On the asymptotic behavior of Hartree-Fock orbitals

Summary Handy et al. have shown that the asymptotic behavior of Hartree-Fock orbitals is controlled by the energy of highest occupied molecular orbital, except for the atomic case in which onlys-orbitals are occupied. However, their proof is not complete at one point. This point is clarified, and a more unified derivation is given. Further, we discuss the preexponential factorr of the leading asymptotic termr exp[–r], where =(–2_HOMO)^1/2 and _HOMO is the orbital energy of HOMO. New results are obtained for linear molecules, and the results of several authors for atoms and non-linear molecules are reproduced.     

Smooth relativistic Hartree-Fock pseudopotentials for H to Ba and Lu to Hg

We report smooth relativistic Hartree-Fock pseudopotentials (also known as averaged relativistic effective potentials) and spin-orbit operators for the atoms H to Ba and Lu to Hg. We remove the unphysical extremely nonlocal behavior resulting from the exchange interaction in a controlled manner, and represent the resulting pseudopotentials in an analytic form suitable for use within standard quantum chemistry codes. These pseudopotentials are suitable for use within Hartree-Fock and correlated wave function methods, including diffusion quantum Monte Carlo calculations.     

Efficient exact-exchange time-dependent density-functional theory methods and their relation to time-dependent Hartree-Fock

A recently introduced time-dependent exact-exchange (TDEXX) method, i.e., a response method based on time-dependent density-functional theory that treats the frequency-dependent exchange kernel exactly, is reformulated. In the reformulated version of the TDEXX method electronic excitation energies can be calculated by solving a linear generalized eigenvalue problem while in the original version of the TDEXX method a laborious frequency iteration is required in the calculation of each excitation energy. The lowest eigenvalues of the new TDEXX eigenvalue equation corresponding to the lowest excitation energies can be efficiently obtained by, e.g., a version of the Davidson algorithm appropriate for generalized eigenvalue problems. Alternatively, with the help of a series expansion of the new TDEXX eigenvalue equation, standard eigensolvers for large regular eigenvalue problems, e.g., the standard Davidson algorithm, can be used to efficiently calculate the lowest excitation energies. With the help of the series expansion as well, the relation between the TDEXX method and time-dependent Hartree-Fock is analyzed. Several ways to take into account correlation in addition to the exact treatment of exchange in the TDEXX method are discussed, e.g., a scaling of the Kohn-Sham eigenvalues, the inclusion of (semi)local approximate correlation potentials, or hybrids of the exact-exchange kernel with kernels within the adiabatic local density approximation. The lowest lying excitations of the molecules ethylene, acetaldehyde, and pyridine are considered as examples.     

Combination of pseudopotentials and density functionals

Semilocal pseudopotentials have been determined for first–row (Li to Ne), second row (Na to Ar), and third-row atoms (K, Ca). Core–valence correlation is included by adjusting the pseudopotentials to experimental energies of ions with a single valence electron. Correlation within the valence shell is taken into account by using the spin–density functional formalism. The approximations involved in this approach are tested for atomic ionization energies as well as binding energies of monohydrides and alkali diatomics, agreement with experiment is usually satisfactory, but in certain applications density functionals should be already included in the fitting of the local part of the pseudopotential. In addition, 3s/3p and 3s/2p basis sets (for first and second row, respectively), designed for use in connection with our pseudopotentials, are given; it is shown that they yield reasonable results for both SCF and correlation energies.     

Coherent states and asymptotic behavior of the symmetric top wave functions

Wave packets which may be considered as an analogue of the coherent states of the top are constructed. New integral representations are obtained for the D-function; making use of those, various types of the asymptotic behaviour of the D-function are obtained in the region of large quantum numbers.     

Projected Hartree-Fock theory

Projected Hartree-Fock (PHF) theory has a long history in quantum chemistry. PHF is here understood as the variational determination of an N-electron broken symmetry Slater determinant that minimizes the energy of a projected state with the correct quantum numbers. The method was actively pursued for several decades but seems to have been abandoned. We here derive and implement a "variation after projection" PHF theory using techniques different from those previously employed in quantum chemistry. Our PHF methodology has modest mean-field computational cost, yields relatively simple expressions, can be applied to both collinear and non-collinear spin cases, and can be used in conjunction with deliberate symmetry breaking and restoration of other molecular symmetries like complex conjugation and point group. We present several benchmark applications to dissociation curves and singlet-triplet energy splittings, showing that the resulting PHF wavefunctions are of high quality multireference character. We also provide numerical evidence that in the thermodynamic limit, the energy in PHF is not lower than that of broken-symmetry HF, a simple consequence of the lack of size consistency and extensivity of PHF.     

Approximate relativistic Hartree-Fock equations and their solution within a minimum basis set of slater-type functions

Relativistic closed-shell atoms are treated by the use of a specific approximation for the small component of the one-electron Dirac spinors. It is assumed that the large and the small component are interconnected by a parameter-dependent relation which is formally analogous to that of the one-electron system. Subject to this constraint, the total energy is varied with respect to the large components. The resulting eigenvalue equations for the large components contain only regular potential terms and reduce to the familiar Hartree-Fock equations in the limit of infinite velocity of light. Analytical solution of these approximate relativistic Hartree-Fock equations is achieved using a minimum basis set of Slater-type functions for the expansion of the radial part of the large components. Total relativistic energies, orbital energies, orbital exponents and mean radii are calculated for the ground states of He, Be, Ne, Mg, Ar, Kr Xe and Cu⁺. Dedicated to Prof. O. E. Polansky on occasion of his 60th birthday.     

Energy-consistent pseudopotentials for quantum Monte Carlo calculations

The authors present scalar-relativistic energy-consistent Hartree-Fock pseudopotentials for the main-group elements. The pseudopotentials do not exhibit a singularity at the nucleus and are therefore suitable for quantum Monte Carlo (QMC) calculations. They demonstrate their transferability through extensive benchmark calculations of atomic excitation spectra as well as molecular properties. In particular, they compute the vibrational frequencies and binding energies of 26 first- and second-row diatomic molecules using post-Hartree-Fock methods, finding excellent agreement with the corresponding all-electron values. They also show their pseudopotentials give superior accuracy than other existing pseudopotentials constructed specifically for QMC. Finally, valence basis sets of different sizes (VnZ with n=D,T,Q,5 for first and second rows, and n=D,T for third to fifth rows) optimized for our pseudopotentials are also presented.     

Energy-adjusted pseudopotentials for the rare earth elements

Nonrelativistic and quasirelativistic energy-adjusted pseudopotentials and optimized (7s6p5d)/[5s4p3d]-GTO valence basis sets for use in molecular calculations for fixed f-subconfigurations of the rare earth elements, La through Lu, have been generated. Atomic excitation and ionization energies from numerical HF, as well as SCF pseudopotential calculations using the derived basis sets, differ by less than 0.1 eV from numerical HF all-electron results. Corresponding values obtained from CI(SD), CEPA-1, as well as density functional calculations using the quasirelativistic pseudopotentials, are in reasonable agreement with experimental data.     

An analysis of core effects on shape-consistent pseudopotentials

Large core (seven-valence electrons) shape-consistent averaged relativistic pseudopotentials (AREP) including core effects have been derived for the halogen series (Cl,Br,I,At). The influence of core effects on the spin-orbit splitting of the halogen and alkali atoms is clearly demonstrated within an all-electron four-component atomic reference calculation by means of a perturbation analysis. In particular, it is shown that AREPs extracted at the Dirac-Coulomb-Fock level, which already include spin-orbit polarization effects, give excellent results for atomic spectroscopy and equilibrium distances of halogen dimers. We also show that in our approach the core effects, included by configuration interaction using the numerical GRASP code, are transferred to the averaged orbital one-electron energy, defined in a perturbational way. This leads to a modification of the extracted AREPs by core effects, which is illustrated by calculations of the first atomic excited states using these AREPs. These results support the validity of including core effects directly in the AREPs extracted in a shape-consistent scheme. The transferability to the atomic excited states as well as to the molecular case is also verified.     

Improved relativistic energy-consistent pseudopotentials for 3d-transition metals

Energy-consistent relativistic pseudopotentials for 3d-transition metals Sc to Ni based on modified valence energies are proposed. The pseudopotentials are adjusted at the finite difference level within the intermediate coupling scheme with respect to multi-configuration Dirac–Hartree–Fock data based on the Dirac–Coulomb Hamiltonian with an estimate of the Breit contributions in quasidegenerate perturbation theory. Typically a few hundred to thousand J levels arising from about 35 to 40 configurations ranging from the anion down to the highly charged cation are considered as references. It is shown that introducing a small common energetic shift of all valence energies reduces the errors in the parameter adjustment considerably. Results of highly correlated atomic and molecular test calculations using large basis sets and basis set extrapolation techniques are presented. To be submitted to Theoretical Chemistry Accounts (special volume on the occasion of Prof. Dr. H. Stoll's 60th birthday)     

Global behavior and asymptotic reduction of a chemical kinetics system with continua of equilibria

We consider a model chemical kinetics system describing the dynamics of species concentrations taking part is consecutive-competitive reaction in a continuopusly stirred tank reactor. Corresponding dynamical system has a continua of equilibria. Particular equilibrium to which the solution of the system tends depends on the initial conditions. The global behavior of the system and its reductions via the invariant manifold and the boundary function methods are studied.AMS Subject Classification: 34D15, 34D35, 37C10     

Hartree-Fock theory for negative ions

It is shown that HF computations which yield ?_i > 0 for an occupied MO do not minimize the HF energy. If ?_i > 0, which frequently occurs in the RHF treatment of negative ions, one can reduce ?_i to zero and simultaneously lower E_HF by an appropriate admixture of a continuum function to the corresponding MO ?. We propose a modification of the HF model that takes these facts into account. Applications to the systems O^2?, N^3?, C^4?, S^2?, O₂^2?, C₂^2? are reported and discussed.     

Soluble LC polyesters and their blend behavior

Poly(p-phenylene terephthalate) with substituents on the hydroquinone or on both monomer units were synthesized with the aim to obtain soluble LC polyesters. Focus was given to phenylalkyl substitution. With one phenylalkyl group in the repeating unit the polyesters are crystalline with melting points > 300°C. With two substituents the crystallinity is very low or the polymers are amorphous. These polyesters are soluble in chloroform forming isotropic solutions. Isotropic films of polyesters obtained from these solutions either crystallize or become anisotropic if heated above the glass transition temperature. The exponent in the Kuhn-Mark-Houwink equation is 0.95. Some of the polyesters form homogenous ternary solutions with polycarbonate. Films of the blends show additivity of modulus with respect to both components.     

N-Aminoarylpyridinium perchlorates and their behavior in diazotization

N-Aminoarylpyridinium salts react with nitrosyl perchlorate to give diazonium dications, which undergo diazo coupling, exchange a diazo group, and are cyclized by the Pschorr method.     

Quasirelativistic energy-consistent 5f-in-core pseudopotentials for pentavalent and hexavalent actinide elements

Quasirelativistic energy-consistent 5f-in-core pseudopotentials modeling pentavalent (5f ^n?2 occupation with n = 2–6 for Pa–Am) and hexavalent (5f ^n?3 occupation with n = 3–6 for U–Am) actinides have been adjusted. Energy-optimized (6s5p4d) and (7s6p5d) valence basis sets contracted to polarized double- to quadruple-zeta quality as well as 2f1g correlation functions have been derived. Corresponding smaller basis sets (4s4p3d) and (5s5p4d) suitable for calculations on actinide(V) and actinide(VI) ions in crystalline solids form subsets of these basis sets designed for calculations on neutral molecules. Calculations using the Hartree–Fock and the coupled-cluster method with single and double excitation operators and a perturbative estimate of triple excitations for actinide pentafluorides show satisfactory agreement with calculations using 5f-in-valence pseudopotentials and experimental data, respectively. However, in the hexavalent case the 5f-in-core approximation seems to reach its limitations except for hexavalent uranium (5f⁰), where results for both uranium hexafluoride and the uranyl ion deviate only slightly from the 5f-in-valence reference data.     

Quasirelativistic energy-consistent 5f-in-core pseudopotentials for divalent and tetravalent actinide elements

Quasirelativistic energy-consistent 5f-in-core pseudopotentials modeling divalent (5f ⁿ⁺¹ occupation with n = 5–13 for Pu–No) respectively tetravalent (5f ^n-1 occupation with n = 1–9 for Th–Cf) actinides together with corresponding core-polarization potentials have been generated. Energy-optimized (6s5p4d) and (7s6p5d) valence basis sets as well as 2f1g correlation functions have been derived and contracted to polarized double, triple, and quadruple zeta quality. Corresponding smaller (4s4p) and (5s5p) respectively (4s4p3d) and (5s5p4d) basis sets suitable for calculations on actinide(II) respectively actinide(IV) ions in crystalline solids form subsets of these basis sets designed for calculations on molecules. Results of Hartree–Fock test calculations for actinide di- and tetrafluorides show a satisfactory agreement with calculations using 5f-in-valence pseudopotentials. Electronic Supplementary Material The online version of this article doi: contains supplementary material, which is available to authorized users.     

Energy-adjustedab initio pseudopotentials for the second and third row transition elements

Summary Nonrelativistic and quasirelativisticab initio pseudopotentials substituting the M^(Z–28)+-core orbitals of the second row transition elements and the M^(Z–60)+-core orbitals of the third row transition elements, respectively, and optimized (8s7p6d)/[6s5p3d]-GTO valence basis sets for use in molecular calculations have been generated. Additionally, corresponding spin-orbit operators have also been derived. Atomic excitation and ionization energies from numerical HF as well as from SCF pseudopotential calculations using the derived basis sets differ in most cases by less than 0.1 eV from corresponding numerical all-electron results. Spin-orbit splittings for lowlying states are in reasonable agreement with corresponding all-electron Dirac-Fock (DF) results.