( ) R kj. = y k y j. y A ( ) z A. y a. z a. Derivatives of the second order electrostatic tensor with respect to the translation of ( ) δ yβ.

Transcription

1 Supporting information Derivatives of R with respect to the translation of fragment along the y and z axis: y = y k y j (S1) z ( = z z k j) (S2) Derivatives of S with respect to the translation of fragment along the y and z axes: S B = c y υ j c µk υ µ a S µυ y a B ψ = c υ j c µk u ψ υ µ a y υ (S3) a S B = c z υ j c µk υ µ a S µυ z a B ψ = c υ j c µk u ψ υ µ a z υ (S4) a Derivatives of the second order electrostatic tensor with respect to the translation of fragment along the y and z axes: T αβ y (S5) 7 = 15R R R 3R 2 R y α β y δ αβ + R α δ yβ + R β δ yα T αβ z (S6) 7 = 15R R R 3R 2 R z α β z δ αβ + R α δ zβ + R β δ zα Derivatives of the third-order electrostatic tensor with respect to the translation of fragment along the y and z axes: 1

2 T γσκ 1 = y y γ σ κ R 9 = 105R R R R 15R 2 R y γ σ κ y R γ δ σκ + R y R σ δ γκ + R y δ γσ + R γ R σ δ yκ + R γ δ yσ + R σ δ yγ + 3R 4 δ ( yγ δ σκ +δ yσ δ γκ +δ yκ δ γσ ) R 9 (S7) T γσκ 1 = z z γ σ κ R 9 = 105R R R R 15R 2 R z γ σ κ z R γ δ σκ + R z R σ δ γκ + R z δ γσ + R γ R σ δ zκ + R γ δ zσ + R σ δ zγ + 3R 4 δ ( zγ δ σκ +δ zσ δ γκ +δ yκ δ γσ ) R 9 (S8) Rotational derivatives of R around the y and z axes: θ y = ( z k z j ) ( x k x COM ) ( x k x j ) z k z COM (S9) θ z = ( x k x j ) ( y k y COM ) ( y k y j ) x k x COM (S10) Rotational derivatives of the electrostatic tensors around the y and z axes for the rotation of fragment : T αβ = T αβ ( x θ y z k x COM ) T αβ z x k z COM (S11) T αβ = T αβ ( y θ z x k y COM ) T αβ x y k x COM (S12) 2

3 T γσκ θ y = T γσκ z ( x k x COM ) T γσκ x ( z k z COM ) (S13) T γσκ θ z = T γσκ x ( y k y COM ) T γσκ y ( x k x COM ) (S14) Rotational derivatives of the dynamic dipole-dipole polarizability tensor: In order to simplify notation the dynamic dipole-dipole polarizability tensor is rewritten as: α = α xx α xy α xz α yx α yy α yz α zx α zy α zz = α 11 α 12 α 13 α 21 α 22 α 23 α 31 α 32 α 33 (S15) The rotational derivative of each element of the dynamic dipole-dipole polarizability tensor around the x axis can be calculated using: lim θ 0 ααβ α αβ θ (S16) where α αβ represents the new value after rotation of the tensor around the x axis. In order to obtain the rotated elements of the tensor one needs to compute: α αβ = Q αm Q βn α mn (S17) In Eq. (S17) the Einstein summation notation is used for m and n. Q represents the x rotation matrix: 3

4 Q= cosθ sinθ 0 sinθ cosθ (S18) αmnrepresents the original elements of the tensor. For example for α 11: α 11= Q 1m Q 1n α mn = Q 11 Q 11 α 11 + Q 11 Q 12 α 12 + Q 11 Q 13 α 13 + Q 12 Q 11 α 21 + Q 12 Q 12 α 22 +Q 12 Q 13 α 23 Q 11 α 31 Q 12 α 32 Q 13 α 33 =α 11 (S19) Since all required elements of Q are 0 except for Q11=1. α 11 α lim 11 α = lim 11 α 11 = 0 (S20) θ 0 θ θ 0 θ Similarly for α 12: α 12 = Q 1m Q 2n α mn = Q 11 Q 21 α 11 + Q 11 Q 22 α 12 + Q 11 Q 23 α 13 + Q 12 Q 21 α 21 + Q 12 Q 22 α 22 +Q 12 Q 23 α 23 Q 21 α 31 Q 22 α 32 Q 23 α 33 (S21) = cosθ α 12 + sinθ α 13 α 12 α lim 11 = lim θ 0 θ cosθ 1 θ 0 θ sinθ α 12 + lim θ 0 θ α =α (S22) Rotational derivatives around the y axis and the z axis are calculated in a similar fashion using the following rotation matrices: cosθ 0 sinθ Q y = sinθ 0 cosθ 4

5 Q z = cosθ sinθ 0 sinθ cosθ Rotational derivatives of the dynamic dipole-quadrupole polarizability tensor: xxx θz = xxy + yxx + xyx xxy θ z = yxy + xyy xxx xxz θ z = xyz + yxz xyx θ z = xyy + yyx xxx xyy θz = yyy xxy xyx xyz θz = yyz xxz xzx θz = xzy + yzx xzy θ z = yzy xzx xzz θ z = yzz yxx θ z = yyx xxx + yxy yxy θ z = yyy yxx xxy yxz θ z = yyz xxz yyx θ z = yyy yxx yxy yyy θz = xyy yxy yyx yyz θ z = xyz yxz yzx θz = yzy xzx yzy θ z = xzy yzx yzz θz = xzz zxx θz = zxy + zyx zxy θz = zyy zxx zxz θz = zyz zyx θ z = zyy zxx zyy θz = zxy zyx zyz θ z = zxz zzx θz = zzy zzy θ z = zzx zzz θz = 0 5

6 xxx θ y = xxz xzx zxx xxy θ y = xzx zxy xxz θ y = xxx zxz xzz xyx θ y = xyz zyx xyy θ y = zyy xyz θ y = xyx zyz xzx θ y = xxx xzz zzx xzy θ y = xxy zzy xzz θ y = xxz + xzx zzz yxx θ y = yxz yzx yxy θ y = yzy yxz θ y = yxx yzz yyx θ y = yyz yyy θ y = 0 yyz θ y = yyx yzx θ y = yxx yzz yzy θ y = yxy yzz θ y = yxz + yzx zxx θ y = xxx zxz zzx zxy θ y = xxy zzy zxz θ y = xxy + zxx zzz zyx θ y = xyx zyz zyy θ y = xyy zyz θ y = xyz + zyx zzx θ y = xzx + zxx zzz zzy θ y = xzy + zxy zzz θ y = xzz + zxz + zzx xxx θ x = 0 xxy θx = xxz xxz θx = xxy xyx θx = xzx xyy θ x = xyz + xzy xyz θx = xzz xyy xzx θ x = xyx xzy θx = xzz xyy xzz θ x = xyz xzy yxx θ x = zxx yxy θ x = yxz + zxy yxz θx = zxz yxy yyx θ x = yzx + zyx yyy θx = yyx + yzy + zyy yyz θ x = yzz + zyz yyy yzx θ x = zzx yyx yzy θ x = zzy + yzz yyy yzz θx = zzz yyz yzy zxx θx = yxx zxy θ x = zxz yxy zxz θ x = yxz zxy zyx θ x = zzx yyx zyy θ x = zyz + zzy yyy zyz θ x = zzz zyy yyz zzx θ x = yzx zyx zzy θx = zzz zyy yzy zzz θ x = yzz zyz zzy 6

7 Optimization of the ethene-ethyne complex The ethene-ethyne complex optimized at the MP2/ G(2df2p) level of theory is shown in Figure S1. The complexes optimized with the EFP method including (structure 1) and not including the E7 dispersion term (structure 2) are very similar and therefore not shown in the figure. The relevant lengths and angles are displayed in Table S1 for all three optimized structures. Overall the inclusion of the E7 dispersion term yields an optimized structure closer to the MP2/ G(2df2p) optimized complex. The EFP energy contributions of structures 1 and 2 of the ethane-ethyne complex are shown in Table S2. The E7 energy of structure 1 represents only 1.4 % of the total EFP energy and 7% of the dispersion energy (these values are calculated in a similar manner as for the water trimer cf text). While these contributions may seem small they are important in order to obtain more accurate structures. Figure S1: Optimized ethene-ethyne complex at the MP2/ G(2df2p) level of theory 7

8 Table S1. Geometric structures for the ethene-ethyne complex optimized with the EFP2 method including dispersion (structure 1) with the EFP2 method not including dispersion (structure 2) and at the MP2/ G(2df2p) level of theory. Structure 1 Structure 2 MP2/ G(2df2p) Distance between the center of the C C2 double bond and C8 (Å) ngle C7-C8-midpoint between C1and C2 (º) 8

9 Table S2. EFP2 energy contributions (in kcal/mol) for the ethene-ethyne complex optimized with and without the E7 contribution. The values in parentheses are the percentage contributions of each term to the total EFP interaction energy as calculated using Eq.(36) for the electrostatic energy. Structure 1 Structure 2 Electrostatic energy (kcal/mol) (40.0%) (39.9%) Exchange-repulsion energy (kcal/mol) 1.78 (33.0%) 1.91 (34.2%) Polarization energy (kcal/mol) (4.7%) (4.8%) E 6 dispersion energy (kcal/mol) (13.2%) -0.74(13.2%) E 7 dispersion energy (kcal/mol) 0.07 (1.4%) N/ E 8 dispersion energy (kcal/mol) (4.4%) (4.4%) Charge transfer energy (kcal/mol) (3.3%) (3.4%) Total EFP energy Sum of the absolute values of each energy contribution to the total EFP energy (Eq.(35)) Optimization of the benzene dimer The benzene dimer optimized at the MP2/ G(2df2p) level of theory is shown in Figure S2. The complexes optimized with the EFP method including (structure 1) and not including the E7 dispersion term(structure 2) are very similar and therefore not shown in the figure. The relevant lengths and angles are displayed in Table S3 for all three optimized structures. Structures 1 and 2 are very similar but the intermolecular distance is 9

10 overestimated by 0.5Å in comparison to MP2. The EFP energy contributions of structures 1 and 2 of benzene dimer are shown in Table S4. The E7 energy of structure 1 represents only 0.3 % of the total EFP energy and 0.5% of the dispersion energy (these values are calculated in a similar manner as for the water trimer cf text). It is interesting to see that in comparison E6 represents 45% of the EFP energy. Figure S2: Optimized benzene dimer at the MP2/ G(2df2p) level of theory Table S3. Geometric structures for the benzene dimer optimized with the EFP2 method including dispersion (structure 1) with the EFP2 method not including dispersion (structure 2) and at the MP2/ G(2df2p) level of theory Structure 1 Structure 2 MP2/ G(2df2p) Distance between the centers of mass of the rings (Å) 10

11 Table S4. EFP2 energy contributions (in kcal/mol) for the benzene dimer optimized with and without the E7 contribution. The values in parentheses are the percentage contributions of each term to the total EFP interaction energy as calculated in Eq.(36) for the electrostatic energy. Structure 1 Structure 2 Electrostatic energy (kcal/mol) 0.15 (2.0%) 0.17 (2.4%) Exchange-repulsion energy (kcal/mol) 2.46(35.0%) 2.52 (34.6%) Polarization energy (kcal/mol) (2.9%) (2.9%) E 6 dispersion energy (kcal/mol) (45.2%) -3.29(45.1%) E 7 dispersion energy (kcal/mol) (0.3%) N/ E 8 dispersion energy (kcal/mol) (15%) (15%) Charge transfer energy (kcal/mol) 0 (0%) 0 (0%) Total EFP energy Sum of the absolute values of each energy contribution to the total EFP energy (Eq.(35)) 11