Supporting Information. I. A refined two-state diabatic potential matrix

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1 Signatures of a Conical Intersection in Adiabatic Dissociation on the Ground Electronic State Changjian Xie, Christopher L. Malbon, # David R. Yarkony, #,* Daiqian Xie,,%,* and Hua Guo,* Department of Chemistry and Chemical Biology, University of New Mexico, Albuquerque, New Mexico 873, USA # Department of Chemistry, Johns Hopkins University, Baltimore, Maryland 8, USA Institute of Theoretical and Computational Chemistry, Key Laboratory of Mesoscopic Chemistry, School of Chemistry and Chemical Engineering, Nanjing University, Nanjing 0093, China % Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 3006, China Supporting Information I. A refined two-state diabatic potential matrix This section focuses on the (limited) changes made in the previously reported (Ref. ) two diabatic state electronic Hamiltonian matrix H d, needed here to describe, low energy, adiabatic dynamics restricted to the ground electronic state. For a review of the ab initio methods used to construct the coupled diabatic states used in this work, including the use of surface hopping trajectories to define the coupled potential energy surfaces and the shifting technique, we direct the reader to Ref.. As outlined in Ref. 3, this shifting procedure improves agreement with experimental data. The two-state diabatic potential matrix (H d, ) used in this work is derived from a 3-state fit (H d,3 ). For a review of the construction of a two-state diabatic potential matrix fit from an H d,n with larger n, we direct the reader to Ref., where the general good agreement between the H d,3 and the corresponding H d, is described. As discussed in Ref. 4, H d, can be used to treat the MAB effect, described briefly in the main text. The H d, used here is fit to a larger ab initio data point set than that needed to construct the original H d,3. This is a consequence of the increased exit-channel out-of-plane geometries sampled in this work. The description of the exit channel, see Fig. Sa is improved compared to that obtained from the original H d,3. An artifactual extremum near the formaldehyde asymptote, φ = 0 is evinced in Fig. Sa. This feature had no bearing on the A and 3 A photodissociation described by the original H d,3. However, the ground state dynamics described here will sample this region. To remove this artifact, we started quasi-classical trajectories on the hydroxymethylformaldehyde transition state and computed ab initio data, energies, energy gradients and derivative couplings for random geometries along each trajectory. This method improved the description of this region of nuclear coordinate space, as is evinced in Figs. Sa. Fig. Sb shows that this feature does not persist for smaller R(O-H), that is regions closer to the well sampled minimum energy conical intersection. In these figures since the H d,n results are based on shifted energetics, to better compare with the ab initio data, the φ = 0 energies are set equal to zero. Table S reports RMS error values for the fits. Each H d,3 is compared with ab initio data and each H d, is compared with a shifted H d,3. S

2 The topography of a conical intersection (CI) is specified by two vectors: the gradient difference (g) vector and the interstate coupling vector (h). Together the vectors define the branching space of the CI seam. Both vectors for the minimum energy crossing (MEX) are shown in Fig. S and norms and components are given in Table S. Mass weighted internal coordinates are used. The largest component of the g vector is the dissociative OH 3 stretch, R, composing ~9% of the vector. The largest component of the h vector is the H COH 3 out-ofplane coordinate, φ, composing ~95% of the vector. Thus, the branching space is well described by the coordinates R and φ. II. Dynamical model In this work, we used a four-dimensional (4D) model to study the overtone-induced dissociation dynamics of the hydroxymethyl radical (CH OH) into the H + H CO(X) channel. The four coordinates used to describe the quantum dynamics are shown in Figure S3. r is the distance between the mass center of CH and O atom, R is the distance between the H CO center of mass and H atom, θ is the angle between R and the inertial axis I a of H CO, and ϕ is the out of plane angle of the H atom. The nuclear Hamiltonian in the diabatic representation is given as follows: ˆ ˆ 0 V V d H = T +, (S-) 0 V V in which the potential matrix is given in the diabatic representation. In the adiabatic representation, the Hamiltonian is given as follows: ˆ a H = Tˆ + W, (S-) where W is the adiabatic potential energy surface of the ground electronic state. The rotationless (J tot =0) kinetic energy operator ˆT is described in coordinates (r, R, θ, ϕ), and its expression can be written as ( h = hereafter) ˆ T = + sinθ + + Kˆ µ R R µ r r µ RR sinθ θ θ sin θ φ momch where µ r = m + m O CH mhm and µ R = m + m H CH O CH O rot, (S-3) are corresponding reduced masses, respectively. The rotational kinetic energy operator K ˆ rot of H CO, which is an asymmetric rotor, can be approximated as ˆ rot ˆ ˆ K = j + j + ˆj, (S-4) I I I a b c a b c S

3 which does not include the vibration-rotation coupling effect. In this 4D model, the H CO moiety is fixed at a planar geometry. While the equilibrium geometry of CH OH( A) is not planar, the corresponding potential energy surface is quite flat with respect to the CH wag angle. 5 As shown in Table S3, the geometries and energies of the A minimum,, A MEX, product H CO(X) and saddle point in this 4D model are close to those in the full 9 dimensions, obtained from H d,. This approximation will introduce some small errors, but is not expected to qualitatively change the results. As shown in our previous study, this model successfully reproduced the experimental translational energy distributions of product H CO(X) in the nonadiabatic photodissociation dynamics of CH OH( A). The energies of MEX and saddle points are 9054 (7005) and 5000 (95) cm - higher than the minimum (Zero-point energy) of CH OH(X), respectively. All the predissociative resonances A~E (shown in Figure in main text) are well below the MEX. In the two-state diabatic representation, there are two (complex) nuclear wavefunctions associated with the two electronic states ( and ): ψ Ψ = ψ. (S-5) and correspond to the ground and first excited electronic states, respectively. While in the single-state adiabatic representation, only ground adiabatic electronic state is included and its 0 (real) nuclear wavefunction is denoted as ψ. Each nuclear wavefunction is expanded in a direct-product basis in a mixed representation: in which i i ψ = Cαβ jm α β jm, i = 0,, (S-6) αβ jm i C αβ jm are the expansion coefficients, α and β denote the DVR (discrete variable representation 6 ) and PODVR (potential optimized DVR 6 ) grid indices for radial coordinates r and R, respectively. Specifically, the Fourier bases e imϕ (m=0, ±, ) were used to represent the wavefunctions of the whole nonadiabatic system, which allows both the odd and even reflection im m symmetries. jm is the spherical harmonics Ne φ m P j ( θ ), where Pj ( θ ) is the associated Legendre polynomial, m is the projection of j onto the I a axis of H CO, and N is the normalized factor. The matrix elements of K ˆ rot in this representation are given by 7 S3

4 ˆ ˆj ˆ z j j x y jm ( + + ) I I I a b c j m m [ j( j + ) m ] [ j( j + ) m ] = + + δ jj δ Ia 4Ib 4Ic + mm j( j + ) m ( m ± ) j( j + ) ( m ± )( m ± ) δ δ 8I b j( j + ) m ( m ± ) j( j + ) ( m ± )( m ± ) δ δ 8I c jj mm ± jj mm ±. (S-7) The initial wave packet, which is prepared as a product of the ground vibrational state wavefunction multiplied by a model dipole in the form of the Taylor expansion along R coordinate, 6 i e, (S-8) i= 0 µ ( R) = c ( R R ) i where R e is the equilibrium of ground state, c i are the fitting coefficients. The dipole moment surface was fitted by Eq. (S-8) from 6 points (R=.5~4.0 bohr) calculated at the low ab initio (DFT) level. The wave packet is propagated in the Chebyshev order (k) domain: 8 Ψ = DHˆ Ψ D Ψ, k, (S-9) k s k k ˆ s with Ψ = DH Ψ 0 where Ψ 0 is the initial wave packet. The Hamiltonians are scaled to the spectral range of (-,) via ˆ ( ˆ + H = H H ) / H, in which the spectral medium ( H ( H H ) / + = + ) and half width ( max min max min s H + = ( H + H ) / ) were determined by the spectral extrema, H max and H min, which can be readily estimated. To avoid reflection, damping functions (D given in Table S4) were used at the edge of the radial grids. The absorption spectra were obtained from the discrete cosine Fourier transform of the Chebyshev autocorrelation function Ck ψ0 ψk, 9 S( E) = ( ) cos( k ) C π H sinϑ δ k,0 ϑ k, (S-0) k = 0 where E is the total energy and ϑ = arccos E is the Chebyshev angle. The numerical parameters for the spectrum calculations are listed in Table S4. To extract the lifetime of the predissociative resonances, a low-storage filter diagonalization method 8 was used to determine their complex energies (E-iΓ/). In this approach, S4

5 the Chebyshev correlation functions C k are used to build a small energy-localized Hamiltonian matrix, from which the complex energies of the resonances are obtained by diagonalization. The distribution of all energetically available ro-vibrational states of the H CO(X) product was obtained from discrete Fourier transformations: 0 A f (E) = π H sinϑ N k=0 ( δ k 0 )e ikϑ C k f, (S-) f in which C are the corresponding Chebyshev cross-correlation functions, which were k calculated as f C = ϕ δ( R R ) ξ, (S-) k f k where ξ k are the kth-order wave packet on the lower adiabat obtained either directly in the adiabatic model or by transforming the corresponding diabatic wave packets, and R was fixed at R =.0 bohr in the product channel H + H CO(X). ϕ f are the ro-vibrational eigenstates of the product H CO(X), which can be separated into two different parities even(ε=) and odd(ε=-) ones. The angular bases for even and odd parities are jm m N cos( mφ ) Pj ( θ ), m = 0,,... even = m N sin( mφ ) Pj ( θ ), m =,... odd (S-3) The matrix elements of K ˆ rot in parity-adapted bases are given by ˆ ˆj ˆ z j j x y jm ( + + ) I I I a b c m δ jj δmm = + { ε j j + δ δ δ + j j + m δ δ I 8I a b ( ) j j m m m [ ( ) ] j j m m + + δ Λ Λ δ δ + + δ Λ Λ δ δ + + m0 jm jm+ j j m m+ m 0 jm jm j j m m { ε j j + δ δ δ j j + m δ δ 8I c j m ( ) j j m m m [ ( ) ] j j m m δm0λ jmλ jm+ δ j jδm m+ + + δm 0Λ jmλ jm δ j jδm m } }, (S-4) ± in which Λ AB= A( A + ) B( B ± ). The numerical parameters used in the product state distributions calculations are the same as those in our previous study, but the Chebyshev propagation is much longer (70,000 steps). S5

6 Table S. Fit information. H d,3 is compared to ab initio data. H d, is compared to a shifted H d,3. Surface RMS Gradient Error (%) RMS Energy Error (cm - ) H d,3 (0399 points) H d, (0399 points) H d,3 (055 points) H d, (055 points) S6

7 Table S. Norms and components of the g and h vectors at the, A MEX, from H d, in mass weighted atomic units. g h Norms CO Stretch CH Stretch CH Stretch OH 3 Stretch COH 3 Bend H CH Bend H x CO (x=,) Bend H COH Out of plane H COH 3 Out of plane S7

8 Table S3. Comparison of equilibrium structures of CH OH( A), H CO(X), the, A MEX, and the saddle point in full-dimensional and 4D models. A minimum H CO(X), A MEX Saddle point Full 4D Full 4D Full 4D Full 4D C-H' /Å C-H /Å O-C /Å O-H'' /Å COH'' /deg HCH' /deg ϕ HH'CO /deg ϕ HCOH'' /deg Energy /cm S8

9 Table S4. Numerical parameters (in a.u.) used in the wave packet calculations for the overtone spectra. Grid/basis range and size: R [., 6.5], N R = 50 ; 0 PODVR for r j =, N = 50 over [0, 80 ]; N ϕ = 99 over [-80, 80 ] max 49 j Damping: D=exp[-0.0(R-4.0) ], for 6.5> R >4.0; =, otherwise Propagation step: 0,000 S9

10 Figure Sa. Improvement of H d,n determined topography at a slice through an exit channel cross section of the ground state potential energy surface. Dashed lines illustrate surface before additional ab initio points are added. Solid lines represent H d,n (n=,3) once ab initio points shown are included in the fit. Figure Sb. Comparison of surfaces vs. ab initio for a path between (equivalent) saddle points for (φ=90, φ=70). Ab initio points shown are not included in fit. For comparison, the zero of energy is shifted to the saddle point (at φ = 0). Ab initio determined saddle point has R(O-H) =.90 au and energy cm -. S0

11 Figure SA. Mass-weighted g vector. Figure SB. Mass-weighted h vector. S

12 Figure S3. The coordinates used in the dynamical calculations. S

13 References: () Xie, C.; Malbon, C.; Yarkony, D. R.; Guo, H. J. Chem. Phys. 07, 46, () Malbon, C. L.; Yarkony, D. R. J. Chem. Phys. 07, 46, (3) Zhu, X.; Malbon, C. L.; Yarkony, D. R. J. Chem. Phys. 06, 44, 43. (4) Malbon, C. L.; Zhu, X.; Guo, H.; Yarkony, D. R. J. Chem. Phys. 06, 45, 34. (5) Johnson, R. D.; Hudgens, J. W. J. Phys. Chem. 996, 00, (6) Light, J. C.; Carrington Jr., T. Adv. Chem. Phys. 000, 4, 63. (7) Zare, R. N. Angular Momentum; Wiley: New York, 988. (8) Guo, H. Rev. Comput. Chem. 007, 5, 85. (9) Guo, H. J. Chem. Phys. 998, 08, 466. (0) Xie, C.; Ma, J.; Zhu, X.; Zhang, D. H.; Yarkony, D. R.; Xie, D.; Guo, H. J. Phys. Chem. Lett. 04, 5, 055. S3