3/30/2005 CHEM 408 - Sp05L11 - 1Ne2 Potentials - Calc & Exptr / Å3456V(r) / kB/K-40-20020406080100QCISD(T) ExptCP corrbasis=aug-cc-VQZCalculated Ne2 Potentialsr / Å3456V(r)/kB / K-40-20020406080100HFMP2 QCISD(T) B3LYP basis=aug-cc-VQZAb Initio Calculations of Intermolecular Interactions • calculating dispersion energies is hard; (BSSE)3/30/2005 CHEM 408 - Sp05L11 - 2• “BSSE - basis set superposition error”: when calculating the interaction energy of an intermolecular complex the obvious waybaabRinterBEAEBAERV∞∞−−−= )()()()(there is an imbalance between the quality of the basis sets used for the complex and the fragments that leads to an overestimation of the intermolecular attraction• this error can be approximately corrected by adding a “counterpoise correction”baabRabRCPBEAEgABEgBAERE∞∞−−−+−=∆ )()()()()(3/30/2005 CHEM 408 - Sp05L11 - 3•non-bonded interactions Velect+VvdWare included for 1↔4+ atom pairs:- atoms separated by >3 bonds have no bonded interaction terms and are treated identically to intermolecular interactions- atoms separated by 3 bonds interact via Vtorsionand may also be included (often with interactions reduced by some factor)• even for MM calculations of isolated molecules intermolecular interactions = “non-bonded interactions” must be included for all but the smallest molecules• recall that bonded interactions are:Vstretch1-2 terms (2 atoms, 1 bond)Vbend1-2-3 terms (2 bonds with shared atom)Vtors1-2-3-4 (4 atoms in 3 bond sequence)123456Modeling Intermolecular InteractionsModeling Intermolecular Interactions3/30/2005 CHEM 408 - Sp05L11 - 41. Electrostatic Interactions• ignore Velect• molecular multipoles• atomic charges• bond dipoles (MMn)• atomic + supplemental charges• distributed multipolesφeland Charge Reps. of N2φelfrom Ψ3-q model5-q model• representations of Velectand source of charges vary greatly with purpose of MM potential • some common choices are:3/30/2005 CHEM 408 - Sp05L11 - 5HyperChem Renderingof N2HF/6-31G*+-3/30/2005 CHEM 408 - Sp05L11 - 6Distributed Multipole Analysisfrom: A. J. Stone & M . Alderton, Mol. Phys. 56, 1047 (1985). +-+qµΘ• q, µ, & Θ, at nuclei plus bond centers provides virtually exact φel3/30/2005 CHEM 408 - Sp05L11 - 7• in most cases ε= 1, but in some cases a different value is used to mimic the effect of an intervening medium• for point charge models (most common case) Velectis simple:∑−=jijijielectrrqqV,sites0||41rrεπε• (“distance-dependent ε”in some older force fields; really unjustified r-1→r-2cheat)• the r-1dependence of Velectmeans that such interactions extend over large distances (10s of Å) and the way that far-separated atoms are treated may affect the results r / Å0 1020304050Velect(r) / kBT02468Vq-q q=.2eVµ-µ µ = 3 D3/30/2005 CHEM 408 - Sp05L11 - 82. “van der Waals” Interactions• attractive dispersion and short-range repulsive interactions are grouped together into “van der Waals” contributions between pairs of atoms (or sites):∑−=jijiijvdWrrvV,sites|)(|rr• vij(r) modeled terms of simple 2- or 3-parameter functionsuijr0εσrm• one of two distance parameters is used: σ - zero crossing pointrm- minimum positionand the energy parameter ε - well depth3/30/2005 CHEM 408 - Sp05L11 - 9• by far the most popular functional form is: ⎪⎭⎪⎬⎫⎪⎩⎪⎨⎧⎟⎠⎞⎜⎝⎛−⎟⎠⎞⎜⎝⎛=mnrrkrvσσε)(6126124)(rCrArrrv −=⎪⎭⎪⎬⎫⎪⎩⎪⎨⎧⎟⎠⎞⎜⎝⎛−⎟⎠⎞⎜⎝⎛=σσεLennard-Jones (6-12) Functionσ6/12=mr34εσ=A64εσ=C• some ffs use a different power for the repulsive part; in general:)/( mnmmnmnnk−⎟⎠⎞⎜⎝⎛−=3/30/2005 CHEM 408 - Sp05L11 - 10• less popular (but more accurate) is the 3-parameter Buckingham (exp-6) function: ⎪⎭⎪⎬⎫⎪⎩⎪⎨⎧⎟⎠⎞⎜⎝⎛−−−−−=66)]1/(exp[66)(rrrrrvmmααααε• the exp-6 function is sometimes used as a 2-parameter function by fixing the value of α(well shape); α~14.5 is comparable to a LJ 6-12 function at low energies• Halgren proposed the 2-parameter “buffered 14-7” potential, which is excellent representation of rare-gas V(r) ()⎟⎟⎠⎞⎜⎜⎝⎛−+⎟⎟⎠⎞⎜⎜⎝⎛+= 212.012.107.007.17777mmmmrrrrrrrvεmr8893.0=σexp repulsion more realistic• the Morse potential (3 param) also good, but seldom used because of its computational expense3/30/2005 CHEM 408 - Sp05L11 - 11• these various functions differ mainly in details of well shaper / rmin0.8 1.0 1.2 1.4v(r)/ε-1.0-0.8-0.6-0.4-0.20.00.20.4LJ (9-6)LJ (12-6)Exp-6Bf (14-7) Potential Function σ/rmin FWHM/rmin Bf (14-7) 0.8894 0.2918 Exp-6 0.8930 0.2992 LJ (12-6) 0.8909 0.3124 LJ (9-6) 0.8736 0.35873/30/2005 CHEM 408 - Sp05L11 - 122b. “Combining Rules”• parameters for VvdW, (σ, ε) are specific to pairs of atoms• the number of parameters that need to be defined in a ff is reduced by specifying only parameters for like atoms (σii, εii) and using “combining rules” to determine interactions of unlike pairs i-j• Lorentz-Berthelot rules:are the most commonly used (and the least accurate) )(21jjiiijσσσ+=2/1)(jjiiijεεε=arithmetic mean geometric mean3/30/2005 CHEM 408 - Sp05L11 - 13%(rmin(comb)-rmin(obs))-8-6-4-202LBcHHGFHKongWHrmin(1)/rmin(2)1.1 1.2 1.3 1.4 1.5%(εcomb-εobs)-4004080Combining Rules LB Lorentz-Berthelot )(21jjiiijσσσ+= jjiiijεεε= FH Fender-Halsey (‘62)a )(21jjiiijσσσ+= ()1112−−−+=jjiiijεεε cHHG Cubic-HHG (‘92)b 2233jjiijjiiijσσσσσ++= ()221214jjiijjiiijεεεεε+= Kong Kong (’73)c 664εσ=C jjiiijCCC666= 12124εσ=C () (){}1313/11213/112131221jjiiijCCC += WH Waldman-Hagler (’93)d 6/1662⎟⎟⎠⎞⎜⎜⎝⎛+=jjiiijσσσ jjiijjiijjiiijεεσσσσε⎟⎟⎠⎞⎜⎜⎝⎛+=66332 (a) B. E. F. Fender and G. D. Halsey Jr., "Second Virial Coefficients of Argon and Krypton," J. Chem. Phys. 36, 1886-1887 (1962). (b) T. A. Halgren, "Representation of van der Waals (vdW) Interactions in Molecular Mechanics Force Fields: Potential Form, Combination Rules, and vdW Parameters," J. Am. Chem. Soc. 114, 7827-7843 (1992). (c) C. Kong, "Combining Rules for Intermolecular Potential Parameters. II Rules for the Lennard-Jones (12-6) Potential and the Morse Potential," J. Chem. Phys. 59, 2464-2467 (1973). (d) M. Waldman and A. Hagler, "New Combining Rules for Rare Gas van der Waals Parameters," J. Comput. Chem. 14, 1077-1084