Franck Condon Factors, Transition Probabilities and Radiative Lifetimes for Hydrogen Molecules and Their Isotopomeres

Transcription

1 INDC(NDS)-457 Franck Condon Factors, Transition Probabilities and Radiative Lifetimes for Hydrogen Molecules and Their Isotopomeres U. Fantz Lehrstuhl für Experimentelle Plasmaphysik, Universität Augsburg, Germany D. Wünderlich Max Planck Institut für Plasmaphysik, Garching, Germany May 2004

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3 Abstract A systematic fundamental molecular data base for all the isotopomeres of hydrogen molecules is presented. The vibrational levels for the electronic states of the molecules up to the principal quantum number n = 4 in each multiplet system are calculated on the basis of potential curves and the reduced mass. From the overlap integral the Franck Condon factors are obtained for all transitions between these levels. For the optically allowed transitions the vibrationally resolved transition probabilities are calculated using the electronic dipole transition moment available in literature. The vibrationally resolved radiative lifetimes of electronic states are then given by a summation. Assuming a vibrational temperature for the vibrational population of the molecules in the ground state and assuming that the Franck Condon principle holds for electronic excitation a relative vibrational population in the excited states is defined. With this population an effective radiative transition probability is calculated. A summation over all optically allowed transitions yields the effective radiative lifetime for an electronic state. The report includes also a compilation of the latest potential curves and electronic dipole transition moment from literature. Calculations are carried out for the homonuclear molecules H 2,D 2,T 2 as well as for the heteronuclear molecules HD, HT, DT which will be all of relevance in cold plasma regions, in particular in the divertor of fusion experiments. Reproduced by the IAEA in Vienna, Austria May 2004

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5 i Contents 1 Introduction 1 2 Theory and Computing Methods Molecular structure of hydrogen The computer program TraDiMo Description of Input Data Potentialcurves Electronic transition dipole moments Discussion of error bars Vibrational Eigenvalues 21 5 Franck Condon Factors Franck Condon factors for H Franck Condon factors for D Franck Condon factors for T Allowed Dipole Transitions Transition Probabilities and Radiative Lifetimes Transition probabilities for H Transition probabilities for D Transition probabilities for T Radiative lifetimes Comparison of Isotopomeres 291 References 297 List of Symbols and Abbreviations 301 List of Tables 309 List of Figures 311

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7 1 Chapter 1 Introduction In the last years the relevance of hydrogen molecules in the edge plasma of fusion experiments has been increasingly recognised due to comparable amounts of molecules and atoms in this cold plasma regions [GR96, FRHC01]. In contrast to laboratory experiments where the molecular gas dissociates and the majority of species in the plasma are still molecules, in fusion experiments molecules are formed by recombination of atoms and ions at the walls. The penetration depth of the molecules depends strongly on the plasma parameters. For example dissociation and ionisation of molecules depends on electron density n e and electron temperature T e. The divertor region has a large dynamic in these plasma parameters due the possibility to operate in different recycling regimes [Sta00]. The extreme cases are recombination at the surfaces which is the attached plasma regime, and the recombination in the volume which is the completely detached plasmas regime. Since the heat load to the targets is drastically reduced in the second case, a detached plasma regime is preferred. Here, temperatures are observed down to tenths of ev, typical values are around a few ev. Molecules which are formed in these cold plasmas can then undergo a variety of processes. In particular the manifold of energy levels in these diatomic molecules opens a variety of further reaction channels. As a consequence hydrogen molecules must be taken into consideration in plasma edge codes, for example B2 EIRENE [SRZ + 92, Rei92]. In addition they must be quantified by diagnostic techniques which are often supported by models, for example collisional radiative models for molecules [SF95, Gre01, WF01]. One of the simplest and most powerful diagnostic methods is the molecular emission spectroscopy which relies in particular on all the manifold of molecular data introduced in this report. In contrast to atoms molecules show isotope effects which means data is needed for hydrogen molecules and their isotopomeres. The reports by Celiberto et al. [CJ95, CLC + 99] represent a compilation of electron impact cross sections of vibrationally excited molecules for selected processes. The book by Janev [JLEP87] and the report by Reiter [Rei00] summarise atomic and molecular data of hydrogen. However, up to now the data base for the fundamental molecular data itself is scarce. This means a systematic compilation of the vibrational levels in electronic states, Franck Condon factors, transition

8 2 Chapter 1. Introduction probabilities and radiative lifetimes is missing. The present report provides an almost complete set of these data up to electronically excited states with principal quantum number n = 4 (in the united atom approximation) for H 2,D 2,T 2 and HD, HT, DT. The heteronuclear molecules are of particular interest since in plasmas with isotope mixtures (D 2 and T 2 as filling gas) not only the homonuclear molecules are of relevance but also the corresponding heteronuclear molecules. Chapter 2 starts with a short review of molecular physics specified for hydrogen. This includes the notation of electronic states, the corresponding energy level diagram and the energy levels in energetic order. The underlaying physics of the computer program TraDiMo is summarised also in this chapter. Since the program is based on the on the solution of the Schrödinger equation using potential curves isotope effects are obtained by changing the reduced mass of the diatomic system. The description of the input data and the references as well as an estimation of the error bars will be presented in Chapter 3. The results for the vibrational energy levels (E v ) in the electronic states up to the principal quantum number n = 4 (in the united atom approximation) sorted by the energy of the electronic state are listed in tables in Chapter 4. Figures will illustrate vibrational eigenfunctions in single and double minima potential curves for selected electronic states as well as for a repulsive state. The Franck Condon factors (FCFs, q p p v v ) for all possible transitions sorted by the isotope are presented in Chapter 5 in which the multiplet systems are treated separately. However, transitions from the ground state to the triplet system are included. Starting from the ground state the sequence is determined by the lower electronic energy level. Allowed dipole transitions are described and illustrated in Chapter 6 applying the same sequence as in Chapter 5. Results for vibrationally resolved transition probabilities A p p v v are listed in Chapter 7. The summation of all possible lower lying vibrational level yields the transition probability of the vibrational level A p p v for an electronic transition. The reciprocal value of a summation over all transition probabilities in an electronic state with vibrational quantum number v yields the vibrational resolved radiative life time τ p v. Assuming a relative vibrational population in the electronically excited state the radiative lifetime of the electronic level τ p eff can be determined in a similar manner. Finally, Chapter 8 will discuss isotope effects in Franck-Condon factors, transition probabilities and radiative lifetimes illustrated with figures. It should be mentioned that all data are included in an electronic numerical database accessible on the IAEA website: However, in order to reduce the amount of tables in this report, tables listed here are limited to the isotopes H 2,D 2,T 2 and principal quantum numbers n<4. The transitions were restricted to changes in the principal quantum number, which means n = 0 except for one transition in the triplet system of the molecules in n = 2 since this transition represents the continuum radiation of the molecules used in a manifold of applications.

9 3 Chapter 2 Theory and Computing Methods 2.1 Molecular structure of hydrogen In principle the spectroscopic notation of electronic energy levels in molecules follows the notation for atoms. However, to simplify the notation the electronic levels are labelled in alphabetical order in energetic sequence, except X is used for the ground state. The two electrons of hydrogen molecules can be arranged to form either a singlet system or a triplet system. The ground state of hydrogen molecules is in the singlet system and is then labelled X 1 Σ + g indicating also the angular momentum and the symmetry and parity of the electron wave function. Capital letters are used for the singlet system and lower case letters for the triplet system (compare with Figure 2.1). In the united atom approximation the electronic states p can be correlated with a principle quantum number n. Due to the possibility of vibrational and rotational motion of molecules each electronic state has vibrational levels labelled with their quantum number v = Each vibrational level has rotational levels with the quantum number J =0.... The book by Herzberg [Her50] gives details for molecular structure of diatomic molecules. The complete spectroscopic notation of the electronic states of hydrogen and the potential curves of H 2,H + 2 and the negative ion H 2 which decays in H and H is given by [Sha71]. Since this report is focussed on the vibrationally resolved quantities the rotational structure is neglected in the following. Figure 2.1 shows the energy level diagram for hydrogen molecules up to the principle quantum number n = 4 up to the d shell (angular momentum two). For simplification, the vibrational levels are not marked. In n = 4 the electronic level with the spectroscopic notation s 3 Σ + g is missing since this state is not identified yet. Due to the presence of double minima potential curves in the singlet systems these levels are labelled with two capital letters. The ionisation energy is ev. The zero level for the energy scale is set to the minimum of the potential curve in the ground state. The b 3 Σ + u level covers a wide range of energy since this is an repulsive state and is therefore the main channel for dissociation into H atoms. The dissociation energy is 4.75 ev. The energies for the minimum of the potential curves of the electronic states

10 4 Chapter 2. Theory and Computing Methods are listed in Table 2.1. The sequence of the following data will be structured according this energetic order separated into singlet and triplet system for all isotopomeres. Singlet Triplet n Notation Energy Notation Energy 1 X 1 Σ + g 0 2 B 1 Σ + u b 3 Σ + u repulsive C 1 Π u c 3 Π u EF 1 Σ + g a 3 Σ + g B 1 Σ + u e 3 Σ + u GK 1 Σ + g d 3 Π u I 1 Π g h 3 Σ + g J 1 g g 3 Σ + g D 1 Π u i 3 Π g H H 1 Σ + g j 3 g B B 1 Σ + u f 3 Σ + u P 1 Σ + g k 3 Π u R 1 Π g p 3 Σ + g no data S 1 g r 3 Π g D 1 Π u s 3 g O 1 Σ + g Table 2.1: Notation and energy [ev] of electronic states (with respect to the potential curve minimum) in energetic order for each multiplet system.

11 2.1. Molecular structure of hydrogen 5 Ion O HH Singlet system B'' D' P D B' GK R S I J n=. 4 3 h Triplet system s 1 Σ + p 1 Σ + p 1 Π d 1 Σ + d 1 Π d 1 u g g u g g s 3 Σ + p 3 Σ + p 3 Π d 3 Σ + d 3 Π d 3 u g g u g g f k d p g r i s j e EF C 2 a c B b X 1 Energy [ev] Figure 2.1: Energy level diagram for hydrogen molecules with respect to the minimum of the potential curve of the ground state X 1 Σ + g.

12 6 Chapter 2. Theory and Computing Methods 2.2 The computer program TraDiMo One of the first systematic calculations for the FCF of hydrogen molecules was presented by Spindler in the year 1969 [Spi69a, Spi69b]. Vibrational wave functions were calculated by numerical solution of the radial Schrödinger equation using either potential curves described by the Morse formula or RKR (Rydberg Klein Rees) potential data. Both potential curves are constructed from spectroscopic data which means molecular constants and vibrational levels are determined from the evaluation of measured spectra. More details about Morse formula and parameters can be obtained from [Her50]. The theory behind RKR potential curves is well summarised in [LeR92] who introduces also his computer code. In 1982 the computer program FCFRKR [TT82] becomes available which is very flexible and calculates FCF as well as transition probabilities and resolves also the rotational structure of molecules. The input data set is based on spectroscopic data, vibrationally and rotationally resolved. These methods are very inconvenient if one wants to calculate an isotope sequence and the corresponding spectroscopic data is not available. In that case the isotope shifts in these spectroscopic data must be calculated first using approximations. In addition, these programs can not handle double minima potential curves and repulsive states, which are of relevance in hydrogen molecules. In order to obtain a complete data set for hydrogen and its isotopomeres the present calculations are based on potential curves and reduced masses. The potential curves (U(r)) are compiled from literature and are described in detail in Chapter 3. The Born Oppenheimer approximation is used, however one should kept in mind that this might be a crude approximation for the light molecule H 2. The reduced mass m r is given by m r =(m 1 m 2 )/(m 1 + m 2 ). The calculation of vibrational wave functions (ψ v (r)) is based on the solution of the Schrödinger equation which yields also the eigenvalues (vibrational energy levels) (E v ): ) ( 2 2 2m r r + U(r) ψ 2 v (r) =E v ψ v (r). (2.1) This second order differential equation is transformed in two ordinary first order differential equations. These equations are solved numerically using the Taylor progression with a fixed step width r. The initial condition is given by the fact that the eigenfunctions approach asymptotically the value zero for r 0: ψ(r min )=ψ 0, ψ (r min ) = 0. The maximum order of the Taylor method as well as the step width are determined by balancing the precision for the calculation against the computing time. It turned out that a Taylor method of the order of three and a step-width of Å for internuclear distances of 0 <r<20 Å fulfill these considerations. In case of the repulsive state b 3 Σ + u the vibrational quantum number is replaced by the energies E above the dissociation limit ψ E (r). The amplitude of the wave functions was normalised to one at r. In order to find the eigenvalues an iteration procedure is used which is based on counting the number of nodes of the wave function for a given energy E. If

13 2.2. The computer program TraDiMo 7 E is below E i, i.e. the energy of the eigenvalue of the vibrational level i in the state of question, the number of nodes is lower than or equal i, ife is above E i, the number of zeros is greater than i. By starting with a given step size for the energy and going through several iterations, each time reducing the step size, the eigenvalues can be determined with arbitrary precision. The precision which is chosen for the eigenvalues is δe cm 1. The overlap integral of two wave functions in different electronic levels yields the Franck Condon factors (q p p v v ) for an electronic transition p to p : 2 q p p v v = 0 ψ p v (r) ψ p v (r) dr. (2.2) For optically allowed transitions the vibrationally resolved transition probabilities A p p v,v can be calculated if the corresponding electronic transition dipole moment D el (r) is known: A p p v v = 16 π ε 0 h λ 3 ψ p v (r) D el (r) ψ p v (r) dr. (2.3) 0 In case of transitions which are coupled with the repulsive state a calculation of FCFs is meaningless. However, transitions to the repulsive states are optically allowed and result in continuum spectra. The corresponding transition probabilities depend on vibrational quantum number of the upper electronic level and the wavelength: A p p v λ (λ) = π 3 c ε 0 h 1 2mr λ 5 E 0 ψ p v (r) D el (r) ψ p E (r) dr 2, (2.4) using the energy of the free particle. Transition probabilities for a vibrational level corresponding to an electronic transition can be obtained from a summation or, in case of the repulsive state, from the transition probability integrated over all wavelengths: A p p v = v A p p v v or A p p v = 0 A p p v,λ (λ) dλ. (2.5) The vibrationally resolved radiative lifetime of an electronically excited state p is then given by the reciprocal value over the sum of all possible transitions into lower electronic states p : 1 τ p v = p A p p v. (2.6) The radiative lifetime for the electronic state τ p is equal to the radiative lifetime of the vibrational level v = 0 under the assumption that the molecule is not

14 8 Chapter 2. Theory and Computing Methods vibrationally excited. An effective radiative lifetime for the electronic state τ p eff can be obtained by assuming a vibrational population for this state: τ p eff = 1 p A p p eff with A p p eff = A p p v n p v v and v n p v =1. (2.7) In thermodynamical equilibrium the population in the vibrational levels is characterised by the Boltzmann distribution and a vibrational temperature T vib can be assigned. However, due to the short lifetime of electronically excited states the assumption of a Boltzmann distribution is not justified. An exceptional case is the ground state where the vibrational levels can thermalise due to their longer lifetime: in case of homonuclear molecules the vibrational levels are metastable states and in case of the heteronuclear molecules radiative transitions are allowed, however the dipole moment of these molecules is very weak. Therefore, a vibrational temperature is assigned to the population in the ground state T vib (X). Under the assumption that the Franck Condon principle holds for electron impact excitation the vibrational population in the ground state is projected into the excited state via the FCFs. This implies that electron impact excitation from the ground state is the dominant excitation channel which is a reasonable assumption for low temperature plasmas. Since the energy gap between vibrational levels decreases with increasing mass of the molecule, the same vibrational temperature for the isotopomeres would result in higher relative vibrational populations for the heavier molecule. From experimental investigations of the isotope effect in vibrational populations in the ground state it is known that the isotopes H 2 and D 2 show similar populations under same experimental conditions [FH98, HFBA01]. Thus, the corresponding vibrational temperature decreases with increasing mass. Following the results obtained in divertor plasmas of fusion experiments [HFBA01] T vib (X) = 5000 K is used for hydrogen. One of the huge advantages of this program is the applicability to bound bound and bound free transitions based on the potential curves, the reduced mass and electronic transition dipole moments. Double minima potential curves are treated as well. Details of the program and applications to low pressure plasmas, in particular for analysing the continuum radiation of hydrogen molecules (a 3 Σ + g b 3 Σ + u transition) are presented in [FSB00].

15 9 Chapter 3 Description of Input Data The potential curves and electronic transition dipole moments were taken from journal literature and are the basis for the calculation of FCFs and transition probabilities for all the isotopomeres of hydrogen molecules. The reduced masses are listed in Table 3.1 in atomic mass units. H HD D HT T DT Table 3.1: Compilation of reduced masses for the isotopomeres in atomic mass units. 3.1 Potential curves The Born Oppenheimer potential curves were compiled from journal literature, where the corresponding electronic wave functions are mostly determined by variational calculations. Preferences were set for the latest data with most data points for a curve. In some cases older references give more data points around the minimum of a potential curve and a newer one includes more data for larger internuclear distances. Then, these data points were put together. Table 3.2 summarises the references for the potential curves. The potential curves were then divided up in sections, each section fitted with polynomial function of tenth order. To avoid oscillations at large internuclear distances a cubic spline is used instead of the polynomial fit. As an maximum value for the internuclear distance 20 Å is used. The procedure resulted in roughly 500 data points for each curve, which means 25 data points are within one Å. The potential curves are shown for the singlet system and the triplet system separately up to the principle quantum number n = 4 in Figures 3.1 and 3.2, respectively. The small tics in the potential curve of the ground state indicate the energy of the vibrational levels for H 2.

16 10 Chapter 3. Description of Input Data Singlet Triplet n Notation Ref. Notation Ref. 1 X 1 Σ + g [Wol93] 2 B 1 Σ + u [SW02] b 3 Σ + u [SW99] C 1 Π u [WS03a] c 3 Π u [SW99, KR77] EF 1 Σ + g [OSW99, WD85] a 3 Σ + g [SW99, KR95] 3 B 1 Σ + u [SW02] e 3 Σ + u [SW99, KR95] GK 1 Σ + g [WD85, WD94] d 3 Π u [SW99] I 1 Π g [KR77, Wol95] h 3 Σ + g [SW99] J 1 g [Wol95, KR82] g 3 Σ + g [SW99] D 1 Π u [WS03a] i 3 Π g [SW99, KR77] H H 1 Σ + g [WD85, WD94, Wol98] j 3 g [Wol95] 4 B B 1 Σ + u [SW02] f 3 Σ + u [SW99] P 1 Σ + g [WD94, DW95] k 3 Π u [SW99] R 1 Π g [Wol95] p 3 Σ + g S 1 g [Wol95, Ryc91] r 3 Π g [SW99] D 1 Π u [WS03a] s 3 g [Wol95, Ryc91] O 1 Σ + g [WD94, DW95] Table 3.2: Compilation of references for the potential curves.

17 3.1. Potential curves s 1 Σ + (O) 4p 1 g Π u (D') 3p 1 Π u (D) H + 2 4d 1 Π g (R) 4p 1 Σ + (B''B) u 4d 1 g (S) 4d 1 Σ + (P) g 14 3d 1 g (J) 3s 1 Σ + (HH) 3d 1 g Π g (I) 3p 1 Σ + (B') 3d 1 Σ + u (GK) g Potential energy [ev] p 1 Σ + u (B) v= s 1 Σ + g (EF) 13 1s 1 Σ + g (X) 14 2p 1 Π u (C) Internuclear distance [Å] Figure 3.1: Potential curves for the singlet system up to n =4.

18 12 Chapter 3. Description of Input Data 18 4d 3 Π g (r) 3d 3 Σ + (g) g 3d 3 g (j) 16 H + 2 4p 3 Π u (k) 3p 3 Π u (d) 4d 3 g (s) 14 3s 3 Σ + (h) 3d 3 g Π g (i) 4p 3 Σ + u (f) 3p 3 Σ + u (e) Potential energy [ev] p 3 Σ + u (b) v= p 3 Π u (c) s 1 Σ + g (X) 2s 3 Σ + g (a) Internuclear distance [Å] Figure 3.2: Potential curves for the triplet system up to n =4.

19 3.2. Electronic transition dipole moments Electronic transition dipole moments Tables 3.3 and 3.4 summarise the references for the electronic transition dipole moments which are used for the calculations of transition probabilities. The data points were interpolated by a linear fit. Singlet Triplet Lower state Upper state Ref. Lower state Upper state Ref. X 1 Σ + g B 1 Σ + u [WS03b] X 1 Σ + g C 1 Π u [WS03a] X 1 Σ + g B 1 Σ + u [WS03b] X 1 Σ + g D 1 Π u [WS03a] X 1 Σ + g B B 1 Σ + u [WS03b] X 1 Σ + g D 1 Π u [WS03a] B 1 Σ + u EF 1 Σ + g [DW95] b 3 Σ + u a 3 Σ + g [SW99] B 1 Σ + u GK 1 Σ + g [DW95] b 3 Σ + u h 3 Σ + g [SW99] B 1 Σ + u I 1 Π g [DW84] b 3 Σ + u g 3 Σ + g [SW99] B 1 Σ + u H H 1 Σ + g [DW95] b 3 Σ + u i 3 Π g [SW99] B 1 Σ + u P 1 Σ + g [DW95] b 3 Σ + u r 3 Π g [SW99] B 1 Σ + u O 1 Σ + g [DW95] C 1 Π u EF 1 Σ + g [DW95] c 3 Π u a 3 Σ + g [SW99] C 1 Π u GK 1 Σ + g [DW95] c 3 Π u h 3 Σ + g [SW99] C 1 Π u I 1 Π g [Wol96] c 3 Π u g 3 Σ + g [SW99] C 1 Π u J 1 g [Wol96] c 3 Π u i 3 Π g [SW99] C 1 Π u H H 1 Σ + g [DW95] c 3 Π u r 3 Π g [SW99] C 1 Π u P 1 Σ + g [DW95] C 1 Π u R 1 Π g [Wol96] C 1 Π u S 1 g [Wol96] C 1 Π u O 1 Σ + g [DW95] EF 1 Σ + g B 1 Σ + u [DW95] a 3 Σ + g e 3 Σ + u [SW99] EF 1 Σ + g D 1 Π u [DW95] a 3 Σ + g d 3 Π u [SW99] EF 1 Σ + g B B 1 Σ + u [DW95] a 3 Σ + g f 3 Σ + u [SW99] a 3 Σ + g k 3 Π u [SW99] Table 3.3: Compilation of references for the electronic transition dipole moments. Part 1.

20 14 Chapter 3. Description of Input Data Singlet Triplet Lower state Upper state Ref. Lower state Upper state Ref. B 1 Σ + u GK 1 Σ + g [DW95] e 3 Σ + u h 3 Σ + g [SW99] B 1 Σ + u I 1 Π g [DW84] e 3 Σ + u g 3 Σ + g [SW99] B 1 Σ + u H H 1 Σ + g [DW95] e 3 Σ + u i 3 Π g [SW99] B 1 Σ + u P 1 Σ + g [DW95] e 3 Σ + u r 3 Π g [SW99] B 1 Σ + u O 1 Σ + g [DW95] GK 1 Σ + g D 1 Π u [DW95] d 3 Π u h 3 Σ + g [SW99] GK 1 Σ + g B B 1 Σ + u [DW95] d 3 Π u g 3 Σ + g [SW99] d 3 Π u i 3 Π g [SW99] d 3 Π u r 3 Π g [SW99] I 1 Π g D 1 Π u [Wol96] h 3 Σ + g f 3 Σ + u [SW99] h 3 Σ + g k 3 Π u [SW99] J 1 g D 1 Π u [Wol96] g 3 Σ + g f 3 Σ + u [SW99] g 3 Σ + g k 3 Π u [SW99] D 1 Π u H H 1 Σ + g [DW95] i 3 Π g f 3 Σ + u [SW99] D 1 Π u P 1 Σ + g [DW95] i 3 Π g k 3 Π u [SW99] D 1 Π u R 1 Π g [Wol96] D 1 Π u S 1 g [Wol96] D 1 Π u O 1 Σ + g [DW95] H H 1 Σ + g B B 1 Σ + u [DW95] f 3 Σ + u r 3 Π g [SW99] B B 1 Σ + u P 1 Σ + g [DW95] k 3 Π u r 3 Π g [SW99] B B 1 Σ + u O 1 Σ + g [DW95] Table 3.4: Compilation of references for the electronic transition dipole moments. Part Discussion of error bars Since the input data is the same for all isotopomeres, except the reduced mass, the discussion can be restricted to calculations for hydrogen. In order to check results of the underlying computer code eigenvalues and FCFs were compared with data from literature. Uncertainties in the numerical algorithm are obtained by using identical potential curves in the code as in literature. This can be done best with potential curves described by the Morse formula. Therefore, a transition in the triplet system is chosen where the Morse parameters are taken from [Spi69a] which gives also the corresponding eigenvalues and FCFs. A comparison of eigenvalues yields a reproducibility which is better than 10 6 of the absolute value. Deviations of FCFs are below 0.1%.

21 3.3. Discussion of error bars Deviation of eigenvalues [%] X 1 Σ + g C 1 Π u a 3 Σ + g B 1 Σ + u Vibrational quantum number v' Figure 3.3: Deviations of eigenvalues using potential curve data from [KW68, KW65] instead of data from [Wol93, SW02, WS03a, SW99]. Since the potential curve determines the eigenvalues and the wave functions, further examinations are focussed on this input parameter. As mentioned in Section 3, the latest data was always chosen. Figure 3.3 shows the deviations of eigenvalues when using older data sources [KW65, KW68, KW65, KW68] instead of the latest data [Wol93, SW02, WS03a, SW99] for the electronic states X 1 Σ + g, B 1 Σ + u, C 1 Π u and a 3 Σ + g, respectively. The absolute value of the eigenvalues refer to the minimum of the corresponding potential curve. It can be seen clearly that the deviations of eigenvalues increase with increasing vibrational quantum number. That means that data taken from older literature have a slightly different shape for the region close the dissociation energy. The decrease of the deviations very close to the dissociation energy results from the dense lying eigenvalues in this region of the potential curve. In a second step, the number of equidistant data points for the potential curves is varied. The consequences on eigenvalues are shown in Figure 3.4 for the electronic states X 1 Σ + g and B 1 Σ + u. Reduction of the number of data points from 500 to 250 and 125 (which means: 1/2 and 1/4) leads clearly to strong deviations for the eigenvalues close to the potential minimum. The decrease of the deviations for higer vibrational levels is due to the decrease of the second derivation of the potential curve in this region which means less data points are needed to reproduce the shape of the curve. A doubling of the number of data points results in discrepancies for the first three to four eigenvalues only but increases the computing time drastically. All in all this means that the chosen number of data points is well balanced against computing time and tolerances for

22 16 Chapter 3. Description of Input Data 3 Fitted data, 125 points B 1 Σ + u Deviation of eigenvalues [%] Fitted data, 250 points Fitted data, 1000 points Vibrational quantum number Deviation of eigenvalues [%] Fitted Data, 250 points Fitted data, 125 points Fitted Data, 1000 points X 1 Σ + g Vibrational quantum number Figure 3.4: Deviations of eigenvalues for the B 1 Σ + u and X 1 Σ + g state using potential curves with different number of data points. eigenvalues. The consequences of deviations of eigenvalues due to different input data bases for the potential curves on the FCFs and transitions probabilities are illustrated in Figures 3.5 and 3.6 for the B 1 Σ + u X 1 Σ + g transition. The deviations are plotted for the first row and the first column of the corresponding matrix. In

23 3.3. Discussion of error bars Deviation of A p'p'' v'v'' Deviation of q p'p'' v'v'' Deviation [%] v''=0 Variation of v' Vibrational quantum number v' 2 Deviation of A p'p'' v'v'' 1 Deviation [%] 0-1 Deviation of q p'p'' v'v'' -2 v'=0 Variation of v'' Vibrational quantum number v'' Figure 3.5: Deviations of q p p v v and p Ap v v for the B 1 Σ + u to X 1 Σ + g transition using potential curve data from [KW68, KW65] instead of data from [Wol93, SW02]. case of different input data sources a strong increase of deviations is obtained for higher vibrational levels. Together with the fact that in this region the deviations of eigenvalues decreases (Figure 3.3) these deviations can be correlated to the deviations in vibrational wave functions. Small differences in eigenvalues

24 18 Chapter 3. Description of Input Data 2 v''=0 Variation of v' 1 Deviation [%] Deviation of q p'p'' v'v'' Deviation of A p'p'' v'v'' Vibrational quantum number v' 4 0 Deviation of q p'p'' v'v'' Deviation [%] Deviation of A p'p'' v'v'' v'=0 Variation of v'' Vibrational quantum number v'' Figure 3.6: Deviations of q p p v v and p Ap v v for the B 1 Σ + u to X 1 Σ + g number of data points for both potential curves. transition using half the drastically change the position of the right turning point of the particle which results in completely different wave functions. In case of the X 1 Σ + g state the scattering of data for higher vibrational quantum numbers can be explained by the fact that the value of the deviation depends on the absolute value of the FCFs

25 3.3. Discussion of error bars 19 and transitions probabilities. This means the precision is determined by the digit rather than by a percentage value. As can be seen in the first row of Table 5.1 the FCF decreases from vibrational quantum number v =9tov = 10 by two orders of magnitude. Deviations in eigenvalues due to the usage of half the number of data points for the potential curves are reproduced in the deviations of the FCFs and transition probabilities as can be seen from a comparison of Figure 3.4 with Figure 3.6. Again a scattering appears for higher vibrational quantum numbers of the X 1 Σ + g state due to the same reasons as discussed above. In summary, these considerations demonstrate clearly that the precision of the calculated values depends strongly on the underlaying potential curve and that the error bars for FCFs and transitions probabilities increase drastically for small values. Therefore, the precision of FCF and transition probabilities is limited to a certain number of digits rather than a percentage value. In case of transition probabilities, the full number of digits is given in order to emphasise the small probabilities. Consequences of high uncertainties for small transition probabilities on radiative lifetimes of electronic states are almost negligible since radiative lifetimes are determined by high transition probabilities.

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27 21 Chapter 4 Vibrational Eigenvalues Figure 4.1 shows the vibrational wave functions for the ground state of molecular hydrogen. For H 2 15 vibrational levels fit into the potential curve curve up to the dissociation limit. The number of vibrational eigenvalues increases with increasing reduced mass of the molecule. In case of double minima potential curves the vibrational eigenvalues alternate between the two curves. In most of these cases the lowest vibrational eigenvalue and wave function is in the left part of the curve which means at lower internuclear distances. One exception is the GK 1 Σ + g state as shown in Figure 4.2. The location of the vibrational eigenfunctions alternates between right and left potential well. In Figure 4.2 and the following tables eigenvalues corresponding to the right well are marked with an asterisk v= Energy [ev] v= v= Internuclear distance [Å] Figure 4.1: Vibrational eigenfunctions of the X 1 Σ + g state of H 2.

28 22 Chapter 4. Vibrational Eigenvalues Energy [ev] v= v=0 * Internuclear distance [Å] Figure 4.2: Vibrational eigenfunctions of the GK 1 Σ + g state of H Energy [ev] Internuclear distance [Å] Figure 4.3: Example of a wave function for a free particle in the repulsive b 3 Σ + u state. Another feature is the repulsive state: the wave function of a free particle is related to the energy as can be seen in Figure 4.3. The eigenvalues for all isotopomeres are listed in the following tables sorted

29 23 by the electronic state (except the repulsive state) in energetic sequence. In order to reduce the size of the tables the number of eigenvalues is limited to 20, except for the ground state. The maximum vibrational quantum number v max is presented also in the tables. The condition for v max is that E vmax < E diss = E(20Å) with a wave function which is definitely inside the potential curve and that the wave function of the next eigenvalue is not in the potential well. However, the complete data set, which means all eigenvalues of the electronic states up to the principal quantum number n = 4 is accessible through the website

30 24 Chapter 4. Vibrational Eigenvalues v H 2 D 2 T 2 HD HT DT v max Table 4.1: Vibrational energy levels E(v) [ev] for the X 1 Σ + g state.

31 25 v H 2 D 2 T 2 HD HT DT v max Table 4.2: Vibrational energy levels E(v) [ev] for the B 1 Σ + u state.

32 26 Chapter 4. Vibrational Eigenvalues v H 2 D 2 T 2 HD HT DT v max Table 4.3: Vibrational energy levels E(v) [ev] for the C 1 Π u state.

33 27 v H 2 D 2 T 2 HD HT DT v max Table 4.4: Vibrational energy levels E(v) [ev] for the EF 1 Σ + g state.

34 28 Chapter 4. Vibrational Eigenvalues v H 2 D 2 T 2 HD HT DT v max Table 4.5: Vibrational energy levels E(v) [ev] for the B 1 Σ + u state.

35 29 v H 2 D 2 T 2 HD HT DT v max Table 4.6: Vibrational energy levels E(v) [ev] for the GK 1 Σ + g state.

36 30 Chapter 4. Vibrational Eigenvalues v H 2 D 2 T 2 HD HT DT v max Table 4.7: Vibrational energy levels E(v) [ev] for the I 1 Π g state. v H 2 D 2 T 2 HD HT DT v max Table 4.8: Vibrational energy levels E(v) [ev] for the J 1 g state.

37 31 v H 2 D 2 T 2 HD HT DT v max Table 4.9: Vibrational energy levels E(v) [ev] for the D 1 Π u state.

38 32 Chapter 4. Vibrational Eigenvalues v H 2 D 2 T 2 HD HT DT v max Table 4.10: Vibrational energy levels E(v) [ev] for the H H 1 Σ + g state.

39 33 v H 2 D 2 T 2 HD HT DT v max Table 4.11: Vibrational energy levels E(v) [ev] for the c 3 Π u state.

40 34 Chapter 4. Vibrational Eigenvalues v H 2 D 2 T 2 HD HT DT v max Table 4.12: Vibrational energy levels E(v) [ev] for the a 3 Σ + g state.

41 35 v H 2 D 2 T 2 HD HT DT v max Table 4.13: Vibrational energy levels E(v) [ev] for the e 3 Σ + u state.

42 36 Chapter 4. Vibrational Eigenvalues v H 2 D 2 T 2 HD HT DT v max Table 4.14: Vibrational energy levels E(v) [ev] for the d 3 Π u state.