Natural spin orbital analysis of diatomic molecular wave functions in terms of generalized diatomic orbitals

The original idea of the model applied to HeH⁺ excited states is: One electron occupies a diatomic orbital similar to the HeH⁺⁺ ground state 1s function. The other electron occupies an orbital which can be represented by a linear combination of functions similar to H ₂ ⁺ excited state functions. One or two screening parameters are variationally optimized to compensate for the smallness of the one-electron basis.CI calculations have been performed for five excited HeH⁺ states covering a wide range of internuclear distances. The CI wave functions have been submitted to a natural spin orbital analysis. The strongly occupied NSOs are compared with the original model functions.Dedicated to Professor Hermann Hartmann on occasion of his 65th birthday on May 4th, 1979.     

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.     

A floating one centre perturbation treatment for H-like molecules

Closed form one centre partial wave perturbation results are obtained through second order in the energy for H-like molecules with arbitrary nuclear charges. The expansion centre for the method is taken at an arbitrary point along the internuclear axis and the zeroth-order wave function is a screened “1s” function centred at the expansion point. Various fixed one centre calculations of other workers for the lsσ states of H and HeH⁺⁺ and for the 2pσ state of HeH⁺⁺ are generated as limiting cases of this more general treatment. The floating one centre perturbation results for the 1sσ states of H₂⁺ and HeH⁺⁺ are used as models for discussing the usefulness of allowing the expansion centre in the one centre method to be a function of nuclear configuration.     

Rates of radiative recombination to form HD+ and HeH+

The rates of radiative recombination to form HD⁺ and HeH⁺ along the ground electronic state potential surface have been calculated from known potentials. These exact quantum calculations are compared with a classical model. For the formation HD⁺, the classical model predicts the rate in the range of 100–1000 K. Below 100 K, the rate is dominated by the effect of low energy shape resonances. For the formation of HeH⁺, the classical model predicts the correct temperature dependence.     

Calculation of a power spectrum and definition of wave functions of an ion Cr3+ in antiferromagnetic crystal Cr2O3 in the model of a crystalline field

The antiferromagnetic crystal Cr₂O₃ contains magnetoactive ions Cr3⁺ and has C3v local symmetry. In this crystal exist cross‐effects, for example linear magnetoelectric effect and others. It is necessary to know the energy levels and wave functions of an ion Cr3⁺ exact simulation of this effect. In the present paper the problem of calculation of the crystalline field potential is solved in the single‐ion model. For Cr₂O₃ the splitting of energy levels are determined and wave functions for ground and exited states are calculated. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2002     

Molecular formation and electron correlation in HeH+

A one-centre CI wave function for HeH⁺ reported by Stuart and Matsen for 0.1 ? R ? 5.0 has been analysed in detail from the viewpoint of molecular formation. Further, by means of a natural orbital analysis, it was possible to obtain some measure of the electron correlation contained within such wave functions for various R values. These effects were illustrated by means of a series of difference maps for the electron density. One- and two-particle expectation values were obtained as a function of R. Thus, it was possible to study several aspects of the influence of the proton on the electron charge cloud as we pass from He through to the united atom Li⁺. The occupation numbers within the natural expansions were compared with those which arise from a similar analysis of a two-centre wave function for HeH⁺. The “character” of such wave functions for HeH⁺, and also for He and Li⁺, were analysed and compared.     

Multiphoton association reaction He + H+ → HeH+ steered by ultra‐short laser pulse

The multiphoton association reaction He + H⁺ → HeH⁺ in the electronic ground state is investigated using the time‐dependent quantum wave packet method. It is shown that the collision pairs He + H⁺ in continuum state transfer into ν = 0 state and then produce stable molecules HeH⁺ through emission of two or three photons. The multiphoton transition takes place via intermediate states, and the transfer probability is determined by the collision energy and the intermediate states. The populations of the intermediate states and ν = 0 state can be controlled by the laser duration. The three‐photon transition is more efficient than the two‐photon transition for producing the molecule HeH⁺ in ν = 0 state. © 2010 Wiley Periodicals, Inc. Int J Quantum Chem, 2011     

Binding/antibinding analyses for diatomic interactions. the HeH2= system

On the basis of the Berlin diagram, the region-functional contribution of the electron density has been quantitatively examined for the Is, 2p, and 2p states of HeH²⁺ system. The binding and antibinding contributions and the dynamic behaviours of the electron density during the interaction processes are discussed in comparison with the previous results for homonuclear H ₂ ⁺ system. The effect of coordinate-dependence of the Berlin diagram on the regional partitioning has also been studied.     

Dissociative recombination of HeH+. I. Rovibrational spectrum of HeH Rydberg states

The repulsive ground electronic state X²Σ⁺ of HeH is strongly coupled to the Rydberg states at small interatomic distances. Such large couplings also occur between some of the Rydberg states. HeH⁺ ions that capture an electron in a Rydberg state end up in separated He and H atoms by indirect predissociation. This paper presents a study of potential functions and pertinent matrix elements involving the lowest electronic states: the ²Σ⁺ states, X, A, C, and D, and the ²Π states B and E. Individual transition rates as well as total radiative and non-radiative lifetimes have been computed for the lowest vibrational and rotational levels. Received: 22 June 1998 / Accepted: 21 August 1998 / Published online: 12 October 1998     

Threshold photoelectrons coincidence spectroscopy of the rare gases Ne,Ar, Kr and Xe

Threshold single photon double ionisation of the rare gases neon, argon, krypton and xenon has been studied in an electron-electron coincidence experiment. A new technique has been used where only near-zero energy electrons are detected and these are measured in coincidence. The spectrometer used here employs the penetrating field technique which provides very high detection efficiency, sensitivity and energy resolution. Relative partial double ionisation cross sections have been measured at threshold for then p⁴(³P,¹D and¹S) states of Ne⁺⁺, Ar⁺⁺, Kr⁺⁺ and Xe⁺⁺ and then sn p⁵ (³P and¹P) states of Ne⁺⁺, Ar⁺⁺ and Kr⁺⁺. The observed relative cross sections are in general agreement with a propensity rule for excitation of these states, except for the case of neon.     

Infrared bound to quasibound vibration-rotation spectrum of HeH+ and its isotopes

The positions and widths of all the bound and rotationally quasibound levels of the ground states of ⁴HeH^-, ⁴HeD′-³HeH^- and ³HeD′ have been calculated using an adiabatic potential. From these theoretical data the positions of bound to quasibound vibration-rotation transitions lying in the range of the carbon dioxide laser 0880 to 1090 cm ¹ have been predicted. Infrared predissociation spectroscopy of Doppler-tuned ion beams has been used to determine the positions and widths of many of the resonances. There is excellent agreement between the theoretical predictions and the experimental measurements.     

Vibrational linestrengths for the ground and first excited electronic states of HeH<Subscript>2</Subscript><Superscript>+</Superscript>

Together with recent improved potential-energy surface calculations for the ground (X) and first excited (Ã) electronic states of HeH₂ ⁺, the electric dipole moment surfaces for each state and the transition dipole moments connecting the two states were evaluated for the entire range of the energy calculations. Using these functions the linestrengths of all dipole-allowed transitions between the bound vibrational levels within each of the two states (XX) and (ÃÃ) as well as between them (ÃX) are evaluated here. These data are believed to be useful both in the experimental search for the yet unobserved molecular spectra of HeH₂ ⁺ and in evaluating theoretical rates for the radiative association or photodissociation processes involving the two lowest electronic states of the ion.Contribution to the Björn Roos Honorary Issue     

Rotational predissociation of HeH+: The effect of nuclear motion on ro-vibrational energies and widths

The approximate diagonal nuclear-motion correction for the X¹Σ⁺ state of HeH⁺ is calculated at selected points on the internuclear axis. The energies (E_υJ) and widthis (Γ_υJ) of the quasibound rotational-vibrational states of ⁴HeH⁺, ³HeH⁺, ⁴ HeD⁺ and ³HeD⁺ in the electronic ground state are calculated in the Born—Oppenheimer approximation using a discrete-states method. The E_υJ and Γ_υJ, corrected approximately for the effects of nuclear motion, are also evaluated.     

Unique orbitals

A method is presented which allows reduction of exact wave functions of one electron diatomic molecules to unique orbitals from which exact wave functions for the particular molecule can be written as LCAO-MO functions. The method is illustrated with applications to the hydrogen molecule-ion and the doubly charged helium hydride molecule-ion. Extension of the method to two electron unique geminals is discussed.

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Zusammenfassung Ein Verfahren wird angegeben, aus exakten Eigenfunktionen zweiatomiger Moleküle mit einem Elektron unique = einzigartige Orbitale zu gewinnen, aus denen sich die exakten Molekülfunktionen als Linearkombinationen ergeben. Anwendungen auf H ₂ ⁺ und HeH⁺⁺ erläutern die Methode. Die Erweiterung auf unique-Elektronengeminale wird diskutiert.

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Résumé On présente une méthode qui permet la réduction des fonctions d'onde exactes de n'im-porte quelle molécule diatomique monoélectronique aux orbitales «uniques», sur base des-quels les fonctions exactes des molécules s'écrivent en combinaisons linéaires. La méthode est illustrée par les exemples de H ₂ ⁺ et HeH⁺⁺. On discute son extension sur des géminals «uniques» d'électrons.

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Taken in part from the Ph. D. thesis of DKH, Northwestern University, 1963. This work was supported by a research grant from the National Science Foundation to North-western University.     

The Kinetics of Uptake of Cu++ Ions in Ionac SR-5 Cation Exchange Resin

The kinetics of sorption of Cu⁺⁺ ions in a typical cation exchange resin (IONAC SR-5) have been studied by two methods; the traditional batch method which involves following the change in Cu⁺⁺ concentration in a solution of finite volume in contact with the resin, and by removing and sectioning individual resin particles to follow the progress of the adsorption front. The results obtained by both methods are consistent. The equilibrium isotherm for this system is of highly favorable (Langmuir) form and the uptake kinetics show a clear transition from linear Fickian behavior at low concentrations to irreversible (shrinking core) behavior at high concentrations. The addition of ammonia to the system has little effect on the equilibrium isotherm but the intraparticle diffusivity is increased by a factor of about four. This may be due to reduction in the degree of hydration of the cuprammonium ion in comparison with uncomplexed Cu⁺⁺.     

Physical nature of the chemical bond

The effect of such variables as size of basis set, number of S and P type functions in the basis set, size of initial exponent in the basis set, size of multiplicative factor in the geometrical progression of exponents, and interatomic distance has been examined by a self-consistent-field calculation with Gaussian orbitals on HeH⁺. A quasi-optimized wave function has been obtained by allowing only the initial exponent and multiplicative factor to vary in the optimization process.     

Variable-screening model for diatomic molecules

An effective hamiltonian method based on a one-electron potential is proposed. The potential is represented by a sum of two coulombic interactions with effective nuclear charges depending upon the internuclear distance. This potential preserves the separability of Schrödinger equation. The method can be usefully applied to various atom (ion)-atom collision problems. Calculations are carried out for some states of HeH⁺ and HeH using one configuration built up from a minimal basis set chosen to ensure correct dissociation.     

Comparative study of the photoassociation reaction of He + X+→ HeX+ (X = H,D): Including multiphoton transitions and dissociations

Using the time-dependent quantum wave packet method, the photoassociation (PA) processes of He + H ⁺→ HeH⁺ and He + D ⁺→ HeD⁺, driven by the sin²-shaped femtosecond laser pulse in the electronic ground state, including multiphoton transitions and dissociations, are investigated for a wide range of initial collision momenta spanning from 1 to 4 a.u. (or for the collision energy roughly in the ranges of 0.009∼0.148 eV and 0.006∼0.089 eV for HeH⁺ and HeD⁺ systems, respectively). It is found that, at some collision momenta, multiphoton transitions to deeply bound states are inevitable to occur and can greatly decrease the PA probability of the target state that selected is the vibrational state v = 6. For the dissociation process, the higher-order (two- and three-photon) dissociations, measured from the target state, tend to be significant at relative high collision energies, which implies that above-threshold dissociations may also be an important loss mechanism in the PA process. In addition, it is also shown that the higher-order dissociation is much stronger for HeH⁺ systems than that for HeD⁺ systems at a given collision momentum, and could be enhanced by the strong transitions among deeply bound states.     

Quantum properties of complete solutions for a new noncentral ring‐shaped potential

We propose a new exactly solvable ring‐shaped potential V(r,θ) = ?(α/r) + (σ/r²) + β cos²θ/(r²sin²θ). The exact bound‐state solutions are presented explicitly. The creation and annihilation operators are established directly from the normalized radial wave functions. We present two important recurrence relations among the diagonal matrix elements and obtain some explicit expressions of mean values of r^k (8 ≥ k ≥ ?11). The exact form of continuum states is also obtained analytically. Comments are made on the calculation formula of phase shifts and the analytical properties of the scattering amplitude. It is interesting to find that the exact form of continuum states will reduce to that of the bound states at the poles of the scattering amplitude. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2005