Dariusz
Kędziera
*a,
Guntram
Rauhut
*b and
Attila G.
Császár
*c
aFaculty of Chemistry, Nicolaus Copernicus University in Torun, 87-100 Torun, Poland. E-mail: teodar@chem.umk.pl
bInstitute for Theoretical Chemistry, University of Stuttgart, Pfaffenwaldring 55, 70569 Stuttgart, Germany. E-mail: rauhut@theochem.uni-stuttgart.de
cLaboratory of Molecular Structure and Dynamics, Institute of Chemistry, ELTE Eötvös Loránd University and MTA-ELTE Complex Chemical Systems Research Group, Pázmány Péter sétány 1/A, H-1117 Budapest, Hungary. E-mail: attila.csaszar@ttk.elte.hu
First published on 11th May 2022
The linear molecular ions H2He+, HHe+2, and He+3 are the central units (chromophores) of certain He-solvated complexes of the H2He+n, HHe+n, and He+n families, respectively. These are complexes which do exist, according to mass-spectrometry studies, up to very high n values. Apparently, for some of the H2He+n and He+n complexes, the linear symmetric tetratomic H2He+2 and the diatomic He+2 cations, respectively, may also be the central units. In this study, definitive structures, relative energies, zero-point vibrational energies, and (an)harmonic vibrational fundamentals, and, in some cases, overtones and combination bands, are established mostly for the triatomic chromophores. The study is also extended to the deuterated isotopologues D2He+, DHe+2, and D2He+2. To facilitate and improve the electronic-structure computations performed, new atom-centered, fixed-exponent, Gaussian-type basis sets called MAX, with X = T(3), Q(4), P(5), and H(6), are designed for the H and He atoms. The focal-point-analysis (FPA) technique is employed to determine definitive relative energies with tight uncertainties for reactions involving the molecular ions. The FPA results determined include the 0 K proton and deuteron affinities of the 4He atom, 14875(9) cm−1 [177.95(11) kJ mol−1] and 15
229(8) cm−1 [182.18(10) kJ mol−1], respectively, the dissociation energies of the He+2 → He+ + He, HHe+2 → HHe+ + He, and He+3 → He+2 + He reactions, 19
099(13) cm−1 [228.48(16) kJ mol−1], 3948(7) cm−1 [47.23(8) kJ mol−1], and 1401(12) cm−1 [16.76(14) kJ mol−1], respectively, the dissociation energy of the DHe+2 → DHe+ + He reaction, 4033(6) cm−1 [48.25(7) kJ mol−1], the isomerization energy between the two linear isomers of the [H, He, He]+ system, 3828(40) cm−1 [45.79(48) kJ mol−1], and the dissociation energies of the H2He+ → H+2 + He and the H2He+2 → H2He+ + He reactions, 1789(4) cm−1 [21.40(5) kJ mol−1] and 435(6) cm−1 [5.20(7) kJ mol−1], respectively. The FPA estimates of the first dissociation energy of D2He+ and D2He+2 are 1986(4) cm−1 [23.76(5) kJ mol−1] and 474(5) cm−1 [5.67(6) kJ mol−1], respectively. Determining the vibrational fundamentals of the triatomic chromophores with second-order vibrational perturbation theory (VPT2) and vibrational configuration interaction (VCI) techniques, both built around the Eckart–Watson Hamiltonian, proved unusually challenging. For the species studied, VPT2 has difficulties yielding dependable results, in some cases even for the fundamentals of the H-containing molecular cations, while carefully executed VCI computations yield considerably improved spectroscopic results. In a few cases unusually large anharmonic corrections to the fundamentals, on the order of 15% of the harmonic value, have been observed.
Despite their overall relevance, surprisingly little is known about the structure, energetics, and especially the spectroscopy and nuclear dynamics of the HmHe+n species, with m = 0, 1, 2, and 3 and n larger than 1. Definitive high-resolution spectroscopic31,37,38,44–55 and accurate first-principles quantum chemical27,43,56–65 information is available for the different isotopologues of HHe+. However, most of the experimental investigations for the larger systems are still limited to mass spectrometry (MS).29,39,66–70 This is unfortunate as detailed structural and dynamic information, especially relevant for astronomy, cannot be obtained from MS measurements. There are only a handful of optical spectroscopic and variational quantum dynamics investigations on the smallest members of the HmHe+n family,27,29,37,38,41,43–59,61–65,71–79 and even the experimental spectroscopic studies contain data not fully understood (vide infra), due to the limited spectral range and the low resolution of some of the measurements.
Most modern electronic-structure computations agree that HHe+2 is a highly stable, linear, symmetric molecular ion and it forms the central unit, the “dopant”, the “chromophore”, of all the HHe+n clusters studied. The situation about the chromophore of the H2He+n and He+n systems is somewhat more complex. For the H2He+n family, the tetratomic ion H2He+2 is the chromophore for n = 3–5, while for n ≥ 8, the triatomic H2He+ ion becomes the chromophore. In between these n values two isomers seem to coexist. As to the He+n family, for the smaller clusters He+3 is the chromophore, but for larger clusters the chromophore is the diatomic He+2 cation. Obviously, it is the stability of the H2He+2 and He+3 ions toward loosing a He which determines which ion is the chromophore. Due to the change in the chromophore, it is just to call the H2He+n and He+n systems chameleonic.
In recent studies37,38,41,55,79 performed on some of the HmHe+n systems it was observed that computing relative energies and especially anharmonic vibrational fundamentals, especially with n ≥ 3, provides challenges for electronic-structure theory. Thus, we decided to redesign the Gaussian basis sets applied for the electronic-structure computations. We report new atom-centered, fixed-exponent, Gaussian-type basis sets for hydrogen and helium, from triple-ζ (TZ) to hextuple-ζ (6Z) quality. With the help of these basis sets, we determine definitive energies for a number of chemical reactions involving the title ions. Following well-established recommendations,80 all the final relative energies reported in this paper carry a conservative uncertainty estimate. Determination of the final, accurate relative energies and their uncertainties is based on the focal-point-analysis (FPA) technique,81,82 which has been employed successfully during a large number of studies of small and medium-sized species for three decades.83–93
In this study, the zero-point vibrational energies (ZPVE) and the vibrational fundamentals of the triatomic and tetratomic chromophores are determined with the help of second-order vibrational perturbation theory (VPT2)94–96 and the vibrational configuration interaction (VCI)97–100 technique, both built around the Eckart–Watson Hamiltonian.101 It must be noted that determining even the vibrational fundamentals of these molecular cations beyond the harmonic-oscillator model proved to be quite challenging. In the cases when the computed results were deemed to be converged and reasonable, vibrational overtones and combination bands are also reported. Since D-substituted molecular ions of the title compounds of this study are also of interest, both experimentally37,55 and theoretically, spectroscopic results are also reported for the global minima of DHe+, DHe+2, D2He+, and D2He+2, as well.
The rest of the paper is structured as follows. Section 2 provides details about the derivation of the MAX family of Gaussian basis sets for H and He, with the cardinal number of the basis X = T(3), Q(4), P(5), and H(6). The most relevant computational details concerning this study are found in Section 3. Sections 4, 5, and 6 deal with the HHe+n, H2He+n, and He+n species, respectively, as well as with their deuterated isotopologues. To facilitate reading of these sections, the FPA tables are placed into an Appendix. Section 7 summarizes the most important conclusions of this study.
As a result, a new family of basis sets, called MA X, with cardinal numbers X = T(3), Q(4), P(5), and H(6), designed similar to the aug-cc-pVXZ family of Gaussian basis sets,103 was constructed employing neutral and positively charged systems consisting only of H and He atoms (see Table 1 for the structure of the primitive and contracted sets). We did not develop a MAD (X = 2) basis as that would appear to be too small to lead to meaningful results for the systems we are interested in.
Atom | Name | Primitive set | Contracted set | No. of basis functions | |
---|---|---|---|---|---|
H and He | MAT | (10s3p2d) | → | [7s3p2d] | 26 |
MAQ | (11s4p3d2f) | → | [8s4p3d2f] | 49 | |
MAP | (12s5p4d3f2g) | → | [9s5p4d3f2g] | 83 | |
MAH | (13s6p5d4f3g2h) | → | [10s6p5d4f3g2h] | 130 | |
H | aug-cc-pVTZ | (6s3p2d) | → | [4s3p2d] | 23 |
aug-cc-pVQZ | (7s4p3d2f) | → | [5s4p3d2f] | 46 | |
aug-cc-pV5Z | (9s5p4d3f2g) | → | [6s5p4d3f2g] | 80 | |
aug-cc-pV6Z | (11s6p5d4f3g2h) | → | [7s6p5d4f3g2f] | 127 | |
He | aug-cc-pVTZ | (7s3p2d) | → | [4s3p2d] | 23 |
aug-cc-pVQZ | (8s4p3d2f) | → | [5s4p3d2f] | 46 | |
aug-cc-pV5Z | (9s5p4d3f2g) | → | [6s5p4d3f2g] | 80 | |
aug-cc-pV6Z | (11s6p5d4f3g2h) | → | [7s6p5d4f3g2f] | 127 |
To ensure the general applicability of the MAX basis sets without biasing them to particular molecular systems, the basis sets were optimized based on the sum of CCSD energies of the He atom, the He2 and H2 molecules, and the He+2 and HHe+ molecular cations. The reference interatomic distances of the diatomic systems, resulting from geometry optimization at the aug-cc-pV5Z CCSD or CCSD(T) levels, were set to 0.7416 Å, 1.0809 Å, and 0.7746 Å for H2, He+2, and HHe+, respectively. In the case of He2, a strange behavior of the aug-cc-pV5Z basis was observed, which we now address. An irregularity in the geometry convergence pattern was observed: the aug-cc-pV5Z CCSD(T) bond length, at 2.98821 Å, is longer than those coming from aug-cc-pVQZ and aug-cc-pV6Z CCSD(T) computations at 2.98612 Å and 2.97975 Å, respectively. One would expect smooth convergence for geometry parameters, just as observed for the other systems investigated. The end result of our related investigation is that the problem lies in the augmentation exponent for the p shell. The original value of the exponent is 0.14. Increasing this single exponent within the extended aug-cc-pV5Z basis visibly shortens the bond length and slightly modifies the interaction energy (see Table 2). It is also important to point out that the reference geometry of the He2 molecule has a negligible impact on the basis-set optimization as the interaction is extremely weak and thus the optimization minimum is “broad”. Finally, we decided to set the reference HeHe distance to 2.9818 Å, which corresponds to the augmentation p exponent of 0.20 (see Table 2). The optimized MAP basis leads to a very similar bond length, 2.9812 Å, with a p augmentation exponent equal to 0.2074, which perfectly fits the aug-cc-pV5Z_p = 0.21 result (see Table 2). It should also be emphasized that this perfect agreement does not hold for basis sets without diffuse functions, with values of 3.1620 Å for cc-pV5Z and 3.1702 Å for the MP basis, which is the MAP basis with deleted augmentation functions. This points toward the importance of the augmentation exponent for the p shell and is a reason why we suggest using its slightly increased value, 0.16, or, in the case of a structure-oriented investigation, even 0.21.
Basis | r e(HeHe) | Interaction energy | ||
---|---|---|---|---|
No-CBS | CBS(4–5) | CBS(5–6) | ||
aug-cc-pVTZ | 3.0154 | −5.93 | ||
aug-cc-pVQZ | 2.9861 | −6.49 | ||
aug-cc-pV5Z | 2.9882 | −6.86 | −7.20 | −7.79 |
aug-cc-pV5Z_p = 0.15 | 2.9868 | −6.90 | −7.33 | −7.69 |
aug-cc-pV5Z_p = 0.16 | 2.9855 | −6.94 | −7.44 | −7.60 |
aug-cc-pV5Z_p = 0.17 | 2.9844 | −6.97 | −7.52 | −7.52 |
aug-cc-pV5Z_p = 0.18 | 2.9834 | −6.99 | −7.58 | −7.47 |
aug-cc-pV5Z_p = 0.19 | 2.9826 | −7.00 | −7.63 | −7.42 |
aug-cc-pV5Z_p = 0.20 | 2.9818 | −7.01 | −7.67 | −7.38 |
aug-cc-pV5Z_p = 0.21 | 2.9812 | −7.01 | −7.69 | −7.36 |
aug-cc-pV5Z_p = 0.22 | 2.9806 | −7.01 | −7.70 | −7.34 |
aug-cc-aug-pV6Z | 2.9798 | −7.14 | ||
MAT | 3.0218 | −5.98 | ||
MAQ | 2.9872 | −6.78 | ||
MAP | 2.9812 | −7.05 | −7.46 | −7.62 |
MAH | 2.9771 | −7.24 |
Table 3 contains the bond lengths of the species of this study optimized at the MAQ FCI level. The only difference which can be considered significant between the aug-cc-pV5Z CCSD(T) and the MAQ FCI structural parameters is for He2, exhibiting one of the weakest bonds, where the bond length at the MAQ FCI level is 0.0018 Å shorter than the aug-cc-pV5Z CCSD(T) value (vide supra).
Molecular species | EC | ES | Symmetry | r e(HHe) | r e(HeHe) | r e(He⋯He) | r e(HH) |
---|---|---|---|---|---|---|---|
H+2 | 1 | X2Σ+g | D ∞h | 1.05681 | |||
H2 | 2 | X1Σ+g | D ∞h | 0.74163 | |||
HHe+ | 2 | X1Σ+ | C ∞v | 0.77457 | |||
He+2 | 3 | X2Σ+u | D ∞h | 1.08117 | |||
H2He+ | 3 | ![]() |
C ∞v | 1.02324 | 1.09798 | ||
He2 | 4 | ![]() |
D ∞h | 2.98005 | |||
HHe+2 | 4 | ![]() |
D ∞h | 0.92475 | |||
C ∞v | 0.77272 | 2.10340 | |||||
H2He+2 | 5 | ![]() |
D ∞h | 1.23731 | 1.06172 | ||
He+3 | 5 | ![]() |
D ∞h | 1.23608 |
It is worth discussing some of the bond lengths of Table 3. The shortest re(HHe) bond lengths, about 0.77 Å for HHe+ and the complex HHe+⋯He, clearly reflect dative, less strong than the usually covalent, two-electron–two-center (2e–2c) bonds, perturbed slightly in the latter case. Significantly longer re(HHe) bond lengths characterize the global minima of HHe+2 and H2He+. These elongated bonds reflect the change in bond length between a (2e–2c) and a one-electron–two-center (1e–2c) bond, as also seen for the case of H2 and H+2.
The possible problem of linear dependency, that may occur during the optimization of basis-set parameters, was eliminated, following the work of Petersson et al.,104 by an expansion, up to six terms, of the Gaussian exponents' logarithms into orthonormal Legendre polynomials. Exponents of the diffuse functions, one for each shell, were optimized based on the E(20)disp dispersion energy correction105 for two interacting dimers of helium (the He4 system) or two hydrogen molecules (the H4 system), the geometries of which were previously optimized at the aug-cc-pV5Z CCSD(T) level of theory. All basis set optimization calculations were supported by the codes ChemTools106 and MOLPRO.107
During the design of the MAX basis sets the correlation-consistent nature of the Dunning-type basis sets103 was preserved. Nevertheless, the number of s-type exponents has been increased both in the primitive and the contracted sets for all the MAX bases of hydrogen and helium. For the MAT basis adding the s exponents leads to better convergence of the anharmonic vibrational contributions. This is a much cheaper alternative than going to a basis set with an increased cardinal number. This behavior is likely connected to a better description of σ bonds. Note that increasing the number of p exponents fails for the challenging case of He+3 and thus has not been pursued.
The composition of the MAX basis sets, along with that of the correlation-consistent basis sets, of H and He is summarized in Table 1 (vide supra). During this study even the MAH basis, definitely close to the CBS limit, could be employed at the FCI level for most of the species (vide infra).
As part of the FPA scheme, extrapolations to the CBS limit need to be performed. These extrapolations are done separately for the Hartree–Fock (HF) level and for the electron-correlation increments. The usual111–113 two-points formulas were employed for this task.
The CBS FCI energies are augmented with so-called “small (auxiliary) corrections”,81,82 in the present case the diagonal Born–Oppenheimer (DBOC)114 and the relativistic115 corrections. Finally, the electronic energies of the molecular species are corrected for the effect of vibrations through the inclusion of ZPVEs. The most significant advantage of the original FPA method, compared to derivative composite approaches (e.g., HEAT116,117), is that FPA is designed to yield appropriate uncertainties for the computed relative energies.80–82
The tables corresponding to the FPA analyses of this study, Tables 8–14, are put into the Appendix. All of them are quoted in the appropriate sections of the main body of the paper; in particular, in Sections 4–6.
Geometry optimizations were performed not only for primary but also for secondary minima, corresponding to He-solvated ions. For example, in the case of HHe+2 the global minimum is linear He–H+–He, with an equilibrium structure of D∞h point-group symmetry, while the secondary minimum is linear He⋯He–H+, with an equilibrium structure of C∞v point-group symmetry (Table 3). The effect of basis set superposition error on the computed distances118 was not investigated.
The diagonal Born–Oppenheimer corrections114 to the electronic energies were estimated at the MAQ CCSD level. These “small corrections”81,82 are expected to be larger than the relativistic ones for the light H- and He-containing systems studied here. Nevertheless, if sufficiently large uncertainties are attached to these computed values than there is little doubt that their effect is taken correctly into account during the FPA study of relative energies of the title molecular cations. As the systems studied are light, the choice of masses used during the DBOC computations does matter for them.60 In this study, the following masses have been used: m(H) = 1.007 825 u, m(D) = 2.014 102 u, and m(4He) = 4.002 603 u.
During the execution of the project, issues have been found with some of the VPT294–96 results, which are usually quite accurate for the fundamentals of semirigid molecules when resonances are absent or not overly strong. Due to these problems, we decided to obtain the ZPVE values and the vibrational fundamentals with the VCI technique, which should be devoid of vibrational resonances. When the difficulties encountered could not be solved either at the VPT2 or VCI levels, the corresponding entries in Tables 4–7 are not provided. In such cases variational nuclear-motion computations utilizing a nuclear Hamiltonian written in internal coordinates121 and a carefully fitted purpose-built PES are required to obtain meaningful results.
Species | Technique | ZPVE |
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a All the values reported correspond to different representations of the PES around the global minimum of HHe+2 obtained at the MAQ CCSD(T) level of electronic-structure theory. | ||||||||||
HHe+2 | Harmonic | 2298.0 | 1139.3 | 953.4 | 1550.1 | |||||
VPT2 | 2267.0 | 896.5 | 883.7 | 1352.0 | — | — | — | — | — | |
VCI | 2254.8 | 957.5 | 882.7 | 1318.6 | 1889.3 | 1739.4 | 1778.2 | 2972.4 | 2086.7 | |
Experiment38 | — | 1315.8 | ||||||||
DHe+2 | Harmonic | 1858.7 | 1139.3 | 711.1 | 1156.0 | |||||
VPT2 | 1829.9 | 930.9 | 670.3 | 1034.6 | — | — | — | — | — | |
VCI | 1825.7 | 975.4 | 670.3 | 1023.3 | 1854.9 | 1356.4 | 1349.8 | 2153.8 | 1848.3 |
Species | Technique | ZPVE |
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a See footnote a to Table 4. b Broad estimates from low-resolution experiments. The experimental uncertainties of the FELIX data of ref. 55 are typically 0.5%. c Due to the multi-photon process of detection, the experimental signal is very broad, shifted and unspecific. Thus, the assignment of the experimental signal was deemed to be tentative in ref. 55. Therefore, no uncertainty is quoted for these data in this table. d Numerical convergence issues, see text. | ||||||||||
H2He+ | Harmonic | 2181.0 | 1935.5 | 719.2 | 988.2 | |||||
VPT2 | 2096.5 | 1860.5 | 655.3 | 741.9 | — | — | — | 1297.2 | 1291.4 | |
VCI | 2092.3 | 1830.1 | 640.0 | 735.4 | —d | —d | —d | 1302.9 | 1243.1 | |
Variational78 | 1832 | 648 | 734 | |||||||
Variational55 | 2090.8 | 1833 | 640.0 | 732 | 1136.1 | 1256.4 | ||||
Experiment55b | — | 1840(9) | 695 | 840 | 1159 | |||||
D2He+ | Harmonic | 1610.1 | 1369.4 | 521.8 | 807.3 | |||||
VPT2 | 1560.9 | 1350.3 | 484.2 | 641.9 | 2455.4 | 929.2 | 970.1 | 1120.8 | 1061.6 | |
VCI | 1559.1 | 1323.8 | 478.3 | 642.6 | 2533.9 | 902.3 | 950.8 | 1158.5 | 1052.2 | |
Variational55 | 1317.5 | 478.4 | 640.9 | 891.1 | 1139.6 | |||||
Experiment55b | — | 1309 | 459 | 670 | 860 | 1145 |
The VPT2 calculations of this study were executed with the help of the MOLPRO99,100,107 package. The vibrational configuration interaction (VCI) facility of MOLPRO has been used extensively to determine ZPVEs and vibrational fundamentals, overtones, and combination bands. While the VPT2 calculations rely on a quartic force field (QFF) approximation122,123 of the potential energy surface (PES) around the reference structure, the VCI treatment employs a multidimensional PES represented by an n-mode expansion, truncated after the 4-mode coupling terms.124 Symmetry has been fully exploited within the electronic structure calculations and within the individual terms of the n-mode expansions.100 The resulting grid representation of the PES has been transformed to polynomials using efficient Kronecker product fitting, with up to 12 monomials per mode.125 Optimized one-mode wave functions (modals) have been determined from vibrational self-consistent field (VSCF) theory based on a primitive basis of 16 distributed Gaussians.126 Subsequent state-specific configuration-selective VCI calculations made use of a basis of single Hartree products, including up to quadruple excitations. A constant μ-tensor has been used throughout. In order to ensure that the VCI results are converged, a large number of consistency checks have been performed with respect to the polynomial fit and the correlation space.
The QTAIM approach127 relies on the fact that critical points of the electron density, ρ, defined by the equation ∇ρ = 0, arise when there is interaction between atoms of the molecule. In other words, presence of a bond path linking the interacting atoms via a bond critical point (BCP) is evidence of an interaction. In all cases studied, BCPs and bond paths have been found where expected, suggesting that simple chemical intuition agrees with the sophisticated analysis of the electron density.
The code AIMAll128 has been used for the QTAIM127 calculations, yielding atomic charges, bond critical points (BCP), and bond paths, supporting a qualitative understanding of some of the quantitative results of this study (for this reason, no quantitative results, apart for charges, are provided). MAQ CCSD wave functions computed with the Gaussian16 code129 have been utilized for these calculations.
The electronic-structure computations utilized the packages MOLPRO,107 CFOUR,130 and MRCC.131 The CCSDTQ geometry optimizations were performed with the MRCC code, employing ultratight convergence criteria in order to avoid the nonzero-force dilemma.108 The CFOUR package was used for the auxiliary relativistic (MVD1) and the DBOC computations. All VPT2 and VCI results reported were obtained with the help of the MOLPRO package. Note that extensive modifications of the default values characterizing the VCI procedure within MOLPRO were required to obtain converged results for almost all of the molecular cations studied.
As detailed in Table 8, the electronic proton affinity (PA) of He, defined by the reaction He + H+ → HHe+, obtained with the FPA approach and the MAX basis-set family, is 16459(7) cm−1, very similar to a previous theoretical estimate, 16 457(8) cm−1,41 and an empirical estimate of 16
456.2(1) cm−1.60 The electron-correlation contribution to this value is unusually small: the CBS HF estimate is 15 673(1) cm−1. It is noteworthy that the reaction energy obtained at the MAT HF level is substantially closer, by more than 50%, to the CBS HF value than the one obtained with the aug-cc-pVTZ basis set, proving the good performance of the MAT basis, at least for cationic He-containing species.
The mass-dependent DBOC corrections to the reaction partners are relatively substantial, at the MAQ FCI level they are +92.1, +100.2, and +92.4 cm−1 for He, HHe+, and DHe+, respectively. The reaction-energy DBOC corrections are, of course, much smaller, −8.1(25) and −0.3(10) cm−1 for HHe+ and DHe+, respectively. The relativistic corrections are −24.7 and −23.9 cm−1 for He and HHe+, respectively, yielding a relativistic correction of −0.8(10) cm−1, which is negligible even at the level of accuracy sought in this study. Overall, the auxiliary corrections to the proton and deuteron affinities of 4He amount to −9(3) and −1(1) cm−1, respectively.
The harmonic ZPVE estimates for HHe+ and DHe+ are 1610(2) and 1248(2) cm−1, obtained at the MAQ FCI level, where the attached uncertainties reflect other harmonic ZPVE values computed with different basis sets (not reported). The VPT2 corrections to these values are −35.2(30) and −21.1(25) cm−1, obtained at the same level, yielding anharmonic ZPVE estimates of 1575(4) and 1227(3) cm−1 for HHe+ and DHe+, respectively. The electronic proton and deuteron affinities must be corrected with these anharmonic ZPVE estimates as the other reaction partners are atoms. Note that (a) the ZPVE values obtained are significantly higher than half of the calculated fundamentals of these diatomic molecular ions, 2911 and 2310 cm−1 for HHe+ and DHe+, respectively; and (b) the anharmonic corrections to the fundamentals are unusually large, approaching 10%.
Overall, our FPA estimate of the 0 K proton affinity of the ground-state 4He atom is 14875(9) cm−1. This computed value compares extremely well with the best available literature results, namely 14
876(12),41 14
873,57 and 14
863(8)133 cm−1. Based on the present FPA analysis, the deuteron affinity of 4He is 15
229(8) cm−1. To the best of our knowledge, there are no experimental values to compare this predicted estimate with.
The QTAIM analysis, see Fig. 1, indicates that the bond critical points are close to the proton for both isomers of HHe+2. Clearly, the larger the charge on the H atom the closer the BCP is to it. Furthermore, in the case of the SM, there is a BCP in between the two He atoms, indicating the presence of a weak interaction. The QTAIM charges have very different numerical values compared to the Mulliken charges reported in ref. 41; nevertheless, the qualitative picture does not change in a significant way. According to QTAIM, the positive charge remains much more localized on the proton and delocalizes much less to the He atom(s). The solvating He acquires only a very small positive charge, it is only +0.004, as shown in the example of the secondary minimum. It is also important to point out that upon solvation by a He atom the charge on the proton remains basically the same, the charge on the proton in HHe+ and the SM of HHe+2 are +0.947 and +0.946, respectively (see Fig. 1).
The T1 diagnostic values of CCSD theory134 are 0.0015, 0.0053, 0.0051, and 0.0040 for He, HHe+, HHe+2(GM), and HHe+2(SM), respectively, obtained with the MAQ basis, suggesting that the electronic structure of all these chemical entities can be described very well with single-reference CC techniques. Furthermore, very fast convergence of the correlation-energy increments in the He + H+ → HHe+ and HHe+ + He → HHe+2 reaction energies is expected, which is indeed the case (see the relevant entries of Tables 8 and 9). The energies of the reactions HHe+2 → HHe+ + He (Table 9) and HHe+2 → H+ + 2He (total atomization energy (TAE), which can be deduced from Tables 8 and 9 by adding the appropriate entries) suggest that the binding of one and two He atoms to a proton is strongly exothermic. The CBS FCI-based FPA value for the TAE of HHe+2 is 21 091(12) cm−1, which should be compared to the electronic TAE of HHe+, 16459(7) cm−1.
The pure electronic dissociation energy of the HHe+2 → HHe+ + He process is 4632(4) cm−1 (Table 9), very similar to its previous estimate, 4631(10) cm−1.41 As expected, convergence of the CC series is very pronounced. The small auxiliary DBOC corrections to the electronic energy of the GM of HHe+2 and DHe+2 are +194.4 and +185.8 cm−1, respectively. Thus, the DBOC corrections to the dissociation reactions are 2.1 and 1.3 cm−1 for HHe+2 and DHe+2, respectively, minuscule values to which a reasonable uncertainty of ±2 cm−1 can be attached. The relativistic correction to the energy of the GM is −48.7 cm−1. Thus, using the relativistic corrections of −23.9 and −24.7 cm−1 for HHe+ and He, respectively, the relativistic correction to the reaction is 0.1(10) cm−1, a completely negligible value. Overall, the auxiliary correction to the first dissociation energy is 2(2) cm−1 for both HHe+2 and DHe+2.
The harmonic ZPVE of HHe+2 is estimated to be 2298 and 2299 cm−1 at the MAQ CCSD(T) and CCSDTQ ≡ FCI levels, respectively (Table 4 contains the MAQ CCSD(T) value). The VCI treatment yields an anharmonic ZPVE estimate of 2254.8 cm−1, and, based also on further, unreported results, we accept 2255(3) cm−1 as the best estimate of the ZPVE of the GM of HHe+2. The value of 2255(3) cm−1 for the ZPVE of GM-HHe+2 is unusual in the sense that it is more than 200 cm−1 higher than half of the sum of the calculated fundamentals. The ZPVE of DHe+2 is 1826(3) cm−1, with characteristics similar to those of HHe+2.
Based on the ZPVE estimates of 1575(4) and 2255(3) cm−1 for HHe+ and HHe+2, respectively, the estimated ZPVE-corrected first dissociation energy of HHe+2 becomes 3948(7) cm−1, comparing favorably with a previous estimate, 3931(20) cm−1.41 As to the first dissociation energy of DHe+2, it can be estimated by using the ZPVE values of 1227(3) and 1826(3) cm−1 for DHe+ and DHe+2, respectively. The final FPA estimate is thus 4033(6) cm−1.
The FPA energy difference between the GM and SM isomers of HHe+2 is substantial on the energy scale of cations containing He-solvated proton, it is 4368(6) cm−1 (see Table 10), without taking into account the ZPVEs. Comparison of this value with the estimate of the first dissociation energy, 4632(4) cm−1, suggests a very shallow minimum and a small He-solvation energy, 264(7) cm−1. It is also of interest to note that while the HF contribution is very substantial for the global minimum, a direct consequence of the two strong bonds within this ion, for the secondary minimum of HHe+2 the solvation energy comes to a much larger extent from electron correlation. The mass-dependent DBOC corrections to the electronic energy of the SM of HHe+2 and DHe+2 are +192.3 and +185.8 cm−1, respectively. Not too surprisingly, these values are almost exactly the same as the sum of the DBOC corrections of the He and (H/D)He+ systems, (+192.3/+184.5) cm−1. This means that the DBOC corrections to the isomerization reaction are 0.0(10) and 1.3(10) cm−1 for HHe+2 and DHe+2, respectively. The relativistic correction to the electronic energy of the SM is −48.6 cm−1. This means that the relativistic correction to the isomerization reaction is 0.1 cm−1, once again a completely negligible quantity. Given the uncertainties of all the other contributions to the izomerization energy, especially that coming from the vibrations (vide infra), the auxiliary corrections can be safely neglected.
Due to the extremely weakly-bound nature of the SM, we did not make an attempt to determine the anharmonic estimate of its ZPVE. The energy difference of the two linear isomers of HHe+2 becomes 3828(40) cm−1 when the harmonic ZPVE correction of 540(40) cm−1 is taken into account.
The dissociation energy of the SM of HHe+2 can be associated with the He-solvation energy of the HHe+ molecular ion. The He-solvation energy thus determined is 124(42) cm−1. Note that the smallness of the He-solvation energies is what allowed the characterization of the dynamics of these molecular ions via action spectroscopy in the infrared region.37,38
Table 4 contains a considerable amount of accurate information about the vibrational fundamentals, overtones, and a combination band, 1 +
3(σ−u), of HHe+2 and DHe+2, obtained with the help of different theoretical models. Note that the potential for HHe+2 along the normal coordinate of ω3 is strongly quartic in nature, with a huge quartic normal-coordinate force constant of F3333 = 6611.3 cm−1. This, associated with the need for high-order monomials within the fitting of the n-mode representation of the PES, leaves some doubt whether a representation of the PES by a QFF, as used within the VPT2 calculations, is appropriate. The strong quartic character of the
3 vibration is also revealed by the fact that the computed values for 2
3 are significantly larger than twice the value of the corresponding fundamental. As a consequence, the results are very sensitive to the potential and the only calculation in good agreement with the experimental value38 for
3 is provided by the VCI calculation, based on the n-mode representation of the PES, which accounts for high-order terms. The sensitivity of the computed fundamentals on the level of theory can be further demonstrated by a comparison of the harmonic frequency for ω1 at 1139.3 cm−1 (MAQ CCSD(T)) and the respective anharmonic values for
1 listed in Table 4. An anharmonicity correction of about 21% (VPT2) would be exceptionally large, while a value of about 15% (VCI) is still very large, but it is closer to the usual range. The expected uncertainty of the fundamentals of HHe+2 obtained by VCI should be about ±5 cm−1. This means that the VPT2 value for the
1(σ+g) fundamental is seriously in error. As to the overtones of HHe+2, 2
02(σ+g) and 2
1(σ+g) are coupled by a strong Darling–Dennison resonance. This is an unexpected observation as the corresponding harmonic energy difference is larger than 370 cm−1. However, the strong anharmonic corrections bring these states close together. Therefore, we do not report VPT2 results for the overtones and the combination band of this system as resonance corrections were found to be large and tend to deteriorate the results in comparison to the VCI values. Differences between these two sets of data were found to be huge and thus we conclude that VPT2 is not capable of predicting the overtones of HHe+2 correctly. It is also expected that the accuracy of the overtones and the combination band computed with VCI is worse than that of the fundamentals, i.e., it would not be surprising if some of the VCI values were in error by as much as 30 cm−1.
In Table 4 spectrocopic results are presented for DHe+2, as well. It is considerably easier, compared to HHe+2, to obtain converged results for DHe+2. As to the fundamentals, the VPT2 and VCI results agree with each other rather well, except for 1(σ+g). In fact, the situation is similar to the case of HHe+2, where the VCI result is some 60 cm−1 higher than the VPT2 result. For DHe+2, the difference is less, about 45 cm−1, but clearly the VPT2 value for
1(σ+g) must be the incorrect one. An inspection of the polynomial coefficients for the one-dimensional cut along this mode reveals that there are fairly strong contributions from the fifth- and seventh-order terms; these terms, of course, are not included in the QFF of the VPT2 calculations. The uncertainties of the VCI fundamentals of DHe+2 are estimated to be ±4 cm−1. While VCI values are presented for the overtones and the
1 +
3(σ−u) combination band, the reported values might have a sizeable uncertainty, up to about 25 cm−1.
Fig. 2 presents the equilibrium structures of the H2He+, H2He+2, and He+3 molecular ions, together with QTAIM atomic charges and BCPs. As observed for HHe+ and HHe+2 (vide supra), the partial charges on the He atoms are relatively small for both H2He+ and H2He+2, only 0.06 and 0.04, respectively. The position of the BCPs clearly confirm that all these systems are molecular ions.
The first dissociation energy of H2He+, toward loosing the He atom, is not large. As Table 11 attests, the pure electronic part is just 2730(3) cm−1. The HF contribution, at the CBS limit, is only about half of this value, 1422(1) cm−1. All energy increments in the FPA hierarchy of HF → MP2 → CCSD → CCSD(T) → CCSDT ≡ FCI are positive and decrease drastically with each consecutive improvement of the level of electron-correlation treatment. In fact, the increment from CCSD(T) to CCSDT (≡FCI) is only 7(1) cm−1, making, not too surprisingly, the gold-standard CCSD(T) level very accurate for this three-electron system.
The availability of high-quality PESs135,137 and related variational estimates55,135 of the rovibrational states of H2He+ allows the tight determination of the ZPVE of this molecular ion (Table 5). According to ref. 55, the ZPVE of H2He+ is 2090.8(20) cm−1. This, in turn, allows the estimation of the accuracy of the VPT2 and VCI ZPVE results for this species, which are 2096 and 2092 cm−1, respectively, both at the MAQ CCSD(T) level. Clearly, for this basically semirigid system both VPT2 and VCI perform very well, as expected123 for a system where the stretching motions dominate for the ZPVE. The suggested ZPVE of H2He+ is 2091(2) cm−1. The tight agreement of the various first-principles results for H2He+ suggest a similarly high accuracy, ±2 cm−1, for the VCI estimate of the ZPVE of D2He+, which is thus 1559(2) cm−1.
The ZPVE corrections to the dissociation reactions H2He+ → H+2 + He and D2He+ → D+2 + He can be calculated utilizing the ZPVEs of H+2 and D+2, 1149.5(1)138 and 815(1) cm−1, respectively (the latter value was obtained with the MAQ basis and a VPT2 treatment). Thus, the ZPVE-corrected dissociation energies are 1789(4) and 1986(4) cm−1 for H2He+ and D2He+, respectively. The so-called FCI PES of ref. 135 exhibits a dissociation energy of D0 = 1794 cm−1 for para-H2He+ and D0 = 1852 cm−1 for ortho-H2He+.55,135 The latter value is higher due to the allowed dissociation to a j = 1 state, where j is the quantum number associated with the rotation of the H+2 product. The nearly perfect agreement of the FPA estimate of the dissociation energy of para-H2He+, 1789(4) cm−1, with the one characterizing the FCI PES of ref. 135, 1794 cm−1, points toward the accuracy of the rovibrational results obtained with the FCI PES reported in ref. 55. Note that the ν1 fundamental of H2He+, 1840(9) cm−1,55 see also Table 5, is right in between the two D0 dissociation energies. This ensured the straightforward spectroscopic observation of this fundamental.55
The fundamentals of H2He+ are 1861(1(σ+)), 655(
2(π)), and 742(
3(σ+)) cm−1 within the VPT2 approximation at the MAQ CCSD(T) level and 1833, 640, and 732 cm−1, respectively, obtained variationally.55 Previous variational results,78 employing a different PES, resulted in fundamentals at 1832, 648, and 734 cm−1, respectively. This is a quite reasonable agreement among the different modeling results and supports the relative rigidity of the structure against single-quantum excitation. The present VCI and the previous variational results also agree very well. The uncertainty of the variational values55,78 is probably a maximum of ±5 cm−1, judged by the accuracy of the D0 value of the PES.
One must note that near resonances have been detected in the VPT2 calculations, e.g., 2ω3 ≈ ω1 (see Table 5), for H2He+. These resonances lead to shifts for several of the calculated vibrational modes, even up to hundreds of cm−1, i.e., huge discrepancies can be observed between the perturbed and the diagonalization-corrected VPT2 results. For the overtones of H2He+, with the exception of 23(σ+), the VCI calculations did not converge with respect to a systematically increased correlation space. For this reason these values are not provided in Table 5.
Convergence problems proved to be much less severe for the deuterated compound. It is expected that the fundamentals and overtones obtained for D2He+via the present VCI and the previous variational treatments are accurate, to better than ±5 cm−1 for the fundamentals and better than ±20 cm−1 for the overtones. Note, the 23(σ+) overtone for D2He+ computed at the VPT2 level deviates by almost 40 cm−1 from the VCI value. However, this arises mainly from a significant resonance correction, while the original perturbed value of 1166.2 cm−1 is in much better agreement with the VCI number, 1159 cm−1. The VCI result of this study and a dependable variational estimate,55 1140 cm−1, are in good agreement with the experimentally measured value, 1145 cm−1.55
Finally, a few words about the experimental estimates of the fundamentals of H2He+ and D2He+.55 The spectroscopic measurements are supported by the quoted first-principles results for 1 (1840(9)55vs. 1833(5) cm−1). Due to substantial technical difficulties related to the measurements, dependable experimental results are still not available for the other two fundamentals, as the “tentative experimental results” published for
2 and
3 in ref. 55 for H2He+ are supported neither by the VPT2 nor by the variational computations. As to D2He+, while the
2 fundamental's position might be approximately correct, it seems very likely that the position of the
3 fundamental is incorrect. Thus, high-resolution spectroscopic studies are needed to settle the position of these two fundamentals both for H2He+ and D2He+.
At the levels of electronic-structure theory employed, CCSD(T) and CCSDT, and with basis sets MAT and MAQ, the solvated linear complex of H2He+2, an assumed secondary minimum on the PES of the [H, H, He, He]+ system, is not a minimum but a second-order saddle point. Thus, this possible isomer was not considered further during this study. Nevertheless, this configuration may become important, even if it is not a minimum, once the true dynamics of H2He+2 will be studied, either experimentally under high resolution or computationally via suitable variational nuclear-motion techniques, close to the first dissociation limit.
The reaction studied for the title species is H2He+2 → H2He+ + He (see Table 12 for the FPA results). As the data of Table 12 show, the dissociation energy is not large, the CBS FCI electronic energy is only 652(3) cm−1. HF theory does not provide a good estimate of this value, as the CBS HF estimate is only 91.3(2) cm−1. Nevertheless, MP2 theory is able to provide most (almost 80%) of the correlation energy and then the convergence of the correlation-energy increments is fast. In fact, the increments beyond CCSDT are negligible, considerably smaller than the assumed uncertainty of the CBS FCI estimate.
While harmonic fundamentals could be determined for H2He+2, see Table 6, attempts to use VPT2 and VCI to treat the unusual spectroscopic characteristics of this relatively weakly bound molecular ion did not result in reasonable and converged fundamentals. Part of the problem is that, for H2He+2, during the calculation of the outer regions of the PES convergence problems within the underlying RHF calculations were observed. Such behavior usually indicates close-lying, intruder electronically excited states. It seems that a variational treatment utilizing an internal-coordinate Hamiltonian and a carefully derived high-level PES, obtained after taking care of the intruder state problem, is needed to estimate accurate anharmonic fundamentals for H2He+2.
For He–H+2–He, the VCI and the VPT2 treatments yield 2311 and 2305 cm−1 for the ZPVE. For He–D+2–He, the VCI and VPT2 ZPVE values are 1738 and 1735 cm−1, respectively. Thus, unlike the fundamentals, the ZPVE results look fairly reliable and consistent and the anharmonic values of 2308(5) and 1737(4) cm−1 are accepted for He–H+2–He and He–D+2–He, respectively.
It is clear that the pure electronic contribution to the binding energy of the second helium to the H+2 core is much smaller, 652(3) cm−1, than that of the first helium, 2730(3) cm−1. Based on the computed values for the triatomic species, the “auxiliary” corrections are neglected for (H2/D2)He+2, they are considered only as contributions of 0(3) cm−1. The ZPVE corrections decrease these values significantly. Our final estimates are 1789(4) cm−1 and 435(6) cm−1 for the attachment of one or two He atoms to the H+2 core. The disparity in these values helps to explain why the chromophore of the H2He+n complexes changes with the increase in n from H2He+2 to H2He+. The first dissociation energy of D2He+2 is 474(5) cm−1.
According to ref. 25 and 31, in ionized helium clusters the core (chromophore) contains either two or three He atoms. The smaller clusters are characterized by the triatomic chromophore, He+3, while the larger ones by the diatomic chromophore, He+2.
All He+n clusters are open-shell species, the ground electronic states of He+2 and He+3 are X2Σ+u and X2Σ+g, respectively. The T1 diagnostic values of CCSD theory, obtained with the help of the MAQ basis, are 0.0039 and 0.0233 for the He+2 and He+3 cations. Thus, the single-reference CC techniques should perform reasonably well even for He+3, though this is by far the largest T1 diagnostic value found during this study.
As seen in Fig. 2, in the case of He+3, the charge on the middle He atom is about twice as large as that on the terminal ones. Thus, the solvating He atoms of the larger He+n complexes will be placed around the middle atom, as expected.
The auxiliary relativistic and DBOC corrections to the dissociation energy are minuscule, only 2.2 and 0.2 cm−1, respectively. Their overall effect can be estimated as 2(2) cm−1, a basically negligible correction.
The ZPVE correction to the dissociation reaction, coming simply from He+2, is −860 cm−1, obtained from VPT2 theory, and it can be considered well determined at the MAQ CCSD level, perhaps with an uncertainty of ±4 cm−1. Thus, the final FPA estimate of the dissociation energy of He+2 is 19 099(13) cm−1. This value is in excellent agreement with the literature value of 19 075(50) cm−1 (2.365 ± 0.006 eV),139 but it has a considerably smaller uncertainty.
At all the high levels of electronic-structure theory employed, CCSD(T), CCSDT, and CCSDTQ, and with basis sets MAT and MAQ, the solvated linear complex He⋯He+2, an assumed second stationary point on the PES of the [He, He, He]+ system, could not be located. We also attempted to locate another possible He-solvated complex, a T-shaped stationary point of C2v point-group symmetry. This stationary point turned out to be, in accordance with previous electronic-structure studies,20,28 a first-order saddle point. Since it is not a secondary minimum on the PES of the [He, He, He]+ system, this stationary point was not considered further.
The dissociation energy of He+3 is simply defined by the reaction He+3 → He+2 + He. As shown in Table 14, the electronic dissociation energy is relatively small, 1490(10) cm−1. The HF limit value is basically achieved with the MAQ basis. The most interesting aspect of the FPA table is that at the HF level He+3 is simply unbound, as shown by the negative HF entries of Table 14. This points toward the uniqueness of He+3 in the set of chromophores investigated. In other words, He+3 is bound by correlation forces. With a ZPVE of 949(4) cm−1 (Table 7), the final FPA estimate of the dissociation energy becomes 1401(12) cm−1. The present FPA estimate compares fairly well with a much less accurate experimental value, 1371(242) cm−1.18
The vibrational wavenumbers obtained for He+3 during this study are listed in Table 7. Except for 1, the agreement of the VPT2 fundamentals with the VCI values is good.
1 shows a strong Fermi resonance with the overtone of
3, i.e., 2
3, which leads to huge corrections within the VPT2 formalism. Furthermore, the leading VCI coefficient for the
1 state is as low as 0.57 and the underlying leading configuration contributes significantly to five different states. This makes it rather difficult to name the transition at 617.1 cm−1 a fundamental transition as it carries no distinct state identity. Formally this mode shows an anharmonicity effect of more than 30%, but such a consideration is at its limit when the state identity has been lost. Note, the one-dimensional potential along the normal coordinate shows extremely strong anharmonic contributions up to high order monomials, which readily explains the difference between the VPT2 and VCI results. The VCI calculations rely on ground-state-based modals as the underlying VSCF calculations did not converge for the vibrational state corresponding to
3. (Note that Rosi and Bauschlicher20 identified the extreme anharmonicity in the one-dimensional potential of the
3 mode.) The VPT2 and VCI results for the 2
02(σ+g) overtone deviate significantly, but this arises mainly from the resonance correction within the VPT2 formalism, while the uncorrected VPT2 value of 447.8 cm−1 is in reasonable agreement with the VCI result. The VCI results for 2
1(σ+g) and 2
3(σ+g) were found to be strongly dependent on the correlation space and are thus numerically not stable. Therefore, they are not reported in Table 7. It'd be interesting to investigate experimentally, under high resolution, the challenging internal dynamics of the He+3 molecular ion, in order to confirm the curious quantum-chemical results of the present study. The uncertainty of the VCI estimates of the fundamentals of He+3 is about ±8 cm−1. The best candidate for a dipole-allowed spectroscopic observation is the
3 fundamental at 443(8) cm−1.
The MAX basis sets have been employed to obtain definitive results for the structure, energetics, and vibrational characteristics of the chromophores of He-solvated HmHe+n complexes with m = 0, 1, and 2, as well as for their deuterated analogues. In order to obtain well-defined uncertainties for the computed relative energies, the focal-point analysis (FPA) approach was employed. For the FPA analysis extensive coupled-cluster electronic-structure computations up to full configuration interaction (FCI) were performed, which usually could be afforded even with the MAH basis. The large set of computations performed allow to estimate the CBS FCI limit for the relative energies. It holds for all the reactions studied that special relativity has no significant effect on the energetics (less than 1 cm−1). The DBOC corrections to the dissociation energies are basically zero for the weakly-bound cases but somewhat more significant, with values of a few cm−1, for the more strongly bound molecular ions. The complete neglect of the “auxiliary” corrections seems to be justifiable for the set of HmHe+n molecular cations studied. Nevertheless, in a number of cases the “auxiliary” corrections have been considered explicitly and added to the CBS FCI values.
HHe+ and HHe+2 are relatively strongly bound, linear molecular ions, with equilibrium bond lengths of 0.775 and 0.925 Å, respectively (obtained at the MAQ FCI level). HHe+2 is the chromophore of the He-solvated HHe+n complexes. The FPA estimates of the proton and deuteron affinities of He are 14875(9) and 15
229(8) cm−1, respectively. The former value agrees with the best experimental value, 14
863(8) cm−1.133 The first dissociation energy of HHe+2 (the reaction energy of the HHe+2 → HHe+ + He process) is 3948(7) cm−1. The isomerization energy between the two linear isomers of HHe+2, the global minimum, He–H+–He, and the secondary minimum, He⋯He–H+, is 3828(40) cm−1, clearly reflecting the instability of the He-solvated He⋯He–H+ complex.
The molecule-like species H2He+, the chromophore of the H2He+n species with smaller n values, also has a linear equilibrium structure. The FPA dissociation energy of the H2He+ → H+2 + He reaction is 1789(4) cm−1. The FPA estimate of the similar dissociation energy of D2He+ 1986(4) cm−1.
The species He+n allow the investigation of the solvation of a positive charge by an increasing number of He atoms. He+3 has a linear equilibrium structure and this is the chromophore of a number of He+n species with smaller n values. An interesting property of the He+3 molecular cation is that it is not bound at the HF level, it is hold together by correlation forces. The FPA estimate of the first dissociation energy of He+3 is 1401(12) cm−1.
The triatomic chromophores investigated are neither particularly strongly bound (they have a low first dissociation energy toward loosing a helium atom) nor they contain heavy elements. Consequently, both the perturbational second-order vibrational perturbation theory (VPT2) and the variational vibrational configuration interaction (VCI) treatments are prone to difficulties, though of different nature. The level of difficulty increases as the dissociation energy decreases, the most problematic molecular ion is clearly He+3. For the class of species investigated, VPT2 turns out to be unreliable to predict the ions' spectroscopic signatures. VPT2 does work in some cases but the importance of high-order potential terms and exceptionally strong resonances lead to clear failures of VPT2 even for the fundamentals of the molecular cations. The occurrence of strong mode couplings and a multitude of resonances within the vibrational structure calculations require highly accurate potential energy surfaces and carefully converged vibrational wavefunctions. VCI seems to perform well for the fundamentals, but larger correlation spaces are needed than in standard calculations. In some cases very low leading VCI coefficients render a clear state assignment problematic and result in a loss of state identities. The most clearcut failure of the VPT2 and VCI treatments is for H2He+2, for which we were not able to report even the anharmonic fundamentals. Sizeable anharmonic corrections, up to 30%, for the vibrational fundamentals is another unique feature of these species.
Zero-point vibrational energy (ZPVE) values of the triatomic and tetratomic molecular cations could be determined in all cases at the variational VCI level, employing PES representations determined at the MAQ CCSD(T) level. With conservative uncertainty estimates, the best ZPVE estimates for the triatomic ions HHe+2, H2He+, and He+3 are 2255(1), 2092(2), and 949(4) cm−1, respectively. The corresponding values for the deuterated isotopologues DHe+2 and D2He+ are 1826(1) and 1559(2) cm−1, respectively. The ZPVE values of the H2He+2 and the D2He+2 ions are 2308(5) and 1737(4) cm−1, respectively.
Basis | ΔEe(HF) | δ[MP2] | δ[CCSD] | ΔEe[FCI] |
---|---|---|---|---|
a The symbol δ denotes the increment in the relative energy (ΔEe) with respect to the preceding level of theory in the hierarchy HF → MP2 → CCSD ≡ FCI. CBS = complete basis set. The basis set extrapolations are described in the text, they are based on the cardinal number X of the bases. Uncertainties are given in parentheses. All energy values are given in cm−1. | ||||
aug-cc-pVTZ | 15601.8 | +587.3 | +226.3 | 16415.4 |
aug-cc-pVQZ | 15660.6 | +569.5 | +231.6 | 16461.7 |
aug-cc-pV5Z | 15670.7 | +561.6 | +232.7 | 16465.0 |
aug-cc-pV6Z | 15672.7 | +554.8 | +235.3 | 16462.8 |
CBS(3–4) | 15668.6 | +556.6 | +235.5 | 16460.7 |
CBS(4–5) | 15672.5 | +553.3 | +233.9 | 16459.6 |
CBS(5–6) | 15673.2(10) | +545.4(60) | +238.8(30) | 16457.4(80) |
MAT | 15643.3 | +577.6 | +222.2 | 16443.1 |
MAQ | 15668.6 | +569.2 | +227.8 | 16465.5 |
MAP | 15671.9 | +560.0 | +232.7 | 16464.5 |
MAH | 15673.1 | +554.8 | +234.9 | 16462.8 |
CBS(3–4) | 15672.0 | +563.0 | +231.9 | 16466.9 |
CBS(4–5) | 15672.5 | +550.3 | +237.8 | 16460.6 |
CBS(5–6) | 15673.4(10) | +547.8(50) | +237.9(30) | 16459.1(70) |
Basis | ΔEe(HF) | δ[MP2] | δ[CCSD] | δ[CCSD(T)] | δ[CCSDT] | δ[CCSDTQ] | ΔEe[FCI] |
---|---|---|---|---|---|---|---|
a See footnote a to Table 8, with the exception that the hierarchy of electronic-structure techniques utilized is HF → MP2 → CCSD → CCSD(T) → CCSDT → CCSDTQ ≡ FCI. | |||||||
aug-cc-pVTZ | 3983.1 | +604.2 | +26.8 | +54.3 | +9.5 | +0.9 | 4678.8 |
aug-cc-pVQZ | 3963.3 | +595.1 | +21.2 | +56.9 | +9.1 | +0.9 | 4646.5 |
aug-cc-pV5Z | 3962.1 | +589.6 | +21.8 | +57.6 | +8.8 | +0.9 | 4640.7 |
aug-cc-pV6Z | 3959.9 | +585.4 | +23.3 | +57.8 | +8.7 | +0.9 | 4636.0 |
CBS(3–4) | 3960.6 | +588.5 | +17.1 | +58.8 | +8.8 | +0.9 | 4634.7 |
CBS(4–5) | 3961.9 | +583.8 | +22.4 | +58.3 | +8.6 | +0.9 | 4635.8 |
CBS(5–6) | 3959.5(20) | +579.6(40) | +25.3(20) | +58.2(10) | +8.5(10) | +0.9(10) | 4632.0(60) |
MAT | 3965.7 | +579.7 | +22.5 | +55.2 | +9.6 | +0.9 | 4633.5 |
MAQ | 3958.5 | +585.8 | +22.1 | +57.1 | +9.1 | +0.9 | 4633.5 |
MAP | 3959.6 | +587.0 | +22.6 | +57.6 | +8.8 | +0.9 | 4636.6 |
MAH | 3959.6 | +584.2 | +23.6 | +57.9 | +8.7 | +0.9 | 4634.8 |
CBS(3–4) | 3957.6 | +590.3 | +21.9 | +58.4 | +8.7 | +0.9 | 4637.7 |
CBS(4–5) | 3959.8 | +588.3 | +23.2 | +58.3 | +8.6 | +0.9 | 4639.0 |
CBS(5–6) | 3959.5(10) | +580.2(30) | +25.0(10) | +58.1(10) | +8.5(10) | +0.9(10) | 4632.3(40) |
Basis | ΔEe(HF) | δ[MP2] | δ[CCSD] | δ[CCSD(T)] | δ[CCSDT] | δ[CCSDTQ] | ΔEe[FCI] |
---|---|---|---|---|---|---|---|
a See footnote a to Table 8, with the exception that the hierarchy of electronic-structure techniques utilized is HF → MP2 → CCSD → CCSD(T) → CCSDT → CCSDTQ ≡ FCI. | |||||||
aug-cc-pVTZ | 3821.7 | +522.0 | +13.8 | +49.1 | +8.4 | +0.8 | 4415.9 |
aug-cc-pVQZ | 3801.1 | +512.2 | +9.6 | +51.3 | +8.1 | +0.8 | 4383.1 |
aug-cc-pV5Z | 3799.3 | +506.2 | +10.4 | +51.8 | +7.8 | +0.8 | 4376.4 |
aug-cc-pV6Z | 3797.1 | +502.0 | +11.9 | +52.0 | +7.7 | +0.8 | 4371.6 |
CBS(3–4) | 3798.3 | +505.1 | +6.6 | +52.8 | +7.8 | +0.8 | 4371.5 |
CBS(4–5) | 3799.0 | +499.9 | +11.3 | +52.4 | +7.6 | +0.8 | 4370.9 |
CBS(5–6) | 3796.6(10) | +496.3(60) | +14.0(20) | +52.3(10) | +7.5(10) | +0.8(10) | 4367.6(80) |
MAT | 3804.0 | +499.5 | +10.3 | +49.9 | +8.5 | +0.8 | 4373.1 |
MAQ | 3796.0 | +503.6 | +10.5 | +51.4 | +8.0 | +0.8 | 4370.4 |
MAP | 3796.8 | +503.8 | +11.2 | +51.9 | +7.8 | +0.8 | 4372.4 |
MAH | 3796.7 | +500.9 | +12.3 | +52.0 | +7.7 | +0.8 | 4370.4 |
CBS(3–4) | 3794.9 | +506.6 | +10.7 | +52.5 | +7.7 | +0.8 | 4373.2 |
CBS(4–5) | 3797.0 | +504.1 | +12.0 | +52.3 | +7.6 | +0.8 | 4373.7 |
CBS(5–6) | 3796.7(10) | +496.8(40) | +13.8(15) | +52.3(10) | +7.6(10) | +0.8(10) | 4367.8(60) |
Basis | ΔEe(HF) | δ[MP2] | δ[CCSD] | δ[CCSD(T)] | δ[CCSDT] | ΔEe[FCI] |
---|---|---|---|---|---|---|
a See footnote a to Table 8, with the exception that the hierarchy of electronic-structure techniques utilized is HF → MP2 → CCSD → CCSD(T) → CCSDT ≡ FCI. | ||||||
aug-cc-pVTZ | 1418.0 | +991.8 | +279.9 | +37.6 | +7.8 | 2735.1 |
aug-cc-pVQZ | 1424.3 | +991.6 | +275.5 | +39.5 | +7.5 | 2738.3 |
aug-cc-pV5Z | 1422.6 | +992.1 | +273.1 | +39.9 | +7.3 | 2735.1 |
aug-cc-pV6Z | 1421.6 | +989.8 | +273.3 | +40.1 | +7.2 | 2732.1 |
CBS(3–4) | 1425.1 | +991.4 | +272.2 | +40.8 | +7.4 | 2737.0 |
CBS(4–5) | 1422.3 | +992.6 | +270.7 | +40.4 | +7.1 | 2733.1 |
CBS(5–6) | 1421.4(10) | +986.7(40) | +273.6(15) | +40.3(5) | +7.1(3) | 2729.1(50) |
MAT | 1419.3 | +972.6 | +277.8 | +38.0 | +7.9 | 2715.6 |
MAQ | 1421.6 | +986.8 | +274.0 | +39.4 | +7.5 | 2729.4 |
MAP | 1421.4 | +990.5 | +273.2 | +39.9 | +7.3 | 2732.5 |
MAH | 1421.6 | +989.4 | +273.2 | +40.1 | +7.2 | 2731.5 |
CBS(3–4) | 1421.9 | +997.2 | +271.3 | +40.5 | +7.3 | 2738.2 |
CBS(4–5) | 1421.4 | +994.4 | +272.4 | +40.5 | +7.1 | 2735.8 |
CBS(5–6) | 1421.6(10) | +987.9(25) | +273.1(10) | +40.3(3) | +7.1(2) | 2730.1(30) |
Basis | ΔEe(HF) | δ[MP2] | δ[CCSD] | δ[CCSD(T)] | δ[CCSDT] | δ[CCSDTQ] | δ[CCSDTQP] | ΔEe[FCI] |
---|---|---|---|---|---|---|---|---|
a See footnote a to Table 8, except that hierarchy is HF → MP2 → CCSD → CCSD(T) → CCSDT → CCSDTQ → CCSDTQP ≡ FCI. Fixed increments taken from a smaller basis set are given in brackets. | ||||||||
aug-cc-pVTZ | 97.7 | +387.1 | +123.4 | +29.4 | +7.2 | +0.3 | +0.004 | 645.1 |
aug-cc-pVQZ | 92.8 | +407.5 | +116.4 | +31.2 | +7.0 | +0.3 | +0.004 | 655.3 |
aug-cc-pV5Z | 92.5 | +405.1 | +115.4 | +31.5 | +6.9 | +0.3 | [+0.004] | 651.7 |
aug-cc-pV6Z | 91.4 | +407.1 | +114.1 | +31.7 | +6.8 | [+0.3] | [+0.004] | 651.3 |
CBS(3–4) | 92.2 | +422.5 | +111.2 | +32.4 | +6.9 | +0.3 | [+0.004] | 665.6 |
CBS(4–5) | 92.4 | +402.5 | +114.3 | +31.9 | +6.7 | +0.3 | [+0.004] | 648.2 |
CBS(5–6) | 91.2(10) | +409.8(15) | +112.2(15) | +31.9(5) | +6.7(3) | [+0.3](10) | [+0.004](4) | 652.1(20) |
MAT | 88.7 | +378.4 | +123.7 | +29.7 | +7.3 | +0.3 | +0.004 | 628.2 |
MAQ | 90.8 | +397.0 | +117.7 | +31.0 | +7.0 | +0.3 | +0.004 | 643.9 |
MAP | 91.3 | +404.7 | +115.3 | +31.5 | +6.9 | +0.3 | [+0.004] | 650.0 |
MAH | 91.3 | +406.7 | +114.0 | +31.7 | +6.8 | [+0.3] | [+0.004] | 650.9 |
CBS(3–4) | 91.1 | +410.6 | +113.2 | +31.9 | +6.8 | +0.3 | +0.004 | 654.1 |
CBS(4–5) | 91.4 | +412.8 | +112.8 | +32.1 | +6.7 | +0.3 | [+0.004] | 656.1 |
CBS(5–6) | 91.3(3) | +409.4(10) | +112.3(10) | +31.9(4) | +6.7(3) | [+0.3](10) | [+0.004](4) | 652.0(10) |
Basis | ΔEe(HF) | δ[MP2] | δ[CCSD] | δ[CCSD(T)] | δ[CCSDT] | ΔEe[FCI] |
---|---|---|---|---|---|---|
a See footnote a to Table 8, with the exception that the hierarchy is HF → MP2 → CCSD → CCSD(T) → CCSDT ≡ FCI. | ||||||
aug-cc-pVTZ | 15966.5 | +2590.5 | +1173.3 | +46.9 | +9.1 | 19786.3 |
aug-cc-pVQZ | 15938.5 | +2717.1 | +1184.7 | +51.5 | +8.9 | 19900.6 |
aug-cc-pV5Z | 15940.8 | +2757.5 | +1174.4 | +52.8 | +8.6 | 19934.0 |
aug-cc-pV6Z | 15940.4 | +2771.6 | +1170.3 | +53.3 | +8.4 | 19944.0 |
CBS(3–4) | 15934.7 | +2809.5 | +1193.1 | +54.8 | +8.7 | 20000.7 |
CBS(4–5) | 15941.2 | +2799.9 | +1163.6 | +54.1 | +8.3 | 19967.1 |
CBS(5–6) | 15940.3(10) | +2790.9(100) | +1164.7(70) | +54.0(10) | +8.2(3) | 19958.2(120) |
MAT | 15922.4 | +2626.4 | +1201.1 | +48.4 | +9.3 | 19807.7 |
MAQ | 15934.3 | +2726.3 | +1181.3 | +51.6 | +8.8 | 19902.3 |
MAP | 15938.7 | +2759.0 | +1173.7 | +52.8 | +8.6 | 19932.8 |
MAH | 15940.1 | +2772.3 | +1169.6 | +53.3 | +8.4 | 19943.8 |
CBS(3–4) | 15935.9 | +2799.2 | +1166.7 | +54.0 | +8.5 | 19964.3 |
CBS(4–5) | 15939.5 | +2793.3 | +1165.7 | +54.1 | +8.3 | 19960.9 |
CBS(5–6) | 15940.4(10) | +2790.5(60) | +1164.1(40) | +53.9(10) | +8.2(3) | 19957.2(65) |
Basis | ΔEe(HF) | δ[MP2] | δ[CCSD] | δ[CCSD(T)] | δ[CCSDT] | δ[CCSDTQ] | δ[CCSDTQP] | ΔEe[FCI] |
---|---|---|---|---|---|---|---|---|
a See footnote to Table 8, with the exception that the hierarchy is HF → MP2 → CCSD → CCSD(T) → CCSDT → CCSDTQ → CCSDTQP ≡ FCI. Fixed increments taken from a smaller basis set are given in brackets. | ||||||||
aug-cc-pVTZ | −783.4 | +1578.0 | +506.3 | +113.6 | +27.2 | +1.4 | +0.004 | 1443.1 |
aug-cc-pVQZ | −789.1 | +1613.7 | +501.7 | +121.1 | +26.8 | +1.5 | +0.005 | 1475.6 |
aug-cc-pV5Z | −786.5 | +1621.6 | +498.3 | +123.0 | +26.2 | +1.5 | [+0.005] | 1484.1 |
aug-cc-pV6Z | −786.8 | +1625.5 | +496.5 | +123.8 | +25.9 | [+1.6] | [+0.005] | 1486.5 |
CBS(3-4) | −789.8 | +1639.7 | +498.3 | +126.5 | +26.5 | +1.6 | +0.005 | 1502.7 |
CBS(4-5) | −786.1 | +1629.9 | +494.7 | +125.1 | +25.6 | +1.6 | [+0.005] | 1490.8 |
CBS(5-6) | −786.8(10) | +1630.9(60) | +494.1(30) | +124.9(10) | +25.5(10) | [+1.6](2) | [+0.005](5) | 1490.1(70) |
MAT | −789.5 | +1574.6 | +510.4 | +116.1 | +27.8 | +1.5 | +0.004 | 1440.8 |
MAQ | −785.8 | +1610.6 | +500.5 | +121.3 | +26.8 | +1.5 | +0.005 | 1475.0 |
MAP | −785.7 | +1621.3 | +498.0 | +123.1 | +26.2 | +1.6 | [+0.005] | 1484.5 |
MAH | −786.5 | +1625.4 | +496.4 | +123.8 | +25.9 | +1.6 | [+0.005] | 1486.6 |
CBS(3-4) | −785.3 | +1636.9 | +493.3 | +125.1 | +26.0 | +1.6 | +0.005 | 1497.7 |
CBS(4-5) | −785.7 | +1632.5 | +495.4 | +125.0 | +25.6 | +1.6 | [+0.005] | 1494.4 |
CBS(5-6) | −786.6(10) | +1630.9(60) | +494.2(30) | +124.8(10) | +25.5(10) | +1.6(1) | [+0.005](5) | 1490.4(70) |
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