Ab Initio Calculations of Intermolecular Interactions. calculating dispersion energies is hard; (BSSE)

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1 V()/k B / K Ab Initio Calculations of Intemolecula Inteactions Calculated Ne 2 Potentials basis=aug-cc-vqz HF MP2 QCISD(T) B3LYP V() / k B /K Ne 2 Potentials - Calc & Expt basis=aug-cc-vqz QCISD(T) Expt CP co / Å calculating dispesion enegies is had; (BSSE) / Å 3/30/2005 CHEM Sp05 L11-1

2 BSSE - basis set supeposition eo : when calculating the inteaction enegy of an intemolecula complex the obvious way V inte ( R) = E( A B) E( A) E( B) ab R a thee is an imbalance between the quality of the basis sets used fo the complex and the fagments that leads to an oveestimation of the intemolecula attaction this eo can be appoximately coected by adding a countepoise coection E CP ( R) = E( A gb) ab R + E( B ga) ab R b E( A) a E( B) b 3/30/2005 CHEM Sp05 L11-2

3 Modeling Intemolecula Inteactions even fo MM calculations of isolated molecules intemolecula inteactions = non-bonded inteactions must be included fo all but the smallest molecules ecall that bonded inteactions ae: V stetch V bend V tos 1-2 tems (2 atoms, 1 bond) tems (2 bonds with shaed atom) (4 atoms in 3 bond sequence) non-bonded inteactions V elect +V vdw ae included fo 1 4+ atom pais: - atoms sepaated by >3 bonds have no bonded inteaction tems and ae teated identically to intemolecula inteactions - atoms sepaated by 3 bonds inteact via V tosion and may also be included (often with inteactions educed by some facto) 3/30/2005 CHEM Sp05 L

4 1. Electostatic Inteactions epesentations of V elect and souce of chages vay geatly with pupose of MM potential some common choices ae: ignoe V elect molecula multipoles atomic chages bond dipoles (MMn) atomic + supplemental chages distibuted multipoles φ el and Chage Reps. of N 2 φ el fom Ψ 3-q model 5-q model 3/30/2005 CHEM Sp05 L11-4

5 HypeChem Rendeing of N 2 HF/-31G* + - 3/30/2005 CHEM Sp05 L11-5

6 Distibuted Multipole Analysis fom: A. J. Stone & M. Aldeton, Mol. Phys. 5, 1047 (1985). q, µ, & Θ, at nuclei plus bond centes povides vitually exact φ el Θ q µ 3/30/2005 CHEM Sp05 L11 -

7 fo point chage models (most common case) V elect is simple: 1 qiq j Velect = 4πε ε 0 sites i j i, j in most cases ε = 1, but in some cases a diffeent value is used to mimic the effect of an intevening medium the -1 dependence of V elect means that such inteactions extend ove lage distances (10s of Å) and the way that fa-sepaated atoms ae teated may affect the esults ( distance-dependent ε in some olde foce fields; eally unjustified -1-2 cheat) / Å 3/30/2005 CHEM Sp05 L11-7 V elect () / k B T V q-q q=.2e V µ-µ µ = 3 D

8 2. van de Waals Inteactions attactive dispesion and shot-ange epulsive inteactions ae gouped togethe into van de Waals contibutions between pais of atoms (o sites): V = v ( ) vdw sites i, j v () modeled tems of simple 2- o 3-paamete functions one of two distance paametes is used: - zeo cossing point m - minimum position and the enegy paamete ε - well depth i j u 0 ε m 3/30/2005 CHEM Sp05 L11-8

9 3/30/2005 CHEM Sp05 L11-9 by fa the most popula functional fom is: = m n k v ε ) ( ) ( C A v = = ε Lennad-Jones (-12) Function 1/ = 2 m 3 A = 4ε C = 4ε some ffs use a diffeent powe fo the epulsive pat; in geneal: /( m) n m m n m n n k =

10 less popula (but moe accuate) is the 3-paamete Buckingham (exp-) function: v( ) = ε exp[ α ( / α m 1)] α α the exp- function is sometimes used as a 2-paamete function by fixing the value of α (well shape); α~14.5 is compaable to a LJ -12 function at low enegies Halgen poposed the 2-paamete buffeed 14-7 potential, which is excellent epesentation of ae-gas V() () m 1.12m v = ε 2 = m m m the Mose potential (3 paam) also good, but seldom used because of its computational expense m exp epulsion moe ealistic 3/30/2005 CHEM Sp05 L11-10

11 these vaious functions diffe mainly in details of well shape Potential Function / min FWHM/ min Bf (14-7) Exp LJ (12-) LJ (9-) v()/ε Bf (14-7) Exp- LJ (12-) -0.8 LJ (9-) / min 3/30/2005 CHEM Sp05 L11-11

12 2b. Combining Rules paametes fo V vdw, (, ε) ae specific to pais of atoms the numbe of paametes that need to be defined in a ff is educed by specifying only paametes fo like atoms ( ii, ε ii ) and using combining ules to detemine inteactions of unlike pais i-j Loentz-Bethelot ules: 1 = ( + 2 ii aithmetic mean ) ε = ( ε iiε 1/ 2 ) geometic mean ae the most commonly used (and the least accuate) 3/30/2005 CHEM Sp05 L11-12

13 some moe complex ules ae: Combining Rules 1 LB Loentz-Bethelot 2 ( ii + ) FH chhg Kong WH Fende-Halsey 1 ( ) ( 2) a 2 ii ii + Cubic-HHG = 2 2 ( 92) b ii + Kong ( 73) c Waldman-Hagle ( 93) d = ε = ε iiε 1 1 = ( ) 1 C = 4ε ii C = C C + ε = 2 ε ii ε ε = 4ε ε 1 1 ( ) ε + ε ii C 12 = 4ε ii 12 ii 1/13 {( C ) ( ) } 1/ C12 1 C 12 = / ii ii = ε = 2 ε iiε ii + 2 (a) B. E. F. Fende and G. D. Halsey J., "Second Viial Coefficients of Agon and Kypton," J. Chem. Phys. 3, (192). (b) T. A. Halgen, "Repesentation of van de Waals (vdw) Inteactions in Molecula Mechanics Foce Fields: Potential Fom, Combination Rules, and vdw Paametes," J. Am. Chem. Soc. 114, (1992). (c) C. Kong, "Combining Rules fo Intemolecula Potential Paametes. II Rules fo the Lennad-Jones (12-) Potential and the Mose Potential," J. Chem. Phys. 59, (1973). (d) M. Waldman and A. Hagle, "New Combining Rules fo Rae Gas van de Waals Paametes," J. Comput. Chem. 14, (1993). %( min(comb) - min(obs) ) %(ε comb -ε obs ) Tests on Rae Gases LB FH Kong WH chhg LB min(1) / min(2) He-Xe Ne-Xe 3/30/2005 CHEM Sp05 L11-13

14 3. Neglect of Non-Additive Tems only electostatic inteactions ae tuly pai-wise additive + + in geneal, the inteaction of N molecules can be witten: i> i V = v + v +... = N i inteactions of pais of mols. i j> i k > j # calcs = N(N-1)/2 N(N-1)(N-2)/ due to the expense of calculating n-body tems, MM ffs almost always use pai-wise additive effective potentials inductive inteactions can be teated with explicit polaizabilities (α) but inductive effects ae usually teated appoximately by using lage effective chages in electostatic calculations k inteactions of tiplets of mols. + i i> i v eff pai-wise additive effective potential 3/30/2005 CHEM Sp05 L11-14

15 even nonpola systems ae non-additive A Potentials A Coexistence Cuve v()/k B / K "Tue" effective Tempeatue, T / K Expt Effective "Tue" / Å Density, ρ / g cm -3 3/30/2005 CHEM Sp05 L11-15

16 4. Reduced Repesentations fo lage systems, educed molecula epesentations ae often employed single-site (typically LJ) models may be good enough fo some puposes NH4+ N2 CCl4 3/30/2005 C10H8 CHEM Sp05 L11-1

17 C N Potentials: d L. A. Giifalco, J. Phys. Chem. 95, 5370 (1991) LJ assume LJ(-12) inteactions between N atoms unifomly distibuted ove a sphee of diamete d Potential Shape v () s = / = 2β d s( s 1) s( s + 1) 1 1 2α 3 s( s 1) s( s + 1) 3 v()/ ε modified fom Fig. 1 of Chen et al., J. Phys. Chem. B 107, (2003) C 0 β = 2 45 d 12 N 2 ε 1 α 3 d = N 2 ε LJ(-12) / 3/30/2005 CHEM Sp05 L11-17

18 Gay-Bene Potential fo Ellipsoidal Molecules J. G. Gay & B. J. Bene, J. Chem. Phys. 74, 331 (1981). Fig. fom Leech v () = 4ε ( Ω ) s ( Ω ) + s 12 s ( Ω ) + s ε, oientation dependent Ω = { uˆ, uˆ, ˆ} i j ûi ˆ û j 3/30/2005 CHEM Sp05 L11-18

19 fo macomolecula simulations, CH2 and CH3 goups ae often teated as single sites = united atoms badykin: Ag-Po-Po-Gly-Phe-Se-Po-Phe-Ag all-atom (MM+) united-atom (OPLS) 3/30/2005 CHEM Sp05 L11-19

20 Example: Simple Wate Models fo Liquid Simulations fom: Jogensen et al., J. Chem. Phys. 79, 92 (1983). µ gas = 1.85 D µ liq ~ 2. D µ / D ( /A, ε/k B /K) (3.17, 78) (3.17, 78) (3.15, 77) (2.9, 157) (3.15, 78) (3.10, 38) Table 4.3 & Figs. fom Leech 3/30/2005 CHEM Sp05 L11-20

Interatomic Forces. Overview

Interatomic Forces. Overview Inteatomic Foces Oveview an de Walls (shot ange ~1/ 6, weak ~0.010.1 e) Ionic (long ange, ~1/, stong ~510 e) Metallic (no simple dependence, ~0.1e) Covalent (no simple dependence, diectional,~3 e) Hydogen