Second-order density matrix for a spin- projected single determinant.
Irreducible tensorial components of two-electron operator and second-order density matrix for spin- projected single-determinantal wave function
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Radioactive transitions in the helium isoelectronic sequence
The principles of the atomic spectrum theory are used to quantitatively analyze radiation transitions in two-electron helium-like atomic systems. Quantum theoretical methods, describing absorption and emission of a single photon in a radiative transition between two stationary states of an atomic system, reproduced the energy level diagram for the low lying states of helium. Reliable values are obtained from accurate variationally determined two-electron nonrelativistic wave functions for radiative transition probabilities of 2 3p states in the helium isoelectric sequence, and for the 2 1s and 2 3s1 states of the helium sequence.
Dalgarno, A.↗
Polarizabilities of two-electron positive ions and Rydberg levels of lithium
Second-order sums are computed for the two-electron positive ions Li(+), Be(2+), and B(3+) by representing the intermediate states as discrete pseudostates. The Hamiltonian for the system is given and translated to semi-Jacobi variables for its derivation in reduced Rydberg units. The interaction potential is developed in its multipole form and employed in second-order perturbation theory. The perturbation is always a multipole-potential term, and the initial and intermediate states are constructed from two-particle Hylleraas basis sets. The resulting pseudostates are shown to represent the continuum with good convergence, and the asymptotic-optical-potential method is applied to the two-body quantities to determine the energy levels of three-electron systems in high Rydberg states. The method is shown to give accurate values for the polarizabilities of the two-electron isoelectronic systems.
Bhatia, A. K.↗
Sum rules and other properties involving resonance projection operators
A sum rule is derived for the auxiliary eigenvalues of an equation whose eigenspectrum pertains to projection operators which describe electron scattering from multielectron atoms and ions. The sum rule's right-hand side depends on an integral involving the target system eigenfunctions. The sum rule is checked for several approximations of the two-electron target. It is shown that target functions which have a unit eigenvalue in their auxiliary eigenspectrum do not give rise to well-defined projection operators except through a limiting process. For Hylleraas target approximations, the auxiliary equations are shown to contain an infinite spectrum. However, using a Rayleigh-Ritz variational principle, it is shown that a comparatively simple aproximation can exhaust the sum rule to better than five significant figures. The auxiliary Hylleraas equation is greatly simplified by conversion to a square root equation containing the same eigenfunction spectrum and from which the required eigenvalues are trivially recovered by squaring.
Berk, A.↗
Interfacing Relativistic and Nonrelativistic Methods: A Systematic Sequence of Approximations
A systematic sequence of approximations for the introduction of relativistic effects into nonrelativistic molecular finite-basis set calculations is described. The theoretical basis for the approximations is the normalized elimination of the small component (ESC) within the matrix representation of the modified Dirac equation. The key features of the normalized method are the retention of the relativistic metric and the ability to define a single matrix U relating the pseudo-large and large component coefficient matrices. This matrix is used to define a modified set of one- and two-electron integrals which have the same appearance as the integrals of the Breit-Pauli Hamiltonian. The first approximation fixes the ratios of the large and pseudo-large components to their atomic values, producing an expansion in atomic 4-spinors. The second approximation defines a local fine-structure constant on each atomic centre, which has the physical value for centres considered to be relativistic and zero for nonrelativistic centres. In the latter case, the 4-spinors are the positive-energy kinetic al ly-balanced solutions of the Levy-Leblond equation, and the integrals involving pseudo-large component basis functions on these centres, are set to zero. Some results are presented for test systems to illustrate the various approximations.
Dyall, Ken↗
Electrochemical Hydrogen Peroxide Generator
Two-electron reduction of oxygen to produce hydrogen peroxide is a much researched topic. Most of the work has been done in the production of hydrogen peroxide in basic media, in order to address the needs of the pulp and paper industry. However, peroxides under alkaline conditions show poor stabilities and are not useful in disinfection applications. There is a need to design electrocatalysts that are stable and provide good current and energy efficiencies to produce hydrogen peroxide under acidic conditions. The innovation focuses on the in situ generation of hydrogen peroxide using an electrochemical cell having a gas diffusion electrode as the cathode (electrode connected to the negative pole of the power supply) and a platinized titanium anode. The cathode and anode compartments are separated by a readily available cation-exchange membrane (Nafion 117). The anode compartment is fed with deionized water. Generation of oxygen is the anode reaction. Protons from the anode compartment are transferred across the cation-exchange membrane to the cathode compartment by electrostatic attraction towards the negatively charged electrode. The cathode compartment is fed with oxygen. Here, hydrogen peroxide is generated by the reduction of oxygen. Water may also be generated in the cathode. A small amount of water is also transported across the membrane along with hydrated protons transported across the membrane. Generally, each proton is hydrated with 3-5 molecules. The process is unique because hydrogen peroxide is formed as a high-purity aqueous solution. Since there are no hazardous chemicals or liquids used in the process, the disinfection product can be applied directly to water, before entering a water filtration unit to disinfect the incoming water and to prevent the build up of heterotrophic bacteria, for example, in carbon based filters. The competitive advantages of this process are: 1. No consumable chemicals are needed in the process. The only raw materials needed are water and oxygen or air. 2. The product is pure and can therefore be used in disinfection applications directly or after proper dilution with water. 3. Oxygen generated in the anode compartment is used in the electrochemical reduction process; in addition, external oxygen is used to establish a high flow rate in the cathode compartment to remove the desired product efficiently. Exiting oxygen can be recycled after separation of liquid hydrogen peroxide product, if so desired. 4. The process can be designed for peroxide generation under microgravity conditions. 5. High concentrations of the order of 6-7 wt% can be generated by this method. This method at the time of this reporting is superior to what other researchers have reported. 6. The cell design allows for stacking of cells to increase the hydrogen peroxide production. 7. The catalyst mix containing a diquaternary ammonium compound enabled not only higher concentration of hydrogen peroxide but also higher current efficiency, improved energy efficiency, and catalyst stability. 8. The activity of the catalyst is maintained even after repeated periods of system shutdown. 9. The catalyst system can be extended for fuel-cell cathodes with suitable modifications.
Tennakoon, Charles L. K.↗
Mixed Cu-simple metal dimers and trimers - CuLi, CuLi2, CuNa, CuK, CuBe, CuBe2, Cu2Be, CuAl, and CuAl2
Theoretical studies of selected diatomic and triatomic molecules containing copper and the simple metals Li, Na, K, Be, and Al are presented, with emphasis on elucidating the nature of the bonding in mixed transition metal-simple metal systems. Large Gaussian basis sets are used in the diatomic calculations, and are used to calibrate the triatomic calculations, in which somewhat smaller Gaussian basis sets are employed. Electron correlation is incorporated using both the single-reference singles plus doubles configuration interaction and coupled pair functional methods. It is found that alkali atoms form very polar sigma bonds with copper, and that the ionicity increases with the inclusion of higher excitations because they improve the electron affinity of copper, which in turn allows a larger negative charge on copper. Aluminum is found to form stronger bonds than beryllium, since it does not have to undergo sp hybridization. Some of the trimers bond by forming three-center three-electron bonds. These multicenter bonds are quite strong even when compared to the two-electron bonds in the dimers or to other bonding mechanisms in the trimers.
Bauschlicher, Charles W., Jr.↗
Observation of Strong Two-Electron-One-Photon Transitions in Few-Electron Ions
We resonantly excite the K series of O5+ and O6+ up to principal quantum number n = 11 with monochromatic x rays, producing K-shell holes, and observe their relaxation by soft-x-ray emission. Some photoabsorption resonances of O5+ reveal strong two-electron–one-photon (TEOP) transitions. We find that for the [(1s2s)15p3/2]3/2;1/2 states, TEOP relaxation is by far stronger than the radiative decay and competes with the usually much faster Auger decay path. This enhanced TEOP decay arises from a strong correlation with the near-degenerate upper states [(1s2p3/2 )14s]3/2;1/2 of a Li-like satellite blend of the He-like Kα transition. Even in three-electron systems, TEOP transitions can play a dominant role, and the present results should guide further research on the ubiquitous and abundant many-electron ions where electronic energy degeneracies are far more common and configuration mixing is stronger.
M Togawa↗