Automatic generation of active coordinates for quantum dynamics calculations: application to the dynamics of benzene photochemistry

A new practical method to generate a subspace of active coordinates for quantum dynamics calculations is presented. These reduced coordinates are obtained as the normal modes of an analytical quadratic representation of the energy difference between excited and ground states within the complete active space self-consistent field method. At the Franck-Condon point, the largest negative eigenvalues of this Hessian correspond to the photoactive modes: those that reduce the energy difference and lead to the conical intersection; eigenvalues close to 0 correspond to bath modes, while modes with large positive eigenvalues are photoinactive vibrations, which increase the energy difference. The efficacy of quantum dynamics run in the subspace of the photoactive modes is illustrated with the photochemistry of benzene, where theoretical simulations are designed to assist optimal control experiments.     

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.     

Intramolecular energy transfer dynamics in 24-mode pyrazine by partitioning technique: A time-dependent approach

We investigate the intramolecular energy transfer dynamics of the S(2) excited electronic state of pyrazine due to radiationless transitions to energetically lower-lying singlet electronic states using a new time-dependent method. The femtosecond decay of S(2) to the S(1) excited state and the picosecond decay of S(2) to the ground electronic state S(0) are studied within an efficient methodology for computing the intramolecular dynamics in multidimensional configurational spaces. Our method is based on partitioning the full configuration space into the (small) subspace of interest Q and the rest, the subspace P. The exact equations of motion for the states in Q, under the influence of P, are derived in the time domain in form of a system of integrodifferential equations. Their numerical solution is readily obtained when the Q space consists of just a few states. Otherwise, the integrodifferential equations for the states in Q are transformed into a (larger) system of ordinary differential equations, which can be solved by a single diagonalization of a general complex matrix. The former approach is applied to study the pyrazine picosecond S(2)→S(0) dynamics and the latter is applied to the study of the ultrafast pyrazine S(2)→S(1) decay dynamics.     

Crossover from bosonic to fermionic features in composite boson systems

We study the quantum dynamics of conversion of composite bosons into fermionic fragment species with increasing densities of bound fermion pairs using the open quantum system approach. The Hilbert space of N-state-functions is decomposed into a composite boson subspace and an orthogonal fragment subspace of quasi-free fermions that enlarges as the composite boson constituents deviate from ideal boson commutation relations. The tunneling dynamics of coupled composite bosons states in confined systems is examined, and the appearance of exceptional points under experimentally testable conditions (densities, lattice temperatures) is highlighted. The theory is extended to examine the energy transfer between macroscopically coherent systems such as multichromophoric macromolecules in photosynthetic light harvesting complexes.     

多组态含时自旋非限制HartreeFock理论

基于自旋非限制HartreeFock理论,发展了自旋非限制多组态含时HartreeFock理论方法来研究激光场中的多电子相关动力学．自旋向上和自旋向下的自旋轨道分别在他们各自的子空间内传播;并通过约化密度矩阵和平均场算符相互作用．分别利用了自旋限制和非限制的多组态含时HartreeFock方法虚时和实时传播计算氦原子基态能量和电离几率．自旋非限制的计算结果与其他报道相吻合．     

Dimension reduction by balanced truncation: application to light-induced control of open quantum systems

In linear control, balanced truncation is known as a powerful technique to reduce the state-space dimension of a system. Its basic principle is to identify a subspace of jointly easily controllable and observable states and then to restrict the dynamics to this subspace without changing the overall response of the system. This work deals with a first application of balanced truncation to the control of open quantum systems which are modeled by the Liouville-von Neumann equation within the Lindblad formalism. Generalization of the linear theory has been proposed to cope with the bilinear terms arising from the coupling between the control field and the quantum system. As an example we choose the dissipative quantum dynamics of a particle in an asymmetric double well potential driven by an external control field, monitoring population transfer between the potential wells as a control target. The accuracy of dimension reduction is investigated by comparing the populations obtained for the truncated system versus those for the original system. The dimension of the model system can be reduced very efficiently where the degree of reduction depends on temperature and relaxation rate.     

基于化学子空间对线性变换稳定的秩估计方法

提出了一个新的化学计量学方法,即线性变换下稳定的化学子空间法,用于二维数据的化学秩估计.该方法的基本思想是具有化学意义的子空间对在一个方向上的二维数据的线性变换是最稳定的.据此,两个新的定量指标投影残差法和子空间夹角法被提出用于衡量两个子空间之间的差异性.对两个近红外数据的分析结果表明,这一方法为二维数据的化学秩估计提供了一个很有价值的工具.     

Improved configuration space sampling: Langevin dynamics with alternative mobility

We present a new and efficient method for determining optimal configurations of a large number (N) of interacting particles. We use a coarse-grained stochastic Langevin equation in the overdamped limit to describe the dynamics of this system and replace the standard mobility by an effective space dependent inverse Hessian correlation matrix. Due to the analogy of the drift term in the Langevin equation and the update scheme in Newton's method, we expect accelerated dynamics or improved convergence in the convex part of the potential energy surface Phi. The stochastic noise term, however, is not only essential for proper thermodynamic sampling but also allows the system to access transition states in the concave parts of Phi. We employ a Broyden-Fletcher-Goldfarb-Shannon method for updating the local mobility matrix. Quantitative analysis for one and two dimensional systems shows that the new method is indeed more efficient than standard methods with constant effective friction. Due to the construction, our effective mobility adapts high values/low friction in configurations which are less optimal and low values/high friction in configurations that are more optimal.     

Some aspects of dynamic chaos in small Lennard-Jones clusters

For the purpose of studying stochastic phenomena in the clusters due to an instability of their trajectories in phase space, an evolution of ensembles of the clusters with the representative points initially located in small regions of phase space is investigated using the molecular dynamics method. A path of a cloud of the representative points through the subspace of phase space available for the clusters, and cloud growth in size and in volume are considered. Specific results are presented for Ar₄. A “cold” cluster (subjected to neither an isomerization nor an evaporation) is examined in details. A brief comparison with a “hot” cluster (both the phenomena are available) is given.     

Detecting coupled collective motions in protein by independent subspace analysis

Protein dynamics evolves in a high-dimensional space, comprising aharmonic, strongly correlated motional modes. Such correlation often plays an important role in analyzing protein function. In order to identify significantly correlated collective motions, here we employ independent subspace analysis based on the subspace joint approximate diagonalization of eigenmatrices algorithm for the analysis of molecular dynamics (MD) simulation trajectories. From the 100 ns MD simulation of T4 lysozyme, we extract several independent subspaces in each of which collective modes are significantly correlated, and identify the other modes as independent. This method successfully detects the modes along which long-tailed non-Gaussian probability distributions are obtained. Based on the time cross-correlation analysis, we identified a series of events among domain motions and more localized motions in the protein, indicating the connection between the functionally relevant phenomena which have been independently revealed by experiments.     

Ab initio calculations on the excited states of Na3 cluster to explore beyond Born-Oppenheimer theories: adiabatic to diabatic potential energy surfaces and nuclear dynamics

We perform ab initio calculation using quantum chemistry package (MOLPRO) on the excited states of Na(3) cluster and present the adiabatic PESs for the electronic states 2(2)E' and 1(2)A(1)', and the non-adiabatic coupling (NAC) terms among those states. Since the ab initio calculated NAC elements for the states 2(2)E' and 1(2)A(1)' demonstrate the numerical validity of so called "Curl Condition," such states closely form a sub-Hilbert space. For this subspace, we employ the NAC terms to solve the "adiabatic-diabatic transformation (ADT)" equations to obtain the functional form of the transformation angles and pave the way to construct the continuous and single valued diabatic potential energy surface matrix by exploiting the existing first principle based theoretical means on beyond Born-Oppenheimer treatment. Nuclear dynamics has been carried out on those diabatic surfaces to reproduce the experimental spectrum for system B of Na(3) cluster and thereby, to explore the numerical validity of the theoretical development on beyond Born-Oppenheimer approach for adiabatic to diabatic transformation.     

Krylov subspace methods for computing hydrodynamic interactions in Brownian dynamics simulations

Hydrodynamic interactions play an important role in the dynamics of macromolecules. The most common way to take into account hydrodynamic effects in molecular simulations is in the context of a Brownian dynamics simulation. However, the calculation of correlated Brownian noise vectors in these simulations is computationally very demanding and alternative methods are desirable. This paper studies methods based on Krylov subspaces for computing Brownian noise vectors. These methods are related to Chebyshev polynomial approximations, but do not require eigenvalue estimates. We show that only low accuracy is required in the Brownian noise vectors to accurately compute values of dynamic and static properties of polymer and monodisperse suspension models. With this level of accuracy, the computational time of Krylov subspace methods scales very nearly as O(N(2)) for the number of particles N up to 10 000, which was the limit tested. The performance of the Krylov subspace methods, especially the "block" version, is slightly better than that of the Chebyshev method, even without taking into account the additional cost of eigenvalue estimates required by the latter. Furthermore, at N = 10 000, the Krylov subspace method is 13 times faster than the exact Cholesky method. Thus, Krylov subspace methods are recommended for performing large-scale Brownian dynamics simulations with hydrodynamic interactions.     

Normal mode partitioning of Langevin dynamics for biomolecules

We propose a novel normal mode multiple time stepping Langevin dynamics integrator called NML. The aim is to approximate the kinetics or thermodynamics of a biomolecule by a reduced model based on a normal mode decomposition of the dynamical space. Our basis set uses the eigenvectors of a mass reweighted Hessian matrix calculated with a biomolecular force field. This particular choice has the advantage of an ordering according to the eigenvalues, which have a physical meaning of being the square of the mode frequency. Low frequency eigenvalues correspond to more collective motions, whereas the highest frequency eigenvalues are the limiting factor for the stability of the integrator. In NML, the higher frequency modes are overdamped and relaxed near their energy minimum while respecting the subspace of low frequency dynamical modes. Our numerical results confirm that both sampling and rates are conserved for an implicitly solvated alanine dipeptide model, with only 30% of the modes propagated, when compared to the full model. For implicitly solvated systems, NML gives a twofold improvement in efficiency over plain Langevin dynamics for sampling a small 22 atom (alanine dipeptide) model and in excess of an order of magnitude for sampling an 882 atom (bovine pancreatic trypsin inhibitor) model, with good scaling with system size subject to the number of modes propagated. NML has been implemented in the open source software PROTOMOL.     

Locating conical intersections relevant to photochemical reactions

A new computerized method for locating conical intersections of interest in photochemistry is presented. The search is based on the Longuet-Higgins phase change theorem (Berry phase) which provides the subspace required for the initial search. The subspace is approximated as a plane containing three stable structures lying on a Longuet-Higgins loop. The search is conducted for a minimum of ΔE, the energy difference between two electronic states. It is started using up to three points within the circle defined by the three structures; symmetry, if relevant, is helpful but not essential. Since a two-dimensional subspace of the large 3N − 6 space is used, the search that uses either Cartesian or internal coordinates is efficient and yields a degeneracy after a few iterations. Given that not all degrees of freedom are included in the search, usually a high lying part of the conical intersection is initially located. The system is subsequently optimized along all coordinates keeping ΔE as close to zero as desired. The method is demonstrated for the symmetric H₃ system and also for the butadiene–cyclobutene–bicyclobutane system in which the three stable structures are not equivalent. The method is general and can be extended to any photochemical system.     

On the relationship between quantum control landscape structure and optimization complexity

It has been widely observed in optimal control simulations and experiments that state preparation is surprisingly easy to achieve, regardless of the dimension N of the system Hilbert space. In contrast, simulations for the generation of targeted unitary transformations indicate that the effort increases exponentially with N. In order to understand such behavior, the concept of quantum control landscapes was recently introduced, where the landscape is defined as the physical objective, as a function of the control variables. The present work explores how the local structure of the control landscape influences the effectiveness and efficiency of quantum optimal control search efforts. Optimizations of state and unitary transformation preparation using kinematic control variables (i.e., the elements of the action matrix) are performed with gradient, genetic, and simplex algorithms. The results indicate that the search effort scales weakly, or possibly independently, with N for state preparation, while the search effort for the unitary transformation objective increases exponentially with N. Analysis of the mean path length traversed during a search trajectory through the space of action matrices and the local structure along this trajectory provides a basis to explain the difference in the scaling of the search effort with N for these control objectives. Much more favorable scaling for unitary transformation preparation arises upon specifying an initial action matrix based on state preparation results. The consequences of choosing a reduced number of control variables for state preparation is also investigated, showing a significant reduction in performance for using fewer than 2N-2 variables, which is consistent with the topological analysis of the associated landscape.     

Theory of heteronuclear decoupling in solid-state nuclear magnetic resonance using multipole-multimode Floquet theory

A formal theory for heteronuclear decoupling in solid-state magic angle spinning (MAS) nuclear magnetic resonance experiments is presented as a first application of multipole-multimode Floquet theory. The method permits a straightforward construction of the multispin basis and describes the spin dynamics via effective Floquet Hamiltonians obtained using the van Vleck transformation method in the Floquet-Liouville space. As a test case, we consider a model three-spin system (I2S) under asynchronous time modulations (both MAS and rf irradiation) and derive effective Hamiltonians for describing the spin dynamics in the Floquet-Liouville space during heteronuclear decoupling. Furthermore, we describe and evaluate the origin of cross terms between the various anisotropic interactions and illustrate their exact contributions to the spin dynamics. The theory presented herein should be applicable to the design and understanding of pulse sequences for heteronuclear and homonuclear recoupling and decoupling.     

Laser control in open molecular systems: STIRAP and Optimal Control

We examine the effect of dissipation on the laser control of a process that transforms a state into a superposed state. We consider a two-dimensional double well of a single potential energy surface. In the context of reactivity, the objective of the control is the localization in a given well, for instance the creation of an enantiomeric form whereas for quantum gates, this control corresponds to one of the transformation of the Hadamard gate. The environment is either modelled by coupling few harmonic oscillators (up to five) to the system or by an effective interaction with an Ohmic bath. In the discrete case, dynamics is carried out exactly by using the coupled harmonic adiabatic channels. In the continuous case, Markovian and non-Markovian dynamics are considered. We compare two laser control strategies: the Stimulated Raman Adiabatic Passage (STIRAP) method and the optimal control theory. Analytical estimations for the control by adiabatic passage in a Markovian environment are also derived.     

An efficient algorithm for solving nonlinear equations with a minimal number of trial vectors: applications to atomic-orbital based coupled-cluster theory

The conjugate residual with optimal trial vectors (CROP) algorithm is developed. In this algorithm, the optimal trial vectors of the iterations are used as basis vectors in the iterative subspace. For linear equations and nonlinear equations with a small-to-medium nonlinearity, the iterative subspace may be truncated to a three-dimensional subspace with no or little loss of convergence rate, and the norm of the residual decreases in each iteration. The efficiency of the algorithm is demonstrated by solving the equations of coupled-cluster theory with single and double excitations in the atomic orbital basis. By performing calculations on H(2)O with various bond lengths, the algorithm is tested for varying degrees of nonlinearity. In general, the CROP algorithm with a three-dimensional subspace exhibits fast and stable convergence and outperforms the standard direct inversion in iterative subspace method.