Dickinson College Dickinson Scholar Student Honors Theses By Year Student Honors Theses 5-22-2016 Numerical Simulations of High-Dimensional Mode-Coupling Models in Molecular Dynamics Kyle Lewis Liss Dickinson College Follow this and additional works at: http://scholar.dickinson.edu/student_honors Part of the Atomic, Molecular and Optical Physics Commons Recommended Citation Liss, Kyle Lewis, "Numerical Simulations of High-Dimensional Mode-Coupling Models in Molecular Dynamics" (2016). Dickinson College Honors Theses. Paper 236. This Honors Thesis is brought to you for free and open access by Dickinson Scholar. It has been accepted for inclusion by an authorized administrator. For more information, please contact scholar@dickinson.edu.
Numerical Simulations of High-Dimensional Mode-Coupling Models in Molecular Dynamics Submitted in partial fulfillment of honors requirements for the Department of Physics and Astronomy, Dickinson College, by Kyle Lewis Liss Advisor: Professor Brett J. Pearson Carlisle, PA May 20, 2016
Abstract A fundamental question in molecular dynamics is the following: Given some bond-localized excitation of a molecule, what will be the pathway and rate of energy flow throughout the molecule s various degrees of freedom? This notion of vibrational energy transfer throughout a molecule is referred to as intramolecular vibrational redistribution (IVR) and has been a long-standing subject of interest in physical chemistry. Historically, IVR has been studied on a case-by-case basis. However, the essence of IVR for any molecular system is an anharmonic potential energy surface that causes dynamics in which the system s many degrees of freedom are coupled in any coordinate system. We perform a general study of anharmonic coupling by examining the dynamics resulting from the lowest-order power series potential that causes coupled motion. Specifically, we consider both quantum and classical simulations starting with a localized excitation in a single vibrational mode. We analyze the quantum case only in two dimensions, including a study of the dynamics for both wavepackets and approximate eigenstates. In the classical case, we study simulations where the initially localized energy is coupled to a large dimension bath. For a specific coupling model and system dimension, we observe energy dephasing into the bath that agrees with results from experimentally observed IVR. However, we find the unexpected result that once the size of the bath reaches a certain critical value, increasing the dimension further causes the energy in the system to remain more localized. ii
Acknowledgements First and foremost, I would like to thank my advisor, Prof. Brett Pearson, for his continued guidance, feedback, and support throughout the year. Without him, this project could not have been possible. I would also like to acknowledge in particular the support of Sahil Nayyar 16, who has been not only a stellar colleague in physics, but also a great friend. Next, I would like to thank Prof. David Mertens and Prof. Lars English for lending me their expertise on coupled oscillators during many useful and productive discussions. I would also like to acknowledge Prof. Robert Boyle and the entire Senior Seminar class for useful comments and feedback throughout the year. Lastly, I would like to thank the entire Dickinson College Department of Physics and Astronomy for providing me with a comforting and congenial learning environment. iii
Contents Abstract ii 1 Introduction 1 2 Theory 6 2.1 Fundamentals of Quantum Mechanics and the Schrödinger Equation... 7 2.1.1 Operators and Expectation Values of Observables......... 8 2.1.2 The Time-Independent (Eigenstate) Approach........... 9 2.1.3 The Time-Dependent (Wavepacket) Approach........... 11 2.1.4 Dispersion in Quantum Mechanics.................. 12 2.1.5 Ehrenfest s Theorem and the Correspondence Between Quantum and Classical Dynamics....................... 14 2.2 Motivating the Form of our Anharmonic Potential............. 17 3 Two-Dimensional Quantum Simulation Methods 19 3.1 The Split-Operator Method......................... 19 3.2 Two-Dimensional Quantum Simulation General Setup........... 21 3.3 Two-Dimensional Quantum Simulation Initial Conditions......... 23 3.3.1 Wavepacket Initial Conditions.................... 23 3.3.2 Eigenstate Initial Conditions..................... 23 4 Two-Dimensional Quantum Simulation Results and Discussion 25 4.1 Wavepacket Initial Condition Results.................... 25 4.2 Eigenstate Initial Condition Results..................... 28 5 High-Dimensional Classical Simulation Methods 35 6 High-Dimensional Classical Simulation Results and Discussion 38 7 Suggestions for Further Research 45 A Lowest-Order Power Series Potential Yielding Undecoupleable Dynamics 46 iv
List of Figures 1 A coupled mass-spring system. The system diagram is shown in (a) and an example of typical motion is shown in (b)................ 2 2 A plot of potential energy against internuclear separation for both a Morse potential (black) and a harmonic potential (dashed red). Note that the harmonic potential provides an accurate approximation only near the minimum...................................... 3 3 A CO 2 molecule and associated equipotential curves for the potential energy as a function of the C-O bond stretches at a fixed bend angle.... 4 4 Plots of the potential energy function in Eq. (3.2.1) and the modulus of our initial Gaussian wavefunction....................... 24 5 Plot of the first moments x and y for typical simulation results. The parameters used were ɛ = 0.00022 and δ = 0.01. Note that the fast timescale 20 fs oscillation in both curves is unresolved in the figure.... 25 6 Energy in the x and y vibrational modes (left) and total system energy (right) associated with the motion in Fig. 5................. 26 7 A plot of the running average of Var(x) as a function of time obtained using Eq. (3.3.2). The averaging is performed to dampen natural oscillations in the variation resulting from wavepacket breathing.............. 27 8 Plot of typical results for the positions as functions of time for a classical simulation of Eq. (1.0.2) with energy initially localized in the x degree of freedom. The parameters used were k x = 1, k y = 1.01, and ɛ = 0.1.... 28 9 Typical results for the for the quantum dynamics of Eq. (3.2.1) using an initial wavefunction given by Eq. (3.3.6). Both plots display the modulus squared of the projection of the wavefunction onto various eigenstates (indicated in the plot legends) as a function of time. Both plots use m x = m y = 100 amu and ɛ = 0.0008......................... 29 10 Plots of both the beat period and beat amplitude as functions of the coupling strength for the case where δ = 0.98. In both plots, data was taken for coupling strengths within the range of 0.00003 ɛ 0.001... 34 11 Close up images of Fig. 9b to enhance the visibility of the fast oscillations in c 01 and the probability coefficients of the off-resonant eigenstates. Both exhibit fast oscillations with a period of roughly 20 fs............ 35 12 The coupling matrix Λ is shown on the left and a schematic of the associated coupling scheme is shown on the right. Note that Λ is symmetric and has zeroes down the main diagonal................... 37 v
13 The energy in the system mode averaged over 200 trials as a function of time for the total dimension numbers N = 25, 35, and 50 (indicated in legend)..................................... 39 14 The normalized Shannon entropy of the entire system averaged over 200 trials as a function of time for the total dimension numbers N = 25, 35, and 50 (indicated in legend).............................. 40 15 A close up image of the first 1800 time units of the system mode energies shown in Fig. 13................................ 41 16 Experimental IVR data obtained by Felker and Zewail for the 1792 cm 1 vibrational mode of anthracene, C 14 H 10. On the vertical axis, P (t) represents the probability strength of the 1792 cm 1 vibrational mode in molecular state space (figure reprinted from [19]; experimental data from [20, 21])..................................... 42 17 A plot of the long-time system mode energies shown in Fig. 13 as a function of the total dimension number. The results were obtained by averaging the system mode energy over the region (time 6875). Results are shown for all dimension numbers sampled in the simulation............ 43 18 Results from a second simulation run for the energy in the system mode as function of time and for the energy localization as a function of total dimension number. Note that both (a) and (b) indicate that the surprising energy localization in the first simulation run is a robust result for our chosen potential................................ 45 vi
1 Introduction The basis of all chemical reactions is the formation and destruction of bonds within molecules. Such bond altering processes result from internuclear vibrations at energies on the order of ev that occur during the reaction. This link between molecular vibrations and chemical reactions motivates the following fundamental question: Given some bond-localized excitation of a molecule, what will be the pathway and rate of energy flow throughout the molecule s various degrees of freedom? This notion of vibrational energy transfer throughout a molecule is referred to as intramolecular vibrational redistribution (IVR) and has been a long-standing subject of interest in physical chemistry [1]. The transfer of energy throughout a large molecule is complicated, and in most molecular systems the dynamics of IVR is not well understood. On the experimental side, it has proven difficult to access the appropriate time and energy scales with sufficient resolution. For theorists, a primary difficulty has been the high dimensionality of molecular state space [2]. Despite challenges, experimentalists and theorists alike have continued to work towards a deeper understanding of IVR in large molecules, primarily because of its supreme importance in predicting the rates of chemical reactions and gaining manipulative power over chemical reactivity [1, 2]. Although there does not exist a general theory characterizing IVR, many successful studies have been performed on a case-by-case basis [3, 4, 5]. It is of interest to advance the knowledge in this field by studying models for molecular dynamics in many dimensions. The most basic approach to studying molecular dynamics is to model a molecule as a system of masses and ideal springs. Using this model reduces molecular motion to a study of linear, coupled oscillators, the dynamics of which are well understood. A classic example of a linear, coupled oscillator system that exhibits energy transfer between its degrees of freedom is shown in Fig. 1a [6]. The system consists of two mass-spring systems, each with spring constant k, that are weakly coupled by a spring with spring constant k 12 k. Applying Newton s Second Law, one can solve for the system s motion relatively easily [7]. A sample solution, with initial energy localization in the x 1 degree of freedom, is shown in Fig. 1b. Observe how the initially localized energy transfers into the x 2 degree of freedom, and back again, as time progresses. We see periodic complete energy transfer, resulting in overall beating motion between x 1 and x 2. Given the motion shown in Fig. 1b, one might believe that the mass-spring system is inherently coupled. It is true that the coordinates x 1 and x 2 are coupled; however, it is possible to consider characteristic motions of this system, called normal modes, that define a coordinate system in which the motion is entirely decoupled. In the normal mode coordinate system, a single degree of freedom displaced from equilibrium oscillates periodically for all time with a characteristic frequency, completely decoupled from the 1
Figure 1: A coupled mass-spring system. The system diagram is shown in (a) and an example of typical motion is shown in (b). (a) The system can be described using two degrees of freedom, which are most naturally chosen as the coordinates x 1 and x 2 shown above. The outer springs have spring constant k. The middle spring, with spring constant k 12 k, provides the coupling (Adapted from [6]). (b) Sample motion for the system shown in (a), where the mass associated with x 1 is released from rest at an arbitrary nonzero amplitude and the mass associated with x 2 is released from rest at x 2 = 0. other normal modes of the system [8]. The concept of a normal mode is particularly significant in the context of energy transfer within systems because energy placed into one normal mode of motion never transfers into any others. The normal mode coordinates for the system displayed in Fig. 1 are ζ 1 = 1 2 (x 1 + x 2 ), ζ 2 = 1 2 (x 1 x 2 ). (1.0.1) The coordinate ζ 1 corresponds to motion where both masses oscillate in phase with the middle spring completely relaxed, while the coordinate ζ 2 corresponds to motion where both masses oscillate out of phase with the center of the middle spring remaining at rest [7]. In more complicated problems, switching to normal coordinates often leads to a considerable simplification. In fact, one can show that for an arbitrary system of N masses constrained to move in one dimension under the influence of linear forces, there exist N normal modes that entirely decouple the system s motion [7]. This is equivalent to asserting that there exists a coordinate system in which the system s potential energy is purely harmonic. An N-atom molecular system moving in three dimensions has 3N motional degrees of freedom. Therefore, a molecule can in general be described using 3N normal coordinates. Three of these normal coordinates involve overall translation of the molecule through space and three of the normal coordinates involve overall rotation of the molecule. Each of these coordinates represents no internal motion of the atoms relative to one another, and hence are irrelevant in the context of vibrational energy transfer throughout the molecule. 2
Thus, we can sufficiently describe the vibrations of an N-atom molecular system using 3N 6 normal coordinates [8]. Note that this reduces to 3N 5 coordinates for a linear molecule, since the molecule is symmetric about the bond axis [8]. The specific motions corresponding to the normal modes of vibration for an arbitrarily configured N-atom molecule are fully understood [8, 9]. Now, if the mass-spring model characterized by normal modes and purely harmonic potentials was an entirely accurate description of a molecule, then the problem of IVR would already be completely solved. This, however, is not the case, which can be seen immediately by considering the interaction between two atoms in a simple diatomic. If a sufficient amount of energy is used to excite vibrations between the two atoms, then the bond can break. Since this dissociation energy is finite, it cannot be the case that molecular vibrations are described by exactly harmonic, bounded potentials. Internuclear separations are more accurately described by Morse potentials, which become unbounded at high energies. A typical example of a Morse potential is shown in Fig. 2. Note that the asymptotic value of the energy in the limit of large internuclear separation represents the dissociation energy. Furthermore, observe that the harmonic approximation is only valid for relatively small displacements from equilibrium. We call such a potential anharmonic. Figure 2: A plot of potential energy against internuclear separation for both a Morse potential (black) and a harmonic potential (dashed red). Note that the harmonic potential provides an accurate approximation only near the minimum. As we move from one-dimensional vibrations to higher-dimensional molecular motion, the relevant potential energy shifts from a basic Morse potential to a complex anharmonic 3
potential energy surface. In Fig. 3a, we show equipotential energy curves for the anharmonic surface describing vibrations of a CO 2 molecule (Fig. 3b) at a fixed bend angle [8]. The potential interaction in each stretch coordinate resembles a Morse potential, where the intersection of the dashed lines represents the minimum point in which each bond is at its equilibrium length. The region labeled O + CO indicates dissociation of the O -C bond; the labels O C + O and O + C + O have analogous meaning. The critical effect of multi-dimensional anharmonicity is that the potential interaction in one degree of freedom may in general depend on the state of the other coordinates. In the context of CO 2, the equipotential curves seen in Fig. 3a will distort as the bend angle changes. The result of such coupling is that there exist no true normal modes for a multi-dimensional anharmonic system. Energy initially localized in a single vibrational degree of freedom may flow into the system s other degrees of freedom. Figure 3: A CO 2 molecule and associated equipotential curves for the potential energy as a function of the C-O bond stretches at a fixed bend angle. (a) Equipotential energy curves for the anharmonic surface describing vibrations of a CO 2 molecule at a fixed bend angle. The axes R 1 and R 2 denote the internuclear separation for the two C-O bonds (Reprinted from [8]). (b) A CO 2 molecule. Note that CO 2 is a linear triatomic molecule, and hence has four modes of motion relevant in vibrational energy transfer (Reprinted from [10]). We have seen that normal mode descriptions provide only an approximation to molecular dynamics, which is characterized by energy coupling between degrees of freedom. The harmonic approximation is valid only when the displacement from equilibrium is small, i.e., when the energies involved are much less than the dissociation energy [6]. 4
However, as we have mentioned, vibrations near the dissociation energy are of primary interest to study, since these are the energies relevant in chemical reactions [1]. Therefore, a general study of anharmonic coupling is both applicable and required to understanding molecular dynamics and IVR. Furthermore, there is a deep connection between anharmonicity and nonlinearity, which makes a study of anharmonic coupling relevant to problems outside the realm of molecular motion. In particular, understanding anharmonicity provides insight into problems in classical mechanics regarding energy transfer within nonlinear, coupled oscillators. Such problems are of interest and have been studied in [11, 12]. In this paper, we analyze multi-dimensional motion resulting from what we show to be the simplest possible anharmonic potential that induces dynamics that cannot be decoupled. In two dimensions, the potential we study is given by V (x, y) = 1 2 k xx 2 + 1 2 k yy 2 + 1 2 ɛ(x2 y + xy 2 ), (1.0.2) where k i represents the strength of the restoring force in the i th coordinate and ɛ is a relatively small parameter that determines the strength of the coupling between the system s degrees of freedom. To make the potential only weakly anharmonic, we would like ɛ to be small in some appropriate sense in comparison to the k i s. Note however that the k i s and ɛ have different units, and so we cannot meaningfully compare their magnitudes directly without making the spatial variables, x and y, dimensionless. This subtlety is briefly addressed in Secs. 3.2 and 5. In general, we study the case where the system s energy is initially localized in a single degree of freedom. The potential given in Eq. (1.0.2) extends easily to an arbitrary number of dimensions; however, one must introduce additional coupling parameters to allow for the possibility of coupling between any two degrees of freedom. In the case of molecular motion and IVR, the dynamics are quantum mechanical and thus described by the time-dependent Schrödinger equation. Nevertheless, in this paper we study both the quantum and classical dynamics of Eq. (1.0.2). In addition to being interesting for its own sake, a classical model is significantly simpler to deal with than a quantum model, yet still provides insight into molecular dynamics. This notion is supported both by Ehrenfest s theorem and the use of the classical expression for energy in the quantum mechanical Hamiltonian. Regardless of whether a classical or quantum model is used, the equations of motion will couple the degrees of freedom and hence capture the essence of the anharmonicity. While there exist analytical methods to study nonlinear ordinary and partial differential equations, results tend to become intractable for three or more dimensions [13, 14]. In this paper, we study dynamical systems in over ten dimensions. Therefore, our work 5
relies heavily on computer simulations. Specifically, we use Matlab to numerically solve ordinary and partial differential equations. In two dimensions, we study primarily the quantum dynamics of Eq. (1.0.2). We analyze cases where the initial wave function is either a Gaussian wavepacket displaced in a single degree of freedom or an eigenstate of the harmonic oscillator. We then transition into a study of higher dimensional anharmonic coupling using the generalization of Eq. (1.0.2). The simulations are performed classically for total dimension numbers 10 N 50. For a specific coupling model and system dimension, we observe energy dephasing into the bath that agrees with results from experimentally observed IVR. However, we find the unexpected result that once the size of the bath reaches a certain critical value, increasing the dimension further causes the energy in the system to remain more localized. This paper is organized is follows. In Sec. 2 we introduce the time-dependent Schrödinger equation and both the eigenstate and wavepacket approaches to studying it. We then discuss dispersion in quantum mechanics and the correspondence between quantum and classical dynamics. Lastly, we provide motivation for the use of the potential seen in Eq. (1.0.2) and its generalization to higher dimensions. In Sec. 3, we develop our numerical technique for solving the time-dependent Schrödinger equation and describe the setup of our two-dimensional quantum simulations. In Sec. 4 we provide the results of our quantum simulations and include a detailed analysis. In Sec. 5, we describe the setup for our high-dimensional classical simulations and in Sec. 6 we discuss the results. In Sec. 7, we summarize our overall findings and discuss possibilities for future work. 2 Theory In this section, we outline the essential and underlying theory relevant in our studies. We begin by discussing the fundamentals of quantum mechanics and the physical meaning of the Schrödinger equation. In particular, we explain expectation values and observables, the traditional time-independent (eigenstate) approach to quantum mechanics, and the less-frequently-employed time-dependent (wavepacket) approach. We then show that the Schrödinger equation is in general dispersive, and briefly discuss dispersion in quantum mechanics in the cases of harmonic and weakly anharmonic potentials. Next, we show that while the Schrödinger equation is dispersive, expectation values in quantum mechanics follow approximately classical trajectories. Specifically, we provide the precise criteria for which classical and quantum dynamics correspond, and justify the claim that for our purposes classical simulations give insight into molecular dynamics. We conclude by motivating our use of potential energies of the form seen in Eq. (1.0.2). 6
2.1 Fundamentals of Quantum Mechanics and the Schrödinger Equation It is well known that classical mechanics breaks down on the atomic scale. Specifically, classical physics assumes that particles have a precisely localized position; however, it has been shown experimentally that atomic-scale particles instead exhibit wavelike properties. The wave theory that we use to describe such particles is called quantum mechanics. Evidently, a study of molecular dynamics at any level is governed by quantum mechanics. Thus, we begin by outlining the fundamentals of quantum theory. The equivalent of Newton s second law in quantum mechanics, i.e., the equation of motion, is called the Schrödinger equation. For a particle of mass m subject to a potential energy V (r), the Schrödinger equation in N space dimensions is given by Ψ(r, t) i t = 2 2m 2 Ψ(r, t) + V (r)ψ(r, t), (2.1.1) where i is the imaginary unit, is Planck s constant, r R n, 2 is the N-dimensional Laplacian, and t represents time. Note that the potential V can in general depend on both position and time; however, in what follows we assume that V = V (r). The solution to the Schrödinger equation, Ψ(r, t), is called the particle s wavefunction and provides statistical information about the outcome of measurements performed on the particle. In particular, we interpret Ψ(r, t) 2 = Ψ (r, t)ψ(r, t) as a position probability density. That is, the probability of finding the particle at a given time t in the infinitesimal volume d N r is Ψ(r, t) 2 d N r. For the probability density interpretation of Ψ(r, t) to make sense, we must have Ψ(r, t) 2 d N r = 1, (2.1.2) where the integral is taken over all space. We call a wavefunction that satisfies Eq. (2.1.2) normalized. It is common to write Eq. (2.1.1) more succinctly as Ψ(r, t) i t where we have defined the Hamiltonian operator = ĤΨ(r, t), (2.1.3) Ĥ = 2 2m 2 + V. (2.1.4) Equation (2.1.3) is called the time-dependent Schrödinger equation (TDSE). 7
2.1.1 Operators and Expectation Values of Observables As mentioned above, Ψ(r, t) 2 represents a position probability density. The average position, which we denote by r, can therefore be computed using r = r Ψ(r, t) 2 d N r (2.1.5) = Ψ (r, t)rψ(r, t)d N r. We call r the expectation value of the position observable. Physically, it represents the average result that a measurement of position would yield if the measurement was performed repeatedly on an ensemble of identically prepared particles. In general, an observable is any quantity that can be measured in an experiment; e.g., position, momentum, and energy. Any observable A has an associated operator Â. In position space, the operator  is defined as the expression such that A = Ψ (r, t)âψ(r, t)dn r. (2.1.6) The right-hand side of Eq. (2.1.6) is merely an inner product, and so we often use the notation Ψ (r, t)âψ(r, t)dn r = Ψ Â Ψ. (2.1.7) In light of Eqs. (2.1.5) and (2.1.6), it is clear that the position operator (in position-space) is given by ˆr = r. (2.1.8) To determine the form of the momentum operator, we note that the inverse relationship between position and momentum implies that taking the Fourier transform of Ψ(r, t) yields a momentum-space representation of the wavefunction. Applying Plancherel s theorem (preservation of L 2 inner products under Fourier transform) and the fact that differentiation in position-space becomes multiplication in Fourier-space, one can show that the momentum operator (in position-space) is given by ˆp =. (2.1.9) i All classical quantities can be expressed in terms of position and momentum, and thus we may substitute Eqs. (2.1.8) and (2.1.9) into the expression for any classical observable to determine its associated operator. For example, the classical expression for the total energy E of a system is given by E = p2 + V (r), (2.1.10) 2m 8
and hence Ê = ˆp2 + V (ˆr). (2.1.11) 2m Substituting Eqs. (2.1.8) and (2.1.9) into Eq. (2.1.11) yields Ê = 2 2m 2 + V (r). (2.1.12) Therefore, in consideration of Eq. (2.1.4), we conclude that the Hamiltonian operator represents a system s total energy. Note that the TDSE holds regardless of the form of the system s Hamiltonian. For example, the Hamiltonian for a system of N particles, each free to move in one dimension, is given by Ĥ = N j=1 2 2m j 2 x 2 j + V (x 1,..., x N ), (2.1.13) where the j th particle has mass m j and position coordinate x j. The appropriate TDSE for such a system would be Eq. (2.1.3) with Ĥ given by Eq. (2.1.13). Hamiltonians of the form given by Eq. (2.1.13) will be particularly important for us. 2.1.2 The Time-Independent (Eigenstate) Approach The standard approach to quantum mechanics is to solve Eq. (2.1.1) using separation of variables. We refer to this as the time-independent, or eigenstate, approach. To use separation of variables, we assume that we can write the solution to Eq. (2.1.1) in the form Ψ(r, t) = φ(t)ψ(r). (2.1.14) Substituting Eq. (2.1.14) into Eq. (2.1.1) and dividing through by Ψ(r, t) yields i 1 dφ(t) φ(t) dt = 2 1 2m ψ(r) 2 ψ(r) + V (r). (2.1.15) As is standard in separation of variables, we conclude that both sides of Eq. (2.1.15) must be a constant. Calling the separation constant E, we find from Eq. (2.1.15) that and dφ(t) dt = ie φ(t) (2.1.16) 2 2m 2 ψ(r) + V (r)ψ(r) = Eψ(r). (2.1.17) Equation (2.1.16) is easily solved to yield φ(t) = e iet/, (2.1.18) 9
where the constant of integration will be absorbed into the solution of Eq. (2.1.17). Using the Hamiltonian operator, we may write Eq. (2.1.17) more succinctly as Ĥψ(r) = Eψ(r). (2.1.19) Equation (2.1.19) is called the time-independent Schrödinger equation (TISE) and cannot be solved until the potential V (r) is specified. If ψ(r) is a solution to the TISE, then Ψ(r, t) = e iet/ ψ(r) is a solution to the TDSE. Assuming Ψ(r, t) is normalized, the expectation value of energy for such a state is given by H = Ψ ĤΨd N r = e iet/ ψ (r)ĥ ( e iet/ ψ(r) ) d N r = ψ (r)eψ(r)d N r = E ψ(r) 2 d N r = E, (2.1.20) which illuminates the reasoning behind the choice of the letter E for the separation constant. From Eq. (2.1.19) we see that E is an eigenvalue of the Hamiltonian operator. The separable solutions that we obtain from the TISE are the associated eigenstates. When combined with their time-dependence [Eq. (2.1.18)], eigenstates of the Hamiltonian operator yield states of definite energy, which is clear from Eq. (2.1.20). In general, the eigenvalues of an operator represent the possible values that a measurement of the associated observable could return. The eigenstates of the operator are the states of definite expectation values (given by the eigenvalues) of the associated observable. The TISE generally admits an infinite collection of solutions {ψ n (r)}, each with an associated E n. The set {E n } of all possible energies is called the spectrum of the Hamiltonian for the particular system. Note that the eigenstates can typically be chosen so that they are orthonormal, meaning that they satisfy ψ i ψ j = δ ij, (2.1.21) where δ ij is the Kronecker delta function. The general solution to Eq. (2.1.1) is then given as a superposition of the separable solutions: Ψ(r, t) = c n ψ n (r)e ient/, (2.1.22) n=1 10
where the c n s are constants chosen to satisfy the initial condition Ψ(r, 0) = c n ψ n (r). (2.1.23) n=1 From Eqs. (2.1.21) and (2.1.23), it follows that c k can be found by projecting the initial wavefunction onto ψ k. That is, c k = Ψ(r, 0) ψ k. (2.1.24) Physically, c n 2 represents the probability of finding energy E n for a measurement of a system in state Ψ(r, t) given by Eq. (2.1.22). Note that Eqs. (2.1.22) and (2.1.23) assume that the collection {E n } is discrete. If the energies instead form a continuous spectrum, the summation in Eq. (2.1.22) is converted to an integral over all possible values of E: 2.1.3 The Time-Dependent (Wavepacket) Approach c(e)ψ E (r)e iet/ de. (2.1.25) The time-independent approach derives its name from the fact that all information about the possible outcomes of an experiment is encoded within the time-independent eigenstate solutions ψ(r) and their associated coefficients in Eq. (2.1.22). While the timeindependent approach is useful for theoretical arguments and for determining the energy spectrum of a given system, it is not particularly enlightening for visualizing the time evolution of the wavefunction. The interesting time evolution of Ψ(r, t) only becomes apparent when taking a sum of separable solutions that each have oscillatory time dependence. Now, to study molecular motion, we must analyze the time evolution of wavepackets (a superposition of waves that together form a localized disturbance) on multi-dimensional potential energy surfaces. To this end, it is more natural for us to take a time-dependent approach to quantum mechanics that more directly reveals the time evolution of the wavefunction. First, notice that we can separate the TDSE as Ψ Ψ = i Ĥ t. (2.1.26) If the Hamiltonian is time-independent, we may integrate Eq. (2.1.26) directly to obtain Ψ(r, t) = e iĥt/ Ψ(r, 0), (2.1.27) where the exponentiation of an operator is defined with respect to its Taylor series expansion. The exponential term in Eq. (2.1.27) is called the time-evolution operator. While 11
the expression for Ψ(r, t) given in Eq. (2.1.27) appears quite compact, the fact that the time-evolution operator is given by an infinite series implies that no analytical solution is immediately apparent. Nevertheless, Eq. (2.1.27) lends itself well to numerical solution. Discretizing the time axis by timesteps t, it follows from Eq. (2.1.27) that given Ψ(r, t) at any time t, we can find the solution at the later time t + t using Ψ(r, t + t) = e iĥ t/ Ψ(r, t). (2.1.28) Thus, specifying an initial wavefunction Ψ(r, 0), we can use Eq. (2.1.28) to propagate Ψ(r, 0) through time and determine Ψ(r, t) at any later time t. Note that Eq. (2.1.28) is exact for any size timestep; however, our numerical technique relies on an approximation that will require a sufficiently small timestep. We discuss the specific numerical implementation of Eq. (2.1.28) in Sec. 3.1. 2.1.4 Dispersion in Quantum Mechanics We now consider the quantum free-particle, which will demonstrate the dispersive nature of the Schrödinger equation and motivate a discussion of Ehrenfest s theorem. The freeparticle in one space dimension is defined by V (x) = 0. From Eq. (2.1.4), the Hamiltonian is simply Ĥ = 2 2m x. (2.1.29) 2 We will solve the Schrödinger equation using the time-independent approach. That is, we seek to solve the TISE, which reads 2 2 ψ(x) = Eψ(x). (2.1.30) 2m x 2 Assuming E > 0, the solutions to Eq. (2.1.30) are complex exponentials (sines and cosines) [8]. In particular, the eigenfunctions of the free particle Hamiltonian are 2 ψ k (x) = e ikx, (2.1.31) where k 2 = 2mE. (2.1.32) 2 From Eq. (2.1.32), we see that the energy eigenvalues associated with the eigenfunctions in Eq. (2.1.31) are given by E = 2 k 2 2m = p2 2m, (2.1.33) where we have defined p = k due to the analogy of Eq. (2.1.33) with the classical expression for the total energy of a free particle. Combining Eq. (2.1.31) with the timedependence derived in Eq. (2.1.18) yields the solution ( Ψ k (x, t) = e i kx k2 2m ). t (2.1.34) 12
Equation (2.1.31) is a plane wave solution and represents a state of definite energy and, since V = 0, momentum. While the eigenvalues form a continuous spectrum, a free particle of definite energy cannot exist because Eq. (2.1.34) is not normalizable. Any physically realizable solution must be a wavepacket formed by a superposition of eigenstates. In fact, since the spectrum is continuous the general solution is determined by Eq. (2.1.25). Allowing < k <, the general solution to the free-particle is given by [8] Ψ(x, t) = c(k)e i ( kx k2 2m t )dk. (2.1.35) Considering Eq. (2.1.34) as a solution to a general wave equation, it is clear that plane wave solutions to the Schrödinger equation satisfy the dispersion relation ω(k) = k2 2m. (2.1.36) Recall from general wave theory that the dispersion relation for a wave in a medium yields the phase velocity, v p, and the group velocity, v g : v p = ω k, v g = ω k. (2.1.37) The phase velocity determines the speed of propagation of individual waves, while the group velocity determines the speed of propagation for a well defined wavepacket. When the group and phase velocities are not equal, we say that the system is dispersive. A wavepacket traveling in a dispersive medium will either compress or spread in time. From Eq. (2.1.36), we see that the phase velocity for plane wave solutions to the Schrödinger equation is k/2m, while the group velocity is given by k/m. Thus, a free particle wavepacket in quantum mechanics is dispersive, and will spread in time. Note that in general a system is dispersive when there is a nonlinear relationship between ω and k. The nonlinearity in Eq. (2.1.36), and hence the dispersive nature of quantum waves, is fundamentally linked to the fact that the Schrödinger equation is second order in space but only first order in time [15]. While the discussion above was for the specific case of the free particle potential, it is in fact true that the Schrödinger equation is dispersive in general [15]. There is, however, the special case of the harmonic oscillator, which is of particular importance for us. The harmonic oscillator is defined in one-dimension by the potential V (x) = 1 2 mω2 x 2, (2.1.38) where m is the particle mass and ω is the natural frequency. For the harmonic oscillator, the natural dispersion of the Schrödinger equation is counteracted by the quadratic 13
nature of the potential [15]. The width of a wavepacket on a parabolic potential remains relatively constant, undergoing only small oscillations in time with no long-term spreading. We can understand this result conceptually by noting that the higher momentum components of the wavepacket must travel further to the potential barrier before the classical turning point. The difference in distance traveled by the high and low momentum components during each half-round trip precisely cancels out the differences in speed, which results in all momentum components making the trip in the same amount of time [15]. The result is that a wavepacket will remain coherent for long times. We can equivalently view the lack of dispersion in a quadratic potential as a result of the fact that the energy eigenvalues for the harmonic oscillator, given by ( E n = ω n + 1 ), n = 0, 1, 2,..., (2.1.39) 2 are equally spaced. The equal spacing in the energies is crucial for periodicity because the beat frequency between two eigenstates with energies E n and E m is given by E n E m /. From Eq. (2.1.39), it then follows that the beat frequency between any two harmonic oscillator eigenstates is an integer multiple of the fundamental frequency, ω. Now, the motion of a wavepacket composed of many eigenstates will in general be periodic with a period given by the least common multiple of the pairwise beat periods between its component eigenstates. Since all states beat at an integer multiple frequency of ω, a harmonic oscillator wavepacket will always be periodic with a frequency no less than ω, and hence will have no net spread over one natural period of oscillation. If we instead have a weakly anharmonic potential, such as the Morse potential shown in Fig. 2, the energy spacings between eigenstates is no longer equal. In particular, the energy spacings get smaller as the energy approaches the dissociation energy. The beat frequencies between eigenstates in a wavepacket will thus not be integer multiples of the fundamental frequency, and so a wavepacket will in general disperse with a revival time on a timescale much longer than the period of the fundamental. While this revival time is long, it is always finite assuming that the wavepacket contains a finite number of eigenstates. Nevertheless, these revival times tend to be on a timescale inaccessible by experiments, especially in the multi-dimensional case when the number of excited eigenstates grows significantly. 2.1.5 Ehrenfest s Theorem and the Correspondence Between Quantum and Classical Dynamics As we have seen in the previous section, the Schrödinger equation is dispersive, which is a key property that distinguishes quantum and classical dynamics. Nevertheless, we will show in this section that there is simultaneously a deep link and correspondence between 14
quantum and classical mechanics. Specifically, we will discuss Ehrenfest s theorem, which is a general relationship between expectation values of quantum mechanical observables and their associated classical equations of motion. We now briefly derive Ehrenfest s theorem. Consider the time dependence of the expectation value of an arbitrary observable represented by the operator Â: d ( ) dt A = Ψ ÂΨ dx. (2.1.40) t Applying the product rule and the shorthand notation defined in Eq. (2.1.7), we then have that d Ψ dt A = t ÂΨ + Ψ Â Ψ + Ψ Â t t Ψ. (2.1.41) In light of the TDSE equation and the fact that the Hamiltonian is Hermitian, we can rewrite Eq. (2.1.41) as d dt A = 1 i ĤΨ ÂΨ + 1 i Ψ ĤΨ + = 1 Ψ [Â, Ĥ] Ψ + i  t Ψ Â t Ψ, (2.1.42) where [Â, Ĥ] = ÂĤ Ĥ is the commutator between  and Ĥ [8]. In most cases, the operator  has no explicit time dependence, and so the second term Eq. (2.1.42) vanishes. We thus arrive at the result [8] For Hamiltonians of the form seen in Eq. (2.1.13), d dt A = 1 Ψ [Â, Ĥ] Ψ. (2.1.43) i 2 [ˆx j, Ĥ] = = i ˆp j, m j x j m j (2.1.44) [ˆp j, Ĥ] = V. i x j (2.1.45) Substituting these relations into Eq. (2.1.43) yields Ehrenfest s theorem: d dt x j = p j d dt p j =, (2.1.46) m j V. (2.1.47) x j For an N-dimensional system, Ehrenfest s theorem produces a set of two equations for each 1 j N. 15
There is a notable correspondence between Ehrenfest s theorem and Hamilton s equations of classical mechanics. Given the Hamiltonian, H, of a classical system, Hamilton s equations tell us that the system s time evolution is determined by dx j dt = H, p j (2.1.48) dp j dt = H, x j (2.1.49) where x j and p j represent the classical position and momentum of the j th particle, respectively. Under suitable conditions, the Hamiltonian of a classical system is simply the total energy. In this case, H may be expressed for a system of N particles as H = N j=1 p 2 j 2m j + V (x 1, x 2,..., x N ), (2.1.50) where m j is the mass the j th particle. When the Hamiltonian takes the form of Eq. (2.1.50), Eqs. (2.1.48) and (2.1.49) yield the equations of motion dx j dt = p j, m j (2.1.51) dp j dt = V. x j (2.1.52) Comparing Eqs. (2.1.46) and (2.1.47) with Eqs. (2.1.51) and (2.1.52), respectively, highlights the correspondence between quantum and classical dynamics. That is, the equations of motion for the expectation values of the position and momentum observables in quantum mechanics exhibit a striking resemblance to the associated classical equations of motion. While there is a relationship between the trajectories of quantum expectation values and their associated classical quantities, the correspondence is not exact. Equation (2.1.46) is exactly analogous to Eq. (2.1.51); however, there is a disparity between Eq. (2.1.47) and Eq. (2.1.52) due to the fact that in general V (x) V ( x ) x j x j, (2.1.53) where we have defined x = (x 1,..., x N ) and x = ( x 1,..., x N ) [8]. For simplicity we now assume our system is one-dimensional. To illuminate the nature of the error in using the classical equations of motion, consider expanding V (x)/ x in an operator power series about the expectation value of x: V (x) x = V ( x ) x + (x x ) 2 V ( x ) x 2 16 + 1 2 (x x )2 3 V ( x ) x 3 +..., (2.1.54)
where n V ( x )/ x n denotes n V (x)/ x n evaluated at x. Keeping terms no higher than quadratic in x, the expectation value of the derivative of the potential may then be approximated by V (x) x V ( x ) x + 1 2 χ 3 V ( x ) x 3, (2.1.55) where we have defined χ (ˆx x ) 2 = x 2 x 2, which represents the overall spread in position of the wavefunction [8]. The first term of the RHS of Eq. (2.1.55) is exactly the RHS of Eq. (2.1.53). Thus, the disparity between the time evolution of quantum expectation values and the associated classical trajectories depends on both the spread in position of the wavefunction as well as the third derivative of the potential energy. The correspondence is exact for up to quadratic potentials because the third derivative of the potential vanishes. For potentials that are not quadratic, the momentum and position expectation values follow the classical trajectories for small width wavefunctions. These qualitative results generalize to higher dimensional potential energy functions Based on Eq. (2.1.55), it follows that the classical-quantum correspondence is strong for weakly anharmonic potentials of the form seen in Eq. (1.0.2). To see this, first note that an approximately harmonic potential is only weakly dispersive, and so an initially compact wavefunction will remain coherent (small width) for long times. Additionally, the third derivative for a cubic anharmonic potential will be a constant and proportional to the strength of the anharmonicity. In combination, these facts show that the second term in Eq. (2.1.55) is relatively small for weakly anharmonic potentials, and hence the use of classical simulations to provide insight into quantum dynamics is justified in our case. 2.2 Motivating the Form of our Anharmonic Potential In this section, we motivate our study of cubic anharmonic potentials of the form seen in Eq. (1.0.2). Specifically, we discuss that when seeking a polynomial expansion for an anharmonic potential about a local minimum, Eq. (1.0.2) is the lowest-order twodimensional function that results in classical dynamics that cannot be decoupled. We begin by noting that we can approximate any potential energy surface near a local minimum using a Taylor expansion. For a two-dimensional potential V (x, y), the expansion takes the form V (x, y) = c 0 + c 1 x + c 2 y + c 3 x 2 + c 4 y 2 + c 5 xy + c 6 x 3 + c 7 y 3 + c 8 x 2 y + c 9 xy 2 +..., (2.2.1) where we have assumed the local minimum to be at the point (0, 0). The c i s are constants determined by the partial derivatives of V (x, y). Because we are assuming that (0, 0) is a minimum, c 1 = c 2 = 0. Moreover, the c 0 term represents an arbitrary offset, and so 17
without loss of generality we can take c 0 = 0. Thus, we may assume an expansion of the form V (x, y) = c 3 x 2 + c 4 y 2 + c 5 xy + c 6 x 3 + c 7 y 3 + c 8 x 2 y + c 9 xy 2 +.... (2.2.2) As we discussed in Sec. 1, the essence of anharmonicity and a general characteristic of molecular systems is coupling between the system s degrees of freedom that cannot be removed by defining a new coordinate system. Therefore, we require our polynomial expansion for the potential energy to result in equations of motions that cannot be decoupled in any coordinate system. Moreover, for simplicity we seek the lowest order polynomial that exhibits such behavior. At first glance, one might expect Eq. (2.2.2) with c i = 0 for all i = 6, 7,... to result in dynamics that cannot be decoupled; however, this is not the case. The coupling that results from the bilinear term c 5 xy in the potential V (x, y) = c 3 x 2 + c 4 y 2 + c 5 xy (2.2.3) can always be removed with a simple coordinate system rotation. In fact, the lowest order terms that induce undecoupleable dynamics are the x 2 y and xy 2 terms (for proofs of these claims, see Appendix). In generalizing Eq. (1.0.2) to higher dimensions, we therefore use a pairwise coupling scheme where coordinate x i is coupled to coordinate x j using a term of the form ɛ ij (x i x 2 j + x 2 i x j ). In N dimensions, the potential is given by [ ] N 1 V (x 1, x 2,..., x N ) = 2 k ix 2 i + 1 ɛ ij (x 2 i x j + x i x 2 2 j), (2.2.4) i=1 where k i determines the natural harmonic frequency in the i th degree of freedom and ɛ ij is a relatively small positive constant that represents the strength of the coupling between coordinates i and j. We assume that the couplings are symmetric, and hence we impose the condition that ɛ ij = ɛ ji. Note that the second summation is taken only over j > i to avoid over-counting. In this paper, we study the dynamics of Hamiltonians of the form Ĥ = H = N i=1 N i=1 2 2m 2 x 2 i j>i + V, (2.2.5) p 2 i + V, (2.2.6) 2m where V is given by Eq. (2.2.4). Equation (2.2.5) with N = 2 is used in the quantum simulations and its classical analog, Eq. (2.2.6), is used in the higher dimensional (N 10) classical simulations. 18
3 Two-Dimensional Quantum Simulation Methods In this section, we discuss the methods for our two-dimensional quantum simulations. We first outline the split-operator method, which is the numerical procedure we use to solve the time-dependent Schrödinger equation. We then discuss the setup for the various quantum simulations that we perform. 3.1 The Split-Operator Method Given some initial wavefunction Ψ(r, 0) and a potential energy function V (r), the goal of any numerical propagation routine is to accurately determine the wavefunction Ψ(r, t) at some later time t, where the time-evolution is determined by the time-dependent Schrödinger equation. We accomplish this task in our quantum simulations using the split-operator propagation routine, which we briefly outline in this section [8]. For the sake of clarity and concreteness, we discuss the split-operator method in the context of propagating a wavefunction through time in two space dimensions x and y. Additionally, a discussion of the two-dimensional case is particularly relevant for us because we perform our quantum simulations exclusively in two-dimensions. Note however that the split-operator method generalizes to an arbitrary number of dimensions. Recall from Eq. (2.1.27) that the solution to the TDSE can be written as Ψ(x, y, t) = e iĥt/ Ψ(x, y, 0), (3.1.1) where Ψ(x, y, 0) is the given two-dimensional initial wavefunction. First, note that we may write the Hamiltonian as Ĥ = ˆT (ˆp x, ˆp y ) + ˆV (ˆx, ŷ), (3.1.2) where ˆT is the kinetic energy operator, ˆV is the potential energy operator, and ˆp x = i x and ˆp y = represent the momentum operators in the x and y directions, respectively. i y The backbone of the split-operator method is to make the approximation e iĥt/ = e i( ˆT + ˆV )t/ e i ˆT t/ e i ˆV t/. (3.1.3) If ˆT and ˆV were numbers and not operators, then Eq. (3.1.3) would be exact. However, since operators need not commute, Eq. (3.1.3) in general induces an error. Specifically, using the definition e iât/ = 1 + ( ) ( ) iât + 1 iât 2 +.... (3.1.4) 2! 19
for an arbitrary operator Â, one can show that the first order term in the error associated with Eq. (3.1.3) is given by Error = t2 2 2 [ ˆT, ˆV ]. (3.1.5) Thus, in light of Eqs. (3.1.1) and (3.1.3), we can propagate a wavefunction Ψ(x, y, t) through a timestep t using Ψ(x, y, t + t) = e i ˆT t/ e i ˆV t/ Ψ(x, y, t), (3.1.6) where an O( t 2 ) error is accrued during the computation. To solve the TDSE numerically using Eq. (3.1.6), we define spatial grids {x j } and {y j }, and choose a sufficiently small fixed timestep t. The first step in the propagation routine is to compute e i ˆV (ˆx j,ŷ j ) t/ Ψ(x j, y j, 0) Ψ 1 (x j, y j, 0). (3.1.7) The computation in Eq. (3.1.7) is straightforward and efficient because e i ˆV (ˆx j,ŷ j ) t/ and Ψ(x j, y j, 0) are both simply numbers for all j, and hence computing e i ˆV (ˆx j,ŷ j ) t/ Ψ(x j, y j, 0) involves only multiplications. Completing the propagation of the wavefunction through one timestep then involves computing Ψ(x j, y j, t) = e i ˆT t/ Ψ 1 (x j, y j, 0). (3.1.8) The computation in Eq. (3.1.8) cannot be conveniently done in position-space since ˆT contains spatial derivatives when represented in the position basis. The solution is to take the Fourier transform in space of Ψ 1 and then perform the computation in the momentum basis. We recover the propagated position-space wavefunction Ψ(x j, y j, t) by taking the inverse Fourier transform. This simplifies the computation because applying the kinetic energy portion of the split time-evolution operator to a wavefunction in momentum space again involves only multiplications. Performing the computation of Eq. (3.1.8) in momentum-space and using a tilde to denote the Fourier transform, we have Ψ(p xj, p yj, t) = e i ( ˆp 2 x j /2m x+ˆp 2 y j /2m y ) t/ Ψ1 (p xj, p yj, 0), (3.1.9) where m x and m y are the masses in the x and y degrees of freedom, respectively, and {p xj } and {p yj } are appropriate discretized momenta axes that will be discussed in more detail in the following section. Recovering the propagated wavefunction Ψ(x j, y j, t) is then merely a matter of taking the inverse Fourier transform of Ψ(p xj, p yj, t). This completes the propagation of Ψ(x, y, 0) through one timestep. We repeat this procedure iteratively to obtain Ψ(x, y, t) at any desired time t. Lastly, note that one can perform an analogous propagation routine, except with Eq. (3.1.3) replaced by e iĥt/ e i ˆV t/2 e i ˆT t/ e i ˆV t/2. (3.1.10) 20
Equation (3.1.10) is called the symmetrized product. Using the symmetrized product, the error associated with the propagation through each timestep is reduced to O( t 3 ) [8]. 3.2 Two-Dimensional Quantum Simulation General Setup We study the dynamics on the two-dimensional potential energy surface V (x, y) = 1 2 m xωxx 2 2 + 1 2 m yωyy 2 2 + 1 2 ɛ(x2 y + xy 2 ), (3.2.1) where m x and ω x represent the mass and natural frequency, respectively, in the x degree of freedom, and similarly for m y and ω y. The parameter ɛ is a positive constant that determines the degree of coupling between the x and y modes of motion. Note that the general shape of the potential in Eq. (3.2.1) is an approximately parabolic bowl, where the slight asymmetry is due to the cubic coupling terms. We perform full quantum simulations for the dynamics on Eq. (3.2.1) using the split-operator method described in Sec. 3.1. For the parameter values in Eq. (3.2.1), we choose ω x such that the natural harmonic period in x is 20 femtoseconds (fs) (10 15 seconds); that is, 2π/ω x = 20 fs. This value is motivated by the fact that 20 fs is a typical timescale for a molecular vibration. To see efficient coupling, the natural frequencies in the vibrational modes must be nearly degenerate. Note however that the vibrational frequencies for molecular systems tend not to be exactly degenerate. Thus, we set ω y = (1+δ)ω x, where δ is a dimensionless number that represents a small detuning between the natural frequencies in x in y. Typically, we use 0.01 δ 0.05 in our simulations, but in certain cases we will use δ 1 (see Secs. 3.3.2 and 4.2). For the masses, we set m x = m y = 100 amu. These masses have no specific physical motivation. Instead, they are chosen because they are found to solve an only partially understood numerical issue that is discussed in more detail at the end of this section. We choose ɛ to be sufficiently small so that the potential is only weakly anharmonic. Note that ɛ and m j ωj 2, j = x, y, have different units, and so we cannot meaningfully compare their magnitudes directly. Nevertheless, we may force the anharmonicity to be weak by choosing ɛ small enough so that ɛ(x 2 y + xy 2 ) m x ωxx 2 2, and similarly for m y ωyy 2 2, over the spatial region relevant in the simulations. To this end, we typically use 0.00005 ɛ 0.001. We perform the split-operator propagation routine in Matlab using a timestep of 0.1 fs. For the sake of computational efficiency and simplicity, we use the built-in Matlab Fourier and inverse Fourier transform functions. Note that we use the symmetrized product described in Eq. (3.1.10) in our simulations to reduce the error accrued during each timestep. 21
Our spatial grids {x j } and {y j } are each defined using 2 8 discrete points equally spaced between 5 Å and 5 Å. The two-dimensional grid is then generated using the meshgrid function in Matlab. For the time axis, the number of points varies accordingly as the run-time of the simulation changes and the timestep is held fixed at 0.1 fs. Our space and time grids are defined in angstroms and femtoseconds, but are converted to atomic units for the purposes of performing the computation. The switch to atomic units is done primarily because = 1 in atomic units. We convert back to angstroms and femtoseconds for plotting. In creating the {p xj } and {p yj } axes, we first note that there exists a maximum samplable momentum value that depends on the minimum wavelength detectable by our spatial grid. This minimum wavelength corresponds to a maximum detectable frequency, which is referred to as the Nyquist frequency [8]. The minimum wavelength we can detect in the x direction, λ x,min, is given by 2 x, where x is the stepsize of the spatial grid for x. The maximum samplable momentum in x, p x,max, is then determined by the De Broglie relation: p x,max = h λ x,min = h 2 x = π x. (3.2.2) Similarly, p y,max = π / y, where y is the stepsize of the spatial grid for y. We choose our momentum axis for x to run from p x,max to p x,max p x, where p x = 2π/ xrange is the stepsize for the axis and xrange = 10Å is the total range of the x-axis. momentum axis for y is defined similarly. The specific value for stepsize of the momenta axes is chosen so that the momentum grid is the same size (in terms of the number of points in each direction) as the spatial grid. We now briefly return to the only partially understood numerical issue that has forced us to choose m x = m y = 100. If all parameters are fixed and the masses are varied, then the energy of the wavepacket increases, but the largest samplable momentum remains the same because the stepsize of the spatial grid remains the same [see Eq. (3.2.2)]. Therefore, if the masses are increased by a sufficient amount, then the energy of the wavepacket exceeds that which can be supplied by purely kinetic energy with a momentum p max. When we near this critical mass, the propagation routine fails and the wavepacket appears stationary for all time. A more physically relevant choice for the masses would be m x = m y = 13m p, where m p = 1836 amu is the mass of one proton. For these masses, we find the aforementioned issue. In principle, one could reduce the stepsize of the spatial grid by a sufficient amount to run the simulations with larger masses. In practice, however, this makes the simulations become too slow to be practical. In the future, we would like to explore possible solutions to this issue. The 22
3.3 Two-Dimensional Quantum Simulation Initial Conditions We perform our two-dimensional quantum simulations using two different sets of initial conditions, which we now discuss in the following two brief subsections. 3.3.1 Wavepacket Initial Conditions The first initial wavefunction we use is a Gaussian centered at x = 1 Å and y = 0 with a FWHM of 1 Å. In this case, we always use δ 0. Plots of the modulus of the initial wavefunction and potential energy landscape are shown in Fig. 4. Note that the initial wavefunction is normalized and centered in a position where the potential energy is only weakly anharmonic. At each timestep of the propagation routine, we compute the first moment and variance in x and y for the wavepacket Ψ(x, y, t). The appropriate equations are shown below for x: x(t) = x Ψ(x, y, t) 2 dxdy (3.3.1) Var(x) = (x x ) 2 Ψ(x, y, t) 2 dxdy. (3.3.2) The equations for y are exactly analogous. We would also like to compute the energy in each degree of freedom. Note however that since the vibrational modes are coupled, the exact energy in each mode is not well defined. Nevertheless, since the potential is only weakly anharmonic, we may obtain a reasonable description of the energy in each mode using the harmonic approximation. That is, we compute the vibrational energies E x = 1 2 m xω 2 x x 2 + p2 x 2m x, (3.3.3) E y = 1 2 m yω 2 y y 2 + p2 y 2m y, (3.3.4) where the computation of the kinetic energies is done using the momentum-space representation of the wavefunction. All integrations are performed numerically and taken over the entire spatial or momentum grid. 3.3.2 Eigenstate Initial Conditions To connect the time-dependent and time-independent approaches to quantum mechanics discussed in Sec. 2, we next perform simulations where the initial wavefunction is chosen to be an eigenstate of the two-dimensional harmonic oscillator defined by the potential energy function V (x, y) = 1 2 m xω 2 xx 2 + 1 2 m yω 2 yy 2. (3.3.5) 23
Figure 4: Plots of the potential energy function in Eq. (3.2.1) and the modulus of our initial Gaussian wavefunction. (a) A plot of the potential energy function using the parameters m x = m y = 100 amu, δ = 0.01, and ɛ = 0.0003. (b) A plot of the modulus of our chosen initial wavefunction, which is a Gaussian centered at (1, 0) with a FWHM of 1 Å. Specifically, we choose the initial wavefunction to be the normalized eigenstate with quantum numbers n x = 0 and n y = 1. The analytical form for this eigenstate is given by ψ 01 = ( mx ω x m y ω ) 1 y 4 2 π 2 2 2 ( my ω y ) 1 2 ye 1 2( mxωx x 2 + myωy y 2 ). (3.3.6) The analytical expression for any harmonic oscillator eigenstate ψ nx,ny is the product of a Gaussian and Hermite polynomials and can be found in [16]. For the anharmonic potential, we use either Eq. (3.2.1) or Eq. (3.2.1) with only one of the two cubic coupling terms included. For δ, in addition to performing simulations with δ 0, we study the case where δ 1. The reason for performing simulations with only one cubic term as well as with δ 1 will be made clear in Sec. 4.2 At each timestep of all simulations, we calculate the modulus squared of the projection of Ψ(x, y, t) onto each eigenstate ψ nx,n y of the harmonic oscillator with n x = 0, 1, or 2 and n y = 0, 1, or 2. These projections are computed using Eq. (2.1.24): 2 c nx,ny = Ψ (x, y, t)ψ nx,ny dxdy. (3.3.7) The integration is performed numerically and taken over the entire spatial grid. Typically, c nx,ny refers to the projection of Ψ onto ψ nx,ny. Note here that for simplicity we use c nx,ny to refer instead to the modulus squared of the projection. 24
4 Two-Dimensional Quantum Simulation Results and Discussion In this section, we present and analyze the results from the two-dimensional quantum simulations described in Sec. 3. 4.1 Wavepacket Initial Condition Results We first discuss the simulation results for the case of an initial Gaussian wavepacket displaced by 1 Å in the x degree of freedom. Typical results for the first moments x and y obtained using Eq. (3.3.1) are shown in Fig. 5. Note that both first moments exhibit fast oscillations of 20 fs that are unresolved in the figure. Observe from Fig. 5 that the motion of the wavepacket, initially displaced in the x degree of freedom, slowly beats into y over roughly the first 1000 fs. This result is indicative of the inescapable coupling between vibrational modes in an anharmonic potential energy surface. In general, energy localized in a single vibrational mode will flow into the other degrees of freedom. Beats occur approximately every 1850 fs. Figure 5: Plot of the first moments x and y for typical simulation results. The parameters used were ɛ = 0.00022 and δ = 0.01. Note that the fast timescale 20 fs oscillation in both curves is unresolved in the figure. 25
While the motion of the wavepacket continually beats back and forth between x and y, notice from Fig. 5 that the amplitude of oscillation in the first moments decays on a long timescale. One might be tempted to attribute this to a loss of energy in the system; however, this is not the case. In fact, while the amplitude of oscillation in the first moments decays in time, the energy in each vibrational mode does not. In Fig. 6a we show the energy in both vibrational modes obtained using the harmonic approximation in Eqs. (3.3.3) and (3.3.4). Observe that the average energy in each mode is roughly constant, and that the energy beats with an 1850 fs period, i.e., precisely the beat period of the first moment oscillations. There is again a 20 fs fast oscillation in both vibrational mode energies that is unresolved in Fig. 6a. This fast oscillation corresponds to breathing in the width of the wavepacket that occurs with each trip of the wavepacket across the potential. The energy flow between the two vibrational modes is such that the total energy of the system (expectation value of the Hamiltonian) is constant in time. We show this explicitly in Fig. 6b. Figure 6: Energy in the x and y vibrational modes (left) and total system energy (right) associated with the motion in Fig. 5. (a) Energy in the x and y vibrational modes associated with the motion in Fig. 5 obtained using the harmonic approximation. Note that the 20 fs oscillation corresponding to the breathing of the wavepacket is unresolved in the figure. (b) A plot of the total energy, H, as a function of time. We can attribute the decay in the first moment oscillations to the spreading and dephasing of the wavepacket with time. Conceptually, the first moments sample the 26
peak of the wavepacket; however, the peak loses meaning after the wavepacket has dephased and dispersed significantly. Dephasing causes multiple peaks to appear in the wavepacket and dispersion causes overall spreading. The net result is that the average position decays. In fact, the first moments will tend to the center of the potential well (in our case, x = 0 and y = 0), as the wavepacket spreads to the cover the well s entirety. We can see the qualitative relationship between the width of the wavepacket and the amplitudes of oscillation of the first moments by examining Fig. 7, which shows a running average of Var(x) as a function of time. We take a running average to dampen fast oscillations in the variance caused by breathing in the width of the wavepacket. Observe that the growth of the variance in x with time mirrors the decaying amplitude of the first moment oscillations. Figure 7: A plot of the running average of Var(x) as a function of time obtained using Eq. (3.3.2). The averaging is performed to dampen natural oscillations in the variation resulting from wavepacket breathing. The slowly growing variance is a manifestation of general wavepacket dispersion on anharmonic potentials that we discussed in Sec. 2.1.4. Although the wavepacket spreads in time, recall that we mentioned in Sec. 2.1.4 that we should expect long (but finite) revival times for a wavepacket composed of a finite number of eigenstates. It is worth noting that carrying the simulation results out to longer time yields an approximate revival at roughly 30 ps (30000 fs). The variance exhibits an associated decrease to nearly 1.5 Å, which clearly indicates an approximate rephasing of the wavepacket. 27
Overall, we see that wave packet dynamics on a two-dimensional anharmonic potential resembles qualitatively what one might from expect from a marble rolling in a slightly asymmetric bowl, except with the added complexity of dispersion and long-time revivals. Indeed, in Fig. 8 we show typical results of classical simulations of Eq. (1.0.2) with energy initially localized in the x degree of freedom. Observe in Fig. 8 that the initially localized energy flows into y, and back again, creating an overall beating motion. While we also observed beating in the quantum case, we see the qualitative difference that the beat amplitude does not decay as in Fig. 5, but instead is perfectly periodic. Nevertheless, the short-time dynamics (when Var(x) is relatively small) of Fig. 5 strongly resemble the classical behavior. This qualitative result demonstrates how classical simulations tend to accurately describe quantum dynamics on timescales short enough that the initial wavefunction is still relatively compact. Figure 8: Plot of typical results for the positions as functions of time for a classical simulation of Eq. (1.0.2) with energy initially localized in the x degree of freedom. The parameters used were k x = 1, k y = 1.01, and ɛ = 0.1. 4.2 Eigenstate Initial Condition Results We now turn to the analysis and discussion of our results for the eigenstate simulations described in Sec. 3.3.2. First, note that if the initial wavefunction Ψ(x, y, 0) = ψ 01 in our 28