ARTICLE pubs.acs.org/JPCA
Ab Initio Adiabatic and Diabatic Energies and Dipole Moments of the CaH+ Molecular Ion Hela Habli,*,† Riadh Dardouri,† Brahim Oujia,† and Florent Xavier Gadea‡ † ‡
Laboratoire de Physique Quantique, Faculte des Sciences de Monastir, Avenue de l’Environnement 5019, Monastir, Tunisie Laboratoire de Chimie et Physique Quantique, UMR5626 du CNRS, Universite de Toulouse, UPS, 118 Route de Narbonne, 31062 Toulouse Cedex 4, France ABSTRACT: The diabatic and adiabatic potential-energy curves and permanent and transition dipole moments of the highly excited states of the CaH+ molecular ion have been computed as a function of the internuclear distance R for a large and dense grid varying from 2.5 to 240 au. The adiabatic results are determined by an ab initio approach involving a nonempirical pseudopotential for the Ca core, operatorial core valence correlation, and full valence configuration interaction. The molecule is thus treated as a two-electron system. The diabatic potential energy curves have been calculated using an effective metric combined to the effective Hamiltonian theory. The diabatic potential-energy curves and their permanent dipole moments for the 1∑+ symmetry are examined and corroborate the high imprint of the ionic state in the adiabatic representation. Taking the benefit of the diabatization approach, correction of hydrogen electron affinity was taken into account leading to improved results for the adiabatic potentials but also the permanent and transition electric dipole moments.
1. INTRODUCTION Alkaline earth hydrides and their corresponding ions have been the object of intense theoretical and experimental interest for many years.1 5 Such molecules have great importance in spectroscopy and astrophysics. They are important constituents of interstellar media6,7 and comets.7,8 Although not really expected, our previous study devoted to the adiabatic states of the CaH+ molecule has revealed the important role played by the ionic state Ca2+H . Noteworthy, production of H by collisions between Ca+* and H* or even Ca* and H+ seems thus possible in astrophysical conditions. These collisions can also lead to charge transfer between Ca and H, which is also an important process in astrophysics. The calculation of such cross-sections needs the evaluation of nonadiabatic couplings if the adiabatic representation is used, or alternatively of the diabatic states and their electronic coupling.9 11 The possibility to perform such crosssection calculations has been an additional motivation for this diabatic study. In addition, the diabatic approach yields physical insight as shown in previous studies of the LiH,11 13 NaH,10 KH,14 RbH,15 and CsH16 systems and opens the way for taking into account efficiently non-Born Oppenheimer effects, either in dynamics (nonadiabatic transitions, surface hopping, ...) or in spectroscopy (nonradiative lifetimes, adiabatic corrections, vibronic shift, ...). Moreover, the diabatic approach can be used to improve the accuracy of the ab initio calculation by performing specific corrections, for example, to the hydrogen electron affinity, surmounting limitations in the actual basis sets, which is among the main constraints in the ab initio calculation. r 2011 American Chemical Society
For LiH, it was shown that this correction is important for the binding energy of the ground state but also for the global vibrational spacings for most of the excited states.12 The undulations with repulsive barriers in the potential energy curves of the highly excited states were analyses in the earlier study of LiH12 and confirmed in several recent works in particular for CsLi.17 These undulations are amplified in the diabatic curves and were related to intrinsic characteristics of the Rydberg states.12 An interest of this work is thus to improve the potential-energy curves and dipole moments thanks to the correction of hydrogen electron affinity, allowed by the use of an efficient diabatic approach. This work is the first diabatic calculation on highly excited states of the CaH+; we treat nearly all the states dissociating below the ionic limit Ca2+H and discuss them. This paper is organized as follows: In section 2, we briefly present the methods used in adiabatic and diabatic approaches. Section 3 is devoted to the presentation and discussion of the diabatic and adiabatic results. The permanent and transition electric dipole moments are reported in section 3.2. Finally, concluding remarks are given in section 4.
2. METHODS 2.1. Adiabatic Approach. The methods used in the ab initio calculation have been presented in detail in the previous Received: May 2, 2011 Revised: September 28, 2011 Published: October 20, 2011 14045
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Figure 1. Diabatic potential energy curves for the eight low-lying 1∑+ states of the CaH+ molecule. The dotted line presents the curve ionic with hydrogen electron affinity correction.
publication devoted to the adiabatic results.18 For consistency, we briefly recall here the main lines. We have used a pseudopotential for the Ca2+ core, supplemented by operatorial core polarization potentials (CPP), with a single cutoff radius, in order to take into account core valence correlation effects. The molecule id thus treated as a two-electron system and full configuration interaction (CI) is easily carried out. The package used is the one developed by the LCPQ in Toulouse (see ref 18 and references there included). 2.2. Diabatic Approach. We have used here the method developed and thoroughly tested for the whole alkali hydride series, first, for the CsH19 molecule and later for LiH,11 13 NaH,10 RbH15 and LiCs17 systems, or mixed alkali ions.20 For each interatomic distance R, the diabatic states are defined as a unitary transformation of a set of adiabatic states. Following alternative effective Hamiltonian theory, a set of reference states, called model space, are projected onto the set of adiabatic states, called target space, and the projection matrix is made unitary by symmetrical orthogonalization,19 23 like in the des Cloiseaux effective Hamiltonin approach. The reference states are taken as adiabatic states at very large distance where all adiabatic states have a clear physical character. For evaluating the overlap matrix between the two sets of multiconfigurational states, we used an effective metric for the overlap matrix between the atomic basis sets at the different interatomic distances involved.19,20 This effective metric ensures that adiabatic states and diabatic states coincide at large internuclear distances, leading to a unit matrix for the adiabatic diabatic unitary transformation. The method therefore combines two important points, an effective metric, which ensures vanishing couplings at large distance, and alternative effective Hamiltonian theory, which ensures that the diabatic states correspond to a unitary transformation of the adiabatic states. In fact, as analyzed in
refs 19 and 20 in this procedure, a crude estimation of the nonadiabatic coupling is performed via the overlap matrix, and the orthonormalization transformation made it as small as possible. It is necessary to specify the common origin, which reflects in the effective metric. We have fixed it here at the heavier calcium atom. More details about the method can be found in refs 19 and 20. It was shown to be very successful for the alkali hydride series, where the particular role played by the ionic state could be emphasized for the potential shapes as well as for the permanent and transition electric dipole ones. Simple estimates of the radial couplings could be also performed assuming vanishing ones in the diabatic representation,11 and cross-section calculations were carried out for the ion pair neutralization for LiH 9,24 and NaH.10 More recently, detailed comparisons were carried out for NaH excitation transfers between our diabatic approach and adiabatic ones using ab initio nonadiabatic couplings, and the agreement was also very encouraging.25 The diabatic study has been performed for 14 states in symmetry 1∑+, including the ionic state Ca2+H and all states below at large distance. The references were set to these adiabatic states at R = 240 au. At this very large distance, the behavior of the curves is completely flat except for the ionic one which behaves as 2/R, and the assignation of the diabatic states is easy.
3. RESULTS AND DISCUSSION 3.1. Diabatic and Adiabatic Results and Corrections. The calculation was carried out with a wide range of internuclear distances varying from 2.5 to 240 au with a step of 0.2 au around the avoided crossings, in order to cover all of the ionic neutral crossings in the 1∑+ symmetry. This is particularly important for the adiabatic representation, around weakly avoided crossings 14046
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Figure 2. Adiabatic potential-energy curves for higher excited 1∑+ with states. The dotted line is the ionic curve improved with constant electron affinity correction of hydrogen.
Table 1. Equilibrium Distance Re (Å) and Potential Well Depth De (cm 1) for the Ground State X 1∑+ of the CaH+ Ionic Molecule Computed with and without Correction of H Electron Affinity and Compared with the Available Results in the Literature states X ∑
1 +
Re
Table 2. Spectroscopic constants for the A, B, C, and D 1∑+ State of the CaH+ Ionic Molecule Computed with and without Correction of Hydrogen Electron Affinity and Compared with the Available Results in the Literature states
Re
De
A ∑ a
6959 7327
this worka this workb
De
1 +
1.86
17 876
this work
2.32 2.33
1.82
18 198
this workb
2.35
6750
1
1.86 1.93
17 744
1 2
2.33
8872
29
2.79
7913
this worka
2.76
8003
this workb
2.08
3
1.94
5
1.92
28
1.88
17 469
B 1∑+
C 1∑+
29
a
Uncorrected ab initio results. b Results including correction of hydrogen electron affinity.
D 1∑+
2.84
6670
4.03
10992
this worka
4.00 4.03
11230 9291
this workb 1
2.61
6659
this worka
6727
this workb
2.61
between the potential curves and also to follow the permanent and transition dipole moments which may additionally present strong variations in these zones. The diabatic representation is less demanding since the potential and the dipole moments present smoother variations. In Figure 1 we present the diabatic potential-energy curves related to the X P 1∑+ adiabatic ones (see Figure 2), which we will note respectively as D1, D2, ..., D14. As expected the avoided crossings in the adiabatic representation are transformed into real crossings in the diabatic one. As in LiH,12 NaH,10 KH,14 RbH15 and CsH,16,19 the diabatic potential-energy curves present various crossings and undulations with repulsive barriers at short distances related to the nature of the Rydberg states that may extend to rather large distances for the highly excited states. In particular, the diabatic states D5 and D7 present potential barriers at long distances. This effect was related in previous studies with
a
1
b
Uncorrected ab initio results. Results including correction of Hydrogen electron affinity.
the Fermi model;26,27 in fact, the H atom is colliding with a Rydberg electron, and the interaction potential is then proportional to the electron-H scattering length and to the Rydberg amplitude. As expected the diabatic curve D1, corresponding to the ionic limit Ca2+H , crosses all of the others ones at different distances. The crossing occurs with the lowest 4s state around 6.13 au, with the 3d state around 5.58 au, and with the 4p state around 8.15 au, the others occurring at much larger distances. At large internuclear distances, the first diabatic curve, corresponding to an ionic system, nicely behaves as 2/R, as expected for an electrostatic interaction between Ca2+ and H . The remaining 14047
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Figure 3. Diabatic potential energy curves for the eight low-lying (a) 1Δ and (b) 3Δ states of the CaH+ molecule.
states are expected to present weaker polarization and van der Waals interactions. In Figure 1 we have shown (dotted line) the D1 energy curve corrected by the hydrogen electron affinity error due to our basis set limitation. Electron affinity (EA) is rather
difficult to calculate accurately because the anion usually needs much larger basis sets and a higher level treatment of the correlation energy compared to the neutral. For H, the accurate computation of H is much more demanding than the one of 14048
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Figure 4. Permanent dipole moment for the 1∑+ states of the CaH+ molecular ion as a function of the internuclear distance R. Alternating solid and dashed lines correspond to X, A, B, ..., P. Enlarged curves at short distances are shown in the inset.
Figure 5. Permanent diabatic dipole moment for the eight low-lying 1∑+ states of the CaH+ molecule, as a function of the internuclear distance. 14049
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Figure 6. Adiabatic permanent dipole moments for states X, A, B, C, and D 1∑+ with (dotted line) and without (solid line) H electron affinity correction, as a function of the internuclear distance R.
Figure 7. Diabatic transition dipole moments for selected 1∑+ states, as a function of the internuclear distance R.
neutral hydrogen; therefore, the error on the energy of H is much larger than the one on the energy of H. However, even in neutral molecules, ionic states play an important role, for LiH the ground state is ionic at equilibrium geometry; for H2 ionic states imprint the excited states, and therefore they require a special attention. Instead of increasing dramatically the basis set, an alternative thanks to the diabatic approach is to correct the ionic charge-transfer state for the EA error we can easily estimate from the large distances where the atomic states can be identified. This
correction leads to a small shift in energy for the D1 curve which induces shifts to larger distances for the crossings with all of the other diabatic states. Although only the D1 state is affected by this correction, diagonalization of the new electronic Hamiltonian matrix leads to new adiabatic states which all may be affected by the correction. In Figure 2 we depict these new adiabatic potentials (dotted lines) and the previous ones. Noticeable differences only arise when the adiabatic states are strongly imprinted by the ionic state. 14050
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Figure 8. Diabatic transition dipole moments for the selected states 1∑+ of CaH+ molecular ion, as a function of the internuclear distance R.
We note that the well depth of the ground and the first excited states are enlarged. These results that use an improved ionic
potential will be labeled improved results. It is interesting to examine how the improvement modifies the spectroscopic 14051
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Figure 9. Adiabatic transition dipole moments for the selected states X A, X B, A B, B D, and C D 1∑+ of CaH+ molecular ion with (dotted line) and without (solid line) hydrogen electron affinity correction, as a function of the internuclear distance R.
constants of the 1∑+ states. We report in Tables 1 and 2 the spectroscopic constants for X, A, B, C, and D 1∑+ states. Thanks to the correction of the ionic curve, the binding energy of the ground state is clearly improved (Table 1). This indicates the important effect of the hydrogen electron affinity in the quality of ab initio calculations for the alkaline earth hydrides. For the A state this correction improves the equilibrium distance (as shown in Table 2), which is underestimated in the uncorrected ab initio calculation, and the same for the other excited states. Clearly the shift of the avoided crossings to larger distances plays here an important role. Our improved results for the spectroscopic constants are also in rather good agreement with the available theoretical works.1 5,28,29 However, to our knowledge, there are not experimental results available for these main spectroscopic constants (equilibrium distance Re and potential well depth De) and definitive conclusions cannot be drawn. In Figure 2 we can also notice that the charge-transfer state dissociating in Ca(4s2) + H+ seems to be very repulsive and generates a series of avoided crossings at rather short distances. This intriguing behavior surprisingly suggests that the interaction between the valence electrons and the proton is highly repulsive. In Figure 3a,b we present the diabatic potential-energy curves for the 1Δ and 3Δ states, respectively. They are quite similar to the adiabatic ones (see ref 18) since the states are rather far in energy and no strong interaction between them is expected. However, a closer inspection shows that the wells are somewhat less deep in the diabatic representation, an indication that interstate couplings are contributing to the adiabatic well depths. 3.2. Diabatic Permanent Electric Dipole Moment. To illustrate the ionic behavior of the excited electronic states, we have presented in Figures 4 and 5 the adiabatic permanent electric dipole moments for the 1∑+ adiabatic and diabatic states, respectively. We have considered a large and a dense range of internuclear distance R because we expected interesting insight
from such global pictures, including all states in a wide range, similarly to the alkali hydrides.10,14 16 First, we examine the diabatic results; for the dipole moment of the ionic diabatic state (D1), we nicely obtain, as expected, an almost straight line which behaves as R, as shown in Figure 5. Here the origin is put in the calcium atom; therefore only the charge on H matters, and the permanent dipole moment behaves as R for Ca2+ + H and as +R for Ca + H+ and vanishes for Ca+ + H states. In the adiabatic representation, this R line is reproduced piecewise, as can be seen in Figure 4, when all states are put together. One after the other, the adiabatic states became ionic, their dipole moment reaching the line and then dropping back as tone of the neighboring states increases, and so on. Clearly the ionic state imprints this behavior and governs the interplay between the avoided crossings in the potentials and the crossings in the permanent dipole moments, and the less avoided the crossings in the potentials, the sharper the variations are in the permanent dipoles. The permanent dipole moments can thus reach considerable values for the highly excited states at large distances. It is possible to improve also the permanent dipole moments by taking advantage of the diabatic representation and correcting only the diabatic ionic potential curve. Again, the whole diabatic adiabatic unitary transformation is affected by the diagonalization, and new permanent, and also transition, dipole moments arise. The improved permanent dipole for the states X, A, B, C, and D are illustrated in Figure 6 with dotted lines, as well as the uncorrected adiabatic ab initio results. We observe that the crossings are clearly shifted to larger internuclear distances and may lead to substantial changes in the absolute values, although the global picture is only slightly shifted. 3.3. Transition Dipole Moment in Adiabatic and Diabatic Representations. We present, in Figures 7 and 8, the variation with the internuclear distance of a selection of transition dipole moments for the 1∑+ states in the diabatic representation. 14052
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The Journal of Physical Chemistry A In Figure 7 we have collected the transition dipoles from the ionic diabatic state to a selection of other diabatic states. As can be seen, the transition to the Ca+(4p) + H is by far, the largest. Figure 8 illustrates a selection of the remaining transitions. In Figure 9 we show how the correction to the ionic curve influences the transition dipoles moments of the adiabatic states. The dotted lines are for the improved transition dipole moments between the lowest adiabatic states. As for the permanent dipole moments, the global shape is only slightly affected, but noticeable differences may be locally quite large. Again this change is due to the change in the eigenvectors when diagonalizing the matrix after the correction of the ionic energy only. Although point charge models lead to vanishing transition moments, the diabatic states are aimed at diagonalizing the removable part of the nonadiabatic coupling, not the electric dipole moment. At large distance, in our method the diabatic states match the adiabatic ones and therefore allowed atomic transitions remain.
4. CONCLUSION A global study has been performed for the calcium hydride ion in a diabatic representation involving many electronic states in various symmetries. Adiabatic potential curves taking benefit to the correction of hydrogen electron affinity in the ionic diabatic state have been presented for the ground and excited 1∑+ states of the CaH+ molecular ion. They were allowed by the use of an efficient diabatic approach. The ab initio calculations involved a nonempirical pseudopotential for the Ca core, operatorial core valence correlation estimations (CPP), and full valence configuration interaction. We performed a diabatic study for this system using variational effective Hamiltonian theory combined to an effective metric. This allows us to identify the avoided crossings in the adiabatic representation with the real crossings in the diabatic representation. We have been interested to the study of all the states dissociating below the ionic limit Ca2+H . In this framework, we have derived the diabatic potential-energy curves and also the permanent and the transition dipole moment of the 1 + ∑ and 1,3Δ symmetry. We observed that, for the diabatic states, the permanent dipole behaves as straight lines which easily identified their physical meaning ( R for Ca2+ + H , +R for Ca + H+), while, for the adiabatic states, we have curves which relay each other to these lines, illustrating their ionic imprint. These results can be readily explained from the physical character of the diabatic states, and they shed light on the interaction between the ionic and neutral species that dominate in the potential curves for the alkaline earth hydride ions as for the alkali hydride series. The analysis of the transition and permanent dipole moment enabled us to check the positions of the avoided crossings identified in the adiabatic representation. Around these avoided crossings the transition dipole moment between two neighboring states changes in behavior. These data open the way to calculation of vibronic shifts and observables such as the radiative and nonradiative lifetimes or the inelastic collision cross-sections and more generally to beyond Born Oppenheimer studies where the electronic couplings are taken into account in the diabatic representation or alternatively the nonadiabatic couplings in the adiabatic representation.12,30 39 ’ AUTHOR INFORMATION Corresponding Author
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*E-mail:
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