THE JOURNAL OF CHEMICAL PHYSICS 141, 234308 (2014)
Self-interaction corrected density functional calculations of Rydberg states of molecular clusters: N,N-dimethylisopropylamine Hildur Gudmundsdóttir,1 Yao Zhang,2 Peter M. Weber,2 and Hannes Jónsson1,2 1 2
Science Institute and Faculty of Physical Sciences VR-III, University of Iceland, 107 Reykjavík, Iceland Department of Chemistry, Brown University, Providence, Rhode Island 02912, USA
(Received 12 September 2014; accepted 4 November 2014; published online 17 December 2014) Theoretical calculations of Rydberg excited states of molecular clusters consisting of N,Ndimethylisopropylamine molecules using a Perdew-Zunger self-interaction corrected energy functional are presented and compared with results of resonant multiphoton ionization measurements. The binding energy of the Rydberg electron in the monomer is calculated to be 2.79 eV and 2.27 eV in the 3s and 3p state, respectively, which compares well with measured values of 2.88 eV and 2.21 eV. Three different stable configurations of the dimer in the ground state were found using an energy functional that includes van der Waals interaction. The lowest ground state energy conformation has the two N-atoms widely separated, by 6.2 Å, while the Rydberg state energy is lowest for a configuration where the N-atoms of the two molecules come close together, separated by 3.7 Å. This conformational change is found to lower the Rydberg electron binding energy by 0.2 eV. The self-interaction corrected functional gives a highly localized hole on one of the two molecules, unlike results obtained using the PBE functional or the hybrid B3LYP functional which give a delocalized hole. For the trimer, the self-interaction corrected calculation gives a Rydberg electron binding energy lowered further by 0.13 eV as compared with the dimer. The calculated results compare well with trends observed in experimental measurements. The reduction of the Rydberg electron binding energy with cluster size can be ascribed to an effective delocalization of the positive charge of the hole by the induced and permanent dipole moments of the neighboring molecules. A further decrease observed to occur on a time scale of tens of ps can be ascribed to a structural rearrangement of the clusters in the Rydberg state where molecules rotate to orient their dipoles in response to the formation of the localized hole. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4902383] I. INTRODUCTION
Rydberg excited states of molecules, i.e., electronic states of molecules where one electron has been promoted to a high energy orbital, are important for practical as well as for fundamental reasons.1 A molecule in such a state can be considered to be a positively charged molecular core surrounded by an electron in a hydrogen-like electronic state, typically with principal quantum number corresponding to n ≥ 3. The interaction of the loosely bound electron with the ion core can be quite complex and reveal information about the structure as well as possibly vibrational and rotational states of the molecule. The binding energy of the Rydberg electron, i.e., the energy difference between the molecule in the Rydbergexcited state and the corresponding cation (with the same quantum numbers in all other degrees of freedom) can be measured experimentally to high precision and can reveal a wealth of information.2 In particular, the Rydberg electron binding energy has been shown to be a quite sensitive probe of molecular structure,3 leading to the development of spectroscopic tools that use Rydberg states for the identification of molecular structure and possibly identification of molecules.4–12 The technique presently provides a fingerprint of a molecular structure and is referred to as Rydberg Fingerprint Spectroscopy (RFS). It has been proven particularly useful in a time resolved photoionization setup where the ki0021-9606/2014/141(23)/234308/7/$30.00
netics and dynamics of even complex molecular reactions can be followed in real time. To advance the method from a fingerprint to a structure determination tool, a technique to invert the spectra or otherwise derive the structure from the measured data is required. A theoretical tool that can predict a RFS spectrum for a given molecular structure is, therefore, needed. Such calculations need to be fast and applicable to large molecules. Rydberg states of molecules have previously been successfully modelled with high-level wave function methods such as CASPT2, but as these methods scale poorly with system size they are only suitable for small molecules. In a previous article,13 we presented an alternative approach involving self-interaction corrected density functional calculations where computational effort scales only weakly with system size, as the number of valence electrons cubed. The method was shown to give good estimates of the Rydberg electron binding energy for several molecules as compared with high-level wave function methods and/or experimental data. The self-interaction correction is an essential component in these calculations because commonly used KohnSham density functionals,14 such as the PBE generalized gradient functional,15 include a self-interaction error that leads to an incorrect long range form of the effective potential of the electrons and an absence of virtual Rydberg states. Furthermore, the self-interaction error can lead to
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over-delocalization of electronic density,16 as illustrated below. While the Perdew-Zunger self-interaction correction (PZ-SIC) was proposed more than 30 years ago,17 it has not become widely used partly because a variational, selfconsistent implementation of PZ-SIC is more challenging than for Kohn-Sham functionals.18, 19 It has, furthermore, been shown only recently that it is important to use complex orbitals in PZ-SIC calculations.20, 21 In the present project, we applied this approach to the calculation of Rydberg excited states of even larger and more complex systems, namely, molecular clusters of N,Ndimethylisopropylamine (DMIPA). Even the lowest lying excited state of this molecule can be characterized as a 3s Rydberg state. Previously, extensive experimental data and detailed interpretations of the measured Rydberg electron binding energy have been presented,10 making this an appropriate system to test the computational approach and at the same time add some quantitative theoretical input to the analysis. The article is organized as follows: In Sec. II, the methodology is briefly presented. The results of calculations are then presented and compared to experimental results. The article concludes with a discussion in Sec. IV. II. METHODOLOGY
Our approach to the calculation of Rydberg excited states using PZ-SIC to Kohn-Sham density functionals has been described in detail in our previous article.13 There, the Rydberg electron binding energy calculated using this approach was found to agree well with results of experiments and/or high-level quantum chemistry calculations for, in particular, trimethyl amine and ammonia. Here, we give only a brief description of the method. The calculations involve two steps. First, an estimate of the Rydberg electron orbital is obtained from a PZ-SIC calculation of the ground state. The unoccupied orbitals are affected by the self-interaction correction through an optimized effective potential estimated with the Krieger-Li-Iafrate (KLI) approximation.22 In the second step, an electron is placed in this frozen Rydberg orbital and a variational density functional theory calculation including PZSIC is performed for the rest of the electrons to obtain an estimate of the total energy of the molecule in the Rydberg state. The PBE generalized gradient approximation functional is used.15 It is important to also use PZ-SIC in the second step for the clusters because, otherwise the hole does not localize properly, as discussed below. This procedure has been implemented in the GPAW software using a real space grid to represent the orbitals.13, 23, 24 This avoids the need for developing local atomic orbitals, which could be problematic for the highly delocalized orbitals of the clusters. Localization of the hole can take place anywhere and is not biased by the placement of atom centered basis functions. Since the computational effort of this approach scales weakly with system size, as the third power of the number of electrons, calculations of Rydberg states of large molecules and clusters of molecules are possible. The grid was 20 Å on the side in calculations of the monomer and dimers, but up to 23 Å in one direction for the trimer. The distance from an atom nucleus to the edge of the grid was at least 5 Å in all cases. A grid mesh size of
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0.2 Å was used. Tests of convergence with respect to grid and mesh size were carried out and gave 0.01 eV difference in the Rydberg binding energy of the dimer by increasing the size of the grid by 3 Å and 0.02 eV difference by using a mesh size of 0.15 Å. As will be demonstrated below, the hole formed when one electron is promoted to a Rydberg orbital becomes highly localized on one of the molecules in the clusters. As a result, there is a near degeneracy as the hole can localize on either of the two molecules in the dimer. The calculations using the KLI procedure have convergence problems when such a degeneracy is present as has been noted previously.25 To circumvent this problem, a simpler procedure was used in the dimer and trimer calculations, where the lowest energy excited state orbital was estimated from the highest occupied orbital of a triplet state calculation. For the DMIPA monomer, we compare the two approaches and find that the overestimation is 0.26 eV. The variation in the binding energy when comparing the monomer to various configurations of dimers and trimers are, however, assumed to be accurately predicted by this approach. A sampling of ground state conformers of the dimer and trimer was carried out using random initial guesses followed by minimization using the BEEF-vdW functional which includes van der Waals interactions.26 While a complete enumeration of all local minima on the energy surface for these clusters was not attempted, the search led to several stable structures which then were used in calculations of the Rydberg electron binding energy to identify trends with changes in cluster size and shape. The experimental data were previously obtained by Deb et al.10 and are reanalyzed here for the present discussion. By comparing experiments where the carrier Argon gas was at 0 ◦ C on one hand and at −30 ◦ C on the other hand, it can be seen that the higher temperature, which leads to more and larger clusters, shows spectra with lower binding energy of the 3s Rydberg electron. This general trend of reduced binding energy with size of the cluster is reproduced well in the calculations as explained below. III. RESULTS
We first discuss the calculated results for the monomer DMIPA molecule and compare with previously published measurements. Then, results for the dimer and, finally, the trimer are presented and compared with experimental measurements of the 3s Rydberg electron binding energy. A. Monomer calculations
The calculated 3s and 3p Rydberg orbitals using the KLI approach and placing the Rydberg electron in virtual orbitals of a PZ-SIC calculation of the ground state are shown in Fig. 1. The excited electron is coming from the lone pair of the N-atom and the orbitals of the excited electron clearly become centered there. These are the first and second excited states of the molecule, already showing clear Rydberg character, although slight deviations from the shape of hydrogen atom orbitals can be seen. By subtracting the energy of the
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J. Chem. Phys. 141, 234308 (2014) TABLE I. Binding energy (in eV) of the Rydberg electron in the 3s and 3p states of the DMIPA molecule. Results using two different estimates of the Rydberg orbital are given for the 3s state: (1) Using virtual orbital obtained with a KLI approximation to the optimized effective potential, which is the more accurate approach; and (2) using the highest occupied orbital in a triplet state calculation. The latter is used in the calculations of clusters because of convergence problems in the KLI when degeneracy is present. The measured values are taken from Deb et al.10 and Gosselin et al.5 Calculated Rydberg orbital
FIG. 1. The 3s and 3p Rydberg orbitals of a DMIPA molecule. The isosurfaces shown correspond to 0.001 Å−3/2 . For the 3p state, the positive and negative lobes are shown with different colors. The location of atoms in the molecule are indicated with black (C-atoms), white (H-atoms), and blue (Natom) spheres.
Rydberg states from the energy of the cation at the same level of theory, the binding energy of the Rydberg electron is obtained. The values can be compared with previously reported measurements of Deb et al.10 and Gosselin et al.5 as summarized in Table I. The agreement is excellent, within 0.1 eV. This level of agreement is consistent with what we have previously found using this theoretical approach for trimethyl amine and ammonia.13 Another, simpler approach to estimate the Rydberg electron orbital is to carry out a PZ-SIC calculation of the triplet ground state. Then, the highest occupied orbital becomes similar to the 3s Rydberg orbital. This approach can only be applied to the lowest excited state. Similar to our previous studies of other molecules, this more approximate approach gives an overestimate of the binding energy of the Rydberg electron, here by 0.17 eV as compared with the measured value, and 0.26 eV as compared to the KLI calculation. This is a significantly simpler calculation since it does not involve the KLI construction of the effective potential and we use this approach to estimate changes in the binding energy as dimers and trimers form. (a)
method 2
Measured
2.79 2.34 2.28 2.21 2.27
3.05 ... ... ... ...
2.88 2.26 2.20 2.16 2.21
3s 3px 3py 3pz 3p (ave)
B. Dimer calculations
Two DMIPA molecules attract each other because of dispersion or van der Waals interaction and also because of the electrostatic field that results from the permanent dipoles of the molecules. In order to estimate the binding energy of a DMIPA dimer, we have used a functional, BEEF-vdW, that includes van der Waals interaction and has been optimized by comparison with a wide range of ab initio and experimental data.26 Stable structures corresponding to local minima on the energy surface were searched by guessing some initial configurations and then minimizing the energy. Here, selfinteraction correction was not applied. Three different structures are shown in Fig. 2. The binding energy of the dimer for these three structures is given in Table II. The lowest energy structure has the N-atoms in the two molecules rather widely separated, by over 6 Å (see Fig. 2(a)). The binding energy is 0.14 eV, consistent with a strong tendency for clustering. Another structure, only slightly higher in energy, however, has the N-atoms separated by less than 4 Å (see Fig. 2(c)). A structure previously obtained using the B3LYP functional,
(b)
6.18 Å
method 1
(c)
4.63 Å
3.74 Å
FIG. 2. Three structures of the DMIPA dimer corresponding to local minima on the ground state energy surface of the BEEF-vdW functional. The lowest energy structure corresponding to a binding energy of 0.14 eV is shown in (a). There, the N-atoms of the two molecules are separated widely, by 6.18 Å. The configuration giving the lowest energy for the 3s Rydberg state is shown in (c). Here, the N-atoms are closer together, separated by only 3.74 Å, and the energy is lowered by strong electrostatic interaction between the positive hole and the partially negative N-atom of the neighboring molecule.
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TABLE II. Energy (in eV) of the dimer and trimer in the ground state with respect to monomers (cluster binding energy), relative energy of the cations, and relative energy of the 3s Rydberg state of clusters with respect to the lowest energy configuration found, and change in the binding energy of the 3s Rydberg electron as compared with the monomer for the various conformers, shown in Figs. 2 and 5. Cluster configuration
Ground state cluster binding
Relative energy of cation
Relative energy of Rydberg st.
Change in 3s electron binding
Dimer, Fig. 2(a) Dimer, Fig. 2(b) Dimer, Fig. 2(c)
0.14 0.09 0.13
0.76 1.26 0.00
0.54 1.32 0.00
−0.03 −0.30 −0.25
Trimer, Fig. 5(a) Trimer, Fig. 5(b)
0.36 0.32
0.00 0.16
0.00 0.12
−0.38 −0.34
which does not include dispersion interaction, by Deb et al.10 turns out to be somewhat higher in energy, by 0.05 eV (see Fig. 2(b)). After placing an electron in the 3s Rydberg orbital, the two lowest energy structures, which are nearly equally stable in the ground state, have significantly different energy, by 0.5 eV. The structure with N-atoms close together is more stable. The reason for this is that excitation of an electron to the Rydberg orbital leaves a highly localized hole, corresponding to a positive charge, on one of the N-atoms, while the other N-atom still carries a partial negative charge as in the isolated molecule. The 3s Rydberg orbital of the lowest energy configuration is shown in Figs. 3(a) and 3(b). The centering of the Rydberg orbital on one of the N-atoms can be seen clearly.
FIG. 4. Spin density of DMIPA dimer cation with the atomic structure that gave the lowest energy Rydberg state, also shown in Fig. 2(c). The calculations using the PBE functional, shown in (a), and B3LYP functional, shown in (b), give spin density that is delocalized over both N-atoms due to the selfinteraction error in these functionals. The PZ-SIC calculation, shown in (c), gives spin density at only one of the N-atoms, corresponding to a localized hole. The rendered surface corresponds to spin density of 0.16 Å−3 .
FIG. 3. The 3s Rydberg orbital of DMIPA clusters. (a) and (b) Dimer with structure shown in Fig. 2(a). (c) and (d) Trimer with structure shown in Fig. 5(a). The 0.01 Å−3/2 (in (a) and (c)) and 0.002 Å−3/2 (in (b) and (d)) isosurfaces of the orbitals are rendered. The localization of the hole at one of the N-atoms is evident from the position of the center of the orbital.
The PBE calculation, however, fails to localize the hole and distributes it nearly evenly over the two N-atoms. This failure of the PBE functional to localize the hole can be seen even clearer from calculations of the ground state of the dimer cation. The spin density, i.e., the difference in density of electrons with spin-up and density of electrons with spin-down, is shown in Fig. 4. While the PBE functional predicts nearly equal spin density on the two N-atoms (see Fig. 4(a)), the self-interaction corrected calculation shows clear localization of the hole on one of the N-atoms (see Fig. 4(c)). An integration of the spin density over Bader volumes27, 28 for each of the N-atoms gives 0.33 and 0.35 electrons from
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the PBE calculation, while the PZ-SIC calculation gives 0.80 on one of the N-atoms and 0.02 on the other. The tendency of commonly used Kohn-Sham functionals to delocalize electronic states results from the self-interaction error in the functionals. This is a well known shortcoming of Kohn-Sham functionals and has, for example, been noted in studies of various types of defects in solids.29, 30 Hybrid functionals,31 where a fraction of exact exchange is mixed in with local density and gradient dependent functionals, have become widely used and suffer somewhat less from the self-interaction error (since the self-interaction in the exact exchange term partially cancels the self-interaction in the Coulomb energy), but a significant self-interaction error remains. The B3LYP functional32 has, in particular, become widely used in molecular calculations. The results of B3LYP calculations for the dimer cation shown in Fig. 4(b) give similarly delocalized a hole as the PBE functional. It is essential to apply a self-interaction correction to these functionals in order to get proper localization of the hole for the DMIPA dimer. The binding energy of the 3s Rydberg electron is estimated to be 0.03–0.30 eV lower for the three configurations of the DMIPA dimer than in the isolated molecule, see Table II. The lowering of the binding energy is larger for the configuration that is most stable in the Rydberg state than in the most stable ground state configuration. This can be understood from the orientation of the dipole moment of the molecule adjacent to the hole and the distance between the N-atoms. The presence of the dipole effectively diffuses the positive charge of the localized hole and thereby lowers the Rydberg electron binding energy, as discussed by Liang et al.9 and Deb et al.10 The energy difference between the cations in the two dimer configurations shown in Figs. 2(a) and 2(c) is even larger than the energy difference between the Rydberg states, 0.76 eV vs. 0.54 eV, since the Rydberg electron partly screens the positive charge at the hole (see Table II). C. Trimer calculations
A limited search for configurations of the trimer were carried out using the BEEF-vdW functional. Two stable configurations were found. In both cases, the three N-atoms are rather close together, as shown in Figs. 5(a) and 5(b). The binding energy of the trimer with respect to three monomers is quite similar in these two configurations, 0.36 eV and 0.32 eV. The binding energy of the Rydberg electron in these trimer configurations is calculated to be 0.09–0.13 eV lower than in the most stable dimer configuration. The calculations, therefore, show that the Rydberg electron binding energy is reduced as the clusters grow. In the 3s Rydberg state, the configuration shown in Fig. 5(a) is 0.12 eV lower in energy than the one in Fig. 5(b) and also gives lower binding energy by 0.13 eV compared to the most stable dimer Rydberg state (see Table II). These trends are in excellent agreement with the experimental results discussed below. In the Rydberg state, the localized hole attracts the partial negative charge of the Natoms of the neighboring molecules. A lower Rydberg electron binding energy results from the ordering of the dipoles of the neighboring molecules, which effectively diffuse the positive charge of the localized hole.
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(a)
(b)
FIG. 5. Two atomic structures of the DMIPA trimer, corresponding to local minima on the BEEF-vdW energy surface for the ground state, as well as isosurface for spin density of the cation. The structure in 5(a) is the same as that shown in Figs. 3(c) and 3(d). The energy of the two structures in the ground state and Rydberg state are given Table II. The spin density is localized on one of the N-atoms, indicating a localized hole on one of the three molecules. The rendered surface corresponds to 0.18 Å−3 , about 10% of the maximum value.
The localization of the hole in the trimer can again be seen from ground state calculations of the cation. Fig. 5 shows the spin density of the cation of both trimer structures. Clearly, the positive charge is localized on one of the N-atoms. Calculations using the PBE functional without self-interaction correction, however, delocalize the positive charge on all three N-atoms, analogous to the results for the dimer.
D. Comparison with experiments
Experimental measurements of the 3s Rydberg electron binding energy were as reported previously.10 The signal obtained in an experiment using Ar carrier gas and the sample reservoir at −30 ◦ C is shown in Fig. 6. A broad feature covering binding energy in the range of 1.8 eV–2.7 eV can be seen. These values are significantly lower than the binding energy of the 3s Rydberg state of the monomer, 2.88 eV, which was previously obtained from experiments using He carrier gas, where clustering does not occur. It is clear that the 3s
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FIG. 6. Experimental binding energy of 3s Rydberg electron in DMIPA clusters as a function of time.10 The color represents the intensity, as indicated by the color bar (arbitrary units). The average calculated binding energy for the two lowest energy ground state structures found for the dimer and the two lowest energy structures found for the trimer are indicated by white dashed curves at t = 0. The shape of the curves was chosen to indicate further lowering of the Rydberg electron binding energy as neighboring molecules reorient their dipole in response to the localized hole.
Rydberg electron binding energy is lower in clusters than in the monomer. Furthermore, by comparing this spectrum to a spectrum taken with the Ar and the sample at a higher temperature, 0 ◦ C, where more and larger clusters are expected to form, it can be seen that the Rydberg electron binding energy of larger clusters is lowered more than that of smaller ones. The evolution of the measured signal as a function of the time delay between the excitation pulse and the ionization pulse shows a distinct lowering of the 3s Rydberg electron binding energy as the cluster evolves in the Rydberg state on a time scale of a few tens of ps.10
IV. DISCUSSION
Detailed comparison between the experimental measurements and the calculated results is difficult because the size distribution of the clusters at the time of ionization is not known. While mass spectra have been recorded in these experiments, the time delay from ionization to recording of the mass spectra is several hundred nanoseconds so dissociation of clusters has likely occurred. Nevertheless, under similar experimental conditions, clusters with up to 10 molecules have been detected.10 It is, therefore, not possible at this time to assign a value of the Rydberg electron binding energy to a specific cluster size. But, it is clear that the trend in the calculated values going from monomer to dimer and then to trimer is such that the binding energy is lowered. The predicted values of the binding energy for the dimer, 2.65 eV, and trimer, 2.43 eV, obtained by averaging the calculated binding energy for the two lowest energy configurations found for the ground state, are shown in Fig. 6 (corresponding to t = 0). This comparison indicates that the experiment includes a rather broad distribution in cluster size including clusters that are significantly larger than the trimer. The top of the measured signal, i.e., high values in Rydberg electron binding energy are
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in good correspondence with the calculations for the dimers. The sequential lowering of the Rydberg electron binding energy as the cluster size increases can be understood from the induced dipole of molecules in response to the formation of the localized hole. The dipoles effectively diffuse the localized charge seen by the Rydberg electron and this leads to reduced binding energy. There is also good correspondence between the calculations and the time evolution of the measured signal. The molecules adjacent to the hole will tend to rotate in order to point their partially negative N-atom towards the hole. The calculations show that such configurations are lower in energy both for the dimer and the trimer, while little preference is for the orientation of the molecules in the ground state. It can, therefore, be expected that a structural rearrangement takes place in the Rydberg state which will, however, involve overcoming energy barriers and thus take time to complete. The results of the calculations for the dimer show that while two configurations with quite different N–N distance are practically equally stable in the ground state, the one with short N– N distance is preferred by 0.5 eV in the 3s Rydberg state and if the dimer structure transforms from one to the other, the binding energy of the Rydberg electron will be reduced by 0.22 eV (see Table II). This kind of lowering of the Rydberg electron binding energy upon configurational changes in the Rydberg state can explain the time evolution of the measured signal and is the basis for the shape of the sketched time evolution of the dimer and trimer in Fig. 6 (white dashed curves). The calculated results presented here support strongly the interpretation of the experimental data presented by Deb et al.10 The calculations presented here of Rydberg electron binding energy using self-interaction corrected density functional theory demonstrate that theoretical studies of even such large systems as the DMIPA trimer can be carried out to help analyze these types of measurements. In principle, larger clusters could be calculated including dynamics and identification of minimum energy paths for thermally activated transitions in the excited state. Such calculations, are, however, outside the scope of the present project. ACKNOWLEDGMENTS
This work was supported by the Icelandic Research Fund and by DTRA, Grant No. HDTRA1-14-1-0008. The calculations were carried out using the Nordic High Performance Computing (NHPC) facility in Iceland. H.J. thanks Susi Lehtola for helpful discussions. 1 R.
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Self-interaction corrected density functional calculations of Rydberg states of molecular clusters: N,N-dimethylisopropylamine Hildur Gudmundsdóttir,1 Yao Zhang,2 Peter M. Weber,2 and Hannes Jónsson1,2 1 2
Science Institute and Faculty of Physical Sciences VR-III, University of Iceland, 107 Reykjavík, Iceland Department of Chemistry, Brown University, Providence, Rhode Island 02912, USA
(Received 12 September 2014; accepted 4 November 2014; published online 17 December 2014) Theoretical calculations of Rydberg excited states of molecular clusters consisting of N,Ndimethylisopropylamine molecules using a Perdew-Zunger self-interaction corrected energy functional are presented and compared with results of resonant multiphoton ionization measurements. The binding energy of the Rydberg electron in the monomer is calculated to be 2.79 eV and 2.27 eV in the 3s and 3p state, respectively, which compares well with measured values of 2.88 eV and 2.21 eV. Three different stable configurations of the dimer in the ground state were found using an energy functional that includes van der Waals interaction. The lowest ground state energy conformation has the two N-atoms widely separated, by 6.2 Å, while the Rydberg state energy is lowest for a configuration where the N-atoms of the two molecules come close together, separated by 3.7 Å. This conformational change is found to lower the Rydberg electron binding energy by 0.2 eV. The self-interaction corrected functional gives a highly localized hole on one of the two molecules, unlike results obtained using the PBE functional or the hybrid B3LYP functional which give a delocalized hole. For the trimer, the self-interaction corrected calculation gives a Rydberg electron binding energy lowered further by 0.13 eV as compared with the dimer. The calculated results compare well with trends observed in experimental measurements. The reduction of the Rydberg electron binding energy with cluster size can be ascribed to an effective delocalization of the positive charge of the hole by the induced and permanent dipole moments of the neighboring molecules. A further decrease observed to occur on a time scale of tens of ps can be ascribed to a structural rearrangement of the clusters in the Rydberg state where molecules rotate to orient their dipoles in response to the formation of the localized hole. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4902383] I. INTRODUCTION
Rydberg excited states of molecules, i.e., electronic states of molecules where one electron has been promoted to a high energy orbital, are important for practical as well as for fundamental reasons.1 A molecule in such a state can be considered to be a positively charged molecular core surrounded by an electron in a hydrogen-like electronic state, typically with principal quantum number corresponding to n ≥ 3. The interaction of the loosely bound electron with the ion core can be quite complex and reveal information about the structure as well as possibly vibrational and rotational states of the molecule. The binding energy of the Rydberg electron, i.e., the energy difference between the molecule in the Rydbergexcited state and the corresponding cation (with the same quantum numbers in all other degrees of freedom) can be measured experimentally to high precision and can reveal a wealth of information.2 In particular, the Rydberg electron binding energy has been shown to be a quite sensitive probe of molecular structure,3 leading to the development of spectroscopic tools that use Rydberg states for the identification of molecular structure and possibly identification of molecules.4–12 The technique presently provides a fingerprint of a molecular structure and is referred to as Rydberg Fingerprint Spectroscopy (RFS). It has been proven particularly useful in a time resolved photoionization setup where the ki0021-9606/2014/141(23)/234308/7/$30.00
netics and dynamics of even complex molecular reactions can be followed in real time. To advance the method from a fingerprint to a structure determination tool, a technique to invert the spectra or otherwise derive the structure from the measured data is required. A theoretical tool that can predict a RFS spectrum for a given molecular structure is, therefore, needed. Such calculations need to be fast and applicable to large molecules. Rydberg states of molecules have previously been successfully modelled with high-level wave function methods such as CASPT2, but as these methods scale poorly with system size they are only suitable for small molecules. In a previous article,13 we presented an alternative approach involving self-interaction corrected density functional calculations where computational effort scales only weakly with system size, as the number of valence electrons cubed. The method was shown to give good estimates of the Rydberg electron binding energy for several molecules as compared with high-level wave function methods and/or experimental data. The self-interaction correction is an essential component in these calculations because commonly used KohnSham density functionals,14 such as the PBE generalized gradient functional,15 include a self-interaction error that leads to an incorrect long range form of the effective potential of the electrons and an absence of virtual Rydberg states. Furthermore, the self-interaction error can lead to
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over-delocalization of electronic density,16 as illustrated below. While the Perdew-Zunger self-interaction correction (PZ-SIC) was proposed more than 30 years ago,17 it has not become widely used partly because a variational, selfconsistent implementation of PZ-SIC is more challenging than for Kohn-Sham functionals.18, 19 It has, furthermore, been shown only recently that it is important to use complex orbitals in PZ-SIC calculations.20, 21 In the present project, we applied this approach to the calculation of Rydberg excited states of even larger and more complex systems, namely, molecular clusters of N,Ndimethylisopropylamine (DMIPA). Even the lowest lying excited state of this molecule can be characterized as a 3s Rydberg state. Previously, extensive experimental data and detailed interpretations of the measured Rydberg electron binding energy have been presented,10 making this an appropriate system to test the computational approach and at the same time add some quantitative theoretical input to the analysis. The article is organized as follows: In Sec. II, the methodology is briefly presented. The results of calculations are then presented and compared to experimental results. The article concludes with a discussion in Sec. IV. II. METHODOLOGY
Our approach to the calculation of Rydberg excited states using PZ-SIC to Kohn-Sham density functionals has been described in detail in our previous article.13 There, the Rydberg electron binding energy calculated using this approach was found to agree well with results of experiments and/or high-level quantum chemistry calculations for, in particular, trimethyl amine and ammonia. Here, we give only a brief description of the method. The calculations involve two steps. First, an estimate of the Rydberg electron orbital is obtained from a PZ-SIC calculation of the ground state. The unoccupied orbitals are affected by the self-interaction correction through an optimized effective potential estimated with the Krieger-Li-Iafrate (KLI) approximation.22 In the second step, an electron is placed in this frozen Rydberg orbital and a variational density functional theory calculation including PZSIC is performed for the rest of the electrons to obtain an estimate of the total energy of the molecule in the Rydberg state. The PBE generalized gradient approximation functional is used.15 It is important to also use PZ-SIC in the second step for the clusters because, otherwise the hole does not localize properly, as discussed below. This procedure has been implemented in the GPAW software using a real space grid to represent the orbitals.13, 23, 24 This avoids the need for developing local atomic orbitals, which could be problematic for the highly delocalized orbitals of the clusters. Localization of the hole can take place anywhere and is not biased by the placement of atom centered basis functions. Since the computational effort of this approach scales weakly with system size, as the third power of the number of electrons, calculations of Rydberg states of large molecules and clusters of molecules are possible. The grid was 20 Å on the side in calculations of the monomer and dimers, but up to 23 Å in one direction for the trimer. The distance from an atom nucleus to the edge of the grid was at least 5 Å in all cases. A grid mesh size of
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0.2 Å was used. Tests of convergence with respect to grid and mesh size were carried out and gave 0.01 eV difference in the Rydberg binding energy of the dimer by increasing the size of the grid by 3 Å and 0.02 eV difference by using a mesh size of 0.15 Å. As will be demonstrated below, the hole formed when one electron is promoted to a Rydberg orbital becomes highly localized on one of the molecules in the clusters. As a result, there is a near degeneracy as the hole can localize on either of the two molecules in the dimer. The calculations using the KLI procedure have convergence problems when such a degeneracy is present as has been noted previously.25 To circumvent this problem, a simpler procedure was used in the dimer and trimer calculations, where the lowest energy excited state orbital was estimated from the highest occupied orbital of a triplet state calculation. For the DMIPA monomer, we compare the two approaches and find that the overestimation is 0.26 eV. The variation in the binding energy when comparing the monomer to various configurations of dimers and trimers are, however, assumed to be accurately predicted by this approach. A sampling of ground state conformers of the dimer and trimer was carried out using random initial guesses followed by minimization using the BEEF-vdW functional which includes van der Waals interactions.26 While a complete enumeration of all local minima on the energy surface for these clusters was not attempted, the search led to several stable structures which then were used in calculations of the Rydberg electron binding energy to identify trends with changes in cluster size and shape. The experimental data were previously obtained by Deb et al.10 and are reanalyzed here for the present discussion. By comparing experiments where the carrier Argon gas was at 0 ◦ C on one hand and at −30 ◦ C on the other hand, it can be seen that the higher temperature, which leads to more and larger clusters, shows spectra with lower binding energy of the 3s Rydberg electron. This general trend of reduced binding energy with size of the cluster is reproduced well in the calculations as explained below. III. RESULTS
We first discuss the calculated results for the monomer DMIPA molecule and compare with previously published measurements. Then, results for the dimer and, finally, the trimer are presented and compared with experimental measurements of the 3s Rydberg electron binding energy. A. Monomer calculations
The calculated 3s and 3p Rydberg orbitals using the KLI approach and placing the Rydberg electron in virtual orbitals of a PZ-SIC calculation of the ground state are shown in Fig. 1. The excited electron is coming from the lone pair of the N-atom and the orbitals of the excited electron clearly become centered there. These are the first and second excited states of the molecule, already showing clear Rydberg character, although slight deviations from the shape of hydrogen atom orbitals can be seen. By subtracting the energy of the
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J. Chem. Phys. 141, 234308 (2014) TABLE I. Binding energy (in eV) of the Rydberg electron in the 3s and 3p states of the DMIPA molecule. Results using two different estimates of the Rydberg orbital are given for the 3s state: (1) Using virtual orbital obtained with a KLI approximation to the optimized effective potential, which is the more accurate approach; and (2) using the highest occupied orbital in a triplet state calculation. The latter is used in the calculations of clusters because of convergence problems in the KLI when degeneracy is present. The measured values are taken from Deb et al.10 and Gosselin et al.5 Calculated Rydberg orbital
FIG. 1. The 3s and 3p Rydberg orbitals of a DMIPA molecule. The isosurfaces shown correspond to 0.001 Å−3/2 . For the 3p state, the positive and negative lobes are shown with different colors. The location of atoms in the molecule are indicated with black (C-atoms), white (H-atoms), and blue (Natom) spheres.
Rydberg states from the energy of the cation at the same level of theory, the binding energy of the Rydberg electron is obtained. The values can be compared with previously reported measurements of Deb et al.10 and Gosselin et al.5 as summarized in Table I. The agreement is excellent, within 0.1 eV. This level of agreement is consistent with what we have previously found using this theoretical approach for trimethyl amine and ammonia.13 Another, simpler approach to estimate the Rydberg electron orbital is to carry out a PZ-SIC calculation of the triplet ground state. Then, the highest occupied orbital becomes similar to the 3s Rydberg orbital. This approach can only be applied to the lowest excited state. Similar to our previous studies of other molecules, this more approximate approach gives an overestimate of the binding energy of the Rydberg electron, here by 0.17 eV as compared with the measured value, and 0.26 eV as compared to the KLI calculation. This is a significantly simpler calculation since it does not involve the KLI construction of the effective potential and we use this approach to estimate changes in the binding energy as dimers and trimers form. (a)
method 2
Measured
2.79 2.34 2.28 2.21 2.27
3.05 ... ... ... ...
2.88 2.26 2.20 2.16 2.21
3s 3px 3py 3pz 3p (ave)
B. Dimer calculations
Two DMIPA molecules attract each other because of dispersion or van der Waals interaction and also because of the electrostatic field that results from the permanent dipoles of the molecules. In order to estimate the binding energy of a DMIPA dimer, we have used a functional, BEEF-vdW, that includes van der Waals interaction and has been optimized by comparison with a wide range of ab initio and experimental data.26 Stable structures corresponding to local minima on the energy surface were searched by guessing some initial configurations and then minimizing the energy. Here, selfinteraction correction was not applied. Three different structures are shown in Fig. 2. The binding energy of the dimer for these three structures is given in Table II. The lowest energy structure has the N-atoms in the two molecules rather widely separated, by over 6 Å (see Fig. 2(a)). The binding energy is 0.14 eV, consistent with a strong tendency for clustering. Another structure, only slightly higher in energy, however, has the N-atoms separated by less than 4 Å (see Fig. 2(c)). A structure previously obtained using the B3LYP functional,
(b)
6.18 Å
method 1
(c)
4.63 Å
3.74 Å
FIG. 2. Three structures of the DMIPA dimer corresponding to local minima on the ground state energy surface of the BEEF-vdW functional. The lowest energy structure corresponding to a binding energy of 0.14 eV is shown in (a). There, the N-atoms of the two molecules are separated widely, by 6.18 Å. The configuration giving the lowest energy for the 3s Rydberg state is shown in (c). Here, the N-atoms are closer together, separated by only 3.74 Å, and the energy is lowered by strong electrostatic interaction between the positive hole and the partially negative N-atom of the neighboring molecule.
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TABLE II. Energy (in eV) of the dimer and trimer in the ground state with respect to monomers (cluster binding energy), relative energy of the cations, and relative energy of the 3s Rydberg state of clusters with respect to the lowest energy configuration found, and change in the binding energy of the 3s Rydberg electron as compared with the monomer for the various conformers, shown in Figs. 2 and 5. Cluster configuration
Ground state cluster binding
Relative energy of cation
Relative energy of Rydberg st.
Change in 3s electron binding
Dimer, Fig. 2(a) Dimer, Fig. 2(b) Dimer, Fig. 2(c)
0.14 0.09 0.13
0.76 1.26 0.00
0.54 1.32 0.00
−0.03 −0.30 −0.25
Trimer, Fig. 5(a) Trimer, Fig. 5(b)
0.36 0.32
0.00 0.16
0.00 0.12
−0.38 −0.34
which does not include dispersion interaction, by Deb et al.10 turns out to be somewhat higher in energy, by 0.05 eV (see Fig. 2(b)). After placing an electron in the 3s Rydberg orbital, the two lowest energy structures, which are nearly equally stable in the ground state, have significantly different energy, by 0.5 eV. The structure with N-atoms close together is more stable. The reason for this is that excitation of an electron to the Rydberg orbital leaves a highly localized hole, corresponding to a positive charge, on one of the N-atoms, while the other N-atom still carries a partial negative charge as in the isolated molecule. The 3s Rydberg orbital of the lowest energy configuration is shown in Figs. 3(a) and 3(b). The centering of the Rydberg orbital on one of the N-atoms can be seen clearly.
FIG. 4. Spin density of DMIPA dimer cation with the atomic structure that gave the lowest energy Rydberg state, also shown in Fig. 2(c). The calculations using the PBE functional, shown in (a), and B3LYP functional, shown in (b), give spin density that is delocalized over both N-atoms due to the selfinteraction error in these functionals. The PZ-SIC calculation, shown in (c), gives spin density at only one of the N-atoms, corresponding to a localized hole. The rendered surface corresponds to spin density of 0.16 Å−3 .
FIG. 3. The 3s Rydberg orbital of DMIPA clusters. (a) and (b) Dimer with structure shown in Fig. 2(a). (c) and (d) Trimer with structure shown in Fig. 5(a). The 0.01 Å−3/2 (in (a) and (c)) and 0.002 Å−3/2 (in (b) and (d)) isosurfaces of the orbitals are rendered. The localization of the hole at one of the N-atoms is evident from the position of the center of the orbital.
The PBE calculation, however, fails to localize the hole and distributes it nearly evenly over the two N-atoms. This failure of the PBE functional to localize the hole can be seen even clearer from calculations of the ground state of the dimer cation. The spin density, i.e., the difference in density of electrons with spin-up and density of electrons with spin-down, is shown in Fig. 4. While the PBE functional predicts nearly equal spin density on the two N-atoms (see Fig. 4(a)), the self-interaction corrected calculation shows clear localization of the hole on one of the N-atoms (see Fig. 4(c)). An integration of the spin density over Bader volumes27, 28 for each of the N-atoms gives 0.33 and 0.35 electrons from
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the PBE calculation, while the PZ-SIC calculation gives 0.80 on one of the N-atoms and 0.02 on the other. The tendency of commonly used Kohn-Sham functionals to delocalize electronic states results from the self-interaction error in the functionals. This is a well known shortcoming of Kohn-Sham functionals and has, for example, been noted in studies of various types of defects in solids.29, 30 Hybrid functionals,31 where a fraction of exact exchange is mixed in with local density and gradient dependent functionals, have become widely used and suffer somewhat less from the self-interaction error (since the self-interaction in the exact exchange term partially cancels the self-interaction in the Coulomb energy), but a significant self-interaction error remains. The B3LYP functional32 has, in particular, become widely used in molecular calculations. The results of B3LYP calculations for the dimer cation shown in Fig. 4(b) give similarly delocalized a hole as the PBE functional. It is essential to apply a self-interaction correction to these functionals in order to get proper localization of the hole for the DMIPA dimer. The binding energy of the 3s Rydberg electron is estimated to be 0.03–0.30 eV lower for the three configurations of the DMIPA dimer than in the isolated molecule, see Table II. The lowering of the binding energy is larger for the configuration that is most stable in the Rydberg state than in the most stable ground state configuration. This can be understood from the orientation of the dipole moment of the molecule adjacent to the hole and the distance between the N-atoms. The presence of the dipole effectively diffuses the positive charge of the localized hole and thereby lowers the Rydberg electron binding energy, as discussed by Liang et al.9 and Deb et al.10 The energy difference between the cations in the two dimer configurations shown in Figs. 2(a) and 2(c) is even larger than the energy difference between the Rydberg states, 0.76 eV vs. 0.54 eV, since the Rydberg electron partly screens the positive charge at the hole (see Table II). C. Trimer calculations
A limited search for configurations of the trimer were carried out using the BEEF-vdW functional. Two stable configurations were found. In both cases, the three N-atoms are rather close together, as shown in Figs. 5(a) and 5(b). The binding energy of the trimer with respect to three monomers is quite similar in these two configurations, 0.36 eV and 0.32 eV. The binding energy of the Rydberg electron in these trimer configurations is calculated to be 0.09–0.13 eV lower than in the most stable dimer configuration. The calculations, therefore, show that the Rydberg electron binding energy is reduced as the clusters grow. In the 3s Rydberg state, the configuration shown in Fig. 5(a) is 0.12 eV lower in energy than the one in Fig. 5(b) and also gives lower binding energy by 0.13 eV compared to the most stable dimer Rydberg state (see Table II). These trends are in excellent agreement with the experimental results discussed below. In the Rydberg state, the localized hole attracts the partial negative charge of the Natoms of the neighboring molecules. A lower Rydberg electron binding energy results from the ordering of the dipoles of the neighboring molecules, which effectively diffuse the positive charge of the localized hole.
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(a)
(b)
FIG. 5. Two atomic structures of the DMIPA trimer, corresponding to local minima on the BEEF-vdW energy surface for the ground state, as well as isosurface for spin density of the cation. The structure in 5(a) is the same as that shown in Figs. 3(c) and 3(d). The energy of the two structures in the ground state and Rydberg state are given Table II. The spin density is localized on one of the N-atoms, indicating a localized hole on one of the three molecules. The rendered surface corresponds to 0.18 Å−3 , about 10% of the maximum value.
The localization of the hole in the trimer can again be seen from ground state calculations of the cation. Fig. 5 shows the spin density of the cation of both trimer structures. Clearly, the positive charge is localized on one of the N-atoms. Calculations using the PBE functional without self-interaction correction, however, delocalize the positive charge on all three N-atoms, analogous to the results for the dimer.
D. Comparison with experiments
Experimental measurements of the 3s Rydberg electron binding energy were as reported previously.10 The signal obtained in an experiment using Ar carrier gas and the sample reservoir at −30 ◦ C is shown in Fig. 6. A broad feature covering binding energy in the range of 1.8 eV–2.7 eV can be seen. These values are significantly lower than the binding energy of the 3s Rydberg state of the monomer, 2.88 eV, which was previously obtained from experiments using He carrier gas, where clustering does not occur. It is clear that the 3s
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FIG. 6. Experimental binding energy of 3s Rydberg electron in DMIPA clusters as a function of time.10 The color represents the intensity, as indicated by the color bar (arbitrary units). The average calculated binding energy for the two lowest energy ground state structures found for the dimer and the two lowest energy structures found for the trimer are indicated by white dashed curves at t = 0. The shape of the curves was chosen to indicate further lowering of the Rydberg electron binding energy as neighboring molecules reorient their dipole in response to the localized hole.
Rydberg electron binding energy is lower in clusters than in the monomer. Furthermore, by comparing this spectrum to a spectrum taken with the Ar and the sample at a higher temperature, 0 ◦ C, where more and larger clusters are expected to form, it can be seen that the Rydberg electron binding energy of larger clusters is lowered more than that of smaller ones. The evolution of the measured signal as a function of the time delay between the excitation pulse and the ionization pulse shows a distinct lowering of the 3s Rydberg electron binding energy as the cluster evolves in the Rydberg state on a time scale of a few tens of ps.10
IV. DISCUSSION
Detailed comparison between the experimental measurements and the calculated results is difficult because the size distribution of the clusters at the time of ionization is not known. While mass spectra have been recorded in these experiments, the time delay from ionization to recording of the mass spectra is several hundred nanoseconds so dissociation of clusters has likely occurred. Nevertheless, under similar experimental conditions, clusters with up to 10 molecules have been detected.10 It is, therefore, not possible at this time to assign a value of the Rydberg electron binding energy to a specific cluster size. But, it is clear that the trend in the calculated values going from monomer to dimer and then to trimer is such that the binding energy is lowered. The predicted values of the binding energy for the dimer, 2.65 eV, and trimer, 2.43 eV, obtained by averaging the calculated binding energy for the two lowest energy configurations found for the ground state, are shown in Fig. 6 (corresponding to t = 0). This comparison indicates that the experiment includes a rather broad distribution in cluster size including clusters that are significantly larger than the trimer. The top of the measured signal, i.e., high values in Rydberg electron binding energy are
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in good correspondence with the calculations for the dimers. The sequential lowering of the Rydberg electron binding energy as the cluster size increases can be understood from the induced dipole of molecules in response to the formation of the localized hole. The dipoles effectively diffuse the localized charge seen by the Rydberg electron and this leads to reduced binding energy. There is also good correspondence between the calculations and the time evolution of the measured signal. The molecules adjacent to the hole will tend to rotate in order to point their partially negative N-atom towards the hole. The calculations show that such configurations are lower in energy both for the dimer and the trimer, while little preference is for the orientation of the molecules in the ground state. It can, therefore, be expected that a structural rearrangement takes place in the Rydberg state which will, however, involve overcoming energy barriers and thus take time to complete. The results of the calculations for the dimer show that while two configurations with quite different N–N distance are practically equally stable in the ground state, the one with short N– N distance is preferred by 0.5 eV in the 3s Rydberg state and if the dimer structure transforms from one to the other, the binding energy of the Rydberg electron will be reduced by 0.22 eV (see Table II). This kind of lowering of the Rydberg electron binding energy upon configurational changes in the Rydberg state can explain the time evolution of the measured signal and is the basis for the shape of the sketched time evolution of the dimer and trimer in Fig. 6 (white dashed curves). The calculated results presented here support strongly the interpretation of the experimental data presented by Deb et al.10 The calculations presented here of Rydberg electron binding energy using self-interaction corrected density functional theory demonstrate that theoretical studies of even such large systems as the DMIPA trimer can be carried out to help analyze these types of measurements. In principle, larger clusters could be calculated including dynamics and identification of minimum energy paths for thermally activated transitions in the excited state. Such calculations, are, however, outside the scope of the present project. ACKNOWLEDGMENTS
This work was supported by the Icelandic Research Fund and by DTRA, Grant No. HDTRA1-14-1-0008. The calculations were carried out using the Nordic High Performance Computing (NHPC) facility in Iceland. H.J. thanks Susi Lehtola for helpful discussions. 1 R.
S. Freund, Rydberg States of Atoms and Molecules, edited by R. F. Stebbings and F. B. Dunning (Cambridge Univ. Press, Cambridge, 1983), pp. 355–392. 2 M. P. Minitti, J. D. Cardoza, and P. M. Weber, J. Phys. Chem. A 110, 10212 (2006). 3 N. Kuthirummal and P. M. Weber, Chem. Phys. Lett. 378, 647 (2003). 4 J. L. Gosselin and P. M. Weber, J. Phys. Chem. A 109, 4899 (2005). 5 J. L. Gosselin, M. P. Minitti, F. M. Rudakov, T. I. Sølling, and P. M. Weber, J. Phys. Chem. A 110, 4251 (2006). 6 J. D. Cardoza, F. M. Rudakov, N. Hansen, and P. M. Weber, J. Electron Spectrosc. Relat. Phenom. 165, 5 (2008). 7 F. Rudakov and P. M. Weber, Chem. Phys. Lett. 470, 187 (2009). 8 J. C. Bush, M. P. Minitti, and P. M. Weber, J. Photochem. Photobiol. A 213, 70 (2010).
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