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Multiple Time�Scales in Classical and Quantum�Classical

Molecular Dynamics

Sebastian Reich�

October �� ����

Abstract

The existence of multiple time scales in molecular dynamics poses interesting and challengingquestions from an analytical as well as from a numerical point of view� In this paper� we considersimpli�ed models with two essential time scales and describe how these two time scales inter�act� The discussion focuses on classical molecular dynamics �CMD� with fast bond stretchingand bending modes and the� so called� quantum�classical molecular dynamics �QCMD� modelwhere the quantum part provides the highly�oscillatory solution components� The analytic res�ults on the averaging over fast degrees of motion will also shed new light on the appropriateimplementation of multiple�time�stepping �MTS� algorithms for CMD and QCMD�

� Introduction

Classical molecular dynamics �CMD� ��� leads to Newtonian equations of motion with fast bondstretching and bending modes and a relatively slow motion in the remaining degrees of freedom�For numerical integration� the Verlet method ���� is typically used with a stepsize that resolves thefast bond stretchingbending modes� However� often one is interested in the computation of slowlyvarying quantities andor time averages and a method such as Verlet can quickly become ine�cientfor long time simulations�

Various approaches have been suggested to improve the classical Verlet method� Among theseare �i� methods based on the explicit elimination of the fast bond stretchingbending modes andthe subsequent integration of the corresponding constrained equations of motion by the SHAKE orRATTLE method ��� � � and �ii� reversible multiple time stepping �MTS� methods � � ��� ��� thatuse di�erent time steps for the fast and slow degrees of freedom�

In appropriate �local� coordinates� the fast bond stretching and bending modes can be reduced toweakly coupled harmonic oscillators whose frequency depends on the slow modes� This dependenceleads to a coupling of the slow and fast modes which� in general� implies that the fast degrees ofmotion cannot be removed from the model without changing its long time dynamics ���� ��� ��� ������� It seems that in those situations only methods based on the idea of MTS can and should beused for enhanced classical molecular dynamics� However� one has to be careful� Straightforwardapplication of a MTS method may lead to wrong results or to unstable computations � � ��� Thisfact is brie�y discussed in x�� An improved approach to multiple time stepping has been suggestedby Garc�ia�Archilla� Sanz�Serna � Skeel in ����� In x�� we consider a variant of this approach���� ��� that is particularly suited for the multiple time scale integration of classical moleculardynamics �CMD��

In recent years� the combination of quantum and classical molecular dynamics has become important� In this paper� we focus on the� so called� quantumclassical molecular dynamics �QCMD�model ���� ��� �� ��� ��� where most of the molecular system is described by classical Newtonianequations of motion while a small but important part is modeled by a timevarying Schr�odingerequation �see x��� Again the fast quantum degrees of freedom and the slow classical degrees offreedom are tightly coupled� In fact� the e�ect of this coupling on the �slow� classical degrees offreedom� which is linked to the BornOppenheimer approximation ����� is easier to understand than

�Department of Mathematics and Statistics� University of Surrey� Guildford� Surrey GU� �XH� UK� E�mail�s�reich�surrey�ac�uk

the corresponding coupling e�ects in classical molecular dynamics� However� as we will show inx�� classical molecular dynamics can be transformed to a system resembling the QCMD model andtheoretical results derived for the QCMD model can also be applied to classical molecular dynamics�This is con�rmed by the numerical simulation of a simple test problem�

Because of the importance of the QCMD model� we also discuss MTS methods for QCMD���� ��� ���� Here it is crucial to observe that the method has to be implemented in an appropriateway and that some of the straightforward implementations can lead to erroneous numerical resultsin the �slow� classical degrees of freedom ���� ��� ���� This aspect is discussed in x��

� Classical MD and Multiple�Time�Stepping

The atomic motion of a molecular system� consisting ofN atoms� is typically described by Newtonianequations of motion of the form

�q � M��p� ���

�p � �rqV�q��mXi��

Gi�q�T kiigi�q� ���

where q � R�N is the vector of all atomic positions� p � R�N the vector of the correspondingmomenta� M � R�N��N the diagonal mass matrix� V�q� the potential energy except for the termscorresponding to bond stretching and bending which are described by the second term on the lefthand side of equation ���� Here the functions fgigi������ �m can either stand for gi�q� � r� r� �bondstretching� or gi�q� � � � �� �bond angle bending�� In either case� Gi�q� � R�N denotes theJacobian of gi�q� and kii the force constant� For compactness of notation� we collect the functionsfgig in the vectorvalued function g� the force constants fkiig in the diagonal matrix K � Rm�m�and the Jacobians fGi�q�g in the matrix G�q� � Rm��N � This gives rise to the equations

�q � M��p� ���

�p � �rqV�q��G�q�TKg�q�� ���

The total energy

H ��

�pTM��p� V�q� � �

�g�q�TKg�q�

is a conserved quantity ��rst integral� along solutions of �������Let us denote the potential energy of the system by U�q�� i�e�

U�q� �� V�q� � �

�g�q�TKg�q��

and the kinetic energy by T �p�� Then the Verlet method ���� can be considered as a compositionmethod ���� based on the exact solution operators of the two systems

�q � rpT �p� � M��p�

�p � �

and

�q � ��

�p � �rqU�q��

Let us denote these solution operators by exp�tLT � and exp�tLU �� respectively� Then one step ofthe Verlet method with a stepsize �t is equivalent to the concatenation

exp��t

�LU � � exp��tLT � � exp��t

�LU ��

Since each solution operator is volume preserving �and even symplectic�� the Verlet method is volumepreserving �symplectic� ����� Furthermore� the method conserves linear and angular momentum andthe timereversibility of the Newtonian equations of motion�

The Verlet method becomes ine�cient if the evaluation of the force �eld is expensive due tolongrange interactions� To enhance the classical Verlet method� a symplectic and timereversiblemultipletimestepping �MTS� method was suggested in � � ��� ���� The idea of this MTS method isamazingly simple� We split the total potential energy U into two terms U� and U� with U� containingall the short range interactions �in particular the bond stretchingbending modes�� Then one stepof a MTS scheme with macro stepsize �t � j�t� j � �� reads�

exp��t

�LU�

� �

����exp��t� LU�

� � exp��tLT � � exp��t�LU�

�� �z �j times

���� � exp��t� LU�

�� ���

This method has the same conservation properties as Verlet� but it is potentially more e�cient sincethe long range forces have to computed less frequently�

Although the idea of ��� is simple� the method has some drawbacks� For certain values ofthe macro stepsize �t� the method can become unstable �meaning a systematic increase in totalenergy� due to resonancesampling phenomena � � ��� Even if there is no systematic increase inthe total energy� the numerical results can be qualitatively di�erent from those obtained from thestandard Verlet method� This e�ect does not occur for systems with strong bond bendingstretchingpotentials� But it can be observed for systems with very light particles if the splitting of theHamiltonian is not done properly� We will also encounter this problem when discussing MTSmethods for the QCMD model�

Example �� We take two harmonic oscillators Hx � �px������qx���� and Hy � �py����������qy����� one of whichhas a very small mass m � ��� �� �� coupled by a quadratic potential W � qxqy�

H ��

��px�

� ����

��py�

� ��

��qx�

� ��

��qy�

� � qxqy�

The equations of motion are

qx � px�

px � �qx � qy�

qy � ���py�

py � �qy � qx�

The last two equations describe a highly oscillatory motion about the equilibrium ��qx� ��� If this solution is usedin the second equation to time�average over the rapidly oscillating force term

F �t� � �qx � qy�t��

we obtain the averaged force hF i � �� Thus� the slow dynamics in the variable �qx� px� is approximately given bythe equations

qx � px�

px � ��

A MTS scheme can be obtained via the splitting

T ��

��px�

� ����

��py�

��

U� ��

��qy�

��

and

U� ��

��qx�

� � qxqy�

We assume that the step�size �t � � in ��� is chosen small enough such that the equations of motion corresponding tothe Hamiltonian T �U� are solved exactly� Next we de�ne the macro step�size �t such that the solutions to T �U�satisfy qy��t� � qy��� and py��t� � py���� i�e� �t � k������� k � �� Thus� instead of sampling a highly oscillatorysolution� we obtain a �ctitious constant solution which� when plugged into the numerical approximation of

px � �qx � qy�t� � F �t�

leads� in general� to a qualitatively wrong approximation of the averaged force hF i� This problem does not occur ifthe splitting

U� ��

��qy�

� � qxqy

�See Fig� � in x� for a more explicit formulation of ����

and

U� ��

��qx�

is used� �

An alternative to the MTS scheme ��� is to replace the Verlet discretization of T �p� �U��q� in theinner loop of ��� by one step with an implicit method �such as the implicit midpoint rule ����� andstepsize �t � �t� However� in addition to the resonance problems of the MTS method ��� ����� onealso� in general� encounters an exponential instability unless the stepsize �t is chosen su�cientlysmall ���� Thus such an approach seems inappropriate for CMD simulations�

� Quantum�Classical Molecular Dynamics

��� The QCMD Model

Various extensions of classical mechanics to also include quantum e�ects have been introduced in theliterature� Here we consider the socalled quantumclassical molecular dynamics �QCMD� model�In the QCMD model �see ���� ��� �� ��� ��� and references therein�� most atoms are described byclassical mechanics� but an important small portion of the system by quantum mechanics� Thisleads to a coupled system of Newtonian and Schr�odinger equations�

For ease of presentation� we consider the case of just one quantum degree of freedom with spatialcoordinate x and mass m and N classical particles with coordinates q � R�N and diagonal massmatrix M � R�N��N � Upon denoting the interaction potential by V �x� q�� we obtain the followingequations of motion for the QCMD model�

�� � � i

�H�q�� �

�q � M��p �

�p � �h��rqH�q��i �rqU�q�

with U�q� the purely classical potential energy and H�q� the quantum Hamiltonian operator typically given by

H�q� � T � V �x� q�� T � � ��

�m�x�

In the sequel� we assume that the quantum subsystem has been truncated to a �nitedimensionalsystem by an appropriate spatial discretization and a corresponding representation of the wavefunction � by a complexvalued vector � � Cd� The discretized quantum operators T� V and H aredenoted by T � Cd�d� V �q� � Cd�d and H�q� � Cd�d� respectively�

The total energy of the system

H �pTM��p

�� h��H�q��i� U�q� ���

is a conserved quantity of the QCMD model� Here

h��H�q��i �� ��TH�q��

and �� denotes the complex conjugate of �� Another conserved quantity is the norm of the vector�� i�e�� h���i � const� due to the unitary propagation of the quantum part�

In the context of this paper� an important conservation property of the QCMD model is relatedto its Hamiltonian structure which implies the symplecticness of the solution operator ���� There aredi�erent ways to consider the QCMD model as a Hamiltonian system with Hamiltonian ���� Herewe basically� follow the presentation given in ���� ���� We decompose the complexvalued vector �into its real and imaginary part� i�e��

� ��p��q� � ip�� � � �

�We use a di�erent scaling in ��� which leads to the scaled canonical structure ��� of phase space�

Then� the equations of motion can be derived from the scaled LiePoisson bracket

fF�Gg � ���fF�Ggq��p� � fF�Ggq�p� ���

i�e��

�f � ff�Hg

describes the time evolution of a function f under the Hamiltonian H� The brackets fF�Ggq��p� andfF�Ggq�p in ��� stand for the canonical bracket of classical mechanics ���� For example� fF�Ggq�p ��rqF �TrpG� �rpF �TrqG�

��� The Limit Behavior of the QCMD Model

It is of interest to consider the limit� �� � for a �xed energy function ���� As explicitly shown byBornemann � Sch�utte in ���� ��� using homogenization techniques� the QCMD model reduces tothe BornOppenheimer approximation if the symmetric matrix H�q� can be smoothly diagonalizedand its �real� eigenvalues fEi�q�t��gi������ �d are pairwise di�erent� This implies that the populationsj�i�t�j�� i � �� � � � � k� corresponding to the eigenvalues Ei�q�t��� of the operator H�q� becomeadiabatic invariants� Without going through a detailed analysis� this can be seen from the followingaveraging argument� Let us assume that there is a matrix valued function Q�q� � Rd�d such that�i� Q�q�TQ�q� � I and �ii� E�q� �� Q�q�H�q�Q�q�T is diagonal� For simplicity� we consider onlyone classical degree of freedom� i�e� �q�p� � �q� p� � R�� Next we introduce the new vector

� �� Q�q�� � Cd

and obtain the transformed QCMD equations of motion

�� � � i

�E�q�� �M��pA�q�� � ���

�q � M��p � ����

�p � �h��Q�q�rqH�q�Q�q�T �i � rqU�q� ����

with the skewsymmetric matrix A�q� �� rqQ�q�Q�q�T � Note that M��pA�q� � �Q�q�Q�q�T � Thefast motion is given by the diagonalized timedependent Schr�odinger equation

�� � � i

�E�q�� ����

and the HellmannFeynman force FHF ����� acting on the classical particles� can be written as

FHF � �h��Q�q�rqH�q�Q�q�T�i � �h��rqE�q��i� h�� �A�q�E�q���i� ����

with the matrix commutator

�A�q��E�q�� � A�q�E�q��E�q�A�q��

We call

FBO � �h��rqE�q��i ����

the BornOppenheimer part of the HellmannFeynman force �����If all �real� elements of the diagonal matrix E�q� are di�erent� then the quantum adiabatic

theorem �� � ��� implies that the transformed �wave� vector ��t� follows the solutions of the reducedsystem ���� and the motion in the classical degrees of freedom is obtained by timeaveraging theHellmannFeynman force ���� over the highly oscillatory solutions� ��t� of ����� For this� it is

�One should really consider the limitM ��� i�e�� should increase the mass of the classical particles� But this isequivalent� after an appropriate rescaling of time� to taking the limit �� ��

�The average is taken over a period of time T such that �i� q�t� � const� and �ii� the Schr�odinger equation ����

undergoes many oscillations� For example� T � p� as �� ��

crucial to observe that the matrix commutator �A�q�E�q�� has zero diagonal entries and� thus� thetimeaverage of h��t�� �A�q�E�q����t�i is approximately zero� Thus we obtain the averaged system

�� � � i

�E�q���

�q � M��p�

�p � �h��rqE�q��i � rqU�q��

Since E�q� is diagonal� the entries �i�t� � C� i � �� � � � � d� of the vector ��t� satisfy j�i�t�j� � const�and

�q � M��p�

�p � �Xi

j�ij�rqEii�q��rqU�q��

These equations are known as the BornOppenheimer approximation for the classical coordinate�����

If eigenvalues of the matrix E�q� cross� then j�i�t�j� �� const�� in general� and the BornOppenheimer approximation breaks down� In this case� the full QCMD model has to be solved�Note that the crossing of eigenvalues cannot be avoided in general�

Example �� Let us consider a simple toy problem with two fast modes and one slow mode�

H�q� �

�q cos� q � ��� q� sin� q ��� �q� sin q cos q��� �q� sin q cos q q sin� q � �� � q� cos� q

��

�cos q sin q� sin q cos q

� �q �� �� q

� �cos q � sin qsin q cos q

and

H ��

�p� �

�q� � h��H�q��i�

q� p � R� � � C�� Note that

Q�q� �

�cos q � sin qsin q cos q

��

E�q� �

�q �� �� q

��

and

A �

�� ��� �

��

Thus the transformed equations of motion are

� � � i

�E�q�� � pA��

q � p�

p � �q � h��B�i� h��C�q��iwith

B �

�� �� ��

and

C�q� � �A�E�q�� �

�� �q � �

�q � � �

��

The Hellmann�Feynman force ���� is given by

FHF � �h��B�i� h��C�q��i� ����

Unless q � ��� �resonance point�� the equations can be averaged and we obtain the Born�Oppenheimer system

� � � i

�E�q���

q � M��p�

p � �q � h��B�i�

Numerical results are presented in x�� �

� Multiple Time Scales in Classical MD

��� A Classical MD Model

We now come back to the CMD model of x�� In particular� we consider a conservative system withHamiltonian

HK ��

�pTM��p� V�q� � K

�g�q�T g�q� ����

where V � Rn � R and g � Rn � Rm� m � n� are nonnegative functions� M � Rn�n is a diagonalmass matrix� and K � � is a parameter�� We are interested in the limit K � � and solutionswith energy HK � c for all K su�ciently large� c � � some given constant� This implies that eachcomponent gi�q�� i � �� � � � �m� of the vector valued function g satis�es

gi�q� �r

�c

K

and� for K ��� suggests to replace the equations of motion

�q � M��p�

�p � �rqV�q��KG�q�T g�q��

G�q� � Rm�n the Jacobian of g�q�� by the constrained system

�q � M��p� �� �

�p � �rqV�q��G�q�T�� ����

� � g�q�� ����

We assume throughout the paper that the m�m matrix G�q�M��G�q�T is invertible�The constrained system can be integrated numerically using the SHAKE or RATTLE method

��� � � which are basically equivalent �� � and lead to a modi�ed Verlet method of type

qn�� � qn ��tM��pn�����

pn���� � pn ��t

�rqV�qn���tG�qn�

T�n�

pn�� � pn���� ��t

�rqV�qn����

� � g�qn����

Although this approach is very appealing� the constrained system does not� in general� re�ect thecorrect limit behavior of the unconstrained system for K � �� There are basically two problems�

Even in the limit K � �� solutions of ���� do not� in general� reduce to solutions of theconstrained system �� ������ This is due to a coupling of the fast oscillations to the slowlyvarying solution components� This coupling gives rise to an additional �correcting� force termin �� ������ See ���� ��� ��� ��� ��� and x��� below�

The approximation gi�q� � � is often too crude unless the force constant K is very large� Infact� the function values gi rapidly oscillate about the minimum of the total energy ����� Thisleads to a modi�ed constrained function in ����� The numerical implementation of these �softconstraints� has been discussed in ���� ���� An equivalent approach �but somewhat easier toimplement� is to modify the force �eld �����

A brief account on the relevant analysis leading to the correcting potential is given in the followingsection� The approach is new in the sense that we show the relation of the unconstrained formulationto the QCMD model� This allows us to restrict the analysis of the limiting behavior to the limitingbehavior of a QCMDlike model �as discussed in x���

�This Hamiltonian corresponds to the general system considered in x� except that all force constants fkiig areassumed to be equal to K and the number of degrees of freedom satis�es n � �N �

��� Reduction of the CMD Model to a QCMD�like Model

The underlying QCMD model is found by a sequence of canonical transformations ��� of phase space�We start with the canonical point transformation introduced by the coordinate transformation

q� �� g�q��

q� �� b�q��

where b � Rn � Rn�m is an appropriate function such that B�q�M��G�q�T � �� Here B�q� �R�n�m�n denotes the Jacobian of the function b�q�� The corresponding conjugate momenta arede�ned via the relation

p �G�q�Tp� �B�q�Tp��

Using this transformation� the Hamiltonian ���� becomes

HK ��

�pT�G�q�M��G�q�Tp� �

�pT�B�q�M��B�q�Tp� � V�q� � K

�qT� q��

��

�pT�A�q�� q��p� �

�pT�C�q�� q��p� �W�q�� q�� �

K

�qT� q�

with A�q�� q�� � G�q�M��G�q�T � C�q�� q�� � B�q�M��B�q�T � and W�q�� q�� � V�q�� Sincethe components fq��igi������ �m� of the vector q� satisfy

q��i �r

�c

K�

we can scale q� by K��� and de�ne �q� �� K���q�� This yields the Hamiltonian

H� ��

�pT�A���q�� q��p� �

�pT�C���q�� q��p� �W���q�� q�� �

��qT� �q� with � �� K����� ����

The equations of motion are generated via the scaled LiePoisson bracket

fF�Gg � ���fF�Gg�q��p

�� fF�Ggq

��p

��

Here fF�Gg�q��p

�and fF�Ggq

��p

�denote again the canonical bracket of classical mechanics� Note

that the constrained dynamics is obtained by setting �q� � � and p� � �� Thus� in local coordinates�the constrained system �� ����� is characterized by the Hamiltonian

Hc ��

�pT�K�q��p� �W �q�� ����

with K�q�� � C��� q�� and W �q�� �W��� q���Without giving a rigorous justi�cation� we now set the �small� term ��q in the Hamiltonian ����

equal to zero� This yields

H� ��

�pT�A��� q��p� �

�pT�K�q��p� �W �q�� �

��qT� �q� ����

which is to be compared to the constrained Hamiltonian ����� Next we introduce the matrix valuedfunction D�q�� � Rm�m by

D�q�� � �A��� q������

This gives rise to another canonical point transformation via the coordinate transformation

x � D�q�����q��

y � q�

and corresponding canonical momenta �px�py� de�ned byD�q��

�� �

��h��q�

D�q�����q�

iTI

�pxpy

��

�p�p�

�� ����

Upon dropping the o�diagonal term of order � in ����� the Hamiltonian ���� becomes

H� ��

�pTxH�y�px �

�pTyK�y�py �W �y� �

�xTH�y�x

with H�y� �D�y�� and K�y��W �y� as de�ned in ����� The corresponding LiePoisson bracket is

fF�Gg � ���fF�Ggx�px � fF�Ggy�py �Next we consider x and px as the real and imaginary part of the complex vector z � Cm� i�e�

z ��p��x� ipx� �

This yields

H� � �zTH�y�z ��

�pTyK�y�py �W �y�

and we write this as a QCMDlike system with total energy

H� � hz�H�y�zi�Hc�y�py��

the �nite dimensional �Schr�odinger operator� H�y�� the �wave function� z� the arti�cial �Plankconstant� �� and the classical �constrained� Hamiltonian ����� The equations of motion are

�z � � i

�H�y�z� ����

�y � �rpyHc�y�py�� ����

�py � �ryHc�y�py��ryhz�H�y�zi� ����

We are interested in the limit � � � which we will discuss in x���� The constrained systemapproximation corresponds to H� � Hc which neglects the �quantum� contributions� We note thatI �� hz� zi is a �rst integral of the system� The same quantity is not necessarily conserved for thesystem with the complete Hamiltonian ����� However� numerical experiments indicate that I isan adiabatic invariant for H� and is conserved over relatively long integration intervals up to small�uctuations� See Example � below� A theoretical investigation of this behavior will be carried outin a forthcoming publication�

Example �� We consider a planar system consisting of six particles with coordinates qi � R� and mass m � �� Theparticles interact with each other through a �sti�� harmonic potential

Vsti� �K

�Xj��

�rj�j�� � ����

rij � jjqi � qj jj� and a �soft� anharmonic potential

Vsoft ��X

j��

�Xi�j��

rij�

For simplicity� the �rst particle is �xed at zero� The force constant K is set to K � ���e���� ���e���� ���e���� We

integrate the equations of motion using the Verlet method with a su�ciently small step�size of �t � ����pK and

compute the eigenvalues of the corresponding matrix H�y� � R���� the entries of the vector z � C�� and I � hz�zi�In Fig� �� we plot the time evolution of I�t� over a time interval T � ���� It can be concluded that the norm of the

vector z is relatively well conserved for our two time�scales CMD model� Numerical results on the time evolution of

the individual entries of the vector z can be found in x���� �

In terms of the original variables �q�p�� the QCMDlike equations �������� can be written as aconstrained QCMDlike system

�z � � i

�H�q�z�

�q � M��p�

�p � �rqV�q��rqhz�H�q�zi �G�q�T��

� � g�q�

with H�q� � �G�q�M��G�q�T �����

�We have H� � H� �O����

0 20 40 60 80 100 1201.2

1.3

1.4

1.5

1.6

K=2.5e+04

0 20 40 60 80 100 1201.2

1.3

1.4

1.5

1.6

K=1.0e+06

0 20 40 60 80 100 1201.2

1.3

1.4

1.5

1.6

K=1.0e+08

Figure �� Time evolution of I�t� over a time interval ��� ���� for di�erent values of the force constantK�

��� The Limiting Behavior of the CMD Model

The results of the previous section indicate that one can reduce the discussion of the limiting behaviorof the CMD model as � � K���� � � to the investigation of the QCMDlike equations ���������

In case that H�y� is a scalar� i�e� H�y� � h�y� � R� no resonances can occur and the �BornOppenheimer� approximation is valid for all y� The averaged equations are

�y � �rpyHc�y�py��

�py � �ryHc�y�py�� jzj�ryh�y��

jzj� � const�

The need for the correcting force term Fc � jzj�ryh�y� was �rst pointed out by Rubin � Ungar

����� It was shown in ���� that the corresponding CMD equations satisfy jz�t�j const� over anexponentially long time interval T � ec��� c � � some constant� if the energy function ���� is realanalytic�

Under the assumption that the fast degree of motion is strongly coupled to a heat bath withtemperature T � the correcting force term is determined by the relation jzj�h�y� � kBT and leads tothe Fixman potential Vc � kBT lnh�y� ���� ����

If a given matrix valued H�y� can be smoothly diagonalized� then we can still apply the �BornOppenheimer� approximation provided the eigenvalues of the matrix H�y� are all di�erent� Thiscase was �rst investigated by Takens in ����� For a recent discussion in terms of homogenizationsee ����� If eigenvalues cross� then the �BornOppenheimer� approximation breaks down as for theQCMD model of x�� See the numerical example below�

The correcting force term vanishes ifH�y� �H � const� This situation occurs if the constrainedHamiltonian ���� corresponds to a system of uncoupled rigid bodies� i�e� G�q�M��G�q�T � const��

��

and has been analysed by Benettin� Galgani � Giorgilli in ����

Example �� �cont�� In Fig� �� we present the eigenvalues of the Schr�odinger matrix H�y� and the occupation

numbers jzi�t�j�� i � �� � � � � �� corresponding to the wave vector z�t�� The force constant was set equal to

K � ���e���� Occupation numbers jzi�t�j� jump when the corresponding eigenvalues undergo or are close to a

� � � resonance �except at t � ������ It should be noted that higher�order resonances do not lead to transitions in the

occupation numbers� This is contrary to what can be expected from the results presented in ���� ���� �

20 21 22 23 24 25 26 27 28 29 300

0.5

1

1.5

2

eige

nval

ues

20 21 22 23 24 25 26 27 28 29 300

0.1

0.2

0.3

0.4

0.5

0.6

0.7

K=2.5e+04

occu

patio

n nu

mbe

rs

Figure �� Time evolution of the eigenvalues of H�y� and the corresponding �occupation numbers�jzi�t�j�� i � �� � � � � �� for K � ���e��� and over the time interval ���� ����

� Multiple Time Stepping for Classical MD

The analysis of x� indicates that in most cases the fast oscillations cannot be eliminated �or ignored� in long term MD simulations� In particular� nonadiabatic transitions and the breakdown ofthe �BornOppenheimer� approximation are unavoidable� The best way out might be an e�cientsimulation of the full system which takes into account the existing multiple time scales� Since thestandard MTS method ��� su�ers from resonance induced instabilities � �� we will discuss a variantof the molli�ed MTS methods� as suggested in ����� that is particularly suited for the CMD model�

��� Projected Multiple�Time�Stepping

Let us come back to highly oscillatory Hamiltonian systems of type

�q � M��p�

�p � �rqV�q��G�q�TKg�q� �

��

Step ��

�pn � pn ��t

�rqU��qn�

Step ��

Integrate the fastlocal system

d

dtq � M��p�

d

dtp � �rqU��q�

using Verlet with a stepsize �t � �tj� j � �� and initial conditions�qn� �pn�� Denote the result by �qn��� �pn����

Step ��

pn�� � �pn�� ��t

�rqU��qn���

Figure �� Standard MultipleTimeStepping

The Hamiltonian is

H�q�p� �pTM��p

�� V�q� � g�q�TKg�q�

and we split the potential energy V into a short range contribution V� and a long range contributionV�� The standard MTS method � � ��� ��� is now de�ned via the splitting

H � T �p� � U��q� � U��q�

with U� � V� and U� � V� � ��g�q�TKg�q�� This leads to the MTS algorithm ��� which is moreexplicitly written out in Fig� ��

This formulation su�ers from resonance induced instabilities � � �� which is caused by an unfortunate sampling of the highfrequency oscillations in q� � g�q�� In ����� Garc�ia�Archilla�Sanz�Serna � Skeel suggested to combine averaging with multipletimestepping� Here we useinformation about the analytical solution behavior of the fast system to obtain the averaged force�eld�

The motion in q� �� g�q� is highly oscillatory with a timeaverage close to zero� To eliminatethe e�ect of the highly oscillatory variable q� on the long range forces in ���� we replace the longrange force �eld F ��q� � �rqU��q� by

�F ��q� � �rqU����q��

which is the gradient of the modi�ed potential energy W �q� �� U����q���The function � is de�ned by the SHAKElike nonlinear system of equations

�q � ��q� � q �M��G�q�T � �

� � g��q�

in the variable � � Rm� Note that � projects the q� � g�q� solution component to zero� Toimplement our approach� we need the Jacobian �q� of �� This requires the computation of thesecond derivative �qqgi�q� of the functions gi� i � �� � � � �m� and the solution of a linear system ofequations� i�e��

d�q � dq �M��G�q�T d��M��mXi��

i �qqgi�q�dq�

� � G��q�d�q

��

with �q � ��q� and d�q � �q��q�dq� or� in other words�

�q��q� � I �M��G�q�T �G��q�

� I �M��

mXi��

i �qqgi�q�

with� �� �G��q�M��G�q�T ��� �

This leads us to the projected MTS scheme of Fig� � ���� ����

Step ��

�qn � ��qn� �

F n � ���q��qn��T r�qU���qn�

Step ��

�pn � pn ��t

�F n

Step ��

Integrate the fast system

d

dtq � M��p�

d

dtp � �rqU��q�

using Verlet with a stepsize �t � �tj� j � �� and initial conditions�qn� �pn�� Denote the result by �qn��� �pn����

Step ��

�qn�� � ��qn��� �

F n�� � ���q��qn����T r�qU���qn���

Step �

pn�� � �pn�� ��t

�F n��

Figure �� Projected MultipleTimeStepping Method

This symplectic scheme avoids the resonance problems typically encountered in the standardMTS method and is useful whenever the evaluation of rqU��q� �longrange forces� is much moreexpensive than the evaluation of rqU��q��

The modi�ed MTS method of Garc�ia�Archilla� Sanz�Serna � Skeel as well as our projected multipletimestepping method have been tested for a box of water� Both methods allow one toincrease the stepsize �t from ��fs to � fs ��fs � �����s� without any additional evaluation of thelongrange forces ����� In fact� the projection method turns out to be more robust than the methodsusing averaging ����� Note that the standard MTS method ��� becomes unstable at �t �fs� It canbe expected that improved projectedaveraged MTS methods will allow one to increase the macrostepsize up to �t ��fs �����

��� A Modi�ed Projection Step

The approximation g�q� � � in the de�nition of the map � might not be suitable for moderatevalues of the force constants and a better approximation to the averaged values of q� � g�q� should

��

be used� As pointed out in ���� ���� the variable q� oscillates about the minimum of the total energyin the direction of q�� This minimum is characterized� by the nonlinear equation

� � G��q�M��r�qU���q�

which replaces the constraint g�q� � �� Thus we introduce the modi�ed projection

�q �� ��q�

by means of

�q �� q �M��G�q�T � �

� � G��q�M��r�qU���q�� �� �

The resulting nonlinear system in the variable � � Rm can be solved by Newton�s method with thesimpli�ed �symmetric� Jacobian

J � �G�q�M��G�q�T �K �G�q�M��G�q�� �

As before� we introduce a modi�ed �averaged� long range potential energy function

W �q� �� U����q���

The evaluation of the gradient

rqW �q� � ��q��q��Tr�qU���q�

requires the computation of �q��q�� i�e�

d�q �� �q��q�dq �

� dq �M��G�q�T d� �M��mXi��

�i �qqgi�q�dq�

and d� is determined by the equation

� � ��q G��q�M��

r�q U���q��d�q�

In terms of the individual functions gi� this results in

� �n M��

r�q U���q��T��q�qgi��q� �Gi��q�M

����q�qU���q�od�q

which includes the computation of the Hessian of U��q�� Thus this approach should only be used ifU� is restricted to nearest neighborhood interactions such as the bond stretchingbending potentialsand the repulsive part of the LennardJones interactions�

The modi�ed projection can be built into the MTS scheme of Fig� � by replacing � by �� Welike to point out that this modi�ed force �eld requires additional force �eld evaluations� However�these additional force �eld evaluations are restricted to nearest neighborhood interactions�

� Multiple Time Stepping for Quantum�Classical MD

A natural extension ���� of the Verlet method to the QCMD equations of motion is given by

�n���� � exp��i

t��H�qn�

��n� ����

Leapfrog

�������������

qn���� � qn ��t

�M��pn�

pn�� � pn ��t h�n�����rqH�qn������n����i ��trqU�qn������

qn�� � qn���� ��t

�M��pn���

����

�n�� � exp��i

t��H�qn���

��n����� ����

Here we have neglected velocity dependent contributions and contributions from the long range potential energyU��q��

��

Even if the matrix exponentials in ���� and ���� are evaluated exactly� the scheme requires a verysmall stepsize� Otherwise the HellmannFeynman forces acting on the classical coordinates willbe wrongly approximated ���� ��� ��� and the behavior of the populations fj�i�t�j�g may not bereproduced correctly �see Example � below�� The same holds true for MTS variants of the abovemethod� as suggested in ���� ���� where the matrix exponential is replaced by an approximationusing j steps of a smaller stepsize �t � �tj�

Example �� We like to demonstrate a potentially dangerous implication of using a large time�step on the preservationof the populations fj�i�t�j�g in an adiabatic regime� Let us consider a simple two dimensional system

�� � � i

�H�t��� ����

� � C�� where the dependence of H on the classical coordinate q is replaced by a time dependence� In particular� wetake

H�t� �

�cos� t � sin� t �� cos t sin t�� cos t sin t sin� t� cos� t

and introduce a new vector � by

� � Q�t��

with

Q�t� �

�cos t � sin tsin t cos t

��

This transformation gives rise to the equation

�� � � i

�E� �A�

with

E ��

�� �� ��

and

A ��

�� ��� �

��

Provided � �� this system satis�es the quantum adiabatic theorem which implies that the populations fj�i�t�j�gare almost constant�

An exponential integrator for the system ���� is given by

�n�� � exp

�� i�t

��Hn��

�exp

�� i�t

��Hn

��n

� QTn�� exp

�� i�t

��E

�Qn��Q

Tn exp

�� i�t

��E

�Qn�n

or� in terms of �� as

�n�� � exp

�� i�t

��E

�Qn��Q

Tn exp

�� i�t

��E

��n� ����

Let us assume that the step�size �t is �accidentally� chosen such that

exp

�� i�t

��E

�� I�

then ���� simpli�es to

�n�� � Qn��QTn�n

which is a second order accurate discretization of the di�erential equation

�� � A� � �Q�t�Q�t�T ��

But this is wrong� The populations fj�i�t��jg are no longer conserved but undergo a systematic drift instead�

We like to point out that this e�ect is due to a unfortunate choice of the step�size �t and may not be

observed generally� Nevertheless� it raises concerns about using a large time�step when integrating a slowly varying

time�dependent Schr�odinger equation� �

Provided that we can neglect the problem mentioned in Example �� a larger macro stepsize �t maybe applied in ��������� if the BornOppenheimer approximation ���� to the HellmannFeynmanforce is used in ����� See ��� for details� However� the formula ���� requires the computation of thederivatives of the diagonalized quantum operatorE�q� which is computational expensive� in general�

��

This can be avoided if an explicit averaging along ��t� is applied to the HellmannFeynman forcein ����� See ���� for details�

Here we suggest a di�erent approach based on a splitting of the Hamiltonian ��� intoH � H��H�

in the following way �����

H� �pTM��p

�and H� � h��H�q��i� U�q� �

Let us write down the corresponding di�erential equations� First for H��

�� � � �

�q � M��p �

�p � � �

next for H��

�� � � i

�H�q�� � ����

�q � � � ����

�p � �h��rqH�q��i �rqU�q� � ����

The solution to H� is just a translation of classical particles with constant momentum p�The intriguing point about the second set of equations is that q is now kept constant� Thus the

vector � evolves according to a timeindependent Schr�odinger equation with constant HamiltonianoperatorH�q� and the update of the classical momentum p is obtained by integrating the HellmannFeynman forces ���� acting on the classical particles along the computed ��t� �plus a constant updatedue to the purely classical force �eld��

Upon computing the eigenvalues of the operator H�q�� the equations �������� can be solvedexactly� However� this is� in general� an expensive undertaking� Therefore we proceed with thefollowing multipletimestepping approach� The �rst step is to consider the identity

exp��tLH�� � exp��tL �H�

� � exp��tLU � � exp��tL �H�

�� �z �j times

� exp��tLU � �

where �t � �tj� j � �� and

H� � h��H�q��i �The second step is to use

exp��tLH� � exp��t

�LH�

� � exp��tL �H�

�� �z �j times

� exp��tLU � � exp��t�LH�

� �O��t�� � ����

The last step is to �nd a symplectic� second order approximation !�t to exp��tL �H�

�� In principle�we can use any symplectic integrator suitable for timedependent Schr�odinger equations with atimeindependent Hamilton operator H�q� �see� for example� ������

Provided that V �q� is diagonal� an e�cient method !�t is obtained by exploiting the naturalsplitting of the quantum operatorH�q� � T �V �q� as used in the symplectic PICKABACK scheme����� This yields two exactly solvable subsystems

H��� � h��T�i and H��� � h��V �q��i �The resulting integrator for QCMD� as presented in Fig� � and �rst suggested in ����� is of secondorder� explicit� symplectic� and conserves the norm of the wavefunction� For the implementation ofother choices for !�t� see �����

The MTS scheme of Fig� � may still require a relatively small macro stepsize �t to insure anaccurate computation of the populations fj�i�t�j�g� Thus it might be useful to consider the followingmodi�cation of the MTS scheme �����

exp��t

�LU � �

����exp��t� LH�

� � exp��tL �H�

� � exp��t�LH�

�� �z �j times

���� � exp��t� LU � � �� �

��

Step ��

qn���� � qn ��t

�M��pn �

pn � pn ��trqU�qn�����

Step ��

k � � � � � j

���������������

�n�k�j � exp

�� i

�t

�T

��n��k���j �

pn�k�j � pn��k���j � �t h �n�k�j �rqV �qn����� �n�k�ji �

�n�k�j � exp

�� i

�t

�T

�exp

�� i

��tV �qn�����

��n�k�j

Step ��

pn�� � pn�� �

qn�� � qn���� ��t

�M��pn�� �

Figure �� MultipleTimeStepping for QCMD

This MTS method resolves the quantum part of the QCMD equations of motion more accuratelythan ���� and is approximately as expensive as ���� if the evaluation of the operator V �q� and itsgradient rqV �q� is not too expensive compared to one integration step with !�t exp��tL �H�

��

Example �� �cont�� Here we numerically integrate the model system from Example � in x���� We use the

symplectic and unitary implicit midpoint rule ���� for the numerical approximation of exp��tL H�

� and implemented

it into the MTS method ����� The Plank constant � is set to � � ����� the macro step�size is �t � ���� and

�t � ���e���� As initial conditions� we take q � �� p � �� and � � ��� ��T � In Fig� �� we plot the occupation numbers

j�i�t�j�� i � �� � and the time evolution of the classical coordinate q�t�� It can be seens that the Born�Oppenheimer

approximation breaks down near q�t� � ���� Next we compare the exact solution obtained from the MTS method

���� with �t � ���� and �t � ���e��� to the approximation obtained using the Verlet�based scheme ��������� with

a step�size �t � ����� The results can be found in Fig� �� The di�erence in the trajectories is due to a �wrong�

pointwise evaluation of the Hellmann�Feynman forces at a macro time step �t in ����� �

In summary� one can say that the design of MTS schemes for the QCMD model will require furtherresearch on an �optimal� choice for the method ��t to approximate exp��tL �H�

�� the splitting ofthe Hamiltonian ���� and the ratio of the stepsizes �t and �t� Provided the eigendecomposition ofthe Hamilton operator H�q� is known� one could also directly integrate the transformed QCMDequations �������� This approach will be discussed in more detail in a forthcoming publication�

Acknowledgement� This work was started while the author was at the Konrad Zuse Center Berlin�

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mbe

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Figure �� Time evolution of the �occupation numbers� j�i�t�j�� i � �� �� and of the classical coordinate q�t� over a time interval ��� ��� using the symplectic MTS method �� ��

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���� S� Reich� Smoothed Dynamics of Highly Oscillatory Hamiltonian Systems� Physica D � � ������ �����

���� S� Reich� Torsion Dynamics of Molecular Systems� Phys� Rev� E ��� ���������� �����

���� S� Reich� Dynamical Systems� Numerical Integration� and Exponentially Small Estimates� Habilitationsschrift�Freie Universit�at Berlin� �����

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���� S� Reich� A modi�ed force �eld for constrained molecular dynamics� Numerical Algorithms� to appear� �����

���� H� Rubin and P� Ungar� Motion Under a Strong Constraining Force� Comm� Pure Appl� Math� �� ������ �����

���� J�P� Ryckaert� G� Ciccotti� H�J�C� Berendsen� Numerical Integration of the Cartesian Equations of Motion of aSystem with Constraints� Molecular Dynamics of n�Alkanes� J� Comput� Phys� ��� �������� �����

���� U� Schmitt and J� Brinkmann� Discrete time�reversible propagation scheme for mixed quantum classical dynamics�Chem� Phys�� ��� ������ �����

���� U� Schmitt� Gemischt klassisch�quantenmechanische Molekulardynamik im Liouville�Formalismus� Ph�D� thesis�in german�� Darmstadt� �����

���� J�M� Sanz�Serna and M�P� Calvo� Numerical Hamiltonian Systems� Chapman and Hall� London� �����

���� F� Takens� Motion Under the In uence of a Strong Constraining Force� in� Global Theory of Dynamical Systems�Lecture Notes Math� �� � �������� �����

���� M� Tuckerman� B�J� Berne� and G�J� Martyna� Reversible Multiple Time Scale Molecular Dynamics� J� Chem�

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���� L� Verlet� Computer Experiments on Classical Fluids� I� Thermodynamical Properties of Lennard�Jones Mo�lecules� Phys� Rev� �� � ���������� �����

���� J� Zhou� S� Reich� and B�R� Brooks� Elastic Molecular Dynamics with Flexible Constraints� submitted� �����

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