TVE 16 040 juni

Examensarbete 15 hpJuni 2016

Numerical approaches to solving the time-dependent Schrödinger equation with different potentials

Lina ViklundLouise AugustssonJonas Melander

Teknisk- naturvetenskaplig fakultet UTH-enheten Besöksadress: Ångströmlaboratoriet Lägerhyddsvägen 1 Hus 4, Plan 0 Postadress: Box 536 751 21 Uppsala Telefon: 018 – 471 30 03 Telefax: 018 – 471 30 00 Hemsida: http://www.teknat.uu.se/student

Abstract

Numerical approaches to solving the time-dependentSchrödinger equation with different potentials

Lina Viklund, Louise Augustsson, Jonas Melander

This project is an immersive study in numerical methods, focusing on quantum molecular dynamics and methods for solving the time-dependent Schrödinger equation. First the Schrödinger equation was solved with finite differences and a basic propagator in time, and it was then concluded that this method is far too slow and compuationally heavy for its use to be justified for this type of problem. Instead pseudo-spectral methods with split-operators were implemented, and this proved to be a far more favourable method for solving, both in regards to time and memory requirements.

Further, the pseudo-spectral methods with split-operators were used to solve the dynamics resulting from the excitation of sodium iodide by an ultra-fast laser pulse. This was modeled as two Schrödinger equations coupled with a potential modeling the laser pulse. The resulting solution made the quantum nature of the system clear, but also the limitations and advantages of different numerical methods.

ISSN: 1401-5757, TVE 16 040 juniExaminator: Martin SjödinÄmnesgranskare: Rikard EmanuelssonHandledare: Sverker Holmgren, Hans Karlsson

Popularvetenskaplig sammanfattning

Allting vi ser i vart dagliga liv, inklusive oss sjalva, ar uppbyggt av atomer och molekyler, och molekylarkvantdynamik beskriver hur atomer och molekyler ror sig och interagerar med varandra. Genom kun-skap och forstaelse i omradet oppnas nya mojligheter att simulera och forutspa resultatet av kemiskareaktioner, vilket ar mycket fordelaktigt inom vissa forskningsomraden.

Det forsta steget till att simulera molekylernas rorelse ar att ta fram matematiska modeller sombeskriver denna rorelse som stammer val overens med verkligheten. Den inom kvantfysiken my-cket beromda Schrodingerekvationen beskriver partiklar som vagfunktioner med viss sannolikhet attbefinna sig i en viss punkt vid ett visst tillfalle. Partiklarna ror sig pa olika energinivaer (potentialer)som kan vara konstanta eller andras over tiden, och beroende pa potential och antal partiklar blirekvationerna olika svara att losa. For vissa specialfall finns ’exakta’ losningar, dvs. fall som gar attlosa ”med papper och penna”, men for det mesta ar modellerna komplexa och saknar exakta losningaraven for mycket sma system.

Ett satt att ga tillvaga for att losa dessa komplexa system ar att diskretisera dem och losa dem nu-meriskt. Detta innebar att tids- och rumsintervallet som systemet loses pa delas upp i ett begransatantal punkter, och problemet loses sedan i varje sadan punkt med hjalp av datorer. Bland det storaantalet numeriska metoder som kan anvandas finns givetvis mer eller mindre noggranna, mer ellermindre tidskravande samt mer eller mindre minneskravande metoder. Det galler alltsa att hitta enmetod som ar tillrackligt noggrann, inte tar for lang tid och inte kraver for mycket minne. Det hararbetet syftar dels till att losa Schrodingerekvationen for ett enpartikelsystem bade i tidsoberoendeoch tidsberoende potentialer, dels till att losa Schrodingerekvationen for ett flerpartikelsystem med entidsberoende potential. Metoderna som anvands analyseras och jamfors med varandra med avseendepa noggrannhet, tid och minne.

Acknowledgements

We would like to thank our supervisors, professor Sverker Holmgren at the Department of InformationTechnology, Uppsala University, and professor Hans Karlsson at the Department of Chemistry, UppsalaUniversity. We greatly appreciate all of your help during this project.

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Contents

1 Introduction 3

2 Theory 32.1 Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.1.1 Harmonic oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.1.2 Morse potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.1.3 Time-dependent potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.1.4 Potential for the NaI-system . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.1.5 Coupling potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2 Model problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3 Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.3.1 Spatial methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.3.2 Methods for time stepping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.4 TDSE for NaI-system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3 Method 11

4 Results 124.1 Model problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124.2 NaI-system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

5 Discussion 19

6 Conclusions 21

A Parameters 23

2

1 Introduction

Quantum molecular dynamics (QMD) describes the time evolution of atomic and molecular interac-tions on the quantum scale. Simulating such dynamics gives an insight into the behaviour of moleculesand atoms, and this knowledge in turn helps to e.g. understand how to better control the outcome ofchemical reactions. Experimental studies of chemical reactions by A. H. Zewail on a type of systemsimilar to the one studied in this paper [1] was rewarded by the Nobel Prize in chemistry in 1999.This project will focus on solving the dynamics resulting from the excitation of sodium iodide (NaI)by an ultra-fast laser pulse, operating on the femto-second scale.

To understand QMD, one needs accurate mathematical models to describe the system of particles. Inorder to simulate these interactions, efficient and accurate numerical methods are needed, since mostsystems of particles are too complex to be described by equations with analytic solutions. Numericalmodels are obtained by using discrete representations of the continuous mathematical models, andthis can be done in many ways, some more efficient and accurate than others.

A model problem (presented in section 2.2) with Morse potential (presented in section 2.1.2) willbe solved using both basic finite differences which are widely used in numerical calculations, andpseudo-spectral methods combined with the split operator technique, which have successfully beenapplied to problems of this type. The complete NaI system will be solved with pseudo-spectral meth-ods combined with the split operator technique. This will be followed by a discussion comparing thetwo methods.

2 Theory

The time-dependent Schrodinger equation (TDSE) is an n-dimensional complex-valued partial differ-ential equation (PDE) describing the motion of particles in space and time. In general the TDSE canbe formulated as

ih∂

∂tΨ(~r, t) = − h2

2m∇2Ψ(~r, t) + V (~r, t)Ψ(~r, t), (1)

where ~r is an n-dimensional vector, t is time, i =√−1, m the mass of the object described, and h is

Planck’s constant divided by 2π. Equation (1) may be rewritten on the form

ih∂

∂tΨ(~r, t) = H(t)Ψ(~r, t), (2)

where H(t) is the time dependent Hamiltonian operator given by

H(t) = − h2

2m∇2 + V (~r, t) = T + V (~r, t), (3)

where T is the kinetic energy operator, and V (~r, t) a time-dependent potential.

A diatomic system with two coupled electronic potentials can be modeled as two PDEs.H1Ψ1(~r, t) + Vc(~r, t)Ψ2(~r, t) = ih

∂

∂tΨ1(~r, t)

H2Ψ2(~r, t) + V †c (~r, t)Ψ1(~r, t) = ih∂

∂tΨ2(~r, t).

(4)

H1 and H2 are Hamiltonian operators containing V1(~r) and V2(~r) respectively. V1(~r) and V2(~r) are

potential energy surfaces corresponding to different states. Vc(t) is a coupling potential and V †c (t) isthe complex conjugate of Vc(t).

3

Further, the inner product of the quantum wave function with its complex conjugate,⟨Ψ∣∣Ψ⟩ =

∫ ∞−∞

Ψ∗Ψdr,

can be interpreted as the total probability density, which should always be preserved. A good numer-ical method conserves the probability density, and this is thus a necessary but not sufficient conditionfor correctness.

When performing calculations on this scale it is convenient to use atomic units. In this unit sys-tem me, the electronic mass, e, the electron charge, and h are all set to unity, and all other units arescaled accordingly.

2.1 Potentials

The choice of potential V (r) decides what system that will be modeled. A potential with an easyanalytic solution can be used for testing the accuracy, convergence and stability of the numericalmethods. A more advanced potential that cannot be solved analytically may be a better approximationof the real problem. Below a number of potentials that will be used in the project are presented.

2.1.1 Harmonic oscillator

The accuracy of the numerical solutions to the Schrodinger equation may be difficult to ascertain, notknowing the true solution before hand. The harmonic oscillator

V (r) =1

2mω2r2, (5)

where m is the mass and ω is the eigenfrequency of the particle, has a known analytic solution. Thiscan be used to test aspects of the numerical methods that are used. This is possible because for atime-independent case we have that

Hψ(r) = Eψ(r).

This means that when the Hamiltonian operates on a stationary wave function, the result will be aconstant E multiplied with the same wave function, where E is the eigenenergy of this stationary state.For a harmonic oscillator potential the formula for these eigenenergies are known,

En =(n+

1

2

)hω,

where n is the quantized energy level. Since these can be calculated analytically, they can be comparedto numerical calculations of the eigenenergies, and thus be used to investigate the numerical methods.Figure 1 gives a visual representation of the potential.

Figure 1: The harmonic oscillator potential.

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The initial values for harmonic oscillations can be approximated using a Gaussian wave-packet, ψ(r),which can be expressed as

ψ(r) = µeα2(r−r0)2

, (6)

where µ, the normalization constant, is given by

µ =1√∫ r1

r0ψ∗(r′)ψ(r′)dr′

and α governs the width of the function.

2.1.2 Morse potential

The Morse potential, visualized in figure 2, can be a better approximation of a potential energy surfacewhen working with diatomic molecules. It is given by

V (r) = De(1− ea(r−r0))2, (7)

where r − r0 is the distance from a potential well at r0, a and De are parameters that relate to the”width” and ”depth” of the potential. These parameters are molecule specific.

Figure 2: The Morse potential.

2.1.3 Time-dependent potential

When the Hamiltonian is time-dependent, an extra time-dependent term is added to V (r). Here thetime dependent part model the systems interaction with a photon. In this case with a laserpulse

V (r, t) = V (r) +Acos(ωpt),

where A is a constant, and ωp is the frequency of the time-dependent perturbation.

2.1.4 Potential for the NaI-system

For the NaI-system described in section 2.4 the ground state potential is given by

Vg(r) =

(A2 +

(B2

r

)8)e−r

ρ − 1

r− 1

2

(λ1 + λ2)

r4− 1

r6− 2λ1λ2

r7+ dE, (8)

5

and the excited state is given by

Ve(r) = Aee−r

β . (9)

These potentials with the parameter values tabulated in table 11 of Appendix A have previously beenshown to give accurate results [8] for this kind of simulation. The potentials can be seen in figure 3.

2.1.5 Coupling potential

The laser pulse’s interactions with the NaI-system can be modeled as a coupling potential. The spatialpart of the coupling potential is given by

Vc,r(r) = A12e−β12(r−r0)2

, (10)

and the temporal part is given by

Vc,t(t) = ηe−αt(t−t0)2cos(ω ∗ (t− t0)), (11)

where η is the strength of the laser pulse and

αt =2 log(2)

ξ2, (12)

where ξ is the width of the pulse.

2.2 Model problem

The model problem that will be used to investigate the numerical methods is a TDSE with onedimension in space, both time-independent and time-dependent. This is given by

ih∂

∂tΨ(r, t) = HΨ(r, t), (13)

where r − r0 is the distance to the center of some potential well at r0, and H the Hamiltonian. Thetime-independent Hamiltonian is given by

H = − h

2m

∂2

∂r2+ V (r). (14)

The time-independent case will be used in order to check properties of the numerical methods, suchas order of accuracy and convergence, as explained in section 2.1.1. In other regards than this, thetime-independent case is not that interesting, since a model of a chemical reactions needs to involve atime-dependent potential. The time-dependent Hamiltonian is given by

H = − h

2m

∂2

∂r2+ V (r, t), (15)

and for this case there is no simple analytic solution. An analytic approximation can be made [4], butfor the time-dependent cases it is more efficient to use numerical methods to calculate the solutions.Reducing equation (1) to equation (13) offers an easy way to test numerical methods without demand-ing powerful computers to calculate the solutions.

2.3 Discretization

Applying numerical methods to a PDE problem requires it to be dicretized. This can be done byintroducing a uniform grid in both time and space

rj = j∆r j = 0, · · · , N ∆r = 1/N

tn = n∆t n = 0, 1, · · ·

6

where rj are the grid points in space, N the number of grid points in space and ∆r the step size inspace. tn are the grid points in time, and ∆t is the step size in time. For each grid point j, n we thenget a solution ψnj that approximates the true solution, ψnj ≈ ψ(rj , tn).

Discretized initial conditions (I.C.) are required to get accurate solutions when using other thanHarmonic potentials. These are obtained by solving the matrix equation

HΨ0(r) = EΨ0(r) (16)

for the eigenvector Ψ0(r). E is the eigenenergy for the initial state.

2.3.1 Spatial methods

The model problem will be solved using both a simple method, the finite difference method with acentered difference, and a more advanced method, the pseudo-spectral method.

The main idea with finite difference methods is to approximate derivatives and solutions to differ-ential equations [9] with linear combinations of function values in neighboring grid points. One of themost basic ways to approximate a second order derivative with finite differences is the centered differ-ence, which uses grid points centered around a middle point in order to approximate the derivative.How many grid points that are used for the centered difference, and how they are weighted dependon the desired order of accuracy. The TDSE approximated with a second order centered difference inspace reads

ih∂

∂tψj = − h2

2m

(ψj+1 − 2ψj + ψj−1

(∆r)2

)+ V (rj)ψj = Hψj , (17)

which is a system of ordinary differential equations, where

H =−ih∆t

2m(∆r)2

2 −1−1 2 −1

. . .. . .

. . .

2 −1−1 2

︸ ︷︷ ︸

T

+i∆t

h

V (r). . .

. . .. . .

V (r)

︸ ︷︷ ︸

V(r)

. (18)

T is the differentiation matrix, here for a centered finite difference with order of accuracy 2, and V(r)is the potential matrix. With H as in equation 18, we have Dirichlet boundary conditions, set to zero.Depending on the order of accuracy of the centered difference T will have more diagonals filled in,corresponding to the coefficients in table 1.

Table 1: Coefficients of the central difference for second order derivative

Accuracy -4 -3 -2 -1 0 1 2 3 4

2 1 -2 1

4 -1/12 4/3 -5/2 4/3 -1/12

6 1/90 -3/20 3/2 -49/18 3/2 -3/20 1/90

8 -1/560 8/315 -1/5 8/5 -205/72 8/5 -1/5 8/315 -1/560

In order to check convergence and order of accuracy when solving problems without a known analyticsolution, the method is used on a problem with successively decreased step size. If the solutionsconverge the method is assumed to converge. To determine the order of accuracy, a reference solutionis computed with a method of high order of accuracy and a large number of grid points. Since themethod is assumed to converge the reference solution is seen as the true solution. Solutions from lower

7

order methods can then be compared with the reference solution in order to determine their order ofaccuracy. The order of accuracy tells how fast the solutions converge. When an n-tupled grid is usedthe relative error decreases by a factor of np where p is the order of accuracy of the method.If a system of PDE:s with dirichlet boundary conditions (B.C.) can be approximated with a systemof identical PDE:s with periodic B.C., a pseudo-spectral method (PSM) may be applied. The PSM’sspatial accuracy is formally infinite [5], the main error in the PSM is the inherited error from themodel approximation. Because of the accuracy of the PSM, equally accurate initial conditions areneeded, these are obtained by solving equation (16), where

H = Tt + V, (19)

Tt is a circulant matrix that replaces the standard finite difference differentiation matrix for the kineticenergy and V is a potential. Since all potentials in our case are analytic, the use of Tt in H is enoughto get equally accurate I.C.:s. The PSM take advantage of the kinetic energy operator being local in

the momentum domain, or k-space, making it a diagonal matrix when discretized, withp2

2mon the

diagonal. Table 2 shows the position, momentum and kinetic energy in the position and momentumdomain. Equation (13) can be reformulated with the help of the discrete Fourier transform (DFT)

ih∂

∂tΨ(rj , t) = Tψ(rj , t) + V (rj)Ψ(rj , t) ⇒ ih

∂

∂tΨ(rj , t) = F−1DkFΨ(rj , t) + V (rj)Ψ(rj , t), (20)

where Dk = diag(p2

2m) is the DFT of the kinetic energy operator. These operations are preferably

done with the fast Fourier transform fft in MATLAB.

Table 2: Position, momentum and kinetic energy in the position and momentum domain (k− space).

Operator Position domain Momentum domain

x x ih ∂∂p

p2

2m −ih ∂2

∂x2p2

2m

2.3.2 Methods for time stepping

For a time-independent Hamiltonian the analytic solution to the TDSE is given by

ψ(t) = e−ihHtψ0, (21)

where ψ0 is the initial state and H is the Hamiltonian matrix. A simple way of time stepping in thiscase is then to use MATLAB’s expm() function, which is used as a propagator. The function of apropagator is the propagation (in time) of the function it is applied to. In equation (21) the expo-nential term is the propagator. In this case the solution is analytic, and thus the order of accuracy isinfinity.

However, if the Hamiltonian is time-dependent, which is the case with a time-dependent potential, itwill include a matrix for the kinetic energy and a matrix for the time-dependent perturbation, whichdo not necessarily commute. In this case the analytic solution is not known. Instead the Magnusexpansion can be used to approximate the time-dependence effect on the solution[6]. If the expansionis truncated after the first term it can be written as

ψn = eΩ(tn,t0)ψn−1, (22)

where

Ω(t, t0) =−ih

∫ t

t0

H(τ)dτ. (23)

8

First-order Magnus expansion is second order accurate in time [2]. Using the first term only will notbe a problem since the Magnus expansion is unitary, meaning that the probability is conserved evenif it is truncated. As in the time-independent case, expm() will be used for propagating, but since theintegral in the exponential is approximated by (H(t) − H(t0))∆t which has order of accuracy 2, asmay be shown by Taylor-expansion. Thus the total order of accuracy in time will be 2.

The calculations are performed in Matlab. For a time-independent Hamiltonian the kinetic energymatrix, the Hamiltonian matrix and the matrix exponential are computed once. They are then usedin a for-loop to do the time stepping. When the Hamiltonian is time-dependent it changes for everytime step. Therefore both the Hamiltonian and the matrix exponential must be computed anew forevery time step. Matrix-vector multiplication takes O(N2) operations and the matrix exponentialoperator takes approximately O(N3) operations. This gives that each time step in the solution hascomputational complexity O(N2) for time-independent potential, and O(N3+N2) for time-dependent.Concerning the memory requirements it is the same at the end stage with both time independent andtime dependent potential. However the time-dependent solution will pre-allocate more memory be-cause it needs to compute the matrix exponential in every loop and a full matrix, O(N2), is needed tostore the matrix exponential. Computations of the matrix exponential are computational expensiveand takes a lot of memory, O(N2).

A more advanced method that addresses this concern is the split-operator method [7]. This methodis based on splitting the Hamiltonian into a kinetic energy part and a potential energy part. Thismeans that the solution to the TDSE can be written as

ψ(r, t) = ei(T+V )∆tψ0. (24)

Since T and V are matrices that do not necessarily commute, ei(T+V )∆t 6= eiT∆teiV∆t in this case, butequation 24 can be approximated as

ψ(r, t+ ∆t) ≈ eiT∆teiV∆tψ(r, t) +O((∆t)2). (25)

In order to achieve a better approximation [2] the kinetic energy matrix can be split into

ψ(r, t+ ∆t) ≈ eiT∆t/2eiV∆teiT∆t/2ψ(r, t) +O((∆t)3). (26)

Pairing the split-operator method with the pseudo-spectral method described in section 2.3.1 makesfor an n-efficient method, since the splitting makes it easy to diagonalize the kinetic energy matrix.It takes O(N logN) operations and O(N) in allocated memory.

Throughout the paper time-stepping using expm() without split-operators will be referred to as simpletime-stepping.

9

2.4 TDSE for NaI-system

The complete NaI system can be modeled as

ih∂

∂t

(ψ1(r, t)ψ2(r, t)

)=

(T + Vg(r) V †c (r, t)Vc(r, t) T + Ve(r)

)(ψ1(r, t)ψ2(r, t)

), (27)

where Vg is the ground state potential matrix, Ve is the first excited state potential matrix, Vc is the

coupling potential matrix, and V †c is the complex conjugate of Vc. Ψ1(r, t) and Ψ2(r, t) are vectorscorresponding to Vg and Ve respectively.

Applying PSM and split operators to equation (27) yields

Ψ(r, tn+1) = F−1e−i

2hΞ∆tFe−

ihW∆tF−1e−

i2h

Ξ∆tFΨ(r, t), (28)

Which when applied from left to right will result in stepping the TDSE one time step forward. Here

we have that Ξ = diag(Dk), with Dk = diag(p2

2m). W is a matrix of potential matrices defined as

W =

(Vg(r) V †c (r, t)Vc(r, t) Ve(r)

)(29)

which is not diagonal. Because Vc(r, t) is not a complex-valued function in the case treated here, we

get Vc(r, t) = V †c (r, t). The pre calculation of e−ihW∆t is desired for increased computational speed

and reduced memory requirements. The calculation of the matrix exponential can be done by

e−ihW∆t = exp

(− i

h

(Vg(r) Vc(r, t)Vc(r, t) Ve(r)

)∆t

)

= exp

(− i

h

(Vg(r) + Ve(r)

)∆t

2

)(cos(√

Q∆t

2h

)(1 00 1

)+ (30)

isin(√Q∆t

2h

)√Q

(Ve(r)− Vg(r) −2Vc(r, t)−2Vc(r, t) Vg(r)− Ve(r)

))= exp

(− i

h

(Vg(r) + Ve(r)

)∆t

2

)γ

which is shown in [8] and [2], and Q = (Ve(r)− Vg(r))2 + 4V 2c (r, t). This results in

Ψ(r, tn+1) = F−1e−i

2hΞ∆tFe−

ih

(Vg(r)+Ve(r))∆tγF−1e−i

2hΞ∆tFΨ(r, t) (31)

which is the final form representing the time stepping of the TDSE when applied from right to left.

Figure 3: The potential surfaces.

10

Solving the TDSE on coupled potential surfaces, as is the case for the NaI-system, give rise to thephenomenon where the probability density function splits over the two surfaces shown in figure 3. Theprobability density on a surface is called a population and, as always when dealing with probabiltydensities, the sum of the populations are unity. When the laser pulse excites the molecule a populationwill appear on the excited state and then move along this surface until it reaches the crossing of theexcited state and the ground state. Here the population will split, so there is a probability both ofthe probability density continuing along the excited state and of it moving back down on the groundstate. If the density continues on its excited state or drift further apart along the ground state, themolecule will separate in to its individual atoms. Depending on which potential surface the densitytraveled along it will either be an ion separation or a separation of neutral atoms. If the moleculeseparates into two atoms the models described will no longer be applicable on the system.

The PSM with split operators assumes a spatial periodicity in the system i.e. that the boundaryconditions can be set as periodic. This is not actually true for this system, an approximation ofperiodic boundary conditions is made at such a distance between the atoms as to make the problemuninteresting and the TDSE sufficiently small.

3 Method

The calculations are set up so that they will be working upwards, beginning with more basic methodssuch as finite differences, to the more advanced pseudo-spectral methods with split operators. Thereason for this is that in this way a thorough comparison between the two methods can be made, andit will be made apparent why pseudo-spectral methods are more suitable for this type of problem,compared with finite differences.

First the Schrodinger equation with a known analytic solution, i.e with harmonic oscillator poten-tial, is solved with finite difference methods. Centered finite differences of different order of accuracyare used. The analytical eigenenergies are compared with the numerically computed eigenenergies.By comparing the change in relative error for different number of grid points, the order of accuracyand convergence can be evaluated. The order of accuracy is analyzed a second time by computing areference solution and comparing the numerical solutions with the reference solution. When the orderof accuracy and convergence are confirmed, the method is used to solve the Schrodinger equation withMorse potential.

The methods for solving the TDSE with a time-dependent potential are then analyzed. A centeredfinite difference of order eight is used for solving in space. Here there is no known analytic solution,so the method’s convergence is assured by solving the problem with a decreased step size in space anda small time step. If the solution converges the method is assumed to converge. In order to analyzethe order of accuracy a reference solution is computed and the numerical solutions are compared withthe reference solution. The computational time, computational work and memory requirements arenoted for future comparison.

The TDSE with time-dependent potential is then solved using pseudo-spectral methods with splitoperators. As with finite differences, the computational time, computational work and memory re-quirements are noted and then compared to the same data for finite differences. For the TDSE withtime-dependent potential and coupled potentials the behaviour of the NaI-molecule is anlyzed, alongwith how different parameters affect the simulation.

11

4 Results

4.1 Model problem

Using the harmonic oscillator potential for the time-independent model problem the potential’s eigenen-ergies were computed both analytically and numerically with finite differences in order to investigateorder of accuracy of the centered difference. The results and the relative errors are shown in Table3 and 4 for a second and fourth order method respectively. It can be seen that the relative errordecreases with a factor 22 when doubling the number of grid points for the second order method, andwith a factor 24 for the fourth order method. It can also be seen that for all methods, the relative errorincreases for higher eigenenergies for the same number of gridpoints. When doing the same analysisfor a sixth and eighth order method it is found that the error decreases with a factor 26 and 28.

Table 3: Comparison of eigenenergies, second order method.

n Analytic Number of gridpoints Second order

Eigenenergy Eigenenergy Relative error %

0 0.525 0.4807 3.850 0.4951 0.98100 0.4988 0.24

1 1.525 1.4001 6.750 1.4755 1.6100 1.4938 0.41

2 2.525 2.2193 11.250 2.4358 2.6100 2.4840 0.64

Table 4: Comparison of eigenenergies, fourth order method.

n Analytic Number of gridpoints Fourth order

Eigenenergy Eigenenergy Relative error %

0 0.525 0.48968 0.6450 0.4998 0.04100 0.5000 0

1 1.525 1.4781 1.4650 1.4984 0.11100 1.4999 0.007

2 2.525 2.4208 3.250 2.4942 0.23100 2.4996 0.02

The discretization error as a function of step size for the solution when solving with finite differencesis shown in figure 4. It is evident that the error decreases more rapidly with step size as the order ofthe method is increased. The slope of the curves give approximately the order of accuracy for eachmethod. These are tabulated in table 5.

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Table 5: The slope of the curves in figure 4 corresponding to the order of accuracy of the method.

Order of accuracy Slope

2 1.9

4 3.8

6 4.8

8 7.3

Figure 4: Discretisation error in the numerical methods depending on the size of the time step.

The solution to the TDSE with harmonic oscillator potential solved with an eighth order centeredfinite difference after 0, 1, 3.4 and 4.5 atomic units of time are shown in Figure 5. The black curve isthe initial value, and the other curves show the solution oscillating back and forth. Using the analyticsolution to the ground state as initial value causes the solution to oscillates vertically.

The solution to the TDSE with Morse potential solved with an eighth order centered difference isshown in Figure 6. The solution oscillates in this case as well, but it displays a more oscillatorybehavior the longer the simulation progress.

Figure 5: Numerical solution to the TDSE using a time-independent harmonic oscillator potential.

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Figure 6: Numerical solution to the TDSE using a time-independent Morse potential.

The TDSE with a time-dependent perturbation to the harmonic oscillator potential was solved withan eighth order centered finite difference and simple time-stepping. Figure 7 shows the discretisationerror as a function of step size. It can be seen that the error decreases with decreased step size. Thecurve has a slope of 7 in the steepest part and a slope of 2 for smaller steps.

Figure 7: Error depending on step size when the TDSE with a time-dependent harmonic oscillatorpotential was solved with an eighth order centered difference.

The effect of adding a time-dependent term to the harmonic oscillator potential is shown in figure 8. Inthe time-independent case the particle moves between two fixed points around the equilibrium, whilein the time-dependent case the particles movement changes with time, following a cosine pattern.

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Figure 8: The movement of a particle in a harmonic oscillator potential. The upper graph shows thetime-independent case and the lower the time-dependent case.

Tabulated in table 6 is the runtime required for solving the Schrodinger equation with both a time-independent and time-dependent Morse potential, i.e. the more realistic model problem, with finitedifferences and the simple method for time-stepping. As can be seen, this takes a very long time, over2 hours in total. Table 6 also shows the allocated memory required to run the simulation.

Table 6: Runtime and memory required to solve the Schrodinger equation with both a time-independent and a time-dependent Morse potential, using an 8th order finite difference method withN = 1024 and ∆t = 0.1.

Potential Runtime [s] Memory required [kb]

Time-independent 19.7463 45140.00

Time-dependent 7.53 · 103 4.95 · 108

Table 7 shows the data for solving the Schrodinger equation with both a time-independent and time-dependent Morse potential using the PSM with split-operators. Compared with the data presented intable 6 these methods are evidently much faster than using finite differences with simple time-stepping.

Table 7: Runtime and memory required to solve the Schrodinger equation with both a time-independent and a time-dependent Morse potential, using the PSM with split-operators, with N =1024 and ∆t = 0.1.

Potential Runtime [s] Memory required [kb]

Time-independent 0.473658 5304.00

Time-dependent 0.59669 1.1 · 105

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4.2 NaI-system

Figure 9 and 10 shows how the populations behave for two different sized grids when the wavelengthof the laser pulse is 360 nm. For N = 512 the fluctuations of the population on the excited statecannot be seen.

Figure 9: The populations when solving with a grid of N = 512.

Figure 10: The populations when solving with a grid of N = 2048.

The intensity of the population distribution over time versus nuclei distance is shown in figure 11.The impact of the perturbing laser pulse is clearly shown as the population intensity can be seentransferring between the figure on the left and the figure on the right.

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Figure 11: The intensity of the populations over time, where the left graph shows the ground stateand the right the excited state, for λ = 360 nm.

Figure 12 and 13 show the same calculations as in figures 10 and 11, but with a laser pulse of higherenergy. Here the change in the populations is more distinct. It can be seen that the probability thatthe molecule will be moving on the excited potential surface is much higher than in the case withλ = 360 nm. In the right graph in Figure 13 it can be seen that there is a probability that themolecule dissociates into two ions. This can bee seen since there is a probability for an increased intermolecular distance for the excited state.

Figure 12: The populations when solving with a grid N = 2048 and λ = 335 nm.

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Figure 13: The intensity of the populations over time, where the left graph shows the ground stateand the right the excited state, for λ = 335 nm.

Figures 14 and 15 once again shows the behaviour of the populations, but this time for a wavelengthof 310 nm. Here the population on the excited state is less intense than in the case with λ = 335,even though λ = 310 translates to a higher energy perturbation. There is still a probability that themolecule dissociates into two ions, even if it is smaller than the previous case.

Figure 14: The populations when solving with a grid N = 2048 and λ = 310 nm.

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Figure 15: The intensity of the populations over time, where the left graph shows the ground stateand the right the excited state, for λ = 310 nm.

5 Discussion

The three first eigenenergies for a harmonic oscillator were computed numerically with centered finitedifferences of different order of accuracy. Since the eigenenergies converged to the analytic values forall methods, all methods were assumed to converge. The relative error in the eigenenergies confirmedthe order of accuracy for a second, fourth, sixth and eighth order method. The error decreased witha factor 22, 24, 26 and 28 respectively when doubling the number of grid points, which was expected.The error decreased with the same factor for all three eigenenergies, but the errors were larger forthe third and second eigenenergies than for the first. This is due to that the eigenvectors containhigher frequencies that are badly represented on the discretisation grid. Looking at the slopes of thelog-log plot over discretisation error as a function of step size, tabulated in table 5, is another way todetermine the order of accuracy. The slopes almost correspond to the theoretic order of accuracy. Itcan be seen in the plot that the error goes to zero for smaller time steps for all methods but at a certainpoint the slope is no longer as steep as before. This especially occurs for the eighth order method forsmall steps. This is because the errors are of the same order of magnitude as the truncation errors.

The wave function in an harmonic oscillator potential oscillated as expected. However, when in-serting the analytic ground state for a harmonic oscillator potential as initial value, the numericalsolution oscillated. This happens because the exact analytic solution is not the exact solution to thediscretized model, due to the error in the discretized model. According to the numerical model theinitial value is not the ground state and the solution will therefore oscillate. To avoid this, the numer-ical solution to the ground state should be used instead. In the Morse potential the wave moved moreirregularly. This is not due to discretization errors, instead it is a real property of the anharmonicityof the Morse potential. If one would simulate long enough the solution would once again return to it’ssmooth appearance.

The time-dependent model problem was solved with an eighth order centered finite difference in spaceand Magnus expansion truncated after the first term in time. The error in the Magnus expansion andthe approximation of the integral in the Magnus expansion is of O((∆t)3). Figure 7 shows the errorin the solution depending on the step size. The slope of the curve is 7 in the steepest part and 2 for

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smaller steps. Since the slope corresponds to the order of accuracy it is probable that the accuracyin time affects the accuracy of the whole solution for small steps. The wave function did harmonicoscillations in the case with time dependent potential as well as shown in figure 8. The differencewas that the oscillations moved in a cosine pattern, just like the time dependent perturbation. It wasexpected that the time dependent potential should affect the wave to move in the same pattern as theperturbation.

In table 8 the data from tables 6 and 7 have been compiled to show a comparison of run time,memory requirements and number of calculations for the methods. It is evident that PSM with split-operators are superior to finite differences with simple time-stepping in every respect. Solving usingPSM with split-operators is more than ten thousand times faster than performing the same simulationwith finite differences and simple time-stepping, and requires about a thousand times less memory.

Table 8: Comparison of runtime, memory requirements and number of operations for finite differenceswith simple time-stepping and PSM with split-operators, when solving the Schrodinger equation witha Morse potential, both time-independent and time-dependent. N = 1024, ∆t = 0.1.

Finite differences Pseudo spectral

Time-indep. Time-dep. Time-indep Time-dep.

Runtime [s] 19.7463 7.53 · 103 0.473658 0.59669

Number of operations/time step O(N2) O(N3) O(NlogN) O(NlogN)

Memory usage [kb] 45140.00 4.95 · 108 5304.00 1.1 · 105

Comparing figure 9 and 10, it can be seen that to be able to see the behaviour of the population onthe excited potential surface, a large enough grid is needed. 512 gridpoints is too small, but 1024gridpoints works well enough. However, for all other simulations of the TDSE with coupled potentialsa grid of 2048 points were used.

A laser pulse with a wavelength corresponding to the difference in energy between the ground- andexcited states can be seen to produce a larger population transfer. Thus the molecule can with ahigher probability be excited with a laser pulse with the correct wavelength, duration and intensity.It can also be seen that there is a probability that the molecule dissociates into two ions for certainwavelengths. This can be seen by comparing figures 11, 13 and 15. A laser pulse with either λ = 360nm or λ = 310 nm give roughly the same population intensity on the excited state, while λ = 335 nmgives a noticeably more intense population on the excited state. For λ = 360 nm there is almost noprobability that the molecule dissociates, while for λ = 310 nm there is a small probability and forλ = 335 nm there is a greater probability. This is because of the quantized levels of energy in themolecule making λ = 335 nm simply fit better.

What is shown in these graphs are how the molecule acts during the perturbation process. At timezero there is a large population on the excited potential surface, at the equilibrium distance from thenuclei. The laser pulse pushes the molecule up to the excited state, causing the large population att = 0, and the molecule then moves toward the crossing of the potential surfaces, as can be seen infigure 3. Once it reaches the crossing, one part of the probability continues to move along the excitedstate, though a considerably larger portion of it instead follows along the ground state surface. Thisgives a nice representation of the quantum characteristics of the wave function. In the classical casethe position of the molecule would have been represented by a line, showing where it can be found,but since we are dealing with the quantum case we instead see the position as the population intensity.

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6 Conclusions

Finite differences paired with a simple time-stepping method such as expm() can be used to solvesimpler models of the TDSE, with adequate results. However, when working with more complex mod-els that include a time-dependent Hamiltonian these simple numerical methods become hopelesslydemanding in regards to compuational time, if one wants accurate results. For these problems it ismuch more suitable to use the pseudo-spectral method with split-operators, which is remarkably fasterwhen comparing with the simpler methods.

There is also a clear advantage to the pseudo-spectral method compared to finite differences in regardsto order of accuracy, since the order of accuracy for the pseudo-spectral method goes to infinity. Forthe time-stepping methods though, there is no difference in the order of accuracy between the split-operator method and the use of expm() with Magnus expansion. However the split-operator methodeliminates the need to use expm() with the use of cleverly diagonalized exponential matrices it can becalculated a lot faster. Another advantage lies in that it pairs nicely with the pseudo-spectral method.

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References

[1] T. S. Rose, M. J. Rosker, A. H. Zewail. Femtosecond realtime probing of reaction. IV. Thereactions of alkali halides. J. Chem. 1989 Dec; 91(12): 7415-7436. DOI:10.1063/1.457266

[2] Tannor. Introduction to Quantum Mechanics: A Time-Dependent Perspective. Sausalito: Uni-versity Science Books; 2007.

[3] Fornberg B. Generation of Finite Difference Formulas on Arbitrarily Spaced Grids. Math Comput.1988 Oct; 51(184): 699–706. DOI:10.1090/S0025-5718-1988-0935077-0

[4] Gasiorowicz S. Quantumm Physics. Third edition. Hoboken, NJ: John Wiley & Sons, Inc.; 2003

[5] Kormann K, Holmgren S, Karlsson H. Accurate time propagation for the Schrodinger equa-tion with an explicitly time-dependent Hamiltonian. J Chem Phys. 2008 May; 128(18).DOI:10.1063/1.2916581.

[6] Magnus W. On the Exponential Solution of Differential Equations for a Linear Operator. CommunPure Appl Math. 1954 Nov; 7(4): 649-673. DOI:10.1002/cpa.3160070404

[7] Strang, G. On the Construction and Comparison of Difference Schemes. SIAM J Numer Anal.1968 Sep; 5(3): 506-517. DOI:10.1137/0705041

[8] Schwendner P. Seyl F. Schinke R. Photodissociation of Ar+2 in strong laser fields. J. Chem. Phys.

(1997); 217: 233-247.

[9] LeVeque R. J. Finite difference methods for ordinary and partial differential equations: steady-state and time-dependent problems. Philadelphia: Soc. for industrial and applied mathematics;2007

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A Parameters

Here the values for the parameters that have been used for the simulations are tabulated, scaledaccording to atomic units.

Table 9: Values for the parameters used for the Harmonic oscillator, equation 5.

Value [au]

ω 0.05

r0 2

α 2

Table 10: Values for the parameters used for the Morse potential, equation 7.

Value [au]

De 12440

a 1.875

r0 2.656

Table 11: Values for the parameters used for the potentials used for the NaI-system, described inequations 8, 10 and 12.

Value [au] Value [au]

A2 101.47 B2 3

ρ 0.66 λ1 2.76

λ2 43.44 ∆E 0.076

A 18.36 β 0.77

A12 0.002 β12 0.19

Ae 499.47 ξ 4.1341 · 103

η 1 · 10−3

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