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Studying molecular quantum dynamics with the multiconfiguration time‐dependent Hartree method

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This review covers the multiconfiguration time‐dependent Hartree (MCTDH) method, which is a powerful and general algorithm for solving the time‐dependent Schrödinger equation. The formal derivation is discussed as well as applications of the method. Recent extensions of MCTDH are treated in brief, namely, MCTDHB and MCTDHF, for treating identical particles (bosons and fermions), and the very powerful multilayer (ML‐MCTDH) formalism. Compact representations of potential energy surfaces (PESs) are also discussed, as the representation of a PES becomes a major bottleneck when going to larger systems (nine or more dimensions) while employing a full‐dimensional, complicated, and nonseparable PES. As applications of MCTDH, we discuss the calculation of photoionization and photoexcitation spectra of the vibronically coupled systems butatriene and pyrazine, respectively, and the infra‐red spectrum of the Zundel cation (protonated water dimer) H5O+2. © 2011 John Wiley & Sons, Ltd.

Figure 1.

Tree structures of wavefunctions. (a) Standard method wavefunction tree, where the wavefunction is expanded directly into a primitive (time‐independent) basis denoted by squares. The circle symbolizes the expansion coefficients. (b) Multiconfiguration time‐dependent Hartree (MCTDH) wavefunction tree where the wavefunction is first expanded into a basis of single‐particle functions (SPFs), which, in turn, are expanded into the primitive basis. (c) MCTDH with mode combinations. The SPFs are expanded in two‐dimensional primitive basis sets. (d) Multilayer MCTDH (ML‐MCTDH) wavefunction tree. The (two‐dimensional) SPFs of the second layer are expanded into (one‐dimensional) SPFs of the third layer which in turn are expanded into the primitive basis. This case is similar to case (c), except that an additional layer is introduced rather than using combined primitives. (e) ML‐MCTDH wavefunction tree with three and four layers and with mode combination.

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Figure 2.

Sketch of excitation in butatriene.

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Figure 3.

Photoionization spectrum of butatriene. The black line depicts the simulation,88 and the red line shows the experiment.98

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Figure 4.

Photoabsorption spectrum of pyrazine. The black line is the result of a very accurate multiconfiguration time‐dependent Hartree (MCTDH) calculation and the red line shows a spectrum generated by a very cheap multilayer MCTDH calculation (see text). The spectra shown are with respect to the energy zero‐point of the Hamiltonian. To compare with experiment they must be shifted to higher energy by 2.48 eV.

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Figure 5.

Multiconfiguration time‐dependent Hartree tree for pyrazine. The numbers indicate the numbers of single‐particle functions (SPFs) and grid points used. As a multiset approach is used there are two entries for each set of combined SPFs, one for S1 and the other for S2.

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Figure 6.

Multilayer multiconfiguration time‐dependent Hartree tree of pyrazine. The numbers of single‐particle functions and grid points used are indicated.

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Figure 7.

Polyspherical coordinate system of H5O+2. The vectors are parameterized by their lengths and spherical angles. The two big circles represent oxygen atoms while the small circles represent hydrogens.

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Figure 8.

Calculated IR‐spectrum of H5O+279 (upper part) and the experimental vibrational predissociation spectrum104 (lower part). Two different lasers are used in the experiment, which explains that there are separate figures for different energy regions. The experiment cannot detect structures below 800 cm−1 because the vibrational energy is then insufficient to dissociate the H5O+2 – Ne van der Waals bond. The simulations show that there is indeed no absorption between 2000 and 3500 cm−1 and above 4000 cm−1. However, there are strong lines near 100 cm−1 and 250 cm−1 (wagging motion).

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Figure 9.

IR‐spectra of H5O+2, D5O+2, HD4O+2, and DH4O+2. The four isotopologues exhibit rather different spectra because the isotopic substitution changes the resonance pattern.

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Theoretical and Physical Chemistry > Reaction Dynamics and Kinetics

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