Article pubs.acs.org/JPCA
Vibrational Calculations of Higher-Order Weakly Bound Complexes: The He3,4I2 Cases Á lvaro Valdés† and Rita Prosmiti*,‡ †
Departamento de Física, Universidad Nacional de Colombia, Calle 26, Cra 39, Edificio 404, Bogotá, Colombia Instituto de Física Fundamental (IFF-CSIC), CSIC, Serrano 123, 28006 Madrid, Spain
‡
ABSTRACT: The structure and relative stability of higher-order He3,4I2 clusters are investigated by carrying out full-dimensional quantum calculations within the multiconfiguration time-dependent Hartree framework. The full interaction between the I2 molecule and the He atoms is based on analytical three-body ab initio He−I2 potentials obtained from high level ab initio calculations plus the He−He interaction. The low-lying minima on the potential surfaces are found to be very close in energy with the He atoms in a ring encircling the dopant for the global minimum structure, while for the local minima one or two of the He atoms prefer the linear arrangements along the I2-axis. Such classical description on the basis of the potential energy landscape is corrected by including anharmonic quantum effects, present in highly floppy systems, by carrying out full dimensional quantum calculations. The potential energy operator was constructed by natural potential fits, while a mode combination scheme was employed to optimize the computational cost of the improved relaxation calculations. The obtained results predict the relative stability of the He3,4I2 isomers at zero temperature and provide benchmark data on binding energies and structural properties of these van der Waals systems. The (2,1) and (2,2), involving two He atoms in the T-shape and one or two He atoms in the linear configurations, respectively, are found to be the most stable isomers, although extremely close in energy with the (3,0) and (4,0) ones as predicted by classical optimizations. Comparison with experimental data on similar systems at low temperatures is also discussed. This analysis indicates once more the importance of quantum delocalization and the need of accurate quantum-mechanical treatments to characterize such doped helium nanosystems.
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INTRODUCTION Small helium clusters doped with molecules have been studied in detail by infrared and microwave spectroscopy1−9 with an additional interest in the recent years as model microsolutions for exploring of molecular superfluidity (see refs 5, 10, and 11). Such van der Waals (vdW) complexes exhibit large amplitude vibrations accessing several minima of the underlying potential energy surface (PES) and thus large regions of the configuration space.12−14 For small clusters we can attempt to establish reasonably complete energy-level schemes from spectroscopy and by assigning the calculated transition frequencies and intensities to investigate several aspects of the multidimensional PESs. For larger systems the difficulty of such theoretical studies increases rapidly, and methods like path-integral and quantum Monte Carlo should be employed.15−19 Although, as theoretical approaches for computing energy levels provides very detailed information it is important to exploit modern techniques taking advantage of their capabilities for carrying out full-dimensional quantum calculations for He clusters with four, five or six atoms.14,20−23 A promising method for dealing with multidimensional systems (up to 24) is the multiconfiguration time-dependent Hartree (MCTDH).24−26 In the present article, we report detailed vibrational calculations for the 9- and 12-dimensional He3,4I2 vdW © 2015 American Chemical Society
complexes within the MCTDH framework. Realistic models potentials including two- and three-body interaction terms parametrized to ab initio CCSD(T)/CBS (coupled-cluster at complete basis set limit) data27−29 are employed to performed the nuclear full-dimensional quantum calculations. Such analytical sum-of-potentials scheme has been found to be in reasonable agreement with interactions energies from ab initio MP2 and CCSD(T) calculations19 for larger HeNI2, with N up to 10, clusters. The present theoretical results were used to characterize multiple isomers of these weakly bound systems, including three and four identical He atoms, to determine their relative stability, and to establish possible connections with experimental data available from LIF and action spectra on similar HeN-dihalogen clusters.6 In previous experimental and theoretical studies on such systems,6,14,30,31 various isomers have been assigned to different structural models related with combinations of linear and Tshaped triatomic isomers,27,32 labeled as (#T,#L) where #T and #L the number of He atoms in these configurations. Quantitative estimates of the involved energetics and of the ensuing geometric features at the quantum level have been Received: October 23, 2015 Revised: December 3, 2015 Published: December 3, 2015 12736
DOI: 10.1021/acs.jpca.5b10398 J. Phys. Chem. A 2015, 119, 12736−12741
Article
The Journal of Physical Chemistry A
He3,4I2 systems. One can see that for the r coordinate the sine discrete variable representation (DVR) basis (sin) was employed, for the Rk=1−3,4 we used the harmonic oscillator DVR basis (HO), while for the angular θk and ϕk degrees of freedom the Legendre (Leg) and exponential (exp) DVR basis, respectively. In the IR method as implemented in the MCTDH code24,33 the relaxation procedure is accelerated by computing the MCTDH vector by diagonalization. In this way the ground or a specific excited state can be converged by selecting the appropriate initial wave function, e.g. the lowest energy eigenvector or the eigenvector with the largest overlap with the initial state. The convergence for excited states is numerically demanding and becomes harder when the states are not well separated; and as we will see later on that is the case for both He3,4I2 even for their lowest excited vibrational states. The number of singleparticle functions (SPFs) for each combination mode, as well as the least populated orbital in the IR MCTDH calculations for the He3,4I2 are also given in Table 1. We should point out that the highly correlated Rk and θk coordinates are contracted to (Rk,θk) combination modes. Also, one can see that for the r I−I distance, only one SPF was found enough to describe the lowest vdW states accurately. Depending now, of the structure of each conformer a variable number of SPFs (up to 8 and 5 for the (Rk,θk) modes, and 16 and 5 for the ϕk ones) was needed to reach convergence in the IR calculations of the lowest vdW states of the He3I2 and He4I2 complexes. The populations of the highest (least populated) natural orbital are 10−5 and 10−3 for the He3I2 and He4I2 complexes, respectively (see Table 1), and such values are accurate enough to converge the binding energy values within 0.0001 cm−1. Hamiltonian Operator: Kinetic and Potential Terms. As in previous studies of such vdW systems30,34 the molecular Hamiltonian for zero total angular momentum (J = 0) is written as
found comparable to the experimental observations for these systems at low temperatures.6,32 In particular, for the triatomic HeI2 the (0,1) isomer is by just 0.2 cm−1 energetically more favorable than the (1,0) one,27 while for the tetraatomic He2I2 cluster the (2,0) is the most stable one, with the (0,2) and (1,1) ones being by only 0.03 and 0.3 cm−1 higher in energy.14 Such comparisons could serve to gain insights on the aspects of the multidimensional PES and based on the balance of the L and T interactions to rule out the effects of the zero-point-energy on the structuring of these clusters. A thorough understanding of smaller helium clusters facilitates the analysis of larger ones, and could provide insights on gas-phase solvation effects of such systems. Also, benchmark results are desirable for quantitative checks of approximated methods designed to deal with higherorder complexes. The article is organized as follows: the next section describes the computational details of the MCTDH method, such as kinetic and potential energy operators, primitive basis sets, and single particle functions and potentials for each combination mode in the improved relaxation (IR) calculations, while in results and discussion we present and discuss the results obtained for the structures and energetics of the He3,4I2 clusters. Finally, we give our present conclusions.
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COMPUTATIONAL DETAILS IN THE MCTDH FRAMEWORK Coordinates and Basis Functions. For such vdW complexes satellite coordinates (r,Rk) were used. The corresponding vectors and coordinates are the r vector pointing from one I atom to the other, while the Rk (k = 1, 2, 3, or 4) are the vectors from the center of mass of the I2 molecule to each of the three or four He atoms. The body-fixed (BF) z-axis lies along the r, and we choose the R3 or R4 vector to lie in the xzplane. The angles associated with the vectors are the polar θk and azimuthal ϕk with ϕ3 or ϕ4 = 0. Thus, for this orientation the relative azimuthal angles φ1,2 = ϕ1,2 − ϕ3 or φ1,2,3 = ϕ1,2,3 − ϕ4, together with the θk angles and the Rk vector lengths, constitute the vibrational coordinates. In Table 1 we list the type, number, and range of the primitive basis sets used in the MCTDH calculations for both
2
j ℏ2 ∂ 2 Ĥ = − + + 2 2m ∂r 2mr 2 −
primitive basis Nr (sin) r-range (Å) NRk=1−4 (HO)
ℏ2
(− m ∑k < l ∇k ·∇l ), which are very small (less than 0.02 ICl
cm−1) in these cases,14 are neglected. The potential V(r,Rk=1−3,4) term is represented as the sum of the three-body HeI2 interactions plus the He−He and I−I ones
35/35 [2.2, 3.5] 41/41
1/1 8/5
Nϕk=1−2 Nϕk=1−3 least orbital population
16/-/5 10−5/10−3
k
