David Lauvergnat

Exact numerical computation of a kinetic energy operator in curvilinear coordinates

The conformation and dynamical behavior of molecular systems is very often advantageously described in terms of physically well-adapted curvilinear coordinates. It is rather easy to show that the numerous analytical expressions of the kinetic energy operator of a molecular system described in terms of n curvilinear coordinates can all be transformed into the following more usable expression: (T) over cap=Sigma(ij)f(2)(ij)(q)partial derivative(2)/partial derivativeq(i)partial derivativeq(j)+Sigma(i)f(1)(i)(q)partial derivative/partial derivativeq(i)+nu(q), where f(2)(ij)(q), f(1)(i)(q), and nu(q) are functions of the curvilinear coordinates q=(...,q(i),...). If the advantages of curvilinear coordinates are unquestionable, they do have a major drawback: the sometimes awful complexity of the analytical expression of the kinetic operator (T) over cap for molecular systems with more than five atoms. Therefore, we develop an algorithm for computing (T) over cap for a given value of the n curvilinear coordinates q. The calculation of the functions f(2)(ij)(q), f(1)(i)(q), and nu(q) only requires the knowledge of the Cartesian coordinates and their derivatives in terms of the n curvilinear coordinates. This coordinate transformation (curvilinear-->Cartesian) is very easy to perform and is widely used in quantum chemistry codes resorting to a Z-matrix to define the curvilinear coordinates. Thus, the functions f(2)(ij)(q), f(1)(i)(q), and nu(q) can be evaluated numerically and exactly for a given value of q, which makes it possible to propagate wavepackets or to simulate the spectra of rather complex systems (constrained Hamiltonian). The accuracy of this numerical procedure is tested by comparing two calculations of the bending spectrum of HCN: the first one, performed by using the present numerical kinetic operator procedure, the second one, obtained in previous studies, by using an analytical kinetic operator. Finally, the ab initio computation of the internal rotation spectrum and wave functions of 2-methylpropanal by means of dimensionality reduction, is given as an original application

Full-dimensional (15-dimensional) quantum-dynamical simulation of the protonated water dimer. I. Hamiltonian setup and analysis of the ground vibrational state.

Quantum-dynamical full-dimensional (15D) calculations are reported for the protonated water dimer (H5O2+) using the multiconfiguration time-dependent Hartree (MCTDH) method. The dynamics is described by curvilinear coordinates. The expression of the kinetic energy operator in this set of coordinates is given and its derivation, following the polyspherical method, is discussed. The potential-energy surface (PES) employed is that of Huang et al. [J. Chem. Phys. 122, 044308 (2005)]. A scheme for the representation of the PES is discussed which is based on a high-dimensional model representation scheme, but modified to take advantage of the mode-combination representation of the vibrational wave function used in MCTDH. The convergence of the PES expansion used is quantified and evidence is provided that it correctly reproduces the reference PES at least for the range of energies of interest. The reported zero point energy of the system is converged with respect to the MCTDH expansion and in excellent agreement (16.7 cm(-1) below) with the diffusion Monte Carlo result on the PES of Huang et al. The highly fluxional nature of the cation is accounted for through use of curvilinear coordinates. The system is found to interconvert between equivalent minima through wagging and internal rotation motions already when in the ground vibrational state, i.e., T=0. It is shown that a converged quantum-dynamical description of such a flexible, multiminima system is possible.

Full dimensional (15-dimensional) quantum-dynamical simulation of the protonated water-dimer III: Mixed Jacobi-valence parametrization and benchmark results for the zero point energy, vibrationally excited states, and infrared spectrum.

Quantum dynamical calculations are reported for the zero point energy, several low-lying vibrational states, and the infrared spectrum of the H(5)O(2)(+) cation. The calculations are performed by the multiconfiguration time-dependent Hartree (MCTDH) method. A new vector parametrization based on a mixed Jacobi-valence description of the system is presented. With this parametrization the potential energy surface coupling is reduced with respect to a full Jacobi description, providing a better convergence of the n-mode representation of the potential. However, new coupling terms appear in the kinetic energy operator. These terms are derived and discussed. A mode-combination scheme based on six combined coordinates is used, and the representation of the 15-dimensional potential in terms of a six-combined mode cluster expansion including up to some 7-dimensional grids is discussed. A statistical analysis of the accuracy of the n-mode representation of the potential at all orders is performed. Benchmark, fully converged results are reported for the zero point energy, which lie within the statistical uncertainty of the reference diffusion Monte Carlo result for this system. Some low-lying vibrationally excited eigenstates are computed by block improved relaxation, illustrating the applicability of the approach to large systems. Benchmark calculations of the linear infrared spectrum are provided, and convergence with increasing size of the time-dependent basis and as a function of the order of the n-mode representation is studied. The calculations presented here make use of recent developments in the parallel version of the MCTDH code, which are briefly discussed. We also show that the infrared spectrum can be computed, to a very good approximation, within D(2d) symmetry, instead of the G(16) symmetry used before, in which the complete rotation of one water molecule with respect to the other is allowed, thus simplifying the dynamical problem.

Automatic computer procedure for generating exact and analytical kinetic energy operators based on the polyspherical approach

We develop a new general code to automatically derive exact analytical kinetic energy operators in terms of polyspherical coordinates. Computer procedures based on symbolic calculations are implemented. Sets of orthogonal or non-orthogonal vectors are used to parametrize the molecular systems in space. For each set of vectors, and whatever the size of the system, the exact analytical kinetic energy operator (including the overall rotation and the Coriolis coupling) can be derived by the program. The correctness of the implementation is tested for different sets of vectors and for several systems of various sizes.

Reactive scattering of highly vibrationally excited oxygen molecules: Ozone formation?

A new ab initio potential energy surface based on an internally contracted multireference configuration–interaction wave function is constructed for the O2(X 3Σg−,υ)+O2(X 3Σg−,υ=0)→O3(X 1A1)+O(3P) reaction with υ>20. The vibrational state-to-state reaction probabilities are calculated with a time independent reactive scattering method. The state selected reactive rate constants calculated with 2D reduced dimensionality theory are very small, suggesting that the reaction of ozone formation is not significant in the O2(X 3Σg−,υ)+O2(X 3Σg−,υ=0) collision.

Quantum dynamics with sparse grids: A combination of Smolyak scheme and cubature. Application to methanol in full dimensionality.

Quantum dynamical approaches based on product-grids are limited to the studies of molecular systems with few degrees of freedom, typically less than ten. Recently, Avila et al. [G. Avila, T. Carrington, J. Chem. Phys., 131 (2009) 174103] have introduced the Smolyak scheme [S.A. Smolyak, Sov. Math. Dokl., 4 (1963) 240], which considerably reduces the size of the grids. This approach has pushed back the present calculation limits on the vibrational spectra of polyatomic molecules. In the present study, we have developed an extension of the standard Smolyak scheme in which this scheme is combined with multidimensional grids, such as cubatures, to obtain new sparse grids. This scheme has been applied to the study of the torsional energy levels of methanol in full dimensionality (12D).

Vibrational computing: Simulation of a full adder by optimal control

Within the context of vibrational molecular quantum computing, we investigate the implementation of a full addition of two binary digits and a carry that provides the sum and the carry out. Four qubits are necessary and they are encoded into four different normal vibrational modes of a molecule. We choose the bromoacetyl chloride molecule because it possesses four bright infrared active modes. The ground and first excited states of each mode form the one-qubit computational basis set. Two approaches are proposed for the realization of the full addition. In the first one, we optimize a pulse that implements directly the entire addition by a single unitary transformation. In the second one, we decompose the full addition in elementary quantum gates, following a scheme proposed by Vedral et al. [Phys. Rev. A 54, 147 (1996)]. Four elementary quantum gates are necessary, two two-qubit CNOT gates (controlled NOT) and two three-qubit TOFFOLI gates (controlled-controlled NOT). All the logic operations consist in one-qubit flip. The logic implementation is therefore quasiclassical and the readout is based on a population analysis of the vibrational modes that does not take the phases into account. The fields are optimized by the multitarget extension of the optimal control theory involving all the transformations among the 2(4) qubit states. A single cycle of addition without considering the preparation or the measure or copy of the result can be carried out in a very competitive time, on a picosecond time scale.

Quantum study of the internal rotation of methanol in full dimensionality (1+11D): a harmonic adiabatic approximation

Abstract The internal rotation of the methanol molecule is studied in full dimensionality i.e., the 12 coordinates are treated explicitly using an adiabatic separation of the 1D-torsional and the 11D-inactive wave functions. The potential energy surface is calculated with the help of quantum chemistry codes and is expanded in a Taylor series up to the second order along the torsional path. Our main results show that the torsional energy levels of this 1+11D-adiabatic model are noticeably different from those of the 1D-models, whether the zero point energy correction along the path is included or not.

Optimal control simulation of the Deutsch-Jozsa algorithm in a two-dimensional double well coupled to an environment.

The quantum Deutsch-Jozsa algorithm is implemented by using vibrational modes of a two-dimensional double well. The laser fields realizing the different gates (NOT, CNOT, and HADAMARD) on the two-qubit space are computed by the multitarget optimal control theory. The stability of the performance index is checked by coupling the system to an environment. Firstly, the two-dimensional subspace is coupled to a small number Nb of oscillators in order to simulate intramolecular vibrational energy redistribution. The complete (2+Nb)D problem is solved by the coupled harmonic adiabatic channel method which allows including coupled modes up to Nb=5. Secondly, the computational subspace is coupled to a continuous bath of oscillators in order to simulate a confined environment expected to be favorable to achieve molecular computing, for instance, molecules confined in matrices or in a fullerene. The spectral density of the bath is approximated by an Ohmic law with a cutoff for some hundreds of cm(-1). The time scale of the bath dynamics (of the order of 10 fs) is then smaller than the relaxation time and the controlled dynamics (2 ps) so that Markovian dissipative dynamics is used.

Wave packet dynamics along bifurcating reaction paths

The problem of bifurcating reaction paths is revisited by wave packet (WP) dynamics. The pitchfork model connecting five stationary points—a reactive, two transition structures and two enantiomeric products—is characterized by a Valley Ridge inflection point (VRI) where WP could leave the standard intrinsic reaction path. We question the role of such a VRI point to determine whether the mechanism is sequential or concerted. WP simulations on two-dimensional minimum energy surfaces are carried out in the benchmark case of the methoxy radical isomerization H3CO→H2COH. The ab initio potential energy surface (PES) is fitted to an analytical model which is bent to analyze the incidence of geometrical parameters on the WP behavior. For each of these generated PES, the WP width in the entrance valley is the main factor which conditions the behavior on the unstable ridge. The WP evolution is also analyzed in terms of nonadiabatic transitions among adiabatic channels along the reaction coordinate. Finally, the loc...