Molecular dynamics and monte carlo simulations

7/27/2019 Molecular dynamics and monte carlo simulations

1/10

MolecularDynamicsandMonteCarloSimulations

UrsulaRthlisberger

Lecture1

Adaptedfrom:AbInitioMolecularDynamics

BasicTheoryandAdvancedMethods

DominiqueMarxandJurgHutter

CambridgeUniversityPress(2009)

Chapter2(pp.1120)


2/10

2Getting started: unifying molecular dynamics andelectronic structure

2.1 Deriving classical molecular dynamicsThe starting point of all that follows is non-relativistic quantum mechanicsas formalized via the time-dependent Schrodinger equation

(2.1)in its position representation in conjunction with the standard Hamiltonian

n2 2 n2 2H =- L : - \JI - L : - \J .I 2MI i 2me 2

n2= - L 2M \?J +He({ri}, {RI})I I (2.2)

for the electronic {ri} and nuclear {RI} degrees of freedom. Thus, onlythe bare electron-electron, electron-nuclear, and nuclear-nuclear Coulombinteractions are taken into account. Here, MI and ZI are mass and atomicnumber of the Jth nucleus, the electron mass and charge are denoted byme and - e, and co is the vacuum permittivity. In order to keep the currentderivation as transparent as possible, the more convenient atomic units (a.u.)will be introduced only at a later stage.

The goal of this section is to derive molecular dynamics of classical pointparticles [25 , 468, 577, 1189], that is essentially classical mechanics, starting

11


3/10

12 Getting started: unifying MD and electronic structurefrom Schrodinger's quantum-mechanical wave equation Eq. (2.1) for bothelectrons and nuclei. As an intermediate step to molecular dynamics basedon force fields, two variants of ab initio molecular dynamics are derived inpassing. To achieve this, two complementary derivations will be presented,both of which are not considered to constitute rigorous derivations in thespirit of mathematical physics. In the first, more traditional route [355]the starting point is to consider the electronic part of the Hamiltonianfor fixed nuclei , i.e. the clamped-nuclei part He of the full Hamiltonian,Eq. (2.2). Next, it is supposed that the exact solution of the correspondingtime-independent (stationary) electronic Schrodinger equation,

(2.3)is known for clamped nuclei at positions {RI }. Here, the spectrum of He isassumed to be discrete and the eigenfunctions to be orthonormalizedJl'"k({ri} ;{RI} )'liz({ri}; {RI}) dr = 8kl (2.4)at all possible positions of the nuclei; J dr refers to integration over alli = 1, . . . variables r = {ri}. Knowing all these adiabatic eigenfunctions atall possible nuclear configurations, the total wave function in Eq. (2.1) canbe expanded

00

({ri},{RI};t) = I:wz({ri};{RI})xz({RI} ;t) (2.5)l= O

in terms of the complete set of eigenfunctions {'liz} of He where the nuclearwave functions {xz } can be viewed to be time-dependent expansion coefficients. This is an ansatz of the total wave function, introduced by Bornin 1951 [179 , 811] for the time-independent problem, in order to separatesystematically the light electrons from the heavy nuclei [180, 771, 811] byinvoking a hierarchical viewpoint.1

Insertion of this ansatz Eq. (2.5) into the time-dependent coupled Schrodinger equation Eq. (2.1) followed by multiplication from the left by1 "The terms of the molecular spectra comprise, as is known, contributions of varying orders of

magnitude; the largest contribut ion originates from t he electron movement around the nuclei ,there then follows a contribution stemming from the nuclear vibrations , and, ultimately, thecontribution arising from the nuclear rotation. The justificat ion of the ex istence of such ahierarchy emanates from the magnitude of the mass of the nuclei, compared to that of theelectrons." Translated by the a uthors from "Die Terme der Molekelspektren setzen sich bekan-ntlich aus Anteilen verschiedener Gri:iBenordnung zusammen ; der gri:iBte Beitrag riihrt von derElektronenbewegung urn die Kerne her, dann folgt ein Beitrag der Kernschwingungen, endlichdie von den Kernrotationen erzeugt en Anteile. Der Grund fiir die Mi:iglichkeit einer solchenOrdnung liegt offensichtlich in der GroBe der Masse der Kerne, verglichen mit der der Elektronen. " Cited from the Introduction of the seminal paper [180] by Born and Oppenheimer from1927.


4/10

2.1 Deriving classical molecular dynamics 13wk({ri} ; {RI}) and integration over all electronic coordinates r leads toa set of coupled differential equations

(2.6)where

c., = j Wk [- ~ 2: 1vi] w, dr+ ~ 2( {j wz - in\71] Wz dr} [-in\71] (2.7)

is the exact nonadiabatic coupling operator. The first term is a matrixelement of the kinetic energy operator of the nuclei, whereas the secondterm depends on their momenta.

The diagonal contribution Ckk depends only on a single adiabatic wavefunction W and as such represents a correction to the adiabatic eigenvalueEk of the electronic Schrodinger equation Eq. (2.3) in this kth state. Asa result, the "adiabatic approximation" to the fully nonadiabatic problemEq. (2.6) is obtained by considering only these diagonal terms ,

C """ n2 j * 2kk = - L......t 2M 1 W V'1 \[! k dr ,I (2.8)the second term of Eq. (2.7) being zero when the electronic wave functionis real, which leads to complete decoupling

(2.9)

of the fully coupled original set of differential equations Eq. (2.6). This,in turn, implies that the motion of the nuclei proceeds without changingthe quantum state , k , of the electronic subsystem during time evolution.Correspondingly, the coupled wave function in Eq. (2.1) can be decoupledsimply

(2.10)into a direct product of an electronic and a nuclear wave function. Notet hat this amounts to taking into account only a single term in the generalexpansion Eq. (2.5).


5/10

14 Getting started: unifying MD and electronic structureThe ultimate simplification consists in neglecting also the diagonal cou

pling terms(2.11)

which defines the famous "Born-Oppenheimer approximation". Thus, boththe adiabatic approximation and the Born- Oppenheimer approximation (introduced in Ref. [180] using a cumbersome perturbation expansion in powersof the mass ratio (m8 /MI) 114 , see also 14 and Appendix VII in Ref. [179])are readily derived as special cases based on the particular functional ansatzEq. (2.5) of the total wave function. In the above simplified presentationsubtleties due to Berry's geometric phase [1329] have been ignored, but theinterested reader is referred to excellent reviews [168, 986, 1642] that coverthis general phenomenon with a focus on molecular systems.

The next step in the derivation of molecular dynamics is the task of approximating the nuclei as classical point particles. How can this be achievedin the framework where a full quantum-mechanical wave equation, Xk, describes the motion of all nuclei in a selected electronic state \Ifk? In orderto proceed, it is first noted that for a great number of physical situationsthe Born- Oppenheimer approximation can safely be applied, but see Section 5.3 for a discussion of cases where this is not the case. Based on thisassumption, the following derivation will be built on Eq. (2.11) being theBorn- Oppenheimer approximation to the fully coupled solution, Eq. (2.6).Secondly, a well-known route to extract semiclassical mechanics from quantum mechanics in general starts with rewriting the corresponding wave function

(2.12)in terms of an amplitude factor Ak and a phase Sk which are both consideredto be real and Ak > 0 in this polar representation, see for instance Refs. [345,996 , 1268]. After transforming the nuclear wave function in Eq. (2.11) fora chosen electronic state k accordingly and after separating the real andimaginary parts , the equations for the nuclei

ask "_1_ (\7 8 )2 E = n 2 " _1_ 'VJAkat + 0 2M I k + k 0 2M AI I I I k (2.13)(2.14)


6/10

2.1 Deriving classical molecular dynamics 15using Rexk and Imxk. I t is noted in passing that this quantum fluid dynamic(or hydrodynamic, Bohmian) representation [169, 1636], Eqs. (2.13)-(2.14) ,can actually be used to solve the time-dependent Schrodinger equation [340,878].

The relation for the amplitude, Eq. (2.14), may be rewritten after multiplying by 2Ak from the left as a continuity equation [345, 996, 1268]

a A ~ " ' 1 ( 2 )~ + L_., M 'VI Ak V'ISk = 0ut I I (2.15)(2 .16)

with the help of the identification of the nuclear probability density Pk =1Xkl 2 = ~ , obtained directly from the definition Eq. (2.12) , and with theassociated current density defined as Jk,I = A ~ ( V ' I S k ) / M I . This continuityequation Eq. (2.16) is independent of nand ensures locally the conservationof the particle probability density 1Xkl 2 of the nuclei in the presence of aflux.

More important for the present purpose is a detailed discussion of there lation for the phase Sk , Eq. (2.13), of the nuclear wave function that isassociated with the kth electronic state. This equation contains one termthat depends explicitly on n, a contribution that vanishes

(2.17)

:: the classical limit is taken as n ---+ 0. Note that a systematic expansion inerms of nwould, instead, lead to a hierarchy of semiclassical methods [562 ,196]. The resulting equation Eq. (2.17) is now isomorphic to the equation of

~ ot ion in the Hamilton- Jacobi formulation [528, 1282] of classical mechanics(2.18)

- - = ~ h t he classical Hamilton function(2.19)

:- a !ri.Yen conserved energy dEkot / dt = 0 and henceaSk to t8 t =- (T + Ek) = - Ek = const. (2.20)


7/10

16 Getting started: unifying MD and electronic structuredefined in terms of (generalized) coordinates {RI} and t heir conjugate canonical momenta {PI}. With the help of the connecting t ransformation

[ Jk ]P I= VISk = MI p ~ (2 .21)the Newtonian equations of motion, PI = -V IVk( {RI} ), corresponding tothe Hamilton- Jacobi form Eq. (2.17) can be read off

dPId t = - \ l iE k or.. BOMIRI( t ) =- \ l Vk ({R I (t)}) (2 .22)

separately for each decoupled electronic state k. Thus, the nuclei move according to classical mechanics in an effective potential, VkBO, which is givenby the Born- Oppenheimer potential energy surface Ek obtained by solvingsimultaneously the time-independent electronic Schrodinger equation for thekth state , Eq. (2.3), at the given nuclear configuration {RI(t)}. In otherwords , this time-local many-body interaction potential due to the quantumelectrons is a function of the set of all cla ical nuclear positions at timet. Since the Born- Oppenheimer total energie in a specific adiabatic electronic state yield directly the forces used in thi Yariant of ab initio molecular dynamics, this particular approach is often called "Born- Oppenheimermolecular dynamics", to be discussed in more detail later in Section 2.3.

In order to present an alternative deriYation . which does maintain aquantum-mechanical time evolution of the electrons and thus does not invoke solving the time-independent electronic chrodinger equation Eq. (2.3)as before, t he elegant route taken in Ref . [1516. 15 17] is followed; see alsoRef. [943]. To this end, the nuclear and electronic contributions to the totalwave funct ion 1> ({r i}. {R I}: t) are eparated directly such that , ultimately,the classical limit can be impo ed for the nuclei only. The simplest possibleform is a product an atz

1>({ri} , {R I}:t) ~ w({ r i} : t) \ ({R I}:t) exp ~ 1: Ee(t') dt'] ' (2.23)where the nuclear and electronic waYe functioto unity at every instant of time. i.e. (\ : t l \ : trespectively. In addition. a pha e factor

are separately normalized= 1 and (w; t lw; t) = 1,

Ee = J *({ri}; t) x*({R I}: t ) He W({r ,}: t \ ({RI} ; t ) drdR (2.24)was introduced at this stage for conYenience -uch t hat the final equations


8/10

20 Getting started: unifying MD and electronic structureelectronic states W

0 0

w({ri},{RI};t) = I:>z(t)Wz({ri};{RI}) (2.33)l= O

with complex time-dependent coefficients {cz ( )}. In this case, the coefficients lcz(tW satisfying :Z:::: 1 Icz(tW = 1 describe explicitly the time evolutionof the populations (occupations) of the different states l whereas the necessary interferences between any two such states are included via the offdiagonal term, ck_cl=# One possible choice for the basis functions {wk} is theinstantaneous adiabatic basis obtained from solving the time-independentelectronic Schrodinger equation

(2.34)where {RI} are the instantaneous nuclear positions at time t that are determined acc6rding to Eq. (2.30). The actual equations of motion in termsof expansion coefficients {ck}, adiabatic energies {Ek}, and nonadiabaticcouplings are presented in Section 2.2.

Here, instead, a further simplification is invoked in order to reduce Ehrenfest molecular dynamics to Born- Oppenheimer molecular dynamics. Toachieve this, the electronic wave function W is restricted to be the groundstate adiabatic wave function Wo of He at each instant of time according toEq. (2.34), which implies lco(tW = 1 and thus a single term in the expansionEq. (2.33). This should be a good approximation if the energy differencebetween Wo and the first excited state W1 is large everywhere compared tothe thermal energy kBT , roughly speaking. In this limit the nuclei moveaccording to Eq. (2.30)

~ E = J 07-ieWo dr =: Eo({RI}) (2.35)on a single adiabatic potential energy surface. This is nothing else than theground state Born- Oppenheimer potential energy surface that is obtainedby solving the time-independent electronic Schrodinger equation Eq. (2.34)for k = 0 at each nuclear configuration {RI} generated during moleculardynamics. This leads to the identification VeE =: Eo = V080 and thus toEq. (2.22) , i.e. in this limit the Ehrenfest potential is identical to the groundstate Born- Oppenheimer (or "clamped nuclei") potential.

As a consequence of this observation, it is conceivable to fully decouple thetask of generating the classical nuclear dynamics from the task of computingthe quantum potential energy surface. In a first step, the global potentialenergy surface Eo, which depends on all nuclear degrees of freedom {RI },


9/10

2.1 Deriving classical molecular dynamics 21is computed for many different nuclear configurations by solving the stat ionary Schrodinger equation separately for all these situations. In a secondstep, these data points are fitted to an analytical functional form to yielda global potential energy surface [1280], from which the gradients can beobtained analytically. In a third step, the Newtonian equations of motionare solved on this surface for many different initial conditions, producing a"swarm" of classical trajectories {R I ( )}. This is , in a nutshell, the basis ofcla ssical trajec tory calculations on global potential energy surfaces as usedve ry successfully to underst and scattering and chemical reaction dynamicsof small systems in vacuum [1284, 1514] .

As already explained in the general introduction , Chapter 1, such approaches suffer severely from the "dimensionality b o t t l ~ n e c k " as the numberof active nuclear degrees of freedom increases. One traditional way out ofthis dilemma, making possible calculations of large systems, is to approxi-mate the global potential energy surface

N NV e E ~ VeFF({RI} ) = :L vl (RI) + L V2(RI ,RJ )

l = l l< JN+ L v3(RI ,RJ ,Rx) + (2.36)

l


10/10

22 Getting started: unifying MD and electronic structureinteractions. This amounts to a dramatic simplification and removes, inparticular, the dimensionality bottleneck since the global potential surfaceis reconstructed from a manageable sum of additive few-body contributions.The flipside of the medal is the introduction of the drastic approximationembodied in Eq. (2.36) , which basically excludes the study of chemical reactions from the realm of computer simulation.

As a result of the derivation presented above, the essential assumptionsunderlying standard force field-based molecular dynamics become very transparent. The electrons follow adiabatically the classical nuclear motion andcan be integrated out so that the nuclei evolve on a single Born- Oppenheimerpotential energy surface (typically, but not necessarily, given by the electronic ground state), which is generally approximated in terms of few-bodyinteractions.

Actually, force field-based molecular dynamics for many-body systems isonly made possible by somehow decomposing the global potential energy.In order to illustrate this point, consider the simulation of N = 500 argonatoms in the liquid phase [395] where the interactions can be described faithfully by additive two-body terms , i.e. ~ F F ( { R I } ) ; : : : : : ; L:::f

Documents

Molecular dynamics and monte carlo simulations