Supporting Information. Local decomposition of imaginary polarizabilities. and dispersion coefficients

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1 Electronic Supplementary Material (ESI) for Physical hemistry hemical Physics. This journal is the Owner Societies 27 Supporting Information Local decomposition of imaginary polarizabilities and dispersion coefficients Ignat arczuk, Balazs Nagy, Frank Jensen, Olav Vahtras, and ans Agren,, KT Royal Institute of Technology, School of Biotechnology, Division of Theoretical hemistry and Biology, SE- 9 Stockholm, Sweden Department of hemistry, Langelandsgade 4, 8 Aarhus, Denmark Institute of Nanotechnology, Spectroscopy and Quantum hemistry, Siberian Federal University, Svobodny pr. 79, 4 Krasnoyarsk, Russia hagren@kth.se

2 Theory. Anisotropic Dispersion Interaction We will here give the dispersion energy in a form which requires a few definitions in order to make the presentation self-contained. onsider two separated systems (molecules or atoms) labeled by and 2. They have each a center-of-mass at a position represented by vectors R and R2, respectively, with artesian representations in a common coordinate system: Xi and X2i for i =, 2, 3. We refer to the vector connecting the two subsystems by R2 = R R2 i.e., the distance between the centers of mass is R2 = R2 We use r for an electronic coordinate associated with system and r2 for electrons associated with system 2. The density-density response function of system is defined by (see e.g. Ref., eq 9.4.2) hhˆ ρ(r ); ρˆ(r )ii ω = X hψ ˆ ρ(r ) Ψn ihψn ˆ ρ(r ) Ψ i hψ ˆ ρ(r ) Ψ i ρ(r ) Ψn ihψn ˆ ~(ω ωn ) ~(ω + ωn ) n > (2) where Ψn is a stationary state of system with n = being the ground state and ~ωn En E is excitation energy from the ground state of system to its n :th excited state. The density operator ρˆ(r ) has the property that its expectation value is the electron density at a point in space. It is given in second-quantized form by expanding it in a given basis set on system, {φp (r )} p= : ρˆ(r ) = X φ p (r ) φq (r )(apα aqα + apβ aqβ ) (3) pq where a pα aqα is an operator representing an excitation of an α-electron (spin + 2 ) from orbital q to p and a pβ aqβ the equivalent for a β-electron (spin 2 ). 2

3 In particular the dispersion energy of two interacting molecules (denoted by and 2) is given by Edisp ~ = 2π dr dr2 r2 dr dr 2 r2 dωhhˆ ρ(r ); ρˆ(r )iiiω hhˆ ρ(r2 ); ρˆ(r 2 )iiiω (4) where the linear response functions are evaluated at an imaginary frequency iω. The intermolecular oulomb potential can further be expanded in multipole moments, e.g., in solid spherical harmonics. Another alternative is a real-valued Taylor expansion to obtain the artesian moments, by expanding around the intermolecular distance vector. Introducing the electronic coordinate vectors relative to the local molecular coordinate frames (see Fig.S) r; = r R and r2;2 = r2 R2 we have r2 = r; r2;2 + R2 Atom Atom 2 r2 r; (5) r2; 2 R2 r R R2 r2 Figure S: Definition of coordinates in section 2. When the first two terms are small compared to the last, as we can assume for long-range 3

4 intermolecular interactions, we may expand the oulomb potential in a Taylor series X = ((r; r2;2 ) R2 )n = r2 r; r2;2 + R2 n= n! R2 () where the derivative operator only acts on the internuclear distance R2 =,, 2 3 X2 X2 X2 (7) Furthermore we can use the binomial theorem to obtain n ((r; r2;2 ) R2 ) = n X n k= k (r; R2 )k ( r2;2 R2 )n k (8) Substituting expansions in Eqs. and 8 in the linear response function in Eq. 2, we can rewrite Eq. 4 as Edisp ~ X X n n n k+n k = ( ) 2π nk n k n!n! k k k k ) iiiω dωhh dr ρˆ(r )(r; R2 ) ; dr2 ρˆ(r )(r ; R2 n k n k ) hh dr ρˆ(r2 )(r2;2 R2 ) ; dr2 ρˆ(r2 )(r2;2 R2 iiiω R2 R2 R2 =R 2 (9) where we can identify linear response functions of second-quantized formulations of multipole moments of various orders. The lowest non-vanishing term is with k = k = and n = n = 2, where we have electrical dipole operators ˆr = drˆ ρ(r)r 4 ()

5 and we can identify the well-known expression for dipole-dipole polarizabilities α(iω) = hhˆr; ˆriiiω () This expansion for a specific orientation of the molecules is an expression which couples the molecular polarizability tensors with second derivatives of the intermolecular distance and which decays as R2 Edisp ~ 2π dω X (2) αij (iω)αkl (iω)tik Tjl (2) ijkl where the summation is over cartesian coordinates in a common coordinate system, and T is the second-order derivative of the oulomb potential i k 2 X2 δik R2 2 3X2 Tik = = 5 i k R R2 X2 X2 2.2 (3) Isotropic dispersion coefficient An isotropic expression for the dispersion energy is obtained by rotational averaging of the two subsystems independently. Molecular tensors such as polarizabilities are usually calculated in a coordinate system fixed by molecular nuclei (body-fix axes) whereas observations are carried out in a coordinate system defined by the observer (lab-fix axes). With µi as the transformation matrix between body-fix (greek indices) and space-fix (roman indices) axes this averaging may be written iso Edisp ~ 2π dω XX (2) αµν (iω)αστ (iω)µi νj ijkl µνστ 5 σk τ l Tik Tjl (4)

6 Detailed expressions for rotational averages of tensors can be found, e.g., in Ref. 2; in our case the average can be written as µi νj = δµν δij 3 (5) Substituting with the isotropic polarizabilities α = X αkk 3 k () we obtain iso (R2 ) Edisp ~ = 2π α (iω)α(2) (iω) X Tik Tik ik ~ = α (iω)α(2) (iω) 2π R2 3~ = α (iω) α(2) (iω)dω π R2 R2 (7) which defines the well-known coefficient for the dispersion interaction. By decomposing the molecular polarizabilities α and α(2) into distributed atomic contributions αi and (2) αj, respectively, where i is an atomic site in molecule and j an atomic site in molecule 2, the atom-atom dispersion coefficient ij for atoms i and j can be written as ij 3~ = π (2) α i (iω) αj (iω)dω (8) where now the distributed polarizabilities for the atoms contributes to the atomic ij dispersion. Note that in order to calculate the total dispersion energy by our scheme LoProp (see below), each atom-atom pair has to be evaluated. For a water dimer with 3 atoms in each molecule, there is thus 3 3 = 9 atom-pairs to consider. This is due to the LoProp polarizability being additive and summing up to the molecular polarizability, and the sum

7 of all LoProp ij elements being equal to the molecular. 2 Basis set dependence Table S: artesian coordinates (in A) of formaldehyde optimized at the B3LYP/-3++G** level -28 O Table S2: artesian coordinates (in A) of formamide optimized at the B3LYP/-3++G** level O N Table S3: artesian coordinates (in A) of glycine optimized at the B3LYP/-3++G** level N O

8 O Diatomics 2 dimer Table S4: Reference structure in artesian coordinates ( A). Bond length was taken from experiment

9 Parallel LoProp aniso Molecular aniso Perpendicular Parallel rotated Figure S2: Scatter plots of the dispersion energy for the 2 2 dimers calculated with the anisotropic molecular and LoProp models versus the reference dispersion energy obtained with SAPT2+/aug-cc-pVT. 9

10 Ry=2. Å Ry=2.2 Å Ry=2.4 Å Ry=2.8 Å.5.5 Ry=3. Å.5.5 Ry=2. Å Molecular aniso LoProp aniso.5.5 Figure S3: Scatter plots of the dispersion energy for the 2 2 dimers calculated with the anisotropic molecular and LoProp models versus the reference dispersion energy obtained with SAPT2+/aug-cc-pVT.

11 3.2 N2 dimer Table S5: Reference structure in artesian coordinates ( A). Bond length was taken from experiment. N.5525 N

12 Parallel LoProp aniso Molecular aniso Perpendicular Parallel rotated Figure S4: Scatter plots of the dispersion energy for the N2 N2 dimers calculated with the anisotropic molecular and LoProp models versus the reference dispersion energy obtained with SAPT2+/aug-cc-pVT. 2

13 Ry=2. Å Ry=2.2 Å Ry=2.4 Å Ry=2. Å Ry=2.8 Å Ry=3. Å Molecular aniso LoProp aniso Figure S5: Scatter plots of the dispersion energy for the N2 N2 dimers calculated with the anisotropic molecular and LoProp models versus the reference dispersion energy obtained with SAPT2+/aug-cc-pVT. 3

14 3.3 O dimer Table S: Reference structure in artesian coordinates ( A). Bond length was taken from experiment O.54 4

15 Parallel LoProp aniso Molecular aniso Perpendicular Parallel rotated Figure S: Scatter plots of the dispersion energy for the O O dimers calculated with the anisotropic molecular and LoProp models versus the reference dispersion energy obtained with SAPT2+/aug-cc-pVT. 5

16 Ry=2. Å Ry=2.2 Å Ry=2.4 Å Ry=2. Å Ry=2.8 Å Ry=3. Å Molecular aniso LoProp aniso Figure S7: Scatter plots of the dispersion energy for the O O dimers calculated with the anisotropic molecular and LoProp models versus the reference dispersion energy obtained with SAPT2+/aug-cc-pVT.

17 4 Benzene dimer Table S7: artesian coordinates (in A) for benzene optimized at the B3LYP/-3++G** level Methane dimer Table S8: artesian coordinates (in A) for methane optimized at the B3LYP/-3++G** level

18 2 2 4 Edisp [kcal/mol] Edisp [kcal/mol] Molecular, iso SAPT(DFT) 8 4 Molecular, aniso SAPT(DFT) ROM Å] [ LoProp, iso SAPT(DFT) 8 ROM Å] 8 9 [ (b) Edisp [kcal/mol] Edisp [kcal/mol] (a) 7 4 LoProp, aniso SAPT(DFT) ROM Å] 8 9 [ (c) 5 7 ROM Å] 8 9 [ (d) Figure S8: Molecular isotropic (a), molecular anisotropic (b), LoProp isotropic (c) and LoProp anisotropic (d) dispersion interaction energies for the 4 4 dimer as a function of monomer distances. 8

19 Pyridine dimer Table S9: artesian coordinates (in A) for pyridine optimized at the B3LYP/-3++G** N References Parr, R. G.; Yang, W. T. Density-Functional Theory Atoms Mol.; 989. (2) Salam, A. Molecular Quantum Electrodynamics; John Wiley & Sons, Inc., 29; pp

20 Edisp [kcal/mol] Edisp [kcal/mol] Molecular, iso SAPT(DFT) ROM Å [ 2 3 Molecular, aniso SAPT(DFT) ] LoProp, iso SAPT(DFT) ROM Å [ ] (b) Edisp [kcal/mol] Edisp [kcal/mol] (a) ROM Å [ 2 3 LoProp, aniso SAPT(DFT) 8 4 ] (c) ROM Å [ ] (d) Figure S9: Molecular isotropic (a), molecular anisotropic (b), LoProp isotropic (c) and LoProp anisotropic (d) dispersion interaction energies for the pyridine pyridine dimer as a function of monomer distances. 2