Optical van-der-Waals forces in molecules: from electronic Bethe-Salpeter calculations to the many-body dispersion model

Molecular forces induced by optical excitations are connected to a wide range of phenomena, from chemical bond dissociation to intricate biological processes that underpin vision. Commonly, the description of optical excitations requires the solution of computationally demanding electronic Bethe-Salpeter equation (BSE). However, when studying non-covalent interactions in large-scale systems, more efficient methods are desirable. Here we introduce an effective approach based on coupled quantum Drude oscillators (cQDO) as represented by the many-body dispersion model. We find that the cQDO Hamiltonian yields semi-quantitative agreement with BSE calculations and that both attractive and repulsive optical van der Waals (vdW) forces can be induced by light. These optical-vdW interactions dominate over vdW dispersion in the long-distance regime, showing a complexity that grows with system size. Evidence of highly non-local forces in the human formaldehyde dehydrogenase 1MC5 protein suggests the ability to selectively activate collective molecular vibrations by photoabsorption, in agreement with recent experiments.

creation operators for the i-th electronic state (whose wave function we will denote as ψ i ). φ 0 is the KS Slater determinant relative to the electronic ground state, and v,c indices run over valence and conduction states, respectively. In the case of (optically most relevant) spin-singlet excitations, the BSE operator can be expressed aŝ Here,D is a diagonal term of the form D vc,v c = ( c − v )δ v,v δ c,c , i being the i-th KS energy eigenvalue.K x is conventionally indicated as the BSE exchange term, and provides a coupling between different occupied-virtual electronic transitions in the system. Assuming that KS orbitals are real-valued, its components are defined as: Finally, the so-called BSE direct term is written as where w(r, r , ω) is the screened Coulomb interaction.

Supplementary Note 2
We report in Supplementary Figure 1 a comparison between MBD and BSE excitation spectra for a set of six molecules, namely methane, ammonia, e-butadiene, naphtalene, pyrazine and 2 Supplementary Figure 1: Comparison between MBD and BSE excitation spectra for different molecules: a) methane, b) ammonia, c) e-butadiene, d) naphtalene, e) pyrazine and f) alle-octatetraene. The methane MBD spectrum is rigidly shifted by +2 eV, while no shift was introduced for other molecules, in order to evidence the overall tendency towards red shifts in MBD spectra. 3 all-e-octatetraene. The chosen molecules cover a relatively broad spectrum of sizes, and configurations. However, generally fair qualitative agreement is consistently found between MBD and BSE results. Overall, MBD spectra tend to be redshifted and squeezed over smaller frequency intervals, although exhibiting analogous features to BSE. Although MBD relies on quasiparticle approximations and simplified atomic excitation spectra, spectral similarity is likely preserved by the relevant role of geometry and symmetries in determining collective charge-displacement modes. In fact, detailed geometrical features are naturally detected by the MBD method, due to the presence of interatomic couplings. On the other hand, we recall that MBD is a coarsegrained model that does not account for delocalized charge transfer, or short-range correlation terms. In principle, one may compare single-atom effective MBD frequencies to the relevant electronic transitions of the system, although a one-to-one correspondence is clearly not possible. We note that in the specific case of H, MBD naturally provides a reasonable exciton energy of ∼ 11.5 eV, while pure PBE Kohn-Sham orbitals underestimate this energy gap by a factor of ∼ 3. We stress in any case that any rigid shift of the MBD exciton spectrum has no influence on optical vdW forces, since these are obtained from energy derivatives.

Supplementary Note 3
While London dispersion (computed from the MBD model as well) is essentially negligible in small molecules, in the case of larger systems it can become relevant. In Supplementary   Figure 2 we analyze a 1D chain composed of 100 C atoms. Here the interchain interaction energy due to London dispersion is larger than the exciton splitting up to the nm scale. At larger distance, though, exciton splitting prevails due to its slower power law decay. One can thus generally assume that London dispersion will acquire more relevance in larger structures, but photoinduced vdW forces will finally prevail in the long range limit. scaling. This is essentially due to a cancellation between first-order perturbative terms in Eq. 5) of the main paper, as a consequence of the vanishing total dipole moment. Second-order perturbative terms provide here the leading contribution to optical vdW interactions. On the other hand, in the case of linear carbyne-like C chains, the energy splitting of plasmon-like modes whose symmetry implies overall dipole cancellation can display R −5 scaling. The difference between benzene and linear chains resides in the overall symmetry of the mode, which determines different cancellation effects.
We analyze here in detail the case of carbyne-like chains. If the plasma-like excitation is longitudinal and antisymmetric (i.e. the wave has an odd number of nodes), then the total dipole is zero. The mode arising in a single chain will be characterized by a given dipole displacement at each atom, which we will indicate through the vector ξ. The elements ξ i (i = 1, .., N for N atoms) of this vector indicate the x (i.e. longitudinal) dipole components at each atom. By extending Eqs. 4) (main text) the splitting of this mode is: We now Taylor expand T starting from the midpoints of the two parallel chains (which we indicate asī andj, respectively), and exploit the antisymmetry of the mode, which implies 6 where x i is the x coordinate of atom i (being xī = 0). Since both ξ i and x i are antisymmetric with respect to the midpoint of the chain, the sum is non-vanishing. By performing an analogous expansion over the other chain, one finds that the leading contribution to the exciton splitting Hence, the interchain coupling is dominated in this case by the second derivative of the dipole-dipole tensor T , that yields the correct R −5 scaling in the long range limit. Hence one ultimately has an effective quadrupole-quadrupole coupling.
In the case of benzene, instead, one can focus for instance on exciton 1 (see Supplementary size were adopted to minimize spurious interactions with periodic replicas, and truncation of the Coulomb interaction for intrafragment direct and exchange terms was enforced in GW and BSE calculations to avoid spurious long-range exchange terms. The GLW package [4,5], and the BSE-simple [6] code (both available within the Quantum Espresso suite) were exploited.
The number of Kohn-Sham (KS) orbitals N bnd adopted in the different systems is reported in Supplementary Table 1 for reference. Calculations were run at the gamma point, expanding the GW polarizability over a basis with N w−prod auxiliary functions, that are automatically generated [5], based on the same plane-wave expansion exploited for the PBE-KS orbitals.
Here the index i runs over all collective MBD frequencies (6 in the diatomic case), and the en- Figure 3 we observe that finite temperatures can cause an effective reduction of the interchain interaction of parallel C chains, which is however negligible at room temperature. Force reduction becomes effective only when T approaches the lowest excitation energy of the system, i.e. beyond the typical melting temperature of most materials.
In Supplementary Figure 3 we analyse the temperature dependence of interfragment energy and force between two parallel C chains containing 100 atoms each. At room temperature the occupation of excited electronic levels is negligible even in the long chain, where relatively low excitation modes are found. The impact on interfragment energy and force is only visible at the 10 4 K scale. A ground-state theory is thus sufficient for the description of interfragment forces in the absence of optical excitations. We remark that phonons were not included in the theory, and are expected to yield larger finite-temperature effects.

Supplementary Note 7
The available MBD produces exciton frequencies, after dagonalization of the MBD Hamiltonian (see Eq. 4), main paper). The code reads molecular geometry and Hirshfeld volume ratios from the input, and associates an effective polarizability and oscillator frequency to each atom.
The interaction tensor is subsequently computed, and exciton frequencies are finally obtained