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

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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 (r; R; t) can be written as a linear combination of the set of orthonormal electronic basis functions φ(r; R). Wave function (r; R; t) gets its time dependence from the atomic motion which will be described in time by trajectory R(t). X j (r; R; t)i = cj(t) jφj(r; R)i (4) j The classical path method result shown in Eq. 5 can be found by inserting (r; R; t) into the time-dependent Schrodinger equation and multiplying from the left by k. 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 j rRφji (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)j H0(r; R) jφj(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 ρ^ = j i h j = jφli hφlj (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 j ρ^j φji = hφk j i h j 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 j Hρ − ρH j φji (13) X hφk j Hρ − ρH j φji = [hφk j H j φli hφl j φji − hφk j φli hφl j H j φji] (14) l X X [H; ρ] = [ρljHkl − ρklHlj] = [aljHkl − aklHlj] (15) l l 4 X _ _ i~akj_ = falj[Vkl − i~R · dkl] − akl[Vlj − i~R · dlj]g (16) l The electronic state populations can be calculated for j=k in Eqn 16, resulting in the form below.
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