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High-Level Ab Initio Predictions for the Ionization Energies, Bond Dissociation Energies, and Heats of Formation of Titanium Oxides and Their Cations (TiO/TiO , n = 1 and 2) n

n+

Yi Pan, Zhihong Luo, Yih-Chung Chang, Kai-Chung Lau, and Cheuk Yiu Ng J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.6b09491 • Publication Date (Web): 02 Dec 2016 Downloaded from http://pubs.acs.org on December 5, 2016

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The Journal of Physical Chemistry

Submit to Journal of Physical Chemistry A High-Level ab initio Predictions for the Ionization Energies, Bond Dissociation Energies, and Heats of Formation of Titanium Oxides and Their Cations (TiOn/TiOn+, n = 1 and 2) Yi Pan,a Zhihong Luo,b Yih-Chung Chang,b Kai-Chung Lau,a,* C. Y. Ng b,* a

Department of Biology and Chemistry, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong b Department of Chemistry, University of California, Davis, CA 95616, USA

Abstract: The ionization energies (IEs) of TiO and TiO2 and the 0 K bond dissociation energies (D0) and the heats of formation at 0 K (∆H°f0) and 298 K (∆H°f298) for TiO/TiO+ and TiO2/TiO2+ are predicted by the wave-function based CCSDTQ/CBS approach. The CCSDTQ/CBS calculations involve the approximation to the complete basis set (CBS) limit at the coupled cluster level up to full quadruple excitations along with the zero-point vibrational energy (ZPVE), high-order correlation (HOC), core-valence electronic (CV), spin-orbit coupling (SO), and scalar relativistic effect (SR) corrections. The present calculations yield IE(TiO) = 6.815 eV and is in good agreement with the experimental IE value of 6.81980 ± 0.00010 eV determined in a two-color laser pulsed field ionization-photoelectron (PFI-PE) study.

The CCSDT and MRCI+Q methods give the best

predictions to the harmonic frequencies: ωe (ωe+) = 1013 (1069) and 1027 (1059) cm−1 and the bond lengths re (re+) = 1.625 (1.587) and 1.621 (1.588) Å, for TiO (TiO+) compared with the experimental values. Two nearly degenerate, stable structures are found for TiO2 cation: TiO2+(C2V) structure has two equivalent TiO bonds while the TiO2+(Cs) structure features a long and a short TiO bond. The IEs for the TiO2+(C2v) ← TiO2 and TiO2+(Cs) ← TiO2 ionization transitions are calculated to be 9.515 and 9.525 eV, respectively, giving the theoretical adiabatic IE value in good agreement with the experiment IE(TiO2) = 9.57355 ± 0.00015 eV obtained in the previous vacuum ultraviolet (VUV)-PFI-PE study of TiO2. The potential energy surface of TiO2+ along the normal vibrational 1

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coordinates of asymmetric stretching mode (ω3+) is nearly flat and exhibits a double-well potential with the well of TiO2+(Cs) situated around the central well of TiO2+(C2v). This makes the theoretical calculation of ω3+ infeasible. For the symmetric stretching (ω1+), the current theoretical predictions over-estimate the experimental value of 829.1 ± 2.0 cm−1 by more than 100 cm−1. This work together with the previous experimental and theoretical investigations supports the conclusion that the CCSDTQ/CBS approach is capable of providing reliable IE and D0 predictions for TiO/TiO+ and TiO2/TiO2+ with error limits ≤ 60 meV. The CCSDTQ/CBS calculations give the predictions of D0(Ti+−O) − D0(Ti−O) = 0.004 eV and D0(O−TiO) − D0(O−TiO+) = 2.699 eV, which are also consistent with the respective experimental determination of 0.00832 ± 0.00010 and 2.75375 ± 0.00018 eV. _________________________________________________________________________________ * Email addresses: Kai-Chung Lau ([email protected]), C. Y. Ng ([email protected])

2


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I.

Introduction Transition metal (M)-containing molecules is known to play an important role in many

scientific fields, such as chemical synthesis, material chemistry, catalysis, and astronomy.1,2 In order to understand the function of these compounds played in these environments, it is essential to measure accurate energetic properties of these molecules, such as ionization energies (IEs), 0 K bond dissociation energies (D0’s), and 0 K heats of formation (∆H°f0), which are known to govern their chemical reactivity. The fact that photoionization and photoelectron spectroscopy is concerned with photoexcitation between electronic states and bond breaking processes induced by electronic transitions, makes high-resolution photoionization and photoelectron spectroscopic techniques uniquely suitable for fundamental study of chemical bonding. In spite of the need for understanding the chemical bonding of M-containing compounds, there have only been a few high-precision experimental3-9 and theoretical10-17 studies focusing on their energetic properties. While the major difficulty for experimental studies is concerned with the low intensity of these species that can be prepared in the gas phase, the main obstacle in theoretical investigations of M-containing species is in accounting for the multi-reference characters resulting from unpaired d-electrons of M atoms involved. The contemporary multi-reference configuration interaction (MRCI) methods are ranked as the state-of-the-art ab initio quantum chemical procedures for structural and energetic calculations for M-containing species.11-14 Perturbative treatments18 and variation methods11-14 have also been applied to the multi-configuration wave-function with further correction of scalar relativistic (SR)19,20 and spin-orbit (SO) effects. In the past few years, we have succeeded in performing highly precise spectroscopic and energetic studies on many diatomic 3d and 4d-transition-metal hydrides, carbides, nitrides, and oxides and their ions (MX/MX+, X = H, C, N, and O) by using the two-color laser photoionization efficiency (PIE) and pulsed field ionization-photoelectron (PFI-PE) methods.21-23

These investigations have

allow the determination of the IE(MX) values as well as spectroscopic constants of the MX+ cations with unprecedented precision.21-26

Thus, these IE and spectroscopic data are expected to serve as

excellent benchmarks for state-of-the-art ab initio quantum calculations of the MX/MX+ systems. The experimental IE value for the FeC/FeC+ system has been compared with high-level quantum chemical predictions by the MRCI method with Davidson correction (MRCI+Q) and semi-core 3



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electron correlation (C-MRCI+Q)11,27 obtained by Tzeli and Mavridis.11(c). We have also compared the experimental IE values with the CCSDTQ/CBS predictions for FeC/FeC+, NiC/NiC+, CoC/CoC+, VC/VC+, VN/VN+ and MoO/MoO+ systems.24-30 The CCSDTQ/CBS calculations involve the approximation to the complete basis set (CBS) limit at the coupled cluster level up to quadruple excitations. The zero-point vibrational energy (ZPVE) corrections, the high-order correlation (HOC) beyond the CCSD(T) wavefunction, the core-valence (CV) electronic corrections (up to CCSDT level), and the SO and SR corrections (up to CCSDTQ level) have been taken into account. Compared with the experimental IE values, the CCSDTQ/CBS IE predictions are found to be highly accurate with discrepancy of about 40 meV.

These comparisons between the theoretical and

experimental IE and D0 values in previous studies27,29,30 suggest that the CCSDTQ/CBS procedures is capable of giving reliable energetic predictions for MX molecules with multi-reference characters. Since the coupled cluster method primarily is a single reference method, the good agreement on the MX/MX+ systems, which often have multi-reference characters, might be fortuitous.

Further

examination of the theoretical CCSDTQ/CBS predictions on MX and triatomic transition metal-containing molecules is justified in order to confirm its general reliability for spectroscopic and energetic predictions for transition metal-containing species. Recently, two-color laser photoionization and PFI-PE experiments on TiO and vacuum ultraviolet (VUV) laser photoionization and PFI-PE spectra for TiO2 have been reported.31,32 The well-resolved rotational transitions for the v+ = 0 and 1 vibrational bands of the TiO+(X2∆) and TiO2+(X2B2) ground states have also been observed. The rotational assignment of these PFI-PE spectra has made possible the direct determination of the IE(TiO) = 6.81980 ± 0.00010 eV and the vibrational spacing of 1047.3 ± 0.8 cm−1 between the v+ = 0 and 1 vibrational levels of TiO+(X2∆).31 For TiO2, IE(TiO2) = 9.57355 ± 0.00015 eV and the respective vibrational spacing of 841.1 ± 0.5 and 249.0 ± 0.5 cm−1 between the v1+ (0 → 1) and v2+ (0 →1) vibrational levels of TiO2+(X2B2) have also been determined.32 In the present quantum theoretical study, we report the CCSDTQ/CBS calculations of IE(TiO), D0(Ti−O), D0(Ti+−O), ∆Hf(TiO), ∆Hf(TiO+), vibrational frequencies (ωe and ωe+), and bond lengths (re and re+) of the TiO(X3∆) and TiO+(X2∆) ground states. We also provide predictions for IE(TiO2), 4


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D0(O−TiO), D0(O−Ti+O), ∆Hf(TiO2), ∆Hf(TiO2+), vibrational frequencies (ωn and ωn+, n=1, 2, 3), the bond lengths (re and re+), and the bond angles (θe and θe+) of the TiO2(X1A1) and TiO2+(X2B2) ground states. We have performed density functional theory (DFT) and MRCI calculations for the sake of comparison with the CCSDTQ/CBS calculations. Based on the comparisons, assessments of the reliability and accuracy of the theoretical methods on TiO/TiO+ and TiO2/TiO2+ systems are made. The MRCI predictions on bond lengths and vibrational frequencies for the ground state (and the excited states) of TiO/TiO+ have been reported previously,14 while the relevant theoretical optimizations and frequency calculations on TiO2/TiO2+ are reported for the first time in the present study.

II.

Theoretical calculations In the coupled cluster calculations of the CCSDTQ/CBS procedure, we have chosen to use the

partially unrestricted implementation, conventionally labeled as ROHF-UCCSD(T). This method is based on restricted open-shell Hartree–Fock (ROHF) orbitals and relaxes the spin restriction throughout the calculation.33,34 The CCSDTQ/CBS calculations involve the approximation to the CBS limit at the CCSD(T) level of theory. The CCSDTQ/CBS method in principle is similar to the Weizmann-n by Martin et al.,35 high-accuracy extrapolated ab initio thermochemistry (HEAT) by Stanton et al.,36 the focal-point analysis approach by Schaefer, Császár and co-workers37 and the Feller-Peterson-Dixon composite method.38 A.

Geometry optimization and extrapolated correlation energy The ground state structures of the TiO(3∆), TiO+(2∆), TiO2(1A1), TiO2+(2B2) have been

optimized at the CCSD(T) level with successively larger basis sets, proceeding from aug-cc-pwCVTZ, to aug-cc-pwCVQZ, to aug-cc-pwCV5Z.39,40

Besides the valence 2s2p electrons on oxygen and

4s3d electrons on Ti, the geometry optimizations also correlated the outer-core 3s3p electrons on Ti. The 1s2s2p electrons on Ti and the 1s electrons on oxygen are kept frozen and uncorrelated. The total CCSD(T) energies are used to extrapolate the CBS limit (Eextrapolated

CBS)

by two different

extrapolation schemes: (i)

A three-point extrapolation scheme41 using the mixed exponential/Gaussian function of the 5



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form:

E(X) = Eextrapolated CBS + B exp[−(X−1)] + C exp[−(X−1)2 ],

(1)

where X = 3, 4, and 5 for aug-cc-pwCVTZ, aug-cc-pwCVQZ, and aug-cc-pwCV5Z, respectively. The CBS energies obtained using Eq. (1) is denoted as CBSwTQ5. (ii)

A two-point extrapolation scheme42,43 using the simple power function involving the reciprocal of X,

E (X) = EextrapolatedCBS +

B , X3

(2)

where X = 4 and 5 for aug-cc-pwCVQZ and aug-cc-pwCV5Z, respectively. CBS energies obtained from Eq. (2), it is denoted as CBSwQ5.

For the extrapolated

Previous calculations on

MX/MX+,24-27,29,30 and other main-group compounds28 reveal that the difference of the extrapolated energetics between two-point and three-point extrapolation schemes are very small, we adopt to use an average of two extrapolated energies in the CCSDTQ/CBS procedure.

B.

Higher-order correlation The higher-order correlation energy (HOC) incorporates higher-order triple and quadruple

excitations, where the full triple excitation effect is estimated by the difference between CCSDT and CCSD(T) energies and the iterative quadruple excitations are estimated as the difference of CCSDTQ and CCSDT energies. The HOC calculations are done at the frozen-core level. The HOC for TiO/TiO+ and TiO2/TiO2+ is represented by the following:

EHOC = ECCSDT/aug-cc-pVTZ – ECCSD(T)/aug-cc-pVTZ + ECCSDTQ/cc-pVDZ – ECCSDT/cc-pVDZ (3)

C.

Scalar relativistic effect 6


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The scalar relativistic (SR) energy is computed using the spin-free, one-electron Douglas-Kroll-Hess (DKH) Hamiltonian.19,20 The calculations are done with the DKH-contracted aug-cc-pV5Z-DK basis sets40,44 at the CCSD(T) level. The SR energetic contributions (ESR) are taken as the difference between electronic energies at the CCSD(T)/aug-cc-pV5Z level without using the DKH Hamiltonian and at the CCSD(T)/aug-cc-pV5Z-DK level with the DKH Hamiltonian. The relativistic effect due to the full triple and quadruple excitations have also been included in the similar manner as described in Eq. (3), except that the aug-cc-pVTZ-DK and cc-pVDZ-DK basis sets are used in the respective CCSDT and CCSDTQ calculations.

D.

Core-valence electronic correction beyond CCSD(T) The electronic correlation contributions between the core and valance electrons and those

within core electrons have already been included in the single-point energy and geometrical optimization calculations at the CCSD(T) level. Additional core-valence electronic correlations (ECV) from the full triple excitations are obtained as given in Eq. (4) with the cc-pwCVTZ basis set.39,40 The outer-core 3s3p electrons on Ti are correlated.

ECV = ECCSDT/cc-pwCVTZ – ECCSD(T)/cc-pwCVTZ − (ECCSDT/cc-pVTZ – ECCSD(T)/cc-pVTZ)

E.

(4)

Spin-orbit coupling and zero-point vibrational energy correction The molecular spin-orbit (SO) coupling (ESO) of the TiO/TiO+ and TiO2/TiO2+ are computed

by first-order perturbation theory. The calculations used an uncontracted aug-cc-pVTZ basis set including the s, p, d and f functions on Ti and the s, p, d functions on O. Spin-orbit matrix elements were computed among the components of the TiO(3∆)/TiO+(2∆) state using the internally contracted MRCI wavefunction.45 The 2s2p electrons on O and the 4s3d electrons on Ti were included in the active space. The atomic ESO values of Ti(3F)/Ti+(4F) are done in similar manner. The atomic ESO for O(3P) is 0.93 kJ/mol.46 The harmonic vibrational frequencies at the CCSDTQ/cc-pVTZ, CCSDT/aug-cc-pVTZ levels are used for the zero-point vibrational energy corrections (∆EZPVE) for TiO/TiO+ and TiO2/TiO2+(C2v), respectively. For the TiO2+(Cs) structure, the CCSDT/cc-pVTZ 7



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harmonic vibrational frequencies are used. In the present work, all the CCSD(T) single-point energy and correlation contribution calculations were performed using the MOLPRO 2010.1 program47 and the CCSDT and CCSDTQ calculations were done with the string-based many-body MRCC program48 interfaced with MOLRPO. The ∆H˚f0 and ∆H˚f298 values for the TiO/TiO+ and TiO2/TiO2+ were evaluated using the atomization scheme49 and the following experimental values:50 ∆H˚f0(O) = 246.8 kJ/mol, ∆H˚f0(Ti)= 470.9 kJ/mol, ∆H˚f298(O) = 249.2 kJ/mol, and ∆H˚f298(Ti) = 473.6 kJ/mol. The 298 K thermal and enthalpy corrections to 0 K energies for elements and compounds are adopted from ref. 49.

III.

Results and discussion At the CCSD(T) and CASSCF levels, the ground electronic state of TiO is predicted to be of

3

∆ symmetry, with the CASSCF dominant electronic configuration of 0.96|(7σ)2(8σ)2(3π)4(1δ)1(9σ)1>.

The valence electrons of TiO consist of the 4s23d2 electrons of Ti and the 2s22p4 electrons of O. The 7σ orbital is dominated by the O(2s) character. The TiO molecule is chosen to lie along the z-axis, the 8σ orbital is formed by the overlap of the Ti(3dz2) and O(2pz) orbitals, with the former being dominant in contribution. The two degenerate π bonding orbitals, 3πx and 3πy, are formed by the overlaps of Ti(3dxz) with O(2px) and Ti(3dyz) with O(2py), respectively, with the O(2p) orbital has dominant contribution in each π orbital. The 1δ molecular orbital consists of the Ti(3dx2-y2) orbital. This prediction agrees with the previous experimental determinations.31,51 The first ionization of TiO would involve the removal of an electron from 9σ orbital, resulting in a 2∆ ground state for TiO+ with an electronic configuration of 0.96|(7σ)2(8σ)2(3π)4(1δ)1>. As the 9σ orbital is dominated by Ti(4s) nonbonding electrons, the TiO+ is expected to have similar bonding character with TiO. The molecular orbitals of TiO2/TiO2+ have been studied at CASSCF/aug-cc-pVTZ level. The ground state of TiO2(1A1) has a C2v symmetry and its main electronic configuration is …1a223b128a125b229a126b22, which is consistent with the previous studies.32,52 The 1a2 and 3b1 orbitals have a π bonding character. The 1a2 consist of O(2px) orbitals and Ti(3dxy) orbital, results in a 8


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Ti−O π bond. The 3b1 orbital is mainly contributed by two O(2px) orbitals and Ti(3dxz) orbital . The 8a1 and 5b2 orbitals have the Ti−O σ bond character. The 8a1 orbital is contributed by O(2pz) orbitals and Ti(3dy2-z2) orbital, while the 5b2 orbital is formed by the O(2py) orbitals and Ti(3dyz) orbital. Both the 9a1 and 6b2 orbitals are primarily contributed from the lone pair of electrons on oxygen atoms. The ionization of TiO2 would remove a non-bonding electron from 6b2 orbital, resulting in TiO2+(2B2) structure with an electronic configuration of …1a223b128a125b229a126b21 and a symmetry-broken

structure

of

TiO2+(2A′)

with

the

electronic

configuration

of …3a"24a"212a'213a'214a'215a'1. Once the non-bonding electron in the 6b2 orbital is removed, the repulsion between two oxygen atoms is reduced and thus interatomic θe+ angle in the TiO2+ becomes smaller (see Table 3).

A.

Equilibrium bond length and harmonic vibration frequency of TiO and TiO+ The predicted bond lengths re (re+) and harmonic vibrational frequencies ωe (ωe+) of TiO(X3∆)

[TiO+(X2∆)] at the CCSD(T), CCSDT, CCSDTQ, MRCI+Q and B3LYP levels are summarized in Table 1.

The calculations with the aug-cc-pwCVXZ basis sets have additionally included the

electronic correlations for the 3s3p4s3d(Ti) and 2s2p(O) electrons.

The CCSD(T) predicted re

values for TiO are in the range from 1.639 to 1.636 Å with the basis set size increased successively from aug-cc-pVTZ to aug-cc-pV5Z.

With the core-valence correlation effect, the bond lengths are

reduced by about 0.016 Å. At the CCSDT level, the re is 1.636 Å (aug-cc-pVTZ) and 1.625 Å (aug-cc-pwCVTZ) while the CCSDTQ/cc-pVTZ level yields a re value of 1.637 Å. Compared with the experimental value of 1.6203 Å,53 inclusion of core-valence correlation in the calculations is important to give a reliable re value. The CCSD(T), CCSDT, and CCSDTQ methods give ωe value of TiO ranging from 999 to 1020 cm−1. All these values are in line with the experimental ωe value of 1009 cm−1.53 Similar observations are found in the coupled cluster predictions for re+ and ωe+: (i) inclusion of core-valence correction would slightly decrease the re+ value; (ii) the ωe+ is almost unvaried with the basis sets and/or correlation effects. The CCSDT/aug-cc-pwCVTZ predicted re+ = 1.587 Å and 9



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ωe+ = 1069 cm−1 are in the best agreement with the experimental values of re+ = 1.583 Å and ωe+ = 1056.1 ± 0.8 cm−1, determined in our recent two-color resonance-enhanced laser photoionization and PFI-PE study on TiO+.31

Another experimental ωe+ of 1045 ± 7 cm−1 has been deduced from the

v=0−14 vibrational levels observed in multiphoton ionization photoelectron spectrum of TiO by Weisshaar and co-workers.54 The small change in re and ωe values of TiO upon adiabatic ionization is consistent with the non-bonding nature of the 9σ orbital, thus the triple bond character is retained in TiO and TiO+. The MRCI+Q/aug-cc-pwCVTZ predictions for re (re+) and ωe (ωe+) are 1.621 (1.588) Å and 1027 (1059) cm−1. The MRCI+Q value of ωe is off from experiment value by 18 cm−1 while re, re+, and ωe+ are in good accord with the experimental values. Our MRCI+Q results are also comparable with these obtained at C-MRCI+Q+DKH2 level (with core-valence electronic correlation and relativistic effect) by Miliordos and Mavridis.14 The MRCI+Q, CCSD(T) and CCSDT results (with aug-cc-pwCVTZ basis sets) on re/re+ and ωe/ωe+ are close, the difference is less than 0.005 Å and 20 cm−1, respectively. The B3LYP method gives close predictions to the experiments: the re (re+) values are just smaller by about 0.01 Å, and the ωe (ωe+) values is larger by about 30 (70) cm−1. It is important to find that our theoretical re (re+) and ωe (ωe+) at CCSD(T)/aug-cc-pwCVXZ and CCSDT/aug-cc-pwCVTZ levels are consistent with the experimental data, suggesting that the core-valence effect plays an important role in the reliable bond length and vibrational frequency determinations of TiO and TiO+.

B. Equilibrium bond lengths, bond angle, and harmonic vibration frequencies of TiO2 and TiO2+ The predicted Ti−O bond lengths re (re+), O−Ti−O bond angles θe (θe+), and all three harmonic vibrational frequencies ω1 (ω1+), ω2 (ω2+) and ω3 (ω3+) of TiO2(X1A1) (TiO2+(C2v) and TiO2+(Cs)) at the CCSD(T), CCSDT, MRCI+Q, and B3LYP levels are summarized in Tables 2 and 3. Similar to the calculations of TiO/TiO+, the electronic correlations for 3s3p4s3d(Ti) and 2s2p(O) electrons have 10


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been

included

in

the

calculations

with

the

aug-cc-pwCVXZ

basis

sets.

The

CCSD(T)/aug-cc-pwCVXZ predictions of re (1.648 to 1.653 Å) are in the best agreement with the experimental value of 1.651 Å.55

The CCSDT/aug-cc-pVTZ value for re=1.662 Å is longer than the

experimental one by ~0.01 Å. The coupled cluster theory predicts θe angle to be around 111.3° to 112.2° and all these values are in line with the experimental value of 111.6°.55

The MRCI+Q

prediction gives re=1.666 Å and θe=112.5° values, which are very close to the CCSDT/aug-cc-pVTZ predictions. There are three vibrational modes in TiO2(TiO2+), including the symmetric Ti−O stretching

ω1(ω1+), the bending ω2(ω2+) and the asymmetric Ti−O stretching ω3(ω3+) modes.

At the

CCSD(T)/aug-cc-pVXZ levels, the vibrational frequencies for ω1, ω2, and ω3 are in the range of 965 to 971, 323 to 324, and 946 to 952 cm−1, respectively.

Including the full triplet electronic correlation

now gives ω1= 984, ω2= 325, and ω3= 952 cm−1, which are very close predictions to the experimental values56,57 of ω1 = 968, ω2 = 323, and ω3 = 944 cm−1, and the deviations are ~16 (ω1), ~2 (ω2) and ~8 cm−1 (ω3).

If the core-valence correlation effect is included at CCSD(T) level, it just slightly

increases the vibrational frequencies by ~24 (ω1), ~15 (ω2), and ~5 cm−1 (ω3). Another experimental value of ω3 = 934.8 cm−1 was observed in the infrared spectra in neon matrix by McIntyre et al.58 The MRCI+Q/aug-cc-pVTZ values (ω1 = 958, ω2 = 323, and ω3 = 936 cm−1) are also in good accord compared the experimental values with deviations of

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