Expectation Values Of Dipole Moment Research Articles

Electric multipole moments calculation with explicitly correlated coupled-cluster wavefunctions.

A method of calculation of expectation values of dipole and quadrupole moments with wavefunctions, corresponding to linearly approximated explicitly correlated coupled-cluster singles and doubles [CCSD(F12)] model has been formulated and implemented. As a part of algorithm, explicitly correlated version of Λ equations has also been derived and implemented. Numerical tests, conducted for sets of molecules, show that explicitly correlated results for expectation values of dipole moment are accurate up to 0.01 a.u. already at a double-ζ level compared to those in the complete basis set limit. The corresponding results for quadrupole moments at double-ζ level are accurate up to 0.1 a.u., while for the triple-ζ bases errors do not exceed 0.01 a.u.

Computational molecular spectroscopy for [formula omitted] NiCN: Large-amplitude bending motion

Three-dimensional potential energy surfaces for the 2 Δ electronic ground state of NiCN have been calculated at the MR-SDCI+ Q+ E rel/[Roos ANO (Ni), aug-cc-pVQZ (C, N)] level of theory together with electric dipole moments and transition moments. From the ab initio data, standard molecular constants have been determined by the second-order perturbation method, rovibronic term values, rotational constants, and dipole moment expectation values have been determined by the variational RENNER program, and averaged structures have been computed with the variational MORBID method. The perturbation method failed because of a severe Fermi resonance between 2 ν 2 and ν 3 . The ab initio calculations yield a linear equilibrium structure with r e ( Ni – C ) = 1.8141 Å and r e ( C – N ) = 1.1665 Å . The zero-point averaged structure, determined as expectation values in terms of MORBID wavefunctions, is bent with 〈 ρ ¯ 〉 0 ≡ 180 ° - 〈 ∠ (Ni–C–N) 〉 0 = 9 ( 5 ) ° , 〈 r (Ni–C) 〉 0 = 1.8159 Å , and 〈 r (C–N) 〉 0 = 1.1705 Å . The bending potential is shallow and gives rise to large-amplitude bending motion. Kingston et al. [C.T. Kingston, A.J. Merer, T.D. Varberg, J. Mol. Spectrosc. 215 (2002) 106–127] have obtained r 0 , Ω = 5 / 2 ( C – N ) = 1.1591 ( 29 ) Å in LIF experiments, and Sheridan and Ziurys [P.M. Sheridan, L.M. Ziurys, J. Chem. Phys. 118 (2003) 6370–6379] obtained the value 1.1590(2) Å from microwave spectra. These experimentally derived r 0 , Ω = 5 / 2 (C–N)-values are ‘too short’ when compared with 〈 r (C–N) 〉 0 but not so seriously short as in the cases of FeNC and CoCN.

Comparison of the polarizabilities and hyperpolarizabilities obtained from finite basis set and finite difference Hartree–Fock calculations for diatomic molecules: III. The ground states of N2, CO and BF**This paper is dedicated to Victor R Saunders, on his official retirement from Daresbury Laboratory.

We investigate the accuracy with which the electric dipole polarizability, αzz, and the hyperpolarizability, βzzz, can be calculated by using the algebraic approximation, i.e. finite basis set expansions, and by means of the finite difference method in calculations for the ground states of the 14 electron systems N2, CO and BF within the Hartree–Fock model at their respective experimental equilibrium geometries. For a well-chosen grid, the finite difference technique can provide Hartree–Fock energy and dipole moment expectation values approaching machine precision which can be used to assess the accuracy of corresponding calculations carried out within the algebraic approximation. The finite field approximation is used to determine polarizabilities and hyperpolarizabilities from finite difference Hartree–Fock dipole moment expectation values. The results are compared with finite basis set calculations of the corresponding quantities which are carried out analytically using coupled perturbed Hartree–Fock theory. For the N2 molecule, the Hartree–Fock polarizability is found to be 14.9512 au within the finite basis set approximation and 14.945 au within the finite difference approach. For the CO molecule, the corresponding results are 14.4668 au and 14.4668 au, whilst for the BF molecule the values are 16.6450 au and 16.6450 au, respectively. The Hartree–Fock hyperpolarizability of the CO molecule is found to be 31.4081 au and 31.411 au within the finite basis set and finite difference approximations, respectively. The corresponding hyperpolarizability values for the BF molecule are 63.9687 au and 63.969 au, respectively.

Radiative transition probabilities, lifetimes and dipole moments for the vibrational levels of the X1Sigma+ ground state of 39K85Rb.

Using a potential energy curve (based primarily on the RKR potential of Amiot and Verges [J. Chem. Phys. 112, 7068 (2000)]) and a dipole moment function (based primarily on ab initio calculations of Park et al. [Chem. Phys. 257, 135 (2000)]), we have calculated radiative transition probabilities (Einstein A coefficients), radiative lifetimes, and dipole moment expectation values involving all vibrational levels (for several rotational quantum numbers) of the X1Sigma+ ground state of 39K85Rb. We observe that the radiative lifetimes of vibrationally excited levels, in particular, are approximately 10(3)-10(6) seconds, far too long to be significant in most ultracold experiments involving 39K85Rb or its isotopomers. Comparison with other molecules (LiH and HF) suggests that simple scaling (A approximately mu2nu3 approximately tau(-1)) will predict similarly long lifetimes for many other heteronuclear molecules, e.g., RbCs.

Non-Markovian solutions for probability amplitudes and dipole-moment expectation values in the case of spontaneous Lyman-α emission

By taking into account the counter-rotating terms, equations of motion for probability amplitudes are derived in the case of spontaneous Lyman-α emission. From these equations non-Markovian solutions for probability amplitudes as well as for dipole-moment expectation values are obtained. All the results are provided with accurate error estimates which are absolutely necessary in a rigorous mathematical treatment.

Finite-time deviations from exponential decay of dipole-moment expectation values in the case of spontaneous emission

For the first time, to the authors' knowledge, a mathematically rigorous method is used for obtaining finite-time deviations from the exponential decay of the dipole-moment expectation values in the case of spontaneous Lyman-α transition in a two-level hydrogenic atom. In calculations counterrotating terms, yielding the frequency shift (Lamb shift) within the scope of the two-level model, are taken into account.

Finite-time deviations from exponential decay in the case of spontaneous emission from a two-level hydrogenic atom.

For the first time to the authors' knowledge a mathematically rigorous method is used for treating finite-time deviations from the exponential decay in the case of spontaneous Lyman-\ensuremath{\alpha} transitions in a two-level hydrogenic atom. First, in the so-called Weisskopf-Wigner model (where the rotating-wave approximation is implied) finite-time deviations with a rigorous validity range, based on accurate error estimations, are derived. Second, in order to obtain the frequency shift (Lamb shift), counter-rotating terms are taken into account by using a projection-operator method developed a decade ago by one of the present authors [J. Seke, Phys. Rev. A 21, 2156 (1980)]. By means of this method, equations of motion for dipole-moment expectation values are derived without employing the usual Born approximation. These equations are solved by using the method of the Laplace transform and its inverse. Again finite-time deviations from the exponential decay of the dipole-moment expectation values with a definite validity range are obtained. Finally, it should be emphasized that all the results presented in this paper contain accurate error estimates which are absolutely necessary in a rigorous mathematical method.

The dipole moment function and spectroscopic constants for the CH + and CD + ion-molecules

Potential energy curves, dipole moment functions, dipole moment expectation values, vibrational and rotational constants are computed for CH + and CD + with a configuration interaction wavefunction.

Stark Effect in Color Centers of Alkali Halides

The stark effect of the $F$ center in NaCl, NaBr, KCl, KBr, KI, and RbCl, the $M$ center in KCl, and the ${Z}_{1}$ center in KCl: Sr was measured using an ac electric field. The fractional change of the absorption coefficient, $\frac{\ensuremath{\Delta}\ensuremath{\alpha}}{\ensuremath{\alpha}}$, at the peak of the $F$ band, normalized to a Lorentz local field of 200 kV/cm, was found to be approximately 2\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}5}$ for NaCl and NaBr, 11\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}5}$ for RbCl, and 17\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}5}$ for KCl, KBr, and KI. The value of $\frac{\ensuremath{\Delta}\ensuremath{\alpha}}{\ensuremath{\alpha}}$ at the peak of the $M$ band in KCl was (3.9\ifmmode\pm\else\textpm\fi{}0.4)\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}5}$ for an applied electric field of 97 kV/cm. A measurement of the change in the absorption coefficient, $\ensuremath{\Delta}\ensuremath{\alpha}$, versus photon energy produced a nonsymmetrical curve. This indicates that the effect observed was not the linear Stark effect, and thus provides additional evidence that Seitz's model is incorrect. The absorption coefficient increased at energies lower than the $M$ band, corresponding to a decrease in the absorption coefficient of the $M$ band. The effect is attributed to the mixing of two nondegenerate energy levels of opposite parity. The optical transition to the level of lower energy is normally forbidden. The ${Z}_{1}$ center in KCl: Sr did not exhibit a detectable Stark effect. It was determined, however, that $\frac{\ensuremath{\Delta}\ensuremath{\alpha}}{\ensuremath{\alpha}}l8\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}6}$ for an applied electric field of 97 kV/cm. If the ${Z}_{1}$ possesses a permanent dipole moment, the difference between the expectation values of dipole moment in the ground state and excited state is less than 1.0\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}20}$ cgs unit.