THE JOURNAL OF CHEMICAL PHYSICS 142, 194307 (2015)

Accurate quantum dynamics calculations of vibrational spectrum of dideuteromethane CH2D2 Hua-Gen Yua) Department of Chemistry, Brookhaven National Laboratory, Upton, New York 11973-5000, USA

(Received 20 April 2015; accepted 11 May 2015; published online 21 May 2015) We report a rigorous variational study of the infrared (IR) vibrational spectra of both CH2D2 and 13 CH2D2 isotopomers using an exact molecular Hamiltonian. Calculations are carried out using a recently developed multi-layer Lanczos algorithm based on the accurate refined Wang and Carrington potential energy surface of methane and the low-order truncated ab initio dipole moment surface of Yurchenko et al. [J. Mol. Spectrosc. 291, 69 (2013)]. All well converged 357 vibrational energy levels up to 6100 cm−1 of CH2D2 are obtained, together with a comparison to previous calculations and 91 experimental bands available. The calculated frequencies are in excellent agreement with the experimental results and give a root-mean-square error of 0.67 cm−1. In particular, we also compute the transition intensities from the vibrational ground state for both isotopomers. Based on the theoretical results, 20 experimental bands are suggested to be re-assigned. Surprisingly, an anomalous C isotopic effect is discovered in the nν5 modes of CH2D2. The predicted IR spectra provide useful information for understanding those unknown bands. C 2015 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4921411]

I. INTRODUCTION

Methane is an important hydrocarbon molecule in many fields such as combustion processes, earth’s atmosphere, and interstellar chemistry.1 It is also the smallest prototype molecule for exploring some crucial fundamental properties in terms of symmetries of physics, nuclear spin symmetry conservation, and parity violation.2 Those studies often require a highly accurate and detailed knowledge of rovibrational levels from high resolution spectroscopy. In the past decades, there have been many experimental and theoretical studies2–18 on the spectroscopy of methane and isotopomers. Although the spectroscopy of CH4 has been well understood at low energies, it is still challenging to undertake the complete band modeling of highly excited vibrational states. The investigations of deuteromethanes (CH3D, CHD3, CH2D2, and CD4) can provide invaluable information.3,19–22 In particular, CH2D2 has many advantages because it is an asymmetric top molecule in contrast to other symmetric top isotopomers CH3D and CD3H. Currently, nearly one hundred vibrational levels3,21 of CH2D2 have been measured and assigned up to 6000 cm−1 with the help of an effective Hamiltonian method and Van Vleck perturbation theory (PT). Although the PT approach3,7–9,23–26 is able to take the anharmonic effects into account, no PT calculation is very successful in giving an accurate description of those sophisticated couplings among modes, especially, at high energies. To overcome this difficulty, an accurate variational calculation is demanded. The full dimensional (FD) calculations of vibrational dynamics of methane have been made a great progression10,15,16,18,26–29 in the past decade. The first full-dimensional variational calculation of the vibrational energies of methane a)E-mail: [email protected]

0021-9606/2015/142(19)/194307/13/$30.00

was carried out by Bowman and co-workers30 using a vibrational self-consistent field configuration interaction method and the normal mode picture. Lately, an extensive FD variational calculation had been calculated by Schwenke and Partridge29 using a direct diagonalization method based on their ab initio SP(T8) potential energy surface.11 Based on this SP(T8) surface, several exact variational calculations have been carried out for the vibrational energies of methane and isotopomers using the advanced Lanczos algorithms15,16 in a set of Radau coordinates.31 The converged calculations showed that the SP(T8) surface is needed to improve in order to achieve a spectroscopic accuracy. In addition, there are several adjusted ab initio potential energy surfaces.28,32–34 Calculations show that those surfaces have similar or large errors.2,18,27,28,32 In particular, some calculations2,18,28,35 also provided infrared (IR) transition intensities by taking the advances of the ab initio dipole moment surfaces (DMSs).28,36 The band intensities are very useful in completely understanding the high resolution spectrum of methane. Furthermore, Császár and co-workers22 have recently reported an exact full dimensional quantum dynamics calculation for CH2D2. The 40 lowest-lying vibrational states up to 3500 cm−1 were well converged by using a Lanczos iterative diagonalization method and Watson normal mode Hamiltonian, based on the SP(T8) surface.11 Recently, Wang and Carrington (WC)17 have refined the SP(T8) surface of Schwenke and Patridge11 by the fitting to experimental vibrational band centers of CH4. The refined WC surface is very accurate with a root mean square deviation of 0.28 cm−1 compared with the 40 reliable experimental frequencies used in fitting procedure. In this work, we will take advantage of this accurate WC surface and the coupled-cluster single double (triple) [CCSD(T)]-F12 DMS of Yurchenko et al.28 By using them, we will carry out an exact FD variational calculation of the IR vibrational spectrum of CH2D2 using our

142, 194307-1

© 2015 AIP Publishing LLC

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194307-2

Hua-Gen Yu

J. Chem. Phys. 142, 194307 (2015)

recently developed multi-layer Lanczos method.35 The rest of the paper is organized as follows: the computational method is presented in Sec. II where the multi-layer Lanczos method will be briefly described. Results and discussions are given in Sec. III. Finally, a short conclusion is in Sec. IV.

where TˆQ(i)(Q) are R-independent. This Hamiltonian has no crossed partial derivative terms between the R and Q variables. Its detailed expression is given in Ref. 38. The eigenvalues of Hˆ are solved using the Lanczos method39 β j+1|ψ j+1⟩ = Hˆ |ψ j ⟩ − α j |ψ j ⟩ − β j |ψ j−1⟩,

II. COMPUTATIONAL METHOD

The quantum dynamics calculations are calculated using the multi-layer Lanczos iteration method.35 In the calculations, the (4 + 1) Radau coordinates31 were used as shown in Fig. 1. The nine internal variables are defined by four radial coordinates R = {r 1,r 2,r 3,r 4} and five angular variables Q = {θ 1, ϕ1, θ 2, ϕ2, θ 3}. In those orthogonal polyspherical coordinates,37 the vibrational molecular Hamiltonian can be written as Hˆ = TˆR (R) + Hˆ Q(Q; R),

(1)

Hˆ Q(Q; R) = TˆQ(Q; R) + V (Q, R),

(2)

with

TˆR (R) =

4 

ˆ i ) − V0(r i )], [ h(r

(3)

i=1

where V (Q, R) is the potential energy surface of the system. ˆ i ) is the one dimensional (1D) Hamiltonian, h(r 2 ∂ 2 ∂ ˆ i) = − ~ r + V0(r i ). h(r 2 ∂r i ∂r 2µi r i i i

(4)

Here, V0(r i ) is a 1D reference potential in r i with its associated reduced mass µi . In Eq. (2), TˆQ is the kinetic energy operator in the angular variables. It only parametrically depends on R through pre-factors. Generally, it can be partitioned as TˆQ(Q; R) =

4  i=1

1 ˆ (i) T (Q), 2µi r i2 Q

(5)

where α j and β j are the mean energy and recursion coefficient of the jth vector, respectively. They are defined as α j = ⟨ψ j | Hˆ |ψ j ⟩, β j+1 = ⟨ψ j+1| Hˆ |ψ j ⟩,

(7)

β1 = 0, |ψ0⟩ = 0.

(8)

with Therefore, the Lanczos recursion reduces the original Hamiltonian matrix to a symmetric tridiagonal form α1   β2  T K =     

β2 α2 β3 0

β3 .. .

0 ..

.

..

..

.

.

βK

     .  β K   α K 

(9)

The vibrational energies are determined by the eigenvalues of the T K matrix with a large subspace K. In the multi-layer Lanczos iteration approach,35 the dipole vibrational transition strength is calculated by using the elegant recursive residue generation method, RRGM.40–42 The transition strength from an initial state i to a final state f is given by 1/2

K −1 r  (E f − E)Πk=1 (λk − E)     , (10) | µiαf | = wαi  lim K   E→ E f Πk=1(λk − E)   where {λk } and {λrk } are the eigenvalues of the tridiagonal matrices T K and TrK , respectively. TrK is the reduced matrix obtained by removing the first row and column from T K . Here, the T K matrix is determined by using an initial Lanczos vector of dipole-vibrational wavefunction,

|ψ0⟩ = µα |ψi (Ei )⟩/wαi , α = x, y, z,

FIG. 1. The (4 + 1) Radau coordinates for CH2D2, where the body-fixed Z -axis is coincident with the Radau vector r4, while the vector r3 lies in the X Z -plane toward the positive X direction.

(6)

(11)

FIG. 2. The calculated vibrational spectrum for the transitions from the vibrational ground state of CH2D2 at T = 298 K.

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194307-3

Hua-Gen Yu

J. Chem. Phys. 142, 194307 (2015)

TABLE I. Calculated vibrational frequencies ν i f in cm−1, dipole transition strengths |µ i f | in D, and band intensities Svib in cm−1 atm−2 at T = 298 K for the transitions from the vibrational ground state of CH2D2, together with a comparison between the perturbation theory calculations3,7 and experimental results.3 ∆ are the errors of this work with respect to experiment. This work Label ZPE ν 4, A1 ν 7, B1 ν 9, B2 ν 5, A2 ν 3, A1 2ν 4, A1 ν 4 +ν 7, B1 ν 2, A1 2ν 7, A1 ν 8, B2 ν 4 +ν 9, B2 ν 7 +ν 9, A2 ν 4 +ν 5, A2 ν 5 +ν 7, B2 2ν 9, A1 ν 3 +ν 4, A1 ν 3 +ν 7, B1 ν 5 +ν 9, B1 2ν 5, A1 ν 3 +ν 9, B2 ν 3 +ν 5, A2 2ν 3, A1 ν 1, A1 ν 6, B1 3ν 4, A1 2ν 4 +ν 7, B1 ν 2 +ν 4, A1 ν 2 +ν 7, B1 ν 4 + 2ν 7, A1 ν 4 +ν 8, B2 3ν 7, B1 2ν 4 +ν 9, B2 ν 7 +ν 8, A2 ν 4 +ν 7 +ν 9, A2 ν 2 +ν 9, B2 2ν 4 +ν 5, A2 2ν 7 +ν 9, B2 ν 8 +ν 9, A1 ν 4 +ν 5 +ν 7, B2 ν 2 +ν 5, A2 ν 3 + 2ν 4, A1 ν 4 + 2ν 9, A1 ν 5 + 2ν 7, A2 ν 3 +ν 4 +ν 7, B1 ν 7 + 2ν 9, B1 ν 5 +ν 8, B1 ν 2 +ν 3, A1 ν 4 +ν 5 +ν 9, B1 ν 3 + 2ν 7, A1 ν 5 +ν 7 +ν 9, A1 ν 3 +ν 8, B2 3ν 9, B2 ν 4 + 2ν 5, A1 ν 3 +ν 4 +ν 9, B2 2ν 5 +ν 7, B1

νi f 8441.18 1032.67 1090.70 1235.71 1331.54 1435.16 2053.59 2123.94 2145.16 2203.22 2234.27 2285.51 2329.02 2364.76 2421.73 2458.23 2468.80 2515.56 2560.18 2658.53 2671.24 2766.12 2855.78 2975.76 3012.19 3065.44 3140.84 3183.01 3209.45 3233.49 3243.04 3306.84 3313.04 3321.51 3375.35 3380.81 3386.03 3439.34 3448.45 3455.38 3473.74 3484.15 3521.55 3530.14 3543.37 3556.14 3561.61 3569.20 3609.50 3626.75 3655.12 3663.37 3679.33 3692.35 3719.30 3748.17

∆

−0.38 −0.48 −0.56 0.13 0.03 −0.57 −0.74 −0.53 0.00 −0.43 −0.47 −0.68 −0.30 −0.56 −0.40 0.11 −0.36 0.19 −0.45 0.11 0.27 −0.07 −0.69 −0.79 −0.67 −0.82 −0.19 −0.71 0.08 −0.63 −0.55 −0.67 −0.16 −0.72

−0.22 −0.57

0.31 −0.49 −0.28 0.09 −0.43 −0.68 −0.20 −0.29 −0.16

|µ i f | 0.015 11 0.053 58 0.061 92 0.022 47 0.047 09 0.012 29 0.003 80 0.025 01 0.014 97 0.041 67 0.024 76 0.003 35 0.000 56 0.007 32 0.004 95 0.003 75 0.005 80 0.003 47 0.004 39 0.003 58 0.003 26 0.008 31 0.038 65 0.033 54 0.001 77 0.001 03 0.003 32 0.000 36 0.001 76 0.002 76 0.002 36 0.003 75 0.000 50 0.000 56 0.001 40 0.000 64 0.001 40 0.001 92 0.001 07 0.000 49 0.001 33 0.002 15 0.001 27 0.001 07 0.000 52 0.001 37 0.001 28 0.001 88 0.002 14 0.000 88 0.001 40 0.001 05 0.000 65 0.000 58 0.000 94

Svib

LMT7 νi f

UBA3 νi f

2.1992 29.2639 44.3921 6.3087 29.8711 2.9157 0.2882 12.6010 4.6361 36.4397 13.1606 0.2456 0.0068 1.2182 0.5663 0.3257 0.7947 0.2895 0.4809 0.3209 0.2757 1.8524 41.7607 31.8329 0.0904 0.0311 0.3296 0.0040 0.0942 0.2321 0.1733 0.4369 0.0078 0.0100 0.0625 0.0131 0.0637 0.1197 0.0375 0.0079 0.0581 0.1535 0.0536 0.0381 0.0089 0.0626 0.0551 0.1195 0.1558 0.0264 0.0675 0.0383 0.0147 0.0118 0.0309

1033.89 1093.49 1235.31 1330.73 1435.72 2058.80 2128.11 2167.65 2183.22 2246.64 2267.29 2332.39 2365.13 2424.60 2461.92 2468.09 2521.70 2557.29 2657.04 2671.70 2766.02 2858.01 2972.07 3008.02 3074.72 3153.74 3202.39 3253.12 3253.13 3290.29 3269.19 3264.92 3333.13 3365.10 3398.63 3390.55 3425.71 3491.99 3459.73 3491.06 3491.46 3466.34 3514.72 3554.78 3562.59 3567.72 3603.90 3589.78 3601.33 3654.73 3679.31 3679.82 3691.94 3702.14 3751.29

1032.18 1090.33 1236.87 1331.49 1434.84 2053.94 2121.63 2145.76 2203.39 2235.53 2285.55 2329.66 2365.14 2422.20 2459.14 2469.13 2515.49 2561.06 2657.69 2671.53 2766.89 2855.40 2975.51 3012.03 3065.29 3142.53 3182.76 3210.87 3234.44 3243.30 3306.81 3313.54 3320.80 3374.53 3380.82 3388.37 3439.33 3451.08 3454.98 3473.85 3484.72 3521.63 3531.73 3542.35 3558.11 3562.07 3569.23 3609.36 3626.93 3656.33 3665.11 3679.92 3692.80 3718.76 3748.78

Experiment3 νi f 1033.053 1091.185 1236.277 1331.409 1435.135 2054.163 2124.678 2145.692 2203.217 2234.692 2285.977 2329.698 2422.025 2458.794 2469.201 2515.449 2560.547 2658.335 2671.684 2855.665 2975.482 3012.260 3066.134 3141.626 3183.683 3210.267 3233.681 3243.745a 3306.765 3313.671a 3375.900 3381.472 3439.494 3449.177a

3484.373 3522.117a

3561.300 3569.689 3609.775 3626.660 3663.806 3680.002 3692.542 3719.591 3748.325

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194307-4

Hua-Gen Yu

J. Chem. Phys. 142, 194307 (2015)

TABLE I. (Continued.) This work Label ν 3 +ν 7 +ν 9, A2 ν 5 + 2ν 9, A2 ν 3 +ν 4 +ν 5, A2 ν 3 +ν 5 +ν 7, B2 2ν 3 +ν 4, A1 2ν 5 +ν 9, B2 ν 3 + 2ν 9, A1 2ν 3 +ν 7, B1 3ν 5, A2 ν 3 +ν 5 +ν 9, B1 ν 1 +ν 4, A1 ν 4 +ν 6, B1 ν 1 +ν 7, B1 4ν 4, A1 2ν 3 +ν 9, B2 ν 6 +ν 7, A1 ν 3 + 2ν 5, A1 3ν 4 +ν 7, B1 2ν 3 +ν 5, A2 ν 2 + 2ν 4, A1 ν 1 +ν 9, B2 ν 6 +ν 9, A2 ν 2 +ν 4 +ν 7, B1 2ν 4 +ν 8, B2 2ν 4 + 2ν 7, A1 3ν 3, A1 2ν 2, A1 ν 1 +ν 5, A2 3ν 4 +ν 9, B2 ν 2 + 2ν 7, A1 ν 4 +ν 7 +ν 8, A2 ν 5 +ν 6, B2 ν 4 + 3ν 7, B1 ν 2 +ν 8, B2 3ν 4 +ν 5, A2 2ν 4 +ν 7 +ν 9, A2 ν 1 +ν 3, A1 4ν 7, A1 2ν 7 +ν 8, B2 ν 2 +ν 4 +ν 9, B2 ν 3 +ν 6, B1 ν 4 +ν 8 +ν 9, A1 ν 2 +ν 7 +ν 9, A2 2ν 4 +ν 5 +ν 7, B2 ν 4 + 2ν 7 +ν 9, B2 2ν 8, A1 ν 3 + 3ν 4, A1 ν 2 +ν 4 +ν 5, A2 ν 2 +ν 5 +ν 7, B2 ν 7 +ν 8 +ν 9, B1 3ν 7 +ν 9, A2 ν 3 + 2ν 4 +ν 7, B1 2ν 4 + 2ν 9, A1 ν 4 +ν 5 + 2ν 7, A2 ν 4 +ν 5 +ν 8, B1 ν 2 + 2ν 9, A1 ν 2 +ν 3 +ν 4, A1 ν 4 +ν 7 + 2ν 9, B1

νi f

∆

3748.34 3780.38 3798.34 3842.24 3881.00 3881.83 3899.02 3927.83 3980.91 3995.74 4005.53 4042.12 4057.48 4068.76 4091.43 4092.17 4093.26 4150.10 4186.03 4196.48 4203.80 4235.70 4242.95 4243.17 4253.16 4262.37 4263.49 4297.37 4317.27 4326.70 4330.54 4331.87 4335.99 4348.03 4395.57 4400.86 4403.57 4410.84 4413.51 4425.68 4426.43 4435.78 4448.82 4471.85 4478.77 4484.98 4496.35 4512.17 4538.86 4539.25 4544.34 4556.40 4557.35 4561.08 4569.89 4600.76 4609.83 4613.32

−0.66

−0.19 −0.03 −0.18 −0.23 −0.26 −0.11 −0.45 −0.10 0.42 0.13

−0.74

−0.78 −0.69 0.21

−0.58 −0.13 −0.07

−0.24 0.88 −0.71

−0.28 −0.18

−0.55

−0.54

|µ i f | 0.000 37 0.000 52 0.000 43 0.000 47 0.000 60 0.000 82 0.001 16 0.000 25 0.000 90 0.000 29 0.002 80 0.001 68 0.003 56 0.002 37 0.005 02 0.001 35 0.000 34 0.000 33 0.000 40 0.002 27 0.005 23 0.002 93 0.001 57 0.000 66 0.000 24 0.002 13 0.000 93 0.001 33 0.004 55 0.001 27 0.000 22 0.005 44 0.000 38 0.006 09 0.000 34 0.000 28 0.003 35 0.003 44 0.003 66 0.001 64 0.000 26 0.002 31 0.000 32 0.000 35 0.001 49 0.002 00 0.001 03 0.000 29 0.000 35 0.000 29 0.000 28 0.000 21 0.002 36 0.000 24 0.000 25 0.000 57 0.000 52 0.000 29

Svib

LMT7 νi f

UBA3 νi f

0.0049 0.0098 0.0066 0.0079 0.0133 0.0248 0.0491 0.0023 0.0306 0.0031 0.2955 0.1077 0.4824 0.2144 0.9694 0.0703 0.0045 0.0042 0.0062 0.2029 1.0795 0.3423 0.0986 0.0173 0.0023 0.1817 0.0347 0.0712 0.8396 0.0658 0.0020 1.2039 0.0057 1.5135 0.0048 0.0032 0.4651 0.4888 0.5549 0.1118 0.0028 0.2221 0.0043 0.0050 0.0938 0.1682 0.0452 0.0037 0.0052 0.0037 0.0033 0.0019 0.2390 0.0024 0.0027 0.0139 0.0115 0.0037

3761.25 3775.14 3798.88 3852.36 3888.82 3874.83 3898.96 3936.44 3978.91 3993.23 4003.91 4040.50 4056.77 4081.66 4094.63 4090.24 4091.88 4170.40 4187.84 4228.15 4201.24 4232.52 4288.60 4274.20 4244.93 4307.05 4266.81 4291.05 4304.29 4334.84 4352.13 4325.88 4305.27 4354.57 4407.00 4388.82 4400.55 4351.41 4415.86 4431.45 4421.61 4482.70 4487.69 4485.88 4459.15 4456.12 4505.84 4526.31 4576.92 4556.42 4515.28 4578.87 4513.07 4550.58 4586.50 4620.89 4634.53 4593.38

3749.11 3781.97 3798.88 3842.95 3880.88 3881.55 3900.47 3928.25 3978.59 3995.88 4005.23 4043.07 4057.95 4066.23 4091.90 4091.29 4091.64 4153.01 4185.61 4197.33 4211.77 4236.17 4240.96 4242.94 4255.09 4262.77 4263.02 4295.64 4331.53 4328.82 4328.92 4331.60 4337.04 4348.15 4401.05 4401.57 4401.81 4415.75 4412.40 4423.72 4425.36 4435.43 4449.14 4477.31 4479.11 4485.57 4496.89 4506.40 4539.76 4540.28 4544.47 4560.06 4558.21 4564.18 4572.48 4595.48 4606.41 4612.33

Experiment3 νi f 3749.000

3842.430 3881.030 3882.009 3928.054 3996.006 4005.635 4042.577 4057.576 4091.009 4092.042

4197.229

4243.947 4263.064 4263.282

4327.277a 4332.000 4348.101

4413.750 4425.549 4436.488

4485.262 4496.531

4539.795

4557.890a

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194307-5

Hua-Gen Yu

J. Chem. Phys. 142, 194307 (2015)

TABLE I. (Continued.) This work Label ν 2 +ν 3 +ν 7, B1 ν 5 + 3ν 7, B2 2ν 4 +ν 5 +ν 9, B1 ν 3 +ν 4 + 2ν 7, A1 ν 8 + 2ν 9, B2 ν 5 +ν 7 +ν 8, A1 2ν 7 + 2ν 9, A1 ν 3 +ν 4 +ν 8, B2 ν 4 +ν 5 +ν 7 +ν 9, A1 2ν 4 + 2ν 5, A1 ν 3 + 3ν 7, B1 ν 2 +ν 5 +ν 9, B1 ν 4 + 3ν 9, B2 ν 3 +ν 7 +ν 8, A2 ν 3 + 2ν 4 +ν 9, B2 ν 5 + 2ν 7 +ν 9, B1 ν 5 +ν 8 +ν 9, A2 ν 7 + 3ν 9, A2 ν 4 + 2ν 5 +ν 7, B1 ν 2 + 2ν 5, A1 ν 3 +ν 4 +ν 7 +ν 9, A2 ν 2 +ν 3 +ν 9, B2 ν 3 + 2ν 4 +ν 5, A2 ν 4 +ν 5 + 2ν 9, A2 ν 3 + 2ν 7 +ν 9, B2 2ν 5 + 2ν 7, A1 ν 5 +ν 7 + 2ν 9, B2 4ν 9, A1 2ν 5 +ν 8, B2 ν 3 +ν 4 +ν 5 +ν 7, B2 ν 3 +ν 8 +ν 9, A1 ν 2 +ν 3 +ν 5, A2 2ν 3 + 2ν 4, A1 ν 4 + 2ν 5 +ν 9, B2 ν 3 +ν 5 + 2ν 7, A2 2ν 3 +ν 4 +ν 7, B1 ν 3 +ν 4 + 2ν 9, A1 ν 5 + 3ν 9, B1 ν 2 + 2ν 3, A1 2ν 5 +ν 7 +ν 9, A2 ν 3 +ν 7 + 2ν 9, B1 ν 3 +ν 5 +ν 8, B1 ν 4 + 3ν 5, A2 2ν 3 + 2ν 7, A1 ν 1 + 2ν 4, A1 ν 3 +ν 4 +ν 5 +ν 9, B1 2ν 4 +ν 6, B1 5ν 4, A1 ν 3 +ν 5 +ν 7 +ν 9, A1 3ν 5 +ν 7, B2 2ν 3 +ν 8, B2 ν 1 +ν 4 +ν 7, B1 2ν 5 + 2ν 9, A1 ν 3 + 3ν 9, B2 ν 1 +ν 2, A1 ν 4 +ν 6 +ν 7, A1 ν 3 +ν 4 + 2ν 5, A1 2ν 3 +ν 4 +ν 9, B2

νi f 4624.74 4632.01 4637.21 4641.26 4651.93 4655.86 4668.26 4676.38 4699.59 4701.54 4713.78 4721.21 4735.73 4738.63 4752.54 4762.46 4765.19 4775.32 4782.33 4797.48 4798.57 4804.55 4815.66 4839.34 4852.79 4854.17 4866.13 4870.64 4879.89 4887.20 4890.79 4896.15 4901.34 4929.73 4947.94 4952.28 4955.09 4966.37 4977.86 4978.63 4987.40 5001.49 5015.43 5022.74 5039.72 5042.88 5059.77 5063.75 5064.65 5070.02 5081.62 5087.77 5101.00 5109.25 5110.18 5122.89 5124.42 5139.92

∆

0.02

−0.28

−1.63

−0.20

0.17

−1.40

|µ i f | 0.000 27 0.000 67 0.000 24 0.000 26 0.000 53 0.001 46 0.001 91 0.000 34 0.000 45 0.000 23 0.001 65 0.000 30 0.000 29 0.000 23 0.000 23 0.000 22 0.000 22 0.000 23 0.000 22 0.000 55 0.000 22 0.000 25 0.000 26 0.000 22 0.001 79 0.000 23 0.000 51 0.002 09 0.000 22 0.000 47 0.001 64 0.000 22 0.001 88 0.000 33 0.000 22 0.000 22 0.000 41 0.000 22 0.000 69 0.000 22 0.000 22 0.000 22 0.000 22 0.001 16 0.001 79 0.000 22 0.000 49 0.000 73 0.000 35 0.000 46 0.000 73 0.000 22 0.001 68 0.000 61 0.000 38 0.000 56 0.000 22 0.000 44

Svib

LMT7 νi f

UBA3 νi f

0.0032 0.0195 0.0026 0.0029 0.0121 0.0928 0.1596 0.0051 0.0091 0.0023 0.1206 0.0040 0.0039 0.0023 0.0023 0.0023 0.0023 0.0025 0.0023 0.0138 0.0023 0.0028 0.0031 0.0022 0.1454 0.0024 0.0117 0.1991 0.0023 0.0103 0.1238 0.0023 0.1630 0.0049 0.0023 0.0023 0.0077 0.0022 0.0225 0.0023 0.0022 0.0022 0.0022 0.0638 0.1518 0.0023 0.0116 0.0251 0.0060 0.0099 0.0254 0.0022 0.1349 0.0178 0.0069 0.0153 0.0023 0.0092

4679.27 4601.08 4613.28 4637.71 4677.33 4654.59 4659.51 4696.04 4687.97 4717.88 4682.34 4713.27 4707.97 4758.27 4723.59 4748.46 4778.65 4784.08 4786.93 4810.05 4792.42 4832.94 4822.76 4805.72 4847.04 4841.80 4876.19 4889.01 4884.37 4885.95 4899.65 4924.28 4910.65 4907.84 4934.95 4967.98 4927.48 4984.29 5021.53 4972.68 4992.10 4999.94 5014.34 5011.11 5026.76 5024.18 5064.00 5079.62 5083.16 5073.56 5098.52 5089.34 5083.93 5117.51 5138.89 5123.44 5125.25 5123.52

4638.63 4635.12 4625.58 4643.77 4655.55 4657.08 4668.22 4676.48 4700.07 4717.51 4700.74 4721.39 4734.85 4739.87 4752.70 4764.05 4768.97 4775.23 4783.03 4796.09 4797.87 4803.60 4839.42 4817.97 4854.31 4855.34 4867.60 4874.38 4881.50 4887.49 4889.93 4895.98 4902.61 4928.61 4949.22 4952.29 4954.82 4968.02 5024.54 4979.73 4989.70 5002.36 5015.18 4977.93 5039.59 5042.37 5063.69 5056.75 5064.36 5070.07 5082.69 5086.82 5101.94 5111.35 5111.12 5121.46 5125.52 5139.04

Experiment3 νi f

4668.242a

4736.010a

4854.415a

4955.293a

5039.548

5111.577a

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194307-6

Hua-Gen Yu

J. Chem. Phys. 142, 194307 (2015)

TABLE I. (Continued.) This work Label ν 2 +ν 6, B1 4ν 4 +ν 7, B1 2ν 3 +ν 7 +ν 9, A2 ν 3 + 2ν 5 +ν 7, B1 ν 1 + 2ν 7, A1 3ν 5 +ν 9, B1 ν 1 +ν 8, B2 ν 6 + 2ν 7, B1 ν 2 + 3ν 4, A1 ν 3 +ν 5 + 2ν 9, A2 2ν 3 +ν 4 +ν 5, A2 3ν 4 +ν 8, B2 ν 6 +ν 8, A2 ν 1 +ν 4 +ν 9, B2 ν 2 + 2ν 4 +ν 7, B1 3ν 4 + 2ν 7, A1 2ν 3 +ν 5 +ν 7, B2 3ν 3 +ν 4, A1 ν 4 +ν 6 +ν 9, A2 2ν 2 +ν 7, B1 4ν 5, A1 ν 1 +ν 7 +ν 9, A2 2ν 2 +ν 4, A1 ν 6 +ν 7 +ν 9, B2 2ν 3 + 2ν 9, A1 ν 3 + 2ν 5 +ν 9, B2 2ν 4 +ν 7 +ν 8, A2 ν 1 +ν 4 +ν 5, A2 3ν 3 +ν 7, B1 4ν 4 +ν 9, B2 2ν 4 + 3ν 7, B1 ν 2 +ν 4 + 2ν 7, A1 ν 2 +ν 4 +ν 8, B2 ν 4 +ν 5 +ν 6, B2 ν 1 +ν 5 +ν 7, B2 ν 2 + 3ν 7, B1 4ν 4 +ν 5, A2 ν 2 +ν 7 +ν 8, A2 ν 5 +ν 6 +ν 7, A2 ν 3 + 3ν 5, A2 3ν 4 +ν 7 +ν 9, A2 2ν 3 +ν 5 +ν 9, B1 ν 4 + 2ν 8, A1 ν 4 + 2ν 7 +ν 8, B2 ν 1 + 2ν 9, A1 ν 1 +ν 3 +ν 4, A1 ν 6 + 2ν 9, B1 ν 4 + 4ν 7, A1 ν 2 + 2ν 4 +ν 9, B2 ν 3 +ν 4 +ν 6, B1 ν 1 +ν 3 +ν 7, B1 2ν 4 +ν 8 +ν 9, A1 3ν 4 +ν 5 +ν 7, B2 ν 3 +ν 6 +ν 7, A1 3ν 3 +ν 9, B2 ν 3 + 4ν 4, A1 2ν 2 +ν 9, B2 ν 2 +ν 4 +ν 7 +ν 9, A2

νi f

∆

5141.93 5150.99 5162.08 5162.22 5170.38 5200.98 5204.87 5206.17 5207.07 5208.95 5222.09 5235.82 5240.73 5250.05 5252.62 5253.64 5256.32 5287.27 5287.46 5291.77 5298.72 5309.79 5314.76 5316.09 5318.24 5320.39 5327.72 5327.72 5328.05 5332.06 5343.09 5354.07 5360.27 5365.29 5378.60 5399.54 5401.15 5404.99 5412.18 5413.38 5417.03 5417.29 5418.48 5421.82 5422.14 5433.23 5440.31 5444.56 5446.17 5456.27 5474.28 5481.14 5487.70 5488.86 5495.17 5495.66 5500.75 5503.36

−0.79

0.52 −0.13 −0.79

−0.12

1.80

−2.03

|µ i f | 0.000 56 0.000 84 0.000 21 0.000 21 0.000 92 0.000 21 0.000 23 0.000 21 0.001 28 0.000 22 0.000 21 0.000 61 0.000 39 0.000 31 0.000 97 0.000 21 0.000 51 0.000 58 0.000 39 0.000 26 0.002 16 0.000 23 0.000 59 0.000 48 0.000 52 0.001 67 0.000 27 0.000 27 0.000 23 0.000 80 0.000 40 0.000 30 0.000 27 0.000 44 0.000 28 0.000 26 0.000 56 0.000 28 0.000 28 0.000 23 0.000 20 0.000 20 0.001 24 0.000 84 0.000 26 0.000 35 0.000 21 0.000 47 0.000 21 0.000 21 0.000 97 0.000 25 0.000 55 0.000 54 0.000 68 0.000 21 0.000 35 0.000 20

Svib

LMT7 νi f

UBA3 νi f

0.0150 0.0338 0.0022 0.0022 0.0413 0.0022 0.0026 0.0021 0.0803 0.0023 0.0022 0.0181 0.0076 0.0046 0.0459 0.0023 0.0130 0.0167 0.0077 0.0033 0.2321 0.0027 0.0175 0.0115 0.0136 0.1398 0.0036 0.0037 0.0027 0.0318 0.0082 0.0047 0.0037 0.0099 0.0040 0.0034 0.0159 0.0039 0.0041 0.0026 0.0021 0.0021 0.0785 0.0362 0.0033 0.0064 0.0022 0.0115 0.0022 0.0022 0.0486 0.0033 0.0156 0.0150 0.0236 0.0023 0.0065 0.0021

5175.17 5178.06 5176.65 5178.61 5137.71 5187.96 5219.11 5168.70 5244.92 5211.73 5219.17 5274.51 5258.38 5231.17 5315.08 5262.31 5266.65 5296.09 5263.08 5384.50 5296.38 5289.53 5342.64 5318.32 5322.53 5310.33 5362.15 5323.39 5337.72 5309.32 5332.35 5371.04 5373.68 5358.87 5376.13 5412.80 5414.44 5433.03 5408.48 5413.31 5403.56 5415.70 5458.77 5435.59 5421.71 5430.85 5448.30 5388.21 5455.30 5452.55 5477.73 5490.07 5503.04 5496.30 5504.08 5511.24 5533.67 5521.24

5144.12 5153.08 5161.80 5162.58 5170.18 5199.29 5204.04 5206.44 5207.48 5210.12 5222.97 5233.34 5243.03 5254.67 5260.63 5265.34 5255.85 5286.90 5291.96 5356.85 5294.19 5296.72 5292.64 5316.05 5320.08 5317.89 5327.76 5326.82 5328.21 5332.51 5353.10 5361.90 5356.95 5364.11 5378.54 5488.06 5403.59 5415.78 5411.25 5412.09 5411.08 5415.79 5420.82 5422.23 5427.04 5439.97 5445.38 5446.68 5446.49 5457.54 5474.39 5486.96 5489.29 5495.31 5499.08 5499.32 5504.58 5512.01

Experiment3 νi f 5142.719

5169.863a 5204.999 5206.958

5256.440a

5318.586a

5446.596a

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194307-7

Hua-Gen Yu

J. Chem. Phys. 142, 194307 (2015)

TABLE I. (Continued.) This work Label 2ν 3 + 2ν 5, A1 ν 7 + 2ν 8, B1 3ν 7 +ν 8, A2 ν 2 +ν 4 +ν 5 +ν 7, B2 ν 1 +ν 5 +ν 9, B1 ν 2 + 2ν 4 +ν 5, A2 2ν 4 + 2ν 7 +ν 9, B2 ν 5 +ν 6 +ν 9, A1 ν 2 + 2ν 7 +ν 9, B2 ν 4 +ν 7 +ν 8 +ν 9, B1 ν 2 +ν 8 +ν 9, A1 ν 3 + 3ν 4 +ν 7, B1 2ν 4 +ν 5 + 2ν 7, A2 2ν 2 +ν 5, A2 3ν 3 +ν 5, A2 ν 1 + 2ν 5, A1 ν 4 + 3ν 7 +ν 9, A2 ν 2 +ν 5 + 2ν 7, A2 3ν 4 + 2ν 9, A1 ν 1 +ν 3 +ν 9, B2 5ν 7, B1 ν 2 +ν 3 + 2ν 4, A1 2ν 7 +ν 8 +ν 9, A1 2ν 5 +ν 6, B1 ν 3 +ν 6 +ν 9, A2 4ν 3, A1 ν 4 +ν 5 +ν 7 +ν 8, A1 ν 2 +ν 3 +ν 4 +ν 7, B1 ν 3 + 2ν 4 + 2ν 7, A1 ν 3 + 2ν 4 +ν 8, B2 ν 2 +ν 5 +ν 8, B1 ν 2 +ν 7 + 2ν 9, B1 ν 2 +ν 4 + 2ν 9, A1 3ν 4 +ν 5 +ν 9, B1 2ν 2 +ν 3, A1 2ν 4 +ν 7 + 2ν 9, B1 2ν 8 +ν 9, B2 ν 4 +ν 8 + 2ν 9, B2 ν 4 +ν 5 + 3ν 7, B2 ν 2 +ν 3 + 2ν 7, A1 4ν 7 +ν 9, B2 3ν 4 + 2ν 5, A1 ν 1 +ν 3 +ν 5, A2 ν 4 + 2ν 7 + 2ν 9, A1 ν 7 +ν 8 + 2ν 9, A2 2ν 4 +ν 5 +ν 7 +ν 9, A1 ν 3 +ν 5 +ν 6, B2 ν 3 +ν 4 +ν 7 +ν 8, A2 ν 5 + 2ν 7 +ν 8, B1 ν 2 +ν 4 +ν 5 +ν 9, B1 ν 3 +ν 4 + 3ν 7, B1 ν 2 +ν 3 +ν 8, B2 ν 2 +ν 5 +ν 7 +ν 9, A1 ν 3 + 3ν 4 +ν 9, B2 ν 5 + 4ν 7, A2 ν 4 +ν 5 +ν 8 +ν 9, A2 3ν 7 + 2ν 9, B1 2ν 4 + 2ν 5 +ν 7, B1

νi f 5505.06 5508.13 5508.71 5510.77 5519.12 5526.30 5526.80 5547.26 5549.05 5559.09 5561.85 5570.30 5571.08 5575.74 5578.65 5580.28 5583.06 5590.04 5593.51 5612.10 5614.05 5628.41 5631.68 5633.69 5639.41 5641.56 5645.23 5649.73 5650.05 5650.81 5652.58 5654.98 5659.23 5661.86 5662.64 5664.30 5668.89 5671.41 5677.89 5680.44 5705.64 5713.47 5720.52 5724.71 5727.88 5730.78 5732.14 5739.19 5739.69 5744.36 5745.35 5747.07 5751.28 5754.33 5763.10 5772.01 5772.54 5773.31

∆

1.30 −0.34

−2.90

|µ i f | 0.000 65 0.001 60 0.000 22 0.001 63 0.000 35 0.000 21 0.000 30 0.000 23 0.001 21 0.000 20 0.000 30 0.000 22 0.000 22 0.000 20 0.000 28 0.000 20 0.000 20 0.000 20 0.000 20 0.000 47 0.000 93 0.001 26 0.000 29 0.000 21 0.000 27 0.000 52 0.000 65 0.000 35 0.000 85 0.000 21 0.000 20 0.000 24 0.001 54 0.000 22 0.000 21 0.000 32 0.000 21 0.000 20 0.000 20 0.000 22 0.000 20 0.000 20 0.000 23 0.000 46 0.000 30 0.000 28 0.000 40 0.000 19 0.000 67 0.000 42 0.000 20 0.000 20 0.000 31 0.000 20 0.000 20 0.000 19 0.000 32 0.000 77

Svib 0.0221 0.1319 0.0025 0.1380 0.0062 0.0023 0.0048 0.0029 0.0764 0.0022 0.0048 0.0026 0.0025 0.0022 0.0042 0.0022 0.0022 0.0022 0.0022 0.0117 0.0453 0.0837 0.0045 0.0023 0.0038 0.0144 0.0224 0.0064 0.0387 0.0023 0.0022 0.0031 0.1269 0.0025 0.0022 0.0056 0.0023 0.0022 0.0021 0.0026 0.0022 0.0022 0.0028 0.0112 0.0048 0.0041 0.0085 0.0021 0.0241 0.0094 0.0022 0.0022 0.0053 0.0021 0.0021 0.0020 0.0054 0.0325

LMT7 νi f

UBA3 νi f

5513.25 5535.60 5494.83 5612.90 5511.46 5552.58 5483.59 5541.61 5572.99 5573.50 5569.91 5593.98 5577.45 5623.13 5596.20 5605.60 5549.44 5659.02 5525.16 5630.38

5510.44 5519.00 5484.89 5666.77 5524.52 5528.52 5498.57 5560.85 5548.09 5563.18 5562.51 5570.18 5586.23 5589.16 5592.03 5610.55 5581.60 5653.26 5578.68 5637.91

5658.74 5642.74 5639.32 5646.75 5662.15 5674.10 5713.20 5662.53 5703.80 5668.31 5713.54 5651.80 5627.80 5738.65 5615.19 5660.20 5691.77

5630.89 5653.98 5645.89 5649.32 5657.46 5673.72 5665.83 5615.01 5705.83 5655.35 5677.06 5650.13 5651.73 5727.99 5646.40 5634.98 5671.10

5753.45 5601.09 5734.82 5719.08 5691.02 5770.99 5712.20 5739.02 5775.73 5737.70 5746.61 5716.88 5785.16 5802.72 5736.08 5683.68 5795.52 5752.66 5813.60

5750.60 5653.08 5731.82 5722.00 5713.89 5750.23 5677.51 5744.49 5762.15 5742.02 5752.88 5733.46 5790.66 5825.01 5771.72 5742.71 5800.32 5773.40 5806.90

Experiment3 νi f

5547.744a 5562.191

5631.314a

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194307-8

Hua-Gen Yu

J. Chem. Phys. 142, 194307 (2015)

TABLE I. (Continued.) This work Label ν 2 + 3ν 9, B2 ν 4 +ν 5 + 2ν 7 +ν 9, B1 ν 5 + 2ν 8, A2 2ν 4 + 3ν 9, B2 ν 1 + 2ν 3, A1 2ν 3 +ν 6, B1 ν 3 + 3ν 4 +ν 5, A2 ν 3 + 2ν 7 +ν 8, B2 ν 2 +ν 4 + 2ν 5, A1 ν 8 + 3ν 9, A1 ν 3 + 2ν 4 +ν 7 +ν 9, A2 ν 4 +ν 7 + 3ν 9, A2 ν 3 + 4ν 7, A1 ν 5 +ν 7 +ν 8 +ν 9, B2 ν 2 + 2ν 5 +ν 7, B1 ν 2 +ν 3 +ν 4 +ν 9, B2 ν 2 +ν 3 +ν 7 +ν 9, A2 2ν 1, A1 2ν 7 + 3ν 9, B2 ν 5 + 3ν 7 +ν 9, A1 ν 4 + 2ν 5 + 2ν 7, A1 2ν 4 +ν 5 + 2ν 9, A2 ν 4 + 2ν 5 +ν 8, B2 ν 3 + 2ν 4 +ν 5 +ν 7, B2 ν 3 +ν 4 +ν 8 +ν 9, A1 ν 3 +ν 4 + 2ν 7 +ν 9, B2 ν 3 + 2ν 8, A1 ν 2 +ν 5 + 2ν 9, A2 2ν 3 + 3ν 4, A1 ν 2 +ν 3 +ν 4 +ν 5, A2 ν 4 +ν 5 +ν 7 + 2ν 9, B2 ν 2 +ν 3 +ν 5 +ν 7, B2 ν 3 +ν 7 +ν 8 +ν 9, B1 ν 5 +ν 8 + 2ν 9, B1 ν 7 + 4ν 9, B1 ν 4 + 4ν 9, A1 ν 1 +ν 6, B1 2ν 5 +ν 7 +ν 8, A2 2ν 4 + 2ν 5 +ν 9, B2 ν 3 + 3ν 7 +ν 9, A2 2ν 5 + 3ν 7, B1 ν 3 +ν 4 +ν 5 + 2ν 7, A2 ν 5 + 2ν 7 + 2ν 9, A2 2ν 3 + 2ν 4 +ν 7, B1 ν 3 + 2ν 4 + 2ν 9, A1 ν 3 +ν 4 +ν 5 +ν 8, B1 2ν 6, A1 ν 2 + 2ν 3 +ν 4, A1 ν 2 + 2ν 5 +ν 9, B2 ν 2 + 2ν 3 +ν 7, B1 ν 4 + 2ν 5 +ν 7 +ν 9, A2 ν 2 +ν 3 + 2ν 9, A1 2ν 4 + 3ν 5, A2 ν 1 + 3ν 4, A1 ν 3 +ν 4 +ν 7 + 2ν 9, B1 5ν 9, B2 2ν 3 +ν 4 + 2ν 7, A1 ν 3 +ν 5 + 3ν 7, B2

νi f 5790.06 5797.48 5798.08 5802.70 5807.03 5815.75 5818.96 5819.27 5819.85 5825.81 5826.85 5836.76 5842.35 5845.27 5851.41 5857.75 5862.02 5863.64 5866.09 5867.41 5872.20 5874.07 5876.26 5877.46 5881.49 5884.63 5896.11 5900.58 5904.91 5917.96 5919.94 5933.24 5933.33 5942.24 5949.24 5951.36 5952.90 5955.15 5957.47 5958.33 5966.49 5969.92 5980.98 5988.44 5991.13 5991.60 5999.08 5999.38 6009.78 6018.48 6020.93 6027.21 6027.95 6028.96 6030.38 6037.16 6039.37 6049.27

∆

0.50

0.10

−0.42

−0.99

0.12

|µ i f | 0.001 02 0.000 58 0.000 20 0.000 28 0.002 51 0.000 19 0.000 23 0.002 03 0.000 73 0.000 20 0.000 20 0.000 19 0.000 19 0.000 38 0.000 19 0.000 18 0.000 17 0.000 24 0.000 31 0.004 52 0.000 55 0.000 23 0.000 63 0.005 27 0.000 97 0.000 22 0.000 58 0.000 31 0.000 26 0.000 20 0.000 20 0.000 20 0.000 20 0.000 21 0.000 20 0.000 20 0.000 73 0.000 35 0.000 41 0.000 20 0.000 58 0.000 30 0.000 20 0.000 48 0.000 54 0.000 20 0.001 18 0.000 27 0.000 20 0.000 20 0.000 20 0.000 28 0.000 20 0.000 63 0.000 26 0.000 35 0.000 34 0.000 20

Svib 0.0568 0.0182 0.0021 0.0044 0.3430 0.0020 0.0030 0.2243 0.0293 0.0022 0.0022 0.0021 0.0020 0.0077 0.0020 0.0018 0.0016 0.0031 0.0054 1.1263 0.0168 0.0030 0.0221 1.5356 0.0524 0.0028 0.0186 0.0052 0.0038 0.0023 0.0023 0.0023 0.0023 0.0024 0.0022 0.0022 0.0298 0.0068 0.0094 0.0023 0.0188 0.0050 0.0022 0.0129 0.0166 0.0023 0.0788 0.0042 0.0022 0.0022 0.0022 0.0045 0.0022 0.0221 0.0037 0.0070 0.0067 0.0022

LMT7 νi f

UBA3 νi f

5834.44 5782.40 5767.53 5727.13 5815.57 5821.74 5837.66 5833.48 5845.81 5879.60 5814.60 5812.96 5757.03 5869.12 5896.29 5864.23 5914.48 5889.44 5884.59 5838.41 5878.36 5827.44 5903.66 5910.56 5914.47 5878.94 5885.72 5926.79 5923.51 5958.00 5907.50 6002.62 5982.21 5980.88 5996.86 5915.25 5865.01 5971.62 5931.86 5929.08 5928.54 5969.27 5973.49 5990.54 5947.02 6017.18 5952.76 6053.19 6023.51

5826.58 5803.48 5738.03 5754.81 5806.56 5813.70 5860.02 5819.28 5830.18 5849.73 5829.99 5818.82 5772.22 5860.90 5863.01 5846.30 5875.52 5870.08 5898.84 5867.94 5889.17 5840.97 5881.36 5878.56 5904.84 5880.61 5870.25 5915.42 5992.03 5949.38 5933.60 5958.94 5971.18 5945.63 5986.36 5951.66 5902.84 5984.00 5950.86 5927.41 5958.48 5961.85 5992.92 5975.31 5921.16 6026.54 5999.07 6033.46 6042.11

6006.48 6055.86 6040.79 6040.63 6021.35

6021.66 6052.32 6041.35 6002.61 6026.89

6043.38 6013.78

6027.74 6015.46

Experiment3 νi f

5806.530

5819.168a

5867.824a

5878.456a

5998.965

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194307-9

Hua-Gen Yu

J. Chem. Phys. 142, 194307 (2015)

TABLE I. (Continued.) This work νi f

Label ν 4 +ν 5 + 3ν 9, B1 ν 3 +ν 5 +ν 7 +ν 8, A1 3ν 4 +ν 6, B1 ν 5 +ν 7 + 3ν 9, A1 ν 3 +ν 8 + 2ν 9, B2 2ν 5 + 2ν 7 +ν 9, B2 ν 3 + 2ν 7 + 2ν 9, A1 2ν 5 +ν 8 +ν 9, A1 ν 3 + 2ν 4 +ν 5 +ν 9, B1 2ν 3 +ν 4 +ν 8, B2 ν 1 + 2ν 4 +ν 7, B1 a Those

6055.24 6056.72 6063.50 6065.57 6067.44 6071.91 6073.02 6082.86 6083.63 6085.65 6100.46

∆

|µ i f | 0.000 84 0.000 20 0.000 28 0.001 29 0.000 20 0.000 20 0.000 21 0.000 24 0.000 54 0.000 79 0.000 38

0.0398 0.0022 0.0046 0.0948 0.0023 0.0022 0.0026 0.0032 0.0169 0.0359 0.0084

UBA3 νi f

6012.95 6079.28

6001.15 6064.52

6046.15

6055.02

Experiment3 νi f

bands are re-assigned.

with the normalization factor wαi . µα are the dipole moment components in the body frame (BF) coordinates. |ψi (Ei )⟩ is an initial vibrational state with an energy Ei . The vibrational wavefunction is calculated using a Green function-guided spectral transform Lanczos (GSTL) method.35 In order to increase the numerical efficiency and to avoid large core memory requirement, the dipole-wavefunction µα |ψi (Ei )⟩ is compacted in a full discrete variable representation (DVR) manner.35 Here, the initial wavefunction is computed in a combined radial DVR/diabatic function basis. In particular, the diabatic functions in Q are compacted and saved in DVR. They are first transformed from their original finite basis representation (FBR) functions and contracted according to a potential threshold Vth. In order to avoid the accession of large core memory, the evaluation of the µα |ψi (Ei )⟩ products is carried out sectorby-sector in radial potential optimised discrete variable representations (PODVRs) rather than a full nine dimension manner at once. Here, the evaluation is also pre-screened in terms of the grid density of the wavefunction so that one can save some central processing unit (CPU) time. For more details, the reader can see Ref. 35. By using the multi-layer Lanczos algorithm, for a given initial state |ψi (Ei )⟩, the transition elements from the ith state to all final states |ψ f ⟩ (with a transition energy νi f = |E f − Ei |) can be calculated with a single Lanczos recursion. No product wavefunction is required. The vibrational band intensities at temperature T from the energy level Ei are calculated by Refs. 4, 5, and 28, Svib =

Svib

LMT7 νi f

functions in which the Lanczos vector |vk ⟩ is expressed as  k |vk ⟩ = Cm,α f m (Q; R0, RV0 )|Rα ⟩, (13) m,α 4 Πi=1 |r α i ⟩

where |Rα ⟩ = refer to the direct-product PO-DVR functions in the radial coordinates, with α being a collective DVR index. The 1D PO-DVRs are calculated using the ˆ lowest eigenstates of the Hamiltonian h(r) in Eq. (4). In this work, the direct-product basis functions are further contracted by discarding those PO-DVRs where the minimum potential energies in their corresponding Rα sectors are larger than a threshold value (Vth). In Eq. (13), f m are the vibrationally diabatic basis functions in the angular variables Q as they are independent of the R coordinates. They are formed by the lowest eigenstates of a reference reduced-dimension Hamiltonian Hˆ Q0 (Q; R0, RV0 ) in Q, namely, 0 Hˆ Q0 (Q; R0, RV0 ) f m (Q; R0, RV0 ) = E m f m (Q; R0, RV0 ),

(14)

with Hˆ Q0 (Q; R0, RV0 ) = TˆQ(Q; R0) + V (Q; RV0 ),

(15)

where R0 are the radial references in the kinetic energy operator. Usually, they are constant as {r i0}. RV0 are the references

8π 310−36 LT0 −E i /k BT e νi f (1 − e−hcν i f /k BT )| µi f |2, 3hcQ v (T) T (12)

where L = 2.686 754 × 1019 cm−3 is the Loschmidt’s number. T0 = 273.15 K is the reference temperature. Q v (T) is the vibrational partition function of CH2D2 (or 13CH2D2). It is very challenging to solve the eigenvalue problem for a nine dimensional system owing to the huge basis in the variational calculations. In this work, we use our two-layer Lanczos technique,43 i.e., the “divide-and-conquer” strategy that solves the eigenvalue problem in a sequential reduced dimensional manner. In such an approach, the Lanczos recurrence in Eq. (6) is carried out in combined grid/diabatic basis

FIG. 3. The errors of calculated vibrational energy levels of CH2D2 relative to the experimental results.3

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194307-10

Hua-Gen Yu

J. Chem. Phys. 142, 194307 (2015)

TABLE II. Calculated vibrational frequencies ν i f and their 13C isotopic shifting −∆E in cm−1, dipole transition strengths |µ i f | in D, and band intensities Svib in cm−1 atm−2 at T = 298 K for the transitions from the vibrational ground state of 13CH2D2. Also listed are the experimental3 and theoretical frequencies of CH2D2 for reference. 13CH

CH2D2 Label ZPE ν 4, A1 ν 7, B1 ν 9, B2 ν 5, A2 ν 3, A1 2ν 4, A1 ν 4 +ν 7, B1 ν 2, A1 2ν 7, A1 ν 8, B2 ν 4 +ν 9, B2 ν 7 +ν 9, A2 ν 4 +ν 5, A2 ν 5 +ν 7, B2 2ν 9, A1 ν 3 +ν 4, A1 ν 3 +ν 7, B1 ν 5 +ν 9, B1 2ν 5, A1 ν 3 +ν 9, B2 ν 3 +ν 5, A2 2ν 3, A1 ν 1, A1 ν 6, B1 ν 2 +ν 4, A1 ν 2 +ν 7, B1 ν 4 +ν 8, B2 ν 7 +ν 8, A2 ν 2 +ν 9, B2 ν 8 +ν 9, A1 ν 2 +ν 5, A2 ν 2 +ν 3, A1 ν 5 +ν 8, B1 ν 3 +ν 8, B2 3ν 5, A2 ν 1 +ν 4, A1 ν 4 +ν 6, B1 ν 1 +ν 7, B1 ν 6 +ν 7, A1 ν 1 +ν 9, B2 ν 6 +ν 9, A2 2ν 2, A1 ν 1 +ν 5, A2 ν 5 +ν 6, B2 ν 2 +ν 8, B2 ν 1 +ν 3, A1 ν 3 +ν 6, B1 2ν 8, A1 ν 1 +ν 2, A1 ν 2 +ν 6, B1 ν 1 +ν 8, B2 ν 6 +ν 8, A2 4ν 5, A1 3ν 3 +ν 5, A2 ν 1 + 2ν 5, A1 2ν 1, A1

Experiment3 1033.053 1091.185 1236.277 1331.409 1435.135 2054.163 2124.678 2145.692 2203.217 2234.692 2285.977 2329.698 2422.025 2458.794 2469.201 2515.449 2560.547 2658.335 2671.684 2855.665 2975.482 3012.260 3183.683 3210.267 3243.745a 3381.472 3449.177a 3569.689 3561.300 3663.806 4005.635 4042.577 4057.576 4092.042

4263.282 4332.000 4348.101 4425.549 4485.262 5111.577a 5142.719 5204.999

This work 8441.18 1032.67 1090.70 1235.71 1331.54 1435.16 2053.59 2123.94 2145.16 2203.22 2234.27 2285.51 2329.02 2364.76 2421.73 2458.23 2468.80 2515.56 2560.18 2658.53 2671.24 2766.12 2855.78 2975.76 3012.19 3183.01 3209.45 3243.04 3321.51 3380.81 3448.45 3473.74 3569.20 3561.61 3663.37 3980.91 4005.53 4042.12 4057.48 4092.17 4203.80 4235.70 4263.49 4297.37 4331.87 4348.03 4403.57 4426.43 4484.98 5110.18 5141.93 5204.87 5240.73 5298.72 5578.65 5580.28 5863.64

2D2

(this work)

νi f

−∆E

|µ i f |

Svib

8408.63 1026.24 1081.18 1228.76 1331.71 1433.37 2041.36 2108.03 2132.93 2189.35 2220.61 2271.22 2312.61 2358.48 2412.32 2445.58 2459.45 2504.56 2553.11 2658.78 2662.48 2764.45 2852.23 2969.95 3002.46 3164.11 3187.96 3223.60 3298.45 3362.61 3428.36 3462.06 3553.66 3545.52 3646.74 3981.20 3993.25 4025.97 4042.06 4072.50 4191.06 4217.94 4235.48 4291.63 4320.38 4326.72 4396.15 4409.74 4458.79 5089.25 5117.02 5185.54 5217.85 5298.94 5583.64 5588.37 5855.01

32.55 6.44 9.52 6.96 −0.16 1.80 12.23 15.91 12.23 13.86 13.65 14.30 16.41 6.28 9.40 12.65 9.35 11.00 7.08 −0.26 8.76 1.67 3.54 5.80 9.73 18.90 21.49 19.43 23.05 18.19 20.10 11.69 15.54 16.09 16.63 −0.28 12.28 16.16 15.42 19.68 12.74 17.76 28.01 5.74 11.49 21.32 7.42 16.69 26.19 20.93 24.91 19.33 22.89 −0.23 −4.99 −8.09 8.63

0.014 69 0.054 28 0.062 16 0.022 87 0.047 90 0.012 08 0.003 88 0.024 05 0.017 33 0.042 28 0.024 42 0.003 47 0.000 54 0.007 43 0.004 81 0.003 70 0.005 82 0.003 62 0.004 47 0.003 63 0.003 33 0.008 40 0.038 70 0.033 79 0.003 15 0.000 53 0.002 84 0.000 53 0.001 64 0.001 96 0.000 42 0.001 29 0.001 45 0.001 55 0.000 92 0.002 85 0.001 64 0.003 56 0.005 14 0.005 23 0.001 38 0.001 75 0.001 28 0.003 45 0.007 26 0.003 45 0.002 31 0.002 01 0.000 98 0.000 80 0.000 32 0.000 38 0.002 25 0.000 48 0.000 20 0.004 47

2.0634 29.7466 44.4583 6.5280 30.8462 2.7971 0.2978 11.5863 6.1704 37.2746 12.7132 0.2618 0.0065 1.2491 0.5301 0.3171 0.7955 0.3133 0.4976 0.3292 0.2873 1.8877 41.7642 32.1840 0.2954 0.0085 0.2436 0.0086 0.0849 0.1232 0.0057 0.0560 0.0696 0.0819 0.0317 0.3042 0.1022 0.4813 1.0093 1.0782 0.0754 0.1212 0.0658 0.4824 2.1392 0.4916 0.2218 0.1693 0.0461 0.0307 0.0051 0.0071 0.2523 0.0120 0.0022 1.1008

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194307-11

Hua-Gen Yu

J. Chem. Phys. 142, 194307 (2015)

TABLE II. (Continued.) 13CH

CH2D2 Label

Experiment3

This work

ν 1 +ν 6, B1 2ν 6, A1

5998.965

5952.90 5999.08

a Those

1 2(1 + δ0n δ0m )

{| j1 j2 j3nm⟩ + (−1) p | j1 j2 j3 − n − m⟩}, (16)

where p refers to the parity of inversion symmetry. | jm⟩ are the orthonormal spherical harmonic basis functions.44 In FBR, the  action of Hˆ Q0 (Q; R0, RV0 ) on a Lanczos vector |vk′ ⟩ = l Clk |l⟩ is performed as Hˆ Q0 |vk′ ⟩ =

4 

1 ˆ (i) T (Q)|l⟩ 2 Q 2µi r i0 l i=1  + U†l ′γV (Qγ ; RV0 )Uγl Clk |l ′⟩, (17) l′

(this work)

νi f

−∆E

|µ i f |

Svib

5937.41 5981.87

15.49 17.21

0.000 43 0.001 06

0.0101 0.0628

bands are re-assigned.

in the potential energy surface. In this work, RV0 are set as the same values as R0. The eigen-equation in Eq. (14) is again solved by a Green function-GSTL method with a functional F( Hˆ Q0 ) = 1/( Hˆ Q0 − EQ). In this step, a non-direct product FBR basis set {|l⟩} is employed so that the singularities of the kinetic energy operator in polar angles can be properly described. As usual, we use a symmetrically adapted FBR (SA-FBR) |l⟩ = | j1 j2 j3nmp⟩ in the angular coordinates. It is written as38 |l⟩ = 

2D2

Clk

l

γ

where U is the collocation matrix between the FBR and DVR ({|γ⟩}) basis sets. All calculations are carried out using the PetroVib program38 implemented with the multi-layer Lanczos algorithm.35 III. RESULTS AND DISCUSSIONS

In the vibrational energy level calculations, the accurate refined WC potential energy surface of CH417 was employed. There is no unphysical well on this surface. Similar to its original SP(T8) surface,11 it is a function of the CH4 Radau variables. The energy values for CH2D2 are evaluated via Cartesian coordinates, which are first transformed from the CH2D2 Radau coordinates. The reference radial coordinates in the ki0 netic energy operators are 2.095 28 a0 for r C−H and 2.011 26 a0 0 for r C−D. In order to be consistent with the convention of the potential energy surface,11,17 the nuclear masses of atoms are 1.007 825 amu for H, 2.014 102 amu for D, 12.0 amu for C, and 13.003 355 amu for 13C. We have used a very large combined basis set in the variational calculations. For the radial coordinates, 10 and 14 PODVR functions are for the C-H/D Radau radial coordinates, respectively. They are contracted from 80 primitive Fourier DVRs ranging from 0.6 a0 to 4.2 a0 with a one-dimensional reference potential. For the angular variables, a non-direct product FBR basis set is employed. It is formed by the largest quantum number jmax = 28 (or 34) for the radial r CH (or r CD)

associated angles. The symmetrically adapted DVR basis with respect to inversion symmetry is used with respect to the SAFBR. The potential threshold value is set at Vth = 3.25 eV for the DVR basis contraction. The primitive DVR basis sizes are 4.78 × 107 (or a basis set of 7.66 × 106 FBR functions) in the angular variables that lead to a basis set of 1.50 × 1012 in total. Seven hundred diabatic basis functions are calculated. Such a large basis set is able to converge all vibrational states up to 6500 cm−1 to better than 0.02 cm−1. The dipole transition strengths are computed using the first order expansion of the ab initio DMSs of Yurchenko et al.28 Using the truncated DMS avoids the unphysical interpolations at large geometry distortions.28 It has been shown that this is a good approximation for the transition from the vibrational ground state of CH4.35,45 In the dipole transition strength calculations, the dipole moment surface is evaluated in a compact DVR basis set according to the vibrational ground state wavefunction. The criteria used for DVR contraction are ϵ a = 10−6 and ϵ b = 2 × 10−4 (see Eqs. (45)–(47) in Ref. 35 for details). The calculated IR spectrum of CH2D2 is shown in Fig. 2. The detailed results are listed in Table I which also gives the previous theoretical results and experimental values available for comparison. The LMT (Lee-Martin-Taylor) results are obtained by Ulenkov et al.3 using a full ab initio force field and perturbation theory based on an effective Hamiltonian, while the UBA (Ulenikov-Bekhtereva-Albert et al.) values are determined by fitting to experimental band centers based on the same approach. The ν1 − ν4 fundamental frequencies in A1 are the CH2 symmetric stretching, CD2 symmetric stretching, CH2 scissoring, and CD2 scissoring modes,3 respectively. ν5 in A2 is the CH2 twisting. ν6 and ν7 in B1 are the CH2 antisymmetric stretching and CH2 rocking modes. ν8 and ν9 in B2 are the CD2 antisymmetric stretching and CH2 wagging. Here, the point group C2V is adapted for the mode assignments. Based on our calculations, 20 experimental bands have been re-assigned owing to their large deviations (>20 cm−1). This suggestion largely relies on the accuracy of the potential energy surface and the exact variational calculations. Basically, the re-assignments belong to two types. One is the exchange of two observed bands. For instance, the band centers at 3449.177 cm−1 and 3522.117 cm−1 are re-assigned to ν8 + ν9 and ν4 + 2ν9 instead of the old assignments ν4 + 2ν9 and ν8 + ν9 in Ref. 3. Such bands are complicated by the clustered bands having strong mode coupling. In this case, they are the ν8 + ν9, ν3 + 2ν4, and ν4 + 2ν9 bands in A1. The unusual mode couplings could cause the perturbation theory to fail in simulations. The other type is owing to a potential incorrect assignment, e.g., for the 5878.456 cm−1 band center. It was assigned to ν1 + ν6 that is predicted at 5952.90 cm−1 in our calculations. Such a big difference, however, is unlikely. On the other

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194307-12

Hua-Gen Yu

FIG. 4. The anomalous C isotopic frequency shifting in nν 5 modes of 13CH D . 2 2

hand, according to calculated band positions and transition intensities, we do find a strong position at 5877.46 cm−1 for ν3 + 2ν4 + ν5 + ν7. Therefore, the observed band at 5878.456 cm−1 is re-assigned to ν3 + 2ν4 + ν5 + ν7 instead of the old ν1 + ν6 one. As a result, both the theoretical position and intensity are consistent with experimental ones. This example also demonstrates that calculating transition strengths is very useful. Figure 3 displays the errors of the variational results with respect to 91 experimental bands available. They are in excellent agreement. The root mean square error is 0.67 cm−1 with only six bands having an error larger than 1 cm−1. Therefore, our calculations also verify that the WC surface is very accurate. Furthermore, the variational study outperforms the perturbation theory calculations. Although Uleikov et al.3 have refined the parameters in the effective Hamiltonian model using experimental levels, their energy levels start to have large deviations once energy is higher than 4200 cm−1. This is because the accuracy of the fitting method strongly depends on the experimental data. They are rather limited at high energies. On the other hand, the spectral density becomes large at high energy range that also makes it difficult to apply the perturbation theory. The 13C atom is an important isotope of carbon. The infrared vibrational spectrum of 13CH2D2 is also computed. Some important bands are given in Table II (see the supplementary material46 for a complete list of energy levels up to 6000 cm−1 and the IR spectrum of 13CH2D2). As expected, most energy levels of 13CH2D2 have a red shifting upon the C isotope replacement. Surprisingly, it was found that the nν5 modes have an unusual blue shifting as shown in Fig. 4. Those frequency shiftings hold for all n = 1 − 4 and are much larger than the variational calculation error (