Effective screening of medium-assisted Van der Waals interactions between embedded particles
EE FFECTIVE SCREENING OF MEDIUM - ASSISTED V AN DER W AALS INTERACTIONS BETWEEN EMBEDDED PARTICLES
Johannes Fiedler ∗ Institute of PhysicsUniversity of FreiburgHermann-Herder-Str. 3, 79104 Freiburg, Germany [email protected]
Michael Walter †‡ FIT Freiburg Centre for InteractiveMaterials and Bioinspired Technologies,Georges-Köhler-Allee 105, 79110 Freiburg, Germany
Stefan Yoshi Buhmann § Institute of PhysicsUniversity of FreiburgHermann-Herder-Str. 3, 79104 Freiburg, GermanyFebruary 3, 2021 A BSTRACT
The effect of an implicit medium on dispersive interactions of particle pairs is discussed and simpleexpressions for the correction relative to vacuum are derived. We show that a single point Gaussquadrature leads to the intuitive result that the vacuum van der Waals C coefficient is screened by thepermittivity squared of the environment evaluated near to the resonance frequencies of the interactingparticles. This approximation should be particularly relevant if the medium is transparent at thesefrequencies. In the manuscript, we provide simple models and sets of parameters for commonly usedsolvents, atoms and small molecules. Van der Waals forces are the fundamental interactions between two neutral and polarisable particles [1, 2, 3]. Theseforces prevail in holding together many materials and play an important role in living organisms, such as geckoswalking on smooth surfaces [4]. They have also found increasing importance in technological applications such asmicroelectromechanical and nanoelectromechanical components [5]. During recent years these forces have beenwell-studied in several experiments [6, 7, 8, 9] and in theory [10, 11, 12, 13].Despite their Coulombic origin dispersive forces are among the weakest forces in nature. Time-dependent perturbationtheory suggests their interpretation as being caused by ground-state fluctuations of the electromagnetic fields. This viewhas been taken in Casimir theory [10] dealing with two dielectric plates in vacuum as well as in colloidal systems [14],namely the stabilisation of hydrophobic suspensions of particles in dilute electrolytes [15]. Alternative approachesderive dispersion forces from position-dependent ground-state energies of the coupled field–matter system [1, 10, 16].These descriptions are restricted to partners interacting in vacuum. Natural systems such as colloids or proteins areoften embedded in and environment such as a solvent or a matrix.The impact of an effective medium on dispersive interactions of two particles A , B is illustrated in Fig. 1. We adoptthe simplification of considering point particles characterised by their frequency-dependent polarizabilities α A , B ( ω ) embedded in an effective medium characterised by its frequency-dependent permittivity ε ( ω ) . In this picture the Van ∗ Department of Physics and Technology, University of Bergen, Allégaten 55, 5020 Bergen, Norway. † Cluster of Excellence livMatS @ FIT ‡ Frauenhofer IWM, MikroTribologie Centrum µ TC, Wöhlerstrasse 11, 79108 Freiburg, Germany § Institut für Physik, Universität Kassel, Heinrich-Plett-Str. 40, 34132 Kassel, Germany a r X i v : . [ c ond - m a t . m e s - h a ll ] F e b PREPRINT - F
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3, 2021 G ( r B , r A , ω ) G ( r A , r B , ω ) ε ( ω ) α A ( ω ) r A α B ( ω ) r B Figure 1: Illustration of two particles (with polarizabilities α A , B ( ω ) located at positions r A , B . The particles areembedded in a medium with permittivity ε ( ω ) (grey area) where a medium excluded area surrounding each particle isformed. Interactions described by the Greens functions G are indicated. See text for details.der Waals potential may be expressed as[13] U ( r A , r B ) = − (cid:126) µ π ∞ (cid:90) d ξ ξ Tr [ α A (i ξ ) · G ( r A , r B , i ξ ) · α B (i ξ ) · G ( r B , r A , i ξ )] , (1)where the Green functions G represent the properties of the field including the medium. One may picturize G asdescribing the interaction between both particles via the exchange of virtual photons. Equation (1) has to be read fromright to left: a virtual photon i ξ is created at position r A and propagates to particle B, which is expressed by the Greenfunction G ( r B , r A , i ξ ) . At this point it interacts with the polarizability of particle B, α B (i ξ ) and is back-scattered toparticle A, again expressed by the Green function G ( r A , r B , i ξ ) , where it interacts with particle A. The sum (integral)over all possible virtual photons yields the total Van der Waals interaction.The presence of a medium as the environment has two distinct effects influencing dispersive interactions between A andB: (I) Deformation of the particle’s electron density:
Caused by the short distances between the considered particlesand the environmental particles, its wave function is modified compared to the one of the free particle [17].This phenomenon is depicted for particle A in Fig. 1 by the blue (probability of presence) for the free particleand the semitransparent blue area for the confined particle. This deforming effect affects the polarizabilities ofboth particles α A , B (i ξ ) in the Van der Waals interaction (1).(II) Screening of the virtual photon’s propagation:
In Figure 1, it can be observed that the virtual photon hasto pass the medium. This can be approximated [18] to lead to a damping by /ε (i ξ ) for each propagationdirection. This process leads to the excess polarizability models [18, 19].Applying the above to Eq. (1) for a bulk material, the medium-assisted Van der Waals interaction between twoparticles A and B embedded in a medium with permittivity ε ( ω ) separated by the distance d in the nonretarded limitreads [18, 13, 14] U vdW ( d ) = − C d , C = 3 (cid:126) π ε ∞ (cid:90) α (cid:63) A (i ξ ) α (cid:63) B (i ξ ) ε (i ξ ) d ξ , (2)with the reduced Planck constant (cid:126) and the vacuum permittivity ε . The α (cid:63) A , B (i ξ ) are understood to be modified by thepresence of the medium.The procedure to estimate medium-assisted dispersion interactions by the integral (2) is challenging in practicalcalculations as the polarizabilities as well as the permittivity have to be known over the full frequency range. Furthermore,this integral can get very complex depending on the environmental medium. For instance, the most-commonly appliedmedium is water, whose currently most exact model consists of 19 damped oscillators, 7 for the infrared and 12 forthe ultraviolet regime and 2 Debye terms are involved in order to match the experimental data in the low-frequencyregime [20].Practical electronic structure calculations in the spirit of the model presented in Fig. 1 describe the environment by apolarizable continuum model (PCM) [21, 22] based on the static permittivity sufficient for ground state calculations.2 PREPRINT - F
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3, 2021 E i j ε ( ω )A B Figure 2: Schematic illustration of the microscopic picture: two ensembles of atoms A and B are embedded in adielectric medium with permittivity ε ( ω ) . The dispersion interaction between the ensembles via the two alternativedescriptions as compact objects (mesoscopic picture, see Fig. 1) or via the pairwise interaction as depicted here E ij isscreened by the solvent medium.The workhorse of electronic structure theory is density functional theory (DFT). The most common functionalapproximations within DFT are known to severely lack the description of dispersion interactions. This can be correctedby modifying the functional [23] or by adding a dispersive correction to the energy. The latter is in the spirit of ourconsiderations applied by several approaches like the semi-empirical Grimme [24, 25] or the Tkatchenko–Scheffler [26,27] models. Such models are directly applicable to describe the presence of an explicit environment where all solventmolecules are resolved. Newer developments even take many body interactions into account[27, 25] and shouldtherefore be capable of including nontrivial environmental screening effects at least partly.[28] An explicit descriptionof the environment is computationally very demanding and requires averaging over many different configurations of theenvironment, e.g. by explicit time propagation.[29]Form the viewpoint of an implicit approach like the PCM, these corrections are based on free-particle interactionsdisregarding the presence of the environment. While appropriate in case that both interacting particles are within thesame cavity [30] this approach disregards screening by an implicit environment. It was therefore suggested to scale thevan der Waals contributions by ε − ( ω ) with ω in the optical range [31].We rationalize this conjecture by presenting an algebraic approximation for the medium-assisted C coefficient in thefollowing. It is based on a one-point Gauss quadrature rule leading to C app6 = C AB6 ε (i ω ) = (cid:18) ε (i ω )1 + 2 ε (i ω ) (cid:19) C vac6 ε (i ω ) , (3)with an averaged main-frequency ω to be determined in what follows. We furthermore show that ε − (i ω ) might bereplaced by ε − ( ω ) in the absence of resonances of the environment at the frequency ω . The prefactor on the right handside of Eq. (3) denotes the transition through the interface between the vacuum bubble and the environmental mediumaccording to the Onsager model [32] which is the most-simplest and commonly used excess polarizability model [18] α (cid:63) (i ξ ) = (cid:18) ε (i ξ )1 + 2 ε (i ξ ) (cid:19) α (i ξ ) . (4)Due to the resulting linearity between the free-space and the approximated van der Waals coefficient, the impact ofexcess polarizabilities can be easily included by means of Eq. (3) and will not be considered explicitly in the following.In this manuscript, we adapt the mesoscopic model to the microscopic models applied, for instance, in DFT simulations.The envisioned medium-assisted situation is depicted in Fig. 2. Two particle ensembles A and B are embedded within amedium with permittivity ε ( ω ) screening the interaction. In models based on electronic structure theory, this interactionis written in the generalised Casimir–Polder form [28] E AB = − (cid:126) π ε ∞ (cid:90) d ξ (cid:90) d r A d r (cid:48) A d r B d r (cid:48) B e ε (i ξ ) | r A − r B | e ε (i ξ ) | r (cid:48) A − r (cid:48) B | χ A ( r A , r (cid:48) A , i ξ ) χ B ( r B , r (cid:48) B , i ξ ) , (5)which describes the total dispersion energy between the systems A and B, expressed by the electronic density-densityresponses χ i for i = A , B . In the presence of a separating medium, these are coupled via the screened Coulomb3 PREPRINT - F
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3, 2021interaction e / (4 πε ε | r A − r B | ) . Applying the dipole approximation to (5), the generalised Casimir–Polder energygets equivalent to the mesoscopic model obtained via macroscopic quantum electrodynamics Eq. (2) E AB = − C AB6 R , (6) C AB6 = 3 (cid:126) π ε ∞ (cid:90) d ξ α A (i ξ ) α B (i ξ ) ε (i ξ ) , (7) α i (i ξ ) = (cid:90) d r d r (cid:48) rr (cid:48) χ i ( r , r (cid:48) , i ξ ) , (8)where α i (i ξ ) denotes the screened polarizability caused by the deformation of the particle’s electron density χ i due tothe presence of the environment, which does not include the effect described by the excess polarizabilities.Typically, this interaction is separated into the pairwise interaction of the constituents (the atoms) of systems A and B,as depicted in Fig. 2 E AB = (cid:88) i,j E i,j = − (cid:88) i,j f ij ( R ij ) C ij R ij , (9)with R ij denoting the distance between atom i of cloud A and atom j of cloud B, and a correction function f ij ( R ij ) totake short-range phenomena into account. This pairwise separation of the dispersion energy corresponds to the Hamakerapproach [33] (or first-order Born series expansion [34]) in macroscopic quantum electrodynamics. Such models arecommonly used in modern van-der-Waals–density-functional-theory simulations with tabled vacuum C -coefficientsfor the different interacting constituents.Interestingly, within this approach the deformation of the particle’s electron density (I), is expressed via a reduction of theparticle’s volume, which, due to the transitivity of the polarizability (8), can directly be expressed by a C def6 -coefficientfor the deformed electron density[26] C def6 = (cid:18) V def V free (cid:19) C free6 , (10)with the reduced particle volume V def and the particle volume and C -coefficient of the free particle, V free and C free6 ,respectively. This assumption is questionable from the macroscopic point of view, as, for instance, the mixing of particlestates near interfaces [35] cannot be expressed in such simple way. Further developments[27, 25] take into accountsimilar problems due to non-additivities of the van der Waals interaction [28].It can be observed that the C -coefficient (7) depends on dispersion of the implicit environmental medium via anintegration along the imaginary frequency axis. This fact motivated us to develop a simple model that takes into accountthe screening of the van der Waals interaction with a similar numerical effort as ordinary DFT simulations in vacuumwould require. As the deformation of the particle’s electron density in commonly considered in the form of Eq. (10), thelocal-field corrections as expressed via excess polarizability models [18] in the form of Eq. (3), we neglect the explicitconsideration of these effects within this manuscript. -coefficients by Gaussian quadrature The integral over the imaginary frequency axis for the C -coefficient (2) can be carried out by using a single-pointGauss quadrature rule [33, 36]. This method approximates the integral I by I = ∞ (cid:90) f ( x ) g ( x )d x ≈ f ( x ) m . (11)The values of x and m are selected such that the integrals I i = ∞ (cid:90) x i g ( x )d x , (12)are exact for i = 0 , , which guarantees that Eq. (11) is exact for constant or linear functions f ( x ) . This gives m = I and x = I /I in agreement with Ref. [36]. 4 PREPRINT - F
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3, 2021Choosing the weight g ( ξ ) = α A (i ξ ) α B (i ξ ) leads to the relevant mean frequency in Eq. (3) ω = (cid:82) ∞ ξα A (i ξ ) α B (i ξ )d ξ (cid:82) ∞ α A (i ξ ) α B (i ξ )d ξ . (13)and Eq. (3) directly. This equation therefore is exact for ε − constant or linear in i ξ . We restrict ourselves to theconsideration of non-retarded interactions with respect to the application of the model in density functional theorysimulations. A generalisation of the model to include retardation effects is possible and reported in Ref. [36]. C -coefficients for single oscillator models The simplest model for the polarizabilities α A , B is that of a single oscillator α A , B (i ξ ) = A A , B ξ/ω A , B ) , (14)with the static value A A , B and the resonance frequency ω A , B . Inserting Eq. (14) into Eq. (13) yields the average mainfrequency ω = 2 π ω A ω B ω A − ω B ln (cid:18) ω A ω B (cid:19) (15)giving ω = 2 ω A /π for ω A = ω B .To illustrate the accuracy of the model assumption (3), we calculated the averaged main frequency (15) for differentsets of resonance frequencies (cid:126) ω A , B ∈ [0 ,
10] eV for particles dissolved in one of the most-complex media water [20].We use the parametrization of ε (i ξ ) for water from Ref. [20] to define the "exact" values of the integral C exact6 = 3 (cid:126) A A A B π ε ∞ (cid:90) d ξ [1 + ( ξ/ω A ) ] [1 + ( ξ/ω B ) ] ε (i ξ ) , (16)The comparison between the vacuum and exact Van der Waals coefficients according to Eq. (3) allows to determine thecorresponding "exact" averaged main-frequency ω exact ε (i ω exact ) = (cid:115) C vac6 C exact6 . (17)Using these values we can determine the relative deviations of the approximated values according to Eqs. (3) and (15)Figure 3: Relative deviations in percent between the approximated averaged frequency ω and the exact main-frequencyaccording to Eq. (3) (top left triangle) and between the approximated and exact van der Waals coefficients (bottom righttriangle). 5 PREPRINT - F
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3, 2021as ω − ω exact ω exact , C app6 − C exact6 C exact6 . (18)These deviations are generally rather small as depicted in Fig. 3. It can be observed that for materials with a dominantresonance in the microwave, optical, ultraviolet or with higher energies the relative error due to main-frequencyapproximation is negligible. Only for materials with a dominant resonance in the radio regime and below are not wellapproximated, which are not very common or realistic materials. In order to discuss the general properties of the square of the inverse permittivity, we consider its frequency dependencein terms of common approximations. The permittivity may be described in a generalised Debye form ε Debye ( ω ) = 1 + (cid:88) D ε D − i ωτ D , (19)or similarly in a generalised Drude form ε Drude ( ω ) = 1 + (cid:88) D ε D ω ω − ω − i ωγ D , (20)where the two approximations get very similar if we identify τ D = 3 /ω D and γ D = 3 ω D (see SI). The sums inEqs. (19) and (20) contain a chosen number of resonators. Generally, the resonator weights ε D tend to decrease withincreasing resonance frequency ω D . -8 -6 -4 -2 Figure 4: Drude and Debye models with two oscillators with (cid:126) ω D = (cid:8) − , (cid:9) eV and weights ε D = { , } ,respectively.Figure 4 shows ε − (i ξ ) in the two approximations for the model case of two resonators only. The ε − (i ξ ) is peakedaround resonance frequencies, but is rather flat in other regions, where it takes the form of a step-like function. Incase that ω does not coincide with a resonance of the environment, ε − (i ξ ) can therefore be approximated by a linearfunction for the main part of the integral (2). The Gaussian quadrature is exact in this case. Figure 4 furthermore shows,that ε − (i ξ ) might also be replaced by / (Re[ ε ( ω )]) which is well measured and tabulated for many solvents andother environments [37]. Thus, the medium-assisted Van der Waals interaction can effectively be treated via a thescreening due to an environmental medium U vdW ( d ) = − C vac6 d ε ( ω )]) , (21)6 PREPRINT - F
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3, 2021with an averaged main-frequency ω [Eqs. (15) and (31)]. In general the screening coefficient can be evaluated from thedielectric function ε ( ω ) by using the Kramers–Kronig relation [38, 39, 13] ε (i ω ) = 1 + 2 π ∞ (cid:90) ω Im ε ( ω ) ω + ω d ω . (22) In the following, we apply the model developed to more realistic scenarios and analyse the deviations between theapproximation and the exact solutions for the medium-assisted van der Waals interactions. In principle, realisticmaterials are described via multi oscillator models, which we can distinguish into two classes according to the resonancefrequencies: resonances in the ultraviolet regime, which are caused by electronic transitions, and resonances in theinfrared regime that are caused by vibrational and rotational modes of the system. To this end, we first consider a twooscillator model with one oscillator within each of these spectral ranges and analyse the model predictions due the ratiobetween the corresponding oscillator strengths. Afterwards, we consider the interaction between real molecules whosepolarizabilities consist of several oscillator models. Finally, we analyse the interaction between atomic compoundsin terms of a Hamaker summation according to the common treatment of van der Waals dispersion forces in densityfunctional theory simulations.
In the previous section and also in Fig. 6, we observe that the resonances of the dielectric response function aredominant in two different spectral ranges — in the infrared and in the ultraviolet range. Hence, we analyse the impactof differently weighted oscillator strengths in our model. We consider a two-oscillator model for the polarizability α (i ξ ) = C IR ξ/ω IR ) + C UV ξ/ω UV ) = C IR (cid:20)
11 + ( ξ/ω IR ) + λ ξ/ω UV ) (cid:21) , (23)with the ratio between the oscillator strengths λ = C UV C IR , (24)being small ( λ (cid:28) ) for infrared-dominant species and large ( λ (cid:29) ) for ultraviolet-dominant species. By insertingEq. (23) into Eq. (13), this leads to an averaged main-frequency ω = 2 π (cid:104) ( ω IR + λω UV ) + ω IR ω UV ( λ + 1) (cid:105) × (cid:34) λω IR ω UV ω IR − ω UV ln (cid:18) ω IR ω UV (cid:19) + ( ω IR + ω UV ) (cid:0) λ ω + ω (cid:1)(cid:35) , (25)satisfying the single-oscillator limits (15). The results for two particles of the some species with the parameters (cid:126) ω IR = 0 . and (cid:126) ω UV = 10 eV embedded in water are depicted in Fig. 5. It can be observed the limits of thedifferent dominant regimes are reproduced ω = 2 π (cid:26) ω UV , for λ (cid:29) λ crit ω IR , for λ (cid:28) λ crit . (26)It can be observed that λ crit is smaller than unity due to the weighted integral (13). To understand this behaviourquantitatively, we define λ crit to be the ratio corresponding to the arithmetical averaged main-frequency ω ( λ crit ) =( ω IR + ω UV ) /π . This can be solved analytically and results in λ crit = √ ρ (1 + ρ )( ρ − (cid:110) ρ / ln ρ + 2 √ ρ − ρ / + (cid:2) ρ ln ρ (cid:0) ρ ln ρ + 1 − ρ (cid:1) + ( ρ − (1 + ρ ) (cid:3) / (cid:111) , (27)with the ratio between the resonance frequencies ρ = ω IR /ω UV (cid:28) being typically much smaller than 1. The resulting λ crit < implies that the averaged main-frequency is typically closer to the ultraviolet resonance unless its resonanceis much smaller than that of the infrared resonance. 7 PREPRINT - F
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3, 2021 -4 -2 -2 -3 -2 -1 -2 -2 Figure 5: Comparison of the averaged main-frequency ω (top figure) obtained via (25) (green lines), the exact resultby solving Eq. (17) (red lines), the fitted to a one-oscillator model (14) (blue lines) and an arithmetically averagedmodel ω = 2 /π ( ω IR + λω UV ) / (1 + λ ) (orange lines) for a two-oscillator polarizability (23) with the parameters (cid:126) ω IR = 0 . and (cid:126) ω UV = 10 eV depending on the ratio between the ultraviolet and infrared oscillator strengths λ . The asymptotes of the single oscillator limits are the horizontal dashed lines. The critical ratio λ crit is drawn bythe vertical dashed grey line. The inset of the upper figure illustrates the dependence of the critical ratio on the ratiobetween the resonance frequencies ρ = ω IR /ω UV . The bottom figure illustrate the resulting normalised van der Waalscoefficients with an inset of corresponding relative errors according to Eq. (18), where furthermore the relative error ofthe single oscillator treatment is depicted via the magenta line.Figure 5 illustrates the relevant dependencies of the two-oscillator model. In addition, to the derived two-oscillatormodel (25), we added a simple weighted averaged main-frequency model ω = 2 π ω IR + λω UV λ , (28)a fit of the two-oscillator model onto a single oscillator model, which can be performed analytically by solving thecorresponding least-square equation and by splitting the van der Waals coefficient into its three single oscillatorcontributions C (cid:126) C / (16 π ε ) = ∞ (cid:90) d ξε (i ξ ) (cid:104) ξ/ω IR ) (cid:105) + 2 λ ∞ (cid:90) d ξε (i ξ ) (cid:104) ξ/ω IR ) (cid:105) · (cid:104) ξ/ω UV ) (cid:105) + λ ∞ (cid:90) d ξε (i ξ ) (cid:104) ξ/ω UV ) (cid:105) . (29)It can be observed, that the model (25) agrees very well with the exact averaged main-frequency. The correspondingasymptotes are governed by the conditions (26) together with Eq. (27) for λ crit . Remarkably, the alternative models(weighted average and fit-to-single-oscillator model) strongly deviate for the predictions of the averaged main-frequency,but matches better the van der Waals coefficients of infrared-dominant materials. For ultraviolet-dominant respondingspecies, the presented model predicts better the C -coefficients than the other models. The optimum over the wholerange is given by the treatment of each single oscillator (29). The oscillator model (14) can easily be extended to several oscillators α A , B (i ξ ) = (cid:88) i A ( i )A , B (cid:16) ξ/ω ( i )A , B (cid:17) , (30)8 PREPRINT - F
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3, 2021leading to an averaged main-frequency ω = 2 π (cid:88) i,j A ( i )A A ( j )B (cid:104) ω ( i )A ω ( j )B (cid:105) (cid:104) ω ( i )A (cid:105) − (cid:104) ω ( j )B (cid:105) ln (cid:32) ω ( i )A ω ( j )B (cid:33) (cid:88) i,j A ( i )A A ( j )B ω ( i )A ω ( j )B ω ( i )A + ω ( j )B − . (31)As a note of caution, we remark that, as evident from the discussion in Sec. 2, our method may lead to less accurateresults for molecules whose relevant transitions span several plateau regions of ε (i ξ ) . In this case, a separate treatmentof each oscillator is recommended, as shown in the previous section.CH NO CO CO N O O O N H S NOCH O 12.3 13.9 13.9 13.4 13.6 14.0 14.7 14.0 11.1 14.1O S 10.3 11.3 11.4 11.0 11.1 11.4 11.9 11.5 9.3 11.5NO 12.8 14.5 14.5 13.9 14.1 14.6 15.2 14.6 11.5 14.7Table 1: Average main-frequencies ω (eV) for different molecule pairs. The corresponding parameters for thepolarizabilities are taken from Refs. [18, 40].In Table 1, averaged main-frequencies are given for different pairs of small molecules. The corresponding polarizabilitiesare taken from Refs. [18, 40]. It can be observed that the averaged main-frequency is mainly located in the energy rangebetween 10 and 15 eV.Due to this reduction of the relevant energy range, the dielectric functions of the solvent can be approximated by asingle UV oscillator model ε app (i ξ ) = ε s − ε ∞ ξ/ω UV ) + ε ∞ , (32)where ε s is the low frequency permittivity and ε ∞ is the permittivity for large frequencies that may contain contributionsof other oscillators at higher frequencies. The dielectric functions of typical solvents are illustrated in Fig. 6. Theparametrization of the alcohols are taken from Ref. [41] and that of water from Ref. [20]. These models have beenfitted to the reduced response model (32) whose resulting parameters are given in table 2. This reduced model is ingood agreement with the original data which can be observed in the inset of Fig. 6. Equation (32) describes a Drudeoscillator without damping, which is sufficient for the description of dispersion interactions.The impact of the model (3) for a selection of interacting particles are given in table 3. The complete list can be found inthe supplementary information. It can be seen that the model well-approximates the Van der Waals interactions betweenboth particles within a deviation of roughly 5-10%. We can expect even better agreement for molecules with dominantexcitations in the optical or low UV range as these frequencies are further apart from the resonances of the solvent. In common density functional theory simulations, the van der Waals interaction between complex molecules isdecomposed via the Hamaker approach (pairwise summation) over the atomic constituents of each molecule [26] C AB6 = (cid:88) a ∈A (cid:88) b ∈B C ab , (33)where A and B denotes the set of atoms of molecule A and B, respectively, with an effective van der Waals interactionbetween atom a and b expressed by C ab . Figure 2 illustrates this decomposition. The effective van der Waals interactionbetween each pair is treated analogously to the screening of the electronic density (10) to be linear in the free-space vander Waals coefficient C ab ∝ C free ,ab . (34)This ansatz allows us to apply the effective screening to the atomic decomposition of the molecules. Commonly,atomic dynamic polarizabilities are treated by a single Lorentz oscillator with an oscillator strength and a resonant9 PREPRINT - F
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3, 2021 -2 -1 WaterMethanol EthanolPropanol Butanol
Figure 6: Dielectric function of water and some common solvent alcohols (ethanol, methanol, butanol and propanol)on the imaginary axis.The crosses in the inset illustrate the approximated dielectric functions due to Eq. (32) with theparameters in table 2. Solvent ε ∞ ε s ω UV Water 1.193 1.766 10.73Ethanol 1.141 1.853 12.29Methanol 1.098 1.766 12.46Butanol 1.154 1.954 11.47Propanol 1.144 1.921 11.52Cyclopentane 1.092 1.938 11.65Cyclohexane 1.096 1.991 11.68Benzene 1.169 2.199 10.07Fluorobenzene 1.145 2.088 10.31Chlorobenzene 1.157 2.264 10.38Bromobenzene 1.173 2.371 10.33Pentane 1.080 1.808 13.16Heptane 1.086 1.857 13.07Glycerol 1.152 2.152 12.10Tetrachloromethane 1.103 2.076 12.13Table 2: Parameters for some solvents according to the reduced single oscillator model (32).frequency (14) that are given by the static polarizability α A (0) and the van der Waals coefficient for the equal particlepairwise interaction in vacuum C AA6 = 3 (cid:126) π ε ∞ (cid:90) d ξ (cid:34) α A (0)1 + (cid:0) ξ/ω A0 (cid:1) (cid:35) = 3 (cid:126) α (0) ω A0 π ε . (35)Thus, the resonance frequency is given by ω A0 = 64 π ε C AA6 (cid:126) α (0) . (36)By inserting this (36) into Eq. (15), one finds the averaged main-frequency ω = 128 πε (cid:126) C AA6 C BB6 C AA6 α (0) − C BB6 α (0) ln (cid:18) C AA6 α (0) C BB6 α (0) (cid:19) , (37)which simplifies to ω = 128 πε C AA6 (cid:126) α (0) , (38)10 PREPRINT - F
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3, 2021Mol. A Medium Mol. B C vac6 ω ε (i ω ) ε app (i ω ) C app6 C exact6 rel. Dev.CH Water CH Water N O 174.42 13.92 1.40 1.41 88.16 81.69 7.91%NO Water H S 117.12 11.53 1.46 1.46 55.02 52.01 5.80%CH Ethanol CH Methanol H S 109.06 11.92 1.45 1.45 52.12 49.35 5.61%N Butanol H S 120.86 11.52 1.55 1.55 50.15 48.08 4.32%O Propanol CH C vac6 (10 Jm ) , the corresponding averaged main-frequency ω (eV) , the exact and approximated (32) dielectricfunctions evaluated at the averaged main-frequency, the approximated Van der Waals coefficient according to Eq. (3) C app6 (10 Jm ) , the exact Van der Waals coefficient according to Eq. (2) C exact6 (10 Jm ) and the relative deviationof the approximated and exact C -coefficients.for two particles of the same species. Table 4 illustrates the averaged main-frequency for different atomic combinationsinvolved in organic particles: carbon, hydrogen, oxygen, nitrogen, sulfur, fluorine, chlorine, bromine and iodine. Theparameters for the polarizabilities are taken from Ref. [42]. It can be seen that the resulting parameters are again in theultraviolet range between 7.25 eV and 15 eV. Thus, the dielectric functions for the different solvents provided in table 2can be used to determine the screening factors. (cid:126) ω (eV) C H O N S F Cl Br IC 7.47 7.44 9.51 8.70 7.76 10.45 8.50 8.34 7.37H 7.44 7.41 9.47 8.66 7.73 10.40 8.46 8.31 7.34O 9.51 9.47 12.36 11.21 9.90 13.69 10.93 10.72 9.36N 8.70 8.66 11.21 10.21 9.05 12.38 9.96 9.77 8.57S 7.76 7.73 9.90 9.05 8.06 10.89 8.83 8.67 7.64F 10.45 10.40 13.69 12.38 10.89 15.23 12.06 11.82 10.28Cl 8.50 8.46 10.93 9.96 8.83 12.06 9.71 9.53 8.37Br 8.34 8.31 10.72 9.77 8.67 11.82 9.53 9.35 8.22I 7.37 7.34 9.36 8.57 7.64 10.28 8.37 8.22 7.26Table 4: The averaged main-frequency for atomic combination of organic particles.The application of this atomic model has to be taken with a grain of salt because polarizabilities are typically non-additive with respect to the constituents of the considered particle [28]. There are several effects which are not coveredby Eq. (33). One of these effects is the rescaling of the free-space van der Waals coefficient due to the particle’svolume (10). Several investigations, in theory [35, 43, 44] and experiment [45, 46], have shown that the largest effect ofa surface (respectively a cavity) results in a spectral shift of the particle’s resonance ∆ ω . Such effect can, for instance,be included within this model by applying a Taylor series expansion to Eq. (15) leading to a shift of the averagedmain-frequency ∆ ω = 2 π (cid:34) ω ∆ ω B − ω ∆ ω A ( ω A − ω B ) ln (cid:18) ω A ω B (cid:19) + ω B ∆ ω A − ω rmA ∆ ω B ω A − ω B + ω − ω − ω A ω B ln ( ω A /ω B )( ω A − ω B ) ∆ ω A ∆ ω B (cid:35) , (39)which has to be considered for the evaluation of the screening factor ε − (i ω + i∆ ω ) . Another effect this modelcan be adapted to is the many-particle interaction behind the pairwise assumption, e.g. the three-body interaction(the Axilrod–Teller potential) that describes the interaction between three polarisable particles. Its strength is givenby [47, 48] C ∝ ∞ (cid:90) α A (i ξ ) α B (i ξ ) α C (i ξ ) ε n (i ξ ) , (40)where n denotes the number of interactions crossing the intermediate medium, e.g. if all three particles belong to threedifferent molecules, than n = 3 , whereas if all constituents belongs to the same molecule than n = 0 . In analogy to11 PREPRINT - F
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3, 2021Eq. (13) an averaged main-frequency can be derived, which reads for three single-oscillator models ω = 2 π ω A ω B ω C ω A + ω B + ω C ω B − ω C ) ( ω A − ω C ) ( ω ) A − ω B ) × (cid:20) ω ln (cid:18) ω B ω C (cid:19) + ω ln (cid:18) ω C ω A (cid:19) + ω ln (cid:18) ω A ω C (cid:19)(cid:21) . (41)Beyond this extension, the model can be adapted to the consideration of higher-order multipoles, e.g. the non-retardeddipole-quadrupole interaction [49] U ( r ) = − C r , (42)with C = 90 (cid:126) c π ε ∞ (cid:90) d ξα (i ξ ) β (i ξ ) , (43)with the scalar dipole and quadrupole polarizabilities, α (i ξ ) and β (i ξ ) , respectively. By, for instance, assuming singleLorentz oscillator models (14) to model each response function, the resulting screening effect can be effectively betreated in analogy to Eqs. (15) and (3).Beyond these extensions, there are some limitations that the model cannot cover: (i) the intermediate regime, whereretardation effects start to play a role are not adaptable, because the potential does not factorises into a part depending onthe polarizabilities and another part only depending on the dielectric function of the medium [50]; and (ii) in situations,where the interacting atomic systems A and B are so close together that the electronic densities start to overlap and amolecule is formed, the derived model fails due to the coupling dipole-electric field interaction Hamiltonian applied inthe whole theory. We have shown that the medium-assisted Van der Waals interaction can effectively be treated as a the screening due toan environmental medium. Single point Gauss quadrature suggests the screening to be the inverse of ε (i ω ) with anaveraged main-frequency ω that depends on the resonances of the interacting molecules. The approximation shouldbe particularly accurate if these resonances are far from the resonances of the medium. Then ε (i ω ) − might even bereplaced by the permittivity evaluated at real frequencies ε ( ω ) − . Application of the approximation proposed for smallmolecules with resonances near to these of the solvents reveal still an accuracy of 90-95%.As embedded molecules in solvents often show their dominating resonances in the optical region, screening of vander Waals interactions by the solvent is greatly suppressed as compared to electrostatic Coulomb interactions. In theextreme, but important case of water, the latter are screened by the factor 1/78 [22] due to its large static polarizability.The permittivity in the optical region is much lower [31], leading to a screening of the van der Waals interaction by ≈ . compared to the vacuum case.Nevertheless, the simple form of our result is useful for adjusting Van der Waals corrections of molecules within implicitsolvents [31]. Furthermore, the screening might affect also molecular dynamics calculations in aqueous environmentsuch as the important problem of protein folding [51], or may resolve some of the discrepancies between simulatedand experimental dielectric constants [52]. The solvents’ permittivity in the optical range is caused by resonances ofthe solvents’ electronic system. As classical force fields do not include electrons, the corresponding screening term ismissing and therefore also the screening term described here. Modification of the bare Coulomb interaction by therelative permittivity has been shown to improve the description of ion-ion interactions considerably.[53] References [1] F London. The general theory of molecular forces.
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A Drude and Debye formulas for a single resonator
There are different possible forms of approximations for the frequency dependent permittivity. One of the most compactis the Debye relaxation formula for a single resonator which is completely determined by the relaxation time τ D ε ( ω ) = ε s − ε ∞ − i ωτ D + ε ∞ (44)with the static permittivity ε s and the permittivity for infinite frequency ε ∞ . This approximation reads at complexfrequencies ε (i ξ ) = ε s − ε ∞ ξτ D + ε ∞ . (45)The Drude form for a single resonance at resonance frequency ω D and width γ D is ε ( ω ) = ( ε s − ε ∞ ) ω D ω D − ω − i ωγ D + ε ∞ (46)and for complex frequencies ε (i ξ ) = ( ε s − ε ∞ ) ω D ω D + ξ + ξγ D + ε ∞ . (47)Figure 7: a) Real, b) imaginary parts of the permittivity in Debye und Drude models with (cid:126) ω D = 1 eV , ε s = 80 , ε ∞ = 1 .c) Permittivities at imaginary frequencies. The vertical lines indicate ω D .Drude and Debye approximations are very similar if one identifies τ D = γ D /ω D and γ D = 3 ω D as shown for a simple“water” model with (cid:126) ω D = 1 eV , ε s = 80 , ε ∞ = 1 in Figure 7.Figure 8 compares the real part of the dielectric function and the corresponding function at the imaginary frequencyaxis. B Table of medium-assisted Van der Waals interaction of small molecules in differentsolvents PREPRINT - F
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3, 2021 -8 -6 -4 -2 Figure 8: Comparison of the dielectric function on the imaginary frequency axis and the corresponding real part for atwo oscillator model with the parameters (cid:126) ω D = (cid:8) − , (cid:9) eV and ε D = { , } .Table 5: Comparison of different molecule combinations (Molecule Aand B) with the vacuum Van der Waals coefficient C vac6 (10 Jm ) ,the corresponding averaged main-frequency ω (eV) , the exact and ap-proximated (32) dielectric functions evaluated at the averaged main-frequency, the approximated Van der Waals coefficient according to Eq. (3) C app6 (10 Jm ) , the exact Van der Waals coefficient according to Eq. (2) C exact6 (10 Jm ) and the relative deviation of the approximated andexact C -coefficients.Mol. A Medium Mol. B C vac6 ω ε (i ω ) ε app (i ω ) C app6 C exact6 rel. Dev.CH Water CH Water NO Water CO Water CO 96.18 12.15 1.45 1.44 46.13 43.47 6.12CH Water N O 147.94 12.34 1.44 1.44 71.38 67.16 6.28CH Water O Water O Water N Water H S 157.89 10.28 1.50 1.49 70.96 68.16 4.10CH Water NO 87.30 12.79 1.43 1.43 42.71 40.05 6.64NO Water CH Water NO Water CO Water CO 111.21 13.67 1.41 1.41 55.82 51.71 7.95NO Water N O 171.27 13.90 1.40 1.41 86.52 80.03 8.11NO Water O Water O Water N Water H S 180.42 11.34 1.47 1.46 84.22 79.64 5.74NO Water NO 101.36 14.45 1.39 1.40 51.97 47.92 8.44CO Water CH Water NO Water CO Water CO 113.25 13.69 1.41 1.41 56.88 52.78 7.76Continued on next page16
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Mol. A Medium Mol. B C vac6 ω ε (i ω ) ε app (i ω ) C app6 C exact6 rel. Dev.CO Water N O 174.42 13.92 1.40 1.41 88.16 81.69 7.91CO Water O Water O Water N Water H S 183.51 11.40 1.47 1.46 85.84 81.27 5.63CO Water NO 103.25 14.46 1.39 1.40 52.94 48.92 8.23CO Water CH O 122.44 13.35 1.42 1.42 60.91 56.62 7.57CO Water O S 129.60 10.96 1.48 1.47 59.71 56.75 5.21CO Water NO 72.38 13.87 1.41 1.41 36.54 33.85 7.93N O Water CH O Water NO O Water CO O Water CO 122.44 13.35 1.42 1.42 60.91 56.62 7.57N O Water N O 188.52 13.58 1.41 1.41 94.38 87.60 7.73N O Water O O Water O O Water N O Water H S 199.10 11.13 1.47 1.47 92.26 87.54 5.39N O Water NO 111.50 14.10 1.40 1.40 56.64 52.41 8.07O Water CH Water NO Water CO Water CO 115.77 13.76 1.41 1.41 58.26 53.88 8.13O Water N O 178.30 14.00 1.40 1.41 90.32 83.41 8.29O Water O Water O Water N Water H S 187.75 11.39 1.47 1.46 87.79 82.90 5.89O Water NO 105.53 14.56 1.39 1.39 54.26 49.96 8.62O Water CH Water NO Water CO Water CO 67.71 14.41 1.39 1.40 34.67 32.00 8.36O Water N O 104.35 14.65 1.39 1.39 53.78 49.57 8.49O Water O Water O Water N Water H S 109.06 11.92 1.45 1.45 51.92 48.85 6.27O Water NO 61.86 15.24 1.38 1.38 32.34 29.74 8.77N Water CH Water NO Water CO Water CO 74.72 13.80 1.41 1.41 37.65 34.97 7.66N Water N O 115.11 14.03 1.40 1.40 58.36 54.13 7.80N Water O Water O Water N Water H S 120.86 11.52 1.46 1.46 56.76 53.74 5.61N Water NO 68.17 14.57 1.39 1.39 35.05 32.43 8.10Continued on next page17
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Mol. A Medium Mol. B C vac6 ω ε (i ω ) ε app (i ω ) C app6 C exact6 rel. Dev.H S Water CH S Water NO S Water CO S Water CO 129.60 10.96 1.48 1.47 59.71 56.75 5.21H S Water N O 199.10 11.13 1.47 1.47 92.26 87.54 5.39H S Water O S Water O S Water N S Water H S 215.59 9.33 1.52 1.52 93.40 90.51 3.19H S Water NO 117.12 11.53 1.46 1.46 55.02 52.01 5.80NO Water CH O 111.50 14.10 1.40 1.40 56.64 52.41 8.07NO Water O S 117.12 11.53 1.46 1.46 55.02 52.01 5.80NO Water NO 66.03 14.66 1.39 1.39 34.03 31.40 8.38CH Ethanol CH Ethanol NO Ethanol CO Ethanol CO 96.18 12.15 1.50 1.50 42.69 40.93 4.30CH Ethanol N O 147.94 12.34 1.50 1.50 66.15 63.27 4.55CH Ethanol O Ethanol O Ethanol N Ethanol H S 157.89 10.28 1.56 1.56 64.90 63.85 1.64CH Ethanol NO 87.30 12.79 1.48 1.48 39.70 37.77 5.11NO Ethanol CH Ethanol NO Ethanol CO Ethanol CO 111.21 13.67 1.46 1.46 52.22 48.88 6.83NO Ethanol N O 171.27 13.90 1.45 1.45 81.08 75.70 7.11NO Ethanol O Ethanol O Ethanol N Ethanol H S 180.42 11.34 1.53 1.53 77.52 74.85 3.57NO Ethanol NO 101.36 14.45 1.44 1.44 48.89 45.39 7.72CO Ethanol CH Ethanol NO Ethanol CO Ethanol CO 113.25 13.69 1.46 1.46 53.22 49.88 6.69CO Ethanol N O 174.42 13.92 1.45 1.45 82.62 77.25 6.95CO Ethanol O Ethanol O Ethanol N Ethanol H S 183.51 11.40 1.53 1.52 79.05 76.36 3.52CO Ethanol NO 103.25 14.46 1.44 1.44 49.82 46.32 7.55CO Ethanol CH O 122.44 13.35 1.47 1.47 56.85 53.47 6.32CO Ethanol O PREPRINT - F
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Mol. A Medium Mol. B C vac6 ω ε (i ω ) ε app (i ω ) C app6 C exact6 rel. Dev.CO Ethanol O S 129.60 10.96 1.54 1.54 54.83 53.27 2.93CO Ethanol NO 72.38 13.87 1.45 1.45 34.23 32.01 6.93N O Ethanol CH O Ethanol NO O Ethanol CO O Ethanol CO 122.44 13.35 1.47 1.47 56.85 53.47 6.32N O Ethanol N O 188.52 13.58 1.46 1.46 88.24 82.78 6.59N O Ethanol O O Ethanol O O Ethanol N O Ethanol H S 199.10 11.13 1.53 1.53 84.81 82.21 3.17N O Ethanol NO 111.50 14.10 1.45 1.45 53.16 49.59 7.19O Ethanol CH Ethanol NO Ethanol CO Ethanol CO 115.77 13.76 1.46 1.46 54.55 50.96 7.04O Ethanol N O 178.30 14.00 1.45 1.45 84.71 78.93 7.32O Ethanol O Ethanol O Ethanol N Ethanol H S 187.75 11.39 1.53 1.52 80.83 77.94 3.71O Ethanol NO 105.53 14.56 1.43 1.44 51.10 47.34 7.94O Ethanol CH Ethanol NO Ethanol CO Ethanol CO 67.71 14.41 1.44 1.44 32.61 30.30 7.64O Ethanol N O 104.35 14.65 1.43 1.44 50.67 46.96 7.90O Ethanol O Ethanol O Ethanol N Ethanol H S 109.06 11.92 1.51 1.51 47.97 45.97 4.35O Ethanol NO 61.86 15.24 1.42 1.42 30.61 28.21 8.49N Ethanol CH Ethanol NO Ethanol CO Ethanol CO 74.72 13.80 1.45 1.46 35.25 33.06 6.64N Ethanol N O 115.11 14.03 1.45 1.45 54.74 51.21 6.90N Ethanol O Ethanol O Ethanol N Ethanol H S 120.86 11.52 1.52 1.52 52.30 50.51 3.55N Ethanol NO 68.17 14.57 1.43 1.44 33.01 30.72 7.47H S Ethanol CH S Ethanol NO S Ethanol CO S Ethanol CO 129.60 10.96 1.54 1.54 54.83 53.27 2.93H S Ethanol N O 199.10 11.13 1.53 1.53 84.81 82.21 3.17H S Ethanol O S Ethanol O S Ethanol N S Ethanol H S 215.59 9.33 1.59 1.59 84.99 84.56 0.50H S Ethanol NO 117.12 11.53 1.52 1.52 50.71 48.89 3.72NO Ethanol CH PREPRINT - F
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Mol. A Medium Mol. B C vac6 ω ε (i ω ) ε app (i ω ) C app6 C exact6 rel. Dev.NO Ethanol CO O 111.50 14.10 1.45 1.45 53.16 49.59 7.19NO Ethanol O S 117.12 11.53 1.52 1.52 50.71 48.89 3.72NO Ethanol NO 66.03 14.66 1.43 1.43 32.07 29.75 7.79CH Methanol CH Methanol NO Methanol CO Methanol CO 96.18 12.15 1.44 1.44 46.37 43.92 5.58CH Methanol N O 147.94 12.34 1.43 1.44 71.84 67.87 5.85CH Methanol O Methanol O Methanol N Methanol H S 157.89 10.28 1.49 1.50 70.63 68.76 2.71CH Methanol NO 87.30 12.79 1.42 1.42 43.10 40.49 6.46NO Methanol CH Methanol NO Methanol CO Methanol CO 111.21 13.67 1.40 1.40 56.65 52.30 8.32NO Methanol N O 171.27 13.90 1.40 1.40 87.95 80.97 8.62NO Methanol O Methanol O Methanol N Methanol H2
Mol. A Medium Mol. B C vac6 ω ε (i ω ) ε app (i ω ) C app6 C exact6 rel. Dev.NO Ethanol CO O 111.50 14.10 1.45 1.45 53.16 49.59 7.19NO Ethanol O S 117.12 11.53 1.52 1.52 50.71 48.89 3.72NO Ethanol NO 66.03 14.66 1.43 1.43 32.07 29.75 7.79CH Methanol CH Methanol NO Methanol CO Methanol CO 96.18 12.15 1.44 1.44 46.37 43.92 5.58CH Methanol N O 147.94 12.34 1.43 1.44 71.84 67.87 5.85CH Methanol O Methanol O Methanol N Methanol H S 157.89 10.28 1.49 1.50 70.63 68.76 2.71CH Methanol NO 87.30 12.79 1.42 1.42 43.10 40.49 6.46NO Methanol CH Methanol NO Methanol CO Methanol CO 111.21 13.67 1.40 1.40 56.65 52.30 8.32NO Methanol N O 171.27 13.90 1.40 1.40 87.95 80.97 8.62NO Methanol O Methanol O Methanol N Methanol H2 S 180.42 11.34 1.46 1.46 84.27 80.42 4.78NO Methanol NO 101.36 14.45 1.38 1.38 53.02 48.50 9.31CO Methanol CH Methanol NO Methanol CO Methanol CO 113.25 13.69 1.40 1.40 57.73 53.38 8.15CO Methanol N O 174.42 13.92 1.40 1.40 89.62 82.64 8.45CO Methanol O Methanol O Methanol N Methanol H S 183.51 11.40 1.46 1.46 85.93 82.06 4.71CO Methanol NO 103.25 14.46 1.38 1.38 54.02 49.50 9.11CO Methanol CH O 122.44 13.35 1.41 1.41 61.69 57.26 7.75CO Methanol O S 129.60 10.96 1.47 1.47 59.63 57.29 4.09CO Methanol NO 72.38 13.87 1.40 1.40 37.13 34.25 8.42N O Methanol CH O Methanol NO O Methanol CO O Methanol CO 122.44 13.35 1.41 1.41 61.69 57.26 7.75N O Methanol N O 188.52 13.58 1.40 1.40 95.73 88.60 8.05N O Methanol O O Methanol O O Methanol N PREPRINT - F
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Mol. A Medium Mol. B C vac6 ω ε (i ω ) ε app (i ω ) C app6 C exact6 rel. Dev.N O Methanol H S 199.10 11.13 1.47 1.47 92.22 88.38 4.34N O Methanol NO 111.50 14.10 1.39 1.39 57.65 53.03 8.72O Methanol CH Methanol NO Methanol CO Methanol CO 115.77 13.76 1.40 1.40 59.17 54.51 8.55O Methanol N O 178.30 14.00 1.39 1.39 91.88 84.39 8.87O Methanol O Methanol O Methanol N Methanol H S 187.75 11.39 1.46 1.46 87.87 83.73 4.94O Methanol NO 105.53 14.56 1.38 1.38 55.40 50.57 9.56O Methanol CH Methanol NO Methanol CO Methanol CO 67.71 14.41 1.38 1.38 35.36 32.38 9.21O Methanol N O 104.35 14.65 1.38 1.38 54.94 50.17 9.51O Methanol O Methanol O Methanol N Methanol H S 109.06 11.92 1.45 1.45 52.12 49.35 5.61O Methanol NO 61.86 15.24 1.36 1.37 33.17 30.11 10.17N Methanol CH Methanol NO Methanol CO Methanol CO 74.72 13.80 1.40 1.40 38.24 35.37 8.11N Methanol N O 115.11 14.03 1.39 1.39 59.37 54.77 8.40N Methanol O Methanol O Methanol N Methanol H S 120.86 11.52 1.46 1.46 56.85 54.27 4.75N Methanol NO 68.17 14.57 1.38 1.38 35.79 32.82 9.05H S Methanol CH S Methanol NO S Methanol CO S Methanol CO 129.60 10.96 1.47 1.47 59.63 57.29 4.09H S Methanol N O 199.10 11.13 1.47 1.47 92.22 88.38 4.34H S Methanol O S Methanol O S Methanol N S Methanol H S 215.59 9.33 1.52 1.53 92.60 91.23 1.50H S Methanol NO 117.12 11.53 1.46 1.46 55.11 52.52 4.94NO Methanol CH O 111.50 14.10 1.39 1.39 57.65 53.03 8.72NO Methanol O S 117.12 11.53 1.46 1.46 55.11 52.52 4.94NO Methanol NO 66.03 14.66 1.38 1.38 34.77 31.78 9.39CH Butanol CH Butanol NO Butanol CO Butanol CO 96.18 12.15 1.53 1.53 41.04 39.03 5.15Continued on next page21
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Mol. A Medium Mol. B C vac6 ω ε (i ω ) ε app (i ω ) C app6 C exact6 rel. Dev.CH Butanol N O 147.94 12.34 1.53 1.52 63.63 60.36 5.41CH Butanol O Butanol O Butanol N Butanol H S 157.89 10.28 1.60 1.60 61.87 60.55 2.18CH Butanol NO 87.30 12.79 1.51 1.51 38.26 36.09 6.02NO Butanol CH Butanol NO Butanol CO Butanol CO 111.21 13.67 1.48 1.48 50.46 46.77 7.89NO Butanol N O 171.27 13.90 1.48 1.48 78.40 72.46 8.18NO Butanol O Butanol O Butanol N Butanol H S 180.42 11.34 1.56 1.56 74.27 71.18 4.35NO Butanol NO 101.36 14.45 1.46 1.46 47.35 43.51 8.83CO Butanol CH Butanol NO Butanol CO Butanol CO 113.25 13.69 1.48 1.48 51.42 47.74 7.73CO Butanol N O 174.42 13.92 1.48 1.48 79.89 73.97 8.01CO Butanol O Butanol O Butanol N Butanol H S 183.51 11.40 1.56 1.56 75.76 72.64 4.29CO Butanol NO 103.25 14.46 1.46 1.46 48.24 44.41 8.63CO Butanol CH O 122.44 13.35 1.49 1.49 54.88 51.13 7.34CO Butanol O S 129.60 10.96 1.57 1.57 52.45 50.60 3.64CO Butanol NO 72.38 13.87 1.48 1.48 33.10 30.65 7.98N O Butanol CH O Butanol NO O Butanol CO O Butanol CO 122.44 13.35 1.49 1.49 54.88 51.13 7.34N O Butanol N O 188.52 13.58 1.49 1.49 85.23 79.19 7.63N O Butanol O O Butanol O O Butanol N O Butanol H S 199.10 11.13 1.57 1.57 81.18 78.13 3.90N O Butanol NO 111.50 14.10 1.47 1.47 51.43 47.50 8.26O Butanol CH Butanol NO Butanol CO Butanol CO 115.77 13.76 1.48 1.48 52.72 48.76 8.13O Butanol N O 178.30 14.00 1.47 1.48 81.92 75.56 8.42O Butanol O Butanol O Butanol N Butanol H S 187.75 11.39 1.56 1.56 77.46 74.12 4.51O Butanol NO 105.53 14.56 1.46 1.46 49.49 45.38 9.07Continued on next page22
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Mol. A Medium Mol. B C vac6 ω ε (i ω ) ε app (i ω ) C app6 C exact6 rel. Dev.O Butanol CH Butanol NO Butanol CO Butanol CO 67.71 14.41 1.46 1.46 31.58 29.04 8.73O Butanol N O 104.35 14.65 1.45 1.46 49.09 45.04 9.01O Butanol O Butanol O Butanol N Butanol H S 109.06 11.92 1.54 1.54 46.08 43.79 5.21O Butanol NO 61.86 15.24 1.44 1.44 29.69 27.09 9.61N Butanol CH Butanol NO Butanol CO Butanol CO 74.72 13.80 1.48 1.48 34.08 31.65 7.66N Butanol N O 115.11 14.03 1.47 1.47 52.95 49.05 7.93N Butanol O Butanol O Butanol N Butanol H S 120.86 11.52 1.55 1.55 50.15 48.08 4.32N Butanol NO 68.17 14.57 1.46 1.46 31.97 29.46 8.53H S Butanol CH S Butanol NO S Butanol CO S Butanol CO 129.60 10.96 1.57 1.57 52.45 50.60 3.64H S Butanol N O 199.10 11.13 1.57 1.57 81.18 78.13 3.90H S Butanol O S Butanol O S Butanol N S Butanol H S 215.59 9.33 1.64 1.64 80.60 79.90 0.87H S Butanol NO 117.12 11.53 1.55 1.55 48.62 46.52 4.52NO Butanol CH O 111.50 14.10 1.47 1.47 51.43 47.50 8.26NO Butanol O S 117.12 11.53 1.55 1.55 48.62 46.52 4.52NO Butanol NO 66.03 14.66 1.45 1.46 31.07 28.53 8.88CH Propanol CH Propanol NO Propanol CO Propanol CO 96.18 12.15 1.51 1.51 42.08 39.80 5.74CH Propanol N O 147.94 12.34 1.51 1.51 65.25 61.54 6.01CH Propanol O Propanol O Propanol N Propanol H S 157.89 10.28 1.58 1.58 63.53 61.82 2.76CH Propanol NO 87.30 12.79 1.49 1.49 39.22 36.78 6.62NO Propanol CH Propanol NO Propanol CO Propanol CO 111.21 13.67 1.47 1.47 51.69 47.64 8.52NO Propanol N O 171.27 13.90 1.46 1.46 80.31 73.80 8.81NO Propanol O PREPRINT - F
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Mol. A Medium Mol. B C vac6 ω ε (i ω ) ε app (i ω ) C app6 C exact6 rel. Dev.NO Propanol O Propanol N Propanol H S 180.42 11.34 1.54 1.54 76.21 72.61 4.96NO Propanol NO 101.36 14.45 1.44 1.45 48.48 44.29 9.46CO Propanol CH Propanol NO Propanol CO Propanol CO 113.25 13.69 1.47 1.47 52.68 48.63 8.34CO Propanol N O 174.42 13.92 1.46 1.46 81.84 75.34 8.62CO Propanol O Propanol O Propanol N Propanol H S 183.51 11.40 1.54 1.54 77.73 74.11 4.88CO Propanol NO 103.25 14.46 1.44 1.45 49.40 45.22 9.25CO Propanol CH O 122.44 13.35 1.48 1.48 56.24 52.09 7.95CO Propanol O S 129.60 10.96 1.55 1.55 53.83 51.64 4.23CO Propanol NO 72.38 13.87 1.46 1.46 33.90 31.22 8.60N O Propanol CH O Propanol NO O Propanol CO O Propanol CO 122.44 13.35 1.48 1.48 56.24 52.09 7.95N O Propanol N O 188.52 13.58 1.47 1.47 87.33 80.68 8.24N O Propanol O O Propanol O O Propanol N O Propanol H S 199.10 11.13 1.55 1.55 83.31 79.73 4.50N O Propanol NO 111.50 14.10 1.45 1.45 52.67 48.38 8.89O Propanol CH Propanol NO Propanol CO Propanol CO 115.77 13.76 1.46 1.46 54.01 49.66 8.76O Propanol N O 178.30 14.00 1.46 1.46 83.92 76.95 9.05O Propanol O Propanol O Propanol N Propanol H S 187.75 11.39 1.54 1.54 79.48 75.61 5.12O Propanol NO 105.53 14.56 1.44 1.44 50.68 46.20 9.71O Propanol CH Propanol NO Propanol CO Propanol CO 67.71 14.41 1.44 1.45 32.34 29.57 9.36O Propanol N O 104.35 14.65 1.44 1.44 50.27 45.85 9.64O Propanol O Propanol O Propanol N Propanol H S 109.06 11.92 1.52 1.52 47.26 44.66 5.81O Propanol NO 61.86 15.24 1.42 1.43 30.40 27.57 10.24N Propanol CH Propanol NO PREPRINT - F
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Mol. A Medium Mol. B C vac6 ω ε (i ω ) ε app (i ω ) C app6 C exact6 rel. Dev.N Propanol CO Propanol CO 74.72 13.80 1.46 1.46 34.91 32.24 8.27N Propanol N O 115.11 14.03 1.46 1.46 54.23 49.96 8.55N Propanol O Propanol O Propanol N Propanol H S 120.86 11.52 1.53 1.53 51.45 49.05 4.91N Propanol NO 68.17 14.57 1.44 1.44 32.74 30.00 9.16H S Propanol CH S Propanol NO S Propanol CO S Propanol CO 129.60 10.96 1.55 1.55 53.83 51.64 4.23H S Propanol N O 199.10 11.13 1.55 1.55 83.31 79.73 4.50H S Propanol O S Propanol O S Propanol N S Propanol H S 215.59 9.33 1.61 1.61 82.84 81.64 1.47H S Propanol NO 117.12 11.53 1.53 1.53 49.89 47.46 5.12NO Propanol CH O 111.50 14.10 1.45 1.45 52.67 48.38 8.89NO Propanol O2