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~ 1Σ+ Electronic State Vibrational Energies of LiH+2 and LiD+2 in the A Wolfgang P. Kraemer* Max-Planck-Institute of Astrophysics, Postfach 1317, D-85741 Garching, Germany
Vladimír Spirko Department of Molecular Modeling, Center for Biomolecules and Complex Molecular Systems, Institute of Organic Chemistry and Biochemistry, Academy of Sciences of the Czech Republic, Flemingovo n.2, 160 10 Prague 6, Czech Republic
bS Supporting Information ABSTRACT: In connection with the recent study of the ground electronic state of the LiH2+ molecular ion (Kraemer, W. P.; Spirko, V. Chem. Phys. 2006, 330, 190), the adiabatic threedimensional double-minimum potential enery surface of the first excited electronic state was evaluated, including its two lowest atomdiatom dissociation channels as well as the threeatom complete fragmentation asymptote. Applying the Sutcliffe Tennyson Hamiltonian for triatomic molecules, the levels of all bound vibrational states and the levels of the states localized in the two energy minimum regions were separately determined. The validity of statistical methods such as the density of states approach and the nearest-neighbor level spacing distribution (NNSD) was tested for the light LiH2+ ion. Special effort was put into investigating possible effects of a tunnelling motion across the proton-transfer barrier on the vibrational level pattern using the NNSD approach.
1. INTRODUCTION Collisions between an alkali atom or ion and a hydrogen molecule resulting in an inelastic scattering event or leading to the formation of a more or less stable association complex is one of the most frequently studied binary reaction processes. Especially the dynamics of the reactions with Li or Li+ as collision partners and the characteristics of the resulting associates have attracted theoreticians and experimentalists in the past because these are suitable small few-electron model systems for which quantitative comparisons can be made between accurate ab initio calculations and experimental results. Following some early theoretical predictions,1 the existence of the stable neutral molecule LiH2 was, for the first time, proved experimentally by mass spectroscopic measurements,2 and after this first existence proof, Hobza and Schleyer made the attempt on the basis of reliable ab initio calculations to supplement some of the experimental findings by a theory-supported understanding.3 This early investigation succeeded to demonstrate that the ground-state complex can only be a very weakly bound van der Waals species and that the excited (2B2)Li*H2 state forms a stable bound complex, and the study tried to verify the experimentally derived ionization potential IPe(LiH2), which is an important ingredient in the lithiumhydrogen chemical reaction network. Later on, classical and quantum dynamical calculations were performed on a restricted collinear potential for the reaction of Li with H2.4 It was suggested in this study that the LiH2 complex r 2011 American Chemical Society
could possibly be of some importance as a reaction intermediate in the hydrogen exchange reaction LiH + H h Li + H2, which may play a role as a possible depletion process for LiH in the primordial gas chemistry. Finally, a detailed study of the association reaction Li* + H2 h LiH + H and its reverse process was performed, investigating in detail the reaction possibilities for different excitation levels of lithium.5 The ionic system LiH+2 on the other hand forms a rather stable molecular complex, which has been the subject of numerous theoretical and experimental studies in the past. An overview of most of the earlier theoretical work was given in a recent ab initio study investigating all bound and the low-lying quasi-bound rovibrational levels of its ground electronic state.6 Since then, two more recent experimental7 and one theoretical study8 were published; all of these are restricted however to focus entirely on ground-state spectroscopic properties. Compared to the single-minimum potential energy surface of this ground state, the topology of the potential surface of the first electronically excited state (hereafter also called the A state) is more complex. Whereas in the ground state the positive charge is essentially Special Issue: Pavel Hobza Festschrift Received: May 27, 2011 Revised: August 10, 2011 Published: September 14, 2011 11313
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The Journal of Physical Chemistry A located on the lithium center (the Li+ ion is very stable with a Helike rare gas electronic configuration), neutral Li predominates in the A state, and its unpaired electron opens a variety of different reaction possibilities. There are two fairly close lying atomdiatom dissociation channels for this state. Along one of these channels, chargequadrupole interaction between Li and H+2 leads to a loosely bound complex, which, due to the weak interaction, essentially preserves the binding characteristics of the H+2 ion. Along the other channel, chargedipole interaction between LiH and H+ leads to the lowest=energy minimum of the excitedstate surface and the formation of the bound LiH+2 ion with an enlarged LiH bond compared to isolated lithium hydride. The two minima are separated by an energy barrier. Complete fragmentation of this excited state gives Li + H + H+. The energies of the two atomdiatom dissociation levels are actually not much different from each other. This is in contrast to the LiH+2 ground state with the positive charge essentially on Li and where, although apart from the lower dissociation channel, Li+ + H2, the second dissociation channel requires breaking of the much stronger neutral hydrogen molecule bond leading to the dissociation products LiH+ + H. Due to the complexity of the potential energy surface of the first excited LiH+2 state, reliable quantitative information about its spectroscopic properties does not exist. It is thus the intention of the present study to close this gap and to provide here in a first step accurate results of the bound vibrational levels in both potential energy minimum areas and to investigate possible tunnelling effects on the levels of all of the bound vibrational states. The levels are evaluated using the SutcliffeTennyson Hamiltonian for triatomic molecules. For the purpose of investigating the tunnelling effect on these levels, statistical methods are applied that have successfully been used in previous studies of the symmetric isomerization process HNN+ h NNH+14 as well as the unsymmetric process HCO+ h COH+.15 Especially the nearest-neighbor level spacing distribution (NNSD) approach has been found in these earlier studies to be useful in analyzing the characteristics of the vibrational level patterns. Especially in the case of the HCO+ h COH+ isomerization process, application of the NNSD analysis clearly demonstrated the effect of the tunnelling motions on the level spacings of the vibrational states just below the isomerization barrier. Whereas for the previously studied strongly bound triatomic systems' statistical methods were needed to characterize the massively large number of states, the question arises here whether statistical approaches can produce stable results for the light LiH+2 ion with the much smaller number of bound states that make safe conclusions possible. Extensive studies have been performed of the bound and quasi-bound states of the closely related corresponding helium compound HeH+2 and their role in the exoergic proton-transfer reaction HeH+ + H f He + H+2 (see refs 9 and 10 and references therein), which takes place on the ground-state potential surface of this ion. The corresponding reaction LiH + H+ f Li + H+2 occurring on the A state potential of LiH+2 is the ionic counterpart of the LiH depletion reaction mentioned above. Like in the helium case, this reaction is in total an exoergic barrierless process, dominated here by the strong long-range chargedipole interaction in the entrance channel. In the case of the lithium reaction, the question is, however, whether the barrier on the A state potential, although lower in energy than the reactant threshold, and the postreaction potential well will influence the reaction dynamics.11 Quantum wave packet calculations using the capture model12 and a quasi-classical trajectory study13 were
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performed determining the thermal rate constants for the LiH depletion reaction on the A state surface, and reasonably good agreement was found between the two approaches. Comparison of these results with an exact quantum scattering calculation are of basic interest. For this purpose, continuum and quasi-bound vibrational states responsible for the depletion reaction of LiH are determined and will be the subject of a separate paper.
2. POTENTIAL ENERGY FUNCTION Internal Jacobi coordinates are used to describe the potential energy surface and to handle the nuclear dynamics calculations. The HH distance is denoted by r with r = |r| (vector r is the vector joining the H nuclei), and R = |R| describes the distance between the center of mass of the HH subunit and the Li atom (R is the vector pointing from the center of mass of HH toward the Li nucleus), whereas θ is the angle between the r and R vectors. Details of the potential enegy surfaces (PES) determination of the electronic ground state of the LiH+2 ion and its first electronically excited state were extensively described in a previous paper.6 A large number of energy points (more than 5000 points altogether) necessary to reproduce the complex topology of the excited-state potential was employed to fit an analytic potential function of the form V ðr, R, θÞ
¼ð
∑ Cðk, l, mÞfexp½ar ðr rr Þgk
k, l, m
fexp½aR ðR Rr Þgl Pm ðcos θÞÞ
¼
∑λ Vλðr, RÞPλ cos θÞ
ð1Þ
When fitting a global potential function, two extra terms + C3R3 + C4R4 were added to ensure correct long-range behavior of the potential. In the above expression, C(k,l,m), ar, aR, rr, Rr, and the coefficients C3 and C4 of the long-range terms are free parameters. Pm(cos θ) are Legendre polynomials. A two step strategy was applied for the fitting process. Initially, the nonlinear parameters ar, aR, rr, and Rr were varied to achieve optimum flexibility of the fitting function, and after fixing them at their optimized values, the remaining linear parameters were determined by linear fits. Unfortunately, due to the very nonharmonic shape of the potential, it was not possible to obtain directly a unique global fit that could reproduce with sufficient accuracy all energy points used in the fit. Therefore, separate fits had to be performed to obtain accurate representations for the sub-barrier parts of each of the two potential wells using different weightings of the data points pertaining to the respective potential wells. The results are collected in the Supporting Information. The fit for the upper potential well (fitted parameter values summarized in Table A1, Supporting Information) turns out to provide a much more accurate reproduction of the corresponding potential range with deviations from the original data points below 10 cm1, whereas fits for the lower-energy minimum (fitted parameter values in Table A2, Supporting Information) had deviations that are about 1 order of magnitude larger. This is partly due to the fact that the Li + H+2 dissociation limit is slightly lower than the energy barrier separating the two potential minima, whereas the LiH + H+ dissociation threshold is considerably higher. The most characteristic features of the topology of the excitedstate potential surface are shown in Figure 1. The figure shows a 11314
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Figure 1. Schematic cut through the potential energy surface for linear geometries of the A ~1Σ+ state of LiH+2 . CAS/MCI energy differences (in cm1) and geometry parameters (in Å) are compared with corresponding results of ref 11.
schematic diagram representing a cut through the potential at linear geometries of the excited state. Energy differences with respect to the lowest potential minimum for characteristic potential points such as the second potential minimum, the two dissociation levels, and the saddle point on the energy barrier together with the corresponding optimized geometry parameters are included and compared with the respective reference values of ref 11 in parentheses. Optimized bond distances of the diatomics at the two dissociation limits, LiH(1Σ+) and H+2 (X2Σ+g ), are in perfect agreement with experimental values (in parentheses), Re(LiH) = 1.598 Å (1.5954 Å) and re(H+2 ) = 1.057 Å (1.0573 Å). For the complex in the lowest-energy minimum and the saddle point on the potential barrier, the optimized geometries are in close agreement (within less than 1%) with the corresponding results of ref 11; only for the loosely bound complex in the upper potential minimum do bond distances not match. The same trend applies for the energy differences. The stabilization energy of the complex in the lower minimum due to the fairly strong chargedipole interaction between LiH and the proton obtained here as 10 405 cm1 compares reasonably well with the 10 606 cm1 of ref 11. For the much weaker chargequadrupole interaction however, the formation energy of the loosely bound complex in the upper potential minimum obtained here as 4307 cm1 differs considerably (by more than 9%) from the 4718 cm1 of ref 11. Optimized geometries and energies quoted in Figure 1 are so-called ab initio values. Because the fit for the upper potential minimum presents a rather accurate representation of the original ab-initio-calculated energy points, the minimum parameters derived from this fit are naturally close to the optimized values in Figure 1 (about 0.1% for the geometry and less than 0.2% for the corresponding De). Even for the less accurate fit representing the lower minimum, the corresponding differences are rather small (about 0.3% for the distances and less than 1% for the minimum energy). Only in the saddle point are parameter deviations larger (on the order of a few percent). Drastic changes of the bond characteristics occur in the region of the energy barrier separating the two potential minima, which is due to the energetically demanding proton-transfer process taking place when moving from one energy minimum to the other. The situation is depicted in Figure 2, showing a plot of the
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Figure 2. Minimum-energy path optimized values of the Jacobi coordinate R as function of r (solid line) together with the corresponding effective potential Vmep(r) (dotted line).
minimum-energy path optimized values of the Jacobi coordinate R as function of the HH-stretching coordinate r together with the corresponding effective minimum-energy path potential Vmep(r). The potential energy surfaces of the two lowest electronic states are energetically well separated, and nonadiabatic couplings between them are very small. This has the consequence that processes with vibronic transitions such as the formation of the X state of the LiH+2 ion in the two-state radiative association reaction þ ~1 0 þ ~1 0 Li þ Hþ 2 f LiH2 ðA A Þ f LiH2 ðX A Þ þ hν
ð2Þ
can only take place via spontaneous photon emission, which reduces the efficiency of this process.
3. VIBRATIONAL DYNAMICS Vibrational energies were determined by employing the corresponding part of the SutcliffeTennyson Hamiltonian for triatomic molecules, which, after integration over the angular coordinates, assumes the form16,17 HST jj Æj0jH
ST 0 J ðr, R, θÞjj 0æjj
"
# 1 ∂2 1 ∂2 1 1 þ jðj þ 1Þ þ ¼ p 2μ1 ∂r 2 2μ2 ∂R2 2μ1 r2 2μ2 R 2 2
þ
∑λ gλ ðj, j0 , 0ÞVλ ðr, RÞ,
þ C3 R 3 þ C4 R 4
ð3Þ
where μ1 and μ2 are the appropriate reduced masses, j is the bending vibration quantum number, and gλ(j,j0 ,0) are the Gaunt coefficients. According to eq 1, Vλ(r,R) represents the potential after performing the SutcliffeTennyson integration. Instead of solving the Schr€odinger equations variationally for the complete full-dimensional Hamiltonian of eq 4, approximate solutions of the nuclear dynamics problem can also be obtained by applying the BornOppenheimer adiabatic separation concept to nuclear motions with largely different energy contents. 11315
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Figure 4. Effective V(r) stretching potentials obtained by adiabatic averaging over the R stretching and θ bending vibrational motions (in cm1).
Figure 3. Global potential (V(r,R,θ)) section for θ = 0 (upper panel) and the effect of bending (V(r,R,θ = 0) V(r,R,θ = 90)) (lower panel). Potential energies in cm1.
For this purpose, the total vibrational wave function is factorized according to Ψðr, R, θÞ ¼ ψðr; R, θÞΦðR, θÞ
ð4Þ
With this ansatz for the wave function, one-dimensional Schr€odinger equations for the high-frequency HH-stretching vibrations are obtained for various sets of R and θ values, and the eigenvalues Ui(R,θ) derived from these equations constitute the effective potential functions for the low-frequency motions for each highfrequency state ψi. The two-dimensional effective Hamiltonian for the low-frequency motions can be further reduced, and after integrating over all angular coordinates, it is finally written in analogy to eq 4 as " ( )# 1 ∂2 1 1 2 ST jðj þ 1Þ þ Hi ¼ p 2μ2 ∂R 2 2μ1 ri2 2μ2 R 2 þ
∑λ gλ ðj, j0 , 0ÞUλi ðRÞ,
þ C3 R 3 þ C4 R 4
ð5Þ
where ri is the effective “equilibrium” bond length of the highfrequency vibrational state ψi. A more general outline of the strategy of successive separations of vibrational motions has been presented previously.18 Apart from full-dimensional variational calculations, appropriate adiabatic separation approximations were also used here. The weak coupling, for instance, between the two stretching modes and the bending, is demonstrated in the lower panel of Figure 3, where the difference between the V(r,R) potentials for θ = 0 and 90 is shown to give an almost perfect plane. This means that the bending motion has practically only little effect on the
shapes of the stretching potentials for different angles. The separation, on the other hand, of the high-frequency r stretching motion from the other two modes turns out to provide only in certain parts of the potential a useful approximation. This is shown in Figure 4, where effective V(r) stretching potentials obtained by adiabatic averaging over the R stretching and θ bending motions are displayed. In the case of perfect separation, these effective potentials would coincide. The plots in Figure 4 indicate however that only in the region of the upper minimum are the effective potentials sufficiently close to each other, which means that in this region, this separation scheme essentially maintaines the true characteristics of the binding situation. For the lower potential well, on the other hand, the couplings between the high-frequency r stretch and especially the R stretching mode are larger, which makes this approximation useless in this case. To some extent, this coupling between the stretching motions along the r and R coordinates is already demonstrated in the plot of Figure 2. In Table 1, the energies of the lower vibrational states localized in the upper potential pocket are collected. Energy values determined by full-dimensional calculations are compared for the j(even) case of LiH+2 with those obtained using the adiabatic approximation described above. Differences between the exact and the approximate numbers turn out to be small; for vb = 0, most are below 1%, and for states with vb > 0, they are larger. For the lower potential, well energies of the vibrational states localized in this potential pocket were obtained from fulldimensional calculations. They are listed in Table 2 together with rotational constants of these states. Details of the full-dimensional calculations can be taken from recent similar studies of triatomic systems.6,14,15 Other than in these previous studies, the present dynamical calculations are hampered by the dramatic dependence of the optimal value of the coordinate R on the HHstretching coordinate r shown in Figure 2. This complicates the construction of the stretching basis set functions. The number of basis set functions necessary in these calculations exceeds the number of bound states by more than an order of magnitude, which is still technically feasible because the number of bound states to be taken into account is small. It may be noted here that the use of simplified models, such as the adiabatic approximation or the minimum-energy path representation, can be helpful to provide a deeper insight into the 11316
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Table 1. Energies of the Lower Vibrational States (J = 0) Localized in the Upper Potential Minimum Region of the Global PES (in cm1)a sib
vr
vb
vR
j(even)
vr
vb
vR
LiH+2
j(odd)
vr
vb
vR
LiH+2
j(even)
vr
vb
vR
LiD+2
j(odd) LiD+2
1
0
0
0
[1597.4]
0
0
0
0.1
0
0
0
[1154.3]
0
0
0
0.0
2 3
0 0
0 0
1 2
273.2 537.7
(271.1) (534.5)
0 0
0 0
1 2
273.3 537.8
0 0
0 0
1 2
216.5 427.8
0 0
0 0
1 2
216.5 427.8
4
0
1
0
621.0
(606.6)
0
1
0
635.3
0
1
0
467.3
0
1
0
467.5
5
0
0
3
793.4
(790.5)
0
0
3
793.6
0
0
3
633.8
0
0
3
633.8
6
0
1
1
878.9
(861.4)
0
1
1
891.8
0
1
1
677.8
0
1
1
678.2
7
1
0
0
2048.4
(2048.4)
1
0
0
2050.0
1
0
0
1510.3
1
0
0
1510.1
a
Excitation energies relative to the zeroth vibrational levels. Zeroth vibrational energies are quoted in brackets. Values in parentheses were obtained using the adiabatic approximation described in the text. b Formal numbering of the vibrational states.
Table 2. Energies of the Lowest j(even) Vibrational States Localized in the Lower Potential Minimium Region of the Global PES (in cm1)a sib
vr
vb
vR
vr
vb
vR
LiH+2
LiD+2
1
0
0
0
0.0
(1.265)
0
0
0
0.0
(0.776)
2
0
0
1
556.2c
(1.265)
0
0
1
425.9c
(0.777)
3
0
1
0
753.4c
(1.298)
0
1
0
537.6c
(0.787)
4
0
0
2
1108.0
(1.263)
0
0
2
847.6
(0.777)
5
1
0
0
1282.8c
(1.286)
0
1
1
950.3c
(0.785)
6 7
0 0
1 2
1 0
1354.3 1554.1
(1.228) (1.320)
1 0
0 2
0 0
1017.4 1108.2
(0.757) (0.800)
a
Excitation energies relative to the zeroth vibrational energy levels, which were obtained as 1336.8 cm1 for LiH+2 and 980.8 cm1 for LiD+2 . Rotational constants of the states are given in parentheses. Energies for the j(odd) states practically coincide with the j(even) numbers. b Formal numbering of the vibrational states. c The corresponding harmonic frequencies for LiH+2 are ωr = 1470.8 cm1, ωb = 340.4 cm1, ωR = 612.4 cm1. c The corresponding harmonic frequencies for LiD+2 are ωr = 1077.8 cm1, ωb = 245.9 cm1, ωR = 462.4 cm1.
vibrational couplings that are rather unusual in the present system. The virtues of these representations become obvious from a brief inspection of Figures 2 and 4. For the lower potential well containing the global minimum of the entire surface, harmonic frequencies were determined and are included in Table 2 for a comparison with the corresponding anharmonic frequencies. It should be noticed in this comparison that in the harmonic approximation, the triatomic ion is strictly treated as a linear system and that therefore the harmonic bending frequency ωb corresponds approximately to 1/2 νb in the table.
4. STATISTICAL CHARACTERISTICS Like in previous studies of the isomerization processes HNN+ h NNH+14 and HCO+ h COH+,15 the main purpose of this study was to examine if in the present light system, the tunnelling motion across the proton-transfer barrier has a visible effect on the statistical characteristics of the vibrational level pattern. According to the picture derived in the previous studies, a noticeable effect can only be expected if there is a significant number of lower states localized in each of the two potential
Figure 5. Energy positions and wave functions of the lowest vibrational states pertaining to the lowest adiabatic stretching potential (in cm1).
wells. For the lower potential well, obviously a number of sufficiently localized states do exist because for all states with an energy lower than the energy level of the upper minimum, tunnelling can practically be excluded. The question is thus whether for the upper potential well sufficiently localized states can be identified. For this purpose, lifetimes of the lowest states within this well were evaluated. Relying on the rather accurate description of the upper potential well within the framework of the adiabatic separation approximation, the lifetimes of these states were determined as lifetimes of hypothetical states located in the one-dimensional wells of the adiabatic potentials of Figure 4, which were modified such that Veff(r < rmin) = V(r) and Veff(r > rmin) = V(rmin), where rmin corresponds to the global minimum re of the entire potential surface. By applying the method described in ref 19, these calculations revealed that the only localized states in the upper potential well are the vr = 0 states (see Figure 5). Within the model used here, their lifetimes were obtained as approximately 1 ps for LiH+2 , 10 ns for LiD+2 , and 1 μs for LiT+2 . The question arising at this point was thus whether the number of these localized states in the upper well is sufficient to cause possibly a visible effect on the standard shapes of the statistical characteristics. It has previously been found14,15 that a useful approach to investigate this problem is to study the NNSD characteristics of the considered state ensemble. This is one of the simplest 11317
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quantities for the local analysis of the vibrational energy data when fitted to the Brody distribution20 PBro ¼ ðω þ 1Þαsω expðαsω þ 1 Þ
ð6Þ
with α = [Γ((ω + 2)/(ω + 1))]ω+1 and where the phenomenological Brody parameter ω is used to semiquantitatively describe the degree of chaos in the quantum dynamical system. For the two extreme cases, ω = 1 and 0, the Brody distribution coincides, respectively, with the Wigner distribution21 π π 2 PWig ¼ s exp s ð7Þ 2 4 which is predicted by the random matrix theory and corresponds to a completely chaotic case, and the Poisson distribution22 PPois ¼ expðsÞ
ð8Þ
corresponding to fully integrable dynamics. In these expressions, the quantity s is defined as s¼
S spacing ¼ D mean spacing
ð9Þ
where spacing means, in the present context, the difference between energies of adjacent levels of the same symmetry and mean spacing is its averaged value. Because of the kinematic effects, the mean spacing parameters are mass-dependent. Therefore, to allow for a direct comparison of the shapes of the NNSDs obtained for different isotopomers, it is suitable to scale the corresponding values of the mean spacing parameters such that Disot sc = Dref, where sc represents the scaling factor and Dref and Disot refer to the reference and actual isotopomer, respectively. Throughout the following discussion, LiH+2 is used as reference the isotopomer. In Figure 6, the numerically determined energy level distributions are plotted as histograms to provide a basic unbiased picture of the actual level distributions. The histograms are compared in the different panels of the figure with representations of the above-defined distribution functions for LiH+2 and its deuteriumand tritium- substituted isotopomers. According to the conclusions drawn from the corresponding analysis of the energy level distribution in Figure 7 of ref 14, the plots in the present Figure 6 show that the range with large s values represents the situation of independent and practically noninteracting vibrational states. Their distribution characteristic is therefore best represented by the Poisson function. In the region of small s spacings, the Wigner function becomes the best representation, which is most clearly seen in the lowest panel of Figure 6 showing the NNSD representations for the LiT+2 isotopomer. There is a small region however at the top of the histogram barrier where the intermediate Brody function seems to provide the best reproduction of the numerically determined energy level distribution. The respective Brody parameter values for the three isotopomers in Figure 6 are 0.740 for LiH+2 , 0.425 for LiD+2 , and 0.250 for LiT+2 . This is interpreted here as an indication that the states of the two potential wells begin to lose their localized character due to the tunnelling effect. As expected, the effect is very small for this light system, but it is still visible, and it shows the non-negligible role of the double-minimum shape of the potential function and its effect on the energies of a few vibrational states close to the top of the proton-transfer barrier.
Figure 6. NNSD representations of the bound vibrational energy levels of LiH+2 , LiD+2 , and LiT+2 .
The global dependence of the vibrational energies on the excitation level acquires a logarithmic shape, which can be described by the logarithmic formula EðNÞ ¼
∑ Ci½ lnðN þ 1Þi
i¼1
N ¼ 0, 1, 2,
...
ð10Þ
where the index n runs from 1 to some number depending on the actual case. The fitted curves for the LiH+2 isotopomers are shown in the upper panel of Figure 7. This representation is particularly 11318
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Figure 7. Calculated vibrational energies of all bound states for LiHH+, LiDD+, and LiTT+ (upper panel) and the corresponding cumulative spectral density functions (lower panel). Labeling of the states by the integer N increasing according to the energy content of the states.
useful for an extrapolation to highly excited levels in large molecular systems with a massively large number of states. Due to the continuously growing energy representations, it is possible to evaluate cumulative spectral density functions W(E), displayed in the lower panel of Figure 7, where W(E) counts the number of states containing energies up to E Z E FðE0 Þ dE0 ð11Þ WðEÞ ¼ 0
The functions W(E) define the state densities F(E) FðEÞ ¼
dWðEÞ dE
ð12Þ
used to determine one of the most important statistical properties, the partition function Q(T) Z ∞ E Q ðTÞ ¼ FðEÞ exp dE ð13Þ kT 0 where T is temperature and k the Boltzmann constant. The upper panel of Figure 7 shows that the curves representing the vibrational energies of the different isotopomers have similar shapes. For a closer comparison of these functions, a simple reduction scheme was introduced ered ¼ E=De
nred ¼ N=Nbound
ð14Þ
Figure 8. Excitation dependence of the LiH+2 , LiD+2 , and LiT+2 vibrational energies in the reduced representation.
where De and Nbound are the dissociation energy and the number of the bound vibrational states, respectively. The upper panel of Figure 8 shows the reduced vibrational energy curves of the three LiH+2 isotopomers over the entire range of nred, and in the lower panel, a comparison of reduced energy curves for different threeatomics is displayed. The close coincidence of these curves seems to suggest that it should be possible to construct a simple universal reduction function with only few parameters, which would allow rather accurate extrapolations of vibrational energies for systems for which the evaluation if high-lying states becomes numerically impossible.
5. CONCLUSIONS As a first step toward a rigorous theoretical attempt to determine the spectroscopic properties of the excited electronic state of the LiH+2 ion, the present study provides, for the first time, predictions of the vibrational energy levels localized essentially in the two potential energy minimum regions and makes an attempt to analyze the level pattern of all bound vibrational states of the ion applying statistical methods. For this purpose, apart from a global potential function including the two low-lying atomdiatom dissociation channels, two additional potential functions are constructed to represent more accurately each of the two minimum-energy regions by separately applying appropriate weighting schemes in the fitting 11319
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The Journal of Physical Chemistry A procedures. With these potential functions for the two minimum regions, full-dimensional variational calculations and, for the upper potential well also, adiabatic separation calculations are performed to determine accurate vibrational energies of the states localized in each of the two potential pockets. The lowest-energy levels obtained from these calculations can be expected to closely coincide with the energies determined from an unbiased complete accurate potential function. Discrepancies of the present results with the complete calculation can be expected however to grow with increasing excitation level. In addition, special emphasis is put here into searching for any indication of the effect of the tunnelling motion across the proton-transfer barrier on the bound vibrational levels determined from the global potential function. The NNSD characteristics of the vibrational level pattern were analyzed. Because only a small number of states are involved in this motion, the effect on the standard shape of the statistical distribution must be expected to be small. According to the picture derived in previous studies,14,15 it follows that depending on the region of the vibrational spectrum, different characteristics prevail. Whereas the low-lying states with larger s acquire a Poisson distribution, the higher states with small s are obviously well represented by the Wigner distribution. The search for an indication of a tunnelling effect has therefore to concentrate on the small area between these two distribution characteristics. The plots collected in Figure 6 show that at small s, spacings where the Poisson curve starts to deviate from the intermediate Brody characteristic provide the best representation of the numerical histogram.
ARTICLE
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’ ASSOCIATED CONTENT
bS
Supporting Information. The fit for the upper potential well (fitted parameter values summarized in Table A1) and fits for the lower-energy minimum (fitted parameter values in Table A2). This material is available free of charge via the Internet at http://pubs.acs.org.
’ AUTHOR INFORMATION Corresponding Author
*Email:
[email protected].
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