Intermolecular force fields and how they can be determined

Transcription

1 Intermolecular force felds and how they can be determned Ad van der Avord Unversty of Njmegen Han-sur-Lesse, December 23 p.1

2 Equaton of state (Van der Waals) of non-deal gas ( p + a )( ) V 2 V b = kt repulson attracton b (egenvolume) a (reduced pressure) Han-sur-Lesse, December 23 p.2

3 Vral expanson (densty ρ = 1/V ) p = kt [ ρ + B 2 (T )ρ 2 + B 3 (T )ρ ] wth B 2 (T ) = 1 2 [ ( exp E(R) ) kt ] 1 4πR 2 dr Han-sur-Lesse, December 23 p.3

4 Intermolecular forces Han-sur-Lesse, December 23 p.4

5 Intermolecular forces 199 / 1912 Renganum, Debye: dpole-dpole, attractve when orentatons are averaged over thermal moton Han-sur-Lesse, December 23 p.4

6 Intermolecular forces 199 / 1912 Renganum, Debye: dpole-dpole, attractve when orentatons are averaged over thermal moton 192 / 1921 Debye, Keesom: dpole (quadrupole) - nduced dpole (attractve) Han-sur-Lesse, December 23 p.4

7 Intermolecular forces 199 / 1912 Renganum, Debye: dpole-dpole, attractve when orentatons are averaged over thermal moton 192 / 1921 Debye, Keesom: dpole (quadrupole) - nduced dpole (attractve) 1927 Hetler & London: Quantum mechancs (QM) covalent bondng for snglet H 2 (S = ) exchange repulson for trplet H 2 (S = 1) Han-sur-Lesse, December 23 p.4

8 Intermolecular forces 199 / 1912 Renganum, Debye: dpole-dpole, attractve when orentatons are averaged over thermal moton 192 / 1921 Debye, Keesom: dpole (quadrupole) - nduced dpole (attractve) 1927 Hetler & London: Quantum mechancs (QM) covalent bondng for snglet H 2 (S = ) exchange repulson for trplet H 2 (S = 1) 1927 / 193 Wang, London: QM dsperson forces (attractve) Han-sur-Lesse, December 23 p.4

9 QM dervaton of ntermolecular forces correspondence wth classcal electrostatcs Han-sur-Lesse, December 23 p.5

10 QM dervaton of ntermolecular forces correspondence wth classcal electrostatcs Intermezzo: (Tme-ndependent) perturbaton theory Han-sur-Lesse, December 23 p.5

11 Schrödnger equaton HΦ = EΦ not exactly solvable. Perturbaton theory Approxmate solutons E k and Φ k Han-sur-Lesse, December 23 p.6

12 Schrödnger equaton HΦ = EΦ not exactly solvable. Perturbaton theory Approxmate solutons E k and Φ k Fnd smpler Hamltonan H () for whch H () Φ () = E () Φ () s solvable, wth solutons E () k and Φ () k Perturbaton H (1) = H H () Wrte H(λ) = H () + λh (1) (swtch parameter λ) H () E () k Φ () k λ 1 H(λ) E k (λ) Φ k (λ) H E k Φ k Han-sur-Lesse, December 23 p.6

13 Expand E k (λ) = E () k + λe (1) k + λ 2 E (2) k +... Φ k (λ) = Φ () k + λφ (1) k + λ 2 Φ (2) k +... Han-sur-Lesse, December 23 p.7

14 Expand E k (λ) = E () k + λe (1) k + λ 2 E (2) k +... Φ k (λ) = Φ () k + λφ (1) k + λ 2 Φ (2) k +... Substtuton nto H(λ)Φ k (λ) = E k (λ)φ k (λ) and equatng each power of λ yelds, after some manpulatons E (2) k = k E (1) k = Φ () k H (1) Φ () k Φ () k H (1) Φ () Φ () H (1) Φ () k E () k E () Used to calculate perturbaton correctons of E () k Han-sur-Lesse, December 23 p.7

15 Frst perturbaton correcton of Φ () k Φ (1) k = k Φ () H (1) Φ () k E () k E () Φ () Han-sur-Lesse, December 23 p.8

16 Frst perturbaton correcton of Φ () k Φ (1) k = k Φ () H (1) Φ () k E () k E () Φ () The second order energy may also be wrtten as E (2) k = Φ () k H (1) Φ (1) k Han-sur-Lesse, December 23 p.8

17 Molecule n electrc feld External potental V (r) = V (x, y, z) Han-sur-Lesse, December 23 p.9

18 Molecule n electrc feld External potental V (r) = V (x, y, z) Partcles wth charge q (nucle q = Z e, electrons q = e) Han-sur-Lesse, December 23 p.9

19 Molecule n electrc feld External potental V (r) = V (x, y, z) Partcles wth charge q (nucle q = Z e, electrons q = e) Hamltonan H = H () + H (1) wth free molecule Hamltonan H () and perturbaton H (1) = n q V (r ) = n q V (x, y, z ) =1 =1 Han-sur-Lesse, December 23 p.9

20 Multpole (Taylor) expanson ( ) V V (x, y, z) = V + x x + y ( V y ) + z ( V z ) +... wth electrc feld F = (F x, F y, F z ) = grad V V (r) = V (x, y, z) = V r F +... Han-sur-Lesse, December 23 p.1

21 Multpole (Taylor) expanson ( ) V V (x, y, z) = V + x x + y ( V y ) + z ( V z ) +... wth electrc feld F = (F x, F y, F z ) = grad V V (r) = V (x, y, z) = V r F +... Perturbaton operator H (1) = qv µ F +... wth total charge q = n q and dpole operator µ = n q r =1 =1 Han-sur-Lesse, December 23 p.1

22 Frst order perturbaton energy (for ground state k = ) E (1) = Φ () H (1) Φ () = Φ () µ F +... Φ () = Φ () µ Φ () F +... = µ F +... Han-sur-Lesse, December 23 p.11

23 Frst order perturbaton energy (for ground state k = ) E (1) = Φ () H (1) Φ () = Φ () µ F +... Φ () = Φ () µ Φ () F +... = µ F +... Energy of permanent dpole µ n feld F. Same as classcal electrostatcs, wth dpole µ. Han-sur-Lesse, December 23 p.11

24 Second order perturbaton energy for neutral molecule (q = ) and feld n z-drecton.e., F = (,, F ) and H (1) = µ z F E (2) = Φ () H (1) Φ () Φ () H (1) Φ () E () E () = Φ () µ z Φ () Φ () µ z Φ () E () E () F 2 Han-sur-Lesse, December 23 p.12

25 Second order perturbaton energy for neutral molecule (q = ) and feld n z-drecton.e., F = (,, F ) and H (1) = µ z F E (2) = Φ () H (1) Φ () Φ () H (1) Φ () E () E () = Φ () µ z Φ () Φ () µ z Φ () E () E () F 2 Same as classcal electrostatcs: E pol = 1 2 αf 2, wth polarzablty Han-sur-Lesse, December 23 p.12

26 α zz = 2 Φ () µ z Φ () Φ () µ z Φ () E () E () Han-sur-Lesse, December 23 p.13

27 α zz = 2 Φ () µ z Φ () Φ () µ z Φ () E () E () The polarzablty α zz can also be obtaned from the nduced dpole moment. The total dpole moment s Φ () + Φ (1) µ z Φ () + Φ (1) = Φ () µ z Φ () + 2 Φ () µ z Φ (1) + Φ (1) µ z Φ (1) Han-sur-Lesse, December 23 p.13

28 α zz = 2 Φ () µ z Φ () Φ () µ z Φ () E () E () The polarzablty α zz can also be obtaned from the nduced dpole moment. The total dpole moment s Φ () + Φ (1) µ z Φ () + Φ (1) = Φ () µ z Φ () + 2 Φ () µ z Φ (1) + Φ (1) µ z Φ (1) The (frst order) nduced dpole moment µ nd s the second term. Wth the frst order wave functon Φ (1) = Φ () H (1) Φ () E () E () Φ () Han-sur-Lesse, December 23 p.13

29 and H (1) = µ z F ths yelds µ nd = 2 Φ () µ z Φ (1) = 2 Φ () µ z Φ () Φ () µ z Φ () E () E () F Han-sur-Lesse, December 23 p.14

30 and H (1) = µ z F ths yelds µ nd = 2 Φ () µ z Φ (1) = 2 Φ () µ z Φ () Φ () µ z Φ () E () E () F As n classcal electrostatcs: µ nd = αf, wth the same formula for the polarzablty α zz as above. Han-sur-Lesse, December 23 p.14

31 and H (1) = µ z F ths yelds µ nd = 2 Φ () µ z Φ (1) = 2 Φ () µ z Φ () Φ () µ z Φ () E () E () F As n classcal electrostatcs: µ nd = αf, wth the same formula for the polarzablty α zz as above. For arbtrary molecules the drecton of the nduced dpole moment µ nd s not parallel to F. The polarzablty α s a second rank tensor wth non-zero elements α xy, etc. Han-sur-Lesse, December 23 p.14

32 and H (1) = µ z F ths yelds µ nd = 2 Φ () µ z Φ (1) = 2 Φ () µ z Φ () Φ () µ z Φ () E () E () F As n classcal electrostatcs: µ nd = αf, wth the same formula for the polarzablty α zz as above. For arbtrary molecules the drecton of the nduced dpole moment µ nd s not parallel to F. The polarzablty α s a second rank tensor wth non-zero elements α xy, etc. For sotropc systems (atoms, freely rotatng molecules) α s dagonal and α xx = α yy = α zz = α. Han-sur-Lesse, December 23 p.14

33 Long range nteractons between two molecules Molecules A and B at dstance R wth no overlap of ther wave functons. Partcles A and j B. Han-sur-Lesse, December 23 p.15

34 Long range nteractons between two molecules Molecules A and B at dstance R wth no overlap of ther wave functons. Partcles A and j B. Hamltonan H = H () + H (1) wth free molecule Hamltonan H () = H A + H B and nteracton operator H (1) = A j B q q j r j Han-sur-Lesse, December 23 p.15

35 Long range nteractons between two molecules Molecules A and B at dstance R wth no overlap of ther wave functons. Partcles A and j B. Hamltonan H = H () + H (1) wth free molecule Hamltonan H () = H A + H B and nteracton operator Same as H (1) = H (1) = A j B q q j r j n q V (r ) n prevous secton wth =1 molecule A n electrc potental of molecule B. V (r ) = j B q j r j Han-sur-Lesse, December 23 p.15

36 Multpole expanson of the nteracton operator r = (x, y, z ), r j = (x j, y j, z j ), R = (,, R) r j = r j r + R and r j = r j Han-sur-Lesse, December 23 p.16

37 A double Taylor expanson n (x, y, z ) and (x j, y j, z j ) of 1 r j = [(x j x ) 2 + (y j y ) 2 + (z j z + R) 2] 1/2 at (x, y, z ) = (,, ) and (x j, y j, z j ) = (,, ) yelds 1 r j = 1 R + z R 2 z j R 2 + x x j + y y j 2z z j R Ths expanson converges when r + r j < R. Han-sur-Lesse, December 23 p.17

38 Substtuton nto H (1) gves, after some rearrangement H (1) = qa q B R + µa z q B R 2 qa µ B z R 2 + µa x µ B x + µ A y µ B y 2µ A z µ B z R 3 wth the total charges q A = A q q B = j Bq j and the dpole operators µ A = A q r µ B = j Bq j r j Han-sur-Lesse, December 23 p.18

39 Substtuton nto H (1) gves, after some rearrangement H (1) = qa q B R + µa z q B R 2 qa µ B z R 2 + µa x µ B x + µ A y µ B y 2µ A z µ B z R 3 wth the total charges q A = A q q B = j Bq j and the dpole operators µ A = A q r µ B = j Bq j r j Ths operator H (1) ncludes the electrostatc nteractons between the charges and dpole moments of the molecules A and B. Hgher (quadrupole) nteractons are neglected. Han-sur-Lesse, December 23 p.18

40 Alternatve forms of the dpole-dpole nteracton operator are µ A x µ B x + µ A y µ B y 2µ A z µ B z R 3 wth the nteracton tensor T xx T xy T xz T = T yz T yy T yz T zx T zy T zz = µa µ B 3µ A z µ B z R 3 = Ths tensor can also be expressed n more general coordnates. = µa T µ B R 3 Han-sur-Lesse, December 23 p.19

41 The solutons of the Schrödnger equatons of the free molecules A and B are H A Φ A k 1 = E A k 1 Φ A k 1 H B Φ A k 2 = E B k 2 Φ B k 2 and of the unperturbed problem H () Φ () K = E() K Φ() K wth Φ () K = ΦA k 1 Φ B k 2 and egenvalues E () K = EA k 1 + E B k 2 Han-sur-Lesse, December 23 p.2

42 Proof H () Φ () K (H = A + H B) Φ A k 1 Φ A k 2 ) ) = (H A Φ A k 1 Φ B k 2 + Φ A k 1 (H B Φ B k 2 ) = (Ek A 1 + Ek B 2 Φ A k 1 Φ B k 2 Han-sur-Lesse, December 23 p.21

43 Proof H () Φ () K (H = A + H B) Φ A k 1 Φ A k 2 ) ) = (H A Φ A k 1 Φ B k 2 + Φ A k 1 (H B Φ B k 2 ) = (Ek A 1 + Ek B 2 Φ A k 1 Φ B k 2 Perturbaton operator (repeated) H (1) = qa q B R + µa z q B R 2 qa µ B z R 2 + µa T µ B R 3 Each term factorzes n A and B operators! Han-sur-Lesse, December 23 p.21

44 The frst order energy s E (1) = Φ () H (1) Φ () = Φ A Φ B H (1) Φ A Φ B Han-sur-Lesse, December 23 p.22

45 The frst order energy s E (1) = Φ () H (1) Φ () = Φ A Φ B H (1) Φ A Φ B Wth the multpole expanson of H (1) one can separate ntegraton over the coordnates (x, y, z ) and (x j, y j, z j ) of the partcles A and j B and obtan E (1) = qa q B R + µa z q B R 2 qa µ B z R 2 + µa T µ B R 3 Han-sur-Lesse, December 23 p.22

46 The frst order energy s E (1) = Φ () H (1) Φ () = Φ A Φ B H (1) Φ A Φ B Wth the multpole expanson of H (1) one can separate ntegraton over the coordnates (x, y, z ) and (x j, y j, z j ) of the partcles A and j B and obtan E (1) = qa q B R + µa z q B R 2 qa µ B z R 2 + µa T µ B R 3 the same as n classcal electrostatcs, wth the permanent multpole moments µ A = Φ A µa Φ A and µ B = Φ B µb Φ B Han-sur-Lesse, December 23 p.22

47 The second order energy s E (2) = K Φ () H (1) Φ () K Φ() K H(1) Φ () E () E () K The ndex K that labels the excted states of the system s a composte ndex K = (k 1, k 2 ). The summaton over K can be splt nto three sums, wth k 1, k 2 = k 1 =, k 2 k 1, k 2 Molecule A excted Molecule B excted Both molecules excted Han-sur-Lesse, December 23 p.23

48 The frst term of E (2) s k 1 Φ A ΦB H(1) Φ A k 1 Φ B ΦA k 1 Φ B H(1) Φ A ΦB E A EA k 1 Han-sur-Lesse, December 23 p.24

49 The frst term of E (2) s k 1 Φ A ΦB H(1) Φ A k 1 Φ B ΦA k 1 Φ B H(1) Φ A ΦB E A EA k 1 The operator H(1) s term-by-term factorzable and the ntegrals n ths expresson can be separated. For example Φ A Φ B µa z µ B z R 3 Φ A k 1 Φ B = ΦA µa z ΦA k 1 Φ B µb z ΦB R 3 = ΦA µa z Φ A k 1 µ B z R 3 Furthermore, one may use the orthogonalty relaton Φ A ΦA k 1 =. Han-sur-Lesse, December 23 p.24

50 The transton dpole moments Φ A µa z Φ A k 1, wth the summaton over k 1, occur n the formula for the polarzablty α A zz. Han-sur-Lesse, December 23 p.25

51 The transton dpole moments Φ A µa z Φ A k 1, wth the summaton over k 1, occur n the formula for the polarzablty α A zz. If one assumes that the polarzablty s sotropc, α A xx = α A yy = α A zz = α A, one fnds for the frst term E (2) (pol. A) = (q B ) 2 αa 2R 4 + 2αA q B µ B z R 5 αa ( µ B x 2 + µ B y µ B z 2 ) 2R 6 Han-sur-Lesse, December 23 p.25

52 Also ths results agree wth classcal electrostatcs. The electrc feld of the pont charge q B at the center of molecule A s ) F = (F x, F y, F z ) = (,, qb and the electrc feld of the permanent dpole moment µ B s ( F = µb x R 3, ) µb y R 3, 2 µb z R 3 The second order nteracton energy E (2) (pol. A) s smply the polarzaton energy 1 2 αa F 2 of molecule A n the electrc feld of the charge and dpole of molecule B. R 2 Han-sur-Lesse, December 23 p.26

53 Analogously, we fnd for the second term, whch ncludes a summaton over the excted states k 2 of molecule B E (2) (pol. B) = (qa ) 2 α B 2R 4 2qA µ A z α B R 5 ( µa x 2 + µ A y µ A z 2 )α B 2R 6 Ths s the classcal energy of polarzaton of molecule B n the feld of A. Han-sur-Lesse, December 23 p.27

54 The thrd term contans the summaton over the excted states of both molecules. All nteracton terms wth the charges q A and q B cancel, because of the orthogonalty relaton Φ A ΦA k 1 =. Only the dpole-dpole term of H (1) s left and we obtan E (2) (dsp) = k 1 k 2 = R 6 k 1 Φ A ΦB H(1) Φ A k 1 Φ B k 2 Φ A k 1 Φ B k 2 H (1) Φ A ΦB k 2 (E A EA k 1 ) + (E B EB k 2 ) Φ A µ A Φ A k 1 T Φ B µb Φ B k 2 2 (E A k 1 E A ) + (EB k 2 E B ) Han-sur-Lesse, December 23 p.28

55 Ths term, the dsperson energy, has no classcal equvalent; t s purely quantum mechancal. It s proportonal to R 6. Han-sur-Lesse, December 23 p.29

56 Ths term, the dsperson energy, has no classcal equvalent; t s purely quantum mechancal. It s proportonal to R 6. It can be easly proved that each of the three second order terms s negatve. Therefore, the nducton and dsperson energes are always attractve. Han-sur-Lesse, December 23 p.29

57 Ths term, the dsperson energy, has no classcal equvalent; t s purely quantum mechancal. It s proportonal to R 6. It can be easly proved that each of the three second order terms s negatve. Therefore, the nducton and dsperson energes are always attractve. For neutral, non-polar molecules the charges q A, q B and permanent dpole moments µ A, µ B are zero, and the dsperson energy s the only second order nteracton. Han-sur-Lesse, December 23 p.29

58 Terms wth hgher powers of R 1 occur as well. They orgnate from the quadrupole and hgher multpole moments that we neglected. Han-sur-Lesse, December 23 p.3

59 Terms wth hgher powers of R 1 occur as well. They orgnate from the quadrupole and hgher multpole moments that we neglected. An approxmate formula, due to London, that s often used to estmate the dsperson energy s E (2) (dsp) 3αA α B 2R 6 I A I B I A + I B Ths formula s found f one assumes that all the exctaton energes Ek A 1 E A and EB k 2 E B are the same, and are equal to the onzaton energes I A and I B. Han-sur-Lesse, December 23 p.3

60 Summary of long range nteractons The nteractons between two molecules A and B can be derved by means of QM perturbaton theory. Han-sur-Lesse, December 23 p.31

61 Summary of long range nteractons The nteractons between two molecules A and B can be derved by means of QM perturbaton theory. The frst order energy equals the classcal electrostatc (Coulomb) nteracton energy between the charges and dpole moments of the molecules. It may be attractve or repulsve, dependng on the (postve or negatve) charges and on the orentatons of the dpole moments. The dpolar terms average out when the dpoles are freely rotatng. Han-sur-Lesse, December 23 p.31

62 The second order energy conssts of three contrbutons. The frst two terms correspond to the classcal polarzaton energes of the molecules n each other s electrc felds. The thrd term s purely QM. All the three contrbutons are attractve. They start wth R 4 terms when the molecules have charges and wth R 6 terms when they are neutral. The dsperson energy, wth the leadng term proportonal to R 6, occurs also for neutral molecules wth no permanent dpole moments. Han-sur-Lesse, December 23 p.32

63 The second order energy conssts of three contrbutons. The frst two terms correspond to the classcal polarzaton energes of the molecules n each other s electrc felds. The thrd term s purely QM. All the three contrbutons are attractve. They start wth R 4 terms when the molecules have charges and wth R 6 terms when they are neutral. The dsperson energy, wth the leadng term proportonal to R 6, occurs also for neutral molecules wth no permanent dpole moments. All of these terms can be calculated when the wave functons Φ A k 1, Φ B k 2 and energes E A k 1, E B k 2 of the free molecules A and B are known, but one should somehow approxmate the nfnte summatons over excted states k 1 and k 2 that occur n the second order expressons. Han-sur-Lesse, December 23 p.32

64 Interactons n the overlap regon Hetler and London (Valence Bond) wave functons for H 2 1s A (r 1 )1s B (r 2 ) ± 1s B (r 1 )1s A (r 2 ) wth the plus sgn for the snglet spn (S = ) functon α(1)β(2) β(1)α(2) and the mnus sgn for the trplet spn (S = 1) functons α(1)α(2) α(1)β(2) + β(1)α(2) β(1)β(2) The total electronc wave functon s antsymmetrc (Paul) Han-sur-Lesse, December 23 p.33

65 Interacton energy E(R) = E H2 2E H Trplet (S = 1) Q(R) = Coulomb ntegral E Q J 1 S 2 J(R) = exchange ntegral Snglet (S = ) Q + J 1 + S 2 S(R) = 1s A 1s B = overlap ntegral R Han-sur-Lesse, December 23 p.34

66 Interacton s domnated by the exchange ntegral J(R), whch s negatve, so that the exchange nteracton s attractve (covalent bondng) n the snglet state and repulsve n the trplet state. Han-sur-Lesse, December 23 p.35

67 Interacton s domnated by the exchange ntegral J(R), whch s negatve, so that the exchange nteracton s attractve (covalent bondng) n the snglet state and repulsve n the trplet state. For He 2 there s only one (snglet) state and the nteracton energy E(R) s purely repulsve: exchange (or Paul) repulson or sterc hndrance. Han-sur-Lesse, December 23 p.35

68 Molecular orbtal pcture Covalent bondng Exchange repulson H H nteracton Han-sur-Lesse, December 23 p.36

69 Molecular orbtal pcture Covalent bondng Exchange repulson Exchange repulson H H nteracton He He nteracton Han-sur-Lesse, December 23 p.36

70 Most stable molecules are closed-shell systems and the exchange energy between them s always repulsve. It depends on the overlap between the wave functons of A and B and decays exponentally wth the dstance R. Han-sur-Lesse, December 23 p.37

71 Most stable molecules are closed-shell systems and the exchange energy between them s always repulsve. It depends on the overlap between the wave functons of A and B and decays exponentally wth the dstance R. In combnaton wth attractve long range nteractons (proportonal to R n ) ths gves rse to a mnmum n E(R). Ths, so-called, non-covalent bondng s much weaker than covalent bondng, except when A and B are (atomc or molecular) ons wth opposte charges (cf. Na + Cl ). Han-sur-Lesse, December 23 p.37

72 Most stable molecules are closed-shell systems and the exchange energy between them s always repulsve. It depends on the overlap between the wave functons of A and B and decays exponentally wth the dstance R. In combnaton wth attractve long range nteractons (proportonal to R n ) ths gves rse to a mnmum n E(R). Ths, so-called, non-covalent bondng s much weaker than covalent bondng, except when A and B are (atomc or molecular) ons wth opposte charges (cf. Na + Cl ). Bndng (merely by the attractve dsperson energy) s weakest when both molecules are neutral and non-polar: pure Van der Waals nteractons. Han-sur-Lesse, December 23 p.37

73 A specal type of nteractons between polar molecules s hydrogen bondng X H Y The bndng manly orgnates from electrostatc (dpolar and quadrupolar) nteractons and the correspondng nducton terms and s strongly drectonal. No specal (HOMO-LUMO, charge-transfer, or weak covalent bondng) nteractons are needed! Han-sur-Lesse, December 23 p.38

74 Exercse: Compute the equlbrum angles of HF HF at R = 2.75 Å and H 2 O H 2 O at R = 2.95 Å from the dpolar and quadrupolar nteractons only. Han-sur-Lesse, December 23 p.39

75 Non-covalent nteractons and hydrogen bondng, n partcular, are very mportant n bology. Alpha helces and beta sheets n protens are stablzed by ntra- and nter-molecular hydrogen bonds, and the double stranded structure of DNA s held together by hydrogen bonds between the base pars. Han-sur-Lesse, December 23 p.4

76 Non-covalent nteractons and hydrogen bondng, n partcular, are very mportant n bology. Alpha helces and beta sheets n protens are stablzed by ntra- and nter-molecular hydrogen bonds, and the double stranded structure of DNA s held together by hydrogen bonds between the base pars. It s essental that a herarchy of nteractons exsts wth bndng energes varyng over several orders of magntude. Interactons n bologcal systems must be suffcently strong to mantan stable structures, but not so strong that they prevent rearrangement processes (DNA replcaton, for nstance). Han-sur-Lesse, December 23 p.4

77 Intermolecular potentals (or force felds) Concept based on Born-Oppenhemer approxmaton (separaton of electronc and nuclear moton) Step 1: Solve electronc Schrödnger equaton H el (r el ; R)Ψ(r el ; R) = E(R)Ψ(r el ; R) for clamped nucle at postons R. Yelds energy E(R). Step 2: Use E(R) as potental energy n solvng Schrödnger equaton for nuclear moton. Yelds bound levels of Van der Waals complexes and scatterng states (cross sectons). Han-sur-Lesse, December 23 p.41

78 Intermolecular potental of a many-body system V = A<B V AB + par V ABC +... A<B<C three-body Han-sur-Lesse, December 23 p.42

79 Intermolecular potental of a many-body system V = A<B V AB + par V ABC +... A<B<C three-body Par potental, n space-fxed (SF) coordnates V AB = V (R AB, Ω A, Ω B, q A, q B ) } Euler angles Ω X = (α X, β X, γ X ) for X = A, B nternal coordnates q X Han-sur-Lesse, December 23 p.42

80 Three angles, the two polar angles of R AB and one of the Euler angles α X (say α A ), can be chosen as overall rotaton angles of the complex A B. The par potental n body-fxed (BF) coordnates s V AB = V (R AB, α B α A, β A, β B, γ A, γ B, q A, q B ) The nternal coordnates q A, q B are often frozen (rgd molecules). Ths s justfed by a Born-Oppenhemer-lke separaton between the fast ntramolecular vbratons (coordnates q A, q B ) and the much slower VRT motons (vbratons, hndered rotatons, tunnelng) of the whole molecules A and B n the complex. Han-sur-Lesse, December 23 p.43

81 Ab nto calculaton of ntermolecular potentals Supermolecule calculatons Symmetry-adapted perturbaton theory (SAPT) Han-sur-Lesse, December 23 p.44

82 Supermolecule calculatons Requrements: E = E AB E A E B 1. Include electron correlaton, ntra- and nter-molecular (dsperson energy = ntermolecular correlaton) 2. Choose good bass, wth dffuse orbtals (and bond functons ) especally to converge the dsperson energy 3. Sze consstency. Currently best method: CCSD(T) 4. Correct for bass set superposton error (BSSE) by computng E A and E B n dmer bass Han-sur-Lesse, December 23 p.45

83 Symmetry-adapted perturbaton theory (SAPT) Combne perturbaton theory wth antsymmetrzaton A (Paul) to nclude short-range exchange effects. Han-sur-Lesse, December 23 p.46

84 Symmetry-adapted perturbaton theory (SAPT) Combne perturbaton theory wth antsymmetrzaton A (Paul) to nclude short-range exchange effects. Advantages: 1. E calculated drectly. 2. Contrbutons (electrostatc, nducton, dsperson, exchange) computed ndvdually. Useful n analytc fts of potental surface. Han-sur-Lesse, December 23 p.46

85 Symmetry-adapted perturbaton theory (SAPT) Combne perturbaton theory wth antsymmetrzaton A (Paul) to nclude short-range exchange effects. Advantages: 1. E calculated drectly. 2. Contrbutons (electrostatc, nducton, dsperson, exchange) computed ndvdually. Useful n analytc fts of potental surface. Advantage of supermolecule method: Easy, use any black-box molecular electronc structure program Han-sur-Lesse, December 23 p.46

86 Problems n SAPT: 1. Paul: AH = HA. Antsymmetrzer commutes wth total Hamltonan H = H () + H (1), but not wth H () and H (1) separately. Has led to dfferent defntons of second (and hgher) order energes. 2. Free monomer wavefunctons Φ A k 1 and Φ B k 2 not exactly known. Use Hartree-Fock wave functons and apply double perturbaton theory to nclude ntra-molecular correlaton, or use CCSD wave functons of monomers. Han-sur-Lesse, December 23 p.47

87 Problems n SAPT: 1. Paul: AH = HA. Antsymmetrzer commutes wth total Hamltonan H = H () + H (1), but not wth H () and H (1) separately. Has led to dfferent defntons of second (and hgher) order energes. 2. Free monomer wavefunctons Φ A k 1 and Φ B k 2 not exactly known. Use Hartree-Fock wave functons and apply double perturbaton theory to nclude ntra-molecular correlaton, or use CCSD wave functons of monomers. Program packages: - SAPT2 for par potentals - SAPT3 for 3-body nteractons Han-sur-Lesse, December 23 p.47