Bemerkung zur Rayleigh-Schrödinger-Störungstheorie

The time-independent Hamiltonians ? ₀ and ?=? ₀ + V have a discrete spectrum, eigenvalues, and eigenvectors E _s ^(o) , ¦s〉^(o) resp. E _s, ¦s〉. If the RS perturbation theory can be applied here then an operator \(\mathfrak{p}\) with the property $$\left| s \right\rangle ^{(n + 1)} = \frac{1}{{n + 1}}\mathfrak{p}\left| s \right\rangle ^{(n)} , E_s^{(n + 1)} = \frac{1}{{n + 1}}\mathfrak{p}E_s^{(n)} $$ exists where ¦s〉⁽ⁿ⁾ and E _s ⁽ⁿ⁾ denote the n-th order corrections of perturbation theory if E _s ^(o) is nondegenerate. In the case of degeneracy the operation \(\mathfrak{p}\) remains defined and can always be used todetermine perturbation corrections of quantum mechanical expressions which are invariant in zerothorder under transformations of the basis in degenerate subspaces of ? ₀. The equations $$\left| s \right\rangle = \sum\limits_n^{0,\infty } {\left| s \right\rangle ^{(n)} = e^\mathfrak{p} \left| s \right\rangle ^{(0)} } , E_s = \sum\limits_n^{0,\infty } {E_s^{(n)} } = e^\mathfrak{p} E_s^{(0)} $$ correspond to a basis transformation where nondegenerate eigenvectors ¦s∝> ^(o) and eigenvalues E _s ^(o) of ? ₀ transform into eigenvectors ¦s∝> and eigenvalues E _s of ?. Examples show the usefulness of this formulation.     

Proton-impact excitation of HeI triplet levels

We investigated the excitation of the λ(1s3d ³ D?1s2p ³ P)=588 nm line of atomic helium by proton and deuteron impact for projectile energies 10 keV≦E _p≦25 keV. In apparent contradiction to Wigner's spin conservation rule, the emission cross section does not vanish. By measuring the intensity of the impact radiation as a function of homogeneous magnetic and electric fields applied to the collision volume, it has been shown thatp- andd-impact excitation of the 1s3d ³ D level of HeI proceeds via 1snl states withl≧3, which populate the 3³ D states by cascade decays. The well-known strong singlet-triplet mixing of these 1snl states enables a population of triplet states in accord with Wigner's rule. Accordingly, we determine the excitation cross section of the 1s4f multiplet from the measured emission cross section of the 588 nm line. The field-dependent signals give evidence that predominantly substates with |m _L|≦1 are excited.     

Regression data processing methods for parallel zero- and first-order processes applied to lactate dehydrogenase subunit determinations

Multipoint data-processing methods developed for parallel zero-order and first-order processes have been evaluated for the simultaneous determination of the heart and muscle subunits of lactate dehydrogenase. Conditions are adjusted such that pyruvate inhibition of heart and muscle subunits of the enzyme follows pseudo-first-order kinetics with apparent rate constants of k_H? 0.7 s^-1 and k_M? 1.4 s^-1 for the heart and muscle subunits, respectively. The influence of these first-order inhibition processes on the zero-order catalytic reaction is used to determine the two subunits in single- and two-component samples. Stopped-flow mixing is used and 250 data points collected during the early part of the reaction are fitted to four different mathematical models for the parallel process in order to determine initial velocity components resulting from each subunit present, and these velocity components are related to enzyme concentration. Enzymes in synthetic samples are determined in the 0–80 nmol l^-1 range with uncertainties of about 1 nmol l^-1 for single- and two-component samples prepared from purified preparations for the enzymes.     

A relativistic C.I. study on theJ=1/2 even spectrum of Sr II

We present a method for performing relativistic CI calculations in ground and excited atomic and ionic levels. An electron occupying a relativistic shellnκ in a given electronic configuration is described by a single numerical four-component Dirac-Fock orbital having the samen and κ quantum numbers to those of the shellnκ. Application of this method yields estimates for the I.P. (88 741 cm^?1) and the core correlation energy (?30916 cm^?1) for Sr II and for the total correlation energy in Sr III (?30916 cm^?1). Core-valence correlation energies for the |core 5s〉 (?4379 cm^?1), |core 6s〉 (?1191 cm^?1) and |core 13s〉 (?32 cm^?1) levels have been calculated for Sr II. Estimates for the total relativistic, Breit, vacuum polarization and self energy corrections for these levels are also reported. Configurations in which the core is not fully occupied have been found to yield significant contributions to the correlation energies of both ground and excited levels. Our results show that full scale relativistic CI calculations using numerical four-component Dirac-Fock orbitals are feasible and provide a useful ab-initio tool for the investigation of atomic properties within the framework of fully relativistic theories.     

Hund's rules

We review the present state of our undertanding of Hund's first and second rules, their domains of validity, and of generalizations in cases where the rules in their original form are invalid. These exceptions occur mainly in atomic configurations with more than one open shell withl ≥ 1, but also in cases with large orbital angular momentaL. We present a derivation of thealternating rule, which is, in some sense, a generalization of Hund's first rule, and present new rules, which generalize Hund's second rule. The importance of the concept ofunnatural parity states for an understanding of these rules is stressed. It is demonstrated that the lowest singlet-triplet average energy for the sameL corresponds to a pair of unnatural parity states (theunnatural parity rule). For sufficiently smalll ₁ andl ₂, the lowest average energy is that of the pair of states with the maximum possibleL (maximum-L rule), though exceptions are found already for moderately large values ofl ₁ andl ₂. Our analysis indicates that the validity of Hund's second rule is to some extent an accidental consequence of the minimisation of an elementary function ofl ₁,l ₂, andL — which does not depend monotonically onL — over states of unnatural parity, and that a more general and more fundamental rule should be formulated in terms of this function. We also discuss the generalisation of Hund's rules to molecules, as well as violations of them, with particular emphasis on the inversion of Hund's first rule by spin-polarization in molecules.     

Wavelet transform of quantum chemical states

By using the technique of integral within an ordered product of operators, we recast classical wavelet transform to a matrix element of the squeezing‐displacing operator U(μ,s) between the mother wavelet vector 〈ψ| and the state vector |f〉 to be transformed, i.e., we propose that 〈ψ|U(μ,s)|f〉 can be considered as a new kind of spectrum for analyzing the quantum state |f〉. In this way, we propose the wavelet‐ transform spectrum for quantum chemical states. As an example, we carry out the numerical calculation of wavelet‐transform spectrum for the binomial state. It seems to us that this kind of spectrum can be used to recognize a variety of quantum chemical states. © 2011 Wiley Periodicals, Inc. Int J Quantum Chem, 2012     

The convective model of cross-flow microfiltration is not inconsistent

The approximate analytical equations of the convective model and of the model of Romero and Davis are compared. Introducing the approximations used by Romero and Davis in the convective model gives the same functional dependence of the average filtration flux, 〈ν〉, on particle radius and solute volume fraction and a different dependence on the shear rate, γ_s. The convective model gives an explicit dependence of 〈ν〉 on transmembrane pressure and non-Newtonian ‘apparent’ viscosity of the cake. It is shown that the pseudoplastic rheological behaviour of concentrated colloidal solutions (cakes) rationalizes the experimentally established dependence 〈nu;〉 ∼ γ^m_s with 0.4 < m < 1.5.     

Fluorescence decay kinetics of acetone vapour at low pressures

Acetone-h₆ and -d₆ were excited by a short UV laser pulse to the nπ* state. Using pressures of 10^?4-10 ^?3 Torr, two distinct decay components were observed - the faster with a decay time of less than 20 ns and the slower of about 5 μs. Increasing the pressure leads to the appearance of two longer-lived decay components, which are apparently absent in the case of isolated molecules. Based on the deuteration effect, excitation wavelength dependence, quenching kinetics and analogy with other molecules, the four decay components are assigned as follows. The fastest component is due to dephasing of the initially excited state, forming a quasi-stationary eigenstate. The second component is due to the radiative decay of the latter states. The third, to decay of triplet states not directly coupled to the initially excited singlet states, and the last to the thermalized triplet state.     

Electron capture by protons from theK-shell of H and Ar

Whenever a collision takes place between charged particles, the first Born approximation for electron capture from hydrogenlike ions (Z _T ,e) by a bare nucleusZ _P , must be modified in order to account for the long-range Coulomb effects. One of the simplest ways to fulfill this requirement is provided by theT-matrix of the following form: $$T_{if}^{(1)} = \left\langle {\Phi _f exp\left\{ { - i\frac{{Z_T (Z_p - 1)}}{\upsilon } ln (\upsilon R + v \cdot R)} \right\}\left| {\frac{{Z_P }}{R} - \frac{{Z_P }}{{r_P }}} \right| exp\left\{ {i\frac{{Z_P (Z_T - 1)}}{\upsilon } ln (\upsilon R + v \cdot R)} \right\}\Phi _i } \right\rangle $$ where Φ's are the usual unperturbed channel states andZ's are the nuclear charges. In this transition amplitude, both initial and final scattering states satisfy the correct asymptotic boundary conditions in their respective channels. In the present paper, detailed computation of theK-shell cross sections is carried out for charge exchange in H⁺-H and H⁺-Ar collisions. The results are in good agreement with experimental data.     

A variation principle for energy differences between states of two different Hamiltonians

A modification of a variation principle due to Delves, is derived which permits the direct calculation of energy differences between states of two different Hamiltonians: [Δ ??] = 〈X₀| ??_x – W_x|X₁〉 – 〈Y₀|??_y – W_y|y₁〉 + 〈X₀| Δ ??|Y₀〉 · 〈X₀| Y₀〉^?1. Δ ?? = ??_y – ??_x, |X₀〉 and |Y₀〉 are the wave functions for the X and Y states and |X₁〉 and |Y₁〉 are functions defined in the text. The principle is applied to a few simple examples.     

Calculations of Jahn-Teller coupling constants in the weak-field basis via the angular overlap model: a simple operator equivalent technique

Within the framework of the angular overlap model the matrix elements of the linear Jahn-Teller operator may easily be calculated in the weak-field basis in terms of simple operator equivalents. The method is applied to the calculation of the 〈M_J|?V/?Q|M_j〉 matrix elements for the |LSJMj〉 ground states of t^x species (x = 1) ? 13) in )^* symmetry.     

Collisional de-excitation of the metastableD-states of Ba+ by He,Ne, N2 and H2

The rate constants 〈σ · υ〉 for collisional de-excitation of the metastable 5D states of Ba⁺ ions have been determined in an ion trap experiment. TheD-states are selectively populated by pulsed laser excitation of the 6P _1/2 or 6P _3/2 state and the decay at different background pressures is monitored by the change in fluorescence intensity of the excited ions. From the pressure dependence of the decay constants we calculate the de-excitation rate constants for different collision partners, averaged over the velocity distribution of the trapped ion cloud. For He, Ne, H₂ and N₂ we obtain in the c.m. energy range of 0.1–0.5 eV: 〈σ·υ〉 (He)=3.0±0.2·10^?13cm³/s, 〈σ·υ〉 (Ne)=5.1±0.4·10^?13cm³/s, 〈σ·υ〉 (H₂)=3.7±0.3·10^?11cm³/s, 〈σ·υ〉 (N₂)=4.4±0.3·10^?11cm³/s. The results can be understood qualitatively by a consideration of the ion-atom and ion-molecules interaction potential.     

Quasi-stationary detection of transient absorption of a dissociating molecule

Using the forth harmonic of a Nd-YAG laser at 266 nm TlI was dissociated and the dissociation was observed by a transient absorption due to a transition from the repulsive potential, which correlates to T1 (6p ² P _3/2) and I, to the repulsive potential with the asymptote T1 (7s) and I. The experiment is performed with quasi-stationary spectroscopy in the nanosecond time domain, but detects transients effects by asymmetric absorption profiles in the frequency domain. Simulations applying Bloch equations with time dependent molecular energies are able to reproduce the observations and give information on the difference potential between the two molecular states involved and the transition dipole moment. The results are also in good agreement with fs experiments from other authors.     

Magnetochiral dichroism as a measure of parity nonconservation in an atomic system

Using as an example a one-electron atom in which parity-mixing interactions are assumed to occur, we consider the emission from states |jm> to states |j′m′>. An applied static magnetic field splits the degeneracy of the magnetic sublevels. It is shown that for any transition between given states for whichm′=m ±1, there is a difference in the intensity of light emitted parallel and antiparallel to the axis of quantization. This is a manifestation of magnetochiral dichroism. There is a direct analogy between this and the forward — backward asymmetry which occurs in the γ-emission from definite nuclear magnetic substates when parity is violated.     

Stark manifolds of barium Rydberg states

In a CW laser-atomic beam experiment Stark manifolds in barium originating from the Rydberg states 6s40f ¹ F ₃, 6s40g ^1,3 G ₄,³ G ₅ and 6s40h ^1,3 H ₅ have been studied. Accurate quantum defect values for higher orbital angular momentum states (l=6, 7) have been determined. The Stark manifolds are also calculated by diagonalization of the energy matrix in the presence of an external electric field. Good agreement between experiment and calculations is obtained.     

Expansion and completeness theorems for operator manifolds

Some expansion and completeness theorems for operator manifolds, which are currently being employed in propagator theory, are derived. It is shown that excitation or ionization operators satisfying the conditions Q|0〉 = |Λ〉 and Q_Λ|0〉 = 0 for general excited states |Λ〉 and reference state |0〉 may be expanded uniquely in particular sets of basis operators. These results are then used to discuss rigorous expressions for fermion propagators.     

CW laser spectroscopy of autoionising 6p 3/2 nl states of Barium

6pnl states of Ba with l ≥ 5, excited with two narrowband, tunable CW lasers have been studied. In the first excitation step 6snl states were populated from metastable states of the 5d ² configuration: to 6s40h (l = 5) states in the absence, and to parabolic 6pnk (n = 30, 35, 40) states in the presence of an electric field. The atomic motion was used to adiabatically reduce this field to zero before applying the second excitation step to high-l 6p _3/2 nl states (Stark-switching). A slow decrease in the autoionisation rate for increasing value of l and, in case of 6pnh, a dependence on the fine-structure state was measured. The narrowest autoionisation linewidth observed is 113(11) MHz, relatively close to the 20 MHz linewidth corresponding to the radiative lifetime of the 6p state of the Ba⁺-ion.     

State selected and oriented molecules: rotationally-resolved photoelectron spectroscopy and dichroism in ionization of symmetric tops

This paper developes a theoretical framework for one-photon, electric dipole ionization of molecules, prepared in a single |JKM〉 rotational state of their ground electronic motion using a hexapole electrostatic field followed by a weak and homogeneous orienting electric field E, by taking also the |J′K′M′〉 eigenstate of the residual photoion into account. This theory has been used to study photoionization in a generic C_3v molecule by the linearly as well as circularly polarized light. Integrated and differential photocurrents plus circular dichroism, obtained from a model calculation, for ionization in two different experimental configurations in 13a ₁ ² orbital of a state selected and oriented CH₃I are discussed. These results show, among other things, that somewhat different propensity rules for |JKM〉 → |J′K′M′〉 transitions are obeyed in different experimental geometries. Also, dichroic effect exists in the photoelectron angular distribution when circularly polarized light is incident perpendicular to E.     

The atomic form factor for vanadium: a treatment including correlation andLS-dependent relativistic effects

The atomic form factor for the ground state of vanadium is evaluated using non-relativistic and “relativistic-corrected” configuration interaction wavefunctions in the |LSM _L M _s 〉 representation. Relaxing the constraints imposed by the Hartree-Fock model results in a very small reduction of the atomic factor at small momentum transfer with a negligible change at higher momentum transfer. Better agreement with the relativistic Hartree-Fock atomic form factor at small momentum transfer is obtained when theLS-dependent relativistic effects are included in the Breit-Pauli approximation. The sensitivity of the atomic form factor to small changes in the magnitude of the expansion coefficients of the configurational functions is also discussed.