Liouville–von Neumann molecular dynamics

Liouville–von Neumann molecular dynamicsJacek Jakowski1,a� and Keiji Morokuma1,2,b�

1Department of Chemistry and Cherry L. Emerson Center for Scientific Computation, Emory University,1515 Dickey Dr., Atlanta, Georgia 30322, USA2Fukui Institute for Fundamental Chemistry, Kyoto University, Sakyo, Kyoto 6006-8103, Japan

�Received 14 January 2009; accepted 19 May 2009; published online 12 June 2009�

We present a novel first principles molecular dynamics scheme, called Liouville–von Neumannmolecular dynamics, based on Liouville–von Neumann equation for density matrices propagationand Magnus expansion of the time-evolution operator. The scheme combines formally accuratequantum propagation of electrons represented via density matrices and a classical propagation ofnuclei. The method requires a few iterations per each time step where the Fock operator is formedand von Neumann equation is integrated. The algorithm �a� is free of constraint and fictitiousparameters, �b� avoids diagonalization of the Fock operator, and �c� can be used in the case offractional occupation as in metallic systems. The algorithm is very stable, and has a very goodconservation of energy even in cases when a good quality conventional Born–Oppenheimermolecular dynamics trajectories is difficult to obtain. Test simulations include initial phase offullerene formation from gaseous C2 and retinal system. © 2009 American Institute of Physics.�DOI: 10.1063/1.3152120�

I. INTRODUCTION

First principles molecular dynamics �MD� methods1–4

are a primary tool for theoretical studies of large systems. Acomplete description of molecular systems requires solvingthe full time-dependent Schrödinger equation for the con-stituent electrons and nuclei. This task is currently unfeasiblefor systems consisting of more than a few atoms and morethan one electronic state, and requires making some approxi-mations on electronic or nuclear equations of motion. It isusually assumed that the motion of electrons �fast� and nuclei�slow� is separable and Born–Oppenheimer �BO� approxima-tion is valid. As a consequence the time-dependentSchrödinger equations are replaced by �a� classical Newton’sequations for nuclei and �b� time-independent Schrödingerequation for electrons. The resulting method is Born–Oppenheimer molecular dynamics �BOMD�,2–5 in which nu-clei are moving classically on the ground state potential en-ergy surface obtained on-the-fly self-consistently fromstationary electronic structure.

In extended Lagrangian methods nuclei are moving clas-sically on the approximation to the BO solution. The bestknown method of this type is Car–Parrinello �CP� MD.1 Astationary approximation to ground �or excited� state BO so-lution is propagated classically with fictitious mass param-eter corresponding to electronic degrees of freedom. For thatpurpose a properties of BO solution such as orthogonality ofwave function, N-representability and idempotence of den-sity matrix or vanishing commutator of Fock and densitymatrix are exploited in extended Lagrangian approaches.1,6,7

It should be noted, that in principle both BOMD and CPmethods can be described by various extended Lagrangians,

but their dynamics are different. Recently, combinations ofboth BOMD and extended Lagrangian approaches are re-ported, in which extended Lagrangian techniques are used toachieve self-consistent field convergence of the electronicstructure in BOMD faster and more accurately.8–10

As noted by Godecker,11 the reason why traditional elec-tronic structure algorithms calculate eigenstates associatedwith discrete levels is probably historical and is related to thefact that the prediction of experimentally observed levelswas the first big success of quantum mechanics. We proposea different paradigm for first principles MD which is basedon direct integration of the time-dependent quantum-mechanical equations of motion for electrons �subject to ini-tial wave function� rather than on solving the stationary BOproblem at every time step.12 For that purpose we employeethe von Neumann equation to describe the evolution of elec-tronic densities.

Introducing time-dependent Hamiltonians enables de-scription of �a� externally driven systems and molecules thatare subject to strong laser field for which linear responseapproximation is not valid,13–19 �b� processes with couplingand energy transfer between nuclear and electronic degreesfreedom,18,20–25 and �c� decoherence of quantum system frompure state to mixed state.26–28 Several methods in which elec-tronic structure is treated explicitly time-dependent quantummechanically exist. The two most widely used classes of firstprinciples MD based on time-dependent Schrödinger equa-tions are Ehrenfest dynamics and surface hoppingdynamics.20,21,29–34 Various implementations of Ehrenfest dy-namics exist, in which electronic structure is propagatedincoherently20,29,32 or coherently.21,35 In the former case elec-tronic structure evolves irreversibly from a pure to a mixedstate. In the latter case propagation of electronic structure istime-reversal due to a unitary propagation of electronic struc-

a�Electronic mail: [email protected]�Electronic mail: [email protected].

THE JOURNAL OF CHEMICAL PHYSICS 130, 224106 �2009�

0021-9606/2009/130�22�/224106/12/$25.00 © 2009 American Institute of Physics130, 224106-1

ture. Our Liouville–von Neumann molecular dynamics�LvNMD� is based on Magnus expansion of time-evolutionpropagator and is similar to time-reversal, coherent ab initioEhrenfest dynamics of Li and co-workers.21,35

It is recognized that a realistic description of dynamics atmetal surfaces must include the effect of nonadiabatic �non-BO� transitions.21,22,25,36 Such effects are often approximatedvia time-independent Mermin electronic energyfunctional.37–39 In this method electronic structure do notevolve adiabatically but isothermally with electrons fraction-ally occupying electronic levels according to Fermi–Diracdistribution. Nuclei are moving in the mean field of freeenergy �electronic energy+electronic entropy� and the sta-tionary time-independent electronic structure is instanta-neously equilibrating at a given electronic temperature.38,40

This method is very different from Ehrenfest and surfacehopping dynamics. In the limit of zero electronic temperatureit becomes a conventional BOMD.

In this work we demonstrate how to perform a time-reversible coherent dynamics of a system in which electronicstructure was initially thermally equilibrated. On the algo-rithmic side, LvNMD method is similar to ab initio Ehren-fest dynamics of Li and co-workers.21,35 However, the Ehren-fest dynamics of Li uses a modified exponential midpointpropagator and diagonalization, and employs a multi-time-step approach. Our LvNMD method avoids diagonalizationand is based on Magnus expansion of time-evolutionoperator.12 This gives a strong physical footing to LvNMDand sets up a base for further improvements. The proposedLvNMD method �a� is general and can be used with anytheories based on an independent-electron model such asHartree–Fock, density functional or tight-binding methods,�b� it does not impose any constraints on electronic degreesof freedom, �c� it avoids diagonalization of Fock operator,and �d� it can describe fractionally and thermally occupiedelectronic states and is therefore suitable for metallic andelectronically hot systems.

In Sec. II we present an outline of theory for ourLvNMD method. For the sake of brevity the presented isspin-restricted version of the method. The generalization to aspin-unrestricted version is straightforward. Additional dis-cussion of theory is presented in Appendixes A and B. In SecIII we present some illustrative examples of applicationwhich include also spin-unrestricted version of our method.Sec IV discusses some practical usage issues. In Sec. V wepresent concluding remarks.

II. THEORY

A. General considerations

The proposed first principles MD method, which we callLvNMD, is based on classical Liouville propagation ofnuclei41 and on von Neumann equation of motion �some-times called quantum-Liouville equation� for P�t� electronicdensities,

ı�dP�t�

dt= �H�t�,P�t�� , �1�

where H�t� is the Hamiltonian operator at time t. Equation�1� is valid regardless of whether density matrix P representsa pure or mixed state with fractional occupancies.

The formal solution to Eq. �1� can be obtained via aunitary time-evolution operator

U�t,t0� = exp�− �ı/���t0

t

H�t��dt�� . �2�

In principle, however, such integration cannot be done di-rectly unless the corresponding Hamiltonians H�t� commuteamong themselves for all values of time.42 The general solu-tion to this problem was given by Magnus and is consideredas a kernel in understanding the problem.43,44 When theMagnus theory is applied to propagate electronic densities,an additional difficulty arises from the fact that Hamiltonianin Eq. �1� is not only time dependent, but it actually dependson the density matrix itself.

In independent-particle Hamiltonian models �tight-biding, Hartree–Fock, density functional theories� the elec-tronic energy can be expressed through electronic densitymatrices as

E = Tr�hP� + 12Tr�G�P�P� + VNN, �3�

where VNN is the nuclear repulsion while h and G�P� are,respectively, one-electron and two-electron Hamiltonians. Inprinciple, density matrix P�t� can be written as

P�t� = CfC† = �i=1

M

fiPi�t� , �4�

where C is M �M matrix of molecular orbital matrix and fthe is diagonal matrix of orbital occupancies f i. DensitiesPi�t� are one-electron densities such as Tr�Pi�t��=1 associ-ated with Ci molecular orbital. Here the one-electron basisfunctions are assumed to be orthonormal. Density matrix Prepresents a pure state when occupancies are equal to 0 or 1,or a mixed state with fractional occupancies. For thermallyexcited electrons45 at electronic temperature Tel occupanciesare related to the Fermi–Dirac distribution and Fermi level�or chemical potential� � as follows:

f��� = �1 + exp��� − ��/�kBTel���−1 �5�

and the total number electrons N is �i f i=N. Equations �1�and �3� are a basis for LvNMD.

It should be stressed that formally the solution to Eq. �1�retains its initial occupancies f i unchanged. It is a conse-quence of unitarity of time-evolution operator U�t� and for-mally guarantees time reversibility of density matrix propa-gation. This fact is well recognized in studies concerning onthe theory of quantum information and decoherencephenomena.28,46,47 As a consequence of independence of f i

on time, the electronic entropy Se associated with densityP�t�,

224106-2 J. Jakowski and K. Morokuma J. Chem. Phys. 130, 224106 �2009�

Se = − kB�i=1

M

�f i log�f i� + �1 − f i�log�1 − f i�� , �6�

is a constant of the motion. This result is very different froma free energy BOMD in which incoherent changes of f i oc-cupancies require explicit treatment of entropy.37,38,48

B. Propagation of density matrix

To propagate the density matrix we use the Fock opera-tor,

F = h + G�P� , �7�

as an effective Hamiltonian H�t� in Eq. �1�. For the densitymatrix and Fock operator we use a real Gaussian atomicorbital �AO� basis set, which has the advantage of beinglocalized and leads to linear scaling algorithms for nonme-tallic systems, but is not orthogonal. We orthogonalize theAO basis via Cholesky decomposition and the density matrixpropagation is performed in the orthogonal basis set.

It should be noted that the density matrix P�t� propa-gated via Eq. �1� is complex. Also one can expect propagatedFock matrices to be complex since a complex P is used toform a two-electron G�P� part of the Fock operator. How-ever, simple derivations show that imaginary part of densitymatrix, I�P�t��, enters into Fock operator and energy expres-sion only through Hartree–Fock exchange operator. In tight-binding or density functional theories that do not have exacttreatment of exchange operator the Fock operator and elec-tronic energy do not depend on imaginary part of densitymatrix. �See Appendix A for discussion of relations betweendensity matrix and Fock operator.� Current implementationof LvNMD, suitable for tight-binding and pure densityfunctional theories, avoids the complex part of the Fockmatrix, energy form and derivatives. It is also worthy to notethat von Neumann equation can be used to describe electrontransport properties with application to molecularelectronics.49

Following Magnus theory, solution to Eq. �1� is giventhrough the exponential time-evolution operator

P�t� = U�t�P�0�U†�t� = exp−ı

�FP�0�exp+

ı

�F , �8�

where F is Magnus integral of Fock operator over time

F�ı��

= �t0

t

dt�F�t��

ı�+

1

2�

t0

t

dt��t0

t�dt � �F�t��

ı�,F�t��

ı�� + ¯

= F�1� + F�2� + ¯ , �9�

and “¯” represents higher order Fock commutator integrals.Symbols F�k� represent terms in Magnus expansion propor-tional to time integral of �k−1� order commutator. Regard-less of the truncation order in the Magnus expansion Eq. �9�,the resulting Magnus operator F is always Hermitian andtime-evolution operator U�t� is always a unitary. See, forexample, Refs. 14, 44, and 50 for the discussion on the con-vergence of Magnus expansion. Comparing Eqs. �4� and �8�,it becomes obvious that scalars f i are constant of motionwhen U�t� is a unitary operator.

Here we make two assumptions about the Magnus ex-pansion. �a� Since motion of nuclei is slow compared to the�t propagation time step, the time dependence of the Fockoperator can be treated as linear over �t. �b� The Magnusexpansion can be truncated to first order. Working expressionfor matrix elements of the averaged Fock operator from firstorder Magnus expansion is

F�,��1� �t1,t0� = �

t0

t1

F�,��t�dt = �F�,��t0� + F�,��t1���t

2, �10�

where �t= t1− t0. Higher order terms in Magnus expansionF�k� are proportional to increasing powers of time step,namely, to ��t�k. Thus, the case of fast moving nuclei can beaddressed by including higher order Magnus terms or bydecreasing the time step �t.

Now, to obtain the propagated density matrix P�t� fromP�0� with the use of F as prescribed in Eq. �8�, one can �a�diagonalize F to obtain the exponential time-evolution op-erator, or �b� apply the Baker–Campbell–Hausdorff �BCH�expansion and thus avoid diagonalization. Here we use thelatter approach. The final expression for the P�t� is

P�t� = P − ıt�F,P� −t2

2!�F,�F,P��

+ ıt3

3!�F,�F,�F,P��� + ¯ , �11�

where P stands for P�0� and F for F�1��t ,0� in Eq. �10�.As it was mentioned, the main difficulty arises from the

fact that the Fock operator depends not only on the positionof nuclei, but its matrix elements directly depend on elec-tronic densities. We solve this issue by improving the Fockoperator iteratively in a Fock refinement step. First the den-sity matrix is propagated using approximate F�t� operator,and then a new improved F�t� is formed which is subse-quently used to perform the density matrix propagation. TheFock refinement is stopped when the final Fock describesadequately the electron-electron repulsion for the electronicstate described with the final P after density propagation. Inpractical calculations we require that error measured as thelargest change in Fock matrix elements for subsequent Fockrefinement is smaller than some preset threshold. Typicallyfor LvN we set the threshold value 10−6 for Fock refinementand 10−8 for the truncation of BCH.

C. Propagation of nuclei

Propagation of nuclei is performed using the velocity-Verlet algorithm.41 To evaluate the forces we use the formu-lation of Schlegel for the gradient of electronic energy.6,7,21

As usually, the gradient is evaluated in AO basis set. Theenergy gradient obtained by differentiating Eq. �3� is equal tothe sum of the nuclear, Hellman–Feynmann �derivative ofAO integrals� and Pulay terms �derivative of overlap�,

224106-3 Liouville–von Neumann molecular dynamics J. Chem. Phys. 130, 224106 �2009�

�E

�R=

�VNN

�R+ Tr� �h�

�R+

1

2

�G�

�RP��

− 2 Tr�F�dU

dRU−1P�� , �12�

where h�, G�, F�, and P� are expressed in AO basis set�notice that h, G, F, and P are in orthogonalized basis set�.Matrices U and U−1 are Cholesky transformation matricesbetween AO and orthogonal basis set �UTU=S�. The matrix��dU /dR�U−1� in the Pulay term �last term in Eq. �12�� rep-resents contribution from the Cholesky orthogonalization

�dU

dRU−1�

�,�=� U−T dS

dRU−1

�,�, � � �

1

2U−T dS

dRU−1

�,�, � = �

0, � � � .� �13�

It should be noted that the standard expression for Pulayterm, Tr�W�dS /dR��, where W is the energy weighted den-sity matrix W= PFP, is not valid for unconverged self-consistent field �SCF� densities case due to nonvanishingcommutator �F , P��0. Moreover, it is not valid for frac-tional occupancy densities �even for converged SCF case,when �F , P�=0� which is related to the fact that P is notidempotent.40,51,52 It is known �see Refs. 6 and 21� that Eqs.�12� and �13� can be used to obtain Pulay forces in the caseof unconverged densities �when �F , P��0�. However, it isnot obvious whether these expressions are also valid for frac-tional occupancies density matrix. In the Appendix B weshow that this expression is also valid for fractionally occu-pied, mixed state densities.

D. Algorithm summary

�1� Initialization: For initial geometry at t=0, setup initialvelocities and do SCF calculation to obtain potentialenergy, nuclear forces, density P�0� and Fock F�0� ma-trices.

�2� Propagate nuclei with velocity-Verlet scheme with timestep �t.

�3� For the new nuclear coordinates transform density to anew AO basis set and form a new Fock operator.

�4� Calculate first order Magnus average operator:F�1�= 1

2 �F�0�+F��t���t.�5� Perform LvN propagation of Magnus average operator:

F�1�=LvN�P�0� ,F�1��.�6� Check the error of F��t� and do Fock refinement if

needed:

• if error is larger than threshold then go to �3�;

• else calculate electronic energy and nuclear forces usingcurrent density and go to �2�.

III. APPLICATIONS

Test simulations were performed with density functionalbased tight-binding method with self-consistent charges�SCC-DFTB�.53–55 In the present work we focus on SCC-DFTB model because �a� this is a preferable method instudying growth of carbon nanotubes and other very largesystems, �b� it treats only valence electrons and neglects coreelectrons, and �c� the working equation for electronic struc-ture is given by Eq. �3�. On the practical side SCC-DFTB isan approximate density functional theory with the use of avalence minimal basis set and all parameters derived fromB3LYP and Perdew-Burke-Ernzerhof �PBE� functionals.

Here we present results for systems �a� benzene, �b� non-reactive scattering of hydrogen atom on a water molecule, �c�

0

2e-05

4e-05

6e-05

8e-05

0.0001

0 200 400 600 800 1000

LvNBOADMP

-0.001

-0.0008

-0.0006

-0.0004

-0.0002

0

0.0002

0.0004

0 500 1000 1500 2000

LvNBOADMP

benzene H2O+H

-0.0001

-8e-05

-6e-05

-4e-05

-2e-05

0

2e-05

0 200 400 600 800 1000

LvNBOADMP

-0.005

-0.004

-0.003

-0.002

-0.001

0

0.001

0 2500 5000 7500 10000

LvNBOADMP

Sc+@C60 30(C2)

FIG. 1. �Color� Conservation of totalenergy as a function of time step num-ber for ADMP, BOMD, and LvNMDwith SCC-DFTB theory. Time step is0.1 fs, Tel=0 K, energy in Eh. Fockconvergence threshold 10−6 was usedfor all LvNMD simulations. Energythreshold for BO is 10−10 for benzeneand water systems, 10−7 for fullerenecomplex and 10−6 for gaseous C2. Seealso Table I for details.

224106-4 J. Jakowski and K. Morokuma J. Chem. Phys. 130, 224106 �2009�

endohedral complex of transition metal and fullerene, �d�highly reactive gas of C2 molecule in the initial phase offullerene growth, and �e� retinaldehyde. The first two sys-tems represent small closed-shell �a� and open-shell �b� testmodels. The last three are much bigger model systems. Sys-tem �d� is very exothermic with large number of bond break-ing and formation occurring. Initial positions of the C2 mol-ecules in this system were randomly generated inside a boxof size 15�15�15 Å. Figure 1 shows conservation of totalenergy for LvNMD in comparison with BOMD and atom-centered density matrix �ADMP� methods. ADMP belongs tothe extended Lagrangian approaches to MD using localizedGaussian basis function and propagating the density matrixwith idempotency constraint on density matrix.6 The bestknown method of this type is CP MD,1 in which the Kohn–Sham molecular orbitals are chosen as the dynamical vari-ables to represent the electronic degrees of freedom in thesystem. The advantages of ADMP comparing with CPmethod1 is that all electrons can be treated explicitly andpseudopotentials are not required.

Figures 1 and 2 show results of LvNMD simulationsperformed at Tel=0 K. All simulations were performed mi-crocanonically with time step 0.1 fs and using SCC-DFTBtheory.53,54,56,57 The initial density matrix for LvNMD wasprepared as corresponding to a pure ground electronic state.

For the open-shell system �b� we used spin-unrestricted elec-tronic structure version of the methods. For ADMP simula-tions, the scalar fictitious mass algorithm was used with fic-titious mass parameter equal to 0.1 amu. In all ADMP casesthe idempotency of density matrix was preserved very welland the density matrix is purified with McWeeny method forevery time step. BOMD simulation for systems �a�–�c� wereperformed with direct inversion of iterative subspace �DIIS�algorithm58,59 and the initial guess wave function taken fromthe final result of the previous time step. In the case of BOdynamics for �d� system, we could not reach SCF conver-gence with the DIIS scheme and instead the quadraticallyconvergent SCF method60 was used.

Figure 3 shows results of LvNMD and BOMD simula-tions performed for metal-fullerene complex �system �c�� atfinite electronic temperature Tel=10 000 K. Similarly Fig. 4shows results of LvNMD and BOMD simulations for retinalat Tel=10 000 K. Conservation of energy in LvNMD case isexcellent. For BOMD we used very tight convergencethreshold 10−10. For BOMD case, we occasionally encoun-tered SCF convergence difficulties. It appears as a small stepin total energy conservations �see, for example, Fig. 4�a��.This, however, does not affect overall BOMD results and, insuch cases, energy and forces can be considered as corre-sponding to values converged with lower SCF threshold. We

-12.57

-12.56

-12.55

-12.54

-12.53

-12.52

0 200 400 600 800 1000

LvNBOADMP

-103.5

-103.49

-103.48

-103.47

-103.46

0 200 400 600 800 1000

LvNBOADMP

FIG. 2. �Color� Potential energy in Eh

as a function of time step number inADMP, BO and LvNMD for C6H6

�left� and Sc+@C60 �right�.

-1e-05

0

1e-05

2e-05

3e-05

4e-05

0 200 400 600 800 1000

LvNBO

0

0.2

0.4

0.6

0.8

1

-0.8-0.6-0.4-0.2 0 0.2 0.4 0.6

(A) (B)

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0 200 400 600 800 1000

-S*Te

Escf(BO)

Epot(LvN)Epot(BO)

(C) (D)

FIG. 3. �Color� LvNMD simulationsof Sc+@C60 at electronic temperatureTel=10 000 K. Time step is 0.1 fs.Fock convergence threshold is 10−6.Panel left �a� shows conservation oftotal energy in Eh as a function of timestep number. Right panel �b� showsinitial molecular orbital occupanciesas a function of orbital energy. Ticsshow current occupancies. Solid lineshows corresponding Fermi–Dirac dis-tribution at Tel=10 000 K. Panel �c�shows comparison of potential energychanges for propagation of nuclei inLvN dynamics and BOMD. ForBOMD case potential energy �Epot� isa sum of SCF energy �ESCF� and elec-tronic entropy �−S�Te�.

224106-5 Liouville–von Neumann molecular dynamics J. Chem. Phys. 130, 224106 �2009�

did not observe difficulties with LvNMD. Panel �c� on Figs.3 and 4 show changes in potential energy for nuclei forLvNMD and BOMD simulations. In BOMD case nuclei aremoving in mean field of free energy �electronic energy+electronic entropy�. In LvNMD case electronic structure ispropagated coherently and the resulting electronic entropy isconstant of the motion �see Sec. II�. For comparison pur-poses changes in BOMD potential energy components,namely, electronic entropy and SCF energy are also shown.Clearly, incoherent changes in Fermi–Dirac weighted occu-pations for BOMD simulations are accounted for in coherentpropagation of electronic densities in LvNMD.

For cases studied in this paper LvNMD method followsBOMD trajectories �coordinates and potential energy� very

closely �see Figs. 2, 3�c�, and 4�c��. The energy oscillationfor LvN dynamics with a 10−6 Fock threshold are similar inmagnitude to BOMD results obtained with a 10−10 energythreshold �see Fig. 1 and Table I�. However, we do not ob-serve energy drift for LvN dynamics. We also tested the sta-bility of LvNMD method for other values of the Fock thresh-old for benzene and fullerene complex dynamics. For athreshold up to 10−4 we did not observe energy drift inLvNMD simulations �see Figs. 5�a�–5�d� for details�. Forthresholds equal to 10−6, 10−5, and 10−4 we got consistentlystable �no energy drift� trajectories over 1000 time steps withconstant energy oscillation of magnitudes 2�10−5, 6�10−5,and 4�10−4 for benzene and 4�10−6, 2�10−5, and 2�10−4 for fullerene complex. Overall, lowering the order of

-2e-05

0

2e-05

4e-05

6e-05

8e-05

0.0001

0 200 400 600 800 1000

LvNBO

0

0.2

0.4

0.6

0.8

1

-0.8-0.6-0.4-0.2 0 0.2 0.4 0.6

(A) (B)

-0.02

0

0.02

0.04

0.06

0 200 400 600 800 1000

-S*Te

Escf(BO)

Epot(BO)Epot(LvN)

(C) (D)

FIG. 4. �Color� Same as Fig. 3 but forretinal system.

TABLE I. Comparison of total energy fluctuation error for BOMD, ADMP and LvN simulations at Tel=0 Kwith 0.1 fs time step for �A� benzene, �B� H2O+H, and �C� complex Sc+@C60. Initial kinetic energy was set to50 mEh in all cases. Max and rms are total energy errors in �Eh calculated with reference to initial energy over1000 steps. Rows 3 and 4 show, respectively, the negative exponent of the LvN convergence threshold andaveraged number of Fock refinement or SCF cycles. BCH iter represents, respectively, average number of termneeded to achieve convergence in BCH expansion. BCH max shows the order of commutator term in BCHexpansion for which the largest correction to matrix element �BCH corr� was observed during the simulation.Last row shows idempotency error after the dynamics in �10−10 calculated as a sum over matrix elements �k,l��P2�k,l− Pk,l�2 /N, where N denotes number of basis functions.

System

BOMD LvN

�A� �B� �C� �A� �B� �C�

Max 17.2 90.0 12.6 11.7 79.8 6.5RMS 10.4 43.4 7.2 23.2 27.7 2.8Threshold 10 10 7 6 6 6No. of cycles 9 8 22 8 9 24Energy drift No Yes Yes No No NoBCH iter. ¯ ¯ ¯ 26.6 25.4 26BCH max ¯ ¯ ¯ 7 8 6BCH corr. ¯ ¯ ¯ 1.31 5.72 0.37Idempotency ¯ ¯ ¯ 16 8 4

224106-6 J. Jakowski and K. Morokuma J. Chem. Phys. 130, 224106 �2009�

the Fock convergence threshold by one order of magnitudecuts the computational cost in half that is the number of Fockrefinement iterations �this is the most expensive part of themethod� decreases twice. See Figs. 5�e�–5�h� for details.

Finally, to compare the behavior of LvNMD and BOMDwith electronic temperature at the dissociation limit we per-formed LvNMD and BOMD simulations for water moleculein which one of the OH bonds was forced to dissociate. Tosee the effect of Fermi–Dirac smearing the electronic tem-perature was exaggerated and set to Tel=30 000 K. Simula-tions were performed with spin-restricted method which isnot suitable for description of dissociation case. The result-ing Mulliken populations calculated at SCF-DFTB level fordifferent stages of OH dissociation are presented in Table II.In both BOMD and LvNMD cases water dissociates to neu-tral H and OH systems with one electron localized on disso-ciating hydrogen �atom 3 in Table II�. However, BOMDleads to unphysical fluctuation of electronic charges in theregion of intermediate OH bond length �maximum at around

3 Å in Table II� whereas LvNMD populations behave verysmoothly. The unphysical maximum of Mulliken populationof BOMD observed at 3 Å is due to the narrowing of highestoccupied molecular orbital-lowest unoccupied molecular or-bital gap and corresponding changes in Fermi–Dirac popula-tions.

Results presented here are only a fraction of all testedsystems and of initial conditions we have investigated. Theseinclude propagation of mixed electronic states with initialdensity matrices prepared at finite and zero electronic tem-peratures. All results obtained are consistent with those pre-sented here.

IV. DISCUSSION

Results presented in Sec. III were obtained with timestep 0.1 fs and with randomly set initial velocities. For sys-tems C6H6, H2O+H, Sc+@C60 initial kinetic energy was setto 50 mEh, and for system 30�C2� it was set to 570 mEh.Authors attribute much worse conservation of energy for30�C2� system to large initial kinetic energy which becomeseven larger as the new C–C bonds are created. Differentchoices for the integration parameters may improve the be-havior of the BO and ADMP dynamics.

In principle, time step in LvNMD is limited by BCHexpansion. Typically for the cases presented here the BCHexpansion in Eq. �11� converges within 20–30 BCH steps�see Table I� with the largest term proportional to seventh orso order of commutator in Eq. �11�. Choosing too large timestep leads to significant increase in a precommutator factor.For example, twofold increase in time step results in the twoorder of magnitude �27� increase in precommutator factor ofterm proportional to seventh order commutator in Eq. �11�.This is especially important as the numerical error �noise�introduced on density matrix from BCH expansion and com-puters precision is equal to the largest matrix element ofcommutator term in BCH times 10−16 �16 digits for doubleprecision numbers�. We did not observe any problems fortime step equal to 0.1 fs. The error was controlled by moni-toring the magnitude of the matrix elements of the correctionterms in BCH expansion and to make sure that the error

TABLE II. Mulliken populations for the dissociation of water to OH and Hat electronic temperature Tel=30 000 K. The dissociating hydrogen is de-noted as atom 3 in the table. R�13� denotes bond length between oxygen anddissociating hydrogen. BOMD and LvNMD simulations were performed atspin-restricted SCC-DFTB level of theory.

Step No. 0 2085 3000R�13�: 0.967 3.053 3.967

BOMD1 O 6.557 24 6.288 90 6.42042 H 0.721 38 0.627 74 0.684 123 H 0.721 38 1.083 66 0.895 51

LvNMD1 O 6.557 24 6.416 50 6.401 802 H 0.721 38 0.689 81 0.688 943 H 0.721 38 0.893 69 0.909 26

-0.006

-0.004

-0.002

0

0.002

0.004

0.006

0 250 500 750 1000

(A) thrsF=10−3

-0.001

-0.0005

0

0.0005

0.001

0 250 500 750 1000

(B) thrsF=10−4

-8e-05

-4e-05

0

4e-05

8e-05

0 250 500 750 1000

(C) thrsF=10−5

-1e-05

0

1e-05

2e-05

3e-05

0 250 500 750 1000

(D) thrsF=10−6

1

1.5

2

2.5

3

0 250 500 750 1000

(E) thrsF=10−3

1

1.5

2

2.5

3

3.5

4

0 250 500 750 1000

(F) thrsF=10−4

2

3

4

5

6

7

0 250 500 750 1000

(G) thrsF=10−5

4

5

6

7

8

9

10

0 250 500 750 1000

(H) thrsF=10−6

FIG. 5. LvNMD simulations of C6H6 with time step dt=0.1 fs for variousvalues of Fock threshold thrsF. Panels �a�–�d� present conservation of totalenergy for Fock refinements threshold 10−3, 10−4, 10−5, 10−6. Panels �e�–�h�show corresponding number of Fock iteration inlvn required to reach con-vergence.

224106-7 Liouville–von Neumann molecular dynamics J. Chem. Phys. 130, 224106 �2009�

introduced by the largest matrix element in BCH expansionis less than preset threshold on density matrix. This is differ-ent than in BOMD case. In BOMD electronic structure isindependent on time and is obtained by minimization of en-ergy. Hence the time step is limited by integration not elec-tronic but nuclear equations of motion. Typical time stepused in BOMD is 1 fs. Another related observation is thatsince the electronic structure in LvNMD is reused hence theerror/noise introduced on the density matrix at the beginningis carried on during dynamics. Same is true in all other time-dependent quantum dynamics schemes such as Ehrenfest dy-namics. This is not the case in BOMD where the informationabout past densities is not needed.

The error on density matrix can be remediated inLvNMD at Tel=0 K by density matrix purification schemes.From Eq. �8� it follows that P�t� stays idempotent if P�0� isidempotent �for Tel=0 K�. However, for the purpose of thispublication we have intentionally chosen not to use purifica-tion of density matrix. This allows to clearly see how mucherror on density matrix is being accumulated during dynam-ics. As can be seen in Table I the error accumulated on den-sity matrix is very small. The idempotency of density matrixis preserved very well.

Another issue is that increasing a time step may not nec-essary lead to lowering the computational cost as the largernumber of Fock refinement steps might be required. For ex-ample simulation of benzene with time step 0.2 fs requiredon average around 20 Fock refinement steps as oppose to 6refinement steps for dt=0.1 fs �the same initial velocitieswere used in both cases�. Also using larger time step mayrequire using higher order magnus terms to account for non-linear changes in time-evolution operator and coupling be-tween various electronic states. The role of higher orderMagnus terms might be more important for metallic systemsin which coupling between electronic states is known to playan important role. This will be a subject of further studies.

Computational cost of current naive LvNMD implemen-tation �as described above� is similar to BOMD which re-flects the fact that the number of LvNMD and SCF iterationsis similar. This cost represents an upper bound of theLvNMD method. Obvious improvements such as Fockextrapolation,58,59,61 progressive convergence threshold onBCH expansion, and time-reversible lossless filterprocessing5 could easily lower computational cost ofLvNMD by an order of magnitude.

V. SUMMARY AND CONCLUSIONS

We propose a rigorous first principles MD scheme calledLvNMD, which is based on integration of Liouvile and vonNeumann equations for density matrix evolution rather thanon following the solution of the time-independent BO prob-lem. The algorithm uses only density matrices, avoids diago-nalization, is very stable, and has a very good conservationof energy. Its capability to treat mixed state density matriceswith fractional occupancies and its explicit time-dependentquantum-mechanical treatment of electrons makes it a verypromising, low cost tool in studies of the electronically ex-cited systems, nonadiabatic processes,20,21,38 quantum deco-

herence of electronic wave function from pure to mixedstate,27,28 molecular electronics,49,62,63 and energyscience.64,65

LvNMD method has been carefully tested againstBOMD and ADMP for variety of systems around the adia-batic surface but not for nonadiabatic cases. The behavior ofpresent method for strongly nonadiabatic cases will be ad-dressed in subsequent studies.

ACKNOWLEDGMENTS

The present research was supported in part by a grantfrom AFOSR �No. FA9550-07-1-0395� and in part by a grantfrom Gaussian Inc. In this work we used SCC-DFTB pro-gram for formation of Fock operator implemented by G.Zheng. J.J. would like to gratefully acknowledge discussionswith B. Schlegel, Xiaosong Li, G. Zheng, J. Tully, D. Marx,N. Doltsinis, and J. Hutter during the preparation of thismanuscript.

APPENDIX A: RELATIONS BETWEEN FOCKAND DENSITY MATRICES IN LvNMD

We will now show using Hartree–Fock theory that theimaginary part of density matrix enters into energy expres-sion and into Fock operator only through the exchange op-erator. This allows to utilize standard “real” electronic struc-ture codes for Fock formation and energy in LvNMD methodfor theories such as DFTB and some DFT that have no exacttreatment of exchange.

1. Dependence of Fock operator on imaginary partof density matrix in LvNMD

We use a real Gaussian type AOs basis set and complex,self-adjoint density matrix P, such that

P = S + ıA , �A1�

where S and A are real symmetric and antisymmetric matri-ces to form Fock operator F. It is clear that for an AO basisset a one-electron h contribution to Fock operator �see Eq.�7�� is always real and symmetric as it depends only on basisset and not on P. The two-electron operator G�P� depends onthe density matrix as follows:

G�P��,� = ���,��

��,�����,���P��,�� = J�,� − K�,�, �A2�

where

J�,� = ���,��

��,���r12−1��,���P��,��,

�A3�K�,� = �

��,��

��,���r12−1���,��P��,��.

2. Coulomb operator

For the Coulomb operator J�,� it follows that

224106-8 J. Jakowski and K. Morokuma J. Chem. Phys. 130, 224106 �2009�

J�,� = ������

��,���r12−1��,���P��,��

+ ������

��,���r12−1��,���P��,��

+ ���=��

��,���r12−1��,���P��,��

= ������

��,���r12−1��,���P��,��

+ ������

��,���r12−1��,���P��,��

+ ���

��,���r12−1��,���P��,��

= ������

���,���r12−1��,���P��,��

+ ��,���r12−1��,���P��,���

+ ���

��,���r12−1��,���P��,��, �A4�

where the notation ��� used in the summation means thatthe summation runs over all � and over all � such that ���. By using permutational symmetry of two-electron inte-grals,

�,�r12−1��,�� = �,��r12

−1��,� = ��,�r12−1�,�� , �A5�

one can rearrange above expression to obtain

J�,� = ������

��,���r12−1��,����P��,�� + P��,���

+ ���

��,���r12−1��,���P��,��. �A6�

The second summation runs over diagonal elements of den-sity matrix, P��,��, which are always real. The first summa-tion runs over nondiagonal elements of self-adjoint densitymatrix. Thus only real parts of P survive the summation inthe parenthesis,

P��,�� + P��,�� = 2R�P��,��� ,

and Coulomb operator J�,� is a purely real matrix.

3. Exchange operator

We will now analyze exchange operator and we willshow that it is always Hermitian in LvNMD. We will com-pare matrix real and imaginary parts of matrix elements K�,�

with corresponding real and imaginary parts of transposedelement K�,�. One can write K�,� and K�,� �see Eq. �A4�� asfollows:

K�,� = ���

��,���r12−1���,��P��,��

+ ������

���,���r12−1���,��P��,��

+ ��,���r12−1���,��P��,��� .

�A7�K�,� = �

��

��,���r12−1���,��P��,��

+ ������

���,���r12−1���,��P��,��

+ ��,���r12−1���,��P��,��� .

Here, one can notice that the first summation is always realwhile the second is complex due to the fact the integrals inparenthesis are not equal �compare with Eq. �A6��. To extractreal and imaginary parts of K�,� and K�,� we use Eq. �A1�.The real part is

R�K�,�� = ���

��,���r12−1���,��S��,��

+ ������

���,���r12−1���,��S��,��

+ ��,���r12−1���,��S��,��� ,

�A8�R�K�,�� = �

��

��,���r12−1���,��S��,��

+ ������

���,���r12−1���,��S��,��

+ ��,���r12−1���,��S��,��� .

It is clear that the real part of K is symmetric �R�K�,��=R�K�,��� by the virtue of Eq. �A5� and symmetry of S. Theimaginary part is

I�K�,�� = ������

���,���r12−1���,��A��,��

+ ��,���r12−1���,��A��,���

= ������

���,���r12−1���,��

− ��,���r12−1���,���A��,��,

�A9�I�K�,�� = �

�����

���,���r12−1���,��A��,��

+ ��,���r12−1���,��A��,���

= ������

���,���r12−1���,��

− ��,���r12−1���,���A��,��,

where we used A��,��=−A��,��. Comparing final expressionsin parenthesis of Eq. �A9� with Eq. �A5� it is clear thatimaginary part of K is antisymmetric. Thus in LvNMD atHartree–Fock theory exchange operator is always Hermitian,

K�,� = K�,�† . �A10�

We have shown that in LvNMD a Coulomb operatoronly depends on real part of density matrix and is hence realand symmetric. The exchange operator is a complex Hermit-ian operator. Its real part can be calculated as in standard

224106-9 Liouville–von Neumann molecular dynamics J. Chem. Phys. 130, 224106 �2009�

time-independent electronic structure codes using only realsymmetric part of densities. The imaginary contribution toexchange operator is a new term that does not show up instandard time-independent electronic structure and it de-pends only on a imaginary part of density matrix. This extraterm can be calculated as a correction to a real-density-basedFock operator.

4. Self-adjoiness of LvNMD propagated densitymatrix

It is easy to see that LvNMD propagation of self-adjointdensity matrix P�0� always leads to self-adjoint matrix P�t�by analyzing formally exact Eq. �11�. Let us note the follow-ing symmetry relations for the commutator: �S� ,S�=A and�A ,A��=A� and �S ,A�=S�, where S and S� are arbitrary sym-metric matrices while A, A�, and A� are arbitrary antisym-metric matrices. Truncation of Magnus expansion in Eq. �9�at any order always leads to self-adjoint operator F. Thisallows to decompose F and P on purely real symmetric �S�and imaginary antisymmetric �A� components and calculateeach symmetry commutators separately.

Commutator summation in Eq. �11� is done recursivelyas follows:

P = P + ı �S + ıA,Ck� , �A11�

where =�t /k is a scalar factor and C is commutator ob-tained from previous �k−1� iteration,

Ck = ı ��S + ıA,Ck−1� . �A12�

For k=1, matrix C is an initial density matrix P and is self-adjoint from the definition

C1 = P = S� + ıA�.

For k�1, matrix C can be expressed as

Ck+1 = S� + ıA�,

where

S� = �A,S�� − �S,A�� ,

A� = �S,S�� − �A,A�� .

Procedure given by Eqs. �A11� and �A12� is performed re-peatedly. At each k step C represents a self-adjoint correctionmatrix to density matrix P. Thus, propagated density matrixis always self-adjoint for any arbitrary truncation order ofcommutator expansion in Eq. �11�.

5. Contribution to energy from imaginary part of P

By noticing that the trace of a product of a symmetric�S� and an antisymmetric �A� matrices vanishes

Tr�S · A� = 0,

one can see that the real part of Fock operator �symmetric�can be contracted only with a real �symmetric� part of den-sity matrix and it contributes to all �one- and two-electron�energy terms. The imaginary part of density matrix �antisym-metric� gives nonzero value only when contracted withimaginary part of exchange operator �antisymmetric�. It can

be treated as correction to exchange energy. The imaginarypart contributes only to exact exchange energy and can becalculated as correction to. This term is absent in DFTB andin density functionals.

APPENDIX B: NUCLEAR FORCES IN LVNMD

In this appendix we show that Schlegel expression forforces6 derived for unconverged densities case ��F , P��0� isalso valid in LvNMD case. Derivations presented in originalpaper were performed with the assumption that the densitymatrix P is idempotent �P2= P�. Here we show that this ex-pression is also valid in case of fractional occupancies�mixed state� density matrix as represented by Eq. �4�.

To distinguish between nonorthogonal basis set of AOsand orthogonalized basis set we use respectively subscript nand o. That is Pn and Fn denotes density and Fock in nonor-thogonal AO basis set while Po and Fo are density and Fockin orthogonalized basis set. In general, orthogonalization isperformed by decomposition of overlap matrix S=LU. ForLöwdin orthogonalization we have L=U=S1/2 while forCholesky orthogonalization L and U are respectively lowerand upper triangular matrices such that L=UT. It follows thatFock operator is

Fn = LFoU , �B1�

density

Pn = U−1PoL−1, �B2�

and electronic energy

E = Tr��ho + 12Go�Po� + VNN = Tr��hn + 1

2Gn�Pn� + VNN.

�B3�

The gradient of electronic energy in AO basis set is

dE

dR= Tr�hn� +

1

2Gn�Pn� + Tr�Fn

dPn

dR� +

dVNN

dR, �B4�

where hn� is derivative of one-electron core hn integrals andGn� is obtained by replacing two-electron AO integrals in Gn

matrix by its derivatives. Let us analyze a second term �Pu-lay forces�. Inserting now Eq. �4� transformed into AO basisset into Pulay term gives

Tr�FndPn

dR� = Tr�Fn

d

dR�

i

M

Pni f i�

= �i

f i Tr�Fn

dPni

dR� + �

i

df i

dRTr�FnPn

i �

= �i

f i Tr�Fn

dPni

dR� , �B5�

where we used the fact that f i is constant during LvNMD dueto unitarity of time-evolution operator U�t� �see Secs. II Aand II B for discussion�. Inserting now Eq. �B2� into the lastresults and differentiating over R leads to

224106-10 J. Jakowski and K. Morokuma J. Chem. Phys. 130, 224106 �2009�

Tr�FndPn

dR� = �

i

f i Tr�FndU−1

dRPo

i L−1 + U−1Poi dL−1

dR� ,

�B6�

where we used the relation Tr�FnU−1�dPo /dR�L−1�=Tr�Fo�dPo /dR��=0. Notice that density matrix Pn dependson R only through U and L transformation matrices. Now,differentiating relation U−1U= I and multiplying the resultfrom the right hand side with U−1 leads to

dU−1

dR= − U−1dU

dRU−1. �B7�

Similarly, from L−1L= I one can obtain

dL−1

dR= − L−1 dL

dRL−1. �B8�

Inserting Eqs. �B7� and �B8� into Eq. �B6� and using Eq.�B2� gives

Tr�FndPn

dR� = − �

i

f i Tr�FnU−1dU

dRPn

i + Pni dL

dRL−1�

= − Tr�FnU−1dU

dR �i

f iPni

+ �i

f iPni dL

dRL−1�

= − Tr�FnU−1dU

dRPn + Pn

dL

dRL−1� . �B9�

Since the second part in the parenthesis is a transpose of thefirst one and by using Tr�AB�=Tr�BA�, hence the final ex-pression for a Pulay force is

Tr�FndPn

dR� = − 2 Tr�FnU−1dU

dRPn� . �B10�

Inserting this into Eq. �B4� gives the final expression for agradient of electronic energy as

dE

dR= Tr�hn� +

1

2Gn�Pn� − 2 Tr�FnU−1dU

dRPn� +

dVNN

dR,

�B11�

which is identical with Eq. �12� of this manuscript as well aswith Eq. 14 in Ref. 6 and Eq. 13 in Ref. 21. Thus we haveproven that Schlegel expression for nuclear gradient is notonly valid for idempotent densities but it also applies tomixed state densities with fractional occupancies.

The main difference between LvNMD and BOMDforces at Tel�0 is that in BOMD case the component ofPulay term Tr�Fo�dPo /dR�� that describes fluctuation of oc-cupations does not vanish and has to be balanced by an elec-tronic entropy term.40,51,52

1 R. Car and M. Parinello, Phys. Rev. Lett. 55, 2471 �1985�.2 D. Marx and J. Hutter, in Modern Methods and Algorithms of QuantumChemistry, edited by J. Grotendorst �NIC, Jülich, 2000�.

3 W. L. Hase, K. Song, and M. S. Gordon, Comput. Sci. Eng. 5, 36 �2003�.4 N. L. Doltsinis, in Computational Nanoscience: Do It Yourself!, edited byJ. Grotendorst, S. Bluegel, and D. Marx �NIC, Jülich, 2006�, Vol. 31, pp.

389–409.5 A. M. N. Niklasson, C. J. Tymczak, and M. Challacombe, Phys. Rev.Lett. 97, 123001 �2006�.

6 H. B. Schlegel, J. M. Millam, S. S. Iyengar, G. A. Voth, A. D. Daniels, G.Scuseria, and M. J. Frisch, J. Chem. Phys. 114, 9758 �2001�.

7 J. M. Herbert and M. Head-Gordon, J. Chem. Phys. 121, 11542 �2004�.8 T. D. Kühne, M. Krack, F. R. Mohamed, and M. Parinello, Phys. Rev.Lett. 98, 066401 �2007�.

9 A. Niklasson, Phys. Rev. Lett. 100, 123004 �2008�.10 J. M. Herbert and M. Head-Gordon, Phys. Chem. Chem. Phys. 7, 3269

�2005�.11 S. Godecker, Rev. Mod. Phys. 71, 1086 �1999�.12 C. Lubich, in Quantum Simulations of Complex Many-Body Systems:

From Theory to Algorithms, edited by J. Groetendorst, D. Marx, and A.Muramatsu �NIC, Jülich, 2002�, Vol. 10, pp. 459–466.

13 R. Kosloff, J. Phys. Chem. 92, 2087 �1988�.14 S. Mukamel, Principles of Nonlinear Optical Spectroscopy, Oxford Se-

ries in Optical and Imaging Sciences �Oxford University Press, NewYork, 1995�.

15 K. Yabana and G. F. Bertsch, Phys. Rev. B 54, 4484 �1996�.16 R. N. Zare, Science 279, 1875 �1998�.17 H. Eshuis, G. G. Balint-Kurti, and F. R. Manby, J. Chem. Phys. 128,

114113 �2008�.18 S. M. Smith, X. Li, A. N. Markevitch, D. A. Romanov, R. J. Levis, and

H. B. Schlegel, J. Phys. Chem. A 109, 10527 �2005�.19 X. Li, S. M. Smith, A. N. Markevitch, D. A. Romanov, R. J. Levis, and

H. B. Schlegel, Phys. Chem. Chem. Phys. 7, 233 �2005�.20 N. L. Doltsinis and D. Marx, J. Theor. Comput. Chem. 1, 319 �2002�.21 X. Li, J. C. Tully, H. B. Schlegel, and M. J. Frisch, J. Chem. Phys. 123,

084106 �2005�.22 J. Tully, Annu. Rev. Phys. Chem. 51, 153 �2000�.23 M. D. Hack and D. G. Truhlar, J. Phys. Chem. A 104, 7917 �2000�.24 B. G. Levine and T. J. Martnez, Annu. Rev. Phys. Chem. 58, 613 �2007�.25 N. Shenvi, H. Cheng, and J. C. Tully, Phys. Rev. A 74, 062902 �2006�.26 E. R. Bittner and P. J. Rossky, J. Chem. Phys. 103, 8130 �1995�.27 E. R. Bittner and P. J. Rossky, J. Chem. Phys. 107, 8611 �1997�.28 W. Zurek, Rev. Mod. Phys. 75, 715 �2003�.29 S. Klein, M. J. Bearpark, B. R. Smith, M. A. Robb, M. Olivucci, and F.

Bernardi, Chem. Phys. Lett. 292, 259 �1998�.30 U. Saalmann and R. Schmidt, Z. Phys. D: At., Mol. Clusters 38, 153

�1996�.31 N. L. Doltsinis and D. Marx, Phys. Rev. Lett. 88, 166402 �2002�.32 A. Bastida, C. Cruz, J. Zuniga, A. Requena, and B. Miguel, Chem. Phys.

Lett. 417, 53 �2006�.33 J. C. Tully, in Dynamics of Molecular Collisions, edited by W. H. Miller,

1976�, Vol. 2, p. 217.34 J. C. Tully, J. Chem. Phys. 93, 1061 �1990�.35 C. M. Isborn, X. Li, and J. C. Tully, J. Chem. Phys. 126, 134307 �2007�.36 A. M. Wodtke, J. C. Tully, and D. J. Auerbach, Int. Rev. Phys. Chem. 23,

513 �2004�.37 N. D. Mermin, Phys. Rev. A 137, 1441 �1965�.38 A. Alavi, J. Kohanoff, M. Parrinello, and D. Frenkel, Phys. Rev. Lett. 73,

2599 �1994�.39 J. Sokoloff, Ann. Phys. 45, 186 �1967�.40 M. Weinert and J. W. Davenport, Phys. Rev. B 45, 13709 �1992�.41 M. E. Tuckerman, B. J. Berne, and G. J. Martyna, J. Chem. Phys. 97,

1990 �1992�.42 P. Pechukas and J. C. Light, J. Chem. Phys. 44, 3897 �1966�.43 W. Magnus, Commun. Pure Appl. Math. 7, 649 �1954�.44 J. A. Oteo and J. Ros, J. Math. Phys. 41, 3268 �2000�.45 Using thermal excitation of electrons allows to approximately include a

“multireference” character of partially filled valence shell into a singledeterminant formalisms �Ref. 39�. It allows hence to study complexestransition metal complexes for which the electronic state is difficult if notimpossible, to determine.

46 L. Szilard, in Quantum Theory and Measurement, Princeton Series inPhysics, edited by J. A. Wheeler and W. H. Zurek �Princeton UniversityPress, Princeton, NJ, 1983�.

47 H.-T. Elze, Phys. Lett. B 369, 295 �1996�.48 N. L. Doltsinis, in Quantum Simulations of Complex Many-Body Sys-

tems: From Theory to Algorithms, edited by J. Groetendorst, D. Marx,and A. Muramatsu �John von Neumann Institute for Computing, Juelich,2002�, Vol. 10, pp. 377–397.

49 K. Burke, R. Car, and R. Gebauer, Phys. Rev. Lett. 94, 146803 �2005�.

224106-11 Liouville–von Neumann molecular dynamics J. Chem. Phys. 130, 224106 �2009�

50 F. Casas, J. Phys. A: Math. Theor. 40, 15001 �2007�.51 R. M. Wentzcovitch, J. L. Martins, and P. B. Allen, Phys. Rev. B 45,

11372 �1992�.52 A. M. Niklasson, J. Chem. Phys. 129, 244107 �2008�.53 G. Zheng, M. Lundberg, J. Jakowski, T. Vreven, M. J. Frisch, and K.

Morokuma, Int. J. Quantum Chem. 109, 1841 �2009�.54 M. Elstner, D. Porezag, G. Jungnickel, J. Elsner, M. Haugk, T. Frauen-

heim, S. Suhai, and G. Seifert, Phys. Rev. B 58, 7260 �1998�.55 M. J. Frisch, G. W. Trucks, H. B. Schlegel et al., GAUSSIAN, Revision

F.02, Gaussian, Inc., Wallingford, CT, 2008.56 J. C. Slater and G. F. Koster, Phys. Rev. 94, 1498 �1954�.57 D. Porezag, T. Frauenheim, T. Köhler, G. Seifert, and R. Kaschner, Phys.

Rev. B 51, 12947 �1995�.58 P. Pulay, Chem. Phys. Lett. 73, 393 �1980�.59 P. Pulay, J. Comput. Chem. 3, 556 �1982�.60 G. B. Bacskay, Chem. Phys. 61, 385 �1981�.61 P. Pulay and G. Fogarasi, Chem. Phys. Lett. 386, 272 �2004�.62 A. Feldman, M. L. Steigerwald, X. Guo, and C. Nuckolls, Acc. Chem.

Res. 41, 1731 �2008�.63 A. Nitzan and M. Ratner, Science 300, 1384 �2003�.64 G. R. Fleming and M. Ratner, Phys. Today 56�7�, 28 �2008�.65 J. Hemminger, G. Fleming, and M. Ratner, U. S. Department of Energy

Report �2007�, http://www.er.doe.gov/bes/reports/files/GC_rpt.pdf.

224106-12 J. Jakowski and K. Morokuma J. Chem. Phys. 130, 224106 �2009�