Baer M., Billing G.D. (eds.) - Advances in Chemical Physics. The Role of Degenerate States in Chemistry, Vol. 124 (2002)(en)
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Appendix C: Propagating the Electronic Wave Function
Acknowledgment
References
I.INTRODUCTION
Knowledge of the underlying nuclear dynamics is essential for the classification and description of photochemical processes. For the study of complicated systems, molecular dynamics (MD) simulations are an essential tool, providing information on the channels open for decay or relaxation, the relative populations of these channels, and the timescales of system evolution. Simulations are particularly important in cases where the Born–Oppenheimer (BO) approximation breaks down, and a system is able to evolve non-adiabatically, that is, in more than one electronic state.
In this chapter, we look at the techniques known as direct, or on-the-fly, molecular dynamics and their application to non-adiabatic processes in photochemistry. In contrast to standard techniques that require a predefined potential energy surface (PES) over which the nuclei move, the PES is provided here by explicit evaluation of the electronic wave function for the states of interest. This makes the method very general and powerful, particularly for the study of polyatomic systems where the calculation of a multidimensional potential function is an impossible task. For a recent review of standard nonadiabatic dynamics methods using analytical PES functions see [1].
Direct dynamics methods are, however, still in their infancy, and have a number of difficulties that need to be solved. One is the sheer size of the problem—all nuclear and electronic degrees of freedom are treated explicitely. A second is the restriction placed on the form of the nuclear wave function as a local, trajectory-based, representation is required. This introduces the problem of including quantum effects into methods that are often based on classical mechanics. For non-adiabatic processes, there is the additional complication of the treatment of the non-adiabatic coupling. In this chapter, we will show how progress has been made in this new and exciting field, highlighting the different problems and how they are being solved. Complimentary reviews on applying direct dynamics to adiabatic problems are given in [2,3].
Interaction with light changes the quantum state a molecule is in, and in photochemistry this is an electronic excitation. As a result, the system will no longer be in an eigenstate of the Hamiltonian and this nonstationary state evolves, governed by the time-dependent Schro¨dinger equation
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ih qt |
ðR; r; tÞ ¼ HðR; rÞ ðR; r; tÞ |
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applying direct molecular dynamics to non-adiabatic systems 357
Central to the description of this dynamics is the BO approximation. This separates the nuclear and electronic motion, and allows the system evolution to be described by a function of the nuclei, known as a wavepacket, moving over a PES provided by the (adiabatic) motion of the electrons.
Coupling between the electronic and nuclear motion can, however, result in the breakdown of the BO approximation, which leads to an effective coupling between the adiabatic states of the system, providing pathways for fast, radiationless, electronic transitions. The wavepacket in non-adiabatic systems, as these are known, must therefore be described as evolving over a manifold of coupled PES. Non-adiabatic coupling is particularly important in regions where the PES are degenerate, or near-degenerate, and it can lead to an interesting topology of the surfaces. Typical features are avoided crossings, where the surfaces seem to repel one another, or conical intersections, where the surfaces meet at a point or seam. While avoided crossings are well established in chemical ideas through the noncrossing rule, it is only in recent years that the importance of conical intersections in photochemistry has emerged [4–8]. The idea of conical intersections has a long history [9–14]. Their general acceptance was delayed by the difficulties in conclusively demonstrating their existence in large molecules, due to the problems in calculating wave functions for excited states.
If the PES are known, the time-dependent Schro¨dinger equation, Eq. (1), can in principle be solved directly using what are termed wavepacket dynamics [15–18]. Here, a time-independent basis set expansion is used to represent the wavepacket and the Hamiltonian. The evolution is then carried by the expansion coefficients. While providing a complete description of the system dynamics, these methods are restricted to the study of typically 3–6 degrees of freedom. Even the highly efficient multiconfiguration time-dependent Hartree (MCTDH) method [19,20], which uses a time-dependent basis set expansion, can handle no more than 30 degrees of freedom.
For larger systems, various approximate schemes have been developed, called mixed methods as they treat parts of the system using different levels of theory. Of interest to us here are quantum-semiclassical methods, which use full quantum mechanics to treat the electrons, but use approximations based on trajectories in a classical phase space to describe the nuclear motion. The prefix quantum may be dropped, and we will talk of semiclassical methods. There are a number of different approaches, but here we shall concentrate on the few that are suitable for direct dynamics molecular simulations. An overview of other methods is given in the introduction of [21].
As mentioned above, the correct description of the nuclei in a molecular system is a delocalized quantum wavepacket that evolves according to the Schro¨dinger equation. In the classical limit of the single surface (adiabatic) case, when effectively h ! 0, the evolution of the wavepacket density
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(amplitude squared) can be simulated by a ‘‘swarm’’ of trajectories, each driven by classical (e.g., Newtonian) mechanics. Note that this does not mean that the nuclei are being treated as classical particles, each is being represented by a set of classical pseudoparticles that together simulate the behavior of the nucleus. Methods based on this approximation are sometimes termed quasiclassical.
A different approach is to represent the wavepacket by one or more Gaussian functions. When using a local harmonic approximation to the true PES, that is, expanding the PES to second-order around the center of the function, the parameters for the Gaussians are found to evolve using classical equations of motion [22–26]. Detailed reviews of Gaussian wavepacket methods are found in [27–29].
To add non-adiabatic effects to semiclassical methods, it is necessary to allow the trajectories to sample the different surfaces in a way that simulates the population transfer between electronic states. This sampling is most commonly done by using surface hopping techniques or Ehrenfest dynamics. Recent reviews of these methods are found in [30–32]. Gaussian wavepacket methods have also been extended to include non-adiabatic effects [33,34]. Of particular interest here is the spawning method of Mart´ınez, Ben-Nun, and Levine [35,36], which has been used already in a number of direct dynamics studies.
In traditional dynamics calculations, the first step is to find a representation of the PES. For accurate calculations, this involves fitting a function to ab initio data, maybe with final adjustments using experimental data. A major hurdle to the calculation of information about the excited state PES of molecules, required for a description of photochemistry, is the development of appropriate quantum chemical methods. Probably the most general method is the complete active space self-consistent field (CASSCF) method [37]. This is a multiconfiguration self-consistent field (MCSCF) method that uses a full configuration interaction (CI) within an active space of the important molecular orbitals. As it is an MCSCF method, both the orbitals and the CI coefficients are optimised. Unlike other, maybe more powerful methods, calculation of analytic gradients is relatively straightforward using CASSCF, which makes it suitable for direct dynamics. Although care is needed in its application, accurate results are possible, particularly when combined with perturbation theory to correct for the missing so-called dynamic electron correlation [38–41].
Techniques have been developed within the CASSCF method to characterize the critical points on the excited-state PES. Analytic first and second derivatives mean that minima and saddle points can be located using traditional energy optimization procedures. More importantly, intersections can also be located using constrained minimization [42,43]. Of particular interest for the mechanism of a reaction is the minimum energy path (MEP), defined as the line followed by a classical particle with zero kinetic energy [44–46]. Such paths can be calculated using intrinsic reaction coordinate (IRC) techniques
applying direct molecular dynamics to non-adiabatic systems 359
[47,48]. For systems in which conical intersections play a role, however, the concept of an IRC must be extended. Due to the topology of the region more than one path may be accessible after crossing from the upper electronic state to the lower one (i.e., the wavepacket may bifurcate). For this situation, the initial relaxation direction (IRD) method has been developed [49,50] to identify the open channels on the ground-state PES moving away from the minimum energy intersection point. The MEP can then be used to explore each channel to the products. For more information on the study of PES critical points using quantum chemistry techniques, see the recent reviews [51,52].
An alternative method that can be used to characterize the topology of PES is the line integral technique developed by Baer [53,54], which uses properties of the non-adiabatic coupling between states to identify and locate different types of intersections. The method has been applied to study the complex PES topologies in a number of small molecules such as H3 [55,56] and C2H [57].
Information about critical points on the PES is useful in building up a picture of what is important in a particular reaction. In some cases, usually thermally activated processes, it may even be enough to describe the mechanism behind a reaction. However, for many real systems dynamical effects will be important, and the MEP may be misleading. This is particularly true in non-adiabatic systems, where quantum mechanical effects play a large role. For example, the spread of energies in an excited wavepacket may mean that the system finds an intersection away from the minimum energy point, and crosses there. It is for this reason that molecular dynamics is also required for a full characterization of the system of interest.
Calculating points on a set of PES, and fitting analytic functions to them is a time-consuming process, and must be done for each new system of interest. It is also an impossible task if more than a few (typically 4) degrees of freedom are involved, simply as a consequence of the exponential growth in number of ab initio data points needed to cover the coordinate space.
For this reason, there has been much work on empirical potentials suitable for use on a wide range of systems. These take a sensible functional form with parameters fitted to reproduce available data. Many different potentials, known as molecular mechanics (MM) potentials, have been developed for ground-state organic and biochemical systems [58–60]. They have the advantages of simplicity, and are transferable between systems, but do suffer from inaccuracies and rigidity—no reactions are possible. Schemes have been developed to correct for these deficiencies. The empirical valence bond (EVB) method of Warshel [61,62], and the molecular mechanics–valence bond (MMVB) of Bernardi et al. [63,64] try to extend MM to include excited-state effects and reactions. The MMVB Hamiltonian is parameterized against CASSCF calculations, and is thus particularly suited to photochemistry.
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A further model Hamiltonian that is tailored for the treatment of nonadiabatic systems is the vibronic coupling (VC) model of Ko¨ppel et al. [65]. This provides an analytic expression for PES coupled by non-adiabatic effects, which can be fitted to ab initio calculations using only a few data points. As a result, it is a useful tool in the description of photochemical systems. It is also very useful in the development of dynamics methods, as it provides realistic global surfaces that can be used both for exact quantum wavepacket dynamics and more approximate methods.
Direct dynamics attempts to break this bottleneck in the study of MD, retaining the accuracy of the full electronic PES without the need for an analytic fit of data. The first studies in this field used semiclassical methods with semiempirical [66,67] or simple Hartree–Fock [68] wave functions to treat the electrons. These first studies used what is called BO dynamics, evaluating the PES at each step from the electronic wave function obtained by solution of the electronic structure problem. An alternative, the Ehrenfest dynamics method, is to propagate the electronic wave function at the same time as the nuclei. Although early direct dynamics studies using this method [69–71] restricted themselves to adiabatic problems, the method can incorporate nonadiabatic effects directly in the electronic wave function.
Major impetus in the field was given by the introduction of the Car– Parrinello method [72–74]. Related to the Ehrenfest dynamics method, this is a very efficient algorithm that propagates the electronic wave function using a fictitious mass to produce classical equations of motion for the expansion coefficients. For full efficiency, however, it requires a plane-wave basis set, which is inefficient for the description of isolated molecules. Recent work using Gaussian functions points the way to the solution of this problem [75]. The method is usually restricted to adiabatic dynamics, although the method has been applied to excited states using a very simple wave function [76]. We shall ignore Car–Parrinello methods in the following.
An important step forward in the study of molecular systems was afforded by the introduction of an efficient propagation algorithm by Helgaker et al. [77] and further improved by Chen et al. [78]. With the large step-size made possible by this method it became feasible to simply reevaluate the electronic wave function at each step, thus opening up all the power of electronic structure calculations for direct BO dynamics. By combining the Helgaker–Chen algorithm with a surface hopping method, a number of dynamics studies of photochemical systems have been made using the MMVB empirical Hamiltonian [79–85]. These studies have allowed us to gain much experience in the behavior of trajectories over coupled PES. The method has then been applied to direct dynamics study using CASSCF wave functions [86,87].
The Gaussian wavepacket based spawning method, mentioned above, has also been used in direct dynamics where it is called ab initio multiple spawning
applying direct molecular dynamics to non-adiabatic systems 361
(AIMS) [88]. The inclusion of quantum effects directly in the nuclear motion may be a significant step, as the motion near a conical intersection is known to be very quantum mechanical.
The present state of the art is not able to use direct dynamics to calculate accurate dynamical properties: For this many trajectories are required, and it is simply too expensive. Even so, as we shall show, mechanistic information can be gained directly from the calculations, extending the minimum energy path picture to include a dynamical term, which is certainly important in the study of excited molecules. A further use, still to be explored fully, is to use the information from direct dynamics trajectories to efficiently generate the PES for more accurate calculations. The ground work for this has been laid by the work of Collins and co-workers [89–93], who developed a scheme to generate a PE function by interpolating information on the surface (the energy, and its first and second derivatives) at a set of points. These points could be generated by direct dynamics, thus sampling only the areas of configuration space important for the system dynamics. The accuracy of the method has been shown recently in state-of-the-art four-dimensional (4D) quantum scattering calculations [94].
By its nature, the application of direct dynamics requires a detailed knowledge of both molecular dynamics and quantum chemistry. This chapter is aimed more at the quantum chemist who would like to use dynamical methods to expand the tools at their disposal for the study of photochemistry, rather than at the dynamicist who would like to learn some quantum chemistry. It tries therefore to introduce the concepts and problems of dynamics simulations, stressing that one cannot strictly think of a molecule moving along a trajectory even though this is what is being calculated.
To demonstrate the basic ideas of molecular dynamics calculations, we shall first examine its application to adiabatic systems. The theory of vibronic coupling and non-adiabatic effects will then be discussed to define the sorts of processes in which we are interested. The complications added to dynamics calculations by these effects will then be considered. Some details of the mathematical formalism are included in appendices. Finally, examples will be given of direct dynamics studies that show how well the systems of interest can at present be treated.
Throughout, unless otherwise stated, R and r will be used to represent the nuclear and electronic coordinates, respectively. Boldface is used for vectors and matrices, thus R is the vector of nuclear coordinates with components Ra. The vector operator $, with components
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where Ma is the mass associated with the coordinate, then the kinetic energy operator can be written
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^n is the nuclear kinetic energy operator, and so all terms describing the Here, T
electronic kinetic energy, electron–electron and electron–nuclear interactions, as well as the nuclear–nuclear interaction potential function, are collected together. This sum of terms is often called the clamped nuclei Hamiltonian as it describes the electrons moving around the nuclei at a particular configuration R.
II. ADIABATIC MOLECULAR DYNAMICS
In this section, the basic theory of molecular dynamics is presented. Starting from the BO approximation to the nuclear Schro¨dinger equation, the picture of nuclear dynamics is that of an evolving wavepacket. As this picture may be unusual to readers used to thinking about nuclei as classical particles, a few prototypical examples are shown.
In the full quantum mechanical picture, the evolving wavepackets are delocalized functions, representing the probability of finding the nuclei at a particular point in space. This representation is unsuitable for direct dynamics as it is necessary to know the potential surface over a region of space at each point in time. Fortunately, there are approximate formulations based on trajectories in phase space, which will be discussed below. These local representations, socalled as only a portion of the PES is examined at each point in time, have a classical flavor. The delocalized and nonlocal nature of the full solution of the Schro¨dinger equation should, however, be kept in mind.