-30 -20 -10 0 10 20 30
0 5 10 15 20 25 30 35
Electron coordinate (au)
Variational approach
(a)
-30 -20 -10 0 10 20 30
0 5 10 15 20 25 30 35
Electron coordinate (au)
Time (fs)
(b)
10-4 10-3 10-2 10-1 100
-30 -20 -10 0 10 20 30
0 5 10 15 20 25 30 35
TDDFT approach
(c)
10-4 10-3 10-2 10-1 100
-30 -20 -10 0 10 20 30
0 5 10 15 20 25 30 35 Time (fs)
(d)
Figure 4.3: Time evolution of the electronic density in H+2 for (a) 256×512, (b) 256×4096, (c) 256×512, and (d) 256×4096 grid points.
which is not correct. When the grid size is increased with 256×1024 grid points (see Fig. 4.4(c)), the situation improves and attains 0%survival probability at 35 fs. But even this does not reproduce the behavior seen for the TDDFT approach. With the remaining sets of grid points, i.e. 256×2048 (see Fig. 4.4(e)) and 256×4096 (see Fig.
4.4(g)), the situation improves and the results of the TDDFT approach are repro-duced. Next we look at the photodissociation probability obtained using TDDFT approach plotted in Figs. 4.4(b), 4.4(d), 4.4(f), and 4.4(h). The TDDFT approach indicates that around 93% of the system has photodissociated and a saturation is attained from 30 fs onwards. The Hartree variational approach for the smallest grid size shows a very poor performance by indicating around 50%photodissociation (see Fig. 4.4(b)). The results gradually improve with increasing number of grid points (see Fig. 4.4(d) and 4.4(f)) and finally an accurate result is obtained for the largest grid size (see Fig. 4.4(h)).
4.4 Discussion
In this section we have presented two different approaches for including absorbing boundaries in the mean-field approximation. Adding the absorbing directly in the
0 5 10 15 20 25 30 35 0.00
0.25 0.50 0.75 1.00
PSU
TDDFT approach Variational approach 0 5 10 15 20 25 30 35
0.00 0.25 0.50 0.75 1.00
0 5 10 15 20 25 30 35 0.00
0.25 0.50 0.75 1.00
0 5 10 15 20 25 30 35 0.00
0.25 0.50 0.75 1.00
0 5 10 15 20 25 30 35
Time (fs)
0.00 0.25 0.50 0.75 1.00
PPD
0 5 10 15 20 25 30 35
Time (fs)
0.00 0.25 0.50 0.75 1.00
0 5 10 15 20 25 30 35
Time (fs)
0.00 0.25 0.50 0.75 1.00
0 5 10 15 20 25 30 35
Time (fs)
0.00 0.25 0.50 0.75 1.00
(a) (c) (e) (g)
(b) (d) (f) (h)
Figure 4.4: Survival and photodissociation probabilities obtained for H+2 using 256×512 (panels (a),(b)), 256×1024 (panels (c),(d)), 256×2048 (panels (e),(f)), and 256×4096 (panels (g),(h)) grid points as function of time.
single-particle equations (termed here as the TDDFT approach) yields results almost independent of the grid size while the variational approach leads to deviations for small grids. Therefore, although the variational approach seems more systematic and sound at first sight, it is inferior to the ad-hoc TDDFT approach. The reason lies in the unphysical increase of the mean-field terms upon absorption of density at the grid boundary. This behavior arises since the variational equations of motion are derived by using a product ansatz for the total wave function. For example, the total wave function is invariant under multiplying two of the single-particle orbitals by factors α and α1, respectively. This can be interpreted such that the magnitudes of the individual orbital norms cannot have physical consequences in the variational approach. Our finding is important in view of the wide-spread use of effective-potential methods. We expect that a similar effect is present in the multiconfiguration time-dependent Hartree [78] and Hartree-Fock [99] methods in the case that only few configurations are included [177].
Chapter 5 Summary
Since its beginning in 1958, the rapid advances in the field of laser technology have made it possible to produce ultra-short laser pulses of high intensities. These laser fields can be so strong that they actually compete with the Coulomb forces that keep the electrons and nuclei bound together, in governing the system dynamics. Inter-action of molecules with such intense fields gives rise to highly nonlinear processes that cannot be explained with a perturbative treatment. A non-perturbative theory is required that treats the dynamics of the electrons and the nuclei fully quantum mechanically. The nonlinear processes observed in laser-matter interactions can be described by solving the time-dependent Schrödinger equation (TDSE). However, the TDSE can be solved exactly only for systems with very few degrees of freedom.
As the degrees of freedom increase, the computational requirements, i.e. required memory and computational time, grow exponentially, making the problem almost intractable.
Various approximation methods have already been proposed to overcome the ex-ponential scaling problem. The Born-Oppenheimer (BO) approximation is one of the oldest and simplest approaches to solve the problem by decoupling electronic and nuclear motion and solving the Schrödinger equation for electron parametrically for each nuclear configuration. In the situation where a system is interacting with a strong field, however, the BO approximation might not accurately describe the system dynamics.
In density functional theory (DFT) calculations, the system density is the basic variable and all the observables are functionals of this density. In most of the earlier static as well as time-dependent DFT calculations, the nuclei were either treated as static or classical. When the system is subjected to strong fields, the electronic and nuclear degrees of freedom are strongly coupled and an equal and unified quantum mechanical treatment of both is desirable. The multicomponent density functional theory (MCDFT) approach for stationary and time-dependent systems is able to treat the electrons and nuclei on equal footing and fully quantum mechanically.
An overview of time-dependent MCDFT (TD-MCDFT) was given in Chapter 2. In TD-MCDFT the basic variables are the electronic and the nuclear densities. All the non-BO effects are present in the solution of the time-dependent multicomponent Kohn-Sham (TD-MCKS) equations as no approximation was made in their deriva-tion. The electronic TD-MCKS equations are not solved parametrically for a fixed nuclear configuration but the information of the nuclear configuration is provided by the TD-MCKS potentials as they are functionals of the nuclear density. Similarly any change in the electronic density is conveyed to the nuclear TD-MCKS potential through the functional dependence on the electronic density. In practice the time-dependent exchange-correlation potentials, which are a part of the KS potentials and which contain the relevant many-body effects, are not easily calculable. They have to be approximated and the success of TD-MCDFT rests on the quality of approximations employed.
The TD-MCDFT was used to describe the coupled electron-nuclei dynamics in the example of a one-dimensional hydrogen molecular ion. In Section 2.3 we discussed and derived an algorithm to calculate the exact KS potentials for this system. Com-plex KS orbitals were first constructed from the quasi-exact density. By inversion of the split-operator propagator, numerically exact KS potentials were obtained. Once the exact KS potentials are known, the exchange-correlation potentials can be easily calculated by substracting the Hartree and external potential contributions from the exact KS potentials. This approach works only for small model systems, but gives insight into the characteristics of the exchange-correlation potential.
In Sections 2.4 and 2.5 the photodissociation process in the hydrogen molecular ion was used to investigate the KS potentials in TD-MCDFT. We considered (i) a suitably prepared state dissociating without external field and (ii) the system in the presence of a strong field. The resulting nuclear KS potential was found to be a piecewise combination of ground-state BO potential energy surface (for small internuclear distances) and excited-state BO potential energy surface (for large internuclear distances).
The Hartree approximation neglects the correlation contributions. The correspond-ing Hartree-only KS potential has an overbindcorrespond-ing character for large internuclear distances. This deficiency affects both the static and time-dependent KS calcula-tions. Therefore, the Hartree approximation fails to describe the photodissociation process even qualitatively. We conclude that correlation effects play an important role in the description of the system dynamics.
The difficulties appearing in the TD-MCDFT approach can be circumvented by using an explicitly correlated ansatz for the many-body wave function in terms of single-particle orbitals. In Chapter 3, we introduced the multiconfiguration time-dependent Hartree (MCTDH) approach for a system of coupled electron and nuclei.
In the MCTDH method, the wave function is written as sum of products of single particle functions (SPFs). A brief introduction to the MCTDH method was given
71 in Section 3.1 and the equations of motion were discussed. We described two one-dimensional model systems, namely the hydrogen molecular ion and the hydrogen molecule. Ground-state properties such as ground-state energies, nuclear and elec-tronic densities were calculated using one SPF (Hartree approach), two to eight SPFs (MCTDH approach), and exact calculations in Section 3.2. It was shown that two to four SPFs were enough to describe the ground-state properties for the hydro-gen molecular ion while four to eight SPFs were required for the hydrohydro-gen molecule due to the additional electron. This indicates that the required number of SPFs increases with number of degrees of freedom.
In Section 3.3 the model hydrogen molecular ion was subjected to laser pulses with long and short wavelengths. This system has been studied extensively in the past.
So our purpose was not to study the model system but to study the applicability of the MCTDH method to a system with coupled electronic and nuclear motion. The model system considered here was rather simple, but very useful as it showed a rich dynamics and exhibited many of the salient features that are characteristic of the behavior of diatomic molecules in intense fields. The failure of the Hartree approach is reflected in the too narrow shape of the ground-state nuclear density. The nuclear potential generated by this state has a large binding potential and in the presence of weak-intensity fields, the molecule is not able to absorb enough energy and overcome this binding force. The results of the MCTDH approach converge towards the exact results with increasing number of SPFs. We have also shown that depending on the observable the number of configurations required for numerically converged results increases for increasing peak intensity of the laser pulse, i.e. the correlation between electronic and nuclear motion is increased by the field. Nevertheless, the number of required SPFs was moderate.
Intense laser fields make the system undergo ionization and dissociation. In an exact calculation, the wave function is expanded in a complete basis set which gives an exact description of all details of the evolving wave function. In the case of MCTDH, the assigned number of finite SPFs does not cover a complete basis set but the cal-culation gives the best possible description of the evolving wave function for a given number of SPFs. With increasing number of SPFs, the basis set becomes complete and the evolution becomes exact. The advantage of using the MCTDH method is that the accuracy of the calculation can be improved systematically while keeping the computational requirements manageable compared to the exact calculations.
When considering a three particle system as the hydrogen molecule, the exact cal-culations become very time consuming and there is a limitation on the grid size.
Also it is interesting to study the additional electron-electron correlation effects in the case of the hydrogen molecule. The MCTDH method makes it feasible to study the dynamics of systems with large grid sizes to capture the ionization and dissociation dynamics. As applications we studied in Section 3.4 the two-center interference effect in high-order harmonic generation (HHG) and lochfraß in the hydrogen molecule. In Section 3.4.1, the two-center interference effect was studied using the Hartree, MCTDH, and exact calculations for the hydrogen molecule. A
minimum in the HHG spectrum was observed for all types of calculations. The HHG spectra obtained from the Hartree and two-SPFs MCTDH calculations reproduced the peaks at the right positions but the magnitudes were not in agreement with the exact calculation. The emission time of the harmonic radiation was studied using the Gabor transform. The Gabor analysis of the exact calculation indicated that the harmonics were supressed in a region around harmonic order 30. The same was verified by integrating the Gabor density over all the emision times. The Hartree approach showed the harmonic suppression at the 35thharmonic order. Such a good qualitative agreement was not necessarily expected from the Hartree approach as it had failed to give reasonable results for the model hydrogen molecular ion. The two-SPFs MCTDH calculation produced the minimum at the right harmonic order as well as the MCTDH calculation with large number of SPFs.
In Section 3.4.2 we studied the lochfraß effect in the hydrogen and deuterium molecules for different pump wavelengths from the ultraviolet to the infrared range and found that the effect is more pronounced at short wavelengths. The interac-tion of these molecules with ultra-short intense fields led them to ionize rapidly and preferentially at large internuclear distances causing the ground state to deplete at those internuclear distances. A superposition of the ground and the first excited vibrational states in the electronic ground state continued to oscillate back and forth after the pulse was over. From this periodic oscillation we could extract the vibrational periods which were in correct correspondence with the energy difference between the two lowest vibrational states. These vibrational wave packets maintain the coherence for an extremely long time. An interesting observation was made by probing the lochfraß state at different time delays. We found additional oscillations with a period shorter than the vibrational periods, presumably due to population of higher vibrational levels.
In the TD-MCDFT and the MCTDH calculations we had to use absorbing bound-aries to avoid reflections from the grid boundbound-aries. In Chapter 4 we have studied the effects of absorbing boundaries on the mean-field calculations using two differ-ent approaches. In the variational approach, the absorbing boundaries were directly added to the (multiparticle) Hamiltonian and the equations of motion were obtained using the variational principle. In the TDDFT approach, the absorbing boundaries are added to the effective potentials. We have shown that the size of the grid has almost no effect on the TDDFT results while the results are grid-size dependent for the variational calculations. The variational mean fields led to incorrect results for small grids. This could be traced back to an effective increase of the mean fields due to renormalization of the interior wave function after the absorption at the grid boundaries. The TDDFT mean fields were not affected by this problem. Thus, it was shown that the TDDFT approach performed better than the variational approach as the TDDFT results are insensitive to the grid size.
We conclude with a note that a proper description of electron-nuclear coupling is necessary, and not only their individual dynamics, to understand the internal working of an atomic/molecular system and to describe its dynamics in the presence
73 of an external field. The Hartree approximation, with its inherant limitation to describe the electron-nuclear coupling, fails to provide a dependable description of the system dynamics in the presence or absence of an external field. With the help of model multiparticle systems, we have presented two approaches, namely TD-MCDFT and MCTDH, that go beyond the Hartree approximation and adequately describe electron-nuclear correlation effects.
Appendix A
Born-Oppenheimer approximation
The main assumption in the Born-Oppenheimer (BO) approximation [15, 17] is that the nuclear and the electronic motion can be separated. The Hamiltonian for the complete system of Ne electrons and Nn nuclei reads
Hb =Tbn(R) +Tbe(r) +cWen(r,R) +Wcnn(R) +Wcee(r). (A.0.1) Here R is the set of nuclear coordinates and r is the set of electronic coordinates.
Tbn/e, cWnn, and cWee represent the nuclear/electronic kinetic energy, nuclear-nuclear Coulomb potential, and electron-electron Coulomb potential. All the terms men-tioned in (A.0.1) can be separated for electrons and nuclei except the termcWen, the electron-nuclei Coulomb potential which depends on both the set of coordinates. If we consider the fact that the nuclei are far more massive then the electrons, it allows us to treat the nuclei as fixed with respect the electron motion. R can be fixed at some value Rα and the electronic eigenstates can be found. This electronic wave function depends only parametrically on R. This approach can be performed for a whole range ofR giving a potential energy curve along which the nuclei move. The separated wave function, with electronic wave function depending parametrically on Ris written as
ψ(r,R) =φ(r;R)χ(R) (A.0.2) Initially we neglectTbn as it is smaller thanTbe. For a fixed nuclear configuration the electronic Hamiltonian is
Hbe=Tbe(r) +Wcee(r) +Wcen(r;R), (A.0.3) such that
Hbeφ(r;R) = Eeφ(r;R). (A.0.4) This is the clamped-nuclei Schrödinger equation. We have omitted cWnn(R) in Eq.
(A.0.3). Its inclusion only shifts the eigenvalues by anR-dependent constant. Here
Ee is the electronic energy eigenvalue for a fixed R. By repeatedly solving the electronic Schrödinger equation for all R, one obtains Ee as a function of R. The next step is to solve the Schrödinger equation for the nuclei by solving
Tbn+cWnn+Ee(R)
χ(R) = Eχ(R) (A.0.5)
The term cWnn +Ee(R) is called the Born-Oppenheimer potential energy surface (BOPES). The eigenvalue E is the total energy of the molecule that includes con-tributions from electrons, nuclear vibrations, and overall rotation and translation of the molecule.
Considering the application of BO approximation to time-dependent systems, the wave function is expanded as
ψ(r,R, t) =X
k
φk(r;R)χk(R, t). (A.0.6)
Substituting Eq. (A.0.6) in the TDSE, we get i∂
∂t X
k
φk(r;R)χk(R, t) = H(t)b X
k
φk(r;R)χk(R, t) (A.0.7) Now multiplying withφ∗l from left and integrating over all the electronic coordinates, we get
i∂χl(R, t)
∂t =
"
− X
α
1 2Mα∇2R
α+hφl|Hbe+cWnn|φli
#
χl(R, t)
−X
k
"
X
α
1
2Mαhφl|∇2R
α|φki+X
α
1
Mαhφl|∇Rα|φki∇Rα
#
χk(R, t). (A.0.8) Eq. (A.0.8) is the time-dependent equation of motion for the nuclear wave packet.
To arrive at Eq. (A.0.8) we have assumed that the electronic functions are or-thonormal. In Eq. (A.0.8) hφl|Hbe+Wcnn|φli are the time-dependent BO potentials.
The last two terms in Eq. (A.0.8) represent the non-adiabatic couplings and are neglected in a BO calculation. These terms are responsible for mixing of different BO electronic states in the absence of an external field. If the system is subjected to an external field then it will induce a transition from one BO surface to another.
To describe this transition, the dipole matrix elements will appear in Eq. (A.0.8).
Appendix B
Derrivation of MCTDH equations of motion
The multi-configurational wave function ansatz reads Ψ(Q1, . . . , Qf, t) =
n1
X
j1=1
. . .
nf
X
jf=1
Aj1...jf(t) Yf κ=1
ϕ(κ)jκ (Qκ, t), (B.0.1) where Q1, . . . , Qf are the particle coordinates, the Aj1...jf denote the MCTDH ex-pansion coefficients, and ϕ(κ)jκ are the nκ single-particle functions (SPFs) for each degree of freedom κ. A simplification is made in the notation by introducing a composite indexJ=(j1. . . jf) and the configurationsΦJ:
AJ =Aj1...jf, ΦJ = Yf κ=1
ϕ(κ)jκ . (B.0.2) The numbers nκ of SPFs that are used to build the configurations can be chosen differently for each degree of freedom.
The linear combination of Hartree products of(f −1)SPFs that do not contain the SPFs for the coordinateQκ,
Ψ(κ)l = X
j1
. . .X
jκ−1
X
jκ+1
. . .X
jf
Aj1...jκ−1ljκ−1...jf ×ϕ(1)j1 . . . ϕ(κ−1)jκ−1 ϕ(κ+1)jκ+1 . . . ϕ(f)jf
= Xκ J
AJlκϕ(1)j1 . . . ϕ(κ−1)jκ−1 ϕ(κ+1)jκ+1 . . . ϕ(f)jf (B.0.3) is defined as the single-hole function. HereJlκ denotes a composite indexJ with the κth entry set at l, and Xκ
J is the sum over the indices for all degrees of freedom excluding theκth.
Using the composite index defined in Eq. (B.0.2) and the single-hole function of Eq.
(B.0.3), we rewrite the multiconfiguration wave function as Ψ =
nκ
X
j=1
ϕ(κ)j Ψ(κ)j . (B.0.4)
The single-hole functions of Eq. (B.0.3) are then used to define the mean-fields hHi(κ)jl =D
Ψ(κ)j |H|Ψ(κ)l E
(B.0.5) wherehHi(κ)jl is an operator acting on theκthdegree of freedom, and density matrices
ρ(κ)jl = D
Ψ(κ)j |Ψ(κ)l E
(B.0.6)
= Xκ J′
Xκ J
A∗J′κ j AJlκ.
To obtain the equations of motion for the system we substitute the multiconfigu-rational ansatz of Eq. (B.0.4) in the Dirac-Frenkel variational principle hδΨ|H − i∂t|Ψi= 0 [133]. The constraints hϕ(κ)j (0)|ϕ(κ)l (0)i=δjl and hϕ(κ)j (t)|i∂t|ϕ(κ)j (t)i= 0 are applied to assure a unique propagation. Variation with respect to the coefficients AJ gives,
δ δAJ
Ψ
Hb −i∂t
Ψ
= 0 hΦJ|Hb|Ψi − hΦJ|iΨ˙i = 0
hΦJ|iΨ˙i = hΦJ|Hb|Ψi hΦJ|i
Xf κ=1
nκ
X
j=1
˙
ϕ(κ)j Ψ(κ)j i+hΦJ|iX
J
A˙JΦJi = hΦJ|Hb|Ψi iA˙J = X
L
hΦJ|Hb|ΦLiAL (B.0.7) Eq. (B.0.7) is the equation of motion for the coefficients.
79 Variation with respect to the SPFs gives,
* δ δϕ(κ)j Ψ
Hb −i∂t
Ψ
+
= 0 hΨ(κ)j |Hb|Ψi − hΨ(κ)j |i∂t|Ψi = 0 hΨ(κ)j |Hb|Ψi − hΨ(κ)j |X
J
iA˙JΦJi − hΨ(κ)j | Xf κ′=1
nκ′
X
l=1
iϕ˙(κ
′) l Ψ(κ
′)
l i = 0
hΨ(κ)j |Hb|Ψi −iX
J
hΨ(κ)j |ΦJiA˙J −ihΨ(κ)j | Xf κ′=1
nκ′
X
l=1
˙
ϕ(κl ′)Ψ(κl ′)i = 0 (B.0.8)
Using the equation of motion for the coefficients, Eq. (B.0.7), and the definition of density matrices, Eq. (B.0.8), we get
hΨ(κ)j |Hb|Ψi −X
J
hΨ(κ)j |ΦJihΦJ|Hb|Ψi −i
nκ
X
l=1
ρ(κ)jl ϕ˙(κ)l = 0
hΨ(κ)j |Hb|Ψi −P(κ)hΨ(κ)j |Hb|Ψi −i
nκ
X
l=1
ρ(κ)jl ϕ˙(κ)l = 0
(1−P(κ))hΨ(κ)j |Hb|Ψi −i
nκ
X
l=1
ρ(κ)jl ϕ˙(κ)l = 0
(1−P(κ))
nκ
X
l=1
hHbijlϕ(κ)l −i
nκ
X
l=1
ρ(κ)jl ϕ˙(κ)l = 0 (B.0.9)
Rearranging Eq. (B.0.9) we get the equations of motion for the SPFs as iϕ˙(κ) = 1−P(κ)
ρ(κ)−1
hHi(κ)ϕ(κ). (B.0.10) Here a vector notation has been adopted for the SPFs withϕ(κ) = (ϕ(κ)1 , . . . , ϕ(κ)nκ)T. Furthermore,P(κ) =Pnκ
j=1|ϕ(κ)j ihϕ(κ)j | is the projector on the space spanned by the SPFs for the κth degree of freedom, ρ(κ) is the density matrix, and hHi(κ) is the matrix of mean fields.
The MCTDH wave function and the equations of motion for the 1D H+2 molecular
ion using nκ=2 are
Ψ(Q1, Q2, t) = X2
j1=1
X2
j2=1
aj1j2
Y2
κ=1
ϕ(κ)jκ (Qκ, t), (B.0.11) iA˙J = X
L
hΦj|H|ΦliAL, (B.0.12)
iϕ˙(κ)n = 1−P(κ)X2
j=1
ρ(κ)−1
nj
X2
l=1
hHi(κ)jl ϕ(κ)l . (B.0.13) The generalization to higher degrees of freedom is straight forward.
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Publications
• C. Jhala and M. Lein, Approaches to Time-Dependent Multicomponent Dy-namics, ISAMP Newsletter 5, 12 (2010).
• C. Jhala and M. Lein,Multiconfiguration time-dependent Hartree approach for electron-nuclear correlation in strong laser fields, Phys. Rev. A 81, 063421 (2010).
• C. Jhala, I. Dreissigacker, and M. Lein, Absorbing boundaries in the mean-field approximation, Phys. Rev. A82, 063415 (2010).
Acknowledgments
I am deeply indebted to my Ph.D. supervisor Prof. Dr. Manfred Lein, whose guidance, stimulating discussions, suggestions, and encouragement helped me at each and every moment during the research and in writing of this thesis.
I am grateful to Prof. Dr. Christiane Koch for agreeing to be the second referee.
I wish to thank my colleagues Ciprian Chirila, Elmar van der Zwan, Ingo Dreissi-gacker, Jost Henkel, Maria Tudorovskaya and Jing Zhao for creating a wonderful and friendly environment in the group. The discussions, general and scientific, I had with them were encouraging as well as exciting. I am grateful to Ingo and Manfred for assistance with the German translation of the thesis summary.
My parents, Ashok and Kunjal, receive my deepest gratitude and love for their constant and unconditional support. I thank my beloved son Tanmay for readily taking every single opportunity to distract me from the work and keeping the home lively and cheerful with his sweet chitchats!