CHAPTER 6 Semiclassical Bohmian Dynamics Sophya Garashchuk,a Vitaly Rassolov,a and Oleg Prezhdob a Department of Chemistry and Biochemistry, University of South Carolina, Columbia, South Carolina bDepartment of Chemistry, University of Rochester, Rochester, New York INTRODUCTION Quantum mechanics lies at the heart of chemistry. It is impossible to under- stand the structure of the Periodic Table, chemical bonding patterns, free ener- gies of chemical reactions, reaction rates and branching ratios, and other chemi- cal phenomena without a quantum-mechanical (QM) description. In particular, the dynamics of molecular systems often involve QM effects such as zero-point energy, tunneling, and nonadiabatic transitions. QM effects are essential for accurate description and understanding of reactions in complex chemical envi- ronments. For example, the zero-point energy stored in the vibrational modes of chemical reactants, products, and transition state species modifies reaction energy barriers. Reaction rates and branching ratios can be affected greatly by such changes. QM tunneling can be critical in proton transfer reactions. However, the conventional methods of solving the time-dependent Schrodinger¨ equation1 scale exponentially with the system size. In addition, the relevant dy- namics occur on a long time-scale.2 Therefore, it is extremely difficult to model QM tunneling in condensed phase chemical systems.3–6 Nonadiabatic dynam- ics involving transitions between different electronic or vibrational energy levels Reviews in Computational Chemistry, Volume 27 edited by Kenny B. Lipkowitz Copyright © 2011 John Wiley & Sons, Inc. 287 288 Semiclassical Bohmian Dynamics is ubiquitous in photochemistry.7–11 In this process, a chemical system is excited by light, undergoes a nonequilibrium evolution, and ultimately relaxes to the ground state. QM tunneling can be viewed as a particular kind of nonadiabatic transition.12 Several multidimensional quantum approaches have been proposed in- cluding those using basis set contractions,13–15 coherent state representations, 16,17 and mixed quantum-classical strategies.18–24 A trajectory representation of large molecular systems carries a special appeal because of several favor- able factors: (1) the initial conditions of a trajectory simulation can be sampled with Monte Carlo techniques. This allows the exponential scaling of the exact wave function or density matrix with system size to be circumvented. (2) Var- ious degrees of freedom involving light and heavy particles can be treated on equal footing, and quantum-classical separation issues can be avoided.25 (3) Wave functions are highly oscillatory close to the classical → 0 limit. As a result, a trajectory description of heavy particles, such as nuclei, is often more appropriate than a grid or a basis set representation. (4) Classical equations of motion are simple to solve. Numerous molecular dynamics techniques26 for propagating classical trajectories are applied routinely to chemical systems composing hundreds of thousands of atoms. Incorporating the dominant QM effects caused by wave function localization27,28 constitutes a challenge for the trajectory methods. Such effects can be considered naturally by representing wave functions in terms of ensembles of trajectories. In comparison, semiclas- sical methods commonly use independent trajectories.29–31 A great interest in the Bohmian interpretation of quantum dynamics has been witnessed during the last decade. In particular, its potential to generate computational tools for solving the time-dependent Schrodinger¨ equation has attracted considerable attention. The Bohmian formulation of quantum dynam- ics promises a better than exponential scaling of the computational effort with system dimensionality. It also offers a convenient approach for mixed quantum- classical descriptions of large chemical systems. The Madelung-de Broglie– Bohm formulation of the time-dependent Schrodinger¨ equation has a long his- tory dating back to the birth of quantum mechanics.32,33 It gained a wider recognition after David Bohm34 used it to develop an alternative interpretation of quantum mechanics. This review describes the semiclassical methodologies inspired by the Bohmian formulation of quantum mechanics. These methods are designed to represent the complex dynamics of chemical systems. The review is constructed as follows: the next section introduces the Madelung-de Broglie–Bohm formalism. This is done by drawing an analogy with classical mechanics and explicitly highlighting the non-classical features of the Bohmian mechanics. The nonclassical contributions to the momentum, energy, and force are introduced. The fundamental properties of Bohmian quan- tum mechanics—the conservation and normalization of the QM probability, the computation of the QM expectation values, properties of stationary states, and behavior at nodes—are discussed. Several ways to obtain the classical limit The Formalism and Its Features 289 within the Bohmian formalism are considered. Mixed quantum-classical dy- namics based on the Bohmian formalism is derived and illustrated with an example involving a light and a heavy particle. At this point, the Bohmian rep- resentation is used as a tool to couple the quantum and classical subsystems. The quantum subsystem can be evolved by either Bohmian or traditional techniques. The Ehrenfest approach is the most straightforward and common quantum- classical approach, and it is the starting point for other quantum-classical for- mulations. The Bohmian formulation of the Ehrenfest approach is used to derive an alternative quantum-classical coupling scheme. This resolves the so-called quantum backreaction problem, also known as the trajectory branching prob- lem. Next, the partial hydrodynamic moment approach to coupling classical and quantum systems is outlined. The hydrodynamic moments provide a con- nection between the Bohmian and phase-space descriptions of quantum me- chanics. The “Independent Trajectory Methods” Section describes approaches based on independent Bohmian trajectories. It includes a discussion of the derivative propagation method, the Bohmian trajectory stability approach, and Bohmian trajectories with complex action. Truncation of these hierarchies at the second order reveals a connection to other semiclassical methods. The focus then shifts toward Bohmian dynamics with globally approximated quantum potentials. Separate subsections are devoted to the global energy-conserving approximation for the nonclassical momentum, approximations on subspaces and nonadiabatic dynamics. Each approach is introduced at the formal theo- retical level and then is illustrated by an example. The Section “Towards Reac- tive Dynamics in Condensed Phase” deals with computational issues including numerical stability, error cancellation, dynamics linearization, and long-time behavior. The numerical problems are motivated and illustrated by consid- ering specific quantum phenomena such as zero-point energy and tunneling. The review concludes with a summary of the semiclassical and quantum- classical approaches inspired by the Bohmian formulation of quantum mechan- ics. The three appendices prove the quantum density conservation, introduce quantum trajectories in arbitrary coordinates, and explain optimization of sim- ulation parameters in many dimensions. THE FORMALISM AND ITS FEATURES The Trajectory Formulation For simplicity, let us start with a derivation of the Bohmian equations in one spatial dimension x for a particle of mass m. For notation clarity, let us use ∇ to denote differentiation with respect to x. Arguments of functions will be omitted where unambiguous. Differentiation (or Derivatives) with respect to a variable other than x will be indicated as a subscript; for example, ∇c denotes differentiation with respect to c. The multidimensional generalization to an 290 Semiclassical Bohmian Dynamics arbitrary coordinate system is given in Appendix B. The conventional form of the time-dependent Schrodinger¨ equation is 2 ∂ (x, t) − ∇2 + V (x, t) = ı [1] 2m ∂t After Madelung, the complex time-dependent wave function is represented in polar form as = ı (x, t) A(x, t) exp S(x, t) [2] where A(x, t) and S(x, t) are real functions. Substitution of Eq. [2] into Eq. [1], division by (x, t), separation into real and imaginary parts, and a few simple manipulations results in a system of two equations ∂S(x, t) 1 =− (∇S(x, t))2 − V − Q [3] ∂t 2m ∂(x, t) 1 =−∇ ∇S(x, t)(x, t) [4] ∂t m In Eq. [3], the term Q denotes what Bohm called the “quantum mechanical potential,” 2 ∇2A(x, t) Q =− [5] 2m A(x, t) The quantum potential Q enters the equation on par with the external “classi- cal” potential V = V(x, t), which is generally a function of x and t as well as Q = Q(x, t). In Eq. [4] (x, t) is the wave function density (x, t) = A2(x, t) [6] With identification of the probability flux as 1 ∗ j(x, t) = (x, t) ∇S(x, t) = (x, t)∇ (x, t) [7] m m Equation [4] becomes the usual continuity equation. Analogy with fluid me- chanics suggests the name “hydrodynamic” formulation of the Schrodinger¨ equation. Note that Eqs. [3] and [4] are formally equivalent to the original Schrodinger¨ Eq. [1] except that the polar form Eq. [2] is problematic at the nodes of the wave function. At the nodes, the phase S is undefined, A(x, t) = 0, The Formalism and Its Features 291 and Q is generally singular. The singularity in Q cancels for excited eigenstates as will be explained. Equations [3] and [4] describe the flow of the