CMISTARK: Python package for the Stark-effect calculation and symmetry classification of linear, symmetric and asymmetric top wavefunctions in dc electric fields
Yuan-Pin Chang, Frank Filsinger, Boris G. Sartakov, Jochen Küpper
CCMIstark : Python package for the Stark-effectcalculation and symmetry classification of linear,symmetric and asymmetric top wavefunctions in dcelectric fields
Yuan-Pin Chang a , Frank Filsinger b,1 , Boris G. Sartakov c , Jochen K¨upper a,d,e, ∗ a Center for Free-Electron Laser Science, DESY, Notkestrasse 85, 22607 Hamburg, Germany b Fritz-Haber-Institut der MPG, Faradayweg 4-6, 14195 Berlin, Germany c General Physics Institute RAS, Vavilov str. 38, 119991, Moscow, Russia d Department of Physics, University of Hamburg, Luruper Chausse 149, 22761 Hamburg, Germany e The Hamburg Center for Ultrafast Imaging, Luruper Chaussee 149, 22761 Hamburg, Germany
Abstract
The Controlled Molecule Imaging group (CMI) at the Center for Free ElectronLaser Science (CFEL) has developed the
CMIstark software to calculate, view,and analyze the energy levels of adiabatic Stark energy curves of linear, sym-metric top and asymmetric top molecules. The program exploits the symmetryof the Hamiltonian to generate fully labeled adiabatic Stark energy curves.
CMIstark is written in Python and easily extendable, while the core numer-ical calculations make use of machine optimized BLAS and LAPACK routines.Calculated energies are stored in HDF5 files for convenient access and programsto extract ASCII data or to generate graphical plots are provided.
Keywords: molecular rotation, linear top molecule, symmetric top molecule,asymmetric top molecule, electric field, Stark effect
1. Program summary
Program title:.
CMIstark
Catalogue identifier:. (to be added in production)
Program summary URL:. (to be added in production)
Program obtainable from:.
CPC Program Library
Licensing provisions:.
GNU General Public License version 3 or later withamendments. See code for details.
No. of lines in distributed program, including test data, etc.:. ∗ Corresponding author
Email address: [email protected] (Jochen K¨upper)
URL: http://desy.cfel.de/cid/cmi (Jochen K¨upper) Present address: Bruker AXS GmbH, Karlsruhe, Germany
Preprint submitted to Computer Physics Communications May 8, 2018 a r X i v : . [ phy s i c s . a t m - c l u s ] S e p o. of bytes in distributed program, including test data, etc.:. Distribution format:. tar.gz
Programming language:.
Python (version 2.6.x, 2.7.x)
Computer:.
Any Macintosh, PC, or Linux/UNIX workstations with a modernPython distribution
Operating system:.
Tested on Mac OS X and a variety of Linux distributions
RAM:.
Classification:.
Atomic and Molecular Physics, Physical Chemistry and Chem-istry Physics
External routines:.
Python packages numpy and scipy; utilizes (optimized) LA-PACK and BLAS through scipy. All packages available under open-source li-censes.
Nature of problem:.
Calculation of the Stark effect of asymetric top moleculesin arbitrarily strong dc electric fields in a correct symmetry classification andusing correct labeling of the adiabatic Stark curves.
Solution method:.
We set up the full M matrices of the quantum-mechanicalHamiltonian in the basis set of symmetric top wavefunctions and, subsequently,Wang transform the Hamiltonian matrix. We separate, as far as possible, thesub-matrices according to the remaining symmetry, and then diagonalize the in-dividual blocks. This application of the symmetry consideration to the Hamilto-nian allows an adiabatic correlation of the asymmetric top eigenstates in the dcelectric field to the field-free eigenstates. This directly yields correct adiabaticstate labels and, correspondingly, adiabatic Stark energy curves. Restrictions:.
The maximum value of J is limited by the available main memory.A modern desktop computer with 16 GB of main memory allows for calculationsincluding all J s up to a values larger than 100 even for the most complex casesof asymmetric tops. Additional comments:.Running time:.
Typically 1 s–1 week on a single CPU or equivalent on multi-CPU systems (depending greatly on system size and RAM); parallelizationthrough BLAS/LAPACK. For instance, calculating all energies up to J = 25 ofindole ( vide infra ) for one field strength takes 1 CPU-s on a current iMac.2 . Introduction Over the last decade, the manipulation of the motion of molecules usingelectric fields has been revitalized [1–5]. Exploiting the Stark effect, largeasymmetric-top polar molecules have been deflected [6], focused [7], and decel-erated [8]. These techniques can be used to spatially separate neutral moleculesaccording to their quantum states [9], structural isomers [7, 10], and clus-ter sizes [11]. These techniques promise advanced applications of well-definedsamples of complex molecules in various research fields, e. g., modern spec-troscopies [12, 13] or the direct imaging of structural and chemical dynam-ics [2, 14–16]. However, successful implementation of these methods requires athorough theoretical understanding of the molecule-field interaction for the in-volved molecular quantum states. Here we provide a well-tested and optimizedprogram package for the calculation and labeling of so called Stark curves, i. e.,the energies of molecules as a function of electric field strength, for generaluse. This software package will benefit the advance of those forthcoming ap-plications, esp. also for complex molecules. Moreover, it allows non-specialistsand newcomers to the field to concentrate on their envisioned applications ofcontrolled molecules.The code presented here is designed to calculate eigenenergies of very cold(on the order of a few Kelvin) ensembles of polar molecules in the presence ofexternal electrostatic fields. The interaction of the molecular dipole momentwith the dc electric field changes the internal energy, and this is called Starkeffect. To quantify this behavior, the eigenvalue problem of the Hamiltonian issolved.
CMIstark does this calculation in terms of numerically diagonalizingthe corresponding Hamiltonian matrix. An efficient method of diagonalizingthe matrix, exploiting underlying physics phenomena, is employed. Moreover,a correct method of correlating eigenvalues to quantum states, i. e., labeling thecalculated energies for all field strengths, is also required for further use in orderto predict or simulate and analyze control experiments.The software package is named
CMIstark . It is developed and maintainedby the Controlled Molecule Imaging (CMI) group at the Center for Free ElectronLaser Science (CFEL), DESY, in Hamburg, based on earlier work by some ofthe authors at the Fritz Haber Institut of the MPG in Berlin.
3. Description
Stark energies are obtained by setting up and diagonalizing the Hamiltonianmatrix for a given electric field strength. The matrix elements can be obtainedanalytically ( vide infra ) and the resulting matrix is diagonalized numericallyto obtain its eigenvalues, corresponding to the energies of the molecular states.First, the matrix is block-diagonalized as far as possible using symmetry con-siderations in order to correctly assign quantum numbers to eigenvalues. Theblock-diagonalization also significantly reduces the overall computation time,which is dominated by the diagonalization. The resulting blocks are diagonal-ized using LAPACK’s dsyevr or zheevr subroutines for real and complex matri-ces, respectively. The following overview section will provide a brief review ofthe main concepts of the above approach.3 .1. Overview
The quantum-mechanical energy of a molecule, E , can be obtained by solvingthe Schr¨odinger equation H Ψ = E Ψ . (1)Neglecting translation, H denotes the Hamiltonian operator in the center-of-mass frame and Ψ is the wavefunction. For a rigid rotor and neglecting nuclearhyperfine-structure effects, the Hamiltonian can be expressed in terms of compo-nents of the total angular moment operator J about the principal axes ( a, b, c ),i. e., J a , J b , J c [17, 18]: H rigid = (cid:126) ( J a I a + J b I b + J c I c ) = h ( A J a + B J b + C J c ) , (2)where h is Planck’s constant, (cid:126) = h/ π , and I a , I b and I c are three principalmoments of inertia of the rotor. By convention, the principal axes of inertia( a, b, c ) are labeled such that I a ≤ I b ≤ I c . Note that, in the program, insteadof moments of inertia we use rotational constants, A, B, C , which in units ofHertz (Hz) are [17, 18]: A = h π I a , B = h π I b , C = h π I c . (3)Molecular rotors are classified in terms of the magnitudes of their inertial mo-ments, or rotational constants, as shown in Table 1. Several quantum numbersare used to denote zero-field wavefunctions and energies of the rotational statesof molecules [17, 18]. The Schr¨odinger equation of H rigid , (1) and (2), of linearrotors and symmetric tops in free space (see Table 1) can be solved analytically,and their eigenfunctions of H rigid are expressed as spherical harmonics | J, M (cid:105) and Wigner D matrices | J, K, M (cid:105) , respectively [17, 18]. J represents the quan-tum number of total angular momentum J , K characterizes the projection of J onto the symmetry axis of the symmetric top, and M is the quantum numbercharacterizing the projection of J onto a space fixed Z -axis. For asymmetrictops K is not a good quantum number and the Schr¨odinger equation of H rigid cannot generally be solved analytically. A numerical calculation uses symmetrictop wavefunctions | J, K, M (cid:105) as a basis set for obtaining asymmetric top eigen-functions | J K a K c M (cid:105) . Here, the quantum number J and two pseudo quantumnumbers K a , K c specify the zero-field rotational states. Finally, we only focuson closed shell molecules, i. e., molecules which do not have unpaired electrons.Therefore, the values of J , K and M are integer.However, a real molecular system is not rigid. It is assumed that all non-rigidity under the experimental conditions (on the order of 1 K) can be describedmoments of inertia rotational constant rotor type I a = 0; I b = I c A = ∞ , B = C linear top I a = I b = I c A = B = C spherical top I a < I b = I c A > B = C prolate symmetric top I a = I b < I c A = B > C oblate symmetric top I a (cid:54) = I b (cid:54) = I c A (cid:54) = B (cid:54) = C asymmetric top Table 1: Types of rotors defined through their inertial parameters.
4y a Hamiltonian representing centrifugal distortion, H d , with correspondingcentrifugal distortion constants [17]. The Hamiltonian for such a nonrigid rotoris thus written as: H rot = H rigid + H d . The further details of H d for each typeof rotor are described in the next sections. In all cases we have implemented thelowest order quartic centrifugal distortion terms. Higher order terms can easilybe added if necessary.The Stark effect of a polar molecule in a dc electric field is dominated bythe interaction (cid:126)µ · (cid:126)ε of the molecule’s dipole moment (cid:126)µ with the field (cid:126)ε . Whilehigher-order effects become relevant in strong field, they can still be neglectedin our case. For instance, the permanent dipole moment of benzonitriles groundstate leads to an energy shift of 300 GHz at 200 kV/cm, but the correspondingeffect due to the polarizability of the very similar non-polar molecule benzeneis only 50 MHz [19], i. e., almost four orders of magnitude smaller.The dipole interaction with the electric field is described by the followingcontribution to the Hamiltonian: H Stark = − ε (cid:88) g = x,y,z µ g φ Z g , (4)where x, y, z represent a molecule-fixed coordinate system, µ g represent thedipole moment components along the molecule-fixed axes x, y, z , and φ Z g arethe direction cosines of the x, y, z axes with reference to the space-fixed X, Y, Z -axes. Z is oriented along the electric-field direction. In the program, the prin-ciple axis system ( a, b, c ) is identified with the molecule-fixed system ( x, y, z ) inrepresentation I r ( x = b, y = c, z = a ) [17, 18]. Note that this definition has theadvantage that the Stark Hamiltonian does not mix states with different valuesof K if the dipole moment is parallel to the molecular a axis. The rotationalHamiltonian in the field ε can thus be written as: H rot, ε = H rot + H Stark .In the program, the Schr¨odinger equation of the Hamiltonian in the field, H rot, ε , is solved numerically. The corresponding Hamiltonian matrix and thestrategy of its diagonalization are described in following sections for each type ofrotor. Finally, the program assigns the calculated rotational energies in the fieldto “adiabatic quantum numbers”, i. e., to the adiabatically corresponding field-free rotor states [20]. To ensure correct assignments, a symmetry classification of H rot, ε and quantum states according the electric field symmetry group [21, 22]is required. In addition to the separation of M and, for the symmetric top, K , this is achieved through an appropriate unitary transformation of H rot, ε following Wang’s method [17, 23]. In a linear polyatomic molecule, the moment of inertia about the principalaxis a is zero whereas the two other moments of inertia along axes b and c areequal: I b = I c = I . The centrifugal distortion Hamiltanion H d takes the form H d = − hD J (5)where D (Hz) is a centrifugal distortion constant. The first-order perturbationenergy, which is the expectation value of H d over field-free linear rotor wave-functions, | J, M (cid:105) , is included in the Hamiltonian matrix. The dipole moment5f a linear molecule is along its symmetry axis z , i. e., µ z = µ and µ x = µ y = 0.Thus the Stark Hamiltonian simply becomes H Stark = − µεφ Z z (6)The non-zero matrix elements for H rot and H Stark in the basis of linear topwavefunctions | J, M (cid:105) , are provided in Appendix A. The Hamiltonian matrix isdiagonalized directly without any further simplification.
A molecule in which two of the principal moments of inertia are equal is asymmetric-top rotor, such as a prolate top ( I a < I b = I c ) and an oblate top( I a = I b < I c ). The figure axis of the molecule, which is parallel to the dipolemoment, must lie along the special principal axis of inertia, i. e., along the a axis for a prolate top and the c axis for an oblate top. The non-rigidity of asymmetric top is taken into account by including the first order perturbationenergy of the corresponding centrifugal distortion Hamiltonian ( H d ) [17] intothe Hamiltonian matrix. The Stark Hamiltonian H Stark of symmetric tops isthe same as that for linear rotors, as shown in Equation 6. The matrix elementsfor H rot and H Stark in the basis of symmetric top wavefunctions, | J, K, M (cid:105) ,are listed in Appendix B. Finally, the strategy for diagonalizing symmetric andasymmetric top Hamiltonian matrices is the same, and is described in the nextsection. Moreover, in the case of the symmetric top, K is a good quantumnumber and an additional factorization into separate K blocks is possible. An asymmetric-top molecule has three non-zero and non-equal principal mo-ments of inertia. As mentioned before, its Schr¨odinger equation even in thefield-free case has no trivial analytical solution for general J , and field-freesymmetric top wavefunctions, | J, K, M (cid:105) , are used as the basis set for the Hamil-tonian matrix H rot, ε [17]. All nonzero matrix elements of H rot, ε in this basis setare listed in Appendix C. Note that, in this Hamiltonian matrix H rot, ε , thereare no off-diagonal matrix elements in M , because M is still a good quantumnumber in the field. Thus, the blocks of each value of M in the matrix, as shownin Figure 1, can be diagonalized separately. As K in the asymmetric top caseand J in the non-zero field case are not good quantum numbers the set of basisfunctions for the block must cover a wide enough range of K and J to ensurethe accuracy of the numerical solution of the eigenvalue problem.A further simplification of the matrix can be obtained by considering thesymmetry properties of the Hamiltonian. As mentioned before, the symme-try classification is required in order to distinguish avoided crossings (betweencurves with the same symmetry) and real crossings (between curves with differ-ent symmetries) and to assign the energy levels correctly. The field-free Hamil-tonian operator ( H rigid and H rot ) belongs to a symmetry group called Fourgroupand it is designated by V ( a, b, c ) (see Appendix D for a detailed introduction).However, symmetric top wavefunctions, which are the natural basis set, do not Here we consider axially symmetric molecules, ignoring molecules which accidentally havean equivalent tensor of inertia. H H H H H MM -1 M +1 M -1 MM +1 M -1 M -1 M M M +1 M +1 Figure 1: Schematic illustration of the factoring of the Hamiltonian matrix into blocks, labeledby H Mi , in terms of M (depicted by dashed lines) and symmetry. Nonzero elements are onlypresent in the shaded blocks. The number of shaded blocks in each M block depends on themolecular properties, i. e., the symmetry and the dipole moment direction; see text for details. belong to the Fourgroup. A transformation to a symmetrized basis is providedby the Wang transformation of the Hamiltonian matrix [17]: H Wang = (cid:101) XHX = (cid:88) i H i (7)where X denotes the Wang transformation matrix and H i denotes a sub-matrixfor each symmetry species i . Thus, the Hamiltonian is expressed in a basis oflinear combinations of symmetric top wavefunctions which obey the Fourgroupsymmetry [17]. For a field-free asymmetric top, its Wang transformed Hamilto-nian matrix, i. e., H Wang, rot , can be factorized into four sub-matrices in termsof the Fourgroup symmetry species, as described in Appendix D.When an external field is applied, the number of sub-matrices in the Hamil-tonian matrix, H Wang, rot, ε , usually reduces depending on the dipole momentdirection in the molecule and on the values of M . As described in Appendix Eand Appendix F, if the molecule’s dipole moment is parallel to one principal axisof inertia, H Wang, rot, ε can be factorized into two sub-matrices for M (cid:54) = 0, andfour for the special case M = 0. If the molecule’s dipole moment is not parallelto any principal axis of inertia, no factorization of H Wang, rot, ε is possible for M (cid:54) = 0. For M = 0 a factorization into two blocks is still possible if one dipolemoment component µ α ( α = a, b, c ) is zero. In any case, the above block diago-nalization ensures that all eigenstates obtained from the diagonalization of eachsub-matrix H Mi (see Figure 1) belong to a same symmetry. This means that allcrossings between eigenstates of each sub-matrix are avoided and that the en-ergy order of these states remains the same adiabatically, at any field strength.As a result, we can sort the resulting states of each H Mi by energy and assignquantum number labels in the same order as for energy-sorted field-free statesof the same symmetry. This yields a correct assignment of “adiabatic quantumnumber labels”, ˜ J ˜ K a ˜ K c ˜ M , to rotational states in the field [20]. In practice, the calculation of the Stark energies is performed for a num-ber of electric field strengths – typically in steps of 1 kV/cm from 0 kV/cm to200 kV/cm – and the resulting energies are stored for later use. The calculated7tark curves and effective dipole moments for lowest-lying rotational quantumstates of OCS (linear rotor), iodomethane (symmetric top), and indole (asym-metric top) are plotted using cmistark plot energies and are shown in Fig-ure 2(a), Figure 2(b), and Figure 2(c), respectively. Here the effective dipolemoment, µ eff , is introduced as: µ eff ( ε ) = − ∂E ( ε ) ∂ε (8)This is the space fixed dipole moment, i. e., the projection of the molecularframe dipole moment onto the field direction. It is extremely useful in furthersimulations on the manipulation of polar molecules with inhomogeneous electricfields, where the force exerted on the molecule is directly proportional to µ eff [24].Firstly, in these figures, the Stark energy curves from different M alwayscross, i. e., M is a good quantum number. Secondly, while the energy curves ofOCS (Figure 2(a) L) and iodomethane (Figure 2(b) L) show relatively simplestructures, those for indole (Figure 2(c) L) show more complicated behavior.Furthermore, for curves of indole of each M (Figure 2(c) L), most crossingsbetween Stark curves are avoided, because the dipole moment of indole is notparallel to any principal axis ( µ a (cid:54) = 0 , µ b (cid:54) = 0 , µ c = 0). Thus, for M (cid:54) = 0 all statesin the field have the same symmetry. For OCS and iodomethane, the sign ofthe effective dipole moment, µ eff , can be negative or positive, depending on thequantum state and field strength, as shown in Figure 2(a) R and Figure 2(b) R.However, for indole, the sign of µ eff is mostly positive, as shown in Figure 2(c) R.The rapid changes of signs and values of µ eff shown in Figure 2(c) R are due toavoided crossings between Stark curves.In order to evaluate the performance, we have calculated the energy curvesfor OCS at 151 field strengths in the range 0 to 150 kV/cm using the differentalgorithms for three types of rotors for all states up to J = 32. This yields thefollowing computation times on a current iMac: • Linear rotor code: 1.0 s • Symmetric-top code: 55 s • Asymmetric-top code: 300 s (3 min 20 s, 150 % CPU utilization)Note that the runtime largely reflects the time spent on diagonalizing the matrix,and thus the size of the matrix. According to the LAPACK benchmark report inLAPACK Users’ Guide [25], for diagonalizing dense symmetric N by N matricesby using dsyevr, the computing time for N = 1000 is about 400 times of thatfor N = 100. However, the computing time for N = 2000 is about 10 times ofthat for N = 1000. In practice, for asymmetric top calculations an increase ofthe maximum J by 10 (e. g., J = 40 →
50 or J = 90 →
4. Installation instructions
CMIstark needs an operational Python installation, the external Pythonpackages, numpy, scipy, PyTables, matplotlib, and a command-line interface tostart the various python scripts provided here.8
50 100 150−40−200204060 ε (kV/cm) ene r g y ( G H z ) ε (kV/cm) L R e ff μ ( D ) (a) OCS ε (kV/cm) ε (kV/cm) ene r g y ( G H z ) L R e ff μ ( D ) (b) iodomethane ε (kV/cm) ene r g y ( G H z ) ε (kV/cm) L R e ff μ ( D ) (c) indoleFigure 2: (L) Stark energies and (R) effective dipole moments ( µ eff ) of OCS, iodomethane,and indole for the M = 0 (black), M = 1 (blue), and M = 2 (red) levels of J = 0 − .2. Obtaining the code The program is available from CPC Program Library, Queen’s University,Belfast, N. Ireland. The latest version of the program can also be obtained fromthe Controlled Molecule Imaging (CMI) group.
Installation is performed by executing the generic Python install command python setup.py install in the unpacked source code directory. This re-quires a write access to the packages directory tree of the Python distribution.Alternatively, on Unix-like systems the provided shell-script user-install.sh can be used to install the program into an user-specified directory, such as $HOME/.python . This method requires the user to define the shell variable
PYTHONHOME to include this directory in the python search path.
5. Documentation
A full documentation is provided within the source code and only brieflysummarized here. To perform a Stark effect calculation the script file cmistark calculate energy is used. Some of its general command-line op-tions are • --
Currently, isotopologues of the following molecules are implemented in cmistark calculate energy , with parameters from the literature as referencedin the code: 3-aminophenol, carbonylsulfide, water, indole, indole(water) ,iodomethane, difluoroiodobenzene, aminobenzonitrile, benzonitrile, iodoben-zene, and sulfurdioxide. Implementing a new molecule is as simple as addinga code block in cmistark calculate energy to define relevant molecular pa-rameters, molecular constants and dipole moment components. For the cis and10 rans conformers of 3-aminophenol [26], this is implemented in the followingway: def three_aminophenol(param):
6. Alternative software
Several programs exist for the simulation of rotationally resolved spectra ofasymmetric top molecules, such as pgopher [27], spfit/spcat [28, 29], qs-tark [30–32], dbsrot [33, 34], krot [35], asyrot [36] and jb95 [37], as wellas programs for automated fitting of high resolution spectra, e. g., based on ge-netic algorithm [38]. Inherently these programs work by calculating the energiesof all states possibly involved in the relevant transitions, i. e., they do performsimilar calculations as
CMIstark . So far, to the best of our knowledge, onlythe programs pgopher [27] and qstark [30–32] can also calculate Stark ener-gies of linear, symmetric, and asymmetric rotors. The program qstark [30–32]allows calculations including quadrupole coupling effects for one nucleus. pgo-pher [27] can take into account some internal motions, such as internal rotations(torsion) or inversion motions, e. g., in NH . These effects will be implementedin future versions of CMIstark . However, they require considerably more in-tricate handling of symmetry properties. However, the available programs arenot well suited for simulations in the controlled molecules field where it is nec-essary to calculate Stark energies in very strong fields and to correctly labellarge numbers of quantum states over the full field-strength regime. For exam-ple, while the program qstark calculates Stark energies essentially correctly instrong fields, labeling problems are known when the off-diagonal elements in the H matrix become sufficiently large [32]. pgopher provides direct access to theStark curves of individual or a few quantum states. However, its graphical/textbased access is not convenient for the calculation and storage of many preciselycalculated Stark curves with a sufficiently large range of quantum states. Notethat, even for relatively small complex molecules, such as benzonitrile or indoleunder conditions of only few K, many thousand Stark curves need to be calcu-lated with J up to 50, with hundreds of energies per curve for specific dc fieldstrengths, and they must be stored for easy retrieval in further calculations.
7. Outlook
The current program has been successfully used in the calculation of Starkenergy maps of various asymmetric top molecules, for instance, benzonitrile [8],11-aminobenzonitrile [39], 3-aminophenol [7, 10], indole, and indole-water clus-ters [11]. Those calculation results from the progam were successfully applied tofit and analyze experimental data on the manipulation of molecules with elec-tric fields. The program was also tested against the energies of lowest rotationalstates from qstark [30–32], with a relative error on the order of 10 − limited bythe numerical precision of slightly different implementations of the Hamiltonianand the matrix diagonalization.The current program will be further improved in several directions. Forexample, for molecules containing large nuclear quadrupole constants the cor-responding quadrupole coupling terms need to be implemented. The challengehere is to still automatically symmetrize the Hamiltonian and to correctly labelthe resulting states. Moreover, especially many of the small molecules employedin electric-field manipulation experiments are open-shell, i. e., they possess elec-tronic (orbital and spin) angular momentum. The respective Hamiltonians couldalso be implemented in CMIstark . We will implement such extensions as theyare relevant for the simulation of our manipulation experiments. We will sup-port third parties to extend our code to their needs, under the provision that itis provided to all users after a reasonable amount of time.
Acknowledgments
We thank Rosario Gonz´alez-F´erez, Bas van der Meerakker, Gerard Meijer,and members of the CFEL-CMI group for helpful discussions. Izan CastroMolina implemented the initial linear and symmetric top calculations. Thiswork has been supported by the DFG priority program 1116 “Interactions inultracold and molecular gases” and by the excellence cluster “The HamburgCenter for Ultrafast Imaging – Structure, Dynamics and Control of Matter atthe Atomic Scale” of the Deutsche Forschungsgemeinschaft.
Appendix A. Matrix elements for linear rotors
For the linear top, the matrix elements of H rigid and H d can be writtenas [17]: (cid:104) J, M | H rigid | J, M (cid:105) = hBJ ( J + 1) , (A.1) (cid:104) J, M | H d | J, M (cid:105) = − hDJ ( J + 1) , (A.2)where B (Hz) and D (Hz) are the corresponding rotational constant and thequartic centrifugal distortion constant, respectively. The matrix elements forthe Stark Hamiltonian H Stark are expressed as following [18]: (cid:104) J + 1 , M | H Stark | J, M (cid:105) = (cid:104) J, M | H Stark | J + 1 , M (cid:105) = − (cid:112) ( J + 1) − M (cid:112) (2 J + 1)(2 J + 3) µε (A.3) Appendix B. Matrix elements for symmetric tops
For the rigid prolate and oblate symmetric top the matrix elements of H rigid can be written as [17]: (cid:104) J, K, M | H rigid | J, K, M (cid:105) = h (cid:2) BJ ( J + 1) + ( A − B ) K (cid:3) (prolate) (B.1) (cid:104) J, K, M | H rigid | J, K, M (cid:105) = h (cid:2) BJ ( J + 1) + ( B − C ) K (cid:3) (oblate) (B.2)12ith the rotational constants A , B , C (Hz). The matrix elements of H d areexpressed as following [17]: (cid:104) J, K, M | H d | J, K, M (cid:105) = − h (cid:2) ∆ J J ( J + 1) + ∆ JK J ( J + 1) K + ∆ K K (cid:3) (B.3)where ∆ J , ∆ JK and ∆ K are the first-order (quartic) centrifugal distortion con-stants (Hz). The matrix elements of H Stark are [18]: (cid:104)
J, K, M | H Stark | J, K, M (cid:105) = − M KJ ( J + 1) µε (B.4) (cid:104) J + 1 , K, M | H Stark | J, K, M (cid:105) = (cid:104) J, K, M | H Stark | J + 1 , K, M (cid:105) = − (cid:112) ( J + 1) − K (cid:112) ( J + 1) − M ( J + 1) (cid:112) (2 J + 1)(2 J + 3) µε (B.5) Appendix C. Matrix elements for asymmetric tops
For the rigid asymmetric top, the matrix elements of H rigid in terms of I r representation [17] can be written as [17, 18]: (cid:104) J, K, M | H rigid | J, K, M (cid:105) = h (cid:20) B + C J ( J + 1) − K ) + AK (cid:21) , (C.1) (cid:104) J, K + 2 , M | H rigid | J, K, M (cid:105) = (cid:104) J, K, M | H rigid | J, K + 2 , M (cid:105) = h ( B − C )4 (cid:112) J ( J + 1) − K ( K + 1) (cid:112) J ( J + 1) − ( K + 1)( K + 2) , (C.2)with the rotational constants A , B , C (Hz). The distortable rotor is describedusing Watson’s A reduction [40]: (cid:104) J, K, M | H d | J, K, M (cid:105) = − h (cid:2) ∆ J ( J ( J + 1)) + ∆ JK J ( J + 1) K + ∆ K K (cid:3) , (C.3) (cid:104) J, K + 2 , M | H d | J, K, M (cid:105) = (cid:104) J, K, M | H d | J, K + 2 , M (cid:105) = − h (cid:20) δ J J ( J + 1) + δ K K + 2) + K ) (cid:21) × (cid:112) J ( J + 1) − K ( K + 1) (C.4) × (cid:112) J ( J + 1) − ( K + 1)( K + 2)with the five linearly independent quartic distortion constants ∆ J , ∆ JK , ∆ K , δ J and δ K (Hz). The contribution of µ a , i. e., the dipole moment component alongthe principal axis of inertia a , is [18, 41]: (cid:104) J, K, M | H a Stark | J, K, M (cid:105) = − M KJ ( J + 1) µ a ε (C.5) (cid:104) J + 1 , K, M | H a Stark | J, K, M (cid:105) = (cid:104) J, K, M | H a Stark | J + 1 , K, M (cid:105) = − (cid:112) ( J + 1) − K (cid:112) ( J + 1) − M ( J + 1) (cid:112) (2 J + 1)(2 J + 3) µ a ε (C.6)13he contribution of µ b is: (cid:104) J, K + 1 , M | H b Stark | J, K, M (cid:105) = − M (cid:112) ( J − K )( J + K + 1)2 J ( J + 1) µ b ε (C.7) (cid:104) J + 1 , K ± , M | H b Stark | J, K, M (cid:105) = ± (cid:112) ( J ± K + 1)( J ± K + 2) (cid:112) ( J + 1) − M J + 1) (cid:112) (2 J + 1)(2 J + 3) µ b ε (C.8)The H Stark matrix elements involving µ c are: (cid:104) J, K ± , M | H c Stark | J, K, M (cid:105) = ± i M (cid:112) ( J ∓ K )( J ± K + 1)2 J ( J + 1) µ c ε (C.9) (cid:104) J + 1 , K ± , M | H c Stark | J, K, M (cid:105) = − i (cid:112) ( J ± K + 1)( J ± K + 2) (cid:112) ( J + 1) − M J + 1) (cid:112) (2 J + 1)(2 J + 3) µ c ε (C.10)Note that the equations above use the representation I r with the phase conven-tion and formalism of Zare [18]. Appendix D. Fourgroup
The symmetry properties of the rotational Hamiltonian, as well as rotationalwavefunctions, of a rigid asymmetric top molecule may be deduced from itsellipsoid of inertia, which is symmetric not only to an identity operation E butalso to a rotation by 180 ◦ , a C operation, about any of its principal axes ofinertia. This set of symmetry operations forms the Fourgroup ( Viergruppe ),which is designated by V ( a, b, c ) [17]. These symmetry operations cause theangular momentum to transform in the following manner [17]: E : J a → J a , J b → J b , J c → J c (D.1) C a : J a → J a , J b → − J b , J c → − J c (D.2) C b : J a → − J a , J b → J b , J c → − J c (D.3) C c : J a → − J a , J b → − J b , J c → J c (D.4)The character table of the Fourgroup can is shown in Table D.2. Appendix E. Wang transformation
The field-free semirigid rotor Hamiltonian operators H rigid + H d describedabove can be symmetrized to belong to the Fourgroup V and every field-free ro-tor wavefunction can be classified according to its behavior under V ( a, b, c ) [17].This symmetry classification is provided in Table E.3.14he symmetrized basis functions constructed by Wang transformation aredefined as [17, 23, 42]: | J, , M, (cid:105) = | J, , M (cid:105) for K = 0 (E.1) | J, K, M, s (cid:105) = 1 √ | J, K, M (cid:105) + ( − s | J, − K, M (cid:105) ) for K (cid:54) = 0 (E.2)where s is 0 (symmetric) or 1 (antisymmetric) and K now takes on only positivevalues. The Wang transformation can be expressed in a matrix form and thetransformation matrix X of order (2 J + 1) can be expressed as: X = X − = ˜ X = 1 √ . . . ... ... − − · · · √ · · · (E.3)The change of basis can be written as Ψ Wang = ˜ X Ψ . For fixed values of J and M , the vector Ψ consists of (2 J + 1) symmetric top basis functions | J, K, M (cid:105) ,whereas Ψ Wang is the vector of new basis functions that contains the (2 J + 1)symmetrized functions | J, K, M, s (cid:105) : Ψ Wang = | J, J, M, (cid:105)| J, ( J − , M, (cid:105) ... | J, , M, (cid:105)| J, , M, (cid:105)| J, , M, (cid:105) ... | J, ( J − , M, (cid:105)| J, J, M, (cid:105) , Ψ = | J, − J, M (cid:105)| J, ( − J + 1) , M (cid:105) ... | J, − , M (cid:105)| J, , M (cid:105)| J, , M (cid:105) ... | J, ( J − , M (cid:105)| J, J, M (cid:105) (E.4)In the new basis the Hamiltonian matrix factorizes into four sub-matrices thatare historically denoted as E + , O + , E − , O − [23, 42]: H Wang, rot = ˜ XH rot X = E + + O + + E − + O − (E.5)These sub-matrices are classified by the eveness and oddness of K and s , asshown in Table E.3. For a single value of J , | J, K, M, s (cid:105) wavefunctions withinV E C a C b C c A 1 1 1 1B a b c Table D.2: character table for the four group V V [17, 43] and thecorrelation is given in Table E.3. Thus, H Wang, rot can also be block-diagonalizedin terms of four symmetry species, A , B a , B b , and B c . This symmetrizationof the basis by the Wang transformation simplifying the numerical evaluationand, most importantly, is necessary for the correct adiabatic labeling of theeigenstates in the electric field. Appendix F. Block diagonalization of the Hamiltonian matrix
The Hamiltonian is block-diagonal in M and the calculations are performedfor each M separately. In the field-free case all M s are degenerate and only M = 0 is calculated. For the symmetric top, K is a good quantum number andthe matrix is always also factorized into separate K blocks. As mentioned inAppendix E, the Hamiltonian matrix of field-free symmetric or asymmetric topscan be block diagonalized into four blocks according to Fourgroup symmetry.An external dc electric field can mix these blocks, but remaining symmetriesallow partial factorization. In Table F.4 we summarize the block diagonalizationof the Hamiltonian matrix in an electric field according to V for all possible casesof non-zero dipole moment directions, i. e., all possible combinations of non-zerodipole-moment components in the principal axes of inertia system. We notethat the remaining symmetry can be higher for M = 0 than for M (cid:54) = 0. Thiscan also be seen from the matrix elements given above, where the ∆ J = 0 Stark-coupling elements are always proportional to M , i. e., these couplings vanish for M = 0.The factorization summarized in Table F.4 can be understood in terms ofthe symmetry properties of the direction cosine φ Z g in (4) [17]. For the case of µ = µ α and M (cid:54) = 0, basis functions of symmetries A and B α are coupled, aswell as those of symmetries B α (cid:48) and B α (cid:48)(cid:48) , where α (cid:54) = α (cid:48) (cid:54) = α (cid:48)(cid:48) (cid:54) = α . However,no coupling between these two subsets exist. The Hamiltonian matrix can thusbe factorized into two blocks, as listed in Table F.4, one (filled square symbol)containing A and B α and the other one (filled diamond symbol) containing B α (cid:48) and B α (cid:48)(cid:48) . For the special case of M = 0, states of symmetries A and B α forany given J are also not coupled, nor are states of symmetries B α (cid:48) and B α (cid:48)(cid:48) coupled [44]. This is due to the vanishing matrix elements (C.5), (C.7), and(C.9) for M = 0. As a result, states of symmetry A in J even ( J odd ) only couplewith those of symmetry B α in J odd ( J even ), etc. This remaining symmetry forthe case µ = µ α is represented by the open square symbol (open circle symbol)in the first line of Table F.4. States of symmetries B α (cid:48) and B α (cid:48)(cid:48) also couplesubmatrix K s J even J odd E + e 0 A ( ee ) B a ( eo ) E − e 1 B a ( eo ) A ( ee ) O + o 0 B b ( oo ) B c ( oe ) O − o 1 B c ( oe ) B b ( oo ) Table E.3: Symmetry classification of asymmetric top wavefunctions | J K a K c , M (cid:105) for represen-tation I r [18]. The symmetry species of each J K a K c is determined by the eveness or oddnessof K a and K c , which is indicated in parentheses in columns 4 and 5. The classification ofWang sub-matrices is also provided.
16n the same manner, and their remaining symmetries are represented by opentriangle and open diamond symbols in Table F.4. Similar behavior is observedfor all cases when the dipole moment is along a principal axis of inertia. Forthe cases with more than one non-zero dipole moment component states of allfour symmetry species A , B a , B b and B c are coupled for M (cid:54) = 0 and only onesymmetry species remains. For M = 0 and one µ α = 0, the Hamiltonian matrixcan be factorized into two blocks. For a dipole moment with components alongall principal axes of inertia no partial Fourgroup symmetry remains in an electricfield. References [1] S. Y. T. van de Meerakker, H. L. Bethlem, N. Vanhaecke, G. Meijer, Chem.Rev. 112 (2012) 4828–4878.[2] F. Filsinger, G. Meijer, H. Stapelfeldt, H. Chapman, J. K¨upper, Phys.Chem. Chem. Phys. 13 (2011) 2076–2087.[3] M. Schnell, G. Meijer, Angew. Chem. Int. Ed. 48 (2009) 6010–6031.[4] M. T. Bell, T. P. Softley, Mol. Phys. 107 (2009) 99–132.[5] J. K¨upper, F. Filsinger, G. Meijer, Faraday Disc. 142 (2009) 155–173.[6] L. Holmegaard, J. H. Nielsen, I. Nevo, H. Stapelfeldt, F. Filsinger,J. K¨upper, G. Meijer, Phys. Rev. Lett. 102 (2009) 023001.[7] F. Filsinger, U. Erlekam, G. von Helden, J. K¨upper, G. Meijer, Phys. Rev.Lett. 100 (2008) 133003.[8] K. Wohlfart, F. Gr¨atz, F. Filsinger, H. Haak, G. Meijer, J. K¨upper, Phys.Rev. A 77 (2008) 031404(R). J even J odd A B a B b B c A B a B b B c µ a µ b µ c E + E − O + O − E − E + O − O + (cid:54) = 0 0 0 (cid:3)(cid:4) (cid:13) (cid:4) (cid:52) (cid:7) ♦(cid:7) (cid:13) (cid:4) (cid:3)(cid:4) ♦(cid:7) (cid:52) (cid:7) (cid:54) = 0 0 (cid:3)(cid:4) (cid:13) (cid:7) (cid:52) (cid:4) ♦(cid:7) (cid:52) (cid:4) ♦(cid:7) (cid:3)(cid:4) (cid:13) (cid:7) (cid:54) = 0 (cid:3)(cid:4) (cid:13) (cid:7) (cid:52) (cid:7) ♦(cid:4) ♦(cid:4) (cid:52) (cid:7) (cid:13) (cid:7) (cid:3)(cid:4) (cid:54) = 0 (cid:54) = 0 0 (cid:3)(cid:4) ♦(cid:4) ♦(cid:4) (cid:3)(cid:4) ♦(cid:4) (cid:3)(cid:4) (cid:3)(cid:4) ♦(cid:4) (cid:54) = 0 (cid:54) = 0 (cid:3)(cid:4) (cid:3)(cid:4) ♦(cid:4) ♦(cid:4) ♦(cid:4) ♦(cid:4) (cid:3)(cid:4) (cid:3)(cid:4) (cid:54) = 0 0 (cid:54) = 0 (cid:3)(cid:4) ♦(cid:4) (cid:3)(cid:4) ♦(cid:4) ♦(cid:4) (cid:3)(cid:4) ♦(cid:4) (cid:3)(cid:4) (cid:54) = 0 (cid:54) = 0 (cid:54) = 0 (cid:3)(cid:4) (cid:3)(cid:4) (cid:3)(cid:4) (cid:3)(cid:4) (cid:3)(cid:4) (cid:3)(cid:4) (cid:3)(cid:4) (cid:3)(cid:4) Table F.4: Symmetries of asymmetric tops in dc electric fields. Different shapes represent thedistinct symmetry species (matrix blocks) for the case of M = 0 (open symbols) and M (cid:54) = 0(filled symbols) http://pgopher.chm.bris.ac.uk . 1828] H. M. Pickett, J. Mol. Spec. 148 (1991) 371–377.[29] H. M. Pickett, SPFIT/SPCAT, programs for fitting and predictions in ro-tational spectroscopy, 1991-2007. URL: http://spec.jpl.nasa.gov .[30] Z. Kisiel, B. A. Pietrewicz, P. W. Fowler, A. C. Legon, E. Steiner, J. Phys.Chem. A 104 (2000) 6970–6978.[31] Z. Kisiel, J. Kosarzewski, B. A. Pietrewicz, L. Pszcz´o(cid:32)lkowski, Chem. Phys.Lett. 325 (2000) 523–530.[32] Z. Kisiel, qstark website, accessed 2013. URL: