Theoretical investigation of intramolecular vibrational

THE JOURNAL OF CHEMICAL PHYSICS 124, 194304 共2006兲

Theoretical investigation of intramolecular vibrational energy redistribution in highly excited HFCO Gauthier Pasin,a兲 Fabien Gatti,b兲 and Christophe Iungc兲 LSDSMS (UMR 5636-CNRS), CC 014, Université Montpellier II, F-34095 Montpellier, Cedex 05, France

Hans-Dieter Meyerd兲 Theoretische Chemie, Universität Heidelberg, Im Neuenheimer Feld 229, D-69120 Heidelberg, Germany

共Received 1 February 2006; accepted 9 March 2006; published online 16 May 2006兲 The present paper is devoted to the simulations of the intramolecular vibrational energy redistribution 共IVR兲 in HFCO initiated by an excitation of the out-of-plane bending vibration 关n␯6 = 2 , 4 , 6 , . . . , 18, 20兴. Using a full six-dimensional ab initio potential energy, the multiconfiguration time-dependent Hartree 共MCTDH兲 method was exploited to propagate the corresponding six-dimensional wave packets. This study emphasizes the stability of highly excited states of the out-of-plane bending mode which exist even above the dissociation threshold. More strikingly, the structure of the IVR during the first step of the dynamics is very stable for initial excitations ranging from 2␯6 to 20␯6. This latter result is consistent with the analysis of the eigenstates obtained, up to 10␯6, with the aid of the Davidson algorithm in a foregoing paper 关Iung and Ribeiro, J. Chem. Phys. 121, 174105 共2005兲兴. The present study can be considered as complementary to this previous investigation. This paper also shows how MCTDH can be used to predict the dynamical behavior of a strongly excited system and to determine the energies of the corresponding highly excited states. © 2006 American Institute of Physics. 关DOI: 10.1063/1.2192499兴 I. INTRODUCTION

Simulations of the intramolecular vibrational-energy redistribution 共IVR兲 in polyatomic molecules are of utmost importance because the IVR process can have a decisive influence on the reactivity of a molecular system.1,2 Consequently, the knowledge of the IVR is an essential prerequisite for initiating chemical reactions via mode-selected vibrational excitations by laser light.3 For instance, at energies above the threshold for a given bond dissociation, vibrational energy is required to flow between vibrational modes into the reaction coordinate. This energy flow can then be the limiting factor in determining the rate of unimolecular dissociation. It is intuitive to think that an initial vibrational excitation leads to a complete redistribution over all vibrational modes in a statistical way as assumed in the RiceRamsperger-Kassel-Marcus 共RRKM兲 theory.4 In the same manner, it is natural to postulate that the IVR time scale is directly linked to the state density. However, sophisticated experimental techniques have been used to show that IVR does not necessarily increase with the state density and that the corresponding mechanism can be subtle and difficult to analyze.5,6 In order to analyze the mine of information present in the experimental data, simple models are not sufficient. Moreover, it is important to dwell upon the fact that these molecular processes are significantly impacted by nuclear quantum mechanical effects. Therefore, it is crucial a兲

Electronic mail: [email protected] Electronic mail: [email protected] c兲 Electronic mail: [email protected] d兲 Electronic mail: [email protected] b兲

0021-9606/2006/124共19兲/194304/11/$23.00

to perform quantal studies which take into account all couplings between modes to correctly describe IVR in molecules. Recently, it was shown that multiconfiguration timedependent Hartree 共MCTDH兲 is a very efficient tool to investigate the IVR in relatively large systems such as HONO 共six degrees of freedom兲,7,8 or toluene9 and fluoroform 共nine degrees of freedom兲.10 This stack of investigations aim at developing a systematic study of the IVR for numerous systems and could offer a precious framework for a synergy between experimentalists and theoreticians in this field. The HONO molecule is characterized by a reversible cis-trans-isomerization which complicates the dynamics in the molecule.11–13 We have confirmed that the cis- → trans-process proceeds faster than the opposite direction and that there are very large differences between the energy redistributions after different initial excitations stressing the strong mode selectivity of HONO. In the fluoroform and toluene systems, the energy flow from CH stretch to the various other modes occurs on very different time scales and therefore multiple IVR pathways. For the fluoroform molecule, we have reinvestigated10 the dominant feature in the CH overtone spectra, i.e., the strong Fermi resonance between the CH stretch and the two FCH bends. New simulations have been performed very high in energy establishing the crucial role played by the FCH bends which were found to constitute an energy reservoir. Note that the IVR of the CH chromophore is of high interest since it can be probed in myriads of molecular environments and has pronounced absorption signals.14–17 For toluene, the situation is even more complicated since it is not clear yet if the complex structure of the experimental CH spectra can be ascribed to the CH

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stretch-bend Fermi resonance 共like for fluoroform兲 or to a coupling with the almost free internal rotation of the methyl group.18,19 In this paper, we focus on the formyl fluoride molecule HFCO which is of particular interest for it displays a relatively low dissociation threshold, about 14 000 cm−1 above the ground state, as compared to formaldehyde 共approximately 28 000 cm−1兲. Furthermore, this system is an excellent prototype to study mode specificity in unimolecular dissociation since the out-of-plane bending mode 共␯6兲 is weakly coupled to the reaction coordinate which lies entirely in the molecular plane 共the transition state is planar with an energy of approximately 17 900 cm−1 above the minimum兲. Indeed, HFCO has been the subject of intensive experimental studies conducted by Moore and co-workers.20–23 Using stimulated emission pumping 共SEP兲 experiments, they have observed and characterized energy levels 共bright states兲 in the range of 13 000– 23 000 cm−1, which correspond to metastable states displaying predissociation widths. These vibrational 共bright兲 states are extremely stable against state mixing in contrast to the general belief that the vibrational quantum states above the dissociation threshold are chaotic. Moreover, one experimental finding seems to indicate that the decoupling of the ␯6 mode from the other modes, and, in particular, from those leading to the dissociation into HF + CO, increases with increasing internal energy of the ␯6 mode. These experimental studies have stimulated Yamamoto and Kato24–26 to determine a sophisticated ab initio global six-dimensional potential energy surface, which includes the dissociation pathway. This potential surface was constructed with the use of 4140 ab initio potential energies computed at the relativistic Hartree-Fock 共RHF兲/MP2 level. It should be stressed that the corresponding ab initio energies were chosen from the potential minimum up to 20 kcal/ mol above the transition state energy 共around 70 kcal/ mol⯝ 3 eV ⯝ 24 000 cm−1 above the minimum兲. There is an energy cutoff at 3.5 eV 共=28 229 cm−1兲 above the minimum in the potential energy surface 共PES兲. Hence the isomerization HFCO→ t-FCOH is not described by this surface. The relevant portions of the PES were reproduced with the rootmean-squares 共rms兲 deviation of 1.5 kcal/ mol 共⯝65 meV ⯝ 520 cm−1兲. Several contributions have then been put forth to calculate the infrared spectrum of HFCO with this PES. For instance in Ref. 27, a quantum mechanical calculation of the 450 vibrational energy levels and eigenvectors from the ground state up to about 7500 cm−1 of excitation energy of HFCO has been reported. This time-independent quantum mechanical study rests on the Davidson scheme coupled to a specific prediagonalization. More recently,28 special emphasis has been placed, within the framework of the same approach, on highly excited states, i.e., the fifth, seventh, and ninth out-of-plane overtones. In particular, an analysis of these overtones in terms of their main zero-order contributions in the normal mode picture has been reported. In this work, we present the simulation of the IVR in HFCO whose out-of-plane mode is initially excited. The Heidelberg package29 of the MCTDH algorithm30–34 is employed to tackle this problem. This study can be seen as complementary to the previous calculations of the eigenstates with the

Davidson algorithm.28 As in Ref. 28, we concentrate on the A⬘ symmetry, i.e., on excitations with an even number of quanta in the out-of-plane mode. This study could be, of course, straightforwardly extended to the other excitations. The technical difficulties in handling this problem stem firstly from deriving and implementing the kinetic energy in polyspherical valence coordinates in the MCTDH package and secondly from the refitting of the six-dimensional potential energy surface to a form which is required for the full efficiency of MCTDH 共see Sec. II A兲. Besides the physical interest of the IVR of HFCO, the goal of this paper is also to show that MCTDH can supply accurate results, although the form of the original PES is not adapted to MCTDH. In particular, one crucial point is the influence of the quality of the refitting on the IVR and the eigenenergies, on which careful emphasis will be placed. In addition, we converge for the first time vibrational eigenvalues very high in energy with MCTDH. The outline of this study is as follows. In Sec. II, a brief description of the theoretical background is given, including an outline of the MCTDH method 共Sec. II A兲, the analytical expression of the kinetic energy operator 共Sec. II C 1兲 in valence polyspherical coordinates and a careful investigation of the convergence of the refitting of the potential surface 共Sec. II C 2兲. Section III is devoted to the simulation of IVR after excitation from 2 to 20 quanta in the out-of-plane mode. It provides a comparison with previous theoretical and experimental works. The paper concludes presenting perspectives for the future.

II. THEORY A. The multiconfiguration time-dependent Hartree method

MCTDH 共Refs. 30–34兲 is a general algorithm to solve the time-dependent Schrödinger equation for distinguishable particles. MCTDH achieves its efficiency from writing highdimensional objects as sums of products of low-dimensional objects. For example, the MCTDH wave function is written as a sum of products of so-called single-particle functions 共SPFs兲. The SPFs, ␸共兵q其 , t兲, may be one- or multidimensional functions and, in the latter case, the coordinate 兵q其 is a collective one, 兵q其 = 兵qk , . . . , ql其. ⌿共q1, . . . ,q f ,t兲 ⬅ ⌿共兵q其1, . . . ,兵q其 p,t兲 n1

np

p

j1

jp

␬=1

= 兺 ¯ 兺 A j1,. . .,j p共t兲兿 ␸共j␬兲共兵q其␬,t兲 = 兺 A J⌽ J ,

␬

共1兲

J

where f denotes the number of degrees of freedom and p the number of MCTDH particles, also called combined modes. There are n␬ SPFs for the ␬th particle. The AJ ⬅ A j1¯j f denote the MCTDH expansion coefficients and the configurations, or Hartree products, ⌽J are products of SPFs, implicitly defined by Eq. 共1兲. The SPFs are finally represented by linear combinations of time-independent primitive basis functions ␹


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Intramolecular vibrational energy redistribution in HFCO

␸共j␬␬ 兲共兵q其␬,t兲

B. The potfit algorithm

N␬

=

兺 c共i ␬兲j 共t兲␹i共␬兲共兵q其␬兲. i =1 ␬

␬ ␬

␬

共2兲

A discrete variable representation35,36 共DVR兲 is mostly used in practice. The MCTDH equations of motion are derived by applying the Dirac-Frenkel variational principle37,38 to the ansatz Eq. 共1兲. They read iA˙J = 兺具⌽J兩H兩⌽L典AL ,

共3兲

L

Potfit is an algorithm which brings a general multidimensional function to product form. The multidimensional function is assumed to be given by its values on a multidimensional product grid. Potfit operates on the grid points only. Despite its name, there is no fitting by continuous functions like polynomials or splines involved. We assume that the values of a potential V are given on a product grid V共兵q其i共1兲, . . . ,兵q其i共p兲兲 = Vi1. . .ip ,

i␸˙ 共␬兲 = 共1 − P共␬兲兲共␳共␬兲兲−1具H典共␬兲␸共␬兲 ,

共4兲

where a vector notation, ␸共␬兲 = 共␸共1␬兲 , . . . , ␸n共␬兲兲T, is used. A dis␬ cussion of these equations, details on the derivation, as well as more general results, can be found in Refs. 31–34 and 39. The MCTDH equations conserve the norm and, for timeindependent Hamiltonians, the total energy. MCTDH contains time-dependent Hartree 共TDH兲 and the standard method 共i.e., propagating the wave packet on the primitive basis兲 as limiting cases.40 MCTDH simplifies to TDH when setting all n␬ = 1. Increasing the n␬ recovers more and more correlation, until finally, when n␬ equals the number of primitive basis functions, the standard method is used. It is important to note that MCTDH uses variationally optimal SPFs, because this ensures fast convergence. The solution of the MCTDH equations of motion requires to build mean fields at every time step. Hence it is of vital importance to use a fast algorithm for this buildup. Here again the trick of writing high-dimensional objects as sum of products of low-dimensional ones is helpful. If the Hamiltonian is of the product form s

p

r=1

␬=1

ˆ = 兺 c 兿 hˆ共␬兲 , H r r

where 兵q其i共␬兲 denotes a grid point of the ␬th one- or multidi␬ mensional grid with 1 艋 i␬ 艋 N␬. Here N␬ denotes the number of grid points of the grid for the ␬th particle and p denotes the number of particles. Similar as in MCTDH, the variable 兵q其 may be a one- or multidimensional 共i.e., collective兲 coordinate. Next we define potential density matrices 共␬兲共␬兲 nm

s

p

r=1

␬=1

ˆ 兩⌽ 典 = 兺 c 兿具␸ 兩hˆ共␬兲兩␸ 典. 具⌽J兩H L r j␬ r l␬

N1

=

N␬−1

兺

Similar equations apply for evaluating the mean fields 具H典共␬兲. 共An alternative to this algorithm is, e.g., the correlation discrete variable representation 共CDVR兲 method of Manthe40兲. Kinetic energy operators may assume a very complicated form when internal curvilinear coordinates are used. However, despite their complicated appearance, they very often are of product form Eq. 共5兲. In particular, kinetic energy operators of nonlinear semirigid systems in curvilinear polyspherical coordinates are always of the desired product form. Turning to potential operators, the situation is less accommodating. A potential may be given by a complicated, nonseparable analytic expression. Then one may use the potfit algorithm,33,41,42 which is briefly described below, to expand the potential into the desired product form.

Np

兺 Vi ¯i

¯

i␬−1=1 i␬+1=1

1

i p=1

␬−1ni␬+1¯i p

⫻Vi1¯i␬−1mi␬+1¯ip

共8兲

and determine their orthonormal eigenvectors, the so called natural potentials, ␷共j␬兲, with components ␷共ij␬兲 as well as their corresponding eigenvalues ␭共j␬兲 called natural weights. The natural weights are assumed to be in decreasing order, ␭共j␬兲 ␬兲 ⬎ ␭共j+1 , and we introduce the notation ␷共j␬兲共兵q其共i ␬兲兲 ª ␷共ij␬兲. We may now approximate the potential as follows: V共兵q其i共1兲, . . . ,兵q其i共p兲兲 p

1

⬇V =

共兵q其i共1兲, 1

app

m1

mp

1

p

. . . ,兵q其i共p兲兲 p

共1兲共p兲共p兲 ¯ 兺 C j ¯j v共1兲兺 j 共兵q其i 兲 ¯ v j 共兵q其i 兲, j =1 j =1 1

p

1

p

1

共9兲

p

with m␬ 艋 N␬. The expansion coefficients C j1¯j p are the overlaps between the potential and the natural potentials N1

C j1. . .j p =

兺 i =1

Np

¯

1

共6兲

N␬+1

兺兺

¯

i1=1

共5兲

where hˆr共␬兲 operates on the ␬th particle only and where the cr are numbers, then the matrix elements of the Hamiltonian can be expressed by a sum of products of monomode integrals,

共7兲

p

1

共p兲 Vi ¯i v共1兲兺 i j ¯ vi j . i =1 1

p

共10兲

p p

1 1

p

Moreover, in order to make the representation as compact as possible, a contraction over the ␯th mode is performed. The following functions are defined D j1. . .j␯−1 j␯+1. . .j p共兵q其i共␯兲兲 = ␯

m␯

兺 C j . . .j . . .j v共i ␯j兲 . j =1 1

␯

␯

p

共11兲

␯ ␯

This allows us to rewrite the potential energy surface as Vapp共兵q其i共1兲, . . . ,qi共p兲兲 =

1

p

m1

m␯−1

m␯+1

mp

1

␯−1

␯+1

p

共1兲共␯−1兲共兵q其i共␯−1兲兲兺 ¯ j 兺=1 j 兺=1 ¯ j兺=1 v共1兲 j 共兵q其i 兲 . . . v j j =1 1

1

␯−1

␯−1

共p兲 ⫻ D j1. . .j␯−1 j␯+1. . .j p共兵q其i共␯兲兲v共j␯+1兲共兵q其i共␯+1兲兲 . . . v共p兲 j 共兵q其i 兲. ␯

␯+1

␯+1

p

p

共12兲


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This contraction over the ␯th mode reduces the number of expansion terms by the factor m␯. The original potential is exactly reproduced 共on the grid points兲 when m␬ = N␬. However, a sufficiently accurate approximation of the potential is usually obtained with much smaller values of m␬. The algorithm produces an approximation which is almost optimal in the L2 sense. The fit can be further refined by specifying a relevant region. The fit is improved in this region by allowing larger deviations outside the relevant region. For more details, discussion of weights, error analysis, etc., the reader is referred to the articles.33,41,42

lence vectors reads R1 = CH, R2 = CF, and R3 = CO. The algorithm, outlined in Ref. 44, gives the kinetic energy operator 关see Eq. 共A1兲 of Ref. 44兴 as 6

1 Tˆ = 兺共pˆ†nGnm pˆm兲, 2 n,m=1

where pˆn denote the momenta conjugate to the six polyspherical coordinates. The matrix G depends on the massmatrix M which in the present case reads

M= C. Hamiltonian 1. Kinetic energy operator

A valence set of three vectors, Ri, allows one to parametrize the configuration of the HFCO molecule.43–45 These three vectors are in turn parametrized by six polyspherical BF BF coordinates 共R1, R2, R3, ␪BF 1 , ␪2 , ␸1 兲, which are depicted in Fig. 1, and three Euler angles which orient the molecule such that R3 is parallel to the z axis and R2 lies in the xz plane. The kinetic energy operator of the HFCO molecule in polyspherical coordinates appears as a particular case of the general expression which is discussed in Ref. 44 and reviewed recently in Ref. 46. Comparing with Eqs. 共4兲 and 共A1兲 of Ref. 44 one has to apply these general equations to the present case characterized by N = 4 atoms and a vanishing total angular momentum, J = 0. The choice of the three va-

3

T=−

M ii ⳵2 − 2 ⳵R2i

兺 i=1

兺冉 2

−

i=1

−

2R23

M i3␰i

冊冉

冊冉

u2i

冊

⳵ ⳵ 2 + u − ⳵␰i ⳵␰i i

冊

兺 i,j=1

冋

1 mC

1 mC

1 1 + mF mC

1 mC

1 mC

1 mC

1 1 + mO mC

册

冊

2,j⫽i

兺 i,j=1

2,j⫽i

兺

i,j=1

冋

冥

共14兲

.

冉

冉

冊

冉

冊

M ij ⳵ ⳵ ⳵ 共ui cos ␸兲 ␰ ju j + ␰ ju j + 2R j ⳵Ri ⳵␰ j ⳵␰ j

冉

冊冉

⳵ M i3 ⳵ 2 ⳵ M ij M i3␰ j ui ⳵ ⳵ sin ␸ ui ␰i + − + sin ␸ ⳵␰i ⳵␸ ⳵␸ R3Ri ⳵␰i 2R j 2R3 u j ⳵Ri

册

冊

2,j⫽i

⳵ ⳵ ⳵ 2 2 ⳵ ⳵ ⳵ 2 u2␰2 cos ␸ + u 1u 2 + u21 u + ⳵␰1 ⳵␰2 ⳵␰1 ⳵␰2 ⳵␰1 ⳵␰2 2

冉

冊

兺

兺冉

M 33␰2i M ii M i3␰i ⳵2 + − − 2R2i u2i 2R23u2i R3Riu2i ⳵␸2

冊

冋兺冉 2,j⫽i i,j=1

冉

M i3ui cos ␸ ⳵ ⳵ ⳵ ⳵ ⳵ 2 M ij␰i ⳵ uj + uj + u2j + u 2R3 ⳵Ri ⳵␰ j ⳵␰ j 2R j ⳵Ri ⳵␰ j ⳵␰ j j

i,j=1

i=1

1 mC

1 1 ⳵2 M ii M 33 ⳵ 2 ⳵ ⳵2 + u − M 共 ␰ ␰ + u u cos ␸ 兲 + + + 12 1 2 1 2 i 2 2 ⳵ R 3⳵ R i R 3R i ⳵ R 1⳵ R 2 R 1R 2 2R3 ⳵␰i ⳵␰i 2Ri

⳵2 ⳵2 u2 + u2 u1 + ⳵␰1⳵␰2 ⳵␰1⳵␰2 2,j⫽i

冤

1 1 + mH mC

In the MCTDH calculations, we have replaced the variBF ables ␪BF i by ␰i = cos ␪i . Indeed, these new variables are advantageous since their conjugate momenta are self-adjointed in an Euclidian normalization.47 Moreover, the following notation is used: ui = 冑1 − ␰2i = sin ␪BF i . The physical wave function was normalized using the volume element, dV = dR1dR2dR3d␰1d␰2d␸, with ␸ ⬅ ␸BF 1 . As this volume element is non-Euclidean for the lengths of the vectors, four so-called extrapotential terms48 appear which are nondifferential, i.e., purely multiplicative. Applying the equations in Ref. 44 provides the final vibrational kinetic energy operator which reads

M ij ⳵ ⳵ ⳵ ⳵ ui␰i sin ␸ + u i␰ i sin ␸ − 2RiR ju j ⳵␰i ⳵␸ ⳵␰i ⳵␸

2

−

u1

冊

2,j⫽i

−

冉

⳵2 uj + ⳵␰ j⳵␰i

+ u 1␰ 1

冋冉

M i3 ⳵ M i3 ⳵ + 2R3 ⳵Ri 2Ri ⳵R3

M 33 cos ␸

+ ␰ iu i

2

兺 i=1

共13兲

兺

i,j=1

兺冉

2,j⫽i i,j=1

冊

冊册

2,j⫽i

兺

i,j=1

−

冉

M 12 2R1R2

冋冉

冉

冉

冊冉

sin ␸

1 M i3␰ j M 12 M 33␰2␰1 − + R 3R iu j u i u 2u 1 R 1R 2 R23

⳵ ⳵ u 1␰ 1u 2␰ 2 ⳵␰1 ⳵␰2

冊

⳵2 ⳵2 uj + uj sin ␸ ⳵␰ j⳵␸ ⳵␰ j⳵␸

冊册

冊册

M i3 cos ␸ ⳵2 uj ␰ iu i 2R3Ri ⳵␰ j⳵␰i

M i3␰ j ⳵2 ⳵2 ␰ iu i + ␰ iu i sin ␸ sin ␸ 2R3Riu j ⳵␰i⳵␸ ⳵␰i⳵␸

M 33␰i M i3 − 2 2R3ui 2R3Riui

冊

⳵ ⳵ cos ␸ ⳵␸ ⳵␸

冊共15兲


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where the symbols M ij denote the matrix elements of M given in Eq. 共14兲. As always within the polyspherical approach,45 the kinetic energy operator can be written as a sum of products of monomode operators. As discussed in Sec. II A, this property is very profitable for MCTDH. In addition, since the Heidelberg MCTDH program package29 is capable to parse all the functions and derivative operators appearing in the KEO, its implementation is straightforward, i.e., without coding a new routine.

2. Zero-order Hamiltonians and potential energy operator

In order to study the energy flow in HFCO when the out-of-plane motion is excited by n quanta, we define the energy in the normal mode i as the expectation value Ei共t兲 = 具⌿共t兲兩hôi 兩⌿共t兲典, where ⌿共t兲 is the wave function of the system at the time t and hôi is the zero-order Hamiltonian associated with the normal mode Qi. Contrary to a description in terms of rectilinear coordinates, the use of polyspherical coordinates makes the definition of these zero-order Hamiltonians is a rather difficult task. Indeed, it is not possible to extract hôi from the system Hamiltonian by simply taking the diagonal part of this operator. Firstly, there is no simple oneto-one correspondence between the polyspherical and the normal coordinates except for ␸. Secondly, taking the diagonal part of the kinetic energy operator would be equivalent to subjecting the system to rigid constraints, i.e., to freezing some bond lengths or angles. If curvilinear coordinates are used, care must be taken when deriving such “constrained” kinetic energy operators and we refer the reader to49,50 for a detailed discussion of the importance of the “correction terms” in rigidly constrained models. In order to avoid the explicit calculation of these exact constrained operators and to obtain a good correspondence between the normal and curvilinear coordinates, a new strategy is adopted here for the internal modes. The method is as follows. As in foregoing articles dedicated to the Jacobi-Wilson approach,27,51–53 we define curvilinear normal modes from a zero-order harˆ o expressed as monic Hamiltonian H

FIG. 1. Valence BF polyspherical coordinates for the HFCO system. R2 BF BF belongs to the xz plane. At the equilibrium: ␸eq = ␲, ␪1eq = 2.23 rad, ␪2eq = 2.15 rad, R1eq = 2.059 au, R2eq = 2.55 au, and R3eq = 2.23 au. 6

ˆ o = 1 兺共q − qeq兲F 共q − qeq兲 + pˆ Go pˆ , H n nm m n nm m n m 2 n,m=1

共16兲

where Go represents the G matrix of Eq. 共13兲 but evaluated at the equilibrium geometry qeq. Here, qn and qm denote the six polyspherical coordinates and the F matrix corresponds to the harmonic approximation for the potential, Fnm = 兩⳵2V / ⳵qn⳵qm兩qeq. One can then proceed along the Wilson G matrix formulation,54 and define curvilinear normal modes 兵Q␣其 in terms of the polyspherical coordinates. To this end one diagonalizes the matrix FGo FGoL = L␻2 ,

共17兲

where ␻2 denotes the diagonal eigenvalue matrix and L is the eigenvector matrix subject to the normalization

TABLE I. Calculated normal mode frequencies of the different vibrational modes and comparison of calculated and vibrational experimental transition energies in cm−1. Frequencies

Harmonica

Mode

obs.b

Davidsonc

MCTDHd

␻1共A⬘兲共CH stretch兲 ␻2共A⬘兲共CO stretch兲 ␻3共A⬘兲共HCO bend兲 ␻4共A⬘兲共CF stretch兲 ␻5共A⬘兲共FCO bend兲 ␻6共A⬙兲共out-of-plane bend兲

3144.7 1852.9 1404.2 1074.9 665.7 1038.1

␯1 ␯2 ␯3 ␯4 ␯5 ␯6

2981.0 1836.9 1342.5 1064.8 662.5 1011.0

3003.2 1821.4 1370.3 1049.5 658.1 1019.2

3003.2 1821.4 1370.3 1049.5 658.1 1019.1

This work, harmonic frequencies determined by the Wilson-Jacobi method 共See Sec. II C 2兲. Experimental values taken from Refs. 21 and 71. c From variational calculations Refs. 27 and 28. d This work, using the MCTDH/FD method. Note that the same PES has been used in both calculations, but the PES was refitted for the use with MCTDH. a

b


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TABLE II. Influence of the number of fit terms on the zero point energy and the 10␯6 level. All other parameters were kept unchanged. The primitive and SPF basis sets are identical to the first ones in Tables III and IV, the number of terms needed to achieve an exact representation of the potential is 共13 ⫻ 14兲40= 7280. The rms error is defined on all relevant points 共i.e., in the current application these are all grid points with a potential energy below 20 000 cm−1 above the potential minimum兲.

Number of termsa

RMS deviationb

Zero pointc

10␯6d

Relative CPU Time

65⫻ 20= 1300 45⫻ 20= 900 45⫻ 15= 675 32⫻ 9 = 288 14⫻ 5 = 70

0.2206共1.78兲 0.4058共3.27兲 0.4147共3.34兲 0.7495共6.05兲 2.5102共20.2兲

4547.72 4547.88 4547.88 4547.64 4548.91

9906.12 9906.16 9905.60 9905.62 9904.65

3.65 3.12 2.62 1.1 1

TABLE IV. Number of SPFs used for the three modes in the MCTDH calculations. The brackets indicate the mode combinations. Number of Quanta

共R1 , cos ␪BF 1 , R 3兲

共R2 , cos ␪BF 2 兲

␸BF 1

2 4 6 8 10 12 14 16 18 20

17 23 24 25 28 28 35 35 35 35

16 22 24 25 25 25 30 30 28 30

10 14 16 18 15 15 16 18 14 16

a

Number of terms of the fit. The first and second number corresponds to the number of terms for mode 共R2 , cos ␪BF 2 兲, and for mode ␸, respectively. The mode 共R1 , cos ␪BF 1 , R3兲 is the contracted mode and hence does not appear here 关cf. Eq. 共11兲兴. b Difference between the original and fitted potential in meV. The values in parenthesis are in cm−1. c in cm−1, obtained by improved relaxation. d in cm−1, obtained with the filter-diagonalization method after a propagation over 330 fs.

共18兲

LTGoL = 1.

The mass and frequency weighted normal coordinates Q␣ are related to the polyspherical coordinates as 3N−6

Q␣ = ␻␣1/2

共qn − qeq 兺 n 兲Ln␣ . n=1

共19兲

ˆ o become sepaThese dimensionless normal coordinates let H o ˆ o = 兺 hˆ , and the one-dimensional operators hô read rable, H i i i

冉

冊

␻i 2 ⳵2 Qi − . hôi = 2 ⳵Q2i

共20兲

Note that these operators depend on all six polyspherical coordinates because of Eq. 共19兲. The frequencies ␻i of the different vibrational modes calculated with this approach are given in Table I. The first vibrational energy levels are also displayed in Table I and compared with those of Refs. 27 and 28. An excellent agreement is obtained. This agreement also reveals that the fit of the PES used for MCTDH provides excellent results for the fundamental levels 共see below for a more detailed discussion regarding the quality of the fit兲. The coordinate ␸ is the only out-of-plane coordinate of the system and it can be shown by symmetry arguments that this coordinate separates from the others in harmonic approximation. Hence ␸ is–except for its normalization—a normal coordinate by its own. In the following we prefer to

use ␸ and the local mode operator hˆ␸ rather than the related normal mode coordinate and operator, respectively. The initial wave packets have been defined in the following way. All terms containing differential operators acting on ␸ were removed and for the remaining terms, ␸ was put to its equilibrium geometry 共␸eq = ␲兲. The ground state of the corresponding five-dimensional operator was obtained by a propagation in negative imaginary time coupled with the Davidson algorithm similar as discussed in Ref. 9 共so called improved relaxation, see also Ref. 34兲. The product of this five-dimensional 共5D兲 wave function with a ␸ orbital then served as the initial wave packet for the propagation. The initial ␸ orbital was chosen as an eigenfunction of a onedimensional local mode operator hˆ␸. Hence by choosing the appropriate eigenstate of hˆ␸ one defines an initial wave packet with an initial excitation of n quanta in the out-ofplane mode. Note that this approach of generating the initial state is well justified since ␸ is only weakly coupled to the five in-plane modes. In fact, the difference between the energy expectation value of the initial wavefunction for zero quanta out of plane bending and the exact ground state energy of the full six-dimensional Hamiltonian 共4540.0 cm−1兲 is only 2.7 cm−1. Let us now turn to the potential energy surface. A full efficiency of the MCTDH algorithm requires the Hamiltonian to be expanded as a sum of products of operators acting on a single degree of freedom or on a combined mode 共cf. Sec. II A兲. As the potential of Yamamoto and Kato24 does not satisfy this requirement, we have applied the scheme potfit9,33,41,42 共cf. Sec. II B兲 to recast the PES as a sum of products of functions of three combined modes sepaBF BF rately, 共R1 , ␪BF 1 , R3兲, 共R2 , ␪2 兲, and ␸1 . The rms error of each expansion was evaluated to estimate the quality of the fit in the relevant region defined in

TABLE III. Parameters for the primitive basis set employed for each degree of freedom. HO denotes a harmonic oscillator 共Hermite兲 DVR.

Primitive basis N Grid length 共a.u.兲

共R1,

cos ␪BF 1 ,

R 3兲

共R2,

cos ␪BF 2 兲

␸BF 1

HO-DVR 10 关1.41,3.35兴

HO-DVR 14 关−0.99, 0.135兴

HO-DVR 10 关1.75,2.93兴

HO-DVR 13 关2.06,3.62兴

HO-DVR 14 关−0.91, −0.055兴

HO-DVR 40 关1.46,4.82兴


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FIG. 2. Evolution of the fraction of the energy in each vibrational mode from 2␯6 up to 20␯6. The spacing between the lines corresponds to two quanta of excitations of the out-of-plane bend mode. The figures correspond to the CH stretching mode 共a兲, the CO stretching mode 共b兲, the HCO bending mode 共c兲, the CF stretching mode 共d兲, the FCO bending mode 共e兲, and the out-of-plane bending mode 共f兲.

Table II. Table II shows the effect of the number of expansion terms on the ground state and the eigenenergy of the 10␯6 level. The observed trends confirm that a large number of expansion terms slows down the numerical resolution of the MCTDH equations. They also highlight that a rms deviation of less than 1 meV 共⯝8.0 cm−1兲 is sufficient to guarantee an accuracy much superior to the fitting error of the original PES. In order to simulate the dynamics, one first has to define primitive basis sets or grids for the various degrees of freedom. The parameters of the grids used are provided in Table III. The number of expansion terms and the rms error in the potential fits are 65⫻ 20= 1300 and 0.2206 meV

共=1.78 cm−1兲, respectively. Finally, the sizes of the singleparticle functions bases are provided in Table IV. Interestingly enough, we have noticed that the structures of the IVR depicted in Figs. 2 are rather insensitive to enlargements of the SPF bases. This indicates that the calculations are well converged with respect to the SPF bases. On the other hand, the IVR 共and in particular the period of the energy transfer to the CO stretch兲 is more sensitive to the sizes of the primitive basis sets, when the initial excitation is large 共n 艌 14, say兲. However, this behavior is not due to a lack of convergence but a consequence of the artificial energy cutoff of the PES which was mentioned in the introduction. When the grids are enlarged, a small portion of the wave packet moves to the cut region of the potential.


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III. RESULTS AND DISCUSSION

To study the energy flow in HFCO when the out-ofplane bend mode is excited by n quanta, we define the fraction in the five in-plane normal mode Qi at time t as Fni 共t兲 =

Ei共t兲 − Ei共t = 0兲 E␸n − E␸0

共21兲

and for the local mode ␸ F␸n 共t兲 =

E␸共t兲 − E␸0 E␸n − E␸0

,

共22兲

where E␸0 represents the zero point energy of hˆ␸, E␸n = E␸共t = 0兲 the energy of the nth excited eigenstate of hˆ␸ and Ei is the expectation value of the normal mode operator hôi which is defined in Sec. II C 2. General features of the dynamics will now be discussed by inspection of the evolution of these fractions. As aforementioned, very stable states corresponding to extreme motions of the C–H out-of-plane bending mode were observed experimentally. The trends of the energy flow in our simulations are consistent with these predictions. At the beginning, the energy transfer is governed by a rapid coupling between the ␯6 bright states and few dark states. This coupling causes a very structured energy flow as depicted in Fig. 2. The amplitude of this flow is always small indicating that at most only weak Fermi resonances appear. This physical behavior of HFCO is thus very different from the one observed for HCF3 in a preceding study.10 For instance, Fig. 1 in Ref. 10 conveys a very different message from Fig. 2 of this paper, it shows a reversible exchange for HCF3 with a very large amplitude between the CH stretch and the FCH bends after excitation of two quanta in the CH bond. For HCF3, the dynamics is indeed dominated by a strong Fermi resonance between these two modes. Very important for our purpose is the fact that this first step of the IVR in HFCO is not strongly affected by the increase of the state density as it can be clearly seen in Fig. 2. Again, the difference between HFCO and HCF3 is remarkably, in Figs. 2 and 3 of Ref. 10, the structure of the energy flow in HCF3 from the CH stretch to the bends quickly disappears for three quanta of excitation and is hardly visible for four quanta. On the other hand, the structure of the IVR of HFCO during this first step of the dynamics is very stable with respect to the number of quanta in the out-of-plane mode from 2␯6 up to 20␯6. It means that no new efficient couplings crop up when n increases and that the effect of the huge increase in the density of states is very limited. To be more specific, the positions of the minima and the maxima of the fraction of the energy in the out-of-plane bending mode are almost identical up to 80 fs as shown in Fig. 2共f兲 and the general structure is very similar up to 200 fs. This characteristic is remarkable but it should be emphasized that this is perfectly consistent with the analysis of the eigenvectors for n = 6, 8, and 10 calculated by means of the Davidson algo-

TABLE V. Analysis of the highly excited out-of-plane overtones up to ten quanta. The results are taken from Ref. 28 and were obtained with the Davidson algorithm along with a contracted primitive basis set of 143,792 functions. The exact eigenstates are overlapped with the first contracted functions, i.e., the anharmonic zero-order states ␯0i defined in Ref. 28 共normal states refined by including the diagonal anharmonicity兲. Shown are the main contributions to the eigenstates. Only projections greater than 0.15 in absolute value are provided. 6␯6 6018.4a

8␯6 7984.9a

0.17 4␯06 +0.77 6␯06 +0.28 共␯01 + 4␯06兲 −0.22 共␯03 + 6␯06兲 +0.18 共␯02 + 4␯06兲 +0.17 共␯04 + 6␯06兲 −0.15 共␯01 + 6␯06兲

−0.25 +0.64 −0.30 −0.24 −0.20 +0.17 −0.15

6␯06 8␯06 共␯01 + 6␯06兲共␯03 + 8␯06兲共␯02 + 6␯06兲共␯04 + 8␯06兲共␯01 + 8␯06兲

10␯6 9948.8a −0.31 +0.51 −0.29 −0.23 −0.20 +0.16 −0.15

8␯08 10␯06 共␯01 + 8␯06兲共␯03 + 10␯06兲共␯02 + 8␯06兲共␯04 + 10␯06兲共␯01 + ␯03 + 8␯06兲

a

Converged energy in cm−1.

rithm in Ref. 28. This detailed analysis is displayed in Table V and also underlines a very striking stability of the main contributions of the eigenvectors. Let us now turn to a more detailed description of the dynamics over this short period. The energy flow to the modes which are linked to the dissociation, such as the CF stretch and the FCO bend, is always very small as depicted in Figs. 2共d兲 and 2共e兲. To illustrate this statement we note that a visual inspection of Fig. 2共e兲 shows that less than 4% of the deposited energy is transferred to the FCO bend after 200 fs even for 20 quanta of initial excitation. Interestingly enough, the energy flow to the CF stretching mode is quasiperiodic 关Fig. 2共d兲兴. On the other hand, at the very beginning of the dynamics 共about 20 fs兲 the HCO bend and the CH stretch are the modes that receive most of the energy from the out-of-plane mode 关Figs. 2共c兲 and 2共a兲兴; these energy flows are quasiperiodic and almost reversible up to 12 quanta. Finally, it is important to notice that an energy flow to the CO stretch also occurs as depicted in Fig. 2共b兲. This step is slower and requires about 100 fs. On a longer time scale and for higher excitations the weaker couplings to all the other modes start to play a role resulting in an irreversible repartition of the energy which is transfered out of the out-of-plane mode. This second step is due to an indirect coupling between the few dark states directly coupled to the out-of-plane mode and the other modes and increases with the state density. The energy flows for n = 4, 8, 10, 12, 14, 18 are depicted in Fig. 3. The present study shows that • The repartition of the energy after a long propagation is not statistical. In particular the CH stretch constitutes an energy reservoir as clearly shown in Figs. 3共d兲–3共f兲. • Moreover, in order to quantify the rate of dissociation, we have performed other calculations with an enlarged R2 grid and a complex absorbing potential55–59 共CAP兲 placed at the end of the R2 grid. In perfect agreement with the experimental findings, the dissociation is very slow. After 600 fs, less than 2.5% and 5% of the wave function have been absorbed by the CAP for n = 18 and


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FIG. 3. Evolution of the fractions of the energy for n␯6 with n = 4 共a兲, 8 共b兲, 10 共c兲, 12 共d兲, 14 共e兲, and 18 共f兲 and for 共1兲 the out-of-plane mode 关thick full line 共1兲兴, 共2兲 the HCO bending mode 关thin full line 共2兲兴, 共3兲 the CH stretching mode 关dashed line 共3兲兴, 共4兲 the CO stretching mode 共triangles兲, 共5兲 the FCO bending mode 共thick dots兲, and 共6兲 the CF stretching mode 共thick full line: the lowest curve兲. Note the different time scale when comparing with Fig. 2.

20 quanta, respectively. Note that a quantitative calculation of the rate constants will be possible with the present approach, when a PES becomes available, which is able to correctly describe the dissociation channel. • Most of the energy is transfered to the CH stretching, the HCO bending and CO stretching modes. However, it should be emphasized that this energy flow does not lead to dissociation since 共i兲 the inspection of the evolution of the wave packet indicates that the small values of the angle ␪BF 1 are favored leading to the opposite direction, i.e., the isomerization HFCO→ t-FCOH and 共ii兲 the energy flow to the CF stretching and FCO bend-

ing modes, which should play a major role when the wave packet passes the transition state, is significantly smaller 共Fig. 3兲. In fact, the role of the dissociation is small and the isomerization, although its barrier lies higher in energy, starts to occur at a similar rate. Unfortunately, the role of this isomerization and the competition between the two reactions cannot be quantified with the present PES. The eigenenergies can be extracted from the propagation of the wave packets with MCTDH by a Fourier transform of the autocorrelation function c共t兲 = 具⌿0 兩 ⌿共t兲典. However, a superior way is to analyze the autocorrelation function with the


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TABLE VI. Vibrational levels obtained with the MCTDH/FD approach and comparison with the results obtained from a time-independent variational calculations and experimental findings. Energies are in cm−1 if not noted otherwise. 兩n␯6典

Variationala

MCTDHb

Np

2 4 6 8 10 12 14 16 18 20

2031.53 4036.23 6018.36 7984.9 9948.7

2030.6 4033.75 6012 7968 9906 11826 13723 15609 17477 19240

1.17 1.6 2.4 4 8 16± 1 29± 1 58± 8 150± 75 —

Expt.c

Katod

reader is referred to the Appendix of Ref. 10. The converged values are listed in Table VI. Even though the energy excitation becomes very high, we observe that N p does not dramatically increase with the number of quanta confirming again the stable behavior of the out-of-plane states. IV. SUMMARY AND OUTLOOK

13 631.7 15 486.3 17 319.0 19 125.2

15528 17382 19236

a

From the variational calculations of Ref. 28. This work, using the MCTDH/FD method. c Experimental values taken from Ref. 21. d Calculated values taken from Ref. 26. The absolute energies were transformed to excitation energies by assuming a ground-state energy of 4548 cm−1. b

filter-diagonalization method.60–65 The corresponding results are shown in Table VI and the comparison with the results obtained from variational calculations with the Davidson algorithm is good. Note, however, that MCTDH yields results slightly lower in energy. This small discrepancy may be traced back to the limited size of the basis set used in the variational calculation. Although this basis was already very large, it was smaller than the primitive basis set used for MCTDH. It is, indeed, a well-known advantage of MCTDH that the size of the primitive basis is not the limiting factor of the calculations. In fact, this small difference is mainly due to the presence of a low energy cutoff at 3.5 eV in the original PES. As explained in the introduction, this low energy cutoff appears since the isomerization HFCO→ t-FCOH, which occurs at about this energy, was not the object of interest of the work of Yamamoto and Kato.24 Contrary to the variational calculation, in which all the grid points were restricted to the energy domain below this cutoff, the grid lengths in the MCTDH simulations are much larger. These large grid lengths were mandatory to probe the dynamics very high in energy. Thereby, several grid points cover the domain of the PES which is cut and the presence of the cutoff slightly lowers the energy levels. Eventually, it should be emphasized that these differences are much smaller than the intrinsic error of the potential energy surface. Moreover, we are confident that the main physical content is correct and identical in both calculations as confirmed by the comparison between the IVR and the analysis of the eigenstates obtained with the Davidson algorithm below. Finally, we have computed the sum ⌺i p2i , where pi denotes the probability of finding the ith exact eigenstate ⌿i in the initial wave function ⌿0, i.e., pi = 兩具⌿i 兩 ⌿0典兩2. As ⌺i pi = 1, we note that the inverse of the sum, N p = 1 / 共⌺i p2i 兲, provides an estimate of the number of eigenstates which have an essential overlap with the initial wave packet.10 For further technical details on how N p is calculated in practise the

The IVR of HFCO initiated by a local mode excitation of the out-of-plane mode has been investigated with the MCTDH approach. Our findings agree with the general trends of experimental20–23 and theoretical predictions.24–26,28 In particular, the highly excited states with respect to the out-of-plane bending mode suffer nearly no state mixing with the other in-plane vibrational states. Furthermore, the stability of the energy process with respect to the number of quanta is very surprising, albeit in perfect agreement with a previous analysis28 of the eigenvectors for n = 6, 8, and 10. We have focused only on the vibrational states of HFCO with an even number of ␯6 quanta which belong to the A⬘ symmetry representation. Concerning the vibrational states with an odd number of ␯6 quanta, we expect similar results. However, since they belong to A⬙ whereas the minimal energy reaction pass lies in A⬘,22 we expect an even lower dissociation rate. Similar SEP experiments to those of HFCO have been carried out on DFCO by Crane et al.66,67 reporting that the stability of the “extreme-motion” states is destroyed probably due to a resonance between the out-of-plane and CO stretching modes. Consequently, further studies should be performed with the MCTDH package to investigate the sensitivity of the energy flow to this Fermi resonance in DFCO. H2CO is also an excellent candidate for future IVR studies. In contrast to HFCO, H2CO is a prototype of systems showing strong intermode couplings and resonances.68–70 Moreover, the competition between the dissociation and the isomerization in H2CO is of particular interest. Another remaining challenge for future work is the effect of rotational motion on the mode-specific phenomena observed for HFCO. Coriolis forces can indeed couple the out-of-plane extreme motion states to the reaction coordinate leading to the dissociation of the molecule. Finally, we are planning to implement the dipole momenta as well as the overall rotation of molecules to carry out simulations in the presence of a time-dependent field and for total J ⬎ 0. We would then be closer to experimental conditions and truly simulate the dynamics during the interaction with laser pulses for systems such as HFCO, HONO, H2CO, HCF3, and H2CS. Work to extend IVR studies to the systems cited above is in progress. ACKNOWLEDGMENTS

Financial support by the Deutsche Forschungsgemeinschaft and by the French Centre National de la Recherche Scientifique is gratefully acknowledged. 1

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