Electronic structure and rovibrational properties of ZnOH in the X ̃ 2 A ′ electronic state: A computational molecular spectroscopy study Tsuneo Hirano, Mounir Ben Dahman Andaloussi, Umpei Nagashima, and Per Jensen Citation: The Journal of Chemical Physics 141, 094308 (2014); doi: 10.1063/1.4892895 View online: http://dx.doi.org/10.1063/1.4892895 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/141/9?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Accurate ab initio ro-vibronic spectroscopy of the X ̃ 2 Π CCN radical using explicitly correlated methods J. Chem. Phys. 135, 144309 (2011); 10.1063/1.3647311 Explicitly correlated treatment of H2NSi and H2SiN radicals: Electronic structure calculations and rovibrational spectra J. Chem. Phys. 135, 074301 (2011); 10.1063/1.3624563 Microwave measurements and ab initio calculations of structural and electronic properties of N -Et-1,2-azaborine J. Chem. Phys. 131, 224312 (2009); 10.1063/1.3270157 Calculation of rovibronic intensities for triatomic molecules in double-Renner-degenerate electronic states: Application to the X ̃ A 2 ″ and A ̃ A 2 ′ electronic states of HO 2 J. Chem. Phys. 130, 224105 (2009); 10.1063/1.3139916 A calculation of the rovibronic energies and spectrum of the B ̃ A 1 1 electronic state of Si H 2 J. Chem. Phys. 123, 244312 (2005); 10.1063/1.2139676

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THE JOURNAL OF CHEMICAL PHYSICS 141, 094308 (2014)

Electronic structure and rovibrational properties of ZnOH in the X˜ 2 A electronic state: A computational molecular spectroscopy study Tsuneo Hirano,1,a) Mounir Ben Dahman Andaloussi,2 Umpei Nagashima,3 and Per Jensen2,b) 1

Department of Chemistry, Faculty of Science, Ochanomizu University, 2-1-1 Otsuka, Bunkyo-ku, Tokyo 112-8610, Japan 2 Physikalische und Theoretische Chemie, Bergische Universität, D-42097 Wuppertal, Germany 3 Nanosystem Research Institute, National Institute of Advanced Industrial Science and Technology, 1-1-1 Umezono, Tsukuba, Ibaraki 305-8568, Japan

(Received 3 June 2014; accepted 31 July 2014; published online 4 September 2014) The three-dimensional ground-state potential energy surface of ZnOH has been calculated ab initio at the MR-SDCI+Q_DK3/[QZP ANO-RCC (Zn, O, H)] level of theory and used as basis for a study of the rovibrational properties carried out by means of the program MORBID (Morse Oscillator Rigid Bender Internal Dynamics). The electronic ground state is 2 A (correlating with 2  + at the linear configuration). The equilibrium structure has re (Zn–O) = 1.8028 Å, re (O–H) = 0.9606 Å, and  e (Zn–O–H) = 114.9◦ . The Zn–O bond is essentially ionic, with appreciable covalency. The bonding character is compared with those of FeOH (quasi-linear) and CsOH (linear). The rovibrationally averaged structural parameters, determined as expectation values over MORBID wavefunctions, are r(Zn–O)0 = 1.8078 Å, r(O–H)0 = 0.9778 Å, and  (Zn– O–H)0 = 117◦ . The Yamada-Winnewisser quasi-linearity parameter is found to be γ 0 = 0.84, which is close to 1.0 as expected for a bent molecule. Since no experimental rovibrational spectrum has been reported thus far, this spectrum has been simulated from the ab initio potential energy and dipole moment surfaces. The amphoteric character of ZnOH is also discussed. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4892895] I. INTRODUCTION

We have already reported computational molecular spectroscopy studies of floppy metal-hydroxides such as X˜ 1  + CsOH1 and X˜ 6 Ai FeOH;2, 3 the former is a very floppy linear molecule for which anomalous, experimentally derived values of the rotation-vibration interaction parameters were reported by Lide and co-workers4–6 in their microwave studies of CsOH (and RbOH) more than 40 years ago, and the latter is a quasi-linear molecule whose potential energy surface (PES) has a small barrier to linearity2, 3 of 273 cm−1 . In recent years, there have been three experimental investigations of zinc hydroxide ZnOH: Mass spectrometry results for this molecule were reported in Ref. 7 and electronbinding-energy spectroscopy of ZnOH− in Ref. 8. The molecular structure of ZnOH, determined by millimeter and microwave gas-phase spectroscopy methods, was reported for the first time by Zack et al.9 to be bent, indicative of covalent bonding. There are only two theoretical studies available, one by Trachtman et al.10 at the MP2 and CCSD(T) levels of theory and the other by Iordanov et al.8 at the DFT/aug-cc-pVQZ level. Our preliminary calculations showed that ZnOH is an essentially ionic molecule with a somewhat weak covalency in its Zn–O bond. The resulting strong directionality in the Zn–O a) E-mail: [email protected] b) Author to whom correspondence should be addressed. Electronic mail:

[email protected]. Tel.: +49 202 439 2468. Fax: +49 202 439 2509.

0021-9606/2014/141(9)/094308/14/$30.00

bond is expected to give a bent structure which can be compared to that of the floppy, quasi-linear FeOH molecule.2, 3 Zinc is the last of the first-row transition metals, and so the Zn atom in ZnOH has a full set of 3d semi-core electrons. Thus, it is interesting to investigate which factors influence whether the structure becomes quasi-linear (FeOH) or bent (ZnOH). In the present paper, we report the three-dimensional potential energy surface (3D PES), calculated at the MRSDCI+Q_DK3/[QZP ANO-RCC (Zn, O, H)] level of theory, and the electronic structure is discussed. Here, MRSDCI+Q_DK3 stands for “internally contracted multireference single and double excitation configuration interaction calculation with Davidson correction11 Q and 3rd order Douglas-Kroll-Hess Hamiltonian.12–14 ” The Davidson correction is made to include the dynamical electron correlation from the quadruple excitations and to recover part of the size-extensivity. The acronym QZP indicates that the basis sets are of quadruple-zeta plus polarization quality, and ANO-RCC denotes the atomic natural orbital basis sets proposed by Roos and co-workers15–17 for relativistic correlation consistent calculations, although the definition of the acronym RCC was not disclosed by the authors. Using the resulting ab initio 3D PES, we calculated rovibrational molecular constants by second-order perturbation theory18, 19 and obtained rovibrational energies in the variational MORBID (Morse Oscillator Rigid Bender Internal Dynamics) approach.20–23 Also, we provide various so-far unobserved rovibrational spectra simulated with the MORBID program system.

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© 2014 AIP Publishing LLC

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Finally, we will discuss the Yamada-Winnewisser quasilinearity parameter24, 25 in the case of the bent ZnOH molecule. II. AB INITIO MOLECULAR ORBITAL CALCULATIONS

The electronic ground state of ZnOH is X˜ 2 A . In contrast to Ai FeOH, there is no quasi-degeneracy problem in ZnOH since for this molecule, all 3d orbitals of Zn are doubly occupied and we expect no electronic degeneracy in the lowest electronic states. For a molecule containing one of the first-row transition metals that start at Mn in the periodic table, we need a relativistic energy correction to obtain a PES of “nearspectroscopic” accuracy. In the present work, we have made this correction in terms of the Douglas-Kroll-Hess third order (DK3) Hamiltonian.12–14 Accordingly, we need basis sets optimized at the DK3 relativistic level3, 26 and so we adopted the valence quadruple zeta plus polarization (QZP) quality contractions of the relativistic atomic natural-orbital-type basis sets ANO-RCC QZP, optimized at the DK3 level by Roos and co-workers15–17, 27 In addition, we appended basis functions with one set of g-functions for O, and one set of f-functions for H, as given in the list of ANO-RCC basis sets.27 Thus, the basis set employed for Zn was (21s, 15p, 10d, 6f, 4g)/[7s, 6p, 4d, 3f, 2g],15 that for O was (14s, 9p, 4d, 3f, 2g)/[5s, 4p, 3d, 2f, 1g],16 and that for H was (8s, 4p, 3d, 1f)/[4s, 3p, 2d, 1f].17 The entire basis set is denoted as [QZP ANO-RCC (Zn, O, H)]. In obtaining the Mulliken charge distribution we used also aug-cc-pVTZ-DK basis sets; for Zn,28 (21s, 17p, 9d, 3f, 2g)/[8s, 7p, 5d, 3f, 2g], for O,29 (11s, 6p, 3d, 2f)/[5s, 4p, 3d, 2f], and for H,29 (6s, 3p, 2d)/[4s, 3p, 2d]. This basis set is denoted as [aVTZ-DK (Zn, O, H)]. All ab initio energy calculations have been done with MOLPRO versions 2006.1 and 2010.130 at the MRSDCI+Q_DK3 level of theory, based on the CASSCF (Complete Active Space Self-Consistent Field) orbitals obtained as the A root under Cs symmetry constraint. The initial-guess orbital for the CASSCF calculation was calculated near the equilibrium structure under Cs symmetry constraint for a bond angle  (Zn–O–H) of 115◦ with a given set of bond lengths r(Zn–O) and r(O–H). Then the CASSCF orbitals were calculated successively at, on one hand,  (Zn– O–H) = 120◦ , 135◦ , 150◦ , 165◦ , 178◦ and, on the other hand, at  (Zn–O–H) = 110◦ , 95◦ , 80◦ , 65◦ using the orbitals at the previous bond angle as the initial guess in order to maintain the correct orbital character. The active space consisted of the 12a –18a and 5a –6a orbitals, originating in Zn 4s and 4p, O 2s and 2p, and H 1s (9 electrons distributed over 9 orbitals) with all the other orbitals optimized as doubly occupied. Thus, we employed a no-core, full-valence CASSCF method. Based on these CASSCF orbitals, a full-valence MRSDCI+Q_DK3 calculation was done. The active space is the same as that in the CASSCF calculation (9 electrons distributed over 9 orbitals), but the 1a –8a and 1a –2a orbitals (originating in the Zn 1s, 2s, 2p, 3s, 3p orbitals and the O 1s orbital) are treated as “core.” The dynamical electron correlation from the Zn 3d orbitals was taken into account by single 6

J. Chem. Phys. 141, 094308 (2014)

and double excitations from the Zn 3d orbitals in the configuration state functions (CSFs) derived from the active space. The number of CSFs in the reference space of the MR-SDCI calculation is 4508 for the equilibrium structure, for example, and the corresponding dimension of the internally contracted (uncontracted) CI matrix reaches 4.1 × 107 (3.2 × 109 ). The dipole moment was calculated as finite electric field derivative of the MR-SDCI+Q_DK3/[QZP ANO-RCC (Zn, O, H)] energy calculated at the electric field of ±0.001 e/a0 . The Mulliken populations were calculated at the MRSDCI_DK3/[aVTZ-DK (Zn, O, H)] level of theory, using lower level basis sets to avoid possible artifacts brought about by too diffuse primitive functions in the [QZP ANO-RCC (Zn, O, H)] basis sets. To compare the bonding character of ZnOH (bent) with those of CsOH (linear1 ) and FeOH (quasi-linear2, 3 ), we require the density matrix and the NOs at the correlated level for these three metal-hydroxides. However, the previous calculations on X˜ 1  + CsOH were done at the RCCSD(T)_DK3 level1 and hence the orbitals and the density matrix there obtained were those of the Hartree-Fock wavefunction (owing to a restriction in MOLPRO 200630 ). Thus, the previous calculations on X˜ 1  + CsOH have been redone at the MRSDCI+Q_DK3 level of theory as shown in the Appendix. The present MR-SDCI+Q_DK3 calculations gave results quite similar to the previous RCCSD(T)_DK3 results given in Ref. 1. For these three hydroxides, the renormalized atomic orbital (AO) coefficients (see Ref. 26 for definition) have been calculated for each NO of the respective MR-SDCI_DK3 wavefunctions.

III. RESULTS AND DISCUSSION A. Geometry and electronic structure

1. Equilibrium geometry and electronic structure

The MR-SDCI+Q_DK3/[QZP ANO-RCC (Zn, O, H)] ab initio calculations of the present work predict the ground electronic state of ZnOH to be 2 A . The ground-state equilibrium structure is “bent” and has re (Zn–O) = 1.8028 Å, re (O–H) = 0.9606 Å, and  e (Zn–O–H) = 114.9◦ . The re (O– H) value for ZnOH is close to those of OH− (re = 0.964317 Å; Ref. 31) and H2 O (re = 0.9578 Å; Ref. 32), and the value of  e (Zn–O–H) is similar to that of H2 O ( e (H–O–H) = 104.5424◦ ; Ref. 32). For ZnOH, the equilibrium bond angle is slightly larger than for H2 O owing to the repulsion between the large positive charges at the Zn and H ends of the molecule. These features indicate that sp3 hybridization at the O atom, which is well established for H2 O, also occurs in ZnOH. The molecular constants determined from the ab initio PES are discussed in Sec. III B. The bending potential along the minimum energy path (MEP) for the bending coordinate ρ¯ = 180◦ −  (Zn–O–H), defined in Fig. 2, is given in Fig. 1. The X˜ 2 A state of ZnOH correlates with a 2  + state at the Zn–O–H linear configuration, where the optimized energy (i.e., the energy obtained by keeping  (Zn–O–H) = 180◦ and letting the bond lengths “relax”) is 2272 cm−1 higher than the equilibrium energy.

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q, y

H

r32 p, -z

Zn

η

G

M

O

r12

FIG. 1. Bending potential along the bending minimum energy path (MEP) of X˜ 2 A ZnOH, calculated at the MR-SDCI+Q_DK3/[QZP ANO-RCC (Zn, O, H)] level of theory. Bending potentials along the MEP for the linear X˜ 1  + CsOH1 and for the quasi-linear case of X˜ 6 Ai and A˜ 6 Ai FeOH3 are shown for comparison. The bending coordinate ρ¯ = 180◦ −  (Zn–O–H) for ZnOH and it is defined analogously for CsOH and FeOH.

The Mulliken populations on the Zn, O, and H atoms, determined at the MR-SDCI_DK3/[aVTZ-DK (Zn, O, H)] level of theory, are 0.60, −0.79, and 0.19 e, respectively. It means that the Zn–O bond is highly polarized. The dipole moment values obtained as finite electric field derivatives of the MR-SDCI+Q_DK3/[QZP ANO-RCC (Zn, O, H)] energy (with the corresponding MR-SDCI_DK3 expectation values in parentheses) are as follows: μe, q = 1.135 (1.172) D and μe,p = 1.755 (1.706) D with the length of the dipole moment vector being |μe | = 2.090 (2.069) D. The predominant CSF in the MR-SDCI_DK3 wavefunction (89% in weight with a CI expansion coefficient of 0.942) of X˜ 2 A ZnOH is

a

FIG. 2. The definitions of the angles ρ, ¯ τ , and η used to describe the geometry of the ZnOH molecule. The axis labeled “a” passes through the Zn nucleus and through M, the center of mass of the OH moiety. This axis is not identical, but very close, to the a principal axis of the molecule. For the dipole moment, we employed the xqp axis system with origin at the nuclear center of mass G. The p and q axes are defined to be in the molecular plane with the p axis parallel to the Zn–O bond, and the x axis is chosen such that the xqp axes form a right-handed axis system. For the orbital names, we employed an xyz axis system whose y axis coincides with the q axis and whose z axis is the p axis reversed. The xqp and xyz systems are body-fixed in the sense that the orientations of the axes are derived geometrically from the instantaneous positions of the nuclei. Such axis systems are customarily used at the ab initio stage of the calculation and here they are further required for defining the dipole moment functions in the MORBID calculations described below. The p axis is parallel to the Zn–O bond and the q axis is at right angle (German: quer) to this bond.

in the xyz axis system of Fig. 2, with minor bonding contribution from Zn 4s (AO coefficient: 0.29). The “ZnOσ ” NO shows appreciable, but not overly large, covalency between Zn and O. Owing to this less pronounced covalent character of the bonding, Zn is situated on one of the lobes expected for an sp3 -type O 2p orbital extending from the O atom as shown in Fig. 3(c). The “ZnOσ ∗ ” NO consists predominantly

-0.140

(a) ZnOσ* (0.997) (b) O 2pπ (1.958) (c) ZnO σ (1.956)

[core] (3d)10 (O2s )2 (OHσ )2 (ZnOσ )2 (O2pπ )2 (ZnOσ ∗ )1 . (1) The xyz axes used in labeling the orbitals are defined in Fig. 2. In Fig. 3, we plot the iso-amplitude-surface images of the natural orbitals (NOs) of the MR-SDCI_DK3 wavefunction and indicate for each NO the energy and occupation number. The numerical values of the renormalized AO coefficients (see Ref. 26 for definition) in these NOs are given in Table I. The NOs “O2s ” [Fig. 3(e)] and “O2pπ ” [Fig. 3(b)] are almost pure O 2s and O 2px atomic orbitals (AOs), respectively, with negligibly small antibonding contributions from the Zn orbitals. The “OHσ ” NO is a bonding σ orbital for OH, mainly consisting of O 2pσ , which in turn comprises the O2p y and O2p orbitals, with a minor in-phase contribution from H z 1s (AO coefficient 0.35). The “ZnOσ ” and “ZnOσ ∗ ” NOs are bonding and antibonding NOs, respectively, between Zn and O, responsible for the character of the Zn–O bond. The main component of the “ZnOσ ” NO is O 2pσ , i.e., O 2pz (AO coefficient: −0.76) and O 2py (AO coefficient: 0.38)

τ

ρ

-0.454

-0.535

(d) OH σ (1.960)

-0.619

(e) O 2s (1.979)

-1.209

Zn

OH

Zn

OH

FIG. 3. (a)–(e) MR-SDCI_DK3 natural orbitals (NOs) of X˜ 2 A ZnOH, visualized in terms of iso-amplitude-surfaces with orbital amplitudes of ±0.06 a0 −3/2 . For each NO, the occupation number is given in parentheses, and the NO energy of the corresponding CASSCF NO (in Eh , see Refs. 26 and 33 for definition) is given underneath the spin configuration.

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TABLE I. Renormalized AO coefficientsa in NOs of the MR-SDCI_DK3 wavefunction of metal hydroxides M-OH (M = Zn, Fe, and Cs) at each equilibrium structure. The labels x, y, and z refer to the xyz axis system defined in Fig. 2, and the M–O bond is placed along the z axis. Occ.b

NO

Energyc

X˜ 2 A ZnOHd OHσ ZnOσ ZnOσ ∗

1.960 1.956 0.997

−0.619 −0.535 −0.140

X˜ 6 A FeOHe OHσ , 2s1s OHσ , 2p1s FeOσ FeOσ ∗ FeOπ FeOπ ∗

1.985 1.967 1.974 0.997 1.974 0.997

−1.131 −0.769 −0.499 −0.166 −0.464 −0.167

X˜ 1  + CsOHg

M 4s

4pz

4py

3dz2

−0.052 0.285 0.913

−0.028 0.109 −0.423

0.034 0.039 −0.019

0.023 −0.031 0.101

4s

4pz

4py

3dz2

0.015 0.037 −0.033

0.051 −0.089 0.135 0.741

−0.308f −0.061 0.159 0.338

1.986 1.984 1.971 1.963

−1.217 −1.008 −0.655 −0.611

CsOπ , x CsOπ , x ∗ CsOπ , y CsOπ , y ∗

1.975 1.971 1.975 1.971

−0.634 −0.336 −0.634 −0.336

3dyz

3dx 2 −y 2

3dyz

3dx 2 −y 2

0.036 0.079 0.473

−0.002 −0.018 −0.031

3dxz

3dxz

H

2s

2pz

2py

−0.109 −0.010 −0.131

0.268 −0.755 0.294

0.727 0.376 −0.057

2s

2pz

2py

0.896 −0.141 −0.109 −0.091

−0.019 0.527 −0.607 0.179

−0.122 0.540 0.611 −0.089

0.115 0.995 5s

CsOsσ CsOsσ ∗ CsOpσ CsOpσ ∗ + OHσ

−0.047 0.106 −0.308

O

0.931 0.333 0.120 0.154

5pz

5px

5py

−0.074 −0.157 0.960 0.337

1s 0.346 −0.077 0.042

2px

1s 0.300 0.817 0.024 0.049

0.876 −0.159

6s

2s

2pz

0.009 −0.023 0.023 0.009

0.312 −0.996 −0.260 −0.137

−0.048 0.058 −0.153 0.758

0.956 −0.300

2px

2px

2py

1s −0.023 0.133 −0.052 0.367

0.229 0.931 0.956 −0.300

0.229 0.931

a

See Ref. 26 for definition. Occupation number. c NO energy in units of Eh (see Ref. 26 for definition). d Equilibrium structure: re (Zn–O) = 1.803 Å, re (O–H) = 0.961 Å, and  e (Zn–O–H) = 114.9◦ (bent). e Equilibrium structure:2, 3 re (Fe–O) = 1.806 Å, re (O–H) = 0.952 Å, and  e (Fe–O–H) = 134.2◦ (quasi-linear). f Coefficient of 3pz AO. g Equilibrium structure (see the Appendix): re (Cs–O) = 2.393 Å, re (O–H) = 0.957 Å, and  e (Cs–O–H) = 180.0◦ (linear). b

of the Zn 4s orbital polarized strongly backwards against O by Zn 4pz (the respective AO coefficients being 0.91 and −0.42) with minor antibonding contribution from O 2pz and comparatively much smaller antibonding contribution from O 2s [Fig. 3(a)]. This NO is singly occupied and, since all the other NOs in the dominant CSF of Eq. (1) are doubly occupied, it is responsible for the spin-multiplicity of 2 (doublet); the unpaired electron is almost completely localized on the Zn atom, more precisely in the Zn 4s-4pz hybrid orbital which is strongly polarized in the direction backwards against the O atom. Thus, both the “ZnOσ ” and “ZnOσ ∗ ” NOs are highly polarized since their electron densities are unevenly distributed between O and Zn. This gives rise to a strong ionic character of the bonding as demonstrated by the Mulliken population and the dipole moment values mentioned above. In consequence, the Zn–O bond can be characterized as an essentially ionic bond with appreciable covalency. The appreciable amount of covalency in the ionic Zn–O bond of X˜ 2 A ZnOH makes us expect that the bending motion, associated with the Zn–O–H bond angle, becomes relatively rigid. Indeed, unlike CsOH and FeOH, ZnOH is not a highly floppy molecule. Nevertheless, its bending motion is a large-amplitude vibration. In Sec. III B 1, this will be discussed quantitatively in terms of the bending force constants.

The amphoteric character (i.e., the capability of reacting chemically either as an acid or a base) of Zn-containing inorganic molecules is well known. In spite of the fact that at the equilibrium geometry, ZnOH consists of Zn+ and (OH)− , the present MR-SDCI+Q_DK3 calculations for X˜ 2 A ZnOH show that the adiabatic dissociation of the O–H bond results in [1  Zn0.57 O−0.57 + 2 S H (neutral)] with a dissociation barrier of 42 600 cm−1 (or 0.194 Eh ), and that of the Zn–O bond results in [1 S Zn (3d10 4s2 ) + X 2 OH] with a similar dissociation barrier of 44 600 cm−1 (or 0.203 Eh ). Thus, we can conclude that ZnOH is amphoteric. To determine the mechanism of the following ZnOHformation reactions starting from the adiabatic dissociation limits, ZnO + H → ZnOH

(2)

Zn + OH → ZnOH,

(3)

or

we have carried out a series of successive ab initio calculations, increasing either the OH bond length r(O· · · H) or the ZnO bond length r(Zn· · · O) from the respective equilibrium value to 20 Å with  (Zn–O–H) fixed to its equilibrium value of 114.9◦ in order to keep the continuity of the relevant bond character. The directions of the actual formation reactions in

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Eqs. (2) and (3) are opposite to those of the processes described in these ab initio calculations. The ZnO + H → ZnOH reaction starts at r(O· · · H) ≈ 2.5 Å. At this point, about half of the electron density on H starts to be transferred to O and so ZnOH is produced. The Zn + OH → ZnOH reaction starts at r(Zn· · · O) ≈ 3 Å where approximately one of the Zn 4s electrons starts to be relocated by 54% to the O atom and by 37% to the Zn 4p orbital. Simultaneously, about 7% of the H 1s electron density is going to be relocated to the O atom, and so the ZnOHproducing reaction is completed. The first and second excited states at the X˜ 2 A equilibrium geometry are the A˜ 2 A and its Renner counterpart B˜ 2 A states. The vertical excitation energies of these states are 29 215 and 35 540 cm−1 , respectively, at the MRSDCI+Q_DK3 level of theory, so that no perturbations from the electronic excited states are expected to be significant in the description of the rovibrational motion for the X˜ 2 A state. On the vertical excitation to the A˜ and B˜ states, about 0.7 and 0.6 e electron density, respectively, are transferred from the O atom of X˜ 2 A ZnOH to the Zn atom of each excited state, while the density on the H atom decreases by only 0.005 e in each excited state. As a result, the direction of the dipole moment vector is reversed. These two excited states converge to an A˜ 2 loose, linear complex, Zn· · · (OH), with re (Zn–O) = 3.777 Å, re (O–H) = 0.972 Å, and the equilibrium energy of 16 153 cm−1 above that of the X˜ 2 A state. The dipole moment (expectation value) is 1.94 D and the Mulliken charges are −0.001 e (Zn), −0.440 e (O), and 0.441 e (H). 2. Geometry and covalent character in M-OH bonds (M = Cs, Fe, and Zn)

The metal–O bond in ZnOH (bent form) manifests itself as being essentially ionic with appreciable covalency as shown above. This situation is quite different from that encountered for the molecules X˜ 6 A FeOH2, 3 and X˜ 1  + CsOH1 which are quasi-linear and linear, respectively. In Table I, the renormalized AO coefficients (see Ref. 26 for definition) are compiled for X˜ 2 A ZnOH, X˜ 6 A FeOH,2, 3 and X˜ 1  + CsOH. Here, to compare the renormalized AO coefficients among the three metal hydroxides at the same MR-SDCI+Q_DK3 level, previous calculations for X˜ 1  + CsOH1 at the RCCSD(T)_DK3 level were redone at the MRSDCI+Q_DK3 level as described in the Appendix. Table I shows that for all metal–O bonds of these three metal hydroxides, the AO coefficients for the metal and oxygen atoms, respectively, in the NOs responsible for the metal– O bonding (MOσ ) are largely unbalanced. Therefore, the metal–O bonds in all three metal hydroxides are essentially ionic. In the case of CsOH,1 the Cs 6s valence electron is transferred almost completely to the O atom of the OH part (the Mulliken charges are Cs0.89 O−1.42 H0.53 ) to give an ionic Cs–O bond. Therefore, both the bonding and the antibonding Cs–O NOs are formed through the Cs inner-shell 5s and 5p electrons with the O 2s and 2p electrons, and Cs 6s components are negligibly small (Table I). In the individual NOs, the covalency (given in terms of the balance in the ratio of AO

J. Chem. Phys. 141, 094308 (2014)

coefficients for the Cs and O atoms) is small but not vanishingly small. However, for each bonding/antibonding NO pair [CsOsσ vs. CsOsσ ∗ , CsOpσ vs. CsOpσ ∗ + OHσ , CsOπ,x vs. CsOπ,x ∗ , and CsOπ,y vs. CsOπ,y ∗ ], both occupation numbers are close to 2, and so the covalency in these NOs is canceled within each pair. Consequently, the Cs–O bond becomes almost purely ionic. In the cases of FeOH and ZnOH, this cancellation does not occur since MOσ is doubly occupied and the corresponding antibonding NO (MOσ ∗ ) is singly occupied. Table I shows that the AO coefficients for metal 4s and 4pz in the ZnOσ NO of ZnOH are 0.29 and 0.11, respectively, and the corresponding AO coefficients in the FeOσ NO of FeOH are 0.16 and 0.11. Hence, the covalency of the Zn–O bond is small but larger than that of the Fe–O bond. As discussed in the previous section (Sec. III A 1), the covalent character of the metal–O bond tends to locate the metal atom in a position corresponding to a bent equilibrium geometry. Therefore, the degree of covalency in the metal– O bond in the three metal hydroxides X˜ 2 A ZnOH, X˜ 6 A FeOH,2, 3 and X˜ 1  + CsOH is shown to be the factor determining whether the metal hydroxide becomes strongly bent, quasi-linear or linear. B. Perturbation-theory analyses

1. Force constants

For the perturbation-theory analyses,18, 19 a total of 539 computed ab initio data points have been used to construct a three-dimensional potential energy function for X˜ 2 A ZnOH. The electronic energies were fitted to a polynomial expansion V (r12 , r32 , θ ) =

5  5 



j =0 k=0 l=0,2,4,6,8

×

1 j f r r k (θ −θe )l j ! k! l! j kl 12 32

(4)

in internal displacement coordinates. e , i = 1 or 3, where r12 is the In Eq. (4), ri2 = ri2 − ri2 instantaneous value of the Zn–O distance (Zn is labeled 1 and the O nucleus is labeled 2), r32 is the instantaneous value of the O–H distance (the proton is labeled 3), and θ is the angle  (Zn–O–H). The quantities r e and θ represent equilibrium e i2 values. The maximum and root-mean-square errors achieved in the fitting of the ab initio points were 1.21 and 0.36 cm−1 , respectively. The most important force constants fjkl obtained in the fitting are given in Table II and a complete list of fjkl -values is available as supplementary data.34 The optie e = re (Zn–O), r32 mized equilibrium bond lengths found for r12 = re (O–H), and θ e = e (Zn–O–H) are included in Table III, and Ee , the value of the potential energy at equilibrium, is included in Table IV. In Sec. III A 1, the Zn–O bond is concluded to be essentially ionic with appreciable covalency. This conclusion was based on the character of the NOs responsible for the bonding. In addition, it is borne out by a comparison between the metal–O stretching force constants of the metal

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TABLE II. Force constants fjkl (Eq. (4)) for X˜ 2 A ZnOH obtained from the ab initio MR-SDCI+Q_DK3 potential energy surface.a f200 /aJ Å−2 f110 /aJ Å−2 f020 /aJ Å−2 f101 /aJ Å−1 f011 /aJ Å−1 f002 /aJ f300 /aJ Å−3 f210 /aJ Å−3 f120 /aJ Å−3 f030 /aJ Å−3 f201 /aJ Å−2 f111 /aJ Å−2 f021 /aJ Å−2 f102 /aJ Å−1 f012 /aJ Å−1 f003 /aJ

3.196390 − 0.043081 8.303185 0.214397 0.159477 0.249030 − 17.141445 − 0.799449 − 0.013906 − 58.650729 − 0.421475 − 0.185719 0.054185 − 0.073878 − 0.140061 − 0.446429

f400 /aJ Å−4 f310 /aJ Å−4 f220 /aJ Å−4 f130 /aJ Å−4 f040 /aJ Å−4 f301 /aJ Å−3 f211 /aJ Å−3 f121 /aJ Å−3 f031 /aJ Å−3 f202 /aJ Å−2 f112 /aJ Å−2 f022 /aJ Å−2 f103 /aJ Å−1 f013 /aJ Å−1 f004 /aJ ···

82.744010 − 0.094652 14.680284 0.161579 380.937561 0.772268 0.137185 − 0.043583 − 0.847044 0.010155 0.413853 − 0.265035 0.246162 0.205181 0.024339

a

Only fjkl -values for terms up to 4th order are listed here. A complete list of fjkl -values is available as supplementary data.34

hydroxides considered. The Zn–O stretching force constant f200 = 3.196 aJ Å−2 is larger than the stretching force constant for the purely ionic Cs–O bond in X˜ 1  + CsOH (1.359 aJ Å−2 ; Ref. 1), and is comparable to that of the Fe–O bond in X˜ 6 A FeOH (3.657 aJ Å−2 ; Ref. 2). The latter bond is ionic but has a small amount of covalency.

The bending force constant f002 = 0.249 aJ is much larger than those of the purely or almost purely ionic metal–O or metal–C bonds; 0.079 aJ for X˜ 1  + CsOH,1 0.086 aJ for X˜ 6 A FeOH,3 0.153 aJ for X˜ 3 CoCN,35 and 0.170 aJ for X˜ 2 NiCN.36 All these molecules exhibit large-amplitude bending motion. Although we have concluded in Sec. III A 1 that the Zn–O bond has an appreciable amount of covalency, the f002 -value for ZnOH is substantially smaller than the bending force constant of the covalently bound H2 O molecule32 (0.706 aJ). Owing to the appreciable covalency of the Zn–O bond, X˜ 2 A ZnOH is less floppy than molecules with a purely or almost purely ionic metal-ligand bond, but it is sufficiently floppy to carry out a large-amplitude bending motion. 2. Second-order perturbation-theory analyses

With the optimized values of the force constants fjkl in Eq. (4) (Table II), we have calculated, by means of perturbation methods,18, 19 values of the standard spectroscopic parameters A0 , B0 , C0 , DJ , DJK , DK , d1 , d2 , α i , ωi , xij ,· · · for ZnOH and ZnOD. The results obtained are given in Tables III and IV. These predicted re , A0 /B0 /C0 , and centrifugal distortion constant values are generally in good agreement with the available values9 derived from spectroscopic experiments. The A0 , B0 , and C0 values for 64 ZnOH, for example, show errors of 7.5%, 0.3%, and 0.3%, respectively. A more detailed discussion of the rovibrational parameters will be given in Sec. III C.

TABLE III. Structural parameters, rotational constants, and centrifugal distortion constants of 64 ZnOH and 64 ZnOD, calculated from the three-dimensional potential energy surface obtained ab initio at the MR-SDCI+Q_DK3/[QZP ANO-RCC (Zn, O, H)] level of theory. 64 ZnOH

Calc.a re (Zn–O)/Å re (O–H)/Å  (Zn–O–H)/deg e r0 (Zn–O)/Å r0 (O–H)/Å  (Zn–O–H)/deg 0 Ae /cm−1 Be /cm−1 Ce /cm−1 Ae /MHz Be /MHz Ce /MHz A0 /MHz B0 /MHz C0 /MHz

64 ZnOD

Expt.9

1.8028 0.9606 114.9

Calc.a

Expt.9

1.8028 0.9606 114.9 1.809(5) 0.964(7) 114.1(5)

24.1558 0.3744 0.3687 724 172 11 223 11 052 800 963 11 198 11 002

Centrifugal distortion constants (s-reduction) 0.0151 DJ /MHz 1.7018 DJK /MHz DK /MHz 897.3504 0.00021 d1 /MHzb − 0.00002 d2 /MHz

745 084(1800) 11 163.4978(51) 10 971.5173(51) 0.0156697(17) 2.1269(28) − 0.0002541(36) − 0.0000399(23)

13.1773 0.3477 0.3387 395 045 10 423 10 155 429 247 10 398 10 098 0.0125 0.6150 294.3519 0.0004 − 0.00005

402 650(170) 10 377.2378(54) 10 085.8528(54) 0.0130081(22) 0.70813(47) − 0.0004498(45) − 0.0000630(21)

In the table, the labels “e” and “0” in the subscript denote “equilibrium structure” and “rovibrational ground state,” respectively. r,  , and A/B/C are bond distances, bond angles, and rotational constants, respectively. b It is remarkable that the observed and calculated values of the parameter d1 have comparable magnitudes but opposite signs. This may indicate a phase difference between the present work and Ref. 9 in the definition of the off-diagonal matrix elements depending on d1 . a

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TABLE IV. Equilibrium energies, rotation-vibration interaction constants, harmonic vibrational wavenumbers, vibrational energies, and dipole moment values of 64 ZnOH and 64 ZnOD, calculated from the three-dimensional potential energy surface obtained ab initio at the MR-SDCI+Q_DK3/[QZP ANORCC (Zn, O, H)] level of theory.a

Ee /Eh α 1, A /cm−1 α 1, B /cm−1 α 1, C /cm−1 α 2, A /cm−1 α 2, B /cm−1 α 2, C /cm−1 α 3, A /cm−1 α 3, B /cm−1 α 3, C /cm−1 ζ 12 ζ 13 ζ 23 ω1 /cm−1 ω2 /cm−1 ω3 /cm−1 x11 /cm−1 x22 /cm−1 x33 /cm−1 x12 /cm−1 x13 /cm−1 x23 /cm−1 g22 /cm−1 ν 1 /cm−1 ν 2 /cm−1 ν 3 /cm−1 ZPVE/cm−1 μe, x b /D μe, q b /D μe, p b /D |μe |b /D

64 ZnOH

64 ZnOD

− 1870.6030688 1.5126 0.0006 0.0009 − 3.1457 − 0.0009 − 0.0003 − 0.3442 0.0028 0.0030 0.930 − 0.341 0.134 3855 695 627 − 86.3 − 10.0 − 3.3 − 8.3 4.5 − 8.3 − 26.9 3681 667 619 2561 0.0(0.0) 1.135(1.172) 1.755(1.706) 2.090(2.069)

− 1870.6030688 0.6146 0.0003 0.0007 − 1.4603 − 0.0007 0.0003 0.0242 0.0028 0.0026 0.962 − 0.209 0.173 2807 505 626 − 45.9 − 6.8 − 3.6 − 3.5 2.8 − 1.7 − 1.9 2715 489 620 1955 0.0(0.0) 1.135(1.172) 1.755(1.706) 2.090(2.069)

a

Ee is the equilibrium energy. α i is a rotation-vibration interaction constant. ζ ij is a Coriolis coupling constant. ωi is a harmonic vibrational wavenumber. xij and g22 are the anharmonic vibrational constants. ν i is the anharmonic vibration wavenumber. ZPVE is the zero-point vibrational energy and μ is the dipole moment. The labels “x,” “q,” and “p” denote the x, q, and p axes defined in Fig. 2. b Dipole moment value at the equilibrium structure, calculated as finite electric-field derivative of the MR-SDCI+Q_DK3 energy. Values calculated as expectation values at the MR-SDCI_DK3 level are given in parentheses. |μe | is the length of the corresponding dipole moment vector.

C. The MORBID calculations

1. The analytical potential energy function

The ab initio points computed for ZnOH are now used to obtain an analytical representation for the electronic-groundstate potential energy surface. By employing this potential energy surface, we then continue to solve variationally (i.e., by diagonalizing a matrix representation of the rotation-vibration Hamiltonian) the associated rotation-vibration Schrödinger equation. For this step of the calculation, we make use of the MORBID (Morse Oscillator Rigid Bender Internal Dynamics20–23 ) program. The ab initio points for ZnOH are used as input for a least-squares fitting aimed at determining the parameter val-

TABLE V. Values of the parameters in Eqs. (5) and (6) for ZnOH, obtained in a least-squares fitting to the computed ab initio energies.a ρ e /deg e /Å r12 e r32 /Å a1 /Å−1 a3 /Å−1 G000 /Eh d G002 G003 G004 G005 G006 G101 G102 G103 G104

64.936(29)b 1.80243(10) 0.960537(37) 1.788c 2.355 − 1870.60305372(42) 7664(20) 735(63) 836(117) 1585(129) 3284(209) − 6503(37) 430(141) − 4795(120) − 899(390)

G011 G012 G013 G014 G200 G201 G020 G021 G110 G111 G112 G030 G031 G040

−3693(25) − 993(89) − 1439(74) 693(243) 25347(35) − 166(78) 37515(30) − 2756(25) − 554(118) 2416(200) 2847(552) 727(39) − 1408(91) 3510(156)

The parameters are given in units of cm−1 unless designated otherwise. The parameters e e and a1 (r32 and a3 ) are associated with the Zn–O (O–H) bond. Equation (5) defines r12 the expansion coefficients Gjkl .

a

b

The standard errors (in units of the least significant digit) are given in parentheses. For parameters held fixed in the least-squares fitting, no standard error is given. d G000 is the equilibrium potential energy value. c

ues in a parameterized, analytical representation of the potential energy surface. This analytical function is taken to be V ( r12 , r32 , ρ) ¯ =



j

Gj kl y1 y3k (cos ρe − cos ρ) ¯ l,

(5)

j kl

where    e yi = 1 − exp − ai ri2 − ri2 .

(6)

Equation (6) expresses the quantities yi , i = 1 or 3, as functions of r12 and r32 , the instantaneous values of the Zn–O and O–H distances, respectively, and the expression involves the e e , i = 1 or 3, where ri2 is the equilibrium parameters ai and ri2 value of ri2 . As the coordinate describing the bending motion we use ρ¯ = π −  (Zn–O–H) with the equilibrium value ρ e . The coefficients used to expand the potential in terms of y1 (associated with the Zn–O bond), y3 (associated with the ¯ are denoted Gjkl . O–H bond), and (cos ρe − cos ρ) The least-squares fitting of the ab initio energies was made in precisely the same manner as detailed, for example, for FeNC in Ref. 37. In the ultimate least-squares fitting, the input consisted of 539 points, and we could successfully determine the values of 27 parameters. The parameters a1 and a3 could not be determined in the fitting and we constrained them as a1 = 1.788 Å−1 and a3 = 2.355 Å−1 . These values were suggested by the perturbation-theory analyses reported above. A standard deviation of 8.6 cm−1 was obtained in the ultimate fitting and Table V lists the optimized parameter values determined. Parameters, whose values could not be determined to be significantly different from zero, were fixed at zero and they are not included in the table. In Figs. 4 and 5, we present the bending potential energy function of ZnOH, which describes the molecule bending with the bond lengths fixed at their respective equilibrium

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FIG. 4. The bending potential energy function of ZnOH, describing the molecule bending with the bond lengths fixed at their respective equilibrium values, plotted as a function of ρ¯ = 180◦ −  (Zn–O–H) (solid curve). The red dotted lines indicate the lowest 10 bending vibrational energies [with the quantum number assignments (0, v2 , 0), v2 = 0, 1, 2, . . . , 9] of 64 ZnOH and for each of the associated vibrational states, the bending wavefunction (consistent with the volume element d ρ) ¯ is depicted as a blue curve. The ordinate scale used for these wavefunctions is arbitrary, but the same for all of them. The wavefunctions and energies presented here result from a rigid-bender calculation in which the molecule is taken to bend with constant, equilibriumvalue bond lengths.

values: ¯ = V ( r12 = 0, r32 = 0, ρ)

6 

G00l (cos ρe −cos ρ) ¯ l.

(7)

l=2

The values of the parameters G002 , G003 , . . . , G006 can be found in Table V. The barrier to linearity of V ( r12 = 0, r32 = 0, ρ) ¯ is determined as 2517 cm−1 . Figure 4 shows the bending wavefunctions for the 10 lowest bending states (0, v2 , 0) [with v2 = 0, 1, 2, . . . , 9] of 64 ZnOH. The four lowest states are below the barrier to linearity with the fourth state (with v2 = 3) being very close to the barrier top. Figure 5 is analogous to Fig. 4 but the bending wavefunctions are now those of 64 ZnOD with v2 = 0, 1, 2, . . . , 12. For the deuterated isotopologue, six states are below the barrier to linearity with the sixth state (v2 = 5) almost coinciding in energy with the barrier maximum. 2. Rotation-vibration energies and structural parameters

Zn has four isotopes with abundances above 1%: 64 Zn (48.6%), 66 Zn (27.9%), 67 Zn (4.1%), and 68 Zn (18.8%). For each isotope, we give in parentheses its natural abundance. In a high-resolution rotational or rotation-vibration spectrum of ZnOH with the Zn isotopes in natural abundance, the transitions of all the corresponding isotopologues Zn16 OH will be observable, and so we do MORBID calculations for these four isotopologues, and for the corresponding isotopologues of ZnOD. In these calculations, we employ the fol-

FIG. 5. The bending potential energy function from Fig. 4 (solid curve) plotted together with the bending wavefunctions for the states (0, v2 , 0) with v2 = 0, 1, 2, . . . , 12 of 64 ZnOD. For each state, the bending energy is indicated by a red dotted line and the corresponding bending wavefunction is drawn as a blue curve. Bending energies and wavefunctions are calculated as described for 64 ZnOH in the caption of Fig. 4.

lowing basis set (see Ref. 20): The stretching problem was prediagonalized with Morse oscillator functions |n1 n3  having n1 + n3 ≤ NStretch = 20. In constructing the final rotationvibration matrices we used the NBend = 21 lowest bending basis functions and the NA = 20 lowest stretching basis functions of A symmetry in the molecular symmetry group38 Cs (M). The resulting vibrational energy spacings are listed in Table VI. ZnOH is unusual in that the ν 2 (bending) and ν 3 (Zn– O stretch) fundamental term values are close in energy. For example, the 64 ZnOH values are 665 and 616 cm−1 , respectively. Consequently, the vibrational energy level patterns of the ZnOH isotopologues have “polyad” structures in which the vibrational levels with a common value of v2 + v3 form a polyad. There are strong Fermi-type interactions within each polyad as reflected by the erratic labeling of the (0, v2 , v3 ) states in Table VI. In the table, the states are labeled by the vibrational quantum numbers of the basis state with the largest contribution to the vibrational eigenfunction and these labels do not follow the simple pattern consistent with E(0, 0, 1) < E(0, 1, 0). For the ZnOD isotopogues, however, E(0, 0, 1) is almost unchanged relative to ZnOH but E(0, 1, 0) is substantially lowered and so E(0, 0, 1) becomes significantly different from E(0, 1, 0). For example, 64 ZnOD has E(0, 0, 1) = 620 cm−1 and E(0, 1, 0) = 488 cm−1 . In consequence, the ZnOD isotopologues do not have polyad-type vibrational energy level pattern and their vibrational-energy labels in Table VI are regular. In Figs. 4 and 5 we recognize the irregular “anharmonicity” effects typical for a bending vibration of a chain molecule governed by a potential energy function with a superable barrier to linearity. Obviously, for the rigid-bender term values of the (0, v2 , 0) states plotted in the figures, the increments

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TABLE VI. Vibrational energy spacingsa (in cm−1 ) for several isotopologues of ZnOH and ZnOD. (v1 , v2 , v3 ) (0, 0, 0) (0,0,1) (0,1,0) (0,0,2) (0,2,0) (0,1,1) (0,3,0) (0,0,3) (0,1,2) (0,2,1) (0,4,0) (0,3,1) (0,0,4) (0,1,3) (0,2,2) (0,5,0) (0,4,1) (0,3,2) (0,0,5) (0,1,4) (0,2,3) (0,5,1) (0,6,0) (0,3,3) (0,0,6) (1,0,0) a b

64 ZnOH

66 ZnOH

67 ZnOH

68 ZnOH

(v1 , v2 , v3 )

64 ZnOD

66 ZnOD

67 ZnOD

68 ZnOD

(2547.3)b 615.9 665.1 1218.2 1254.6 1319.1 1769.7 1834.6 1890.1 1967.0 2252.1 2403.8 2449.8 2522.7 2609.9 2779.5 2919.7 3018.6 3067.9 3154.6 3249.6 3353.5 3490.1 3582.1 3617.1 3677.1

(2546.2)b 614.5 664.5 1216.1 1252.6 1317.1 1769.1 1830.2 1886.6 1963.6 2251.7 2400.3 2444.0 2517.6 2605.0 2778.6 2917.5 3011.5 3061.7 3148.0 3243.2 3351.4 3487.9 3577.0 3608.8 3677.1

(2545.7)b 613.8 664.2 1215.1 1251.7 1316.2 1768.8 1828.1 1884.9 1962.0 2251.5 2398.6 2441.2 2515.2 2602.7 2778.2 2916.4 3008.0 3058.8 3144.9 3240.1 3350.3 3486.9 3574.4 3605.0 3677.2

(2545.3)b 613.1 664.0 1214.0 1250.8 1315.3 1768.5 1826.0 1883.2 1960.4 2251.3 2396.9 2438.5 2512.8 2600.5 2777.8 2915.4 3004.6 3055.9 3141.8 3237.1 3349.3 3485.9 3571.7 3601.4 3677.0

(0, 0, 0) (0,1,0) (0,0,1) (0,2,0) (0,1,1) (0,0,2) (0,3,0) (0,2,1) (0,1,2) (0,4,0) (0,0,3) (0,3,1) (0,5,0) (0,2,2) (0,1,3) (0,4,1) (0,0,4) (0,6,0) (0,3,2) (1,0,0) (0,5,1) (0,2,3) (0,1,4) (0,7,0) (0,0,5) (0,4,2)

(1946.0)b 487.8 619.8 955.9 1106.6 1231.2 1397.7 1575.2 1717.3 1799.0 1834.4 2018.2 2151.8 2187.0 2320.2 2422.7 2432.0 2521.5 2632.1 2708.6 2783.3 2792.7 2918.4 2951.3 3021.3 3040.9

(1945.0)b 487.6 617.8 955.7 1104.4 1227.3 1397.5 1573.0 1713.2 1798.8 1828.7 2015.9 2151.4 2182.8 2314.2 2420.5 2424.5 2521.0 2627.9 2708.6 2780.1 2787.4 2910.6 2950.8 3012.1 3036.7

(1944.5)b 487.6 616.8 955.7 1103.4 1225.5 1397.4 1571.9 1711.2 1798.7 1825.9 2014.8 2151.3 2180.8 2311.3 2419.3 2420.9 2520.8 2625.8 2708.6 2778.3 2785.1 2906.8 2950.6 3007.6 3034.7

(1944.0)b 487.5 615.9 955.6 1102.4 1223.7 1397.3 1570.9 1709.3 1798.6 1823.3 2013.8 2151.1 2178.8 2308.5 2417.3 2418.5 2520.6 2623.8 2708.6 2776.4 2783.1 2903.2 2950.4 3003.3 3032.7

Calculated using the MORBID program with the potential energy parameters from Table V. Zero point energy.

(v2 ) =E(0, v2 + 1, 0) − E(0, v2 , 0) initially decrease with increasing v2 as we start from v2 = 0. They reach a minimum value for the two states energetically closest to the barrier maximum whereas for states with energies above the top of the barrier, (v2 ) increases with increasing v2 . This behavior is also found for the term values of Table VI, obtained in “fully coupled” MORBID calculations that take into account the stretching vibrations and the bend-stretch interaction. For example, for 64 ZnOH we calculate from Table VI that (v2 ) = 665, 590, 515, 482, 527, and 711 cm−1 , respectively, for v2 = 0, 1, 2, 3, 4, 5. The minimum value of 482 cm−1 is attained for v2 = 3. Effective rotational constants Aeff , Beff , and Ceff are given in Table VII for the vibrational ground state and the fundamental levels of the eight isotopologues 64 ZnOX, 66 ZnOX, 67 ZnOX, and 68 ZnOX with X = H or D. These Aeff , Beff , and Ceff values are derived [by solving Eqs. (11-66), (11-67), and (11-68) of Ref. 38] from term values with J = 0 and 1 calculated with the MORBID program from the potential energy parameters in Table V. There are experimentally derived values available for the rotational constants in the vibrational ground states of 64 ZnOH and 64 ZnOD.9 The MORBID values of A0 , B0 , C0 deviate 2.4%, 0.2%, and 0.3%, respectively, from the corresponding experimental values for 64 ZnOH. For 64 ZnOD, the corresponding deviations are 2.0%, 0.1%, and 0.2%. The effective rotational constants for ZnOH show trends similar to those already noted for TeOH in Ref. 39: The val-

ues of Aeff are strongly dependent on the off-axis position of the lightest nucleus H and 10–100 times larger than those of Beff and Ceff . The value obtained for Aeff is critically dependent on details of the bending potential energy function and the level of the theoretical treatment of the bending motion. Thus, the theoretical value of the A0 constant from Table III (which results from a perturbation-theory treatment of the effects of the bending motion) differs 7.5% from the experimental value for 64 ZnOH, whereas as mentioned above, the corresponding, theoretical MORBID value (resulting from a more accurate, variational treatment of the effects of the bending motion) differs by 2.4% only. In addition, Aeff changes drastically upon excitation of the bending motion (ν 2 ) and the O–H stretching motion (ν 1 ). Beff and Ceff are expected to be most sensitive to the excitation of the Zn–O stretching motion ν 3 , but owing to the polyad structure of the vibrational energy level pattern of the ZnOH isotopologues with heavy mixing of basis functions, the effect of vibrational excitation on the rotational constants is somewhat irregular. As done previously for a number of triatomic molecules including FeNC,37 FeCN,33 CoCN,35 BrCN+ ,40 NiCN,36 FeOH,3 and CsOH,1 we carry out vibrational averaging of the structural parameters for ZnOH and ZnOD in their electronic ground states. The averaged values are computed as expectation values involving the relevant rovibrational wavefunctions obtained in the MORBID calculation. In defining the structural parameters, we make use of the angles η, τ , and ρ¯ introduced in Fig. 2 together with the a axis which passes through

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TABLE VII. Effective rotational constants (in cm−1 ) for several isotopologues of ZnOH, calculated with the MORBID program using the potential energy parameters in Table V. (v1 , v2 , v3 )

Aeff

Beff

Ceff

64 ZnOH

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

25.4606a 23.7701 30.1156 27.0032

0.3732a 0.3727 0.3736 0.3708

0.3669a 0.3661 0.3668 0.3642

66 ZnOH

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

25.4600 23.7684 30.2111 26.9060

0.3708 0.3703 0.3713 0.3684

0.3646 0.3639 0.3646 0.3619

67 ZnOH

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

25.4597 23.7648 30.2550 26.8613

0.3697 0.3692 0.3702 0.3673

0.3635 0.3628 0.3635 0.3608

68 ZnOH

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

25.4594 23.7322 30.2965 26.8191

0.3686 0.3653 0.3691 0.3662

0.3624 0.3589 0.3625 0.3597

64 ZnOD

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

13.6924b 12.9929 15.8050 13.6683

0.3465b 0.3464 0.3472 0.3438

0.3370b 0.3364 0.3367 0.3344

66 ZnOD

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

13.6916 12.9922 15.8012 13.6702

0.3442 0.3441 0.3449 0.3416

0.3348 0.3343 0.3345 0.3322

67 ZnOD

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

13.6913 12.9919 15.7994 13.6712

0.3431 0.3430 0.3438 0.3405

0.3338 0.3332 0.3334 0.3312

68 ZnOD

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

13.6909 12.9915 15.7976 13.6721

0.3421 0.3419 0.3427 0.3394

0.3327 0.3322 0.3325 0.3302

Experimental values:9 A0 = 24.8533(600) cm−1 , B0 = 0.37237420(17) cm−1 , and C0 = 0.36597042(17) cm−1 . Quantities in parentheses are quoted uncertainties in units of the last digit given. b Experimental values:9 A0 = 13.4310(57) cm−1 , B0 = 0.34614739(18) cm−1 , and C0 = 0.33642784(18) cm−1 . Quantities in parentheses are quoted uncertainties in units of the last digit given. a

TABLE VIII. Expectation values (in Å unless otherwise indicated) of geometrical parameters for 64 ZnOH and 64 ZnOD, calculated with MORBID. The variable r12 is the Zn–O distance, and r32 is the O–H distance. (v1 , v2 , v3 )K

r12 

r32 

a r12 cos η r32 cos τ  ρ/deg ¯

64 ZnOH

(0,0,0)0 (0,0,0)1 (1,0,0)0 (1,0,0)1 (0,1,0)0 (0,1,0)1 (0,0,1)0 (0,0,1)1 (0,0,2)0 (0,0,2)1 (0,2,0)0 (0,2,0)1 (0,1,1)0 (0,1,1)1 (0,3,0)0 (0,0,3)0 (0,1,2)0 (0,2,1)0 (0,4,0)0 (0,0,4)0 (0,1,3)0 (0,3,1)0 (0,2,2)0

1.8078 1.8080 1.8096 1.8097 1.8074 1.8072 1.8161 1.8171 1.8186 1.8244 1.8154 1.8107 1.8121 1.8136 1.7993 1.8325 1.8229 1.8179 1.7939 1.8402 1.8295 1.8209 1.8245

0.9778 0.9780 1.0098 1.0096 0.9788 0.9792 0.9779 0.9781 0.9780 0.9783 0.9785 0.9800 0.9795 0.9792 0.9776 0.9779 0.9789 0.9800 0.9782 0.9774 0.9791 0.9781 0.9805

1.8071 1.8073 1.8088 1.8090 1.8068 1.8066 1.8154 1.8164 1.8180 1.8237 1.8148 1.8101 1.8115 1.8129 1.7987 1.8318 1.8223 1.8173 1.7932 1.8395 1.8289 1.8204 1.8239

0.4442 0.4420 0.4398 0.4378 0.4650 0.4626 0.4586 0.4535 0.4917 0.4707 0.4861 0.4727 0.4744 0.4835 0.5919 0.4834 0.4923 0.4822 0.6185 0.4923 0.4985 0.5565 0.4886

63(10) 64(10) 63(10) 64(10) 60(17) 60(17) 63(10) 64(10) 63(10) 64(10) 55(23) 57(22) 60(17) 60(17) 48(28) 63(10) 60(17) 55(23) 44(32) 63(10) 60(17) 48(28) 55(23)

64 ZnOD

(0,0,0)0 (0,1,0)0 (0,0,1)0 (1,0,0)0 (0,2,0)0 (0,1,1)0 (0,0,2)0 (0,3,0)0 (0,2,1)0 (0,1,2)0 (0,0,3)0 (0,4,0)0 (0,3,1)0 (0,2,2)0 (0,1,3)0 (0,0,4)0

1.8079 1.8059 1.8189 1.8092 1.8033 1.8169 1.8303 1.7997 1.8140 1.8282 1.8419 1.7943 1.8097 1.8251 1.8397 1.8528

0.9731 0.9739 0.9731 0.9980 0.9744 0.9740 0.9732 0.9743 0.9746 0.9742 0.9733 0.9733 0.9747 0.9748 0.9743 0.9733

1.8054 1.8036 1.8165 1.8067 1.8012 1.8146 1.8279 1.7978 1.8119 1.8259 1.8394 1.7927 1.8078 1.8230 1.8373 1.8504

0.4614 0.4814 0.4643 0.4588 0.5084 0.4825 0.4671 0.5459 0.5080 0.4835 0.4699 0.6023 0.5450 0.5078 0.4842 0.4739

65(8) 63(15) 65(8) 65(8) 60(19) 63(15) 65(8) 56(23) 60(19) 63(15) 65(8) 51(27) 56(23) 60(19) 63(15) 65(8)

a Quantities in parentheses are quantum-mechanical uncertainties δ ρ¯ (see text) in units of degrees.

the Zn nucleus and the center of mass M of the OH fragment; the nuclear center of mass is also located on this axis. This axis is not identical, but very close, to the a principal axis38 of the molecule.35, 37 We denote by r12 cos η(r32 cos τ ) the projection of the Zn–O(O–H) distance onto the a axis, and Table VIII lists values of the averaged projections for selected rovibrational states of 64 ZnOH and 64 ZnOD. The motivation for undertaking the calculations of vibrationally averaged structural parameters is discussed at some length in Refs. 37 and 35. Table VIII also presents averaged values of r12 , r32 , and ρ¯ (see Fig. 2 for definition). (v2 ,2 ) n (v2 ,2 ) |ρ¯ |bend  (n = 1 or 2). Here, We define ρ¯ n  =bend the normalized bending basis wavefunction with the largest contribution to the vibrational state in question is designated (v2 ,2 ) as bend . According to quantum mechanics, the average ρ ¯

 has an uncertainty of δ ρ¯ = ρ¯ 2  − ρ ¯ 2 . In computing the expectation values in Table VIII, we proceeded precisely as detailed for 6 FeNC and 3 CoCN in Refs. 37 and 35. Among the molecules whose rovibrationally averaged structures we have studied, the majority have linear equilibrium structures [FeNC,37 FeCN,33 CoCN,35 BrCN+ ,40 NiCN,36 CsOH1 ]. For these linear molecules, the average structural parameters vary with vibrational excitation in the same manner: Excitation of an A–B stretching vibration causes an elongation of the rovibrationally averaged A–B bond length, and excitation of the bending mode causes ρ ¯ to increase and the projected bond lengths to decrease, the largest decrease being found for the bond connecting the two lightest nuclei. In Ref. 36, we discussed

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the reasons for this variation in detail, using NiCN as example. The molecule FeOH, for which we reported theoretical results in Refs. 2 and 3, is slightly bent at equilibrium. Its electronic ground state is Renner-degenerate with two components, X˜ 6 A and A˜ 6 A , that are exactly degenerate at linearity but slightly split at bent geometries. The X˜ 6 A (A˜ 6 A ) potential energy function has a barrier to linearity3 of 273(266) cm−1 and only the bending ground state is energetically below the barrier; already the first excited bending state has an energy larger than the barrier maximum. In Ref. 3, we reported rovibrationally averaged structural parameters for FeOH calculated from a potential energy surface that we obtained as the average of the X˜ 6 A and A˜ 6 A surfaces. For FeOH, we found the usual variation of the averaged bond lengths with excitation of the stretching vibrations and that ρ ¯ depends significantly on the quantum number K (which measures the projection of the total angular momentum on the moleculefixed z axis). The K-dependence of ρ ¯ is caused by the fact that FeOH, with its low barrier to linearity, is an intermediate case between the “standard” linear molecule and the “standard” bent molecule. For the standard bent molecule, we describe the bending vibrational states by the quantum number v2bent that we also use for FeOH and ZnOH (and call v2 in the present work), while the states of a standard linear molecule are described by v2lin = 2v2bent + K and 2 = K (see Eqs. (17119) and (17-120) of Ref. 38). Thus, states with different K values typically belong to different vibrational states of a standard linear molecule. ¯ decreases from 39◦ to 35◦ between the For FeOH,3 ρ 0 (0,0,0) and (0,1,0)0 states, this decrease being accompanied by a slight increase from 0.7320 Å to 0.7439 Å of the O–H bond length projection. These changes can be easily explained by noting that in FeOH, the (0,0,0) state is energetically below the barrier to linearity while the (0,1,0) state is above the barrier top. Thus, in the (0,1,0) state, it becomes easier for the molecule to approach the linear geometry. In ZnOH, these effects are also present and more pronounced than in FeOH. For the (0, v2 , 0)0 states of 64 ZnOH ¯ varies as 63◦ , 60◦ , 55◦ , 48◦ , with v2 = 0, 1, 2, 3, and 4, ρ and 44◦ , respectively. These changes can be rationalized by considering Fig. 4: Also for ZnOH, it becomes easier for the molecule to approach the linear geometry with increasing v2 . ¯ = 55◦ ) and v2 = 3 (ρ ¯ = 48◦ ) states, Between the v2 = 2 (ρ r32 cos τ  increases from 0.4861 Å to 0.5919 Å, a change of 22% (Table VIII). This is another indication of the molecule’s increased ability to move near the linear geometry. Finally, we note from Table VIII that for ZnOH, the averaged structural parameters do not depend critically on the quantum number K. ZnOH has a barrier to linearity sufficiently high that it behaves like a standard bent molecule so that its bending states are well described by the quantum number v2bent = v2 .

3. Dipole moments and intensities

At each molecular geometry considered in the ab initio calculation of the electronic energy, we have also calculated

J. Chem. Phys. 141, 094308 (2014)

the components of the molecular dipole moment for the electronic ground state of ZnOH. In defining these components, we employ the p and q axes defined in Fig. 1 of Ref. 22 and already mentioned in the caption of Fig. 2. The Cartesian axis system xqp has origin in the nuclear center of mass and is right-handed. The nuclei are in the qp plane and the x axis is perpendicular to this plane. The p axis is parallel to the Zn–O bond, pointing in the direction O → Zn. The q axis is perpendicular to the p axis and oriented such that the q coordinate of the H nucleus is positive. In the ab initio calculation, the dipole moment components along the p and q axes are computed as μ¯ p =  elec |μp | elec el and μ¯ q =  elec |μq | elec el , respectively, where  elec is the ground-state electronic wavefunction of ZnOH and the subscript “el” indicates integration over the electronic coordinates only. We express μ¯ q and μ¯ p as parameterized functions of the nuclear coordinates, where the parameter values are obtained by fitting to the computed ab initio values of the molecular dipole moments:  (q) j k μ¯ q ( r12 , r32 ,ρ) ¯ = sin ρ¯ μj kl r12 r32 (cos ρe −cos ρ) ¯l j kl

(8) and μ¯ p ( r12 , r32 , ρ) ¯ =



(p)

j

k μj kl r12 r32 (cos ρe − cos ρ) ¯ l,

j kl

(9) TABLE IX. The electric dipole moment parameters of ZnOH obtained by fitting the analytical functions of Eqs. (8) and (9) through the calculated ab initio values. μ¯ p (p)

μ¯ q (q)

1.29327(16)

(q)

− 0.61753(91)

μ002 /D

(q)

− 0.0510(16)

(q) μ003 /D (q) μ004 /D (q) μ100 /D (q) μ101 /D (q) μ102 /D (q) μ010 /D (q) μ011 /D (q) μ200 /D (q) μ020 /D (q) μ110 /D (q) μ111 /D

− 0.0602(44)

μ000 /D

1.70028(24)a

μ000 /D

(p) μ001 /D (p) μ002 /D (p) μ003 /D (p) μ004 /D (p) μ100 /D (p) μ101 /D (p) μ102 /D (p) μ010 /D (p) μ011 /D (p) μ012 /D (p) μ200 /D (p) μ201 /D (p) μ020 /D (p) μ021 /D (p) μ111 /D (p) μ300 /D (p) μ030 /D (p) μ031 /D

1.0419(10)

μ001 /D

0.2906(37) − 0.0762(31)

a

0.038(10) Å−1

4.9563(44)

Å−1

− 1.9400(58)

Å−1

1.687(16)

Å−1

− 0.5795(34)

Å−1

0.4374(69)

Å−1

− 0.775(12)

Å−2

0.324(31)

Å−2

− 1.108(67)

Å−2

0.644(20)

Å−2

− 0.167(47)

Å−2

− 0.48(21)

Å−3

− 1.04(30)

Å−3

0.84(12)

Å−3

− 1.39(28)

− 0.0162(51) Å−1

− 0.2634(23)

Å−1

0.6729(77)

Å−1

0.096(12)

Å−1

0.6154(18)

Å−1

− 0.7439(36)

Å−2

− 0.097(21)

Å−2

− 0.496(11)

Å−2

− 1.271(83)

Å−2

0.38(19)

Quantities in parentheses are standard errors in units of the last digit given.

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where the μj kl and the μj kl are expansion coefficients while e , i = 1 or 3. ri2 = ri2 − ri2 (q) (p) Values for the μj kl and μj kl parameters are determined by fitting Eqs. (8) and (9) through the 539 available ab initio values of μ¯ q and μ¯ p , respectively. We used 14(19) parameters to fit μ¯ q (μ¯ p ) with a standard deviation of 0.0016(0.0024) D. Table IX lists the parameter values obtained.

4. Spectral simulations

The rovibrational spectra of X˜ 2 A ZnOH and ZnOD have been simulated with the MORBID program system. The simulated spectra are shown in Figs. 6 and 7; they involve all states with J ≤ 20 and cover the wavenumber regions 0– 4000 cm−1 for ZnOH and 0–3000 cm−1 for ZnOD. The ZnOH and ZnOD molecules are assumed to be in thermal equilibrium at an absolute temperature of T = 12 K. In the figures, the rotation-vibration transitions are depicted as sticks. The height of each stick is the integrated absorption coefficient I(f ← i). For an electric dipole transition from an initial state i (with energy Ei and rovibronic wavefunction ψ i ) to a final

FIG. 7. The predicted infrared spectra of 64 ZnOD (red), 66 ZnOD (blue), 67 ZnOD (black), and 68 ZnOD (green) with the Zn isotopes in natural abundance for J ≤ 20 and T = 12 K in the wavenumber region below 3000 cm−1 . Note the different ordinate scales on the three displays.

state f (with energy Ef and rovibronic wavefunction ψ f ), I(f ← i) is given by38 I (f ← i) =

 8π 3 NA  νif exp −

Ei kT

  1 − exp −

hc νif kT



3hcQ ×S(f ← i).

This expression involves the partition function  Q= gw exp(−Ew /kT )

(10)

(11)

w

for which the summation runs over all rovibronic states of the molecule. The quantity S(f ← i) is the line strength of the electric dipole transition   S(f ← i) = gns |ψf |μA |ψi |2 , (12) mi , mf A=X,Y,Z

FIG. 6. The predicted infrared spectra of 64 ZnOH (red), 66 ZnOH (blue), 67 ZnOH (black), and 68 ZnOH (green) with the Zn isotopes in natural abundance for J ≤ 20 and T = 12 K in the wavenumber region below 4000 cm−1 . Note the different ordinate scales on the four displays.

where the nuclear spin statistical weight is denoted as gns . Furthermore, in Eqs. (10)–(12) the transition wavenumber is ν˜ if = (Ef − Ei )/(hc), the total degeneracy of the state with the energy Ew is denoted gw , and the components of the molecular dipole moment operator in the space-fixed XYZ axis system are called (μX , μY , μZ ), while NA , k, h, and c are the Avogadro constant, the Boltzmann constant, the Planck constant, and the speed of light in vacuum, respectively. Finally, mi and mf are the initial-state and final-state projections, respectively, of the angular momentum onto the space-fixed Z axis in units of ¯.

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In the course of the MORBID intensity calculation, the dipole moment vector is transformed from the body-fixed xqp axis system described above to a molecule-fixed axis system xyz determined from the Eckart-Sayvetz conditions (see, for example, Refs. 20–23) and different from the xyz axis system defined in the caption of Fig. 2. The xyz axis system employed by MORBID is of the type customarily used in spectroscopic applications and it is a principal axis system at equilibrium.

J. Chem. Phys. 141, 094308 (2014) TABLE X. Yamada-Winnewisser quasi-linearity parameter γ 0 and the bond angle deviation from linearity ρ ¯ 0 (= 180◦ −  (X–Y–Z)0 ) in the rovibronic ground states of various triatomic molecules XYZ. Molecule Linear

In Eq. (13), E0 is the energy of the rovibrational ground state, and E(v2bent = 0, Ka = 1) and E(v2bent = 1, Ka = 0) are selfexplanatory (see also Eqs. (17-119) and (17-120) of Ref. 38 together with the text accompanying them). As is discussed in Refs. 24 and 25, we have in the limiting case of a rigidly linear, harmonically bending molecule, that the ratio [E(v2bent = 0, Ka = 1) − E0 ]/[E(v2bent = 1, Ka = 0) − E0 ] ≈ 1/2 so that γ 0 ≈ −1. In the other limiting case of a rigidly bent molecule the energy ratio is very small and so γ 0 ≈ +1. The MORBID value of the numerator energy E(v2bent = 0, Ka = 1) − E0 in Eq. (13) is 25.83 cm−1 for 64 ZnOH. This is the lowest energy with J = Ka = 1;

BrCN+

3

CoCN NiCN 6 FeCN 6 FeNC 1  + CsOH 6 A FeOH 2 A ZnOH 2

IV. SUMMARY AND CONCLUSION

We have calculated a three-dimensional PES for the electronic ground state of ZnOH at the MR-SDCI+Q_DK3/[QZP ANO-RCC (Zn, O, H)] level of ab initio theory. The electronic ground state is confirmed to be 2 A which correlates to a 2  + state at linear configuration. The equilibrium structure of the X˜ 2 A state is nonlinear. The electronic structure is discussed, and the Zn–O bond is concluded to be essentially ionic but with an appreciable covalency [Fig. 3(c)] resulting in a bent structure with  (Zn–O–H) = 114.9◦ . The O–H bond is similar to those of OH− and H2 O. The Mulliken charge distribution is obtained as Zn0.60 O−0.79 H0.19 , and orbital analyses show that the unpaired electron is almost entirely localized on the Zn atom [Fig. 3(a)]. We have shown that in metal hydroxides, the degree of covalency in the metal–O bond is the crucial factor that determines whether the metal hydroxide is strongly bent (ZnOH), quasi-linear (FeOH), or linear (CsOH). Comparison of the bending force constants leads us to conclude that ZnOH is less floppy than the molecules whose metal–O bonds are purely ionic or nearly ionic, but it is sufficiently floppy for the bending motion to acquire a large amplitude. The amphoteric character of X˜ 2 A ZnOH is shown by the dissociation process; the dissociation energies to either ZnO + H or Zn + OH are of similar magnitude (42 600 and 44 600 cm−1 , respectively). The rovibrational properties of ZnOH have been determined using the ab initio three-dimensional PES by second order perturbation theory and by the MORBID method. In the MORBID analyses, so far unobserved rovibrational spectra have been simulated. Quasi-linearity is often quantified by the so-called Yamada-Winnewisser quasi-linearity parameter24   E v2bent = 0, Ka = 1 − E0  γ0 = 1 − 4 ×  bent . (13) E v2 = 1, Ka = 0 − E0

2

Quasi-linear Bent

ρ ¯ 0 a /deg

γ0

Reference

8(4) 8(5) 9(5) 10(5) 13(7) 17(9) 39(14) 63(10)

− 1.00 − 1.06 − 1.06 − 1.07 − 1.11 − 1.07 0.10 0.84

40b 35b 36b 33b 37b 1 3 This study

a Quantities in parentheses are quantum-mechanical uncertainties δ ρ¯ (see text) in units of degrees. b Calculated from the vibrational energies reported in the given reference.

the state with this energy is an excited rotational state in the vibrational ground state for a molecule that is bent at equilibrium. It can be shown that the term value is close to the sum of the Aeff and Ceff values for the (0,0,0) state given in Table VII. The denominator energy is the (0,1,0) vibrational energy, given as 665.1 cm−1 in Table VI. Thus, we obtain γ 0 = 0.84 for 64 ZnOH and similarly 0.88 for 64 ZnOD. Now, we have obtained γ 0 -values for three types of prototypical molecules: The strongly bent molecule exemplified by X˜ 2 A ZnOH, the quasi-linear molecule exemplified by X˜ 6 A FeOH,2, 3 and the linear molecule exemplified by X˜ 1  + CsOH.1 In Table X, we have compiled the γ 0 values for several triatomic molecules that we have studied previously. The list confirms that, as suggested by Yamada and Winnewisser,24 γ 0 is near −1 for a linear molecule, near 0 for a quasi-linear molecule, and near +1 for a bent molecule. Note that even for linear molecules with ρ ¯ 0 deviating substantially from 0◦ (which is indicative of a bending motion with appreciable amplitude), we obtain γ 0 ≈ −1. We hope that the results of the present work will be of assistance in experimental spectroscopic investigations of ZnOH. ACKNOWLEDGMENTS

This work was supported in part by the Deutsche Forschungsgemeinschaft (DFG) and the Fonds der Chemischen Industrie. APPENDIX: AB INITIO CALCULATIONS ON CsOH

The equilibrium structure and properties of X˜ 1  + CsOH are re-calculated at the MR-SDCI +Q_DK3/[QZP+g ANO-RCC (Cs,41 O,16 H17 )] level of theory, with active space over Cs 6s, O 2s and 2p, and H 1s (8 electrons over 6 orbitals) including dynamical electron correlation for electrons in the Cs 4d, 5s, and 5p orbitals by single and double excitations from these orbitals in the CSFs generated from the active space; re (Cs–O) = 2.3932 Å, re (O–H) = 0.9568 Å, and  e (Cs–O–H) = 180.0◦ . From this equilibrium structure and the rotation-vibration interaction constants α i calculated

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previously1 the rotational constant B0 is obtained as 5502.72 MHz (with 0.03% deviation from the experimental value42 of 5501.1612(76) MHz). The dipole moment calculated as the finite electric field derivative of the MR-SDCI+Q_DK3 energy is μe = 6.780 D. The Mulliken charges, calculated at the MR-SDCI_DK3/[Sapporo-DKH3-DZP-2012 + 1s1p (Cs), Sapporo-DZP-2012 + 1s1p (O, H)] level of theory are 0.888 e (Cs), −1.416 e (O), and 0.528 e (H). Here, the basis set series ‘Sapporo’ comprise all-electron, segmented, contracted Gaussian type functions (CGTF).43–46

1 T.

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Electronic structure and rovibrational properties of ZnOH in the X̃²A' electronic state: a computational molecular spectroscopy study.

The three-dimensional ground-state potential energy surface of ZnOH has been calculated ab initio at the MR-SDCI+Q_DK3/[QZP ANO-RCC (Zn, O, H)] level ...
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