Influence of RennerâTeller Coupling between Electronic States on H+

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A: Kinetics, Dynamics, Photochemistry, and Excited States

The Influence of Renner-Teller Coupling between Electronic States on H+CO Inelastic Scattering Steve Alexandre Ndengue, Richard Dawes, Fabien Gatti, and Hua Guo J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.8b05235 • Publication Date (Web): 13 Jul 2018 Downloaded from http://pubs.acs.org on July 16, 2018

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The Journal of Physical Chemistry

The Influence of Renner-Teller Coupling Between Electronic States on H+CO Inelastic Scattering Steve Ndengué,† Richard Dawes,∗,† Fabien Gatti,‡ and Hua Guo¶ Department of Chemistry, Missouri University of Science and Technology, Rolla, MO 65409, USA, Institut des Sciences Moléculaires d’Orsay, CNRS, Université Paris-Sud/Paris Saclay, F-91405 Orsay, France, and Department of Chemistry and Chemical Biology, University of New Mexico, Albuquerque, NM 87131, USA E-mail: [email protected]

∗ To

whom correspondence should be addressed of Chemistry, Missouri University of Science and Technology, Rolla, MO 65409, USA ‡ Institut des Sciences Moléculaires d’Orsay, CNRS, Université Paris-Sud/Paris Saclay, F-91405 Orsay, France ¶ Department of Chemistry and Chemical Biology, University of New Mexico, Albuquerque, NM 87131, USA † Department

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ACS Paragon Plus Environment

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Abstract We examine the excitation of carbon monoxide from its rovibrational ground state via collisions with a hydrogen atom. Calculations employ the Multi-Configuration Time-Dependent Hartree method and treat the nonadiabatic dynamics with the inclusion of both the ground and the Renner-Teller coupled first excited electronic states. For this purpose, a new set of recently presented global HCO Potential Energy Surfaces (PESs) which cover the 0–3 eV range of energy are used. The results obtained here considering only the ground state (without the Renner-Teller coupling) are in qualitative agreement with those available in the literature. The Renner-Teller effect is known to have an important effect on the spectroscopy of the system, and its inclusion and effects on the dynamics for the processes described in this paper are fairly significant also. The results of this study indicate that for certain very particular initial conditions, rather dramatic effects can be observed.

Introduction Carbon monoxide (CO) is after molecular hydrogen (H2 ) the second most abundant molecule in the universe. Currently, the understanding of chemical and physical conditions in low temperature astrophysical environments such as diffuse clouds 1 or protoplanetary disks 2,3 is deduced primarily from observations of these molecules: typically absorption in the UV and near-IR or emission in the far-IR and millimeter wavelengths are recorded. For example the In f rared Space Observatory (ISO) detected a large number of CO rovibrational lines 4 towards Orion Peak 1 and 2 or the |∆v| = 1 rovibrational bands of CO around 4.7 µm are frequently detected with the Cryogenic high resolution Infrared Echelle Spectrograph (CRIRES) on the Very Large Telescope (VLT). 2,3 As the level populations of molecules in most low-density environments depart from local thermodynamic equilibrium (LTE), the prediction of spectral line intensities depends on the magnitude of collisional excitation rate coefficients of those molecules with the dominant species in these environments: H2 , H and He. Because of the importance of the HCO complex in astrophysics/astrochemistry or in atmo2


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spheric chemistry and combustion applications, numerous studies have been devoted to the interaction of carbon monoxide with hydrogen. This includes an accurate description of the intermolecular potential energy surface between the two fragments, 5–11 bound and quasi-bound states, 7,8,12–16 spectroscopy of the formyl radical (HCO), the photodissociation of HCO, 7,13,16,17 the scattering, 18–25 transport properties for collisions of CO with hydrogen 26 with the PES of Song et al 9 and even the recombination 12,27–29 of CO and H. In several instances, theoretical predictions were backed by experimental studies 8,30 and a better and more accurate understanding of the system was obtained. In an effort to produce a more quantitative agreement with theory, we recently presented a new set of potential energy surfaces (PESs) for the ground and two excited states of the formyl radical. 11 The agreement of spectroscopic parameters and data 11 obtained from those fully ab initio PES is excellent, in better agreement with experimental values than the results from previous work, thus confirming their reliability for quantum dynamics studies on the HCO complex. The impacts of including Renner-Teller (RT) coupling in the Hamiltonian can be subtle and interpretation of perturbations relies on accurate underlying PESs. In this work, we will focus on the electronically nonadiabatic inelastic scattering between H and CO. Several studies have been performed in recent years on H+CO inelastic scattering as mentioned previously. While it is well-known and documented that HCO exhibits a seam of electronic state degeneracy governed by Renner-Teller coupling, with a minimum approximately 1730 cm−1 above the CO+H asymptote, to our knowledge no inelastic scattering calculations have been reported assessing its impact on scattering processes. Even for the ground electronic state, the difficulty of performing accurate and reliable ro-vibrational inelastic scattering calculations for the H+CO system has been reported already. 19,25 The main goal of this work is to address the impact of the Renner-Teller coupling by looking at specific rovibrational excitations using a time-dependent approach: the Multi-Configuration Time-Dependent Hartree (MCTDH) method. We recommend the MCTDH method to address calculations for similar systems as the computational cost is more affordable than traditional time-independent methods. The MCTDH method has matured throughout the years as an accurate and efficient way of

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studying the dynamics of systems with high dimensionality. It was recently presented that it could also be used to compute resonances, 11,31 study cases with a relatively high density of states, 32 or even diverse collision partners. 33 In fact, within its implementation in the Heidelberg package, it is straightforward to include more than one electronic state and access fully quantum mechanically relatively high energies in the description of the dynamical process and as such it fits perfectly the purpose of this work. This work is organized as follows: first we will describe briefly the electronic structure procedure followed to obtain the PESs used in this work. Then we will go through the quantum dynamical theory by discussing the Hamiltonian, the MCTDH method, the PES representation and the wavepacket analysis. The next section will present results of our calculations and discuss the impact of the Renner-Teller coupling. We will conclude this work with a general summary and future directions of this investigation.

Electronic structure Despite having only 15 electrons and three nuclear degrees of freedom, the electronic structure of the low-lying states of the formyl radical is remarkably complex. The ground X 2 A0 state has a bent equilibrium structure (θ = 124.95°), 34 but becomes degenerate with the low-lying excited A-state (T0 = 9297 cm−1 ) 34 at linear geometries, together forming a Π-state. The two states interact through the RT coupling mechanism which is strongest near linearity. The equilibrium geometry of the A2 A00 state is linear which results in a highly rotationally excited CO product fragment upon photodissociation from the ground state. 35–37 The dissociation energy (D0 ) to form ground state H + CO fragments is 5083 ± 8 cm−1 according to the latest ATcT database. 38 In our PESs, the minimum on the A-state (minimum of degeneracy seam and barrier to linearity) is 8606 cm−1 above the minimum on the ground state. This places the seam 1733 cm−1 above the asymptote (De = 6873 cm−1 ). Using our PESs we obtain calculated values for D0 and T0 of 5159 cm−1 and 9280 cm−1 respectively, which are both close to the experimental values. 11 At

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higher energy along the collinear intersection seam of HCO, the X 2A0 state has a conical intersection with the B2A0 state, the two states abruptly switching character. 11,39 A similar intersection occurs along the analogous seam of collinear HOC, while a third intersection for bent geometries is avoided for near-equilibrium values of rCO . The X− and B−states are well separated near the equilibrium geometry of the X−state, but the avoided crossing of those states causes a pronounced barrier along the minimum energy dissociation path (HCO → CO + H). The barrier on our PES (1140 cm−1 above dissociation) is close to that reported in a coupled-cluster based study by Song et al. (1138 cm−1 ). 9 The electronic structure method employed to generate data used to fit the PES was the explicitly correlated version of the internally contracted multireference configuration interaction method (icMRCI-F12). 40 The Molpro electronic structure package was used for all of the reported calculations. 41 The employed reference was taken from an 18-doublet-state generalized dynamically weighted 42–47 CASSCF calculation with a full valence active space which provides robust convergence throughout. The MRCI calculations included all electrons in the correlation treatment and added a scaled rotated Davidson correction [(AE)-MRCI(QR*)-F12/CVTZ-F12]. Additional details are given in Ref. ( 11). PESs fit to 8000 electronic structure data were obtained using the interpolating moving least squares (IMLS) method. 11,48–50 As reported previously, the PESs are remarkably accurate, yielding an rms error of only 7.6 cm−1 for the 15 experimentally assigned bound vibrational levels. The resonances are in similarly close agreement with experiment. Without the Davidson correction the rmse is greater than 100 cm−1 .

Theory Hamiltonian Jacobi coordinates, which is the most appropriate coordinate representation to describe scattering problems are used in this work where we follow the derivation performed by Petrongolo 51 or used recently by Lin and Guo 52 or Defazio et al, 53 for example, to derive the molecular Hamiltonian. Some additional modifications inspired by Zhang et al 54 are included to accommodate the general 5



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case of a varying projection of the electronic angular momentum. The H–CO Jacobi coordinates and the Body-Fixed (BF) frame are defined by choosing the z-axis to be along the R vector also referred to as the R − embedding and the molecular plane as the x − z plane. This R − embedding representation is best suited for scattering problems, and in particular for MCTDH calculations as it allows a straightforward projection of the scattering product wavefunction onto the rovibrational states of CO. The total (electronic+nuclear) Hamiltonian representing the system is expressed in atomic units as (¯h = 1 hereafter): 1 ∂2 1 ∂2 − + Tˆrot + Hˆ el Hˆ = − 2µR ∂ R2 2µr ∂ r2

(1)

where the R and r coordinates represent the H–CO and C–O distances respectively with µR and µr their corresponding reduced masses. Tˆrot is given by

B+b + − B (Lˆ z2 − 2Jˆz Lˆ z ) sin2 γ +B(Jˆ+ + Jˆ− )Lˆ z cotγ,

0 Tˆrot

Tˆrot =

(2)

with γ the angle between the two Jacobi vectors and the rotational constants B and b are

b=

1 2µr r2

B=

1 . 2µR R2

(3)

0 operator has the usual form The Tˆrot

0 Tˆrot = (B + b)ˆj† · ˆj + B(Jˆ2 − 2Jˆz2 − Jˆ+ jˆ− − Jˆ− jˆ+ )

(4)

ˆ2 ˆj† · ˆj = − 1 ∂ sinγ ∂ + Jz sinγ ∂ γ ∂ γ sin2 γ

(5)

∂ jˆ± = −cotγ Jˆz ± ∂γ

(6)

with

and

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Jˆ2 , Jˆz and Jˆ± are the squared total angular momentum, projection of J onto the BF z-axis, and the raising/lowering operators respectively. Lˆ z is the electronic angular momentum operator along z couples the two states: the x and y components are usually ignored in the the BF z-axis, and LAX

derivation. 6

Hˆ el |φX i = VX (R, r, γ) |φX i

(7)

Hˆ el |φA i = VA (R, r, γ) |φA i

(8)

z |φA i Lˆ z |φX i = iLAX

z |φX i Lˆ z |φA i = −iLAX

Lˆ z2 |φX i = LXzz |φX i

Lˆ z |φA i = LAzz |φA i

(9) (10)

Diabatic electronic functions are used to avoid complex arithmetic in the calculations. The diabatic electronic functions write 1 |φ± i = √ (|φX i ± i |φA i) 2

(11)

which are also eigenfunctions of Lˆ z with eigenvalues z |φ± i Lˆ z |φ± i = ±LAX

(12)

1 1 Lˆ z2 |φ± i = (LXzz + LAzz ) |φ± i + (LXzz − LAzz ) |φ∓ i . 2 2

(13)

and

In the diabatic representation, the nuclear wavefunction is written as 

 |ψ+ i |ψi =  , |ψ− i

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(14)


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and the Hamiltonian is expressed as 



Hˆ ++ Hˆ +−  Hˆ =  , ˆ ˆ H+− H−−

(15)

where we have

Hˆ ±± =

1 ∂2 1 ∂2 0 − + Tˆrot 2µR ∂ R2 2µr ∂ r2 B+b 1 zz zz z + −B (L + LA ) ± 2LAX sin2 γ 2 X VX +VA + 2 −

(16)

and VX −VA 1 Hˆ +− = + 2 2

B+b − B (LAzz − LXzz ). sin2 γ

(17)

The MCTDH algorithm The Multi-Configuration Time-Dependent Hartree 55–58 (MCTDH) method is an algorithm to solve the time dependent Schrödinger equation which can be considered as a time-dependent version of the complete active space self-consistent field (CASSCF), but for the nuclei and time-dependent : the variationally optimal active space changes in time and adapt along the propagation. Within this method the wavefunction Ψ(Q,t) of the system is written as a sum of products of variationally optimal single particle functions (SPFs), forming a time-dependent orthonormal basis set. SPFs are low dimensional functions (between one and four degrees of freedom): when they contain more than one degree of freedom (DOF) as it is the case in this work, the combined coordinates Qκ = q1,κ , . . . , qdκ ,κ that comprises dκ physical DOF are introduced.

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The ansatz of the MCTDH wavefunction reads

Ψ(q1 , . . . , q f ,t)

≡ Ψ(Q1 , . . . , Q p ,t), np

n1

p

(κ)

= ∑ · · · ∑ A j1 ,..., j p (t) ∏ ϕ jκ (qκ ,t), j1

jp

κ=1

= ∑ A J ΦJ ,

(18)

J

where f denotes the number of degrees of freedom and p the number of MCTDH particles (combined modes). The AJ ≡ A j1 ,..., j p denotes the MCTDH expansion coefficients and the configuration or Hartree products ΦJ are products of SPFs defined in relation (18). The SPFs are finally represented by linear combinations of time-independent primitive basis functions

(κ) ϕ jκ (Qκ ,t) =

N1,κ

Nd,κ

(κ)

(κ)

∑ · · · ∑ c jκ l1···ld (t)χl1

l1 =1

ld =1

(κ)

(q1,κ ) · · · χld (qd,κ ),

(19)

(κ)

usually within a Discrete Variable Representation (DVR), 59,60 here χli (qi , κ), with time dependent coefficients c jκ l1 ···ld . The MCTDH equations of motion are derived by applying the DiracFrenkel variational principle. 55–57

Potential Energy Surface representation The Hamiltonian operator as described in Section Hamiltonian is implemented in the Heidelberg MCTDH package. The kinetic energy operator for a triatomic molecule in Jacobi coordinates is already in the sum-of-product (SOP) form required by the program and thus no modifications were necessary. The Potential Energy Surfaces described in Section Electronic structure is implemented as a set of Fortran programs and is not in that specific format and thus needed to be re-expressed in a form that would be convenient for the MCTDH dynamics. As dynamics calculations usually require long propagation times, and are typically repeated for several values of the total angular momentum J, and generally do not require an accuracy of 0.1 cm−1 or less in the surface representation, it is often advantageous for inelastic scattering applications to represent the surfaces using 9



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a Legendre polynomial expansion to speed up the calculations. This is done instead of using the POTFIT algorithm, 61,62 an MCTDH package algorithm commonly used to transform the PES into an SOP form. The Legendre expansion of the potential in this form writes λmax

Vi (R, r, γ) =

∑ Viλ (R, r)Pλ (cosγ)

(20)

λ =0

where the maximum order λmax of the Legendre polynomials was selected to be 50 and i identifies the states X and A. Adding higher expansion orders did not significantly improve the fit and the average error of the projection of the potentials on the Legendre polynomials is about 1 cm−1 in the fitted range which is also the range in which the dynamics is performed. The subsequent 2D functions Viλ (R, r) were then fitted using the POTFIT algorithm with a fitting RMSE of less than 10−2 cm−1 . Figures 1 and 2 represent cuts of the PESs of the X and A states. Figure 1 shows a 2D plot of the X state PES at linearity. Figure 2 shows 1D cuts of the diabatic X and A states PESs at linearity with CO at its equilibrium position r = 2.13 bohr. We see in the figure around R = 4 bohr indications of the crossing between an excited Σ state and the lower Π state. At smaller values of the R distance, the X and A states which are degenerate components of the Π state are clearly identified. The barrier to linearity for the X state is 8,606 cm−1 and this plays a role in the dynamics as we will discuss later. In Figure 2, a cut obtained from a POTFIT of the PESs (in full lines) is plotted along with the Legendre expansion PES (lines with points). An artificial plateau at 5 eV, a region well beyond the dynamic range of interest, was applied to the PES as shown in Figure 2. The two representations are visually identical but some small differences arise when we look closely. These differences however, as is presented in the Supplementary Material where we compare some state-to-state transition probabilities obtained with the two representations, are sufficiently small that they do not to affect the calculations reported in this work. This verification is essential because the Renner-Teller effect arises near linearity where the two surfaces become degenerate along a 2D seam. A poor description of that degeneracy is likely to affect the results of the calculations: luckily this is not the case in this work and we can safely work with a Legendre

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expansion of the PESs.

4

3

2.8

2.6 1

r(bohrs)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


2.4

0.5 0.25

2.2

2 1.5

2

1.8 2.5

3

3.5

4

4.5 R(bohrs)

5

5.5

6

6.5

Figure 1: 2D cuts of the X state adiabatic PES at linearity. The energies on contours are reported in eV.

Wavepacket propagation and analysis The initial wavepacket in the one-state calculation is defined as the product of a Gaussian wavepacket, a vibrational eigenfunction of the CO molecule (the vibrational ground state) and an associated Legendre polynomial (the P00 (cosγ) = 1 in the specific case of the rovibrational excitation from the ground rovibrational state).

Φ0 (R, r, γ) = χ0 (R)ϕv0 j0 (r)Pjk0 (cosγ)

11


(21)


45000 40000

VA

35000 30000 V(cm−1)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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25000 20000 15000 10000 5000 VX

0 2

3

4

5

6 R(bohrs)

7

8

9

10

Figure 2: 1D cuts of the X and A adiabatic PESs at linearity with r = 2.13 bohr, the equilibrium distance of CO. The full line is from the POTFIT representation of the PESs and the line with points is the Legendre expansion. Note that the Legendre expansion PESs were cut at 5 eV, as can be seen from the figure. More details about the Figure is available in the text. The initial wave-packet was started at 13 bohr (center of the Gaussian wave-packet) with an energy distribution determined from the initial momentum of the Gaussian such that it overlaps the 0-4 eV total energy range: this corresponds to a value of p = −0.19 a.u. where the negative sign indicates that initial wave-packet starting far from the interaction region moves towards it. The initial wavepacket in the two-state case is similarly the product of a Gaussian, a vibrational wavefunction of the CO diatom and an associated Legendre polynomial. However here the initial function is a vector with component in each of the states such that the initial wavepacket is localized  on the  +|Φ0 i , −|Φ0 i where the signs selected will ensure that the initial wavefunction will be on the ground state PES. adiabatic ground state of the system. The initial wavefunction thus writes |ψ0 i =

 √1  2

This was further verified by checking the evolution of the adiabatic populations with time.

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Details of numerical calculations A summary of the parameters used for the Primitive and SPF basis in our calculations is presented in Table 1. The Complex Absorbing Potential (CAP) used in this study reads :

−iW (R) = −iη(R − R0 )n Θ(R − R0 ).

(22)

Here, η = 5.073812 × 10−6 au, Θ the Heaviside step function, R0 = 12 a0 the starting position of the CAP, and n = 4 the monomial order of the CAP. Table 1: Parameters of the primitive basis used for the scattering calculations. Sine-DVR and Leg-DVR denote respectively the Sine DVR and Extended Legendre DVR. The units are bohr and radians. k is an integer. R r γ, k Primitive basis FFT Sine-DVR KLeg-DVR Number of points 256 96 128 Range 1.5–20.0 1.5–3.5 0–2π,-30–30 Size of SPF basis 15–35 10–20 15–45

Results and discussion Convergence of the scattering calculations We performed several tests to ensure convergence of the scattering calculations: increasing the size of the primitive and SPF bases, extending the grid, placing the initial wavepacket farther from the interaction region. Based on these tests, the parameters of Table 1 were selected and provide a converged description of the dynamical processes. The effect of the PES on the accuracy of the calculations was also tested and a more detailed discussion of this aspect is done in the Supplementary Material.

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Fluxes, probabilities and electronic state populations Once the parameters of the calculations are defined and the expansion order of the PES is chosen, one can investigate selected transition probabilities comparing results with and without the inclusion of the RT coupled state. First vibrational transitions from the ground rovibrational state were studied, tracking flux to excited vibrational states with all rotational states included, i.e. (v = 0, j = 0) → (v = n), n = 0, 1 and 2, for various values of the total angular momentum J. The corresponding probabilities are the flux into a particular channel relative to the total. Figure 3 shows a comparison of the wavepacket fluxes from the (v = 0, j = 0) state on the ground state PES to vibrational states as a function of energy with and without the inclusion of the RT coupled state for total angular momenta values J = 0, 5, 20 and 50. The figures show no significant impact of the RT coupling on the net vibrational transition probabilities (summed over the detailed rotational transitions) beyond the slight differences seen in the figures. We then looked at state-to-state transition probabilities, once again from the (v = 0, j = 0) state on the ground state PES to rovibrationally excited states. Figure 4 shows a comparison of selected state-to-state transitions at J = 5 and 30 with and without the RT coupling. Plots of these and other transitions for other values of the total angular momentum show similar behavior to that depicted in the figures. Compared to the vibrational transitions mentioned above (which involves summing over the rotational states), here more marked differences are noted between the calculations with and without the RT effect. Notable changes from the RT coupling are observed in the intensity and/or the position of some of the peaks (resonances) which can be expected from the known effect on spectroscopic levels. 11 To gain further insight into the process and the origin of the differences in the transition probabilities, we examined in Figure 5 the adiabatic X and A states populations as a function of time for J= 5, 20 and 70. The figure shows that for collisions with the CO (v = 0, j = 0, m = 0) ground state, about 99% of the wavepacket remains on the X state during the propagation while less than 1% flows onto the A state (and later moves back to the X state as dissociation occurs along the electronic ground state). In the figure it is seen that most of the wavepacket is rapidly absorbed in the asymptote beginning at about 80 fs. These results make 14


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25

(J=0)

(v=0,j=0) to v’=0 (v=0,j=0) to v’=1 (v=0,j=0) to v’=2 (v=0,j=0) to v’=0 - RT (v=0,j=0) to v’=1 - RT (v=0,j=0) to v’=2 - RT

20 (J=5)

15

Flux [au]

Flux [au]

25


20

10

5

15

10

5

0

0 0

0.5

1

25

1.5 Energy [eV]

2

2.5

3

0

(J=20)

0.5

1

25


20

(J=50)

15

10

5

1.5 Energy [eV]

2

2.5

3


20

Flux [au]

Flux [au]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


15

10

5

0

0 0

0.5

1

1.5 Energy [eV]

2

2.5

3

0

0.5

1

1.5 Energy [eV]

2

2.5

3

Figure 3: Fluxes for selected transitions from the rovibrational ground state to vibrational states 0, 1 and 2 for J = 0, 5, 20 and 50 with Renner-Teller (RT) effect and without (see text for description). it clear why the overall effect is small for these scattering conditions. The topography of the PESs dictates how the intersection seam is accessed, and the intersection seam and associated coupling is localized to a small angular range with high potential energy. Despite the relatively small portion of the wavepacket (1%) that moves onto the excited state, the impact on detailed state-to-state cross-sections is quite significant. The difference between the fluxes and transition probabilities then must arise from the additional terms appearing in the Hamiltonian from the Renner-Teller effect (see for example equations (4) or (16)). Although the wavepacket stays mainly on the X state, it still experiences additional kinetic terms added to the Hamiltonian, from the Renner-Teller effect.

15



1

(J=5)

(v=0,j=0) to (v’=0,j’=2) (v=0,j=0) to (v’=0,j’=2) - RT

0.8

0.6

Flux [au]

Flux [au]

1

(v=0,j=0) to (v’=0,j’=1) (v=0,j=0) to (v’=0,j’=1) - RT

0.8

0.4

0.2

(J=5)

0.6

0.4

0.2

0

0 0

0.5

1

0.5

1.5 Energy [eV]

2

2.5

3

0

0.5

1

0.5

(v=0,j=0) to (v’=0,j’=1) (v=0,j=0) to (v’=0,j’=1) - RT

0.4

(J=30)

0.3

0.2

0.1

1.5 Energy [eV]

2

2.5

3

(v=0,j=0) to (v’=0,j’=2) (v=0,j=0) to (v’=0,j’=2) - RT

0.4

Flux [au]

Flux [au]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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(J=30)

0.3

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0 0

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Figure 4: (0,0)→(0,1) and (0,2) transition probabilities for J = 5, 30. Calculations without the Renner-Teller coupled state are in purple and with the coupled state are in green.

The wavepacket transition between the X and the A states within the RT effect occurs at linearity where the two electronic states are degenerate, offering to the wavepacket a large seam of possibility for transitioning between the two electronic states. It is thus interesting to understand why only a small portion of the wave packet crosses from the X state to the A state as seen in the results of Figures 3, 4 and 5. Here we consider the ground state CO (v = 0, j = 0) initial condition, but will extend our discussion later. In Figure 3 of the Supplementary Material, the average position and standard deviation of the angular degree of freedom γ of the wavepacket was tracked during the propagation for J = 6, 12 and 56. The plot shows that the wavepacket initially moving 16


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1.000000

X -- J=5 A -- J=5 X -- J=20 A -- J=20 X -- J=70 A -- J=70

0.100000

Population

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


0.010000

0.001000

0.000100

0.000010 0

20

40

60

80

100

Time [fs]

Figure 5: X and A states populations as a function of time for J = 5, 20 and 70. (Note the y axis logscale). The initial state on the figure is (v = 0, j = 0, m = 0). The energy of the propagation covers the 0–3 eV range. along the X state experiences an attractive gradient from the interaction region after approximately 50 fs. The attraction is toward the HCO well with the equilibrium position at γ ≈ 55◦ . For these conditions, the wavepacket is guided by the well and the significant barrier toward linearity and hence doesn’t significantly explore linearity where the RT coupling effect is large. This is depicted in Figure 3 of the Supplementary Material for J = 6, 12 and 56. For J = 56, and in general for larger values of the total angular momentum (corresponding to a large impact parameter in the classical view), the wavepacket undergoes scattering, barely experiencing the effect of the attractive part of the potential. Hence, only a small fraction of the wavepacket crosses through the A state and back before dissociating along the X state. Another way to consider scattering of H with ground state CO (v = 0, j = 0), is that since the rotational part of the CO wavefunction is a constant for j = 0, with no orientational preference, the fraction of the total angular range for which the RT

17



coupling effect is significant is small, even before considering the guiding effect of the potential which is relatively repulsive in that range.

State-to-state cross-sections The state-to-state cross sections for the (v = 0, j = 0) → (v = n, j = l) transitions follow what has been described here, that is we notice small but noticeable differences between one electronic state and two electronic state calculations. The wavepacket propagation for a single state calculation is relatively affordable computationally: a shared-memory wavepacket propagation with 8 processors takes 1 to 7 days (the shortest time being for the propagation with the highest value of the total angular momentum) for 99.6% of the wavepacket to be absorbed by the CAP. Hence we were able to run propagations for each total angular momentum value between 0 and 150. The wavepacket propagation in the two-state case is significantly more expensive: a shared memory wavepacket propagation with 16 processors takes 1 to 10 weeks for 99.6% of the wavepacket to be absorbed by the CAP. Because of the increased cost and resource limitations, we ran wavepacket propagations for only selected values of J accounting for about 1/3 of the full 151 values used in the one-state calculations. The transition probabilities for intermediate J were then carefully obtained from the J-interpolation algorithm presented in Appendix A of Ref. 63, introducing only a small uncertainty to the results. The various probabilities are then summed to obtain the state-to-state cross-sections. Figure 6 presents some selected state-to-state excitation cross-sections. It is clear from the figure that the cross-sections reproduce the even ∆J propensity 23 (related to the nearly homonuclear nature of the CO diatom) obtained with the Bowman-Bittman-Harding (BBH) surface, 20 which was absent on the initial Werner-Keller-Schinke (WKS) surface calculations 64 but recovered later 9 with the Song-Avoird-Groenenboom (SAG) surface (which is actually the WKS surface with a more accurate description of the long range interaction). Considering the effect of RT coupling, the apparently small differences in the fluxes and probabilities (Figure 4) sum to yield rather significant differences in the cross-sections (as much as 40% for some energies, see Figure 6). We also observe for some transitions differences in the position of some peaks that are far greater on the 18


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energy scale than any spectroscopic perturbations. The magnitude of the changes is reduced with increasing vibrational excitation. The cross-sections also display some other interesting features. For instance, not surprisingly, cross-sections with and without the RT coupling are most similar at low energies insufficient to access the collinear intersection seam. Another interesting feature in the cross-sections are pronounced oscillations observed on the v = 0 → v = 1, 2 transition crosssections which are absent from v = 0 → v = 0 transitions. These oscillations are significant for the calculations both with and without RT coupling (which does impact the phase). Since the initial state is j = 0, with no orientational preference, as discussed above, very little of the wavepacket finds the collinear alignment and little flux to the A state is observed. However, it is possible that there is a significant orientational dependence to the vibrational excitation cross-section. In other words, the small part of the total cross-section corresponding to vibrational excitation is selectively related to the part of the wavepacket that does explore the collinear region of the seam. In a separate study of photodissociation in this system to appear in 2018 65 we have shown that strong interference effects occur for wavepackets propagating on the ground electronic state after passing through the collinear seam.

Scattering from excited rotational states The previous sections describing scattering from the initial rovibrational ground state showed a relatively small impact of the RT effect on the state-to-state transition probabilities and typically only about 1% of the wavepacket found its way onto the excited electronic state, although some more pronounced differences were noted in Figure 6 comparing the state-to-state cross-sections. This was partly rationalized in terms of the constant spherical CO rotational wavefunction j = 0, and the fact that RT coupling effect is only relevant for a small range of angles near linearity. We thus explored the effect of starting with rotationally excited initial states (with their non-constant rotational wavefunctions) by looking at selected state-to-state transition probabilities for various combinations of initial rovibrational states with specific projections of the magnetic quantum number m = −10, −1, 0 and j = 10, 30. Figure 7 shows the evolution of the adiabatic population of the 19



14

(0,0) ➞ (0,1) (0,0) ➞ (0,2) (0,0) ➞ (0,3) (0,0) ➞ (0,4)

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Figure 6: Selected state-to-state excitation cross-sections (0, 0) → (v, j), (v = 0, 1 and 2). The lines with symbols are 2 electronic state calculations while the light lines are single state calculations. The cross-section are given in Å2 . 20


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X and A states as a function of time for various initial rovibrational states. As seen in the figure, beginning from a pure, particular rotational state of CO, while still only a modest portion of the wavepacket reaches the A state, the relative change is rather dramatic. Compared to the j = 0 result, for certain initial states, about five times more amplitude reached the A state (5 %). As seen in the figure, for the same rotational quantum number j this proportion varies with the magnetic quantum number m, and is largest for small values of the total angular momentum J (classically corresponding to a head-on collision). These results are roughly what one would anticipate intuitively. In fact some guidance can be obtained by considering photodissociation experiments which yield inverted (highly excited) rotational distributions of the CO product for wavepackets coming down from the A state. 37 In fact those distributions demonstrate interference effects resulting in oscillations in the product distribution. It is an intriguing avenue for future research to explore scattering beginning with superpositions of initial state wavepackets corresponding to molecular alignment. Thus it may be possible to achieve quantum control of scattering product state distributions.

1.000000

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X -- j=30,m=-1,J=1 A -- j=30,m=-1,J=1 X -- j=30,m=0,J=1 A -- j=30,m=0,J=1

0.100000

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Population

Population

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0.000010 0

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Time [fs]

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Time [fs]

Figure 7: X and A states populations as a function of time for J=1 and 30. (Note the y axis logscale). The initial state on the left figure are (v = 0, j = 10, m = −10) and (v = 0, j = 10, m = 0) and on the right figure are (v = 0, j = 30, m = −1) and (v = 0, j = 30, m = 0). The energy of the propagation covers the 0–3 eV range.

21



Conclusion We present new calculations of the inelastic scattering of H with CO, exploring the effect of the Renner-Teller coupling (between the ground and first excited electronic state) on the state-to-state probabilities and cross-sections using a recent set of accurate PESs. The MCTDH algorithm proved to be relatively cost effective and reliable in this application. The results show that state-to-state transition probabilities for calculations including the coupling between states typically show small but significant differences compared with one-state calculations. However, these apparently small differences sum into larger differences recorded in the cross-sections with significant changes in peak positions and intensities. For collisions involving the CO ground state v = 0, j = 0, the cause of these differences was assigned mainly to the kinetic energy terms associated with the RennerTeller coupling and not to bifurcation of the wavepacket onto the A-state. For these conditions only a small fraction (less than 1%) of the wavepacket temporarily populates the A state. This is due to the constant spherical rotational wavefunction for the j = 0 rotational state of CO and the small angular range near linearity where the Renner-Teller effect is significant. Moreover the potential energy is relatively high toward this region, further reducing the fraction of the wavepacket that experiences coupling near the seam of degeneracy. Intriguingly however, with initial conditions of a rotationally excited initial state of CO (corresponding to a favorably biased rotational wavefunction), and small total angular momentum (classically corresponding to a head-on collision), five times more population reached the A-state (5%). It is also noteworthy that products of photodissociation from the A-state exhibit oscillatory behavior in their distributions and a highly rotationally excited CO fragment. This has recently been understood as an interference effect stemming from the topography of the PESs. An avenue for further research is to explore the possibility of quantum control of scattering products, perhaps employing initial superposition wavepackets corresponding to molecular alignment. This study has given insight into which scattering processes and quantities are most effected by the Renner-Teller effect. A more comprehensive study on the net effects on state-to-state cross-sections and collision rates for state distributions relevant to certain environments is underway. 22


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Supporting Information A document of Supporting Information provides: 1) evidence of the convergence of the scattering calculations and PES representation, 2) some additional insight regarding the angular range explored by the wavepacket.

Acknowledgments This research was supported by the U.S. Department of Energy Office of Science, Office of Basic Energy Science Award No. DE-SC0010616 to R.D. and DE-SC0015997 to H.G. S.A.N. thanks Hans-Dieter Meyer for discussions on the implementation of the Hamiltonian within the MCTDH package.

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Influence of RennerâTeller Coupling between Electronic States on H+

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