Surpassing the Temple Lower Bound

A non-traditional derivation of the Temple lower bound formula shows how the Temple bound can be improved using expectation values of H ³. Improved upper bounds result as a byproduct as well. Since expectation values of H ² are already quite difficult in analytical calculations we also introduce an approximate lower bound which avoids the need for H ³ but which is rigorous only in a limiting sense. Examples of the two bounds are given using the lithium and perturbed hydrogen atoms.     

Molecular calculations for HeH<Superscript>+</Superscript> with two-center correlated orbitals

A two-center correlated orbital approach was used to calculate the electronic ground state energy for the HeH⁺ molecular ion. The wavefunctions were constructed from the exact solution of the Schrödinger equation for the HeH⁺⁺ problem in prolate-spheroidal coordinates taken together with a Hylleraas type correlation factor. With a simple single term wavefunction, we obtained ground state energy of ?2.95308691 hartree without any variational parameters in the calculation. When a two-configuration-state wavefunction was used and effective charges were allowed to be adjusted, we found an energy of ?2.97384868 hartree, which is to be compared with ?2.97869074 hartree obtained by an 83 term configuration interaction wavefunction or ?2.97364338 hartree by an ab initio calculation (at the MP4(SDQ)/6-311++G(3df, 3dp) level) using the well-known “canned” code.     

Bounds to electronic expectation values for atomic and molecular systems

A generalization of a method to calculate lower bounds to expectation values of non‐negative observables is presented. We consider bounds to three electronic expectation values 〈r²〉, 〈r〉, and 〈r^?1〉 in the helium atom as an example. For both 〈r²〉 and 〈r〉, we are able to calculate improved lower bounds. The lower bound to 〈r^?1〉 does not improve, but we are able to calculate an upper bound which is much closer to the expectation value than the lower bound. Although our generalization allows for improved bounds and/or upper bounds, these bounds to general observables are much less precise than energy bounds and even the expectation values calculated from variational wave functions. © 2010 Wiley Periodicals, Inc. Int J Quantum Chem, 2011     

Computing upper and lower bounds using Monte Carlo methods

We computed lower bounds for some ground and excited states of the helium atom using variational Monte Carlo and the Weinstein, Temple, and Stevenson formulas. For these systems, the Temple and Stevenson bounds are approximately the same and have similar standard deviations. The Weinstein bounds are much further from the actual energies and have much larger standard deviations. We also investigated the reliability of these lower bound formulas when nonorthogonal wave functions are used. The Temple formula has been shown to produce an accurate lower bound even under such circumstances, while the Weinstein and Stevenson formulas are shown to yield incorrect results. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2002     

Radial and angular correlation in heliumlike ions

In the framework of nonrelativistic variational formalism a new type of basis set is proposed, to estimate separately the effect of radial and angular correlations on the ground‐state energy for helium isoelectronic sequence H^? to Ar¹⁶⁺. Effect of radial correlation is incorporated by using multiexponential functions arising from product basis sets suitably formed out of Slater‐type one‐particle orbitals. The angular correlation can be switched on by incorporating an expansion in terms of basis involving interparticle coordinates. With a set of six‐term Slater‐type one‐particle basis and five‐term interparticle expansion, the ground‐state energy of helium is estimated as ?2.9037236 (a.u.) compared with the multiterm variational estimates ?2.9037244 (a.u.) due to Pekeris and Thakkar and Smith and Drake. Matrix elements of different operators in the ground state have been calculated and found to be in good agreement with available accurate results. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem, 2003     

Mononukleare und mehrfach verbrückte dinukleare Phthalocyaninate(1–/2–) des Yttriums durch Solvenz‐kontrollierte Kondensation; kleine Solvenz‐Cluster als Liganden

Mononuclear and Multiply Bridged Dinuclear Phthalocyaninates(1–/2–) of Yttrium by Solvent Controlled Condensation; Small Solvent Clusters as Ligands Green chlorophthalocyaninato(2–)yttrium(III), [Y(Cl)pc^2–] forms when yttrium chloride is heated with o‐phthalonitrile in 1‐chloronaphthalene. Black cis‐di(chloro)phthalocyaninato(1‐)yttrium(III), ^cis[Y(Cl)₂pc^–] is obtained as a stable intermediate by partial reduction. Both complexes are soluble in many O‐donor solvents and pyridine. The solubility in water is remarkable: [Y(Cl)pc^2–] dissolves with green, ^cis[Y(Cl)₂pc^–] with red‐violet color. Typical absorptions of the pc^2– ligand are observed at 14800 and 29700 cm^–1. A solvent dependent monomer‐dimer equilibrium is found for the pc^– radical. The monomer with absorptions at 12100 and 19900 cm^–1 is favored in non‐polar solvents, while in polar solvents the dimer with absorptions at 8700, 13200 and 18600 cm^–1 is preferred. cis‐Tri(dimethylformamide)chlorophthalocyaninato(2–)yttrium(III) etherate ( 1 ) crystallises from a solution of [Y(Cl)pc^2–] in MeOH/dmf, cis‐tetra(dimethylsulfoxide)phthalocyaninato(2–)yttrium(III) chloride etherate methanol disolvate ( 2 ) from thf/dmso, μ‐di(chloro)‐μ‐di〈di(pyridine)(μ‐water)〉di(phthalocyaninato(2–)‐ yttrium(III)) ( 5 ) from py, and cis‐(chloro)pyridine(triphenylphosphine oxide)phthalocyaninato(2–)yttrium(III) semi‐etherate ( 3 ) is obtained from a solution of [Y(Cl)pc^2–] and triphenylphosphine oxide in py. 1 condenses in MeOH yielding a (1 : 1)‐mixture ( 4 ) of μ‐di(chloro)di(〈trans‐(diwaterdimethanol)〉〈dimethanol〉phthalocyaninato(2–)yttrium(III)) ( 4 a ) and μ‐di(chloro)di(dimethylformamide〈dimethanol〉phthalocyaninato(2–)yttrium(III)) ( 4 b ); co‐ordinatively bound solvent clusters are in brakets. The structures of 1 – 5 have been established by X‐ray crystallography. Apart from 3 with hepta‐co‐ordinated yttrium, the metal ion prefers octa‐co‐ordination, and the bond arrangement around Y³⁺ is always a distorted quadratic antiprism. In the dinuclear complexes obtained by solvent controlled condensation both antiprisms share an edge by two μ‐Cl atoms in 4 , while in 5 the antiprisms are face‐shared by two trans positioned μ‐Cl atoms and μ‐O atoms, respectively. In 5 , the bent ^b〈{py}₂(μ‐H₂O)〉 cluster is stabilised by a combined interplanar bonding of pyridine by short N…H–O bonds (d(N…O) = 2.664(7) Å; 2.81(2) Å) and strong van‐der‐Waals interactions with the ecliptic pc^2– ligands. 4 a and 4 b contain the dimeric methanol cluster 〈(MeOH)₂〉, and 4 a in addition the cyclic heterotetrameric trans‐diwaterdimethanol cluster, trans^c〈(H₂O)₂(MeOH)₂〉. The neutral clusters co‐ordinatively bound to the Y atom are compared with structurally established cluster‐anions of type 〈(OMe)(MeOH)〉^–, linear ^l〈(OMe)(MeOH)₂〉^–, cyclic ^c〈(OH)₃(H₂O)₃〉^3–, ^b〈{H₂O}₂(μ‐O)〉^2–, and ^b{H₂O}₂(μ‐F)〉^–.     

Convergence of the partial wave expansion of the He ground state

The configuration interaction (CI) method, using a very large Laguerre orbital basis, is applied to the calculation of the He ground state. The largest calculations included a minimum of 35 radial orbitals for each ? ranging from 0 to 12, resulting in basis sets in excess of 400 orbitals. The convergence of the energy and electron–electron δ‐function with respect to J (the maximum angular momenta of the orbitals included in the CI expansion) were investigated in detail. Extrapolations to the limit of infinite angular momentum using expansions of the type ΔX_J = A_X[J + 1/2]^?p + B_X[J + 1/2]^?p?1 + …, gave an energy accurate to 10^?7 Hartree and a value of 〈δ〉 accurate to about 0.5%. Improved estimates of 〈E〉 and 〈δ〉, accurate to 10^?8 Hartree and 0.01%, respectively, were obtained when extrapolations to an infinite radial basis were done prior to the determination of the J → ∞ limit. Round‐off errors were the main impediment to achieving even higher precision, since determination of the radial and angular limits required the manipulation of very small energy and 〈δ〉 differences. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2007     

Configuration interaction benchmark for Be ground state

A set of 432 energy-optimized Slater-type radial orbitals together with spherical harmonics up to ? = 30 is used to approximate the corresponding full configuration interaction (CI) expansion for Be ground state. An analysis of radial and angular patterns of convergence for the energy yields a basis set incompleteness error of 8.7 μhartree of which 85% comes from radial basis truncations for ? ≤ 30. Select-divide-and-conquer CI (Bunge in J Chem Phys 125:014107, 2006; Bunge and Carbó-Dorca in J Chem Phys 125:014108, 2006) produces an energy upper bound 0.02(1) μhartree above the full CI limit. The energy upper bound E = ?14.6673473 corrected with these two truncation energy errors yields E = ?14.6673560 a.u. (Be) in fair agreement with the latest explicitly correlated Gaussian results of E = ?14.66735646 a.u. (Be). The new methods employed are discussed. It is acknowledged that at this level of accuracy traditional atomic CI has reached a point of diminishing returns. Modifications of conventional (orbital) CI to seek for significantly higher accuracy without altering a strict one-electron orbital formalism are proposed.     

Upper Bound to the Expectation Value of the Squared Hamiltonian

An upper bound to the expectation value of the squared Hamiltonian H ² is derived which relies on replacing products of certain operators with products of the matrix representations of said operators to reduce the computational demands of H ². An example is given which shows the strength of the bound and an application with the Temple lower bound is shown.     

High‐precision Hy–CI variational calculations for the ground state of neutral helium and helium‐like ions

Hylleraas–configuration interaction (Hy–CI) method variational calculations with up to 4648 expansion terms are reported for the ground ¹S state of neutral helium. Convergence arguments are presented to obtain estimates for the exact nonrelativistic energy of this state. The nonrelativistic energy is calculated to be ?2.9037 2437 7034 1195 9829 99 a.u. Comparisons with other calculations and an energy extrapolation give an estimated nonrelativistic energy of ?2.9037 2437 7034 1195 9830(2) a.u., which agrees well with the best previous variational energy, ?2.9037 2437 7034 1195 9829 55 a.u., of Korobov (Phys Rev A 2000, 61, 64503), obtained using the universal (exponential) variational expansion method with complex exponents (Frolov, A. M.; Smith, V. H. Jr. J Phys B Atom Mol Opt Phys 1995, 28, L449). In addition to He, results are also included for the ground ¹S states of H^?, Li⁺, Be⁺⁺, and B⁺³. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2002     

Hylleraas method for many‐electron atoms. I. The Hamiltonian

A general expression for the nonrelativistic Hamiltonian for n‐electron atoms with the fixed nucleus approximation is derived in a straightforward manner using the chain rule. The kinetic energy part is transformed into the mutually independent distance coordinates r_i, r_ij, and the polar angles θ_i, and φ_i. This form of the Hamiltonian is very appropriate for calculating integrals using Slater orbitals, not only of states of S symmetry, but also of states with higher angular momentum, as P states. As a first step in a study of the Hylleraas method for five‐electron systems, variational calculations on the ²P ground state of boron atom are performed without any interelectronic distance. The orbital exponents are optimized. The single‐term reference wave function leads to an energy of ?24.498369 atomic units (a.u.) with a virial factor of η = 2.0000000009, which coincides with the Hartree–Fock energy ?24.498369 a.u. A 150‐term wave function expansion leads to an energy of ?24.541246 a.u., with a factor of η = 1.9999999912, which represents 28% of the correlation energy. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2005     

Diffusion Monte Carlo study of correlation in the hydrogen molecule

The diffusion Monte Carlo (DMC) method shows that correlation in H₂ produces a set of three spatial changes: (i) an enhancement in the electron density distribution n( r ) in the left and right anti‐binding regions that include separately the immediate vicinity of each of the two nuclei, (ii) a reduction in n( r ) in the binding region intervening between the two nuclei as a counterbalance, and (iii) a concomitant increase in the equilibrium internuclear separation. It is stressed that the correlation energy E^c (= T^c + V^c) for diatomic molecules be defined by the difference in the total energy between the exact and the Hartree–Fock (HF) variational calculations that are performed at individually optimized internuclear separations. It is this definition that makes it possible to involve a significant contribution from a correlation‐induced change in the equilibrium internuclear separation as part of the correlation energy and to relate (i) and (ii) to (iii) in consistency with the electrostatic theorem. The present calculations fulfill the virial theorem to an accuracy of ?V/T = 2.00 for DMC and ?V^HF/T^HF = 2.000 for HF. The present correlation energy E^c = ?0.0408 hartree is not only in good agreement with the most accurate value previously reported, but also can be analyzed into all its components in accordance with the correlational virial theorem 2T^c + V^c = 0. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2007     

Conformation study of poly(p‐dioxanone) fibers by polarized Raman spectroscopy,X‐ray diffraction,and conformation analysis

The molecular orientation distribution of poly(p‐dioxanone) (PPDX) uniaxially oriented commercial fibers was determined by polarized Raman spectroscopy and X‐ray diffraction. The order parameters 〈P₂₀₀〉 and 〈P₄₀₀〉 of the orientation distribution function were determined by polarized Raman spectroscopy. For the C?O stretching band, the values of 〈P₂₀₀〉 and 〈P₄₀₀〉 obtained are equal to ?0.40 ± 0.04 and 0.28 ± 0.04, respectively. These results clearly indicate that the carbonyl groups are highly oriented perpendicular to the fiber axis. X‐ray diffraction led to a fiber repeat value of 0.628 nm for these samples, and to 〈P₂₀₀〉 and 〈P₄₀₀〉 values of 0.93 and 0.82, respectively, for the c‐axis orientation, indicating a high orientation in the draw direction of the fibers. A Monte‐Carlo conformational search led to 20 low‐energy conformations, but only one of these was found compatible with both the fiber repeat and the angle between the C?O bond and the fiber axis. This conformation, a 2₁ helix with a tg^?ttg^? succession of torsion angles, is proposed as the existing conformation in the crystalline state. © 2008 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 46: 406–417, 2008     

A procedure for obtaining lower bounds for ground state energies of atoms and molecules

Using Löwdin's partition method we have re-derived the D. Weinstein lower bound, E > H ₁₁ - . By the same method, plus the assumption that the calculated first excited state energy is lower than a certain weighted average of approximate energies of all the excited states, we have derived a moderately better lower bound.     

On optimization procedures based upon the atomic thomas-fermi energy

For the total atomic Thomas-Fermi (TF) energy many expressions in terms of the kinetic and potential energy contributions can be given. Thirty of these expressions exhibit either a maximum or a minimum if some variational approximation to the TF function is used. Three of these expressions, to note E, G, and J (see text) have been used in an optimization procedure, in which four two-parameter and two three-parameter approximate TF functions have been considered. One-parameter functions cannot be optimized, as the one parameter must be fixed to ensure proper normalization. It is found that optimization of E and G give reasonable and similar results, whereas the results of optimization of J are generally not very impressive. Where possible, expectation values of the type 〈rⁿ〉 (with n = ?1, 1, 2, and 3) have been calculated from ten approximate TF functions. A new estimate of the exact atomic TF energy, as well as of the derivative of the TF function at the origin, has been obtained.     

Resonance states of atomic anions

We study destabilization of an atom in its ground state with decrease of its nuclear charge. By analytic continuation from bound to resonance states, we obtain complex energies of unstable atomic anions with nuclear charge that is less than the minimum “critical” charge necessary to bind N electrons. We use an extrapolating scheme with a simple model potential for the electron, which is loosely bound outside the atomic core. Results for O^2? and S^2? are in good agreement with earlier estimates. Alternatively, we use the Hylleraas basis variational technique with three complex nonlinear parameters to find accurately the energy of two‐electron atoms as the nuclear charge decreases. Results are used to check the less accurate one‐electron model. © 2001 John Wiley & Sons, Inc. Int J Quant Chem 82: 255–261, 2001     

Some remarks on the perturbation and variation–perturbation calculations of the free energy for quantum systems

A generalization of the Gibbs–Bogoliubov inequality F ? F₀ + 〈H ? H₀〉₀ for the free energy F is studied which leads to a variation principle for this quantity that may be of importance in certain computational applications to quantum systems. This approach is coupled with a study of the perturbation expansion of the free energy for a canonical ensemble with H = H₀ + λV in the general case when H₀ and V do not commute. The second- and high-order derivatives of the free energy with respect to the perturbation parameter λ are calculated. From the second-order term is finally obtained a second-order correction to the previous variational minimum for the free energy.     

Constrained self-consistent-field wave functions with improved long-range behavior

The recent suggestion that the long-range behavior of energy-optimized Gaussian basis sets can be improved by augmenting them with a Gaussian chosen to satisfy a constraint involving a linearly averaged position moment is explored. Calculations indicate that the high-order moments 〈r^k〉, with k > 4, in He, Be, and Li^?, and 〈x^kz^L?k〉, with L > 4 and k ≤ L, in H₂ are improved by the constraint, but that lower-order moments and dipole polarizabilities are not. In H₂, the higher moments with a given L improve by different amounts for different k, and, hence, the multipole moments do not improve. The basis-set superposition error in He? He and Be? Be interaction energy calculations decreases if the internuclear distance is large enough. Thus, the constraint procedure improves the very long range behavior of the self-consistent-field wave functions. © 1992 John Wiley & Sons, Inc.     

Coumarinyl azoimidazolyl complexes of osmium(II) hydridocarbonyls: spectroscopic and structural characterization,oxidation catalysis,photovoltaic effect and density functional theory computation

Os(II) hydridocarbonyl complexes of coumarinyl azoimidazoles, [Osh(CO)(PPh₃)₂(CZ‐4R‐R′)]^0/+ ( 3 , 4 ) (CZ‐R‐H = 2‐(coumarinyl‐6‐azo)‐4‐substituted imidazole or 1‐alkyl‐2‐(coumarinyl‐6‐azo)‐4‐substituted imidazole), were characterized from spectroscopic data and the single‐crystal X‐ray data for one of the complexes, [Osh(CO)(PPh₃)₂(CZ‐4‐Ph)] ( 3c ) (CZ‐4‐Ph = 2‐(coumarinyl‐6‐azo)‐4‐phenylimidazolate), confirmed the structure. The complexes show higher emission (quantum yield ? = 0.0163–0.16) and longer lifetime (τ = 1.4–10.3 ns) than free ligands (? = 0.0012–0.0185 and τ = 0.685–1.306 ns). Cyclic voltammetry shows quasi‐reversible metal oxidation at 0.67–0.94 V for [Os(III)/Os(II)] and 1.21–1.36 V for [Os(IV)/Os(III)] and subsequent azo reductions (?0.68 to ?0.95 V for [? N?N? ]/[? N N? ]^? and irreversible < ?1.2 V for [? N N? ]^?/[? N? N? ]^2?) of the chelated coumarinyl azoimidazole. The complexes are photostable and show better photovoltaic power conversion efficiency than free ligands. Also, the complexes were used as catalysts for the oxidation of primary/secondary alcohols to aldehydes/ketones using oxidizing agents like N‐methylmorpholine N‐oxide, t‐BuOOH and H₂O₂. Density functional theory computation was carried out from the optimized structures and the data obtained were used to interpret the electronic and photovoltaic properties. Copyright © 2016 John Wiley & Sons, Ltd.