Uncertainty Quantification in First-Principles Predictions of Harmonic Vibrational Frequencies of Molecules and Molecular Complexes
Holden L. Parks, Alan. J. H. McGaughey, Venkatasubramanian Viswanathan
UUncertainty quantification in first-principlespredictions of harmonic vibrational frequenciesof molecules and molecular complexes
Holden L. Parks, Alan. J. H. McGaughey, and Venkatasubramanian Viswanathan ∗ Department of Mechanical Engineering, Carnegie Mellon University, Pittsburgh,Pennsylvania 15213, USA
E-mail: [email protected] a r X i v : . [ phy s i c s . c h e m - ph ] D ec bstract Accurate prediction of molecular vibrational frequencies is important to identifyspectroscopic signatures and reaction thermodynamics. In this work, we develop amethod to quantify uncertainty associated with density functional theory predictedharmonic vibration frequencies utilizing the built-in error estimation capabilities of theBEEF-vdW exchange-correlation functional. The method is computationally efficiencyby estimating the uncertainty at nearly the same computational cost as a single vibra-tional frequency calculation. We demonstrate the utility and robustness of the methodby showing that the uncertainty estimates bounds the self-consistent calculations of sixexchange correlation functionals for small molecules, rare gas dimers, and molecularcomplexes from the S22 dataset. Ten rare-gas dimers and the S22 dataset of molecu-lar complexes provide a rigorous test as they are systems with complicated vibrationalmotion and non-covalent interactions. Using coefficient of variation as a uncertaintymetric, we find that modes involving bending or torsional motion and those dominatedby non-covalent interactions are found to have higher uncertainty in their predictedfrequencies than covalent stretching modes. Given the simplicity of the method, webelieve that this method can be easily adopted and should form a routine part of DFT-predicted harmonic frequency analysis.
Accurate prediction of molecular vibrational frequencies by ab initio methods is importantin many areas of chemistry and physics.
Calculating the enthalpy and free energy of areaction, for example, requires zero-point energy (ZPE) and finite temperature contributions,both of which depend on vibrational frequencies. The accuracy of the frequencies dependson the level of theory used to model the system, convergence criteria, and if anharmoniccorrections have been included. A widely-used method to predict vibrational frequenciesis density functional theory (DFT). The accuracy of a DFT calculation depends strongly2n the choice of the exchange-correlation (XC) functional. There has been much workdevoted to identifying the best XC functional and DFT calculation parameters for accuratelypredicting the frequencies of small-to medium-sized molecules.
Effort has also beenextended to predicting frequencies of weakly-bonded systems.
Patton and Pederson and Tao and Perdew showed that DFT calculations of rare-gas dimers at the generalized-gradient (GGA) level correct the overbinding introduced by the local density approximation(LDA) functional. This body of results has shown that DFT-predicted frequencies at theGGA level may be accurate to within tens of meV of experimental frequencies for manysmall molecules. However, for complex vibration modes, the sensitivity of the predictions tothe choice of XC functionals remains unclear.One naïve way to estimate the uncertainty associated with DFT-predicted frequenciesis to perform the calculation with multiple XC functionals. The selection of functionals issomewhat arbitrary, however, and the calculation must be performed multiple times makingit computationally inefficient. The Bayesian error estimation functional with van der Waalscorrelation (BEEF-vdW) is a GGA-level XC functional that can systematically estimateuncertainty in DFT predictions. It possesses built-in uncertainty estimation capabilitiesin the form of an ensemble of GGA XC functionals that are calibrated to reproduce theerror observed between experimental measurements and DFT predictions. Using the BEEF-vdW ensemble is computationally efficient compared to performing many calculations usingdifferent XC functionals, as results for thousands of XC functionals are obtained non-self-consistently by using one self-consistent calculation. BEEF-vdW has been applied to quantifyuncertainty in magnetic ground states heterogeneous catalysis, electrocatalysis, and mechanical properties of solid electrolytes. The BEEF-vdW model space includescontributions from the non-local vdW-DF2 XC functional. It therefore offers potentialimprovements compared to other GGA-level functionals in describing van der Waals forces,which are traditionally not well described by DFT. In this work, we present a computationally efficient method to estimate the uncertainty3f harmonic vibrational frequencies of non-periodic systems such as molecules and molecularcomplexes. The harmonic approximation is valid at temperatures well below the dissociationlimit, which are on the order of , K for most systems considered herein but can be aslow as
K for the rare-gas dimers.
Frequency ensembles are calculated by solving aneigenvalue problem for an ensemble of Hessian matrices that depend on the second derivativesof the energy of the system with respect to its nuclear coordinates. The formulation of theeigenvalue problem is presented in Section 2.1. The ensemble of Hessians is calculated froman ensemble of energies determined with BEEF-vdW, which is described in Section 2.2.Computational details are provided in Section 2.3.We first apply our method to a set of small benchmark molecules in Section 3.1 todetermine if it can capture the uncertainty associated with choosing an XC functional. Wealso make comparisons to experimental measurements. We then consider a case study ofthe ten rare-gas dimers studied by Patton and Pederson and Tao and Perdew in Section3.2. Rare-gas dimers offer a simple test of the ability of the BEEF-vdW XC functionaland the BEEF-vdW ensemble to describe weakly-bonded systems. In Section 3.3, we studythe non-covalently bonded complexes in the S22 dataset. Because some XC functionalsare known to perform well on rare-gas dimers but poorly on larger complexes (and viceversa), the S22 dataset offers an extensive test of the ability of BEEF-vdW to describecomplicated vibrations in non-covalently bonded systems. We compare the predictions forall the molecules and molecular complexes in Section 3.4. In considering all the results,we utilize coefficient of variation as a metric to quantify uncertainty and should that thecoefficient of variation is largest for modes that involve bending or torsion or whose bonds arenon-covalent. Localized stretching modes tend to have lower uncertainty in their predictedfrequencies. The spread of the frequencies can also be used to quantify the uncertaintyassociated with quantities derived from the frequencies (e.g., Gibbs free energy or ZPE).Given the simplicity of the developed method, we believe that this should form a routinepart of DFT-predicted harmonic frequency analysis.4
Methods
In this section, we discuss a method for predicting harmonic vibrational frequencies bycomputing energies using DFT. Let N be the number of atoms in a molecule, and let i and j index the atoms so that ≤ i, j ≤ N . α and β denote the Cartesian directions [i.e., α, β = x (1) , y (2) , z (3) ]. We make the harmonic approximation, in which a Taylor seriesexpansion of the system’s potential energy, E , is truncated after the second-order term. In this approximation, the force on atom i in the α -direction, F αi , is proportional to thedisplacement of every other atom in the system from its equilibrium position. Newton’s 2ndlaw for atom i in the α -direction is thus m i ¨ x αi = F αi = − (cid:88) β (cid:88) j (cid:32) ∂ E∂x αi ∂x βj (cid:33) x βj = − (cid:88) β (cid:88) j Φ αβij x βj , (1)where m i is the mass of atom i . The second derivatives in eq 1 are called the harmonicforce constants, Φ αβij . By assuming temporally periodic atomic motions, the set of equationsrepresented by eq 1 for all atoms in all directions can be converted into the eigenvalueproblem ω e = He . (2)Here, ω is a vibrational frequency, e is a vector of length N describing the motion of theatoms, i.e. the mode shape, and H is the 3 N × N Hessian matrix. The elements of theHessian are related to the harmonic force constants by5 ∗ ( i − α, ∗ ( j − β = 1 √ m i m j Φ αβij . (3)The harmonic force constants are calculated by numerically approximating the secondderivative in eq 1 using a central finite difference of the energies with respect to perturbationsof the equilibrium structure. Using the shorthand E ( x αi ± h, x βj ± h ) = E i ± h,j ± h , where h is the perturbation magnitude of the atomic displacement, the central difference formulasare Φ αβij ≈ h (cid:18) E i + h,j + h + E i − h,j − h − E i + h,j − h − E i − h,j + h (cid:19) , if i (cid:54) = j − (cid:80) k (cid:54) = i Φ αβik , if i = j. (4)Note that we do not directly calculate the harmonic force constants where i = j . Thispractice reduces numerical error associated with the eggbox effect by enforcing conservationof momentum (i.e., translational invariance). The harmonic force constants are symmetricalwith respect to permutation of the atomic and direction indices (i.e., Φ αβij = Φ βαji ), so thatwe only need to calculate the upper triangular portion of the Hessian. Other molecule-specific symmetries can further reduce the number of force constants to be calculated, butwe do not consider these here.The harmonic force constants can also be calculated with a finite difference of the forceon atom i . We use the energies because, as will be explained in Section 2.2, BEEF-vdWprovides uncertainty estimation in the system energy and not in the atomic forces.
The energy of a system can be predicted using DFT and takes the form E = E KE + E ions + E e − e + E xc , (5)where E is the electronic energy of the system used in eq 4. E KE is the kinetic energy, E ions
6s the potential energy of the electrons due to Coulombic interactions with the ions, and E e − e is the potential energy of the electrons due to Coulombic interactions with other electrons,all of which can be calculated exactly. E xc is the exchange-correlation energy, whose valueis approximated by the chosen XC functional.BEEF-vdW is an XC functional at the GGA level. The BEEF-vdW model space takesthe form E xc = (cid:88) m =0 a m E GGA − x m + α c E LDA − c + (1 − α c ) E PBE − c + E nl − c , (6)where a m and α c are multiplicative factors, E LDA − c is a correlation contribution from thelocal Perdew-Wang LDA correlation, E PBE − c is a correlation contribution from the PBEsemi-local correlation, E nl − c is a correlation contribution from the vdW-DF2 non-localcorrelation, and E GGA − x m is the contribution to the exchange energy given by E GGA − x m = (cid:90) (cid:15) UEGx [ n ( r )] B m { s [ n ( r ) , ∇ n ( r )] } d r . (7)In eq 7, n ( r ) and ∇ n ( r ) are the electron density and its gradient, s is a function thatparameterizes n and ∇ n , (cid:15) UEGx is the exchange energy density of the uniform electron gas, and B m is the m th Legendre polynomial. To determine the optimal BEEF-vdW XC functional,Wellendorff et al. fit a m and α c to energetic and structural data describing bonding in avariety of chemical and condensed matter systems. These parameters were regularized toprevent overfitting to the training data.BEEF-vdW provides a systematic approach to estimating uncertainty in a DFT energycalculation by using an ensemble of XC functionals around the optimal BEEF-vdW XC func-tional. A self-consistent DFT calculation is first performed using the optimal parameters,yielding a converged electron density. This density is then used with distributions of a m and α c to non-self-consistently generate an ensemble of energies using eq 6. The distributionsof a m and α c are tuned such that the spread of the ensemble energies reproduces the er-7ors observed when comparing the experimental training data to BEEF-vdW self-consistentpredictions using the optimal XC functional.BEEF-vdW provides an ensemble of energies rather than a single energy for a DFTcalculation. This ensemble can be propagated in eq 4 to obtain an ensemble of numericalderivatives ∂ E/∂x αi x βj and Hessians. The eigenvalue problem can be solved for each Hessianin the ensemble to determine the ensemble of frequencies. Self-consistent DFT calculations were performed with the real-space projector-augmentedwave method as implemented in GPAW.
The BEEF-vdW XC functional was usedwith 2000 ensemble functionals for each calculation. Using more than 2000 functionals hasbeen found to have little effect on the standard deviation of the ensemble energy values.
We used a real-space grid spacing of . Å for the rare-gas dimers and . Å otherwise.Molecules were surrounded by vacuum in cubic boxes. Box lengths were determined sothat adding eight additional real-space grid points along each axis changed the energy ofthe relaxed structure by no more than . eV. This criterion resulted in a box length ofat least Å for each system. Equilibrium geometries were determined by relaxing thestructure so that each atom experienced a force of less than . eV / Å. Starting geometriesfor the S22 dataset were obtained from the Benchmark Energy and Geometry DataBase. All single-point calculations were converged so that the energy variation between the finalthree iterations was less than − eV.To obtain perturbed energies for numerical estimation of the second derivative in eq 4,atomic displacements ( h ) of at most ± . Å were applied. The . Å displacement sizeled to variations in the harmonic force constants on the order of . eV / Å and at most . eV / Å with respect to force constants calculated with displacements as small as . Å. A smaller displacement leads to smaller energy variations, which require more stringentconvergence criteria and longer computation time, but can yield more accurate numerical8erivatives. Numerical variation in the force constants will impact the calculated frequencies,but this effect is suppressed by two factors: (i) the requirement that the force constants satisfyconservation of momentum (eq 4), and (ii) the low sensitivity of the Hessian to the eigenvalueproblem because it is Hermitian. A comparison of the effect of numerical uncertainty in theforce constants to the BEEF-vdW uncertainty for two vibrational frequencies (one high andone low) of the benzene-ammonia complex, a member of the S22 dataset, is shown in Figure1. The numerical uncertainty histograms were generated by adding draws from a Gaussiandistribution N (0 , . eV / Å to the force constants calculated using the BEEF-vdW XCfunctional. This process was repeated , times to ensure converged error estimates.For both frequencies, the spread due to the numerical uncertainty in the force constants issmaller than the ensemble spread, indicating that in considering the latter we may ignorethe effects of the former.For larger molecules and molecular complexes, the atomic motions described by the eigen-vectors in eq 2 are often delocalized. To analyze the atomic motions in terms of the movementof local groups of atoms, they were transformed from Cartesian to internal coordinates. Theinternal coordinates were determined using the MolMod Python package. Bond length,bend angle, and dihedral angle internal coordinates, corresponding to stretching, bending,and torsional modes, were considered. The atomic motions were then mapped onto the in-ternal coordinates by following the decomposition method outlined by Boatz and Gordon. The components of stretch, bend, and torsion internal coordinate motion for each mode sumsto unity within a numerical error of at most . . As such, the relative contribution of eachtype of internal coordinate motion can be compared across different modes. Intermolecularmodes are excluded from this mapping procedure, as their motion does not correspond tomotion of the internal coordinates. 9 a)(b) Figure 1: Numerical uncertainty (blue histogram, right axis scale), BEEF-vdW ensemble(gray histogram, left axis scale), and BEEF-vdW prediction (dotted black line) for (a) highand (b) low frequency vibration of the benzene-ammonia complex. Histograms in both figuresare normalized to have a total area of unity. The spread of frequencies due to numericaluncertainty in both cases is smaller than the BEEF-vdW ensemble spread.10
Results
To test the effectiveness of the proposed method, we analyze the estimated uncertainty due tochoice of XC functional in predictions of vibrational frequencies of small molecules. We firstexamine a set of eight small molecules: H , N , CO, CO , H O, NH , H CO, each of whichhas only stretching and bending modes. We then compare the results to those for two largermolecules, HCOOH and C H , that have more complicated modes (e.g., a torsional mode inthe case of C H ). We designate these ten molecules as our “benchmark” set. Frequencieswere predicted using the PBE, RPBE, PBEsol, PW91, and optPBE-vdW, andBEEF-vdW functionals self-consistently.Results for the eight small molecules are presented in Table 1 and for HCOOH and C H in Table 2. Experimental values and BEEF-vdW ensemble statistics are also shown. Totest for normality in the BEEF-vdW ensemble, we use the skew and kurtosis to calculatethe Jarque-Bera (JB) statistic, given by JB = n (cid:104) S + K (cid:105) , where n = 2000 is thesample size of the ensemble, S is the sample skew, and K is the sample kurtosis. Underthe null hypothesis that the ensemble is Gaussian, JB is approximately described by a χ (2) distribution for large n . We choose to reject the null hypothesis at a confidence level( α = 0 . ), corresponding to a χ value of approximately 6 (i.e., we label the ensemble asnon-Gaussian if JB > ). We use the standard deviation ( σ ) and coefficient of variation,COV = σ/µ , where µ is the mean of the distribution, of the ensembles as measures ofuncertainty due to choice of XC functional. A low COV indicates that the functionals tendto agree in their frequency predictions, while a high COV indicates disagreement. In Section3.4 we find that a COV of . reasonably separates ensembles with low and high uncertainty.As a representative example, consider the asymmetric stretching mode of NH . TheBEEF-vdW ensemble of frequencies, predictions using other functionals, and the experimen-tal value ( . meV ) are plotted in Figure 2. The internal coordinate decomposition of11 able 1: Harmonic vibrational frequencies for eight small molecules in meV aspredicted with the PBE, RPBE, PBEsol, PW91, optPBE-vdW, and BEEF-vdWXC functional. Experimental values and the BEEF-vdW ensemble statistics arealso listed. Standard deviation values are in meV. COV, skew, and kurtosis aredimensionless. Stretch Bend Tors. Expt. PBE RPBE PBEsol PW91 optPBE BEEF µ σ
COV Skew Kurt. JB-vdW -vdWH O 0 1 0 204.3 CO 0.04 0.96 0 147.7 able 2: Harmonic vibrational frequencies in meV as predicted with the PBE,RPBE, PBEsol, PW91, optPBE-vdW, and BEEF-vdW XC functionals for C H and HCOOH. Experimental values and the BEEF-vdW ensemble statistics arealso listed. Standard deviation values are in meV. COV, skew, and kurtosis aredimensionless. Stretch Bend Tors. Expt. PBE RPBE PBEsol PW91 optPBE BEEF µ σ
COV Skew Kurt. JB-vdW -vdWC H the mode in Table 1 indicates that it is a pure stretching mode. We report a BEEF-vdWvalue of . meV and an ensemble standard deviation of . meV. The ensemble boundsthe experimental frequency to within one standard deviation, demonstrating that DFT canaccurately predict this frequency. In addition, the ensemble bounds the frequencies pre-dicted by the other XC functionals to within one standard deviation, which indicates thatthe ensemble can reproduce the predictions of other XC functionals. Based on the low COVof the BEEF-vdW ensemble ( . ), there is also little disagreement in ensemble functionalsin predictin this frequency. The ensemble has low skew ( − . ) and kurtosis ( . ). Thesevalues yield a JB of . , so that the ensemble is Gaussian.In the Supporting Information, we propagate the uncertainty estimates provided by theNH frequency ensembles for all NH modes to quantify uncertainty in NH ZPE. We alsoshow how ensembles for N , H , and NH can be used to quantify uncertainty in predictingthe vibrational entropy for the Haber process, 3H +N → NH .13he data in Table 1 indicate that most of the 23 modes of the small molecules are purestretching or pure bending. There are only a few modes with internal coordinate mixing,e.g., the third mode for H CO, which is . stretching and . bending. In terms of meanabsolute error (MAE) in comparison to the experimental measurements, BEEF-vdW is thebest XC functional for predicting vibrational frequencies. The experimental frequencies for16 of the 23 frequencies are bounded to within one standard deviation of the BEEF-vdWensemble, with the largest deviation ( . σ ) coming from the sixth mode of H CO. Onaverage, the predictions via BEEF-vdW deviate . σ from the experimental values.The predictions of the other XC functionals are also generally bounded by the BEEF-vdW ensemble to within one standard deviation, with each XC functional predicting atmost three frequencies outside of ± σ . The average deviation is smallest for PW91 ( . σ ),followed by optPBE-vdW ( . σ ), PBE ( . σ ), RPBE ( . σ ), and PBEsol ( . σ ).The largest COV for an ensemble is . , for the lowest frequency mode of NH . The lowCOV values indicate agreement among the ensemble functionals in predicting the frequencies.Each ensemble has relatively low skew and kurtosis values, the highest values being a skewof − . for the third H CO mode and a kurtosis of . for the first NH mode. Eightmodes have JB > and are non-Gaussian. Each of these eight modes is a pure bend modeor a mix of bending and stretching.The results presented in Table 1 indicate that for small molecules with simple vibrationalmotion, the BEEF-vdW ensemble can bound both the spread of predictions of other XCfunctionals and experimental frequencies. To test the robustness of the BEEF-vdW ensemblefor molecules containing torsional vibrations, we next apply the method to HCOOH andC H and the results are presented in Table 2. By the JB test, 12 of the 21 modes havenon-Gaussian ensembles, none of which correspond to pure stretch modes.Based on the MAE, BEEF-vdW yields the frequencies closest to the experimental mea-surements. For 15 of the 21 modes, the experimental frequencies are bounded to withinone standard deviation of the BEEF-vdW prediction, with an average deviation of . σ .14igure 2: Experimental and DFT predictions of the harmonic vibrational frequency of theasymmetric stretch mode of NH , with the BEEF-vdW ensemble shown in gray. The darkergray indicates a range of ± σ around the mean of the BEEF-vdW ensemble mean.Amongst the XC functionals considered, PW91 again has the smallest deviation ( . σ ) andPBEsol again has the largest ( . σ ). These deviations are smaller in comparison to thosefor the eight smaller molecules and seem to indicate better agreement among XC functionalsfor predicting of these frequencies. The ensembles for several modes, however, indicate thisis not always the case. The two lowest frequency modes for both HCOOH and C H haveensemble COV higher than the highest value reported in Table 1, with a maximum of . for the C H torsional mode. This torsional mode also has the largest standard deviation, . meV, of the modes presented in both tables. While the ensemble for this mode eas-ily bounds the experimental and other DFT-predicted frequencies, its high COV indicatesdisagreement among GGA-level functionals in predicting this frequency. This mode is alsonoteworthy because some 5% of the ensemble functionals predict a imaginary frequencies.These these imaginary frequencies correspond to negative solutions to the eigenvalue prob-lem in eq 2 and indicate that some ensemble XC functionals predict that this mode is notstable. These values are not included in calculation of any ensemble statistics. There is alsoa significant fraction of ensemble functionals, over 15%, which predict very small (nonzerobut less than 2 meV) frequencies. These findings are in agreement with previous analysisshowing the difficulty in describing the torsional energy landscape of C H within GGA-level15FT. In Section 3.1, we determined that the proposed methodology can reasonably bound fre-quencies both from experiment and predictions using other XC functionals for a benchmarkset of ten molecules. We next consider ten rare-gas dimers built from He, Ne, Ar, and Kr.These systems are diatomic molecules and therefore have only one stretching mode. Rare-gas dimers are bonded by dispersion interactions and have been studied to determine theapplicability of DFT to simple van der Waals systems. Patton and Pederson and Tao andPerdew found that while GGA functionals corrected the overbinding tendency of the LDAfunctional, they overestimated the interaction strength when the outer electron shell con-sisted of s electrons (as in He ) and underestimated interaction strength when the outer shellconsisted of p electrons (as in Ne ). The rare-gas dimers offer an interesting case study forboth the BEEF-vdW XC functional, which explicitly models non-local interactions, and theBEEF-vdW ensemble, as the vibrational frequencies are small enough that the uncertaintyin the prediction approaches the magnitude of the frequency itself ( ∼ meV). Table 3: Harmonic vibrational frequencies for rare-gas dimers in meV as pre-dicted with the PBE, RPBE, optPBE-vdW, vdW-DF2, and BEEF-vdW XCfunctionals. BEEF-vdW ensemble statistics are also presented. Standard devia-tion values are in meV. COV, skew, and kurtosis are dimensionless. Experimentalvalues are from Ogilvie and Wang.
The PBE and TPSSh values from Taoand Perdew include BSSE corrections. Expt. PBE PW91 TPSSh PBE RPBE optPBE vdW BEEF µ σ
COV Skew Kurt. JB-vdW -DF2 -vdWHe
16e compare the BEEF-vdW ensemble of frequencies for each molecule to those predictedby the PBE, RPBE, optPBE-vdW, vdW-DF2, and BEEF-vdW XC functionals in Table3. Experimental values are from Ogilvie and Wang. We also include DFT values fromPatton and Pedersen and Tao and Perdew. The tendency of GGA functionals to correctthe severe overestimation of frequencies by LDA is consistent with our PBE and RPBEfrequencies. For example, Tao and Perdew reported a frequency of . meV for He using LDA, while our PBE value of . meV for the same quantity is much closer to theexperimental value.The three vdW functionals tested, optPBE-vdW, vdW-DF2, and BEEF-vdW, overesti-mate the frequency for each dimer. BEEF-vdW in particular performs poorly. For example,it predicts a frequency of . meV for HeNe, which has an experimental value of . meV,and overpredicts the experimental frequencies by an average of . meV. The previously-observed tendency of GGA functionals to overestimate s -shell electron interaction energyand underestimate p -shell electron interaction energy is also reproduced in our results.The exceptions are the vdW functionals, which consistently overestimate the vibrationalfrequency for every dimer. This tendency can also be observed in considering the bindingenergy for each dimer, which is calculated from the electronic energy difference between thedimer and its constituent monomers. A table of binding energies can be found in the Sup-porting Information. In considering the MAE in comparison to the experimental frequencies,the most accurate predictions come via Tao and Perdew using the meta-GGA level TPSShfunctional ( . meV MAE). The best GGA-level functional is PBE ( . meV MAE forTao and Perdew’s values, . meV MAE in this work).Statistics for the BEEF-vdW ensembles are presented on the right side of Table 3. Bythe JB test, most notably, none of the ensembles are Gaussian. With the exception of He ,the experimental value for each frequency is bounded to within two standard deviations ofthe BEEF-vdW result, with an average deviation of . σ . For each dimer, the majorityof the frequencies in the ensemble overestimate the experimental value. The frequency17redictions from the other tested XC functionals are also bounded by the ensemble to withintwo standard deviations, but agreement between BEEF-vdW and the other functionals isworse than for the benchmark molecules in Section 3.1. The smallest average deviationfrom the BEEF-vdW predictions came from optPBE-vdW ( . σ ), followed by vdW-DF2( . σ ), PW91 via Patton and Pederson ( . σ ), RPBE ( . σ ), PBE ( . σ ), PBE via Taoand Perdew ( . σ ), and TPSSh via Tao and Perdew ( . σ ). This higher spread is alsopresent in the BEEF-vdW ensembles. The COV of every ensemble is higher than the COVof any frequency ensemble in Section 3.1, with the exception of the C H torsional mode.The high COV values and consistent overprediction of the experimental frequencies reflectthe difficulty that GGA-level XC functionals have in consistently and correctly describingthe van der Waals interactions in rare-gas dimers. The rare-gas dimers from Section 3.2 may not be representative of non-covalent interactionsin larger systems.
As such, we next apply our methodology to S22, a set of 22 non-covalently bonded molecular complexes. The complexes in S22 are bonded by hydrogenbonds, dispersion bonds, or a mix of the two. The size of each complex varies between 6 and30 atoms. The complexes in S22 contain both intermolecular and intramolecular vibrationaldegrees of freedom. Note that there are always six intermolecular vibrational modes for eachcomplex. By predicting their vibrational frequencies, it will be possible to examine a givenXC functional’s ability to describe both covalent and non-covalent interactions in the samesystem.The results for the vibrational frequencies of the water dimer are presented in Table4 with internal coordinate decomposition for each mode. We include frequencies from Xuand Goddard predicted with the GGA functional BLYP and the hybrid functionalB3LYP. From Dunn et al., we include frequencies predicted with MP2 theory us-ing the aug-cc-pVDZ basis set. The results for the entire S22 dataset are available in the18upporting Information.BEEF-vdW predicts the highest intramolecular frequencies (modes 7 to 12), with valuescomparable to the B3LYP frequencies reported by Xu and Goddard. With respect to theexperimental intramolecular frequencies reported by Fredin et al., BEEF-vdW is the bestfunctional out of the six tested in this work, with an MAE of 3.8 meV. The best resultscome via Xu and Goddard using B3LYP. A comparison of intermolecular frequencies isnot possible because, with the exception of mode 4, the experimental measurements havenot been corrected for anharmonic effects, and therefore the reported frequencies are notharmonic. We do note, however, that BEEF-vdW strongly overestimates the intermolecularfrequencies in comparison to the other XC functionals. All six experimental intramolecularfrequencies are bounded to within one standard deviation of the BEEF-vdW value, with anaverage deviation of . σ . Ensembles for the intramolecular modes all have COV lowerthan . , which is consistent with ensembles for H O reported in Table 1. Ensembles forintermolecular modes have relatively high COV, comparable to the COV values of the C H torsional mode and the rare-gas dimer modes, with the highest (1.22) coming from the lowestfrequency mode. Intermolecular modes of other S22 complexes also tend to have higher en-semble COV values. All six intermolecular ensembles for the water dimer are non-Gaussian,while this is true of only two of the six intramolecular modes. As with the C H torsionalmode, some BEEF-vdW ensemble functionals predict unstable imaginary frequencies formode 1 (458 functionals) and mode 2 (23 functionals). Challenges in describing interactionof small molecule clusters with different XC approximations has been previously noted. Based on the decomposition of the normal modes in terms of stretching, bending, and tor-sional motion for every molecule and complex considered, we now examine the relationshipbetween the physical motion of the mode and the spread of its BEEF-vdW ensemble, asmeasured by COV. In Figure 3a, the bending component is plotted against the stretching19 a)(b)
Figure 3: Coefficient of variation of the BEEF-vdW ensemble for each mode as functions of(a) stretching and bending character and (b) bending and torsional character. Modes fromthe benchmark, rare-gas dimer, and S22 datasets are all included.20 able 4: Water dimer vibrational frequencies (meV) predicted with various XCfunctionals. Modes 1 through 6 are intermolecular, while modes 7 through12 are intramolecular. BLYP and B3LYP values were predicted with aug-cc-pVTZ(-f ) basis sets, and MP2 values with the aug-cc-pVDZ basis set. MP2values were multiplied by a scale factor by Dunn et al. and are rescaled here.All experimental intermolecular modes except mode 4 are not corrected foranharmonic effects. ME and MAE are calculated for intramolecular frequenciesonly.
Mode Stretch Bend Tors Expt. PBE RPBE PBEsol PW91 optPBE BLYP B3LYP MP2 BEEF µ σ
COV Skew Kurt. JB-vdW -vdW1 10.9 component with marker color determined by the magnitude of the COV. Figure 3b is similar,with the bending component plotted versus the torsional component. The data plotted inFigures 3a and 3b indicate that COV tends to (i) increase as the component of bendingor torsion increases and (ii) decrease as the stretching component increases. The rare-gasdimers, which have high COV stretch modes, are an exception.The COV is plotted as a function of the BEEF-vdW ensemble mean frequency in Figure4. The physical character of the mode is indicated by the marker color, with red, green, andblue representing pure stretch, bend, and torsion. The intermolecular modes for the S22complexes are marked in black. The COV decreases as the mean frequency increases, withthe S22 intermolecular modes having the largest spreads. Modes with frequencies higher than200 meV are almost entirely stretch modes with low COV. Torsional and bending modes,rare-gas dimer stretching modes, and intermolecular modes tend to have frequencies lowerthan 200 meV and higher COV.The COV for all modes, organized by dataset and molecule, are provided in Figure 5.Within each dataset, the molecules and complexes are listed in order of increasing mass. Ananalogous plot with a logarithmic vertical axis is provided in the Supporting Information.21igure 4: The COV as a function of BEEF-vdW ensemble mean frequency, with the physicalcharacter of the mode indicated by the marker color. Pure stretch, bend, and torsion modesare red, green, and blue. S22 intermolecular modes are black.From Figures 4 and 5, a general ordering of mode types based on their COV can be observed.Stretching modes tend to have the lowest COV, followed by bending modes, torsional modes,and intermolecular modes. For every molecule in S22, nearly all intermolecular modes havehigher COV than any other mode. The relatively high COV values for bending and torsionalmodes, in contrast with the stretching modes, indicates disagreement among XC functionalsin describing these vibrations. As stretching modes are typically localized while bending andtorsional modes are not, our results suggest that XC functionals at the GGA level havedifficulty consistently describing a molecule’s non-local vibrational behavior. The stretchingmodes of the rare-gas dimers have high COV and are an exception to this rule. These systemsare bonded by van der Waals interactions, however, and the high COV values correctly reflectdisagreement in XC functionals in describing non-covalent interactions. This inconsistencyalso causes the high COV values of the intermolecular S22 modes.Based on Figure 5 there is a separation of modes at a COV value of . , which is markedby a horizontal dashed line. Modes above this boundary include the high COV stretchmodes of the rare-gas dimers, most of the S22 intermolecular modes, and some bending andtorsional modes from C H and several S22 complexes. Based on this separation, we proposea criterion of a COV equal to . for disagreement among XC functionals in predicting the22igure 5: The COV for all modes in each of the three datasets considered in this work. Thephysical character is indicated by the color of the marker. Molecules within each dataset(benchmarks, rare-gas dimers, and S22) are listed in order of increasing mass.23requency. For ensembles with COV larger than . , extra care should be taken in choosingan XC functional for the frequency prediction. We presented a computationally efficient method to quantify the uncertainty due to thechoice of the XC functional at the GGA level in DFT predictions of harmonic vibrationalfrequencies. To test the robustness of this method, we considered three sets of molecules ofvarying size and complexity. The first set consisted of small benchmark molecules (Tables 1and 2), none of which were larger than 8 atoms. We found that the BEEF-vdW ensemblebounds most experimental frequencies and the frequency predictions of six other XC func-tionals to within one standard deviation (e.g., Figure 2). We then applied the method to tenrare-gas dimers (Table 3) and the S22 dataset (Table 4) of molecular complexes, which offereda variety of systems that differed in their bonding environments (i.e., covalent, hydrogen,van der Waals) and physical nature of the their vibrational modes (i.e., stretch, bend, andtorsion, and intermolecular). Our results show that frequency predictions for modes withdelocalized motion, such as torsion and bending, and modes involving non-covalent bondsare more sensitive to choice of XC functional in comparison to predictions for localized andcovalent stretch modes (Figures 3, 4, and 5). Our proposed method can therefore be usedto link DFT uncertainty to the physical behavior of the system.
Acknowledgement
H. L. P. acknowledges support from a Presidential Fellowship at Carnegie Mellon Univer-sity and helpful feedback from Dilip Krishnamurthy. V. V. acknowledges support from theNational Science Foundation under award CBET-1554273.24 eferences (1) Cramer, C. J.
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The PBE and TPSSh binding energies from Tao andPerdew include BSSE corrections. Molecule Expt. PBE PBE RPBE optPBE-vdW vdW-DF2 PW91 BEEF-vdW TPSSh He In Figure S1 we plot the COV on a logarthmic vertical scale for all modes considered in thiswork, organized by dataset and molecule. There is an analogous plot without logarithmicscale in the main text.Figure S1: The coefficient of variation, plotted on a log scale, for all modes in each of the threedatasets considered in this work, with modes separated by molecule. The physical characteris indicated by the color of the marker. Molecules within each dataset (benchmarks, raregas dimers, and S22) are listed in order of increasing mass.3
The vibrational degrees of freedom in a non-periodic system contribute to its free energy,enthalpy, ZPE, and entropy. If the system considered is at a temperature well below its dis-sociation limit, then the use of the harmonic approximation is justified and the vibrationalcomponents of these thermodynamic quantities may be determined using harmonic modefrequencies obtained from the method outlined in the main text. We use ω k to denote thefrequency associated with the k th vibrational mode and (cid:15) k to denote the energy associatedwith this mode, so that (cid:15) k = ¯ hω k , where ¯ h is the reduced Planck constant.The Gibbs free energy of a system is given by G = H − T S, (S.2)where H is the enthalpy, T is the temperature, and S is the entropy. We assume that themolecules of interest are ideal gases. The enthalpy of an idea gas is H = E elec + E ZP E + Z T C p dT, (S.3)where E elec is the electronic energy (the direct output of a DFT calculation), E ZP E is theZPE, and C p is the constant pressure specific heat. The ZPE is E ZP E = 12 X k (cid:15) k . (S.4)For an ideal gas, C p is given by C p = k B + C v,rot + C v,trans + C v,vib , (S.5)where k B is the Boltzmann constant and comes from converting the constant volume specificheat to the constant pressure specific heat, C v,rot is the constant-volume rotational specificheat, C v,trans is the constant-volume translational specific heat, and C v,vib is the constant-4olume vibrational specific heat. For an ideal gas, C v,trans = 3 k B / and C v,rot is k B if thesystem is linear and k B / otherwise. The vibrational specific heat is given by C v,vib = X k exp (cid:16) (cid:15) k k B T (cid:17) k B (cid:15) k /T exp (cid:16) (cid:15) k k B T (cid:17) − . (S.6)Thus the integrals of C v,trans , C v,rot , and k B are constants scaled by T . The integral of C v,vib becomes Z T C v,vib dT = X k (cid:15) k exp (cid:16) (cid:15) k k B T (cid:17) − . (S.7)The entropy is given by S = − k B ln PP + S rot + S trans + S elec + S vib , (S.8)where P is the pressure and P is standard pressure, atm. Using the ideal gas approx-imation, the rotational ( S rot ), translational ( S trans ) , and electrical ( S elec ) contributions toentropy depend only on some combination of the temperature, symmetry number, and totalspin state of the molecule (i.e., they do not depend on the ω k ). The vibrational entropycomponent S vib is S vib = X k (cid:15) k exp (cid:16) (cid:15) i k B T (cid:17) − ln (cid:20) − exp (cid:18) − (cid:15) k k B T (cid:19)(cid:21) . (S.9)By calculating the ensemble of vibrational frequencies detailed in the main text, it ispossible to determine ensembles of thermodynamic quantities. For example, in Figure S2a,we plot the ensemble for the ZPE of NH , which is calculated with eq S.4 using the frequenciesfor NH from the main text. The distribution bounds the predictions of the other testedXC functionals and has a small standard deviation of . eV. In Figure S2b, we plot theensemble for the entropy of reaction at standard temperature and pressure ( . K and + N → , a crucial process for the industrial productionof ammonia. The entropy of reaction is given by ∆ S = S NH − (cid:18) S N + 32 S H (cid:19) , (S.10)where S NH , S N , and S H are calculated using eq S.8. The entropy ensemble has standarddeviation of . × − meV/K. The small standard deviations for both distributions indicatethat there is little variation in predicted ZPE and entropy values between different XCfunctionals. However, here we have only considered thermodynamic ensembles calculatedfrom frequencies of N , H , and NH , all of which are simple molecules. Larger molecules,such as the S22 molecular complexes, could yield thermodynamic ensembles with variedbehavior. 6 a)(b) Figure S2: Distributions for (a) the zero-point energy of NH , with standard deviation . eV and (b) entropy of reaction at standard temperature and pressure ( . K and bar)for the Haber process, with standard deviation . × − meV/K. Experimental values arecalculated from experimental frequencies from Shimanouchi. This section contains results for the S22 dataset of molecular complexes. For each vibra-tional mode, the tables contain internal coordinate decompositions, with S, B, and T stand-ing for stretch, bend, and torsion component; predictions using the PBE, RPBE, PBEsol,PW91, optPBE-vdW, and BEEF-vdW XC functionals; and BEEF-vdW ensemble statistics µ (mean), σ (standard deviation), COV (coefficient of variation), skew, Kurt. (kurtosis), JB(Jarque-Bera statistic), and Imag. (number of ensemble functionals predicting imaginaryfrequencies for this mode). Imaginary frequencies have zero real part, but these values arenot included in calculations of ensemble statistics. Table S2: CH dimer harmonic vibrational frequencies. S B T PBE RPBE PBE PW91 optPBE BEEF µ σ
COV Skew Kurt. JB Imag.sol -vdW -vdW1 1.5 0.0 0.0 1.7 2.2 37.2 36.3 17.5 0.48 0.0 -0.56 26.1 10212 2.4 0.0 2.3 2.3 4.3 37.8 32.0 20.1 0.63 -0.02 -0.85 60.8 8743 4.5 1.6 3.1 3.1 5.0 38.4 26.3 22.2 0.84 0.24 -1.14 127.0 5964 6.0 2.5 4.0 5.1 7.6 38.7 23.0 21.9 0.95 0.53 -1.01 176.8 3265 6.8 4.6 4.9 6.3 7.7 39.2 21.3 21.7 1.02 0.67 -0.87 213.4 1176 8.7 7.1 6.8 8.9 9.5 39.4 23.7 19.3 0.81 0.76 -0.47 213.0 167 0.0 1.0 0.0 158.4 159.7 155.4 159.1 161.6 171.2 162.9 10.5 0.06 -0.2 0.23 17.5 08 0.0 1.0 0.0 158.8 160.1 156.0 159.6 161.9 171.2 163.4 10.3 0.06 -0.18 0.22 15.1 09 0.0 1.0 0.0 158.9 160.1 156.0 159.6 162.0 171.4 163.7 10.3 0.06 -0.18 0.21 14.8 010 0.0 1.0 0.0 159.2 160.2 156.4 159.9 162.2 171.5 163.8 10.3 0.06 -0.18 0.21 14.7 011 0.0 1.0 0.0 159.2 160.3 156.5 160.0 162.3 171.6 164.0 10.3 0.06 -0.18 0.22 14.3 012 0.0 1.0 0.0 159.7 160.7 156.9 160.5 162.9 172.6 164.5 10.3 0.06 -0.18 0.2 14.2 013 0.0 1.0 0.0 186.2 187.2 183.9 186.9 188.4 196.3 190.5 7.8 0.04 -0.11 0.12 5.5 014 0.0 1.0 0.0 186.3 187.2 183.9 187.0 188.5 196.5 190.6 7.8 0.04 -0.11 0.11 5.0 015 0.0 1.0 0.0 187.2 187.9 184.9 187.9 189.3 197.1 191.3 7.7 0.04 -0.11 0.12 5.3 016 0.0 1.0 0.0 187.2 187.9 185.0 187.9 189.3 197.3 191.4 7.7 0.04 -0.11 0.11 4.8 017 1.0 0.0 0.0 368.5 367.6 367.1 369.4 367.5 365.4 372.3 6.3 0.02 -0.06 0.09 1.8 018 1.0 0.0 0.0 368.5 367.6 367.1 369.5 367.5 365.7 372.4 6.3 0.02 -0.06 0.09 1.9 019 1.0 0.0 0.0 382.4 380.9 382.1 383.0 379.7 376.8 385.2 7.9 0.02 -0.09 0.08 3.0 020 1.0 0.0 0.0 382.4 381.0 382.2 383.0 379.9 376.9 385.3 7.9 0.02 -0.09 0.08 3.0 021 1.0 0.0 0.0 382.6 381.0 382.2 383.2 380.0 377.1 385.5 7.9 0.02 -0.09 0.08 3.0 022 1.0 0.0 0.0 382.6 381.2 382.3 383.2 380.1 377.3 385.6 7.9 0.02 -0.09 0.08 3.1 023 1.0 0.0 0.0 382.8 381.8 382.7 383.6 380.6 377.8 386.2 8.0 0.02 -0.08 0.07 2.8 024 1.0 0.0 0.0 382.9 381.9 382.8 383.7 380.7 378.0 386.3 8.0 0.02 -0.08 0.07 2.8 0 able S3: NH dimer harmonic vibrational frequencies. S B T PBE RPBE PBE PW91 optPBE BEEF µ σ
COV Skew Kurt. JB Imag.sol -vdW -vdW1 2.1 0.5 0.1 2.6 0.9 27.5 30.1 23.5 0.78 0.17 -1.09 108.4 7292 11.5 9.1 10.8 12.3 9.8 31.7 23.3 24.3 1.04 0.58 -0.98 191.8 2723 14.3 11.6 15.2 15.1 11.9 32.3 21.7 24.4 1.12 0.71 -0.85 228.9 214 16.4 13.4 18.8 16.2 14.8 33.2 23.2 24.1 1.04 0.69 -0.83 217.9 05 26.1 22.2 29.0 25.9 22.6 39.8 30.5 20.6 0.67 0.8 -0.54 237.7 06 52.3 46.0 57.3 52.5 48.8 55.9 45.6 20.8 0.46 0.21 -0.97 93.2 07 0.0 1.0 0.0 126.4 129.2 123.2 126.3 128.1 135.4 130.0 13.0 0.1 -0.37 0.67 83.2 08 0.0 1.0 0.0 129.5 131.3 127.6 129.6 130.7 137.1 132.3 12.6 0.1 -0.35 0.58 68.2 09 0.0 1.0 0.0 198.6 200.4 195.4 199.2 201.2 208.4 204.1 8.9 0.04 -0.14 0.14 8.2 010 0.0 1.0 0.0 200.8 202.0 198.3 201.3 202.9 209.2 205.4 8.7 0.04 -0.13 0.18 8.6 011 0.0 1.0 0.0 201.1 202.3 198.4 201.6 203.3 210.2 206.0 8.7 0.04 -0.13 0.14 7.2 012 0.0 1.0 0.0 202.1 202.9 199.8 202.5 203.9 210.2 206.6 8.6 0.04 -0.13 0.15 7.4 013 1.0 0.0 0.0 418.3 418.1 416.4 419.3 417.4 420.7 425.0 7.5 0.02 -0.06 0.05 1.5 014 1.0 0.0 0.0 418.6 418.2 417.1 419.6 417.6 421.0 425.1 7.5 0.02 -0.06 0.04 1.4 015 1.0 0.0 0.0 433.1 432.7 432.3 434.0 431.7 433.9 439.7 8.5 0.02 -0.08 0.06 2.3 016 1.0 0.0 0.0 433.2 432.7 432.4 434.0 431.8 434.0 439.9 8.5 0.02 -0.08 0.06 2.3 017 1.0 0.0 0.0 435.8 433.8 436.3 436.5 433.0 434.9 440.5 8.4 0.02 -0.08 0.06 2.3 018 1.0 0.0 0.0 435.8 433.8 436.4 436.6 433.0 434.9 440.5 8.4 0.02 -0.08 0.06 2.3 0 able S4: H O dimer harmonic vibrational frequencies.
S B T PBE RPBE PBE PW91 optPBE BEEF µ σ
COV Skew Kurt. JB Imag.sol -vdW -vdW1 17.0 13.2 17.2 14.2 13.4 13.5 26.3 25.2 0.96 0.49 -1.0 164.7 4582 19.7 15.3 21.3 18.8 17.4 25.3 23.3 25.3 1.09 0.73 -0.77 228.9 233 21.1 18.5 22.3 20.0 17.8 35.7 26.1 25.6 0.98 0.63 -0.9 199.7 04 23.4 19.2 27.0 23.4 20.9 39.8 31.1 23.6 0.76 0.64 -0.79 188.6 05 45.9 38.4 49.0 44.0 42.1 47.7 40.3 22.6 0.56 0.51 -0.9 154.6 06 79.5 69.1 88.2 80.2 72.2 74.6 64.8 21.1 0.33 -0.32 -0.47 52.5 07 0.0 1.0 0.0 196.9 198.7 193.9 196.8 199.0 204.6 201.6 9.2 0.05 -0.15 0.19 10.5 08 0.0 1.0 0.0 200.0 201.1 198.2 200.4 201.0 206.9 203.7 9.1 0.04 -0.16 0.17 10.9 09 1.0 0.0 0.0 440.4 447.8 431.6 440.3 444.6 456.5 457.0 9.2 0.02 -0.06 -0.03 1.4 010 1.0 0.0 0.0 460.3 459.9 460.7 461.7 457.9 465.6 466.6 8.2 0.02 -0.06 0.03 1.2 011 1.0 0.0 0.0 470.0 469.6 470.6 471.3 467.8 475.3 476.9 8.8 0.02 -0.06 0.03 1.3 012 1.0 0.0 0.0 472.5 471.9 473.2 473.8 469.8 476.6 478.7 8.6 0.02 -0.07 0.04 1.6 0 able S5: Ethene-ethyne dimer harmonic vibrational frequencies. S B T PBE RPBE PBE PW91 optPBE BEEF µ σ
COV Skew Kurt. JB Imag.sol -vdW -vdW1 4.5 3.6 4.0 2.9 0.3 20.4 18.0 8.9 0.49 -0.1 -0.65 38.1 9992 4.7 3.7 6.0 4.1 3.2 23.5 15.2 13.3 0.87 0.29 -1.16 140.1 5513 8.5 5.9 7.5 8.3 9.1 26.9 14.3 13.8 0.97 0.67 -0.88 213.1 1144 10.4 8.7 11.2 10.2 9.8 27.2 14.6 13.2 0.91 0.73 -0.75 225.8 145 12.5 10.4 13.3 12.0 10.5 36.9 23.3 19.0 0.82 0.9 -0.33 280.6 46 76.8 74.3 77.6 77.2 75.5 85.9 73.1 17.2 0.24 -0.72 1.44 346.5 17 0.35 0.64 0.0 76.8 74.3 77.7 77.2 75.6 85.9 73.2 17.0 0.23 -0.64 1.02 223.3 18 0.01 0.0 0.99 93.7 92.8 93.4 93.8 93.2 101.7 92.6 14.6 0.16 -0.66 1.81 419.1 09 0.05 0.95 0.0 93.9 92.8 93.6 93.9 93.2 101.8 92.7 14.5 0.16 -0.63 1.58 340.5 010 0.0 1.0 0.0 99.7 99.8 98.6 100.1 101.1 109.9 100.7 12.4 0.12 -0.44 0.91 134.3 011 0.0 0.0 1.0 116.8 115.7 116.2 117.3 116.8 124.8 116.6 11.3 0.1 -0.28 0.47 44.8 012 0.0 0.0 1.0 117.4 117.1 116.3 117.7 117.7 125.3 118.0 11.5 0.1 -0.28 0.37 36.9 013 0.0 0.0 1.0 128.6 128.2 128.0 128.9 128.6 135.3 129.5 9.0 0.07 -0.2 0.26 19.0 014 0.0 1.0 0.0 148.9 149.2 147.3 149.5 150.6 157.6 151.8 7.7 0.05 -0.11 -0.01 4.0 015 0.35 0.65 0.0 166.0 166.0 164.6 166.4 167.0 169.8 167.6 3.7 0.02 -1.41 3.21 1517.7 016 0.0 1.0 0.0 176.4 177.0 173.9 177.2 178.9 187.0 180.5 8.7 0.05 -0.13 0.13 7.5 017 0.65 0.35 0.0 203.0 201.9 203.2 203.4 203.2 205.4 206.0 2.1 0.01 0.89 1.75 519.5 018 1.0 0.0 0.0 248.5 247.3 248.3 249.0 248.2 245.3 250.1 4.7 0.02 -0.07 0.04 1.7 019 1.0 0.0 0.0 379.4 378.8 377.9 380.4 378.7 377.0 384.2 6.8 0.02 -0.07 0.08 2.4 020 1.0 0.0 0.0 380.9 380.4 379.4 381.9 380.0 378.1 385.7 7.1 0.02 -0.08 0.08 2.6 021 1.0 0.0 0.0 388.3 387.6 387.3 389.2 387.0 385.3 393.0 7.5 0.02 -0.08 0.06 2.7 022 1.0 0.0 0.0 391.7 391.1 390.6 392.7 390.5 388.7 396.5 7.5 0.02 -0.09 0.07 2.8 023 1.0 0.0 0.0 410.9 413.2 406.9 412.1 412.4 412.3 419.2 7.2 0.02 -0.08 0.05 2.6 024 1.0 0.0 0.0 424.5 425.1 421.9 425.7 425.1 424.3 431.4 7.4 0.02 -0.08 0.05 2.4 0 able S6: Ethene dimer harmonic vibrational frequencies. S B T PBE RPBE PBE PW91 optPBE BEEF µ σ
COV Skew Kurt. JB Imag.sol -vdW -vdW1 2.8 3.8 5.7 0.0 3.0 19.1 17.4 9.0 0.52 -0.14 -0.62 39.0 9952 2.8 3.8 5.7 0.5 6.4 20.7 14.2 10.7 0.75 0.07 -1.14 109.5 6733 5.6 3.9 8.0 5.6 8.7 26.6 13.8 13.8 1.01 0.45 -1.18 182.7 2494 5.8 4.6 8.2 5.7 8.8 26.7 14.3 12.2 0.85 0.71 -0.79 218.9 615 8.1 7.0 10.5 7.0 11.3 36.0 21.1 20.7 0.98 0.83 -0.57 256.2 146 8.1 7.0 10.5 7.0 11.6 36.1 23.1 19.1 0.83 0.91 -0.32 284.3 07 0.0 1.0 0.0 99.5 99.8 98.3 99.8 100.9 109.5 100.7 12.3 0.12 -0.44 0.9 133.2 08 0.0 1.0 0.0 100.0 100.2 98.9 100.3 101.8 110.4 101.6 12.1 0.12 -0.42 0.82 115.4 09 0.0 0.0 1.0 115.8 115.1 115.3 116.3 116.1 123.8 115.8 11.5 0.1 -0.3 0.5 51.0 010 0.0 0.0 1.0 116.3 115.4 115.5 116.8 116.7 124.3 116.2 11.5 0.1 -0.31 0.52 53.5 011 0.0 0.0 1.0 116.5 116.6 115.5 116.8 117.0 124.6 117.4 11.7 0.1 -0.29 0.4 41.8 012 0.0 0.0 1.0 116.5 116.6 116.1 116.8 117.2 124.6 117.5 11.6 0.1 -0.29 0.39 39.7 013 0.0 0.0 1.0 127.7 127.7 127.0 128.1 127.9 134.5 128.7 9.3 0.07 -0.22 0.29 22.7 014 0.0 0.0 1.0 128.8 128.2 128.8 129.1 129.2 135.5 129.6 9.2 0.07 -0.21 0.28 21.6 015 0.0 1.0 0.0 148.9 149.2 147.3 149.5 150.6 157.4 151.8 7.6 0.05 -0.12 0.0 4.8 016 0.0 1.0 0.0 148.9 149.2 147.3 149.5 150.6 157.4 151.8 7.6 0.05 -0.12 0.01 4.6 017 0.3 0.7 0.0 165.4 165.7 163.8 166.0 166.5 169.2 167.2 3.7 0.02 -1.43 3.25 1563.7 018 0.32 0.68 0.0 166.4 166.3 165.3 167.0 167.6 169.9 168.1 3.4 0.02 -1.5 3.61 1834.4 019 0.0 1.0 0.0 176.5 177.2 173.9 177.3 179.1 186.9 180.7 8.6 0.05 -0.13 0.15 7.7 020 0.0 1.0 0.0 176.5 177.2 173.9 177.3 179.1 186.9 180.7 8.6 0.05 -0.13 0.14 7.6 021 0.69 0.31 0.0 202.8 202.0 203.0 203.4 202.9 204.8 205.6 2.1 0.01 0.98 2.11 693.8 022 0.68 0.32 0.0 203.2 202.3 203.3 203.7 203.4 205.5 206.1 2.2 0.01 1.03 2.09 714.4 023 1.0 0.0 0.0 379.3 378.5 377.4 380.4 378.8 377.2 384.2 6.9 0.02 -0.07 0.07 2.2 024 1.0 0.0 0.0 379.3 378.5 377.4 380.4 378.8 377.3 384.3 6.9 0.02 -0.07 0.07 2.2 025 1.0 0.0 0.0 380.8 380.1 378.7 381.8 380.1 378.2 385.6 7.2 0.02 -0.08 0.06 2.4 026 1.0 0.0 0.0 380.9 380.2 379.0 381.9 380.1 378.4 385.9 7.2 0.02 -0.07 0.06 2.2 027 1.0 0.0 0.0 388.2 387.2 386.7 389.2 387.1 385.4 393.0 7.6 0.02 -0.08 0.06 2.6 028 1.0 0.0 0.0 388.2 387.2 386.7 389.2 387.1 385.5 393.0 7.6 0.02 -0.08 0.06 2.6 029 1.0 0.0 0.0 391.6 390.6 389.9 392.6 390.4 388.7 396.3 7.6 0.02 -0.08 0.07 2.7 030 1.0 0.0 0.0 391.6 390.7 390.1 392.6 390.5 389.0 396.6 7.6 0.02 -0.08 0.06 2.7 0 able S7: Formamide dimer harmonic vibrational frequencies. S B T PBE RPBE PBE PW91 optPBE BEEF µ σ
COV Skew Kurt. JB Imag.sol -vdW -vdW1 7.8 7.4 8.2 8.0 6.0 11.3 10.3 6.8 0.66 -0.26 -1.15 132.6 6942 17.5 15.9 19.2 17.7 15.2 14.1 14.0 6.7 0.47 -0.81 -0.34 230.9 5453 18.7 16.9 19.6 18.9 17.8 19.5 15.8 9.1 0.58 -0.43 -1.0 145.5 3294 21.8 17.8 24.1 22.4 20.1 21.3 18.9 8.8 0.46 -0.2 -0.46 30.5 945 22.4 20.5 25.4 22.6 21.0 29.8 23.1 11.4 0.49 0.28 -0.09 27.4 56 28.2 22.2 33.3 28.9 26.5 33.5 26.1 10.9 0.42 0.51 -0.27 94.0 17 0.0 0.0 1.0 61.4 56.2 65.1 61.8 58.9 66.2 52.9 21.2 0.4 -0.45 -0.83 124.6 08 0.0 0.0 1.0 63.2 58.0 66.9 63.6 60.6 67.4 55.8 19.1 0.34 -0.33 -0.85 95.5 09 0.06 0.94 0.0 74.1 72.8 74.9 74.3 73.9 75.6 73.7 7.6 0.1 0.33 3.31 949.3 010 0.06 0.94 0.0 77.8 75.6 79.8 78.1 76.8 78.7 75.9 7.1 0.09 0.12 3.39 964.6 011 0.0 0.0 1.0 103.6 95.9 110.8 104.4 98.5 101.0 94.1 12.6 0.13 -0.36 0.43 58.9 012 0.0 0.0 1.0 106.8 99.7 112.8 107.5 102.3 104.9 98.4 11.5 0.12 -0.3 0.45 46.1 013 0.0 0.0 1.0 124.8 124.3 124.8 125.2 124.1 132.2 125.3 8.9 0.07 -0.38 0.01 48.8 014 0.0 0.0 1.0 126.6 125.5 127.7 127.0 125.6 133.1 126.2 8.9 0.07 -0.44 0.04 65.3 015 0.3 0.7 0.0 132.2 131.0 132.5 132.7 132.3 134.2 132.7 5.6 0.04 -0.22 0.73 60.6 016 0.35 0.65 0.0 133.2 131.7 133.9 133.7 133.2 134.8 133.5 5.5 0.04 -0.13 0.95 80.6 017 0.58 0.42 0.0 161.5 158.7 163.4 161.8 160.0 159.4 160.8 1.7 0.01 -0.63 9.95 8390.4 018 0.54 0.46 0.0 163.2 160.2 164.7 163.5 161.9 160.9 162.2 1.9 0.01 -0.82 7.42 4815.5 019 0.18 0.82 0.0 169.8 170.1 168.7 170.4 170.7 176.8 172.9 5.0 0.03 -0.19 -0.67 50.3 020 0.17 0.83 0.0 169.9 170.2 169.4 170.4 170.8 176.9 173.1 4.8 0.03 -0.11 -0.76 51.6 021 0.32 0.68 0.0 195.9 196.9 194.0 196.2 197.4 200.8 198.9 4.6 0.02 -1.36 1.88 912.0 022 0.1 0.9 0.0 197.5 197.8 195.9 198.0 199.0 203.9 201.0 5.9 0.03 -0.62 0.03 130.3 023 0.61 0.39 0.0 209.2 208.3 209.9 209.4 208.2 210.7 213.1 2.8 0.01 0.79 1.64 433.1 024 0.8 0.2 0.0 213.3 211.9 215.0 213.3 211.5 212.2 216.1 3.1 0.01 0.74 0.87 247.2 025 1.0 0.0 0.0 357.3 356.1 355.9 358.6 357.8 354.6 362.4 7.4 0.02 -0.09 0.08 3.1 026 1.0 0.0 0.0 358.0 356.7 356.6 359.1 358.6 356.4 364.3 7.4 0.02 -0.09 0.08 3.1 027 0.99 0.01 0.0 380.8 399.2 358.3 380.0 390.5 401.8 410.6 8.6 0.02 -0.07 0.02 1.8 028 1.0 0.0 0.0 388.4 403.9 369.4 387.9 396.5 406.8 414.7 8.4 0.02 -0.07 0.02 1.8 029 1.0 0.0 0.0 443.9 444.6 442.1 444.6 442.2 445.8 450.6 7.8 0.02 -0.07 0.06 1.8 030 1.0 0.0 0.0 444.1 444.8 442.2 444.7 442.2 445.9 450.8 7.8 0.02 -0.07 0.06 1.8 0 able S8: Formic acid dimer harmonic vibrational frequencies. S B T PBE RPBE PBE PW91 optPBE BEEF µ σ
COV Skew Kurt. JB Imag.sol -vdW -vdW1 9.6 9.0 9.7 9.5 8.3 15.6 10.4 7.1 0.68 -0.24 -1.19 137.0 5832 21.8 19.7 23.1 22.1 21.0 22.7 13.8 8.5 0.61 -0.52 -1.11 192.5 2033 23.5 22.2 25.3 23.6 21.3 25.5 19.7 7.2 0.37 -0.55 -0.43 116.2 94 26.6 23.6 32.3 27.2 25.4 32.6 24.4 7.7 0.31 0.08 -0.66 39.1 05 32.8 31.3 34.4 32.9 30.8 34.7 29.6 7.6 0.26 0.53 0.42 108.1 06 37.0 31.4 47.8 38.2 34.4 39.7 33.1 8.1 0.24 0.59 -0.09 118.0 07 0.12 0.88 0.0 83.2 81.6 85.1 83.6 82.5 83.4 81.9 2.5 0.03 -0.96 3.9 1575.2 08 0.15 0.85 0.0 88.7 86.0 92.7 89.3 86.9 87.2 85.4 2.3 0.03 -1.05 3.94 1663.9 09 0.0 0.0 1.0 122.0 116.6 124.4 122.6 117.9 117.8 112.3 11.2 0.1 -0.27 0.13 25.6 010 0.0 0.0 1.0 122.7 117.9 125.5 123.5 118.9 120.5 115.1 10.3 0.09 -0.24 0.22 22.8 011 0.0 0.0 1.0 129.6 127.2 139.3 130.4 127.3 134.4 127.8 9.1 0.07 -0.29 0.0 27.7 012 0.0 0.0 1.0 135.0 130.6 145.1 136.0 131.2 136.4 130.0 9.1 0.07 -0.36 0.02 43.7 013 0.79 0.21 0.0 152.7 147.8 159.4 153.5 149.4 146.7 147.5 1.9 0.01 -1.49 9.14 7705.0 014 0.69 0.31 0.0 153.0 148.4 159.4 153.8 149.8 147.5 148.2 1.7 0.01 -0.9 9.09 7149.3 015 0.09 0.91 0.0 167.4 167.2 165.9 168.0 168.2 171.9 169.5 5.7 0.03 0.06 -0.18 3.8 016 0.15 0.85 0.0 167.9 167.5 167.7 168.4 168.4 173.8 170.3 5.9 0.03 0.16 -0.37 20.4 017 0.14 0.86 0.0 176.2 174.3 179.2 176.7 175.0 178.1 175.6 5.0 0.03 -0.14 -0.58 34.0 018 0.3 0.7 0.0 179.9 176.8 183.3 180.7 178.1 178.9 177.3 4.4 0.02 -0.17 -0.33 19.1 019 0.81 0.19 0.0 200.7 202.3 192.6 200.1 201.2 203.5 207.2 3.8 0.02 0.69 0.5 181.8 020 0.88 0.12 0.0 211.9 210.9 212.9 212.1 210.3 210.8 214.3 4.2 0.02 0.43 0.04 61.1 021 0.99 0.01 0.0 336.9 365.4 279.5 332.5 356.2 367.6 374.5 7.1 0.02 -0.09 0.08 3.5 022 1.0 0.0 0.0 354.1 367.7 306.4 350.8 366.4 368.2 375.2 7.1 0.02 -0.09 0.08 3.4 023 1.0 0.0 0.0 370.0 369.4 368.2 371.2 370.3 390.3 389.6 10.2 0.03 -0.02 -0.08 0.6 024 1.0 0.0 0.0 370.7 379.9 368.4 371.7 374.3 401.5 400.6 9.7 0.02 -0.04 -0.03 0.7 0 able S9: Benzene methane harmonic vibrational frequencies. S B T PBE RPBE PBE PW91 optPBE BEEF µ σ
COV Skew Kurt. JB Imag.sol -vdW -vdW1 0.1 0.0 0.2 0.5 0.1 13.2 11.7 6.4 0.54 -0.13 -0.69 45.5 9642 0.8 0.2 3.7 0.9 0.2 13.5 10.1 7.3 0.72 0.01 -1.13 107.0 7543 2.6 1.0 4.0 1.2 2.1 13.5 7.8 7.6 0.97 0.41 -1.15 166.7 2894 4.0 2.8 7.4 3.6 7.0 35.7 19.3 19.7 1.02 0.82 -0.8 276.0 255 4.6 3.1 7.6 3.8 7.2 36.9 19.5 19.8 1.01 0.8 -0.83 272.3 66 6.0 5.3 8.2 5.6 10.2 37.7 22.7 18.6 0.82 0.93 -0.37 302.9 37 0.0 0.0 1.0 49.0 48.8 48.6 49.2 48.9 53.7 48.7 7.9 0.16 -0.54 4.23 1588.5 08 0.0 0.0 1.0 49.0 48.9 48.6 49.2 48.9 53.7 48.8 7.8 0.16 -0.25 3.18 861.3 09 0.06 0.94 0.0 74.5 74.2 74.0 74.8 75.0 77.4 72.2 9.5 0.13 -2.68 9.26 9539.8 010 0.06 0.94 0.0 74.5 74.2 74.0 74.8 75.0 77.4 75.3 3.2 0.04 -0.84 3.67 1360.0 011 0.0 0.0 1.0 82.1 82.1 81.6 82.3 81.9 90.0 81.7 8.2 0.1 -0.04 -1.2 121.1 012 0.0 0.0 1.0 87.3 86.5 87.5 87.5 86.7 92.0 88.9 7.5 0.08 0.95 1.78 561.1 013 0.0 0.0 1.0 103.6 103.3 103.2 103.8 103.4 111.1 103.0 10.9 0.11 -0.55 0.13 102.6 014 0.0 0.0 1.0 103.6 103.4 103.2 103.9 103.5 111.4 103.7 11.3 0.11 -0.14 -0.25 12.1 015 0.0 0.0 1.0 118.5 118.0 118.1 118.9 118.4 120.2 116.1 7.2 0.06 -1.26 2.08 892.7 016 0.0 0.0 1.0 118.5 118.0 118.1 118.9 118.5 125.8 118.0 8.8 0.07 -0.76 0.46 207.5 017 0.0 0.0 1.0 122.0 121.4 121.7 122.4 121.9 126.0 120.3 8.1 0.07 -0.9 1.22 392.2 018 1.0 0.0 0.0 123.3 122.0 122.8 123.5 122.4 126.8 124.9 3.4 0.03 0.52 0.27 95.5 019 0.0 1.0 0.0 123.4 123.0 124.4 123.9 124.2 129.2 127.1 3.8 0.03 1.2 2.06 833.3 020 0.54 0.46 0.0 128.3 127.7 128.4 128.6 128.2 129.3 129.1 3.5 0.03 0.81 4.32 1773.8 021 0.53 0.47 0.0 128.4 127.8 128.4 128.7 128.2 129.4 130.0 3.8 0.03 1.32 3.08 1375.5 022 0.09 0.91 0.0 141.2 141.5 139.7 141.7 142.7 146.4 142.0 5.4 0.04 -0.96 0.25 314.5 023 0.23 0.77 0.0 144.1 144.2 142.9 144.5 145.0 149.0 145.6 5.1 0.03 -0.66 0.66 182.3 024 0.23 0.77 0.0 144.1 144.2 143.0 144.6 145.0 149.3 145.9 5.0 0.03 -0.65 0.57 168.9 025 0.0 1.0 0.0 158.6 159.8 155.6 159.4 161.5 160.7 159.4 6.5 0.04 -1.55 3.04 1563.7 026 0.0 1.0 0.0 159.2 160.3 156.4 160.0 162.1 171.1 163.5 10.0 0.06 -0.19 0.28 19.1 027 0.0 1.0 0.0 159.3 160.5 156.4 160.0 162.2 171.4 163.9 10.1 0.06 -0.15 0.05 7.2 028 0.0 1.0 0.0 165.4 164.0 163.6 166.1 164.1 171.6 167.1 7.4 0.04 0.22 1.0 99.4 029 0.91 0.09 0.0 166.4 165.6 169.4 166.3 167.3 174.6 172.9 4.7 0.03 0.95 1.13 404.9 030 0.29 0.71 0.0 181.8 181.4 181.2 182.4 182.8 187.1 184.5 4.6 0.02 -0.06 1.33 149.1 031 0.29 0.71 0.0 181.9 181.5 181.3 182.5 182.8 187.2 184.9 4.3 0.02 0.39 0.02 51.8 032 0.0 1.0 0.0 186.8 187.6 184.3 187.5 188.8 195.9 190.4 6.2 0.03 -0.45 -0.78 119.2 033 0.0 1.0 0.0 186.8 187.6 184.3 187.5 188.8 196.0 190.6 6.0 0.03 -0.38 -0.92 118.7 034 0.71 0.29 0.0 197.2 195.7 198.6 197.5 196.3 196.7 199.4 2.4 0.01 1.66 4.73 2783.8 035 0.71 0.29 0.0 197.3 195.8 198.7 197.5 196.4 196.8 199.5 2.4 0.01 1.67 4.75 2805.5 036 1.0 0.0 0.0 368.3 367.5 366.5 369.2 367.5 365.5 372.5 6.3 0.02 -0.06 0.09 1.9 037 1.0 0.0 0.0 382.1 380.6 381.6 382.5 378.9 376.1 384.7 7.9 0.02 -0.09 0.08 3.0 038 1.0 0.0 0.0 382.2 380.7 381.8 382.8 379.2 376.5 385.1 7.9 0.02 -0.09 0.08 3.1 039 1.0 0.0 0.0 383.2 382.4 382.0 383.8 382.0 379.8 387.4 7.5 0.02 -0.2 0.19 16.1 040 1.0 0.0 0.0 383.3 382.6 382.1 384.3 383.1 380.2 388.4 7.4 0.02 -0.09 -0.0 2.5 041 1.0 0.0 0.0 384.4 383.6 383.1 385.4 383.2 381.2 388.9 7.3 0.02 -0.07 0.06 1.9 042 1.0 0.0 0.0 384.6 383.7 383.2 385.5 383.5 381.4 389.4 7.5 0.02 -0.01 0.04 0.2 043 1.0 0.0 0.0 386.3 385.5 384.9 387.3 384.9 383.1 390.7 7.3 0.02 -0.07 0.09 2.2 044 1.0 0.0 0.0 386.4 385.6 385.0 387.4 385.0 383.2 391.0 7.3 0.02 -0.06 0.1 2.2 045 1.0 0.0 0.0 387.5 386.8 386.1 388.5 386.1 384.4 392.0 7.3 0.02 -0.08 0.08 2.7 0 able S10: Benzene ammonia harmonic vibrational frequencies. S B T PBE RPBE PBE PW91 optPBE BEEF µ σ
COV Skew Kurt. JB Imag.sol -vdW -vdW1 0.9 0.6 0.9 1.6 0.3 13.2 9.8 7.2 0.74 0.04 -1.11 103.4 7462 1.5 1.8 1.5 3.3 3.9 13.5 7.7 7.6 0.98 0.44 -1.17 179.7 3013 4.0 2.3 4.0 3.9 4.1 13.8 7.5 7.5 1.0 0.49 -1.16 191.9 574 5.1 3.4 5.1 4.9 8.8 32.4 20.2 20.8 1.03 0.76 -1.0 276.3 75 9.0 5.7 9.0 8.6 9.8 38.4 22.1 21.3 0.96 0.61 -1.25 252.3 36 13.6 10.1 13.6 14.7 11.6 39.5 27.2 20.1 0.74 0.79 -0.69 244.8 07 0.0 0.0 1.0 49.0 48.9 49.0 49.2 49.0 53.7 49.4 9.1 0.18 0.35 3.32 961.1 08 0.0 0.0 1.0 49.0 48.9 49.0 49.2 49.0 53.9 49.7 9.3 0.19 0.6 2.92 828.9 09 0.06 0.94 0.0 74.4 74.2 74.4 74.8 75.0 77.3 72.4 9.3 0.13 -2.69 9.38 9745.0 010 0.06 0.94 0.0 74.5 74.2 74.5 74.8 75.0 77.4 75.3 3.3 0.04 -0.51 4.14 1512.9 011 0.0 0.0 1.0 82.5 82.3 82.5 82.9 82.4 90.1 81.9 8.3 0.1 -0.09 -1.21 125.1 012 0.0 0.0 1.0 87.3 86.5 87.3 87.5 86.7 92.2 89.0 7.6 0.09 0.95 1.63 519.1 013 0.0 0.0 1.0 103.8 103.5 103.8 104.2 103.7 111.2 103.3 10.8 0.1 -0.57 0.15 109.8 014 0.0 0.0 1.0 103.9 103.6 103.9 104.3 103.8 111.6 104.1 11.3 0.11 -0.18 -0.17 12.9 015 0.0 0.0 1.0 118.7 118.1 118.7 119.2 118.6 120.1 116.3 7.3 0.06 -1.46 3.48 1718.0 016 0.0 0.0 1.0 118.7 118.2 118.7 119.2 118.7 125.8 118.3 8.7 0.07 -0.79 0.53 229.6 017 0.0 0.0 1.0 122.2 121.5 122.2 122.7 122.1 126.1 120.5 8.2 0.07 -1.02 1.73 599.8 018 1.0 0.0 0.0 123.2 121.9 123.2 123.4 122.3 126.7 122.7 7.4 0.06 -1.6 3.7 1990.4 019 0.0 1.0 0.0 123.4 123.0 123.4 123.9 124.2 129.1 126.3 4.5 0.04 0.82 0.79 274.7 020 0.06 0.94 0.0 127.4 127.7 127.4 127.1 128.0 129.4 128.4 4.0 0.03 0.87 2.47 761.4 021 0.54 0.46 0.0 128.4 127.7 128.4 128.6 128.2 129.4 129.7 4.2 0.03 0.96 2.22 717.0 022 0.48 0.52 0.0 128.4 130.6 128.4 128.7 128.8 137.4 133.2 6.8 0.05 0.62 -0.79 181.6 023 0.09 0.91 0.0 141.3 141.6 141.3 141.9 142.8 146.4 142.6 5.9 0.04 -0.64 0.03 135.2 024 0.23 0.77 0.0 144.1 144.2 144.1 144.6 145.0 149.1 145.7 5.0 0.03 -0.66 0.62 176.5 025 0.23 0.77 0.0 144.2 144.2 144.2 144.6 145.1 149.2 146.0 5.1 0.04 -0.29 0.48 47.0 026 0.0 1.0 0.0 165.4 163.9 165.4 166.1 163.9 160.7 162.9 3.8 0.02 -1.8 6.3 4382.5 027 0.91 0.09 0.0 166.2 165.6 166.2 166.2 167.4 174.6 172.9 4.7 0.03 0.89 0.8 317.1 028 0.29 0.71 0.0 181.7 181.4 181.7 182.3 182.7 187.1 184.7 4.3 0.02 0.43 0.08 62.2 029 0.29 0.71 0.0 181.9 181.5 181.9 182.4 182.8 187.2 184.9 4.2 0.02 0.53 0.03 95.0 030 0.71 0.29 0.0 197.1 195.6 197.1 197.3 196.2 195.9 196.9 3.2 0.02 -2.66 9.68 10171.2 031 0.71 0.29 0.0 197.1 195.7 197.1 197.4 196.3 196.0 197.2 2.9 0.01 -2.24 6.4 5088.2 032 0.0 1.0 0.0 200.1 201.6 200.1 200.7 202.1 208.9 206.3 6.6 0.03 0.83 0.05 228.0 033 0.0 1.0 0.0 200.2 201.9 200.2 200.7 202.3 209.8 206.8 6.7 0.03 0.78 -0.03 200.7 034 1.0 0.0 0.0 383.5 382.4 383.5 384.3 382.0 380.5 387.9 7.1 0.02 -0.08 0.08 2.5 035 1.0 0.0 0.0 384.6 383.6 384.6 385.4 383.0 381.5 388.8 7.2 0.02 -0.08 0.08 2.6 036 1.0 0.0 0.0 384.7 383.8 384.7 385.6 383.2 381.7 389.3 7.2 0.02 -0.08 0.09 2.7 037 1.0 0.0 0.0 386.5 385.5 386.5 387.2 384.8 383.3 390.7 7.2 0.02 -0.08 0.09 2.8 038 1.0 0.0 0.0 386.5 385.6 386.5 387.3 385.0 383.5 391.2 7.3 0.02 -0.08 0.09 2.8 039 1.0 0.0 0.0 387.6 386.8 387.6 388.4 386.1 384.6 392.1 7.3 0.02 -0.08 0.09 2.9 040 1.0 0.0 0.0 420.0 418.9 420.0 420.7 418.2 421.7 426.1 7.7 0.02 -0.07 0.05 1.6 041 1.0 0.0 0.0 434.5 433.4 434.5 435.0 432.3 434.7 440.2 8.5 0.02 -0.08 0.06 2.5 042 1.0 0.0 0.0 435.4 433.5 435.4 436.2 432.7 435.4 442.2 8.9 0.02 -0.07 0.04 1.7 0 able S11: Benzene water harmonic vibrational frequencies. S B T PBE RPBE PBE PW91 optPBE BEEF µ σ
COV Skew Kurt. JB Imag.sol -vdW -vdW1 1.4 0.2 1.5 1.3 0.1 13.0 9.1 7.1 0.78 0.22 -1.12 120.9 6602 3.7 0.4 3.7 3.7 3.2 13.8 7.7 7.3 0.95 0.5 -1.07 179.0 2483 7.3 4.3 9.1 5.1 4.3 14.2 7.9 7.2 0.91 0.53 -1.04 182.3 364 10.4 6.7 10.1 10.2 11.3 30.9 21.2 21.6 1.02 0.67 -1.2 267.1 45 16.4 7.3 18.6 16.6 14.7 32.4 24.1 21.7 0.9 0.49 -1.43 250.8 16 26.8 24.2 30.1 26.8 26.5 41.1 30.2 21.8 0.72 0.51 -1.13 192.7 07 0.0 0.0 1.0 48.9 48.8 48.5 49.1 48.9 53.9 50.0 10.1 0.2 0.59 2.46 620.8 08 0.0 0.0 1.0 49.1 49.0 48.7 49.3 49.2 54.1 51.1 11.0 0.22 0.85 1.87 534.6 09 0.06 0.94 0.0 74.4 74.2 74.0 74.7 74.9 77.2 72.6 9.2 0.13 -2.62 9.59 9943.5 010 0.06 0.94 0.0 74.5 74.2 74.0 74.8 75.0 77.3 75.4 3.7 0.05 0.88 7.66 5142.0 011 0.0 0.0 1.0 82.9 82.6 82.4 83.2 82.7 90.2 82.0 8.3 0.1 -0.11 -1.19 122.3 012 0.0 0.0 1.0 87.3 86.5 87.5 87.5 86.7 92.9 89.2 7.7 0.09 0.95 1.52 493.5 013 0.0 0.0 1.0 104.1 103.8 103.7 104.5 103.9 111.8 103.6 10.8 0.1 -0.59 0.18 117.0 014 0.0 0.0 1.0 104.3 103.8 103.8 104.6 104.1 112.1 104.3 11.3 0.11 -0.16 -0.24 13.7 015 0.0 0.0 1.0 119.0 118.3 118.5 119.4 118.9 120.0 116.4 7.1 0.06 -1.29 2.24 974.7 016 0.0 0.0 1.0 119.0 118.4 118.5 119.5 118.9 126.2 118.4 8.7 0.07 -0.8 0.52 233.5 017 0.0 0.0 1.0 122.4 121.6 122.0 122.9 122.2 126.6 120.7 8.0 0.07 -0.92 1.27 418.8 018 1.0 0.0 0.0 123.2 121.9 122.9 123.4 122.3 126.7 125.0 3.5 0.03 0.5 0.09 84.6 019 0.0 1.0 0.0 123.5 123.0 124.2 124.0 124.2 129.0 127.1 3.9 0.03 1.19 1.83 753.7 020 0.54 0.46 0.0 128.2 127.7 128.3 128.6 128.0 129.3 129.1 3.6 0.03 0.92 4.32 1837.4 021 0.54 0.46 0.0 128.4 127.7 128.5 128.7 128.2 129.9 130.0 4.0 0.03 1.33 2.83 1257.0 022 0.09 0.91 0.0 141.4 141.7 139.9 142.0 142.8 146.4 142.2 5.3 0.04 -0.98 0.26 323.8 023 0.23 0.77 0.0 144.2 144.2 143.1 144.6 145.0 149.1 145.7 5.0 0.03 -0.67 0.62 179.7 024 0.23 0.77 0.0 144.2 144.3 143.1 144.7 145.1 149.1 145.9 4.9 0.03 -0.56 0.11 106.5 025 0.0 1.0 0.0 165.5 163.8 163.7 166.1 163.9 160.7 162.9 3.8 0.02 -1.8 6.34 4424.2 026 0.91 0.09 0.0 166.2 165.6 169.1 166.2 167.4 174.6 172.9 4.7 0.03 0.9 0.81 323.2 027 0.28 0.72 0.0 181.7 181.3 181.0 182.3 182.7 187.0 184.7 4.3 0.02 0.45 0.02 68.6 028 0.29 0.71 0.0 181.9 181.4 181.3 182.5 182.8 187.1 184.9 4.2 0.02 0.51 0.04 86.6 029 0.71 0.29 0.0 197.0 195.5 195.3 197.3 196.1 195.7 196.1 3.8 0.02 -1.95 3.77 2454.6 030 0.69 0.31 0.0 197.1 195.6 198.3 197.4 196.3 195.9 198.6 1.6 0.01 0.68 1.15 266.0 031 0.03 0.97 0.0 198.0 199.7 198.5 198.3 200.1 205.5 205.1 6.4 0.03 1.01 0.52 362.3 032 1.0 0.0 0.0 383.8 382.7 382.4 384.7 382.3 380.7 388.0 7.1 0.02 -0.08 0.08 2.6 033 1.0 0.0 0.0 384.9 383.8 383.5 385.8 383.4 381.8 389.1 7.2 0.02 -0.08 0.09 2.8 034 1.0 0.0 0.0 385.0 383.9 383.6 385.9 383.6 381.8 389.3 7.2 0.02 -0.08 0.08 2.6 035 1.0 0.0 0.0 386.7 385.7 385.2 387.6 385.1 383.5 390.9 7.3 0.02 -0.08 0.09 2.9 036 1.0 0.0 0.0 386.8 385.8 385.4 387.7 385.3 383.6 391.1 7.3 0.02 -0.08 0.09 2.7 037 1.0 0.0 0.0 387.8 387.0 386.4 388.8 386.3 384.7 392.2 7.3 0.02 -0.08 0.09 2.9 038 1.0 0.0 0.0 457.3 460.1 454.6 457.8 455.8 464.6 466.3 8.7 0.02 -0.07 0.03 1.5 039 1.0 0.0 0.0 471.4 471.8 471.4 472.2 468.1 476.0 478.4 8.9 0.02 -0.07 0.03 1.5 0 able S12: Benzene HCN harmonic vibrational frequencies. S B T PBE RPBE PBE PW91 optPBE BEEF µ σ
COV Skew Kurt. JB Imag.sol -vdW -vdW1 3.4 3.5 5.6 2.6 0.2 13.4 8.1 7.4 0.91 0.36 -1.16 154.4 4532 3.5 3.7 5.6 3.1 0.2 13.7 7.2 7.5 1.04 0.57 -1.07 202.9 1443 7.0 5.7 7.5 6.8 7.8 13.9 8.8 6.1 0.7 0.75 -0.53 210.8 174 7.1 6.8 7.7 6.9 8.0 27.7 14.9 13.1 0.88 0.83 -0.58 255.9 55 10.1 7.3 10.4 10.1 11.1 28.3 17.5 11.3 0.64 0.94 -0.1 297.2 16 49.0 49.0 48.6 49.2 49.0 54.0 48.6 7.3 0.15 -1.15 4.6 2206.4 07 0.0 0.0 1.0 49.0 49.0 48.7 49.2 49.1 54.1 48.7 7.1 0.15 -0.94 3.3 1199.7 08 0.06 0.94 0.0 74.4 74.2 73.9 74.7 74.9 77.3 72.6 8.9 0.12 -2.94 11.67 14234.6 09 0.06 0.94 0.0 74.4 74.2 74.0 74.8 75.0 77.4 74.4 6.1 0.08 -4.19 29.28 77271.0 010 0.0 0.0 1.0 83.3 83.0 82.9 83.7 83.0 90.3 81.3 10.3 0.13 -1.29 3.82 1771.7 011 0.0 0.0 1.0 87.3 86.6 87.5 87.6 86.8 93.1 86.9 10.4 0.12 0.21 -0.05 15.2 012 0.23 0.06 0.71 89.7 90.7 88.9 89.9 88.5 99.7 90.0 12.6 0.14 0.1 -0.7 43.7 013 0.0 1.0 0.0 90.1 90.8 89.2 90.2 88.9 99.9 92.5 10.1 0.11 0.63 0.06 134.2 014 0.0 0.0 1.0 104.6 104.1 104.2 104.9 104.3 112.2 104.1 10.7 0.1 -0.57 0.2 111.9 015 0.0 0.0 1.0 104.7 104.1 104.3 105.1 104.4 112.3 104.7 11.2 0.11 -0.17 -0.21 13.5 016 0.0 0.0 1.0 119.3 118.5 118.9 119.7 119.1 120.0 116.7 7.0 0.06 -1.31 2.35 1032.1 017 0.0 0.0 1.0 119.3 118.5 118.9 119.8 119.1 126.5 118.7 8.6 0.07 -0.81 0.56 245.8 018 0.0 0.01 0.99 122.6 121.8 122.3 123.2 122.2 126.6 120.9 7.9 0.07 -0.93 1.3 428.7 019 1.0 0.0 0.0 123.1 121.8 123.0 123.3 122.4 126.8 125.1 3.5 0.03 0.48 0.06 76.6 020 0.0 0.99 0.01 123.6 123.1 124.2 124.1 124.3 129.3 127.2 3.9 0.03 1.18 1.74 713.7 021 0.54 0.46 0.0 128.3 127.7 128.4 128.6 128.1 129.3 129.2 3.6 0.03 0.96 4.29 1837.4 022 0.54 0.46 0.0 128.4 127.7 128.4 128.6 128.2 130.0 130.1 4.0 0.03 1.35 2.76 1245.5 023 0.09 0.91 0.0 141.5 141.7 140.1 142.1 142.9 146.6 142.3 5.4 0.04 -0.98 0.28 329.1 024 0.23 0.77 0.0 144.3 144.3 143.2 144.7 145.1 149.2 145.8 5.0 0.03 -0.67 0.63 183.0 025 0.23 0.77 0.0 144.3 144.3 143.2 144.7 145.2 149.4 146.0 4.9 0.03 -0.57 0.1 107.4 026 0.0 1.0 0.0 165.5 163.7 163.7 166.1 163.8 160.8 162.9 3.8 0.02 -1.8 6.38 4474.0 027 0.91 0.09 0.0 166.2 165.7 169.1 166.3 167.4 174.8 172.9 4.7 0.03 0.89 0.79 316.4 028 0.28 0.72 0.0 181.8 181.3 181.2 182.3 182.7 187.2 184.7 4.3 0.02 0.45 -0.0 68.6 029 0.28 0.72 0.0 181.9 181.4 181.2 182.4 182.8 187.3 185.0 4.2 0.02 0.55 0.0 101.3 030 0.71 0.29 0.0 196.9 195.4 198.2 197.1 196.0 195.8 198.4 1.6 0.01 0.7 1.11 264.5 031 0.71 0.29 0.0 197.0 195.4 198.3 197.2 196.0 195.9 198.4 1.6 0.01 0.7 1.16 277.9 032 1.0 0.0 0.0 262.9 260.9 263.2 263.5 262.2 259.8 264.3 5.4 0.02 -0.07 -0.01 1.6 033 1.0 0.0 0.0 384.0 382.9 382.6 384.9 382.5 380.8 388.1 7.1 0.02 -0.08 0.08 2.6 034 1.0 0.0 0.0 385.0 384.0 383.7 385.9 383.6 381.9 389.2 7.1 0.02 -0.08 0.08 2.7 035 1.0 0.0 0.0 385.1 384.1 383.8 386.1 383.7 381.9 389.4 7.2 0.02 -0.08 0.08 2.7 036 1.0 0.0 0.0 386.8 385.9 385.4 387.6 385.3 383.6 391.0 7.2 0.02 -0.08 0.09 2.9 037 1.0 0.0 0.0 386.9 385.9 385.4 387.8 385.4 383.6 391.2 7.3 0.02 -0.08 0.08 2.7 038 1.0 0.0 0.0 387.9 387.1 386.5 388.8 386.5 384.7 392.2 7.3 0.02 -0.08 0.08 2.9 039 1.0 0.0 0.0 410.6 413.8 405.1 411.5 412.9 412.4 420.9 7.8 0.02 -0.08 0.02 2.0 0 able S13: Benzene dimer (parallel) harmonic vibrational frequencies. S B T PBE RPBE PBE PW91 optPBE BEEF µ σ
COV Skew Kurt. JB Imag.sol -vdW -vdW1 0.0 1.1 0.7 0.2 0.3 13.4 11.5 6.4 0.55 -0.08 -0.74 48.4 9512 0.1 1.3 1.4 0.7 0.6 13.6 10.1 7.1 0.7 0.05 -1.05 92.6 7673 0.4 1.4 1.5 2.8 4.7 13.7 8.4 7.5 0.89 0.35 -1.15 151.8 4514 1.2 1.5 3.3 3.2 4.9 14.1 7.5 7.3 0.98 0.6 -0.97 197.3 1385 1.5 2.8 4.4 3.8 6.7 14.6 8.0 6.9 0.86 0.7 -0.81 217.4 116 2.3 3.5 4.6 4.1 8.2 14.8 10.6 5.2 0.5 0.81 -0.01 220.3 47 0.0 0.0 1.0 48.8 48.7 48.4 49.0 48.8 53.9 48.3 7.5 0.16 -1.36 6.06 3669.0 38 0.0 0.0 1.0 48.9 48.7 48.5 49.1 48.8 54.0 48.4 7.4 0.15 -1.1 4.07 1785.8 29 0.0 0.0 1.0 48.9 48.8 48.6 49.2 48.9 54.1 48.5 7.2 0.15 -0.88 2.79 907.6 110 0.0 0.0 1.0 48.9 48.8 48.6 49.2 49.1 54.1 48.6 7.3 0.15 -0.97 3.39 1273.7 011 0.06 0.94 0.0 74.4 74.2 73.9 74.7 74.9 77.2 71.7 10.1 0.14 -2.61 8.85 8806.8 012 0.06 0.94 0.0 74.5 74.2 74.0 74.7 74.9 77.3 72.1 9.6 0.13 -2.69 9.26 9542.1 013 0.06 0.94 0.0 74.5 74.2 74.0 74.8 75.0 77.3 75.2 3.2 0.04 -0.96 4.86 2278.7 014 0.06 0.94 0.0 74.5 74.2 74.0 74.8 75.0 77.4 75.3 3.2 0.04 -0.94 4.63 2079.2 015 0.0 0.0 1.0 81.3 81.4 80.6 81.5 81.3 90.0 81.4 8.3 0.1 0.0 -1.18 115.2 016 0.0 0.0 1.0 81.9 81.9 81.3 82.3 82.0 90.1 81.7 8.2 0.1 -0.06 -1.19 119.8 017 0.0 0.0 1.0 87.1 86.3 87.3 87.4 86.5 92.0 88.5 7.5 0.08 0.93 1.89 584.1 018 0.0 0.0 1.0 87.1 86.4 87.3 87.4 86.5 92.6 88.8 7.7 0.09 0.92 1.68 517.2 019 0.0 0.0 1.0 102.9 102.8 102.3 103.2 102.8 110.9 102.4 11.1 0.11 -0.52 0.04 91.0 020 0.0 0.0 1.0 103.0 102.8 102.4 103.3 103.0 111.4 102.6 11.0 0.11 -0.52 0.04 91.2 021 0.0 0.0 1.0 103.2 103.1 102.6 103.6 103.3 111.4 103.3 11.4 0.11 -0.15 -0.28 14.1 022 0.0 0.0 1.0 103.4 103.2 103.0 103.8 103.4 111.7 103.6 11.3 0.11 -0.15 -0.26 13.6 023 0.0 0.0 1.0 118.0 117.6 117.5 118.5 118.0 120.0 115.8 7.5 0.06 -1.24 1.95 829.4 024 0.0 0.0 1.0 118.1 117.6 117.6 118.5 118.1 120.1 115.9 7.4 0.06 -1.24 1.95 828.9 025 0.0 0.0 1.0 118.2 117.7 117.7 118.7 118.2 125.7 117.6 8.9 0.08 -0.75 0.45 203.7 026 0.0 0.0 1.0 118.2 117.7 117.7 118.7 118.3 125.8 117.8 8.9 0.08 -0.75 0.46 207.7 027 0.0 0.0 1.0 121.6 121.0 121.1 122.1 121.5 125.9 120.0 8.2 0.07 -0.89 1.17 378.1 028 0.0 0.0 1.0 121.7 121.1 121.2 122.2 121.6 126.1 120.1 8.2 0.07 -0.89 1.18 379.9 029 1.0 0.0 0.0 123.3 122.0 122.8 123.5 122.4 126.7 124.8 3.4 0.03 0.54 0.31 104.6 030 1.0 0.0 0.0 123.4 122.0 122.8 123.6 122.5 126.7 124.9 3.4 0.03 0.53 0.3 102.3 031 0.0 1.0 0.0 123.4 122.9 124.4 123.9 124.1 129.1 126.9 3.7 0.03 1.22 2.12 867.8 032 0.0 1.0 0.0 123.4 123.0 124.5 123.9 124.1 129.1 127.0 3.7 0.03 1.22 2.11 864.0 033 0.55 0.45 0.0 128.3 127.7 128.3 128.5 128.1 129.2 129.1 3.5 0.03 0.79 4.4 1821.7 034 0.55 0.45 0.0 128.4 127.7 128.4 128.6 128.2 129.3 129.1 3.5 0.03 0.79 4.39 1813.1 035 0.55 0.45 0.0 128.4 127.7 128.4 128.6 128.2 129.5 129.9 3.8 0.03 1.33 3.19 1434.2 036 0.55 0.45 0.0 128.4 127.8 128.5 128.7 128.2 129.5 130.0 3.8 0.03 1.33 3.2 1445.0 037 0.11 0.89 0.0 141.1 141.5 139.6 141.6 142.5 146.1 141.9 5.4 0.04 -0.96 0.2 308.9 038 0.11 0.89 0.0 141.2 141.5 139.7 141.6 142.6 146.2 142.0 5.4 0.04 -0.95 0.19 306.9 039 0.24 0.76 0.0 144.0 144.1 142.8 144.4 144.8 148.8 145.5 5.1 0.03 -0.66 0.62 176.3 040 0.24 0.76 0.0 144.0 144.1 142.8 144.4 144.9 148.9 145.6 5.0 0.03 -0.66 0.62 176.5 041 0.24 0.76 0.0 144.1 144.2 142.9 144.4 144.9 148.9 145.7 5.0 0.03 -0.55 0.08 100.4 042 0.24 0.76 0.0 144.1 144.2 143.0 144.5 145.0 148.9 145.8 5.0 0.03 -0.55 0.07 99.9 043 0.89 0.11 0.0 165.3 163.9 163.5 166.0 164.1 160.5 162.8 3.8 0.02 -1.81 6.34 4443.8 044 0.89 0.11 0.0 165.3 164.0 163.5 166.0 164.2 160.6 162.9 3.8 0.02 -1.81 6.27 4371.8 045 0.0 1.0 0.0 166.4 165.5 169.4 166.3 167.2 174.3 172.8 4.7 0.03 0.9 0.83 328.0 046 0.0 1.0 0.0 166.5 165.5 169.5 166.3 167.2 174.3 172.9 4.7 0.03 0.9 0.83 325.1 047 0.27 0.73 0.0 181.8 181.4 181.2 182.4 182.7 186.8 184.7 4.3 0.02 0.47 0.05 73.4 048 0.27 0.73 0.0 181.9 181.4 181.2 182.4 182.7 186.9 184.7 4.2 0.02 0.47 0.05 73.7 049 0.27 0.73 0.0 181.9 181.4 181.2 182.4 182.8 187.0 184.9 4.1 0.02 0.57 0.06 108.6 050 0.27 0.73 0.0 181.9 181.5 181.3 182.4 182.8 187.0 185.0 4.1 0.02 0.57 0.06 109.8 051 0.7 0.3 0.0 197.3 195.7 198.6 197.5 196.3 195.8 198.5 1.7 0.01 0.72 1.11 273.1 052 0.7 0.3 0.0 197.3 195.7 198.6 197.5 196.4 195.8 198.6 1.7 0.01 0.72 1.1 272.1 053 0.7 0.3 0.0 197.3 195.7 198.7 197.6 196.4 195.8 198.7 1.7 0.01 0.72 1.11 275.1 054 0.7 0.3 0.0 197.4 195.7 198.7 197.6 196.5 195.9 198.7 1.7 0.01 0.72 1.1 273.0 055 1.0 0.0 0.0 383.1 382.1 381.7 384.1 381.4 379.8 387.2 7.1 0.02 -0.08 0.08 2.5 056 1.0 0.0 0.0 383.2 382.2 381.8 384.1 381.5 379.9 387.2 7.1 0.02 -0.08 0.08 2.5 057 1.0 0.0 0.0 384.2 383.4 382.8 385.2 382.4 380.9 388.2 7.2 0.02 -0.08 0.08 2.6 058 1.0 0.0 0.0 384.3 383.4 382.9 385.2 382.6 381.0 388.3 7.2 0.02 -0.08 0.08 2.6 059 1.0 0.0 0.0 384.4 383.5 382.9 385.4 382.7 381.0 388.5 7.2 0.02 -0.08 0.08 2.6 060 1.0 0.0 0.0 384.5 383.5 383.0 385.5 382.7 381.1 388.6 7.2 0.02 -0.08 0.08 2.7 061 1.0 0.0 0.0 386.1 385.3 384.7 387.0 384.3 382.7 390.1 7.3 0.02 -0.08 0.08 2.8 0 able S14: Benzene dimer (t-shape) harmonic vibrational frequencies. S B T PBE RPBE PBE PW91 optPBE BEEF µ σ
COV Skew Kurt. JB Imag.sol -vdW -vdW1 0.4 0.2 0.7 0.5 0.1 12.7 12.1 5.9 0.49 -0.02 -0.66 36.8 10282 1.1 1.6 1.7 0.9 0.1 12.8 12.2 5.9 0.48 -0.04 -0.65 35.6 10083 1.8 1.7 2.2 1.8 0.5 13.6 8.4 7.5 0.89 0.3 -1.18 145.5 5154 4.0 2.7 4.8 3.9 6.4 13.6 7.9 6.4 0.81 0.85 -0.52 264.6 235 4.3 2.9 5.9 4.0 6.7 14.2 8.2 6.5 0.79 0.77 -0.68 233.9 96 4.5 3.3 5.9 4.1 6.8 14.2 9.9 5.4 0.55 0.95 -0.05 302.6 37 0.0 0.0 1.0 49.0 48.9 48.6 49.2 48.9 53.7 48.5 7.3 0.15 -1.21 5.19 2736.8 38 0.0 0.0 1.0 49.0 48.9 48.7 49.2 49.0 53.7 48.6 7.2 0.15 -1.01 3.64 1444.9 39 0.0 0.0 1.0 49.1 48.9 48.9 49.3 49.1 53.9 48.7 7.1 0.15 -0.89 3.01 1018.2 110 0.0 0.0 1.0 49.1 49.0 48.9 49.3 49.2 54.0 48.8 7.1 0.15 -0.94 3.28 1193.0 011 0.06 0.94 0.0 74.4 74.2 73.9 74.7 75.0 77.3 72.1 9.5 0.13 -2.76 10.18 11187.2 012 0.06 0.94 0.0 74.5 74.2 73.9 74.8 75.0 77.4 72.3 9.3 0.13 -2.77 10.04 10955.1 013 0.06 0.94 0.0 74.5 74.2 74.0 74.8 75.0 77.4 75.3 3.2 0.04 -0.86 3.72 1400.2 014 0.06 0.94 0.0 74.5 74.3 74.1 74.8 75.1 77.4 75.3 3.2 0.04 -0.84 3.72 1386.4 015 0.0 0.0 1.0 82.1 82.2 81.6 82.5 82.0 90.0 81.7 8.2 0.1 -0.05 -1.21 123.4 016 0.0 0.0 1.0 82.4 82.2 81.8 82.6 82.0 90.2 81.9 8.3 0.1 -0.08 -1.21 123.6 017 0.0 0.0 1.0 87.3 86.5 87.5 87.5 86.7 92.2 88.8 7.6 0.09 0.94 1.76 554.0 018 0.0 0.0 1.0 87.3 86.5 87.6 87.6 86.7 92.4 89.1 7.6 0.09 0.94 1.69 535.0 019 0.0 0.0 1.0 103.5 103.3 103.0 103.8 103.2 111.3 102.9 10.9 0.11 -0.55 0.1 100.0 020 0.0 0.0 1.0 103.6 103.4 103.1 103.9 103.5 111.4 103.2 10.8 0.1 -0.55 0.12 103.0 021 0.0 0.0 1.0 103.9 103.4 103.4 104.1 103.6 111.4 103.9 11.2 0.11 -0.16 -0.25 14.3 022 0.0 0.0 1.0 104.0 103.6 103.5 104.3 103.7 111.8 104.0 11.2 0.11 -0.16 -0.25 13.8 023 0.0 0.0 1.0 118.4 118.0 118.0 118.9 118.3 120.0 116.1 7.3 0.06 -1.26 2.09 894.7 024 0.0 0.0 1.0 118.7 118.0 118.2 119.1 118.5 120.2 116.2 7.2 0.06 -1.26 2.09 896.2 025 0.0 0.0 1.0 118.7 118.1 118.3 119.1 118.5 125.9 118.1 8.7 0.07 -0.77 0.5 221.3 026 0.0 0.0 1.0 118.8 118.2 118.3 119.2 118.6 126.0 118.2 8.7 0.07 -0.78 0.52 226.8 027 0.0 0.0 1.0 122.1 121.4 121.7 122.6 121.9 126.0 120.3 8.1 0.07 -0.91 1.22 398.0 028 0.0 0.0 1.0 122.2 121.5 121.8 122.6 122.0 126.3 120.6 8.0 0.07 -0.92 1.29 422.6 029 0.85 0.15 0.0 123.2 121.8 122.7 123.4 122.2 126.7 124.9 3.4 0.03 0.51 0.2 89.5 030 1.0 0.0 0.0 123.4 122.0 122.8 123.5 122.4 126.7 125.0 3.4 0.03 0.5 0.18 85.3 031 0.15 0.85 0.0 123.4 122.9 124.2 123.9 124.2 129.2 127.0 3.8 0.03 1.19 1.99 799.3 032 0.0 1.0 0.0 123.4 123.0 124.4 123.9 124.2 129.3 127.1 3.8 0.03 1.18 1.93 774.5 033 0.53 0.47 0.0 128.3 127.6 128.2 128.5 128.1 129.3 129.0 3.5 0.03 0.85 4.25 1748.0 034 0.53 0.47 0.0 128.3 127.7 128.3 128.5 128.1 129.4 129.1 3.5 0.03 0.89 4.32 1817.9 035 0.54 0.46 0.0 128.4 127.7 128.4 128.7 128.2 129.4 129.9 3.9 0.03 1.34 3.05 1368.0 036 0.53 0.47 0.0 128.4 127.7 128.4 128.7 128.2 129.8 130.0 3.9 0.03 1.36 2.96 1344.9 037 0.09 0.91 0.0 141.1 141.5 139.4 141.6 142.5 146.4 141.8 5.4 0.04 -0.95 0.2 306.5 038 0.09 0.91 0.0 141.3 141.6 139.8 141.8 142.7 146.4 142.1 5.3 0.04 -0.98 0.26 323.3 039 0.22 0.78 0.0 143.9 144.0 142.6 144.3 144.8 149.1 145.4 5.1 0.03 -0.65 0.57 168.0 040 0.22 0.77 0.0 144.1 144.2 142.9 144.6 145.0 149.2 145.6 5.0 0.03 -0.65 0.56 169.3 041 0.23 0.77 0.0 144.1 144.2 142.9 144.6 145.0 149.2 145.8 5.0 0.03 -0.56 0.08 103.4 042 0.23 0.77 0.0 144.1 144.3 143.1 144.6 145.0 149.2 146.0 4.9 0.03 -0.56 0.09 104.4 043 0.0 1.0 0.0 165.3 163.7 163.4 166.0 163.9 160.7 162.7 3.8 0.02 -1.8 6.25 4331.4 044 0.0 1.0 0.0 165.4 164.0 163.6 166.2 164.1 160.8 163.0 3.8 0.02 -1.79 6.28 4354.5 045 0.91 0.09 0.0 166.3 165.6 169.2 166.3 167.2 174.7 172.7 4.7 0.03 0.89 0.81 317.9 046 0.91 0.09 0.0 166.4 165.6 169.3 166.3 167.3 174.8 173.0 4.7 0.03 0.89 0.79 313.9 047 0.29 0.71 0.0 181.8 181.3 181.1 182.3 182.6 187.1 184.5 4.3 0.02 0.47 0.02 73.3 048 0.29 0.71 0.0 181.8 181.3 181.1 182.3 182.7 187.2 184.7 4.2 0.02 0.47 0.03 73.3 049 0.29 0.71 0.0 181.8 181.4 181.1 182.4 182.7 187.3 184.9 4.1 0.02 0.56 0.03 105.1 050 0.29 0.71 0.0 182.0 181.5 181.3 182.5 182.9 187.3 185.1 4.2 0.02 0.56 0.02 103.8 051 0.71 0.29 0.0 197.1 195.5 198.3 197.4 196.1 195.9 198.4 1.7 0.01 0.69 1.11 260.7 052 0.71 0.29 0.0 197.2 195.5 198.5 197.4 196.2 195.9 198.5 1.7 0.01 0.7 1.07 257.3 053 0.72 0.28 0.0 197.3 195.7 198.5 197.5 196.3 195.9 198.6 1.7 0.01 0.7 1.09 263.8 054 0.71 0.29 0.0 197.3 195.7 198.6 197.5 196.4 195.9 198.7 1.7 0.01 0.7 1.08 261.6 055 1.0 0.0 0.0 383.2 382.3 381.6 384.0 381.6 380.0 387.4 7.1 0.02 -0.08 0.08 2.4 056 1.0 0.0 0.0 383.2 382.3 382.0 384.3 381.9 380.2 387.6 7.1 0.02 -0.08 0.08 2.4 057 1.0 0.0 0.0 384.2 383.3 382.7 385.0 382.3 380.7 388.1 7.2 0.02 -0.08 0.08 2.5 058 1.0 0.0 0.0 384.2 383.5 382.9 385.4 382.9 381.3 388.6 7.2 0.02 -0.08 0.08 2.6 059 1.0 0.0 0.0 384.5 383.6 383.0 385.5 383.2 381.4 388.9 7.2 0.02 -0.08 0.08 2.5 060 1.0 0.0 0.0 384.7 383.9 383.3 385.5 383.2 381.6 389.0 7.2 0.02 -0.08 0.07 2.4 061 1.0 0.0 0.0 386.0 385.2 384.6 386.9 384.3 382.7 390.1 7.3 0.02 -0.08 0.08 2.7 0 able S15: Pyrazine dimer harmonic vibrational frequencies. S B T PBE RPBE PBE PW91 optPBE BEEF µ σ
COV Skew Kurt. JB Imag.sol -vdW -vdW1 2.2 1.0 1.5 1.1 2.0 12.7 10.7 6.1 0.57 -0.08 -0.8 55.2 9232 2.5 1.3 3.0 1.5 3.2 12.9 9.1 6.8 0.75 0.11 -1.14 111.6 6743 2.5 2.1 3.3 2.6 4.6 13.1 7.6 7.3 0.95 0.51 -1.02 173.7 2944 3.7 2.2 5.4 3.6 7.5 13.4 7.6 7.0 0.92 0.69 -0.79 210.1 785 4.7 3.1 5.5 4.5 8.2 14.1 8.4 7.1 0.85 0.71 -0.76 216.6 196 5.1 4.0 6.0 4.8 9.4 14.7 10.9 5.6 0.52 0.73 -0.01 177.2 17 0.0 0.0 1.0 38.3 38.6 37.2 38.4 39.0 45.6 37.1 12.0 0.32 -0.95 0.91 370.5 08 0.0 0.0 1.0 38.7 38.9 37.8 39.0 39.6 46.2 37.9 11.4 0.3 -0.8 0.48 231.0 09 0.0 0.0 1.0 50.5 50.2 50.1 50.7 50.7 55.2 50.5 6.5 0.13 -0.68 2.28 586.3 010 0.0 0.0 1.0 50.6 50.4 50.2 50.9 50.8 55.3 50.7 6.4 0.13 -0.58 1.51 299.6 011 0.06 0.94 0.0 72.9 72.7 72.3 73.2 73.4 75.4 73.5 4.5 0.06 -3.4 28.35 70806.3 012 0.06 0.94 0.0 73.0 72.7 72.4 73.3 73.5 75.7 73.8 4.4 0.06 -3.32 27.7 67616.9 013 0.08 0.92 0.0 86.3 85.8 86.1 86.5 86.4 87.7 85.1 5.6 0.07 -2.25 4.85 3643.8 014 0.08 0.92 0.0 86.4 85.9 86.3 86.7 86.6 87.9 85.4 5.5 0.06 -2.32 5.28 4114.8 015 0.0 0.0 1.0 93.9 93.1 94.1 94.1 93.2 96.5 92.2 5.4 0.06 -0.24 -0.51 40.5 016 0.0 0.0 1.0 93.9 93.1 94.1 94.1 93.2 96.6 92.4 5.3 0.06 -0.26 -0.43 38.0 017 0.0 0.0 1.0 95.3 95.3 94.6 95.7 95.4 104.4 98.0 9.1 0.09 0.62 -0.28 134.3 018 0.0 0.0 1.0 95.4 95.4 94.8 95.9 95.6 105.5 98.3 9.2 0.09 0.59 -0.35 126.1 019 0.0 0.0 1.0 112.8 112.7 112.3 113.3 112.8 120.0 112.1 9.2 0.08 -0.72 0.03 172.3 020 0.0 0.0 1.0 113.0 112.8 112.4 113.3 112.9 120.9 112.5 9.1 0.08 -0.74 0.08 184.1 021 0.0 0.0 1.0 118.4 117.8 118.1 118.9 118.2 123.2 117.1 8.3 0.07 -0.85 0.39 251.9 022 0.0 0.0 1.0 118.5 117.9 118.1 119.0 118.4 123.4 117.5 8.3 0.07 -0.87 0.47 273.3 023 0.0 0.0 1.0 119.5 118.9 119.1 119.9 119.3 125.4 118.6 8.1 0.07 -0.75 0.35 195.7 024 0.0 0.0 1.0 119.6 119.0 119.3 120.1 119.5 125.8 119.0 8.1 0.07 -0.76 0.39 204.3 025 0.18 0.82 0.0 124.1 123.4 124.0 124.4 124.3 126.2 125.4 3.3 0.03 0.98 1.11 419.7 026 0.18 0.82 0.0 124.2 123.5 124.0 124.6 124.4 126.4 125.7 3.4 0.03 0.99 1.08 423.6 027 0.93 0.07 0.0 125.8 124.4 127.0 126.1 124.9 126.4 127.6 3.0 0.02 1.57 3.93 2106.3 028 0.93 0.07 0.0 126.0 124.5 127.2 126.3 125.2 127.1 127.9 3.0 0.02 1.62 3.95 2168.3 029 0.5 0.5 0.0 131.2 130.6 131.4 131.6 131.4 132.2 132.0 2.6 0.02 0.76 2.8 843.8 030 0.5 0.5 0.0 131.4 130.7 131.4 131.8 131.5 132.4 132.3 2.5 0.02 0.73 2.77 819.1 031 0.41 0.59 0.0 140.6 139.7 140.8 140.9 140.2 141.8 140.3 2.2 0.02 -1.23 3.4 1473.6 032 0.41 0.59 0.0 140.6 139.7 141.0 140.9 140.4 141.8 140.5 2.1 0.02 -1.23 3.43 1486.3 033 0.34 0.66 0.0 150.6 149.5 149.9 151.0 149.7 141.9 147.5 3.4 0.02 -0.14 -1.08 104.6 034 0.34 0.66 0.0 150.7 149.7 150.2 151.1 150.0 142.0 147.8 3.5 0.02 -0.16 -1.1 108.7 035 0.98 0.02 0.0 152.6 150.4 157.2 152.7 151.1 154.0 155.1 2.6 0.02 -0.09 1.79 271.1 036 0.98 0.02 0.0 153.0 150.5 157.3 153.0 151.4 154.2 155.4 2.6 0.02 -0.04 1.62 220.0 037 0.07 0.93 0.0 164.5 164.6 163.1 165.2 166.1 171.5 167.7 4.3 0.03 -0.07 -1.07 97.3 038 0.07 0.93 0.0 164.7 164.7 163.3 165.4 166.2 171.5 168.1 4.2 0.03 -0.09 -1.09 102.7 039 0.29 0.71 0.0 173.3 172.8 173.1 173.9 174.0 178.2 175.9 3.9 0.02 0.7 0.32 173.5 040 0.29 0.71 0.0 173.5 172.9 173.1 174.1 174.1 178.3 176.3 4.0 0.02 0.68 0.26 159.8 041 0.27 0.73 0.0 181.5 181.0 181.2 182.0 182.2 186.1 183.7 3.3 0.02 -0.05 -0.68 39.4 042 0.27 0.73 0.0 181.5 181.0 181.4 182.0 182.2 186.2 184.1 3.3 0.02 -0.08 -0.66 38.6 043 0.84 0.16 0.0 189.7 187.3 192.6 190.0 187.8 186.6 191.0 2.9 0.02 0.73 0.53 200.0 044 0.84 0.16 0.0 190.0 187.5 192.7 190.3 188.1 186.7 191.2 2.9 0.02 0.73 0.51 197.7 045 0.68 0.32 0.0 193.5 192.1 194.9 193.8 192.6 192.8 195.2 1.8 0.01 0.66 1.58 353.1 046 0.68 0.32 0.0 193.6 192.1 195.1 193.9 192.8 193.1 195.5 1.7 0.01 0.63 1.53 326.9 047 1.0 0.0 0.0 381.0 380.3 379.0 381.7 379.4 378.2 385.4 7.0 0.02 -0.08 0.07 2.7 048 1.0 0.0 0.0 381.0 380.3 379.1 381.8 379.8 378.4 385.7 7.0 0.02 -0.08 0.07 2.8 049 1.0 0.0 0.0 381.1 380.3 379.3 382.0 380.0 378.5 385.8 7.0 0.02 -0.08 0.07 2.4 050 1.0 0.0 0.0 381.1 380.4 379.4 382.1 380.1 378.6 385.9 7.0 0.02 -0.08 0.07 2.4 051 1.0 0.0 0.0 382.8 382.2 380.9 383.5 381.3 380.0 387.3 7.1 0.02 -0.08 0.08 2.8 052 1.0 0.0 0.0 382.8 382.2 380.9 383.7 381.6 380.2 387.5 7.1 0.02 -0.08 0.08 2.8 053 1.0 0.0 0.0 383.5 382.9 381.6 384.2 382.3 380.9 388.1 7.1 0.02 -0.08 0.07 2.8 054 1.0 0.0 0.0 383.5 382.9 381.7 384.5 382.4 381.0 388.3 7.1 0.02 -0.08 0.08 2.8 0 able S16: Phenol dimer harmonic vibrational frequencies. S B T PBE RPBE PBE PW91 optPBE BEEF µ σ
COV Skew Kurt. JB Imag.sol -vdW -vdW1 1.3 1.5 1.8 2.0 0.9 9.4 8.5 4.5 0.53 -0.17 -0.79 62.5 9542 3.7 3.1 4.0 3.9 2.6 10.8 8.2 5.2 0.64 -0.22 -1.1 117.2 7763 6.8 5.9 6.1 4.7 3.5 12.3 7.6 6.4 0.85 0.22 -1.29 154.8 4594 9.4 6.6 9.1 8.1 7.7 13.7 8.3 6.1 0.74 0.33 -0.98 116.9 1765 12.2 8.1 10.6 9.9 9.3 14.1 9.1 6.3 0.7 0.42 -0.71 101.4 596 15.2 12.1 15.9 13.4 13.3 14.6 11.2 5.6 0.5 0.17 -0.69 48.7 127 0.0 0.0 0.99 28.2 27.8 28.4 28.1 27.9 31.6 23.8 9.8 0.41 -0.58 -1.08 207.9 98 0.01 0.01 0.98 29.6 28.0 28.6 28.4 28.4 31.7 27.1 6.6 0.24 -0.98 1.44 490.7 39 0.02 0.01 0.97 43.7 44.2 43.5 43.7 43.6 48.9 39.0 13.9 0.36 -0.3 -1.39 191.2 210 0.07 0.86 0.07 49.2 48.8 49.5 49.5 49.6 52.5 46.9 9.5 0.2 -1.95 4.05 2636.0 011 0.01 0.06 0.94 50.2 50.1 50.0 50.4 50.3 52.6 49.7 7.2 0.14 -1.55 6.75 4603.8 012 0.08 0.9 0.03 52.4 51.2 51.6 50.6 50.4 54.8 50.3 6.9 0.14 -1.56 6.44 4263.7 013 0.0 0.02 0.97 53.3 51.7 52.3 51.6 51.4 55.0 54.0 8.5 0.16 -0.19 0.8 66.2 014 0.0 0.01 0.99 61.6 61.5 61.5 61.8 61.3 65.9 58.8 8.3 0.14 -1.39 2.37 1111.2 015 0.07 0.29 0.64 63.6 61.5 62.9 63.3 61.6 66.0 60.4 7.8 0.13 -0.64 0.29 144.6 016 0.14 0.62 0.23 64.4 63.9 64.2 64.6 64.5 66.6 62.3 7.5 0.12 -1.02 0.78 399.3 017 0.16 0.71 0.12 64.6 64.1 64.5 64.8 64.7 66.9 66.6 5.0 0.08 1.44 2.84 1357.7 018 0.1 0.9 0.0 75.7 72.7 75.3 76.0 76.1 78.1 71.1 6.9 0.1 -0.12 -1.18 120.5 019 0.11 0.89 0.0 75.8 75.4 75.5 76.3 76.2 78.2 76.4 4.7 0.06 -0.31 7.84 5154.7 020 0.01 0.04 0.95 82.7 75.9 84.5 84.3 77.9 80.8 79.1 7.4 0.09 0.33 0.73 80.9 021 0.0 0.0 1.0 84.6 84.1 86.3 84.8 84.3 89.3 83.6 6.9 0.08 -0.45 0.23 72.8 022 0.0 0.0 1.0 87.9 84.3 88.4 85.6 84.6 89.7 84.2 7.5 0.09 -0.18 0.05 11.5 023 0.0 0.0 0.99 91.8 91.3 91.7 92.1 91.3 98.9 91.1 7.6 0.08 -0.73 0.21 183.2 024 0.01 0.0 0.99 92.2 91.5 92.2 93.0 91.5 99.2 91.6 8.0 0.09 -0.44 0.2 66.8 025 0.01 0.0 0.98 98.7 98.3 98.1 99.0 98.8 99.3 96.0 8.9 0.09 -0.42 0.51 81.0 026 0.63 0.37 0.01 99.8 98.5 100.5 99.6 99.1 100.0 96.5 8.9 0.09 -0.46 0.48 88.0 027 0.6 0.38 0.02 100.8 98.7 101.3 99.9 99.2 107.5 102.7 8.0 0.08 0.34 0.65 73.7 028 0.01 0.0 0.98 103.9 99.5 101.9 101.0 100.1 108.1 103.2 7.9 0.08 0.3 0.72 73.0 029 0.0 0.0 0.99 107.8 106.9 107.4 108.0 107.0 115.2 108.3 8.4 0.08 0.19 -0.28 18.8 030 0.01 0.01 0.99 112.8 107.1 110.0 108.3 107.8 115.8 109.1 8.4 0.08 0.15 -0.33 16.8 031 0.0 0.0 1.0 116.7 116.1 116.2 117.0 116.3 123.2 115.5 8.5 0.07 -0.77 0.35 208.0 032 0.0 0.0 1.0 118.9 116.2 117.5 117.2 116.7 123.3 116.0 8.4 0.07 -0.77 0.36 208.6 033 0.0 0.0 1.0 119.5 118.5 119.2 119.5 118.6 124.4 118.1 8.4 0.07 -0.5 -0.56 110.5 034 0.32 0.67 0.0 122.7 118.8 119.5 120.0 119.6 125.0 118.7 8.2 0.07 -0.54 -0.49 116.5 035 0.4 0.57 0.03 123.3 121.9 122.5 122.9 122.6 126.5 124.7 4.0 0.03 1.29 2.13 935.9 036 0.33 0.4 0.27 124.5 122.7 122.7 123.1 122.9 127.1 125.1 4.0 0.03 1.28 1.9 841.7 037 0.64 0.36 0.0 126.7 125.9 126.6 126.9 126.2 127.3 127.9 3.3 0.03 1.23 3.79 1698.8 038 0.38 0.33 0.29 128.6 126.2 126.9 127.0 126.4 127.6 128.3 3.4 0.03 1.27 3.29 1440.4 039 0.43 0.57 0.0 132.6 132.1 131.8 132.6 132.5 134.5 133.2 3.5 0.03 -0.74 2.43 673.5 040 0.32 0.42 0.26 134.2 132.2 132.1 132.9 132.9 134.6 133.5 3.4 0.03 -0.53 1.42 261.6 041 0.15 0.85 0.0 141.3 141.4 140.0 141.7 142.6 145.3 142.1 5.3 0.04 -1.12 1.1 518.7 042 0.14 0.86 0.0 141.9 142.1 140.6 142.5 143.2 146.6 142.6 5.1 0.04 -1.13 1.15 539.5 043 0.2 0.8 0.0 143.5 143.7 141.6 143.9 144.1 148.1 144.1 5.2 0.04 -1.12 1.22 539.9 044 0.21 0.77 0.02 144.1 144.0 142.1 144.0 144.5 148.7 145.0 5.1 0.04 -0.84 0.72 276.3 045 0.19 0.76 0.05 144.5 144.2 143.2 144.5 144.9 149.0 145.9 5.1 0.04 -0.83 0.69 269.6 046 0.3 0.7 0.01 148.8 147.2 149.5 148.0 148.9 150.1 148.1 4.9 0.03 -1.28 1.87 840.4 047 0.66 0.34 0.0 152.4 151.1 153.9 152.4 151.0 151.6 153.3 1.9 0.01 -0.65 11.82 11791.3 048 0.66 0.33 0.01 156.4 154.2 158.0 156.1 154.8 154.7 155.9 1.8 0.01 -3.05 23.08 47470.0 049 0.04 0.95 0.0 163.9 164.0 162.2 164.6 165.0 162.8 163.9 4.0 0.02 -0.8 0.65 249.2 050 0.14 0.86 0.0 164.3 164.2 163.4 164.8 165.2 164.2 164.7 3.7 0.02 -0.43 0.29 69.0 051 0.84 0.16 0.0 168.2 166.1 170.8 168.1 166.6 172.5 173.0 4.1 0.02 0.86 0.83 303.4 052 0.7 0.3 0.0 169.2 167.4 171.5 168.6 168.2 173.4 173.7 4.1 0.02 0.9 0.98 349.8 053 0.32 0.68 0.0 181.1 180.4 180.6 181.0 181.4 185.2 183.7 3.5 0.02 0.75 0.36 200.1 054 0.33 0.67 0.0 181.3 180.5 180.9 181.6 181.7 185.3 183.9 3.5 0.02 0.74 0.38 196.6 055 0.36 0.64 0.0 183.7 183.1 183.6 184.2 184.1 187.9 186.5 3.3 0.02 0.8 0.47 232.2 056 0.39 0.61 0.0 184.5 184.1 184.6 185.0 184.7 188.4 187.1 3.1 0.02 0.88 0.67 297.6 057 0.71 0.28 0.0 197.3 195.9 198.5 197.6 196.0 195.3 198.4 1.9 0.01 0.66 0.76 194.1 058 0.72 0.28 0.0 197.9 196.3 199.4 198.1 196.8 196.0 199.1 2.0 0.01 0.68 0.78 206.7 059 0.71 0.29 0.0 199.0 197.2 200.3 198.6 197.8 197.1 200.1 1.9 0.01 0.67 0.79 200.6 060 0.71 0.29 0.0 199.2 197.4 200.6 199.2 198.0 197.4 200.4 1.9 0.01 0.68 0.79 204.4 061 1.0 0.0 0.0 382.2 381.9 380.6 383.1 381.3 379.3 386.9 7.2 0.02 -0.08 0.08 2.7 0 able S17: 2-pyridoxine 2-aminopyridine harmonic vibrational frequen-cies. S B T PBE RPBE PBE PW91 optPBE BEEF µ σ
COV Skew Kurt. JB Imag.sol -vdW -vdW1 3.1 3.3 3.5 3.1 2.6 10.0 9.2 4.6 0.49 -0.44 -0.49 85.3 9202 5.0 4.7 5.2 4.9 4.3 11.0 8.4 5.9 0.7 -0.18 -1.23 135.7 6393 9.8 8.8 10.4 9.8 9.1 13.1 9.2 6.4 0.7 0.07 -0.98 81.6 3304 11.7 10.7 12.6 11.8 11.3 14.4 11.1 5.8 0.52 -0.06 -0.6 31.5 1905 14.3 13.7 14.7 14.5 13.5 16.0 13.0 5.1 0.39 -0.37 0.23 51.2 1086 18.3 14.5 20.6 18.6 17.1 17.0 14.8 5.1 0.34 -1.02 1.4 510.5 487 0.0 0.0 1.0 22.4 22.0 22.6 22.4 22.1 27.1 20.8 7.9 0.38 -0.39 -0.31 58.0 208 0.0 0.0 0.99 24.4 24.2 24.7 24.5 24.2 28.9 23.4 7.2 0.31 -0.41 -0.15 59.1 89 0.0 0.0 1.0 48.4 47.8 48.7 48.6 47.9 52.1 41.4 14.2 0.34 -0.8 -0.72 255.7 510 0.0 0.01 0.99 50.0 49.8 49.9 50.2 49.8 54.3 48.2 8.1 0.17 -1.23 3.69 1634.0 111 0.01 0.15 0.84 50.9 51.0 51.7 51.1 51.4 54.4 50.1 7.6 0.15 -1.15 3.15 1266.7 012 0.07 0.81 0.12 53.9 52.7 54.4 54.0 53.5 59.3 53.7 7.3 0.14 -1.29 4.1 1959.1 013 0.09 0.9 0.01 58.2 57.0 59.0 58.4 57.9 63.7 58.4 8.5 0.14 -0.85 2.05 591.4 014 0.0 0.01 0.99 62.5 61.9 62.6 62.7 62.1 67.7 60.8 8.2 0.13 -1.08 1.88 683.2 015 0.0 0.0 1.0 63.5 63.1 63.5 63.7 63.1 68.4 62.7 7.3 0.12 -0.61 1.57 329.3 016 0.14 0.86 0.0 67.7 66.9 68.0 67.9 67.6 68.5 67.5 4.7 0.07 -1.77 6.05 4092.2 017 0.15 0.84 0.01 69.2 68.4 69.3 69.4 69.2 70.1 70.0 4.0 0.06 0.06 4.19 1465.5 018 0.18 0.82 0.0 75.0 74.4 74.9 75.3 75.2 76.3 74.9 4.3 0.06 -1.17 4.65 2257.5 019 0.09 0.91 0.0 78.0 77.4 77.9 78.3 78.4 79.8 78.1 4.7 0.06 0.13 4.36 1590.5 020 0.0 0.0 1.0 88.8 86.6 88.8 89.0 88.0 92.7 84.9 8.5 0.1 -0.17 -0.79 61.5 021 0.0 0.0 1.0 89.2 88.2 89.1 89.5 88.3 93.5 87.6 7.8 0.09 -0.47 -0.3 81.5 022 0.0 0.0 1.0 93.0 89.7 93.2 93.3 90.6 95.4 89.0 8.3 0.09 -0.4 -0.61 85.1 023 0.0 0.0 1.0 94.1 92.4 94.1 94.4 92.7 99.9 92.5 7.8 0.08 -0.57 -0.28 114.4 024 0.01 0.05 0.94 95.8 93.5 101.1 96.5 93.6 100.1 94.6 8.2 0.09 -0.15 0.28 14.0 025 0.79 0.2 0.01 102.0 100.2 103.2 102.2 101.1 100.6 99.8 6.5 0.07 -0.19 1.93 323.0 026 0.0 0.0 0.99 103.2 102.6 103.4 103.5 102.8 103.7 101.2 6.7 0.07 -0.3 0.89 96.5 027 0.0 0.0 1.0 103.3 102.6 105.1 103.6 102.9 110.4 105.8 7.5 0.07 0.26 0.55 47.6 028 0.69 0.3 0.01 104.8 103.6 106.0 105.0 104.3 110.8 107.5 6.6 0.06 0.43 1.33 211.3 029 0.0 0.0 1.0 112.8 111.4 112.8 113.3 112.3 119.0 112.3 7.8 0.07 -0.18 -0.28 17.2 030 0.0 0.0 1.0 115.6 115.2 114.9 115.9 115.6 122.1 115.2 7.7 0.07 -0.68 0.24 161.1 031 0.0 0.0 1.0 119.2 118.7 118.6 119.7 119.0 122.7 117.5 7.3 0.06 -1.05 1.1 469.0 032 0.14 0.84 0.02 120.1 119.6 119.7 120.5 120.0 123.7 118.7 7.2 0.06 -0.6 -0.44 135.3 033 0.0 0.02 0.98 120.3 119.8 119.8 120.7 120.5 124.3 120.1 7.3 0.06 -0.44 -0.46 81.1 034 0.3 0.69 0.01 121.9 120.9 122.4 122.2 122.0 126.8 123.6 4.9 0.04 0.65 0.21 144.0 035 0.69 0.3 0.0 125.7 121.7 126.4 126.0 124.0 127.1 125.0 4.8 0.04 0.96 0.71 347.7 036 0.15 0.08 0.77 128.2 124.6 129.3 128.8 125.0 128.2 127.2 4.2 0.03 1.07 1.62 601.4 037 0.56 0.3 0.13 129.4 128.6 130.2 129.7 128.9 129.2 129.7 3.7 0.03 0.68 1.57 356.8 038 0.34 0.61 0.05 130.2 129.4 135.4 130.6 130.3 132.1 131.7 3.4 0.03 0.51 1.4 248.6 039 0.41 0.59 0.0 136.1 135.5 136.1 136.6 136.2 137.9 136.7 3.3 0.02 -1.31 4.24 2074.7 040 0.27 0.73 0.0 138.7 138.6 137.9 139.1 139.4 142.8 139.8 4.6 0.03 -0.92 0.93 353.0 041 0.15 0.85 0.0 140.4 140.3 139.0 140.9 141.3 144.8 141.2 4.8 0.03 -0.94 0.27 302.8 042 0.14 0.86 0.0 141.5 141.8 140.0 142.0 142.7 147.2 143.3 5.6 0.04 -0.64 -0.06 135.2 043 0.38 0.62 0.0 149.9 148.3 150.4 150.4 149.7 149.0 149.6 2.7 0.02 -1.94 7.12 5476.1 044 0.64 0.36 0.0 155.4 153.2 157.9 155.6 154.2 156.2 155.9 2.0 0.01 -2.54 16.03 23558.3 045 0.32 0.68 0.0 161.6 160.3 160.4 162.0 160.9 157.0 159.9 2.3 0.01 -0.08 -0.4 15.4 046 0.6 0.4 0.0 164.0 162.9 165.6 164.2 163.9 165.0 165.2 1.2 0.01 -0.38 0.26 53.9 047 0.58 0.42 0.0 166.5 164.5 168.0 166.8 165.6 169.3 168.4 2.9 0.02 1.07 1.09 478.8 048 0.26 0.74 0.0 168.7 168.2 168.4 169.3 169.6 174.3 172.7 3.3 0.02 0.48 -0.4 88.6 049 0.38 0.62 0.0 178.6 177.6 178.7 179.1 178.8 181.4 180.2 2.4 0.01 -0.22 -0.54 39.9 050 0.37 0.63 0.0 178.9 178.1 179.1 179.4 179.3 182.1 181.2 2.6 0.01 0.47 -0.21 77.5 051 0.51 0.49 0.0 183.6 182.3 184.6 184.0 183.1 184.4 185.1 1.9 0.01 1.29 3.23 1420.4 052 0.58 0.42 0.0 185.2 183.0 187.1 185.4 183.9 186.5 186.4 2.0 0.01 0.87 1.56 456.2 053 0.75 0.25 0.0 192.3 190.7 193.6 192.6 191.0 190.8 193.2 1.6 0.01 0.31 -0.15 35.0 054 0.7 0.3 0.0 192.5 191.3 193.9 192.7 191.8 191.6 194.4 1.7 0.01 0.74 1.45 357.3 055 0.59 0.41 0.0 198.7 197.1 199.4 199.0 197.9 197.5 199.3 1.5 0.01 -0.1 0.2 6.3 056 0.59 0.41 0.0 200.6 199.0 201.4 200.9 200.0 200.7 202.2 1.3 0.01 0.27 0.95 98.8 057 0.19 0.8 0.01 203.3 202.8 203.2 203.7 203.8 205.8 206.2 3.1 0.01 0.35 -0.03 41.4 058 0.54 0.46 0.0 207.5 205.9 209.2 207.9 206.3 209.8 211.6 3.7 0.02 1.26 2.25 951.0 059 0.99 0.01 0.0 345.1 368.1 314.8 344.8 356.7 373.4 380.0 8.6 0.02 -0.18 -0.02 10.5 0 able S18: Indole benzene stack harmonic vibrational frequencies. S B T PBE RPBE PBE PW91 optPBE BEEF µ σ
COV Skew Kurt. JB Imag.sol -vdW -vdW1 0.3 0.1 0.3 1.1 0.8 9.2 7.9 4.4 0.55 -0.26 -0.77 71.8 9082 0.7 0.3 1.4 1.6 3.0 9.8 7.7 5.4 0.71 0.05 -1.05 92.3 7303 1.3 1.1 2.2 2.0 3.5 13.1 7.9 7.3 0.92 0.42 -1.11 163.1 4354 2.1 1.8 3.2 3.4 6.5 13.6 7.8 7.1 0.91 0.55 -0.96 179.5 1915 3.4 2.7 4.8 3.7 7.9 14.5 8.4 6.7 0.81 0.6 -0.83 178.8 546 3.7 3.0 5.1 4.0 8.6 14.8 9.9 6.1 0.62 0.42 -0.58 86.3 207 0.0 0.0 1.0 25.7 25.3 26.0 25.8 25.9 30.7 22.6 10.4 0.46 -0.58 -0.76 157.9 108 0.0 0.0 1.0 29.5 29.2 29.7 29.6 30.0 33.7 27.6 8.6 0.31 -1.06 0.62 406.6 79 0.17 0.82 0.01 48.6 46.8 48.3 48.7 48.1 51.7 40.0 13.8 0.34 -0.62 -0.9 195.2 310 0.0 0.0 1.0 48.9 48.6 48.5 49.2 48.7 54.1 48.2 8.0 0.17 -1.58 6.57 4429.0 011 0.0 0.0 1.0 48.9 48.8 48.6 49.3 49.0 54.2 48.5 7.5 0.16 -1.27 4.95 2578.6 012 0.0 0.0 1.0 49.6 48.8 50.6 50.4 49.2 56.1 49.7 7.8 0.16 -0.81 2.54 756.2 013 0.0 0.0 1.0 51.6 51.2 52.4 51.9 51.3 59.2 53.2 9.4 0.18 -0.05 0.36 11.4 014 0.28 0.72 0.0 66.5 66.2 66.3 66.7 66.6 67.3 65.7 6.8 0.1 -2.62 10.14 10853.6 015 0.0 0.0 1.0 70.6 70.0 70.6 70.8 70.1 73.7 69.4 6.6 0.1 -2.08 7.62 6271.2 016 0.04 0.17 0.8 74.2 73.6 74.0 74.4 73.6 75.6 72.2 5.2 0.07 -1.38 4.24 2132.4 017 0.15 0.65 0.2 74.3 73.8 74.0 74.5 74.4 77.4 73.6 5.1 0.07 -1.34 3.29 1504.2 018 0.06 0.94 0.0 74.5 74.2 74.2 74.8 74.9 77.4 74.8 4.1 0.05 -1.59 5.74 3588.9 019 0.06 0.94 0.01 74.5 74.2 74.4 74.8 75.0 77.5 76.2 5.1 0.07 0.73 2.74 804.9 020 0.0 0.0 1.0 81.5 81.7 80.8 82.1 81.6 90.1 81.5 8.1 0.1 -0.05 -1.15 111.9 021 0.0 0.0 1.0 86.5 86.2 86.4 86.7 86.1 92.6 84.4 8.2 0.1 -0.41 -0.77 104.2 022 0.0 0.0 1.0 87.1 86.4 87.3 87.5 86.5 93.7 86.8 8.4 0.1 -0.3 -0.55 55.1 023 0.0 0.0 1.0 89.5 89.2 89.2 89.8 89.4 93.9 89.9 8.0 0.09 0.17 0.57 37.0 024 0.0 0.0 1.0 93.3 92.6 93.4 93.6 92.6 97.7 93.1 8.2 0.09 -0.01 0.76 48.4 025 0.57 0.43 0.0 93.9 93.1 94.3 94.1 93.6 98.8 95.3 7.7 0.08 -0.25 1.78 284.8 026 0.0 0.0 1.0 101.8 101.3 101.5 102.2 101.5 108.9 100.5 9.3 0.09 -0.78 0.3 212.5 027 0.0 0.0 1.0 102.7 101.9 102.3 103.1 102.3 110.1 101.5 9.6 0.09 -0.62 -0.19 129.1 028 0.0 0.0 1.0 103.0 102.9 102.3 103.6 102.8 110.8 103.2 9.3 0.09 -0.12 -0.66 40.7 029 0.0 0.0 1.0 103.1 103.0 102.7 103.6 103.1 111.2 104.3 9.1 0.09 0.26 -0.5 43.5 030 0.17 0.83 0.0 107.1 106.5 106.9 107.4 107.3 111.4 108.1 7.3 0.07 -0.03 1.2 121.0 031 0.23 0.77 0.0 109.7 109.1 109.8 110.0 109.8 111.8 110.0 6.4 0.06 -0.08 2.64 581.5 032 0.0 0.0 1.0 112.1 111.6 111.6 112.6 112.0 120.1 113.0 7.1 0.06 -0.5 1.03 172.9 033 0.0 0.0 1.0 117.0 116.5 116.6 117.5 116.9 120.4 116.0 8.0 0.07 -0.77 0.48 218.4 034 0.0 0.0 1.0 118.1 117.6 117.4 118.7 117.9 124.9 117.5 8.2 0.07 -0.5 -0.38 95.9 035 0.0 0.0 1.0 118.1 117.7 117.4 118.7 118.0 125.1 118.2 8.0 0.07 -0.24 -0.85 79.6 036 0.0 0.0 1.0 121.6 121.0 121.0 122.3 121.4 125.9 120.2 7.5 0.06 -0.48 -0.38 89.2 037 1.0 0.0 0.0 123.3 122.0 122.8 123.5 122.4 126.2 124.7 3.6 0.03 0.36 1.5 230.9 038 0.0 0.99 0.0 123.4 123.0 124.4 123.9 124.1 126.8 126.0 3.4 0.03 0.73 2.3 619.4 039 0.64 0.36 0.0 125.2 124.4 125.4 125.4 124.7 129.1 127.2 3.7 0.03 0.93 2.58 843.2 040 0.53 0.47 0.0 128.3 127.7 128.3 128.6 128.1 129.3 128.9 3.4 0.03 0.39 3.07 835.9 041 0.54 0.46 0.0 128.4 127.7 128.4 128.7 128.2 129.7 129.4 3.6 0.03 0.44 2.41 547.4 042 0.31 0.69 0.0 130.8 130.8 130.0 131.1 131.0 133.1 131.7 4.0 0.03 0.11 0.87 67.1 043 0.36 0.64 0.0 134.2 133.8 134.2 134.5 134.0 137.9 135.3 4.0 0.03 -0.43 0.69 101.1 044 0.35 0.65 0.0 137.7 137.4 137.1 138.0 137.9 140.2 138.3 3.9 0.03 -1.24 2.21 920.9 045 0.16 0.84 0.0 141.0 141.3 139.6 141.5 142.1 146.2 141.8 5.3 0.04 -0.95 0.35 311.0 046 0.09 0.91 0.0 141.1 141.4 139.7 141.7 142.5 146.7 142.5 5.4 0.04 -0.75 -0.09 186.5 047 0.22 0.78 0.0 143.9 144.0 142.8 144.4 144.8 148.8 145.2 4.8 0.03 -0.92 0.89 345.3 048 0.22 0.78 0.0 144.1 144.1 142.9 144.5 144.9 149.0 145.6 4.9 0.03 -0.66 0.25 151.4 049 0.36 0.64 0.0 147.9 147.1 148.0 148.2 147.8 149.3 148.3 3.4 0.02 -1.37 3.58 1695.8 050 0.28 0.72 0.0 152.6 152.3 151.8 153.1 153.1 154.7 153.2 3.3 0.02 -1.32 2.97 1319.1 051 0.64 0.36 0.0 157.5 156.2 158.3 157.7 156.6 157.7 158.2 2.5 0.02 -2.13 13.81 17418.1 052 0.0 1.0 0.0 165.3 163.9 163.4 166.1 164.1 160.7 163.0 2.8 0.02 -0.59 0.04 117.9 053 0.28 0.72 0.0 165.9 165.3 165.4 166.3 166.0 163.6 165.4 2.1 0.01 -0.37 0.37 56.8 054 0.91 0.09 0.0 166.4 165.5 169.5 166.4 167.2 171.5 171.0 2.9 0.02 0.72 0.82 228.6 055 0.68 0.32 0.0 167.9 166.3 169.9 167.9 167.2 174.3 172.9 4.0 0.02 0.82 0.26 229.0 056 0.51 0.48 0.0 175.2 173.9 176.0 175.3 174.1 175.8 177.0 2.5 0.01 1.83 4.94 3144.7 057 0.52 0.48 0.0 178.9 177.7 180.2 179.2 178.4 180.9 181.0 2.0 0.01 1.02 2.62 919.4 058 0.29 0.71 0.0 181.8 181.3 181.1 182.3 182.7 186.0 184.3 3.4 0.02 0.14 -0.57 34.1 059 0.29 0.71 0.0 181.9 181.4 181.3 182.5 182.7 186.9 185.3 3.5 0.02 0.83 0.36 238.4 060 0.45 0.55 0.0 183.7 182.7 184.0 184.0 183.7 187.1 186.4 2.9 0.02 1.3 1.79 826.0 061 0.6 0.4 0.0 186.5 185.3 187.4 186.7 185.7 187.3 189.0 2.1 0.01 1.4 3.33 1579.8 0 able S19: Indole benzene t-shaped harmonic vibrational frequencies. S B T PBE RPBE PBE PW91 optPBE BEEF µ σ
COV Skew Kurt. JB Imag.sol -vdW -vdW1 0.1 1.2 0.8 0.5 0.2 8.1 7.4 4.2 0.56 -0.22 -0.67 54.2 9182 1.2 1.4 1.6 0.8 0.5 10.3 7.7 5.7 0.73 0.04 -1.16 112.2 7033 1.6 1.4 1.7 1.6 2.6 12.3 7.7 7.1 0.92 0.45 -1.07 163.5 4114 3.2 3.1 4.6 3.1 5.2 13.0 7.9 6.9 0.88 0.58 -0.85 174.6 2215 4.9 4.4 5.6 4.9 6.5 13.9 8.0 6.9 0.86 0.64 -0.84 195.8 846 6.2 5.0 7.2 6.1 8.1 14.2 9.9 5.8 0.59 0.55 -0.32 108.8 237 0.0 0.0 1.0 25.8 25.6 25.9 25.8 25.1 30.3 22.7 9.9 0.44 -0.62 -0.55 152.0 78 0.0 0.0 1.0 29.4 29.2 29.5 29.5 29.1 33.0 27.3 8.5 0.31 -1.13 0.95 498.8 49 0.17 0.83 0.0 48.6 48.6 48.3 48.8 48.7 51.8 41.6 13.5 0.33 -0.96 -0.3 314.1 110 0.0 0.0 1.0 48.9 49.0 48.6 49.1 49.0 53.9 48.4 7.8 0.16 -1.56 6.62 4458.6 011 0.0 0.0 1.0 49.1 49.1 48.8 49.4 49.3 54.1 48.8 7.4 0.15 -1.21 4.77 2388.9 012 0.0 0.0 1.0 51.4 51.0 51.4 51.6 49.8 55.9 50.1 7.7 0.15 -0.82 2.5 744.8 013 0.0 0.0 1.0 55.5 52.9 57.9 55.6 51.4 63.2 54.2 9.7 0.18 -0.14 -0.03 6.9 014 0.28 0.72 0.0 66.5 66.2 66.4 66.7 66.7 67.4 66.3 6.5 0.1 -2.69 12.28 14977.6 015 0.0 0.0 1.0 70.8 70.3 71.0 71.0 70.3 73.8 69.7 6.6 0.09 -2.11 7.86 6634.3 016 0.19 0.81 0.0 74.2 73.8 74.0 74.5 73.8 75.6 72.5 5.1 0.07 -1.51 4.84 2719.1 017 0.06 0.94 0.0 74.4 74.1 74.0 74.7 74.5 77.2 73.8 5.0 0.07 -1.48 4.02 2074.5 018 0.06 0.94 0.0 74.5 74.2 74.2 74.8 74.9 77.3 74.9 4.0 0.05 -1.56 5.82 3634.0 019 0.0 0.0 1.0 74.6 74.2 74.8 74.8 75.0 77.9 76.5 5.3 0.07 0.79 2.14 591.8 020 0.0 0.0 1.0 83.2 82.9 82.7 83.5 83.1 90.1 82.2 8.1 0.1 -0.16 -1.18 124.4 021 0.0 0.0 1.0 86.8 86.5 86.5 87.1 86.4 93.3 84.8 8.3 0.1 -0.42 -0.8 113.5 022 0.0 0.0 1.0 87.3 86.6 87.6 87.6 86.8 93.8 87.0 8.4 0.1 -0.3 -0.58 59.3 023 0.0 0.0 1.0 89.8 89.4 89.3 90.0 89.4 93.9 90.3 8.0 0.09 0.26 0.53 45.7 024 0.0 0.0 1.0 93.6 92.8 93.7 93.9 92.9 97.6 93.4 8.1 0.09 0.04 0.79 52.1 025 0.57 0.43 0.0 93.9 93.1 94.3 94.1 93.7 98.7 95.6 7.6 0.08 -0.16 1.67 239.5 026 0.0 0.0 1.0 102.1 101.5 101.8 102.5 101.8 109.1 101.0 8.9 0.09 -0.76 0.38 202.3 027 0.0 0.0 1.0 102.9 102.1 102.8 103.4 102.5 109.7 102.1 9.1 0.09 -0.62 -0.06 127.3 028 0.0 0.0 1.0 104.4 104.0 104.0 104.7 104.3 110.6 103.9 9.1 0.09 -0.24 -0.56 45.4 029 0.0 0.0 1.0 104.5 104.0 104.1 104.9 104.5 111.4 105.0 9.0 0.09 0.13 -0.53 28.6 030 0.17 0.83 0.0 107.1 106.5 106.9 107.4 107.4 112.2 108.5 7.4 0.07 -0.05 0.94 74.3 031 0.22 0.78 0.0 109.6 109.1 109.6 110.0 109.9 112.4 110.4 6.6 0.06 0.04 2.02 340.7 032 0.0 0.0 1.0 112.3 111.8 111.8 112.7 112.1 119.9 113.2 7.0 0.06 -0.36 0.86 105.5 033 0.0 0.0 1.0 117.1 116.5 116.6 117.5 117.0 119.9 116.2 7.6 0.07 -0.7 0.43 177.8 034 0.0 0.0 1.0 119.1 118.5 118.7 119.6 119.1 124.8 118.1 7.8 0.07 -0.6 -0.17 122.3 035 0.0 0.0 1.0 119.2 118.5 118.8 119.6 119.1 124.8 118.9 7.9 0.07 -0.37 -0.71 87.9 036 0.0 0.01 0.99 122.5 121.8 122.2 123.1 122.2 126.5 120.8 7.3 0.06 -0.59 -0.19 117.9 037 1.0 0.0 0.0 123.2 121.8 123.0 123.3 122.4 126.7 124.9 3.6 0.03 0.34 1.15 148.9 038 0.0 0.99 0.01 123.5 123.0 124.1 124.0 124.3 126.8 126.1 3.5 0.03 0.74 1.81 456.6 039 0.64 0.36 0.0 125.1 124.4 125.0 125.3 124.6 128.9 127.3 3.8 0.03 0.89 1.89 560.8 040 0.53 0.47 0.0 128.3 127.6 128.3 128.6 128.0 129.2 129.0 3.5 0.03 0.56 2.89 800.3 041 0.54 0.46 0.0 128.4 127.6 128.4 128.6 128.1 130.0 129.4 3.8 0.03 0.58 2.03 455.6 042 0.29 0.71 0.0 130.7 130.7 129.7 131.1 131.2 133.5 131.8 4.1 0.03 0.16 0.54 33.0 043 0.36 0.64 0.0 134.3 134.0 134.0 134.6 134.0 137.5 135.4 3.9 0.03 -0.43 0.84 122.0 044 0.35 0.65 0.0 137.5 137.4 136.8 137.9 137.8 140.1 138.3 3.8 0.03 -1.23 2.32 953.1 045 0.16 0.84 0.0 140.9 141.2 139.4 141.4 142.1 146.3 141.8 5.2 0.04 -0.99 0.53 348.1 046 0.09 0.91 0.0 141.5 141.7 140.0 141.9 142.8 146.5 142.7 5.4 0.04 -0.81 -0.05 218.1 047 0.23 0.77 0.0 144.2 144.2 143.1 144.5 145.0 148.9 145.5 4.7 0.03 -0.93 0.95 361.6 048 0.23 0.77 0.0 144.3 144.3 143.1 144.7 145.2 149.1 145.8 4.9 0.03 -0.64 0.24 143.4 049 0.36 0.64 0.0 147.9 147.2 147.8 148.3 148.0 149.5 148.6 3.3 0.02 -1.44 3.81 1903.0 050 0.27 0.73 0.0 152.5 152.4 151.6 153.0 153.1 155.1 153.3 3.3 0.02 -1.24 2.76 1151.0 051 0.65 0.35 0.0 157.6 156.4 158.3 157.7 157.0 157.8 158.5 2.4 0.02 -2.33 14.85 20179.5 052 0.0 1.0 0.0 165.5 163.6 163.7 166.1 163.6 160.4 162.9 2.7 0.02 -0.5 0.01 81.9 053 0.27 0.73 0.0 165.8 165.3 165.2 166.2 166.0 163.6 165.4 2.1 0.01 -0.32 0.3 41.7 054 0.91 0.09 0.0 166.2 165.7 169.0 166.3 167.1 171.6 171.0 2.9 0.02 0.61 0.56 152.0 055 0.68 0.32 0.0 167.8 166.4 169.7 167.9 167.4 174.5 172.9 3.9 0.02 0.8 0.23 215.8 056 0.51 0.49 0.0 175.3 174.2 175.8 175.5 174.4 175.8 177.1 2.5 0.01 1.67 4.3 2473.8 057 0.52 0.48 0.0 178.9 177.8 180.2 179.1 178.5 180.7 181.1 2.0 0.01 0.99 2.71 940.1 058 0.29 0.71 0.0 181.8 181.3 181.1 182.3 182.6 185.9 184.2 3.4 0.02 0.11 -0.54 28.5 059 0.29 0.71 0.0 181.8 181.4 181.2 182.4 182.7 186.7 185.2 3.4 0.02 0.82 0.37 236.1 060 0.44 0.56 0.0 183.5 182.8 183.8 183.9 183.7 187.1 186.4 2.9 0.02 1.28 1.74 795.3 061 0.6 0.4 0.0 186.2 185.1 187.1 186.5 185.6 187.2 188.9 2.1 0.01 1.38 3.14 1457.4 0 able S20: Uracil dimer h-bonded harmonic vibrational frequencies. S B T PBE RPBE PBE PW91 optPBE BEEF µ σ
COV Skew Kurt. JB Imag.sol -vdW -vdW1 2.8 2.6 3.2 2.7 1.3 9.9 8.6 4.0 0.47 -0.45 -0.53 89.7 9442 5.9 5.5 6.3 6.0 5.3 10.3 8.2 5.2 0.63 -0.35 -1.02 127.0 6743 8.3 7.7 9.2 8.3 7.4 11.7 8.7 4.9 0.56 0.07 -0.68 40.5 3684 10.5 9.4 11.6 10.7 10.3 11.7 9.3 5.0 0.54 -0.13 -0.65 40.1 2655 11.4 10.8 12.2 11.7 11.0 12.3 10.8 4.2 0.39 -0.43 0.84 121.9 1896 17.6 16.0 19.1 17.8 17.5 16.7 14.6 4.7 0.32 -1.4 1.64 881.8 1187 0.0 0.0 1.0 19.2 18.8 19.8 19.3 18.3 23.6 18.3 7.4 0.41 -0.53 -0.0 94.4 628 0.0 0.0 1.0 20.7 20.2 21.5 20.8 19.8 24.7 19.0 7.6 0.4 -0.56 -0.14 106.4 229 0.0 0.0 1.0 22.1 21.4 22.5 22.1 21.4 26.2 20.4 7.5 0.37 -0.26 -0.63 55.9 1110 0.0 0.0 1.0 23.2 22.4 23.8 23.3 22.3 27.2 22.3 6.2 0.28 0.11 -0.19 7.4 511 0.19 0.81 0.0 47.8 47.3 48.1 47.9 47.8 49.1 44.9 7.7 0.17 -2.45 7.24 6363.2 012 0.2 0.8 0.0 49.6 48.8 50.2 49.7 49.6 50.2 45.7 7.6 0.17 -1.95 4.14 2695.3 013 0.0 0.0 1.0 50.1 49.6 50.4 50.3 49.6 54.9 50.5 6.2 0.12 -0.41 4.19 1520.8 014 0.0 0.0 1.0 50.2 49.6 50.4 50.3 49.6 55.0 51.2 5.6 0.11 -0.43 4.76 1952.6 015 0.06 0.94 0.0 62.7 62.5 62.4 62.9 63.1 64.2 62.4 4.8 0.08 -3.16 13.07 17548.6 016 0.15 0.85 0.0 63.8 63.2 64.0 64.1 64.0 64.6 63.1 4.7 0.07 -3.11 10.61 12590.5 017 0.19 0.81 0.0 66.3 65.6 66.5 66.5 66.3 67.8 65.9 3.5 0.05 -2.14 8.55 7611.8 018 0.15 0.85 0.0 66.6 65.9 67.1 66.9 66.8 68.0 66.4 3.1 0.05 -2.19 11.87 13347.7 019 0.25 0.75 0.0 68.9 68.3 69.2 69.1 68.7 69.6 68.5 3.0 0.04 -1.78 5.98 4038.4 020 0.2 0.8 0.0 70.4 69.6 71.1 70.6 70.0 71.2 69.6 3.2 0.05 -1.33 3.18 1426.1 021 0.0 0.0 1.0 82.7 81.4 83.7 82.9 81.6 89.3 80.4 9.5 0.12 -0.22 -1.3 157.1 022 0.0 0.0 1.0 82.8 81.7 83.7 82.9 81.7 89.4 80.7 9.3 0.12 -0.19 -1.33 159.0 023 0.0 0.0 1.0 87.2 86.6 87.4 87.4 86.6 92.1 85.7 7.8 0.09 -0.71 0.14 168.1 024 0.0 0.0 1.0 87.2 86.7 87.5 87.4 86.6 92.1 85.8 7.7 0.09 -0.64 -0.07 135.3 025 0.0 0.0 1.0 92.2 90.8 93.4 92.5 90.9 92.9 90.4 5.7 0.06 -1.25 2.82 1179.8 026 0.0 0.0 1.0 92.3 91.0 93.5 92.6 91.0 93.1 90.9 5.1 0.06 -1.03 3.08 1145.0 027 0.73 0.27 0.0 94.4 93.2 95.4 94.5 93.7 96.0 94.3 6.3 0.07 0.46 1.92 379.7 028 0.73 0.27 0.0 94.6 93.5 95.6 94.8 94.0 96.0 94.9 6.0 0.06 0.54 2.64 678.4 029 0.0 0.0 1.0 98.5 97.9 98.6 98.9 98.3 104.8 99.5 8.4 0.08 0.29 -0.34 37.6 030 0.0 0.0 1.0 98.6 97.9 98.7 98.9 98.4 104.8 100.0 8.1 0.08 0.29 -0.13 28.5 031 0.0 0.0 1.0 108.9 103.5 114.2 109.8 104.9 107.3 103.1 7.9 0.08 0.54 -0.54 121.9 032 0.0 0.0 1.0 112.0 106.9 115.6 112.9 108.5 110.7 105.3 8.3 0.08 0.17 -0.93 81.4 033 0.55 0.45 0.0 117.7 116.6 118.2 117.8 117.3 117.0 114.2 6.3 0.06 -1.26 0.82 581.6 034 0.44 0.56 0.0 117.8 116.8 118.3 118.0 117.5 117.5 114.5 6.5 0.06 -1.19 0.65 504.7 035 0.0 0.0 1.0 117.9 117.0 119.3 118.4 117.6 121.0 119.4 2.5 0.02 0.05 0.56 27.0 036 0.0 0.0 1.0 118.2 117.1 120.7 118.8 117.7 121.1 119.6 2.6 0.02 0.34 1.0 121.8 037 0.3 0.7 0.0 120.3 119.3 121.1 120.6 120.2 125.5 122.9 3.7 0.03 1.0 0.06 335.9 038 0.43 0.57 0.0 120.6 119.5 121.4 120.8 120.4 125.5 123.1 3.7 0.03 1.04 0.24 367.0 039 0.49 0.51 0.0 132.3 131.5 132.4 132.7 132.1 131.4 132.3 2.5 0.02 -0.31 4.63 1817.7 040 0.48 0.52 0.0 132.5 131.7 132.6 132.8 132.3 131.6 132.4 2.6 0.02 -0.35 4.33 1605.5 041 0.64 0.36 0.0 145.6 143.5 147.6 145.7 144.9 142.9 145.2 1.8 0.01 -1.27 6.43 3978.9 042 0.62 0.38 0.0 145.8 143.7 147.7 145.9 145.1 143.3 145.4 1.7 0.01 -1.48 8.53 6793.7 043 0.37 0.63 0.0 151.6 150.9 152.1 152.0 151.7 154.4 153.0 2.8 0.02 0.24 -0.62 51.1 044 0.37 0.63 0.0 151.7 151.1 152.2 152.1 151.9 154.6 153.3 2.7 0.02 0.34 -0.62 69.3 045 0.33 0.67 0.0 165.9 165.7 165.3 166.2 166.6 168.7 167.3 3.1 0.02 -1.02 0.54 369.4 046 0.32 0.68 0.0 166.0 165.9 165.3 166.3 166.8 169.1 167.5 3.1 0.02 -0.97 0.36 323.9 047 0.32 0.68 0.0 170.0 169.2 169.8 170.5 170.4 172.0 171.8 3.1 0.02 0.03 -0.2 3.5 048 0.33 0.67 0.0 170.1 169.3 170.1 170.5 170.5 172.3 172.0 3.2 0.02 0.14 -0.13 7.7 049 0.6 0.4 0.0 174.3 172.4 176.7 174.5 173.5 176.0 176.4 2.4 0.01 1.68 3.26 1828.3 050 0.62 0.38 0.0 174.4 172.5 177.1 174.6 173.5 176.3 176.7 2.4 0.01 1.57 3.06 1602.4 051 0.47 0.53 0.0 187.5 186.4 188.7 187.8 187.3 188.1 188.3 2.5 0.01 -0.3 -0.53 52.9 052 0.33 0.67 0.0 188.0 186.6 189.4 188.5 187.6 188.9 189.1 2.7 0.01 0.04 -0.46 18.1 053 0.75 0.25 0.0 200.9 199.8 202.0 201.2 200.2 199.6 202.8 2.3 0.01 0.58 0.61 143.1 054 0.71 0.29 0.0 201.2 200.0 202.3 201.5 200.4 200.1 203.1 2.1 0.01 0.73 1.12 280.7 055 0.56 0.44 0.0 209.2 207.7 210.2 209.3 207.7 207.0 210.3 3.5 0.02 0.2 0.08 13.8 056 0.73 0.27 0.0 209.5 208.0 211.2 209.5 208.2 208.6 211.3 3.6 0.02 0.4 0.58 81.4 057 0.85 0.15 0.0 212.5 210.6 214.9 212.6 210.8 209.2 213.9 3.7 0.02 0.87 0.68 293.4 058 0.84 0.16 0.0 213.3 211.4 215.6 213.3 211.5 210.4 214.9 3.6 0.02 0.79 0.63 243.0 059 0.99 0.01 0.0 370.2 384.6 346.8 369.3 378.0 386.1 393.1 7.1 0.02 -0.11 0.12 4.9 060 0.99 0.01 0.0 377.6 387.6 356.9 376.9 384.7 386.1 393.2 7.0 0.02 -0.09 0.08 3.0 061 1.0 0.0 0.0 388.2 387.6 386.1 389.1 387.1 391.1 397.7 7.6 0.02 -0.15 0.08 8.2 0 able S21: Uracil dimer stack harmonic vibrational frequencies. S B T PBE RPBE PBE PW91 optPBE BEEF µ σ
COV Skew Kurt. JB Imag.sol -vdW -vdW1 1.0 1.2 1.3 0.5 2.9 8.8 7.6 4.3 0.57 -0.11 -0.87 66.6 8642 2.4 1.9 2.0 1.1 3.4 8.8 7.2 4.5 0.63 -0.05 -0.97 79.5 7123 3.3 3.0 4.1 2.9 5.5 9.2 6.8 5.1 0.75 0.19 -1.07 107.9 4704 4.1 3.4 4.5 3.5 6.0 10.5 7.2 5.0 0.7 0.33 -1.04 126.7 2875 5.4 3.9 6.7 4.8 7.9 10.7 7.7 5.3 0.68 0.38 -0.87 112.8 1826 6.1 5.2 8.3 5.1 9.6 11.8 9.2 4.8 0.53 -0.0 -0.51 21.6 957 0.01 0.02 0.97 18.3 17.5 19.7 18.2 19.5 23.0 17.0 8.7 0.51 -0.49 -0.7 120.8 418 0.02 0.02 0.96 20.0 18.0 20.8 19.4 20.5 24.7 18.6 8.3 0.45 -0.5 -0.55 109.9 209 0.03 0.0 0.97 20.4 20.3 20.9 20.5 21.2 25.1 19.5 8.2 0.42 -0.47 -0.46 90.8 1110 0.04 0.01 0.95 20.9 20.6 22.7 20.9 21.8 25.6 21.2 6.7 0.32 -0.29 -0.12 29.0 211 0.18 0.76 0.06 46.2 46.0 46.1 46.2 46.1 48.4 42.6 9.3 0.22 -2.11 4.82 3421.9 012 0.18 0.76 0.06 46.5 46.2 46.2 46.3 46.1 48.4 43.0 8.5 0.2 -2.0 4.37 2923.0 013 0.03 0.04 0.94 48.3 47.6 48.8 48.5 48.4 53.9 48.4 8.4 0.17 -1.64 6.35 4256.5 014 0.03 0.04 0.93 48.8 48.0 49.4 49.0 49.0 54.4 48.8 8.1 0.17 -1.26 4.03 1877.6 015 0.08 0.9 0.03 62.6 62.3 62.3 62.6 62.5 64.2 59.2 8.2 0.14 -1.62 2.72 1489.2 016 0.07 0.89 0.04 62.7 62.4 62.3 62.7 62.5 64.2 59.4 8.0 0.13 -1.51 1.74 1008.5 017 0.19 0.79 0.01 65.0 64.7 64.9 65.1 64.9 66.8 64.1 5.0 0.08 -2.53 8.8 8593.8 018 0.2 0.78 0.02 65.1 64.7 65.0 65.2 65.0 66.9 64.3 4.8 0.08 -2.48 9.17 9059.7 019 0.21 0.78 0.01 66.9 66.6 67.2 67.2 67.1 68.6 66.4 3.9 0.06 -1.03 2.19 755.1 020 0.22 0.76 0.01 67.2 66.8 67.2 67.3 67.2 68.6 66.5 3.8 0.06 -1.05 2.2 770.1 021 0.01 0.04 0.95 70.3 68.8 71.0 70.1 69.7 77.7 73.3 9.1 0.12 0.65 -0.75 187.5 022 0.01 0.04 0.95 70.6 68.9 71.4 70.7 70.1 78.2 73.5 9.2 0.12 0.63 -0.82 187.6 023 0.0 0.01 0.98 84.0 82.2 83.8 84.1 83.9 91.2 82.0 9.8 0.12 -0.45 -1.0 149.2 024 0.0 0.01 0.98 84.2 82.7 84.5 84.2 84.4 91.7 82.5 9.7 0.12 -0.46 -0.95 143.9 025 0.0 0.0 0.99 87.5 87.3 87.7 87.9 87.1 91.8 86.3 7.9 0.09 -0.66 0.17 146.5 026 0.01 0.0 0.99 87.7 87.4 87.7 87.9 87.2 92.0 86.7 7.8 0.09 -0.66 0.24 152.3 027 0.01 0.0 0.98 90.7 89.5 91.5 91.0 89.4 92.6 90.4 5.3 0.06 0.25 1.56 224.9 028 0.1 0.03 0.87 92.1 90.3 93.6 92.4 91.4 92.7 91.2 5.2 0.06 0.1 1.8 272.8 029 0.72 0.23 0.05 93.5 92.2 94.6 93.7 93.0 96.1 94.9 6.1 0.06 1.02 2.3 787.4 030 0.66 0.22 0.12 93.6 92.3 94.8 93.8 93.2 98.3 95.7 6.5 0.07 0.83 1.41 395.2 031 0.01 0.0 0.99 97.7 97.1 97.9 98.1 97.2 104.6 100.2 6.7 0.07 0.81 0.26 226.2 032 0.01 0.0 0.99 98.1 97.5 98.3 98.5 97.7 105.1 100.6 6.9 0.07 0.78 0.15 202.3 033 0.01 0.01 0.98 115.8 115.3 115.3 116.4 115.7 116.1 112.7 6.3 0.06 -1.23 0.59 536.8 034 0.02 0.01 0.96 116.0 115.3 115.5 116.4 115.8 116.1 112.8 6.3 0.06 -1.23 0.58 529.5 035 0.59 0.38 0.03 116.6 115.6 117.4 116.6 116.2 120.9 118.7 2.6 0.02 -0.03 -0.64 34.3 036 0.61 0.38 0.01 116.7 115.7 117.4 116.7 116.3 121.0 118.8 2.6 0.02 -0.01 -0.62 31.8 037 0.22 0.77 0.01 119.6 118.7 120.1 120.0 119.7 124.7 122.0 3.5 0.03 1.01 0.25 345.8 038 0.22 0.77 0.01 119.8 118.8 120.3 120.1 119.9 124.7 122.1 3.5 0.03 1.01 0.27 349.3 039 0.51 0.49 0.0 131.6 130.7 131.8 131.9 131.4 130.9 131.7 2.4 0.02 -0.53 3.97 1407.3 040 0.5 0.5 0.0 131.9 130.8 131.8 132.0 131.4 131.0 131.7 2.4 0.02 -0.5 4.05 1452.4 041 0.63 0.37 0.0 143.2 141.2 144.1 143.0 141.7 140.4 142.3 2.1 0.01 -1.94 9.66 9037.7 042 0.64 0.36 0.0 143.4 141.3 144.5 143.3 142.0 140.6 142.5 2.1 0.01 -1.96 9.74 9181.8 043 0.33 0.66 0.0 148.1 147.7 148.5 148.4 148.8 152.6 150.9 3.1 0.02 0.32 0.57 61.1 044 0.35 0.65 0.0 148.1 147.8 148.7 148.5 148.9 152.8 151.1 3.1 0.02 0.35 0.53 63.1 045 0.32 0.68 0.01 165.1 164.2 164.4 165.3 165.0 165.7 165.3 2.1 0.01 -1.29 1.63 779.4 046 0.28 0.72 0.01 165.3 164.5 164.5 165.5 165.7 167.1 166.1 2.4 0.01 -1.18 0.96 543.2 047 0.36 0.64 0.01 168.3 167.3 168.5 168.4 168.6 173.2 171.4 4.4 0.03 0.24 -0.33 28.8 048 0.32 0.67 0.01 169.1 167.8 168.7 169.1 169.4 173.5 171.9 4.4 0.03 0.09 -0.24 7.2 049 0.32 0.68 0.0 170.3 169.9 170.2 170.6 171.2 175.7 173.8 4.0 0.02 0.7 -0.03 164.5 050 0.37 0.63 0.01 170.6 170.0 171.0 170.8 171.2 175.7 174.1 3.8 0.02 0.81 0.18 221.4 051 0.47 0.52 0.01 180.6 179.6 181.4 180.8 180.5 182.8 183.2 2.7 0.01 0.79 0.38 222.4 052 0.48 0.52 0.01 180.8 179.8 181.8 181.0 180.6 182.9 183.4 2.6 0.01 0.88 0.61 291.6 053 0.74 0.26 0.0 200.7 199.2 202.0 201.2 199.7 199.4 202.4 2.2 0.01 0.8 1.15 324.4 054 0.74 0.26 0.0 200.9 199.4 202.0 201.2 199.8 199.4 202.5 2.1 0.01 0.83 1.26 360.9 055 0.84 0.16 0.0 210.1 208.2 211.9 210.2 207.5 206.6 210.5 4.5 0.02 0.38 -0.09 47.6 056 0.83 0.17 0.0 211.1 209.1 213.0 211.2 208.6 207.7 211.7 4.4 0.02 0.37 -0.07 46.3 057 0.83 0.16 0.0 215.8 214.2 217.9 216.2 213.9 213.0 217.0 4.5 0.02 0.39 -0.08 50.3 058 0.83 0.17 0.0 216.2 214.7 218.5 216.7 214.5 213.5 217.4 4.4 0.02 0.39 -0.06 51.4 059 1.0 0.0 0.0 388.4 387.7 386.8 389.5 387.3 386.3 393.3 7.0 0.02 -0.08 0.07 2.6 060 1.0 0.0 0.0 388.4 387.7 387.1 389.5 387.5 386.3 393.3 7.0 0.02 -0.08 0.07 2.6 061 1.0 0.0 0.0 393.9 393.5 392.4 394.8 392.4 391.2 398.6 7.2 0.02 -0.08 0.07 2.4 0 able S22: Adenine-thymine stack harmonic vibrational frequencies. S B T PBE RPBE PBE PW91 optPBE BEEF µ σ
COV Skew Kurt. JB Imag.sol -vdW -vdW1 1.3 0.6 1.7 0.5 3.0 7.9 7.1 4.1 0.57 -0.19 -0.89 78.3 8962 2.4 1.7 3.5 2.2 3.9 8.1 7.0 4.4 0.62 -0.05 -0.98 80.8 7753 3.1 2.1 4.6 3.1 5.6 8.9 7.0 4.6 0.65 0.08 -1.04 92.3 6224 4.0 2.9 5.5 3.8 6.9 10.0 7.4 4.8 0.65 0.08 -1.1 103.8 4835 4.8 4.0 7.1 4.4 9.1 10.8 8.2 4.9 0.59 0.04 -1.05 91.7 3766 6.4 4.8 8.0 6.1 10.8 11.1 9.2 4.9 0.53 -0.12 -0.84 63.7 2997 0.05 0.02 0.94 13.4 13.2 13.9 13.7 15.0 18.6 13.4 7.2 0.54 -0.23 -1.01 101.3 2118 0.06 0.01 0.93 16.4 16.9 16.6 16.5 16.4 23.3 16.4 8.7 0.53 -0.27 -0.97 103.9 1179 0.02 0.01 0.97 19.1 18.2 19.3 19.2 19.5 23.9 18.1 8.6 0.47 -0.46 -0.72 113.0 6310 0.02 0.01 0.98 19.5 19.0 20.6 19.5 21.4 31.1 21.7 10.8 0.5 -0.23 -0.98 97.8 3511 0.01 0.01 0.99 25.8 25.5 26.0 25.7 26.5 35.9 25.6 11.0 0.43 -0.37 -0.94 119.2 2212 0.12 0.85 0.03 33.2 33.5 32.8 33.2 33.9 36.4 30.5 9.9 0.33 -1.28 0.7 590.7 1613 0.06 0.92 0.02 33.3 33.6 33.0 33.4 34.2 38.7 33.3 8.8 0.27 -1.27 1.96 857.6 1014 0.0 0.01 0.99 36.1 35.5 36.4 36.1 36.3 39.6 35.0 8.5 0.24 -1.11 2.15 798.7 515 0.0 0.02 0.97 36.7 37.0 37.3 36.9 37.2 41.5 36.9 8.9 0.24 -0.69 1.26 292.5 416 0.21 0.76 0.04 46.5 46.3 46.5 46.5 46.5 47.7 42.5 10.5 0.25 -0.96 0.78 355.6 217 0.02 0.02 0.96 48.3 47.8 48.5 48.5 48.3 53.6 45.8 10.9 0.24 -0.9 0.82 325.0 018 0.05 0.13 0.82 49.4 48.5 52.0 48.7 55.6 56.3 48.7 11.2 0.23 -1.02 0.72 387.4 019 0.19 0.8 0.01 55.7 55.3 55.6 55.7 56.6 63.1 53.0 12.1 0.23 -1.03 0.5 372.8 020 0.06 0.21 0.73 60.1 59.8 59.9 59.9 59.7 64.4 56.1 11.2 0.2 -1.17 1.15 565.1 021 0.17 0.64 0.19 63.5 62.2 63.4 63.5 63.1 66.2 59.2 9.0 0.15 -0.83 0.12 228.3 022 0.11 0.47 0.42 63.9 63.3 63.9 63.9 63.7 67.0 63.2 8.1 0.13 -0.32 -0.04 33.7 023 0.06 0.32 0.62 64.4 63.6 65.2 64.4 64.4 69.9 65.7 6.8 0.1 -0.01 -0.0 0.0 024 0.22 0.71 0.06 65.3 64.9 66.0 65.7 65.4 70.1 67.3 6.4 0.1 0.07 -0.44 17.5 025 0.01 0.02 0.97 69.5 68.3 69.3 68.9 68.2 74.1 69.6 7.3 0.11 0.3 -0.14 32.5 026 0.02 0.08 0.9 70.7 69.4 70.7 70.2 69.8 74.6 71.2 7.0 0.1 0.55 -0.37 113.0 027 0.14 0.76 0.1 73.0 72.5 72.7 73.0 72.9 75.4 73.9 6.7 0.09 0.28 -0.28 32.4 028 0.32 0.67 0.02 74.4 73.8 74.5 74.6 74.5 76.0 76.5 5.9 0.08 0.73 0.12 178.3 029 0.0 0.0 0.99 80.2 79.7 80.3 80.4 79.7 83.5 79.5 6.1 0.08 0.29 -0.46 46.1 030 0.01 0.01 0.98 81.5 81.1 81.3 81.6 80.8 85.2 81.3 6.1 0.07 0.38 -0.0 48.3 031 0.0 0.01 0.99 82.4 81.6 82.8 82.6 81.8 86.9 83.4 5.7 0.07 0.37 0.27 51.1 032 0.71 0.28 0.01 88.0 87.0 88.8 88.1 87.4 87.1 87.5 4.1 0.05 -0.02 4.61 1770.7 033 0.8 0.19 0.01 89.6 88.2 90.6 89.7 89.1 88.8 89.4 4.2 0.05 -0.07 3.58 1069.1 034 0.0 0.0 0.99 90.0 88.8 90.8 90.4 89.1 93.5 91.2 4.6 0.05 0.02 1.83 278.7 035 0.0 0.04 0.96 92.8 91.9 93.2 93.1 91.8 96.3 93.3 4.4 0.05 0.4 1.48 237.1 036 0.43 0.56 0.01 97.4 96.3 97.9 97.5 96.7 96.9 95.9 4.7 0.05 0.28 0.68 64.3 037 0.0 0.0 0.99 97.6 96.4 98.0 97.9 96.9 100.8 98.2 4.6 0.05 0.32 0.99 114.6 038 0.0 0.0 0.99 101.3 100.4 101.3 101.6 101.1 108.6 102.4 6.1 0.06 -0.07 -0.98 80.8 039 0.03 0.13 0.84 107.7 107.1 106.8 108.1 107.5 110.2 106.3 6.1 0.06 -0.31 -0.73 76.5 040 0.19 0.68 0.13 107.8 108.0 108.1 108.3 107.7 114.6 110.0 5.4 0.05 -0.48 -0.0 77.3 041 0.21 0.78 0.0 113.2 112.4 113.5 113.6 113.3 115.7 113.2 4.7 0.04 0.04 -0.06 0.9 042 0.0 0.0 1.0 115.5 115.2 115.1 115.7 115.3 117.2 116.3 4.7 0.04 0.47 0.76 122.5 043 0.47 0.46 0.06 116.4 115.7 116.5 116.7 116.5 123.5 119.1 5.4 0.05 0.29 -0.3 35.4 044 0.18 0.68 0.14 122.7 122.7 122.0 123.2 123.8 125.6 124.0 4.6 0.04 -0.93 1.69 525.4 045 0.44 0.56 0.0 123.2 122.8 123.4 123.4 124.2 128.3 125.3 4.7 0.04 -0.49 0.22 83.8 046 0.0 0.63 0.37 127.4 127.7 125.9 127.9 128.9 131.3 128.4 5.4 0.04 -0.64 -0.63 169.2 047 0.49 0.51 0.0 131.4 130.6 132.0 131.5 130.9 135.2 132.3 3.4 0.03 -0.72 0.74 220.6 048 0.53 0.47 0.0 138.1 136.7 139.1 138.2 137.3 135.9 137.8 1.4 0.01 -3.33 23.92 51382.3 049 0.57 0.4 0.03 139.1 136.9 140.7 139.4 137.8 137.6 139.3 1.8 0.01 -0.85 9.67 8036.1 050 0.46 0.54 0.01 143.9 143.3 144.7 144.6 144.3 145.2 145.3 1.7 0.01 -1.02 4.12 1758.2 051 0.78 0.22 0.0 149.5 147.4 151.0 149.4 147.9 146.6 148.8 1.6 0.01 -1.41 7.55 5416.9 052 0.45 0.55 0.0 150.6 149.4 151.7 150.9 150.5 150.8 151.5 1.3 0.01 -2.67 21.95 42530.7 053 0.46 0.54 0.0 152.4 151.6 153.2 152.8 152.3 153.7 153.6 1.2 0.01 0.04 0.19 3.5 054 0.72 0.28 0.0 161.5 159.5 162.9 161.7 159.7 156.0 160.0 2.8 0.02 -0.34 -0.85 98.6 055 0.29 0.66 0.04 164.0 162.4 163.6 164.5 162.8 160.8 162.8 2.3 0.01 -0.77 0.98 276.9 056 0.64 0.36 0.0 164.4 164.0 166.1 164.6 164.9 165.9 165.8 2.2 0.01 -1.84 5.24 3419.7 057 0.64 0.36 0.0 166.0 164.3 166.3 166.0 165.0 167.4 167.0 2.1 0.01 -1.58 4.88 2816.1 058 0.22 0.76 0.02 167.6 167.6 166.9 168.2 168.9 168.7 169.0 2.3 0.01 -0.83 1.43 401.2 059 0.05 0.95 0.0 169.3 169.0 168.5 170.0 169.8 170.8 170.9 2.5 0.01 -0.41 -0.21 58.7 060 0.49 0.51 0.0 170.5 169.4 171.9 170.6 170.4 172.8 173.0 3.1 0.02 0.27 -0.22 28.2 061 0.47 0.52 0.0 170.7 170.2 172.2 171.2 171.7 174.1 174.0 2.9 0.02 0.39 0.01 51.2 0 able S23: Adenine-thymine Watson-Crick harmonic vibrational fre-quencies. S B T PBE RPBE PBE PW91 optPBE BEEF µ σ
COV Skew Kurt. JB Imag.sol -vdW -vdW1 3.2 2.7 3.1 2.9 2.2 8.5 7.6 3.7 0.49 -0.38 -0.57 74.4 9502 3.8 3.3 4.2 3.9 3.1 9.7 7.7 4.3 0.56 -0.41 -0.83 113.7 8193 7.6 6.9 8.6 7.8 7.3 10.4 8.3 4.9 0.59 -0.23 -0.85 78.5 6384 8.3 7.4 8.9 8.3 7.5 11.8 8.8 4.9 0.56 -0.24 -0.78 70.3 5065 12.6 11.2 13.0 12.7 12.0 13.0 9.7 5.2 0.53 -0.48 -0.76 126.7 3686 13.3 12.3 13.9 13.6 12.7 14.2 11.4 4.9 0.43 -0.91 0.23 282.0 2737 0.0 0.0 0.99 13.8 13.2 15.6 13.8 13.5 17.4 13.6 6.2 0.46 -0.34 -0.33 47.4 1938 0.0 0.0 1.0 17.2 17.8 16.8 17.5 18.0 23.2 16.7 8.2 0.49 -0.25 -0.77 69.9 1109 0.0 0.0 1.0 19.6 19.2 19.8 19.7 19.2 24.7 18.8 7.9 0.42 -0.27 -0.6 54.7 6310 0.0 0.0 1.0 21.2 20.5 21.8 21.2 20.1 31.5 22.4 10.1 0.45 -0.07 -1.07 96.6 3811 0.0 0.0 1.0 27.4 27.0 27.6 27.4 26.6 35.6 26.6 10.6 0.4 -0.28 -1.03 114.6 2412 0.07 0.92 0.02 34.7 34.7 34.4 34.8 35.1 38.2 31.3 9.9 0.32 -0.96 -0.09 309.1 1213 0.0 0.0 1.0 35.9 35.5 36.0 36.0 35.3 39.2 33.9 8.2 0.24 -1.18 1.76 722.1 814 0.0 0.02 0.97 36.2 35.7 36.3 36.3 35.5 39.8 35.2 7.8 0.22 -1.08 2.16 779.1 415 0.14 0.86 0.0 37.9 36.9 39.0 38.0 37.5 40.5 37.7 8.7 0.23 -0.43 1.05 153.9 216 0.2 0.78 0.02 48.8 46.6 49.6 48.9 48.4 49.8 42.8 10.1 0.24 -0.48 -0.28 82.1 117 0.0 0.02 0.98 49.5 48.1 50.0 49.7 48.7 54.4 46.8 10.2 0.22 -0.74 0.4 197.7 018 0.01 0.0 0.99 53.3 48.9 57.4 53.9 50.1 57.3 49.7 9.7 0.2 -0.9 0.75 316.8 019 0.21 0.79 0.0 57.0 56.4 58.8 57.2 56.8 58.4 55.8 8.3 0.15 -0.85 1.5 430.8 020 0.17 0.83 0.0 64.2 62.7 64.4 64.4 62.0 64.9 59.9 7.8 0.13 -1.28 2.57 1095.4 021 0.2 0.74 0.06 64.6 63.6 65.0 64.8 64.0 65.8 62.7 5.7 0.09 -1.11 2.2 812.4 022 0.01 0.04 0.94 64.9 64.0 66.4 64.9 64.6 66.8 65.3 5.4 0.08 0.01 1.46 178.3 023 0.24 0.76 0.0 66.2 65.6 66.5 66.4 66.3 68.7 67.2 5.4 0.08 0.18 0.36 22.0 024 0.0 0.0 1.0 70.4 69.5 71.0 70.5 69.3 73.7 69.8 6.1 0.09 -0.17 -0.61 40.3 025 0.0 0.0 1.0 70.5 69.7 72.1 70.8 69.4 75.4 71.4 6.2 0.09 0.32 -0.63 67.3 026 0.17 0.82 0.01 73.9 73.4 74.3 74.1 73.9 77.2 74.7 5.8 0.08 -0.06 -0.04 1.3 027 0.29 0.71 0.0 76.5 75.5 77.5 76.8 76.4 77.8 77.5 5.4 0.07 0.33 -0.17 38.0 028 0.0 0.0 1.0 80.4 79.6 80.9 80.6 79.6 83.0 79.9 5.3 0.07 0.08 -0.65 37.5 029 0.0 0.0 1.0 82.7 82.0 83.2 82.9 81.7 85.5 81.9 5.3 0.06 -0.02 -0.39 13.0 030 0.74 0.26 0.0 88.4 86.5 89.3 88.5 87.5 87.1 84.2 5.7 0.07 0.04 -0.23 4.8 031 0.26 0.05 0.69 90.7 87.3 91.8 90.9 87.9 89.1 88.0 4.2 0.05 -0.42 3.47 1063.8 032 0.56 0.11 0.33 90.8 89.4 92.2 91.0 89.5 91.0 90.3 4.5 0.05 -0.21 2.3 455.9 033 0.0 0.02 0.98 93.1 89.6 94.1 93.6 90.2 93.4 91.8 4.6 0.05 -0.02 1.66 229.3 034 0.0 0.02 0.98 93.9 92.5 97.6 94.6 92.5 96.6 93.7 4.4 0.05 0.38 1.64 270.8 035 0.0 0.0 1.0 97.5 96.4 98.2 97.8 96.1 97.3 95.9 4.4 0.05 0.29 1.24 156.3 036 0.4 0.6 0.0 97.6 96.8 100.7 97.9 97.2 100.9 98.3 4.6 0.05 0.42 1.33 204.4 037 0.0 0.0 1.0 100.6 100.0 103.0 100.9 99.2 109.0 102.4 6.2 0.06 0.14 -0.87 70.0 038 0.0 0.04 0.96 108.6 108.3 107.9 108.8 107.8 110.2 106.8 6.1 0.06 -0.51 -0.57 113.6 039 0.2 0.8 0.0 109.4 108.4 110.4 109.7 109.2 114.7 110.4 5.5 0.05 -0.83 0.79 282.1 040 0.22 0.78 0.0 113.5 112.5 113.8 113.8 113.5 116.9 113.1 5.7 0.05 -0.65 0.78 191.6 041 0.0 0.0 1.0 117.9 116.9 118.0 118.3 117.6 118.9 116.1 5.8 0.05 0.11 -0.26 9.7 042 0.43 0.48 0.08 118.4 117.5 118.5 118.7 118.2 121.0 118.7 6.2 0.05 0.23 -0.67 55.1 043 0.24 0.64 0.12 122.7 118.1 122.8 123.2 120.3 125.2 120.8 5.9 0.05 0.09 -0.41 16.4 044 0.14 0.18 0.69 125.6 122.8 125.8 125.9 123.6 126.1 124.6 5.1 0.04 -0.65 1.18 255.9 045 0.35 0.42 0.24 125.7 124.1 126.9 126.7 125.3 128.1 125.9 4.9 0.04 -0.25 0.44 36.2 046 0.0 0.6 0.4 127.3 127.6 130.9 128.0 128.7 130.6 128.5 4.9 0.04 -0.33 -0.97 114.5 047 0.49 0.51 0.0 130.8 130.1 135.7 131.0 130.0 136.1 132.2 4.1 0.03 -0.17 -0.08 10.5 048 0.52 0.48 0.0 139.0 137.5 140.2 139.4 138.2 138.3 138.7 1.6 0.01 -3.28 22.72 46593.0 049 0.38 0.59 0.03 141.8 140.5 141.7 142.1 141.6 138.9 141.2 2.1 0.02 -1.92 8.47 7212.2 050 0.77 0.23 0.0 147.9 145.0 150.7 148.3 146.6 145.7 147.5 1.5 0.01 -0.91 4.6 2039.8 051 0.77 0.23 0.0 149.2 147.6 152.1 149.3 147.9 146.7 149.3 1.4 0.01 -0.79 4.86 2179.5 052 0.34 0.66 0.0 152.2 150.5 152.4 152.6 151.6 151.2 152.1 1.3 0.01 -0.53 1.83 372.3 053 0.56 0.44 0.0 152.6 151.7 154.3 152.9 152.3 153.8 154.0 1.3 0.01 0.12 0.02 5.1 054 0.63 0.37 0.0 162.4 160.6 163.1 162.6 161.0 157.8 161.1 2.7 0.02 -0.48 -0.64 111.7 055 0.67 0.32 0.01 164.0 161.9 163.9 164.1 162.5 160.4 163.0 2.2 0.01 -0.5 0.28 88.7 056 0.38 0.58 0.04 164.4 164.1 166.6 164.8 165.0 166.1 166.3 1.8 0.01 -1.63 5.78 3662.3 057 0.65 0.35 0.0 166.9 165.0 167.3 167.1 165.6 167.7 167.7 1.6 0.01 -1.68 7.75 5937.9 058 0.05 0.95 0.0 169.4 169.1 168.8 170.1 169.7 169.4 170.2 2.0 0.01 -0.49 -0.17 82.7 059 0.48 0.52 0.0 170.3 169.5 172.0 170.5 170.6 171.3 172.0 1.9 0.01 -0.71 1.03 256.0 0 eferences (1) Tao, J.; Perdew, J. 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