Visualization of electron correlation in a series of helium S states

Electron correlation has been studied for a series of helium S states represented by a variety of wavefunctions, the best of which are accurate Hylleraas—Kinoshita functions. The states studied are the ground state, the lowest excited ¹S and ³S states, and the (2s)² and (2p)² doubly-excited ¹S states. Primary data is obtained from graphs of the conditional probability density as a function of the radial distance r₂ and the interelectronic angle θ₁₂, given that r₁ is fixed at various distances. Such graphs make clear the extent to which characteristics such as angular and radial correlations, and Fermi and Coulomb holes, are consequences of the relative motion of electrons in two-electron atoms.     

Atoms-in-molecules from the Stockholder Partition of the Molecular Two-electron Distribution

The effective one-electron distributions of bonded atoms obtained from the “stockholder” partition of the molecular two-electron density are reported. These two-electron stockholder (S) atoms are compared with their one-electron analogs represented by the corresponding Hirshfeld (H) one-electron stockholder pieces of the molecular electron density. The influence of the exchange (Fermi) and Coulomb correlation between electrons on the resultant shapes of bonded atoms is investigated The vertical (for the fixed molecular electron density) and horizontal (involving the electron density displacement) correlation influences on the two-electron stockholder atoms are examined. The two sets of bonded stockholder atoms in the near-dissociation bond-elongated diatomics are compared for different approximations of the electron correlation effects. The cluster components in atomic resolution of the S-partitioning scheme are investigated for illustrative homonuclear and heteronuclear diatomics: H₂, LiH, HF, LiF, and N₂. This framework facilitates an understanding of the origins of the observed differences between the S and H variants of Atoms-in-Molecules. With the exception of hydrogen atoms, especially in light molecules, the two sets of bonded atoms were found to be practically identical. For H₂ and LiH the S atoms were shown to exhibit a distinctly higher degree of the bonding character, compared to their H analogs. The main electron correlation effects have been found to be well represented already at the exchange-only level, e.g., in the unrestricted Hartree–Fock (UHF) theory. An inclusion of the extra vertical Coulomb correlation exerts a marginal moderating influence on the ionic/covalent composition of the chemical bond already predicted by the UHF approximation, in the direction of a slightly more covalent (less ionic) bond character. The horizontal shifts of the molecular density due to Coulomb correlation, relative to the UHF reference, often act in the opposite direction.     

Approximate ansatz for the expansion of the spherically averaged wave function in terms of interelectronic separation r12 for the Hookean atom,atomic ions,and the H2 molecule

For the two‐electron Hookean atom, it is first emphasized that, for a specific force constant k = 1/4, the ground‐state wave function has a simple dependence on the interelectronic separation r₁₂, namely, (1 + ½r₁₂)exp(??r). For this two‐electron model, therefore, the study of Rassolov and Chipman on the electron–electron cusp conditions on the spherically averaged wave function for the N electron atomic ions can be generalized to all orders in the interelectronic separation r₁₂. This Hookean model has therefore been used to give some justification for an ansatz for the spherically averaged wave function in atomic ions with N electrons for N ≥ 2. Several approximate two‐electron wave functions satisfying the Rassolov and Chipman conditions were tested and found to give excellent results. Another ansatz has been tested numerically on the ground state of two‐electron atomic ions and the H₂ molecule. Finally, for the Hookean atom a partial differential equation that is essentially for the pair correlation density is given in the Appendix . © 2003 Wiley Periodicals, Inc. Int J Quantum Chem 95: 21–29, 2003     

Ionisation of small molecules by state-selected Ne* (3P0, 3P2) metastable atoms in the 0.06 < E < 6 eV energy range

The velocity dependence and absolute values of the total ionisation cross section for the molecules H₂, N₂, O₂, NO, CO, N₂O, CO₂, and CH₄ by metastable Ne* (³P₀) and Ne* (³P₂) atoms at collision energies ranging from 0.06 to 6.0 eV have been measured in a crossed beam experiment. State selection of the two metastable states of Ne* was obtained by optical pumping with a cw dye laser. We observe a strongly different velocity dependence at collision energies below about 1 eV for the ionisation cross section of the systems Ne*H₂, N₂, CO, and CH₄, and the systems Ne*O₂, NO, CO₂, and N₂O, respectively. The first group shows an increasing cross section in this energy range, similar to the Ne*Ar system, while the second group shows a very flat behaviour. This behaviour correlates with the difference in character (π or σ^b) of the orbital of the electron that is removed from the target molecule. For the molecules H₂, N₂, CO, and CH₄ an electron from a σ^b orbital is removed from the molecule, whereas for O₂, NO, N₂O, and CO₂ an outer π-ortibal electron is involved. For the systems Ne* (³P₀, ³P₂)H₂ we have derived the imaginary part of the optical potential by assuming a real potential similar to the theoretically calculated ground state NaH₂ potential of Botschwina et al. The resonance width Γ(r) as a function of the internuclear distance r shows a saturation at small r (r < 2.8 Å) for both the Ne*(³P₀)H₂ and the Ne*(³P₂)H₂ interaction. This supports previous conclusions of Verheijen et al. and Kroon et al. Reliable values for the absolute value of the total ionisation cross section have been obtained by performing a careful calibration of the density—length product of the supersonic secondary beam. The results are in good agreement with the values of West et al. for experiments without state selection. The total ionisation cross sections for molecules with π-type ionisation orbitals, with their larger spatial extent, in general are larger than those for molecules with σ^b-type ionisation orbitals.     

The collisional behaviour of the spin orbit states of the lead atom,Pb(63P1,2), studied by time-resolved attenuation of atomic resonance radiation

A kinetic study of lead atoms in the spin orbit states, Pb(6³P₁) and Pb Pb(6³P₂), 0.969 and 1.320 eV, respectively, above the 6³P₀ ground state, has been carried out by atomic absorption spectroscopy. The electronically excited lead atoms were generated by the pulsed irradiation of lead tetraethyl and monitored photoelectrically by time-resolved attenuation of resonance radiation. The decay of the two atomic states has been studied in the presence of He, Ar, H₂, D₂, N₂, O₂, CO, NO, CO₂, N₂O, CH₄, C₂H₄, C₂H₂ CF₄, SF₆, and PbEt₄, and rate constants for the collisional quenching by these gases are reported. The resulting data are compared with those for the deactivation of other atomic spin orbit states of comparable energy. In general, the higher energy state, Pb(6³P₂), is found to be deactivated more rapidly. It would appear that the magnitude of the electronic energy to be transferred on collision governs the rates of quenching, at least where a weak interaction potential is involved, and that for most gases, deactivation of Pb(6³P₂) proceeds via Pb(6³P₁).     

Simple extended STO basis sets. Helium

The two-parameter function, φ = (C₁ + C₂r^n?1) exp (?ζr), (n = 2–5), has been used as a basis function to determine the independent particle model energy of two-electron atomic systems in their ground state. The best energy is found for n = 3 (He—B³⁺) and for n = 4 (H^?). Our energy values are significantly close to Hartree-Fock results.     

Extremal electron pairs

Extremal pair functions for an n-electron wave function of a closed-shell state are defined as linear combinations of spin-orbital-product pair functions that make some functionals (e.g., r²₁₂ or r⁻¹₁₂) extremal. They are related to the natural spin geminals in the uncorrelated limit and are useful both for an analysis of wave functions in view of an understanding of the chemical bond and for the treatment of electron correlation. Numerical examples are shown and discussed for He₂ as well as the 10-electron systems Ne, HF, H₂O, NH₃, and CH₄. © 1996 John Wiley & Sons, Inc.     

原子配分参数、配分连接性指数及其应用

结构决定性质，性质反映结构，这是化学、生物学等学科的一条基本规律。因此，分子的微观结构与性质之间存在着密切关系。物质的理化性质、生物活性等数据的获取，迄今主要来自于实验。如能建立结构与性能之间的数量关系用以估算与预测分子的性质，这无疑是一项十分有意义的工作。拓扑指数法以其计算简单、准确性高、应用范围广而在上述领域中发挥重要作用犤１～４犦。拓扑指数是对分子结构进行的定量描述，使分子之间的结构差异定量化。自Wiener提出第一个拓扑指数（W）犤５犦以来，迄今已有２００余种拓扑指数问世。其中一部分能够有效地…     

Optimal Hylleraas wave functions

Hylleraas wave functions composed of the optimally combinedN terms (2 N 20) are presented for two-electron atoms with nuclear chargesZ = 1 (H^–), 2(He), 3(Li⁺), 5(B³⁺), and 10(Ne⁸⁺). The spherically-averaged electron density (r) and electron-pair densityh(r ₁₂) are constructed in a simple and analytical functional form from the 20-term functions. Comparison of several one- and two-electron moments r ^k and r ₁₂ ^k shows that the present density functions have near-exact accuracy.     

Spatial correlation of atomic electrons: He

For any two-electron wavefunction whose angular dependence is given in terms of the spherical harmonics of the individual electrons and/or o₁₂, where o₁₂ is the interelectronic angle. a transformation is generated which reduces |ψ(r₁, r₂, o₂, o₁, o₁₂ )|² to |ψ(r₁, r₂, o₁₂)|². Geometric aspects of electron correlation are analyzed in terms of this resulting three-coordinate function, with specific application to a number of wave functions for doubly-excited helium. Angular correlation has been studied by integrating |ψ(r₁, r₂, o₁₂)|² over its radial dependence to yield p(o₁₂), the probability density function for the interelectronic angle. Trends of p(o₁₂) for doubly-excited states of the helium atom are related to a number of quantities including energies and autoionization widths. These trends can be rationalized in terms of a simple classical model. The full spatial correlation involving |ψ(r₁, r₂, o₁₂)|² is explored by the use of three-dimensional graphs for some of these states.     

On the behaviour of the wave function for the 23S state of the two-electron atom in the region where both electron–nuclear distances are small

The expansion of the wave function for the 2³S state of the two-electron atom in the neighbourhood of the singularity at r₁ = r₂ = 0 is considered. The restrictions imposed on the variational functions by this expansion are discussed. For the 2³S state of He, Li⁺, N⁵⁺ the behaviour of the variational function based on the Fock expansion in the neighbourhood of this singularity is investigated. The agreement of the variational coefficients with the theoretical coefficients is satisfactory. The calculated values of E and 〈δ(r₂)〉 for He, Li⁺, N⁵⁺ are given.     

Accurate ab initio potential energy surface and vibration‐rotation energy levels of hydrogen peroxide

The accurate ground‐state potential energy surface of hydrogen peroxide, H₂O₂, has been determined from ab initio calculations using the coupled‐cluster approach in conjunction with the correlation‐consistent basis sets up to septuple‐zeta quality. Results obtained with the conventional and explicitly correlated coupled‐cluster methods were compared. The core–electron correlation, scalar relativistic, and higher‐order valence–electron correlation effects were taken into account. The adiabatic effects were also discussed. The vibration–rotation energy levels of the H₂O₂, D₂O₂, and HOOD isotopologues were predicted, and the experimental vibrational fundamental wavenumbers were reproduced to 1 cm^?1 (“spectroscopic”) accuracy. © 2012 Wiley Periodicals, Inc.     

Kinetics of Depletion of Electronically Excited Ti Atom by CH4, C2H2, C2H4 and C2H6

The kinetics of depletion of ground state Ti(a³F) and electronically excited state Ti(a⁵F) upon interactions with CH₄, C₂H₂, C₂H₄, and C₂H₆ are studied in a fast-flow reactor at a He pressure of 0.70 Torr. No depletion of ground state Ti(a³F) was observed upon interaction with all hydrocarbons studied here. Two alkanes, CH₄ and C₂H₆, were also quite inert for depletion of the excited state Ti(a⁵F), On the other hand, C₂H₂ and C₂H₄ deplete the excited state Ti(a⁵F) very efficiently. Rate constants were determined to be (266 ± 86) and (476 ± 88) × 10^?12 cm³s^?1 for Ti(a⁵F) + C₂H₄ and Ti(a⁵F) + C₂H₂, respectively. These large rate constants compared with the ground state Ti were explained by an electron donor-acceptor interaction model that works in the interaction between C₂H₄ or C₂H₂ and the excited state with unfilled 4s orbital.     

Photoinduced Electron Transfer in Tetrathiafulvalene−Porphyrin−Fullerene Molecular Triads

The two molecular triads 1a and 1b consisting of a porphyrin (P) covalently linked to a fullerene (C₆₀) electron acceptor and tetrathiafulvalene (TTF) electron‐donor moiety were synthesized, and their photochemical properties were determined by transient absorption and emission techniques. Excitation of the free‐base‐porphyrin moiety of the TTF−P_{2 H}−C₆₀ triad 1a in tetrahydro‐2‐methylfuran solution yields the porphyrin first excited singlet state TTF−¹P_{2 H}−C₆₀, which undergoes photoinduced electron transfer with a time constant of 25 ps to give TTF−P_{2 H}^.+−C₆₀^.−. This intermediate charge‐separated state has a lifetime of 230 ps, decaying mainly by a charge‐shift reaction to yield a final state, TTF^.+−P_{2 H}−C₆₀^.−. The final state has a lifetime of 660 ns, is formed with an overall yield of 92%, and preserves ca. 1.0 eV of the 1.9 eV inherent in the porphyrin excited state. Similar behavior is observed for the zinc analog 1b . The TTF‐P_Zn^.+−C₆₀^.− state is formed by ultrafast electron transfer from the porphyrinatozinc excited singlet state with a time constant of 1.5 ps. The final TTF^.+−P_Zn−C₆₀^.− state is generated with a yield of 16%, and also has a lifetime of 660 ns. Although charge recombination to yield a triplet has been observed in related donor‐acceptor systems, the TTF^.+−P−C₆₀^.− states recombine to the ground state, because the molecule lacks low‐energy triplet states. This structural feature leads to a longer lifetime for the final charge‐separated state, during which the stored energy could be harvested for solar‐energy conversion or molecular optoelectronic applications.     

Comment on a perturbation treatment of the ground state of two-electron atoms using the coordinates r< and r>

A perturbation variation treatment of two-electron atoms using exp (–αr_< – βr_>) as the zeroth order wave function is presented. The parameters α and β are variationally determined and the results are compared with the “physical” choice α = Z, β = Z – 1, and with Z^?1 theory. The energy is given through fifth order in the perturbation.     

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     

Quenching mechanism of the UO2 2+ excited state by phenols and related compounds in aqueous solutions

Quenching rate constants,k _q, of the uranyl excited state by phenolic compounds were determined in H₃PO₄ 1M by luminiscence techniques. A plot of logk _q vs. ΔG₂ for the photo-induced electron transfer reaction rendered a slope value of 4.0 eV⁻¹ which was much smaller than that predicted from the Rehm-Weller correlation (16.9 eV⁻¹). This value is similar to those found for aromatic quenchers that fail to interact with the UO₂ ²⁺ ground state, and quench the uranyl luminescence through a non-radiative donor-acceptor complex formation.     

One-Electron Pseudo-Potential Determination of Stable Isomers of the Li*Ar n Excited Clusters: Absorption Spectra

We present pseudo-potential calculations of geometrical structures of stable isomers of LiAr _n clusters with both an electronic ground state and excited states of the lithium atom. The Li atom is perturbed by argon atoms in LiAr _n clusters. Its electronic structure obtained as the eigenfunctions of a single-electron operator describing the electron in the field of a Li⁺Ar _n core, the Li⁺ and Ar atoms are replaced by pseudo-potentials. These pseudo-potentials include core-polarization operators to account for the polarization and correlation of the inert core with the valence Lithium electron [J Chem Phys 116, 1839 1]. The geometry optimization of the ground and excited states of LiAr _n (n = 1–12) clusters is carried out via the Basin-Hopping method of Wales et al. [J Phys Chem 101, 5111 2; J Chem Phys 285, 1368 3]. The geometries of the ground and ionic states of LiAr _n clusters were used to determine the energy of the high excited states of the neutral LiAr _n clusters. The variation of the excited state energies of LiAr _n clusters as a function of the number of argon atoms shows an approximate Rydberg character, corresponding to the picture of an excited electron surrounding an ionic cluster core, is already reached for the 3s state. The result of optical transitions calculations shows that the absorption spectral features are sensitive to isomer structure. It is clearly the case for transitions close to the 2p levels of Li which are distorted by the cluster environment.     

Transition state geometry in radical abstraction reactions: comparison of interatomic distances in the intersecting parabolas and Morse curves models with quantum-chemical calculations

Interatomic distances in the transition state were estimated for the reactions of radical abstraction: H^· + H₂, H^· + HCl, H^· + CH₄, N^·H₂ + NH₃, HO^· + H₂O, HO₂ ^· + HOOH, and C^·H₃ + SiH₄. The calculation was performed by the quantum-chemical density functional method or coupled clusters method (QCH), as well as by the methods of intersecting parabolas (IPM) and Morse curves (IMM), using experimental data (activation energies and reaction enthalpies). The results of the latter two methods are close to the quantum-chemical calculation and differ only by the increment a: r(IPM or IMM) = a + r(QCH), where a = –4.5·10^–12 m for IPM and a = +1.9·10^–12 m for IMM.