IWittig's approach to Landau-Zener made even simpler
on-pair formation in the multiphoton photodissociation of HCl studied by observation of H+ and Cl- ions by 3D-imaging
M. Poretskiy,a) A. I. Chichinin,a,b,c) C. Maul, a) K.-H. Gerickea)
a) Institut für Physikalische und Theoretische Chemie, Technische Universität Braunschweig,
Hans-Sommer-Straße 10, 38106 Braunschweig, Germany
ab)Institute of Chemical Kinetics and Combustion, Institutskaya,3, Novosibirsk, 630090Russiabc)Novosibirsk State University, Pirogova Str.,2, Novosibirsk, 630090 Russia
Imaging The Landau-Zener formula was derived in 1932 and since then has remained an important tool in molecular dynamics and spectroscopy. In 2005, C. Witting proposed a simple derivation of the formula,[1] it looks very useful and probably will be widely used. In this note a new derivation of Landau-Zener formula is proposed.[2] Several shortcomings of the Witting's derivation are overcome; now it is really "a single line" derivation.of positive ions proved to be a very powerful method for the investigation of photodissociation, whereas ion imaging of negative ions has only rarely been used. In contrast to positive ions, negative ions can result only from ion-pair photodissociation channels. Therefore, imaging of negative ions exclusively describes these ion-pair channels.
Let there be two nonperturbed diabatic states, labeled 1 and 2, and the energy difference between them is whereis velocity, is time, and and are the slopes of the potential energy curves. We search for the perturbed wave function in the form , where and are eigen functions of the nonperturbed Hamiltonian. The perturbation is given by time-independent off-diagonal matrix elements , and . Putting the wave function into the time-dependent Schrödinger equation yields the differential equation
, (1)
where is a complex function, and all other parameters are real. We assume that and the aim is to find . After dividing (1) by tB and integrating over one obtains: ,. (2)
here the integral at the right side is denoted as . From Eq. (1) it follows that the expression in square brackets has no pole at . The integral may be calculated from the relation , where is the contour integral, see figure. The integral may be obtained from integral of Eq. (2) by replacing real by complex variable ():
, (3)
where the upper and the lower sign correspond to semicircle in upper and lower part of the complex plane, respectively. According to Eq. (1), , hence this term vanishes. Substituting the result from Eq. (3) in Eq. (2) gives . Since is a wave function coefficient, it should be ; this condition makes one of the signs impossible. Finally, we obtain the Landau-Zener formula for the probability of nonadiabatic transition: \[$.$.
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The purpose of this work is twofold: first, to modify the positive ion imaging setup for simultaneous detection of positive (H+, HCl+, Cl+) and negative ions (Cl-) produced by multi-photon ionization and fragmentation of simple molecules (HCl in the present study) and second, to study the dynamics of the ion-pair formation in the case of HCl. The new imaging setup includes a double-sided time-of-flight mass-spectrometer with 3-dimensional (3D) delay-line detectors1,2, which are able to determine the velocity of both positive and negative photoions from the same laser pulse:
The HCl ion-pair photodissociation channel starts from a resonant two-photon absorption to an intermediate state IS1. A third photon excites an unknown state IS2, from which we suggest the system to undergo a nonadiabatic transition to the V1Σ+ state, from which the molecule finally dissociates:
HCl (X1Σ+ ) HCl*(IS1) HCl**(IS2) → HCl**( V1Σ+) → H+ + Cl-. (1)
In our work the 3D velocity distributions of Cl- ions were registered via excitation of IS1= V1Σ+(v=8,9,10,11,12,13, J=0), E1Σ+(v=0, J=0), and g3Σ-(0+,v=0, J=0) states. In all cases, the speed distributions, β parameters of anisotropy, and the dependencies of [Cl-] vs squared laser intensity (I2) were determined. Such dependencies were obtained for the H+, HCl+, and Cl+ ions also. All observed β parameters are close to 2, indicating the symmetry of the state IS2 to be 1Σ+. The dependence of [Cl- ] vs I2 has a maximum. For intermediate values of <I2>. This behavior we explain by the increasing role of the following additional pathway for large laser intensities:
… → HCl**(IS2) →HCl**( V1Σ+) HCl***(IS3)→ H*(n=2) + Cl H+ + e- + Cl, (2)
which reduces the [Cl- ]. The nature of the intermediate states IS1, IS2, and IS3 is discussed.
1. A.I. Chichinin, T.Einfeld, C.Maul, and K.-H.Gericke, Rev. Sci. Instr., 73 1856 (2002).
2.C. Wittig, J. Phys. Chem. B, 109, 8428 (2005).A.I. Chichinin, S.Kauczok, K.-H. Gericke, and C.Maul, Int. Review Phys. Chem. 28, 607 (2009).
2. A. I. Chichinin, J. Phys. Chem. B, 117, 6018 (2013).