Solvent effects and more (Part I): Continuum models G´erald MONARD
Th´ eorie - Mod´ eliation - Simulation UMR 7565 CNRS - Universit´ e de Lorraine Facult´ e des Sciences - B.P. 239 54506 Vandœuvre-les-Nancy Cedex - FRANCE http://www.monard.info/
Outline 1. Introduction to solvent effects 2. The QM scaling problem 3. The implicit solvent model 4. Models baseds on multipole expansion 5. The Self-Consistent Reaction Field (SCRF)
Introduction to solvent effects Solvation (IUPAC) Any stabilizing interaction of a solute (or solute moiety) and the solvent [. . . ]. Such interactions generally involve electrostatic forces and van der Waals forces, as well as chemically more specific effects such as hydrogen bond formation. How to model solvent effects when solute can be small ...
or large ?
Two kinds of solvent effects Short range â e.g. hydrogen bonds, molecular reorientation in the presence of ions, etc. â Specific solvation mainly concentrated in the first solvation shells. â Must be described by explicit solvent molecules. Long range â or ”macroscopic”, it involves the screening of charges (solvent polarization). â The long range part is responsible for generating a (macroscopic) dielectric constant different from 1. â Can be described through the use of a dielectric continuum (implicit interaction). â The dielectric continuum only accounts for averaged solvent effects.
Different solvation schemes:
Fully solvated molecule â Short + Long range effects â Explicit solvent participation â “accurate” â ((very) very) slow
Supermolecule
Continuum method
â Aggregate: solute + small number of solvent molecules
â Average/Average solvent effects
â Short range only
â no short range
â No long range
â no explicit solvent participation
â # of solvent molecules?
â fast & “accurate”
The QM scaling problem energy of a water cluster (3-21G basis set) 3500
B3LYP/6-31G* BLYP/6-31G* CCSD(T)/6-31G* MP2/6-31G* HF/6-31G*
3000
wall clock CPU time (seconds)
3000
wall clock CPU time (seconds)
energy of a water cluster (6-31G* basis set) 3500
B3LYP/3-21G BLYP/3-21G CCSD(T)/3-21G MP2/3-21G HF/3-21G
2500
2000
1500
1000
500
2500
2000
1500
1000
500
0
0 0
50
100 number of water molecules
150
200
0
â Wall clock time limit: 1 hour â Intel(R) Xeon(R) CPU E5620 2.40GHz (8 cores) 32Gb RAM
3500
150
200
B3LYP/6-311+G** BLYP/6-311+G** CCSD(T)/6-311+G** MP2/6-311+G** HF/6-311+G**
3000
wall clock CPU time (seconds)
â Gaussian G09.B01 (NProcShared=4, Mem=8Gb, MaxDisk=36Gb)
100 number of water molecules
energy of a water cluster (6-311+G** basis set)
â (H2 O)n water cluster (n from 1 to 216) â 1 energy calculations
50
2500
2000
1500
1000
500
0 0
50
100 number of water molecules
150
200
Quantum Chemistry is CPU intensive Theoretical CPU scaling order for different QM methods QM method semiempirical DFT ab initio MP2 Full CI
Scaling O(N 3 ) O(N 3 ) O(N 4 ) O(N 5 ) O(expN )
The (H2 O)n example: n max in 1/2 hour (4 cores)
3-21G 6-31G* 6-311+G**
HF 216 96 32
BLYP 128 96 32
B3LYP 128 96 28
MP2 32 24 16
CCSD(T) 8 4 4
How to solve the QM scaling problem? â Moore’s Law: CPU power doubles every 18 months + doubling a molecular system is possible: 3 O(N 3 ) scaling: every 18x3 months = 4.5 years 3 O(N 4 ) scaling: every 6 years 3 O(N 5 ) scaling: every 7.5 years, etc.
â Parallelism is not a valid option in the long run 3 Good speeds-up are difficult to obtain (Amdahl’s Law) 3 non linear scaling of the “standard” algorithms
+ limit the number of atoms: use continuum models + change the methods: use approximate quantum methods 3 semiempirical QM methods 3 molecular mechanics (MM) force fields 3 combined QM/MM methods
+ change the algorithms 3 Linear scaling algorithms
An Implicit Solvent Model
â The solvent can be described in first approximation as a uniform polarizable medium with a dielectric constant of εs ; â The solute M is placed in a suitable shaped hole in the medium.
M
Thermodynamic Background â The Solvation free energy ∆Gsol is the free energy change to transfer a molecule from vacuum to solvent. â It can be approximated by: ∆Gsol = ∆Gcav + ∆Gvdw + ∆Gelec â ∆Gcav is the free energy required to form the solute cavity (> 0); â ∆Gvdw is the van der Waals interaction between the solute and the solvent (mainly a dispersive term, < 0); + ∆Gcav + ∆Gvdw : steric or non-electrostatic contributions â ∆Gelec is the electrostatic component.
∆Gsol Continuum models differ by â How the size and shape of the cavity is defined (spherical, ellipso¨ıdal, molecular, etc) â How the dispersion contributions are calculated â How the charge distribution of M is represented (multipole expansions, apparent surface charges, etc) â How the solute is described (QM or MM) â How the dielectric medium is described
Cavitation energy The cavity can have different shapes â Spherical â Ellipso¨ıdal â Molecular: e.g. van der Waals surface, solvent accessible surface (SAS), isodensity surface
Cavitation energy Experimental fact â ∆Gcav + ∆Gdis change proportionnaly to the surface area atoms
∆Gcav + ∆Gdis =
∑
ξi Si
i
â where ξi are an empirical atomic parameters, and Si are fractional contributions to the SAS by atoms i â Continuum solvation models mainly differs by the way they calculate ∆Gelec
The Classical Electrostatic Model â Poisson(-Boltzmann) equation: −ε(r)∇2 Φ(r) = 4πρ(r) ε(r) = 1 ε(r) = εs ρ(r) = 0 Φ(r)
for r ∈ Vin (inside the cavity) for r ∈ Vout (outside the cavity) for r ∈ Vout (charge distribution confined in the cavity) the total electrostatic potential
â ∆Gelec is obtained from two independent calculations: ∆Gelec = Φsol (r) Φ0 (r)
ε = 1 in Vin ε = 1 in Vin
1 2
Z r∈Vin
h i ρ(r) Φsol (r) − Φ0 (r)
ε = εs in Vout ε = 1 in Vout
Some Models Based on Multipole Expansion The Born Model (1920) A charge q inside a spherical cavity of radius a 1 1 q2 ∆Gelec = − 1− 2 εs a a q
the radius of the cavity the point charge at the center of the cavity
Some Models Based on Multipole Expansion The Kirkwood Model (1934) â Generalization to a discrete charge distribution ∆Gelec = −
1 ∞ l (1 + l)(εs − 1) (Mlm )2 ∑ (1 + l)εs + 1 a2l +1 2 l∑ =0 m=−l
â where Mlm is the m component of the multipolar moment of order l describing the charge distribution, and calculated at the center of the spherical cavity.
Some Models Based on Multipole Expansion The Onsager Model (1936) A dipole moment µ inside a spherical cavity of radius a ∆Gelec = − µ α
εs − 1 µ 2 2εs + 1 a3
εs − 1 2α −1 1− 2εs + 1 a3
the dipole moment the isotropic dipolar polarizability
Some Models Based on Multipole Expansion The Generalized Born Approximation (1956 & 1990) ∆Gelec = − where fBG =
1 1 N N qi qj 1− ∑ 2 εs ∑ i j fGB q rij2 + αi αj e −Dij
and Dij = αi :
rij2 4αi αj
effective radii of atom i
+ SMx models (Cramer & Truhlar) + various MM implementation
Some Models Based on Multipole Expansion The MPE Model: Rivail et al. (1973) â MPE: MultiPole Expansion â Generalization of the Kirkwood model to a quantum wavefunction 0
∆Gelec = −
∞ l 0 0 1 ∞ l M m f mm Mlm0 0 ∑ ∑ ∑ ∑ 2 l =0 m=−l l 0 =0 m0 =−l 0 l ll
â Mlm : components of the charge distribution (multicentered) 0
â fllmm : reaction field factors 0 they depend only on the shape of the cavity and the dielectric constant of the solvent.
The Self-Consistent Reaction Field (SCRF) When the solute is polarizable (e.g., when using a QM method): â The solute charge distribution polarizes the solvent â A charge surface density is induced along the surface of the cavity: σ (rs ) = rs : n(rs ):
(εs − 1) ~ out (εs − 1) ~ in ∇Φ (rs ).n(rs ) = ∇Φ (rs ).n(rs ) 4π 4πεs
a point of the surface of the cavity the normal vector to the cavity surface on that point
â This charge surface density polarizes back the solute â Which in turn polarizes the solvent, etc. â Self-converging process: the Self-Consistent Reaction Field (SCRF)
The Self-Consistent Reaction Field (SCRF) â Spherical and ellipsoidal cavities −→ analytical solutions â Molecular shaped cavities −→ numerical solutions â Perturbation of the molecular hamiltonian: H = H0 + Vσ Z
Vσ (r) =
σ (rs ) drs |r − rs |
â Iterative procedure + included in the SCF â Many implementation: PCM (Polarizable Continuum Model), COSMO, . . .
Some SCRF Models in Quantum Chemistry The PCM Model: Tomasi et al. (1981) â PCM: Polarizable Continuum Model â The surface of the cavity is divided into tesserae, each with an area ∆Sk containing a charge qk â Point charges on the cavity surface: qk = −σ (rk )∆Sk â qk charges are incorporated into the core hamiltonian Vσ (r) = ∑ k
â Different cavity shapes can be used
qk |r − rk |
Selected Reviews â Tomasi, J. and Persico, M., Chem. Rev., 1994, 94, 2027–2094; Tomasi, J., Theor. Chem. Acc., 2004, 112, 184–203; Tomasi, J.; Mennucci, B. and Cammi, R., Chem. Rev., 2005, 105, 2999–3093 â Cramer, C. J. and Truhlar, D. G., Acc. Chem. Res., 2008, 41(6), 760–768; Klamt, A.; Mennucci, B.; Tomasi, J.; Barone, V.; Curutchet, C.; Orozco, M. and Luque, F. J., Acc. Chem. Res., 2009, 42(4), 489–92; Cramer, C. J. and Truhlar, D. G., Acc. Chem. Res., 2009, 42(4), 493–497 â Monard, G. and Rivail, J.-L., In Handbook of Computational Chemistry, Leszczynski, J., 2012
Some Illustrative Examples Curutchet, C.; Cramer, C. J.; Truhlar, D. G.; Ruiz-L´opez, M. F.; Rinaldi, D.; Orozco, M. and Luque, F. J., J. Comput. Chem., 2003, 24, 284–297 Comparison of SCRF Continuum Models â PCM and MPE behaves quite identically â Predicted solvation free energies < 0.5kcal/mol compared to experiment Cappelli, C.; Corni, S. and Tomasi, J., J. Phys. Chem. A, 2001, 105(48), 10807–10815 Solvent effects on trans/gauche conformational equilibria of substituted chloroethanes Abul Kashem Liton, M.; Idrish Ali, M. and Tanvir Hossain, M., Comput. Theor. Chem., 2012, 999, 1–6 pKa calculations for trimethylaminium ion
