P.
Jasik
^{a},
J. E.
Sienkiewicz
*^{a},
J.
Domsta
^{b} and
N. E.
Henriksen
^{c}
^{a}Faculty of Applied Physics and Mathematics, Gdańsk University of Technology, Narutowicza 11/12, 80-233 Gdańsk, Poland. E-mail: jes@mif.pg.gda.pl
^{b}Institute of Applied Informatics, State University of Applied Sciences in Elbląg, Wojska Polskiego 1, 82-300 Elbląg, Poland
^{c}Department of Chemistry, Technical University of Denmark, Building 207, DK-2800 Kgs. Lyngby, Denmark
First published on 6th June 2017
The adiabatic potential energy curves of the ^{1}Σ^{+} and ^{1}Π 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 1^{1}Π 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 1^{1}Π state is explained for the first time in detail.
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 Leininger^{2} in 2006.
In 1935–1936, Crawford and Jorgensen^{3,4} analysed the LiH band spectra. Since then, many notable studies have been undertaken. Among them, in 1962 Singh and Jain^{5} 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 moments^{28,29} 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, Mulliken^{35} 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 Velasco^{37} 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 Velasco^{37} 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. Zewail^{38} 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 1^{1}Π state and compare them with the measurements of Velasco.^{37} Conclusions are given in the last section.
H^{el}Ψ^{el}_{i}(;R) = E^{el}_{i}(R)Ψ^{el}_{i}(;R). | (1) |
H^{el} = T^{el} + V, | (2) |
(3) |
(4) |
(5) |
(6) |
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. Feller^{47} 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.
Atomic asymptotes | Experiment Moore^{50} | Theory Boutalib^{6} | Theory Gadea^{2} | Theory present |
---|---|---|---|---|
Li(2p) + H(1s) | 14904 | 14905 | 14898 | 14904 |
Li(3s) + H(1s) | 27206 | 27210 | 27202 | 27202 |
Li(3p) + H(1s) | 30925 | 30926 | 30920 | 30921 |
Li(3d) + H(1s) | 31283 | 31289 | 31279 | 31276 |
Li(4s) + H(1s) | 35012 | 35018 | 35007 | 35016 |
Li(4p) + H(1s) | 36470 | 36475 | 36465 | 36464 |
Li(4d) + H(1s) | 36623 | 37590 | 36626 | 36617 |
(7) |
Rovibrational energies E(v,J) depend on E^{el}(R) as well as on vibrational v and rotational J quantum numbers. They are the solutions of the time-independent nuclear Schrödinger equation
H^{nuc}_{J}Ψ^{nuc}_{v,J}(R) = E(v,J)Ψ^{nuc}_{v,J}(R), | (8) |
(9) |
The effective potential U_{J}(R) forms a barrier for J > 0 with a maximum U_{J}(R_{J}), at the internuclear distance R_{J}, which can easily be estimated. Any rovibrational state with a positive energy E(v,J) lower than U_{J}(R_{J}) 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 U_{J}(R_{J}), any bound state is not possible. Following Way and Stwalley,^{52} we introduce a critical value of the rotational quantum number J_{c}, which obeys the two following inequalities:
E(v,J_{c}) < U_{Jc}(R_{Jc}) | (10) |
E(v,J_{c} + 1) > U_{Jc+1}(R_{Jc+1}). | (11) |
In other words, for a given v, the state with the energy E(v,J_{c}) is the last of the quasibound states series supported by the barrier, and the state with the energy E(v,J_{c} + 1) already belongs to the continuum. By solving eqn (8) we obtain E(v,J_{c}) and estimate E(v,J_{c} + 1). The differences E(v,J_{c}) − E(0,0) and E(v,J_{c} + 1) − E(0,0) may refer to the last observed and the first unobserved rotational predissociation experimental results, respectively.
We start our consideration from the time-dependent Schrödinger equation written in the following form:
(12) |
By definition, the wavepacket is a coherent superposition of stationary states (e.g. Tannor^{53}), which may be represented in the following form consisting of two contributions from the discrete and continuous parts of the spectrum:
(13) |
(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 R_{max} for the particular state labeled by J, in accordance with the effective potential energy U_{J} from eqn (7), is calculated as
(15) |
We determine the discrete spectrum by the inverse Fourier transform of S(t)^{54} as follows:
(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 R_{max} to be equal to 100a_{0}. There are 2^{12} points in the integration grid. To avoid the diffraction between the outgoing waves and the incoming ones due to bouncing from the boundary at R_{max}, a negative imaginary potential is placed at 90a_{0}. This potential smoothly absorbs the wavepacket near the boundary.^{55} A normalized Gaussian-shaped wavepacket Φ is initially centered at 6.15a_{0} and possesses the half-width equal to 0.95a_{0}.
Fig. 1 Adiabatic potential energy curves of LiH: 1–8^{1}Σ^{+} states (solid lines), 1^{1}Π state (dashed line). |
To benchmark our electronic structure calculations, bond lengths R_{e}, dissociation energies D_{e}, vibrational constants ω_{e} and electronic term energies T_{e} are compared with other theoretical and experimental results in Table 2. For the ground state our position of R_{e} agrees exactly with the theoretical value of Gadea and Leininger^{2} 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 D_{e} of our results and the experimental data of Stwalley et al. In the case of 1^{1}Π, our results of R_{e} and D_{e} agree within 0.01a_{0} and 2 cm^{−1} with the experimental data of Velasco, respectively. All the theoretical results indicate the existence of a double well for the 3^{1}Σ^{+} 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 1^{1}Π, the excellent consistency between our results and the experimental data is better than for any previous theoretical results.
State | Dissociation limit | Author | R _{e} | D _{e} | ω _{e} | T _{e} |
---|---|---|---|---|---|---|
a MSDFT. b MS-CASPT2. | ||||||
1^{1}Σ^{+} | Li(2s) + H(1s) | Present (theory) | 3.003 | 20327 | 1391 | |
Dulick 1998 (exp.)^{56} | 3.014 | 20286 | 1405 | |||
Stwalley 1993 (exp.)^{1} | 3.015 | 20288 | 1407 | |||
Grofe 2017 (theory)^{26} | 3.024^{a} | 17930^{a} | ||||
3.001^{b} | 19753^{b} | |||||
Bande 2010 (theory)^{25} | 3.013 | 20333 | ||||
Aymar 2009 (theory)^{23} | 3.002 | 20167 | 1398 | |||
Gadea 2006 (theory)^{2} | 3.003 | 20349 | ||||
Dolg 1996 (theory)^{14} | 3.000 | 20123 | 1391 | |||
Boutalib 1992 (theory)^{6} | 3.007 | 20174 | ||||
2^{1}Σ^{+} | Li(2p) + H(1s) | Present (theory) | 4.866 | 8687 | 260 | 26544 |
Stwalley 1993 (exp.)^{1} | 4.906 | 8679 | ||||
Grofe 2017 (theory)^{26} | 4.724^{a} | 7662^{a} | 23551^{a} | |||
4.250^{b} | 8469^{b} | 26132^{b} | ||||
Bande 2010 (theory)^{25} | 5.173 | 8679 | 26584 | |||
Aymar 2009 (theory)^{23} | 4.820 | 8698 | 241 | |||
Gadea 2006 (theory)^{2} | 4.862 | 8687 | ||||
Boutalib 1992 (theory)^{6} | 4.847 | 8690 | 26390 | |||
Vidal 1982 (theory)^{13} | 4.910 | 8686 | 244 | |||
1^{1}Π | Present (theory) | 4.50 | 286 | 226 | 34945 | |
Velasco 1957 (exp.)^{37} | 4.49 | 284 | 216 | |||
Aymar 2009 (theory)^{23} | 4.52 | 251 | 243 | |||
Vidal 1982 (theory)^{13} | 4.50 | 289 | ||||
3^{1}Σ^{+} | Li(3s) + H(1s) | Present (theory) | 3.821 | 1270 | 540 | 46259 |
10.172 | 8438 | 293 | 39092 | |||
Huang 2000 (exp.)^{57} | — | — | ||||
10.140 | 8469 | |||||
Grofe 2017 (theory)^{26} | 4.016^{a} | −1371^{a} | 47425^{a} | |||
4.001^{b} | 1129^{b} | 45570^{b} | ||||
9.449^{a} | 7662^{a} | 38553^{a} | ||||
9.997^{b} | 8711^{b} | 42021^{b} | ||||
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 | 46109 | |||
10.206 | 8444 | 38942 |
Fig. 2 displays spacings between successive rovibrational levels of the 1^{1}Π state. Our first set of values was obtained by solving^{58}eqn (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}
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 1^{1}Π 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. |
Fig. 3 The discrete spectrum calculated from eqn (16). |
The peaks in the spectrum (Fig. 3) were obtained by solving the time-dependent Schrödinger equation^{55} (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 U_{J} (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 J_{c}, 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 1^{1}Π 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 = R_{max}.
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 R_{max}. 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.
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 1^{1}Π 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.
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