zeta. Received: 7 July 2000 / Accepted: 21 September 2000 / Published online: 21 December 2000     

Self-consistent-field – Hartree–Fock method with finite nuclear mass corrections

We have upgraded a Self-consistent-field – Hartree–Fock routine to include a finite nuclear mass correction for molecules developed in our laboratory. The new routine can handle isotopomers without calculating any nuclear kinetic energy matrix element. Tests on H₂, LiH, HF, F₂, and H₂O isotopomers indicate the equivalence of our correction to the standard diagonal adiabatic correction. A further original application to C₂H₆ illustrates the usefulness of the method for polyatomic molecules. The resulting molecular orbitals carry the nuclear mass signature, exemplified with Koopmans ionization potentials.From the Proceedings of the 28th Congreso de Químicos Teóricos de Expresión Latina (QUITEL 2002)     

The basis set convergence of the Hartree–Fock energy for H3 +, Li2 and N2

 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. Received: 14 February 2000 / Accepted: 12 April 2000 / Published online: 18 August 2000     

Adiabatic integration formula for the correlation energy functional of the Hartree–Fock density

An adiabatic integration formula for the quantum chemistry correlation energy functional of the Hartree–Fock density, E _c ^QC[n], is presented. The functional E _c ^QC[n] is meant to be added to the completed Hartree–Fock energy to produce the exact ground-state energy of the system under consideration. The initial slope of the integrand in this connection formula is identified as a second-order energy and an explicit expression for the initial slope of the integrand is presented. Our expression should be useful for arriving at new improved approximations to E _c ^QC[n]. Previous numerical results by Huang and Umrigar (1997) Phys Rev A 56:290, for two-electron densities are proved, and a generalization to more than two electrons is presented. Results obtained by means of the present density functional theory correlation energy functionals, when used to approximate the initial slope in our adiabatic integration formula for E _c ^QC[n], are compared against exact numbers. Received: 10 September 1998 / Accepted: 3 February 1999 / Published online: 21 June 1999     

All-electron scalar relativistic basis sets for the elements Rb–Xe

Segmented all-electron relativistically contracted (SARC) basis sets are presented for the elements ₃₇Rb–₅₄Xe, for use with the second-order Douglas–Kroll–Hess approach and the zeroth-order regular approximation. The basis sets have a common set of exponents produced with established heuristic procedures, but have contractions optimized individually for each scalar relativistic Hamiltonian. Their compact size and loose segmented contraction, which is in line with the construction of SARC basis sets for heavier elements, makes them suitable for routine calculations on large systems and when core spectroscopic properties are of interest. The basis sets are of triple-zeta quality and come in singly or doubly polarized versions, which are appropriate for both density functional theory and correlated wave function theory calculations. The quality of the basis sets is assessed against large decontracted reference basis sets for a number of atomic and ionic properties, while their general applicability is demonstrated with selected molecular examples.     

Revised relativistic basis sets for the 5d elements Hf–Hg

The relativistic double-zeta (dz) and triple-zeta (tz) basis sets for the 5d elements Hf–Hg have been revised for consistency with the recently optimized 4f basis sets. The new dz basis sets have 24 s functions instead of 22 s functions, and the new tz basis sets have 30 s functions instead of 29 s functions. New contraction patterns have been determined, including the 6p orbital.     

Coupling coefficients “symmetry dilemma” in the restricted open‐shell Hartree–Fock method

Two opposite conclusions are known on the symmetry of the vector coupling coefficients (VCCs), a_mn and b_mn, in the restricted open-shell Hartree-Fock (ROHF) method. The first one states that the VCCs are symmetric for all spectroscopic terms, i.e., a_mn = a_nm and b_mn = b_nm. An opposing statement is that the "non-Roothaan" terms, arising from the degenerate open-shell electronic configuration γ^N, can be characterized by non-symmetric VCCs matrices only: ‖a_mn‖ ≠ ‖a_nm,‖ ‖b_mn‖ ≠ ‖b_nm‖. This article presents a detailed analysis of the VCCs symmetry problem. A general approach to the VCCs determination has been developed leading to non‐symmetric VCCs for γ^N systems with γ ≥ 3. The main purpose of this work is to eliminate the contradiction arising in the ROHF theory when the latter is applied to highly symmetric open‐shell molecules and atoms. This revised version was published online in July 2006 with corrections to the Cover Date.     

A Hartree–Fock approach to the Steklov eigenproblem for a two-electron atom in an s2 state

Two decades ago, in a brief note (Hinze and Hamacher, J Chem Phys 92:4372–4373, 1990), Hinze and Hamacher made a conjecture that the Hartree–Fock method may be applied to determine approximate solutions to a non-standard (Steklov-type) eigenproblem encountered in the non-relativistic eigenchannel R-matrix method. Later, this thread was mathematically pursued further by the present author (Szmytkowski, Phys Rev A 61:022725, 2000; erratum 66:029901, 2002). In the present paper, which is dedicated to the memory of Professor Jürgen Hinze, we make an attempt to elucidate the idea underlying the aforementioned works. To focus on the essence and avoid obscuring mathematical details, we consider the simplest system which is a two-electron atom in an s² state. Variational principles for pertinent Steklov eigenvalues (i.e., eigenvalues of a two-electron Dirichlet-to-Neumann integral operator) and for their reciprocals (i.e., eigenvalues of a two-electron Neumann-to-Dirichlet integral operator) are used to derive a radial integro-differential eigensystem of a Hartree–Fock type, in which both a Lagrange multiplier in an integro-differential equation and a Steklov eigenvalue appearing in a boundary condition are to be determined simultaneously at a fixed total energy of the atom. Mathematical similarities and (particularly) differences between the problem considered here and that of a spherically confined two-electron atom are highlighted.     

Reduced–size polarized basis sets for calculations of molecular electric properties. III. Second–row atoms

Reduced–size polarized (ZmPolX) basis sets are developed for the second–row atoms X = Si, P, S, and Cl. The generation of these basis sets follows from a simple physical model of the polarization effect of the external electric field which leads to highly compact polarization functions to be added to the chosen initial basis set. The performance of the ZmPolX sets has been investigated in calculations of molecular dipole moments and polarizabilities. Only a small deterioration of the quality of the calculated molecular electric properties has been found. Simultaneously the size of the present reduced–size ZmPolX basis sets is about one-third smaller than that of the usual polarized (PolX) sets. This reduction considerably widens the range of applications of the ZmPolX sets in calculations of molecular dipole moments, dipole polarizabilities, and related properties.     

An improved method for tracking and reducing the void volume in nano HPLC–MS with micro trapping columns

Tandem mass spectrometry coupled to HPLC is the state of the art technique in proteomic research. Here we describe a highly sensitive nano liquid chromatography system (nano HPLC) for analysis of protein digests. Using preconcentration in a column-switching set-up, we were able to inject large sample volumes (250 µL) without significant loss of sensitivity. The major problem with this type of preconcentration is usually the occurrence of void volumes. In order to diagnose void volumes a simple and easy test was developed by which the UV trace and the pressure profile in the separation column were monitored. Part by part replacement of connection tubing restored a void volume-free system. A major pre-requisite for handling samples in the femtomol range was found to be the use of protein/peptide-saturated columns tryptic digests of cytochrome C were injected directly onto the reversed-phase nano separation column (75 µm inner diameter) and the separation results were compared with chromatograms obtained from separations using column switching. By using column switching we were able to inject large sample volumes in a short time period without losing resolution.     

Correlation consistent, Douglas–Kroll–Hess relativistic basis sets for the 5p and 6p elements

The diatomic carbon molecule has a complex electronic structure with a large number of low-lying electronic excited states. In this work, the potential energy curves (PECs) of the four lowest lying singlet states ( $X^{1} \Sigma^{ + }_{g}$ , $A^{1} \Pi_{u}$ , $B^{1} \Delta_{g}$ , and $B^{\prime1} \Sigma^{ + }_{g}$ ) were obtained by high-level ab initio calculations. Valence electron correlation was accounted for by the correlation energy extrapolation by intrinsic scaling (CEEIS) method. Additional corrections to the PECs included core–valence correlation and relativistic effects. Spin–orbit corrections were found to be insignificant. The impact of using dynamically weighted reference wave functions in conjunction with CEEIS was examined and found to give indistinguishable results from the even weighted method. The PECs showed multiple curve crossings due to the $B^{1} \Delta_{g}$ state as well as an avoided crossing between the two $^{1} \Sigma^{ + }_{g}$ states. Vibrational energy levels were computed for each of the four electronic states, as well as rotational constants and spectroscopic parameters. Comparison between the theoretical and experimental results showed excellent agreement overall. Equilibrium bond distances are reproduced to within 0.05 %. The dissociation energies of the states agree with experiment to within ~0.5 kcal/mol, achieving “chemical accuracy.” Vibrational energy levels show average deviations of ~20 cm^?1 or less. The $B^{1} \Delta_{g}$ state shows the best agreement with a mean absolute deviation of 2.41 cm^?1. Calculated rotational constants exhibit very good agreement with experiment, as do the spectroscopic constants.     

An improved method for the asymmetric protonation of enolates with chiral α-sulfinyl alcohols/trifluoroethanol

Enantioselective protonation of 2-methyl tetralone enolate using a stoichiometric amount of 2-sulfinyl alcohol (S,R_s)-1 as a chiral proton donor yields the corresponding chiral ketone with a high level of enantioselectivity, provided that the chiral proton donor (CPD) is regenerated with an achiral proton source (APS). In contrast, stereoselectivity was only moderate if catalytic CPD (0.2 equiv.) and an APS are used.     

Validity of <Emphasis Type="Italic">B</Emphasis>-splines as a universal basis set for atomic Hartree–Fock–Roothaan calculations

The validity of B-splines as a universal basis set for atomic Hartree–Fock–Roothaan calculations is studied. In order to accomplish our aim, the ground-state energies of neutral atoms He–Xe, cations Li ⁺–Xe ⁺, and anions H ^-–I ^- with the nuclear charge Z=54 are calculated by the Hartree–Fock–Roothaan method with the B-spline sets. All radial functions of the atoms and singly charged ions are expanded by common B-spline sets regardless of atomic systems and symmetries of atomic orbitals. The energies obtained by the best B-spline set are in excellent agreement with ten-digit numerical Hartree–Fock results.     

Pd–Fe/TiO2 catalysts for phenol degradation with in situ generated H2O2

This study deals with the degradation of phenol over Pd–Fe/TiO₂ catalysts at mild conditions in the presence of in situ generated H₂O₂ from oxygen and formic acid. This catalytic system demonstrated interesting ability to oxidize phenol by Fenton process in a one-pot reaction without the addition of ferrous ion. Lower Pd content catalysts, despite producing a higher hydrogen peroxide amount for bulk purposes, did not reach the same efficiency as the 5Pd–5Fe catalyst in phenol degradation. A close interaction between Pd and iron oxide species is necessary to obtain high active catalysts. These results highlight the advantage of in situ generation of H₂O₂, for oxidation reactions with respect to conventional Fenton process.