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Stereodynamics of the reaction H + LiH (v = 0, j = 0) → H2 + Li and its isotopic variants
Computational and Theoretical Chemistry 965 (2011) 107–113
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Stereodynamics of the reaction H + LiH (v = 0, j = 0) ? H2 + Li and its isotopic variants Yufang Liu ⇑, Xiaohu He, Deheng Shi, Jinfeng Sun Department of Physics, Henan Normal University, Xinxiang 453007, China
a r t i c l e
i n f o
Article history: Received 14 November 2010 Received in revised form 22 January 2011 Accepted 22 January 2011 Available online 28 January 2011 Keywords: Quasi-classical trajectory H + LiH reaction Vector correlation Reactive probability
a b s t r a c t Quasi-classical trajectory (QCT) method is used to calculate the stereodynamics of the reactions H + LiH (v = 0, j = 0) ? H2 + Li and its isotopic variants based on the ground electronic state potential energy surface (PES) reported by Prudente et al. [14]. The reactive probabilities of the title reactions are computed. We also observed the changes of vector correlations and four generalized polarization-dependent differential cross-sections (PDDCSs) at different collision energies, and we compared the stereodynamics among different isotopic variants of the title reactions. The product polarization distribution of the title reactions exhibits distinct difference which may arise from different mass combinations or kinetic energies. Crown Copyright Ó 2011 Published by Elsevier B.V. All rights reserved.
1. Introduction The LiH molecules and its ionic variants received extensive attention because this system is of potential importance in the primordial cosmic chemistry [1–4]. In the standard Big Bang hypothesis model, the LiH molecules and its ionic variants are parts of the few molecular species which take part in the entire gas-phase chemical reaction network [3]. The relevance of LiH molecules may be limited by the small abundance of Li molecular species which is thought to exist in the recombination era, but researches have already shed light on a large portion of the gas-phase chemistry of LiH and of its positive ion [3,4]. Theoretical studies on the spectroscopic, polarizability and mono-bichromatic electron dynamics of the LiH molecules have been reported [5–8]. Experimental studies about the formation reaction of LiH of various electronic states also have been reported [9,10]. Bililign et al. [9] conducted laser induced pump–probe far-wing scattering experiments to study the photochemical reaction Li + H2 ? LiH + H. Chen et al. [10] observed the nascent rotational population distribution of LiH in a Li with H2 reaction by using a pump–probe technique. However, the experimentally imperative data of the depletion reaction of LiH is indeed still insufficient. The LiH2 system is of interest for some physical reasons. In the LiH2 system, there are only five electrons. Two are core electrons, the other three are valence electrons, which are involved in the ground state. The LiH2 chemical system can be deemed to the next ⇑ Corresponding author. Tel.: +86 373 3329350; fax: +86 373 3329297. E-mail address: [email protected] (Y. Liu).
electronically simple neutral triatomic system (after H3) exhibiting bound diatomic asymptotes [11]. In addition, this system is one of the simplest alkali metal-dihydrogen partners. The alkali metal atoms also present a high density of nearby excited electronic states into which they can be easily promoted. During the past decade, the LiH2 chemical system has been the subject of a large number of studies on its potential energy surfaces (PESs) [11–20], on the sub-reactive and reactive dynamics and on the formation or depletion reactions of LiH molecules [11,14,17,19–36]. Clarke et al. [12] firstly obtained the ground state potential energy surface (PES) of LiH2 system for the collinear arrangement by using the extensive valence bond (VB) non-orthogonal configuration interaction (CI) method. Dunne, Murrell and Jemmer (DMJ) [13] reported the first three-dimensional LiH2 PES by fitting the CI ab initio points to an analytical function within the many-body expansion (MBE) framework. Prodente et al. [14] improved the ground electronic potential of Dunne et al. [13] by performing accurate ab initio full CI calculations and fitting them to a slightly modified function. In this paper, quasi-classical trajectory (QCT) calculations are performed for the reaction H + LiH ? H2 + Li and its isotopic variants based on the PES reported by Prodente et al. [14]. The potential energy result reported by Wernli et al. [11] refered that there is no energy barrier along the reaction path, which is opposite to that of Prodente et al. [14]. The result of Wernli et al. [11] is applied to calculate the rate coefficient of the formation and depletion reaction of the LiH molecule [20]. The reaction H + LiH ? Li + HH in our study is barrier-less and highly exoergic. Although quantum [12] and classical methods [12,13,16] have been performed in the aforementioned papers,
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only the analytical potential energy surface reported by Dunne et al. [13] has been employed in some theoretical studies by using time-dependent wave-packet dynamics methodologies [24– 27,33,34]. Clarke et al. [12] have performed classical and timedependent wave-packet calculations only for the linear geometry by using the spin-coupled valence bond (SCVB) non-orthogonal configuration interaction (CI) method. Bodo et al. [23] calculated the interaction between LiH and H. In view of examining the strength of the coupling between the impinging atom and the rovibrational LiH states in low energy collision regimes, a Coupled Cluster (CC) approach was used. Defazio et al. [27] reported rate constant and mechanism of the reactions H + LiH ? H2 + Li and H + LiH0 ? H0 + LiH by performing a quantum–mechanical investigation. Padmandan and Mahapatra [24–26,33,34] formed a series of time-dependent wave-packet dynamical calculations for the H + LiH scattering based on the PES of Dunne et al. [13] The posiþ tive ionic variant of LiH2, LiH2 , is also of astrophysical interest and diffusely studied [17,21,22,30–32,36]. The aforementioned papers have dealt with the scalar properties such as rate constant, reactive probability and resonances in reactive scattering. Vector properties, such as velocities and angular momenta, possess magnitude dynamical information [37–40]. The works of Fano and Macek [41] and Herschbach et al. [42–44] directly led to comprehensive interests [45] in vector correlation or molecular polarized distribution in the reaction A + BC ? AB + C. As far as we know, there are very rare vector correlation (stereodynamics) studies for the depletion reaction of LiH. Here we focus on the dynamics of the reaction H + LiH ? H2 + Li and its isotopic variants:
H þ LiH ! Li þ HH;
ð1Þ
H þ LiD ! Li þ HD;
ð2Þ
D þ LiH ! Li þ DH;
ð3Þ
D þ LiD ! Li þ DD:
ð4Þ
Quasi-classical trajectory (QCT) method is applied to study the stereodynamics of the four reactions based on the PES reported by Prudente, Marques and Maniero (PMM) [14]. 2. Theory The theoretical details of QCT method are specified elsewhere [37–39,45–52]. The Hamilton equations were numerically integrated in three dimensions with six order symplectic integration [52,53]. The integration step size was chosen as 0.1 fs. The initial vibrational and rotational quantum numbers were selected as v = 0 and j = 0, respectively. For one collision energy, a batch of 50,000 trajectories was run, and the statistical uncertainty is no more than 2.0%. The collision energy was chosen as less than 2.6 eV. The maximum impact parameters bmax for different collision energies and different isotopic variants are listed in Table 1. The dynamics of the four reactions presented in Eqs. (1)–(4) were calculated. The reactive probability is defined as P r ¼ NNreact , where N react and tot N tot are the numerals of reactive and total trajectories, respectively. The center-of-mass (CM) frame was chosen in this calculation. k 0 and k are the reactant relative velocity and product relative velocity, respectively. k is parallel to the z-axis and y-axis is perpendic0 ular to the xz-plane containing k and k vectors. The final angular 0 momentum is j , with its polar and azimuthal angles being hr and 0 /r , respectively. The angle between k and k was defined as the scattering angle. 0 The function Pðhr Þ used to describe the k j correlation can be expanded in a series of Legendre polynomials as [37,38]:
Pðhr Þ ¼
1X ðmÞ ð2m þ 1Þa0 Pm ðcos hr Þ: 2 m
ð5Þ
Table 1 Maximum impact parameters. Collision energy (eV)
0.043 0.108 0.216 0.324 0.434 0.65 0.867 1.08 1.3 1.52 1.73 1.95 2.17 2.39 2.6
bmax (Å) H + LiH
H + LiD
D + LiH
D + LiD
4.58 4.42 4.01 3.67 3.40 3.38 3.43 3.48 3.46 3.40 3.42 3.38 3.36 3.31 3.29
5.40 4.60 3.98 3.61 3.25 3.21 3.23 3.22 3.15 3.12 3.13 3.07 2.95 2.92 2.87
3.71 3.97 3.75 3.52 3.36 3.40 3.43 3.47 3.43 3.43 3.43 3.38 3.44 3.36 3.42
4.63 4.28 3.92 3.57 3.35 3.20 3.22 3.20 3.24 3.25 3.22 3.12 3.12 3.12 3.1
The dihedral angle distribution function Pð/r Þ describing 0 0 k k j vector correlation can be expanded as a Fourier series [38]. The /r distribution can be written as:
! X X 1 Pð/r Þ ¼ 1þ an cosðn/r Þ þ bn sinðn/r Þ ; 2p ev en;nP2 odd;nP1
ð6Þ
where
an ¼ 2hcosðn/r Þi;
ð7Þ
bn ¼ 2hsinðn/r Þi:
ð8Þ
A set of generalized polarization-dependent differential crosssections (PDDCSs) is used to describe the full three-dimensional 0 0 angular distribution associated with k k j correlation in the CM frame. The fully correlated CM angular distribution can be presented as a joint density function which is written as the sum [46]:
Pðcos h; cos hr ; /r Þ ¼
1 X k 1 X 2p drkq ð2k þ 1Þ C ðh ; / Þ ; 4p k¼0 q¼k r dx kq r r
ð9Þ
where ð1=rÞðdrlq =dht Þ are the generalized PDDCSs. Many photoinitiated bimolecular reactions are only sensitive to multipole moments k = 0 and k = 2. In this article, ð2p=rÞðdr00 =dxt Þ, ð2p=rÞðdr20 =dxt Þ, ð2p=rÞðdr22þ =dxt Þ and ð2p=rÞðdr21 =dxt Þ are calculated, and we use P00, P20, P22+ and P21 for short. 3. Results and discussion The reactive probabilities for the title reactions are depicted in Fig. 1a. The four probabilities for the title reactions all peak at the collision energy of 0.22 eV and show similar trend on the whole. From the comparison in Fig. 1a, one can find that the difference among them is very obvious. In the whole investigated collision energy range, the probability values of the four reactions decrease in the order of reaction (3), (4), (1), and (2). A noticeable reality is that the impinging atom of reactions (3) and (4) is deuterium while that of reactions (1) and (2) is hydrogen, so it can be concluded from Fig. 1a that the impinging atom mass plays a influential role in a barrier-less reaction. Fig. 1b shows the reactive probability comparison of our QCT result and other quantum results only for the reaction H + LiH ? Li + HH. There are considerable discrepancies among the three results. Generally, our result is comparatively close to the result based on the PES of Dunne et al. [13]. Both the two quantum results are reported in Ref. [14]. The reaction H + LiH ? Li + HH is a barrier-less reaction. The reactive probability of it peaks at a low
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Fig. 1. (a) Energy-dependent reactive probabilities of the four reactions. (b) Reactive probability comparison of our QCT result and other quantum results. Both the two quantum results are from Ref. [14]. The solid square symbol is our QCT result; the cycle symbol is the result based on the PES of Prudente et al. [14] and the triangle symbol is the result based on the PES of Dunne et al. [13].
collision energy (about 0.2 eV) and decreases with increasing the value of collision energy in the energy range 0.2–2.6 eV, which is similar with another barrier-less reaction F + HBr ? HF + Br [54,55]. Since that the effect of zero-point energy has been considered in our calculation, it can be concluded that our QCT result is more reliable. The result based on the PES of Prudente et al. [14] has very high numeral value, which is not consistent with the barrier-less property of this reaction. Figs. 2–5 depict the comparison of Pðhr Þ and Pð/r Þ for the title reactions of various collision energies. Pðhr Þ and Pð/r Þ are probability density functions describing the probability density distribution of reaction products. Fig. 2a shows the comparison of Pðhr Þ distributions of the four reactions at the collision energy 0 0.043 eV. Pðhr Þ reflects k j vector correlation, namely the rotational alignment of the product. The Pðhr Þ distributions for different reactive products are relatively broad and symmetric, with peaks at hr ¼ p=2, which also demonstrates that the product rotational angular momentum vector prefers to be perpendicular to the initial relative velocity direction. The Pðhr Þ peaks of reactions (1),
(2), and (4) are almost the same, while that the Pðhr Þ peak of reaction (3) is the highest one, which can prove the rotational angular momentum vector of the product from reaction (3) mostly prefer to be perpendicular to the k vector. Pð/r Þ of the four reactions for collision energy at 0.043 eV de0 0 picted in Fig. 2b reflects distributions of k k j correlation. They tend to be symmetric at /r ¼ p, reflecting the strong polarization of angular momentum. Fig. 2b shows that distributions for reactions (1), (2), and (4), Pð/r Þ peak at /r ¼ p=2 and /r ¼ 3p=2, and the peak of reaction (3) at /r ¼ 3p=2 is obviously higher than other three ones. It is obvious that the molecular products of reaction (3) are preferentially oriented along the negative direction of the CM y-axis. For reactions (1), (2), and (4), the degree of the products scattered with /r 6 p are almost the same with those which are scattered with /r P p at such a low collision energy. Figs. 3–5 depict the Pðhr Þ and Pð/r Þ distributions of the four reaction for the collision energies at 0.43 eV, 0.87 eV and 2.6 eV, respectively. In Fig. 3a the distributions of Pðhr Þ are similar to those in Fig. 2a. Fig. 3b shows that the Pð/r Þ distributions peak at /r ¼ p=2
Fig. 2. Pðhr Þ and Pð/r Þ distributions for the four reactions for the collision energy of 0.043 eV.
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Fig. 3. Pðhr Þ and Pð/r Þ distributions for the four reactions for the collision energy of 0.43 eV.
Fig. 4. Pðhr Þ and Pð/r Þ distributions for the four reactions for the collision energy of 0.87 eV.
Fig. 5. Pðhr Þ and Pð/r Þ distributions for the four reactions for the collision energy of 2.6 eV.
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and /r ¼ 3p=2, and for reactions (1), (2), and (4) the peaks at /r ¼ 3p=2 is higher than that at /r ¼ p=2, namely the products prefer to distribute along the negative direction of the CM y-axis. For reaction (3) the Pð/r Þ peak at /r ¼ p=2 is higher than that at /r ¼ 3p=2. In Fig. 4a the Pðhr Þ distributions are similar to those in Fig. 3a, and the peaks are higher than those in Fig. 3a. In Fig. 4b the four Pð/r Þ distributions peak at /r ¼ p=2 and /r ¼ 3p=2 and the peaks at /r ¼ p=2 are higher than those at /r ¼ 3p=2. The Pðhr Þ and Pð/r Þ distributions for collision energy of 2.6 eV depicted in Fig. 5 is similar to those in Fig. 3 and 4, but the peaks are higher. It can be concluded from Figs. 2–5 that the Pðhr Þ distribution of reaction (3) reaches a higher peak than that of the other three reactions at various collision energies and the product molecules are more preferentially along the positive direction of the CM y-axis with a increase of collision energy. It is easily to find that the differences of Pðhr Þ and Pð/r Þ among the four reactions are becoming smaller with the increasing collision energies. 0 0 The generalized PDDCSs describe k k j correlation and the scattering direction of the product molecules. The PDDCS P00, which is proportional to the differential cross-section (DCS), only describes the angular distribution of the product molecule. It is not associated with the product orientation and alignment. The trajectory of P20 shows the opposite trend to that of P00, which indicates the degree of alignment of the product rotational angular 0 0 momentum j perpendicular to k. The k k scattering plane is not determined by these limiting scattering angles, so the value of these PDDCSs with q–0 should be zero at the extremities of backward and forward scattering [39,46]. The representation of PDDCSs with q–0 for scattering away from the extreme forward and backward directions may be more attractive and can provide abundant dynamic information [39,46] on the /r dihedral angle distribution. Figs. 6–8 show the PDDCSs of the four reactions for different collision energies of 0.043 eV, 0.43 eV and 2.6 eV. In Figs. 6a and 6d, the P00 shows that the product molecules are scattered sideways for reactions (1) and (4). The P00 of reaction (2) which is depicted in Fig. 6b shows a product scattering nearly all the same in the whole scattering angle range, while Fig. 6c presents that the product molecules of reaction (3) are strongly scattered backward.
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The P20 of reactions (1), (2), and (4) are close to zero, i.e., the degree 0 of alignment of the product rotational angular momentum j perpendicular to k is completely weak. The P20 of reaction (3) in 0 Fig. 6c shows that j is strongly aligned perpendicular to k in the scattering angle range 90–180°. Since the PDDCS P20 is only related to the PES and mass factor, i.e., cos2 b ¼ ma mc =ðma þ mb Þðmb þ mc Þ for the reaction A þ BC ! AB þ C [40], we believe that the differences among the four P20 trajectories are mainly due to different mass factors, i.e., isotopic effects. The PDDCSs with q–0 of the reactions (1), (2), and (4) are almost zero in the whole hr range. Only the P22+ of reaction (3) has a downward direction peak at hr ¼ 150 which exhibits unusual and quite strong polarization of product molecules. Fig. 7 presents the PDDCSs of the title reactions for the collision energy of 0.43 eV. The four P00 indicate that the product molecules are strongly scattered forward (in the scattering range 0–45°) and backward (in the scattering range 135–180°). The product scattering intensity of the four reactions are almost the same with each 0 other. The four P20 show that j is weakly aligned perpendicular to k in the scattering angle ranges 0–45° and 135–180°. The PDDCSs with q–0 of the four reactions are all nearly to be zero. Fig. 8 shows the PDDCSs of the title reactions at the collision energy 2.6 eV. The product molecules of the four reactions are scattered forward in the scattering angle range 0°30°. There is just a little difference among the forward scattering degree of the four reactions. The figure shows that products of reaction (3) are scattered most strongly. The four P20 depicted in Fig. 8 indicate that 0 j is weakly aligned perpendicular to k in the scattering angle range 0–30°. The PDDCSs with q–0 in Fig. 8 show isotropic distributions of products at a high collision energy of 2.6 eV. Some previous works have mentioned that the product rotational angular momentum vector distribution is sensitive to the mass factor [39]. Figs. 2–5 have clearly show that there are obvious differences of Pðhr Þ and Pð/r Þ distributions for different reactions at low collision energy, but at middle and high collision energies the differences become smaller, which directly demonstrates that the product vector distribution is also sensitive to the collision energy. In summery both the mass factor and collision energy can affect
Fig. 6. The PDDCSs P00, P20, P22+ and P21 for the four reactions for collision energy of 0.043 eV.
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Fig. 7. The PDDCSs P00, P20, P22+ and P21 for the four reactions for collision energy of 0.43 eV.
Fig. 8. The PDDCSs P00, P20, P22+ and P21 for the four reactions for collision energy of 2.6 eV.
the vector polarization distribution of product molecules. At low collision energy, the effects of trajectory and different binding energy in the LiH or LiD molecule are noticeable, so the difference of vector correlation among the four reactions is obvious. But at middle and high collision energy, such effects are visibly diminished. Different to the results of vector correlations, we could converge the numerical values of the four reaction probabilities at the lowest collision energy of 0.043 eV. At such a low energy level, the isotopic effect does not increase the efficiency of the four reactions (the reactive probabilities are almost same), i.e., the kinematic effect is inconspicuous. With an increase in the collision energy, the difference of reactive probabilities becomes notable, namely the kine-
matic effect becomes conspicuous. When comparing reaction (1) and (2), one can easily find that the reactive probability of former is bigger that of latter at relative high collision energy (more than 0.22 eV). Similar result can be got from the comparison of reaction (3) and (4). It can be concluded that it is the deuterium atom substitution makes the reaction taking place more difficult. The mass combination type of the reactions we studied is LightLight-Heave (LLH). Han and co-workers have studied about the product polarization of the LLH mass combination reaction in detail, and found that the product molecules are scattered forwardly based on a so-called ‘attractive’ PES at middle and high collision energies [39]. According to the definition in Ref. [39], the PES
Y. Liu et al. / Computational and Theoretical Chemistry 965 (2011) 107–113
which is applied in this paper is ‘attractive’. Our calculation results in Figs. 6–8 show good agreement with the theory of Han et al. [39]. 4. Conclusion In this paper the stereodynamics calculations of the reactions H + LiH ? H2 + Li, H + LiD ? HD + Li, D + LiH ? DH + Li and D + LiD ? D2 + Li are performed by using QCT method based on the PES reported by Prudente, Marques and Maniero [14]. The reactive probabilities of the four reactions for different collision energies are calculated and compared with other two quantum results. The reactive probability values of the four reactions reach the peak at a low collision energy (i.e., approximately 0.22 eV). Pðhr Þ distributions, Pð/r Þ distributions and generalized PDDCSs of the four reactions for different collision energies are discussed in detail. The calculations indicate that the peak of Pðhr Þ of reaction (3) is the highest one, and that the difference of Pðhr Þ between reaction (3) and other three reactions is decreasing with an increase of the collision energy. The difference of Pð/r Þ distributions among the reactions is also decreasing with an increase of the collision energy. Pð/r Þ distributions also show that the product molecules are scattered with /r 6 p more easily than with /r P p in a higher collision energy for the four reactions. It can be concluded that, in the Pðhr Þ and Pð/r Þ distributions, isotopic effect is more difficult to emerge at a higher collision energy. The P00 of different reactions indicate that the degree of forward scattering increases with collision energy, while that the degree of backward scattering decreases. The P20 of different reactions indicate that the degree of 0 alignment of the product rotational angular momentum j is preferentially perpendicular to k in the scattering angle range 0–45° in a high collision energy and the isotopic effects is not obvious. The P22+ and P21 of the reactions for different collision energies show a comprehensive isotropic distribution of products. Acknowledgements This work was supported by the National Natural Science Foundation of China (Grant No. 60977063), the Innovation Scientists and Technicians Troop Construction Projects of Henan Province, China (Grant No. 084100510011) and the Innovation Talents of Institution of Higher Education of Henan Province, China (Grant No. 2006KYCX002). Thanks for Professor K.L. Han for his supply for The QCT Code. References [1] [2] [3] [4] [5] [6]
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Computational and Theoretical Chemistry journal homepage: www.elsevier.com/locate/comptc
Stereodynamics of the reaction H + LiH (v = 0, j = 0) ? H2 + Li and its isotopic variants Yufang Liu ⇑, Xiaohu He, Deheng Shi, Jinfeng Sun Department of Physics, Henan Normal University, Xinxiang 453007, China
a r t i c l e
i n f o
Article history: Received 14 November 2010 Received in revised form 22 January 2011 Accepted 22 January 2011 Available online 28 January 2011 Keywords: Quasi-classical trajectory H + LiH reaction Vector correlation Reactive probability
a b s t r a c t Quasi-classical trajectory (QCT) method is used to calculate the stereodynamics of the reactions H + LiH (v = 0, j = 0) ? H2 + Li and its isotopic variants based on the ground electronic state potential energy surface (PES) reported by Prudente et al. [14]. The reactive probabilities of the title reactions are computed. We also observed the changes of vector correlations and four generalized polarization-dependent differential cross-sections (PDDCSs) at different collision energies, and we compared the stereodynamics among different isotopic variants of the title reactions. The product polarization distribution of the title reactions exhibits distinct difference which may arise from different mass combinations or kinetic energies. Crown Copyright Ó 2011 Published by Elsevier B.V. All rights reserved.
1. Introduction The LiH molecules and its ionic variants received extensive attention because this system is of potential importance in the primordial cosmic chemistry [1–4]. In the standard Big Bang hypothesis model, the LiH molecules and its ionic variants are parts of the few molecular species which take part in the entire gas-phase chemical reaction network [3]. The relevance of LiH molecules may be limited by the small abundance of Li molecular species which is thought to exist in the recombination era, but researches have already shed light on a large portion of the gas-phase chemistry of LiH and of its positive ion [3,4]. Theoretical studies on the spectroscopic, polarizability and mono-bichromatic electron dynamics of the LiH molecules have been reported [5–8]. Experimental studies about the formation reaction of LiH of various electronic states also have been reported [9,10]. Bililign et al. [9] conducted laser induced pump–probe far-wing scattering experiments to study the photochemical reaction Li + H2 ? LiH + H. Chen et al. [10] observed the nascent rotational population distribution of LiH in a Li with H2 reaction by using a pump–probe technique. However, the experimentally imperative data of the depletion reaction of LiH is indeed still insufficient. The LiH2 system is of interest for some physical reasons. In the LiH2 system, there are only five electrons. Two are core electrons, the other three are valence electrons, which are involved in the ground state. The LiH2 chemical system can be deemed to the next ⇑ Corresponding author. Tel.: +86 373 3329350; fax: +86 373 3329297. E-mail address: [email protected] (Y. Liu).
electronically simple neutral triatomic system (after H3) exhibiting bound diatomic asymptotes [11]. In addition, this system is one of the simplest alkali metal-dihydrogen partners. The alkali metal atoms also present a high density of nearby excited electronic states into which they can be easily promoted. During the past decade, the LiH2 chemical system has been the subject of a large number of studies on its potential energy surfaces (PESs) [11–20], on the sub-reactive and reactive dynamics and on the formation or depletion reactions of LiH molecules [11,14,17,19–36]. Clarke et al. [12] firstly obtained the ground state potential energy surface (PES) of LiH2 system for the collinear arrangement by using the extensive valence bond (VB) non-orthogonal configuration interaction (CI) method. Dunne, Murrell and Jemmer (DMJ) [13] reported the first three-dimensional LiH2 PES by fitting the CI ab initio points to an analytical function within the many-body expansion (MBE) framework. Prodente et al. [14] improved the ground electronic potential of Dunne et al. [13] by performing accurate ab initio full CI calculations and fitting them to a slightly modified function. In this paper, quasi-classical trajectory (QCT) calculations are performed for the reaction H + LiH ? H2 + Li and its isotopic variants based on the PES reported by Prodente et al. [14]. The potential energy result reported by Wernli et al. [11] refered that there is no energy barrier along the reaction path, which is opposite to that of Prodente et al. [14]. The result of Wernli et al. [11] is applied to calculate the rate coefficient of the formation and depletion reaction of the LiH molecule [20]. The reaction H + LiH ? Li + HH in our study is barrier-less and highly exoergic. Although quantum [12] and classical methods [12,13,16] have been performed in the aforementioned papers,
2210-271X/$ - see front matter Crown Copyright Ó 2011 Published by Elsevier B.V. All rights reserved. doi:10.1016/j.comptc.2011.01.034
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only the analytical potential energy surface reported by Dunne et al. [13] has been employed in some theoretical studies by using time-dependent wave-packet dynamics methodologies [24– 27,33,34]. Clarke et al. [12] have performed classical and timedependent wave-packet calculations only for the linear geometry by using the spin-coupled valence bond (SCVB) non-orthogonal configuration interaction (CI) method. Bodo et al. [23] calculated the interaction between LiH and H. In view of examining the strength of the coupling between the impinging atom and the rovibrational LiH states in low energy collision regimes, a Coupled Cluster (CC) approach was used. Defazio et al. [27] reported rate constant and mechanism of the reactions H + LiH ? H2 + Li and H + LiH0 ? H0 + LiH by performing a quantum–mechanical investigation. Padmandan and Mahapatra [24–26,33,34] formed a series of time-dependent wave-packet dynamical calculations for the H + LiH scattering based on the PES of Dunne et al. [13] The posiþ tive ionic variant of LiH2, LiH2 , is also of astrophysical interest and diffusely studied [17,21,22,30–32,36]. The aforementioned papers have dealt with the scalar properties such as rate constant, reactive probability and resonances in reactive scattering. Vector properties, such as velocities and angular momenta, possess magnitude dynamical information [37–40]. The works of Fano and Macek [41] and Herschbach et al. [42–44] directly led to comprehensive interests [45] in vector correlation or molecular polarized distribution in the reaction A + BC ? AB + C. As far as we know, there are very rare vector correlation (stereodynamics) studies for the depletion reaction of LiH. Here we focus on the dynamics of the reaction H + LiH ? H2 + Li and its isotopic variants:
H þ LiH ! Li þ HH;
ð1Þ
H þ LiD ! Li þ HD;
ð2Þ
D þ LiH ! Li þ DH;
ð3Þ
D þ LiD ! Li þ DD:
ð4Þ
Quasi-classical trajectory (QCT) method is applied to study the stereodynamics of the four reactions based on the PES reported by Prudente, Marques and Maniero (PMM) [14]. 2. Theory The theoretical details of QCT method are specified elsewhere [37–39,45–52]. The Hamilton equations were numerically integrated in three dimensions with six order symplectic integration [52,53]. The integration step size was chosen as 0.1 fs. The initial vibrational and rotational quantum numbers were selected as v = 0 and j = 0, respectively. For one collision energy, a batch of 50,000 trajectories was run, and the statistical uncertainty is no more than 2.0%. The collision energy was chosen as less than 2.6 eV. The maximum impact parameters bmax for different collision energies and different isotopic variants are listed in Table 1. The dynamics of the four reactions presented in Eqs. (1)–(4) were calculated. The reactive probability is defined as P r ¼ NNreact , where N react and tot N tot are the numerals of reactive and total trajectories, respectively. The center-of-mass (CM) frame was chosen in this calculation. k 0 and k are the reactant relative velocity and product relative velocity, respectively. k is parallel to the z-axis and y-axis is perpendic0 ular to the xz-plane containing k and k vectors. The final angular 0 momentum is j , with its polar and azimuthal angles being hr and 0 /r , respectively. The angle between k and k was defined as the scattering angle. 0 The function Pðhr Þ used to describe the k j correlation can be expanded in a series of Legendre polynomials as [37,38]:
Pðhr Þ ¼
1X ðmÞ ð2m þ 1Þa0 Pm ðcos hr Þ: 2 m
ð5Þ
Table 1 Maximum impact parameters. Collision energy (eV)
0.043 0.108 0.216 0.324 0.434 0.65 0.867 1.08 1.3 1.52 1.73 1.95 2.17 2.39 2.6
bmax (Å) H + LiH
H + LiD
D + LiH
D + LiD
4.58 4.42 4.01 3.67 3.40 3.38 3.43 3.48 3.46 3.40 3.42 3.38 3.36 3.31 3.29
5.40 4.60 3.98 3.61 3.25 3.21 3.23 3.22 3.15 3.12 3.13 3.07 2.95 2.92 2.87
3.71 3.97 3.75 3.52 3.36 3.40 3.43 3.47 3.43 3.43 3.43 3.38 3.44 3.36 3.42
4.63 4.28 3.92 3.57 3.35 3.20 3.22 3.20 3.24 3.25 3.22 3.12 3.12 3.12 3.1
The dihedral angle distribution function Pð/r Þ describing 0 0 k k j vector correlation can be expanded as a Fourier series [38]. The /r distribution can be written as:
! X X 1 Pð/r Þ ¼ 1þ an cosðn/r Þ þ bn sinðn/r Þ ; 2p ev en;nP2 odd;nP1
ð6Þ
where
an ¼ 2hcosðn/r Þi;
ð7Þ
bn ¼ 2hsinðn/r Þi:
ð8Þ
A set of generalized polarization-dependent differential crosssections (PDDCSs) is used to describe the full three-dimensional 0 0 angular distribution associated with k k j correlation in the CM frame. The fully correlated CM angular distribution can be presented as a joint density function which is written as the sum [46]:
Pðcos h; cos hr ; /r Þ ¼
1 X k 1 X 2p drkq ð2k þ 1Þ C ðh ; / Þ ; 4p k¼0 q¼k r dx kq r r
ð9Þ
where ð1=rÞðdrlq =dht Þ are the generalized PDDCSs. Many photoinitiated bimolecular reactions are only sensitive to multipole moments k = 0 and k = 2. In this article, ð2p=rÞðdr00 =dxt Þ, ð2p=rÞðdr20 =dxt Þ, ð2p=rÞðdr22þ =dxt Þ and ð2p=rÞðdr21 =dxt Þ are calculated, and we use P00, P20, P22+ and P21 for short. 3. Results and discussion The reactive probabilities for the title reactions are depicted in Fig. 1a. The four probabilities for the title reactions all peak at the collision energy of 0.22 eV and show similar trend on the whole. From the comparison in Fig. 1a, one can find that the difference among them is very obvious. In the whole investigated collision energy range, the probability values of the four reactions decrease in the order of reaction (3), (4), (1), and (2). A noticeable reality is that the impinging atom of reactions (3) and (4) is deuterium while that of reactions (1) and (2) is hydrogen, so it can be concluded from Fig. 1a that the impinging atom mass plays a influential role in a barrier-less reaction. Fig. 1b shows the reactive probability comparison of our QCT result and other quantum results only for the reaction H + LiH ? Li + HH. There are considerable discrepancies among the three results. Generally, our result is comparatively close to the result based on the PES of Dunne et al. [13]. Both the two quantum results are reported in Ref. [14]. The reaction H + LiH ? Li + HH is a barrier-less reaction. The reactive probability of it peaks at a low
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Fig. 1. (a) Energy-dependent reactive probabilities of the four reactions. (b) Reactive probability comparison of our QCT result and other quantum results. Both the two quantum results are from Ref. [14]. The solid square symbol is our QCT result; the cycle symbol is the result based on the PES of Prudente et al. [14] and the triangle symbol is the result based on the PES of Dunne et al. [13].
collision energy (about 0.2 eV) and decreases with increasing the value of collision energy in the energy range 0.2–2.6 eV, which is similar with another barrier-less reaction F + HBr ? HF + Br [54,55]. Since that the effect of zero-point energy has been considered in our calculation, it can be concluded that our QCT result is more reliable. The result based on the PES of Prudente et al. [14] has very high numeral value, which is not consistent with the barrier-less property of this reaction. Figs. 2–5 depict the comparison of Pðhr Þ and Pð/r Þ for the title reactions of various collision energies. Pðhr Þ and Pð/r Þ are probability density functions describing the probability density distribution of reaction products. Fig. 2a shows the comparison of Pðhr Þ distributions of the four reactions at the collision energy 0 0.043 eV. Pðhr Þ reflects k j vector correlation, namely the rotational alignment of the product. The Pðhr Þ distributions for different reactive products are relatively broad and symmetric, with peaks at hr ¼ p=2, which also demonstrates that the product rotational angular momentum vector prefers to be perpendicular to the initial relative velocity direction. The Pðhr Þ peaks of reactions (1),
(2), and (4) are almost the same, while that the Pðhr Þ peak of reaction (3) is the highest one, which can prove the rotational angular momentum vector of the product from reaction (3) mostly prefer to be perpendicular to the k vector. Pð/r Þ of the four reactions for collision energy at 0.043 eV de0 0 picted in Fig. 2b reflects distributions of k k j correlation. They tend to be symmetric at /r ¼ p, reflecting the strong polarization of angular momentum. Fig. 2b shows that distributions for reactions (1), (2), and (4), Pð/r Þ peak at /r ¼ p=2 and /r ¼ 3p=2, and the peak of reaction (3) at /r ¼ 3p=2 is obviously higher than other three ones. It is obvious that the molecular products of reaction (3) are preferentially oriented along the negative direction of the CM y-axis. For reactions (1), (2), and (4), the degree of the products scattered with /r 6 p are almost the same with those which are scattered with /r P p at such a low collision energy. Figs. 3–5 depict the Pðhr Þ and Pð/r Þ distributions of the four reaction for the collision energies at 0.43 eV, 0.87 eV and 2.6 eV, respectively. In Fig. 3a the distributions of Pðhr Þ are similar to those in Fig. 2a. Fig. 3b shows that the Pð/r Þ distributions peak at /r ¼ p=2
Fig. 2. Pðhr Þ and Pð/r Þ distributions for the four reactions for the collision energy of 0.043 eV.
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Fig. 3. Pðhr Þ and Pð/r Þ distributions for the four reactions for the collision energy of 0.43 eV.
Fig. 4. Pðhr Þ and Pð/r Þ distributions for the four reactions for the collision energy of 0.87 eV.
Fig. 5. Pðhr Þ and Pð/r Þ distributions for the four reactions for the collision energy of 2.6 eV.
Y. Liu et al. / Computational and Theoretical Chemistry 965 (2011) 107–113
and /r ¼ 3p=2, and for reactions (1), (2), and (4) the peaks at /r ¼ 3p=2 is higher than that at /r ¼ p=2, namely the products prefer to distribute along the negative direction of the CM y-axis. For reaction (3) the Pð/r Þ peak at /r ¼ p=2 is higher than that at /r ¼ 3p=2. In Fig. 4a the Pðhr Þ distributions are similar to those in Fig. 3a, and the peaks are higher than those in Fig. 3a. In Fig. 4b the four Pð/r Þ distributions peak at /r ¼ p=2 and /r ¼ 3p=2 and the peaks at /r ¼ p=2 are higher than those at /r ¼ 3p=2. The Pðhr Þ and Pð/r Þ distributions for collision energy of 2.6 eV depicted in Fig. 5 is similar to those in Fig. 3 and 4, but the peaks are higher. It can be concluded from Figs. 2–5 that the Pðhr Þ distribution of reaction (3) reaches a higher peak than that of the other three reactions at various collision energies and the product molecules are more preferentially along the positive direction of the CM y-axis with a increase of collision energy. It is easily to find that the differences of Pðhr Þ and Pð/r Þ among the four reactions are becoming smaller with the increasing collision energies. 0 0 The generalized PDDCSs describe k k j correlation and the scattering direction of the product molecules. The PDDCS P00, which is proportional to the differential cross-section (DCS), only describes the angular distribution of the product molecule. It is not associated with the product orientation and alignment. The trajectory of P20 shows the opposite trend to that of P00, which indicates the degree of alignment of the product rotational angular 0 0 momentum j perpendicular to k. The k k scattering plane is not determined by these limiting scattering angles, so the value of these PDDCSs with q–0 should be zero at the extremities of backward and forward scattering [39,46]. The representation of PDDCSs with q–0 for scattering away from the extreme forward and backward directions may be more attractive and can provide abundant dynamic information [39,46] on the /r dihedral angle distribution. Figs. 6–8 show the PDDCSs of the four reactions for different collision energies of 0.043 eV, 0.43 eV and 2.6 eV. In Figs. 6a and 6d, the P00 shows that the product molecules are scattered sideways for reactions (1) and (4). The P00 of reaction (2) which is depicted in Fig. 6b shows a product scattering nearly all the same in the whole scattering angle range, while Fig. 6c presents that the product molecules of reaction (3) are strongly scattered backward.
111
The P20 of reactions (1), (2), and (4) are close to zero, i.e., the degree 0 of alignment of the product rotational angular momentum j perpendicular to k is completely weak. The P20 of reaction (3) in 0 Fig. 6c shows that j is strongly aligned perpendicular to k in the scattering angle range 90–180°. Since the PDDCS P20 is only related to the PES and mass factor, i.e., cos2 b ¼ ma mc =ðma þ mb Þðmb þ mc Þ for the reaction A þ BC ! AB þ C [40], we believe that the differences among the four P20 trajectories are mainly due to different mass factors, i.e., isotopic effects. The PDDCSs with q–0 of the reactions (1), (2), and (4) are almost zero in the whole hr range. Only the P22+ of reaction (3) has a downward direction peak at hr ¼ 150 which exhibits unusual and quite strong polarization of product molecules. Fig. 7 presents the PDDCSs of the title reactions for the collision energy of 0.43 eV. The four P00 indicate that the product molecules are strongly scattered forward (in the scattering range 0–45°) and backward (in the scattering range 135–180°). The product scattering intensity of the four reactions are almost the same with each 0 other. The four P20 show that j is weakly aligned perpendicular to k in the scattering angle ranges 0–45° and 135–180°. The PDDCSs with q–0 of the four reactions are all nearly to be zero. Fig. 8 shows the PDDCSs of the title reactions at the collision energy 2.6 eV. The product molecules of the four reactions are scattered forward in the scattering angle range 0°30°. There is just a little difference among the forward scattering degree of the four reactions. The figure shows that products of reaction (3) are scattered most strongly. The four P20 depicted in Fig. 8 indicate that 0 j is weakly aligned perpendicular to k in the scattering angle range 0–30°. The PDDCSs with q–0 in Fig. 8 show isotropic distributions of products at a high collision energy of 2.6 eV. Some previous works have mentioned that the product rotational angular momentum vector distribution is sensitive to the mass factor [39]. Figs. 2–5 have clearly show that there are obvious differences of Pðhr Þ and Pð/r Þ distributions for different reactions at low collision energy, but at middle and high collision energies the differences become smaller, which directly demonstrates that the product vector distribution is also sensitive to the collision energy. In summery both the mass factor and collision energy can affect
Fig. 6. The PDDCSs P00, P20, P22+ and P21 for the four reactions for collision energy of 0.043 eV.
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Fig. 7. The PDDCSs P00, P20, P22+ and P21 for the four reactions for collision energy of 0.43 eV.
Fig. 8. The PDDCSs P00, P20, P22+ and P21 for the four reactions for collision energy of 2.6 eV.
the vector polarization distribution of product molecules. At low collision energy, the effects of trajectory and different binding energy in the LiH or LiD molecule are noticeable, so the difference of vector correlation among the four reactions is obvious. But at middle and high collision energy, such effects are visibly diminished. Different to the results of vector correlations, we could converge the numerical values of the four reaction probabilities at the lowest collision energy of 0.043 eV. At such a low energy level, the isotopic effect does not increase the efficiency of the four reactions (the reactive probabilities are almost same), i.e., the kinematic effect is inconspicuous. With an increase in the collision energy, the difference of reactive probabilities becomes notable, namely the kine-
matic effect becomes conspicuous. When comparing reaction (1) and (2), one can easily find that the reactive probability of former is bigger that of latter at relative high collision energy (more than 0.22 eV). Similar result can be got from the comparison of reaction (3) and (4). It can be concluded that it is the deuterium atom substitution makes the reaction taking place more difficult. The mass combination type of the reactions we studied is LightLight-Heave (LLH). Han and co-workers have studied about the product polarization of the LLH mass combination reaction in detail, and found that the product molecules are scattered forwardly based on a so-called ‘attractive’ PES at middle and high collision energies [39]. According to the definition in Ref. [39], the PES
Y. Liu et al. / Computational and Theoretical Chemistry 965 (2011) 107–113
which is applied in this paper is ‘attractive’. Our calculation results in Figs. 6–8 show good agreement with the theory of Han et al. [39]. 4. Conclusion In this paper the stereodynamics calculations of the reactions H + LiH ? H2 + Li, H + LiD ? HD + Li, D + LiH ? DH + Li and D + LiD ? D2 + Li are performed by using QCT method based on the PES reported by Prudente, Marques and Maniero [14]. The reactive probabilities of the four reactions for different collision energies are calculated and compared with other two quantum results. The reactive probability values of the four reactions reach the peak at a low collision energy (i.e., approximately 0.22 eV). Pðhr Þ distributions, Pð/r Þ distributions and generalized PDDCSs of the four reactions for different collision energies are discussed in detail. The calculations indicate that the peak of Pðhr Þ of reaction (3) is the highest one, and that the difference of Pðhr Þ between reaction (3) and other three reactions is decreasing with an increase of the collision energy. The difference of Pð/r Þ distributions among the reactions is also decreasing with an increase of the collision energy. Pð/r Þ distributions also show that the product molecules are scattered with /r 6 p more easily than with /r P p in a higher collision energy for the four reactions. It can be concluded that, in the Pðhr Þ and Pð/r Þ distributions, isotopic effect is more difficult to emerge at a higher collision energy. The P00 of different reactions indicate that the degree of forward scattering increases with collision energy, while that the degree of backward scattering decreases. The P20 of different reactions indicate that the degree of 0 alignment of the product rotational angular momentum j is preferentially perpendicular to k in the scattering angle range 0–45° in a high collision energy and the isotopic effects is not obvious. The P22+ and P21 of the reactions for different collision energies show a comprehensive isotropic distribution of products. Acknowledgements This work was supported by the National Natural Science Foundation of China (Grant No. 60977063), the Innovation Scientists and Technicians Troop Construction Projects of Henan Province, China (Grant No. 084100510011) and the Innovation Talents of Institution of Higher Education of Henan Province, China (Grant No. 2006KYCX002). Thanks for Professor K.L. Han for his supply for The QCT Code. References [1] [2] [3] [4] [5] [6]
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