Non-covalent force fields computed ab initio Supermolecule calculations Symmetry-adapted perturbation theory (SAPT)
Supermolecule calculations Requirements: E = E AB E A E B. Include electron correlation, intra- and inter-molecular (dispersion energy = intermolecular correlation). 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
Symmetry-adapted perturbation theory (SAPT) Combine perturbation theory with antisymmetrization A (Pauli) to include short-range exchange effects. Advantages:. E calculated directly.. 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
Problems in SAPT:. Pauli: AH = HA. Antisymmetrizer commutes with total Hamiltonian H = H (0) + H (), but not with H (0) and H () separately. Has led to different definitions of second (and higher) order energies.. Free monomer wavefunctions Φ A k and Φ B k 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: - SAPT for pair potentials - SAPT3 for 3-body interactions
Most difficult: dispersion interactions First ab initio calculation of He He binding curve: Phys. Rev. Letters, 5 (970) H.F. Schaefer, D.R. McLaughlin, F.E. Harris, and B.J. Alder page 988: D e =.0 K P. Bertoncini and A. C. Wahl page 99: D e =.4 K Present value: D e =.0 K = 7.65 cm 0. kj/mol 0 3 ev 3.5 0 5 Hartree
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, (CH 4 ), (C H ), (benzene) 40-00 % dipole-induced dipole CH 4 HF, H O benzene, etc. 5-00 % dipole-dipole (H CO), etc. 0-40 % hydrogen bonded (NH 3 ), (H O), (HCOOH), etc. 3-0% Some VdW dimers, (benzene) for example, not bound B97 best, B3LYP worst Often best results without BSSE correction, or smaller basis sets Right for wrong reason
Basic problems with DFT. Exchange repulsion Incorrect asymptotic behavior of one-electron potential: v(r) exp( αr) instead of /r In intermolecular Coulomb energy no self-term present self-exchange spurious attraction. Dispersion Intrinsically non-local: cannot be described by local LDA or semi-local GGA methods
DFT with dispersion explicitly included vdw-df: M. Dion, H. Rydberg, E. Schroder, D.C. Langreth, B.I. Lundqvist, Phys. Rev. Lett. 9 (004) 4640 0 0 40 Ar Ar interaction E int (cm ) 60 80 00 0 40 60 SAPT(DFT)/PBE0 SAPT(DFT)/B97 CCSD(T)/CBS Benchmark vdw DF 3.5 4.0 4.5 5.0 5.5 6.0 6.5 R (Å) From: R. Podeszwa and K. Szalewicz, Chem. Phys. Lett. 4 (005) 488
DFT with dispersion included semi-empirically Becke, Johnson, and others: - Include atom-atom C 6 R 6 C 8 R 8 C 0 R 0 term in addition to the semi-local correlation energy - Approximate C 6, C 8, C 0 from atomic squared multipole moment expectation values and (empirical) static polarizabilities - Empirically optimized damping parameters
Alternative: SAPT-DFT Implemented by G. Jansen et al. (Delaware) (Essen) and K. Szalewicz et al. 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
Caution! SAPT-DFT requires XC potential that is good in inner region and has correct /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.
Ar Ar interaction From: R. Podeszwa and K. Szalewicz, Chem. Phys. Lett. 4 (005) 488 0 0 40 E int (cm ) 60 80 00 0 40 60 SAPT(DFT)/PBE0 SAPT(DFT)/B97 CCSD(T)/CBS Benchmark vdw DF 3.5 4.0 4.5 5.0 5.5 6.0 6.5 R (Å)
A. Heßelmann, G. Jansen, M. Schütz, J. Chem. Phys. (005) 0403 (benzene) 5 GTOs (aug-cc-pvqz), extrapolation to basis set limit MP 4.4 5. 0.3 kj/mol CCSD(T) 6.7.8.4 DF-SAPT-DFT 7.6.9.7 standard DFT unbound metastable unbound
Adenine-Thymine (G. Jansen et al.) DF-SAPT-DFT up to aug-cc-pvqz level E [kj/mol] -00-50 0 50 00 50 E [kj/mol] -00-50 0 50 00 50 E () el E () el E () exch E () exch E () ind E () ind E () exch-ind E () exch-ind E () disp E () disp E () exch-disp E () exch-disp δ(hf) DFT-SAPT MP δ(hf) DFT-SAPT MP E int CCSD(T) E int CCSD(T)
Pair interactions in water trimer 40 30 0 exch kcal/mole 0 0 disper -0-0 -30 elec induct total
Many-body interactions (per hydrogen bond) Trimer Tetramer pair 3-body total pair 3-body 4-body total 0 - - -3-4 -5-6 -7 0 - - -3-4 -5-6 -7 0 - - -3-4 -5-6 -7 Pentamer pair 3-body 4-body 5-body total kcal/mole
Illustration: New 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, 35, 49 (007)
New 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 MP level 50 carefully selected water dimer geometries Estimated uncertainty < 0.07 kcal/mol (same as best single-point calculations published)
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 0.09 kcal/mol for E < 0) and simplicity needed for molecular simulations
How to test non-covalent force fields?
Molecular beam spectroscopy of Van der Waals molecules
W.L. Meerts, Molecular and Laser Physics, Nijmegen
Intermolecular potential Cluster (Van der Waals molecule) quantum levels, i.e., eigenstates of nuclear motion Hamiltonian Van der Waals spectra
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)
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
Method for molecule-molecule dimers H O H O, NH 3 NH 3, etc. Hamiltonian H = T A + T B h µr R R + (J j A j B ) µr + V (R, ω A, ω B ) Monomer Hamiltonians (X = A, B): T X = A X (j X ) a + B X (j X ) b + C X (j X ) c Basis for bound level calculations χ n (R) D (J) MK (α, β, 0) 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
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 v {E, E, (), () } C 3v {E, (3), (3), (), (3), (3) }
Example: H O P I operation frame rotation point group operation () R z (π) = C z C z E R y (π) = C y σ xz reflection () R x (π) = C x σ yz reflection permutation frame rotation + point group rotation permutation-inversion frame rotation + reflection Hence: P I-group point group
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 P I(C 3v ) = {E, (3), (3), (), (3), (3) } + inversion tunneling (umbrella up down) P I(D 3h ) = P I(C 3v ) {E, E } Also (), (3), (3) and E are feasible Additional feasible P I-operations observable tunneling splittings in spectrum
For H O H O PI group G 6 = {E, P } {E, P 34 } {E, P AB } {E, E } Equilibrium geometry has C s symmetry 8-fold tunneling splitting of rovib levels
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
Water dimer tunneling pathways Acceptor Tunneling Donor-Acceptor Interchange Tunneling Donor (Bifurcation) Tunneling
B E Water dimer A tunneling levels (J = K = 0) B + E + A + Acceptor tunneling Interchange tunneling Bifurcation tunneling
0 5 (H O) levels computed from various potentials (J=K=0) B R. S. Fellers et al., J. Chem. Phys. 0, 6306 (999) E A B E A cm 0 B E A B E A B E 5 0 B + E + + A A + B E + + A B + E + + A B + E + + A B + E + + A Experiment ASP W ASP S NEMO3 MCY
Acceptor tunneling a(k=0) + a(k=) = 6.58 (ab initio) 3.9 (experiment) Rotational constants 0.47 0.4 + B,A E +,E + A,B H O dimer A = (B+C)/ = 7.34 7.44 B+C = 0.43 0.4 0.37 0.37 B +,A E,E + A +,B tunneling levels from CC-pol (red) experiment (black) 0.66 0.65 B E A B +,A A +,B B,A + + 0.5 A,B a 0.54 0 a 0.68 0.7..3 E +,E E,E + 0.69 0.75 B + E + + A J=K=0 J=K= J=K=
H O dimer intermolecular vibrations from CC-pol (red) cm 60 40 0 00 80 calculated experiment calculated experiment K = 0 K = 0 K = K = DT AT AW DT DT AT AW DT AT AW DT AW DT experiment (black) 60 40 0 0 GS GS GS GS
Acceptor tunneling D O dimer a(k=0) a(k=) a(k=).4 0.79.7 (ab initio).77 0.6.3 (experiment) Rotational constants A = (B+C)/ = 4.7 4.7 B+C = 0.380 0.36 a(k=) 0.034 0.07 0.03 0.07 B +,A E +,E + A,B B,A + E,E + + A,B tunneling levels from CC-pol (red) experiment (black) 0.04 0.036 B E A a(k=) 0.04 0.036 0.038 0.033 B +,A + A,B B +,A A +,B E +,E E,E + a(k=0) 0.043 0.039 B + E + + A J=K=0 J=K= J=K=
40 calculated experiment calculated experiment K = 0 K = 0 K = K = D O dimer 0 DT DT DT intermolecular vibrations from CC-pol (red) experiment (black) cm 00 80 60 AT AW DT AT AW DT AT AW DT AW DT 40 0 0 GS GS GS GS
0 00 Second virial coefficient of water vapor B(T) [cm 3 /mol] 400 600 800 000 00 Experiment CC pol SAPT 5s VRT(ASP W)III 400 600 800 00 300 400 500 600 700 800 900 000 T [K]
Water trimer tunneling pathways 4 Torsional tunneling (flipping) 3 B 6 C 5 dud A uud Bifurcation (donor) tunneling
Water trimer tunneling levels (J = 0)
H O trimer torsional levels cm 00 k 09.4 08.3 3 87. 75 75.9 ± 65.6 50 56. 47.6 3.5 5 ±.7 9.4 4.6 8.4 0 0 Experiment CC pol TTM. VRT(ASP W)III
D O trimer torsional levels cm k 00 ± 98. 3 90.3 75 5. 06. (k=0) 0. 94. 8.6 64.6 50 3 4. 35.4 5 ± 8.0 5. 6.3 0 ± 0 8.5 8.0 9.8 7.3.4 Experiment CC pol TTM. VRT(ASP W)III
MD simulations of liquid water, T = 98 K
3.5 CC-pol, B CC-pol+NB CC-pol+NB(ind) Experiment.6. g OH 0.8 0.4 g OO.5 0 r [Angstroms] 3 4 5 6 7.6. g HH 0.5 0.8 0.4 0 r [Angstroms] 3 4 5 6 7 0 r [Angstroms] 3 4 5 6 7 Atom-atom radial distribution functions
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.8 with pure pair potential
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
Latest development Water pair potential with flexible monomers: -dimensional X. Huang, B. J. Braams, and J. M. Bowman, J. Phys. Chem. A 0 (006) 445 Improved version (J. Chem. Phys., accepted) From CCSD(T) calculations in aug-cc-pvtz basis. 0 000 30 000 water dimer geometries. Accurate fit in terms of symmetry-adapted polynomials of scaled atom-atom distances. Produces even more accurate VRT levels of (H O) and (D O). Also acceptor splitting a 0 + a accurate in d calculations. Good VRT levels of (H O) 3 and (D O) 3 when 3-body interactions added.