- Email: [email protected]
Dynamical effects on the photo-detachment spectra of Li4−
10 January 2002
Chemical Physics Letters 351 (2002) 289±294 www.elsevier.com/locate/cplett
Dynamical eects on the photo-detachment spectra of Li4 C. Ashman a, S.N. Khanna b
a,*
, M.R. Pederson
b
a Department of Physics, Virginia Commonwealth University, P.O. Box 842000, Richmond, VA 23112-2000, USA Center For Computational Materials Science ± 6392, Naval Research Laboratory, Washington, DC 20375-5000, USA
Received 17 October 2001; in ®nal form 31 October 2001
Abstract It is shown that the observed sharp peaks in the photodetachment spectra of Li4 can be understood by including the eect of cluster dynamics. Theoretical electronic structure studies show that Li4 is marked by doublet rhombus and quartet distorted tetrahedral isomers. By including dynamical eects within a classical picture, it is shown that the observed spectrum can be explained in terms of transitions from rhombus ground state to the singlet neutral con®gurations. A vibrational analysis is used to identify the modes leading to the observed peaks. Our studies underscore the importance of dynamical eects on the photodetachment spectra. Ó 2002 Published by Elsevier Science B.V.
Despite more than a decade of extensive research into the physical, chemical, electronic and magnetic properties of small clusters, the basic issue of experimentally determining their geometries remains a challenging problem. The clusters are too small for spectroscopic studies while too large for microscopic determination of their geometries. One therefore has to resort to indirect approaches and the negative ion photoelectron spectroscopy has emerged as a powerful technique [1]. Here, one starts with an anion, crosses it with a laser of ®xed frequency (energy), and measures the energy of the photo-detached electron. The peaks in the photoelectron spectra then correspond to transitions from the ground state of the anion with a multiplicity of M to the neutral cluster states with
*
Corresponding author. Fax: +1-804-828-7073. E-mail address: [email protected] (S.N. Khanna).
multiplicity M 1. Since, the photo-detachment process is fast, the transitions occur to the neutral cluster with the same geometry as that of the anion i.e. all transitions are vertical. By comparing the experimental peaks against the theoretically calculated excitation energies, it is then possible to identify the geometry and the spin multiplicity of the ground state of the anion. Indeed, over the past few years, photoelectron spectra of numerous anionic clusters have been experimentally measured [2] and these combined with sophisticated ab initio theories have been used to derive information on the ground state geometries and the spin multiplicities of numerous clusters [3]. Despite the above success, the photoelectron spectra of some of the simplest cases still remain somewhat puzzling. One such case is Li4 observed by Bowen and co-workers [4]. These authors measured the photoelectron spectra of Li4 , Na4 , and K4 and showed that while the spectra of Na4
0009-2614/02/$ - see front matter Ó 2002 Published by Elsevier Science B.V. PII: S 0 0 0 9 - 2 6 1 4 ( 0 1 ) 0 1 3 8 3 - 5
290
C. Ashman et al. / Chemical Physics Letters 351 (2002) 289±294
and K4 are very broad, the spectrum of Li4 is marked by two sharp peaks at about 0.76 and 0.86 eV above the anion. There is also a minor peak at 2.1 eV and a broadening around 1.00 eV. Theoretical studies [5,6] show that the ground states of Na4 and K4 are linear while the ground state of neutral Na4 and K4 are rhombi. It is argued that the large dierence in the geometry of the anion and the neutral ground states leads to the broadening of the observed spectrum. For Li4 , the sharp peaks suggest that the ground state structure of Li4 is dierent from the other alkali's and is probably close to the rhombus, the ground state for neutral Li4 . While the ab initio calculations do lead to a rhombus ground state structure, the corresponding excitation energies cannot reproduce the peak at 0.86 eV. This has led to numerous theoretical investigations as to the origin of the spectra. Rao et al. [5] were the ®rst to propose a possible explanation. They suggested that the ground state of Li4 consist of two nearly degenerate isomers with spin doublet rhombus and spin quartet tetrahedral con®gurations. Their calculated spectra for the rhombus doublet structure lead to singlet and triplet neutral states at 0.70 and 1.80 eV above the anion ground state. For the spin quartet tetrahedral con®guration, they ®nd the transition to the triplet neutral at 0.94 eV. According to these authors, the two peaks at 0.76 and at 0.86 eV in the observed spectrum arise due to the lowest transition from the rhombus and tetrahedron con®gurations. That the two major peaks come from dierent structures was a novel idea. Bauschlicher [6] attempted to verify the validity of this idea using dierent levels of theory. He, however, found that the disagreement between the calculated and experimental transition energies became larger as they improved the level of the theory. There are also other problems with such an explanation. Both Rao et al. [5] and Bauschlicher [6] ®nd the tetrahedron quartet state to be 0.13± 0.16 eV above the rhombus ground state which corresponds to a temperature of around 1800 K. That clusters with such dierent energies co-exist is highly unlikely. In addition to the rhombus and the tetrahedron, the anion has Y shaped and linear con®gurations close to the ground state. However, none of the transitions can reproduce the peak at
0.86 eV. This raises an interesting dilemma. Can the peaks in the photodetachment spectra arise from other reasons? In this Letter we propose a novel explanation of the photodetachment spectra of Li4 . We show that the peaks at 0.76 and 0.86 eV arise from the same structure namely the rhombus con®guration. However, while the peak at 0.76 eV corresponds to the vertical transition from the ground state of the rhombus, the peak at 0.86 eV arises due to dynamical eects. Using a simple classical dynamics, we are able to quantitatively reproduce the experimental spectra. We then invoke a quantum picture to identify the vibrational mode leading to this peak. In addition to the main peaks, our calculated transition energies reproduce the minor peaks at 2.1 eV and the broadening around 1.0 eV. We show that the dynamical eects can lead to peaks in the photo-detachment spectra and their inclusion is necessary to quantitatively compare them with theory. The theoretical studies were carried out using a linear combination of atomic orbitals molecular orbital (LCAO-MO) approach [7] within a density functional formalism [8]. The gradient corrected functional proposed by Perdew et al. [9] was used to incorporate the exchange correlation eects. The molecular orbitals of the cluster are expressed as a sum of Gaussian functions of various angular momentum states centered at the atomic sites. The hamiltonian matrix elements are calculated by numerical integration over a mesh of points. The actual computations were carried out using a set of programs called NRLMOL which were developed by Pederson and Jackson [10]. The basis set for Li consisted of 6s, 4p, and 3d functions with an auxiliary d function built by contracting a set of 10 primitive Gaussians [11]. For each structure, the geometry was optimized by moving atoms in the directions of forces until the forces were lower than the threshold value of 3:0 10 4 a.u. No symmetry constraints were imposed. The ®nal structures therefore include possible Jahn±Teller distortions [12]. In Fig. 1 we show the ground state and the lowest energy structures for Li4 . The ground state is a near C2v tetrahedral structure with a quartet spin multiplicity. Note that the structure does not
C. Ashman et al. / Chemical Physics Letters 351 (2002) 289±294
291
and the Fig. 1. Ground state geometry, binding energy, and the transition energies of the anionic Li4 isomers. Bond lengths are in A energies are in eV. The binding energies within parenthesis include the zero point energies.
have a tetrahedral symmetry as assumed in earlier studies. Only 0.02 eV above this ground state is a spin doublet rhombus con®guration. The apex angle in the rhombus is 128°. In addition to these structures, the cluster is marked by a linear con®guration 0.19 eV above the ground state and a Y shaped con®guration, that is 0.20 eV above the ground state. In the linear structure, the extra charge is mostly localized at the end atoms and the Li±Li distance is around 3.0 a.u. We found that the potential energy hypersurfaces are fairly ¯at i.e. small changes in bond lengths lead to minor variations in energy. Since the energy dierence between various structures are small, a more meaningful comparison between dierent structures should include the zero point energy. To this end, we calculated the vibrational frequencies for each structure and used the frequencies to calculate the zero point energy. Note that the dierence between the distorted tetrahedral and rhombus structures is only 0.01 eV. In view of the accuracy of the calculations, the two structures are therefore quasi degenerate. In
the earlier calculations of Rao et al. [5] using Hartree±Fock approximation and a fourth order Moller±Plesset perturbation theory, they found the rhombus to be the ground state and the tetrahedron structure was 0.13 eV above the ground state. Similar ordering of the rhombus and distorted tetrahedron was also obtained by Bauschlicher who used coupled cluster singles and doubles approach. Since these energy dierences are at the limit of theoretical accuracy, it can be concluded that irrespective of the theoretical approach, the two structures are isomers. As mentioned before, the photodetachment spectra of Li4 is characterized by major sharp peaks at 0.76 and 0.86 eV, a minor peak at 2.1 eV and a broadening around 1.0 eV. In order to understand the origin of these peaks, we computed the vertical transition energies to the neutral con®gurations with multiplicities one higher and one lower than the anionic ground states. These are given in Fig. 1. Note that the lowest transition from the rhombus anion to the spin singlet neutral has an energy of 0.79 eV which is close to the ®rst peak
292
C. Ashman et al. / Chemical Physics Letters 351 (2002) 289±294
in the photodetachment spectrum at 0.76 eV. However, none of the remaining transitions can account for the peak at 0.86 eV. The transition from the distorted tetrahedron to the neutral triplet is at 1.0 eV. Where does this peak then originate? The energy hypersurfaces are fairly ¯at and small energies can lead to large distortions, as previously noted. A calculation of the vibrational frequencies indicated that the cluster is characterized by low energy vibrational modes. Can the cluster dynamics produce peaks in the photodetachment spectra? To analyze such a possibility, we studied the energy hypersurface for large distortions of the major and minor axes of the rhombus on a grid of 0.02 a.u. At each displacement, we calculated the vertical transition energies to the spin singlet and triplet con®gurations. We found that several of the singlet transitions correspond to an energy of 0.87 eV! To explore whether these could lead to a peak, we assumed a temperature corresponding to 0.02 eV (i.e. 232 K). The vibrating cluster can attain con®gurations that dier from the ground state by 0.02 eV. The kinetic energy in various con®gurations, however, is different. To incorporate this, we assumed a classical picture where the cluster samples all con®gurations within 0.02 eV of the anionic rhombus ground state but with a velocity determined by the kinetic energy. The singlet transitions from each grid point was then weighted proportional to the time spent in each state. This time is inversely proportional to the velocity that can be determined from the square root of the kinetic energy obtained by the energy dierence between the local energy and the total energy. Using such an approach, we recalculated the excitation spectra from the rhombus ground state to the singlet neutral state and the results are shown in Fig. 2. Here, each transition was broadened with a Gaussian of half width 0.014 eV. Note that the spectra are characterized by two peaks at 0.79 and 0.87 eV that are close to the experimental values of 0.76 and 0.86 eV, respectively. A comparison with the experimental spectra reveals that even the relative intensities of the main peaks are reproduced by the theoretical spectra. This is a remarkable result since it demonstrates that the sharp peaks in the photodetachment spectra can also originate from the
Fig. 2. The calculated photodetachment spectra from the doublet rhombus to the singlet neutral con®gurations assuming a temperature of 0.02 eV.
dynamical features. This would happen when portions of the energy hypersurfaces of the anion ground state and the resulting neutral con®guration are parallel. In this case dierent geometries lead to the same transition energies and consequently lead to a narrow peak in the spectrum. In the same vein, the absence of such parallel hypersurfaces would lead to a broadening of the peaks. This would happen when the ground state and vertically excited energy hypersurface have dierent shapes as the structure is distorted. Indeed, the transitions from the rhombus doublet to the triplet falls under this category since the ground state of the neutral triplet is a tetrahedral con®guration. This, then, would not produce a sharp peak. The minor feature around 1.0 eV and a minor peak at 2.1 eV then arise from the transitions from the rhombus doublet to neutral triplet (1.05 eV) and from the anion quartet to neutral triplet (1.0 eV) and quintet (2.10) eV. We would like to add that we have assumed a cluster temperature of 0.02 eV. We also carried out calculations at other temperatures and found that the relative intensity of the peaks changes with temperature. While the above classical picture reproduces the experimental results, one can ask about the detailed vibrational modes leading to the theoretical sharp peak at 0.87 eV. To this end we carried out calculation of the vibrational modes for each cluster and calculated the transition energies to the neutral structures by moving atoms in the direc-
C. Ashman et al. / Chemical Physics Letters 351 (2002) 289±294
293
Fig. 3. The vertical excitations from the doublet rhombus to singlet neutral for the vibrational modes corresponding to frequencies of (a) 85 cm 1 and (b) 244 cm 1 and from the distorted tetrahedron to triplet neutral (c) and quintet neutral (d).
tion of the normal modes up to the energy cuto. In Fig. 3 we show the corresponding transitions from the rhombus doublet to the neutral singlet and from the tetrahedral quartet to the neutral triplet and quintet. Note that the 0.87 eV feature can be associated with the transitions from a vibrational mode at 244 cm 1 . Also, the broad feature around 1.0 eV and the peak at 2.1 eV arise from the 190 cm 1 vibrational modes of the distorted tetrahedral quartet. An analysis of the vibrational modes indicated that none of one's in Fig. 3 is infra red active. The details about other vibrational modes will be given in a separate Letter. To summarize, the present studies show that the current practice of associating the main peaks in the photo-detachment spectra to the vertical transitions from the ground state structures only may not be correct and the dynamical eects can lead to sharp peaks. For the case of Li4 , the inclusion of these dynamical eects allows one to provide a quantitative explanation of the observed spectra. It is shown that the two main peaks arise from the rhombus structure while the minor peaks originate in the rhombus and distorted tetrahedral structures. This is dierent from the currently accepted picture that the two main peaks arise due to dierent isomers. The present results also provide insight into the observation that sharper peaks in
photodetachment spectra originate when the anion and the neutral have similar geometries. This happens because the energy hypersurfaces are parallel and even though the clusters may be vibrating, dierent con®gurations lead to the same transition energy. We hope that the present work will stimulate a more detailed investigations of the eects of cluster dynamics in interpreting the experimental photo-detachment spectra. Finally, as we mentioned before, the relative intensity of the main peaks in Li4 spectra depend on the cluster temperature. We hope that this work will stimulate experimental studies at dierent temperatures to verify our prediction.
Acknowledgements S.N.K. and C.A. are thankful to Department of Energy (DE-FG02-96ER45579) for ®nancial support.
References [1] O. Cheshnovsky, K.J. Taylor, J. Conceicao, R.E. Smalley, Phys. Rev. Lett. 64 (1990) 1785. [2] L.-S. Wang, H. Wu, Z. Phys. 203 (1998) 45.
294
C. Ashman et al. / Chemical Physics Letters 351 (2002) 289±294
[3] G.L. Gutsev, S.N. Khanna, P. Jena, Phys. Rev. B 62 (2000) 1604. [4] H.W. Sarkas, S.T. Arnold, J.H. Hendricks, K.H. Bowen, J. Chem. Phys 102 (1995) 2653. [5] B.K. Rao, P. Jena, A.K. Ray, Phys. Rev. Lett. 76 (1996) 2878. [6] C.W. Bauschlicher Jr., Chem. Phys. Lett. 260 (1996) 309. [7] W.J. Hehre, L. Radom, P. von, R. Schleyer, J.A. Pople, Ab Initio Molecular Orbital Theory, Wiley, New York, 1986.
[8] W. Kohn, L.J. Sham, Phys. Rev. 140 (1965) A1133. [9] J.P. Perdew, K. Burke, M. Ernzerhof, Phys. Rev. Lett. 77 (1996) 3865. [10] K.A. Jackson, M.R. Pederson, Phys. Rev. B 42 (1990) 3276. [11] D.V. Porezag, M.R. Pederson, Phys. Rev. A 60 (1999) 2840. [12] J.L. Martins, J. Buttet, R. Car, Phys. Rev. B 31 (1985) 1804.
Chemical Physics Letters 351 (2002) 289±294 www.elsevier.com/locate/cplett
Dynamical eects on the photo-detachment spectra of Li4 C. Ashman a, S.N. Khanna b
a,*
, M.R. Pederson
b
a Department of Physics, Virginia Commonwealth University, P.O. Box 842000, Richmond, VA 23112-2000, USA Center For Computational Materials Science ± 6392, Naval Research Laboratory, Washington, DC 20375-5000, USA
Received 17 October 2001; in ®nal form 31 October 2001
Abstract It is shown that the observed sharp peaks in the photodetachment spectra of Li4 can be understood by including the eect of cluster dynamics. Theoretical electronic structure studies show that Li4 is marked by doublet rhombus and quartet distorted tetrahedral isomers. By including dynamical eects within a classical picture, it is shown that the observed spectrum can be explained in terms of transitions from rhombus ground state to the singlet neutral con®gurations. A vibrational analysis is used to identify the modes leading to the observed peaks. Our studies underscore the importance of dynamical eects on the photodetachment spectra. Ó 2002 Published by Elsevier Science B.V.
Despite more than a decade of extensive research into the physical, chemical, electronic and magnetic properties of small clusters, the basic issue of experimentally determining their geometries remains a challenging problem. The clusters are too small for spectroscopic studies while too large for microscopic determination of their geometries. One therefore has to resort to indirect approaches and the negative ion photoelectron spectroscopy has emerged as a powerful technique [1]. Here, one starts with an anion, crosses it with a laser of ®xed frequency (energy), and measures the energy of the photo-detached electron. The peaks in the photoelectron spectra then correspond to transitions from the ground state of the anion with a multiplicity of M to the neutral cluster states with
*
Corresponding author. Fax: +1-804-828-7073. E-mail address: [email protected] (S.N. Khanna).
multiplicity M 1. Since, the photo-detachment process is fast, the transitions occur to the neutral cluster with the same geometry as that of the anion i.e. all transitions are vertical. By comparing the experimental peaks against the theoretically calculated excitation energies, it is then possible to identify the geometry and the spin multiplicity of the ground state of the anion. Indeed, over the past few years, photoelectron spectra of numerous anionic clusters have been experimentally measured [2] and these combined with sophisticated ab initio theories have been used to derive information on the ground state geometries and the spin multiplicities of numerous clusters [3]. Despite the above success, the photoelectron spectra of some of the simplest cases still remain somewhat puzzling. One such case is Li4 observed by Bowen and co-workers [4]. These authors measured the photoelectron spectra of Li4 , Na4 , and K4 and showed that while the spectra of Na4
0009-2614/02/$ - see front matter Ó 2002 Published by Elsevier Science B.V. PII: S 0 0 0 9 - 2 6 1 4 ( 0 1 ) 0 1 3 8 3 - 5
290
C. Ashman et al. / Chemical Physics Letters 351 (2002) 289±294
and K4 are very broad, the spectrum of Li4 is marked by two sharp peaks at about 0.76 and 0.86 eV above the anion. There is also a minor peak at 2.1 eV and a broadening around 1.00 eV. Theoretical studies [5,6] show that the ground states of Na4 and K4 are linear while the ground state of neutral Na4 and K4 are rhombi. It is argued that the large dierence in the geometry of the anion and the neutral ground states leads to the broadening of the observed spectrum. For Li4 , the sharp peaks suggest that the ground state structure of Li4 is dierent from the other alkali's and is probably close to the rhombus, the ground state for neutral Li4 . While the ab initio calculations do lead to a rhombus ground state structure, the corresponding excitation energies cannot reproduce the peak at 0.86 eV. This has led to numerous theoretical investigations as to the origin of the spectra. Rao et al. [5] were the ®rst to propose a possible explanation. They suggested that the ground state of Li4 consist of two nearly degenerate isomers with spin doublet rhombus and spin quartet tetrahedral con®gurations. Their calculated spectra for the rhombus doublet structure lead to singlet and triplet neutral states at 0.70 and 1.80 eV above the anion ground state. For the spin quartet tetrahedral con®guration, they ®nd the transition to the triplet neutral at 0.94 eV. According to these authors, the two peaks at 0.76 and at 0.86 eV in the observed spectrum arise due to the lowest transition from the rhombus and tetrahedron con®gurations. That the two major peaks come from dierent structures was a novel idea. Bauschlicher [6] attempted to verify the validity of this idea using dierent levels of theory. He, however, found that the disagreement between the calculated and experimental transition energies became larger as they improved the level of the theory. There are also other problems with such an explanation. Both Rao et al. [5] and Bauschlicher [6] ®nd the tetrahedron quartet state to be 0.13± 0.16 eV above the rhombus ground state which corresponds to a temperature of around 1800 K. That clusters with such dierent energies co-exist is highly unlikely. In addition to the rhombus and the tetrahedron, the anion has Y shaped and linear con®gurations close to the ground state. However, none of the transitions can reproduce the peak at
0.86 eV. This raises an interesting dilemma. Can the peaks in the photodetachment spectra arise from other reasons? In this Letter we propose a novel explanation of the photodetachment spectra of Li4 . We show that the peaks at 0.76 and 0.86 eV arise from the same structure namely the rhombus con®guration. However, while the peak at 0.76 eV corresponds to the vertical transition from the ground state of the rhombus, the peak at 0.86 eV arises due to dynamical eects. Using a simple classical dynamics, we are able to quantitatively reproduce the experimental spectra. We then invoke a quantum picture to identify the vibrational mode leading to this peak. In addition to the main peaks, our calculated transition energies reproduce the minor peaks at 2.1 eV and the broadening around 1.0 eV. We show that the dynamical eects can lead to peaks in the photo-detachment spectra and their inclusion is necessary to quantitatively compare them with theory. The theoretical studies were carried out using a linear combination of atomic orbitals molecular orbital (LCAO-MO) approach [7] within a density functional formalism [8]. The gradient corrected functional proposed by Perdew et al. [9] was used to incorporate the exchange correlation eects. The molecular orbitals of the cluster are expressed as a sum of Gaussian functions of various angular momentum states centered at the atomic sites. The hamiltonian matrix elements are calculated by numerical integration over a mesh of points. The actual computations were carried out using a set of programs called NRLMOL which were developed by Pederson and Jackson [10]. The basis set for Li consisted of 6s, 4p, and 3d functions with an auxiliary d function built by contracting a set of 10 primitive Gaussians [11]. For each structure, the geometry was optimized by moving atoms in the directions of forces until the forces were lower than the threshold value of 3:0 10 4 a.u. No symmetry constraints were imposed. The ®nal structures therefore include possible Jahn±Teller distortions [12]. In Fig. 1 we show the ground state and the lowest energy structures for Li4 . The ground state is a near C2v tetrahedral structure with a quartet spin multiplicity. Note that the structure does not
C. Ashman et al. / Chemical Physics Letters 351 (2002) 289±294
291
and the Fig. 1. Ground state geometry, binding energy, and the transition energies of the anionic Li4 isomers. Bond lengths are in A energies are in eV. The binding energies within parenthesis include the zero point energies.
have a tetrahedral symmetry as assumed in earlier studies. Only 0.02 eV above this ground state is a spin doublet rhombus con®guration. The apex angle in the rhombus is 128°. In addition to these structures, the cluster is marked by a linear con®guration 0.19 eV above the ground state and a Y shaped con®guration, that is 0.20 eV above the ground state. In the linear structure, the extra charge is mostly localized at the end atoms and the Li±Li distance is around 3.0 a.u. We found that the potential energy hypersurfaces are fairly ¯at i.e. small changes in bond lengths lead to minor variations in energy. Since the energy dierence between various structures are small, a more meaningful comparison between dierent structures should include the zero point energy. To this end, we calculated the vibrational frequencies for each structure and used the frequencies to calculate the zero point energy. Note that the dierence between the distorted tetrahedral and rhombus structures is only 0.01 eV. In view of the accuracy of the calculations, the two structures are therefore quasi degenerate. In
the earlier calculations of Rao et al. [5] using Hartree±Fock approximation and a fourth order Moller±Plesset perturbation theory, they found the rhombus to be the ground state and the tetrahedron structure was 0.13 eV above the ground state. Similar ordering of the rhombus and distorted tetrahedron was also obtained by Bauschlicher who used coupled cluster singles and doubles approach. Since these energy dierences are at the limit of theoretical accuracy, it can be concluded that irrespective of the theoretical approach, the two structures are isomers. As mentioned before, the photodetachment spectra of Li4 is characterized by major sharp peaks at 0.76 and 0.86 eV, a minor peak at 2.1 eV and a broadening around 1.0 eV. In order to understand the origin of these peaks, we computed the vertical transition energies to the neutral con®gurations with multiplicities one higher and one lower than the anionic ground states. These are given in Fig. 1. Note that the lowest transition from the rhombus anion to the spin singlet neutral has an energy of 0.79 eV which is close to the ®rst peak
292
C. Ashman et al. / Chemical Physics Letters 351 (2002) 289±294
in the photodetachment spectrum at 0.76 eV. However, none of the remaining transitions can account for the peak at 0.86 eV. The transition from the distorted tetrahedron to the neutral triplet is at 1.0 eV. Where does this peak then originate? The energy hypersurfaces are fairly ¯at and small energies can lead to large distortions, as previously noted. A calculation of the vibrational frequencies indicated that the cluster is characterized by low energy vibrational modes. Can the cluster dynamics produce peaks in the photodetachment spectra? To analyze such a possibility, we studied the energy hypersurface for large distortions of the major and minor axes of the rhombus on a grid of 0.02 a.u. At each displacement, we calculated the vertical transition energies to the spin singlet and triplet con®gurations. We found that several of the singlet transitions correspond to an energy of 0.87 eV! To explore whether these could lead to a peak, we assumed a temperature corresponding to 0.02 eV (i.e. 232 K). The vibrating cluster can attain con®gurations that dier from the ground state by 0.02 eV. The kinetic energy in various con®gurations, however, is different. To incorporate this, we assumed a classical picture where the cluster samples all con®gurations within 0.02 eV of the anionic rhombus ground state but with a velocity determined by the kinetic energy. The singlet transitions from each grid point was then weighted proportional to the time spent in each state. This time is inversely proportional to the velocity that can be determined from the square root of the kinetic energy obtained by the energy dierence between the local energy and the total energy. Using such an approach, we recalculated the excitation spectra from the rhombus ground state to the singlet neutral state and the results are shown in Fig. 2. Here, each transition was broadened with a Gaussian of half width 0.014 eV. Note that the spectra are characterized by two peaks at 0.79 and 0.87 eV that are close to the experimental values of 0.76 and 0.86 eV, respectively. A comparison with the experimental spectra reveals that even the relative intensities of the main peaks are reproduced by the theoretical spectra. This is a remarkable result since it demonstrates that the sharp peaks in the photodetachment spectra can also originate from the
Fig. 2. The calculated photodetachment spectra from the doublet rhombus to the singlet neutral con®gurations assuming a temperature of 0.02 eV.
dynamical features. This would happen when portions of the energy hypersurfaces of the anion ground state and the resulting neutral con®guration are parallel. In this case dierent geometries lead to the same transition energies and consequently lead to a narrow peak in the spectrum. In the same vein, the absence of such parallel hypersurfaces would lead to a broadening of the peaks. This would happen when the ground state and vertically excited energy hypersurface have dierent shapes as the structure is distorted. Indeed, the transitions from the rhombus doublet to the triplet falls under this category since the ground state of the neutral triplet is a tetrahedral con®guration. This, then, would not produce a sharp peak. The minor feature around 1.0 eV and a minor peak at 2.1 eV then arise from the transitions from the rhombus doublet to neutral triplet (1.05 eV) and from the anion quartet to neutral triplet (1.0 eV) and quintet (2.10) eV. We would like to add that we have assumed a cluster temperature of 0.02 eV. We also carried out calculations at other temperatures and found that the relative intensity of the peaks changes with temperature. While the above classical picture reproduces the experimental results, one can ask about the detailed vibrational modes leading to the theoretical sharp peak at 0.87 eV. To this end we carried out calculation of the vibrational modes for each cluster and calculated the transition energies to the neutral structures by moving atoms in the direc-
C. Ashman et al. / Chemical Physics Letters 351 (2002) 289±294
293
Fig. 3. The vertical excitations from the doublet rhombus to singlet neutral for the vibrational modes corresponding to frequencies of (a) 85 cm 1 and (b) 244 cm 1 and from the distorted tetrahedron to triplet neutral (c) and quintet neutral (d).
tion of the normal modes up to the energy cuto. In Fig. 3 we show the corresponding transitions from the rhombus doublet to the neutral singlet and from the tetrahedral quartet to the neutral triplet and quintet. Note that the 0.87 eV feature can be associated with the transitions from a vibrational mode at 244 cm 1 . Also, the broad feature around 1.0 eV and the peak at 2.1 eV arise from the 190 cm 1 vibrational modes of the distorted tetrahedral quartet. An analysis of the vibrational modes indicated that none of one's in Fig. 3 is infra red active. The details about other vibrational modes will be given in a separate Letter. To summarize, the present studies show that the current practice of associating the main peaks in the photo-detachment spectra to the vertical transitions from the ground state structures only may not be correct and the dynamical eects can lead to sharp peaks. For the case of Li4 , the inclusion of these dynamical eects allows one to provide a quantitative explanation of the observed spectra. It is shown that the two main peaks arise from the rhombus structure while the minor peaks originate in the rhombus and distorted tetrahedral structures. This is dierent from the currently accepted picture that the two main peaks arise due to dierent isomers. The present results also provide insight into the observation that sharper peaks in
photodetachment spectra originate when the anion and the neutral have similar geometries. This happens because the energy hypersurfaces are parallel and even though the clusters may be vibrating, dierent con®gurations lead to the same transition energy. We hope that the present work will stimulate a more detailed investigations of the eects of cluster dynamics in interpreting the experimental photo-detachment spectra. Finally, as we mentioned before, the relative intensity of the main peaks in Li4 spectra depend on the cluster temperature. We hope that this work will stimulate experimental studies at dierent temperatures to verify our prediction.
Acknowledgements S.N.K. and C.A. are thankful to Department of Energy (DE-FG02-96ER45579) for ®nancial support.
References [1] O. Cheshnovsky, K.J. Taylor, J. Conceicao, R.E. Smalley, Phys. Rev. Lett. 64 (1990) 1785. [2] L.-S. Wang, H. Wu, Z. Phys. 203 (1998) 45.
294
C. Ashman et al. / Chemical Physics Letters 351 (2002) 289±294
[3] G.L. Gutsev, S.N. Khanna, P. Jena, Phys. Rev. B 62 (2000) 1604. [4] H.W. Sarkas, S.T. Arnold, J.H. Hendricks, K.H. Bowen, J. Chem. Phys 102 (1995) 2653. [5] B.K. Rao, P. Jena, A.K. Ray, Phys. Rev. Lett. 76 (1996) 2878. [6] C.W. Bauschlicher Jr., Chem. Phys. Lett. 260 (1996) 309. [7] W.J. Hehre, L. Radom, P. von, R. Schleyer, J.A. Pople, Ab Initio Molecular Orbital Theory, Wiley, New York, 1986.
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