7144
J. Phys. Chem. A 2000, 104, 7144-7150
Quantum Simulations of Nen-OH+ Clusters Markus Meuwly† Department of Chemistry, UniVersity of Durham, South Road, Durham, DH1 3LE England ReceiVed: April 11, 2000
Structures and dynamics of Nen-OH+ clusters are investigated using both adiabatically corrected Ne-OH+ dimer potential energy surfaces and a three-dimensional interaction potential. The first rigorous test of the previously developed vibrational adiabatic approximation to construct potential energy surfaces for clusters comprising vibrationally excited monomers against calculations encompassing all coordinates is presented. Good agreement between the different approaches and with experiment is found. Frequency shifts, ∆V(n), for the OH+ stretching excitation as a function of the number, n, of neon atoms are presented. Under the assumption that the shift for the first completely filled solvation ring is representative an infrared transition of OH+ in a neon matrix at around 2845 cm-1 is predicted. This value includes many-body contributions inferred from recent studies on Nen-HN2+.
1. Introduction With the advent of sophisticated experimental techniques detailed investigations of small molecular clusters have become possible.1,2 These studies provide useful information about intermolecular interactions, structures, and energetics of the systems. As a paradigm for stepwise solvation, small aggregates studied under high-resolution constitute an important class of systems for which controlled experiments are possible.3-5 This development opens possibilities to address questions of real chemical and biochemical relevance such as the influence of a solvent on the thermodynamics and dynamics of enzymes6 or effects on reaction rates modified by solvent dynamics.7 In this regard the studies performed on charged species have been particularly fruitful.4,8-11 Usually the clusters consist of an infrared chromophore that is surrounded by structureless atoms or small molecules. In cases where it is possible to follow specific characteristics as a function of the number of solvent moieties around the infrared chromophore, stepwise solvation effects can be investigated. Some of the systems are small enough to allow accurate theoretical work. This is necessary to devise and test approximate methods. For this purpose, several methodologies have been developed.12-14 One of the conceptssthe adiabatic correction of a potential energy surfaceshas been specifically designed to include the effect of the monomer vibration into the interaction potential.15 However, this approach has not yet been tested against fully dimensional calculations. It is one of the aims of the present work to assess the usefulness of the decoupling approach. For this, the Ne-OH+ ionic complex constitutes an appropriate system. The adiabatically corrected surfaces for the ground and first vibrationally excited state of OH+ have been constructed and shown to yield satisfactory agreement with experiment.16 Ab initio calculations of the three-dimensional interaction potential are possible and the ro-vibrational problem can be accurately solved. In addition, Ne-OH+ is a good candidate for further experiments on microsolvation of ionic chromophores. To this † Present address: Departement fu ¨ r Chemie und Biochemie, Universita¨t Bern, Freiestrasse 3, CH-3000 Bern 9, Switzerland.
end the vibrational red shift of the OH+ frequency as a function of the number of neon atoms surrounding the ion is calculated. The three-dimensional potential is used to calculate minimum energy structures and quantum mechanical ground state energies of Nen-OH+ clusters for n e 6. This work is structured as follows. In the next section the previously calculated, two-dimensional potential energy surface is extended to include the OH+ coordinate. Then, the fundamentals on the three-dimensional surface are calculated and compared with results on the adiabatically corrected potential V0(R,θ) and V1(R,θ). The third part is devoted to the investigation of complexation induced red shifts, structures, and energetics of the Nen-OH+ clusters. 2. Interaction Potential The present work uses a standard Jacobi coordinate system in which r is the OH distance, R is the distance from the center of mass of OH+, and θ is the angle between the two distance vectors. It is advantageous to choose the grid on which energies are calculated such as to minimize the effort in subsequent bound state calculations. Evaluation of the necessary angular integrals is most stable if Gauss-Legendre points are used. In addition, the representation of the interaction potential is simplified. Thus, calculations at angles corresponding to an 8-point quadrature (θ ) 16.20°, 37.19°, 58.31°, 79.44°, 100.58°, 121.71°, 142.82°, and 163.80°) were performed. The OH+ bond lengths were r ) 1.76a0, 1.948a0, and 2.17a0; r ) 1.948a0 is the optimized OH+ bond length and the two other points correspond to the inner and outer turning point of the V ) 0 state of the free monomer. The grid of Jacobi radii R included 16 points between 3.8a0 and 18a0. It should be noted that the Gauss-Legendre grid in θ is different from the equidistant one used previously. In particular, the smaller angles, around which the fundamental excitations are concentrated, are more densely covered. All ab initio calculations were carried out with the Gaussian 94 suite of programs.17 Electronic structure calculations at the QCISD(T) level of theory employed the previously used Ahlrichs valence triple-ζ basis set augmented by diffuse and polarization functions from Dunning’s aug-cc-pVTZ contrac-
10.1021/jp001380d CCC: $19.00 © 2000 American Chemical Society Published on Web 07/06/2000
Quantum Simulations of Nen-OH+ Clusters
J. Phys. Chem. A, Vol. 104, No. 30, 2000 7145
tion.18,19 As proposed by Boys and Bernardi, the interaction energies were counterpoise corrected.20 To verify the suitability of QCISD(T) an additional onedimensional cut through the potential was calculated at the CCSD(T) level of theory. Here, r ) 1.948a0 and θ ) 0 were chosen. The results differ by less than 5 cm-1 over the entire range of R values. The interaction in the Ne-OH+ complex is described by
E(R,r,θ) ) V(R,r,θ) + VOH+(r)
(1)
where VOH+(r) is the potential curve for isolated OH+ and V(R,r,θ) is the counterpoise corrected potential energy surface for the complex. The OH+ potential curve has been calculated using the same basis set and correlation treatment as those for the Ne-OH+ complex. To be useful in subsequent bound state calculations, the discrete energy points have to be represented as a continuous function. It is convenient to represent the interaction potential V(R,r,θ) as an expansion in Legendre polynomials and radial strength functions
V(R,r,θ) )
∑λ Vλ(R,r) Pλ(θ)
(2)
Projecting the coefficients Vλ(R,r) is most stable if the interaction potential is given at Gauss-Legendre points in θ. The radial strength functions are subsequently represented using a reproducing kernel (RKHS).21 (A code to construct the interaction potential is available from the author upon request.) Cuts through V(R,r,θ) for the r ) 1.76a0 and r ) 2.17a0 are displayed in Figure 1. The global minimum of the interaction between Ne and OH+ is found at r ) 1.980a0, R ) 5.013a0, and θ ) 0 and has a well depth of 1251.6 cm-1. 3. Bound State Calculations In the following the three-dimensional bound state problem for Ne-OH+ is considered. This gives the opportunity to directly assess the validity of the vibrational adiabatic correction.15 To allow a fair comparison, the bound state calculations on the two adiabatically corrected potentials V0(R,θ) and V1(R,θ) have to be repeated because the angular grid previously used for the ab initio calculations was different. The form and parameters of the correction function are the ones reported earlier.19 Bound state close-coupling calculations are carried out using BOUND.22 After expanding the total wave function in a product of angular and radial functions, the resulting coupled equations in R are solved numerically using a log-derivative propagator.23,24 All calculations are performed in space-fixed representation. The reduced mass of the complex is 9.189189mu. The coupled equations are propagated from Rmin ) 2.0 to Rmax ) 15.0 Å and extrapolated to zero step-size from log-derivative interval sizes of 0.02 and 0.04 Å using Richardson h4 extrapolation. The nomenclature for the quantum numbers involved is identical to the one used earlier.16,19 Thus, a state is described + by (V1Vlp b Vs) where V1 gives the number of quanta in the OH bond, Vlp labels the bending together with the parity p of the b state, and Vs is the stretching coordinate. To characterize the fundamentals in a useful way, rotational BV and DV constants are calculated from a fit of energy levels EV(J) to a standard diatomic Hamiltonian
EV(J) ) EV + BV[J(J + 1) - K2] - DV[J(J + 1) - K2]2
(3)
Figure 1. Contour plot of the Ne-OH+ potential energy surface V(R,r,θ) from QCISD(T) calculations for r ) 1.76a0 (above) and 2.17a0 (below). Between the lowest contour (-800 cm-1 for r ) 1.76a0 and -1300 cm-1 for r ) 2.17a0) and -400 cm-1 the spacing is 100 cm-1; above it is 50 cm-1. Contours on the repulsive wall are drawn at 100, 200, 500, and 1000 cm-1.
Here, K is the projection of the OH+ angular momentum j on the intermolecular axis. A. Two-Dimensional Calculations. For the two-dimensional calculations on the adiabatically corrected potentials, the rotational constants for OH+ were b0 ) 16.422 86 cm-1 and b1 ) 15.695 26 cm-1.25 Monomer rotational levels up to j ) 25 were included in the basis set, which converges energies to better than 10-4 cm-1. A general observation is that all bound states are found deeper in the well than the previous calculations.19 This is probably due to the finer grid in θ on which the present ab initio calculations have been carried out. The overall effect is about 10 cm-1 for all states. However, shifts in the relatiVe energies are considerably smaller. As an example, the stretching vibration (0 00 1) is found at 154.1 cm-1 instead of 151.2 cm-1. These results are not too surprising as with the present potentials the low-angle region, to which the fundamentals are most sensitive, is better covered. In addition, the spline interpolation previously used overestimated the angular curvature in the potential because only information about the potential at θ ) 0 and 30° was available.19 B. Three-Dimensional Calculations. Three-dimensional close-coupling calculations require means to evaluate matrix elements between monomer vibrational wave functions. Here, the quadrature scheme proposed by Schwenke and Truhlar is adopted.26 The three distances r on which ab initio calculations for Ne-OH+ were performed are chosen as quadrature points. Anharmonic OH+ wave functions required to calculate the
7146 J. Phys. Chem. A, Vol. 104, No. 30, 2000
Meuwly
TABLE 1: Relative Energies (in cm-1) and Rotational B and D Constants (in cm-1) of the Fundamentals of Ne-OH+ a (0 00 0) exp. (ref 16) B D/10-6 2-d B D/10-6 3-d B D/10-6
(0 00 1)
(0 11e 0)
(0 11f 0)
(0 20 0)
0.25648 2.24 0.2516 2.5 0.2517 2.4 (1 00 0)
154.4 247.4 247.4 290.6 0.2401 0.2507 0.2515 0.2284 2.9 3.4 3.3 3.8 158.1 253.8 253.8 297.6 0.2403 0.2505 0.2513 0.2285 2.8 3.2 3.2 3.6 (1 00 1)
(1 11e 0)
(1 11f 0)
(1 20 0)
exp. (ref 16) 2786.5 327 327 B 0.26044 D/10-6 1.71 2-d 2792.0 173.1 322.3 322.3 328.1 B 0.2592 0.2482 0.2552 0.2558 0.2366 D/10-6 2.2 2.5 2.9 2.9 2.8 3-d 2800.8 178.5 319.0 319.0 337.2 B 0.2566 0.2456 0.2534 0.2540 0.2340 D/10-6 2.0 2.3 2.6 2.6 2.7 a Energies are given with respect to the ground state of Ne-OH+ (V1) except for (1 00 0), which is given with respect to (0 00 0); the labeling (V1Vlp b Vs) is the one employed for semirigid triatomics.
integration weights are computed using the LEVEL program.27 Points and weights constructed in this manner evaluate constant, linear, and quadratic potential terms over r exactly. To carry out three-dimensional bound state calculations, monomer rovibrational basis functions (V1,j) have to be specified. They included states up to V1 ) 2 and j ) 11. The energies of these states are calculated by solving the one-dimensional Schro¨dinger equation with the QCISD(T) potential V(r) using LEVEL.27 Convergence studies including higher (V1,j) basis functions changed the energies by less than 10-4 cm-1. The ground state (0 00 0) of Ne-OH+ is found at -839 cm-1 below dissociation to Ne + OH+. This amounts to a zero point energy of roughly 400 cm-1. Calculations on V0(R,θ) yield a slightly smaller dissociation energy also because the well depth is reduced by 35 cm-1. First stretching (1 00 1) and bending (0 11e/f 0) excitations lie 158 and 254 cm-1 higher in energy, which compares to 154 and 247 cm-1 on V0(R,θ), respectively. For V1 ) 0 the only rotational constants available from experiment are those for the ground state. The values resulting from calculations on the adiabatically corrected surface underestimate B and overestimate D. Once the three-dimensional calculation is performed, the values of B and D change slightly and move in the appropriate direction. Additional comparison between quantities calculated on V0(R,θ) and V(R,r,θ) is given in Table 1. One possibility to assess approximate ro-vibrational energies for levels correlating with V1 ) 1 is to perform bound state calculations including V1 g 1 monomer basis functions only. Although this might seem to be a drastic procedure, mixing between the V1 ) 0 and V1 ) 1 states can be expected to be small. In second-order perturbation theory matrix elements between V1 ) 0 and V1 ) 1 are of the order of 1 cm-1. More accurate procedures involve the determination of scattering resonances and fitting them to a Breit-Wigner form.28,29 The lowest state, correlating with (1 00 0), is calculated at 1961 cm-1 above the ground state. This gives an energy difference of 2800 cm-1 between Ne-OH+ (V1 ) 1) and NeOH+ (V1 ) 0). The experimentally measured value is 2786.5 cm-1. Consequently, the vibrational red shift is underestimated
by 13.5 cm-1. Excitation of the intermolecular stretch (1 00 1) is found 179 cm-1 higher followed by the Π (1 11 0) (319 cm-1) bending vibration. The experimental value for the Π bending frequency is 327 cm-1 whereas a calculation on the adiabatically corrected potential V1(R,θ) yields 322 cm-1. Given the considerably reduced computational effort to construct V0(R,θ) and V1(R,θ) compared to V(R,r,θ), the results justify an approximate treatment of the intramolecular degree of freedom r. This is even more so as no truly spectroscopically accurate interaction potential can be expected from a pure ab initio calculation. Under these circumstances V0(R,θ) and V1(R,θ) constitute a useful working hypothesis and a good starting point for further refinement together with future experimental data in the spirit of ref 30. 4. Nen-OH+ Clusters In the previous section the reliability of the vibrational adiabatic approximation has been established. Consequently, further explorations of Nen-OH+ clusters based on the V0(R,θ) and V1(R,θ) potential energy surfaces can be undertaken. Of immediate interest to experiment is the complexation induced red shift of the OH+ frequency, ∆V(n), as a function of the number of neon atoms, n. Furthermore, the three-dimensional interaction potential allows one to investigate structural and dynamic characteristics including all coordinates involved. In the following the interaction between Nen-OH+ is assumed to be pairwise additive. The Ne-Ne interaction has been thoroughly investigated by Aziz and co-workers.31 Thus, the total intermolecular potential in a Nen-OH+ cluster is then given by n
Vtot(n) )
∑ i)1
n
VNei-OH+(Ri,r,θi) +
VNe-Ne(sij) ∑ i
