Self-consistent calculation of 1s2s1S state wave function of helium

A simple self consistent variation perturbation method in the coupled Hartree–Fock scheme has been proposed to calculate 1s2s ¹S state of the He atom. The present paper deals with an 1s2s ¹S wave function in which all the relevant orthogonality conditions are imposed in successive stages. The resulting wave functions together with some interesting features are discussed.     

Self-consistent calculation of 1s2s3S state of helium and heliumlike ions

In this paper we present a simple method within the coupled Hartree–Fock framework to calculate the 1s2s ³S state of helium and heliumlike ions. The results are in very good agreement with those obtained by the use of multiterm correlated wave functions. Some interesting observations associated with the wave function are presented.     

Relativistic calculation of autoionization widths of 1s2s2pjJ, 1s2s2 2S1/2, and 1s2pJ 2pj′J levels

The autoionization widths of levels 1s 2s 2p_jJ, 1s2s^{2 2}S_1/2, and 1s2p_j2p_j′ J have been calculated for ions with Z = 6–30. The calculation has been carried out in intermediate coupling. The decay amplitudes have been calculated in a relativistic approximation.     

Precision energies and hyperfine structures of the 5s5p 3 P 0,1,2 0 and 5s6s 3 S 1 levels in In II

Absolute frequencies of hyperfine components of the 230.6 nm (5s ^{2 1} S ₀?5s5p ³ P ₁ ⁰ ), 193.6 nm (5s5p ³ P ₀ ⁰ ?5s6s ³ S ₁), 197.7 nm (5s5p ³ P ₁ ⁰ ?5s6s ³ S ₁) and 207.9 nm (5s5p ³ P ₂ ⁰ ?5s6s ³ S ₁) transitions in In II emitted from a hollow-cathode source have been measured using a high-resolution, scanning échelle monochromator. The measured frequencies of these four transitions have been used to determine the energies and hyperfine interaction constants of the 5s5p ³ P ₀ ⁰ ,³ P ₁ ⁰ ,³ P ₂ ⁰ and 5s6s ³ S ₁ levels in In II. The hyperfine interaction constants for the dominant isotope¹¹⁵In are found to be: 5s5p ³ P ₁ ⁰ A=0.2322(2) cm^?1,B=?0.0159(9) cm^?1 5s5p ³ P ₂ ⁰ A=0.1699(4) cm^?1,B= 0.021 (6) cm^?1 5s6s ³ S ₁ A=0.4022(4) cm^?1,B= 0.002 (2) cm^?1. The absolute frequency of the very narrow, strongly forbidden In II 236.5 nm (5s ^{2 1} S ₀?5s5p ³ P ₀ ⁰ ) transition, which has been proposed as a candidate for a new optical frequency standard, is found to be 42275.986(7) cm^?1.     

The reactions of S(3PJ) atoms with thiiranes

The reactions of ground-state S(³P_J) atoms with thiirane, methylthiirane, and trans-2,3-dimethylthiirane have been studied by flash photolysis-VUV kinetic absorption spectroscopy. From the analysis of the S(³P_J) decay plots the following rate constants were determined: (1.4 ± 0.2) × 10¹³, (2.7 ± 0.3) × 10¹³ and (4.0 ± 0.2) × 10¹³ (in cm³ mol^?1 s^?1 units) for thiirane, methylthiirane and trans-2,3-dimethylthiirane, respectively, showing an upward trend with increasing methylation.     

Effect of different subsets on convergence patterns of hyperspherical harmonic expansion for the S states of the helium atom

Effects of different subsets on convergence patterns of hyperspherical harmonic (HH) expansions for the low-lying ¹S and ³S states of the helium atom have been investigated with the correlation-function-hyperspherical-harmonic-generalized-Laguerre-function (CFHHGLF) method by successively introducing HH subsets with the fixed three-dimensional angular momentums (l) into the atomic wave functions. The eigenenergies given by the HH subsets of l=0, 1, 2, and 3 are in good agreement with the best s-, sp-, spd-, and spdf limits of variational configuration interaction (CI) calculations, respectively. The final eigenenergies of the ground state as well as the examined low-lying excited ¹S and ³S states are quite close to the exact Hylleraas CI (HCI) values at the sixth decimal place. Moreover, l=0 and l≠0 expansion results also tell us that it is not necessary to take into account too many HHs at the given l, especially for higher l, and that the more the absolute electron correlation energies the bigger l it takes to obtain precise eigenenergies. © 1997 John Wiley & Sons, Inc. Int J Quant Chem 64 : 661–668, 1997     

Electronic structures of the S2O and S3 isomers: an ab initio CI study

The electronic structures of the S₂O and S₃ isomers have been dealt with by the multireference double-excitation (MRD) configuration-interaction (CI) calculations, using contracted [5s3p1d] and [4s2p1d] basis functions for the S and O atoms, respectively. The ground-state geometries for the SOS (symmetric chain), S₂O (symmetric ring) and SSO (unsymmetric chain) are optimized, and their vertical singlet excitation energies are calculated. It is found that SSO is the most stable of the three isomers and that the ground state (¹A₁) of the S₂O (ring) is correlated with the excited states of SOS (2¹A₁) and SSO (3¹A). The chain and ring isomers of S₃ have been treated in a similar manner. Energetics for the ring closure of the O₃, SO₂, SSO and S₃ chain molecules are discussed on a unified ground.Dedicated to Professor J. Koutecký on the occasion of his 65th birthdayPresented at the 5th International Congress on Quantum Chemistry, Montréal, August 1985     

Doubly excited3Se,3De and3Ge states of two-electron atomic systems

Summary Time-dependent perturbation theory has been applied to calculate the doubly excited triplet statesNsns:³S^e,Npnp:³D^e andNdnd:³G^e (N=2, 3, 4,n=N+1, ... ,5) for He, Li⁺, Be²⁺ and B³⁺. A time-dependent harmonic perturbation causes simulataneous excitation of both the electrons with a change of spin state. The doubly excited energy levels have been identified as the poles of an appropriately constructed linearized variational functional with respect to the driving frequency. In addition to the transition energies, effective quantum numbers of these doubly excited states have been calculated and analytic representations of their wave functions are obtained. These are utilized to estimate the Coulomb repulsion term for these states which checks the consistency of the wave functions. These wave functions may also be used for calculating other physical properties of the systems.     

Synthesis and Stereochemistry of (1S,3S,4S,7R11R)-3-(4-Nitrophenyl)-11-aza-2,6-dioxatricyclo[5,3,1,04,11]undecane

The reaction of (1S,2S)-2-amino-1-(4-nitrophenyl)-1,3-propanediol with glutaraldehyde has been studied. It has been established on the basis of AM1 and PM3 calculations and ¹H NMR spectra recorded in the presence of the shift reagent Eu(fod)₃ that (1S,3S,4S,7R,11R)-3-(4-nitrophenyl)-11-aza-2,6-dioxatricyclo[5,3,1,0^4,11]undecane is formed as the result of the reaction.     

Time-dependent Hartree-Fock calculations for the excited ‘S’ states of lithium isoelectronic sequence

Time-dependent HF (RPA) calculations have been performed to estimate the transition energies and excited state wavefunctions of Li, Be⁺, B²⁺ and C³⁺ for the transitions 1s ²2s ² S 1s ² ns ² S (n = 3,4, ... 8). The excitation energies and excited state wavefunctions are extracted from the position of the poles of a linearised variational functional. The excitation energies are in excellent agreement with those obtained spectroscopically. The excited state wavefunctions are utilised to find the matrix elements of different operators and the cusp values.     

(R,R)-2,3-Butanediol and (s)-pinanediol allylboronates in chiral synthesis of (2S,3S)-3-methyl-5-hexen-2-ol

(R,R)-Butanediol (dichloromethyl)boronate ( 1 ) with 1 equiv. allylmagnesium halide yields (R,R)-2,3-butanediol (1S)-(1-chloro-3-butenyl)boronate ( 3 ) together with the diallylated product (R,R)-2,3-butanediol (1-allyl-3-butenyl)boronate ( 4 ). The formation of 4 is unprecedented in reactions of α-chloroboronic esters with Grignard reagents. With methylmagnesium bromide 3 yielded (R,R)-2,3-butanediol (1S)-(1-methyl-3-butenyl)boronate ( 5 ), which failed to hydrolyze with water. Hydrolysis of 3 yielded impure α-chloroboronic acid, which was esterified with pinacol and treated with methylmagnesium bromide to form 6 , which with (dichloromethyl)lithium followed by methylmagnesium bromide yielded diastereomeric boronic esters 7 and 8 . Oxidation by hydrogen peroxide yielded (2S,3S)- and (2R,3S)-3-methyl-5-hexen-2-ol ( 9 and 10 , ees unknown). Treatment of (s)-pinanediol allylboronate ( 11 ) with (dichloromethyl)lithium at −100°C followed by zinc chloride at up to 25°C has proceeded in the normal way to form (s)-pinanediol (1S)-(1-chloro-3-butenyl)-boronate ( 12 ), which has been elaborated via 13 , 14 , and 15 to (2S,3S)-3-methyl-5-hexen-2-ol ( 9 ) in 95% de.     

The spin-allowed and spin-forbidden 5s 2 1 S 0-5s5p 1 P 1,3 P 1 transitions in strontium isoelectronic sequence

Relativistic multiconfiguration Dirac-Fock (MCDF) transition energies and oscillator strengths are determined for both the spin-allowed 5s ^{2 1} S ₀-5s5p ¹ P ₁ and the spin-forbidden 5s ^{2 1} S ₀-5s5p ³ P ₁ transitions in the strontium isoelectronic sequence. The modest relativistic configuration mixing to represent intravalence correlation is combined with a polarization model to account for valence-core electron correlations. The multiconfiguration Dirac-Fock calculations are performed in an average level scheme; however for neutral strontium and singly ionized yttrium a thorough comparison of the average and the optimal level schemes is presented. The average level scheme, though less accurate for the neutral end of the sequence, avoids the convergence problems encountered for highly ionized systems, where the 5s 5p ³ P ₁,¹ P ₁ states are raised owing to the collapse of the 4d _{3/2, 5/2} spin-orbitals in the isoelectronic sequence and, thus, allows us to extend our study to multiple charged ions (throughW ³⁶⁺). Since for such systems there is practically no difference between the results of the average and the optimal level versions of MCDF calculations, we believe that our average level predictions of ionization energies and oscillator strengths for states with total angular numberJ=0 andJ=1 are of comparable quality to those that could be obtained with an optimal level scheme.This study was supported by the Pedagogical Academy of Kraków Statutory Activity Grant No BS-29/91     

Lifetimes of the 4s4p 3 P 1 and 4s3d 1 D 2 states of Ca I

The lifetimes of the 4s4p ³ P ₁ and 4s3d ¹ D ₂ metastable states of Ca have been studied using the time-of-flight technique. Two kinds of observations were performed. First, the exponential decay of the fluorescence, using a (continuous) dc discharge for excitation and then the velocity distribution of the radiating atoms, using a pulsed discharge, were measured. From the combined results of these measurements the lifetimes were derived. The lifetimes of the 4s4p ³ P ₁ and 4s3d ¹ D ₂ states of Ca are determined to be 0.57±0.03 ms and 1.5±0.4 ms, respectively.     

Wavelength and isotopic shift measurements of the3S(3s 4s) −3P(3s 3p) MgI transitions

The wavelengths of the³S(3s 4s) ?³P(3s 3p) MgI transitions have been measured for the²⁴Mg isotope with a precision of ±2 × 10^?7; moreover the isotope shifts of the above transitions have been measured for the three natural Mg isotopes as well as the hyperfine constants of the³S(3s 4s) level of²⁵Mg. The measurements have been performed in a metastable atomic beam.     

Synthesis of(+)-(2S, 6S)-trans-α-Irone and of(-)-(2S, 6S)-trans-γ-Irone

A 3:1 mixture of (+)-(2S, 6S)-trans-α-irone ((+)-1) and (?)-(2S, 6S)-trans-γ-irone (?)-2) has been synthesized with ca. 70% e. e. by the ene reaction of (?)-(S)-3 and but-3-yn-2-one.     

Synthesis of stereoregular poly[(2S,3S)-benzyl β-3-methylmalate]

The asymmetric lactone (3 S, 4 R)-3-methyl-4-benzyloxycarbonyl-2-oxetanone ( 6 ) was anionically polymerized to give an insoluble, crystalline, highly isotactic polymer with (2 S, 3 S)-benzyl β-3-methylmalate repeating units. Solubility was achieved by copolymerization of 6 with the recemic (R, S)-butyl malolactonate ( 7 ). The semicrystalline copolymer was characterized (M̄_n = 107 000, T_g = 29,6°C, T_m = 161°C, [α] = 1,5 deg · dm⁻¹ · g⁻¹ · cm³) and its stereosequence investigated by ¹³C NMR.     

Intercombination decay of 3s3p 3 P 1 0 in MgI-like Ni and Cu

The intercombination transition 3s^{2 1}S₀ ?3s3p³P ₁ ⁰ in Ni¹⁶⁺ and Cu¹⁷⁺ has been studied by beam-foil spectroscopic methods. Decay curve analysis yields lifetime values of (12.0±1.0)ns and (8.8±0.6)ns for Ni and Cu in agreement with various predictions.     

Triple differential cross sections in the vicinity of the (2s 2)1 S state of helium

Triple differential cross sections have been measured in the vicinity of the (2s ²)¹ S autoionising state of helium, following impact by 200 eV electrons. The scattered electron detector was set at an angle of ?12° (anti-clockwise) and the forward and backward ejected electron angular ranges scanned. The direct ionisation cross section at an ejected electron energy of 33.5 eV has been obtained and the results for the resonant ionisation of the¹ S state are presented in the Shore/Balashov parametrisation. These measurements are compared with previous experimental data and emphasise the need for new detailed theoretical calculations on the autoionisation process.     

Long-range interaction between 1s and 2s or 2p hydrogen atoms

Long-range interaction energy between two hydrogen atoms has been computed in the second order of the perturbation theory. All states of the system arising when one of the atoms is in the 1s and the other in the 2s or 2p state have been considered. The energy represented by a series expansion in inverse powers of the internuclear distance, R, has been computed up to the terms in R^?8. The results are believed to give reliable interaction energies for R > 15 a.u. Accurate interaction energy for two ground-state hydrogen atoms has also been obtained up to the terms in R^?10. Results for the B′ ¹∑ state are employed to discuss the experimental ground-state dissociation energy of H₂, D₂, and HD. For H₂ all values of the dissociation energy obtained from various experimental absorption limits, by using the computed potential energy curve to separate off the effect of rotation, are shown to be satisfactorily consistent. The resulting total energy of H₂ is, however, higher than the most accurate theoretical value.     

Dendron to Central Core S1–S1 and S2–Sn (n>1) Energy Transfers in Artificial Special Pairs Containing Dendrimers with Limited Numbers of Conformations

Two dendrimers consisting of a cofacial free‐base bisporphyrin held by a biphenylene spacer and functionalized with 4‐benzeneoxomethane (5‐(4‐benzene)tri‐10,15,20‐(4‐n‐octylbenzene)zinc(II)porphyrin) using either five or six of the six available meso‐positions, have been synthesized and characterized as models for the antenna effect in Photosystems I and II. The presence of the short linkers, ‐CH₂O‐, and long C₈H₁₇ soluble side chains substantially reduces the number of conformers (foldamers) compared with classic dendrimers built with longer flexible chains. This simplification assists in their spectroscopic and photophysical analysis, notably with respect to fluorescence resonance energy transfer (FRET). Both steady‐state and time‐resolved spectroscopic measurements indicate that the cofacial free bases and the flanking zinc(II)–porphyrin antennas act as energy acceptor and donor, respectively, following excitation in either the Q or Soret bands of the dendrimers. The rate constants for singlet electronic energy transfer (k_EET) extracted from the S₁ and S₂ fluorescence lifetimes of the donor in the presence and absence of the acceptor are ≤ (0.1–0.3)×10⁹ and ～2×10⁹ s^?1 for S₁→S₁ (range from a bi‐exponential decay model) and about 1.5×10¹² s^?1 for S₂→S_n (n>1). Comparisons of these experimental data with those calculated from Förster theory using orientation factors and donor–acceptor distances extracted from computer modeling suggest that a highly restricted number of the many foldamers facilitate energy transfer. These foldamers have the lowest energy by molecular modeling and consist of one or at most two of the flanking zinc porphyrin antennas folded so they lie near the central artificial special pair core with the remaining antennas located almost parallel to and far from it.