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The paper surveys recent progress in the mathematical modelling and simulation of essential molecular dynamics. Particular emphasis is put on computational drug design wherein time scales of $msec$ up to $min$ play the dominant role. Classical long-term molecular dynamics computations, however, would run into ill-conditioned initial value problems already after time spans of only $psec=10^{-12} sec$. Therefore, in order to obtain results for times of pharmaceutical interest, a combined deterministic-stochastic model is needed. The concept advocated in this paper is the direct identification of metastable conformations together with their life times and their transition patterns. It can be interpreted as a {\em transfer operator} approach corresponding to some underlying hybrid Monte Carlo process, wherein short-term trajectories enter. Once this operator has been discretized, which is a hard problem of its own, a stochastic matrix arises. This matrix is then treated by {\em Perron cluster analysis}, a recently developed cluster analysis method involving the numerical solution of an eigenproblem for a Perron cluster of eigenvalues. In order to avoid the 'curse of dimension', the construction of appropriate boxes for the spatial discretization of the Markov operator requires careful consideration. As a biomolecular example we present a rather recent SARS protease inhibitor.
In molecular dynamics applications there is a growing interest in mixed quantum-classical models. The article is concerned with the so-called QCMD model. This model describes most atoms of the molecular system by the means of classical mechanics but an important, small portion of the system by the means of a wavefunction. We review the conditions under which the QCMD model is known to approximate the full quantum dynamical evolution of the system. In most quantum-classical simulations the {\em Born-Oppenheimer model} (BO) is used. In this model, the wavefunction is adiabatically coupled to the classical motion which leads to serious approximation deficiencies with respect to non-adiabatic effects in the fully quantum dynamical description of the system. In contrast to the BO model, the QCMD model does include non-adiabatic processes, e.g., transitions between the energy levels of the quantum system. It is demonstrated that, in mildly non-adiabatic scenarios, so-called {\em surface hopping} extensions of QCMD simulations yield good approximations of the non-adiabatic effects in full quantum dynamics. The algorithmic strategy of such extensions of QCMD is explained and the crucial steps of its realization are discussed with special emphasis on the numerical problems caused by highly oscillatory phase effects.
Recently, a novel concept for the computation of essential features of the dynamics of Hamiltonian systems (such as molecular dynamics) has been proposed. The realization of this concept had been based on subdivision techniques applied to the Frobenius--Perron operator for the dynamical system. The present paper suggests an alternative but related concept that merges the conceptual advantages of the dynamical systems approach with the appropriate statistical physics framework. This approach allows to define the phrase ``conformation'' in terms of the dynamical behavior of the molecular system and to characterize the dynamical stability of conformations. In a first step, the frequency of conformational changes is characterized in statistical terms leading to the definition of some Markov operator $T$ that describes the corresponding transition probabilities within the canonical ensemble. In a second step, a discretization of $T$ via specific hybrid Monte Carlo techniques is shown to lead to a stochastic matrix $P$. With these theoretical preparations, an identification algorithm for conformations is applicable. It is demonstrated that the discretization of $T$ can be restricted to few essential degrees of freedom so that the combinatorial explosion of discretization boxes is prevented and biomolecular systems can be attacked. Numerical results for the n-pentane molecule and the triribonucleotide adenylyl\emph{(3'-5')}cytidylyl\emph{(3'-5')}cytidin are given and interpreted.
The Car-Parrinello (CP) approach to ab initio molecular dynamics serves as an approximation to time-dependent Born-Oppenheimer (BO) calculations. It replaces the explicit minimization of the energy functional by a fictitious Newtonian dynamics and therefore introduces an artificial mass parameter $\mu$ which controls the electronic motion. A recent theoretical investigation shows that the CP-error, i.e., the deviation of the CP--solution from the BO-solution {\em decreases} like $\mu^{1/2}$ asymptotically. Since the computational effort {\em increases} like $\mu^{-1/2}$, the choice of $\mu$ has to find a compromise between efficiency and accuracy. The asymptotical result is used in this paper to construct an easily implemented algorithm which automatically controls $\mu$: the parameter $\mu$ is repeatedly adapted during the simulation by choosing $\mu$ as large as possible while pushing an error measure below a user-given tolerance. The performance and reliability of the algorithm is illustrated by a typical example.
The function of many important biomolecules is related to their dynamic properties and their ability to switch between different {\em conformations}, which are understood as {\em almost invariant} or {\em metastable} subsets of the positional state space of the system. Recently, the present authors and their coworkers presented a novel algorithmic scheme for the direct numerical determination of such metastable subsets and the transition probability between them. Although being different in most aspects, this method exploits the same basic idea as {\sc Dellnitz} and {\sc Junge} in their approach to almost invariance in discrete dynamical systems: the almost invariant sets are computed via certain eigenvectors of the Markov operators associated with the dynamical behavior. In the present article we analyze the application of this approach to (high--friction) Langevin models describing the dynamical behavior of molecular systems coupled to a heat bath. We will see that this can be related to theoretical results for (symmetric) semigroups of Markov operators going back to {\sc Davies}. We concentrate on a comparison of our approach in respect to random perturbations of dynamical systems.
In Quantum Chemistry the field of Laser--Assisted Molecular Control'' has received a considerable amount of attention recently. One key problem in this new field is the simulation of the dynamical reaction of a molecule subjected to external radiation. This problem is described by the Schrödinger equation, which, after eigenfunction expansion, can be written in the form of a large system of ordinary differential equations, the solutions of which show a highly oscillatory behaviour. The oscillations with high frequencies and small amplitudes confine the stepsizes of any numerical integrator -- an effect, which, in turn, blows up the simulation time. Larger stepsizes can be expected by averaging these fast oscillations, thus smoothing the trajectories. Standard smoothing techniques (averaging, filtering) would kill the whole process and thus, lead to wrong numerical results. To avoid this unwanted effect and nevertheless speed up computations, this paper presents a quasiresonant smoothing algorithm (QRS). In QRS, a natural splitting parameter $\delta$ controls the smoothing properties. An adaptive QRS--version (AQRS) is presented which includes an error estimation scheme for choosing this parameter $\delta$ in order to meet a given accuracy requirement. In AQRS $\delta$ is permanently adapted to the solution properties for computing the chemically necessary information'' only. The performance of AQRS is demonstrated in several test problems from the field Laser--Assisted Selective Excitation of Molecules'' in which the external radiation is a picosecond laser pulse. In comparison with standard methods speedup factors of the order of $10^2$ are observed.
In molecular dynamics applications there is a growing interest in mixed quantum-classical models. The {\em quantum-classical Liouville equation} (QCL) describes most atoms of the molecular system under consideration by means of classical phase space density but an important, small portion of the system by means of quantum mechanics. The QCL is derived from the full quantum dynamical (QD) description by applying the Wigner transform to the classical part'' of the system only. We discuss the conditions under which the QCL model approximates the full QD evolution of the system. First, analysis of the asymptotic properties of the Wigner transform shows that solving the QCL yields a first order approximation of full quantum dynamics. Second, we discuss the adiabatic limit of the QCL. This discussion shows that the QCL solutions may be interpretated as classical phase space densities, at least near the adiabatic limit. Third, it is demonstrated that the QCL yields good approximations of {\em non-adiabatic quantum effects,} especially near so-called {\em avoided crossings} where most quantum-classical models fail.
This paper presents a mathematical derivation of a model for quantum-classical molecular dynamics (QCMD) as a {\em partial} classical limit of the full Schrödinger equation. This limit is achieved in two steps: separation of the full wavefunction and short wave asymptotics for its ``classical'' part. Both steps can be rigorously justified under certain smallness assumptions. Moreover, the results imply that neither the time-dependent self-consistent field method nor mixed quantum-semi-classical models lead to better approximations than QCMD since they depend on the separation step, too. On the other hand, the theory leads to a characterization of the critical situations in which the models are in danger of largely deviating from the solution of the full Schrödinger equation. These critical situations are exemplified in an illustrative numerical simulation: the collinear collision of an Argon atom with a harmonic quantum oscillator.
The interaction potential of molecular systems which are typically used in molecular dynamics can be split into two parts of essentially different stiffness. The strong part of the potential forces the solution of the equations of motion to oscillate on a very small time scale. There is a strong need for eliminating the smallest time scales because they are a severe restriction for numerical long-term simulations of macromolecules. This leads to the idea of just freezing the high frequency degrees of freedom (bond stretching and bond angles). However, the naive way of doing this via holonomic constraints is bound to produce incorrect results. The paper presents a mathematically rigorous discussion of the limit situation in which the stiffness of the strong part of the potential is increased to infinity. It is demonstrated that the average of the limit solution indeed obeys a constrained Hamiltonian system but with a {\em corrected soft potential}. An explicit formula for the additive potential correction is given and its significant contribution is demonstrated in an illustrative example. It appears that this correcting potential is definitely not identical with the Fixman-potential as was repeatedly assumed in the literature.
It was revealed that the QCMD model is of canonical Hamiltonian form with symplectic structure, which implies the conservation of energy. An efficient and reliable integrator for transfering these properties to the discrete solution is the symplectic and explicit {\sc Pickaback} algorithm. The only drawback of this kind of integrator is the small stepsize in time induced by the splitting techniques used to discretize the quantum evolution operator. Recent investigations concerning Krylov iteration techniques result in alternative approaches which overcome this difficulty for a wide range of problems. By using iterative methods in the evaluation of the quantum time propagator, these techniques allow for the stepsize to adapt to the coupling between the classical and the quantum mechanical subsystem. This yields a drastic reduction of the numerical effort. The pros and cons of both approaches as well as the suitable applications are discussed in the last part.
A theoretical investigation of the dynamic properties of integrated optical Er--doped waveguide lasers is presented. It includes the construction of a physical model and of numerical techniques which allow reliable simulations of the dynamical behaviour of the laser signal depending on essential parameters of the laser device and on its external, time--dependent pump radiation. Therefore, a physical theory is developed which describes the propagation of light and its interaction with the active substrate in the laser cavity. This is realized in two steps. First, a {\em fundamental model} based on Maxwell's equations and on rate equations for the transitions in the active medium is constructed. Since this turns out to prohibit reliable simulations, it is, in a second step, reformulated via averaging in time and space which suppresses the fluctuations on the fastest time scales but represents them correctly. For this {\em reduced model} reliable and efficient simulation techniques using adaptive control schemes are designed and implemented. We apply the linear--implicit Euler discretization with extrapolation in time and a multilevel quadrature scheme in space. Finally, the model is justified in comparison with experimental observations in four cases of technological relevance.
The article surveys the development of novel mathematical concepts and algorithmic approaches based thereon in view of their possible applicability to biomolecular design. Both a first deterministic approach, based on the Frobenius-Perron operator corresponding to the flow of the Hamiltonian dynamics, and later stochastic approaches, based on a spatial Markov operator or on Langevin dynamics, can be subsumed under the unified mathematical roof of the transfer operator approach to effective dynamics of molecular systems. The key idea of constructing specific transfer operators especially taylored for the purpose of conformational dynamics appears as the red line throughout the paper. Different steps of the algorithm are exemplified by a trinucleotide molecular system as a small representative of possible RNA drug molecules.
In molecular dynamics applications there is a growing interest in including quantum effects for simulations of larger molecules. This paper is concerned with {\em mixed quantum-classical} models which are currently discussed: the so-called QCMD model with variants and the time-dependent Born-Oppenheimer approximation. All these models are known to approximate the full quantum dynamical evolution---under different assumptions, however. We review the meaning of these assumptions and the scope of the approximation. In particular, we characterize those typical problematic situations where a mixed model might largely deviate from the full quantum evolution. One such situation of specific interest, a non-adiabatic excitation at certain energy level crossings, can promisingly be dealt with by a modification of the QCMD model that we suggest.
\noindent In molecular dynamics applications there is a growing interest in so-called {\em mixed quantum-classical} models. These models describe most atoms of the molecular system by the means of classical mechanics but an important, small portion of the system by the means of quantum mechanics. A particularly extensively used model, the QCMD model, consists of a {\em singularly perturbed}\/ Schrödinger equation nonlinearly coupled to a classical Newtonian equation of motion. This paper studies the singular limit of the QCMD model for finite dimensional Hilbert spaces. The main result states that this limit is given by the time-dependent Born-Oppenheimer model of quantum theory---provided the Hamiltonian under consideration has a smooth spectral decomposition. This result is strongly related to the {\em quantum adiabatic theorem}. The proof uses the method of {\em weak convergence} by directly discussing the density matrix instead of the wave functions. This technique avoids the discussion of highly oscillatory phases. On the other hand, the limit of the QCMD model is of a different nature if the spectral decomposition of the Hamiltonian happens not to be smooth. We will present a generic example for which the limit set is not a unique trajectory of a limit dynamical system but rather a {\em funnel} consisting of infinitely many trajectories.
Statistical methods for analyzing large data sets of molecular configurations within the chemical concept of molecular conformations are described. The strategies are based on dependencies between configurations of a molecular ensemble; the article concentrates on dependencies induces by a) correlations between the molecular degrees of freedom, b) geometrical similarities of configurations, and c) dynamical relations between subsets of configurations. The statistical technique realizing aspect a) is based on an approach suggested by {\sc Amadei et al.} (Proteins, 17 (1993)). It allows to identify essential degrees of freedom of a molecular system and is extended in order to determine single configurations as representatives for the crucial features related to these essential degrees of freedom. Aspects b) and c) are based on statistical cluster methods. They lead to a decomposition of the available simulation data into {\em conformational ensembles} or {\em subsets} with the property that all configurations in one of these subsets share a common chemical property. In contrast to the restriction to single representative conformations, conformational ensembles include information about, e.g., structural flexibility or dynamical connectivity. The conceptual similarities and differences of the three approaches are discussed in detail and are illustrated by application to simulation data originating from a hybrid Monte Carlo sampling of a triribonucleotide.
Models for occupation dynamics in discrete quantum systems lead to large or even infinite systems of ordinary differential equations. Some new mathematical techniques, developed for the simulation of chemical processes, make a numerical solution of countable systems of ordinary differential equations possible. Both, a basic physical concept for the construction of such systems and the structure of the numerical tools for solving them are presented. These conceptual aspects are illustrated by a simulation of an occupation process from spectroscopy. In this example the structures of rotation spectra observed in infrared spectroscopy are explained and some possibilities for an extension of the model are shown.
A quasiresonant smoothing algorithm for the fast analysis of selective vibrational excitation.
(1993)
One key problem in modern chemistry is the simulation of the dynamical reaction of a molecule subjected to external radiation. This is described by the Schrödinger equation, which, after eigenfunction expansion, can be written in form of a system of ordinary differential equations, whose solutions show a highly oscillatory behaviour. The oscillations with high frequencies and small amplitudes confine the stepsizes of any numerical integrator -- an effect, which, in turn, blows up the simulation time. Larger stepsizes can be expected by averaging these fast oscillations, thus smoothing the trajectories. This idea leads to the construction of a quasiresonant smoothing algorithm (QRS). In QRS, a natural and computationally available splitting parameter $\delta$ controls the smoothing properties. The performance of QRS is demonstrated in two applications treating the selective excitation of vibrational states by picosecond laser pulses. In comparison with standard methods a speedup factor of 60--100 is observed. A closer look to purely physically motivated quasiresonant approximations such as WFQRA shows some additional advantages of the above smoothing idea. Among these the possibility of an adaptive formulation of QRS via the parameter $\delta$ is of particular importance.
An analysis of relaxation oscillations in local Er-doped optically pumped lasers is reported. It is based on a time dependent rate equation model for a quasi-two-level-system with wavelength dependent emission- and absorption cross-sections. For the first time a numerically reliable simulation of the characteristic laser behaviour was possible: the onset and decay of the oscillations, the time-dependent repetition period and the steady state signal output power. The characteristic waveguide parameters, as the erbium-concentration profile, the polarization dependent pump- and signal mode intensity profiles, the scattering losses, the cavity length and the front and rear reflectivities were all taken into account. The basic formulas are general and can also be used for Er-doped fiber lasers. Mathematically the problem can be characterized as a large boundary value problem, which can approximately be replaced by a stiff initial value problem of ordinary differential equations. The used algorithmic replacement procedure is motivated and discussed in detail. Here, pump- and signal evolution versus time are presented for an planar Er-diffused $\rm Ti$:$\rm LiNbO_{3}$ waveguide laser. The numerically obtained results show a nearly quantitative agreement with experimental investigations. Simultanously they supply knowledge about non-measureable (space-dependent population dynamic of the Er-atoms) and till today not measured data (dynamical response of the laser by a sharp peak in the external pump).
This paper makes use of statistical mechanics in order to construct effective potentials for Molecular Dynamics for systems with nonstationary thermal embedding. The usual approach requires the computation of a statistical ensemble of trajectories. In the context of the new model the evaluation of only one single trajectory is sufficient for the determination of all interesting quantities, which leads to an enormous reduction of computational effort. This single trajectory is the solution to a corrected Hamiltonian system with a new potential $\tilde{V}$. It turns out that $\tilde{V}$ can be defined as spatial average of the original potential $V$. Therefore, the Hamiltonian dynamics defined by $\tilde{V}$ is smoother than that effected by $V$, i.e. a numerical integration of its evolution in time allows larger stepsizes. Thus, the presented approach introduces a Molecular Dynamics with smoothed trajectories originating from spatial averaging. This is deeply connected to time--averaging in Molecular Dynamics. These two types of {\em smoothed Molecular Dynamics} share advantages (gain in efficiency, reduction of error amplification, increased stability) and problems (necessity of closing relations and adaptive control schemes) which will be explained in detail.
The paper studies Hamiltonian systems with a strong potential forcing the solutions to oscillate on a very small time scale. In particular, we are interested in the limit situation where the size $\epsilon$ of this small time scale tends to zero but the velocity components remain oscillating with an amplitude variation of order ${\rm O}(1)$. The process of establishing an effective initial value problem for the limit positions will be called {\em homogenization} of the Hamiltonian system. This problem occurs in mechanics as the problem of realization of holonomic constraints, in plasma physics as the problem of guiding center motion, in the simulation of biomolecules as the so called smoothing problem. We suggest the systematic use of the notion of {\em weak convergence} in order to approach this problem. This methodology helps to establish unified and short proofs of the known results which throw light on the inherent structure of the problem. Moreover, we give a careful and critical review of the literature.
A hybrid Monte Carlo method with adaptive temperature choice is presented, which exactly generates the distribution of a mixed-canonical ensemble composed of two canonical ensembles at low and high temperature. The analysis of resulting Markov chains with the reweighting technique shows an efficient sampling of the canonical distribution at low temperature, whereas the high temperature component facilitates conformational transitions, which allows shorter simulation times. \\The algorithm was tested by comparing analytical and numerical results for the small n-butane molecule before simulations were performed for a triribonucleotide. Sampling the complex multi-minima energy landscape of these small RNA segments, we observed enforced crossing of energy barriers.
The paper presents the concept of a new type of algorithm for the numerical computation of what the authors call the {\em essential dynamics\/} of molecular systems. Mathematically speaking, such systems are described by Hamiltonian differential equations. In the bulk of applications, individual trajectories are of no specific interest. Rather, time averages of physical observables or relaxation times of conformational changes need to be actually computed. In the language of dynamical systems, such information is contained in the natural invariant measure (infinite relaxation time) or in almost invariant sets ("large" finite relaxation times). The paper suggests the direct computation of these objects via eigenmodes of the associated Frobenius-Perron operator by means of a multilevel subdivision algorithm. The advocated approach is different to both Monte-Carlo techniques on the one hand and long term trajectory simulation on the other hand: in our setup long term trajectories are replaced by short term sub-trajectories, Monte-Carlo techniques are just structurally connected via the underlying Frobenius-Perron theory. Numerical experiments with a first version of our suggested algorithm are included to illustrate certain distinguishing properties. A more advanced version of the algorithm will be presented in a second part of this paper.
The aim of this work is to study the accuracy and stability of the Chebyshev--approximation method as a time--discretization for wavepacket dynamics. For this frequently used discretization we introduce estimates of the approximation and round--off error. These estimates mathematically confirm the stability of the Chebyshev--approximation with respect to round--off errors, especially for very large stepsizes. But the results also disclose threads to the stability due to large spatial dimensions. All theoretical statements are illustrated by numerical simulations of an analytically solvable example, the harmonic quantum oszillator.
The Car-Parrinello method for ab-initio molecular dynamics avoids the explicit minimization of energy functionals given by functional density theory in the context of the quantum adiabatic approximation (time-dependent Born-Oppenheimer approximation). Instead, it introduces a fictitious classical dynamics for the electronic orbitals. For many realistic systems this concept allowed first-principle computer simulations for the first time. In this paper we study the {\em quantitative} influence of the involved parameter $\mu$, the fictitious electronic mass of the method. In particular, we prove by use of a carefully chosen two-time-scale asymptotics that the deviation of the Car-Parrinello method from the adiabatic model is of order ${\rm O}(\mu^{1/2})$ --- provided one starts in the ground state of the electronic system and the electronic excitation spectrum satisfies a certain non-degeneracy condition. Analyzing a two-level model problem we prove that our result cannot be improved in general. Finally, we show how to use the gained quantitative insight for an automatic control of the unphysical ``fake'' kinetic energy of the method.
{\footnotesize In classical Molecular Dynamics a molecular system is modelled by classical Hamiltonian equations of motion. The potential part of the corresponding energy function of the system includes contributions of several types of atomic interaction. Among these, some interactions represent the bond structure of the molecule. Particularly these interactions lead to extremely stiff potentials which force the solution of the equations of motion to oscillate on a very small time scale. There is a strong need for eliminating the smallest time scales because they are a severe restriction for numerical long-term simulations of macromolecules. This leads to the idea of just freezing the high frequency degrees of freedom (bond stretching and bond angles) via increasing the stiffness of the strong part of the potential to infinity. However, the naive way of doing this via holonomic constraints mistakenly ignores the energy contribution of the fast oscillations. The paper presents a mathematically rigorous discussion of the limit situation of infinite stiffness. It is demonstrated that the average of the limit solution indeed obeys a constrained Hamiltonian system but with a {\em corrected soft potential}. An explicit formula for the additive potential correction is given via a careful inspection of the limit energy of the fast oscillations. Unfortunately, the theory is valid only as long as the system does not run into certain resonances of the fast motions. Behind those resonances, there is no unique limit solution but a kind of choatic scenario for which the notion ``Takens chaos'' was coined. For demonstrating the relevance of this observation for MD, the theory is applied to a realistic, but still simple system: a single butan molecule. The appearance of ``Takens chaos'' in smoothed MD is illustrated and the consequences are discussed.}
The topic of the present paper bas been motivated by a recent computational approach to identify chemical conformations and conformational changes within molecular systems. After proper discretization, the conformations show up as almost invariant aggregates in reversible nearly uncoupled Markov chains. Most of the former work on this subject treated the direct problem: given the aggregates, analyze the loose coupling in connection with the computation of the stationary distribution (aggregation/disaggregation techniques). In contrast to that the present paper focuses on the inverse problem: given the system as a whole, identify the almost invariant aggregates together with the associated transition probabilites. A rather simple and robust algorithm is suggested and illustrated by its application to the n-pentane molecule.
Recently, a novel concept for the computation of essential features of Hamiltonian systems (such as those arising in molecular dynamics) has been proposed. The realization of that concept was based on subdivision techniques applied to the Frobenius--Perron operator for the dynamical system. The present paper suggests an alternative but related concept based on statistical mechanics, which allows to attack realistic molecular systems. In a first step, the frequency of conformational changes is characterized in statistical terms leading to the definition of some Markov operator $T$ that describes the corresponding transition probabilities within the canonical ensemble. In a second step, a discretization of $T$ via hybrid Monte Carlo techniques (based on short term subtrajectories only) is shown to lead to a stochastic matrix $P$. With these theoretical preparations, an identification algorithm for conformations is applicable (to be presented elsewhere). Numerical results for the n-pentane molecule are given and interpreted.
The function of many important biomolecules comes from their dynamic properties and their ability to switch between different {\em conformations}. In a conformation, the large scale geometric structure of the molecule is understood to be conserved, whereas on smaller scales the system may well rotate, oscillate or fluctuate. In a recent article [J. Comp. Phys., 151,1 (1999)], the present author and coworkers demonstrated that (a) conformations can be understood as almost invariant sets of some Markov chain being defined via the Hamiltonian system governing the molecular dynamics and that (b) these sets can efficiently be computed via eigenvectors of the corresponding Markov operator. The persent manuscript reviews the mathematical modelling steps behind the novel concept, includes a rigorous analytical justification of this approach and especially of the numerical details of the algorithm, and illustrates its performance when applied to realistic molecular systems.
Uncoupling-coupling Monte Carlo (UCMC) combines uncoupling techniques for finite Markov chains with Markov chain Monte Carlo methodology. UCMC aims at avoiding the typical metastable or trapping behavior of Monte Carlo techniques. From the viewpoint of Monte Carlo, a slowly converging long-time Markov chain is replaced by a limited number of rapidly mixing short-time ones. Therefore, the state space of the chain has to be hierarchically decomposed into its metastable conformations. This is done by means of combining the technique of conformation analysis as recently introduced by the authors, and appropriate annealing strategies. We present a detailed examination of the uncoupling-coupling procedure which uncovers its theoretical background, and illustrates the hierarchical algorithmic approach. Furthermore, application of the UCMC algorithm to the $n$-pentane molecule allows us to discuss the effect of its crucial steps in a typical molecular scenario.