Article pubs.acs.org/JPCA
Limits of the Creation of Electronic Wave Packets Using TimeDependent Density Functional Theory Shampa Raghunathan and Mathias Nest* Theoretische Chemie, Technische Universität München, Lichtenbergstrasse 4, 85747 Garching, Germany ABSTRACT: Explicitly time-dependent density functional theory (TDDFT) has often been suggested as the method of choice for controlling the correlated dynamics of many electron systems. However, it is not yet clear which control tasks can be achieved reliably and how this depends on the functionals used. In this article, we show that the control task of creating a simple wave packet, having a population of 50% in the excited state, can indeed be achieved if a certain condition is fulfilled. This result is in contrast to the observation that a full population inversion is extremely difficult to achieve. In addition, we identify a rule to predict when TDDFT produces the correct wave packet. To illustrate our findings, we study the molecules Li2C2, Li7OH, and B2N2CO using two different functionals as well as time-dependent Hartree−Fock (TDHF). To assess the performance of TDDFT and TDHF, we compare with timedependent configuration interaction calculations.
T
ime-dependent density functional theory (TDDFT)1 has become one of the most widely used methods to describe the real-time evolution of correlated many-electron systems. In principle, TDDFT is an exact theory, but, as in the case of ground state DFT, the exact exchange-correlation (xc) potential vxc is not known. Moreover, it can be shown2 that in the case of TDDFT, this potential depends on the entire history of the system, leading to memory terms in the equations of motion.3 As a result, most applications of real-time TDDFT are done in the adiabatic approximation, where the instantaneous density, ρ(t), is plugged into standard ground state xc functionals. Obviously, there is no guarantee that these functionals will generate the correct dynamics, although they have been found to work well in some cases, such as the nonlinear dynamics of electrons in metal clusters,4−8 and molecules.9−11 On the other hand, we found in a previous paper,12 as others before us,2,13 that adiabatic TDDFT is not able to perform the control task of a state-to-state transition, if simple parametrized laser pulses are used. So to paraphrase Burke et al.,13 although in ground state DFT there is a general consensus on which properties are captured by which functional, in TDDFT, we are still exploring even which properties are captured at all. All these considerations are about electron dynamics in the fixed nuclei approximation. Propagation times of more than 20 fs would require relieving this constraint,14,15 but even pure electron dynamics poses enough open questions. In this article, we address one of them, and show that even though a full population inversion is nearly impossible, equal populations of about 50% can be obtained reliably if a certain condition is fulfilled. In other words, we are considering here the coherent control task of creating a simple wave packet containing the ground state |0⟩ and one excited state |n⟩ with equal amplitude. Such a target state is not as far away from the ground state as in the © 2012 American Chemical Society
case of a complete population inversion. Thus, one can expect that TDDFT performs somewhat better for this less demanding task. We would like to emphasize that all eigenstates appearing in this paper refer to the true, interacting system of electrons. A wave packet as described above has the almost trivial timedependence |Ψ(t )⟩ = (e−iE0t / ℏ|0⟩ + e−iEnt / ℏ|n⟩)/ 2
(1)
which leads to easily observable features during a propagation. Three such features can be obtained from the time-dependent dipole moment ⟨μ⟩(t ) =
μ00 + μnn 2
+ 2μ0n cos((En − E0)t /ℏ)
(2)
where μ00 and μnn are the permanent dipole moments of the ground and excited states, respectively, and μ0n is the corresponding transition dipole moment. Equation 2 shows that information on whether the desired wave packet has been created can be extracted from the long time average of the oscillation (μ00 + μnn)/2, the amplitude 2μ0n, and the frequency (En − E0)/ℏ. In addition, all methods should show a similar energy uptake during the laser pulse. The usage of these four quantities means that we take a rather practical approach to the problem of excited states in TDDFT. We will not discuss here whether the states involved are actually stationary states according to the TDDFT equations of motion nor the nature of the Kohn−Sham determinant. We are aware that the four quantities mentioned above do not define a state uniquely, but of course, they should agree at least to within the limits that are Received: May 16, 2012 Revised: July 25, 2012 Published: July 26, 2012 8490
dx.doi.org/10.1021/jp3047483 | J. Phys. Chem. A 2012, 116, 8490−8493
The Journal of Physical Chemistry A
Article
dipole moment of the wave packet is shown as black line in Figure 1.
inherent if different electronic structure methods are used. This is our definition to decide whether wave packets that have been created are similar: All four quantities should agree at least qualitatively. In the following, we will compare these quantities obtained from TDDFT calculations for three molecules (Li2C2, Li7OH, and B2N2CO) using two xc functionals and time-dependent Hartree−Fock (TDHF) with time-dependent configuration interaction (TDCI) calculations. Details on the methods can be found, for example, in ref 12. To solve the time-dependent Kohn−Sham equations, we have used the Octopus code,16,17 in which a real-space, pseudopotential-based approach is used. In all calculations, we used norm-conserving, nonlocal, ionic Troullier−Martins18 pseudopotentials to model the electron−ion interactions. Two xc functionals are employed: the local-density approximation (LDA) with the Perdew−Zunger parametrization19 and the extended hybrid functional combined with Lee−Yang− Parr correlation functional (X3LYP).20 For the TDCI calculations, we used our own code21,22 with a 6-31G** basis set. We always performed two TDCI calculationsone with only single excitations and one with additional double excitations from an active spaceto make sure that the results are qualitatively stable. Only results with doubles included are reported. In the CISD calculations, the character of the target excited states is dominated by single excitations for all three molecules. In TDCI, it is very simple to achieve the desired control task: If a resonant excitation can be achieved, then this laser pulse is equivalent to a π/2-pulse, obeying the relation π f0 μ0n σ = (3) 2
Figure 1. Dipole z-component of Li2C2 as a function of time computed using different TD theoretical methods. Frequencies, amplitudes, and long time average are very similar for similar laser pulses. Half of the applied photon energy is absorbed for all TD calculations.
Will a similar pulse create a similar wave packet in TDDFT? Linear-response TDDFT excitation energies cannot be directly used in real-time TDDFT as an input for thehopefully resonantlaser frequency. Instead, we performed a frequency scan, that is, we repeated the calculation with several different laser frequencies and then identified those to which the molecule reacts strongly (in the sense of energy uptake), which indicates proximity to a resonance. The second parameter of the laser pulse, its field strength, has then been determined by the condition that half the photon energy is absorbed. Ideally, this leads to a population of 0.5 on the excited state. The optimum laser parameters are collected in Table 1 for various TD methods. TDDFT needs slightly higher field strength than TDCI, but this is a molecule-specific result and not generally valid.
among field strength, transition dipole moment, and duration. The pulse has to be just long enough to avoid multiphoton excitations. In the derivation, the laser pulse has a sine-square envelope between t = 0 and t = 2σ: f(t ) = f 0 sin 2(πt /2σ ) cos(ωt )
(4)
Table 1. Laser Parameters Used for Wave Packet Creation in Li2C2, Li7OH, and B2N2CO in Various Time-Dependent Theoriesa
In general, the excited state in the wave packet has been chosen according to two criteria: first, it should have a large transition dipole moment, and second, the excitation energy should not be too small because then a near-resonant laser pulse would require a too long propagation time. To evaluate whether a wave packet that is similar to the desired one has been created, we compare the frequency, amplitude, and average of the timedependent dipole moment (eq 2) from TDDFT calculations with those obtained by TDCI. We begin by looking at the molecule Li2C2. The orientation is such that the molecular axis is the z-axis. To form the wave packet, we choose the state |n = 8⟩ (see eq 1) because it has a large transition dipole moment of μ08,z = −2.399 ea0 at the TDCI level. Because of symmetry, the permanent dipole moments of both states are zero along the z direction. The TDCI results reported here were obtained choosing the doubly excited determinants from a (10, 10) active space, that is, the highest 10 valence electrons are distributed over 10 spatial orbitals. The desired wave packet can easily be created with TDCI because the excitation energy, E8 − E0 = 4.3 eV, is known, and after choosing a sufficiently long pulse of 20 fs (σ = 10 fs), eq 3 immediately gives the required field strength. Here, sufficiently long means that at the end, 51% of the population is in the ground state, and 47% is in the excited state. Here, we chose a z-polarized laser pulse. The resulting time-dependent
Li2C2
a
Li7OH
B2N2CO
methods
ω
f 0,z
ω
f 0,x
ω
f 0,z
CISD LDA X3LYP TDHF
4.3 5.2 4.8 5.3
0.001 584 0.002 0.001 85 0.001 80
2.4 2.4 2.3 3.0
0.001 44 0.000 17 0.000 13 0.000 50
6.9 6.0 6.3 7.5
0.013 1 0.007 0.007 0.008
Units of ω and f0,q are eV and Eh/ea0, respectively.
We can see from Figure 1 that the time-dependent z dipoles behave similarly for all methods, including TDHF. It should be noted that the propagation during the final few femtoseconds is practically field-free because of the sine-squared envelope of the laser pulse. All three characteristicsfrequency, amplitude, and long time averageare comparable. The differences among the curves is not larger than what can be expected from three quite different electronic structure methods. These three quantities are not sufficient to prove rigorously that the same states have been populated, but because they are quite similar, we believe that the states are similar, too. Moreover, the laser pulses that created these wave packets have been quite similar. Therefore, this is an example in which it is possible to achieve a control 8491
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task with TDCI and TDDFT in an analogous way. In addition, there is no significant difference between the two functionals. For Li7OH, the situation is slightly different. Again, the excited state in our target wave packet is |n = 8⟩, according to the CI calculation with doubles in a (14, 14) active space. This excited state has a relatively high transition dipole moment with μ08,x = 1.316 ea0, where x means roughly the vector from the center of the metal cluster to the OH adsorbate. If this transition is actually done, the permanent dipole moment should change from about −0.7ea0 to 0ea0. For the excitation of Li7OH, we use an x-polarized laser of 40 fs (σ = 20 fs). The parameters for the laser pulse used in TDCI again follow immediately from the CI calculation and the π/2 condition. In practice, at the end, 50% of the population is in the ground state, and 45% is in the target excited state. To perform the same control task in TDDFT and TDHF, we have followed the same strategy as in the case of Li2C2; however, we found that the field strength determined from the condition of absorption of half a photon gave poor results. Therefore, we made an additional intensity scan to get the dipole moments into agreement. Figure 2 shows the best agreement we could
Figure 3. Dipole z component of B2N2CO as a function of time computed using different TD theoretical methods. For TDDFT calculations, absorption energies are >1 photon energy.
creating a simple wave packet in three different molecules and with two different functionals was mixed. For the highly symmetric molecule Li2C2, similar laser pulses were found to achieve the control task. For Li7OH, this was slightly more difficult. Apart from the violation of the half-photon-absorption condition, the long time average also does not agree very well. Amplitude and frequency of the oscillating wave packet matched much better. The most difficult case was B2N2CO, in which barely the oscillation frequency is in agreement. The question that now arises is, is there a rule behind this? There seems to be no connection between the excitation energy and the quality of the TDDFT results. But the present results indicate that there is a connection between the quality with which the control task is performed and the difference between the permanent dipole moments of the ground and excited states: A larger difference leads to worse results. At first sight, this suggests a connection to the well-known charge transfer problem in TDDFT; however, only a short-range charge transfer is required in B2N2CO, and one would expect that the hybrid functional X3LYP performs much better in this case. As in our previous study,12 we found, that the performance is independent of the functional and that TDHF behaves like TDDFT. Further studies are required to elucidate whether a connection really exists and to determine which changes to the TDDFT scheme are required to improve its suitability for coherent control. Even if one stays within the adiabatic approximation, specific functionals for real-time applications might yield significant improvements.
Figure 2. Dipole x component of Li7OH as a function of time computed using different TD theoretical methods. For TDDFT calculations, absorption energies are
