Anharmonic force fields and spectroscopic constants of H 2 AsO: An ab initio study

2.1 Vibration-rotation interaction constants

For asymmetric top molecules, the effective rotation constant along the b axis can be expressed as, X v = X e - ∑ i α i X ( v i + 1 2 ) + ⋯(2.1)

Where X is the rotational constant including A, B and C. α r X is vibration-rotation interaction constants, the expression is, - α r X = ( 2 X e 2 / ω r ) [ ∑ ξ 3 ( a r ( b ξ ) ) 2 / 4 I ξ + ∑ s ≠ r ( ζ r , s ( b ) ) 2 ( 3 ω r 2 + ω s 2 ) / ( ω r 2 - ω s 2 )(2.2) + π ( c / h ) 1 / 2 ∑ s φ rrs a s ( bb ) ( ω r / ω s 3 / 2 ) ]

In Equation (2.2), ζ r , s ( b ) is coriolis coupling constants, which indicates the coupling between Q_r coordinates and Q_s coordinates rotating around b axis and can be expressed as, ζ r , s ( b ) = ∑ i ( L ir ( β ) L is ( γ ) - L ir ( γ ) L is ( β ) )(2.3)

For asymmetric top molecules, when the approximate degeneracy of two vibration dynamics (ω _r  = ω _s ) occurs, in Equation (2.2), this part will be replaced with ( ζ r , s ( b ) ) 2 ( 3 ω r 2 + ω s 2 ) / ( ω r 2 - ω s 2 ) → - 1 2 ( ζ r , s ( b ) ) 2 ( ω r - ω s ) 2 / ω s ( ω r + ω s )(2.4)

This substitution ignores the first order term, resulting in the inaccuracy of the eigenenergy calculation, and the diagonalization of the vibration Hamiltonian matrix will solve such problems.

Centrifugal distortion tensor are reported by Wilson and Howard [28] and expressed as, ∑ α β γ δ 1 4 τ α β γ δ J α J β J γ J δ(2.5) where τ α β γ δ = - ( ℏ 4 / 2 hc I α I β I γ I δ ) ∑ k a k ( α β ) a k ( γ δ ) / λ k(2.6)

The quartic centrifugal distortion constant can be expressed as, D J = - 1 32 [ 3 τ bbbb + 3 τ cccc + 2 ( τ bbcc + 2 τ bcbc ) ](2.7) D K = D J - 1 4 [ τ aaaa - ( τ aabb + 2 τ abab ) - ( τ ccaa + 2 τ caca ) ] D JK = - D J - D K - 1 4 τ aaaa R 5 = - 1 32 [ τ bbbb - τ cccc - 2 ( τ aabb + 2 τ abab ) + 2 ( τ ccaa + 2 τ caca ) ] R 6 = 1 64 [ τ bbbb + τ cccc - 2 ( τ bbcc + 2 τ bcbc ) ] δ J = - 1 16 ( τ bbbb - τ cccc )

The sextic centrifugal distortion constants can be expressed as, H J = Φ 600 + 2 Φ 204(2.8) H JK = Φ 420 - 12 Φ 204 + 2 Φ 024 + 16 σ Φ 006 - 16 ( R 5 - 2 σ R 6 ) S 111 + 8 ( 2 A ′ 0 - B ′ 0 - C ′ 0 ) S 111 2 H KJ = Φ 240 + 10 3 Φ 420 - 30 Φ 204 - 10 3 H JK H K = Φ 060 - 7 3 Φ 420 + 28 Φ 204 + 7 3 H JK h J = Φ 402 + Φ 006 h JK = Φ 222 - 10 Φ 006 + 4 σ Φ 204 + 2 ( D JK - 2 σ δ J - 4 R 6 ) S 111 - 4 ( B ′ 0 - C ′ 0 ) S 111 2 h K = Φ 042 + 4 3 σ Φ 024 + ( 9 + 32 3 σ 2 ) Φ 006 + 4 [ D K - 2 3 σ R 5 + 2 ( 1 + 8 3 σ 2 ) R 6 ] S 111 + ( 6 + 10 3 σ 2 ) ( B ′ 0 - C ′ 0 ) S 111 2

The defined constant In Equation (2.8) is a linear combination of seven deterministic constants of ten six centrifugal distortion constants.

For asymmetric top molecules, the anharmonic correction constants can be divided into diagonal and non-diagonal parts, which can be expressed as, χ rr = 1 16 φ rrrr - 1 16 ∑ s φ rrs 2 ( 8 ω r 2 - 3 ω s 2 ) / ω s ( 4 ω r 2 - ω s 2 )(2.9)

Non-diagonal parts can be expressed as, χ rs = 1 4 φ rrss - 1 4 ∑ l φ rrl φ lss / ω l - 1 2 ∑ l φ rsl 2 ω l ( ω l 2 - ω r 2 - ω s 2 ) / Δ rsl(2.10) + [ A e ( ζ rs ( a ) ) 2 + B e ( ζ rs ( b ) ) 2 + C e ( ζ rs ( c ) ) 2 ] ( ω r / ω s + ω s / ω r )

Where Δ rsl = ( ω r + ω s + ω l ) ( ω r + ω s - ω l ) ( ω r - ω s + ω l ) ( ω r - ω s - ω l )(2.11)

The ith fundamental frequency of H₂AsO can be obtained by the ith harmonic frequency (ω _i ) and anharmonic constant (χ _ij ) shown in Table 5 as follow, ν i = ω i + 2 χ ii + 1 2 ∑ i ≠ j χ ij(2.12)

The calculation detail of the spectroscopic constants and anharmonic force field of H₂AsO is the same as that of Refs. 13–16, therefore, we will not repeat too much in this article.

It has been theoretically and experimentally shown that ground state H₂AsO is a nonplanar, pyramidal molecule of C_s point group symmetry [2, 7]. Employing the analytic gradients of molecular energy, the H₂AsO geometries were optimized by MP2, B3LYP, B3P86, and B3PW91 methods with aug-cc-pVQZ and aug-cc-pV5Z basis sets considering the C_s point group symmetry. The geometries of H₂AsO can be described by five parameters: two bond lengths of As–O (R_As–O) and As–H (R_As–H), two bond angles of H–As–O (Φ_HAsO) and H–As–H (Φ_HAsH), and a dihedral angel or out-of-plane angle (Θ_opp). In order to compare with the existing experimental or theoretical data, we choose the out-of-plane angle to represent the nonplanar structure. Of course, the dihedral angel can be easily derived from the corresponding out-of-plane angle.

The obtained equilibrium geometries, dipole moments and total energies of H₂AsO are listed in Table 1, as well as the experimental data [7] and previous theoretical results [9]. In this work, the B3PW91/aug-cc-pV5Z results are in accord with the experimental data which can be derived from the ground-state rotational constants of Table 3. The discrepancy is only 0.003 Å, 0.013Å, 1.065 Å, 1.618°, and 1.328° for R_As–O, R_As–H, Φ_HAsO, Φ_HAsH, and Θ_opp, respectively. The comparison between the equilibrium molecular geometries and effective r₀ structure should be noticeable for the same reason with the reference [18]. The values of the dipole moments and total energies of the H₂AsO molecule at the B3PW91/aug-cc-pV5Z theoretical level are also listed in Table 1, which are 2.97 Debye and –2312.3 a.u., respectively. The optimized equilibrium geometry of the H₂AsO molecule at the B3PW91/aug-cc-pV5Z level can be found in Fig. 1.

OPEN IN VIEWER

Fig.1

The optimized geometry of H₂AsO at B3PW91/aug-cc-pV5Z theoretical level.

The optimized equilibrium geometry of H₂XO (X = As, P, N) are showed in Table 2. The experimentally derived HXH (X = As, P, N) bond angles of H₂AsO and H₂PO as well as the theoretical bond angles of H₂NO are 101.8°, 102.56° and ∼121° [2, 29–33]. There are four groups around the central atom X: a double bond XO, two single bonds HX, and an unpaired electron of the X atom in H₂XO molecules. According to Valence shell electron pair repulsion (VSEPR) theory [34], the bond angle of the H₂XO molecule should be less than 120° [19]. The bond angle of φ_HXO (X = N, P, As) markedly diminishes and the bond length of r_HX and r_XO (X = N, P, As) distinctly enlarges. Therefore, the covalent bond of the H₂AsO molecule is the weakest one.

The rotational constants of the H₂AsO molecule are showed in Table 3. The relationship between ground-state rotational constants values (A₀, B₀, C₀) and equilibrium rotational constants (A_e, B_e, C_e) can be expressed in Equation (2.1).

The difference between the ground-state rotational constants of the H₂AsO molecule at the B3PW91/aug-cc-pV5Z and the experimental values are 0.57%, 0.04%, and 0.26%, respectively. Therefore, one can easily conclude from Equation (2.1) that the ground-state equilibrium rotational constants (A_e = 3.989, B_e = 0.427, C_e = 0.412) of H₂AsO at the B3PW91/aug-cc-pV5Z level are reliable.

3.3 Vibrational frequencies

Molecular vibrational effects play a key role in electron scattering and various molecular properties such as the dipole moment [35–37]. The H₂AsO molecule has six vibrational modes: AsH symmetric stretch ν₁, HAsH symmetric bend ν₂, AsO stretch ν₃, HAsH wag ν₄, AsH antisymmetric stretchν₅, and HAsH antisymmetric bend or rock ν₆.

The ω₂ of H₂AsO at B3PW91/aug-cc-pV5Z is fairly in agreement with the corresponding experimental value with the errors higher than 2.828 cm^–1 and better than previous higher-level theories B3LYP, QCISD, CCSD(T), and CCSD with 6–31G**, 6–31++G, and aug-cc-pVTZ basis sets [7], respectively. The deviations between ω₁, ω₃, and ω₄ of H₂AsO at the B3PW91/aug-cc-pV5Z and the corresponding experimental data are just 4.4034%, 1.0711%, and 4.5618%, respectively.

3.4 Anharmonic constants

Table 5 summarizes the anharmonic constants of H₂AsO, most of which can be regarded as the predicted values since the corresponding experiment data have not been observed. The experimental anharmonic constant of H₂AsO is yielded from the assignments of the observed energy levels from the emission spectra [38, 39]. The anharmonic corrections seem to be estimated empirically, for example, χ₃₃ = (ν₃–ω₃)/2. When one substitutes the BPW91/aug-cc-pV5Z result in Table 5 to this empirical expression, χ₃₃ =−7.9825 cm^–1. However, when taking 1 4 ( χ 31 + χ 32 + χ 34 + χ 35 + χ 36 ) into account in Equation (2.1), χ₃₃ should be –4.341 cm^–1. The neglect of 1 4 ( χ 31 + χ 32 + χ 34 + χ 35 + χ 36 ) may cause a large error. Hence, it should be careful when the empirical anharmonic correction is used.

3.5 Vibration-rotation interaction constants

Table 6 collects the vibration–rotation interaction constants α i X (X = A, B, C; i = 1–6). To our knowledge, the theoretical and experimental studies of vibration-rotation interaction constants of the H₂AsO molecule have not been reported. Comparing the calculated results of DFT methods with that of MP2 method, the results of DFT methods generally yield vibration–rotation constants with the same sign and magnitude. It is needed to be pointed out that the calculated vibration–rotation interaction constants results by the MP2 method show the obvious gap with the DFT data. Based on the above calculation and analysis of spectroscopic constants for H₂AsO, the results using the BPW91/aug-cc-pV5Z can be expected to be accurate.

3.6 Centrifugal distortion constants

The equilibrium quartic and sextic centrifugal distortion constants of the H₂AsO molecule are listed in Tables 7 and 8. There are no corresponding experimental and theoretical values of centrifugal distortion constants of the H₂AsO molecule as yet. The calculated equilibrium quartic and sextic centrifugal distortion constants of the H₂AsO molecule at BPW91/aug-cc-pV5Z level are expected to provide the useful information for the future experiment.

3.7 Force constants

The cubic and quartic force fields of H₂AsO showed in Tables 9 and 10 can be produced by vibration–rotation interaction constants, Coriolis coupling constants and centrifugal distortion constants [19]. There are also no corresponding experimental or theoretical reports up to now. The force constant values from the B3P86 and B3PW91 methods employing aug-cc-pVQZ and aug-cc-pV5Z basis sets are very close to each other. The cubic and quartic force constants of the H₂AsO molecule at B3PW91/aug-cc-pV5Z theoretical level are expected to be trustworthy.

3.8 Coriolis coupling constants

The Coriolis coupling plays an important role in understanding the energy distribution, molecular structure, and dynamics processes of molecules in the micro-level. It especially needs to point out that the Coriolis coupling causes energy transfer between different vibration modes that are originally independent of each other, which will lead to an important influence on the infrared spectrum and the Raman spectrum behavior of the molecule. The calculated Coriolis coupling constants of H₂AsO in Table 11 gives an index of the coupling strengths between the normal coordinates Q_i and Q_j rotating along the a (x), b(y) and c (z) axis. It can be found from Table 11 that all the values obtained from B3LYP, B3PW91 and B3P86 methods, especially B3PW91 and B3P86 methods have the same order of magnitude and close to each other.