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Water "dielectric" is really nonlocal linear response (fwd)
Sender: bushb@merck.com
Subject: WSN: Water "dielectric" is really nonlocal linear response
Several correspondents have replied to a query regarding "local" or
spatially varying dielectric response of water. Some of these
note that classical dielectric constant is a MACROscopic quantity.
While this is correct, I believe that there is a valid MICROscopic
quantity, observable (in principle) and calculable by simulation,
that reduces to the macroscopic dielectric constant in the appropriate limit.
The "observable" is the (free) energy of solvation of a solute
molecule placed into the solvent. More particularly, assume that
(a) the solute is rigid, and (b) there exist UNCHARGED or DIFFERENTLY
CHARGED solute molecules of the same "shape". Then the observable
is the *difference* in free energies of solvation as the charges are
"turned on" or "mutated".
In the linear-response limit the solvation free energy due to a set of charges
(Q) distributed any way within the solute is: Delta_Energy = -(1/2) Q' A Q ,
where (Q) is formally a vector (partial charges or multipoles on various
"sites" (atoms, lonepairs, bond centroids, basis set functions, etc.) and
(A) is a symmetric matrix.
The matrix (A) is a *microscopic* quantity that summarizes completely
the linear response of the medium to the perturbation set up by the solute
charges (while the "molecular shape" remains constant).
A physical interpretation of (A) is that any solute charges (Q) set up a
polarization (P) of the solvent which itself interacts back on the charges.
Indeed, one might write (A) as -(K' R K) and (P) as (R K) Q so that
Delta_Energy = -(1/2) (K' R K) Q .
Here (K) is the "Coulomb kernel" (K(i,j) = 1/dist(i,j)) and
(R) is some self-consistent response function. This is a completely general
formulation of the linear response.
The response function (R) need not be a "local" function of each point in space.
In the macroscopic limit, however, (R) is usually taken as local (averaged
over sufficiently large cells). Then, (K Q) is the field set up by the
solute "at a point in the solvent", and (R)(K Q) is the polarization of
the solvent (at the same point) created by (K Q) ** and (self-consistently)
by the polarization of the rest of the solvent **.
Calculation of (A) could proceed in various ways by free-energy perturbation,
turning on each possible *pair* of point charges in the solute. More
reasonably, one could turn on each possible single point charge, equilibrate
the system (solvent), and record the *potential* created by the solvent at each
solute site (including the diagonal matrix element at the charged site itself).
In principle, (A) could be derived from a single simulation of the solvent
around the *uncharged* solute; A(i,j) derives from the correlation of potentials
seen at site (i) and site (j) created by the spontaneous fluctuations of the
solvent charge distribution, as well as from the mean-field response.
Again, this approach does not necessarily assign a scalar "effective dielectric
constant" to each region of the solvent. But it does completely summarize the
dielectric response of the solvent system.
-- Bruce_Bush@merck.com