Transcript Continuum Representations of the Solvent

Continuum Representations of
the Solvent
pp. 502 - 512 (Old Edition)
Eva Zurek
Surface Types
• van der Waals Surface: is constructed from
the overlapping vdW spheres of the atoms
• Molecular Surface: is traced out by the
inward-facing part of the probe sphere as it
rolls on the vdW surface of the molecule.
Usually defined using a water molecule
(sphere, radius 1.4 A) as the probe.
• Contact surface: consists of regions where
the probe is in contact with the vdW surface.
• Re-entrant surface: regions occur where
there are crevices too narrow for the probe
molecule to penetrate.
• The Accessible Surface: is the surface traced
by the center of the probe molecule.
How to Model the Solvent?
• Solvent molecules are directly involved in the reaction; the
solvent molecules are tightly bound to the solute 
explicit solvation.
Solvent
Solute
Solvent
• The solvent provides a ‘bulk medium’; the dielectric
properties of the solvent are of primary importance 
continuum solvation models.
Solute
Dielectric
continuum; e
The Free Energy of Solvation
DGsol  the free energy change to transfer a molecule
from vacuum to solvent.
• DGsol = DGelec + DGvdw + DGcav (+ DGhb)
•
Electrostatic
component.
Van der Waals interaction
between solute and solvent.
An explicit hydrogen
bonding term.
Free energy required to form the
solute cavity. Is due to the entropic
penalty due to the reorganization of
the solvent molecules around the
solute and the work done in creating
the cavity.
The Born Model
• Born, 1920: the electrostatic component of the free energy of
solvation for placing a charge in a spherical solvent cavity.
• The solvation energy is equal to the work done to transfer the ion
from vacuum to the medium. This is the difference in work to charge
the ion in the two environments.
q  1 
DGel ec   1 
2a
e
e
2
q a
• Ionic radii from crystal structures is used.
• Only relevant for species with a formal charge.
The Generalized Born (GB)
Equation, Classical
• Consider a system of N particles with radii ai and charges qi in a
medium of relative permittivity e.
qiq j 1  1  N qi2
Gel   
 1 
2  e i 1 ai
i 1 j i erij
N
N
qiq j  1  N N qiq j 1  1  N qi2
Gel   
 1  
 1 


e i 1 j i rij
2  e i 1 ai
i 1 j i rij
N
N
 1  N N qiq j 1  1  N qi2
DGel   1  
 1 
 e i 1 j i rij
2  e i 1 ai
Implementations of GB
• The GB equation has been incorporated into MM calculations by Still et al.
where:
1  1  N N qiq j
DGel 
1  
2  e i1 j 1 f (rij ,aij )
f (rij ,aij ) 
r
2
ij
 aij2 e D  where aij 
a a  and D 
i
j
2
rij
2a 
2
ij
• When i = j this expression returns the Born equation; for two charges close
together the expression returns the Onsager result (ie. a dipole where rij  ai,
aj); for two charges very far apart (rij  ai, aj)) it is close to sum of Coulomb
and Born expressions.
• Advantage: the expression can be differentiated analytically; therefore fast
geometry optimizations!
Gruesome Detail You Really
Don’t Want to Know So I
Won’t Talk About It
• A rather complex procedure is used to determine the Born radii in
Still’s implementation. In short,
The Born radius of an atom corresponds to the radius that
would return the electrostatic energy of the system according to the
Born equation if all of the other molecules in the system were
uncharged.
• In Cramer and Truhlar’s QM approach the radius of the atom is a
function of the charge on the atom.
The Onsager Model
• Onsager, 1936: considers a polarizable dipole with polarizability a at
the center of a sphere.
• The solute dipole induces a reaction field in the surrounding medium
which in turn induces an electric field in the cavity (reaction
field)which interacts with the dipole.

a
e
Classical Onsager
 RF
2e  1

2e  1a 3
Energy of a dipole in anE field  RF = -  RF
Work done assembling charge distribut ion
=
DGel ec
 RF
2
 RF
e  1 2


2
2e  1a3
• If the species is charged an appropriate Born term must be added.
• Other Models: A point dipole at the center of a sphere (Bell), A
quadrupole at the center of a sphere (Abraham), multipole expansion to
represent the solute, ellipsoidal and molecular cavities.
Quantum Onsager  SelfConsistent Reaction Field
• The reaction field is a first-order perturbation of the Hamiltonian.
Htot  H0  HRF
HRF
ˆ
 
2e  1
ˆ
3  
2e  1a
2e  1
2
DGel ec   Htot   0 H0 0 

3
22e  1a
Correction factor corresponding
to the work done in creating the
charge distribution of the solute
within the cavity in the medium
The Cavity in the Onsager Model
• Spherical and ellipsoidal cavities may be used
• Advantage: analytical expressions for the first and second derivatives may be
obtained
• Disadvantage: this is rarely true!
• How does one define the radius value?
– For a spherical molecule the molecular volume, Vm can be found:
3Vm
MW
; Vm 
4N A

– Estimate by the largest interatomic distance
– Use an electron density contour Radii
– Often the radius obtained from these procedures is increased to account for the
fact that a solvent particle can not approach right up to the molecule
a3 
The Polarizable Continuum
Method (PCM)
• The van der Waals radii of the atoms are used to determine the cavity
surface.
• The surface is divided into a number of small surface elements with
area DS.
• If Ei is the electric field gradient at pt i due to the solute then an initial
charge, qi is assigned to each element via: q   e  1E DS
i
4e 
i
• The potential due to the point charges, s(r) is found, giving a new
electric field gradient. The charges are modified until they converge.
• The solute Hamiltonian is modified: H  Ho  s (r )
• After each SCF new values of qi and s(r) are computed.
The PCM Approach
1
DGel   Hd   0 H00d   (r)(r)dr
2
• Problems:
Work done in creating the charge
distribution within the cavity in
the dielectric medium.
– Since the continuous charge distribution is discretized when the electrostatic
potential due to the charges on the surface elements is calculated for a given
element i, the charge on i must be excluded otherwise the charges would diverge
– The contribution of the charge on this surface element is found separately using
the Gauss theorem
– Due to the fact that the wavefunction extends outside the cavity the sum of the
charges on the surface is not equal and opposite to the charge of the solute 
outlying charge error
– The charge distribution may be scaled so that this is true
The Conductor-like Screening
Model (COSMO)
• The dielectic is replaced with a conductor. To pass from a conductor to the
dielectric an empirical factor is introduced. Thus,
e  1
DEel (dielectric) 
DE (conductor) ; 0  x  2
e  x  el
• For the classical case the energy of the system is, where C is the Coulomb matrix,
Bi represents the interaction between two unit charges placed at the position of
the solute charge Qi and the apparent charge q,and An represents the interaction
between two unit charges at q and qn.
1
1
QCQ  QBq  qAq
2
2
E(q) = Bq  Aq  0
E(q) 
q   A 1 BQ
1
DE   QBA1 BQ
2
• These eq’s may also be derived from the boundary condition of vanishing
potential on the surface of a conductor ( = 0).
Applications
• The effect of solvent upon energetics and equilibria ie, the tautomeric
equilibria of 2-pyridone:
H
H
H
N
H
2-Pyridone
H
H
H
O
H
H
N
O
H
2-Hydroxypyridone
• The calculated free energy differences were calculated as being -0.64
kcal/mol, 0.36 kcal/mol and 2.32 kcal/mol in gas phase, cyclohexane
(non-polar) and acetonitrile (polar) in good comparison with
experiment.
• The medium was found to have a greater influence on the keto rather
than the enol form since the keto tautomer is more polar.