Ab initio study of Hg(1S0)⋯H2(1Σg+) van der Waals complex

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Chemical Physics 349 (2008) 32–36 www.elsevier.com/locate/chemphys

Ab initio study of Hg(1S0)  H2ð1Rþ g Þ van der Waals complex Michal Ilcˇin a,*, Vladimı´r Lukesˇ a, Viliam Laurinc a, Stanislav Biskupicˇ b a b

Department of Chemical Physics, Slovak University of Technology, Radlinske´ho 9, SK-812 37 Bratislava, Slovak Republic Department of Physical Chemistry, Slovak University of Technology, Radlinske´ho 9, SK-812 37 Bratislava, Slovak Republic Received 28 November 2007; accepted 15 January 2008 Available online 19 January 2008 Dedicated to Professor Hans Lischka on the occasion of his 65th birthday.

Abstract The supermolecular CCSD(T) ab initio calculations of potential energy surface for the electronic ground state of van der Waals (vdW) complex formed from a mercury atom with hydrogen molecule are presented. Our results indicate the linear geometry (Jacobi coordi˚ and R = 4.12 A ˚ ) vdW system with the relative small depth of De = 59.6 cm1. The physical origin of the stanates are rH–H = 0.743 A bility of the studied vdW structure was analyzed by symmetry adapted perturbation theory. The separation of its interaction energy shows that the dispersion interaction is ca seven-times stronger than the induction and ca five-times higher than the electrostatic energy. Finally, the temperature dependence of the coefficient of diffusion was simulated from the calculated potential energy surface using collisional model. Theoretical value of 0.51 cm2 s1 very well corresponds with the available experimental value (0.53 cm2 s1) for T = 238 K. Ó 2008 Elsevier B.V. All rights reserved. Keywords: IIb metals; Hydrogen molecule; Intermolecular perturbation theory; Weak interaction; Ab initio; Transport properties

1. Introduction The metals (Me) of IIb group and its compounds have been studied extensively as toxic chemicals in environment or as important substances occurring in many catalytic and absorption processes [1–5]. The intense interest in photosensitization where a photoexcited metal atom reacts with other molecules has been revived in recent years and it features prominently in photochemistry. To understand the mechanism and dynamics of reactions involving Hg, Cd or Zn atoms, large efforts have been devoted to the simple gas phase reactions with hydrogen molecule [6–9]. In this context, the stable MeH2 compounds have been observed and measured by Fourier transform infrared spectroscopy [10]. Prior to the experimental discovery, Bernier and Millie *

Corresponding author. Tel.: +421 2 593 25 741; fax: +421 2 524 93 198. E-mail address: [email protected] (M. Ilcˇin). 0301-0104/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.chemphys.2008.01.021

[11] have predicted that the ground state geometry in singlet state of HgH2 is linear and could be stabilised in matrices. Next theoretical calculations based on the coupled cluster CCSD(T), Møller–Plesset (MP) and complete active space self consistent field (CASSCF) theories were oriented on the obtaining of accurate global potential surfaces (PES) of MeH2 species [12]. The quality of the PES was affirmed by good agreement with the available experimental spectra. Contrary to the fact that the chemically reactive events in the laser-excited half-collision experiments are supposed to start via the excitation of electronic ground state van der Waals (vdW) Me  H2 complexes, theoretical and experimental studies are minimally devoted to the characterisation of the weak interaction. We can mention the work of Boatz et al. [13] where the linear configuration with ˚ is reported the distance R (in Jacobi coordinates) of 4 A for Cd  H2 complex. The intermolecular H–H distance ˚ corresponds to the equilibrium distance (rH–H) of 0.76 A

M. Ilcˇin et al. / Chemical Physics 349 (2008) 32–36

of the non-interacting hydrogen molecule. The similar ˚ was equilibrium intersystem distance for Hg  H2 of 4 A mentioned in work of Breckenridge et al. [7]. The above mentioned ab initio studies were performed using the supermolecular (SM) approach [14] where the interaction energy is obtained as the difference between the value of the energy of the complex EAB and the sum of the energies of its constituents (EA + EB) DE ¼ EAB  EA  EB

ð1Þ

Although the SM approach is conceptually and computationally simple, it cannot offer a detailed picture of the interaction forces. On the other hand, only the intermolecular perturbation theory (I-PT) [15] allows direct calculations of electrostatic (Eelst), exchange-penetration (Eexch), dispersion (Edisp) and induction (Eind) contributions that provide a physical interpretation of the interactions between the monomer units of a complex X ðnijÞ X ðnijÞ X ðnijÞ X ðnijÞ Eels þ Eind þ Edisp þ Eexch Eint ¼ X ðnijÞ þ Eother þ    ð2Þ The superscript n in Eq. (2) denotes the order of the perturbation VAB and i (j) indicates the order of the Møller–Plesset fluctuation potential for the A (B) system. Despite the above mentioned ab initio calculations of IIb atoms with hydrogen atoms and/or molecule there is still natural demand for the detailed PES mapping and the analysis of the energy contributions stabilising the vdW systems. Encouraged by earlier theoretical studies, the purpose of this work is to provide basis set superposition error (BSSE)-free characterization of Hg  H2 vdW system at the supermolecular CCSD(T) theoretical level. Available computational facilities allowed us to obtain the reliable interaction potential and the calculation of selected SAPT interaction energy contributions up to the fourth order of perturbation expansion (n = 2, see Eq. (2)). The physical origin of vdW structure stability versus the structure of vdW transition state from the SAPT results will be discussed. Finally, on the basis of the obtained theoretical PES, the simulation of the temperature dependence of coefficients of diffusion will be performed.

33

X = D, T, Q) were employed for the hydrogen atoms [20]. In order to improve the effects of the basis set on the quality of interaction energy calculations, we have applied also the set of modified midbond functions [3s3p2d2f] of Tao and Pan (with the exponents sp: 0.9, 0.3, 0.1; d: 0.6, 0.2; f: 0.6, 0.2) [21]. These bond functions were fixed at the center of the axis defined by atoms. A system of Jacobi coordinates (rH–H, R, h) was used to describe the geometry of the studied complex. The coordinates rH–H, R and h represent intramolecular H–H distance, the distance from metal to the centre of H2 molecule and the Jacobi angle (h = 0° in linear arrangement of Hg  H2), respectively. The interaction energy was determined for 10 values of h (0°, 10°, 20°, 30°, 40°, 50°, 60°, 70°, 80° and 90°) in the range of R from 3.0 to ˚ . The total number of 410 points was evaluated. Based 7.0 A on the calculated PES, the spectroscopic parameters were evaluated from Dunham expansion [22] for effective diatomics. 2.2. Diffusion coefficients The coefficient of diffusion D12 for low density atomic gas can be calculated using collisional model as [23] qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 3 3 pk T ðm1 þ m2 Þ=2m1 m2 D12 ¼ ð3Þ ð1;1Þ 8 ppr2 X ðT  Þ 12

12

where m1, m2 are atomic weights, k is Boltzmann’s constant, r is the collision diameter for low energy collisions (it is the value of interatomic distance where the potential function equals to zero), p is pressure, T is the absolute temperature and T* is the reduced temperature (T* = kT/De where De is the well depth of potential energy curve of the interaction ð1;1Þ between atom 1 and atom 2) and X12 ðT  Þ is a collision integral. General formula for calculating the collision integrals X(l,s)*(T*) is [24] Z 1 1 E Xðl;sÞ ðT  Þ ¼ eT  Esþ1 QðlÞ ðE Þ dE ð4Þ sþ2 ðs þ 1Þ!T 0

2.1. Quantum chemical calculations

where E* is reduced kinetic energy (E* = E/De) and Q(l)* are reduced cross-sections Z 1 2 ðlÞ  Q ðE Þ ¼ ð1  cosl vÞb db ð5Þ l 0 1  12 1þð1Þ 1þl

All I-PT calculations were performed by SAPT program codes [16] interfaced to the Gaussian 03 program package [17]. Gaussian 03 was used for SM calculations as well. The supermolecular BSSE was determined via the counterpoise method of Boys and Bernardi [18]. The presented HF interaction energy terms were developed using dimer-centered basis sets of the constituent monomers. In the present study the relativistic small core STUTTGART RSC 1997 ECP (denoted next as ST97) [19] pseudopotential was used. Augmented correlation consistent basis sets (aug-cc-pVXZ,

with b* being the reduced impact parameter (b* = b/r). The angle of deflection v(E*, b*) depends on the potential function. For the calculation of the collision integral we use the computer code presented in Appendix 12 of work [23] which is based on the method illustrated in Ref. [25]. In mono-atomic gas (or mixture of mono-atomic gases) the resulting motion after collision of two atoms depends besides the initial motion vectors of the colliding atoms also on the potential function which depends on the interatomic distance only. In the gas containing molecules consisting of two or more atoms, the motion after collision

2. Calculation details

M. Ilcˇin et al. / Chemical Physics 349 (2008) 32–36

34

with such molecules depends on much more parameters (e.g. the relative angular displacement, the phase of the rotation at the moment of collision). In order to use the expressions valid for mono-atomic gases, the potential function has to be averaged in order to depend only on the distance. One possible way is to calculate the coefficients of diffusion for all angular displacements (D12(h)). Consequently, the resulting coefficient of diffusion is obtained by angle averaging through whole space Z 2p Z p 1 D12 ¼ D12 ðhÞ sin h dh d/ 4p 0 0 Z p=2 ¼ D12 ðhÞ sin h dh: ð6Þ 0

3. Results and discussion 3.1. Features of the PES In Fig. 1, two-dimensional PES calculated for rH–H = ˚ using aug-cc-pVTZ+ST97 basis sets is presented 0.743 A for the studied system. The distance r was obtained from the optimal geometry for the non-interacting hydrogen molecule using CCSD(T)/aug-cc-pVTZ calculations. The evaluated potential energy points were fitted to the following general functional form: V ðR; hÞ ¼

5 X 6 X

k

C 2k;k ½ð1  ea2 ðRR0 Þ Þ  1P 2k ðcos hÞ

ð7Þ

k¼0 k¼1

The symbols P2k denote Legendre polynomials. The obtained parameters are collected in Table 1S. The residual sum of squares is 0.23 cm2 (standard deviation of 0.02 cm1), the average absolute deviation of the fit is 0.02 cm1 and the maximum absolute deviation is 0.08 cm1. Some parameters did not improve the quality of the fit. Their values were fixed at 0 cm1. The investigation of the BSSE-corrected PES for the used aug-cc-pVTZ+ST97 basis set shows the occurrence

R (Å)

6

-10

Table 1 The BSSE-corrected CCSD(T) optimal distances (rH–H, R), well depths (De), real harmonic vibration frequencies (xe) and anharmonicity coefficients (xexe) for linear and perpendicular structures Basis set:

aug-ccpVDZ+ST97

aug-ccpVTZ+ST97

aug-ccpVQZ+ST97

Jacobi angle: ˚) rH–H (A ˚) R (A

0°

90°

0°

90°

0°

90°

0.762 4.10 60.6 59.2 14.5

0.762 4.23 39.8 47.6 14.3

0.743 4.12 55.9 57.0 14.5

0.743 4.25 37.6 46.1 14.1

0.742 4.13 56.0 57.4 14.7

0.742 4.25 37.2 46.2 14.4

De (cm1) xe (cm1) xexe (cm1)

of two extremal points. The optimal stable structures occur ˚. for the linear geometries (C1v symmetry) at R = 4.12 A On the other hand, the vdW barriers with energy differences 18.3 cm1 are located in perpendicular geometries ˚ . The position of our barrier (C2v symmetry) at R = 4.25 A ˚ is ca 0.51 A higher than the CCSD(T) distance localized by Li et al. [12] on PES along the exit channel for HgH2. These calculations were not BSSE-corrected. Moreover, Fig. 1 shows that the shape of the valley between linear and perpendicular structures is practically isotropic, i.e. the lowest values of energy in the valley occur almost at the same value of R independent on the angle. To estimate the effect of the weak interaction and basis set selection on the intermolecular H2 distance rH–H, the additional calculations for linear and perpendicular orientations were performed. The differences between the equilibrium rH–H distances obtained for the interacting and isolated molecules are practically zero for all three types of used aug-cc-pVXZ (X = D, T, Q) basis sets. The same fact for ab inito PES in vdW area is indicated by Li et al. [12]. The optimal R and rH–H distances with relevant well depths and spectroscopic parameters are collected in Table 1. The CCSD(T) equilibrium bond distances rH–H behave monotonically with the basis set extension for the hydrogen atoms. The extrapolated complete basis set (CBS) limit is ˚ [26]. identical with the experimental value of 0.741 A Although, the further evaluated quantities characterizing vdW system are also sensitive to the above mentioned basis set extension, their behavior is more oscillating than monotonic. As shown in Table 1, there is no significant discrepancy between the results obtained using aug-ccpVTZ+ST97 and aug-cc-pVQZ+ST97 basis sets.

-20 5

3.2. Decomposition of the interaction energy

-30 -40 -50

4 50

0

30

60

100

90

θ (deg) Fig. 1. Contour plot of the calculated interaction potential (for rH–H = ˚ ) at the supermolecular CCSD(T)/aug-cc-pVTZ+ST97 level of 0.743 A theory for Hg  H2 system. All energies are in cm1.

The next step of this work is to discuss the physical origin of the stability of the studied vdW structure. The simple separation of SM interaction energy shows that the Hartree–Fock (HF) interaction energy (DESCF) has a strong character while the interaction correlation energy repulsive  CCSDðTÞ has a stabilisation effect. Using the I-PT DEcorr decomposition, we can analyse and estimate the influence of the fundamental components for the interaction energy in the linear and perpendicular stationary points (see

M. Ilcˇin et al. / Chemical Physics 349 (2008) 32–36

Fig. 2). All separated SAPT energies calculated at HF level of theory indicate higher absolute magnitudes for the linear ð200Þ orientation. The electrostatic induction energy Eind resp (index resp. indicate the inclusion of response effects, see e.g. Ref. [14]) is higher than for the pure HF electrostatic ð100Þ term Eels . With respect to the orientation of angle h, the ð100Þ

ð200Þ

ratio Eels : Eind resp is 1.6 for the linear configuration while for the perpendicular orientation it is equal to 3.0. The repulsive exchange contributions included in Heitler–Lon ð100Þ ð11Þ ð12Þ HL don energy EHL E ¼ E þ E þ E þ    [14] exch exch exch exch exch ð200Þ and Eexch-ind resp represent an important counterpart to attractive HF electrostatic and induction energies. In the case of the perpendicular structure these energies exhibit the maximal negative magnitude which is compensated by the interaction correlation energy. The dominant part of the interaction correlation energy naturally originates from the dispersion energy. The importance of the dispersion vs. induction interaction is demonð200Þ ð200Þ strated by the ratio Edisp : Eind resp which has the value of 7.2 for the linear configuration and 1.31 for the T-shape one. Fig. 2 clearly shows the importance of the dispersion vs. induction interaction indicated by the ratios ð200Þ ð200Þ Edisp : Eind resp (1.18). Next, we can see that the higher orders  of dispersion energies  contributing to the ð2Þ ð2Þ ð200Þ ð12Þ ð22Þ Edisp Edisp ¼ Edisp þ Edisp þ Edisp dispersion energy are ð2Þ

practically constant (ca 8–9% of Edisp Þ for both investigated

a

150

Energy [cm cm-1]

0˚ 100

50

90˚

(100) E els HL Δ E exch (200) Eind_resp (200) Eexch-ind_resp Δ E SCF

35

orientations. On the other side, the electrostatic correlation ð12Þ energy Eels is very small and positive for linear structure. The repulsive dispersion exchange-correlation energy   ð200Þ

ð200Þ

Eexch-disp is ca 0.8 (1.1)-times higher than Eexch-ind resp for h = 0° (90°). The sum of the selected SAPT contributions (156 cm1 for 0° and 94 cm1 for 90°) agrees well with the interaction correlation energy DECCSDðTÞ (147 cm1 corr 1 for 0° and 94 cm for 90°). It seems that the higher orders of interaction correlation terms will play small role in the stabilisation of the studied vdW structure. 3.3. Diffusion coefficients simulations

Based on the PES fitted function (see Table 1S and Fig. 1) and the Eqs. (3)–(6), the coefficients of diffusion were calculated for atmospheric pressure. In the applied model, the hydrogen molecule was considered rigid during the collisions, its geometry was fixed in equilibrium geo˚ for aug-cc-pVTZ basis set). With metry (rH–H = 0.743 A respect to the shape of PES and the qualitatively identical shapes of the energy cuts on PES, the angular dependence of the coefficients of diffusion D12(h) for the studied system is negligible. The obtained isothermal curves are practically constant, independent on angle changes. Based on Eq. (6), the coefficient of diffusion D12 averaged over whole space angle increases with temperature (T1.68, see Fig. 3). In the case of the lowest investigated temperature T = 273 K, the experimental value is available. The comparison of the theoretical value for T = 273 K (0.51 cm2 s1) with the available experimental value of 0.53 cm2 s1 [27] (see solid circle in Fig. 3) indicates very small difference of 4%. With respect to the applied theoretical model and the fact that the measurements of the coefficient of diffusion are strongly dependent on the used experimental technique (experimental error might be over 5%), we can conclude very good agreement between the experimental and theoretical value.

2

-1

-50

Averaged diffusion coeffcient (cm s )

0

b Energy [cm-1]

10

0˚

90˚

0 -100

-150

(1 (12) E els-resp (200) Edisp (2) Edisp (200) Eexch-disp CCSD(T) Δ E corr

Fig. 2. The relevant SAPT contributions to the interaction supermolecular SCF energy (a) and interaction correlation energy (b) for the studied vdW system. Calculated using aug-cc-pVTZ+ST97 basis set for rH–H = ˚ and R = 4.12 A ˚ (h = 0°) and R = 4.25 A ˚ (h = 90°). 0.743 A

2.5

2.0

1.5

1.0

0.5 300

400

500

600

700

Temperature (K)

Fig. 3. The temperature dependence of the theoretical averaged diffusion coefficients (open squares). The experimental value (solid circle) was taken from Ref. [27].

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M. Ilcˇin et al. / Chemical Physics 349 (2008) 32–36

4. Conclusions The two-dimensional ab initio potential energy surface for the interaction of a mercury atom with a hydrogen molecule was evaluated at the CCSD(T) level including BSSE-correction. The stable geometry occurs for the linear orientation. The energy barrier is located in perpendicular orientation. The physical origin of the stability of the vdW structure was calculated using SAPT. The interaction energies were separated into four fundamental components: electrostatic, exchange-penetration, induction and dispersion. The analysis of these components reveals that the HF interaction energy is ruled by attractive electrostatic and induction enerð200Þ gies. The stabilisation effect of Eind resp energy is more compensated by the repulsive contributions included in ð200Þ ð200Þ ð200Þ Eexch-ind resp term than in the case of Edisp and Eexch-disp energies. Our calculations show that the dispersion interaction is ca two-times stronger that the induction and approximately four-times stronger than the electrostatic energy. The theoretical coefficient of diffusion is in good agreement with the experimental value with respect to the applied theoretical model and the experimental error. Acknowledgements The work has been supported by Slovak Grant Agency (Project Nos. 1/3566/06 and 1/3036/06) and by Science and Technology Assistance Agency under the Contract No. APVT-20-005004. Appendix A. Supplementary material Linear and non-linear coefficients of the fitted function (Eq. (7)) are available in Table 1S. Supplementary data associated with this article can be found, in the online version, at doi:10.1016/j.chemphys.2008.01.021. References [1] E. Stokstad, Science 303 (2004) 34. [2] R.P. Mason, F.M.M. Morel, H.F. Hemond, Water, Air, Soil Pollut. 26 (1996) 1.

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