Electronic structure and time-dependent description of rotational predissociation of LiH

P. Jasik a, J. E. Sienkiewicz *a, J. Domsta b and N. E. Henriksen c
aFaculty of Applied Physics and Mathematics, Gdańsk University of Technology, Narutowicza 11/12, 80-233 Gdańsk, Poland. E-mail: jes@mif.pg.gda.pl
bInstitute of Applied Informatics, State University of Applied Sciences in Elbląg, Wojska Polskiego 1, 82-300 Elbląg, Poland
cDepartment of Chemistry, Technical University of Denmark, Building 207, DK-2800 Kgs. Lyngby, Denmark

Received 31st March 2017 , Accepted 6th June 2017

First published on 6th June 2017


The adiabatic potential energy curves of the 1Σ+ and 1Π states of the LiH molecule were calculated. They correlate asymptotically to atomic states, such as 2s + 1s, 2p + 1s, 3s + 1s, 3p + 1s, 3d + 1s, 4s + 1s, 4p + 1s and 4d + 1s. A very good agreement was found between our calculated spectroscopic parameters and the experimental ones. The dynamics of the rotational predissociation process of the 11Π state were studied by solving the time-dependent Schrödinger equation. The classical experiment of Velasco [Can. J. Phys., 1957, 35, 1204] on dissociation in the 11Π state is explained for the first time in detail.


1 Introduction

During the past twenty years, the physics of diluted gases have seen major advances in two fields, namely laser cooling of atomic and molecular samples and femtosecond chemistry. In both cases, appropriate frequency and phase-shaped laser light are used to control the system. In this context, two fundamental processes, i.e., photoassociation and photodissociation, or in other words formation and breaking of the chemical bond by light, have attracted the attention of theoreticians, as well as experimentalists. In particular, photodissociation of diatomic or small polyatomic molecules is an ideal field for investigating molecular dynamics at a high level of precision.

Homonuclear and heteronuclear alkali metal molecules, including LiH, are valuable for theoreticians, mainly because they have a simple electronic structure, being two-valence electron systems. They can serve as convenient prototypes to test theoretical methods, which can be further applied to more complicated molecular systems. Besides that, knowledge of interatomic adiabatic potential energy curves of diatomic systems is essential for the understanding of several processes such as photodissociation, photoassociation, cooling and trapping. An extensive survey on the spectroscopy and structure of LiH was published in 1993 by Stwalley and Zemke,1 and this was followed by a study by Gadea and Leininger2 in 2006.

In 1935–1936, Crawford and Jorgensen3,4 analysed the LiH band spectra. Since then, many notable studies have been undertaken. Among them, in 1962 Singh and Jain5 applied the Rydberg–Klein–Rees method to obtain energies of the low excited states of LiH. Gadea and coworkers calculated potential energy curves,2,6–8 radial couplings,9 nonadiabatic energy shifts,10 as well as LiH formation by radiative association in ion collisions.11 Results of several other calculations, including semiempirical and ab initio approaches to describe important physical and chemical properties of LiH are available.12–26 Calculations related to LiH are also used in the description of the formation of ultracold polar molecules in a single quantum state (e.g. Côté et al.27). Special investigations have been devoted to dipole moments28,29 and the ionic states of LiH.30–32 Tung et al.33 and Holka et al.34 performed very accurate calculations of the ground and some excited state potential curves.

LiH was also intensively explored in time-dependent studies. Again, being only a four-electron molecule makes it a convenient example for molecular dynamics calculations. In 1936, Mulliken35 noted that the change in the internuclear separation may cause a rearrangement in the distribution of the density of electrons. Recently, the LiH molecule was used in a computational study using the time-dependent multiconfiguration method.36

The aim of our work was to provide accurate potential energy curves and to use them to explain a classical experiment by Velasco37 on rotational predissociation. We chose to solve the time-dependent Schrödinger equation (TDSE) with a probe wavepacket placed on the effective interatomic potential possessing a centrifugal barrier. This approach made it possible to compare rovibrational spacings with the results derived from the experiment by Velasco37 and with those calculated directly from the electronic structure. Our work was also motivated by the case of the NaI molecule intensively studied by A. Zewail38 and later by others.39–42 The NaI dimer shows similar behavior to LiH in creating ionic bonds and is a well-studied prototype molecule in femtochemistry, particularly in relation to the dynamics of unimolecular reactions.

In Section II, the appropriate model of the electronic structure is defined, leading to an algorithm for the calculation of some low-excited singlet Σ+ and Π states. Later, we describe the theoretical backgrounds of rotational predissociation and molecular dynamics. We explain how the obtained adiabatic potentials can be used in the theoretical treatment of the rotational predissociation proccess. In particular, we present a method for the calculation of the dynamics of predissociation of molecules starting with a given coherent wavepacket. In Section III, we present the rotational predissociation results for the 11Π state and compare them with the measurements of Velasco.37 Conclusions are given in the last section.

2 The model

2.1 Electronic structure

We consider the interaction between the lithium (atom A) and hydrogen (atom B) under the assumption that the molecular state is a composition of the electronic adiabatic states Ψeli([r with combining right harpoon above (vector)];R), i = 1, 2, 3,…, which depend on the positive variable R, i.e. on the separation between the nuclei of these atoms. The applied notation indicates that our considerations are restricted to such eigenstates, which are independent of the direction of the vector joining the nuclei. In other words, electronic wave functions possess spherical symmetry with respect to the nuclear coordinates. Our calculations are based on the Born–Oppenheimer approximation, i.e. the solutions of the following time-independent Schrödinger equation:
 
HelΨeli([r with combining right harpoon above (vector)];R) = Eeli(R)Ψeli([r with combining right harpoon above (vector)];R).(1)
Here, the separation parameter R is kept fixed, vector [r with combining right harpoon above (vector)] represents all the electronic coordinates, Hel is the electronic Hamiltonian of a diatomic system. Thus, Ψeli([r with combining right harpoon above (vector)];R) describes the ith eigenstate of the Hamiltonian, Eeli(R) are the corresponding eigenvalues, also called adiabatic potentials. The Hamiltonian of the system can be written as
 
Hel = Tel + V,(2)
where Tel stands for the kinetic energy operator of the valence electrons and V represents the operator of the interaction between the valence electrons, the Li-core and the nucleus of H. In the present approach only the valence electrons are treated explicitly, which allows for an adequate description of electron correlation at low computational cost. The lithium core is represented by an angular momentum-dependent pseudopotential. The latter is obtained as
 
image file: c7cp02097j-t1.tif(3)
Here, VA describes the Coulomb and exchange interaction as well as the Pauli repulsion between the valence electrons and the lithium core. We use the following semi-local energy-consistent pseudopotentials:
 
image file: c7cp02097j-t2.tif(4)
where QA = 1 denotes the net charge of the lithium core, PAl is the projection operator onto the Hilbert subspace of angular symmetry l with respect to the Li+-core. The parameters BAl,k and βAl,k define the semi-local energy-consistent pseudopotential. The second interaction term in eqn (3) is the polarization term that describes, among others, core-valence correlation effects and is calculated as
 
image file: c7cp02097j-t3.tif(5)
where αA = 0.1915a0 is the dipole polarizability of the lithium core43 and [F with combining right harpoon above (vector)]A is the electric field generated at its site by the valence electrons. For the latter we use the following formula:
 
image file: c7cp02097j-t4.tif(6)
where δA is the cutoff parameter, which equals 0.831a0−2 (value taken from Fuentealba et al.43). The third term in eqn (3) represents the Coulomb interaction between the valence electrons and the hydrogen nucleus. The fourth term stands for the repulsion between the valence electrons, whereas the last term describes the interaction between the lithium core and hydrogen nucleus. Since the lithium atomic core and the hydrogen nucleus are well separated, we choose a simple point-charge Coulomb interaction in the latter case. More detailed characteristics of the applied potentials are given in the papers by Czuchaj and coworkers44,45 and Dolg.46

The core electrons of the Li atom are represented by the pseudopotential ECP2SDF,43 which was formed from the uncontracted (9s9p8d3f) basis set. The basis for the s and p orbitals, which comes with this pseudopotential, is enlarged by the functions for d and f orbitals given by D. Feller47 and assigned by cc-pV5Z. Additionally, our basis set was augmented by four s short-range correlation functions (1979.970927, 392.169555, 77.676373, 15.385230), four p functions (470.456384, 96.625417, 19.845562, 4.076012), four d functions (7.115763, 3.751948, 1.978298, 1.043103) and four f functions (2.242072, 1.409302, 0.885847, 0.556818). Also, we added to the basis the following set of diffuse functions: two s functions (0.010159, 0.003894), two p functions (0.007058, 0.002598), two d functions (0.026579, 0.011581) and two f functions (0.055000, 0.027500). The numbers in parenthesis are the coefficients of the exponents of the primitive Gaussian orbitals. The basis set for the hydrogen electron is the standard cc-pV5Z basis.48

The spin–orbit coupling (SO) contributes insignificantly to the energy of our system, so we do not take it into account. To calculate the adiabatic potential energy curves of the LiH diatomic molecule, we use the multiconfigurational self-consistent field/complete active space self-consistent field (MCSCF/CASSCF) method and the multi-reference configuration interaction (MRCI) method. All the calculations are performed by means of the MOLPRO program package.49 Using these computational methods we obtained adiabatic potential energy curves for singlet Σ+ and Π states, which correlate to the Li(2s) + H(1s) ground atomic asymptote and the Li(2p) + H(1s), Li(3s) + H(1s), Li(3p) + H(1s), Li(3d) + H(1s), Li(4s) + H(1s), Li(4p) + H(1s) and Li(4d) + H(1s) excited atomic asymptotes, respectively. The quality of our calculations can be confirmed by the comparison with experimental and theoretical asymptotic energies for different electronic states, which is shown in Table 1. Our asymptotic energies for ground and excited states are in very good agreement with experimental and other theoretical values. In particular, a perfect match is found between our result and the experimental value for the Li(2p) energy level.

Table 1 Comparison of asymptotic energies with other theoretical and experimental results. Energies are shown in cm−1 units
Atomic asymptotes Experiment Moore50 Theory Boutalib6 Theory Gadea2 Theory present
Li(2p) + H(1s) 14[thin space (1/6-em)]904 14[thin space (1/6-em)]905 14[thin space (1/6-em)]898 14[thin space (1/6-em)]904
Li(3s) + H(1s) 27[thin space (1/6-em)]206 27[thin space (1/6-em)]210 27[thin space (1/6-em)]202 27[thin space (1/6-em)]202
Li(3p) + H(1s) 30[thin space (1/6-em)]925 30[thin space (1/6-em)]926 30[thin space (1/6-em)]920 30[thin space (1/6-em)]921
Li(3d) + H(1s) 31[thin space (1/6-em)]283 31[thin space (1/6-em)]289 31[thin space (1/6-em)]279 31[thin space (1/6-em)]276
Li(4s) + H(1s) 35[thin space (1/6-em)]012 35[thin space (1/6-em)]018 35[thin space (1/6-em)]007 35[thin space (1/6-em)]016
Li(4p) + H(1s) 36[thin space (1/6-em)]470 36[thin space (1/6-em)]475 36[thin space (1/6-em)]465 36[thin space (1/6-em)]464
Li(4d) + H(1s) 36[thin space (1/6-em)]623 37[thin space (1/6-em)]590 36[thin space (1/6-em)]626 36[thin space (1/6-em)]617


2.2 Rotational predissociation

When the adiabatic potential Eel(R) of the singlet state 1Λ is obtained from the solution of eqn (1), the effective potential energy may be written in the following form (e.g. Landau and Lifshitz):51
 
image file: c7cp02097j-t5.tif(7)
where Λ is the component of the sum over all the electron angular momenta on the diatomic axis, JΛ is the rotational quantum number of the molecule and μ is the reduced mass of the nuclei.

Rovibrational energies E(v,J) depend on Eel(R) as well as on vibrational v and rotational J quantum numbers. They are the solutions of the time-independent nuclear Schrödinger equation

 
HnucJΨnucv,J(R) = E(v,J)Ψnucv,J(R),(8)
where the nuclear Hamiltonian is obtained as
 
image file: c7cp02097j-t6.tif(9)

The effective potential UJ(R) forms a barrier for J > 0 with a maximum UJ(RJ), at the internuclear distance RJ, which can easily be estimated. Any rovibrational state with a positive energy E(v,J) lower than UJ(RJ) has a finite lifetime before it will be decomposed due to a quantum tunneling effect. These states are called quasibound states and formally belong to the continuum. What is important is that during their lifetimes they can be regarded as bound states. When the energy E(v,J) exceeds the barrier maximum UJ(RJ), any bound state is not possible. Following Way and Stwalley,52 we introduce a critical value of the rotational quantum number Jc, which obeys the two following inequalities:

 
E(v,Jc) < UJc(RJc)(10)
and
 
E(v,Jc + 1) > UJc+1(RJc+1).(11)

In other words, for a given v, the state with the energy E(v,Jc) is the last of the quasibound states series supported by the barrier, and the state with the energy E(v,Jc + 1) already belongs to the continuum. By solving eqn (8) we obtain E(v,Jc) and estimate E(v,Jc + 1). The differences E(v,Jc) − E(0,0) and E(v,Jc + 1) − E(0,0) may refer to the last observed and the first unobserved rotational predissociation experimental results, respectively.

2.3 Molecular dynamics

The time-dependent approach that is mathematically equivalent to the time-independent one can be regarded as a complementary tool and is often used in studying photodissociation processes. Here, it serves as an alternative and quite illustrative method for testing the results of our structural calculations.

We start our consideration from the time-dependent Schrödinger equation written in the following form:

 
image file: c7cp02097j-t7.tif(12)
for each JJc separately, where Φ(R,t) is the time-dependent wavepacket moving on the effective potential energy curve UJ(R) (eqn (7)) and HnucJ is the nuclear Hamiltonian given in eqn (9).

By definition, the wavepacket is a coherent superposition of stationary states (e.g. Tannor53), which may be represented in the following form consisting of two contributions from the discrete and continuous parts of the spectrum:

 
image file: c7cp02097j-t8.tif(13)
where
image file: c7cp02097j-t9.tif
and
image file: c7cp02097j-t10.tif
are the energy-dependent coefficients, squares of these coefficients form the spectral distribution of Φ normalized to 1, e−ıE(v,J)t/ħ and e−ıEt/ħ are the time evolution factors, and Ψnucv,J(R) and ΨE,J(R) are eigenfunctions of HnucJ(R). The wavepacket Φ(R;t) is a solution of eqn (12) and its initial shape at t = 0 is taken as a Gaussian function of arbitrary half-width placed on the effective potential energy curve. The wavepacket moves away from its starting location due to the Newtonian force −dUJ/dR. This process is described by the time-dependent autocorrelation function
 
image file: c7cp02097j-t11.tif(14)

In our case, the autocorrelation function describes evolution of the initial nuclear wavepacket in the excited electronic state. The time-dependent population in the range till Rmax for the particular state labeled by J, in accordance with the effective potential energy UJ from eqn (7), is calculated as

 
image file: c7cp02097j-t12.tif(15)

We determine the discrete spectrum by the inverse Fourier transform of S(t)54 as follows:

 
image file: c7cp02097j-t13.tif(16)

In our calculations of the autocorrelation function (eqn (14)), the propagation time is 150 ps, which is sufficient for the estimation of the integral in eqn (16). In eqn (15), we set the value of Rmax to be equal to 100a0. There are 212 points in the integration grid. To avoid the diffraction between the outgoing waves and the incoming ones due to bouncing from the boundary at Rmax, a negative imaginary potential is placed at 90a0. This potential smoothly absorbs the wavepacket near the boundary.55 A normalized Gaussian-shaped wavepacket Φ is initially centered at 6.15a0 and possesses the half-width equal to 0.95a0.

3 Results and discussion

Our results of the calculated adiabatic potential energy curves of 1–81Σ+ and 11Π states are presented in Fig. 1. Several characteristic avoided crossings are visible, particularly the double one at 5 and 20a0 between the curves of the 31Σ+ and 41Σ+ states. Although not very pronounced, there are avoided crossings between 11Σ+ and 21Σ+ at 7.5a0 and 21Σ+ and 31Σ+ at 10a0. Additionally, in Fig. 1, the ionic character of the molecular bond is clearly visible for larger R.
image file: c7cp02097j-f1.tif
Fig. 1 Adiabatic potential energy curves of LiH: 1–81Σ+ states (solid lines), 11Π state (dashed line).

To benchmark our electronic structure calculations, bond lengths Re, dissociation energies De, vibrational constants ωe and electronic term energies Te are compared with other theoretical and experimental results in Table 2. For the ground state our position of Re agrees exactly with the theoretical value of Gadea and Leininger2 and reasonably with the experimental values of Stwalley et al.1 and Dulick et al.56 We also find good agreement within 40 cm−1 between the well depths De of our results and the experimental data of Stwalley et al. In the case of 11Π, our results of Re and De agree within 0.01a0 and 2 cm−1 with the experimental data of Velasco, respectively. All the theoretical results indicate the existence of a double well for the 31Σ+ state, but this is not confirmed by the only available experiment by Huang et al.57 A key observation is that for the state of interest, namely 11Π, the excellent consistency between our results and the experimental data is better than for any previous theoretical results.

Table 2 Spectroscopic parameters Re [a0], De, ωe, and Te [cm−1] for the ground and low-excited states of the LiH molecule
State Dissociation limit Author R e D e ω e T e
a MSDFT. b MS-CASPT2.
11Σ+ Li(2s) + H(1s) Present (theory) 3.003 20[thin space (1/6-em)]327 1391
Dulick 1998 (exp.)56 3.014 20[thin space (1/6-em)]286 1405
Stwalley 1993 (exp.)1 3.015 20[thin space (1/6-em)]288 1407
Grofe 2017 (theory)26 3.024a 17[thin space (1/6-em)]930a
3.001b 19[thin space (1/6-em)]753b
Bande 2010 (theory)25 3.013 20[thin space (1/6-em)]333
Aymar 2009 (theory)23 3.002 20[thin space (1/6-em)]167 1398
Gadea 2006 (theory)2 3.003 20[thin space (1/6-em)]349
Dolg 1996 (theory)14 3.000 20[thin space (1/6-em)]123 1391
Boutalib 1992 (theory)6 3.007 20[thin space (1/6-em)]174
21Σ+ Li(2p) + H(1s) Present (theory) 4.866 8687 260 26[thin space (1/6-em)]544
Stwalley 1993 (exp.)1 4.906 8679
Grofe 2017 (theory)26 4.724a 7662a 23[thin space (1/6-em)]551a
4.250b 8469b 26[thin space (1/6-em)]132b
Bande 2010 (theory)25 5.173 8679 26[thin space (1/6-em)]584
Aymar 2009 (theory)23 4.820 8698 241
Gadea 2006 (theory)2 4.862 8687
Boutalib 1992 (theory)6 4.847 8690 26[thin space (1/6-em)]390
Vidal 1982 (theory)13 4.910 8686 244
11Π Present (theory) 4.50 286 226 34[thin space (1/6-em)]945
Velasco 1957 (exp.)37 4.49 284 216
Aymar 2009 (theory)23 4.52 251 243
Vidal 1982 (theory)13 4.50 289
31Σ+ Li(3s) + H(1s) Present (theory) 3.821 1270 540 46[thin space (1/6-em)]259
10.172 8438 293 39[thin space (1/6-em)]092
Huang 2000 (exp.)57
10.140 8469
Grofe 2017 (theory)26 4.016a −1371a 47[thin space (1/6-em)]425a
4.001b 1129b 45[thin space (1/6-em)]570b
9.449a 7662a 38[thin space (1/6-em)]553a
9.997b 8711b 42[thin space (1/6-em)]021b
Aymar 2009 (theory)23 3.830 1267 390
10.150 8361 390
Gadea 2006 (theory)2 3.821
10.181 8453
Boutalib 1992 (theory)6 3.825 1277 46[thin space (1/6-em)]109
10.206 8444 38[thin space (1/6-em)]942


Fig. 2 displays spacings between successive rovibrational levels of the 11Π state. Our first set of values was obtained by solving58eqn (8). The second set comes from the appropriate differences between the positions of peaks in the absorption spectrum obtained from eqn (16) and presented in Fig. 3. These two sets agree very well with each other. Moreover, there is also very good agreement with the experimental values of Velasco.37


image file: c7cp02097j-f2.tif
Fig. 2 Differences ΔE(v,J′,J) = E(v,J′) − E(v,J) between rovibrational levels with the same vibrational quantum number v of the 11Π state. Three series of differences are drawn for v = 0, 1 and 2. Each difference is specified by (J,J′). The black lines were derived from the calculated rovibrational levels. The red lines were derived from the experimental data of Velasco.37 The green lines represent our results obtained from the absorption spectrum shown in Fig. 3.

image file: c7cp02097j-f3.tif
Fig. 3 The discrete spectrum calculated from eqn (16).

The peaks in the spectrum (Fig. 3) were obtained by solving the time-dependent Schrödinger equation55 (eqn (12)) in combination with eqn (16). Here, we are not interested in the intensity of the peaks and the precise shape of the initial wavepacket is unimportant. The set of effective potentials UJ (eqn (7)) spans J from 1 to 10. The broadened peak labeled by v = 0 and J = 9 is the last in the series since J = 9 is a critical value Jc, discussed in Section 2.2. Its half-width (FWHM) is equal to 2.7 cm−1. The last very broad peak with J = 10 illustrates the situation where the depth of the effective potential is too shallow to allow for existence of any bound vibrational level. The last and already broadened peak observed by Velasco was assigned as v = 0 and J = 8. In his analysis, he correctly foresaw the existence of an unobserved peak labeled by v = 0 and J = 9 before the molecule breaks off due to high rotations. However, his prediction of the existence of two other missing peaks in the spectrum, namely with v = 1, J = 6 and v = 2, J = 3, is not confirmed by our results. The broadening of the peak with v = 0 and J = 9 shown by our calculation is due to quantum tunneling through the centrifugal barrier.

The last figure (Fig. 4) shows the results for the time-dependent population of the 11Π state for the same initial condition. For J = 10, no bound states are supported by the effective potential and the drop in population around 2.5 ps shows the time it takes for the continuum wavepacket to reach R = Rmax.


image file: c7cp02097j-f4.tif
Fig. 4 Time-dependent population of the wavepacket placed on the effective potential UJ(R) (J = 1,…,10) for the electronic energy of the 11Π state. All the lines refer to the same initial conditions at t = 0 of the wavepacket.

In all the cases, the population is close to the one within the first approximately 2.5 ps, since any continuum part of the wavepacket needs this time to reach Rmax. Furthermore for low values of J, the population is close to the one within the time window of 15 ps, meaning that essentially all parts of the wavepacket can be represented by bound states. For J = 9, the wavepacket consists of a continuum as well as a (quasi-) bound part. The quasibound part decays through tunneling, giving rise to the slow exponential decay with a decay constant of 2.4 ps. On the basis of the time-energy uncertainty principle, we can estimate that this lifetime should give rise to a line width of approximately 2 cm−1. This is in good agreement with the spectrum in Fig. 3.

4 Conclusions

To describe the rotational predissociation process of the LiH molecule, we started by calculating the low-lying adiabatic potential energy curves, with particular emphasis on the 11Π state. Our spectroscopic parameters are in very good agreement with the experimental values. Having the potential curve of 11Π state, we calculated the rovibrational levels. The differences between these successive levels were compared with those derived from the experimental data of Velasco. The agreement again was very good, which means that the shape of the first excited electronic state 11Π is reliable. On the other hand, since our difference (Te) between the potential wells of 11Π and of the ground state 11Σ+ is around 50 cm−1 larger than the experimental value of Stwalley et al., the direct comparison with the spectrum of Velasco shows a small systematic shift.

To gain insight into the complementary time-dependent approach, we solved the time-dependent nuclear Schrödinger equation. The solution shows the evolving wavepacket originally placed on the effective potential curve. The spectrum was calculated as a Fourier transform of the autocorrelation function. The differences between the successive peaks in the spectrum were compared with those of Velasco and ours obtained in the time-independent approach. All three sets of values are in very good agreement. Our results for the time-dependent population of the 11Π state explain in detail the rotational predissociation mechanism of the LiH molecule. A challenge for experimentalists would be to detect in real time (via pump–probe spectroscopy) the predissociation due to quantum tunneling through the centrifugal barrier.

Acknowledgements

This work was partially supported by the COST action XLIC (CM1204) of the European Community. The calculations were carried out using the resources of the Academic Computer Centre in Gdańsk.

References

  1. W. C. Stwalley and W. T. Zemke, J. Phys. Chem. Ref. Data, 1993, 22, 87 CrossRef CAS .
  2. F. X. Gadea and T. Leininger, Theor. Chem. Acc., 2006, 116, 566 CrossRef CAS .
  3. F. H. Crawford and T. Jorgensen Jr, Phys. Rev., 1935, 47, 932 CrossRef CAS .
  4. F. H. Crawford and T. Jorgensen Jr, Phys. Rev., 1936, 49, 745 CrossRef CAS .
  5. N. L. Singh and D. C. Jain, Proc. Phys. Soc., London, 1962, 79, 753 CrossRef CAS .
  6. A. Boutalib and F. X. Gadea, J. Chem. Phys., 1992, 97, 1144 CrossRef CAS .
  7. M. E. Casida, F. Gutierrez, J. Guan, F. X. Gadea, D. Salahub and J. P. Daudey, J. Chem. Phys., 2000, 113, 7062 CrossRef CAS .
  8. H. Beriche and F. X. Gadea, Eur. Phys. J. D, 2016, 70, 2 CrossRef .
  9. F. X. Gadea and A. Boutalib, J. Phys. B: At., Mol. Opt. Phys., 1993, 26, 61 CrossRef CAS .
  10. F. Gemperle and F. X. Gadea, Europhys. Lett., 1999, 48, 513 CrossRef CAS .
  11. A. S. Dickinson and F. X. Gadea, Mon. Not. R. Astron. Soc., 2000, 318, 1227 CrossRef CAS .
  12. B. O. Roos and A. J. Sadlej, J. Chem. Phys., 1982, 76, 5444 CrossRef CAS .
  13. C. R. Vidal and W. C. Stwalley, J. Chem. Phys., 1982, 77, 883 CrossRef CAS .
  14. M. Dolg, Theor. Chem. Acc., 1996, 93, 141 CAS .
  15. F. A. Gianturco and P. Gori Giorgi, Phys. Rev. A: At., Mol., Opt. Phys., 1996, 54, 4073 CrossRef CAS .
  16. F. A. Gianturco, P. Gori Giorgi, H. Berriche and F. X. Gadea, Astron. Astrophys., Suppl. Ser., 1996, 117, 377 CAS .
  17. P. C. Stancil and A. Dalgarno, Astrophys. J., 1997, 479, 543 CrossRef CAS .
  18. A. K. Sharma and S. Chandra, J. Phys. B: At., Mol. Opt. Phys., 2000, 33, 2623 CrossRef CAS .
  19. E. Bodo, F. A. Gianturco and R. Martinazzo, Phys. Rep., 2003, 384, 85 CrossRef CAS .
  20. S. Bubin and L. Adamowicz, J. Chem. Phys., 2004, 121, 6249 CrossRef CAS PubMed .
  21. R. Fondermann, M. Hanrath and M. Dolg, Theor. Chem. Acc., 2007, 118, 777 CrossRef CAS .
  22. J. R. Trail and R. J. Needs, J. Chem. Phys., 2008, 128, 204103 CrossRef CAS PubMed .
  23. M. Aymar, J. Deiglmayr and O. Dulieu, Can. J. Phys., 2009, 87, 543 CrossRef CAS .
  24. I. L. Cooper and A. S. Dickinson, J. Chem. Phys., 2009, 131, 204303 CrossRef PubMed .
  25. A. Bande, H. Nakashima and H. Nakatsuji, Chem. Phys. Lett., 2010, 496, 347 CrossRef CAS .
  26. A. Grofe, Z. Qu, D. G. Truhlar, H. Li and J. Gao, J. Chem. Theory Comput., 2017, 13, 1176 CrossRef CAS PubMed .
  27. R. Côté, E. Juarros and K. Kirby, Phys. Rev. A: At., Mol., Opt. Phys., 2010, 81, 060704 CrossRef .
  28. M. Cafiero and L. Adamowicz, Phys. Rev. Lett., 2002, 88, 033002 CrossRef PubMed .
  29. F. M. Fernandez, J. Chem. Phys., 2009, 130, 166101 CrossRef PubMed .
  30. P. Decleva and A. Lisini, J. Phys. B: At. Mol. Phys., 1986, 19, 981 CrossRef CAS .
  31. S. Magnier, J. Phys. Chem., 2004, 108, 1052 CrossRef CAS .
  32. M. Cheng, J. M. Brown, P. Rosmus, R. Linguerri, N. Komiha and E. G. Myers, Phys. Rev. A: At., Mol., Opt. Phys., 2007, 75, 012502 CrossRef .
  33. W.-C. Tung, M. Pavanello and L. Adamowicz, J. Chem. Phys., 2011, 134, 064117 CrossRef PubMed .
  34. F. Holka, P. G. Szalay, J. Fremont, M. Rey, K. A. Peterson and V. G. Tyuterev, J. Chem. Phys., 2011, 134, 094306 CrossRef PubMed .
  35. R. S. Mulliken, Phys. Rev., 1936, 50, 1028 CrossRef CAS .
  36. M. Nest, F. Remacle and R. D. Levine, New J. Phys., 2008, 10, 025019 CrossRef .
  37. R. Velasco, Can. J. Phys., 1957, 35, 1204 CrossRef CAS .
  38. A. H. Zewail, Femtochemistry, Ultrafast dynamics of the chemical bond, World Scientific Publishing Co. Pte. Ltd, Singapore, 1994, vol. I and II Search PubMed .
  39. M. Grønager and N. E. Henriksen, J. Chem. Phys., 1996, 104, 3234 CrossRef .
  40. M. Grønager and N. E. Henriksen, J. Chem. Phys., 1998, 109, 4335 CrossRef .
  41. H. Dietz and V. Engel, J. Phys. Chem. A, 1998, 102, 7406 CrossRef CAS .
  42. K. B. Møller, N. E. Henriksen and A. H. Zewail, J. Chem. Phys., 2000, 113, 10477 CrossRef .
  43. P. Fuentealba, H. Preuss, H. Stoll and L. Von Szentpály, Chem. Phys. Lett., 1982, 89, 418 CrossRef CAS .
  44. E. Czuchaj, F. Rebentrost, H. Stoll and H. Preuss, Theor. Chem. Acc., 1998, 100, 117 CrossRef CAS .
  45. E. Czuchaj, M. Krośnicki and H. Stoll, Chem. Phys., 2003, 292, 101 CrossRef CAS .
  46. M. Dolg, Effective Core Potentials, in Modern Methods and Algorithms of Quantum Chemistry, ed. J. Grotendorst, NIC Series, 2000, vol. 3, p. 507 Search PubMed .
  47. Unofficial set from D. Feller, see http://www.molpro.net.
  48. T. H. Dunning Jr, J. Chem. Phys., 1989, 90, 1007 CrossRef .
  49. H.-J. Werner, P. J. Knowles, G. Knizia, F. R. Manby, M. Schütz, P. Celani, W. Györffy, D. Kats, T. Korona, R. Lindh, A. Mitrushenkov, G. Rauhut, K. R. Shamasundar, T. B. Adler, R. D. Amos, A. Bernhardsson, A. Berning, D. L. Cooper, M. J. O. Deegan, A. J. Dobbyn, F. Eckert, E. Goll, C. Hampel, A. Hesselmann, G. Hetzer, T. Hrenar, G. Jansen, C. Köppl, Y. Liu, A. W. Lloyd, R. A. Mata, A. J. May, S. J. McNicholas, W. Meyer, M. E. Mura, A. Nicklaß, D. P. O'Neill, P. Palmieri, D. Peng, K. Pflüger, R. Pitzer, M. Reiher, T. Shiozaki, H. Stoll, A. J. Stone, R. Tarroni, T. Thorsteinsson and M. Wang, MOLPRO, version 2012.1, is a package of ab initio programs, 2012, see http://www.molpro.net Search PubMed .
  50. C. E. Moore, Atomic energy levels as derived from the analysis of optical spectra-Hydrogen through Vanadium, Circular of the National Bureau of Standards, 467, U. S. Government Printing Office, Washington, 1949, vol. I Search PubMed .
  51. L. D. Landau and E. Lifshitz, Quantum Mechanics, Pergamon, New York, 1965 Search PubMed .
  52. K. R. Way and W. C. Stwalley, J. Chem. Phys., 1973, 59, 5298 CrossRef CAS .
  53. D. J. Tannor, Introduction to quantum mechanics: a time-dependent perspective, University Science Books, Sausalito, 2007 Search PubMed .
  54. P. Bilingsley, Probability and measure, John Wiley & Sons, New York, Chichester, Brisbane, Toronto, Singapore, 1995 Search PubMed .
  55. B. Schmidt and U. Lorenz, WavePacket: A Matlab package for numerical quantum dynamics. I: Closed quantum systems and discrete variable representations, Comput. Phys. Commun., 2017, 213, 223 CrossRef CAS ; B. Schmidt and C. Hartmann, WavePacket: A Matlab package for numerical quantum dynamics. II: Open quantum systems and optimal control, 2017, manuscript in preparation.
  56. M. Dulick, K.-Q. Zhang, B. Guo and P. F. Bernath, J. Mol. Spectrosc., 1998, 188, 14 CrossRef CAS PubMed .
  57. Y. L. Huang, W. T. Luh, G. H. Jeung and F. X. Gadea, J. Chem. Phys., 2000, 113, 683 CrossRef CAS .
  58. R. J. Le Roy, LEVEL: a computer program for solving the radial Schrdinger equation for bound and quasibound levels, J. Quant. Spectrosc. Radiat. Transfer, 2017, 186, 167 CrossRef CAS .

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