Ro-vibrational Eigenstates - American Chemical Society

Article pubs.acs.org/JPCA

Iterative Diagonalization in the Multiconfigurational Time-Dependent Hartree Approach: Ro-vibrational Eigenstates Robert Wodraszka* and Uwe Manthe* Theoretische Chemie, Fakultät für Chemie, Universität Bielefeld, Universitätsstraße. 25, D-33615 Bielefeld, Germany ABSTRACT: A scheme to efficiently calculate ro-vibrational (J > 0) eigenstates within the framework of the multiconfigurational time-dependent Hartree (MCTDH) approach is introduced. It employs a basis of MCTDH wave packets which is generated in the calculation of vibrational (J = 0) eigenstates via existing MCTDH-based iterative diagonalization approaches. The subsequent ro-vibrational calculations for total angular momenta J > 0 use direct products of these wave packets and the Wigner rotation matrices. In this ro-vibrational basis, the Hamiltonian matrix can be computed and diagonalized with minor numerical effort for any value of J. Accurate ro-vibrational states are obtained if the number of iterations in the J = 0 calculations and the basis set sizes in the MCTDH wave function representation are converged. Test calculations studying CH2D show that ro-vibrational eigenstates for moderately large J can be converged within wavenumber accuracy with the same MCTDH basis sets and only slightly increased iteration counts compared to purely vibrational (J = 0) calculations. If large J’s are considered or very high accuracies are required, the number of iterations required to obtain convergence increases significantly. Comparing the theoretical results with experimental data for the out-of-plane bend, symmetric stretch, and antisymmetric stretch fundamentals, the accuracy of the ab initio potential energy surface employed is investigated. functions.27 However, while these schemes can efficiently provide moderate numbers of vibrational (or ro-vibrational) wave functions even for high-dimensional systems, they are not designed to obtain large numbers of ro-vibrational levels. In recent work, Fábri et al.28 employed a basis consisting of products of vibrational (J = 0) eigenstates and Wigner rotation matrices to efficiently obtain large numbers of ro-vibrational eigenstates for the ketene molecule. Inspired by the idea of Fábri et al., the present work develops a scheme to efficiently compute large numbers of ro-vibrational eigenstates within the MCTDH framework. In this approach, a basis of vibrational MCTDH wave functions is generated by a block Lanczos-type iteration sequence using the J = 0 propagator. The J > 0 Hamiltonian is represented in a direct product basis spanned by these MCTDH wave functions obtained in the iteration sequence and the Wigner rotation matrices, and the resulting Hamiltonian matrix is diagonalized. Accurate ro-vibrational energies can be obtained provided a sufficient number of iterations is used in the iteration sequence and sufficiently many single-particle functions are employed in the MCTDH representation of the wave function. The ro-vibrational states of CH2D are calculated as an example to investigate the new approach, and the rather floppy umbrella (out-of-plane bending) mode is studied in particular. The convergence of the approach will be tested by comparison with reference results and by internal checks which do not require a reference solution. Finally, the computed ro-vibrational levels are compared to recent experimental results,29,30 and the accuracy of the ab initio potential energy surface employed is discussed.

I. INTRODUCTION The accurate calculation of vibrational and ro-vibrational states is an important subject in theoretical chemistry and chemical physics (see, e.g., ref 1 for a recent review). If rigid molecules are considered, normal coordinates, the Eckart-Watson Hamiltonian, and vibrational self-consistent field (VSCF)2−4 based approaches as the vibrational configuration interaction (VCI)4 scheme or related perturbational or coupled cluster approaches provide an efficient description (see, e.g., refs 5−7.). Curvilinear coordinates provide a more flexible description of the system dynamics which can also account for large amplitude motion. Dynamical calculations based on curvilinear coordinates require a more system-specific strategy, and truncationdiagonalization approaches or iterative techniques are used to obtain numerically efficient schemes. CH4 served as a benchmark system studied by work developing these techniques,8−10 and the results reported for the fluxional CH5+11 system present an impressive demonstration of the achievements. The time-dependent multiconfigurational Hartree (MCTDH) approach12,13 provides a framework to accurately describe quantum dynamics of a high-dimensional system. Considering vibrational states of molecular systems showing large amplitude motion, the accurate calculation of tunneling splittings of the ground14−17 and vibrationally excited16,18 states of malonaldehyde and the detailed investigation of the vibrational states of the protonated water dimer H5O+2 19−21 are prominent examples of applications. A number of techniques have been introduced to obtain vibrational energies via MCTDH propagation in real or imaginary time: Lanczos-type ̂ iterative diagonalization of the e−βH operator22 and block-Lanczos type extensions of the scheme which employ imaginary and real time propagation of state-averaged MCTDH wave functions,18,23 filter diagonalization of correlation functions obtained by MCTDH propagation,24,25 the improved relaxation approach,26 and block relaxation (subspace iteration) of state-averaged MCTDH wave © 2013 American Chemical Society

Special Issue: Joel M. Bowman Festschrift Received: January 31, 2013 Revised: April 4, 2013 Published: April 8, 2013 7246

dx.doi.org/10.1021/jp401129t | J. Phys. Chem. A 2013, 117, 7246−7255

The Journal of Physical Chemistry A

Article

The present work is based on the iterative diagonalization approaches described in refs 18 and 23 and introduces a scheme to efficiently compute ro-vibrational (J > 0) eigenstates. Since the presence of multiple wells does not alter the basic formalism, the approach will be described only for the single well case. The extension to the multiwell case is straightforward: one would just have to replace the summations with respect to the P stateaveraged MCTDH wave functions by summations running over all P state-averaged MCTDH wave functions in all W wells. Considering state-averaged MCTDH wave functions consisting of P wave packets, the scheme of refs 18 and 23 works as follows: Starting with the initial seed vectors ψ(0) P , further sets of wave functions ψ(n) P are generated using the block Lanczos-type iteration sequence

The article is organized as follows. Section II introduces the new approach and reviews the background theory. The details of the CH2D system are described in section III and the numerical results are presented and discussed in section IV. Concluding remarks (section V) complete the article.

II. THEORY A. MCTDH Approach. The multiconfigurational timedependent Hartree (MCTDH) approach12,13,31 provides an efficient method for the accurate simulation of multidimensional quantum dynamics. Within the state-averaged MCTDH scheme,27 the wave function ansatz reads n1

Ψp(q11 , q21 , ..., qd1 , t ) =

j1 = 1

φ j1; d(qd1 , d

t ),

nd

∑ ··· ∑ A 1j ,...,j ,p(t )·φ j1;1(q11, t )·...· jd = 1

p = 1, ..., P

1

d

n−1 P

1

ψ̃p(n) = Â ψp(n − 1) −

p−1

ψp(n) = Cp(n)(ψp̃ (n) −

κ

κ

i=1

∑ ψq(n)⟨ψq(n)|ψp̃ (n)⟩) (4)

q=0

Here, ψ(n) P denotes the pth wave packet which has been generated ̂ ̂ in the nth iteration step. The operator Â is chosen as e−iHte−βH. (n‑1) (i) (̂ n) P is a projector onto the SPF basis of Â ψP . The αq are given by the set of linear equations

Nκ

∑ A j2;;κi (t )·χi2;κ (qκ1)

(3)

i=0 q=1

(1)

A product basis of time-dependent single-particle functions (SPFs) φ1;κ ji is used to represent a set of P wave packets Ψp simultaneously. The corresponding time-dependent expansion coefficients are denoted by A1ji,···,jd,m. In the original MCTDH scheme,12,13 each φ1;κ ji is represented employing an underlying time-independent basis or grid (typically using a discrete variable representation (DVR)32−34 or a fast Fourier transform (FFT)-scheme35): φ j1; κ (qκ1 , t ) =

(n) (i) ψq

∑ ∑ αq(i)P ̂

(n) ⟨P ̂ ψr (j)|Â ψp(n − 1)⟩ =

(2)

n−1 P

∑ ∑ αq(i)⟨P(̂ n)ψr(j)|P(̂ n)ψq(i)⟩, i=0 q=1

This scheme can be seen as a two-layer representation, the superscripts 1 and 2 denote the upper and lower layer, respectively. In the multilayer extension of the MCTDH approach,36,37 the φ1;κ jκ can be multidimensional functions which themselves are represented using a MCTDH expansion. While the numerical example presented uses the simple MCTDH scheme, the approach introduced in the following sections can also employ multilayer MCTDH calculations. For the evaluation of potential energy matrix elements appearing in the equations of motion, the present work employs the correlation DVR (CDVR) approach.37−40 Specific schemes for the efficient integration of the MCTDH equations of motion have been developed.41,42 Here the CMF2 scheme of ref 42 is used. The present work focuses on the calculation of ro-vibrational eigenstates. Within the MCTDH framework, all rotational degrees of freedom are usually described by a single combined coordinate and Wigner functions are used as the time-independent basis in the representation of the single-particle function.43,44 The Wigner functions are denoted by DJMK (α,β,γ) or |JMK⟩ where J denotes the quantum number associated with the total (nuclear) angular momentum, M and K are the quantum numbers associated with the projection of the total angular momentum on the space-fixed and body-fixed z-axis, respectively, and (α,β,γ) are the Euler angles. B. Iterative Diagonalization for J = 0. The MCTDH approach can be employed to calculate eigenstates of a Hamiltonian Ĥ . Several methods have been developed for this purpose, for ̂ example, Lanczos-type iterative diagonalization of the e−βH 22 operator, block relaxation (subspace iteration) of state-averaged MCTDH wave functions,27 or the improved relaxation approach.26 Recently, the iterative Lanczos-type diagonalization approach was extended to a block Lanczos-type approach using state-averaged MCTDH wave functions.18 Reference 23 introduced a multiwell iterative diagonalization procedure for the calculation of vibrational eigenstates for systems showing multiwelled potential energy surfaces.

j = 0...n − 1, r = 1...P

(5)

The states ψ(n) P then serve as a basis for the diagonalization of the Hamiltonian Ĥ (or a bijective function of it). The best results for the eigenenergies of the Hamiltonian Ĥ are obtained by ̂ −βĤ (j) 18 diagonalizing e−iHt in the basis of ϕ(j) ψP . To this end, p = e the generalized eigenvalue problem N−1 P

∑

̂

∑ ⟨ϕp(j)|e−iHt|ϕq(n)⟩cq(n,m) =

n=0 q=1

N−1 P

∑ ∑ εm⟨ϕp(j)|ϕq(n)⟩cq(n,m) n=0 q=1

(6)

is solved. Here, the denote the eigenvector components, εm are the eigenvalues, and N is the number of iterations employed. The vibrational eigenstates ψvib,N calculated using N iteration m steps thus read c(n) q,m

N−1 P

ψmvib, N =

∑ ∑ cq(n,m) ϕq(n) n=0 q=1

(7)

and the corresponding eigenenergies Evib,N are obtained via εm = m vib,N e‑iEm t. It should be noted that the iterative diagonalization scheme tends to converge the vibrational energy levels from bottom to top: typically all lower energy states have to be reasonably well converged before higher energy states can be accurately calculated. C. Ro-Vibrational Coupling within the Iterative Diagonalization Approach. The iterative diagonalization approach described above can favorably be combined with the ideas employed by Fábri et al.28 to calculate ro-vibrational (J > 0) eigenstates using basis sets taken from J = 0 calculations. Rovibrational wave functions showing the quantum numbers J and M are represented by products of the vibrational functions |ϕ(n) p ⟩ 7247



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compared to the numerical effort required to obtain the ϕ(n) p using the iterative diagonalization approach, and the matrix elements ⟨JMK|f k(Jx̂ , Jŷ , Jẑ )|JMK′⟩ can be easily calculated analytically using standard angular momentum algebra. To obtain ro-vibrational levels for a specific J requires the diagonalization of a (2J + 1)·N· P-dimensional matrix. For matrix dimensions not exceeding a few 10 000, the numerical effort of the diagonalization is also comparatively small. Thus, once vibrational levels have been calculated for J = 0 by the MCTDH-based iterative diagonalization scheme described in section II B, the present approach allows one to compute ro-vibrational levels for many relevant J values without requiring significant additional numerical effort. However, to obtain accurate results, convergence of the computed energy levels with respect to the number of iterations N and the size of the SPF basis sets is required. The J-dependence of the convergence behavior is a central question which will be addressed in the results section.

generated in the iterative diagonalization procedure for J = 0 (eq 4) and the appropriate Wigner functions |JMK⟩ N−1

|Ψ JM ⟩ =

P

K =J JM (n) cnpK |ϕp ⟩|JMK ⟩

∑∑ ∑

(8)

n = 0 p = 1 K =−J

The above expansion utilizes all N·P vibrational wave functions generated in the iteration sequences of eq 4. It should be emphasized that the number of converged J = 0 eigenstates which could be obtained using this N·P-dimensional basis is, in general, much smaller than N·P. In this respect the present approach differs from the one of Fábri et al.,28 which employs a basis of converged J = 0 vibrational eigenstates. The full ro-vibrational Hamiltonian Ĥ can be split into a part already considered in the J = 0 calculation, Ĥ J=0, and a remaining part T̂ vibrot which includes the kinetic energy terms resulting from the rotational motion and the ro-vibrational coupling:

̂ Ĥ = HĴ = 0 + Tvibrot

(9)

Furthermore, T̂ vibrot can be decomposed into a sum of products ̂ depending solely on the internal (vibrational) of operators tkvib coordinates and functions f k depending only on the components of the angular momentum operator, Jx̂ , Jŷ , and Jẑ : ̂ = Tvibrot

∑ tvibk̂ ·fk (Jx̂ , Jŷ , Jẑ )

III. SYSTEM DETAILS The coordinate system employed for the description of the CH2D radical is based on a Radau construction. After separation of the translational motion of the whole system, one Radau vector which connects the canonical point with the D-atom (r1) and two Radau vectors which connect the canonical point with the hydrogens (r2,r3) are used to describe the radical. The vectors r1 and r2 define the body-fixed frame. The z-axis lies along r1 and the y-axis lies along r1 × r2. To describe these embedding vectors in the body-fixed frame, mass-weighted lengths r1 and r2, and the enclosed angle θ are employed. The remaining Radau vector which points to one hydrogen is parametrized by the stereographic coordinates r3, s, t utilizing the “north pole” projection.45 The overall rotational motion is described by the body-fixed angular momentum operator J.̂ The kinetic energy operator corresponding to this coordinate system is taken from the general expression derived in a previous work.45 The explicit form reads

(10)

k

Matrix elements of Ĥ with respect to the basis function |ϕ(n) p ⟩|JMK⟩ used in eq 8 read ⟨JMK |⟨ϕp(n)|Ĥ |ϕq(m)⟩|JMK ′⟩ = ⟨ϕp(n)|HĴ = 0|ϕq(m)⟩·δKK +

′

∑ ⟨JMK |fk (Jx̂ , Jŷ , Jẑ )|JMK ′⟩ k

k ̂ |ϕq(m)⟩ ·⟨ϕp(n)|tvib

(11)

The matrix elements of Ĥ J=0 are constructed employing the eigenstate basis ψvib,N defined by eq 7 and the corresponding i energies Evib,N using i ⟨ψi vib, N |HĴ = 0|ψjvib, N ⟩ = E ivib, N ·δij

(12)

T̂ = −

Transforming eq 12 to the eigenstate representation, one obtains ⟨JMK |⟨ψi vib, N |Ĥ |ψjvib, N ⟩|JMK ′⟩ = E ivib, N ·δij·δKK + ′ N−1

·∑

P

× sin θ

∑ ⟨JMK |fk (Jx̂ , Jŷ , Jẑ )|JMK ′⟩

∂ ∂θ

2 1 1 ∂2 L̂ − + 2 ∂r32 2r32 sin θ

1 ̂2 2 2 2 {J + Jŷ − 2(Jx̂ Lx̂ + Jŷ L̂y) + (Lx̂ + L̂y )} 2r12 x 1 ⎛ cot2 θ csc 2 θ ⎞ ̂ 2 2 ⎟(J − 2Jẑ Lẑ + Lẑ ) + ⎜ 2 + 2 ⎝ r1 r2 2 ⎠ z cot θ ̂ ̂ {J J + Jẑ Jx̂ − 2(Jx̂ Lẑ + Jẑ Lx̂ ) + 2r12 x z i ∂ +(Lx̂ Lẑ + Lẑ Lx̂ )} + 2 (Jŷ − L̂y) r1 ∂θ (14)

+

k

N−1 P

∑ ∑ ∑ c(pn,i) *⟨ϕp(n)|tvibk̂ |ϕq(m)⟩cq(m,j )

n=0 p=1 m=0 q=1

1 ∂2 1 ∂2 1⎛ 1 1 ⎞ 1 ∂ − − ⎜ 2 + 2⎟ 2 2 2 ∂r1 2 ∂r2 2 ⎝ r1 r2 ⎠ sin θ ∂θ

(13)

It should be stressed that the Evib,N appearing in the above i equations are the vibrational energies numerically calculated for J = 0 using N iterations. Their values might differ significantly from the exact vibrational energies for small iteration counts N or highly vibrationally excited states i. Equation 13 is the working equation of the present approach. It defines the Hamiltonian matrix which is then diagonalized by standard techniques to compute the ro-vibrational energies and eigenfunctions. The Hamiltonian matrix of eq 13 can be constructed for many relevant values of J with minor numerical effort: the numerical k̂ (m) costs of calculating the matrix elements ⟨ϕ(n) p |tvib|ϕq ⟩ and transforming them to the eigenstate representation are small

Here, L̂ denotes the angular momentum operator associated with the Radau vector r3. L̂ 2 and the components Lx, Ly, and Lz can be expressed explicitly by the stereographic coordinates s and t: ⎛ ∂2 1 ∂2 ⎞ L2 = − (1 + s 2 + t 2)⎜ 2 + 2 ⎟(1 + s 2 + t 2) 4 ⎝ ∂s ∂t ⎠ 7248

(15)



Article

⎡⎛ ⎞ ⎛− (1 − s 2 + t 2)⎞ 2st ⎛ 2t ⎞⎤ ⎢⎜ ⎟∂ ⎜ ⎟∂ 1 ⎜ ⎟⎥ 2 2 L̂ = ⎢⎜− (1 + s − t )⎟ − ⎜ ⎟ + ⎜− 2s ⎟⎥ 2 st 2i ⎢⎜ ⎟ ∂s ⎜ ⎟ ∂t ⎝ 0 ⎠⎥⎦ ⎠ ⎝ ⎠ ⎣⎝ − 2t − 2s (16)

Please note that the volume element v = 4r12r22r32 sin θ/(1 + s2 + t2)2 has been incorporated into the kinetic energy operator of eq 14 (i.e., T̂ → √vT̂ (1/√v)). The present study employs the potential energy surface of Medvedev et al.46 To test the convergence of the iterative diagonalization approach of sections II B and II C with the SPF basis employed, four different basis sets (B0−B3) of systematically increasing size are used. The corresponding SPF basis set sizes and the underlying grid representations are given in Table 1. Eight wave packets (P = 8) are newly generated in each iteration step in all calculations which employ β = 100 au for the propagation in imaginary time and t = 50 au in the real time propagation step. Accurate reference data for the total rotational quantum numbers J = 0, J = 3, and J = 9 is obtained by state averaged MCTDH calculations using the block-relaxation algorithm.27 The corresponding basis set sizes are chosen sufficiently large

Table 1. Wave Function Representations Employed in the Iterative Diagonalization Calculations SPF basis mode

B0

B1

B2

B3

grid type

grid points

r1 r2 θ r3 s t

2 2 4 2 4 6

3 3 6 3 6 8

4 4 8 4 8 12

5 5 10 5 10 16

Hermite-DVR Hermite-DVR Legendre-DVR Hermite-DVR FFT [−1.5 au,1.5 au] FFT [−1.5 au,1.5 au]

16 16 50 16 64 64

Figure 1. The ro-vibrational energies corresponding to the vibrational ground state and the first vibrationally excited state are studied for J values of 0, 3, and 9 in panels a, b, and c, respectively. The RMSE values of the energies computed by the iterative approach (relative to the accurate reference results, see text for details) are displayed as a function of the number of iterations for different SPF bases. For comparison, the RMSE of the rigid-rotor approximation is also indicated. 7249



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geometry (taken from the potential energy surface) is diagonalized to obtain the eigenvalues Ix, Iy, and Iz and the rotational energies are calculated via diagonalizing Ĥ rot = Jx̂ 2/ (2Ix) + Jŷ 2/(2Iy) + Jẑ 2/(2Iz) in a basis of Wigner functions |JMK⟩. For J = 3, the rigid-rotor approximation results in a RMSE of 1.9 cm−1. Diagonalizing the exact ro-vibrational Hamiltonian, a higher accuracy is obtained after five iteration steps. Considering that the RMSE for the B0 basis mainly results from an inaccurate description of the vibrational states (the J = 0 results in Figure 1a show a similar RMSE), one finds that the errors resulting from an inaccurate description of the rotational motion are below 1 cm−1 already after five iterations. Since four to five iterations are required to demonstrate convergence for the purely vibrational (J = 0) calculations, such results for the ro-vibrational energies can be obtained with essentially no additional numerical effort. Increasing the angular momentum to J = 9, the RMSE of the rigid rotor model rises to 15.7 cm−1 which indicates strong ro-vibrational coupling. For these large couplings, the present approach is less efficient than for small J values. However, it can account for such a coupling strength yielding subwavenumber accuracies if larger numbers of iterations and bigger SPF basis sets are used. The dependence of the ro-vibrational energies on the number of iterations can also be used to estimate the importance of the ro-vibrational coupling. The J = 0 results presented in Figure 1a show that the purely vibrational energies are converged after four iterations. The impact of ro-vibrational coupling on the eigenenergies can thus be estimated by studying the dependence of the energy on the iteration count for iteration counts above four. For J = 3 (Figure 1b), the energies change by much less than 1 cm−1 per iteration step indicating the moderate effect for the ro-vibrational coupling. In contrast, an energy change above 10 cm−1 per iteration can be found in Figure 1c for J = 9 which demonstrates the strength of the ro-vibrational coupling in this domain. Thus, the iterative scheme allows one to at least estimate the importance of ro-vibrational coupling even if a converged result could not be achieved. In Figure 1, the accuracy of the iterative results was evaluated by comparison with accurate reference results. However, in practical applications it is important that the accuracy can also be

to guarantee converged ro-vibrational eigenstates and eigenenergies.

IV. RESULTS The ro-vibrational states of the CH2D radical are employed as an example in the present work. The vibrational ground state, the out-of-plane bending fundamental mode, and the symmetric and antisymmetric stretch fundamental modes, respectively, are investigated. The corresponding bands in the infrared absorption spectrum of CH2D have recently been studied experimentally.29,30 A. Convergence of the J = 0 and J > 0 Calculations. To study the iterative approach introduced in sections II B and II C, the convergence of the calculated ro-vibrational energies with the number of iterations and the size of the basis sets employed is investigated. First, ro-vibrational states corresponding to the vibrational ground state and the first vibrationally excited state are considered. Results obtained by the iterative approach are compared to accurate reference results obtained by block relaxation. The root-mean-square errors (RMSEs) of all computed ro-vibrational levels for total angular momenta J = 0, J = 3, and J = 9 are displayed in Figure 1 as a function of the number of iterations employed. To study the dependence on the basis set size, results for all four basis sets (B0−B3) described in the previous section are plotted. Figure 1a shows the rapid convergence of the computed vibrational energies with respect to the number of iterations. After three to four iteration steps, convergence is reached for all basis sets. The accuracies obtained for the four basis sets increase significantly with increasing basis set size: RMSEs of 0.8, 0.03, 0.01, and 0.002 cm−1 are found for the basis sets B0, B1, B2, and B3, respectively. Figures 1 panels b and c demonstrate that the number of iterations required to obtain convergence increases significantly if rotationally excited states are considered. This is hardly surprising considering that the vibrational wave functions added during the iterative procedure are optimized toward the J = 0 eigenstates. The number of iterations required to compute accurate ro-vibrational energies increases strongly with J. For J = 3, RMSEs below 1 cm−1 or 0.1 cm−1 can be obtained after 5 or 18 iterations, respectively, if sufficiently large SPF bases are used. For J = 9, already about 60 iterations are required to converge the ro-vibrational energies to wavenumber accuracy. In contrast, the impact of the SPF basis set size on the accuracy of the computed rotational energies is moderate. The calculation of ro-vibrational energies for J = 3 does not require significantly larger SPF bases than the J = 0 calculations. The converged RMSE for the small B0 basis is 1 cm−1, just 20% larger than for J = 0. For the larger basis sets B1 to B3, RMSE errors below 0.1 cm−1 are found after 25 iterations. For larger rotational excitation, J = 9, the effect of the SPF basis is more pronounced. Results converged with respect to the number of iterations could be computed only for the smallest basis B0. Here the RMSE increases to above 10 cm−1. The convergence curves shown in Figure 1c indicate that the next larger basis, B1, can also yield only an accuracy of 1−2 cm−1. Only the larger basis sets B2 and B3 allow one to obtain subwavenumber accuracy for J = 9. It is interesting to relate the accuracies discussed above to the strength of the ro-vibrational coupling. To indicate this strength, a straight line is drawn in Figure 1 panels b and c at the RMSE resulting from a simple rigid-rotor approximation. To compute this number, the moment of inertia tensor at the equilibrium

Figure 2. The average values of all ro-vibrational energies corresponding to the vibrational ground state and the first vibrationally excited state are displayed for J = 3 as a function of the number of iterations for different SPF bases. 7250



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tested without comparison to accurate reference results. This can be achieved by plotting the mean energy of all relevant rovibrational states instead of the RMSEs (with respect to some reference results) as a function of the iteration count. In Figure 2,

the averaged values of all relevant eigenenergies are shown as a function of the number of iterations. Results for J = 3 and the four basis sets B0 to B3 are displayed. Investigating the convergence with respect to the number of iterations, one finds that 1 cm−1 accuracy requires about five iterations and that 0.1 cm−1 accuracy is achieved after less than 20 iterations. Analyzing the convergence with respect to the SPF basis set sizes, basis sets B1 to B3 show converged mean energies differing by less than 0.1 cm−1, and the energy obtained with the B0 basis is about 1 cm−1 larger. All these findings are consistent with the results obtained from Figure 1b. This demonstrates that the accuracy achieved by the iterative approach can be reliably estimated without requiring any reference results. While up to now only the vibrational ground state and the first vibrationally excited state showing out-of-plane bending excitation have been considered, also excitations in high frequency modes should now be considered. Here, the symmetric and antisymmetric CH stretch fundamentals for J = 3 will serve as examples. The corresponding mean excitation energies, i.e., the average of the energies of the ro-vibrational states showing this vibrational excitation minus the ground state energy (for the given value of J), are displayed as a function of the number of iterations in Figures 3 and 4, respectively. Since an accurate description of the stretching excitation requires larger SPF bases, only results for the larger bases B1 to B3 are displayed. After about 50 iterations, convergence with respect to the iteration count is achieved for all basis sets with an accuracy much better than 0.1 cm−1. The converged energies obtained with the bases B1 to B3 differ by less than 1 cm−1, and the differences between the largest basis sets B2 and B3 are about 0.1 cm−1. It should be noted that the excitation energies for the B2 basis in Figures 3 and 4 are even slightly smaller than the ones obtained for the larger B3 basis. This nonvariational dependence of computed energies on the SPF basis set size results from CDVR quadrature errors. B. Comparison with Experiment. To estimate the errors of the calculated ro-vibrational energies introduced by the potential energy surface employed, the theoretical results are compared to recent experimental results obtained from IR spectroscopy.29,30 The theoretical results are calculated using the iterative diagonalization approach with the largest SPF basis set and the largest number of iterations considered in the previous section. First, combination differences, that is, differences between frequencies of transitions having one state in common, associated with the out-of-plane bend fundamental are considered. These combination differences correspond to rotational excitation energies of the out-of-plane bending excited vibrational state. The experimental and calculated values are presented in Table 2 and the assignments of the ro-vibrational states are indicated. All differences between the calculated and the experimental results are below 1 cm−1. Thus, subwavenumber accuracy is achieved in the theoretical simulations of the rotational motion based on the

Figure 3. The average values of all ro-vibrational excitation energies corresponding to the fundamental excitation of the symmetric stretch vibration are displayed for J = 3 as a function of the number of iterations for different SPF bases.

Figure 4. The average values of all ro-vibrational excitation energies corresponding to the fundamental excitation of the antisymmetric stretch vibration are displayed for J = 3 as a function of the number of iterations for different SPF bases.

Table 2. Combination Differences (in cm−1) of the Out-of-Plane Bend Fundamental: Experimental Data ΔEexp,29 Calculated Values ΔEcalc, and Results of the Rigid Rotor Approximation ΔErr J′

Ka′

Kc′

J″

Ka″

Kc″

ΔEexp

ΔEcalc

ΔEcalc − ΔEexp

ΔErr

ΔErr − ΔEexp

6 6 7 8 8 9 9

1 3 1 3 1 3 1

5 4 6 6 7 7 8

5 5 6 7 7 8 8

1 1 1 1 1 1 1

5 4 6 6 7 7 8

83.1 87.4 97.7 111.6 112.1 125.1 126.3

82.6 86.9 97.1 110.9 111.4 124.3 125.5

−0.5 −0.5 −0.6 −0.7 −0.7 −0.8 −0.8

85.4 90.2 100.5 115.0 115.4 128.9 130.1

2.3 2.8 2.8 3.4 3.3 3.8 3.8

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Table 3. Experimental29 and Calculated Ro-vibrational Excitation Energies (in cm−1) of the Out-of-Plane Bend Fundamental, ΔEexp and ΔEcalc, Respectively, and the Rotational Contributions ΔErot to These Excitation Energies (See Text for Details) J′

Ka′

Kc′

J″

Ka″

Kc″

ΔEexp

ΔEcalc

ΔEcalc − ΔEexp

ΔErot exp

ΔErot calc

5 6 8 7 8 9 4 7 5 5 4 6 6 3 7 8 9 2 6 3 3 3 4 4 4 4 5 4 5 6 5 5 6 7 7 6 6 7 6 7 8 7 7 7 7 8 8 8 7 8 9 8 9 8 9 8

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

4 5 6 6 7 8 4 5 5 5 4 6 6 3 7 8 9 2 4 2 0 1 3 1 2 0 2 1 3 3 0 1 5 4 3 1 2 5 1 6 5 2 3 1 2 3 6 4 1 7 4 2 5 1 7 0

5 6 8 7 8 9 4 7 5 5 4 6 6 3 7 8 9 2 6 2 2 2 3 3 3 3 4 3 4 5 4 4 5 6 6 5 5 6 5 6 7 6 6 6 6 7 7 7 6 7 8 7 8 7 8 7

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

4 5 6 6 7 8 4 5 5 5 4 6 6 3 7 8 9 2 4 2 0 1 3 1 2 0 2 1 3 3 0 1 5 4 3 1 2 5 1 6 5 2 3 1 2 3 6 4 1 7 4 2 5 1 7 0

555.9 557.1 557.2 557.6 557.8 558.0 560.1 560.4 560.5 560.8 560.8 560.9 561.0 561.3 561.3 561.8 562.3 562.4 564.8 599.7 609.6 610.3 613.9 618.1 620.5 626.5 626.6 626.6 631.5 636.0 642.4 642.4 643.9 647.0 650.8 651.4 651.6 655.9 657.6 658.7 659.7 659.8 660.8 666.6 666.6 667.3 669.2 670.1 672.2 673.4 674.1 675.2 679.9 681.0 682.9 686.1

541.8 543.0 543.2 543.6 543.8 543.9 546.0 546.3 546.4 546.7 546.7 546.8 546.8 547.2 547.2 547.6 548.2 548.3 550.7 585.4 595.3 595.9 599.4 603.7 606.1 612.0 612.1 612.2 617.0 621.5 628.0 628.0 629.2 632.3 636.2 636.9 637.1 641.2 643.2 644.0 645.0 645.2 646.2 652.2 652.2 652.7 654.4 655.5 658.0 658.5 659.5 660.8 665.3 667.0 668.1 672.5

−14.1 −14.1 −14.0 −14.0 −14.0 −14.1 −14.1 −14.1 −14.1 −14.1 −14.1 −14.1 −14.2 −14.1 −14.1 −14.2 −14.1 −14.1 −14.1 −14.3 −14.3 −14.4 −14.5 −14.4 −14.4 −14.5 −14.5 −14.4 −14.5 −14.5 −14.4 −14.4 −14.7 −14.7 −14.6 −14.5 −14.5 −14.7 −14.4 −14.7 −14.7 −14.6 −14.6 −14.4 −14.4 −14.6 −14.8 −14.6 −14.2 −14.9 −14.6 −14.4 −14.6 −14.0 −14.8 −13.6

−4.6 −3.4 −3.3 −2.9 −2.7 −2.5 −0.4 −0.1 0.0 0.3 0.3 0.4 0.5 0.8 0.8 1.3 1.8 1.9 4.3 39.2 49.1 49.8 53.4 57.6 60.0 66.0 66.1 66.1 71.0 75.5 81.9 81.9 83.4 86.5 90.3 90.9 91.1 95.4 97.1 98.2 99.2 99.3 100.3 106.1 106.1 106.8 108.7 109.6 111.7 112.9 113.6 114.7 119.4 120.5 122.4 125.6

−4.7 −3.5 −3.3 −2.9 −2.7 −2.6 −0.5 −0.2 −0.1 0.2 0.2 0.3 0.3 0.7 0.7 1.1 1.7 1.8 4.2 38.9 48.8 49.4 52.9 57.2 59.6 65.5 65.6 65.7 70.5 75.0 81.5 81.5 82.7 85.8 89.7 90.4 90.6 94.7 96.7 97.5 98.5 98.7 99.7 105.7 105.7 106.2 107.9 109.0 111.5 112.0 113.0 114.3 118.8 120.5 121.6 126.0

PES of Medvedev et al. For comparison, also values resulting from the rigid-rotor approximation (based on the equilibrium geometry)

are given in Table 2. Here, larger errors ranging between 2.3 and 3.8 cm−1 are found. 7252



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Table 4. Experimental30 and Calculated Ro-vibrational Excitation Energies (in cm−1) of the Symmetric C−H Stretch Fundamental, ΔEexp and ΔEcalc, Respectively, and the Rotational Contributions ΔErot to These Excitation Energies (See Text for Details) J′

Ka′

Kc′

J″

Ka″

Kc″

ΔEexp

ΔEcalc

ΔEcalc − ΔEexp

ΔErot exp

ΔErot calc

1 1 1 0 1 1 1 2 2 2 3 3 3

1 0 1 0 1 1 0 1 0 1 1 1 0

0 1 1 0 1 0 1 2 2 1 3 2 3

2 2 2 1 1 1 0 1 1 1 2 2 2

1 0 1 0 1 1 0 1 0 1 1 1 0

1 2 2 1 0 1 0 1 1 0 2 1 2

3044.1 3047.3 3048.7 3056.1 3063.2 3067.7 3075.1 3082.1 3083.7 3086.7 3089.9 3096.6 3091.2

3028.0 3031.2 3032.6 3039.9 3046.9 3051.5 3058.8 3065.8 3067.4 3070.3 3073.5 3080.2 3074.7

−16.1 −16.1 −16.1 −16.2 −16.3 −16.2 −16.3 −16.3 −16.3 −16.4 −16.4 −16.4 −16.5

−21.5 −18.3 −16.9 −9.5 −2.4 2.1 9.5 16.5 18.1 21.1 24.3 31.0 25.6

−21.4 −18.2 −16.8 −9.5 −2.5 2.1 9.4 16.4 18.0 20.9 24.1 30.8 25.3

Table 5. Experimental30 and Calculated Ro-vibrational Excitation Energies (in cm−1) of the Antisymmetric C−H Stretch Fundamental, ΔEexp and ΔEcalc, Respectively, and the Rotational Contributions ΔErot to These Excitation Energies (See Text for Details) J′

Ka′

Kc′

J″

Ka″

Kc″

ΔEexp

ΔEcalc

ΔEcalc − ΔEexp

ΔErot exp

ΔErot calc

1 1 0 2 1 1 2 2 1 2 2 2 3 3 2 2 3 3 3

0 1 0 0 0 1 1 2 1 0 2 1 0 1 2 2 2 2 3

1 1 0 2 1 0 1 0 1 2 1 2 3 3 1 0 2 1 1

2 2 1 2 1 1 2 2 0 1 2 1 2 2 1 1 2 2 2

1 0 1 1 1 0 0 1 0 1 1 0 1 0 1 1 1 1 2

2 2 1 1 0 1 2 1 0 1 2 1 2 2 0 1 1 2 0

3137.8 3143.6 3145.1 3149.1 3152.3 3164.1 3167.0 3169.6 3171.4 3172.7 3175.7 3178.5 3181.7 3184.8 3190.2 3193.2 3197.2 3207.5 3209.5

3120.2 3126.0 3127.5 3131.4 3134.6 3146.4 3149.2 3151.8 3153.6 3154.9 3157.8 3160.6 3163.8 3166.9 3172.2 3175.3 3179.2 3189.4 3191.5

−17.6 −17.6 −17.6 −17.7 −17.7 −17.7 −17.8 −17.8 −17.8 −17.8 −17.9 −17.9 −17.9 −17.9 −18.0 −17.9 −18.0 −18.1 −18.0

−20.5 −14.7 −13.2 −9.2 −6.0 5.8 8.7 11.3 13.1 14.4 17.4 20.2 23.4 26.5 31.9 34.9 38.9 49.2 51.2

−20.4 −14.6 −13.1 −9.2 −6.0 5.8 8.6 11.2 13.0 14.3 17.2 20.0 23.2 26.3 31.6 34.7 38.6 48.8 50.9

accurate fits to the rotator models presented in the experimental work. For the rotational contributions ΔErot to the excitation energies, differences between the experimental and calculated values are always below 1 cm−1. The differences do not even exceed 0.2 cm−1 for transitions showing small ΔErot below 10 cm−1. Considering that the inaccuracies of the PES result in a relative error of 2−3% in the vibrational frequencies, the relative effect of these inaccuracies on the rotational energies and the ro-vibrational couplings is found to be of roughly similar size. Tables 4 and 5 show similar results for excitations of the symmetric and antisymmetric CH stretch fundamentals. Here, experimental results for J values up to three are available.30 The findings are very similar to the ones discussed for the out-of-plane rot bend excitation. The differences between ΔErot exp and ΔEcalc do not −1 exceed 0.4 cm .

Experimental and calculated excitation energies of ro-vibrational transitions exciting the out-of-plane bending vibration are presented in Table 3. One immediately notes that all differences between the calculated and experimental results are about 14 cm−1. Since this difference is largely independent of the rotational excitation, it presumably results from an inaccurate description of the vibrational excitation. This interpretation is consistent with results of Medvedev et al.:46 investigating the accuracy of the PES by vibrational (J = 0) calculations for CH3, they found and error of 14.8 cm−1 for the umbrella fundamental. To study effects of the rotations and the ro-vibrational coupling in more detail, one thus wants to correct for the inaccurate description of the vibrational motion. To this end, we subtract the excitation energies for J = 0, ΔEJ=0, from the rovibrational excitation energies ΔE: ΔErot = ΔE − ΔEJ = 0

(17)

V. CONCLUSIONS An efficient scheme to calculate large numbers of ro-vibrational states within the framework of the MCTDH approach was

=0 The theoretical value for ΔEJ=0, ΔEJcalc = 546.5 cm−1, could be taken directly from the calculations. The experimental value, =0 ΔEJcalc = 560.5 cm−1, is only indirectly available via the highly

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(7) Seidler, P.; Christiansen, O. Automatic Derivation and Evaluation of Vibrational Coupled Cluster Theory Equations. J. Chem. Phys. 2009, 131, 234109. (8) Yu, H.-G. Converged Quantum Dynamics Calculations of Vibrational Energies of CH4 and CH3D Using an ab Initio Potential. J. Chem. Phys. 2004, 121, 6334−6340. (9) Schwenke, D. W. Towards Accurate ab Initio Predictions of the Vibrational Spectrum of Methane. Spectrochim. Acta, Part A 2002, 58, 849−861. (10) Wang, X.-G.; Carrington, T., Jr. A Contracted Basis-Lanczos Calculation of Vibrational Levels of Methane: Solving the Schrodinger Equation in nine Dimensions. J. Chem. Phys. 2003, 119, 101−117. (11) Wang, X.-G.; Carrington, T., Jr. Vibrational Energy Levels of CH5+. J. Chem. Phys. 2008, 129, 234102. (12) Meyer, H.-D.; Manthe, U.; Cederbaum, L. S. The Multiconfigurational Time-Dependent Hartree Approach. Chem. Phys. Lett. 1990, 165, 73−78. (13) Manthe, U.; Meyer, H.-D.; Cederbaum, L. S. Wave-Packet Dynamics within the Multiconfiguration Hartree Framework: General Aspects and Application to NOCl. J. Chem. Phys. 1992, 97, 3199−3213. (14) Coutinho-Neto, M. D.; Viel, A.; Manthe, U. The Ground State Tunneling Splitting of Malonaldehyde: Accurate Full Dimensional Quantum Dynamics Calculations. J. Chem. Phys. 2004, 121, 9207−9210. (15) Hammer, T.; Coutinho-Neto, M. D.; Viel, A.; Manthe, U. Multiconfigurational Time-Dependent Hartree Calculations for Tunneling Splittings of Vibrational States: Theoretical Considerations and Application to Malonaldehyde. J. Chem. Phys. 2009, 131, 224109. (16) Hammer, T.; Manthe, U. Intramolecular Proton Transfer in Malonaldehyde: Accurate Multilayer Multi-configurational TimeDependent Hartree Calculations. J. Chem. Phys. 2011, 134, 224305. (17) Schroeder, M.; Gatti, F.; Meyer, H.-D. Theoretical Studies of the Tunneling Splitting of Malonaldehyde Using the Multiconfiguration Time-Dependent Hartree Approach. J. Chem. Phys. 2011, 134, 234307. (18) Hammer, T.; Manthe, U. Iterative Diagonalization in the StateAveraged Multi-configurational Time-Dependent Hartree Approach: Excited State Tunneling Splittings in Malonaldehyde. J. Chem. Phys. 2012, 136, 054105. (19) Vendrell, O.; Gatti, F.; Lauvergnat, D.; Meyer, H.-D. Dynamics and Infrared Spectroscopy of the Protonated Water Dimer. Angew. Chem. Int. Ed. 2007, 46, 6918. (20) Vendrell, O.; Gatti, F.; Lauvergnat, D.; Meyer, H.-D. FullDimensional (15-Dimensional) Quantum-Dynamical Simulation of the Protonated Water Dimer. I. Hamiltonian Setup and Analysis of the Ground Vibrational State. J. Chem. Phys. 2007, 127, 184302. (21) Vendrell, O.; Gatti, F.; Meyer, H.-D. Full Dimensional (15 Dimensional) Quantum-Dynamical Simulation of the Protonated Water-Dimer IV: Isotope Effects in the Infrared Spectra of D(D2O)2+, H(D2O)2+, and D(H2O)2+ Isotopologues. J. Chem. Phys. 2009, 131, 034308. (22) Manthe, U.; Matzkies, F. Iterative Diagonalization within the Multi-configurational Time-Dependent Hartree Approach: Calculation of Vibrationally Excited States and Reaction Rates. Chem. Phys. Lett. 1996, 252, 71−76. (23) Wodraszka, R.; Manthe, U. A Multi-configurational TimeDependent Hartree Approach to the Eigenstates of Multi-well Systems. J. Chem. Phys. 2012, 136, 124119. (24) Beck, M. H.; Meyer, H.-D. Efficiently Computing Bound-State Spectra: A Hybrid Approach of the Multi-configuration TimeDependent Hartree and Filter-Diagonalization Methods. J. Chem. Phys. 2001, 114, 2036−2046. (25) Gatti, F.; Beck, M.; Worth, G. A.; Meyer, H. D. A Hybrid Approach of the Multi-configuration Time-Dependent Hartree and Filter-Diagonalisation Methods for Computing Bound-State Spectra. Application to HO2. Phys. Chem. Chem. Phys. 2001, 3, 1576−1582. (26) Meyer, H.-D.; Le Quere, F.; Leonard, C.; Gatti, F. Calculation and Selective Population of Vibrational Levels with the Multiconfiguration Time-Dependent Hartree (MCTDH) algorithm. Chem. Phys. 2006, 329, 179−192.

introduced. In this approach, MCTDH wave functions describing the vibrational motion for J = 0 are generated by a block Lanczos-type iteration sequence. Representing the rovibrational Hamiltonian in a product basis consisting of these MCTDH wave functions and Wigner rotation matrices, ro-vibrational levels for many relevant J values are computed. In principle, computing the J > 0 data requires minor numerical effort compared to the J = 0 calculation. Thus, valuable information on ro-vibrational couplings can be obtained essentially free of costs. However, if highly accurate ro-vibrational energies are required for large J, the test calculations studying the CH2D molecule indicate that many more iterations and slightly larger MCTDH basis set sizes are required for convergence compared to purely vibrational (J = 0) calculations. The ro-vibrational energies obtained with the present scheme on the ab initio PES of Medvedev et al.46 were compared with experimental results.29,30 Here the ro-vibrational transitions corresponding to fundamental excitations in the out-of-plane bending and the symmetric and antisymmetric C−H stretching modes were investigated. While the vibrational excitation energies differed by 10 to 20 cm−1 due to inaccuracies of the PES employed, subwavenumber accuracy was obtained for the rotational excitation energies even for large J values. The present approach thus offers a perspective to calculate ro-vibrational states in polyatomic systems showing large amplitude motion or multiwelled potential energy surfaces. As demonstrated in work on the tunneling splittings in malonaldehyde or the vibrational levels in the protonated water dimer, the MCTDH approach facilitates the accurate calculations of vibrational (J = 0) states in high dimensionality. Using the scheme presented, future MCTDH calculations might be able to investigate ro-vibrational states of floppy polyatomic molecules, at least for moderate values of J.

■

AUTHOR INFORMATION

Corresponding Author

* E-mail: [email protected]; [email protected]. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS Financial support by the Deutsche Forschungsgemeinschaf t is gratefully acknowledged.

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REFERENCES

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