Molecular Dynamics with Quantum Transitions (MDQT): a Match Made in Phase Space

Molecular Dynamics with Quantum Transitions (MDQT): A Match Made in Phase Space

Kristin Benke & Morgan Hammer

December 4, 2014

1 Introduction

Molecular dynamics (MD) is a field of study in computational chemistry concerned with describing and simulating the motion of atoms and molecules in time. Such simulations give information about time-dependent physical and chemical processes as well as provide statistical information such as diffusion coefficients, heat capacities, or time-correlation functions for a chemical system of interest. To ease computational cost, most of these MD simulations are conducted treating the system classically. However, in systems where quantum effects such as tunneling, quantum interference, zero point energy are significant such as for low temperature systems or systems involving hydrogen atoms, using fully classical MD can give inaccurate results. Molecular dynamics methods can be modified to describe select quantum effects. The methods described by Tully provide a modified molecular dynamics method to incorporate electronic transitions.

In quantum mechanics, all the information about nuclei and electrons is contained within a wave function obtained by solution of the Schrodinger¨ equation. Manipulation of such a wave function requires treatment of a number of degrees of freedom that is not reasonably tractable with the computational power currently available for all but the smallest of systems. The nuclear-electronic wave function requires digestion before it can be fed into any algorithm to extract interesting information. First, the nuclear and electronic degrees of freedom can be decoupled in accordance with the Born-Oppenheimer approximation. Because the nuclei move on such a slower time-scale than the electrons and if a molecule remains on only its electronic ground state, the nuclei can be treated as classical particles moving through an effective potential of their electron clouds. This gives rise to classical MD in which atoms are reduced to point charge masses connected by “springs” (bonds). Now instead of being described by a high-dimensional wave function, each atom is just assigned 3 position coordinates and 3 momenta vector components, so the complexity is signiﬁcantly reduced. The particles that give rise to interesting quantum effects can still be described by the appropriate wave function.

We know from classical mechanics that position and momentum can be described using Hamilton’s equations of motion as shown below where position and momentum can be assigned to q and p.

∂H q˙ = (1) ∂p ∂H p˙ = − (2) ∂q Therefore, to solve the time evolution of a system in classical mechanics one requires only an appropriate set of initial positions and momenta for the atoms and a force-ﬁeld to describe their interactions (vibrations, torsions, van der Waals forces, etc.). With this information in hand, Hamilton’s equations can be integrated to describe how the set of atomic positions and momenta evolve over time.

1 However, because of the assumptions made moving from a terribly intractable fully quantum description of molecular dynamics to a wholly classical one, dynamical phenomena that lie outside the classical limit cannot be properly accounted for without including quantum effects in some manner. Such problematic phenomena include tunneling, photoexcitation, bond breaking/formation, and nonadiabaticity to name a few.

One prominent class of solutions to reuniting quantum and classical treatments of molecular dynamics came in the form of “surface-hopping” methods. The classical nuclei would propagate on a given electronic potential energy surface until some switching criterion was met, based on the nonadiabatic coupling between electronic states, that would cause the nuclei to switch the electronic potential energy surface on which they are propagating. Based on the magnitude and direction of the nonadiabatic coupling at the time of the switch, the nuclear velocities would be adjusted accordingly to conserve energy. Therefore the classical equations of motion would be solved for the nuclei and the time-dependent Schrodinger¨ equation would be solved for the electrons in the system self-consistently.

A variety of criteria were proposed to provide a method of switching that for an ensemble of trajectories reproduced the proper time-dependence of electronic state populations while providing dynamics that was not simply for a single trajectory an averaging of movement on both potential surfaces. In an attempt to reconcile these, John Tully proposed in 1990 his “fewest-switches” algorithm for Molecular Dynamics with Electronic Transitions.

In section 2 we develop the foundation of the surface-hopping formalism and further justify the use of the fewest-switches method and compare it to other methods. In section 3 we discuss the results of the implementation of Tully’s fewest-switches surface hopping algorithm for a single avoided crossing. Finally in section 4 we discuss the more generalized Molecular Dynamics with Quantum Transitions as well as look into an example of MDQT applied to proton transfer in solution.

2 Derivations

Several methods exist that rely on mixed quantum-classical trajectories, such as surface hopping and the Ehrenfest approach which is a single trajectory time dependent Hartree-Fock method. The most distinguishing characteristic of the MDET method is its fewest switches criteria. The criteria that govern switching behavior affect the effective potential energy surface on which the the trajectories evolve. Consider the case of a set of two possible potential energy curves. For a single trajectory approach, the trajectory can be taken to undergo zero switches and follow a single potential energy surface, which is composed of an average of the two potential energy surfaces (Fig. 1(b)). The same result is achieved when the time ∆t between switches becomes inﬁnitesimal. The system behaves as if both trajectories are followed simultaneously, and what results is an averaged trajectory weighted by the occupation of each electronic state, given by a11(t) and a22(t) (Fig. 1(c)). When a trajectory evolves on a single averaged potential energy surface, as happens in the two previous cases, important information is lost and such a trajectory is never actually realized. An averaged trajectory cannot hope to represent the full picture of a set of two drastically diverging trajectories. This “averaging” problem can be avoided by requiring that trajectories split into branches, as is achieved in the fewest switches method (Fig. 1(d)).

2 Figure 1: (a) Probability of occupation of 2 states in a region of strong coupling. (b) Effective potential (dashed) with two potential energy surfaces for a single trajectory method (black). (c) Rapidly switching trajectory (dashed) for 2 potential energy surfaces (black). (d) Fewest switches trajectories (dashed) for 2 potential energy surfaces (black).

The atomic and electronic motion is described by the total Hamiltonian in Eq. (3) giving

H = TR + H0(r, R) (3)

where T is the kinetic energy and H0 is the electronic Hamiltonian. The Born-Oppenheimer approximation is used to ignore the nuclear kinetic energy operator. The electronic wave function ψ(r, R, t) can be written as a linear combination of the set of orthonormal electronic basis functions φ(r, R). Wave function ψ(r, R, t) gets its time dependence from the atomic motion which will be described in time by trajectory R(t).

X |ψ(r, R, t)i = cj(t) |φj(r, R)i (4) j The classical path method result shown in Eq. 5 can be found by inserting ψ(r, R, t) into the time-dependent Schrodinger equation and multiplying from the left by ψk.

X ˙ i~c˙k = cj(Vkj − i~R · dkj) (5) j

3 This result represents a set of differential equations that can be integrated numerically over electronic coordinate r to determine the amplitude cj of each electronic state for a given atomic trajectory, R(t). In this equation is the nonadiabatic coupling strength dkj(R) which is

dkj(R) = hφk | ∇Rφji (6) where the gradient is taken over atomic coordinates, R. The second term on the right hand side of Eq. 5 is obtained using the chain rule as shown below in Eq. 7. The matrix elements of the electronic Hamiltonian are represented below by Vij(R). The terms Vkj and djk both affect the probability of a electronic transition between states j and k.

∂φj ∂R ∂φj φk = φk (7) ∂t ∂t ∂R

Vij = hφi(r, R)| H0(r, R) |φj(r, R)i (8) Eq. 5 can be described in an alternative but equivalent form using density matrix notation as shown below in Eq. 16. This alternative form follows from the von Neumann equation (Eq. 9) which involves implementation of the quantum density operator ρ. The quantum mechanical version of the classical density operator is equivalent to the projection operator for the wave function as shown below.

∂ρ i = [H, ρ] (9) ~ ∂t X ρˆ = |ψi hψ| = |φli hφl| (10) l

This operator can be applied to state j and left multiplied by state k to obtain the off-diagonal elements ρkj, which are equivalent to akj in Eq. 12. The akj terms can also be written in terms of the coefﬁcients of the electronic states ∗ ck and ck.

∗ ρkj = hφk | ρˆ| φji = hφk | ψi hψ | φji = ckcj (11)

∗ akj = ckcj (12) The von Neumann equation can be expanded to obtain the density matrix notation form as shown below, relying on calculation of the commutator and applying the chain rule again on the left side as with the time-dependent Schrodinger¨ equation.

[H, ρ] = hφk | Hρ − ρH | φji (13)

4 X ˙ ˙ i~akj˙ = {alj[Vkl − i~R · dkl] − akl[Vlj − i~R · dlj]} (16) l The electronic state populations can be calculated for j=k in Eqn 16, resulting in the form below.

X a˙ kk = bkl (17) l6=k

2 ∗ ∗ ˙ bkl = Im(aklVkl) − 2Re(aklR · dkl) (18) ~ The density matrix form is useful for an initial electronic wavefunction in a mixed state, while Eq. 5 can be used if the electronic wavefunction begins in a pure state.

With this architecture, Tully’s method can finally be described. The method involves four main steps: (1) selection of initial conditions for the first trajectory; (2) integration of classical equations of motion and electronic amplitudes; (3) calculation of switching probabilities, gkj; and (4) iteration of steps (2) and (3) until the trajectory is finished. These steps are guided by the ultimate goal of selection of the best trajectory in a self-consistent way.

(1) The initial conditions must be selected to best represent the experiment that will be simulated. All atoms must be assigned an initial position and momentum, and the electronic states assigned values of ajk, elements of the density matrix.

(2) Simulations begin with integration of the classical equations of motion corresponding to the Vkk of the currently occupied potential energy surface and Eqs. 5 or 16, depending on the occupation of states in the initial wave function. The step size ∆t must be chosen so that a linear approximation of each of these equations holds. This is particularly important in areas of strong interaction where the nonadiabatic coupling can change rapidly.

∆tbjk gkj = (19) akk (3) At each integration point, the occurrence of a switch is determined based on comparison of the switching probability gkj (Eq. 19) to a random number ζ between 0 and 1. A switch will occur if gkj is larger than ζ.

(4) If a switch occurs in step (3), that trajectory is assigned to the new potential energy surface Vk0k0 .To maintain conservation of energy, the atomic velocity must be adjusted on the new potential energy surface in the direction of dkk0 at the new coordinate R. Additionally for a switch to a greater potential energy surface, if the velocity adjustment is lower than the velocity reduction required (reduction because for conservation of energy, if potential energy is increased, kinetic energy is decreased) then the switch is not carried out. If a switch does not occur, the trajectory returns to step (2) for the next time interval. This process is repeated for a swarm of trajectories until experiment-speciﬁc criteria is reached for completion.

3 Example: Single-Avoided Crossing

To provide a baseline test of the effectiveness of the “fewest-switches” algorithm, Tully tested the method for three two-level test cases against an “accurate” quantum method and results obtained using the Landau-Zener approximation. The accurate quantum method involved propagation of a gaussian wavepacket by fast fourier transform (FFT). For

5 simplicity and brevity, we will focus on the simplest test case, the simple avoided-crossing model.

Avoided crossings occur in regions where two states potential energy surfaces would intersect but by symmetry they are forced to avoid this intersection (when the intersection is symmetry-allowed, this point is known as a conical intersection). The Hamiltonian in the diabatic representation of the simple avoided-crossing model given is

V11(x) = sgn(x)A[1 − exp(−sgn(x)Bx)] (20)

V22(x) = −V11(x) (21)

2 V12(x) = V21(x) = Cexp(−Dx ) (22) For this model, the parameters were chosen to be A=0.01, B=1.6, C=0.005, and D=1.0, in atomic units. Diagonalization of this diabatic representation yields two adiabatic potential energy surfaces of the form

q 2 2 U±(x) = ± V11(x) + V12(x). (23) These surfaces and the nonadiabatic coupling strength between them are shown in ﬁgure ζ. We have reproduced the adiabatic potential energy curves alongside this plot. As expected, the coupling asymptotically becomes zero far away from the avoided crossing region and rapidly increases near the avoided crossing. Trajectories were started in the negative asymptotic region (which we assume to mean x=-10a.u. given that is the far left of the plot shown) and was propagated until it left the “interaction region” −10 < x < 10.

Figure 2: (a) Adiabatic potential surfaces (black) and nonadiabatic coupling strength (dashed) for the simple avoided crossing. (b) Our reproduction of the adiabatic surfaces.

A variety of different initial k values were used and 2000 trajectories were conducted for each k value. For a single trajectory the particle was considered to be “transmitted” through the avoided crossing if it ended to the right

6 of the crossing and “reﬂected” if it ended on the left side of the crossing. The transmission probabilities are reported for each k value by dividing the number of trajectories in which transmission was successful by the total number of trajectories.

Figure 3: (a) transmission probability for trajectories started on lower state (b) reﬂection probability for trajectories started on lower state (c) transmission probability for trajectories started on upper state. FFT method results depicted as full circles. MDET results depicted as open circles. Landau-Zener results depicted as dashed lines.

The transmission probabilities from Tully’s method are plotted against those obtained by the FFT wave packet and Landau-Zener methods in ﬁgure epsilon. Tully’s MDET method gives quantitatively comparable results to the fully quantum FFT wave packet method over the full range of energies.

4 Generalized MDQT as applied to Proton-transfer

Four years after the publication of the MDET paper, Tully and Sharon Hammes-Schiffer extended the method to describe proton transfer in solution, thus generalizing MDET to Molecular Dynamics with Quantum Transitions (MDQT). Proton transfer is a process that contains signiﬁcant quantum effects for a nucleus, such as tunneling and zero-point energy, both of which are signiﬁcant due to the relatively lighter mass of hydrogen, which cannot be produced using classical methods. The proton transfer is represented by a general scheme shown below

7 − + AH − B A − H B (24) where AH-B can represent an OH-N complex. Parameters were chosen to model the system after a typical amine-phenol complex in liquid methyl chloride as the solvent. The motion of atoms A and B are treated classically, while the hydrogen atom is treated like a harmonic oscillator that oscillates between the two atoms A and B. This system gives a rigorous test of MDQT because it possesses both adiabatic and nonadiabatic qualities as well as signiﬁcant effects due to zero point energy and tunneling.

In this case, the “surface hops” represent transitions between vibrational energy levels of the hydrogen atom as it moves along a single electronic Born-Oppenheimer potential energy surface. The shape of double well on which the hydrogen atom moves is affected by coordinate of hydrogen with respect to A and B. Three different potential well shapes are shown in Fig. 4. In Fig. 4(a), the reactant side is deeper, so the state has more reactant carrier than product carrier. The opposite is true in Fig. 4(c) where the deeper well is closer to the product side. Because each well has similar depth in Fig. 4(b), the state has approximately equal product and reactant character. The rate constants for this proton transfer reaction were calculated using an adiabatic method and a nonadiabatic method, using 2 or 4 of the lowest states for each.

Figure 4: Double-welled potential curves for hydrogen motion.

8 As we have demonstrated, MDQT provides a promising framework for incorporating quantum effects into molecular dynamics simulations. Tully’s “fewest-switches” algorithm gives a means to not force all possible switches to be averaged in a single trajectory, but instead forces that an ensemble of trajectories average to provide the correct state populations consistent with time-dependent quantum mechanics. Tully demonstrated that for several test systems that MDQT can be quantitatively competitive with an accurate fully quantum numerical method. The method was shown to be even applicable to treating proton transfer. Sharon Hammes-Schiffer has continued to work on developing MDQT methods for application to the study photoinduced electron-proton transfer (photo-EPT).

5 Acknowledgements

We would like to thank Sharon Hammes-Schiffer and Alexander Soudackov for their help and guidance in this endeavor.

6 References

(1) Tully, J.C. Molecular dynamics with electronic transitions. J. Chem. Phys. 1990, 93, 1061-1071.

(2) Hammes-Schiffer, S. Proton Transfer in solution: Molecular dynamics with quantum transitions. J. Chem. Phys. 1994, 101, 4657-4667.

(3) Cramer, C. J.: Essentials of Computational Chemistry: Theories and Models, 2nd ed.; Wiley: West Sussex, 2004.

(4) Schatz, G.C.; Ratner, M.A.: Quantum Mechanics in Chemistry, 2nd ed.; Dover: Mineola, New York, 2002.