Spin-orbit Effects on the Structure of Haloiodomethane Cations CH₂XI⁺ (X=F, Cl, Br, and I)

Abstract

The importance of including spin-orbit interactions for the correct description of structures and vibrational frequencies of haloiodomethanes is demonstrated by density functional theory calculations with spin-orbit relativistic effective core potentials (SO-DFT). The vibrational frequencies and the molecular geometries obtained by SO-DFT calculations do not match with the experimental results as well as for other cations without significant relativistic effects. In this sense, the present data can be considered as a guideline in the development of the relativistic quantum chemical methods. The influence of spin-orbit effects on the bending frequency of the cation could well be recognized by comparing the experimental and calculated results for $CH_2BrI$ and $CH_2ClI$ cations. Spin-orbit effects on the geometries and vibrational frequencies of $CH_2XI$ (X=F, Cl, Br, and I) neutral are negligible except that C-I bond lengths of haloiodomethane neutral is slightly increased by the inclusion of spin-orbit effects. The $^2A^{\prime}$ and $^2A^{{\prime}{\prime}}$ states were found in the cations of haloiodomethanes and mix due to the spin-orbit interactions and generate two $^2E_{1/2}$ fine-structure states. The geometries of $CH_2XI^+$ (X=F and Cl) from SO-DFT calculations are roughly in the middle of two cation geometries from DFT calculations since two cation states of $CH_2XI$ (X=F and Cl) from DFT calculations are energetically close enough to mix two cation states. The geometries of $CH_2XI^+$ (X=Br and I) from SO-DFT calculations are close to that of the most stable cation from DFT calculations since two cation states of $CH_2XI$(X=Br and I) from DFT calculations are energetically well separated near the fine-structure state minimum.

Introduction

Quantum chemical calculations could provide results comparable to experimental measurements in many instances, but still need improvement in the accuracy and reliability. One such direction is the inclusion of spin-orbit interactions in Hamiltonian. The importance of relativity for the description of heavy elements is well recognized,1-3 and the spinorbit interaction is present in the relativistic formalism based upon Dirac operator for the electron. Spin-orbit and other relativistic effects required for the accurate description of valence states of molecules containing heavy elements can be represented as two-component relativistic effective core potentials (RECPs) derived from Dirac-Coulomb Hamiltonian based all-electron calculations of atoms.4 Scalar relativistic effects are routinely included in electronic-structure calculations through the use of RECPs, but spin–orbit interactions are usually omitted when deriving optimized structures partly because of the assumption that their influence on the molecular structures is negligible and partly due to computational difficulties.5,6 Even when spin–orbit terms are available in RECPs, the usual treatment involves perturbational inclusion of these terms after the variational determination of orbitals and structures. Quantum chemical calculations employing RECPs with spin–orbit operators from the beginning of ab initio calculations have been available.7,8 Spin– orbit density functional theory (SO-DFT) method available in NWCHEM is particularly useful for the present purpose of demonstrating spin–orbit effects on geometries since the structures can be optimized with both spin–orbit interactions and electron correlations included.9,10 Relativistic effects on geometries of molecules containing heavy elements can be comparable or larger than electron correlation effects and make an unexpected shape to be a local minimum. We found this influence of spin-orbit interactions on the geometry for polyhalomethanes CH2ClI+ and CH2BrI+ cations in collaborations with the group of Prof. M. S. Kim.11-13

Polyhalomethanes have been of considerable interest in relation to ozone depletion in the atmosphere.14-16 The photochemistry of iodine containing polyhalomethanes such as CH2ClI, CH2BrI, and CH2I2 have been actively investigated in this context for the role in the ozone depletion in the troposphere,15,17,18 and also from the point of view of fundamental science for the importance of these molecules as prototypes of the reaction control via electronic state selection. 19-21 Another important aspect in the fundamental study of these iodine containing molecules is the extent of the influence of relativistic effects on their structures and molecular properties.11,12 A conventional approach to handle this effect in the electronic structure calculation is to adopt RECPs for heavy elements. Spin-orbit interaction is usually omitted as noted above. The same approach is generally adopted to analyze the results from the photoelectron spectroscopy (PES) which provides low resolution spectroscopic data for cations.22,23 Zero kinetic energy (ZEKE) photoelectron spectroscopy24,25 and mass-analyzed threshold ionization (MATI) spectroscopy26 are offspring of PES and provide much higher resolution spectroscopic data for cations. The vibrational spectra of the halomethane cations have been recorded using the MATI spectrometry.11-13 SO-DFT calculations for spectroscopic constants of mixed dihalomethanes have been performed using shape-consistent RECPs with effective one-electron spin-orbit operator to confirm and improve the phenomenological assignments.

In order to generalize and understand interesting spin-orbit effects obtained for CH2ClI+ and CH2BrI+, we extend the study to include CH2FI+ and CH2I2+. The vibrational frequencies of CH2XI (X=F, Cl, Br, and I) neutral and cation were obtained from SO-DFT calculations. The shapes and energy levels of molecular orbitals from DFT calculations and the energy levels of molecular spinors from SO-DFT calculations were analyzed to reveal the origin of characteristic spin-orbit effects on the X-C-I bond angle.

 

Computational Details

Optimized molecular geometries of haloiodomethane CH2XI neutral and cation (X=F, Cl, Br, and I) were calculated at the DFT level of theory employing RECPs including spin-orbit terms. The RECP employed for the present calculations is expressed by the following form within the framework originated from Lee et al.,27

where |ljm〉〈ljm| represents a two-component projection operator. Molecular spinors which are obtained as oneelectron wave functions of the Hamiltonian containing the above RECP have two components. The RECP, UREP which is referred to SOREP here can be expressed as the sum of the scalar relativistic effective core potential (AREP), UAREP, and the effective one-electron spin-orbit operator,28 USO, as

Values calculated with AREP include scalar relativistic effects, and those with SOREP contain spin-orbit effects as well as scalar relativistic effects. The spin-orbit effects (ΔSO) are defined as the difference between AREP and SOREP values calculated with the same basis set at a given level of theory.

SO-DFT calculations were done for the CH2XI (X=F, Cl, Br, and I) neutral and cation in the ground electronic state with the B3LYP29 (Becke's three parameter hybrid functional 30 with the Lee-Yang-Parr correlation functional31) functional using NWCHEM.9,10 RECPs with the effective spin-orbit potentials and the basis sets for the valence shell reported by Christiansen and coworkers32,33 were used for halogen atoms. To this basis set of F, Cl, Br, and I, the diffuse and polarization functions reported by Lee et al.5 were added to make halogen basis sets consistent with the large basis sets used for other atoms as shown in Table 1. AREP and SOREP calculations were performed to observe the magnitude of spin-orbit effects. The step size in the calculation of the numerical Hessian was optimized to produce reliable vibrational frequencies. Equilibrium geometries and vibrational frequencies of the neutral and the cation were calculated.

Table 1.aRECPs and basis sets used for halogen atoms are in Refs. 32 and 33 and spin-orbit terms in the same references were used when needed.

Six sets of DFT calculations were performed for CH2ClI to test the effects of the type of functionals. The applied hybrid functionals were B3LYP,29 ACM30 (Becke’s adiabatic connection method, also known as B3PW91) and PBE034,35 (Perdew-Burke-Ernzerhof parameter-free functional) functionals. Additional DFT calculations use local density approximation (LDA), generalized gradient approximation (GGA), and hybrid meta-GGA (HMGGA) of the exchange-correlation functionals to gauge the reliability of the hybrid functionals. The applied LDA and GGA functionals are SVWN5 (Slater exchange36 and Vosko-Wilk-Nusair correlation functional (VWN)37), BP86 (Becke exchange38 and Perdew correlation functional39), respectively. The applied HMGGA functionals is M0540 (Minnesota 2005). It is noted that RECPs used here are all generated using relativistic HF results and may introduce some inconsistency when used with DFT methods. We assume this error to be small enough at least for the purposes of the present study.

 

Results and Discussion

Test Calculations for CH2ClI. DFT calculations with several functionals were performed for CH2ClI and optimized geometrical parameters are listed in Table 2. There are the functional dependencies on the bond lengths of CH2ClI neutral, while that on the bond angles are small as shown in Table 2. Spin-orbit effects on the geometries of CH2ClI neutral are negligible except on the C-I bond lengths for which spinorbit effects slightly increase the bond length. The most pronounced discrepancy from experimental geometrical parameters is the C-I bond length. All calculated C-I bond lengths of CH2ClI from DFT and SO-DFT calculations are overestimated compared to the experimental ones. The C-I bond lengths of CH2ClI from HF and SO-HF calculations are closer to the experimental one. The C-I bond lengths of CH2ClI from DFT calculations have rather small variation with the class of the functional employed among LDA, GGA and HGGA. LDA yields shortest one followed by HGGA and then GGA. Roughly speaking, the C-I bond lengths of CH2ClI from DFT calculations with functionals including HF exchange contribution, which are hybrid functionals, are indirectly proportional to the amount of the HF exchange contribution to the functionals. The C-I bond lengths of CH2ClI from DFT calculations with functionals including HF exchange contribution are 2.181 (B3LYP with 20% HF exchange), 2.163 (ACM or B3PW91 with 20% HF exchange), 2.154 (PBE0 with 25% HF exchange), 2.152 (M05 with 28% HF exchange), and 2.151 (BHLYP with 50% HF exchange). The hybrid functional with the larger HF exchange results in the shorter C-I bond length and closer to experimental one. The vibrational frequencies of CH2ClI neutral from DFT calculations with hybrid functional are overestimated compared to experimental ones. Since HF results for energies and vibrational frequencies are not reliable, we may select DFT and hybrid functionals among DFT as the method of choice for chloroiodomethane.

Table 2.aThe set 1 basis sets and RECPs in Table 1 were used. bMicrowave spectroscopic data from Ref.42

Table 3.aThe set 1 basis sets and RECPs in Table 1 were used.

Two stationary states, 2A' and 2A'', of the chloroiodomethane cation corresponding to removal of HOMO‒1 a' and HOMO a'', respectively, from the neutral molecule can be obtained in AREP-HF and AREP-DFT calculations. The optimized structures of the two cation states indicate that structural change from the neural one is larger for the 2A' state than for the 2A'' state, It appears that antibonding interaction for the a' orbital is larger than that in the a'' orbital as can be seen in Figure 1. Reflecting this larger geometry changes, the most stable cation of CH2ClI is the 2A' state generated by removal of an electron from the second highest occupied a' orbital (Fig. 1). Qualtitative changes from neutral to the two cation states in Table 3 are essentially similar for all methods. One exception is the C-I bond length. HF gives longer bond lengths for cation states while LDA and GGA produce shorter ones for the cation states. It appears that electron correlations shorten the C-I bond lengths, which is also observed in all-electron calculations.41 It is noted that this trend is opposite to that in the neutral. The 2A' and 2A'' states mix due to the spin-orbit interactions and generate two 2E1/2 fine-structure states. The energy differences of two cation states from AREP calculations for CH2ClI+ is about 1200 cm‒1. Energetically close two states of chloroiodomethane cation mix substantially to generate the fine-structure states with geometries somewhere between two AREP states. The geometries of CH2ClI from SO-DFT calculations are roughly in the middle of two cation geometries especially in terms of the Cl-C-I bond angle. In this regard, HF and all DFT functional considered afford the same trend. This has been the subject of several reports. We try to observe the general pattern of this behavior by considering other members of CH2XI+, The performance of HF and various DFT methods for the cation is similar to that for the neutral case and we proceed to use B3LYP with AREP and SOREP for other members of CH2XI.

Figure 1.Shapes and energy levels (energy levels) of molecular orbitals (molecular spinors) from the B3LYP (SO-B3LYP) calculations for CH2XI (X=F, Cl, Br, and I). For each molecule orbital energy levels are on the left column and molecular spinor energy levels are open the right. Each dotted line connecting an orbital with a spinor imply spin-orbit coupling.

Spin-orbit Effects on the Vibrational Frequencies of Haloiodomethane Cations (CH2XI+ (X=F, Cl, Br, and I)). Spin-orbit effects on the geometries and vibrational frequencies of CH2XI (X=F, Cl, Br, and I) neutral are negligible as shown in Tables 4 and 5. The C-I bond length of haloiodomethane neutral is slightly increased by the inclusion of spin-orbit effects. Calculations incorporating spin-orbit terms still need to achieve more accurate theoretical results. The calculated C-I bond lengths of CH2ClI and CH2I2 are overestimated in comparison to the available experimental ones.42,43 The calculated C-I bond lengths of CH2BrI and CH2I2 are longer than the available CASPT2 results.44,45 DFT calculations with the B3LYP functional using RECPs by Christiansen et al.32,33 seem to overestimate the C-I bond lengths, but the degree of overestimate is not so large as to affect the reliability of the present study. The C-X bond lengths of CH2XI (X=F, Cl, Br, and I) increase on descending the halogen group in the periodic table, as expected from the increasing size of halogen atoms. The C-I bond lengths of CH2XI (X=F, Cl, Br, and I) decrease on descending the halogen group in the periodic table. The X-C-I bond angle of CH2XI (X=F, Cl, Br, and I) increases on descending the halogen group in the periodic table probably due to the increasing steric repulsion between X and I. In order to reduce the steric repulsion between X and I, the shorter C-X bond length induces the longer C-I bond length. The X-C-I bond angle of CH2I2 is also larger than that of CH2FI indicating that the steric repulsion between Two I atoms is larger than that between F and I atoms.

The shapes and energy levels of molecular orbitals and energy levels of molecular spinors of CH2XI (X=F, Cl, Br, and I) neutral are shown in Figure 1. For CH2XI (X=F and Cl), HOMO and HOMO−1 have a'' and a' symmetries, respectively. For CH2BrI and CH2I2, HOMO and HOMO−1 have a' and a'' symmetries, respectively. The a' and a'' correspond largely to the iodine 5p nonbonding orbtials, parallel n(I5p||) and perpendicular n(I5p⊥), respectively, to the molecular plane as shown in Figure 1. The a' orbitals are two inplane nonbonding orbitals, n(I5p||) and n(Xnp||). The in-plane nonbonding orbitals of two halogens have small antibonding interactions with each other The antibonding characteristics of a' orbitals for CH2XI (X=F, Cl, Br, and I) increase on descending the periodic table. Thus the energy levels of molecular orbitals increase for the lower members of the halogen group in the periodic table. The a'' orbitals have the perpendicular nonbonding orbital, n(I5p⊥). The energy levels of molecular orbitals are all similar in CH2XI (X=F, Cl, Br, and I). As a result, the destabilization of a' orbitals induces the symmetry alternation of HOMO and HOMO−1 in CH2XI (X=Br and I) compared with CH2XI (X=F and Cl). Thus the gap in energy levels of HOMO and HOMO−1 in CH2ClI (299 cm−1) and CH2BrI (379 cm−1) are closer than that in CH2FI (1302 cm−1) and CH2I2 (1813 cm−1). The HOMO and HOMO−1 levels in DFT calculations shift due to spin-orbit interactions and generate highest occupied molecular spinor (HOMS) and HOMS−1 in SO-DFT calculations. Actually all energy levels are affected by spin-orbit interactions. The closer the energy levels in DFT calculations, the larger the spin-orbit splitting in SO-DFT calculations. The splitting of HOMS and HOMS−1 energy levels due to spin-orbit effects are 4751 cm−1 for CH2FI, 4448 cm−1 for CH2ClI, 3956 cm−1 for CH2BrI, and 5402 cm−1 for CH2I2.

Table 4.aThe set 1 basis sets and RECPs in Table 1 were used. bMicrowave spectroscopic data from Ref.42 cB3LYP/Sadlej-PVTZ calculations from Ref.46 dCASSCF/ANO-RCC calculations from Ref.44 eCASPT2/ANO-RCC calculations from Ref.44 fBLYP calculations from Ref.47 gMP2 calculations from Ref.48 hQCISD calculations from Ref.49 iCASPT2L1 calculations from Ref.45 jRef.43

Table 5.aThe set 1 basis sets and RECPs in Table 1 were used. bMulliken notation. cRaman spectroscopic data from Ref.50 dRaman spectroscopic data from Ref.51 eB3LYP/Sadlej-PVTZ data from Ref.46 fCASSCF/ANO-RCC calculations from Ref.44 gCASPT2/ANO-RCC calculations from Ref.44 hRaman spectroscopic data from Ref.19 iRaman spectroscopic data from Ref.52

Similar analysis is possible for lower lying halogen orbitals. For CH2XI (X=F, Cl, and Br), HOMO−2 and HOMO−3 have a' and a'' symmetries, respectively. For CH2I2, HOMO−2 and HOMO−3 have a'' and a' symmetries, respectively. The a' and a'' orbitals correspond to the antibonding combination of mainly nobonding orbitals parallel and perpendicular to the molecular plane, respectively, as shown in Figure 1. The a' orbitals have two in-plane orbitals a(I5p||) and a(Xnp||) in antibonding combination. The antibonding characters of a' orbitals for CH2XI (X=F, Cl, Br, and I) increase on descending the halogen group in the periodic table in line with increase in the energy levels of molecular orbitals. The a'' orbitals comprise the perpendicular nonbonding orbital, n(I5p⊥), n(Xnp||) and small contributions from the CH2 fragment. The perpendicular nonbonding orbitals of two halogens have slight antibonding interactions with each other. The antibonding characters of a'' orbitals for CH2XI (X=F, Cl, Br, and I) also increase on descending the halogen group in the periodic table, and the energy levels of molecular orbitals rise on descending the halogen group in the periodic table. The extent of the destabilization of a'' orbitals is faster than that of a' orbitals, which results in the symmetry alternation of HOMO−2 and HOMO−3 in CH2I2 in comparison to CH2XI (X=F, Cl, and Br). The destabilization of HOMO−2 and HOMO−3 is faster than that of HOMO and HOMO−1. For CH2I2, the energy levels of HOMO, HOMO−1, HOMO−2, and HOMO−3 are quite close.

The spin-orbit splitting of the HOMO−2 and HOMO−3 levels to the HOMS−2 and HOMS−3 levels strongly depends on the X halogen atom. The energy levels of HOMO−2 and HOMO−3 for CH2XI (X=F and Cl) are similar to those of HOMS−2 and HOMS−3, reflecting small spin-orbit splittings of F and Cl. The HOMO−2 and HOMO−3 levels for CH2XI (X=Br and I) in DFT calculations split further due to the spin-orbit interactions and generate more separated HOMS−2 and HOMS−3 in SO-DFT calculations. The closer the energy levels in DFT calculations, the larger the spin-orbit splitting in SO-DFT calculations. For CH2I2, the spin-orbit splitting between the HOMO and HOMO−1 levels decrease the HOMO−1 level and that of the HOMO−2 and HOMO−3 levels increase the HOMO−2 level. Thus, the HOMS−1 level is mainly from spin-orbit splitting of HOMO−2 and the HOMS-2 level is mainly from spin-orbit splitting of HOMO−1 since the energy levels of HOMO, HOMO−1, HOMO−2, and HOMO−3 are close as shown in Figure 1. The above interpretation of the spin-orbit splitting is an attempt to identify major factors. Since all molecular orbitals and also molecular spinors are mixtures of X and I atomic orbitals, additional orbital interactions play a role in determining accurate positions of energy levels.

Two stationary states, 2A' and 2A'', of haloiodomethane cations were found from AREP calculations, and the geometries and vibrational frequencies of CH2XI+ molecules are summarized in Tables 6 and 7. The stable cation of haloiodomethane except CH2ClI from AREP calculations is generated by the removal of an electron from the HOMO. The most stable cation of CH2FI is the 2A'' state generated by the removal of an electron from the highest occupied a'' orbital. The most stable cation of CH2XI (X=Br and I) is the 2A' state generated by the removal of an electron from the highest occupied a' orbital while the most stable cation of CH2ClI is the 2A' state generated by the removal of an electron from the second highest occupied a' orbital. The 2A' state has much smaller X-C-I angles than the neutral molecule in all CH2XI, implying that the shape of the cation in this state is somewhat different from the neural molecule. On the contrary, the 2A' state of the cation has the X-C-I angle similar to the neural molecule. Vibrational frequencies of the cation states do not differ too much for the X-C-I bending mode (ν6 in Tables 5 and 7) except for CH2FI. The two 2A' and 2A'' states mix due to the spin-orbit interactions and generate two 2E1/2 states. The energy differences of two cation states from AREP calculations for CH2FI, CH2ClI, CH2BrI, and CH2I2 are 834, 1225, 2768, and 3584 cm−1 in Table 5. Energetically closer two states of cations of haloiodomethanes mix more substantially than energetically less close two states of cations. The geometries of CH2XI+ (X=F and Cl) from SO-DFT calculations are roughly in the middle of two cation geometries from DFT calculations since two cation states of CH2XI (X=F and Cl) from DFT calculations are energetically close enough to mix two cation states. In contrast, the geometries of CH2XI (X=Br and I) from SODFT calculations are close to those of more stable cations from DFT calculations since two cation states of CH2XI (X=Br and I) from DFT calculations are energetically not so close enough to produce well mixed fine-structure states. In this sense, large spin-orbit induced shape change is predicted for CH2FI and CH2ClI, but not for CH2BrI and CH2I2. One confirmation of this is the X-C-I bending mode in Table 7 since this ν6 bending mode reflects bonding property of the X-C-I bond angle. Energetics of spin-orbit effects will be important in analyzing PES data, but this is not attempted here.

Table 6.aThe set 1 basis sets and RECPs in Table 1 were used. bB3LYP/Sadlej-PVTZ data from Ref.46

Table 7.aThe set 1 basis sets and RECPs in Table 1 were used. bMulliken notation. cMATI spectroscopic data from Ref.11,12 dMATI spectroscopic data from Ref.13

 

Conclusion

The importance of including spin-orbit interactions for the correct description of structures and vibrational frequencies of haloiodomethanes is demonstrated by DFT calculations with spin-orbit relativistic effective core potentials. The vibrational frequencies and the molecular geometries obtained by SO-DFT calculations do not match with the experimental results as well as for other cations without significant relativistic effects. In this sense, the present data can be considered as a guideline in the development of the relativistic quantum chemical methods. The influence of spin-orbit effects on the bending frequency of the cation could be recognized by comparing the experimental and calculated results.

Spin-orbit effects on the geometries and vibrational frequencies of CH2XI (X=F, Cl, Br, and I) neutral are negligible except that C-I bond lengths of haloiodomethane neutral is slightly increased by the inclusion of spin-orbit effects. The 2A' and 2A'' states were found in the cations of haloiodomethanes and mix due to the spin-orbit interactions and generate two 2E1/2 fine-structure states. The energy differences of two cation states from AREP calculations for CH2FI+, CH2ClI+, CH2BrI+, and CH2I2 + are 834, 1225, 2768, and 3584 cm‒1. The geometries of CH2XI+ (X=F and Cl) from SO-DFT calculations are roughly in the middle of two cation geometries from DFT calculations since two cation states of CH2XI (X=F and Cl) from DFT calculations are energetically close enough to mix two cation states. The geometries of CH2XI+ (X=Br and I) from SO-DFT calculations are close to that of most stable cation from DFT calculations since two cation states of CH2XI (X=Br and I) from DFT calculations are energetically not close enough near the fine-structure state minimum to extensively mix two cation states.