Group theoretical treatment of the planar internal rotation problem in (HF)2

The HF dimer is believed to exhibit an internal rotation tunneling process between two planar but nonlinear equilibrium configurations, during which tunneling the roles of the hydrogen-bonded and the free hydrogen atom are interchanged. This process can be represented schematically with labeled atoms as H_lF_aH₂F_b ? F_aH_lF_bH₂, and gives rise to a permutation-inversion group G₄ containing four operations. In the present work the vibration-rotation-tunneling problem in (HF)₂ is treated group theoretically in three ways: (i) by allowing tunneling only through a trans planar C_2h intermediate, (ii) by allowing tunneling only through a cis planar C_2v intermediate, and (iii) by considering the trans and cis tunneling processes both to occur, though not necessarily with the same probability. The molecular symmetry groups used for these treatments are (i) the point group C_2h, (ii) the point group C_2v, and (iii) a double group, which might be thought of as $\text{G}{}_4{}^2\text{= C}{}_{\mathrm{2h}}{}^2\text{= C}{}_{\mathrm{2v}}{}^2$. Nonplanar tunneling paths are not considered, since the internal axis method (IAM) coordinate system used here cannot easily be adapted to nonplanar internal rotation motions in this molecule. Various-details of energy level diagrams, symmetry species for operators, selection rules for spectroscopic transitions, and statistical weights are presented for the (HF)₂ tunneling problem, as well as some speculation on the general question of when point groups, permutation-inversion groups, or double groups are preferable for treating large-amplitude vibrational motion problems.     

The 3540-Å electronic transition of malonaldehyde,a tunneling hydrogen-bonded molecule

Revised and more complete vibrational assignments are made for the 3540-Å $\text{π}{}^1\text{← n}$ band system of malonaldehyde. The $\text{0}{}^{\mathrm{+}}\text{0}{}^{\mathrm{?}}$ tunneling splitting is found to be 19 ± 11 cm^?1 for the $\text{nπ}{}^1$ state and this represents a 7-cm^?1 decrease relative to the ground electronic state. The tunneling splitting and the Franck-Condon envelope of intensities in the 185-cm^?1 upper-state progression suggest that the ${}^1\text{B}{}_1\text{(nπ}{}^1\text{)}$ state is significantly less tightly hydrogen-bonded than the ground ¹A₁ state.     

Paзлoжeниe пo 1Z для чapтpи-фoкoвcкoй и кoppeляциoннoй энepгии, peлятивиcтcкич пoпpaвoк, дипoльнoгo мaтpичнoгo элeмeнтa

Energies and dipole matrix elements have been calculated for He, Li, Be, B, C, N, O, F, and Ne-like ions (configurations 1s²2sⁿ¹2pⁿ²?1s²2s^n1?12pⁿ²⁺¹). The Hartree-Fock energy, the correlation energy, and relativistic corrections were taken into account. Relativistic corrections were obtained by computing the entire quantity H_B. Numerical results are presented for energies of the terms in the form

$\text{E=E}{}_0\text{Z}{}^2\text{+ ΔE}{}_1\text{Z + ΔE}{}_2\text{+}\text{1}\text{Z}\text{ΔE}{}_3\text{+}\text{α}{}^2\text{4}\text{(E}{}_0{}^{\mathrm{p}}\text{Z}{}^4\text{+ ΔE}{}_1{}^{\mathrm{p}}\text{Z}{}^3\text{)}$

, and for the fine structure of the terms in the form

$\text{〈1s}{}^2\text{2s}{}^{\mathrm{n1}}\text{2p}{}^{\mathrm{n2}}\text{LSJ|H}{}_{\mathrm{Б}}\text{|1s}{}^2\text{2s}{}^{\mathrm{n1\prime }}\text{2p}{}^{\mathrm{n2\prime }}\text{L′S′J〉=(?1)}{}^{\mathrm{L+S\prime +J}}\text{LSJ}\text{S′L′1}\text{×}\text{α}{}^2\text{4}\text{(Z?A)}{}^3\text{[E}{}^{\mathrm{(0)}}\text{(Z ? B)+}\text{E}{}_{\mathrm{c0}}\text{]+(?1)}{}^{\mathrm{L\; +\; S\prime \; +\; J}}\text{LSJ}\text{S′L′2}\text{α}{}^2\text{4}\text{(Z?A)}{}^3\text{E}{}_{\mathrm{cc}}$

. Dipole matrix elements are required for calculation of oscillator strengths or transition probabilities. For the dipole matrix elements, two terms of the expansion in $\text{1}\text{Z}$ have been obtained. Numerical results are presented in the form P(a, a′) = (a/Z)[1 + (τ/Z)].     

Space-time symmetries and the gravitational-neutrino field equations in general relativity

We consider a neutrino field with geodesic rays in interaction with a gravitational field admitting a Killing vector field n^μ. It is found that for solutions of the Einstein-Weyl field equations the neutrino field ξ^A and the neutrino flux vector l^μ are restricted by the equations: $\text{L}{}_{\mathrm{n}}\text{ξ}{}^{\mathrm{<mtext>A\; =\; ?</mtext><mtext>1</mtext><mtext>2</mtext>}}\text{is}\text{ξ}{}^{\mathrm{A}}$ and $\text{L}{}_{\mathrm{n}}\text{l}{}^{\mathrm{\mu }}\text{= 0}$, whereas s is a real constant. In the case of pure radiation neutrino fields these equations become: $\text{L}\text{ξ}{}^{\mathrm{A}}\text{=}\text{case1}\text{2}\text{(p ? is)ξ}{}^{\mathrm{A}}$, $\text{L}{}_{\mathrm{n}}\text{l}{}^{\mathrm{\mu }}\text{= pl}{}^{\mathrm{\mu }}$, where p and s are in general real functions of the coordinates.     

Predissociation of the Schumann-Runge bands of O2

The predissociation line broadening in the Schumann-Runge bands of O₂ is interpreted through an ab initio calculation of the pertinent repulsive potential energy curves and spin-orbit matrix elements. The ab initio results provide an overall qualitative picture of the predissociation which is further refined through a detailed comparison of calculated level shifts and widths with experimental data. The position of the dominant repulsive curve is also deduced by a deperturbation of the level shift in the second vibrational difference. The predissociation is dominated by the ⁵Π_u state crossing the $\text{B}{}^3\text{Σ}{}_{\mathrm{u}}{}^{\mathrm{?}}$ state around 1.875 Å with a spin-orbit matrix element of 65 cm^?1. The ¹Π_u and ³Π_u states have small spin-orbit matrix elements and play only minor roles in the predissociation. The calculated and experimental widths are in good agreement for low and high vibrational levels. The apparent experimental widths between v = 5 and 11 are shown to be inconsistent with the theoretical analysis, the difference probably being due to line blending.     

Singular perturbation potentials

This is a perturbative analysis of the eigenvalues and eigenfunctions of Schrödinger operators of the form ?Δ + A + λV, defined on the Hilbert space L²(Rⁿ), where $\text{Δ = Σ}{}_{\mathrm{i=1}}{}^{\mathrm{n}}\text{?}{}^2\text{?X}{}_{\mathrm{i}}{}^2$, A is a potential function and V is a positive perturbative potential function which diverges at some finite point, conventionally the origin. λ is a small real or complex parameter. The emphasis is on one-dimensional or separable problems, and in particular the typical example is the “spiked harmonic oscillator” Hamiltonian, $\text{?d}{}^2\text{dx}{}^2\text{+ x}{}^2\text{+}\text{l(l + 1)}\text{x}{}^2\text{+ λ|x|}{}^{\mathrm{?\alpha }}$, where α is a positive constant. When this kind of perturbation is very singular, the first-order Rayleigh-Schrödinger perturbative correction, (u₀, Vu₀), where u₀ is the unperturbed eigenfunction, diverges. This analysis constructs explicitly calculable terms in a modified perturbation series to a finite order, by using linear operator theory in concert with approximation methods for differential equations. Along the way a connection between a W-K-B type approximation and Bessel functions is exploited.     

Transverse spin correlation function of the one-dimensional spin-12 XY model

The transverse spin pair correlation function p^x_n=<S^x_mS^x_m+n>=<S^x_mS^x_m+n> is calculated exactly in the thermodynamic limit of the system described by the one-dimensional, isotropic, spin-$\text{1}\text{2}$, XY Hamiltonian

$\text{H=?2J}\text{∑}\text{l=1}\text{N}\text{(S}{}^{\mathrm{x}}{}_{\mathrm{l}}\text{S}{}^{\mathrm{x}}{}_{\mathrm{l+1}}\text{+S}{}^{\mathrm{y}}{}_{\mathrm{l}}\text{S}{}^{\mathrm{y}}{}_{\mathrm{l+1}}\text{)}$

. It is found that at absolute zero temperature (T = 0), the correlation function ρ^x_n for n ≥ 0 is given by

$\text{ρ}{}^{\mathrm{x}}{}_{\mathrm{2p}}\text{=}\text{1}\text{4}\text{2}\text{π}{}^{\mathrm{2p}}\text{Π}\text{j=1}\text{p?1}\text{4j}{}^2\text{4j}{}^2\text{?1}{}^{\mathrm{2p?2j}}\text{if}\text{n=2p}$

,

$\text{ρ}{}^{\mathrm{x}}{}_{\mathrm{2p+1}}\text{=±}\text{1}\text{4}\text{2}\text{π}{}^{\mathrm{2p+1}}\text{Π}\text{j=1}\text{p}\text{4j}{}^2\text{4j}{}^2\text{?1}{}^{\mathrm{2p+2j}}\text{if}\text{n=2p+1}$

, where the plus sign applies when J is positive and the minus sign applies when J is negative. From these the asymptotic behavior as n → ∞ of |?^x_n| at T = 0 is derived to be $\text{|ρ}{}^{\mathrm{x}}{}_{\mathrm{n}}\text{| ～}\text{a}\text{n}$ with a = 0.147088?. For finite temperatures, ρ^x_n is calculated numerically. By using the results for ?^x_n, the transverse inverse correlation length and the wavenumber dependent transverse spin pair correlation function are also calculated exactly.     

The visible absorption spectrum of NO2: A three-mode nuclear dynamics investigation

The vibronic band origins of the visible absorption spectrum of NO₂ are calculated theoretically with the aid of a simple model Hamiltonian for the coupled electronic and vibrational motions. Including all three vibrational modes in the calculation and using ab initio values of the relevant parameters, we obtain satisfactory qualitative agreement with experiment. In particular, the observed high density and irregular intensity distribution of the band origins is reproduced correctly by the calculation. The present results confirm unambiguosly that the anomalous vibronic structure of the ${}^2\text{B}{}_2\text{←}{}^2\text{A}{}_1$ transition is caused by strong nonadiabatic interactions between the ²B₂ and ²A₁ electronic states of NO₂. They also show that simple deconvolution procedures, which are often used to deperturb irregular spectra, are not applicable to the ${}^2\text{B}{}_2\text{←}{}^2\text{A}{}_1$ transition of NO₂. To further explore the strength of the nonadiabatic effects in NO₂, we calculate the mixing of the different electronic species in the vibronic eigenstates and compare it to several relevant experimental quantities.     

The 7Li(d, n0)8Be reaction with polarized 800 keV deuterons

The analysing power of the ${}^7\text{Li}\text{(}\text{d}\text{, n}{}_0\text{)}{}^8\text{Be}$ reaction for vector and tensor polarization of an 800 keV deuteron beam, as well as the relative cross section for the unpolarized beam were measured at 7 to 9 angles between 0° and 160°, using a thick target. Analysis in terms of (l, s, J^π) matrix elements shows that two intermediate states with $\text{J}{}^{\mathrm{\pi }}\text{=}\text{3}\text{2}{}^{\mathrm{+}}$ and $\text{J}{}^{\mathrm{\pi }}\text{=}\text{5}\text{2}{}^{\mathrm{?}}$ present, strongly interfering with each other. Assignments to known ⁹Be levels and to threshold resonances as suggested by Hackenbroich and Seligman are briefly discussed. The magnitude of the vector analysing power makes the reaction interesting as a monitor for the vector polarization of low-energy deuteron beams.     

Matthiessen's rule and its variation for cyclotron resonance width in two and three dimensions

Based on the proper connected diagram expansion, we calculated cyclotron resonance widths Γ_n associated with neighboring Landau states (n, n +1) for free electrons in interaction with more than one kind of impurities. In 3D usual Matthiessen's rule Γ_n=Γ⁽¹⁾_n+Γ⁽²⁾_n+…where Γ⁽ⁱ⁾_n represent widths calculated separately for each kind, is obtained. In 2D a new rule: is obtained.     

l-type doubling transitions in Δ states of OCS

Molecular beam electric resonance spectroscopy has been used to measure the l-type doubling transitions in carbonyl sulfide. Transitions in J = 4 to J = 20 have been observed for the (02²0) vibrational state and for J = 1 and J = 2 for the (03¹0) state. The data has been analyzed to give the v = 2 energy separation $\text{E}{}_{\mathrm{<mtext>\Delta </mtext>}}{}^0\text{- E}{}_{\mathrm{<mtext>\Sigma </mtext>}}{}^0\text{= ?5.7861(2) + 8.36(1) × 10}{}^{\mathrm{?5}}\text{J(J + 1)}\text{cm}{}^{\mathrm{?1}}$, and the vibrational dependence of q to be 86.52(9)(v₂ + 1) KHz. The dipole moment of the (02²0) vibrational state is 0.6936(3) D.     

New potential energy function for alkali halide molecules

A new repulsive term in the ionic interaction potential for computing the lattice energy, the rotational constant (α₀) and the vibrational constant (w₀x₀) of alkali halide molecules has been proposed as $\text{ψ(r) = Af}{}^{\mathrm{n}}\text{r}{}^{\mathrm{-n}}\text{e}{}^{\mathrm{<mtext>?r</mtext><mtext>\lambda </mtext>}}$, which apart from being generalised, dimensionally homogenous and physically sound yields better values for lattice energy, α₀ and w₀x₀ than the previously reported potentials models.     

Infrared hole burning in matrix-isolated 1,2-difluoroethane with a tunable diode laser

Infrared hole burning within the absorption profile of the ν₁₇ vibrational transition of 1,2-difluoroethane matrix-isolated in solid Ar, Kr, and N₂ was observed. These measurements allowed the determination of the homogeneous and inhomogeneous linewidth ($\text{Δ}\text{ν}\text{?}{}_{\mathrm{<mtext>h</mtext>}}\text{= 0.002}\text{cm}{}^{\mathrm{?1}}$ at 2.5 K in Kr, $\text{Δ}\text{ν}\text{?}{}_{\mathrm{<mtext>i</mtext>}}\text{= 0.25}\text{cm}{}^{\mathrm{?1}}$). The temperature dependence of the homogeneous linewidth is explained in terms of vibrational relaxation as well as dephasing processes. A detailed analysis of the changes in the absorption profile with irradiation and calculation of the potential energy surface for rotation of the molecule in the matrix cage suggest a reorientation of the molecules in the matrix to be the cause of the observed hole burning.     

The ϱ and A2 exchange amplitudes in meson-baryon scattering for ϱlab between 2.5 and 200 GeV/c

We present a parametrization of the ? and A₂ exchange amplitudes which fits all the available data on the reactions $\text{π}{}^{\mathrm{?}}\text{p}\text{→ π}{}^0\text{n}\text{, π}{}^{\mathrm{?}}\text{p}\text{→}\text{K}{}^0\text{n}\text{,}\text{and K}{}^{\mathrm{+}}\text{n}\text{→}\text{K}{}^0\text{p}$ in the momentum range from 2.5 to 200 GeV/c and for momentum transfers up to t = ?2.0 (GeV/c)².     

Observed and calculated interactions between valence states of the NO molecule

No perturbation between two valence states of NO has ever been identified, although many valence-Rydberg and several Rydberg-Rydberg perturbations have been extensively studied. The first valence-valence crossing to be experimentally documented for NO is reported here and occurs between the ¹⁵N¹⁸O B²Π (v = 18) and B′²Δ (v = 1) levels. No level shifts larger than the detection limit of 0.1 cm^?1 are observed at the crossings near $\text{J = 6.5 [B}{}^2\text{Π}\text{(F}{}_1\text{) ～ B′}{}^2\text{Δ}\text{(F}{}_2\text{)]}$ and $\text{J = 12.5 [B}{}^2\text{Π}\text{(F}{}_1\text{) ～ B′}{}^2\text{Δ}\text{(F}{}_1\text{)]}$; two crossings involving higher rotational levels could not be examined. Semi-empirical calculations of spin-orbit and Coriolis perturbation matrix elements indicate that although the electronic part of the $\text{B}{}^2\text{Π}\text{～ B′}{}^2\text{Δ}$ interaction is large, a small vibrational factor renders the ¹⁵N¹⁸O B (v = 18) ? B′ (v = 1) perturbation unobservable. Semi-empirical estimates are given for all perturbation matrix elements of the operators $\text{Σ}{}_{\mathrm{i}}\text{a}\text{?}{}_{\mathrm{i}}\text{l}{}_{\mathrm{i}}\text{·s}{}_{\mathrm{i}}$ and $\text{B(L}{}_{\mathrm{\pm }}\text{S}{}_{\mathrm{?}}\text{? J}{}_{\mathrm{\pm }}\text{L}{}_{\mathrm{?}}\text{)}$ which connect states belonging to the configurations $\text{(σ2p)}{}^2\text{(π2p)}{}^4\text{(π}{}^1\text{2p)}$, $\text{(σ2p)(π2p)}{}^4\text{(π}{}^1\text{2p)}{}^2$, and $\text{(σ2p)}{}^2\text{(π2p)}{}^3\text{(π}{}^1\text{2p)}{}^2$.     

Recombination in electron-hole droplets in polar and non-polar indirect band gap semiconductors and carrier concentration power law

The dependence of the recombination relaxation rate on the carrier concentration, n, is examined in detail. It is found that in polar indirect band gap semiconductors the phonon-induced recombination rate varies as $\text{n}{}^{\mathrm{<mtext>4</mtext><mtext>3</mtext>}}$ at low temperatures. This is a new power law. On the other hand, in the non-polar materials, the relaxation rate varies as a linear combination of n², $\text{n}{}^{\mathrm{<mtext>5</mtext><mtext>3</mtext>}}$ and $\text{n}{}^{\mathrm{<mtext>4</mtext><mtext>3</mtext>}}$ terms. We find the experimental evidence for the occurence of $\text{n}{}^{\mathrm{<mtext>5</mtext><mtext>3</mtext>}}$ and $\text{n}{}^{\mathrm{<mtext>4</mtext><mtext>3</mtext>}}$ contributions for the first time.     

A direct method for modifying certain phase-integral approximations of arbitrary order

The approximate solution of the differential equation $\text{d}{}^2\text{ψ}\text{dz}{}^2\text{+ Q}{}^2\text{(z)ψ = 0}$ by a general modification of certain phase-integral approximations of arbitrary order is considered. A consistent modification of these higher-order phase-integral approximations is derived on the assumption that one has found a function Q_mod²(z) which makes the modified first-order approximation good at certain singular points of Q²(z), where the unmodified approximation would break down.     

Replica variables,loop expansion,and spectral rigidity of random-matrix ensembles

The replica trick of statistical mechanics is used to derive integral representations of n-point Green's functions both for the GOE and the EGOE. These integral representations are particularly suited for perturbative evaluation (loop expansion). Using the one-loop correction to the GOE one-point function, it is found that the density of states at the edge of the semicircle scales is $\text{～N}{}^{\mathrm{<mtext>?1</mtext><mtext>3</mtext>}}\text{?}\text{(N}{}^{\mathrm{<mtext>2</mtext><mtext>3</mtext>}}\text{δ}\text{)}$ where N is the dimension of the matrix ensemble. For the n-point functions with n ≥ 2, the existence of the microscopic limit to all orders in N^?1 is proved by decomposing the integration variables into massive (i.e., macroscopic) and massless (microscopic) components. Evaluation of the EGOE two-point function to leading order in the inverse local distance variable yields the first analytic evidence that the long-range correlations of EGOE spectra are similar to the GOE but not-stationary.     

The spectrum and structure of the He2 molecule: Characterization of the triplet states associated with the UAO's 4pσ, 5pσ, and 6pσ

The emission spectrum of the He₂ molecule has been rephotographed in the ～3200–4100 Å region and the $\text{4pσ g}{}^3\text{Σ}{}_{\mathrm{g}}{}^{\mathrm{+}}\text{→ 2s a}{}^3\text{Σ}{}_{\mathrm{u}}{}^{\mathrm{+}}$, $\text{5pσ k′}{}^3\text{Σ}{}_{\mathrm{g}}{}^{\mathrm{+}}\text{→ 2s a}{}^3\text{Σ}{}_{\mathrm{u}}{}^{\mathrm{+}}$, and $\text{6pσ n}{}^3\text{Σ}{}_{\mathrm{g}}{}^{\mathrm{+}}\text{→ 2s a}{}^3\text{Σ}{}_{\mathrm{u}}{}^{\mathrm{+}}$ transitions analyzed. The g, k′, and n states, which have not been reported previously, are characterized through v = 1, 2, and 1, respectively. Several small accidental perturbations have been observed in the rotational manifolds of the k′ and n states.     

On the mobility of a very pure semiconductor including the low impurity concentration limit

The low temperature mobility μ limited by charged impurities is calculated by solving the equation for the relaxation rate previously derived. The calculated μ behaves like $\text{μ = 2.03 κ}{}^2\text{(k}{}_{\mathrm{B}}\text{T)}{}^{\mathrm{<mtext>3</mtext><mtext>2</mtext>}}\text{e}{}^{\mathrm{?3}}\text{z}{}^{\mathrm{?2}}\text{n}{}_{\mathrm{s}}{}^{\mathrm{?1}}\text{m}{}^{\mathrm{<mtext>1</mtext><mtext>?1</mtext><mtext>2</mtext>}}$ In $\text{[38.2κ}{}^2\text{m}{}^{\mathrm{<mtext>1</mtext><mtext>1</mtext><mtext>2</mtext>}}\text{(k}{}_{\mathrm{B}}\text{T)}{}^{\mathrm{<mtext>5</mtext><mtext>2</mtext>}}\text{/z}{}^2\text{e}{}^4\text{h}\text{?}\text{n}{}_{\mathrm{s}}\text{]}$ for lowest concentrations n_s<10¹¹cm^?3 for Ge and

$\text{μ = 0.360}\text{h}\text{?}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{κ(k}{}_{\mathrm{B}}\text{T)}{}^{\mathrm{<mtext>1</mtext><mtext>4</mtext>}}\text{(ze)}{}^{\mathrm{?1}}\text{n}{}_{\mathrm{s}}{}^{\mathrm{<mtext>?1</mtext><mtext>2</mtext>}}\text{m}{}^{\mathrm{<mtext>1</mtext><mtext>?3</mtext><mtext>4</mtext>}}$

for intermediate concentrations n_s ～ 10¹²?10¹⁴cm^?3.