Noncovalent interactions between molecules. spectra of weakly bound molecular clusters

Transcription

1 Noncovalent interactions between molecules and spectra of weakly bound molecular clusters Ad van der Avoird Theoretical Chemistry

2 Outline: Preamble: Born-Oppenheimer approximation What are non-covalent interactions? Quantum mechanical derivation How to compute intermolecular force fields ab initio? How to test intermolecular force fields? Van der Waals molecules, spectra Illustration: ab initio force field for water, applications Molecular collisions, scattering resonances

3 Useful concepts: Molecular force fields (MM calculations) Interatomic / intermolecular forces (potential energy surfaces) Forces on atoms in solids Equilibrium structure, force constants of a molecule Chemical reaction paths (from QM calculations) Exist only in the Born-Oppenheimer approximation

4 Born-Oppenheimer (adiabatic) approximation Step 1: Solve electronic Schrödinger equation H e φ(r;r) = E e (R) φ(r;r) for nuclei fixed at positions R. Involves neglect of nuclear kinetic energy T n. Step 2: Solve nuclear Schrödinger equation [T n +E e (R)]χ(R) = Eχ(R) with potential energy surface E e (R) vibrations, rotations, (phonons, librations), chemical reaction dynamics, molecular collisions.

5 Alternative for step 2 Molecular dynamics (MD): solve nuclear motions classically on potential surface E e (R).

6 Derivation of the Born-Oppenheimer approximation Exact (non-relativistic) Hamiltonian H = T n +T e +V(r,R) with T e = i 2 2m 2 i and T n = A 2 2M A 2 A V(r,R) = A>B Z A Z B e 2 R A R B i,a Z A e 2 r i R A + i>j e 2 r i r j Electronic Hamiltonian (with the nuclei fixed at positions R) H e = T e +V(r,R)

7 The total Schrödinger equation reads Expand the total wave function H Ψ(r,R) = EΨ(r,R) Ψ(r,R) = k φ k (r;r)χ k (R) in solutions φ k (r;r) of the electronic Schrödinger equation H e φ k (r;r) = E k (R)φ k (r;r) and substitute it into the total Schrödinger equation. Multiply by the function φ k (r;r) from the left and integrate over the electronic coordinates r. The electronic Hamiltonian H e is diagonal φ k (r;r) H e φ k (r;r) (r) = δ k k E k(r) and the electronic wave functions are orthogonal φ k (r;r) φ k (r;r) (r) = δ k k

8 This yields a set of coupled eigenvalue equations for the nuclear wave functions [T n +E k (R) E] χ k (R) = k [F n ] k k χ k(r) Coupling between different electronic states k,k [F n ] k k (R) = φ k (r;r) T n φ k (r;r) (r) T n δ k k occurs through the nuclear kinetic energy operator T n. When this coupling is neglected one obtains the Born-Oppenheimer nuclear Schrödinger equation for electronic state k [T n +E k (R)]χ k (R) = Eχ k (R)

9 The (non-adiabatic) coupling terms are [F n ] k k (R) = A 2 2M A [ 2 φ k ( ) A φ k (r) A + φ k ( 2 A φ ) ] k (r) They are small because of the large nuclear masses M A in the denominator. In the first term one may write φ k ( ) A φ k (r) = φ ] k [ A,H e φk (r), E k (R) E k (R) which shows that the coupling is small only when the electronic energies E k (R) and E k (R) are well separated. This is normally the case, and the Born-Oppenheimer approximation holds. For certain geometries R the energies E k (R) and E k (R) may be equal: two (or more) electronic states are degenerate. The Born- Oppenheimer approximation breaks down.

10 Breakdown of the Born-Oppenheimer approximation Examples: Open-shell systems: radicals, molecules in excited states Degeneracies at symmetric structures Jahn-Teller, Renner-Teller distortions (Conical) intersections of different potential surfaces important in photochemistry Metals

11 Jahn-Teller effect in Benzene + Degenerate ground state E 1g Sixfold symmetry distorted by Jahn-Teller coupling with normal modes of e 2g symmetry π molecular orbitals

12 Potential surfaces Two adiabatic potentials corresponding to the E 1g state as functions of the two ν 6 e 2g normal mode coordinates Red circle shows the vibrational zero-point level dynamic Jahn-Teller effect

13 Conical intersection between excited S 1 and ground state S S 1 potentials of ethene C 2 H 4 S torsion angle wagging angle Fast non-radiative transition to ground state through non-adiabatic coupling prevents UV radiation damage in DNA

14 Metals Aluminium E F Energy (ev) States / ev unit cell

15 For metals Interatomic potential Energy Distance R electron-phonon (non-adiabatic) coupling

16 How were intermolecular forces discovered? Han-sur-Lesse, December 212 p. 1

17 Equation of state of a gas Ideal gas pv = kt Non-ideal gas (Van der Waals, 1873) ( p+ a )( ) V 2 V b = kt repulsion b (eigenvolume) attraction a (reduced pressure) Han-sur-Lesse, December 212 p. 2

18 Virial expansion (density ρ = 1/V ) p = kt [ ρ+b 2 (T)ρ 2 +B 3 (T)ρ ] with B 2 (T) = 1 2 [ exp ( E(R) ) kt 1 ] 4πR 2 dr E(R) is the interaction energy between two atoms/molecules as function of their distance R Han-sur-Lesse, December 212 p. 3

19 Intermolecular forces Han-sur-Lesse, December 212 p. 4

20 Intermolecular forces 199 / 1912 Reinganum, Debye: dipole-dipole, attractive when orientations are averaged over thermal motion Han-sur-Lesse, December 212 p. 4

21 Intermolecular forces 199 / 1912 Reinganum, Debye: dipole-dipole, attractive when orientations are averaged over thermal motion 192 / 1921 Debye, Keesom: dipole (quadrupole) - induced dipole (attractive) Han-sur-Lesse, December 212 p. 4

22 Intermolecular forces 199 / 1912 Reinganum, Debye: dipole-dipole, attractive when orientations are averaged over thermal motion 192 / 1921 Debye, Keesom: dipole (quadrupole) - induced dipole (attractive) 1927 Heitler & London: Quantum mechanics (QM) covalent bonding for singlet H 2 (S = ) exchange repulsion for triplet H 2 (S = 1) Han-sur-Lesse, December 212 p. 4

23 Intermolecular forces 199 / 1912 Reinganum, Debye: dipole-dipole, attractive when orientations are averaged over thermal motion 192 / 1921 Debye, Keesom: dipole (quadrupole) - induced dipole (attractive) 1927 Heitler & London: Quantum mechanics (QM) covalent bonding for singlet H 2 (S = ) exchange repulsion for triplet H 2 (S = 1) 1927 / 193 Wang, London: QM dispersion forces (attractive) Han-sur-Lesse, December 212 p. 4

24 QM derivation of intermolecular forces correspondence with classical electrostatics Han-sur-Lesse, December 212 p. 5

25 QM derivation of intermolecular forces correspondence with classical electrostatics Intermezzo: (Time-independent) perturbation theory Han-sur-Lesse, December 212 p. 5

26 Schrödinger equation HΦ = EΦ not exactly solvable. Perturbation theory Approximate solutions E k and Φ k Han-sur-Lesse, December 212 p. 6

27 Schrödinger equation HΦ = EΦ not exactly solvable. Perturbation theory Approximate solutions E k and Φ k Find simpler Hamiltonian H () for which H () Φ () = E () Φ () is solvable, with solutions E () k and Φ () k Perturbation H (1) = H H () Write H(λ) = H () +λh (1) (switch parameter λ) H () E () k Φ () k λ 1 H(λ) E k (λ) H E k Φ k (λ) Φ k Han-sur-Lesse, December 212 p. 6

28 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 212 p. 7

29 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 +... Substitution into H(λ)Φ k (λ) = E k (λ)φ k (λ) and equating each power of λ yields, after some manipulations E (2) k = i k E (1) k = Φ () k H (1) Φ () k Φ () k H (1) Φ () i Φ () i H (1) Φ () k E () k E () i Used to calculate perturbation corrections of E () k Han-sur-Lesse, December 212 p. 7

30 First perturbation correction of Φ () k Φ (1) k = i k Φ () i H (1) Φ () E () k E () i k Φ () i Han-sur-Lesse, December 212 p. 8

31 First perturbation correction of Φ () k Φ (1) k = i k Φ () i H (1) Φ () E () k E () i k Φ () i The second order energy may also be written as E (2) k = Φ () k H (1) Φ (1) k Han-sur-Lesse, December 212 p. 8

32 Molecule in electric field External potential V(r) = V(x,y,z) Han-sur-Lesse, December 212 p. 9

33 Molecule in electric field External potential V(r) = V(x,y,z) Particles i with charge q i (nuclei q i = Z i e, electrons q i = e) Han-sur-Lesse, December 212 p. 9

34 Molecule in electric field External potential V(r) = V(x,y,z) Particles i with charge q i (nuclei q i = Z i e, electrons q i = e) Hamiltonian H = H () +H (1) with free molecule Hamiltonian H () and perturbation H (1) = n q i V(r i ) = n q i V(x i,y i,z i ) i=1 i=1 Han-sur-Lesse, December 212 p. 9

35 Multipole (Taylor) expansion ( ) V V(x,y,z) = V +x x +y ( V y ) +z ( V z ) +... with electric field F = (F x,f y,f z ) = grad V V(r) = V(x,y,z) = V r F +... Han-sur-Lesse, December 212 p. 1

36 Multipole (Taylor) expansion ( ) V V(x,y,z) = V +x x +y ( V y ) +z ( V z ) +... with electric field F = (F x,f y,f z ) = grad V V(r) = V(x,y,z) = V r F +... Perturbation operator H (1) = qv µ F +... with total charge q = n q i and dipole operator µ = n q i r i i=1 i=1 Han-sur-Lesse, December 212 p. 1

37 First order perturbation 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 212 p. 11

38 First order perturbation energy (for ground state k = ) E (1) = Φ () H (1) Φ () = Φ () µ F +... Φ () = Φ () µ Φ () F +... = µ F +... Energy of permanent dipole µ in field F. Same as classical electrostatics, with dipole µ. Han-sur-Lesse, December 212 p. 11

39 Second order perturbation energy for neutral molecule (q = ) and field in z-direction i.e., F = (,,F ) and H (1) = µ z F E (2) = i = i Φ () H (1) Φ () i Φ () i H (1) Φ () E () E () i Φ () µ z Φ () i Φ () i µ z Φ () E () E () i F 2 Han-sur-Lesse, December 212 p. 12

40 Second order perturbation energy for neutral molecule (q = ) and field in z-direction i.e., F = (,,F ) and H (1) = µ z F E (2) = i = i Φ () H (1) Φ () i Φ () i H (1) Φ () E () E () i Φ () µ z Φ () i Φ () i µ z Φ () E () E () i Same as classical electrostatics: E pol = 1 2 αf2, with polarizability F 2 Han-sur-Lesse, December 212 p. 12

41 α zz = 2 i Φ () µ z Φ () i Φ () i µ z Φ () E () i E () Han-sur-Lesse, December 212 p. 13

42 α zz = 2 i Φ () µ z Φ () i Φ () i µ z Φ () E () i E () The polarizability α zz can also be obtained from the induced dipole moment. The total dipole moment is 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 212 p. 13

43 α zz = 2 i Φ () µ z Φ () i Φ () i µ z Φ () E () i E () The polarizability α zz can also be obtained from the induced dipole moment. The total dipole moment is 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 (first order) induced dipole moment µ ind is the second term. With the first order wave function Φ (1) = i Φ () i H (1) Φ () E () E () i Φ () i Han-sur-Lesse, December 212 p. 13

44 and H (1) = µ z F this yields µ ind = 2 Φ () µ z Φ (1) = 2 Φ () µ z Φ () i Φ () i µ z Φ () i E () i E () F Han-sur-Lesse, December 212 p. 14

45 and H (1) = µ z F this yields µ ind = 2 Φ () µ z Φ (1) = 2 Φ () µ z Φ () i Φ () i µ z Φ () i E () i E () F As in classical electrostatics: µ ind = αf, with the same formula for the polarizability α zz as above. Han-sur-Lesse, December 212 p. 14

46 and H (1) = µ z F this yields µ ind = 2 Φ () µ z Φ (1) = 2 Φ () µ z Φ () i Φ () i µ z Φ () i E () i E () F As in classical electrostatics: µ ind = αf, with the same formula for the polarizability α zz as above. For arbitrary molecules the direction of the induced dipole moment µ ind is not parallel to F. The polarizability α is a second rank tensor with non-zero elements α xy, etc. Han-sur-Lesse, December 212 p. 14

47 and H (1) = µ z F this yields µ ind = 2 Φ () µ z Φ (1) = 2 Φ () µ z Φ () i Φ () i µ z Φ () i E () i E () F As in classical electrostatics: µ ind = αf, with the same formula for the polarizability α zz as above. For arbitrary molecules the direction of the induced dipole moment µ ind is not parallel to F. The polarizability α is a second rank tensor with non-zero elements α xy, etc. For isotropic systems (atoms, freely rotating molecules) α is diagonal and α xx = α yy = α zz = α. Han-sur-Lesse, December 212 p. 14

48 Long range interactions between two molecules Molecules A and B at distance R with no overlap of their wave functions. Particles i A and j B. Han-sur-Lesse, December 212 p. 15

49 Long range interactions between two molecules Molecules A and B at distance R with no overlap of their wave functions. Particles i A and j B. Hamiltonian H = H () +H (1) with free molecule Hamiltonian H () = H A +H B and interaction operator H (1) = i A j B q i q j r ij Han-sur-Lesse, December 212 p. 15

50 Long range interactions between two molecules Molecules A and B at distance R with no overlap of their wave functions. Particles i A and j B. Hamiltonian H = H () +H (1) with free molecule Hamiltonian H () = H A +H B and interaction operator Same as H (1) = H (1) = i A j B q i q j r ij n q i V(r i ) in previous section with i=1 molecule A in electric potential of molecule B. V(r i ) = j B q j r ij Han-sur-Lesse, December 212 p. 15

51 Multipole expansion of the interaction operator r i = (x i,y i,z i ), r j = (x j,y j,z j ), R = (,,R) r ij = r j r i +R and r ij = r ij Han-sur-Lesse, December 212 p. 16

52 A double Taylor expansion in (x i,y i,z i ) and (x j,y j,z j ) of 1 r ij = [(x j x i ) 2 +(y j y i ) 2 +(z j z i +R) 2] 1/2 at (x i,y i,z i ) = (,,) and (x j,y j,z j ) = (,,) yields 1 r ij = 1 R + z i R 2 z j R 2 + x ix j +y i y j 2z i z j R This expansion converges when r i + r j < R. Han-sur-Lesse, December 212 p. 17

53 Substitution into H (1) gives, after some rearrangement H (1) = qa q B R + µa z q B R 2 qa µ B z R 2 + µa xµ B x +µ A yµ B y 2µ A z µ B z R 3 with the total charges q A = i A q i q B = j Bq j and the dipole operators µ A = i A q i r i µ B = j Bq j r j Han-sur-Lesse, December 212 p. 18

54 Substitution into H (1) gives, after some rearrangement H (1) = qa q B R + µa z q B R 2 qa µ B z R 2 + µa xµ B x +µ A yµ B y 2µ A z µ B z R 3 with the total charges q A = i A q i q B = j Bq j and the dipole operators µ A = i A q i r i µ B = j Bq j r j This operator H (1) includes the electrostatic interactions between the charges and dipole moments of the molecules A and B. Higher (quadrupole) interactions are neglected. Han-sur-Lesse, December 212 p. 18

55 Alternative forms of the dipole-dipole interaction operator are µ A xµ B x +µ A yµ B y 2µ A z µ B z R 3 with the interaction 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 = This tensor can also be expressed in more general coordinates. = µa T µ B R 3 Han-sur-Lesse, December 212 p. 19

56 The solutions of the Schrödinger equations 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 B k 2 = E B k 2 Φ B k 2 and of the unperturbed problem H () Φ () K = E() K Φ() K with Φ () K = ΦA k 1 Φ B k 2 and eigenvalues E () K = EA k 1 +E B k 2 Han-sur-Lesse, December 212 p. 2

57 Proof H () Φ () K = ( H A +H B) Φ A k 1 Φ B k 2 = ( H A Φ A k 1 ) Φ B k2 +Φ A k 1 ( H B Φ B k 2 ) = ( E A k 1 +E B k 2 ) Φ A k1 Φ B k 2 Han-sur-Lesse, December 212 p. 21

58 Proof H () Φ () K = ( H A +H B) Φ A k 1 Φ B k 2 = ( H A Φ A k 1 ) Φ B k2 +Φ A k 1 ( H B Φ B k 2 ) = ( E A k 1 +E B k 2 ) Φ A k1 Φ B k 2 Perturbation 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 factorizes in A and B operators! Han-sur-Lesse, December 212 p. 21

59 The first order energy is E (1) = Φ () H (1) Φ () = Φ A Φ B H (1) Φ A Φ B Han-sur-Lesse, December 212 p. 22

60 The first order energy is E (1) = Φ () H (1) Φ () = Φ A Φ B H (1) Φ A Φ B With the multipole expansion of H (1) one can separate integration over the coordinates (x i,y i,z i ) and (x j,y j,z j ) of the particles i A and j B and obtain E (1) = qa q B R + µa z qb R 2 qa µ B z R 2 + µa T µ B R 3 Han-sur-Lesse, December 212 p. 22

61 The first order energy is E (1) = Φ () H (1) Φ () = Φ A Φ B H (1) Φ A Φ B With the multipole expansion of H (1) one can separate integration over the coordinates (x i,y i,z i ) and (x j,y j,z j ) of the particles i A and j B and obtain E (1) = qa q B R + µa z qb R 2 qa µ B z R 2 + µa T µ B R 3 the same as in classical electrostatics, with the permanent multipole moments µ A = Φ A µa Φ A and µ B = Φ B µb Φ B Han-sur-Lesse, December 212 p. 22

62 The second order energy is E (2) = K Φ () H (1) Φ () K Φ() K H(1) Φ () E () E () K The index K that labels the excited states of the system is a composite index K = (k 1,k 2 ). The summation over K can be split into three sums, with k 1, k 2 = k 1 =, k 2 k 1, k 2 Molecule A excited Molecule B excited Both molecules excited Han-sur-Lesse, December 212 p. 23

63 The first term of E (2) is 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 212 p. 24

64 The first term of E (2) is 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) is term-by-term factorizable and the integrals in this expression can be separated. For example Φ A Φ B µa z µb z R 3 Φ A k 1 Φ B = ΦA µa z Φ A k 1 Φ B µb z Φ B R 3 = ΦA µa z Φ A k 1 µ B z R 3 Furthermore, one may use the orthogonality relation Φ A ΦA k 1 =. Han-sur-Lesse, December 212 p. 24

65 The transition dipole moments Φ A µa z Φ A k 1, with the summation over k 1, occur in the formula for the polarizability α A zz. Han-sur-Lesse, December 212 p. 25

66 The transition dipole moments Φ A µa z Φ A k 1, with the summation over k 1, occur in the formula for the polarizability α A zz. If one assumes that the polarizability is isotropic, α A xx = αa yy = αa zz = αa, one finds for the first 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 2 +4 µ B z 2 ) 2R 6 Han-sur-Lesse, December 212 p. 25

67 Also this results agree with classical electrostatics. The electric field of the point charge q B at the center of molecule A is ) F = (F x,f y,f z ) = (,, qb and the electric field of the permanent dipole moment µ B is ( ) F = µb x R 3, µb y R 3, 2 µb z R 3 The second order interaction energy E (2) (pol.a) is simply the polarization energy 1 2 αa F 2 of molecule A in the electric field of the charge and dipole of molecule B. R 2 Han-sur-Lesse, December 212 p. 26

68 Analogously, we find for the second term, which includes a summation over the excited 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 2 +4 µ A z 2 )α B 2R 6 This is the classical energy of polarization of molecule B in the field of A. Han-sur-Lesse, December 212 p. 27

69 The third term contains the summation over the excited states of both molecules. All interaction terms with the charges q A and q B cancel, because of the orthogonality relation Φ A ΦA k 1 =. Only the dipole-dipole term of H (1) is left and we obtain E (2) (disp) = 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 212 p. 28

70 This term, the dispersion energy, has no classical equivalent; it is purely quantum mechanical. It is proportional to R 6. Han-sur-Lesse, December 212 p. 29

71 This term, the dispersion energy, has no classical equivalent; it is purely quantum mechanical. It is proportional to R 6. It can be easily proved that each of the three second order terms is negative. Therefore, the induction and dispersion energies are always attractive. Han-sur-Lesse, December 212 p. 29

72 This term, the dispersion energy, has no classical equivalent; it is purely quantum mechanical. It is proportional to R 6. It can be easily proved that each of the three second order terms is negative. Therefore, the induction and dispersion energies are always attractive. For neutral, non-polar molecules the charges q A,q B and permanent dipole moments µ A, µ B are zero, and the dispersion energy is the only second order interaction. Han-sur-Lesse, December 212 p. 29

73 Terms with higher powers of R 1 occur as well. They originate from the quadrupole and higher multipole moments that we neglected. Han-sur-Lesse, December 212 p. 3

74 Terms with higher powers of R 1 occur as well. They originate from the quadrupole and higher multipole moments that we neglected. An approximate formula, due to London, that is often used to estimate the dispersion energy is E (2) (disp) 3αA α B 2R 6 I A I B I A +I B This formula is found if one assumes that all the excitation energies Ek A 1 E A and EB k 2 E B are the same, and are equal to the ionization energies I A and I B. Han-sur-Lesse, December 212 p. 3

75 Summary of long range interactions The interactions between two molecules A and B can be derived by means of QM perturbation theory. Han-sur-Lesse, December 212 p. 31

76 Summary of long range interactions The interactions between two molecules A and B can be derived by means of QM perturbation theory. The first order energy equals the classical electrostatic (Coulomb) interaction energy between the charges and dipole moments of the molecules. It may be attractive or repulsive, depending on the (positive or negative) charges and on the orientations of the dipole moments. The dipolar terms average out when the dipoles are freely rotating. Han-sur-Lesse, December 212 p. 31

77 The second order energy consists of three contributions. The first two terms correspond to the classical polarization energies of the molecules in each other s electric fields. The third term is purely QM. All the three contributions are attractive. They start with R 4 terms when the molecules have charges and with R 6 terms when they are neutral. The dispersion energy, with the leading term proportional to R 6, occurs also for neutral molecules with no permanent dipole moments. Han-sur-Lesse, December 212 p. 32

78 The second order energy consists of three contributions. The first two terms correspond to the classical polarization energies of the molecules in each other s electric fields. The third term is purely QM. All the three contributions are attractive. They start with R 4 terms when the molecules have charges and with R 6 terms when they are neutral. The dispersion energy, with the leading term proportional to R 6, occurs also for neutral molecules with no permanent dipole moments. All of these terms can be calculated when the wave functions Φ A k 1,Φ B k 2 and energies E A k 1,E B k 2 of the free molecules A and B are known, but one should somehow approximate the infinite summations over excited states k 1 and k 2 that occur in the second order expressions. Han-sur-Lesse, December 212 p. 32

79 Interactions in the overlap region Heitler and London (Valence Bond) wave functions for H 2 1s A (r 1 )1s B (r 2 )±1s B (r 1 )1s A (r 2 ) with the plus sign for the singlet spin (S = ) function α(1)β(2) β(1)α(2) and the minus sign for the triplet spin (S = 1) functions α(1)α(2) α(1)β(2) + β(1)α(2) β(1)β(2) The total electronic wave function is antisymmetric (Pauli) Han-sur-Lesse, December 212 p. 33

80 Interaction energy E(R) = E H2 2E H Triplet (S = 1) Q(R) = Coulomb integral E Q J 1 S 2 J(R) = exchange integral Singlet (S = ) Q + J 1 + S 2 S(R) = 1s A 1s B = overlap integral R Han-sur-Lesse, December 212 p. 34

81 Interaction is dominated by the exchange integral J(R), which is negative, so that the exchange interaction is attractive (covalent bonding) in the singlet state and repulsive in the triplet state. Han-sur-Lesse, December 212 p. 35

82 Interaction is dominated by the exchange integral J(R), which is negative, so that the exchange interaction is attractive (covalent bonding) in the singlet state and repulsive in the triplet state. For He 2 there is only one (singlet) state and the interaction energy E(R) is purely repulsive: exchange (or Pauli) repulsion or steric hindrance. Han-sur-Lesse, December 212 p. 35

83 Molecular orbital picture Covalent bonding Exchange repulsion Han-sur-Lesse, December 212 p. 36

84 Molecular orbital picture Covalent bonding Exchange repulsion Exchange repulsion H H interaction He He interaction Han-sur-Lesse, December 212 p. 36

85 Most stable molecules are closed-shell systems and the exchange energy between them is always repulsive. It depends on the overlap between the wave functions of A and B and decays exponentially with the distance R. Han-sur-Lesse, December 212 p. 37

86 Most stable molecules are closed-shell systems and the exchange energy between them is always repulsive. It depends on the overlap between the wave functions of A and B and decays exponentially with the distance R. In combination with attractive long range interactions (proportional to R n ) this gives rise to a minimum in E(R). This, so-called, non-covalent bonding is much weaker than covalent bonding, except when A and B are (atomic or molecular) ions with opposite charges (cf. Na + Cl ). Han-sur-Lesse, December 212 p. 37

87 Most stable molecules are closed-shell systems and the exchange energy between them is always repulsive. It depends on the overlap between the wave functions of A and B and decays exponentially with the distance R. In combination with attractive long range interactions (proportional to R n ) this gives rise to a minimum in E(R). This, so-called, non-covalent bonding is much weaker than covalent bonding, except when A and B are (atomic or molecular) ions with opposite charges (cf. Na + Cl ). Binding (merely by the attractive dispersion energy) is weakest when both molecules are neutral and non-polar: pure Van der Waals interactions. Han-sur-Lesse, December 212 p. 37

88 A special type of interactions between polar molecules is hydrogen bonding: X H Y exchange repulsion Van der Waals Binding energy E hydrogen bonding dispersion electrostatic (dipole dipole, etc.) induction (dipole induced dipole, etc.) dispersion Distance R No special (HOMO-LUMO, charge-transfer, or weak covalent bonding) interactions are needed! Han-sur-Lesse, December 212 p. 38

89 Exercise: Compute the equilibrium angles of HF HF at R = 2.75 Å and H 2 O H 2 O at R = 2.95 Å from the dipolar and quadrupolar interactions only. Han-sur-Lesse, December 212 p. 39

90 Non-covalent interactions and hydrogen bonding, in particular, are very important in biology. Alpha helices and beta sheets in proteins are stabilized by intra- and inter-molecular hydrogen bonds, and the double stranded structure of DNA is held together by hydrogen bonds between the base pairs. Han-sur-Lesse, December 212 p. 4

91 Non-covalent interactions and hydrogen bonding, in particular, are very important in biology. Alpha helices and beta sheets in proteins are stabilized by intra- and inter-molecular hydrogen bonds, and the double stranded structure of DNA is held together by hydrogen bonds between the base pairs. It is essential that a hierarchy of interactions exists with binding energies varying over several orders of magnitude. Interactions in biological systems must be sufficiently strong to maintain stable structures, but not so strong that they prevent rearrangement processes (DNA replication, for instance). Han-sur-Lesse, December 212 p. 4

92 Non-covalent force fields computed ab initio Supermolecule calculations Symmetry-adapted perturbation theory (SAPT)

93 Supermolecule calculations Requirements: E = E AB E A E B 1. Include electron correlation, intra- and inter-molecular (dispersion energy = intermolecular correlation) 2. Choose good basis, with diffuse orbitals (and bond functions ) especially to converge the dispersion energy 3. Size consistency. Currently best method: CCSD(T) 4. Correct for basis set superposition error (BSSE) by computing E A and E B in dimer basis

94 Symmetry-adapted perturbation theory (SAPT) Combine perturbation theory with antisymmetrization A (Pauli) to include short-range exchange effects. Advantages: 1. E calculated directly. 2. Contributions (electrostatic, induction, dispersion, exchange) computed individually. Useful in analytic fits of potential surface. Advantage of supermolecule method: Easy, use any black-box molecular electronic structure program

95 Problems in SAPT: 1. Pauli: AH = HA. Antisymmetrizer commutes with total Hamiltonian H = H () + H (1), but not with H () and H (1) separately. Has led to different definitions of second (and higher) order energies. 2. Free monomer wavefunctions Φ A k 1 and Φ B k 2 not exactly known. Use Hartree-Fock wave functions and apply double perturbation theory to include intra-molecular correlation, or use CCSD wave functions of monomers Many-body SAPT. Program packages: - SAPT2 for pair potentials - SAPT3 for 3-body interactions

96 Most difficult: dispersion interactions First ab initio calculation of He He binding curve: Phys. Rev. Letters, 25 (197) H.F. Schaefer, D.R. McLaughlin, F.E. Harris, and B.J. Alder page 988: D e = 12. K P. Bertoncini and A. C. Wahl page 991: D e = 11.4 K Present value: D e = 11.1 K = 7.65 cm 1.1 kj/mol 1 3 ev Hartree

97 Can one use DFT methods? Reviews by Johnson & DiLabio, Zhao & Truhlar Many different functionals tested with different basis sets type dimer mean error in D e Van der Waals Rg 2, (CH 4 ) 2, (C 2 H 2 ) 2, (benzene) % dipole-induced dipole CH 4 HF, H 2 O benzene, etc % dipole-dipole (H 2 CO) 2, etc. 1-4 % hydrogen bonded (NH 3 ) 2, (H 2 O) 2, (HCOOH) 2, etc. 3-2% Some VdW dimers, (benzene) 2 for example, not bound B971 best, B3LYP worst Often best results without BSSE correction, or smaller basis sets Right for wrong reason

98 Basic problems with DFT 1. Exchange repulsion Incorrect asymptotic behavior of one-electron potential: v(r) exp( αr) instead of 1/r In intermolecular Coulomb energy no self-term present self-exchange spurious attraction 2. Dispersion Intrinsically non-local: cannot be described by local LDA or semi-local GGA methods

99 DFT with dispersion explicitly included vdw-df: M. Dion, H. Rydberg, E. Schroder, D.C. Langreth, B.I. Lundqvist, Phys. Rev. Lett. 92 (24) Ar Ar interaction E int (cm 1 ) SAPT(DFT)/PBE SAPT(DFT)/B97 2 CCSD(T)/CBS Benchmark vdw DF R (Å) From: R. Podeszwa and K. Szalewicz, Chem. Phys. Lett. 412 (25) 488

100 DFT+D methods Compute interaction energy with standard DFT Add atom-atom C n r n long range dispersion terms with n = 6,... Switch between short range correlation contained in DFT and the long range terms with a smooth (parameterized) switch function Parameters from atomic electron densities, polarizabilities, etc. (Becke & Johnson, Tkatchenko & Scheffler) Empirically optimized parameters (Yang, Grimme)

101 Alternative: SAPT-DFT Implemented by G. Jansen et al. (Essen) and K. Szalewicz et al. (Delaware) First order SAPT energy (electrostatic + exchange) computed with monomer densities and density matrices from Kohn-Sham DFT Second-order SAPT energy (induction, dispersion + exchange) from (time-dependent) coupled perturbed Kohn-Sham response functions Only Hartree-Fock like expressions from Many-Body SAPT needed better scaling

102 Caution! SAPT-DFT requires XC potential that is good in inner region and has correct 1/r behavior for r Coupled time-dependent DFT must be used for (frequencydependent) density-density polarizabilities α(r,r,ω) Both groups, K. Szalewicz (Delaware) and G. Jansen (Essen), further improved efficiency by implementation of density fitting.

103 Ar Ar interaction From: R. Podeszwa and K. Szalewicz, Chem. Phys. Lett. 412 (25) E int (cm 1 ) SAPT(DFT)/PBE SAPT(DFT)/B97 2 CCSD(T)/CBS Benchmark vdw DF R (Å)

104 A. Heßelmann, G. Jansen, M. Schütz, J. Chem. Phys. 122 (25) 1413 (benzene) GTOs (aug-cc-pvqz), extrapolation to basis set limit MP kj/mol CCSD(T) DF-SAPT-DFT standard DFT unbound metastable unbound

105 Adenine-Thymine (G. Jansen et al.) DF-SAPT-DFT up to aug-cc-pvqz level E [kj/mol] E [kj/mol] E (1) el E (1) el E (1) exch E (1) exch E (2) ind E (2) ind E (2) exch-ind E (2) exch-ind E (2) disp E (2) disp E (2) exch-disp E (2) exch-disp δ(hf) DFT-SAPT MP2 δ(hf) DFT-SAPT MP2 E int CCSD(T) E int CCSD(T)

106 Pair interactions in water trimer exch kcal/mole 1 disper elec induct total

107 Many-body interactions (per hydrogen bond) Trimer Tetramer pair 3-body total pair 3-body 4-body total Pentamer pair 3-body 4-body 5-body total kcal/mole

108 How to test non-covalent force fields?

109 Molecular beam spectroscopy of Van der Waals molecules

110 W.L. Meerts, Molecular and Laser Physics, Nijmegen

111 Intermolecular potential Cluster (Van der Waals molecule) quantum levels, i.e., eigenstates of nuclear motion Hamiltonian Van der Waals spectra

112 Nuclear motion Hamiltonian H = T +V for normal (= semi-rigid) molecules single equilibrium structure small amplitude vibrations Use rigid rotor/harmonic oscillator model For (harmonic) vibrations Wilson GF-matrix method frequencies, normal coordinates Rigid rotor model fine structure (high resolution spectra)

113 Nuclear motion Hamiltonian H = T +V for weakly bound complexes (Van der Waals or hydrogen bonded) multiple equivalent equilibrium structures (= global minima in the potential surface V) small barriers tunneling between minima large amplitude (VRT) motions: vibrations, internal rotations, tunneling (more or less rigid monomers) curvilinear coordinates complicated kinetic energy operator T

114 Method for molecule-molecule dimers H 2 O H 2 O, NH 3 NH 3, C 6 H 6 C 6 H 6, etc. Hamiltonian H = T A +T B h2 2µR 2 (J j R 2R+ A j B ) 2 2µR 2 +V(R,ω A,ω B ) Monomer Hamiltonians (X = A, B): T X = A X (j X ) 2 a +B X (j X ) 2 b +C X (j X ) 2 c Basis for bound level calculations χ n (R) D (J) MK (α,β,) m A,m B D (j A) m A k A (ω A ) D (j B) m B k B (ω B ) j A,m A ;j B,m B j AB,K

115 Wave function expansion Ψ k (R,ω A,ω B,Θ,Φ) = i i c ik yields matrix eigenvalue problem Hc k = E k c k with H ij = i H j dimension 3 for water dimer Lanczos/Davidson iterative methods lowest 2 eigenvalues E k and eigenvectors c k Start with trial vectors x () k Calculate Hx () k x (1) k Calculate Hx (n 1) Iterate until x (n) k k x (n) k converged c k,e k

116 The permutation-inversion (PI) symmetry group For semi-rigid molecules Use Point Group of Equilibrium Geometry to describe the (normal coordinate) vibrations N.B. This point group is isomorphic to the P I group, which contains all feasible permutations of identical nuclei, combined with inversion E. Molecule Point group P I group C 2v {E,E,(12),(12) } C 3v {E,(123),(132), (12),(13),(23) }

117 Example: H 2 O P I operation frame rotation point group operation (12) R z (π) = C 2z C 2z E R y (π) = C 2y σ xz reflection (12) R x (π) = C 2x σ yz reflection permutation frame rotation + point group rotation permutation-inversion frame rotation + reflection Hence: PI-group point group

118 For floppy molecules/complexes multiple equivalent minima in V low barriers: tunneling between these minima is feasible. additional feasible P I-operations Example NH 3 semi-rigid NH 3 PI(C 3v ) = {E,(123),(132), (12),(13),(23) } + inversion tunneling (umbrella up down) PI(D 3h ) = PI(C 3v ) {E,E } Also (12),(13),(23) and E are feasible Additional feasible PI-operations observable tunneling splittings in spectrum

119 For H 2 O H 2 O PI group G 16 = {E,P 12 } {E,P 34 } {E,P AB } {E,E } Equilibrium geometry has C s symmetry 8-fold tunneling splitting of rovib levels

120 Illustration: Ab initio water potential Tested by spectroscopy on dimer and trimer Used in MD simulations for liquid water R. Bukowski, K. Szalewicz, G.C. Groenenboom, and A. van der Avoird, Science, 315, 1249 (27)

121 Polarizable water pair potential: CC-pol From CCSD(T) calculations in aug-cc-pvtz + bond function basis Extrapolated to complete basis set (CBS) limit at MP2 level 251 carefully selected water dimer geometries Estimated uncertainty <.7 kcal/mol (same as best single-point calculations published)

122 CC-pol: Analytic representation Site-site model with 8 sites (5 symmetry distinct) per molecule Coulomb interaction, dispersion interaction, exponential overlap terms: first-order exchange repulsion, second-order exchange induction + dispersion Extra polarizability site for induction interaction Long range R n contributions computed by perturbation theory, subtracted before fit of short range terms Good compromise between accuracy of reproducing computed points (rmsd of.9 kcal/mol for E < ) and simplicity needed for molecular simulations

123 Water cluster spectra (far-infrared, high-resolution) from Saykally group (UC Berkeley) Used for test of potential: Pair potential Dimer VRT levels Dimer spectrum Pair + 3-body potential Trimer VRT levels Trimer spectrum

124 Water dimer tunneling pathways Acceptor Tunneling Donor-Acceptor Interchange Tunneling Donor (Bifurcation) Tunneling

125 B 2 E 2 Water dimer A 2 tunneling levels (J = K = ) B 1 + E 1 + A 1 + Acceptor tunneling Interchange tunneling Bifurcation tunneling

126 2 15 (H 2 O) 2 levels computed from various potentials (J=K=) B 2 R. S. Fellers et al., J. Chem. Phys. 11, 636 (1999) E A 2 B 2 E A 2 cm 1 1 B 2 E A 2 B 2 E A 2 B 2 E 5 B 1 + E + + A 1 A 2 + B 1 E + + A 1 B 1 + E + + A 1 B 1 + E + + A 1 B 1 + E + + A 1 Experiment ASP W ASP S NEMO3 MCY

127 H 2 O dimer tunneling levels from CC-pol-8s Acceptor tunneling a(k=) + a(k=1) = (ab initio) (experiment) Rotational constants A = (B+C)/2 = B+C = B,A1 1 E +,E + A,B1 1 + B,A2 2 E,E + + A,B2 2 potential ( Szalewicz et al.) and experiment a(k=) B 2 E A 2 a(k=1) B,A1 1 + A,B1 1 E+,E + B,A2 2 + A E,E +,B B 1 + E + + A 1 J=K= J=K=1 J=K=2

128 Acceptor tunneling a(k=) a(k=1) a(k=2) D 2 O dimer tunneling levels (ab initio) (experiment) Rotational constants A = (B+C)/2 = B+C = a(k=2) B 1 +,A1 E + A +,E,B 1 1 B 2,A 2 + E A,E + + 2,B 2 from CC-pol-8s potential and experiment a(k=) B 2 E A 2 B 1 + E + + A 1 a(k=1) B +,A E +,E A,B1 1 B +,A A E,E +,B2 2 J=K= J=K=1 J=K=2

129 β A 36 Acceptor Tunneling A + 1,B+ 1,E β A β A α potential 3 α A 2,B 2,E α

130 χ A 12 Donor-Acceptor Interchange Tunneling χ A 5 A + 1 /A χ D χ A potential χ D B + 1 /B χ D

131 χ A E states localized 15 χ A 5 E x χ D χ A potential χ D E y 15 χ D

132 6 6 3 Donor (Bifurcation) Tunneling β A β A potential 3 β D β D A + 1 /B+ 1 /A 2 /B 2 /E+ /E

133 Intermolecular vibrations

134 α 36 Donor torsion (DT) α 36 Wave functions Acceptor twist (AT) β D 18 Acceptor wag (AW) α DT overtone γ A β A Stretch 18 γ A α DT + AW β A γ A R β A

135 H 2 O dimer intermolecular vibrations from CC-pol-8s potential cm calculated experiment calculated experiment K = K = K = 1 K = 1 DT 2 AT AW DT DT 2 AT AW DT AT AW DT AW DT and experiment GS GS GS GS

136 D 2 O dimer intermolecular calculated experiment calculated experiment K = K = K = 1 K = 1 DT 2 2 DT 2 DT vibrations from CC-pol-8s potential and cm AT AW DT AT AW DT AT AW DT AW DT experiment 4 2 GS 2 1 GS 2 1 GS 1 2 GS 1 2

137 2 Second virial coefficient of water vapor B(T) [cm 3 /mol] Experiment CC pol SAPT 5s VRT(ASP W)III T [K]

138 Water trimer tunneling pathways 4 Torsional tunneling (flipping) 3 B 1 6 C 5 dud A 2 uud Bifurcation (donor) tunneling

139 Water trimer tunneling levels (J = )

140 H 2 O trimer torsional levels cm 1 1 k ± ± Experiment CC pol TTM2.1 VRT(ASP W)III

141 D 2 O trimer torsional levels cm 1 k 1 ± (k=) ± ± Experiment CC pol TTM2.1 VRT(ASP W)III

142 MD simulations of liquid water, T = 298 K

143 3 2.5 CC-pol, 2B CC-pol+NB CC-pol+NB(ind) Experiment g OH g OO 1.5 r [Angstroms] g HH r [Angstroms] r [Angstroms] Atom-atom radial distribution functions

144 Conclusions CC-pol pure ab initio Predicts dimer spectra better than semi-empirical potentials fitted to these spectra Second virial coefficients in excellent agreement with experiment CC-pol + 3-body potential gives good trimer spectrum Simulations of liquid water with CC-pol + N-body forces predict the neutron and X-ray diffraction data equally well as the best empirical potentials fitted to these data Important role of many-body forces in liquid water Nearly tetrahedral coordination: 3.8 hydrogen bonds, only 2.8 with pure pair potential

145 General conclusion CC-pol, with the accompanying many-body interaction model, provides the first water force field that recovers the dimer, trimer, and liquid properties well

146 Resonances (quasi-bound states) in molecular collisions

147 Why are resonances important? See: D. W. Chandler, J. Chem. Phys, 132, 1191 (21) on shape/orbiting resonances: Long life time of collision complex Enable recombination reactions A + B AB Probe long range behavior of the potential Early ( ) observations in scattering for H-Hg by Scoles et al. and for H-X, H 2 -X with X = Ar, Kr, Xe by Toennies et al. (E 8 cm 1 ) Also observed in dimer spectra, e.g. for He-HF by Nesbitt et al. Anomalous line broadening seen in Ar-CH 4 by Roger Miller, due to resonances, explained by calculations of AvdA et al.

148 Difficult to observe resonances in molecular (crossed) beam scattering experiments Collision energies too high Difficult to scan the collision energy Energy spread too large too much averaging

149 New possibilities: Stark-decelerated molecular beams velocity-tuning and state-selection Gerard Meijer, Berlin

150 Crossed beam setups Bas van de Meerakker, Gerard Meijer: Berlin/Nijmegen

151 Textbook example of scattering resonances Square well potential E a E 3 x E 2 E 1 V

152 Scattering wave function at energy E ψ = 2Ae iϕ sin(k x) for x a, with k = e ikx +e i(kx+2ϕ) for a < x, with k = 2mE Match functions and log-derivatives at x = a 2m(E +V ) Phase shift ϕ = arctan [ ] k tan(k a) k π 2 Weight in well region A 2 = k 2 k 2 +V cos 2 (k a) Life time τ = 1 v ϕ k = ϕ E

153 E / V E / V E / V Life time τ Amplitude 2 in well Phase shift / π

154 Molecule-molecule collisions, theory Use same Hamiltonian as in bound state calculation on dimers Use same angular basis Instead of using radial basis, solve coupled differential equations in R by propagator method (log-derivative, Numerov, Airy, etc.)

155 Time independent quantum scattering theory Channels, molecule-molecule (asymptotic, uncoupled) n = v A j A m A k A v B j B m B k B Plane wave in center-of-mass frame Ψ pw n = n e ik n R, E = 2 k 2 n 2µ +ǫ n, k n = k n k Expansion in spherical waves for large R e ik nr 2π = Y LML ( R) e i(k n R L π 2 ) e i(k nr L π 2 ) i L Y LML ( k) ik n LM R L

156 Coupled channels method Expand wave function in nlm L = n Y LML ( R) Ψ nlml = 1 R n L M L U n L M L ;nlm L (R) n L M L Substitution into Schrödinger equation ( yields matrix problem Coupling matrix 2 d2 R 1 2µ dr2r+ H E ) U (R) = W(R)U(R) Ψ nlml = W n L M L ;nlm L (R) = 2µ 2 n L M L H E nlm L

157 Coupled-channel calculations Solve coupled channel equations U (R) = W(R)U(R) by propagation Match U(R) to asymptotic form S-matrix Calculate cross sections, integral and differential S-matrix eigenvalues e 2iϕ j phase shifts ϕ j Eigenphase sum ϕ = j ϕ j, jumps by π at resonances Resonance life time τ = 1 v ϕ k = ϕ E

158 Shape / orbiting resonances Centrifugal barrier V(R) + L(L+1) / (2 µ R 2 ) R V(R)

159 Feshbach resonances (multichannel) Atom-diatom (j = diatom rotation, b = rotational constant) j = 1 V(R,θ) + b j(j+1) R j = V(R,θ)

160 Example 1 OH-X scattering with X = He, Ne, Ar, Kr, Xe

161 OH-X cross sections relative cross section / % relative cross section / % relative cross section / % He Ne Xe /2f /2f 3/2e 5/2e 5/2e 3/2e 5/2e 3/2e 5/2f mean collision energy / cm -1

162 More OH-X cross sections He Ne Xe relative cross section / % relative cross section / % relative cross section / % /2e 1/2f 1/2e 1/2f 1/2e 1/2f /2f 3/2f 5/2f 5/2e 3/2e 3/2e 5/2f 5/2e 3/2f 3/2e 5/2f 5/2e /2e 3/2f mean collision energy / cm -1

163 Example 2 NH 3 -He scattering

164 NH 3 monomer Inversion tunneling (umbrella up down) semi-rigid NH 3 PI(C 3v ) symmetry + inversion tunneling PI(D 3h ) symmetry also E feasible

165 Rotation-inversion Hamiltonian H A = H rot +H inv H rot = 2 2 j2 x I xx (ρ) + j2 y I yy (ρ) + j2 z I zz (ρ) H inv = 2 2 g(ρ) 1/2 ρ I 1 ρρ (ρ)g(ρ) 1/2 ρ +V inv(ρ) Moments of inertia ( ζ = m ) N 3m H +m N I xx (ρ) = I yy (ρ) = 3m H r 2 NH ( 12 sin 2 ρ+ζcos 2 ρ ) I zz (ρ) = 3m H r 2 NH sin2 ρ I ρρ (ρ) = 3m H r 2 NH ( cos 2 ρ+ζsin 2 ρ ) Metric tensor diagonal, determinant g(ρ) = I xx (ρ)i yy (ρ)i zz (ρ)i ρρ (ρ)

166 Potential V inv (ρ) and vibration-inversion levels v ±

167 NH 3 rotationinversion levels (jk ± ) Energy (cm 1 ) ortho para

168 Stark effect in para-nh 3 11 Energy 11 + Electric field 11 state (low-field seeking) can be decelerated Scattering with velocity-tuned, pure 11 para-nh 3 or ND 3

169 NH 3 -He potential surface Hodges and Wheatley, J. Chem. Phys. 114, 8836 (21) New potential calculated by CCSD(T)-F12 method in our group R e (a ) D e (cm 1 ) Hodges ours

170 NH 3 -He potential Expanded in spherical harmonics V(R,ρ,θ,φ) = L,M v LM (R,ρ)Y LM (θ,φ) v LM (R,ρ) expressed analytically in R,ρ Correct R n dependence for each L Θ (Degrees) R (a )

171 Coupled-channel calculations for NH 3 -He NH 3 inversion explicitly included (basis v 2 = ±,1 ± ) Apply full PI(D 3h ) symmetry (of collision complex) paranh 3 -He has E,E symmetries, parity ± Calculate cross sections σ jk ± 11, dσ, phase shifts dω

172 Elastic and inelastic cross sections, orbiting resonances 1 3 Elastic J = 4,6 J = 5 Cross sections (A 2 ) Inelastic J = 4 J = 5 J = 4 J = Energy (cm 1 )

173 Bound state calculations with R max = 2, 25, 3 a 1 J = J = 1 J = 2 J = 3 J = 4 Energy (cm 1 ) Diss. limit R < 2 R < 25 R < 3 R < 2 R < 25 R < 3 R < 2 R < 25 R < 3 R < 2 R < 25 R < 3 R < 2 R < 25 R < 3

174 Phase shift, life times Phase shift / π J = J = 1 J = 2 J = 3 J = 4 J = 5 J = Energy (cm 1 ) Life time τ J = J = 1 J = 2 J = 3 J = 4 J = 5 J = Energy (cm 1 )

175 Our potential Hodges-Wheatley potential Elastic J = 4,6 J = Elastic J = 3 Cross sections (A 2 ) Inelastic J = 4 J = 5 J = 4 J = 5 Cross sections (A 2 ) Inelastic J = 4 J = 4, Energy (cm 1 ) Energy (cm 1 )

176 Elastic and inelastic cross sections, Feshbach resonances 4 3 Elastic open 22 + open 21 open 21 + open Cross sections (A 2 ) 2 Inelastic ( 15) Energy (cm 1 )

177 Inelastic cross sections, Feshbach and shape resonances 4 22 open 22 + open 21 open 21 + open 3 Inelastic ( 8) Cross sections (A 2 ) 2 Inelastic ( 8) 1 Inelastic ( 2) Inelastic ( 2) Energy (cm 1 )

178 Phase shift, life times 5 Phase shift / π J = J = 1 J = 2 J = 3 J = 4 J = 5 J = channel open 1 Lifetime τ [5.31 ps] Energy [cm 1 ] J = J = 1 J = 2 J = 3 J = 4 J = 5 J = Energy [cm 1 ]

179 Scattering wave functions at E = cm 1 (J = 2, E symmetry) Wave function 2 2 x Wave function R (L=) (L=1) 2 2 (L=1) (L=1) (L=2) (L=3) 2 2 (L=3) (L=4) R

180 Scattering wave functions at E = cm 1 (J = 3, E symmetry) Wave function 2 2 x Wave function R (a ) (L=1) 2 1 (L=1) (L=2) 2 2 (L=2) 1 1 (L=3) 2 1 (L=3) (L=4) 2 2 (L=4) (L=4) (L=5) 2 1 (L=5) R (a )

181 Conclusions Good agreement between theory and velocity-tuned experiments for OH-X scattering, with X = He, Ne, Ar, Kr, Xe Many orbiting and Feshbach resonances found in NH 3 -He scattering between and 12 cm 1 Life times much larger than normal collision times Feshbach resonances should be experimentally observable

182 Acknowledgements Theory Krzysztof Szalewicz (Delaware) Claude Leforestier (Montpellier) Koos Gubbels, Gerrit Groenenboom (Nijmegen) Experiments Rich Saykally (Berkeley) Ludwig Scharfenberg, Moritz Kirste, Gerard Meijer (Berlin) Bas van de Meerakker (Nijmegen) Alexander von Humboldt Stiftung (Research Award 27)