Approximate Sum Rule Research Articles

Theoretical evaluation of the radiative lifetimes of LiCs and NaCs in the A1Σ+ state

Calculations of the adiabatic potential energy curves and the transition dipole moments between the ground (A1Σ+) and the first excited (A1Σ+) states have been determined for the LiCs and NaCs molecules. The calculations are performed using an ab initio approach based on non-empirical pseudopotentials for Cs+, Li+ and Na+ cores, parameterized l-dependent polarization potentials and full configuration interaction calculations. The potential energy curves and the transition dipole moment are used to estimate the radiative lifetimes of the vibrational levels of the A+Σ+ state using the Franck–Condon (FC) approximation and the approximate sum rule method. The radiative lifetimes associated with the A+Σ+ state are presented here for the first time. These data can help experimentalists to optimize photoassociative formation of ultracold molecules and their longevity in a trap or in an optical lattice.

Regularity of non-stationary subdivision: a matrix approach

In this paper, we study scalar multivariate non-stationary subdivision schemes with integer dilation matrix M and present a unifying, general approach for checking their convergence and for determining their Hölder regularity (latter in the case M = mI, m ge 2). The combination of the concepts of asymptotic similarity and approximate sum rules allows us to link stationary and non-stationary settings and to employ recent advances in methods for exact computation of the joint spectral radius. As an application, we prove a recent conjecture by Dyn et al. on the Hölder regularity of the generalized Daubechies wavelets. We illustrate our results with several examples.

Approximation order and approximate sum rules in subdivision

Several properties of stationary subdivision schemes are nowadays well understood. In particular, it is known that the polynomial generation and reproduction capability of a stationary subdivision scheme is strongly connected with sum rules, its convergence, smoothness and approximation order. The aim of this paper is to show that, in the non-stationary case, exponential polynomials and approximate sum rules play an analogous role of polynomials and sum rules in the stationary case. Indeed, in the non-stationary univariate case we are able to show the following important facts: (i) reproduction of N exponential polynomials implies approximate sum rules of order N; (ii) generation of N exponential polynomials implies approximate sum rules of order N, under the additional assumption of asymptotical similarity and reproduction of one exponential polynomial; (iii) reproduction of an N-dimensional space of exponential polynomials and asymptotical similarity imply approximation order N; (iv) the sequence of basic limit functions of a non-stationary scheme reproducing one exponential polynomial converges uniformly to the basic limit function of the asymptotically similar stationary scheme.

Magnetic dipole and electric quadrupole hyperfine constants for V II 3d4, 3d3 4s and 3d2 4s2J = 1–5 states

Relativistic configuration interaction (RCI) calculations have been done for hyperfine constants and Lande g-values for 69 even parity V II states. Our results are in good agreement with the few recent experimental magnetic dipole results. We suggest that several Lande g-values need to be re-measured. Our electric quadrupole constants seem to be the first ones available. Analysis and methodology improvements include the introduction of an approximate sum rule for hyperfine constants, and a way of producing a more diagonal reference basis function set. The importance of mixing of 3dn−1 4s basis members with near degenerate 3dn or 3dn−2 4s2 is emphasized.

Polyakov Loop and the Hadron Resonance Gas Model

The Polyakov loop has been used repeatedly as an order parameter in the deconfinement phase transition in QCD. We argue that, in the confined phase, its expectation value can be represented in terms of hadronic states, similarly to the hadron resonance gas model for the pressure. Specifically, L(T)≈1/2[∑(α)g(α)e(-Δ(α)/T), where g(α) are the degeneracies and Δ(α) are the masses of hadrons with exactly one heavy quark (the mass of the heavy quark itself being subtracted). We show that this approximate sum rule gives a fair description of available lattice data with N(f)=2+1 for temperatures in the range 150 MeV<T<190 MeV with conventional meson and baryon states from two different models. For temperatures below 150 MeV different lattice results disagree. One set of data can be described if exotic hadrons are present in the QCD spectrum while other sets do not require such states.

Coupled channels optical theorem and non-elastic cross section sum rule

Within a coupled channels framework a theorem, similar to the optical theorem, is proved which provides a new formula for total non-elastic (inelastic, rearrangement and absorption) cross sections. An approximate cross section energy sum rule is also developed in terms of the trace of the imaginary part of an energy independent, but non-local optical potential. Corrections to sum rule are found small indicating that the result is a good approximation. The theorem and sum rule are applied to the Lane model describing charge exchange reactions.

Analysis of new charmless strange B decay data leaves high [formula omitted] and [formula omitted] still unexplained

The question of whether the anomalously large B→Kη′ and B→Kη′X (inclusive) branching ratios are consistent with the standard model or a sign of new physics is still open and in the experimental court. The nature of the extra hadron X in B→Kη′X is completely unknown. Directions for investigation and analysis of inclusive data are suggested as well as for comparable inclusive data on B→KηX.New data confirming the prediction BR(K±ρ0)=BR(K±ω) support flavor-topology B→VP sum rules based on the OZI rule and emphasize the sharp contrast with the failure of the analogous B→PP sum rule due to the anomalously large BR(B→Kη′). Confirmation of OZI validity in B-decay analyses for VP final states suggests improving data on the analogous neutral decay difference BR(B0→K0ρ0)−BR(B0→K0ω) which measures tree-penguin interference and possible direct CP violation.A successful approximate isospin sum rule is rearranged and reinterpreted to pinpoint tree-penguin interference in B→Kπ and B→Kρ transitions. The magnitude of the interference is shown to still be at the statistical noise level with present data.

Josephson coupling energy and c-axis sum rules in Bi2Sr2Ca2Cu3O10

Polarized reflectivity measurements were performed on a textured Bi2Sr2Ca2Cu3O10 polycrystal. The far infrared c-axis optical conductivity exhibits similarities with other multilayer superconductors. Namely, a broad band develops around 450cm−1 and nearby phonon anomalies are observed as temperature is decreased. In agreement with lattice dynamics calculation, two oxygen bond-bending modes are found at 362cm−1 and 392cm−1. In contrast with bilayer compounds, this latter phonon mode grows in intensity as temperature decreases. In order to describe quantitatively the appearance of the 450cm−1 peak in the superconducting state, we used the Josephson superlattice model to fit experiments. In this model, the multilayer superconductor is considered as a stack of superconducting layers with non-equivalent intralayer and interlayer Josephson coupling constants. The temperature dependence of the interlayer Josephson coupling energy is extracted from the fit and compared to a possible change in c-axis kinetic energy deduced from an approximate tight-binding sum rule.

Sum rules for free energy and frequency distribution of DNA dinucleotides

The large discrepancy between the values of the free energy for DNA dinucleotides (or dimers) measured by different teams has raised a debate, yet unsettled. Here the free energy is fitted by a three parameter empiric formula derived in the framework of the crystal basis model of genetic code. Approximate sum rules are derived and compared satisfactorily with the data. On the basis of theoretical and phenomenological arguments, a relation between the correlation functions of dimer distribution and the free energy is assumed. From consistency conditions, sum rules are derived. A check of these conditions with different samples of experimental data is performed, allowing us to argue on the self-consistency and the reliability of the different sets of experimental data.

Approximate tight-binding sum rule for the superconductivity-related change ofc-axis kinetic energy in multilayer cuprate superconductors

We present an extension of the c-axis tight-binding sum rule discussed by Chakravarty, Kee, and Abrahams [Phys. Rev. Lett. 82, 2366 (1999)] that applies to multilayer high-${T}_{c}$ cuprate superconductors (HTCS) and use it to estimate---from available infrared data---the change below ${T}_{c}$ of the c-axis kinetic energy, $〈{H}_{c}〉,$ in ${\mathrm{YBa}}_{2}{\mathrm{Cu}}_{3}{\mathrm{O}}_{7\ensuremath{-}\ensuremath{\delta}}$ $(\ensuremath{\delta}\ensuremath{\approx}0.45,0.25,0.07),$ ${\mathrm{Bi}}_{2}{\mathrm{Sr}}_{2}{\mathrm{CaCu}}_{2}{\mathrm{O}}_{8},$ and ${\mathrm{Bi}}_{2}{\mathrm{Sr}}_{2}{\mathrm{Ca}}_{2}{\mathrm{Cu}}_{3}{\mathrm{O}}_{10}.$ In all these compounds $〈{H}_{c}〉$ decreases below ${T}_{c}$ and except for ${\mathrm{Bi}}_{2}{\mathrm{Sr}}_{2}{\mathrm{CaCu}}_{2}{\mathrm{O}}_{8}$ the change of $〈{H}_{c}〉$ is of the same order of magnitude as the condensation energy. This observation supports the hypothesis that in multilayer HTCS superconductivity is considerably amplified by the interlayer tunneling mechanism.

Energy and radiative properties of the low-lying NaRb states

D 1 P ‐B 1 P transition dipole moments, as well as nonadiabatic L-uncoupling matrix elements between the examined 1 P and four lowest 1 S 1 states for both 23 Na 85 Rb and 23 Na 87 Rb isotopomers. The relevant MPPT ab initio matrix elements and energy curves were converted by means of the approximate sum rule to radiative lifetimes and L-doubling constants ( q factors! for the particular rovibronic levels of the B 1 P and D 1 P states. The theoretical lifetimes agree well with their experimental counterparts for both B 1 P and D 1 P states. The q factor estimates obtained in the singlet-singlet approximation are in good agreement with the experimental ones for the D 1 P(1<v8<12;7<J8<50) levels, exhibiting a pronounced difference for the B 1 P state. Considerably better agreement was achieved by accounting for the spin-orbit perturbation effect caused by the near-lying c 3 S 1 state. Relative intensity distributions in the D 1 P!X 1 S 1 dispersed fluorescence spectra excited by fixed Ar 1 laser lines were measured for v8(J8)50(44), 1(104), 4(25), 6(44,120), 10(36), and 12(50) D 1 P levels. The experimental intensities and term values were simultaneously embedded in the nonlinear least-square fitting procedure to refine theD 1 P potential.

The origin of Λ-doubling effect for the B 1∏ and D 1∏ states of NaK

The origin of Λ-doubling effect (q factors) for the regularly perturbed NaK B 1∏ and D 1∏ states has been investigated by means of ab initio many-body multipartitioning perturbation theory calculation of the electronic L-uncoupling matrix elements between the examined ∏1 and five lowest ∑+1 states for both Na3923K and Na4123K isotopomers. The hypothesis of pure precession was found to be valid for the B 1∏–A 1∑+ and D 1∏–C 1∑+ pairs of the interacted states, while the unique perturber approximation works properly for the D 1∏ state and completely breaks down for the B 1∏ state. The theoretical rovibronic q-factor values estimated for both states by means of the approximate sum rule agree well with their experimental counterparts and demonstrate high sensitivity to vibrational and rotational quantum numbers.

An Approximate Sign Sum Rule for the Transmission Amplitude through a Quantum Dot

We study the phase of the transmission amplitude through a disordered quantum dot in the Coulomb blockade regime. We calculate the phase dependence on gate voltage for a disorder configuration. We show that over a ``period,'' consisting of a resonance and a transmission valley, the total phase change is 0 (mod $2\ensuremath{\pi}$). Deviations from this sum rule are small in the parameter (level spacing and/or charging energy). The disorder-averaged phase-phase correlation function is found showing interaction-induced correlations between phases at different gate voltages.

The Das-Mathur-Okubo sum rule for the charged pion polarizability in a chiral model

The Das-Mathur-Okubo (DMO) sum rule for the polarizability of charged pions is evaluated for the Nambu-Jona-Lasinio model Lagrangian in both its minimal and extended forms. A comparison is made with the results obtained using the same sum rule from chiral perturbation theory ( χPT), approximate QCD sum rule calculations, explicit calculations on the lattice by Wilcox, and using the semi-empirical Kapusta-Shuryak spectral densities. The χPT results from Compton scattering are also given. We point to a delicate cancellation between the intrinsic and recoil contributions to α π ± in the DMO sum rule approach that can lead to calculated polarizabilities of either sign.

Approximate Sum Rules of CKM Matrix Elements from Quasi-Democratic Mass Matrices

To extract sum rules of CKM matrix elements, eigenvalue problems for quasi-democratic mass matrices are solved in the first order perturbation approximation with respect to small deviations from the democratic limit. Mass spectra of up and down quark sectors and the CKM matrix are shown to have clear and distinctive hierarchical structures. Numerical analysis shows that the absolute values of calculated CKM matrix elements fit the experimental data quite well. The order of the magnitude of the Jarlskog parameter is estimated by the relation $|J| \approx \sqrt{2}(m_c/m_t + m_s/m_b)|V_{us}|^2|V_{cb}|/4$.

Exact sum rule for transversely polarized DIS

We have shown how, working in a field-theoretic framework, one can derive expressions for the even moments of the valence parts of ${g}_{1,2}(x)$. These expressions cannot be written as matrix elements of local operators and do not coincide with the analytic continuation to $n=$ even integer of the OPE results. Just as for the OPE one can in some cases argue that the hadronic matrix elements should be small, leading to approximate sum rules for the moments of the valence parts of ${g}_{1,2}(x)$. But, most importantly, for the case $n=2$ we have proved rigorously that the hadronic matrix element vanishes, yielding an exact sum rule. We have argued that the convergence properties of this sum rule are good and are a further test of QCD.

Optical absorption in the substituted phenylene-based conjugated polymers: Theory and experiment

We investigate theoretically and experimentally the effects of (2,5) chemical substitution on the optical absorption in the phenylene-based conjugated polymers. Theoretically, substitution destroys both the charge-conjugation symmetry and spatial symmetry that characterize the unsubstituted materials. Within Coulomb-correlated theoretical models, the effect of broken charge-conjugation symmetry alone on the underlying electronic structure and on the absorption spectrum is rather weak. When both broken spatial symmetry and broken charge-conjugation symmetry are taken into account, a strong effect on the electronic structure of polyphenylene derivatives is found. In spite of the strong effect of the broken symmetries on the electronic structure, the effect on the optical-absorption spectrum is weak. This surprising result is a consequence of the subtle nature of the configuration interaction in the substituted polyphenylenes within Coulomb-correlated models. We demonstrate numerically an approximate sum rule that governs the strength of an absorption band at 3.7 eV in the absorption spectra of poly(para-phenylene vinylene) (PPV) derivatives. Although substitution can make a previously forbidden transition weakly allowed, the latter acquires strength from a ``finite-size band'' at about the same energy, and not from a higher-energy band at 4.7 eV, as has been previously claimed. It is further predicted that the 3.7-eV band is polarized predominantly along the polymer-chain axis. We have measured the polarization dependence of the optical absorption in an oriented-substituted PPV film. We found that the two lowest-energy absorption bands are polarized predominantly parallel to the chain axis, while the band at 4.7 eV is polarized predominantly perpendicular to the chain axis. These results are in excellent agreement with the theory.

Size- and shape-dependent optical properties of assembled semiconductor spheres

The radiative corrections of nanostructures are investigated by applying a nonlocal theory to a model of semiconductor spheres which are assembled one-, two- and three-dimensionally. An approximate sum rule for radiative corrections and the relation between finite and infinite lattices are given. It is shown that radiative correction strongly depends on the shape as well as the size. For a linear chain, the radiative correction gets saturated when the system length becomes comparable to the wavelength of resonant light. For cubic systems, the radiative width deviates from the size linearity for a surprisingly small size. The two-dimensional case is marginal. This peculiar shape dependence originates from the dipole-dipole interaction.

Approximate sum rule for diatomic vibronic states as a tool for the evaluation of molecular properties

The approximate sum rule < v 1| fΔ U n g| v 1 > ≈ Σ v 2 < v 1| f| v 2 > < v 2| g| v 1> ( E v1 - E v2 ) n , where f, g are operators which couple the vibrational levels v 1, v 2 of two interacting electronic states and Δ U= U 1— U 2 is the difference of the corresponding internuclear potentials, is numerically tested for radiative lifetimes, electronic polarizabilities, fine-structure constants and Landé factors for the vibronic states of the NaK molecule. The approximation is demonstrated to have high accuracy and efficiency for non-diagonal systems, especially for those with significant continuum contributions. Simple ways of estimating the sum rule accuracy are given.

Approximate sum rule for diatomic vibronic states.

The simple expression 〈v'\ensuremath{\Vert}f\ensuremath{\Delta}${\mathit{U}}^{\mathit{n}}$g\ensuremath{\Vert}v'〉 is offered tg are some operators which couple the vibrational levels v' and v'' of two interacting electronic states and \ensuremath{\Delta}U=U'-U'' is the difference of the corresponding internuclear potentials. The rigorous quantum proofs as well as semiclassical justification of this formula are presented and its absolute accuracy is estimated in a closed functional form. Invariant properties and spectroscopic applications of the sum rule, including the evaluation of lifetimes, fine-structure (\ensuremath{\Lambda}-doubling) constants, and Land\'e factors, are discussed.