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2014 | OriginalPaper | Buchkapitel

3. Molecular Beam Electric Field Deflection: Theoretical Description

verfasst von : Sven Heiles, Rolf Schäfer

Erschienen in: Dielectric Properties of Isolated Clusters

Verlag: Springer Netherlands

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Abstract

After having explained the experimental realization of beam deflection experiments in Chap. 2 , this chapter will introduce various interpretation schemes of the experimental results. Depending on the experimental conditions and the studied system either the rigid rotor model or the floppy cluster assumption must be applied. For the rigid rotor model perturbation theory methods, classical and quantum chemical simulations are discussed and their performance is compared. The latter two methods require a model of the geometric structure of the cluster and the corresponding dielectric properties. Therefore, a very brief introduction of unbiased structure search routines and quantum chemical computations is given. For floppy or thermally activated clusters the simple Langevin-Debye model is introduced while a few more sophisticated methods are discussed at the end of the chapter.

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Fußnoten
1
This is a very good approximation for collimated clusters obtained from hard supersonic expansions. In particular, for our experimental setup \(\textit{v}_x\) is typically \(\sim \)1200 m/s at \(300\) K. After collimation to \(250\) \(\upmu \)m the clusters expand freely for \(\sim \)2.5 m and a typical full width at half maximum of 1.5 mm is measured (see molecular beam profiles). This corresponds to \(\textit{v}_y\) and \(\textit{v}_z\) on the order of \(\approx 1\) m/s.
 
2
At least for atoms and non-linear molecules.
 
3
An in-depth discussions on the influence of the velocity distribution on the experimental results can be found in [7, 8].
 
4
For possible functional forms of \(f_i(\eta _i)\) see for example Ref. [13].
 
5
Some of the properties can be deduced from simple model calculations which maybe give an reasonable agreement with experimental findings. Here, we will concentrate on the most generally applicable approach using ab initio methods. For the prediction of \(\alpha \) by simple model calculations see Refs. [3, 34].
 
6
All following considerations will only be valid for closed shell atoms and molecules. The procedure is very similar for open-shell systems but the resulting equations are somewhat more complicated. For open shell systems the reader is referred to Ref. [31].
 
7
The larger the basis the smaller will be \(\epsilon ^{{\small {\mathrm{HF}}}}\). If \(\epsilon ^{{\small {\mathrm{HF}}}}\) does not change when the basis set is increased the HF limit is reached, i.e. the best result the HF approximation can offer. In a similar way, predictions from other quantum chemical methods can be improved.
 
8
This is only of importance for molecular systems.
 
9
For all what follows we define the principle moments of inertia to fulfill the relation \(I_a\ge I_b \ge I_c\). Please note that this different from the definition most commonly used. In terms of moments of inertia a spherical rotor will have \(I_a=I_b=I_c\). Breaking this symmetry will lead to a prolate (\(I_a=I_b>I_c\)) and an oblate (\(I_a>I_b=I_c\)) rotor which both are classified to be symmetric tops. Note that the symmetry axis of a prolate top is \(c\) and for an oblate rotor it is \(a\). For an asymmetric rotor the moments of inertia fulfill the relations \(I_a\ne I_b\ne I_c\).
 
10
For a system with only two energy levels Eq. 3.14 simplifies to Eq. 1.​1.
 
11
In cluster physics a well known approximation to these equations is the Jellium model. In this model, the valence electrons are treated explicitly and the nuclear charge that balances the valence electrons is assumed to be distributed uniformly. While this approximation is not restricted to DFT it was mainly used in this scientific community. For an in-depth review of this method see [50].
 
12
A systematic improvement of the results is achieved by increasing the basis set size but so far there is no strict way to systematically account for xc effects.
 
13
In order to compare to experiments like beam deflection measurements this conclusion is valid. In the case of high resolution spectroscopy techniques, highly correlated methods and large basis sets have to be used in order to obtain the accuracy needed to describe the experimental findings. See Ref. [72] for an example.
 
14
Even though we are introducing \(\hbar \), we are still using the classical approximation, since \(J\), \(K\), and \(M\) are not restricted to be integers. We only have the restrictions, \(J\ge 0\), \(-J\le M\le J\) and \(-J\le K\le J\).
 
15
Throughout the book the rotational constants have the dimension of an energy. A conversion into wavenumbers, commonly used for rotational constants, is possible by multiplying with \(100/(h\cdot c)\).
 
16
The same result is obtained when using quantum mechanical instead of classical FOPT calculations [74].
 
17
An analytic dipole moment distribution function for a symmetric rotor in the low field limit can be obtained, too (Sascha Schäfer, private communications). However, the influence of this shape correction is rather small.
 
18
For a spherical cluster the value of \(\zeta (\kappa )\) is \(2/9\) and not \(1/3\) what would be expected in a canonical ensemble in thermal equilibrium.
 
19
There are different definitions of the Euler-angels. Here we follow the definitions used in [89].
 
20
The same approach can easily be applied to spherical and symmetric rotors by making use of the symmetry of the moment of inertia tensor. Hence, this discussion can be considered as universally valid.
 
21
The dot represents the time derivative of the corresponding quantity.
 
22
In case of a symmetric rotor an analytical solution is found as described in [91, 92].
 
23
The simulation time employed must be sufficient so that all quantities have converged.
 
24
For a symmetric rotor this would be the symmetry axis.
 
25
Situations exist in which the time-average of \(\mu _a\) will be zero. However, this will be due to a special choice of \(I_c\), \(I_a\), \(J\) and \(\epsilon _{{\small \mathrm{rot}}}\) and is not true in general.
 
26
These observations for an asymmetric rotor will be discussed in Sect. 4.​2 for the case study of Ge-Clusters.
 
27
Here we only consider the case of a prolate symmetric rotor. For an oblate top similar relations can be obtained but the moment of inertia relation \(I_c=I_b\) must be used.
 
28
This is an assumption which breaks down for some asymmetric rotors as discussed below.
 
29
At the point were the states cross this is not true, since the \(\chi \)-values can vary considerably.
 
30
See Ref. [100] for an in-depth discussion and definition of the \(\left| J,K,M \right\rangle \)-Eigenfunctions.
 
31
Additionally, the polarizability will change due to an thermal expansion of the cluster [103, 104]. However, as we will see this effect is small compared to the influence of \({\mu }_0\) and only is important for non-polar clusters.
 
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Metadaten
Titel
Molecular Beam Electric Field Deflection: Theoretical Description
verfasst von
Sven Heiles
Rolf Schäfer
Copyright-Jahr
2014
Verlag
Springer Netherlands
DOI
https://doi.org/10.1007/978-94-007-7866-5_3

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