On-the-Fly ab Initio Semiclassical Dynamics of Floppy Molecules

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On-the-fly Ab initio Semiclassical Dynamics of Floppy Molecules: Absorption and Photoelectron Spectra of Ammonia Marius Wehrle, Solene Oberli, and Jiri Vanicek J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.5b03907 • Publication Date (Web): 30 Apr 2015 Downloaded from http://pubs.acs.org on May 2, 2015

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The Journal of Physical Chemistry

On-the-fly Ab Initio Semiclassical Dynamics of Floppy Molecules: Absorption and Photoelectron Spectra of Ammonia Marius Wehrle, Solène Oberli, Jiří Vaníček * Laboratory of Theoretical Physical Chemistry, Institut des Sciences et Ingénierie Chimiques, Ecole Polytechnique Fédérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland * [email protected]

electronic absorption spectrum, photoelectron spectrum, ammonia, thawed Gaussian approximation, direct semiclassical dynamics, on-the-fly ab initio semiclassical dynamics

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Abstract We investigate the performance of on-the-fly ab initio (OTF-AI) semiclassical dynamics combined with the thawed Gaussian approximation (TGA) for computing vibrationally resolved absorption and photoelectron spectra. Ammonia is used as a prototype of floppy molecules, whose potential energy surfaces display strong anharmonicity. We show that despite complications due to the presence of large amplitude motion, the main features of the spectra are captured by the OTF-AI-TGA, which— by definition—does not require any a priori knowledge of the potential energy surface. Moreover, the computed spectra are significantly better than those based on the popular global harmonic approximation. Finally, we probe the limit of the TGA to describe higher-resolution spectra, where long time dynamics is required.


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I. INTRODUCTION Ab initio semiclassical methods have proven to provide a powerful tool for decoding molecular spectra:1-8 Not only do the classical trajectories underlying such methods help visualize the relevant physical and chemical processes, but, in contrast to popular ab initio classical molecular dynamics, nuclear quantum effects are taken into account at least approximately.

Since the propagation

requires only a local knowledge of the potential energy surface (PES), an on-the-fly ab initio (OTFAI) scheme can be applied, circumventing the construction of global PESs. The overall cost, however, limits almost all semiclassical methods to small systems. Challenged by this limitation, we showed on the example of oligothiophenes that the efficiency of Heller’s thawed Gaussian approximation (TGA)9 combined with the OTF-AI scheme permits calculating vibrationally resolved spectra of large systems with up to 105 degrees of freedom (DOFs).10 Furthermore, we showed that this OTF-AI-TGA can sometimes capture qualitatively the effect of transitions to PESs with a double-well character, such as in the emission spectra of oligothiophenes the equilibrium geometries of which are twisted in the ground state and planar in the excited state. Within the OTF-AI-TGA, the nuclear vibrational wave function is described by a single Gaussian wave packet (GWP) whose time-dependent width is propagated using the local harmonic approximation of the PES. Since the short-time dynamics is often well described by a thawed GWP, the OTF-AI-TGA has been developed with the purpose of computing broad vibronic spectra. Here, we focus on how the OTF-AI-TGA performs on floppy molecules, the spectra of which depend strongly on large amplitude motions, and hope to find an improvement over the standard approach based on the more drastic, global harmonic approximation for the PESs.

When the

vibronic spectra of floppy molecules are calculated within the global harmonic approximation,11-18 the choices of the coordinate system and reference structures are crucial. Furthermore, to produce satisfactory results, the global harmonic approximation often had to be combined with anharmonic potentials for the floppy DOFs,14,19 requiring a preliminary exploration of the PESs, which is rather 3 ACS Paragon Plus Environment


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cumbersome if not impossible for larger molecules. In contrast, the OTF-AI-TGA completely avoids this “prescreening” and treats all DOFs on an equal footing; yet, the anharmonicity is captured partially, and, if needed, the PES can be studied afterwards by extracting the most important DOFs using, e.g., the approach proposed by us in ref 10. We use ammonia (NH3) as a prototype of a floppy molecule, on which, due to its small size, different and rather accurate levels of theory for the OTF-AI scheme can be compared, permitting to separate the errors due to electronic structure evaluation from those due to the dynamical approximation. The ← (S ← S ) absorption spectrum of ammonia is dominated by a long progression resulting from the activation of its well-known umbrella vibration,20-22 along which the PES of the state has a shape of a double well with two degenerate minima with pyramidal geometry of C3v symmetry. In contrast, the trigonal planar equilibrium configuration of the excited state belongs to the D3h point group. Such a non-planar to planar electronic transition involves a substantial displacement between equilibria and, as a result, induces a large-amplitude nuclear motion exploring anharmonic regions of the excited PES. The spectral broadening is caused by the short lifetime of the state, which has been assigned to strong pre-dissociation.23,24 While we do not explore it here, the photodissociation of NH3 due to the quasi-bound nature of the FranckCondon region of and its influence on the diffuse character of the spectrum have been studied by full-dimensional nonadiabatic quantum dynamics calculations.25-29 We also probe the limits of the OTF-AI-TGA’s capability to describe long-time propagation required for higher-resolution vibronic spectra such as the ← photoelectron spectrum of NH → NH + that shows both a strong and weak progressions (see Edvardsson et al.30 and references therein), and has served as a benchmark for spectra calculations of floppy molecules.14,19 The relevant dynamics is similar to the dynamics responsible for the absorption spectrum: The equilibrium geometry of the ionic ground state is also planar with D3h symmetry and the main vibration induced by the excitation is the umbrella motion resulting in a long, intense progression. In contrast to the absorption spectrum, however, another, weak progression appears. 4 ACS Paragon Plus Environment

Theoretical

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studies using the multiconfiguration time-dependent Hartree (MCTDH) quantum dynamics method combined with high-level ab initio PESs31-33 assigned this weak progression to the totally symmetric stretching motions.31,32 Our OTF-AI-TGA calculations of absorption and photoelectron spectra are compared with different global harmonic models in order to evaluate the effect of the global harmonic approximation on the peak spacing and progressions appearing in the spectra.

Experimental

absorption and photoelectron spectra serve as benchmarks. II. THEORY A. Absorption and photoelectron spectra calculations In the time-dependent approach to electronic spectroscopy, popularized by Heller,34 the spectrum is obtained, using the electric-dipole approximation and time-dependent perturbation theory, as the Fourier transform

σ (ω ) =

2π ω C µµ ( t ) e iωt 3h c ∫

(1)

of the dipole time autocorrelation function Cµµ (t ) := Tr[ρˆinit µˆ (t ) µˆ ], whose shape is modulated by the nuclear motion induced by the electronic excitation. Whereas the NH3 absorption spectrum arises from the light-induced nuclear motion on the electronic surface, the photoelectron spectrum is determined by the nuclear motion on the cationic surface generated by the ejection of an electron from the neutral molecule. In the low temperature limit and within the Franck-Condon approximation the time autocorrelation function becomes 2

C µµ (t ) ≈ µ ge C (t ) e

iE g t / h

,

(2)

where is the transition dipole moment between the ground and excited electronic states, Eg is the energy of the vibrational ground state of the ground electronic state, and ˆ

C(t ) = 〈ψ init | e−iHet /h |ψ init 〉 is the wavepacket autocorrelation function of the initial nuclear state init , stationary in the ground after excitation. electronic state, but which starts to evolve with the excited-state Hamiltonian 5 ACS Paragon Plus Environment

(3)


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B. On-the-fly ab initio TGA Within the TGA,9,35 the nuclear GWP is guided by a classical trajectory feeling the full anharmonicity of the potential, while the width of the GWP is propagated using a time-dependent " effective potential eff (#) given by the local harmonic approximation of the potential constructed at

the current normal mode coordinate #" of the central trajectory,

(

Vefft ( q ) = V |qt + grad qV |qt

) ⋅ ( q − q ) + 12 ( q − q ) T

t T

t

⋅ Hess qV |qt ⋅( q − q t ).

(4)

Here |%& , grad% |%& , and Hess% |%& are the potential energy, its gradient, and Hessian evaluated at the center # " of the GWP at time ,. The quadratic nature of the effective potential ensures that the wave packet remains a Gaussian of the form

  ψ t (q) = N 0 exp −(q − qt )T ⋅ At ⋅ (q − qt ) + ( pt )T ⋅ (q − qt ) + γ t  , i h



(5)



where N0 is a normalization constant, - " = (#" , 0" ) denotes the phase-space coordinates of the GWP’s center, At is a complex symmetric width matrix, and the real part of the complex number 1 " represents an overall phase factor while its imaginary part guarantees normalization of " (#) for , ≥ 0. For further details about the GWP propagation using the TGA, see Supporting Information and refs 9,10,35. Vibrational normal mode coordinates # provide an appropriate choice of coordinates for propagating the TGA wave packet as well as for constructing the multidimensional global harmonic PESs. Let N denote the number of atoms in the molecule and let vector 4ref contain the 3N Cartesian coordinates of a reference structure, for which we used one of the two degenerate ground state equilibrium geometries of NH3. Cartesian coordinates 4 of a general configuration are transformed into the 36 normal mode coordinates q about 4ref by the equation

q = OT m1/2 (ξ − ξref ) ,

(6)

where m is the 36 × 36 diagonal mass matrix and O is the 36 × 36 orthogonal matrix diagonalizing

the

mass-scaled

Cartesian

Hessian

evaluated

at

489: ,

i.e.,

;< =⁄ Hess? |?ref =⁄ ; = @ and @ = diag(A , … ,AC ) is the diagonal matrix containing 6 ACS Paragon Plus Environment

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the normal mode frequencies. Even though we assume the center of mass of the reference geometry 4ref to be at the origin, this is not required from the general configuration 4. As a result, q still contains three translational and three rotational DOFs, all six of them with zero frequencies. To remove the uninteresting translations and rotations from q, the transformation from the ab initio Cartesian coordinates 4 to the vibrational normal-mode coordinates is done in three steps:10 First, to eliminate translational DOFs, the configuration 4 is shifted to the center-of-mass frame by 4 ≔ 4 − G; more explicitly, for each atom a, the vector 4H containing its three coordinates is shifted as C 4H ≔ 4H − 4IJK , where 4IJK ≔ (∑C HM =H 4H )/(∑HM =H ) . Second, to minimize the rovibrational

coupling, each 4H is rotated by a rotation O to the Eckart frame36, 4H ≔ O4H , which is equivalent to minimizing the distance (in mass-scaled coordinates) between 4′′ and 4ref ;37,38 for this we employed the quaternion formalism.39 Third, the transformation is completed by applying eq 6 to 4′′ instead of 4 and by replacing O with its 36 × (36 − 6) submatrix L, which amounts to dropping the translational and rotational DOFs. All together, the transformation of the geometry, gradient, and Hessian from Cartesian to 36 − 6 vibrational normal mode coordinates q is achieved by

q = LT m1/2 [ R(ξ − δ ) − ξref ] grad qV = gradξV ⋅ RT m−1/2 L −1/2

HessqV = L m T

(7)

R ⋅ HessξV ⋅ R m T

−1/2

L,

where we slightly abused notation by using the notation O also for the block-diagonal 36 × 36 matrix consisting of N identical 3 × 3 blocks which apply the same rotation O to each atom. In the OTF-AI scheme, the geometries, gradients, and Hessians of the excited PES along the trajectory are evaluated in Cartesian coordinates by an ab initio electronic structure package and transformed to normal mode coordinates using eq 7. C. Global harmonic potential construction When computing vibronic spectra, it is popular to approximate the relevant PESs globally by multidimensional harmonic oscillators; we refer to this as the global harmonic approximation and will show its limitations below. The reason for the widespread use of the global harmonic 7 ACS Paragon Plus Environment


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approximation is the existence of efficient exact algorithms allowing the treatment of even large molecules in this setting.18,40,41 However, one can also (and we do) evaluate the global harmonic spectra with the TGA since it, too, is exact in globally quadratic PESs. The harmonic model for the PES i expressed in vibrational normal mode coordinates about the reference geometry 4ref has the general form Vi (q ) = V0,i +

1 T ( q − di ) ⋅ Ki ⋅ ( q − di ) , 2

(8)

with a minimum ,R at position di and the force matrix Ki given by

V0,i = Vi |qi −Erea di = −Ki−1 ⋅ gradqVi |qi +qi Ki = HessqVi |qi , where the energy R |%S , gradient grad% R |%S , and Hessian Hess% R |%S are the results of ab initio calculations at geometry 4R and transformed into normal mode coordinates #R using eq 7. Finally,

Trea = (UR − #R )< ⋅ WR ⋅ (UR − #R ) is the rearrangement energy. The global harmonic models are usually based on the following choices for 4ref and 4R :11,18 In the ground state global harmonic approximation, W is diagonal and U = XY 0 since the ground state equilibrium geometry serves as a reference state (4ref = 4 ). As for the excited state, there are two natural yet different choices for the harmonic PES: The adiabatic harmonic PES corresponds to ab initio data computed at the excited state equilibrium geometry 4 , whereas the vertical harmonic PES is constructed using ab initio data evaluated on the excited state at the ground state equilibrium geometry 4 . We constructed all three harmonic PESs; all of them had only real frequencies.

III. COMPUTATIONAL METHODS Dunning’s correlation consistent basis set aug-cc-pVTZ was used in all ab initio calculations. The Molpro2012.142 package was used for CASPT2 and GAUSSIAN0943 for B3LYP, CCSD, and MP2 calculations (for precise specifications see Supporting Information). The initial GWP was the ground vibrational eigenstate of the harmonic fit to the PES at one of the two degenerate equilibrium 8 ACS Paragon Plus Environment

(9)

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geometries of the state. The GWP was propagated with a time step of 8 a.u. ≈ 0.2 fs for 1000 steps (a total time of 8000 a.u ≈ 193.5 fs) for absorption spectra and for 2000 steps (a total time of 16000 a.u. ≈ 387.0 fs) for photoelectron spectra. The resulting spectra were shifted (Supporting Information Table S 2) to eliminate the systematic errors of ab initio vertical excitation energies and broadened by a phenomenological (inhomogeneous) Gaussian with half-width at half-maximum of either 200 cm-1 ≈ 0.025 eV (for absorption spectra) or 50 cm-1 ≈ 0.006 eV (for photoelectron spectra). Further details are found in Supporting Information.

IV. RESULTS AND DISCUSSION D. Absorption spectrum A single long progression due to the umbrella motion of NH3 is observed in the experimental absorption spectrum (see Figure 1), which exhibits substantial broadening, especially in the higher energy region, largely due to photodissociation. Such diffuse spectra depend only on short-time dynamics,34 during which the OTF-AI-TGA GWP remains trapped in the quasi-bound region of , making the comparison with global harmonic models straightforward; in all our calculations, the broadening was taken into account phenomenologically. Figure 1 compares the experimental spectrum44 with spectra calculated with the adiabatic harmonic, vertical harmonic, and our OTF-AI-TGA approach using CASPT2 and B3LYP levels of theory. In the adiabatic harmonic model, the stretching modes are overly excited due to their coupling to the bending mode in the Cartesian representation, which results in unphysical progressions.12,14,19 Furthermore, the adiabatic harmonic model is very sensitive to the ab initio level of theory: Small changes in equilibrium geometries have a dramatic effect on the spectrum. The vertical harmonic model suffers much less from these two problems and, in addition, it obviously provides a better description of the Franck-Condon region important for spectra calculations.12,15,45 Still, it is clear that neither of the two harmonic models can reproduce the anharmonic peak spacing. In contrast, the local harmonic approximation employed in the OTF-AI-TGA captures partially the 9 ACS Paragon Plus Environment


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anharmonicity of the PES, resulting in an almost perfect peak spacing in the OTF-AI-TGA spectrum (numerical values of peak energies are listed in the Supporting Information Table S 3). Furthermore, the progression is longer than in the vertical harmonic spectrum and the relative intensity distribution matches well the experiment. Finally, by comparing the B3LYP and CASPT2 calculations we conclude that the ab initio level of theory affects more the intensity distribution than the peak spacing.

Figure 1: Absorption spectrum of NH3: Comparison of the experimental spectrum44 recorded at the temperature of 175 K with the spectra computed with the OTF-AI-TGA, vertical harmonic (VH), and adiabatic harmonic (AH) models within the B3LYP and CASPT2 ab initio methods. All spectra are rescaled so that the highest spectral peak in each spectrum is of unit intensity. E. Photoelectron spectrum Figure 2 compares the experimental photoelectron spectrum30 with spectra calculated using the OTF-AI-TGA, vertical harmonic, and adiabatic harmonic models within the CCSD and MP2 levels of theory. As expected, the adiabatic harmonic model fails completely, while the vertical harmonic spectrum contains both the intense and weak progressions, but the main progression is too short and 10 ACS Paragon Plus Environment

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the peak spacing is poor.

Compared with the vertical harmonic model, the OTF-AI-TGA

significantly improves both the peak spacing and intensity distribution of the main progression (numerical values of peak energies are listed in the Supporting Information Table S 3). Yet, the spectral intensities obtained with the OTF-AI-TGA combined even with the expensive CCSD method decay faster with increasing distance from the highest peak than do the experimental intensities because the TGA cannot capture the nonlinear spreading of the wavepacket due to anharmonicities in the exact potential. Here this effect is stronger than in the absorption spectra since the photoelectron spectra depend on much longer dynamics, allowing the nonlinearities to accumulate. Interestingly, this affects mainly the spectral intensities and not the peak positions, implying that in the photoelectron dynamics of ammonia the local harmonic approximation for the PES remains accurate for the phase of the thawed Gaussian but not for its density. Also, the weak progression is hidden: The OTF-AI-TGA spectrum exhibits noise and negative intensities, which appear, to a much lower extent, also in the lower-resolution absorption spectrum (see Figure 1). The origin of these negative intensities goes back to the local harmonic approximation, which results in a time-dependent effective potential [eq 4] and, therefore, nonconservation of energy—positive intensities in spectra are only guaranteed for time-independent Hamiltonians.

To explore this

phenomenon in more detail, one can write the total energy of the GWP as a sum of the classical energy Tcl along the central trajectory and semiclassical energy Tsc due to the finite size of the nuclear wavepacket (derivation is shown in Supporting Information). Figure 3 shows that in the OTF-AI propagation scheme, Tcl is conserved exactly since the central trajectory feels the exact, time-independent potential, while Tsc is growing with time. As a result, the total GWP energy is not conserved, which is reflected as noise in the spectrum. This effect does not appear in quadratic PESs (for which the local harmonic approximation does not change the exact potential), and is also negligible for short times.



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Figure 2: Photoelectron spectrum of NH3: Comparison of the experimental spectrum30 with the spectra computed with the OTF-AI-TGA, vertical harmonic (VH), and adiabatic harmonic (AH) models using MP2 and CCSD ab initio methods. All spectra are rescaled so that the highest spectral peak in each spectrum is of unit intensity.

Figure 3: Time dependence of the classical (Tcl ) and semiclassical (Tsc ) energies for the OTF-AITGA and vertical harmonic (VH) model using CCSD level of theory. The total GWP energy TGWP is the sum of Tcl evaluated along the central trajectory and Tsc resulting from the nuclear GWP width. The TGA is exact for quadratic PESs and, consequently, TGWP is constant in the VH model. For


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general potentials, however, Tcl stays constant, whereas Tsc increases with time due to the time dependence of the effective potential.

V. CONCLUSION In conclusion, we used ammonia as a prototype of floppy molecules in order to probe the ability of the OTF-AI-TGA to depict large amplitude nuclear motions. Additionally, we pushed the OTF-AITGA method to its limit by performing long time propagation in order to assess the accuracy of this approach in describing higher-resolution spectra. The absorption and photoelectron spectra were calculated within the OTF-AI-TGA method and compared with experiment and popular global harmonic approaches. As expected, the adiabatic harmonic model is not usable for floppy molecules with strongly anharmonic PESs, whereas the vertical harmonic model reproduces the main spectral features. The accuracy of the spectra is further significantly improved with the less severe local harmonic approximation employed in the OTF-AI-TGA. We found that in ammonia the simple OTFAI-TGA is adequate even in the case of large amplitude motions: Both the peak spacing and intensity distributions of experimental spectra are well reproduced. Two drawbacks of the method are that, in contrast to, for example, methods employing Gaussian bases,46-49 OTF-AI-TGA cannot describe wavepacket splitting and that in the limit of long-time propagation necessary for higherresolution spectra, the local harmonic approximation used in the OTF-AI-TGA induces noise in the spectra that can hide weak progressions. Yet, the computed spectra are better than spectra obtained with the popular global harmonic approaches. Finally, although the OTF-AI-TGA is obviously less accurate than the full quantum dynamical calculation on analytical surfaces, it is much less expensive and does not require any prescreening of the PESs.

ASSOCIATED CONTENT Supporting Information



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

Details of electronic structure calculations, equilibrium geometries, details of the TGA propagation and spectra calculation, peak energies, propagation of the width matrix and phase of the GWP, and the derivation of the GWP energy. This material is available free of charge via the Internet at http//pubs.acs.org.

AUTHOR INFORMATION Corresponding Author *Email: [email protected] Notes The authors declare no competing financial interests. Author Contributions The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript.

ACKNOWLEDGMENT This research was supported by the Swiss National Science Foundation (NSF(CH)) with Grant No. 200020_150098 and National Center of Competence in Research (NCCR) Molecular Ultrafast Science and Technology (MUST), and by the EPFL. The authors would like to thank Miroslav Šulc for discussions. ABBREVIATIONS DOF, degree of freedom; GWP, Gaussian wave packet; MCTDH, multiconfiguration time-dependent Hartree; OTF-AI, on-the-fly ab initio; PES, potential energy surface; TGA, thawed Gaussian approximation.

REFERENCES


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