List of Papers. This thesis is based on the following papers, which are referred to in the text by their Roman numerals.

Transcription

1

2

3 List of Papers This thesis is based on the following papers, which are referred to in the text by their Roman numerals. I II III A. Nissen and G. Kreiss. An optimized perfectly matched layer for the Schrödinger equation, Commun. Comput. Phys. 9: , The ideas were developed in close collaboration between the authors. The author of this thesis performed part of the analysis and all the computations. The manuscript was written in close cooperation between the authors. K. Kormann and A. Nissen. Error control for simulations of a dissociative quantum system, In Numerical mathematics and advanced applications: 2009, pp , Springer-Verlag, Berlin, The author of this thesis was mainly responsible for the parts regarding the PML. Both authors performed the computations, wrote the manuscript and drew conclusions in close collaboration. A. Nissen, H. O. Karlsson and G. Kreiss. A perfectly matched layer applied to a reactive scattering problem, J. Chem. Phys. 133:054306, The author of this thesis implemented the methods, performed all the computations and had the main responsibility for preparing the manuscript. The ideas were developed in collaboration between the authors. 1 With permission from Global Science Press. 2 With kind permission from Springer Science and Business Media. 3 With permission from American Institute of Physics.

4 IV V VI A. Nissen, G. Kreiss and M. Gerritsen. High order stable finite difference methods for the Schrödinger equation, Technical report , Department of Information Technology, Uppsala University, (Submitted) A. Nissen, G. Kreiss and M. Gerritsen. Stability at nonconforming grid interfaces for a high order discretization of the Schrödinger equation, Technical report , Department of Information Technology, Uppsala University, (Submitted) The ideas were developed in close collaboration between the authors. The author of this thesis had the main responsibility for the theoretical development of the stability analysis, performed most of the detailed analysis and all the computations. The manuscript was written in close cooperation between the authors. M. Gustafsson, A. Nissen and K. Kormann. Stable difference methods for block-structured adaptive grids, Technical report , Department of Information Technology, Uppsala University, The ideas were developed in close collaboration between the authors in this paper. The author of this thesis contributed mainly in the development of the numerical method and wrote a part of the manuscript. The computations were performed in close collaboration between the authors. Reprints were made with permission from the publishers.

5 Contents 1 Introduction The time-dependent Schrödinger equation The Born-Oppenheimer approximation Bound states and non-bound states Numerical methods and computational challenges Boundary treatment for open systems Absorbing layers in quantum dynamics Absorbing layers and absorbing boundary conditions in the numerical analysis community Perfectly matched layers Spatial adaptivity Adaptive mesh refinement Summation-by-parts operators Summary of papers Paper I Paper II Paper III Paper IV Paper V Paper VI Discussion and outlook Summary in Swedish Acknowledgements Bibliography

6

7 1. Introduction The investigation of the dynamics of chemical reactions, both from the theoretical and experimental side, has drawn an increasing interest since Ahmed H. Zewail was awarded the 1999 Nobel Prize in Chemistry for his work on femtochemistry. On the experimental side, new techniques such as femtosecond lasers and attosecond lasers enable laser control of chemical reactions. Numerical simulations serve as a valuable complement to experimental techniques, not only for validation of experimental results, but also for simulation of processes that cannot be investigated through experiments. With increasing computer capacity, more and more physical phenomena fall within the range of what is possible to simulate. Also, the development of new, efficient numerical methods further increases the possibilities. The focus of this thesis is twofold; numerical methods for open quantum mechanical systems and methods that can handle problems with large variations in spatial scales. Firstly, we consider numerical methods for open quantum mechanical systems, in particular chemical reactions involving dissociative states. An open quantum mechanical system is a system in interaction with its surroundings. It could be a nano-electric device, into which electrons enter and leave, where the positions of the electrons can only be described as a time-dependent probability distribution. Dissociative chemical reactions are reactions where molecules break up into smaller components. The dissociation can occur spontaneously, e.g. by radioactive decay, or be induced by adding energy to the system, e.g. in terms of a laser field. Quantities of interest can for instance be the reaction probabilities of possible chemical reactions. Secondly, we consider methods that can deal with coexistence of spatial regions with very different physical properties. One application of interest is long-range molecules, where the atoms are affected by chemical attractive forces that lead to fast movement in the region where they are close to each other and exhibits a relatively slow motion of the atoms in the long-range region. The ability of the method to adapt to different scales is important in the study of more complex chemical systems. The key equation in the mathematical modeling of quantum mechanics is the Schrödinger equation, which is defined on an infinite spatial domain. Most numerical models involve the introduction of a finite computational domain and boundary conditions. In the context of dissociative problems we focus specifically on the boundary treatment. In order to avoid the introduction of additional errors that are due to an ad hoc imposition of boundary conditions, 7

8 we use an analytic approach to impose the boundary conditions. Papers I, II and III in this thesis are based on the perfectly matched layer (PML) technique, which stands on a well-founded mathematical ground, while it is also very similar to methods employed in the field of chemical physics. We investigate the limitations and quantify the errors introduced by the PML. In order to deal with different spatial scales, the computational domain can be decomposed into smaller subdomains where the computational work is distributed according to the work needed to resolve the physical features in each subdomain. Our approach for the domain decomposition is based on the summation-by-parts-simultaneous approximation term (SBP-SAT) framework, a finite difference methodology for which robust and accurate approximations often can be achieved. In papers IV and V we consider the numerical treatment of the boundaries surrounding the computational domain and of the artificial boundaries between subdomains. The accuracy and robustness of the numerical method is investigated. In paper VI the SBP-SAT methodology is extended to multiblock-domains. The outline of the thesis is as follows. The underlying equations and problem formulation are described in section 2. Numerical techniques and challenges are reviewed in section 3. In section 4, we describe different ways of imposing absorbing boundary conditions for the Schrödinger equation. Section 5 deals with adaptive discretizations in space and the numerical treatment across interior grid boundaries. A summary of the papers included in the thesis is given in section 6, and section 7 concludes with a discussion and an outline of possible future work. 8

9 2. The time-dependent Schrödinger equation Fundamental processes at the atomic level cannot be modelled by Newtonian mechanics, but require a quantum mechanical description. Theoretical understanding of such processes, like the dynamics of chemical reactions, can be obtained by solving the time-dependent Schrödinger equation (TDSE), i h ψ(x,t) t = Hψ(x,t), (2.1) where the Hamiltonian of the system, H = 2m h2 +V, consists of a kinetic and a potential energy operator. h is the reduced Planck s constant and m is the mass of the system. The TDSE (2.1) was introduced in 1926 in a series of papers by Erwin Schrödinger [72, 74, 71, 73] and gives a comprehensive description of a physical system through the space- and time-dependent wavefunction ψ(x, t). Although the physical interpretation of ψ(x, t) is not trivial, all obtainable information is accessible through it. However, the information is limited due to the uncertainty principle. Instead of considering the complexvalued ψ(x,t) directly, the probability density ψ(x,t) 2 gives the likelihood of finding the system in a particular state for the given x at time t. In a oneparticle system, x denotes the position of the particle in space. For a larger system, e.g. a polyatomic molecule, x describes the spatial relation between the particles, usually in terms of distances and angles between the particles. Analytic solutions are in most cases not obtainable for the TDSE, and therefore we need to turn to numerical simulations in order to obtain approximate solutions. However, to directly solve the TDSE numerically is unfeasible for nearly all real systems. A seemingly small carbon dioxide molecule (CO 2 ), consists of 25 particles (3 nuclei and 22 electrons) and gives rise to a problem with 75 degrees of freedom. In addition, the time-scales for the relatively light electrons and the heavy nuclei differ with several orders of magnitude. Due to the high requirements of computational resources in order to solve the TDSE for both nuclei and electrons, the interaction induced by the electrons is in practical computations reduced to potential energy surfaces through the Born-Oppenheimer approximation, which will be described in the following section. 9

10 2.1 The Born-Oppenheimer approximation The Born-Oppenheimer approximation was derived by Born and Oppenheimer [14] in 1927, and is still a crucial tool in quantum chemistry and chemical physics. By separating the nuclear and the electronic parts of the wavefunction and solving the Schrödinger equation for each part separately, the dimensionality of the problem can be reduced significantly. The Born-Oppenheimer approximation is derived starting from the time-independent Schrödinger equation H(x, X)Ψ(x, X) = EΨ(x, X), (2.2) where the full molecular Hamiltonian can be written as H(x,X) = h2 2 e,i i 2m + j>i j>i e 2 x i x j + i h2 2 N,i 2M i + Z i Z j e 2 X i X j Z j e 2 i j x i X j T e +V e + T N +V N +V en, (cf. [81]). x and X refer to the electron and nuclear coordinates, respectively, e to the elementary charge and Z i to the charge on nucleus i. m is the electron mass, M i the mass of nucleus i, and e and N refer to the electron and nuclear momenta, respectively. By assuming that the full wavefunction Ψ(x, X) can be separated into an electronic part at a fixed nuclear position X, ϕ(x;x), and a nuclear part, χ(x), we get Ψ(x, X) = ϕ(x; X)χ(X), (2.3) where Ψ(x,X) is an energy eigenfunction in the full coordinate space and a solution to (2.2). In the first step, the electronic eigenvalue problem, H e ϕ(x;x) = E(X)ϕ(x;X), H e = T e +V e +V en, is solved for various values of X, yielding E(X) as a function of the nuclear coordinates X. Substituting (2.3) into (2.2) leads to the time-independent Schrödinger equation for the nuclear wavefunction, H(x,X)χ(X) [T N + E(X) +V N (X)]χ(X), by neglecting the two terms with derivatives of the electronic wavefunction with respect to the nuclear coordinates. The omission of these terms is known as the Born-Oppenheimer approximation. Once the effective potential under which the nuclei move, V (X) = E(X) +V N (X), is determined, the Born- Oppenheimer approximation allows us to solve (2.1) separately for the nuclear coordinates. In terms of the CO 2 example in the previous section, the influence of the 22 electrons has been modeled into the potential V, leaving 9 degrees of 10

11 freedom corresponding to the 3 nucei remaining. Moreover, 5 of these involve translation and rotation of the linear CO 2 molecule, and thus only 4 degrees of freedom are sufficient to describe the internal motion of the atoms. The task of computing the effective potentials, or potential energy surfaces, is cumbersome and an important branch of quantum chemistry [24]. However, here we are interested in solving the nuclear TDSE and we will assume that the potential energy surfaces are given. 2.2 Bound states and non-bound states The TDSE (1) is defined on an infinite spatial domain, which needs to be truncated and supplied with boundary conditions and initial data for the purpose of numerical simulation. When a system involves particles that do not separate unless sufficient energy is added there are so-called bound states, i.e. localized eigenfunctions with corresponding discrete eigenvalues. At low energy only bound states are involved in the dynamics of the system, and the wavefunction will also be localized and naturally confined to a smaller domain. For example, the internal vibration of a diatomic molecule can be modeled by a harmonic oscillator. Then the range of the most probable distance between the two particles corresponds to the distance between the turning points of the harmonic oscillator. Hence, the domain can be truncated closely beyond the turning points, where the localized probability distribution is vanishingly small, with e.g. periodic boundary conditions. In many cases the wave function is localized in space and exhibits fast oscillations compared to the length scale of the domain of interest. For such problems it is important to include a sufficiently large region to capture interesting phenomena, and yet to resolve the high spatial frequencies. For computational efficiency it is convenient with a high grid density in areas with high frequency modes and a lower grid density where the frequency components are slower. Different ways of adapting the grid to the physical properties are considered in this thesis. Many interesting problems involve not only bound states, but also the continuous spectrum of the Hamiltonian. Investigating chemical reactions involving dissociation, where compounds break up into smaller subsystems e.g. by interaction with a laser field, is one example of interest. Another is calculating the energies of resonance states. These correspond to eigenfunctions which are not square integrable, unlike the eigenfunctions corresponding to the discrete spectrum of the bound states. In the time-dependent setting, solutions with energies in the continuous spectrum lead to waves that travel out from the designated computational domain, as the distance between the scattered particles increases. As a result of the outgoing waves, reflections are generated unless adequate boundary conditions are used. In this thesis we discuss different boundary models employed to absorb outgoing waves for the time- 11

12 dependent Schrödinger equation, and how they fit in with commonly used spatial and temporal discretizations. Although our focus is on time-dependent dynamics, we stress that similar techniques are of importance and can be used also for time-independent problems. In that case they can e.g. provide a means to find the energies of the resonance states. 12

13 3. Numerical methods and computational challenges Applying the Born-Oppenheimer approximation to the full description of a molecule, where both the spatial coordinates of the electrons and the nuclei are accounted for, considerably reduces the dimensionality of the problem. However, the dimensionality of the nuclear TDSE increases with the number of atoms in the molecule, and a straightforward discretization will result in a very high number of degrees of freedom. This is due to the so-called curse of dimensionality, caused by the exponential increase in grid-points as a function of the number of independent variables. As of today, only problems up to a few spatial dimensions are within reach of direct solution of the TDSE. Thus, in order to undertake large-scale problems, further approximations have been necessary. Approaches used to tackle high-dimensional problems include Multi Configuration Time Dependent Hartree (MCTDH) [57] and Hagedorn wave packets [29]. However, these methods will not be addressed further here. In this thesis we focus on how to expand the range of simulations based directly on the TDSE. Efficient numerical methods are essential. Moreover, they must be incorporated into a powerful parallel implementation framework [37, 15]. Discretization of (2.1) usually follows the method of lines, i.e. the spatial discretization is first introduced and the resulting system of ODEs can be solved by some suitable time-stepping method. The standard spatial discretizations in chemical physics are based on pseudospectral methods [31], e.g. the Fourier method and sinc-dvr [22]. Pseudospectral methods capture the dispersion relation correctly and accurate solutions can often be obtained with relatively few grid points. However, pseudospectral methods are global approximations, which in one spatial dimension leads to a full spatial discretization matrix. In the multi-dimensional case, the matrix is not full, but block-structured according to the independent variables. The popularity of pseudospectral methods is due to that efficient implementations can be achieved in combination with the fast Fourier transform (FFT). An alternative approach to pseudospectral methods is finite difference methods. The locality of finite differences is advantageous for parallel implementations, since the communication between processors is reduced, compared to the pseudospectral case. However, unlike for pseudospectral methods, the dispersion relation is not accurately captured for finite difference methods. Gray and Goldfield [34] propose dispersion-fitted finite difference methods, especially for problems where low accuracy is tolerable. 13

14 Moreover, finite differences are flexible in terms of boundary conditions and spatial adaptivity, whereas pseudospectral methods are not. Large variations in the solution to the Schrödinger equation are often concentrated to small parts of the full computational domain, and spatial adaptivity can thus serve as a way of enhancing the computational efficiency. The global nature of the basis functions makes it difficult to combine a pseudospectral approach with an adaptive spatial discretization, whereas the local support of the finite difference stencils makes them well suited to combine with local mesh refinement. We will elaborate on this topic in chapter 5. Finite element methods have also been used for the Schrödinger equation, although to a lesser extent than finite difference methods and pseudospectral methods. Efficient implementations are more difficult to achieve for finite element methods. On the other hand, the stability properties can be advantageous. For example, Antoine and Besse employ a finite element discretization in space in order to preserve the stability for their numerical scheme, see [4]. For an adaptive finite element discretization of the TDSE in one spatial dimension, Dörfler derived a posteriori error estimates [27]. As already mentioned in the context of open systems, a computational challenge is to truncate the unbounded domain of (2.1), without destroying the accuracy of the model. In the next section we will present boundary treatment procedures that address this issue. As a first step we will briefly describe some methods for temporal discretization and how restrictions can be inflicted upon them by the boundary treatment. Time-propagation methods developed and used for the TDSE have been tailored to share some properties with the physical system, to be detailed below. If there is no explicit time-dependence in the Hamiltonian operator H, the solution to (2.1) with initial condition ψ(x,t 0 ) is given exactly at time t by ψ(x,t) = U(t t 0 )ψ(x,t 0 ), where U(t t 0 ) = e ih(t t 0)/ h (3.1) is the evolution operator. Primarily, the numerical approximation of the evolution operator should be unitary. Unitarity implies norm-preservation of the wavefunction, i.e. the integral ψ(x,t) ψ(x,t)dx = 1, (3.2) holds for all times. Here, denotes the complex conjugate. Suppose that the wavefunction describes the state of a single particle, then (3.2) corresponds to the probability of finding the particle anywhere in space being equal to one. Secondly, the TDSE is time-reversible and that should be true also for the numerical time-propagation. By retaining the exponential form of (3.1) for 14

15 the temporal discretization, both unitarity and time-reversibility can be preserved. Besides methods based on an exponential form, partitioned Runge- Kutta methods are commonly used to propagate the Schrödinger equation in time. Partitioned Runge-Kutta methods own the advantage of being symplectic, a property which is described in [69]. A discussion of time-propagation methods is found in [51]. In [46], different propagators for the TDSE with an explicitly time-dependent Hamiltonian are compared. Kormann et al. [47] developed an h, p-adaptive Magnus-Lanczos algorithm with global error control for systems with explicitly time-dependent Hamiltonians. They use the truncated Magnus expansion to average the Hamiltonian operator. Instead of direct application of the matrix exponential of the discretized Hamiltonian, a subspace spanned by a few dominating eigenvectors is generated by the Lanczos algorithm [25], and used as a substitute for the discretized Hamiltonian in (3.1). This method is used in both Paper II and III along with a high-order finite difference spatial discretization and a perfectly matched layer (PML) for boundary truncation. A three-state system with explicit time-dependence in terms of a laser pulse is treated with the full propagator in Paper II. In Paper III the time-independent version is applied to a dissociative problem for which the Hamiltonian is only spatially dependent. However, the Hamiltonian with PML boundary treatment is no longer Hermitian. This is generally the case when including boundary modeling for dissociative states. Hence, the use of numerical methods developed for bound states are not always directly applicable to dissociative problems. For example, explicit partitioned Runge-Kutta methods are not available for Hamiltonians with complex entries [45]. In Paper II and III, the Lanczos algorithm is substituted by the Arnoldi algorithm for general matrices to enable the application to dissociative problems. The SBP-SAT spatial discretizations considered in Papers IV-VI do not give rise to a Hermitian discretized Hamiltonian in the regular l 2 norm. However, in most cases it is Hermitian in specific norms compatible with the SBP operators and Lanczos iteration can be used if the scalar product in the algorithm is based on the SBP norms. In Paper IV-VI the Lanczos algorithm is used for the temporal propagation in the numerical experiments for which we have the symmetry described above. 15

16

17 4. Boundary treatment for open systems Consider the one-dimensional Schrödinger equation with compactly supported initial conditions and a decay condition at infinity, i h ψ(x,t) t = 2m h2 2 ψ(x,t) +V (x,t)ψ(x,t) = Hψ(x,t), x 2 lim x ψ(x,t) = 0, ψ(x,0) = ψ 0 (x). (4.1) In order to perform numerical simulations, the infinite domain needs to be truncated to a finite computational domain and closed with boundary conditions. We are looking for boundary conditions for which the solution on the computational domain coincides with the solution on the infinite domain. This is obtained by truncating the domain beyond the compact support of the initial data at a point where the potential V is constant, and imposing boundary conditions that absorb waves that leave the computational domain. At large, there are two types of boundary treatments that can be used for this purpose, absorbing boundary conditions (ABCs) and absorbing layers. An ABC is imposed exactly at the numerical boundary, while an absorbing layer is a buffer zone where the equation is modified so that waves are damped immediately outside the domain of interest. The absorbing layer is then terminated with a stable boundary condition, at some point outside the interior domain where the outgoing waves are sufficiently damped. A two-dimensional computational domain surrounded by an absorbing layer is illustrated in figure

18 Figure 4.1: Computational domain (dark gray) extended with an absorbing layer (light gray). Most methods for absorbing boundary treatment in the field of chemical physics are based on absorbing layers. In the field of numerical analysis, on the other hand, the work on absorbing layers for the TDSE is rather limited compared to the work on ABCs [87]. Also in the cases where similar approaches have been used, there are surprisingly few references between the two communities, as Hein et al. [39] point out. Increased communication between the two communities is of potential benefit for the development of efficient boundary treatment models for problem settings of interest in the quantum chemistry community. In this section we review the techniques of absorbing boundary treatment that have been used for the Schrödinger equation, from the quantum chemist s and the numerical analyst s point of view, respectively. 4.1 Absorbing layers in quantum dynamics The vast majority of absorbing boundary methods in quantum dynamics are based on absorbing layers. The most common approach is the complex absorbing potential (CAP), where a complex potential iw(x) is added to the operator on the right hand side of (4.1), 18 i h ψ(x,t) t = h2 2 ψ(x,t) 2m x 2 +V (x,t)ψ(x,t) iw(x)ψ(x,t).

19 W(x) is typically zero everywhere, except near the numerical boundary where energy is absorbed by letting the real part of W(x) be positive. The CAP method has been used extensively in both time-dependent and time-independent scattering problems [61, 66, 75], which to a large part is due to the simplicity of the method and its compatibility with pseudospectral methods. Much effort has been devoted to finding effective and optimized CAPs for different applications, by minimizing the reflection and the transmission coefficients [83, 67, 63]. The reflection and transmission coefficients describe the relation between the amplitudes of an incident wave and the waves that are reflected on and transmitted through the CAP, respectively. By minimizing the sum of the two the absorption is maximized. However, the reflection of the CAP increases with increasing frequency, while the transmission is reduced with increasing frequency. Thus, a CAP can only be optimized for certain frequencies. Also, numerical reflections arise from the region where the CAP is active, and pollute the solution in the interior. A transmission-free CAP was derived by Manolopoulos [52, 32], who used a pole at the end of the absorbing region to eliminate all transmission. Since only minimization of the reflection coefficient is required, the transmission-free CAP is not as frequency-dependent as a regular CAP. An elegant and mathematically rigorous approach for dealing with the continuum in the eigenvalue spectrum is complex scaling of the spatial coordinate [1, 6, 76]. Complex scaling involves an analytical continuation into the complex plane, x xe iθ, (4.2) where θ is a positive and real constant. Simon [76] showed that the resulting complex scaled Hamiltonian, H θ (x) = H(xe iθ ), has complex eigenvalues that are identified as resonances. The method of complex scaling provides a means of dealing with both bound states and resonance states through the same formalism [59], since the eigenfunctions of H θ are square-integrable. Although most frequently used in time-independent settings [16, 58], Bengtsson et al. [9] recently used complex scaling for the time-dependent Schrödinger equation. The scaling of the Hamiltonian through (4.2) implies scaling of a potential energy surface. This might be difficult, for instance if the potential is given as a set of ab initio points instead of an analytic expression. Exterior complex scaling (ECS) circumvents this issue to a large extent, by introducing the rotation into the complex plane beyond some x = x 0, where the potential is close to constant. For instance, if the potential is given as a set of points in the interaction region, but analytically continued in the asymptotic region, it is possible to perform the coordinate transformation for the potential for x x 0. McCurdy et al. [56] recommend ECS for practical use of complex scaling in time-dependent problems. A similar approach to ECS is smooth exterior 19

20 scaling (SES). The difference between ECS and SES consists of the transition to the complex plane being simply rotated or rotated through a smooth transition function. In the continuous setting, the SES interface is non-reflecting, whereas the ECS interface is not. From a numerical point of view, the smoothness of the transition function is of importance in order to avoid numerical reflections from the absorbing layer. This is discussed in Paper III. The relation between CAP and SES is investigated both in [68] and [42]. 4.2 Absorbing layers and absorbing boundary conditions in the numerical analysis community In the numerical analysis community, most of the effort on absorbing boundary techniques for the TDSE has been directed towards absorbing boundary conditions. A recent and extensive review on absorbing boundary techniques for the TDSE, mainly treating ABCs, was written by Antoine et al [3]. A shorter comparison is given by Yevick et al [86]. Also, the dissertations of Jiang [41], and Schädle [70], deal with absorbing boundary conditions for the Schrödinger equation. An exact transparent boundary condition for (4.1), is a boundary condition for which the solution on the computational domain coincides exactly with the infinite space solution on the computational domain. It is in general difficult to find an exact transparent boundary condition in the case of a space- and timedependent potential. However, an exact condition can be derived by assuming that V (x,t) = V is constant for x x 0. This condition can for instance be expressed as the Dirichlet-to-Neumann (DtN) map n ψ(x,t) = e π 4 i π e ivt d dt t 0 ψ(x,τ)e iv τ t τ dτ, (4.3) see [3]. The imposition of (4.3) at x = ±x 0 solves the problem of reducing the infinite domain of (4.1) to [ x 0,x 0 ] completely. However, (4.3) is a global condition in time for the one-dimensional problem. In the multi-dimensional case, the corresponding relation is global both in time and space. In order to avoid the global coupling, (4.3) is often approximated to a local condition in time, which naturally introduces errors. Also, numerical errors are generated as an effect of the spatial discretization. The Crank-Nicolson scheme is frequently used to discretize (4.1) on the truncated domain [ x 0,x 0 ] with absorbing boundary conditions. Crank-Nicolson is unconditionally stable, unitary and time-reversible, which makes it suitable for solving the Schrödinger equation. However, Crank-Nicolson is rather expensive due to it being an implicit method. In Paper II, we investigate the performance of the Crank-Nicolson scheme compared to that of the Magnus-Arnoldi propagator [47], and conclude 20

21 that the latter is much more efficient, at least in the case of an explicitly time-dependent Hamiltonian. Even though the Crank-Nicolson discretization is unconditionally stable in the interior, caution needs to be taken in the process of imposing the boundary conditions. An unfortunate discretization of the boundary conditions may lead to the full discretization being unstable, which was discovered by Mayfield [55] and Baskakov and Popov [7]. An alternative approach is to first discretize (4.1) and thereafter find an exact transparent boundary condition for the semi-discrete or fully discrete formulation of (4.1). Antoine and Besse [4] used a conform finite element subspace for the spatial discretization in order to preserve the stability of their fully discrete numerical scheme including absorbing boundary condition, which was constructed for the semi-discrete equation in time. The approach of Ehrhardt and Arnold [28] was to first discretize both in space and time, using the Crank-Nicolson scheme. An exact transparent boundary condition was then derived for the fully discretized scheme, retaining the stability of the interior scheme. The trade-off for their non-reflecting boundary condition is the lack of flexibility. The discretization needs to be uniform, and extension to more than one dimension is a difficult matter Perfectly matched layers The approach we employ for boundary treatment throughout this thesis is the perfectly matched layer (PML). The PML is an absorbing layer, for which a modified set of equations are solved in the layer. The significant feature of the PML in contrast to a general absorbing layer is the perfect matching, which guarantees that the continuous PML is non-reflecting at the interface to the interior domain for all frequencies, and all incoming angles in the multidimensional case. However, discretization introduces numerical reflections at the interface. The PML method was developed for Maxwell s equations by Berenger in 1994 [10]. It has since then become the standard method in electromagnetics, and is widely used for hyperbolic problems in general [5, 8, 77]. The PML approach has been used also for the Schrödinger equation [2, 23], but the extent of the investigations is far from the extent of the analysis of PMLs for hyperbolic equations. A common approach for the TDSE is the modal ansatz PML [38], where the starting point of the PML derivation is modal solutions in the frequency domain. An alternative way is to introduce a coordinate change [20], as in the case of ECS and SES. For most hyperbolic problems, the modified equations in the frequency domain can not be transformed back into the time domain without the introduction of auxiliary variables, which leads to additional equations in the system. However, no auxiliary variables need to be introduced for the Schrödinger equation. Thus, when solving the TDSE with PML no additional equation needs to be solved. Extra computational effort may be required, compared 21

22 to the solution with ABC, due to the additional grid points in the layer. On the other hand, compared to ABC the PML formulation is easier to extend to multi-dimensional problems. Application of PMLs have also been made to the nonlinear TDSE by Zheng in [87] and by Dohnal [26], who constructed a PML for a system of twodimensional coupled nonlinear Schrödinger equations with mixed derivatives. Due to the mixed derivatives, waves with opposite phase and group velocities are supported. For many hyperbolic problems, this would lead to unstable layers, in the sense of exponentially growing solutions for the continuous problem, see [8]. Here, it only leads to a stability condition on the layer parameters. However, stability analysis for Schrödinger PMLs is an area of research which needs more consideration. In order to derive a PML for (4.1), we need to make the assumption that the potential is independent of x, i.e. V (x,t) = V (t). For the multi-dimensional case, the requirement is that the potential in the layer is independent of the normal direction in which the PML is imposed, but it may depend on other spatial variables. By performing the coordinate transformation x x + e iγ x ±x 0 σ(ω)dω, x x 0 (4.4) in (4.1), we arrive at the PML equation ( i h ψ(x,t) t = 2m h2 1 1+e iγ σ(x) x ψ( x 0 d,t) = ψ(x 0 + d,t) = 0, ψ(x,0) = ψ 0 (x), 1 1+e iγ σ(x) ) ψ(x,t) x +V (t)ψ(x, t), (4.5) on the domain [ x 0 d,x 0 +d], after domain truncation and imposing Dirichlet boundary conditions. The absorption function σ(x) is a nonnegative, real function in [ x 0 d, x 0 ] [x 0,x 0 +d], and zero in [ x 0,x 0 ], and γ should be in the range 0 γ π 2 for well-posedness, as discussed in Paper I. Perfect matching means that there are no reflections at the interface. This is achieved if σ(±x 0 ) = 0, since it implies continuity of ψ and ψ x at the interface. In the continuous setting, the damping introduced by the PML depends only on the value of the integral in (4.4), and on the parameter γ. Hence, a large value of the integral results in a large damping, and allows the use of a thin layer. However, the case is different for the discretized problem and σ(x) needs to be chosen carefully in order to avoid numerical reflections. Thus, a balance between a sufficiently steep slope of σ(x), such that the width of the PML can be kept small, and a sufficiently smooth function in order to avoid numerical reflections is advisable. Such an optimization process was performed by Sjögreen and Petersson for Maxwell s equation [77]. We employ a similar approach in Paper I for optimization of the Schrödinger PML. 22

23 Equation (4.5) can be written as a complex symmetric expression, see [42]. This form, often used for SES, is easily implemented with high-order finite differences in space, which is exploited in Paper I-III. In contrast to the SES approach the potential is not analytically continued into the complex plane. Except for this difference the PML and the SES approaches are equivalent, as discussed in Paper I. 23

24

25 5. Spatial adaptivity Many problems in chemical physics demonstrate large local variations of spatial scales where a uniform discretization either leads to unneccessary refinement in parts of the computational domain or to unresolved physical features in the fine scale regions. For applications that exhibit such variety in the spatial scales, local grid refinement can lead to considerable savings of computer memory and computational time. Also, the limit of feasible problems can be pushed further, especially for problems of high dimensionality. The development of techniques for spatial adaptivity is an important branch of scientific computing and a considerable amount of work has been done for instance in computational fluid dynamics. Spatial adaptivity for the TDSE has been investigated to a much lesser extent. One reason for this may be the commonly used pseudospectral methods for the spatial discretization, which due to the global nature of the basis functions are difficult to combine with local mesh refinement. The majority of the work done on adaptive grid based discretizations for the TDSE is either based on a coordinate transformation for the continuous equation, or on a combination of semi-classical and quantum mechanical methods. Fattal et al. [30] developed a method where a mapping of the wave function optimizes the phase space of the new wave function to fit with the Fourier method, and applied it in the context of time-independent problems including a Coulomb potential. Their approach is very similar to the complex scaling coordinate transformation described in the previous chapter, with the distinction that the coordinate transformation in the mapped Fourier method is along the real line instead of along a complex contour. Kokoouline et al. [44] extended the method to a broader class of long-range potentials. The mapped Fourier method was later extended to time-dependent problems by Kleinekathöfer and Tannor [43] and applied in numerical simulations up to three dimensions. They claim that a three-dimensional calculation would be out of reach for the original Fourier method, but also remark on the limitations of the method when it comes to dealing with dissociative problems, as well as problems for which the spectral components of the wave packet change in time. In the time-independent setting similar methods have been developed that also include the treatment of resonances [85, 21], and could possibly be extended to time-dependent dissociative problems. Pettey and Wyatt [62] use a combination of a semi-classical and a quantum mechanical approach. In their hybrid method a uniform, fixed grid fol- 25

26 lows the wave packet by moving the boundary points surrounding the fixed mesh. The boundary conditions are determined using a semi-classical trajectory method and the uniform grid is discretized using a fourth order finite difference method. In order to deal with more general situations and distribute the grid points according to the temporal evolution of the wave packet, a more flexible technique would be advantageous. Local approximations such as finite difference methods or finite element methods could suit this purpose well. In the following section we describe adaptive mesh refinement techniques that have been successful for example for problems in computational fluid dynamics. We continue by describing the methodology that we suggest as a building block in an adaptive mesh refinement framework for the TDSE. The work on spatial adaptivity for the TDSE in this thesis has primarily been focused on the numerical aspects, with an efficient parallel implementation of a dynamically evolving adaptive discretization in mind. 5.1 Adaptive mesh refinement Adaptive mesh refinement (AMR) techniques use computational resources efficiently by adapting the computational grid dynamically in time. Grid points are automatically added in regions where the solution changes rapidly, and removed from parts of the computational domain that no longer need a higher grid resolution. When discretizing complicated domains flexible approaches that can adapt to complex geometries have been successful. One way is to use unstructured meshes to represent the geometry of the domain [84]. An alternative technique is overlapping grids [40], where the domain is discretized using structured composite grids that overlap. Typically, boundary-fitted curvilinear grids are used to discretize regions close to boundaries. If the domains are simple, as is the case for the TDSE, a structured approach is beneficial. Structured meshes can be very efficient since the relations between neighboring grid points can be inferred from a less complicated data structure than in the case of unstructured meshes [64]. Additionally, unstructured meshes can often lead to loss in accuracy in comparison to structured meshes [60]. Berger and Oliger [13] developed an approach for structured adaptive mesh refinement (SAMR) based on the idea of multiple component grids for finite difference methods. Each grid point for which the numerical solution does not meet a user-defined local error tolerance is marked and regions that cover the marked points are refined. In the original method the alignment of the refined regions could vary with respect to the underlying grid patches. Berger and Colella [12] proposed a modified method where the boundaries of the refined regions are required to be parallel to the underlying grid. To allow for simpler data structures and reduce overhead for grid-fitting, block-structured versions 26

27 of SAMR have been developed [78]. In the block-structured approach, refinement is carried out concurrently for all points in a, typically rectangular, grid patch if the local error tolerance is not met for any point belonging to that grid patch. The block decomposition in Paper VI is based on a block-structured approach, where each grid block contains the same number of grid points. Such a division of grid blocks facilitates the load balancing. From a computer science perspective, block structured AMR has many advantages that can lead to an efficient implementation if the communication between processors can be limited. From a numerical analysis point of view a high order accurate discretization that leads to a stable numerical approximation is desirable. Berger [11] used normal mode analysis to investigate stability at a two-dimensional grid interface for the advection-diffusion equation for different finite difference approximations of low order. For high order discretizations this type of analysis often becomes very techniqual and it may be unfeasible to derive analytic stability conditions. Alternatively, a numerical procedure can be used. Thuné [82] presented an algorithm for automatic stability investigation using normal mode analysis for hyperbolic initialboundary value problems. Summation-by-parts operators are finite difference operators with special boundary closures that in combination with the simultaneous approximation term often lead to stable approximations via the energy method [17]. Also, stability-, accuracy-, and conservation-properties shown for a single block can often be extended to a multi-block configuration [53]. SBP-SAT discretizations for grid interfaces have been considered by Carpenter et al. [18, 19] and Lindström and Nordström [50] to mention a few. An advantage of the SBP- SAT method is that the stability results are independent of the order of the spatial discretization as long as the operators possess certain so-called SBP properties, described in section 5.2. Also, the SBP operators contain onesided approximations close to the interface, which limit the communication between grid blocks to the SAT terms that couple adjacent grid blocks with each other. A drawback is that the operators are of lower order close to boundaries and interfaces. In our case the order of accuracy close to boundaries is only half the order of accuracy in the interior. A straightforward application of the energy method leads to error estimates that imply the lower convergence order. However, earlier results have shown that the accuracy of the numerical solution is often one or two orders higher than the order the boundary approximation would suggest [36, 35]. Svärd and Nordström [80] proved that two orders of accuracy are recovered for numerical approximations of equations with second derivatives for which uniform stability can be shown. In Paper IV and Paper V we investigate the accuracy for SBP-SAT discretizations of the TDSE using normal mode analysis. In this case uniform stability estimates cannot be derived. However, we show that two orders of accuracy are gained both for the boundary case with homogeneous Dirichlet boundary conditions and for the situation with interfaces. 27

28 To prove stability of a grid interface coupling using normal mode analysis may become very technical, or even unfeasible, especially for high order approximations. However, if the grid coupling is proven stable by e.g. the energy method, normal mode analysis can be used to consider the accuracy of the grid interface coupling. For accuracy it suffices to restrict the analysis to a limited region of Laplace space near the origin, as opposed to the stability investigation where the whole right half-plane needs to be considered. The use of SBP operators in AMR has been very limited. Until recently it has not been known how to couple adjoining nonconforming grid blocks, i.e. grid blocks for which collocation points across grid boundaries do not match, in a stable way using SBP operators. Recently, Mattsson and Carpenter [53] derived interpolation and projection operators that together with commonly used SBP operators preserve stability between nonconforming grid blocks. In order to show stability for a multiblock SBP-SAT configuration, corners with different refinement levels, illustrated by A and B in figure 5.1, need to be treated properly. Kramer et al. [48] considered stability and accuracy for grids in two dimensions with corners illustrated by A, for possible extension to AMR. They derived conditions for stability at corners and showed that the order of accuracy at the corner points is considerably lower compared to the accuracy order at grid points away from corners. By imposing the SAT interfaces denoted by 2, 3, 4 and 5, a stable discretization can be shown for the corner point B, using the interpolation operators from [53] for 4 and 5. However, in a multiblock configuration with several layers with different refinement levels this solution is not optimal with respect to load balancing. Continuing the interfaces that start in the corner (corresponding to 2 and 3) may lead to grid blocks with only a few grid points in the outer layer of the multiblock configuration grid. In order to avoid this situation it is important to be able to treat the junction, illustrated by J in figure 5.1, in a stable way. In paper VI we consider the imposition of interface 1 for a stable coupling at the junction J. 5.2 Summation-by-parts operators Summation-by-parts operators are finite difference approximations that satisfy a summation by parts formula. First developed by Kreiss and Scherer [49], the idea is that the summation by parts rule mimics the corresponding integration by parts formula for the continuous operators. The operators in [49] approximate first derivatives. Since then, SBP operators have been constructed for instance by Strand [79] for first derivatives and by Mattsson and Nordström [54] for second derivatives. If boundary conditions are imposed using the simultaneous approximaion term method, it is often possible to show stability of the numerical scheme. We will demonstrate the SBP-SAT technique by a simple example below. 28