Present address: ]Max Planck Institute for the Structure and Dynamics of
Matter, Luruper Chaussee 149, 22761 Hamburg, Germany

# Long-range dispersion forces between molecules subject to ultra-short
optical pulses from ab initio calculations

Micael J. T. Oliveira  [  nanomat/QMAT/CESAM, University of Liège, B-4000,
Liège, Belgium  European Theoretical Spectroscopy Facility http://www.etsf.eu
Ksenia Komarova  Theoretical Physical Chemistry, MolSys, University of Liège,
B-4000, Liège, Belgium  Françoise Remacle  Theoretical Physical Chemistry,
MolSys, University of Liège, B-4000, Liège, Belgium  Matthieu J. Verstraete
<EMAIL_ADDRESS>nanomat/QMAT/CESAM, allée du 6 août 19 B5a,
University of Liège, B-4000, Liège, Belgium  European Theoretical Spectroscopy
Facility http://www.etsf.eu

###### Abstract

London-van der Waals dispersion forces are a fundamental component of
condensed matter systems, biological processes, and self-assembly. In this
letter we propose a method to calculate the C6 coefficients that characterize
dispersion forces in the non-retarded regime for molecules in a coherent
superposition of excited states. Several ultrafast femtosecond pump-probe
schemes are investigated. We apply the method to LiH molecules and show that
their London-van der Waals interaction can change dramatically after the
interaction with the pump pulse. The pulse modulates the C6 coefficients, and
the interplay between polarization, orientation of the molecules, and the
dipole fields gives rise to a rich variety of combinations.

The London-van der Waals dispersion forces arising from instantaneously
induced dipoles in molecules are a key ingredient in a wide range of phenomena
in physics, chemistry, and biology. Among these, one finds the condensation of
non-polar gases, the self-assembly of nanostructures, and the folding and
dynamics of proteins Israelachvili (2011). Therefore, the ability to control
and manipulate dispersion forces between atoms and molecules is of great
importance. Because those dispersion interactions depend crucially on the
electronic properties of the molecular systems, a simple route to achieve this
would consist in manipulating their electronic states. Up to now, the vast
majority of experimental and first-principles studies of dispersion forces
focused interacting systems in their electronic ground-state. However,
experimental measurements have been made of the van der Waals (vdW)
interaction between Rydberg atoms Béguin et al. (2013) and the problem of
calculating the interaction between excited atoms or molecules has attracted
strong attention recently Power and Thirunamachandran (1995); Sherkunov
(2005); Donaire et al. (2015); Barcellona et al. (2016).

The recent development of ultra-short optical pulses has given researchers
unprecedented control over electronic degrees of freedom Vrakking (2014).
These pulses, tailored in their frequency and envelope, allow the generation
of a strongly out of equilibrium population of electronic states. Since the
generation of the electron wave-packet happens before any significant movement
of the atomic nuclei, this could be used to steer chemical reactions Remacle
and Levine (2006).

In this Letter, we show for the first time how the London interaction
coefficients between molecules can be modified by an ultrafast laser pump
pulse. We describe our method, which employs either standard quantum chemistry
or time-dependent density functional theory (TD-DFT) to compute the excited
states. We show how this method can be used to control the dispersion
interaction between LiH molecules subject to pulses of Infrared (IR) or
Ultraviolet (UV) light. We find that dispersion coefficients can increase by
orders of magnitude in a superposition of states, and even become negative.

There are many methods available to solve the time-dependent Schrödinger
equation describing the interaction of a molecule with a pump pulse. One
option is to do so numerically in a basis of multi-determinant field-free
electronic states that diagonalize the stationary electronic Hamiltonian. The
eigenvalues and wavefunctions of the field-free electronic states can be
obtained using any suitable electronic structure method: configuration
interaction (CI) Krause et al. (2005); Sonk and Schlegel (2012); Raghunathan
and Nest (2012); Bauch et al. (2014), coupled-cluster Sonk et al. (2011); Sonk
and Schlegel (2011), CAS-SCF Eade and Robb (1981); Jorgensen et al. (1981);
Roos (1987); Werner (1987); Mignolet et al. (2011), etc. As an alternative,
one could use methods with explicit propagation of orbitals in time, such as
TD-DFT Marques et al. (2012), TD-CI Rohringer et al. (2006), or TD-CAS Sato
and Ishikawa (2013); Miyagi and Madsen (2014); Remacle et al. (2007); Nest et
al. (2008).

The London interaction energy between two molecules $A$ and $B$ in the non-
retarded regime can be expanded in terms of the inter-molecular distance $R$.
The coefficients of the expansion are known as Hamacker constants Hamaker
(1937), and the leading term $-C_{6}^{\mathrm{AB}}/R^{6}$ can be expressed
using the dynamical polarizability tensors $\bm{\alpha}$ of the molecules, at
imaginary frequencies $i\omega$ McLachlan (1963). The polarizability
$\bm{\alpha}$ can be obtained either using linear response theory as a sum
over states involving the transition dipole moments and the energy
differences. Alternatively, it can be obtained by solving explicitly the time-
dependent equation for the molecule with an impulsive perturbation. The
polarizability is then obtained from the induced dipole moment as a function
of time Strubbe et al. (2012).

The case of the polarizability of a molecule in which one or more excited
states are populated is a less studied one. This problem is relevant in the
context of transient spectroscopy, where the interaction of a molecule with a
probe is preceded by the application of a pump pulse. As in the ground-state
case, the dynamical polarizabilities can be either calculated through explicit
real-time propagation of the time-dependent equations De Giovannini et al.
(2013), or expressed as a sum over states Perfetto and Stefanucci (2015).

In this work we have chosen to solve the time-dependent Schrödinger equation
describing the interaction of the system with the pump pulse numerically in
the basis of the multi-determinant field-free electronic states, and to
compute the polarizabilities at any given time-delay $T$ after the pump pulse
as a sum over states Perfetto and Stefanucci (2015):

$\alpha(i\omega,T)=\sum_{\alpha\beta\gamma}c^{*}_{\alpha}(T)c_{\beta}(T)d_{\alpha\gamma}d_{\gamma\beta}\\\
\times\left(\frac{1}{i(\omega+\eta)-E_{\gamma}+E_{\alpha}}-\frac{1}{i(\omega+\eta)+E_{\gamma}-E_{\beta}}\right)\,,$
(1)

where $c$ are the expansion coefficients of the coherent superposition of
states, $d$ are the transition dipole moments, and $E$ are the eigenenergies
of the states. The quantity $\eta$ is introduced to avoid the divergence of
some terms at zero frequency, and it corresponds to assuming that the excited
states have a finite lifetime Sakurai (1993). Calculating $\alpha$ through the
Laplace transform of an explicit real-time propagation De Giovannini et al.
(2013) gives the same results, to within numerical accuracy, as using Eq. 1.
The choice between the two methods is thus one of convenience.

Figure 1: Schematic representation of different geometrical arrangements of
molecules subject to ultra-short optical pump pulses: (a) Two molecules with
fixed relative orientations with respect to each other and to the pump pulse.
(b) Set of randomly oriented molecules. (c) Set of molecules aligned with
respect to the pump pulse.

Before proceeding with the calculation of the Hamacker constants, we consider
the possible experimental set ups where these can be measured. In particular,
one needs to consider how the molecules are oriented with respect to each
other and to the pump pulse, that is, with respect to the lab frame. A
schematic representation of the different cases considered is shown in Fig. 1.
In the first case, a single measurement is done and the relative orientation
of the molecules with respect to each other and to the lab frame are known and
remain fixed (Fig. 1(a)). This is probably the most difficult case to set up
experimentally. The second case considered is a sample of randomly oriented
molecules (Fig. 1(b)). In this case it is necessary to perform a rotational
average of the Hamacker constants over all possible orientations of the
molecules with respect to each other and to the lab frame. This yields the
well known expression:

$C_{6}^{\mathrm{AB}}=\frac{3}{\pi}\int_{0}^{\infty}\\!\mathrm{d}\omega\,\left\langle\bm{\alpha}^{\mathrm{A}}(i\omega)\right\rangle\left\langle\bm{\alpha}^{\mathrm{B}}(i\omega)\right\rangle\,,$
(2)

where the brackets denote rotational averaging. In the case of molecules in
their ground-state one obtains the simple analytical expression
$\left\langle\bm{\alpha}(i\omega)\right\rangle=\mathrm{Tr}[\bm{\alpha}(i\omega)]/3$
Andrews (2004). This expression is not valid in the presence of the pump
pulse, as the polarizabilities of the molecules will now also depend on their
relative orientation with respect to the lab frame: molecules with different
orientations will interact differently with the pump and will thus be driven
to a different superposition of states. This means that the rotational average
needs to be performed numerically to sample all possible relative
orientations.

We also consider a gas of molecules which have been aligned (e.g. as in Refs.
Stapelfeldt and Seideman (2003); Artamonov and Seideman (2012)) prior to the
pump pulse (Fig. 1(c)). Their interaction will now only depend on relative
positions, and experiments yield the corresponding partial average. This leads
to a simple analytical expression and only needs to consider a single
orientation of the molecules in the lab frame. (Formulae and derivations in
the Supplemental Information (SI)). In all cases we assume that the distance
between the molecules is large enough that there is no overlap between their
electronic densities, but still small enough so that the long-range
interaction is not in the retarded regime.

As a test system we have chosen the LiH molecule. Heteronuclear diatomics have
well-separated excited states which obey simple selection rules. The $\Sigma$
excited states exhibit different polarities and the $\Pi$ states have a
polarized electron density far from the internuclear axis Remacle et al.
(2007); Nest et al. (2008). Using the pump frequency and polarization we can
target specific excited states or superpositions. The LiH field-free states
are determined using CAS-SCF Eade and Robb (1981); Jorgensen et al. (1981);
Roos (1987); Werner (1987), which for LiH is close to the full CI limit
(details are the same as in Ref. Oliveira et al. (2015)). We choose three pump
pulses, with the same envelope, and different intensities and carrier
frequencies, so as to excite coherent superpositions of states with different
features in the beating electronic density, either along the molecular axis or
with a perpendicular component. The first pulse has an IR frequency which will
populate the 1$\Sigma$ state by a two photon process (3.1 eV) when the
polarization is parallel to the molecular axis. With a perpendicular
polarization the 1$\Pi$ transition is also accessible with two photons, given
the pulse bandwidth, but it is further from resonance. The two other pulses
carry UV frequencies resonant with the 1$\Sigma$ and 1$\Pi$ states (labeled
1$\Sigma$ and 1$\Pi$, resp.). As for the IR pulse, the character of the
populated states depends on the pulse polarization having a component parallel
($\Sigma$) or perpendicular ($\Pi$) to the molecular axis. Note that $\Sigma$
states can also be populated with a perpendicularly polarized pulse, through
further transitions from populated $\Pi$ states. All the pulse intensities are
chosen such that no significant populations of excited states occur above the
1$\Pi$ excited state, and we avoid ionizing regimes which would complicate the
dynamics. Detailed pulse parameters are in the SI.

All our calculations were carried out for the equilibrium geometry of the LiH
ground-state, as no significant motion of the nuclei should occur within the
time-delays considered. When calculating the polarizabilities, we have endowed
each electronic wave-packet with the same lifetime $\tau=$ 50 fs. This
lifetime is large enough to characterize the time evolution of the London
interaction, while processes causing dephasing of the electronic coherence,
like luminescence, fragmentation, or non-adiabatic coupling induced by nuclear
motion, should occur on longer time scales Ajay et al. (2016); Nikodem et al.
(2016, 2017). We note that the choice of the lifetime can have a significant
effect when calculating the polarizabilities: Terms in Eq. 1 where
$\gamma=\alpha$ or $\gamma=\beta$ will have a divergence when
$\omega\rightarrow 0$ and $\eta=2/\tau\rightarrow 0$ and will dominate the
integral of Eq. 2 when the lifetime is sufficiently large. Calculations
showing the dependence of the C6 on the lifetime are included in the SI.

Figure 2: Averaged C6 a system of randomly oriented LiH molecules after
interacting with an IR optical pump pulse and two UV pump pulses resonant with
the 1$\Sigma$ and 1$\Pi$ excited states, as a function of the time delay.

The orientation averaged C6, as a function of the time delay, are presented in
Fig. 2 for a system of randomly oriented LiH molecules subject to the IR pump
pulse and to the UV pump pulses resonant with the 1$\Sigma$ and 1$\Pi$ states.
The pump pulse builds a coherent superposition of the electronic states
Remacle et al. (2007). The oscillations as a function of the delay $T$
originate in the expansion coefficients $c_{\alpha}(T)$: after the pump pulse,
the populations of the different excited states remain constant, but the
coefficients acquire a phase $c_{\alpha}(T)=c_{\alpha}(0)e^{-iE_{\alpha}T}$.
Because the C6 coefficients depend on the product of polarizabilities, their
dependence on $T$ can be written as a linear combination of terms of the
following form: $e^{-i(E_{\alpha}-E_{\beta}+E_{\gamma}-E_{\delta})T}$. In Fig.
2 the main frequencies in the C6 oscillations are E${}_{1\Sigma}-$EGS and
2(E${}_{1\Sigma}-$EGS) for the IR pump pulse (the $\Pi$ state is further from
resonance); 2(E${}_{1\Sigma}-$EGS) for the 1$\Sigma$ pulse; and
E${}_{1\Sigma}+$E${}_{1\Pi}-2$EGS for the 1$\Pi$ pulse.

Figure 3: Average C6 coefficient of a system of aligned LiH molecules, as a
function of delay time and angle between pump polarization direction and
molecular axis. Upper panel: pump resonant with 1$\Sigma$; Lower panel:
resonant with 1$\Pi$.

The amplitude of the predicted C6 coefficients can be several orders of
magnitude larger than the ground state value of 243 a.u., and the time
averaged value of the C6 is quite different depending on the pump pulse. We
therefore expect a large measurable effect in the vdW forces when the
molecules are subject to specific pump pulses. This configuration (random
orientations) is the most representative for gas phase experiments. Even in
the pessimistic case of a short lifetime (tens or hundreds of fs), in light
atom molecular dimers the resulting forces should have time to influence
intermolecular distances, vibrational frequencies and other observables. The
net classical momentum $F\cdot\delta t$ gained is equivalent to that of the
ground state London force over many picoseconds. Longer lived electronic
superpositions will have the double effect of increasing further the C6
(integrating the divergence further towards $i\omega=0$) and allowing for more
time to move atoms subjected to enhanced vdW forces. Experiments by Despre et
al.Despré et al. (2015) recently demonstrated sustained coherence in the
presence of atomic motion, over at least tens of femtoseconds. Recent
calculations suggest electronic coherence can survive well over 100 fs in LiH
in the presence of nuclear motion Nikodem et al. (2016).

(a) (b) (c) (d) (e) (f)

Figure 4: C6 values between two LiH molecules as a function of delay time, for
selected relative orientations co-planar with the field polarization
$\vec{E}$. (a) The orientations of the molecules with respect to $\vec{E}$ and
to each other are given by the angles $\theta_{A}$, $\theta_{B}$, and
$\theta_{AB}$. (b)-(c) Pump resonant with the 1$\Sigma$ state. (d)-(e) Pump
resonant with the 1$\Pi$ state. (f) IR pump which populates both states. In
(b)-(e) only one molecule is rotated with respect to the lab frame, with the
other angles fixed. In (f) the orientations of both molecules are fixed, while
their relative position varies.

As a second example, we consider in Fig. 3 the C6 of a gas of aligned LiH
molecules, as a function of the time delay and of the angle between the
polarization and the collective molecular axis. The amplitude of the
oscillations is naturally largest when the population of the targeted excited
state is maximized. For the 1$\Sigma$ pump pulse, this corresponds to
$\theta=0$ and $\theta=180$, while for the 1$\Pi$ case the pulse polarization
should be perpendicular to the molecules ($\theta=90$). This setup is also
experimentally within reach, using a pre-pulse which sets up the molecules in
a rotational wave packet Stapelfeldt and Seideman (2003); Artamonov and
Seideman (2012).

In Fig. 4 we plot C6 values, as a function of the time-delay, between two LiH
molecules for different relative orientations and pump pulses. The aim is to
decompose two main effects that contribute to the orientationally averaged
results: the nature of the states present in the coherent superposition of
electronic states and the relative orientations of the molecules over which
the averages are done. To simplify the analysis, we only consider cases where
the pulse polarization and the two molecules are co-planar. The relative
orientations can then be specified by three angles in the lab frame set by the
field polarization: the polar angles of the main molecular axis of each
molecule, and the polar angle of the vector from one center of mass to the
other (Fig. 4(a)). We will now decompose the dependencies of the net C6
response on symmetry, geometry, and frequency resonance. In Fig. 4(b) the C6
values are plotted as a function of the angle between the first molecule and
the pump pulse. The population of the 1$\Sigma$ excited state of the second
molecule is maximized, with its axis parallel to the pump polarization, and a
1$\Sigma$ resonant carrier frequency. As above, the C6 values oscillate as a
function of time, but the sign and amplitude of the oscillations depend
strongly on the orientation of the first molecule. When the molecules are
parallel ($\theta_{A}=0^{\circ}$), they are driven to the same superposition
of excited states, with a significant population ($\sim 20\%$) in the
1$\Sigma$ state: the interaction between the molecules is then maximal. When
the first molecule is perpendicular ($\theta_{A}=90^{\circ}$), no $\Sigma$
states are populated and the interaction is much smaller. When the molecules
are anti-parallel ($\theta_{A}=180^{\circ}$), the interaction is again large,
but the populations and phases of the wave-packets are different. Therefore,
contrary to the first impression conveyed by Fig. 4(b), an average over
$\theta_{A}$, for each time delay, will give a finite value. This also
explains why the amplitude of the oscillations is smaller (but not zero) in
the randomized case of Fig. 2 than in Fig. 4. In Fig. 4(c) we also keep
molecule B fixed, but perpendicular to the polarization direction. As the
frequency is further from resonance, the 1$\Pi$ state is weakly populated. The
dependence of the C6 values with respect to the orientation of the first
molecule is quite different, and there are long time beatings. The shorter
beatings correspond to twice $E_{1\Sigma}-E_{GS}$ and the longer ones to twice
$E_{1\Pi}-E_{1\Sigma}$. The next two cases (Figs. 4(d)-(e)) are similar, but
populating the 1$\Pi$ instead of the 1$\Sigma$ state. This 1$\Pi$ population
is maximized when the molecule is perpendicular to the pump polarization. In
the last panel (Fig. 4(f)) we show a case where the IR pulse populates the
1$\Sigma$ state of molecule A ($\theta_{A}=0^{\circ}$) and the 1$\Pi$ state of
molecule B ($\theta_{B}=90^{\circ}$). The C6 shows a maximum for relative
orientation of the molecules at angles of 45 and 135∘.

In this Letter we present a method to calculate the C6 coefficients of
molecules subject to ultra-short laser pulses. Using LiH molecules as a test
case, we show that, by driving them to an appropriate superposition of excited
states, it is possible to considerably increase their London-van der Waals
interaction. The origin of the increase is clear in the integral form of the
C6 for partially occupied states, and should carry over to other molecules. A
denser set of excited states in other molecules may be detrimental if it
induces quick decoherence, or beneficial if more transitions participate in
the buildup of the interaction. The effects shown here should be measurable
experimentally. We have addressed several experimental configurations
(geometries, molecule alignments, laser frequencies), and our results are
quite robust. As in any pump-probe experiment, electronic coherence is key,
and we use conservative hypotheses for the wave packet lifetime. The
efficiency of experimental probes of the modified C6 will depend on their
sensitivity to atomic positions or (better) acquired momentum. We have finally
shown that it is possible to change selectively the interaction of molecules
that have specific relative orientations. This could be exploited to control
the self assembly of nanostructures, or to steer chemical reactions including
steric or docking effects.

This work was funded by the Belgian Fonds de la Recherche Fondamentale
Collective, through project number 2.4545.12 “Control of attosecond dynamics:
applications to molecular reactivity”. F. R. acknowledges support from the
Fonds National de la Recherche Scientifique (Belgium) and partial support by
the U.S. Department of Energy (DOE), Office of Science, Basic Energy Sciences
(BES) under Award #DE-SC0012628. Computer time was provided on Zenobe by the
Walloon Region under GA 1117545, and by PRACE on ARCHER (ThermoSpin and ACEID
3IP RI-312763). The authors thank COST Actions MP1306 EUSpec and CM1204 XLIC
for support.

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