Theoretical Study on Rotational Constants of CH₃O/CD₃O Induced by Geometrical Isotope Effect

Abstract

We analyzed the rotational constants of CH₃O and CD₃O by using the multi-component molecular orbital method, which takes into account the quantum effect of proton and deuteron directly. We clearly observed the difference in C–H and C-D bond lengths due to the anharmonicity of the potential. The rotational constants of CH₃O and CD₃O based on the optimized structures including H/D geometrical isotope effect showed good agreement with the experimental value. We found that the geometrical changes induced by the H/D isotope effect influence the spectroscopic properties, such as rotational constants.

1 Introduction

It is of great importance to determine the geometrical structure of an isolated polyatomic molecule. Recently, the possibility of the comparison in the experimental spectroscopic observations and ab initio theoretical calculation has been shown due to the achievements of both accurate measurements and calculations [1,2,3]. Small molecules with high symmetry are more favorable because of the small number of structural parameters. It is expected that the equilibrium structure and potential energy surface can be precisely elucidated by these methods.

The methoxy radical (CH₃O) is one of the typical molecules and has been extensively investigated, which is known to be important in environmental chemistry and combustion. There have been excellent studies of rotationally resolved high-resolution spectroscopy in the vapor phase based on the supersonic jet, for example, electronic transitions in the near ultraviolet region [4,5,6,7], vibrational transitions in the infrared region [8,9], and pure rotational transitions in the microwave region [10,11,12,13]. The accurate rotational constants at the zero-vibrational level in the ground state were obtained from those measurements. It is necessary to obtain the constant values of isotopic species to derive the geometrical structure of CH₃O (C_3v symmetry, three structural parameters). The geometrical parameters, such as bond lengths and bond angles, were accurately determined by the experimentally obtained rotational constants A and B( = C) of ¹²CH₃O, ¹³CH₃O, and ¹²CD₃O. Generally, the experimentally obtained bond lengths are the averaged values (r₀) including the anharmonicity of the potential. On the other hand, the equilibrium bond length (r_e) is obtained from the ab initio calculation. The difference between r₀ and r_e becomes larger when the mass of nucleus is lighter, such as proton. For CH₃O, it has been experimentally found that the averaged C–H bond length (r₀(C–H)) is longer than that of C-D (r₀(C-D)) by 3.67 mÅ [13]. Although there are also some theoretical studies of ab initio calculation with geometrical optimization [14,15,16,17], the description of geometrical difference induced by H/D is difficult by the use of conventional molecular orbital (MO) approach based on the Born-Oppenheimer approximation.

To take into account such geometrical changes by H/D isotope effect, we developed the multi-component molecular orbital (MC_MO) approach and found the important phenomena induced by the geometrical changes due to the anharmonicity of the potential [18,19,20,21].

In this study, the geometrical changes of CH₃O and CD₃O by MC_MO method were analyzed. We also calculated the rotational constants of CH₃O and CD₃O based on the optimized structure and compared our result with the experimental values to clearly show the important relation between rotational constants and local C–H/C-D bond length difference.

2 Computational Details

We calculated the optimized geometries of CH₃O and CD₃O by using conventional MO and MC_MO methods under the MP2 level of theory. The 6-31G (3df,3pd) electronic basis function was selected to reproduce the experimental parameters. The protons and deuterons in CH₃O and CD₃O were treated as quantum waves, as well as the electrons under the fields of the C and O nuclear point charges. The [1s1p] and [1s1p1d] Gaussian-type functions (GTFs) were employed for protonic and deuteronic basis functions. For detailed analysis of quantum behavior of protons and deuterons, the exponent (α) value in GTF variable parameter for protons and deuterons was optimized with respect to the total energy. The position of protons and deuterons was determined from the expectation value of [1s1p] and [1s1p1d] GTFs.

3 Results and Discussion

We optimized the geometrical parameters of CH₃O and CD₃O by using the conventional MO and MC_MO methods. In the MC_MO calculation, the [1s1p] and [1s1p1d] GTFs were employed for the protonic and deuteronic basis functions. The optimized geometries of CH₃O and CD₃O obtained by the conventional MO and MC_MO calculations were shown in Figure 1.

Figure 1.

 Optimized geometrical parameters of CH₃O and CD₃O obtained by conventional MO and MC_MO methods under the MP2/6-31G (3df,3pd) level of theory. The bond distance and angle are shown in Å and degrees. Experimental values are taken from in Ref. 13.

We could not observe the geometrical difference between CH₃O and CD₃O because the equilibrium interatomic distance was obtained by the conventional MO method. We found the geometrical difference of the C–H and C-D as 1.1217 and 1.1125 Å by the use of the MC_MO calculation with [1s1p] protonic and deuteronic GTFs. However, the calculated C–H and C-D distances overestimated because the C–H and C-D bond distance of experiment was 1.1070 and 1.1033 Å, respectively. The C–H and C-D bond distances were improved as 1.1119 and 1.1067 Å by the use of [1s1p1d] GTFs for proton and deuteron in the MC_MO calculation. This result means that the electron-nuclear correlation energy is sufficiently treated by the use of [1s1p1d] GTFs. The C-D bond distance was 0.0052 Å shorter than C–H as the H/D isotope effect. The difference of C–H and C-D estimated from rotational constant was 0.0037 Å [13]. Although the C–H and C-D bond difference obtained by the MC_MO calculation was slightly larger than the experiment value, we clearly found the geometrical difference of C–H and C-D bond lengths. In addition, we observed the H/D isotope effect for C–O bond distance and H-C–H/D-C-D angle although these geometrical changes induced by H/D substitution were not considered in the geometrical parameter analysis from the rotational constant measurement. For example, the C–O bond distance in CH₃O is longer than that in CD₃O in our calculation. This result indicates that the H/D isotope effect influences not only the geometry due to the anharmonicity of the potential but also the electronic structure of molecule.

To explain the details of geometrical change by H/D isotope effect, we analyzed the distribution of protonic and deuteronic wavefunctions and atomic charge. We show in Table 1 the optimized α values in [1s1p] and [1s1p1d] GTFs for proton and deuteron in CH₃O and CD₃O.

Table 1. Optimized exponent (α) values of [1s1p] and [1s1p1d] GTFs for proton and deuteron in CH₃O and CD₃O.

	[1s1p]	[1s1p1d]
Proton
αs	20.5979	17.8354
αp	21.6640	18.2216
αd		31.3760
Deuteron
αs	30.2414	26.1496
αp	31.6952	28.3866
αd		47.8794

In the case of [1s1p] GTFs, the wavefunction of deuteron was more localized than that of proton because the α values for proton and deuteron were about 20 and 30, respectively. We obtained the same trend in protonic and deuteronic wavefunctions in the case of [1s1p1d] GTFs. The calculated atomic charges in CH₃O and CD₃O by using the MC_MO calculation with the [1s1p1d] GTFs are shown in Table 2.

Table 2.  Mulliken atomic charges and spin densities in CH₃O and CD₃O.

	MC_MO		MO
	CH3O	CD3O	CH3O
	Mulliken atomic charges
O	−0.462	−0.462	−0.453
C	−0.332	−0.251	−0.006
H	0.265	0.238	0.153
	Mulliken atomic spin densities
O	1.031	1.032	1.036
C	−0.110	−0.110	−0.119
H	0.026	0.026	0.028

The atomic charges of H and D in CH₃O and CD₃O obtained by the MC_MO calculations were 0.265 and 0.238, respectively. This result indicates that the electron density around deuteron is larger than that around proton. The difference of distribution of protonic and deuteronic wavefunctions influences the electrons around proton and deuteron. The change of atomic charge is not only observed for the H and D. The atomic charge of C in CH₃O (−0.332) was larger than that in CD₃O (−0.251). The atomic charge difference of C in CH₃O and CD₃O is one of the reasons to show the geometrical difference of C–O bond distance in CH₃O and CD₃O. Because the C–O bond in CD₃O becomes weaker by the decrease of the atomic charges of C, the C–O bond distance in CD₃O becomes longer. We clearly demonstrated that the reorganization of geometrical parameters and electronic structure of CH₃O and CD₃O is observed due to the difference of nuclear quantum effect of H and D. It is noted that the direct observation of the geometrical changes, such as C–O and angle, is difficult because the geometrical parameters are determined from the rotational constants under the symmetry rule and some spectroscopic assumptions in experiment.

We finally estimated the rotational constants of CH₃O and CD₃O based on the optimized geometry by MC_MO calculation. The calculated rotational constants are shown in Table 3. The rotational constants of CH₃O obtained by the conventional MO and experiment show large deviation because the geometry determined by the conventional MO is the equilibrium structure. The difference of rotational constants of CH₃O and CD₃O in conventional MO is based on the mass difference of H and D although the optimized structures of CH₃O and CD₃O are the same. By the use of the MC_MO method with [1s1p1d] GTFs, the calculated rotational constants of CH₃O showed good agreement with the experimental one. The main reason for the difference in rotational constants between conventional MO and MC_MO is the difference of optimized structures, i.e., r_e and r₀. Due to the accurate estimation of geometrical change by the deuteration, we reproduced the rotational constants of CD₃O. We clearly demonstrated that the geometrical changes induced by the H/D isotope effect influence the rotational constants of CH₃O and CD₃O.

Table 3.  Rotational constants of CH₃O and CD₃O obtained by conventional MO and MC_MOmethods.

	CH3O		CD3O
	A(MHz)	B(MHz)	A(MHz)	B(MHz)
MO	159.341	27683.20	79.732	22071.91
MC_MO ([1s1p])	151.979	27514.43	77.247	21899.25
MC_MO ([1s1p1d])	155.553	27699.61	78.469	22041.28
Expt.(Ref. 13)	154.670	27930.38	78.338	22194.07

4 Conclusions

We analyzed the geometrical parameters and rotational constants of CH₃O and CD₃O by using the MC_MO method, which takes into account of the quantum effect of the proton and deuteron directly. The C-D bond distance in CD₃O was about 0.005 Å shorter than C–H in CH₃O. This result showed a good agreement with the experimental result. We also found the H/D isotope effect in C–O bond distance in CH₃O and CD₃O due to the electronic structure change by the different distribution of protonic and deuteronic wavefunctions. The rotational constants of CH₃O and CD₃O based on the optimized structure by MC_MO method show good agreement with the experimental ones. We clearly found that the H/D isotope effect influences not only the geometrical parameters but also the spectroscopic properties, such as rotational constants.

The activities of INAMORI Frontier Research Center are supported by Kyocera Corporation.

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