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The formic acid catalyzed gas-phase reaction between H(2)O and SO(3) and its reverse reaction are respectively investigated by means of quantum chemical calculations at the CCSD(T)//B3LYP/cc-pv(T+d)z and CCSD(T)//MP2/aug-cc-pv(T+d)z levels of theory. Remarkably, the activation energy relative to the reactants for the reaction of H(2)O with SO(3) is lowered through formic acid catalysis from 15.97 kcal  mol(-1) to -15.12 and -14.83 kcal  mol(-1) for the formed H(2)O⋅⋅⋅SO(3) complex plus HCOOH and the formed H(2)O⋅⋅⋅HCOOH complex plus SO(3), respectively, at the CCSD(T)//MP2/aug-cc-pv(T+d)z level. For the reverse reaction, the energy barrier for decomposition of sulfuric acid is reduced to -3.07 kcal  mol(-1) from 35.82 kcal  mol(-1) with the aid of formic acid. The results show that formic acid plays a strong catalytic role in facilitating the formation and decomposition of sulfuric acid. The rate constant of the SO(3)+H(2)O reaction with formic acid is 10(5) times greater than that of the corresponding reaction with water dimer. The calculated rate constant for the HCOOH+H(2)SO(4) reaction is about 10(-13) cm(3)  molecule(-1)  s(-1) in the temperature range 200-280 K. The results of the present investigation show that formic acid plays a crucial role in the cycle between SO(3) and H(2)SO(4) in atmospheric chemistry.

The approximately analytical scattering state solutions of the l-wave Schrödinger equation for the Eckart potential are carried out by a proper approximation to the centrifugal term. The normalized radial wavefunctions of l-wave scattering states on the 'k/2π scale' are presented and the calculation formula of phase shifts is derived. It is interesting to find that the energy levels of the continuum states reduce to those of the bound states at the poles of the scattering amplitude. We consider and verify two special cases: the l = 0 and the s-wave Hulthén potential.

The DKP equation with Dirac oscillator potential for spin-0 particles has been studied when both space-space noncommutativity and momentum-momentum noncommutativity are considered. The exact wave functions and corresponding energy levels have been found. Due to the noncommutative effect, the energy spectrum is not degenerate.

The exact solutions of the Duffin–Kemmer–Petiau (DKP) oscillator for spin-0 particles have been studied in noncommutative phase space. The results show that due to the noncommutative effect, the energy spectrum of the DKP oscillator for spin-0 particles is no longer degenerate. In addition, we obtain the nonrelativistic limit of the energy spectrum.

In this work, the generalized Dirac oscillator in cosmic string space-time is studied by replacing the momentum p μ with its alternative p μ + mωβf μ (x μ ). In particular, the quantum dynamics is considered for the function f μ (x μ ) to be taken as cornell potential, exponential-type potentialand singular potential. For cornell potential and exponential-type potential, the corresponding radial equations can be mapped into the confluent hypergeometric equation and hypergeometric equation separately. The corresponding eigenfunctions can be represented as confluent hypergeometric function and hypergeometric function.The equations satisfed by the exact energy spectrum have been found. For singular potential, the wave function and energy eigenvalue are given exactly by power series method.

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A spinless particle coupled covariantly to a uniform magnetic field parallel to the string in the background of the rotating cosmic string is studied. The energy levels of the electrically charged particle subject to the Klein–Gordon oscillator are analyzed. Afterwards, we consider the case of the position-dependent mass and show how these energy levels depend on the parameters in the problem. Remarkably, it shows that for the special case, the Klein–Gordon oscillator coupled covariantly to a homogeneous magnetic field with the position-dependent mass in the rotating cosmic string background has the similar behaviors to the Klein–Gordon equation with a Coulomb-type configuration in a rotating cosmic string background in the presence of an external magnetic field.