Investigation of Nonadiabatic Effects for the Vibrational Spectrum of a

Aug 18, 2017 - For these new vibrational calculations, potentials based on explicitly correlated wave functions are used where, in addition, adiabatic...
3 downloads 17 Views 3MB Size
Download PDF
Article pubs.acs.org/JPCA

Investigation of Nonadiabatic Effects for the Vibrational Spectrum of a Triatomic Molecule: The Use of a Single Potential Energy Surface with Distance-Dependent Masses for H3+ Ralph Jaquet* and Mykhaylo V. Khoma Theoretische Chemie, Universität Siegen, D-57068 Siegen, Germany ABSTRACT: On the basis of first-principles, the influence of nonadiabatic effects on the vibrational bound states of H3+ has been investigated using distance-dependent reduced masses and only one single potential energy surface. For these new vibrational calculations, potentials based on explicitly correlated wave functions are used where, in addition, adiabatic corrections and relativistic contributions are taken into account. For the first time, several different fully distance-dependent reduced mass surfaces in three dimensions have been incorporated in the vibrational calculations.

■

INTRODUCTION H3 has attracted both experimental spectroscopists and theoreticians for quite some time.1−5 Highly accurate calculations for small molecules (mostly diatomics and triatomics containing hydrogen atoms), solving the electronic and nuclear parts of the Schrödinger equation, are used as benchmarks. Deviations from experiment are a hint for the deficiency of the theoretical approach. In the case of the IR spectroscopy of H3+, deviations from experiment are due to different sources: (a) For highly excited rovibrational states the underlying potential energy surface (PES) might be not adequate enough. This is not related to the ab initio energy points (including all potential-like corrections, that is, adiabatic corrections and relativistic contributions) but to the quality of the fitting of these data to get a global surface with the same absolute quality as known for the given ab initio points (better than 0.1 cm−1). When we talk about quality of the PES we do not mean the root-mean-square (RMS) error but the maximum deviation at each single energy point, and, still, we do not really know the quality of the interpolation because there are never enough data points. Tests with analytical model potentials for H3+ have shown that B-spline6 will reproduce intermediate points at least to 0.1 cm−1, only if a dense grid of 20 Bohr distances. Now we are going to use for the reduced masses full distance-dependent mass surfaces in three dimensions. In the past, nonadiabatic contributions to low-lying rovibrational states had been simulated by using different constant masses for rotational and vibrational motions. That this assumption is reasonable as a first-order correction has been proven theoretically.13−23 Recent calculations24 have shown that for transition frequencies higher than the barrier of

+

© 2017 American Chemical Society

Received: May 16, 2017 Revised: August 17, 2017 Published: August 18, 2017 7016

DOI: 10.1021/acs.jpca.7b04703 J. Phys. Chem. A 2017, 121, 7016−7030

Article

The Journal of Physical Chemistry A linearity in H3+ the strategy with different fixed masses for rotation and vibration does not improve on average the deviation from experiment. The correct way would be to take into account that all masses or reduced masses used in the rovibrational Hamiltonian are distance-dependent: This can be done based on rigid theory7−10,12 or empirically.26 The distance dependence means that the contribution or coupling of the electronic mass to the nuclear motion changes with internuclear distance. In the present work we use the Born−Oppenheimer PES based on the Gaussian Geminal calculations of Pavanello and Adamowicz,27,28 with and without diagonal adiabatic and relativistic29 corrections. This is the first time that the nuclear motion of a triatomic molecule is treated on a single PES where the nonadiabatic effects have been fully included based on first principle. Because all kinetic energy operator terms do include mass prefactors that depend explicitly on three internuclear coordinates (we use the three Jacobi coordinates r, R, and θ), our original code30,31 had to be totally reorganized. Only the discrete variable representation (DVR) of the potential energy was kept the same as in the original code. The structure of this paper is as follows: (1) We present how the perturbative treatment of electron−nucleus coupling will lead to effective electron mass contributions, which have to be added to the nuclear mass. The effective electron mass contributions result from the coupling of the electronic ground state with 499 excited electronic states calculated “separately by perturbation theory”. (2) For this treatment derivative matrix elements (gradients with respect to nuclear coordinates of excited electronic wave functions coupled to the electronic ground-state wave function) are needed. (3) Some typical plots of the different distance-dependent masses in three dimensions are presented. (4) The different effective masses have to be used in a newly derived kinetic energy operator, which is much more complicated than for traditional constant masses. The accuracy of the matrix representations of the new additional kinetic energy matrix elements needs greater attention. (5) The basis set parameters and masses are presented. (6) First applications for the vibrational bound states of H3+ for two different potentials will be shown. The differences in applications for effective masses based on different sets of excited electronic wave functions will be discussed. At the end, the most important aspects will be summarized.

Figure 1. H3+−nonadiabatic corrections (ALL) to the effective mass μz2 plotted in Jacobi coordinates: R(H−H 2 ), r(H 2 ), θ = 90°. Contributions of the electron mass, Δme, to the proton mass for the effective reduced mass of the diatom μz2(r, vib)(H2/H2+) in H3+ within the vibrational kinetic energy operator in r. Near the energy minimum the value for the contribution of the electron mass to μvib r is 0.3607me. 0 means nuclear mass and +1.0 means atomic mass for two protons in H3+. All 499 excited electronic states contributed to the effective nuclear mass.

Figure 2. H3+−nonadiabatic corrections (EXCL2) to the effective mass μz2 (as in Figure 1), but the electronic coupling of the ground state with the first excited electronic state (EXCL2) is excluded to get the effective mass for vibration in r.

■

METHODS AND RESULTS Perturbative Calculations of Mass Contributions Based on Coupling of Electron Nuclear Motion: Atoms, Diatoms, and Triatoms. In the following, we present the ideas of Herman and Asgharian,13 who provided a way of calculating mass corrections. We explain the strategy for the kinetic energy operator of a single nucleus, which helps us to calculate the mass corrections for a general diatomic or triatomic molecule. The effective mass contribution of one nucleus (e.g., nucleus 1) along its Cartesian motion (X1, Y1, Z1) can be calculated in the following way. We start with the conventional expression for the kinetic energy operator (ℏ= 1) TN1 = −

1 (∇2X1 + ∇Y21 + ∇2Z1) = TX1 + TY1 + TZ1 2m1

Figure 3. H3+−nonadiabatic corrections (EXCL23) to the effective mass μz2 (as in Figure 1), but the electronic coupling of the ground state with the next two excited electronic states (EXCL23) is excluded to get the effective mass for vibration in r.

(1) 7017

DOI: 10.1021/acs.jpca.7b04703 J. Phys. Chem. A 2017, 121, 7016−7030

Article

The Journal of Physical Chemistry A

Figure 7. H3+−nonadiabatic corrections (EXCL23) to the effective mass μx2 (as in Figure 3).

Figure 4. H3+−nonadiabatic corrections (ALL) to the effective mass μx3 (as in Figure 1).

Figure 8. H3+−nonadiabatic corrections (EXCL23) to the effective mass μz3 (as in Figure 3).

Figure 5. H3+−nonadiabatic corrections (EXCL2) to the effective mass μx3 (as in Figure 2).

Figure 9. H3+−nonadiabatic shifts in cm−1 of vibrational frequencies (J = 0, MBB potential) up to dissociation limit using different mass definitions: DD = distance-dependent (EXCL23: first and second excited states excluded); DD-av = distance-dependent, but averaged (see eq 27); VM: constant vibrational mass (Moss); AT = atomic mass.

Figure 6. H3+−nonadiabatic corrections (EXCL23) to the effective mass μx3 (as in Figure 3).

The action of TX1 on a general nonadiabatic electron nuclear wave function ΨNAD of a molecule (X⃗ : set of nuclear coordinates; x⃗: set of electronic coordinates; φλ(e) = electronic (e) wave function of state λ; χλk(n) = nuclear (n) wave function in the electronic state λ) with ΨNAD(X⃗ , x ⃗) =

∑ Cλkφλ(e)(x ⃗ ; X⃗ )χkλ (n)(X⃗ ) λk

TX1ΨNAD(X⃗ , x ⃗) = −

1 2m1

∑ Cλk(φλ(x ⃗ ; X⃗ )(∇2X χkλ (X⃗ )) 1

λk

+ χkλ (X⃗ )(∇2X1φλ(x ⃗ ; X⃗ )) (2)

+ 2(∇X1φλ(x ⃗ ; X⃗ ))(∇X1χkλ (X⃗ )))

is 7018

(3)

DOI: 10.1021/acs.jpca.7b04703 J. Phys. Chem. A 2017, 121, 7016−7030

Article

The Journal of Physical Chemistry A

Figure 12. H3+−nonadiabatic shifts in cm−1 of vibrational frequencies (J = 0, PAVA-potential: electronic, adiabatic, and relativistic contributions (EAR)) up to dissociation limit using different mass contributions: ALL: all excited states up to n = 500, EXCL2: state 2 = first excited state excluded, EXCL23: first and second excited states excluded. For further details see the text.

Figure 10. H3+−numerical deviations (in cm−1) from correct degeneracy for vibrational frequencies of E′-symmetry (MBB potential) for a simplified Hamiltonian with the same (averaged) mass corrections for distance-dependent μ2 and μ3. See eq 27.

are interested only in lowest order effects of order me/m1, we obtain ΔEλ(2) k (X1) = −

1 m12

∑

⟨λk|∇eX1λ′∇nX1(∑k ′ |k′⟩⟨k′|)⟨λ′|∇eX1λ∇nX1k⟩ (Eλ ′ − Eλ)

λ ′≠ λ

(6)

Using the closure rule in summation on k′, changing the order of summation and integration, we get for the three Cartesian degrees of a single nucleus ΔEλ(2) k = −

D λ(X1) 1 = m1 m1

λ ′ k ′≠ λk

∑ λ ′≠ λ

⟨λ|∇X1|λ′⟩⟨λ′|∇X1|λ⟩ (Eλ ′ − Eλ)

(8)

the energy contribution (followed from the sum over states formula) from Tmix(X1) results from an additional kinetic energy operator for the adiabatic electronic state λ

The superscripts indicate that the ∇X1 operators apply only to the electronic (e) or nuclear (n) wave functions. The derivative operator for the nuclear motion is represented in Cartesian form, so no approximation with respect to the later use of internal coordinates within the nuclear motion problem is included. Following Herman and Asgharian,13 the energy contribution from Tmix(X1) in second-order perturbation theory for the state λk is defined as

∑

λ ′≠ λ i = X1, Y1, Z1

⟨λ|∇ie |λ′⟩∇in ⟨λ′|∇ie |λ⟩∇in k (Eλ ′ − Eλ)

In the derivation of ΔE(2) λk we neglected in the moment the smaller terms like ∇i ⟨λ′|∇ei |λ⟩ and ⟨λ′|∇ei |λ⟩⟨λ′|∇ej |λ⟩, i ≠ j. Introducing the functions Dλ(X1)

The largest contribution of nonadiabatic effects stems from the mixed derivative (third contribution in the sum). We write this as 1 − ∇eX1∇nX1 = T mix(X1) m1 (4)

1 m12

∑ ∑

k

(7)

Figure 11. H3+−nonadiabatic shifts in cm−1 of vibrational frequencies (J = 0, PAVA-potential: electronic (E) versus electronic, adiabatic, and relativistic contributions (EAR)) up to dissociation limit: EXCL23: first and second excited states excluded. For further details see the text.

ΔEλ(2) k (X1) = −

1 m12

T Xeff,1 λ = −

λ 1 D (X1) 2 ∇X1 m1 m1

(9)

χλk(X⃗ ).

acts only on the nuclear wave function The final total kinetic energy operator for nucleus 1 and coordinate X1 for the adiabatic electronic state λ is Teff,λ X1

λ TXnew, =− 1

⟨λk|∇eX1λ′∇nX1k′⟩⟨λ′k′|∇eX1λ∇nX1k⟩

2D λ(X1) ⎞ 2 1 ⎛ 1 ⎜⎜1 + ⎟⎟∇X1 = − ∇2X 2m1 ⎝ m1 ⎠ 2m1X1 1

(10)

X m1 1

with = m1 + Δm1, where Δm1 is an effective mass correction that is position- and orientation-dependent. Now X m1 1 can be calculated for every isotope, where only the constant mass m1 has to be exchanged. Because the mass contributions are calculated in Cartesian coordinates, contributions for the

(Eλ ′ k ′ − Eλk ) (5)

By neglecting the difference in the molecular nuclear energies against the electronic energies in the denominator, that is, we 7019

DOI: 10.1021/acs.jpca.7b04703 J. Phys. Chem. A 2017, 121, 7016−7030

Article

The Journal of Physical Chemistry A Table 1. Nuclear and Distance-Dependent Masses: J = 0 Vibrational Band Origins in cm−1a state

nuclear

EXCL23

E′ A′1 A′1 E′ E′ A′1 E′ A′1 A′2 A′1 E′ E′ A′1 E′ A′1 E′ A′1 E′ A′2 A′1 E′ E′ A′1 E′ A′2 E′ A′1 E′ A′1 E′ A′1 E′ A′1 E′ A′2 A′1 E′ E′ A′1 E′ E′ A′1 A′2 E′ A′1 E′ E′ A′2 A′1 E′ E′ A′1 A′1 A′1 E′ A′1 A′2 E′ E′ A′2 E′

2521.595 3178.671 4778.680 4998.469 5554.844 6262.863 7006.721 7286.343 7493.650 7770.107 7871.105 8488.967 9002.479 9113.954 9252.977 9654.735 9969.975 9998.297 10211.457 10594.355 10646.466 10863.752 10924.355 11324.363 11530.464 11659.476 11815.732 12080.641 12147.829 12304.511 12383.126 12478.560 12591.791 12698.756 12833.539 13290.295 13319.586 13396.510 13406.512 13593.546 13703.702 13726.560 13757.969 14056.510 14199.899 14219.462 14479.610 14567.839 14667.732 14892.034 14903.122 14911.130 14941.787 15081.675 15124.078 15169.773 15192.189 15216.286 15337.464 15375.267 15792.617

2521.392 3178.399 4778.318 4998.068 5554.392 6262.334 7006.216 7285.816 7493.057 7769.489 7870.492 8488.283 9001.845 9113.326 9252.205 9654.020 9969.266 9997.524 10210.678 10593.535 10645.656 10863.081 10923.673 11323.465 11529.624 11658.668 11814.882 12079.800 12146.830 12303.704 12382.469 12477.764 12590.913 12697.814 12832.591 13289.321 13318.601 13395.588 13405.625 13592.651 13702.988 13725.795 13757.032 14055.423 14198.928 14218.452 14478.653 14566.744 14666.638 14890.918 14902.251 14910.258 14940.589 15080.643 15123.138 15168.748 15191.173 15215.237 15336.412 15374.155 15791.585

Δ

EXCl2

(0.003)

2521.390 3178.396 4778.304 4998.058 5554.387 6262.328 7006.203 7285.784 7493.018 7769.473 7870.472 8488.275 9001.823 9113.307 9252.195 9653.989 9969.228 9997.508 10210.622 10593.511 10645.628 10863.056 10923.630 11323.451 11529.593 11658.636 11814.854 12079.746 12146.815 12303.677 12382.422 12477.734 12590.863 12697.789 12832.519 13289.284 13318.565 13395.548 13405.582 13592.607 13702.945 13725.736 13756.990 14055.400 14198.878 14218.414 14478.589 14566.707 14666.602 14890.885 14902.199 14910.190 14940.564 15080.591 15123.085 15168.676 15191.118 15215.193 15336.349 15374.075 15791.527

(−0.016) (−0.007) (0.026)

(−0.018) (−0.019) (0.003) (−0.002) (0.006)

(−0.018) (0.048) (−0.032) (0.002) (0.031) (0.004) (0.035) (0.012)

(−0.013) (−0.028) (−0.007) (0.108)

(−0.040) (0.013) (0.022)

(−0.065) (0.090)

(0.113)

(−0.027) (−0.005) (0.040) 7020

Δ (0.019)

(0.014) (0.027) (0.025)

(0.030) (0.033) (0.008) (−0.010) (0.035)

(0.041) (0.048) (0.036) (0.005) (−0.002) (0.029) (0.042) (0.044)

(0.045) (−0.028) (0.001) (0.100)

(0.036) (0.021) (−0.004)

(0.004) (0.072)

(0.112)

(−0.004) (−0.036) (0.039)

ALL 2521.352 3178.365 4778.183 4997.950 5554.319 6262.266 7006.039 7285.566 7492.688 7769.304 7870.301 8488.172 9001.510 9113.084 9252.101 9653.678 9968.960 9997.246 10210.231 10593.281 10645.399 10862.778 10923.244 11323.310 11529.068 11658.258 11814.486 12079.330 12146.686 12303.312 12381.999 12477.390 12590.533 12697.482 12832.062 13288.959 13318.277 13395.050 13405.141 13592.079 13702.559 13725.180 13756.420 14055.211 14198.353 14217.950 14478.174 14566.006 14666.028 14890.397 14901.785 14909.700 14940.379 15079.976 15122.547 15168.134 15190.396 15214.703 15335.645 15373.506 15790.911

Δ (0.085)

(0.130) (0.143) (0.077)

(0.158) (0.207) (0.093) (0.028) (0.176)

(0.187) (0.163) (0.276) (0.017) (−0.029) (0.263) (0.233) (0.141)

(0.206) (0.047) (0.045) (0.371)

(0.337) (0.101) (0.046)

(0.232) (0.296)

(0.397)

(−0.003) (−0.146) (0.119)

DOI: 10.1021/acs.jpca.7b04703 J. Phys. Chem. A 2017, 121, 7016−7030

Article

The Journal of Physical Chemistry A Table 1. continued state

nuclear

EXCL23

A′1 A1′ A′2 A′1 A′2 E′ A′1 E′ E′ A′1 E′ A′2 E′ A′1 E′ E′ E′ A′1 A′2 E′ A′1 E′ A′1 E′ A′1 E′ A′1 A′2 E′ A′1 A′2 A′2 E′ E′ E′ A′1 A′2 E′ A′1 E′ A′1 A′1 E′ E′ E′ A′1 A′2 E′ E′ E′ A′1 A′2 E′ A′1 E′ A′2 E′ E′ A′1 E′ A′2

15879.431 15889.885 15889.885 15926.580 15971.092 16020.385 16220.984 16266.954 16456.787 16460.623 16573.864 16603.131 16678.215 16717.408 16734.923 16875.167 16918.891 17082.528 17091.747 17237.906 17296.079 17407.388 17445.047 17465.461 17597.375 17625.154 17688.311 17696.453 17706.917 17765.017 17835.507 17865.210 17877.497 17976.848 18239.798 18264.643 18335.706 18362.501 18372.775 18458.912 18479.265 18600.997 18604.214 18734.516 18812.650 18819.861 18897.141 18944.993 19063.247 19102.595 19127.811 19204.175 19227.297 19289.406 19292.144 19294.913 19313.678 19437.139 19442.553 19536.252 19759.903

15878.311 15888.751 15888.762 15925.495 15970.060 16019.293 16219.963 16265.981 16455.666 16459.501 16572.814 16602.145 16676.993 16716.133 16733.701 16874.033 16917.651 17081.385 17090.538 17236.718 17294.882 17406.234 17443.868 17464.292 17596.130 17624.001 17687.034 17695.317 17705.702 17763.745 17834.331 17864.029 17876.342 17975.639 18238.597 18263.481 18334.472 18361.236 18371.438 18457.461 18477.770 18599.674 18603.020 18733.292 18811.356 18818.626 18895.943 18943.782 19061.951 19101.254 19126.425 19202.827 19225.903 19288.120 19290.734 19293.675 19312.321 19435.825 19441.271 19534.874 19758.557

Δ

EXCl2 15878.262 15888.701 15888.663 15925.442 15969.994 16019.249 16219.882 16265.918 16455.588 16459.428 16572.732 16602.099 16676.950 16716.070 16733.649 16873.963 16917.610 17081.315 17090.475 17236.654 17294.813 17406.171 17443.800 17464.219 17596.060 17623.929 17686.982 17695.254 17705.638 17763.673 17834.252 17863.960 17876.250 17975.581 18238.499 18263.381 18334.398 18361.170 18371.383 18457.395 18477.691 18599.605 18602.925 18733.214 18811.289 18818.537 18895.872 18943.680 19061.876 19101.178 19126.323 19202.749 19225.836 19288.020 19290.655 19293.592 19312.259 19435.761 19441.188 19534.800 19758.472

(−0.006) (0.086) (−0.001) (−0.058) (−0.034) (0.102) (0.016) (−0.022)

(−0.041) (0.035) (−0.086) (0.096)

(0.029)

(−0.053) (−0.003) (0.063)

(0.005) (0.114)

(0.000) (0.031) (−0.021)

(0.036) (−0.024) (0.065)

(−0.032) (−0.038) (0.005) (0.013) (0.003)

7021

Δ

(0.008) (0.097) (−0.024) (−0.090) (0.015) (0.119) (0.010) (−0.014)

(−0.012) (0.067) (−0.083) (0.104)

(0.029)

(−0.101) (0.044) (0.013)

(0.035) (0.135)

(−0.024) (0.066) (−0.025)

(0.010) (−0.017) (0.076)

(−0.019) (−0.048) (0.028) (0.058) (0.026)

ALL 15877.800 15888.315 15888.110 15924.985 15969.309 16018.681 16219.220 16265.466 16454.829 16458.782 16571.938 16601.283 16676.577 16715.163 16732.961 16873.447 16916.867 17080.708 17089.719 17236.050 17294.097 17405.669 17443.220 17463.384 17595.564 17623.323 17686.553 17694.458 17705.103 17762.725 17833.452 17863.004 17875.070 17974.786 18237.883 18262.764 18333.492 18360.673 18370.723 18456.534 18476.757 18598.893 18602.163 18732.642 18810.261 18817.796 18894.841 18942.958 19061.097 19100.296 19125.365 19201.662 19225.075 19287.220 19289.532 19292.639 19311.610 19434.970 19440.172 19533.921 19757.430

Δ

(0.181) (0.412) (−0.127) (−0.126) (0.289) (0.459) (0.111) (0.055)

(0.118) (0.325) (−0.077) (0.395)

(0.139)

(−0.431) (0.179) (0.126)

(0.225) (0.539)

(0.047) (0.352) (−0.056)

(0.118) (0.124) (0.272)

(0.106) (−0.123) (0.154) (0.177) (0.138)

DOI: 10.1021/acs.jpca.7b04703 J. Phys. Chem. A 2017, 121, 7016−7030

Article

The Journal of Physical Chemistry A Table 1. continued state

nuclear

EXCL23

A′1 E′ A′1 E′ E′

19782.121 19787.380 19826.087 19850.790 19898.593

19780.793 19786.078 19824.679 19849.408 19897.346

Δ

Δ

EXCl2 19780.709 19785.979 19824.596 19849.335 19897.256

(0.044) (−0.098) (0.046)

Δ

ALL

(0.040) (−0.086) (0.058)

19780.083 19785.345 19823.818 19848.405 19896.305

(0.232) (−0.053) (0.146)

a

For E′ states the numerical lifting of degeneracy is given (PAVA-EAR: BO potential and adiabatic corrections from ref 28 and relativistic corrections from ref 29).

In the case of a triatomic molecule, we have to calculate for the three nuclei nine Cartesian contributions for a given nuclear arrangement. Subtracting the contributions of the center-ofmass and representing the kinetic energy in Jacobi coordinates we end up with an internal kinetic energy expression in Cartesian coordinates of Jacobi vectors r ⃗ = (rx, ry, rz) and R⃗ = (Rx, Ry, Rz) (with r ⃗ oriented on the z-axis and R⃗ in the (z, x)plane) ⎛ ⎞ ⎛ 2 ∇r2y ∇2R y ∇r2 ⎞ 1 ⎛ ∇2R ∇2R ⎞ 1 ∇r T 3D Ψ = ⎜ − ⎜⎜ xx + y + zz ⎟⎟ − ⎜⎜ xx + y + zz ⎟⎟⎟Ψ ⎜ 2 μ μ2 μ2 ⎠ μ3 μ3 ⎠⎟ 2 ⎝ μ3 ⎝ 2 ⎝ ⎠

(13) 1 μx,y,z 2 (μ x 2

and −1

Figure 13. H3 −numerical deviations (in cm ) from correct degeneracy for vibrational frequencies of E′ symmetry (E = BOPAVA-potential). ALL: all excited states up to n = 500; EXCL2: state 2 = first excited state excluded; EXCL23: first and second excited states excluded. For further details see the text. +

1

1 + m x ) are the reduced masses for the “diatomic” 2 1 1 = m x + m x + m x ) for the “triatomic” based on 3 1 2 couplings in x, y, z-motion. The meaning of mx1, my1, ...

derivative and so on was defined in eq 10. We derived for the first time33 that the new form of the kinetic energy operator using distance-dependent reduced masses for vibration (total angular momentum J = 0) expressed in Jacobi coordinates r, R, θ is Tv ≡ Tav + Tbv

relative motion in molecules are easily calculated for any internal coordinates. In the case of homonuclear diatomic molecules, we introduce the functions Avib(R) (for vibration: the Z axis coincides with the molecular axis) and B r o t (R) (f or ro tat io n) m *m (μR = m 1 + m2 , m1 = m2 ) vib

1 = mx 1 1 ( μx,y,z 3 μ3x

Tva = − −

1 ⎛ ∂2 2 ∂⎞ 1 ⎛ ∂2 2 ∂ ⎞ + ⎟ ⎟− z⎜ 2 + xz ⎜ 2μ2 ⎝ ∂r r ∂r ⎠ 2μ3 ⎝ ∂R2 R ∂R ⎠ 1 ⎛⎜ 1 1 ⎞⎛ ∂ 2 ∂ ⎞ + zx 2 ⎟⎟⎜ 2 + cot θ ⎟ ⎜ x 2 2 ⎝ μ2 r ∂θ ⎠ μ3 R ⎠⎝ ∂θ

(14)

2

A (R ) 2 = m1 μR

∑ λ ′≠ λ

Brot (R ) 2 = m1 μR

and

⟨λ|∇Z |λ′⟩⟨λ′|∇Z |λ⟩ , (Eλ ′ − Eλ)

∑ λ ′≠ λ

⟨λ|∇X |λ′⟩⟨λ′|∇X |λ⟩ (Eλ ′ − Eλ)

⎛ ω yx c ⎞ c ∂2 ⎞ c ∂ ∂ 1⎛c ∂ + 2 + ⎜ 22 + 32 ⎟ cot θ − 4 Tvb = − ⎜ 1 ⎟ ⎝ r R ∂R ∂θ 2 ⎝ r ∂r R ∂R ∂θ ⎠ R ⎠ (15)

(11)

with

with an effective perturbational contribution in the kinetic energy operator for the relative motion of vibration and rotation (without center of mass contributions)

c1 = ω2xz + ω2yz ,

c3 = ω3yx − ω3xz sin 2 θ ,

Avib(R ) ⎞ 2 1 ⎡⎛ ⎢⎜1 + ⎟∇Z Teff (diatomic) = − 2μR ⎢⎣⎝ m1 ⎠ ⎤ ⎛ Brot (R ) ⎞ 2 + ⎜1 + ⎟(∇X + ∇Y2 )⎥ ⎥⎦ m1 ⎠ ⎝

c 2 = ω3yx + ω3xz(3 cos2 θ − 1),

1 sin 2 θ cos2 θ = + , μ3xz μ3x μ3z

c4 = ω3zx sin 2θ

(16)

1 sin 2 θ cos2 θ = + μ3zx μ3z μ3x (17)

(12)

ωnαβ = (μnα )−1 − (μnβ )−1 ,

Here the index λ refers to the electronic ground state and is omitted for A and B. Applications for H2+ and H2 are presented in ref 12. As mentioned before for a single nucleus, the mass corrections for diatomic and triatomic molecules will not include the moment couplings of different gradient components.

n = 2, 3,

α, β = x, y, z (18)

Note that in eq 14 we have four different reduced masses instead of two in the case of conventional constant masses. 1 After the substitution Ψ → rR Ψ (to get rid of r2R2 volume elements), the operator Tav transforms into a new expression without first derivatives 7022

DOI: 10.1021/acs.jpca.7b04703 J. Phys. Chem. A 2017, 121, 7016−7030

Article

The Journal of Physical Chemistry A

Table 2. Nuclear, Vibrational (VM: Moss43) Mass, and Distance-Dependent Masses (with Averaged Frequencies for E′ States): J = 0 Vibrational Band Origins in cm−1a,b ν1

ν2

l

Γ

nuclear

VMc

Δd

EXCL23

Δd

Δd(EXCl2)

Δd(ALL)

0 1 0 0 1 2 0 0 0 1 1 2 0 0 3 1 1 0 1 2 2 0 0 3 0 0 1 1 4 2 0 0 2 1 2 3 3 1 1 0 0 0 0 1 4 2 1 2 0 1 1 3 0 5 0 0 3 0 2 1

1 0 2 2 1 0 3 3 3 2 2 1 4 4 0 3 3 4 3 2 2 5 5 1 5 5 4 4 0 3 6 6 3 4 3 2 2 5 5 6 7 7 7 5 1 4 5 4 7 6 6 3 8 0 6 8 3 6 4 6

1 0 0 2 1 0 1 3 3 0 2 1 0 2 0 1 3 4 3 0 2 1 3 1 3 5 0 2 0 1 0 2 3 4 3 0 2 1 3 4 1 3 3 3 1 0 5 2 3 0 2 1 0 0 6 2 3 6 4 4

E′ A1′ A′1 E′ E′ A′1 E′ A′1 A′2 A′1 E′ E′ A′1 E′ A′1 E′ A′1 E′ A′2 A′1 E′ E′ A′1 E′ A′2 E′ A′1 E′ A′1 E′ A′1 E′ A′1 E′ A′2 A′1 E′ E′ A′1 E′ E′ A′1 A′1 A′2 E′ A′1 E′ E′ A′2 A′1 E′ E′ A′1 A′1 A′1 E′ A′1 A′2 E′ E′

2521.595 3178.671 4778.680 4998.469 5554.844 6262.863 7006.721 7286.343 7493.650 7770.107 7871.105 8488.967 9002.479 9113.954 9252.977 9654.735 9969.975 9998.297 10211.457 10594.355 10646.466 10863.752 10924.355 11324.363 11530.464 11659.476 11815.732 12080.641 12147.829 12304.511 12383.126 12478.560 12591.791 12698.756 12833.539 13290.295 13319.586 13396.510 13406.512 13593.546 13703.702 13726.560 13726.560 13757.969 14056.510 14199.899 14219.462 14479.610 14567.839 14667.732 14892.034 14903.122 14911.130 14941.787 15081.675 15124.078 15169.773 15192.189 15216.286 15337.464

2521.301 3178.292 4778.155 4997.896 5554.206 6262.131 7005.974 7285.563 7492.786 7769.230 7870.230 8488.013 9001.567 9113.040 9251.915 9653.701 9968.944 9997.183 10210.330 10593.195 10645.318 10862.749 10923.361 11323.120 11529.245 11658.312 11814.525 12079.402 12146.464 12303.338 12382.154 12477.397 12590.530 12697.406 12832.175 13288.917 13318.198 13395.205 13405.251 13592.256 13702.588 13725.432 13725.432 13756.613 14055.011 14198.517 14218.020 14478.222 14566.281 14666.180 14890.479 14901.822 14909.870 14940.156 15080.186 15122.646 15168.303 15190.714 15214.805 15335.963

(−0.294) (−0.379) (−0.525) (−0.573) (−0.638) (−0.732) (−0.747) (−0.780) (−0.864) (−0.877) (−0.875) (−0.954) (−0.912) (−0.914) (−1.062) (−1.034) (−1.031) (−1.114) (−1.127) (−1.160) (−1.148) (−1.003) (−0.994) (−1.243) (−1.219) (−1.164) (−1.207) (−1.239) (−1.365) (−1.173) (−0.972) (−1.163) (−1.261) (−1.350) (−1.364) (−1.378) (−1.388) (−1.305) (−1.261) (−1.290) (−1.114) (−1.128) (−1.128) (−1.356) (−1.499) (−1.382) (−1.442) (−1.388) (−1.558) (−1.552) (−1.555) (−1.300) (−1.260) (−1.631) (−1.489) (−1.432) (−1.470) (−1.475) (−1.481) (−1.501)

2521.390 3178.399 4778.318 4998.076 5554.396 6262.334 7006.203 7285.816 7493.057 7769.489 7870.501 8488.292 9001.845 9113.324 9252.205 9654.021 9969.266 9997.521 10210.678 10593.535 10645.665 10863.057 10923.673 11323.481 11529.624 11658.667 11814.882 12079.784 12146.830 12303.702 12382.469 12477.746 12590.913 12697.808 12832.591 13289.321 13318.608 13395.602 13405.625 13592.655 13702.934 13725.795 13725.795 13757.032 14055.443 14198.928 14218.445 14478.642 14566.744 14666.638 14890.950 14902.206 14910.258 14940.589 15080.643 15123.082 15168.748 15191.173 15215.250 15336.414

(−0.204) (−0.272) (−0.362) (−0.393) (−0.448) (−0.529) (−0.518) (−0.527) (−0.593) (−0.618) (−0.604) (−0.675) (−0.634) (−0.630) (−0.772) (−0.714) (−0.709) (−0.776) (−0.779) (−0.820) (−0.801) (−0.695) (−0.682) (−0.882) (−0.840) (−0.809) (−0.850) (−0.856) (−0.999) (−0.809) (−0.657) (−0.814) (−0.878) (−0.948) (−0.948) (−0.974) (−0.978) (−0.908) (−0.887) (−0.891) (−0.768) (−0.765) (−0.765) (−0.937) (−1.067) (−0.971) (−1.016) (−0.968) (−1.095) (−1.094) (−1.084) (−0.916) (−0.872) (−1.198) (−1.032) (−0.996) (−1.025) (−1.016) (−1.036) (−1.050)

(−0.214) (−0.275) (−0.376) (−0.418) (−0.471) (−0.535) (−0.530) (−0.559) (−0.632) (−0.634) (−0.648) (−0.709) (−0.656) (−0.651) (−0.782) (−0.741) (−0.747) (−0.807) (−0.835) (−0.844) (−0.859) (−0.720) (−0.725) (−0.930) (−0.871) (−0.843) (−0.878) (−0.894) (−1.014) (−0.849) (−0.704) (−0.847) (−0.928) (−0.989) (−1.020) (−1.011) (−1.043) (−0.948) (−0.930) (−0.940) (−0.807) (−0.824) (−0.824) (−0.979) (−1.128) (−1.021) (−1.058) (−1.019) (−1.132) (−1.130) (−1.151) (−0.959) (−0.940) (−1.223) (−1.084) (−1.049) (−1.097) (−1.071) (−1.091) (−1.097)

(−0.285) (−0.306) (−0.497) (−0.584) (−0.596) (−0.597) (−0.720) (−0.777) (−0.962) (−0.803) (−0.883) (−0.898) (−0.969) (−0.916) (−0.876) (−1.071) (−1.015) (−1.139) (−1.226) (−1.074) (−1.161) (−1.056) (−1.111) (−1.191) (−1.396) (−1.227) (−1.246) (−1.297) (−1.143) (−1.331) (−1.127) (−1.287) (−1.258) (−1.344) (−1.477) (−1.336) (−1.412) (−1.484) (−1.371) (−1.490) (−1.328) (−1.380) (−1.380) (−1.549) (−1.468) (−1.546) (−1.562) (−1.459) (−1.833) (−1.704) (−1.753) (−1.485) (−1.430) (−1.408) (−1.699) (−1.729) (−1.639) (−1.793) (−1.582) (−1.746)

7023

DOI: 10.1021/acs.jpca.7b04703 J. Phys. Chem. A 2017, 121, 7016−7030

Article

The Journal of Physical Chemistry A Table 2. continued ν1

ν2

l

Γ

nuclear

VMc

Δd

EXCL23

Δd

Δd(EXCl2)

Δd(ALL)

3 2 4 4 2 2

3 5 2 2 5 5

3 1 0 2 3 3

A′2 E′ A′1 E′ A′1 A′2

15375.267 15792.617 15879.431 15889.885 15926.580 15971.092

15373.681 15791.117 15877.858 15888.289 15925.047 15969.601

(−1.586) (−1.500) (−1.573) (−1.596) (−1.533) (−1.491)

15374.155 15791.565 15878.311 15888.756 15925.495 15970.060

(−1.112) (−1.052) (−1.120) (−1.129) (−1.085) (−1.032)

(−1.192) (−1.110) (−1.169) (−1.203) (−1.138) (−1.098)

(−1.761) (−1.765) (−1.631) (−1.672) (−1.595) (−1.783)

a

PAVA-EAR: BO potential and adiabatic corrections from ref 28 and relativistic corrections from ref 29. The assignment is taken from Table IV in ref 28. bALL, EXCL2, EXCL23: see explanation in the text. cPresent results, identical to ref 28. dΔ = ν−ν(nuclear).

Tva = −

1 ∂2 1 ∂2 1 ⎡⎢ 1 1 ⎤⎥ − − + z xz 2μ2 ∂r 2 2μ3 ∂R2 2 ⎢⎣ μ2x r 2 μ3zx R2 ⎥⎦

⎡ ∂2 ∂ ⎤ ⎢ 2 + cot θ ⎥ ∂θ ⎦ ⎣ ∂θ

c ∂ c c ⎞ 1 ⎛ c1 ∂ ⎜ + 2 − 12 − 22 ⎟δk ′ kδj ′ j ⎝ 2 r ∂r R ∂R r R ⎠ xz xz ⎛ω ω cos 2θ ⎞ 1 + j(j + 1)⎜ 22 + 3 2 ⎟δk ′ kδj ′ j ⎝ r ⎠ 2 R xz 2 ⎛ xz ω cos 2θ ⎞ k ω −2 − δk ′ k ⎜ 22 + 3 2 ⎟⟨j′k| sin θ|jk⟩ ⎠ 2⎝ r R

⟨Tvb⟩ = −

(19)

and operator Tbv transforms into ⎛ ω yx c ⎞ c ∂2 c ∂ ⎞ 1⎛ ∂ Tvb = − ⎜W + ⎜ 22 + 32 ⎟ cot θ − 4 + 42 ⎟ ⎝ r 2⎝ ∂θ R ∂R ∂θ R ⎠ R ∂θ ⎠ (20)

W=

c1 ∂ c ∂ c c + 2 − 12 − 22 r ∂r R ∂R r R

yx c ⎞ ∂ 1⎛ω − δk ′ k ⎜ 22 + 32 ⎟⟨j′k| cot θ |jk⟩ ∂θ 2⎝ r R ⎠ ω3zx sin 2θ ⎛ 1 ∂ ∂ 1⎞ ⎜− − δk ′ k + 2 ⎟⟨j′k| |jk⟩ ⎝ R ∂R ∂θ 2 R ⎠

(21)

(25)

Tbv

In the conventional approach with constant masses would not exist. We reformulate Tv as the sum Tv = T̅ av + T̅ bv to get a simplified form with identical mass-prefactors for the radial and angular momentum part (μz2 for r, μxz 3 for R) Tv̅ a = −

⟨Tav ⟩

is similar to the kinetic energy expression for constant masses assuming different effective vibrational (and rotational masses)31 ⎡ 1 ∂2 1 ∂2 ̂ − Tvib = δk , k ′δj , j ′⎢⎢ − vib 2 2 r vib 2μ3R ∂R ⎣ 2μ2 ∂r

1 ∂2 1 ∂2 1 ⎡⎢ 1 1 ⎤⎥ − − + z xz 2μ2 ∂r 2 2μ3 ∂R2 2 ⎢⎣ μ2z r 2 μ3xz R2 ⎥⎦

⎡ ∂2 ∂ ⎤ ⎢ 2 + cot θ ⎥ ∂θ ⎦ ⎣ ∂θ

⎛ ⎞⎤ 1 1 ⎟⎥ + j(j + 1)⎜ + vib ⎜ r vib 2 ⎟⎥ 2μ3R R2 ⎠⎦ ⎝ 2μ2 r

(22)

⎞ ⎛ 1 1 ⎟ − δk , k ′k 2⟨j′k|sin−2 θ|jk⟩⎜ + vib ⎟ ⎜ r vib 2 2μ3R R2 ⎠ ⎝ 2μ2 r

yx c ⎞ 1 ⎡⎛ ω ∂ Tv̅ b = − ⎢⎜ 22 + 32 ⎟ cot θ ⎝ ⎠ 2⎣ r ∂θ R

⎛ ω xz ω xz cos 2θ ⎞⎛ ∂ 2 ∂ ⎞⎤ + ⎜ 22 + 3 2 ⎟⎜ 2 + cot θ ⎟⎥ ⎝ r ⎠⎝ ∂θ ∂θ ⎠⎦ R c ∂ c c ⎞ 1 ⎛ c1 ∂ ⎜ + 2 − 12 − 22 ⎟ 2 ⎝ r ∂r R ∂R r R ⎠ ⎛ ⎞ c c 1 ∂ ∂ − ⎜− 4 + 42 ⎟ 2 ⎝ R ∂R R ⎠ ∂θ

(26)

A detailed derivation of the former expressions for distancedependent masses is presented in ref 33. For test reasons we used an even more simplified, symmetrized form

−

sym =− Tvib

(23)

Assuming that the prefactors of differential operators can be αα treated as constants (like c1, ωyx 2 , μ2,3′, etc.), we can derive a more compact effective kinetic energy expression: after the integration of the expression for Tv over Euler angles (using Wigner rotation functions) and the θ angle (using associated Legendre functions Pkj (cos θ)), we obtain the effective ⟨Tv⟩

⎛ ⎞ 1 1 ⎟⎟ + j(j + 1)⎜⎜ + 2 2μ3 (r , R , θ )R2 ⎠ ⎝ 2μ2 (r , R , θ )r (27)

where a common distance-dependent (DD) mass-correction factor (compared to a constant mass (CoM)) was used for μ2 and μ3

⟨Tva⟩ = ⟨kj|Tva|k′j′⟩ ⎤ ⎡ ⎛ 1 1 1 ∂2 1 ∂2 1 ⎞ = − ⎢ z 2 + xz 2 − j(j + 1)⎜⎜ z 2 + xz 2 ⎟⎟⎥δk ′ kδj ′ j μ3 ∂R 2 ⎢⎣ μ2 ∂r μ3 R ⎠⎥⎦ ⎝ μ2 r −

k 2 ⎛⎜ 1 1 ⎞⎟ −2 + ⎜ ⎟⟨j′k|sin θ|jk⟩δk ′ k 2 ⎝ μ2z r 2 μ3xz R2 ⎠

1 1 1 1 ∂2 ∂2 − 2 μ2 (r , R , θ ) ∂r 2 2 μ3 (r , R , θ ) ∂R2

μ2DD = μ2CoM + μ3DD = μ3CoM

(24) 7024

1 Δm(r , R , θ ), 2 2 + Δm(r , R , θ ) 3 DOI: 10.1021/acs.jpca.7b04703 J. Phys. Chem. A 2017, 121, 7016−7030

Article

The Journal of Physical Chemistry A Δm (r, R, θ) represents an average mass correction for the rz and Rz, Rx motion

perturbative approach for the mass correction does not represent an adequate description. Distance-Dependent Masses for H3+. The six different distance-dependent mass corrections μx2, μy2, μz2, μx3, μy3, and μz3 have been fitted individually by B-splines,6 and from them other needed combinations are produced. A polynomial fit (with hyperspherical coordinates) for an extended geometrical region, as performed for the potential or the diagonal adiabatic corrections to guarantee the D3h symmetry,25 had difficulties in reproducing the input data with sufficient accuracy. In Figures 1−8 we present some typical mass corrections (for θ = 0 and 90°) that have the largest impact on the vibrational states of H3+. As said before, the molecule is defined in the z−xplane, with r fixed to the z-direction. As presented in eqs 19 and zx 20, μz2, μx2, μxz 3 , and μ3 show the largest distance-dependent behavior (large values for Δme ≫1.0 are cut off). In the two cases μz2 and μx3 (both for θ = 90; because R lies in the z−x zx plane, μx3 contributes to the expressions for μxz 3 and μ3 (see eq 17)), we show the differences of mass contributions including the coupling of the ground state to all excited states (ALL), without the coupling of the electronic ground state with the first (EXCL2), and first and second excited (EXCL23) electronic states. Near the potential minimum (r = 1.65, R = 1.43 Bohr, θ = 90) the mass corrections are similar for the cases “ALL”, EXCl2, or EXCL23. At larger R values the coupling of the first and excited states with the ground states strongly increases and becomes singular. The smoothest effective mass surfaces are obtained by neglecting the contributions of the two first excited states. Because we are mostly interested in the influence of the effective masses on vibrational states near the potential minimum, there should be only a small influence by the excluded excited states (states 2 and 3). Near the potential minimum region H3+ has an equilateral triangular form: θ = 90, r in z-direction, and R in x-direction. The largest influence on the vibrational eigenstates comes from μz2 of the vibrational kinetic energy operator for r (diatomic) xz and μx3 (within μzx 3 and μ3 ) of the vibrational kinetic energy operator for R at θ = 90. At R = 1.43, r = 1.65 Bohr, the electron contributes ∼0.36me to the mass of the two nuclei for x μz2 ≈ μvib r , and the mass contribution for the three nuclei for μ3 vib ≈ μR is also ∼0.36 me. For all other geometries different from equidistant triangular form, the values for μz2 and μx3 are different (compare Figure 1 with Figure 4). The plots of the mass contributions show a complicated behavior for larger R values: The mass contributions result from coupling of the electronic ground state with all excited states (in the present work: 499 excited states). At larger R values the electronic ground state wave function changes its behavior, representing in some special geometrical regions the arrangement H2+ + H, which has a totally different mass correction than the wave function representing the arrangement H2 + H+. Therefore, (compare Figure 1 to Figure 3), μz2 (from r small to large) increases to values representing μ2(H2) as long as r is shorter than 2.5 Bohr. Beyond r = 2.5 Bohr the mass contributions represent μ2(H2+). If one excludes the coupling of the ground state with the first excited state then the mass contributions stay finite. This behavior is clearly not representable by a mass analysis based on Mullikan population analysis.26 Whereas the change in the electronic wave function (representing H2 + H+ or H2+ + H) is seen easily in the dramatic change of the mass contributions at very large R values, at intermediate regions R > 6 Bohr the changes in mass contributions cannot be easily interpreted by simple pictures

⎛ ⎞ 3 Δm(r , R , θ ) = ⎜2Δmrz + (ΔmR z + ΔmR x )⎟ /5 ⎝ ⎠ 2

Derivative Matrix Elements. The derivative matrix elements, needed for the computation of the distancedependent masses, are calculated within a coupled-perturbed full-CI (CP-FCI) method34−36 using eqs 7 and 8 for one nucleus or eq 11 for the two nuclei in a diatomic molecule or similar formulations for three nuclei in a triatomic molecule. We calculated the coupling of 500 electronic states, which seems to provide converged contributions of mass corrections at least for the electronic ground state. This program package is the same that is used for all nuclear Cartesian derivatives of the electronic wave function needed to calculate all adiabatic and nonadiabatic (first and second derivatives) corrections. As said before, in the case of diatomics where we have only one internal nuclear degree of freedom, nuclear derivatives of the electronic wave function can be easily represented by the Cartesian derivatives. In the case of triatomics, like H3+, derivatives with respect to the Jacobi internal coordinates are represented by linear combinations of the Cartesian derivatives. Other internal nuclear coordinates would rely on other linear combination of the same Cartesian derivatives; recalculation of the electronic matrix elements is not needed. We tested different basis sets and finally used the (7s3p1d)/ [5s3p1d] basis set of Augspurger and Dykstra.37 We reached convergence for the A and B factors (see eq 11 for diatomics, i.e., H2) using the matrix elements of the type ⟨λ|∇Z|λ′⟩, when adding up the contributions to a maximum of 499 electronically excited states. Convergence tests have been performed for different geometries using up to 999 excited states. To check whether for H3+ the correction factors for the masses are converged, we calculated at R(H−H2) = 100 Bohr, that is, for the asymptotic arrangement H+ + H2 or H + H2+, the factors A(r) and B(r) that are needed to describe correctly distancedependent masses for H2 and H2+. With these mass corrections the rovibrational eigenstates of the diatomics are in agreement within 0.01 to 0.001 cm−1 compared with the most accurate theoretical data in the literature.8,10,12 We hope that the mass corrections are of similar quality for H3+ at all nuclear arrangements. In the case of H2, nearly half of the mass correction factor is caused by electronic wave functions related to electronic bound states; the rest comes from wave functions describing electronic continuum states. This can be seen as a disadvantage of the sum-over-states formulation given in eqs 5−7. If the basis set is too compact but still good enough for the description of the ground-state energy, then the mass correction factors are too small because there are not sufficient appropriate continuum functions available. A diffuse basis set will lead to a very large set of adequate continuum functions, but one does not know in advance which excited states are needed for the different nuclear Cartesian derivatives. Symmetry selection of the excited states would be helpful but has not been implemented up to now. In the case of H3+ most of the excited electronic states are repulsive and will result in small mass correction factors. This is different from H2, where the effective mass correction can be larger than an electron mass, resulting from relatively strong bound excited states. Large mass corrections for H3+ do only occur near the avoided crossing at large R and r ≈ 2.5 Bohr: In that case, the 7025

DOI: 10.1021/acs.jpca.7b04703 J. Phys. Chem. A 2017, 121, 7016−7030

Article

The Journal of Physical Chemistry A describing the arrangements H2 + H+ or H2+ + H. This will be analyzed in more detail in a future work. As long as we are investigating vibrational states below the physical dissociation energy (H2(v = 0,j = 0,1) + H+) there is no explicit transition to the next higher electronic state possible. The potential energy near the geometric region of avoided crossing is fairly high (around 35 000−45 000 cm−1) and will hopefully not influence the bound ro-vibrational energies of H3+ near the potential minimum region; the situation is different for resonances. As said before, the use of distancedependent reduced masses needed a completely new code for the calculation of ro-vibrational states because the structure of the DVR-representation for the different kinetic energy operator terms became much more complicated. This will be explained in the next chapter. Vibrational Hamilton Operator, Basis Functions, Matrix Elements, and Matrix-Vector Products. H3+ is a floppy molecule, and because standard perturbative methods38 are not adequate, we choose basis functions and compute eigenvalues of a matrix representation of the Hamiltonian operator. We use the Jacobi coordinates r(H−H), R(H−H2), and the angle θ between the two Jacobi vectors. The body-fixed z axis is along the r vector and the body-fixed x axis is in the plane of the three nuclei. The derivation of the conventional (constant-mass) kinetic energy operator (KEO) is already presented in refs 30 and 31. In the present work, we split the Hamiltonian H in two parts: H = H(CoM) + Tadd(DDM), that is, into the original Hamiltonian with constant masses (CoM) H(CoM) and the distance-dependent mass (DDM) kinetic energy operator Tadd(DDM). Only a rough overview will be given to clarify the important parts of the additional new kinetic energy contributions. The conventional vibrational Hamiltonian30,31 is split into contributions describing a spherical oscillator (SO) and the rest. The Hamiltonian with constant masses H(CoM) is given as (ℏ = 1)

ψ (SO) = Nnje−z 2

z =

1 ∂2 2μ2 ∂r ⎡ 1 ∂ f ⎞ ∂ ⎤ ⎛ sin θ ⎥ + ⎜V − R2⎟ , − B (r )⎢ ⎣ sin θ ∂θ ∂θ ⎦ ⎝ 2 ⎠ 1 2μ2 r 2 (28)

θ);

μ3 R2 (31)

ℏ

1 ∂ zc 1 ∂ (xz)c ∂ ∂ ω2 (r , R , θ) − ω3 (r , R , θ) + ... 2 ∂r 2 ∂R ∂r ∂R 1 1 , ω2zc(r , R , θ) = z − μ2 (r , R , θ) μ2 (CoM) 1 1 ω3(xz)c(r , R , θ) = xz − μ3 (r , R , θ) μ3(CoM)

sym,DD Tvib,add =−

with ⎡ 1 ∂ f ∂ ⎤ 1 ∂2 − B (R )⎢ sin θ ⎥ + R2 , 2 ⎣ sin θ ∂θ ∂θ ⎦ 2μ3 ∂R 2 1 B (R ) = 2μ3 R2 (29)

Hso = −

(32) ∂

∂

The symmetric evaluation of expressions like ∂x G(x) ∂x is described in the work of Light and Carrington.40 Because the wave function within the Lanzcos algorithm is represented in DVR for r and FBR for R, θ, we have now a more complicated sequence of matrix−matrix and matrix−vector products for the product of the Hamiltonian with the wave function. To get a 1 symmetric matrix representation for ω3(xz)c(r , R , θ ) R2 we split

The new contributions to the kinetic energy operator are given as (see eqs 24 and 25) Tadd(DDM) = (⟨Tva⟩ + ⟨Tvb⟩) − T(CoM)

/2 j + 1 j + 1/2 z Ln /2 − j /2(z 2)P jk(cos

j+1/2 where Nnj is a normalization constant, Ln/2−j/2 (z2) is a k generalized Laguerre polynomial, and Pj (cos θ) is an associated Legendre function (k is a quantum number for the projection of the total angular momentum on the body-fixed z-axis). f is a harmonic oscillator force constant. The allowed values of the vibrational quantum numbers are n = j, j + 2, j + 4, ... The eigenvalues of Hso are (n + 3/2) f /μ3 . See ref 30 for more information. Note that on this basis matrix elements of the full Hamiltonian are simple and finite, despite the singularities at θ = 0 and π and the singularity at R = 0. The eigenfunctions of Hso are multiplied by tridiagonal Morse discrete variable representation functions39 for r. The basis functions are parity adapted so that separate calculations for even values of j (and n) and for odd values of j (and n) are possible. The use of a nondirect product spherical oscillator basis is useful because it is appropriate for dealing with the singularity in the KEO.30 It is of key importance that the nondirect product character of the basis does not preclude the efficient evaluation of matrix-vector products, required to use an iterative eigensolver, by sequential summation.30 Nondirect product spherical oscillator functions have important advantages. First, they, compared to direct product spherical oscillator functions, reduce the spectral range of the Hamiltonian matrix and thereby reduce the number of Lanczos iterations required to achieve converged energy levels. Unless a nondirect product basis is used, singularities inflate the spectral range. Second, a nondirect product basis describes as compactly as possible the wave functions near singularities. To solve the eigenvalue problem for all needed eigenstates within the Lanczos approach we prefer to solve a symmetric eigenvalue problem. In the case of H(CoM) there was no problem to provide the Hamiltonian matrix representation for the kinetic energy terms in a symmetric form: The matrix representation for the potential was given in DVR for (r, R, θ), the matrix representation for the kinetic energy of r was also given in DVR, whereas for R, θ we used a finite basis representation (FBR). Details are described in refs 30 and 31. In the case of T(DDM) the situation is a little bit more difficult. The new additional kinetic energy terms in r and R, including distance-dependent masses (see eqs 24 and 25), are now represented in the following way (ℏ = 1,k = 0)

H(CoM) = T(CoM) + V = Hso −

B (r ) =

2

(30)

The exact eigenfunctions of Hso are 7026

DOI: 10.1021/acs.jpca.7b04703 J. Phys. Chem. A 2017, 121, 7016−7030

Article

The Journal of Physical Chemistry A the expression into two pieces: The reason is that 1/R2 is not well represented in DVR but in FBR. Whereas the matrix representation for ∂ of the Morse oscillator functions was

Basis Set Parameters and Masses. The tridiagonal Morse parameters and the spherical oscillator force constant are the same as those of refs 30 and 31. Our standard basis (called basis B1) consists of Nr = 36 DVR basis functions with r points in the range [0.78, 7.8 Bohr]; the Gauss-Laguerre grid of order 1/2 has NR = 95 points between R = 0.08 Bohr and R = 9.5 Bohr, the Gauss-Legendre grid Nθ = 18 points. Basis B1 would be appropriate for J < 40 calculations for H3+ isotopomers.32 In the present work and in the past different constant masses have been used: nuclear (NM), atomic (AM), vibrational (VM), and so on with mH(NM) = 1836.15268277237me, m H ( AM ) = m H ( NM ) + 2/ 3m e , an d m H (V M ) = 1836.62762707857me. In the case of mH(VM) there is a mass shift of ca. 0.475me.28,43 Differences between energies computed using unequal constant vibrational and rotational masses, and their experimental counterparts are expected to be smallest for small total angular momentum J and wave functions concentrated close to the minimum; in comparison with known experimental data (low energy regime), calculations with nuclear masses show too large deviations. On the contrary, for the highest bound states near the dissociation limit atomic masses might be more appropriate. Because now distance-dependent masses are available, one can investigate the quality of different constant mass assumptions for vibration and rotation. The energy conversion factor between atomic units and cm−1 is 219 474.631.44 Results and Discussion. In the present work calculations only for vibrational states with total angular momentum J = 0 had been performed. Rovibrational eigenstates of H3+ and its various isotopologues have been analyzed in recent years by several groups5,18,28,29,46−53 using constant masses. Our recent works were mostly related to rovibrational transitions for energies above the barrier to linearity.24,29 In the present paper variational calculations for two different PESs have been investigated: MBB-PES (Meyer, Botschwina, and Burton54) and PAVA−PES (Pavanello et al.27,28). For the MBB-surface54 we investigate the differences between different constant mass concepts as nuclear, atomic, and vibrational masses based on eq 26 and results based on averaged distancedependent mass corrections (see eq 27). For the PAVA−PES (only electronic contributions: called PAVA-E) based on Gaussian Geminals of Pavanello et al.27,28 (similar studies have been performed with our former BO-PES based on Gaussian Geminals, too25,29), we compared different distancedependent mass corrections within eqs 24 and 25. Similar investigations had been performed by including adiabatic28 and relativistic29 corrections (called PAVA-EAR). What do we expect when we use distance-dependent masses? We expect improvement in the direction to correct experiment values. But as long as not all main contributions to the description of vibrational (and rovibrational) frequencies are taken into account there can still exist a deviation to experiment. An internal test of our calculations is to show how good the degeneracy of the E-states of the homonuclear H3+ is reproduced. For other heteronuclear isotopomers the quality of only rovibrational frequencies compared with experiment can be taken as a reference of quality. Like the diagonal adiabatic corrections per nucleus, the effective mass of each nucleus does not have to be identical at every nuclear arrangement. The global effect of the effective mass of each nucleus integrated over all arrangements should lead to the correct degeneracies of eigenstates (which would be in

∂r

easily derived (see ref 39), the matrix representation of ∂ ∂R needed the introduction of an auxiliary basis of ψ(SO)/z. So, ω(xz)c (r, R, θ) was represented in DVR based on the auxiliary 3 basis: This is different from what was needed for the potential evaluation. The matrix elements needed for the original (constant mass) Hamiltonian are already presented in refs 30 and 31 Here we will introduce the ones needed for representing distancedependent mass kinetic energy terms. The DDM are represented as (with G2,3(r, R, θ) = ωΩ2,3(r, R, θ), Ω = x, y, z, zc, etc.) θ

2

G Ω → δα ′ , α ∑ Tβj ′n ′ ∑ Tγj ′[G Ω((r )α ), (R )β ), θγ )]θ Tγ2jTβj n = (G Ω)nα′ j ′ , nj β

γ

(33) Ω

l i k e (G )r 2 (r , R , θ ) =

Terms

ω2Ω(r ,

R,

1 θ) 2 r

and

(G Ω)R2 (r , R , θ ) = ω3Ω(r , R , θ ) 2 can be treated like G Ω R before (see eq 33) or are represented as a product of two matrices 1

ω3Ω(r , R , θ )

⎛1⎞ 1 → ((G3Ω)nα′ j ′ , nj )† (⎜ 2 ⎟ δ ) 2 ⎝ R ⎠n ′ j ′ , nj j ′ j R

⎛1⎞ 1 Ω ω (r , R , θ ) → (⎜ 2 ⎟ δ )† (G3Ω)nα′ j ′ , nj 2 3 ⎝ R ⎠n ′ j ′ , nj j ′ j R

−

(35)

∂ zc ∂ ω 2 (r , R , θ ) ∂r ∂r →

r

∑ m′

−

(34)

Tα ′ m ′(∑ (r Dm ′ m(G2zc)nα′ j ′ , nj r Tαr mDm ′ m)) m

(36)

∂ (xz)c ∂ ω3 (r , R , θ ) ∂R ∂R →

∑ (R Fn′ n ∑ ((G3(xz)c)nα′ j ′ ,nj R Fn′ n)) n′

(37)

n

with r

Dm ′ m = R

∫ dr φm′(r) ∂∂r φm(r),

F n′n =

∫ dR χn′ (R) ∂∂R χn (R)

(38)

φm(r1) is a tridiagonal Morse basis function. χn(R) is a short notation for the Laguerre function, as described in refs 30 and 31. RTβn, θTkγj, and rTα′m′ are matrix elements of the matrices that, for each coordinate, are used to evaluate the quadratures. We use the Lanczos algorithm41,42 to calculate eigenvalues by evaluating matrix−vector products. Because of the interdependency of the basis indices the order of the sums required to evaluate matrix−vector products is important. All matrix elements except an additional one (because of distancedependent masses) can be evaluated as efficiently as before.30,31 ∂ ∂ In the case of − ∂r ω2zc(r , R , θ ) ∂r = (G2zc)nα′ j ′ , n ′ j ′ we need one summation more in R in the evaluation of the matrix−vector products: This reduces the speed of evaluation by a factor NR (NR = number of grid points or functions for the variable R). 39

7027

DOI: 10.1021/acs.jpca.7b04703 J. Phys. Chem. A 2017, 121, 7016−7030

Article

The Journal of Physical Chemistry A agreement with nuclear permutation inversion symmetry), although for fixed geometries the effective mass of the three nuclei is mostly different. Assuming identical effective masses at each arrangement is a simplified model. Once the individual (2),x,y,z or μx,y,z reduced mass μx,y,z 2 (DD) = μ2(CoM) + Δme 3 (DD) = μ3(CoM) + Δme(3),x,y,z have been calculated (with all contributions of the excited electronic states), the electronic for 2 or 3 nuclei (within μ2 or mass contributions Δm(2,3),x,y,z e μ3) have to be assumed identical. This does not allow extracting absolute information of those effective masses for each single nucleus that contributes to the internal r ⃗ and R⃗ motion. As presented in previous calculations,12 similar absolute information is also not available for H2+ or HD+ and H2 or HD, but the rovibrational frequencies using distance-dependent masses are strongly improved in the direction of experimental frequencies. Now, we do expect the same for H3+ and its isotopomers. Because for the MBB potential a relatively simple potential ansatz exists and frequencies for many different constant masses had already been published30(including informations about resonances55), we show in Figure 9 for the MBB potential nonadiabatic shifts (J = 0) for the vibrational frequencies. We compare results using atomic (AT), vibrational (VM), and distance-dependent (DD) masses relative to calculations with nuclear (NU) masses: ν(DD) − ν(NU). Two versions for the distance-dependent masses had been tested. (1) DD-av is based on a mass-average: Δm1 = Δm2 = Δm3

(

3

nonadiabatic shifts is shown. Detailed values (vibrational band origins and the numerical lifting of degeneracies) are shown in Table 1. Including all excited electronic states (especially the first excited state) for the effective nuclear mass corrections has the strongest influence, especially when the frequencies approach the dissociation limit. Because the mass corrections become large (singular) in some regions of arrangement when the first excited state is included, we numerically fixed the largest correction to 1.2me. To improve the physical description we have to combine the effective nuclear mass approach with coupled surface calculations. In Figure 13 we present deviations from the correct degeneracy of the vibrational frequencies (PAVA-potential) using different numbers of excited states (ALL, EXCL2, EXCL23). The numerical lifting of the correct degeneracy in the case of including the first excited electronic state shows that the treatment is not appropriate and needs, as said before, the explicit coupling of the lowest-lying PESs. In Table 2 band origins (including assignments) and nonadiabatic shifts for different mass corrections and in comparison with the constant vibrational mass (based on the work of Moss43) are presented. As is clearly seen, the constant vibrational mass, compared with DDM corrections including all states, Δm(ALL), is simulating the nonadiabatic shifts for the band origins quite reasonably, but compared with the mass corrections Δm(EXCL2) and Δm(EXCL23) the nonadiabatic shifts with constant vibrational mass are larger by 0.2 to 0.3 cm−1. These aspects have to be further investigated, especially when total angular momentum J ≠0 calculations allows us to compare with experimental transition frequencies.

)

= 2Δmrz + 2 (ΔmR z + ΔmR x ) /5 should ensure by definition that the three masses have an identical variation with respect to position (using the kinetic energy operator given in eq 27; this does not mean that the assumption is physically correct. (2) Different individual distance-dependent masses are used as needed for eq 24 (excluding terms from eq 25). Within our investigations “identical looking” geometries (i.e., identical BOpotential value) with permuted nuclei that could be optimally described by different Jacobi geometries (r3 = r12, R3 = R(12− 3), θ3 or r2 = r31, R2 = R(31−2), θ2 or r1 = r23, R1 = R(23−1), θ1) are treated in the present work only by one fixed set of Jacobi coordinates (say r1, R1, θ1). We have not found up to now a simple relation of the distance-dependent masses for “identical looking” geometries: This can lead to a small numerical lifting of degeneracies in the eigenvalue solutions. For the parity even (A′1, E′ = E′g) and odd (A′2, E′ = E′u) we show in Figure 9 the variations of nonadiabatic shifts of vibrational band origins up to the dissociation limit. In Figure 10 we present deviations from correct degeneracy of the vibrational frequencies using mass-averaged distance-dependent masses. As one can clearly see, the degeneracy of the frequencies with E′-symmetry shows up to 30 000 cm−1 mostly a lifting of 0 Rovibrational States of H3+. J. Phys. Chem. A 2013, 117, 9493−9500. (32) Jaquet, R. Investigation of the Highest Bound Ro-Vibrational States of H3+, DH2+, HD2+, D3+, and T3+: Use of a Nondirect Product Basis to Compute the Highest Allowed J > 0 States. Mol. Phys. 2013, 111, 2606−2616. (33) Jaquet, R.; Khoma, M. The Kinetic Energy Operator for Distance-Dependent Effective Nuclear Masses: Derivation for a Triatomic Molecule. J. Chem. Phys. 2017, accepted. (34) Osamura, Y.; Yamaguchi, Y.; Schaefer, H. F. Generalization of Analytic Energy Derivatives for Configuration Interaction Wave Functions. Theor. Chim. Acta 1987, 72, 71−91; Theor. Chim. Acta 1987, 72, 93−122. (35) Pulay, P. Analytical Derivative Methods in Quantum Chemistry. Adv. Chem. Phys. 1987, 69, 241−286. (36) Jaquet, R.; Rö hse, R. Rovibrational Energy Levels and Transitions for H3+ Computed from a New Highly Accurate Potential Energy Surface. Mol. Phys. 1995, 84, 291−302. (37) Augspurger, J. D.; Dykstra, C. E. Geometrical Dependence of the Electrical Properties of H3+. J. Chem. Phys. 1988, 88, 3817−3825. (38) Papousek, D.; Aliev, M. R. Molecular Vibrational-Rotational Spectra; Elsevier: Amsterdam, 1982. (39) Wei, H.; Carrington, T., Jr. The Discrete Variable Representation for a Triatomic Hamiltonian in Bond Length-Bond Angle Coordinates. J. Chem. Phys. 1992, 97, 3029−3037. (40) Light, J. C.; Carrington, T., Jr Discrete Variable Representations and their Utilization. Adv. Chem. Phys. 2000, 114, 263−310. (41) Cullum, J. K.; Willoughby, R. A. Lanczos Algorithms for Large Symmetric Eigenvalue Computations; Birkhäuser: Boston, 1985.

Ralph Jaquet: 0000-0003-1711-6942 Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS Support from HRZ-Siegen (HORUS-Cluster) is gratefully acknowledged. We thank T. Carrington, V. Staemmler, and W. Kutzelnigg for help and many stimulating discussions. This work is supported by the Deutsche Forschungsgemeinschaft.

■

REFERENCES

(1) Oka, T. Physics, Chemistry and Astronomy of H3+ - Introductory Remarks. Philos. Trans. R. Soc., A 2006, 364, 2847−2853. (2) Oka, T. Spectroscopy and Astronomy: H3+ from the Laboratory to the Galactic Center. Faraday Discuss. 2011, 150, 9−22. (3) Kutzelnigg, W.; Jaquet, R. Explicitly Correlated Potential Energy Surface of H3+, Including Relativistic and Adiabatic Corrections. Philos. Trans. R. Soc., A 2006, 364, 2855−2876. (4) Rychlewski, J. Explicitly Correlated Wave Functions in Chemistry and Physics; Kluwer Academic Publisher: Dordrecht, The Netherlands, 2003. (5) Tennyson, J. Perspective: Accurate Ro-Vibrational Calculations on Small Molecules. J. Chem. Phys. 2016, 145, 120901. and references therein. (6) Ayouz, M.; Dulieu, O.; Guérout, R.; Robert, J.; Kokoouline, V. Potential Energy and Dipole Moment Surfaces of H3− Molecule. J. Chem. Phys. 2010, 132, 194309−11. and references to B-spline techniques used for fitting the H3−-system (7) Kutzelnigg, W. Which Masses are Vibrating or Rotating in a Molecule? Mol. Phys. 2007, 105, 2627−2647. (8) Jaquet, R.; Kutzelnigg, W. Non-Adiabatic Theory in Terms of a Single Potential Energy Surface. The Vibration-Rotation Levels of H2+ and D2+. Chem. Phys. 2008, 346, 69−76. (9) Pachucki, K.; Komasa, J. Nonadiabatic Corrections to the Wave Function and Energy. J. Chem. Phys. 2008, 129, 034102. (10) Pachucki, K.; Komasa, J. Nonadiabatic Corrections to Rovibrational Levels of H2. J. Chem. Phys. 2009, 130, 164113. (11) Komasa, J.; Piszczatowski, K.; Lach, G.; Przybytek, M.; Jeziorski, B.; Pachucki, K. Quantum Electrodynamics Effects in Rovibrational Spectra of Molecular Hydrogen. J. Chem. Theory Comput. 2011, 7, 3105−3115. (12) Jaquet, R.; Khoma, M. V. Nonadiabatic Investigations of RoVibrational Frequencies within the Systems H+2 , H2, and Prospects for H3+: Use of Distance-Dependent Effective Masses. Mol. Phys. 2012, 110, 669−683. (13) Herman, R. M.; Asgharian, A. Theory of Energy Shifts Associated with Deviations from Born-Oppenheimer Behavior in 1 Σ State Diatomic Molecules. J. Mol. Spectrosc. 1966, 19, 305−324. (14) Bunker, P. R.; Moss, R. E. Breakdown of Born-Oppenheimer Approximation - Effective Vibration-Rotation Hamiltonian for a Diatomic Molecule. Mol. Phys. 1977, 33, 417−424. (15) Bunker, P. R.; Moss, R. E. The Breakdown of the BornOppenheimer Approximation: the Effective Vibration-Rotation Hamiltonian for a Diatomic Molecule. Mol. Phys. 1977, 33, 417−424. (16) Moss, R. E. On the Adiabatic and Non-Adiabatic Corrections in the Ground Electronic State of the Hydrogen Molecular Cation. Mol. Phys. 1996, 89, 195−210. (17) Jaquet, R. Quantum Nuclear Dynamics of the H3+-System and Its Isotopomers. Prog. Theor. Chem. Phys. 2003, 13, 503−554. (18) Jaquet, R. The Influence of Non-Adiabaticity on the Nuclear Motion in the H3+ Molecule. Chem. Phys. Lett. 1999, 302, 27−34. (19) Polyansky, O. L.; Tennyson, J. Ab Initio Calculation of the Rotation-Vibration Energy Levels of H+3 and its Isotopomers to Spectroscopic Accuracy. J. Chem. Phys. 1999, 110, 5056−5064. 7029

DOI: 10.1021/acs.jpca.7b04703 J. Phys. Chem. A 2017, 121, 7016−7030

Article

The Journal of Physical Chemistry A (42) Paige, C. C. Computational Variants of the Lanczos Method for the Eigenproblem. J. Inst. Math. Appl. 1972, 10, 373−381. (43) Moss, R. E. On the Adiabatic and Nonadiabatic Corrections in the Ground Electronic State of the Hydrogen Molecular Cation. Mol. Phys. 1996, 89, 195−210. (44) Mohr, P. J.; Taylor, B. N.; Newell, D. B. CODATA Recommended Values of the Fundamental Physical Constants: 2006. Rev. Mod. Phys. 2008, 80, 633−730. (45) Tennyson, J. Variational Calculations of Rotation-Vibration Spectra In Computational Molecular Spectroscopy; Jensen, P., Bunker, P. R., Eds.; Wiley: 2000; pp 305−323. (46) Röhse, R.; Kutzelnigg, W.; Jaquet, R.; Klopper, W. Potential Energy Surface of the H3+ Ground-State in the Neighborhood of the Minimum with MicroHartree Accuracy and Vibrational Frequencies derived from it. J. Chem. Phys. 1994, 101, 2231−2243. (47) Jaquet, R.; Cencek, W.; Kutzelnigg, W.; Rychlewski, J. SubMicroHartree Accuracy Potential Energy Surface for H+3 Including Adiabatic and Relativistic Effects. II. Rovibrational analysis for H3+ and D3+. J. Chem. Phys. 1998, 108, 2837−2846. (48) Jaquet, R. First-Principles Rovibrational Analysis of the H3+Molecule. Spectrochim. Acta, Part A 2002, 58, 691−725. (49) Schiffels, P.; Alijah, A.; Hinze, J. Rovibrational States of H3+. Part 1: The Energy Region below 9000 cm−1 and Modeling of the Non-Adiabatic Effects. Mol. Phys. 2003, 101, 175−188. (50) Schiffels, P.; Alijah, A.; Hinze, J. Rovibrational States of H3+. Part 2: The Energy Region between 9000 cm−1 and 13 000 cm−1 Including Empirical Corrections for the Non-Adiabatic Effects. Mol. Phys. 2003, 101, 189−209. (51) Velilla, L.; Lepetit, B.; Aguado, A.; Beswick, J. A.; Paniagua, M. The H3+ Rovibrational Spectrum Revisited with a Global Electronic Potential Energy Surface. J. Chem. Phys. 2008, 129, 084307. (52) Viegas, L. P.; Alijah, A.; Varandas, A. J. C. Accurate Ab Initio Based Multisheeted Double Many-Body Expansion Potential Energy Surface for the three Lowest Electronic Singlet States of H3+. J. Chem. Phys. 2007, 126, 074309. (53) Alijah, A. Accurate Calculations and Assignments of AboveBarrier States of H3+ up to 15300 cm−1. J. Mol. Spectrosc. 2010, 264, 111−119. (54) Meyer, W.; Botschwina, P.; Burton, P. Ab Initio Calculation of Near-Equilibrium Potential and Multipole Moment Surfaces and Vibrational Frequencies of H3+ and its Isotopomers. J. Chem. Phys. 1986, 84, 891−900. (55) Mandelshtam, V. A.; Taylor, H. S. The Quantum Resonance Spectrum of the H3+ Molecular Ion for J = 0. An Accurate Calculation Using Filter Diagonalization. J. Chem. Soc., Faraday Trans. 1997, 93, 847−860.

7030

DOI: 10.1021/acs.jpca.7b04703 J. Phys. Chem. A 2017, 121, 7016−7030