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Ab Initio Characterization of the Electrostatic Complexes Formed by H2 Molecule and Cr+, Mn+, Cu+, and Zn+ Cations Denis G. Artiukhin,†,§ Evan J. Bieske,‡ and Alexei A. Buchachenko*,¶ †

Department of Chemistry, Moscow State University, Moscow 119991, Russia School of Chemistry, The University of Melbourne, Parkville, VIC 3010, Australia ¶ Skolkovo Institute of Science and Technology, 100 Novaya Street, Skolkovo, Odintsovsky District, Moscow Region 143025, Russia ‡

S Supporting Information *

ABSTRACT: Equilibrium structures, dissociation energies, and rovibrational energy levels of the electrostatic complexes formed by molecular hydrogen and first-row S-state transition metal cations Cr+, Mn+, Cu+, and Zn+ are investigated ab initio. Extensive testing of the CCSD(T)-based approaches for equilibrium structures provides an optimal scheme for the potential energy surface calculations. These surfaces are calculated in two dimensions by keeping the H−H internuclear distance fixed at its equilibrium value in the complex. Subsequent variational calculations of the rovibrational energy levels permits direct comparison with data obtained from equilibrium thermochemical and spectroscopic measurements. Overall accuracy within 2−3% is achieved. Theoretical results are used to examine trends in hydrogen activation, vibrational anharmonicity, and rotational structure along the sequence of four electrostatic complexes covering the range from a relatively floppy van der Waals system (Mn+···H2) to an almost a rigid molecular ion (Cu+···H2).

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INTRODUCTION The hydrogen molecule, the most abundant molecular species in the universe, is relatively inert, as the cost for dissociation is more than 430 kJ/mol (more precisely,1 36 118 cm−1). On the one hand, H2 serves as an important inert medium and marker for adsorption and porosity, and on the other as a test molecule for bond activation and cleavage, particularly in catalysis and hydrogen storage applications. Both aspects are related to intermolecular interactions−weak physical and strong chemical, the understanding of which has already opened a way toward controlled hydrogen activation.2 In this regard, transition metals (TM) that have close-lying electronic configurations with distinct chemical properties are particularly appealing and important activation centers.3 Gas-phase interactions of the hydrogen molecule with the bare metal ions provide an important reference for understanding the chemistry of more complex systems and materials.4 The reaction TM+ + H2 → TMH+ + H is generally endothermic and requires activation by either kinetic or electronic energy of the metal ion.5 Guided-ion beam experiments on the reaction with a number of transition metal and lanthanide cations have been performed (see reviews 5−7 and most recent publications 8 and 9), frequently accompanied by theoretical calculations of the potential energy surfaces (PESs) or energetic properties. These studies reveal strong state selectivity of the reactions and have allowed the estimation of related thermochemical parameters. At low energy, three-body collisions may stabilize the adductsso-called “electrostatic complexes” TM+···H2, which, © XXXX American Chemical Society

as a rule, correspond to the lowest energy structure of the triatomic system. Known exceptionsinsertion of Sc+ and Zr+ into the dihydrogen bondwere attributed to curve crossing and may not proceed from the ground-state reactants.10,11 The thermochemistry of complexes and higher clusters formed by different transition metal ions was characterized by temperature-dependent equilibrium measurements by Bowers and coworkers.10,12−14 Binding energies of the complexes so determined constitute a reference point for organometallic complexation and can be compared to adsorption enthalpies on TM centers. However, hydrogen bond activation, usually quantified as the elongation of the internuclear H−H separation Δr or as the vibrational frequency shift ΔνHH, is better characterized by spectroscopic methods. Infrared (IR) photodissociation spectroscopy has been applied during the last decades to a variety of electrostatic complexes,15,16 including those of the TM+ cations Cr,17 Mn,18 Zn,19 and Ag.20 In this method, the spectrum is obtained by detecting TM+ photofragments following excitation of the nHH = 0 → nHH = 1 transition of the H2 molecule, forbidden in a free molecule, but allowed in the presence of the polarizing cation. The spectrum allows one to discriminate the complexes formed by para (p) Special Issue: Piergiorgio Casavecchia and Antonio Lagana Festschrift Received: December 29, 2015 Revised: February 24, 2016

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DOI: 10.1021/acs.jpca.5b12700 J. Phys. Chem. A XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry A

the internuclear separation of the H2 molecule, R is the vector pointing from the H2 center-of-mass to the TM+ ion, and θ is the angle between r and R. The total rovibrational Hamiltonian describing an electrostatic complex in three dimensions (3D) is written (in atomic units, a.u.) as

and ortho (o) spin isomers of H2 or D2 and provides the shift ΔνHH and, owing to rotational resolution, structural data (although Δr parameter cannot usually be determined by fitting the rotational lines 16,21 ). Binding energies cannot be determined, unless the dissociation energy is accidentally slightly above the nHH = 1 energy such that there is a fragmentation onset for higher rovibrational levels.17 Overall, these complementary experimental studies accompanied by the large body of theoretical data on equilibrium structures and binding energies allow one to explore the correlation between H2 binding and activation, so indispensable for understanding and optimizing catalytic and storage materials. Recent reviews15,16 contain most of the relevant references. Importantly, electrostatic complexes exhibit nontrivial correlation, at odds with a naı̈ve premise that their properties are mostly determined by the long-range chargequadrupole electrostatic and charge-induced dipole induction forces that are independent of the nature of the ion. It is true that this zero-order approximation dictates the equilibrium Tshaped structure for cationic complexes and linear structure for anionic complexes, as well as the other topological properties of the potential energy surfacea long-range maximum and saddle in the collinear geometry of the cationic complexes. However, exchange and chemical interactions complicate the relation between the dissociation energy and vibrational shift (bonding and activation). This is perhaps the most important reason for continued interest in these simple systems. Another important point concerns the theoretical approaches. The electrostatic complexes are floppy systems, and to characterize them quantitatively, at a level of accuracy matching experimental requirements, the harmonic approximation is not sufficient. From a practical perspective, this means that the ab initio theory should go beyond the simple and well-automated optimization and normal-mode analysis procedures toward much more demanding strategies involving generation of full PESs and sophisticated corrections to anharmonicity. Our recent work22 provides more detailed arguments and supporting references. To our knowledge, such theoretical strategies have not yet been explored for electrostatic complexes involving TM cations. The first goal of the present paper is to test highly correlated ab initio coupled cluster methods for PES calculation and variational energy level calculations with the reduced dimensionality. The second goal is to apply and assess these approaches against the available experimental data for the electrostatic complexes of the first-row S-state transition metal cations, namely, Cr+,12,17 Mn+,13,18 Zn+,13,19 and Cu+.14 Availability of the precise spectroscopic estimation of the dissociation energy makes the chromium complex especially important in this respect and justifies the particular attention paid to it here. In what follows, we first introduce a modified vibrational decoupling approximation that allows us to consider a twodimensional PES by fixing the internuclear distance of the hydrogen fragment. Then we assess the ab initio approaches for equilibrium structures of the complexes identifying the most important factors and suggest an optimal approach for PES calculations. The PESs and results of the variational calculations are presented and discussed afterwards.

(J − j)2 j2 1 ∂2 1 ∂2 − + + + U (r ) Ĥ = − 2 2 2 2μ ∂R 2m ∂r 2μR 2mr 2 + V (r , R , θ )

(1)

where J and j are the rotational angular momenta of the complex and of the H2 diatom, respectively, whereas μ and m are the corresponding reduced masses. Note that the total PES is represented here as the sum of the unperturbed intermolecular H2 potential U and intermolecular interaction V. The problem can be simplified if one takes into account that the rapid vibration of the H2 moiety is weakly coupled to the slower intermolecular motions. The latter are described by the effective two-dimensional (2D) Hamiltonian of the form (J − j)2 1 ∂2 + + beff j 2 + Veff (R , θ ) Ĥ eff = − 2μ ∂R2 2μR2

(2)

where the effective H2 rotational constant beff and 2D interaction potential Veff depend on the approximation adopted to decouple fast motion. Two simple ways were normally considered for vibrationally elastic scattering23 and vibrational predissociation dynamics,24 the vibrationally diabatic and adiabatic decoupling approximations, VDD and VAD. The first approach implies that an ion moves in the mean field created by the vibrating H2 fragment in its particular state nHH, so that Veff = VnHH = ⟨χnHH|V(r, R, θ)|χnHH⟩r and beff = bnHH = ⟨χnHH| r−2|χnHH⟩/2m, where χnHH(r) is the vibrational wave function of the free H2 molecule determined by the potential U. This approximation was used in many previous studies of the H2 electrostatic complexes with atomic anions and cations.22,25−27 It not only adequately approximates the full 3D approach for nHH = 0 (except, perhaps, the Be+···H2 case, see below), but can also be used to compute the nHH = 0 → nHH = 1 excitation spectrum of a complex directly using different Hamiltonians for initial and final levels.25 Note that the final levels are metastable with respect to vibrational predissociation and that diabatic decoupling provides the means to describe them within the L2 variational technique. The VAD approximation assumes that the diatomic fragment vibrates so rapidly that the partner does not feel its length variations. The effective potential can therefore be taken at the single mean distance r, most naturally, at equilibrium rem, Veff = V(rem, R, θ) and beff = bem = (2mrem2)−1. The subscript m is introduced here to distinguish the equilibrium properties of the isolated diatom from those of the diatom in the complex. The VAD approximation is generally less accurate than the VDD one, but it requires the PES defined in the 2D (R, θ) subspace only, an important practical advantage when demanding ab initio calculations are required. To balance accuracy and computational expense, we modify the VAD approach, using it not with the molecular equilibrium rem value, but with the re value for the equilibrium configuration of the whole complex. In this way, the influence of the intermolecular interaction on the diatom should partially be accounted for. To test this approximation, we applied it to the H2 electrostatic complexes already studied using the 3D and

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VARIATIONAL METHODS FOR ENERGY LEVELS The geometry of the TM+···H2 complex is conveniently described using Jacobi coordinates (r, R, θ). Here r denotes B


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The Journal of Physical Chemistry A VDD approaches, namely, Al+,26 B+,27 and Be+.22 The results of the variational calculations for the dissociation energies D0 at J = 0 within different decoupling approximations are compared with the accurate 3D values in Table 1. In accord with previous

structure of a complex is found to determine the re value and H2 vibrational frequency shift, which are not available from 2D variational calculations. At the second stage, the calculations of the 2D PES in R and θ coordinates are performed at fixed re value. The first step was also used here for testing the ab initio schemes. The ground states of the complexes that have A′ symmetry in the Cs point group and correlate to the ground states of the ions Cr+(3d5, 6S), Mn+(3d54s, 7S), Cu+(3d10, 1S), and Zn+(3d104s, 2S) were investigated. Equilibrium Structures. All calculations were performed with the MOLPRO (ver. 2010.1) program package.29 In all cases, the partially spin-restricted coupled-cluster method including single and double excitations and noniterative triple excitations [CCSD(T)]30 was employed with the restricted Hartree−Fock reference. We used the all-electron option (i.e., no core pseudopotential) and accounted for the second-order Douglas−Kroll−Hess (DK) scalar relativistic correction.31,32 The augmented correlation-consistent polarized valence and core−valence basis sets aug-cc-pVnZ-DK and aug-cc-pwCVnZDK (for brevity, AVnZ and AWCVnZ, respectively) optimized by Balabanov and Peterson33 for use with DK correction were employed with n = T, Q, and 5. For hydrogen, aug-cc-pVnZ set34 with the same cardinal number n was chosen. Extensions by the two sets of the bond functions placed in the midpoint of R distance, 3s3p2d35 (bf1) and 3s3p2d2f1g36 (bf2), were tried, as well as the two extrapolations to the complete basis set (CBS) limit through the sequence of n = T, Q, 5 calculations, by the two-parameter (n + 1/2)−4 and three-parameter “mixed” exp[−(n − 1)] − exp[−(n − 1)2] formulas.37,38 Within the large core (lc) option compatible, by construction, with the AVnZ bases the 1s22s22p63s23p6 TM shells were not correlated, small core (sc) option compatible with the AWCVnZ sets kept in core the 1s22s22p6 TM shells, whereas all electrons were correlated if no core (nc) option is chosen. Both equilibrium structure search and point-by-point PES calculations included the counterpoise (CP) correction to the basis set superposition error,39 so that not only the resulting interaction energies but also optimized geometry parameters were subjected by CP correction. The global minimum for all complexes corresponds to a Tshaped (θ = π/2) configuration and is characterized by the re and Re values, as well as by the well depth (equilibrium binding energy) De. A selection of these quantities is provided as Supporting Information to the present paper. Most extensive testing was performed for the Cr+···H2 complex, and we briefly describe here the main outcome. The lc correlation treatment is not sufficient, and at least sc option should be used. Regardless of the basis optimization (valence or weighted core−valence) and inclusion of the bond functions, the series of n = T, Q, 5 calculations approach the same (within 4 cm−1 for De and 0.001 Å for distances) CBS limit if the three-parameter mixed extrapolation is used. A more restrictive two-parameter (n + 1/2)−4 extrapolation behaves worse, but still defines the CBS limit to better than 1% accuracy. This finding indicates that saturating the finite basis by expanding its core−valence subset or by adding bond functions one indeed accelerates the convergence to the CBS limit without introducing systematic error. This is particularly remarkable for bond functions, which are known to cause excessive basis set superposition error. Indeed, the CP corrections for De amount to 350, 392, and 248 cm−1 for the AVTZ, AVQZ, and AV5Z sequences, whereas for the same bases with the bf1 set added they vary as 2103, 1583, and 716

Table 1. Dissociation Energiesa for M+···pH2 Complexes Calculated by the Three-Dimensions Method and under Different Vibrational Decoupling Approximations (cm−1) M+ cation

re, Å

3D

VDD

VAD-rem

VAD-re

Al+ B+ Be+

0.746 0.757 0.768

472.5 1280.2 2677.9

469.4 (1) 1254.2 (2) 2596.1 (3)

439.0 (7) 1155.6 (10) 2431.2 (9)

444.8 (6) 1214.1 (5) 2616.6 (2)

a

Values in parentheses are the (negative) percentage errors with respect to 3D result. Each complex is also characterized by the equilibrium re value (rem = 0.741 Å).

experience, the VAD-rem approximation is not sufficiently accurate even for the Al+ complex, in which H2 is essentially unperturbed (if judged by the bond elongation Δr = rem − re). The VAD-re approximation is also of little help in this case. However, its accuracy is improved for the ions that perturb the H2 fragment more strongly. Accuracy of the VDD approach, which performs extremely well for weak perturbations, follows an opposite trend and is surpassed by the VAD-r e approximation for the Be+···H2 complex. Note that all approximations underestimate the dissociation energy. This behavior reflects the fact that the 3D interaction PES of the electrostatic complexes V is strongly repulsive along the r coordinate, and the complex is stabilized by the diatomic potential energy U. As a result, the motion of the ion is correlated with the vibrations of hydrogen fragment. The more freedom the variational method provides for stretching the H− H bond, the more possibility the complex has to explore the low-energy regions of the interaction PES. The same reasoning explains why elongation of the H2 bond (“activation”) in the electrostatic complexes is accompanied by a red shift of its frequency. Alhough the VAD-re approach may not seem to be very accurate, these preliminary tests provide clear hints on the sign and magnitude of any discrepancy. In what follows, we will use this 2D approximation without an explicit “VAD-re” reference. Other details of the variational calculations were the same as described elsewhere.25 Most importantly, full account of symmetry leads to the rovibrational energy levels being classified as belonging to Jpipj blocks, where J is the total angular momentum of the complex (and simultaneously its rotational angular momentum, as far as spin-rotational coupling is ignored). The parity index pi = ± is connected to the inversion parity p, where p = pi(−1)J, while the permutation parity pj defines the parity of j (pj = + for even j, and pj = − for odd j). The latter is directly related to the symmetry of the nuclear spin wave function of the monomer, namely, pj = + for para hydrogen (pH2) and ortho deuterium (oD2), and pj = − for ortho hydrogen (oH2) and para deuterium (pD2). Note that this classification has a one-to-one relation to the J K a K c designation for energy levels of asymmetric tops.28

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AB INITIO CALCULATIONS The variational approach to the nuclear problem presented above determines the two-stage strategy for the ab initio electronic structure calculations. At first, the equilibrium C


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The Journal of Physical Chemistry A Table 2. Calculated Parametersa for the Equilibrium T-Shaped Structures of the TM+···H2 Complexes method

re(Δ)

Re

De

νHH (Δ)

νDD(Δ)

D0h(H2)

D0h(D2)

4091 (312) 4087 (313)

2893 (222) 2892 (220)

2525 2544

2741 2760

Cr+ AV5Z AWCVQZ/bfb GVB44 MCPF43 B3LYP12

0.763 (0.021) 0.764 (0.022) 0.77 (0.03) 0.760 (0.018) 0.780 (0.038)

1.988 1.986 2.34 2.094 1.965

3274 3293 1252 2448 3442

4045 (356)c

2924

Mn+ AV5Z AWCVQZ/bfb B3LYP/SVP13 B3LYP/TZVP18 PBE1PBE/TZVP18

0.749 (0.007) 0.750 (0.008) 0.768 (0.026)

2.658 2.645 2.511 2.57 2.56

905 911 1193

4290 4291 4258 4037 4036

(113) (109) (143)c (124)d (125)e

3034 (81) 3034 (78)

583 585 888 538d 570e

676 679

3805 3784 3882 4021

(598) (615) (511) (503)

2691 (424) 2676 (436)

5285 5315 4501 4529 5666

5588 5616

Cu+ AV5Z AWCVQZ/bfb CCSD(T)45 MP245 MP214 MCPF45 B3LYP14

0.786 (0.044) 0.787 (0.045) 0.780 (0.038) 0.772 (0.030) 0.78 (0.04) 0.779 (0.037) 0.79 (0.05)

1.666 1.665 1.702 1.739 1.70 1.720 1.70

6334 6357 5474 5166 6331 5089 6505

5805 Zn+

AV5Z AWCVQZ/bfb B3LYP/SVP13 MPW1B9519

0.757 0.757 0.774 0.758

(0.015) (0.015) (0.032) (0.016)

2.261 2.264 2.237 2.246

1595 1602 2120

a

4191 4188 4172 3995 b

(211) (211) (229)c (235)f

2964 (151) 2962 (150) 2825 (169)g

1105 1169 1728 1599f

1246 1294 1680g c

Distances in angstroms; frequencies and energies in inverse centimeters. Frequencies calculated with the bond functions omitted. With respect to experimental H2 and D2 vibrational frequencies. dFrequencies scaled by a factor of 0.9405 to reproduce experimental H2 vibrational frequency. e Frequencies scaled by a factor of 0.9418 to reproduce experimental H2 vibrational frequency. fFrequencies scaled by a factor of 0.9364 to reproduce experimental H2 vibrational frequency. gFrequencies scaled by a factor of 0.9524 to reproduce experimental D2 vibrational frequency.

cm−1 (CCSD(T)-DK/sc scheme). Extrapolation of both series to CBS limit, where the basis set superposition error vanishes, give the same results, justifying the use of bond functions in combination with the standard CP correction. Calculations that correlate all electrons (nc option) of Cr+ with suboptimal AVnZ sets revealed noticeable weakening of the bonding properties: core correlation leads to Re elongation by 0.3% and De reduction by 1.1%. However, our approach misses high-order excitations in the coupled cluster method.40 Recently, Hobza and co-workers have investigated their effect on noncovalent binding energies.41 For the small six-electron ion-neutral species studied therein, the (positive) error of the CCSD(T) approach with respect to the full configuration interaction limit has been estimated as 0.1%, but for the transition-metal system under consideration it might be larger. On the basis of this analysis, the CCSD(T)-DK/sc/ AWCVQZ/bf1 scheme (hereafter for brevityAWCVQZ/bf) was chosen for PES calculations. First, it reproduces the geometric parameters within 0.002 Å and dissociation energy within 0.5% (worst case being Zn+···H2) with respect to the sc CBS limit. Second, some underestimation of the dissociation energy (in all cases except Cr+···H2) plus the positive contributions from higher excitations may partly cancel the core correlation effect. The simpler CCSD(T)-DK/sc/AV5Z (AV5Z) scheme was also used for checking purposes. The results for equilibrium T-shaped structures are summarized in Table 2. Presented are the optimized re and Re values, equilibrium binding energies D e , harmonic frequencies νHH and νDD of the H2 and D2 vibrations in the TM+ complexes with hydrogen and deuterium molecules and

harmonic estimation of dissociation energies D0h. For the parameters characterizing the molecular fragment, their deviations Δ from those for bare molecule are given in parentheses. The latter are taken as calculated within the same ab initio scheme. For instance, for bare molecules the AV5Z scheme gives re = 0.742 Å, νHH = 4402.9 cm−1, and νDD = 3114.7 cm−1, in close agreement with the reference data 0.7414 Å, 4401.21 cm−1, and 3115.50 cm−1, respectively.42 The AWCVQZ/bf and AV5Z calculations generally agree well with each other. Note that the harmonic frequency analysis, which cannot be made in the MOLPRO package in the presence of dummy centers, was performed for the AWCVQZ/bf scheme with the bond functions omitted. The deviations between them are largest for the Cu+ complex by absolute magnitude, but, relatively, are still well below 1%. For the Zn+ complex, strong sensitivity of the bending frequency on the basis set size was detected. For this reason, the AWCVQZ/ bf and AV5Z D0h values differ much more than the De ones. Dihydrogen vibrational frequency shifts ΔνHH calculated here agree with the measured ones within 3% (see Discussion Section below), justifying applicability of the harmonic approximation for high-frequency vibrational mode of the complex. Further, ΔνHH values are linearly correlated with H− H bond elongations Δre for all the complexes calculated within the same ab initio scheme. This confirms that both parameters are applicable to gauging hydrogen activation (different experimental methods used to probe molecular hydrogen in organometallic complexes and other materials gives one parameter or the other2). D


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PESsa small maximum and saddle, both located at relatively long-range at θ = 0 (Y = 0). These features are typical for electrostatic complexes formed by cations and reflect an interplay between the lowest-order charge−quadrupole electrostatic and charge-induced dipole induction long-range interactions.22

Table 2 also provides comparison with theoretical results available from the literature. The few single- and multireference calculations available for Cr+ and Cu+ complexes tend to underestimate the bonding and perturbation of hydrogen molecule, likely due to insufficient basis set and/or recovery of the electron correlation. In contrast, all the density functional theory (DFT) calculations systematically overestimate them. The ability of the TM+ cation to bind and activate the hydrogen molecule increases along the Mn+, Zn+, Cr+, and Cu+ sequence. The reasons are well-understood.12,14−16 The halffilled 4s orbital in Mn+ and Zn+ increases the repulsion and suppresses the electron donation from the H2 moiety with respect to ions having 3dN outer shell with empty 4s orbital (Cr+ and Cu+). All else being equal, a 3d10 configuration (Zn+ and Cu+) is more favorable for bonding than 3d5 one (Cr+ and Mn+) facilitating back-donation to an antibonding hydrogen orbital. Potential Energy Surfaces. The 2D PESs were obtained using the AWCVQZ/bf scheme at fixed re values indicated in Table 2, and with seven values of θ (0, 30, 45, 60, 75, 85, and 90°) and 27−28 values of R from 1.4 to 20 Å. The CPcorrected interaction energies were represented as the Legendre polynomial expansion

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VARIATIONAL CALCULATIONS Variational VAD-re calculations were performed for all Jpipj symmetry blocks for J = 0−5. Up to 30 radial and 20 angular basis functions25 were used to ensure convergence of the five lowest levels in each block to within 0.01 cm−1. Vibrationally averaged distances R were determined from expectation values of R−2 in accord with their spectroscopic definition. Energy Levels. Figure 2 shows the Cartesian contour plots of the wave functions corresponding to four lowest vibrational levels of the 0++ block of the Cu+···H2 complex labeled as k = 0−3 in ascending order in energy. For this strongly bound complex, the nodal patterns justify the assignment of levels in terms of stretching and bending excitations. The levels k = 1 and 2 correspond to single and double stretching excitations ns = 1 and 2, whereas the k = 3 level corresponds to doubly excited bending excitation nb = 2 (by symmetry, the singly excited bending level does not appear in the 0++ block). Note that the second node of this level appears symmetrically in the negative X plane (or at π − θ Jacobian angle) not shown in the figure, whereas the single node of the bending fundamental would appear at X = 0 (θ = π/2). Although the Cu+ complex can be regarded as semirigid, the Mn+···H 2 complex, representing the weak-bonding extreme, is much more floppy, as follows from wave function contours represented on the same scale in Figure 3. The ground-state wave function extends over a larger angular domain than that of Cu+···H2, while the intermolecular excitations further delocalize the wave functions. Approximate (ns, nb) assignments, however, can still be made: the k = 1−3 levels shown in the figure all correspond to stretching excitations. Table 3 presents the energies E and averaged distances R for the lowest level of the complexes containing pH2, oH2, oD2, and pD2 molecules, classified by the (ns, nb) quantum numbers. Note that the levels of the complexes containing pH2 and oD2 appear in the 0++ and 1++ symmetry blocks, whereas those of oH2 and pD2 are in the 1+− and 0+− blocks.

Nλ

Veff (R , θ ) =

∑ Aλ(R)Pλ(cos θ) λ=0

(3)

where Aλ functions were determined by numerical integration and interpolated by cubic splines. The Nλ parameters were chosen to provide the smallest possible deviation from the ab initio points in the vicinity of the minimum. The cumulative root-mean-square deviations for ∼20 ab initio points lying below the harmonic zero-point energy were less than 0.4 cm−1. For the Cr+···H2 complex, a AV5Z PES was also generated in the same way. The FORTRAN routines for generating the PESs are available upon request to the corresponding author. As examples, contour plots of the Cu+···H2 and Mn+···H2 2D PESs are plotted, in Cartesian coordinates X = R cos θ, Y = R sin θ, in Figure 1. Using the symmetry with respect to the Y = 0 axis, we plot the former at the negative X half-plane (left panel) and the latter at the positive one (right panel). Aside from differences in equilibrium distances and well depth, the figure emphasizes the existence of two more stationary points on the

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DISCUSSION In this section, we first compare our results with available experimental data for the binding and activation of the hydrogen molecule in the complexes. Then we briefly consider vibrational anharmonicity issues with regards to dissociation energy estimations and implications for rotationally resolved IR photodissociation spectroscopy. Comparison with Experimental Data. Variational results can be directly compared with experimental observables. A few words should first be said regarding complexes containing distinct spin isomers. In Table 3, all energies are related to the lowest dissociation limit, TM+ + pH2(nHH = 0, j = 0) or TM+ + oD2(nDD = 0, j = 0). However, since the complexes containing oH2 and pD2 can only dissociate to give the rotationally excited j = 1 diatomic fragment, their dissociation energies should be corrected by the rotational energy of the product molecule, 118.67 cm−1 for H2 (j = 1) and 59.83 cm−1 for D2 (j = 1). Therefore, the spin isomers that dissociate to j = 1 diatomic

Figure 1. Contour plots of the Cu+···H2 (negative X half-plane) and Mn+···H2 (positive X half-plane) 2D PES in Cartesian (X,Y) coordinates. E


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with oD2 and pD2 differ less, both by statistical factor and energy difference. As a result, both complexes are normally observed in spectroscopic experiments. Equilibrium constant measurements, not specific to spin isomer, should also reflect predominantly the properties of the more abundant isomers. In Table 4, we compare the results of our calculations with existing experimental data. Presented are dissociation energies and averaged distances obtained from the variational calculations, frequency shifts, and stretching and bending frequencies νs and νb calculated at equilibrium under the harmonic approximation. Generally, agreement with experimental data is very good. Spectroscopic vibrational energy shifts are reproduced within 5 cm−1, validating the accuracy of the ab initio approaches and the harmonic estimation of this quantity. Averaged interfragment distances deduced from rotational analysis and from variational calculations agree with one another to within 0.01 Å. However, the analysis above showed that these quantities are not sensitive to the detail of ab initio scheme, in contrast to the binding or dissociation energy. Inspection of calculated and experimental values presented in Table 4 indicates that calculated values may be more accurate than those deduced from the equilibrium measurements; the theoretical dissociation energies always fall within the error bars, not far from the median. A stringent test is provided by the Cr+···oD2 complex,17 for which the onset of dissociation at a particular rovibrational level makes it possible to estimate D0 to within 20 cm−1. At this benchmarking scale, our calculations underestimate the dissociation energy by 2.2%, well outside the error bars. Variational calculations were also performed with the less-accurate AV5Z PES to give D0 = 2769 cm−1. The zeropoint energies (ZPEs) for the two PESs differ by only 4 cm−1, so the major source of difference is the underestimation of interaction energy. Considerations presented in the preceding sections demonstrate that the AWCVQZ/bf scheme emulates the CBS limit for the small-core CCSD(T) method reasonably well. Correlation of the core pushes the interaction energy up, whereas higher cluster excitations work in the opposite direction. However, the error of the two-dimensional VAD-re approximation may be even larger. As follows from Table 1, it amounts to ca. 60 cm−1 for the Be+···H2 complex, which has a similar dissociation energy to Cr+···D2. It is therefore impossible to distinguish the various factors responsible for the 64 cm−1 difference between calculated and measured Cr+··· oD2 dissociation energies without elaborate 3D PES calculations. Vibrational Anharmonicity. It is instructive to assess the validity of previous harmonic estimations for dissociation energies of the TM+···H2 complexes. In doing so, we took the ZPEs calculated variationally for the TM+···pH2 complexes as the reference for three simple approximations. The first one (“harmonic”) customarily uses the stretching and bending frequencies νs and νb obtained by solving the normal mode problem ab initio to estimate the ZPE. Second (“harmonic effective”) uses the effective frequencies evaluated by fitting the lowest variational energy levels to one-dimensional quadratic anharmonic expansions, whereas the third (“anharmonic effective”) approximation uses the same together with quadratic anharmonic corrections. (Since no doubly excited bending level was found for the Mn+···pH2 complex, in the two latter cases the same fundamental bending frequency was used.) The percentage overestimation of the actual ZPE is shown in the Figure 4. The standard harmonic approximation works reasonably well only for the strongly bound Cu+···H2 complex.

Figure 2. Cartesian contour plots of the vibrational wave functions of the four lowest levels from the 0++ block of the Cu+···H2 complex. Dashed and solid lines represent negative and positive values, respectively. Approximate (ns, nb) assignments to stretching and bending excitations are given as described in the text.

Figure 3. Cartesian contour plots of the vibrational wave functions of the four lowest levels from the 0++ block of the Mn+···H2 complex. Dashed and solid lines represent negative and positive values, respectively. Approximate (ns, nb) assignments to stretching and bending excitations are given as described in the text.

fragments appear to be more strongly bound. This leads to further enrichment of the complexes containing oH2 and pD2 in ligand-exchange reactions.46 Rotationally resolved IR photodissociation spectra allow one to distinguish the complexes containing the two spin-isomers. However, cation, anion, and neutral complexes containing pH2 are seldom observed due to their low abundance.47 Relative abundances of the complexes F


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Table 3. Lowest Energy Levelsa of the TM+ Complexes with the Spin-Forms of H2 and D2 Classified within the Decoupled (ns, nb) Approximation TM+···pH2

(ns, nb) E

TM+···oH2 R

E

(0,0) (1,0) (0,1) (2,0)

−2602.2 510.4 832.3 959.7

2.037 2.108 2.080 2.190

−2537.2 512.1 682.6 963.7

(0,0) (1,0) (2,0) (0,1)

−638.5 179.7 326.8 481.0

2.740 2.891 3.077 2.786

−559.9 183.1 333.7 247.1

(0,0) (1,0) (0,1) (2,0)

−5374.6 775.6 1122.7 1487.5

1.701 1.746 1.733 1.796

−5314.1 776.3 992.3 1489.3

(0,0) (1,0) (2,0) (0,1)

−1220.9 288.4 532.5 549.7

2.329 2.435 2.562 2.365

−1148.5 291.1 537.8 351.1

TM+···oD2 R Cr+ 2.037 2.107 2.083 2.189 Mn+ 2.736 2.886 3.068 2.809 Cu+ 1.700 1.746 1.733 1.795 Zn+ 2.327 2.433 2.559 2.376

TM+···pD2

E

R

E

R

−2784.0 388.6 601.7 745.0

2.022 2.071 2.051 2.124

−2752.5 388.9 534.0 745.7

2.022 2.070 2.051 2.124

−702.6 139.6 262.1 312.2

2.711 2.813 2.929 2.750

−667.3 140.3 263.6 221.7

2.710 2.811 2.927 2.759

−5642.3 579.8 813.9 1126.2

1.690 1.722 1.712 1.755

−5612.6 579.9 751.8 1126.6

1.690 1.722 1.712 1.755

−1317.2 220.0 416.9 373.7

2.308 2.380 2.462 2.336

−1283.6 220.5 417.8 294.6

2.308 2.380 2.461 2.339

a

For vibrational ground levels (0,0), the energy E is given with respect to dissociation limits, whereas for excited levels the excitation energies are given with respect to the ground state (in cm−1). Vibrationally averaged distances R are in Ångströms.

Table 4. Comparisona of Calculated Properties of TM+···H2 and TM+···D2 Complexes with Experimental Data complex

source

Cr+···H2

this work exptl12 this work exptl17 this work exptl13 exptl18 this work this work exptl14 this work this work exptl13 this work exptl19

Cr+···D2 Mn+···H2

Mn+···D2 Cu+···H2 Cu+···D2 Zn+···H2 Zn+···D2

R 2.037 2.022 2.023 2.736(2.740) 2.73 2.710 1.701(1.700)

D0

ΔνHH/DD

νs

νb

2602(2656) (2660 ± 175) 2784(2812) 2848 ± 9 639(679) (660 ± 140)

313

601

896

220 215 109

433

634

213

431

112 78 615

153 866

305 1236

436 211

619 350

874 630

150 155

251

446

703(727) 5375(5433) (5386 ± 350) 5642(5672) 1121(1267) (1310 ± 140) 1317(1343)

1.690 2.329(2.327) 2.321 2.32

a Theoretical results are presented for complexes containing both spin-isomers: The pH2/oD2 value is given first, whereas, if it differs significantly, the oH2/pD2 value follows second in parentheses. Experimental results for R and D0 are presented in the same way: values without parentheses are attributed to pH2/oD2, with − to oH2/pD2 complexes. The R values are in Ångströms; energies and frequencies are in inverse centimeters.

Rotational Structure of Infrared Photodissociation Spectra. To assess the present theoretical predictions for rotational energy level structure, the lower rovibrational levels of the TM+···H2 and TM+···D2 complexes were fitted using a Watson A-reduced Hamiltonian, 49 as used to fit the experimental rovibrational transition frequencies obtained from IR photodissociation spectra.16 Calculated and measured effective rotational constants are listed in Table 5. Where comparisons between experimental and theoretical spectroscopic data are possible, the agreement is within 1−2%. For example, the calculated B and ΔJ constants for Cr+···D2 (1.073 cm−1and −3.3 × 10−5 cm−1) are in good accordance with

The use of effective frequencies and effective anharmonic corrections improves the results making them accurate to better than 10%, validating the anharmonic decoupling approximations, such as, for example, the vibrational self-consistent field method recently tested for the main-group electrostatic complexes.48 Figure 4 shows some deviations from the expected trend “stronger bonding−smaller anharmonicity”, in particular, for the Zn+ complex. It is not surprising since the large amplitude bending motion of these floppy systems is poorly described within the normal mode approximation, the accuracy of which may depend on the details of the interaction anisotropy for each particular system. G


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ab initio at the level appropriate for direct comparison with binding energies extracted from equilibrium measurements and spectroscopic data from IR photodissociation measurements. The main outcomes can be summarized as follows. (i) Systematic testing of the ab initio CCSD(T)-based schemes for equilibrium structure of the Cr+ complex (supported by selected comparison for other TM+ complexes) revealed a strong effect of the semicore and core correlation. Extrapolation to the complete basis set limit showed that n = T, Q, 5 sequences of the aug-ccpVnZ and aug-cc-pwCVnZ sets approach the same limit even if augmented by the bond functions. These results justify the use of bond functions as a legitimate and efficient means for basis set saturation and, implicitly, add extra certainty in the counterpoise procedure to correct the basis set superposition error. (ii) A modified vibrationally adiabatic decoupling scheme was proposed for handling the rovibrational problem, whereby the bond length of rapidly vibrating H 2 fragment is fixed at its equilibrium distance in the whole complex, and the remaining problem of the coupled van der Waals vibrations and overall rotation is solved variationally in two dimensions. The tests performed for selected electrostatic complexes of the main-group cations indicated that the accuracy of this approximation with respect to full 3D method is ∼6% for weakly bound complexes, but that the error margin decreases as the interaction energy increases. For strongly bound complexes with binding energy below ca. 3000 cm−1 the proposed approach may outperform other well-known decoupling approximations. (iii) Two-dimensional PESs of the four complexes were computed using the aug-cc-pwCVQZ-DK basis supplemented by the set of bond functions, keeping 1s22s22p6 TM shells in core. This scheme was found to emulate well the complete basis set limit at equilibrium configuration. The topology of all PESs is similar, being primarily determined by the lowest long-range charge-quadrupole electrostatic and charge-induced dipole induction forces independent of the nature of the ion. However, the binding energy and H2 activation markers (vibrational frequency shift and bond elongation) increase substantially along the Mn, Zn, Cr, and Cu sequence, reflecting the occupation of, first, the 4s orbital and, second, the 3d orbitals. (iv) The ab initio PESs were used in variational calculations of the lowest rovibrational energy levels of the complexes of the distinct spin isomers of H2 and D2. We found that

Figure 4. Percentage overestimation of the true ZPE for different approximations as explained in the text. True ZPE values obtained in variational calculations are specified in the legend.

experimental values derived from the IR spectrum (1.082 cm−1and −3.27 × 10−5 cm−1). As noted previously,21 the large-amplitude bending motion affects the effective A rotational constant. Assuming that the H2 molecule is undistorted, for a TM+···H2 complex with a rigid C2v structure, the A constant equals the rotational constant b of the diatomic fragment (A = b), whereas for a complex with no barrier to internal rotation of H2, A = 2b. For intermediate anisotropies of the PES, the value of A lies between these two extremes. The A constants for the TM+···H2 and TM+···D2 complexes follow the expected trend with A decreasing as D0 increases. At one extreme, the strongly bound Cu+···H2 and Cu+···D2 complexes, which have short intermolecular bonds, have A constants that are close to the b constants of H2 and D2, respectively, whereas the A constants of the Mn+···H2 and Mn+···D2 complexes with longer, weaker intermolecular bonds lie between b and 2b (b is 59.3 and 29.9 cm−1 for H2 and D2, respectively). Note that the present 2D ab initio approach can be combined with experimental data providing an initial guess for the A constant, which cannot be extracted directly from analysis of the IR spectra, and for the difference between B and C constants. This approach should lead to a refinement of the rotational structure parametrization.

■

CONCLUSIONS The structure, bonding, and rovibrational spectroscopic properties of the electrostatic complexes formed by the firstrow transition metal (TM) cations (Cr+, Mn+, Cu+, and Zn+) with molecular hydrogen and deuterium have been investigated

Table 5. Theoretical Spectroscopic Constantsa for TM+···H2 and TM+···D2 Complexes Obtained by Fitting Calculated Ka = 0 and 1, J = 0−5 Energy Levels Derived from the PESs Developed in the Current Work to a Watson A-Reduced Hamiltonian

a

complex

A

B

C

B

ΔJ, 1 × 10−5

ΔJK, 1 × 10−3

Cr+···H2 Cr+···D2 Mn+···H2 Mn+···D2 Cu+···H2 Cu+···D2 Zn+···H2 Zn+···D2

62.94 30.47 77.50 34.65 57.49 28.20 70.88 32.78

2.075 1.096 1.149 0.608 2.961 1.550 1.588 0.832

1.984 1.049 1.104 0.590 2.782 1.457 1.518 0.802

2.029 1.073(1.082) 1.127(1.152) 0.599 2.872 1.504 1.553 0.817(0.827)

−10.3 −3.3(−3.27) −15.3(−14.8) −3.9 −14 −3.8 −15.6 −4.0(−3.79)

−2.17 −0.6 −0.44 −0.64 −2.5 −0.69 −3.64 −1.1

Experimental spectroscopic constants17−19 are given in parentheses. Units for all parameters are inverse centimeters. H


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(4) Armentrout, P. B.; Beauchamp, J. L. The Chemistry of Atomic Transition-Metal Ions - Insight into Fundamental Aspects of Organometallic Chemistry. Acc. Chem. Res. 1989, 22, 315−321. (5) Elkind, J. L.; Armentrout, P. B. State-Specific Reactions of Atomic Transition Metal Ions with H2, HD, and D2: Effects of d Orbitals on Chemistry. J. Phys. Chem. 1987, 91, 2037−2045. (6) Armentrout, P. B. Periodic Trends in the Reactions of Atomic Ions with Molecular Hydrogen. Int. Rev. Phys. Chem. 1990, 9, 115− 148. (7) Weisshaar, J. C. Control of Transition-Metal Cation Reactivity by Electronic State Selection. Adv. Chem. Phys. 1992, 82, 213−262. (8) Hinton, C. S.; Citir, M.; Armentrout, P. B. Guided Ion Beam and Theoretical Study of the Reactions of Os+ with H2, D2, and HD. J. Chem. Phys. 2011, 135, 234302. (9) Li, F. X.; Hinton, C. S.; Citir, M.; Liu, F. Y.; Armentrout, P. B. Guided Ion Beam and Theoretical Study of the Reactions of Au+ with H2, D2, and HD. J. Chem. Phys. 2011, 134, 024310. (10) Bushnell, J. E.; Kemper, P. R.; Maître, P.; Bowers, M. T. Insertion of Sc+ into H2: The First Example of Cluster-Mediated σBond Activation by a Transition Metal Center. J. Am. Chem. Soc. 1994, 116, 9710−9718. (11) Bushnell, J. E.; Kemper, P. R.; Van Koppen, P.; Bowers, M. T. Mechanistic and Energetic Details of Adduct Formation and σ-Bond Activation in Zr+(H2)n Clusters. J. Phys. Chem. A 2001, 105, 2216− 2224. (12) Kemper, P. R.; Weis, P.; Bowers, M. T. Cr+(H2)n Clusters: Asymmetric Bonding from a Symmetric Ion. Int. J. Mass Spectrom. Ion Processes 1997, 160, 17−37. (13) Weis, P.; Kemper, P. R.; Bowers, M. T. Mn+(H2)n and Zn+(H2)n Clusters: Influence of 3d and 4s Orbitals on Metal-Ligand Bonding. J. Phys. Chem. A 1997, 101, 2809−2816. (14) Kemper, P. R.; Weis, P.; Bowers, M. T.; Maître, P. Origin of Bonding Interactions in Cu+(H2)n Clusters: An Experimental and Theoretical Investigation. J. Am. Chem. Soc. 1998, 120, 13494−13502. (15) Dryza, V.; Poad, B. L. G.; Bieske, E. J. Attaching Molecular Hydrogen to Metal Cations: Perspectives from Gas-Phase Infrared Spectroscopy. Phys. Chem. Chem. Phys. 2012, 14, 14954−14965. (16) Dryza, V.; Bieske, E. J. Non-Covalent Interactions Between Metal Cations and Molecular Hydrogen: Spectroscopic Studies of M+−H2 Complexes. Int. Rev. Phys. Chem. 2013, 32, 559−587. (17) Dryza, V.; Bieske, E. J. The Cr+−D2 Cation Complex: Accurate Experimental Dissociation Energy, Intermolecular Bond Length, and Vibrational Parameters. J. Chem. Phys. 2009, 131, 164303. (18) Dryza, V.; Poad, B. L. J.; Bieske, E. J. Spectroscopic Study of the Benchmark Mn+-H2 Complex. J. Phys. Chem. A 2009, 113, 6044− 6048. (19) Dryza, V.; Bieske, E. J. Structure and Properties of the Zn+-D2 Complex. J. Chem. Phys. 2009, 131, 224304. (20) Dryza, V.; Bieske, E. J. Infrared Spectroscopy of the Ag+-H2 Complex: Exploring the Connection Between Vibrational Band-Shifts and Binding Energies. J. Phys. Chem. Lett. 2011, 2, 719−724. (21) Nesbitt, D. J.; Naaman, R. On the Apparent Spectroscopic Rigidity of Floppy Molecular Systems. J. Chem. Phys. 1989, 91, 3801− 3809. (22) Artiukhin, D. G.; Kłos, J.; Bieske, E. J.; Buchachenko, A. A. Interaction of the Beryllium Cation with the Molecular Hydrogen and Deuterium. J. Phys. Chem. A 2014, 118, 6711−6720. (23) Gianturco, F. A.; Serna, S.; Delgado-Barrio, G.; Villarreal, P. On the Coupling of Rovibrational Motions in He-Li2 Inelastic Collisions. J. Chem. Phys. 1991, 95, 5024−5035. (24) Buchachenko, A. A.; Baisogolov, A. Yu.; Stepanov, N. F. Decoupling Approximations for Quantum Vibrational Predissociation Dynamics: Tests on the ”Low-Level” Fermi Golden Rule Approaches for Some Rare Gas − Cl2, ICl Complexes. J. Comput. Chem. 1996, 17, 919−930. (25) Buchachenko, A. A.; Grinev, T. A.; Kłos, J.; Bieske, E. J.; Szczȩsń iak, M. M.; Chałasiński, G. Ab Initio Potential Energy and Dipole Moment Surfaces, Infrared Spectra, and Vibrational Predis-

the standard harmonic normal-mode analysis can achieve an accuracy better than 10% only for strongly bound semirigid complexes such as Cu+···H2. The use of effective harmonic frequencies and anharmonic corrections deduced by fitting the vibrational energy levels improves the ZPE estimation. (v) Our calculations reproduce the dissociation energies derived from complex formation equilibrium measurements and intermolecular distances and H2/D2 vibrational energy shifts determined from spectroscopic studies. Our predictions for the Cu+ complexes, which are less amenable to IR photodissociation measurements due to their large dissociation energies, are deemed to be the most accurate to date. However, the best benchmark for theory is provided by the dissociation energy of the Cr+···oD2 complex 2484 cm−1 measured with an accuracy of 0.7%.17 Our calculations underestimate the dissociation energy by 64 cm−1 (or 2.2%) without clear attribution of the discrepancy to either electronic or nuclear part of the theory. More elaborate approaches, first of all, with 3D PES and energy-level calculations, should be used to generate theoretical data of comparable quality to the spectroscopic data.

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ASSOCIATED CONTENT

* Supporting Information S

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpca.5b12700. Abbreviations, references, the testing of ab initio schemes on the stationary points of the TM+···H2 electrostatic complexes at the T-shaped (θ = π/2) configuration. (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Phone: +007(915) 3185181. Present Address §

Theoretische Organische Chemie, Organisch-Chemisches Institut and Center for Multiscale Theory and Computation, Westfälische Wilhelms-Universität Münster, Corrensstraße 40, 48149 Münster, Germany.

Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by Russian Foundation for Basic Research under Project No. 14-03-00422 and the Australian Research Council’s Discovery Project funding scheme (Project Nos. DP110100312 and DP120100100).

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REFERENCES

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