# fitcharge (c46b2)

The Charge and Drude Polarizability Fitting

By V.Anisimov and G.Lamoureux, December 2004
Editions By E. Harder 2007

The commands of this section solve the task of charge fitting to
QM electrostatic potential (ESP) maps. In the case of classical Drude
polarizable systems both ESP fitted charges and atomic polarizabilities
will be determined in the single fitting step. The polarizability
determination is based on Drude charge fitting to the series of perturbed
ESP maps obtained in presence of perturbation charges. See DRUDE.DOC for
a description of the classical Drude polarizable model. The citations given
in the references section give further details about the charge fitting
procedure. See FITCHARGE test for the practical sample of charge fitting
and Drude polarizability determination.

The fitcharge routine can be used for charge fitting for the additive
model. A single unperturbed QM ESP is used in this case.

The program supports lone-pairs in either additive or Drude
polarizable model. The QM ESP maps and fitcharge instruction set are
independent of the presence of lone pairs.

* Syntax | Syntax of charge fitting commands
* Introduction | Introduction to charge fitting
* Function | Purpose of the commands
* Example | Input example
* Limitations | Known limitations

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Syntax of charge fitting commands

[SYNTAX FITCharge - charge fitting]

FITCharge { [EQUIvalent atom-selection]
[RESTraint [PARAbolic|HYPErbolic]
[BHYP real] [DHYP real] [FLAT real] [DFLAt real]
]
atom-selection-1 atom-selection-2
[NITEr int] [TOLErance real] [COUNter int]
NCONf int UPOT int UOUT int NPERt int [int]
[TEST] UPPOt int UCOOrd int [ALTInput]
[UPRT int] [ASCAle real] [RDIPole real] [VTHOLe] }
}

atom-selection ::= see » select

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Introduction to charge fitting

Unrestrained charge fitting:
The electrostatic properties of a molecular mechanics model with
Drude polarizabilities are represented by atomic partial charges {q_i}
and Drude charges {delta_i}. The Drude charges are related to atomic
polarizabilities {alpha_i = q_i^2/k_D}, where k_D is a uniform
harmonic coupling constant between each atom and its Drude particle.
These charges are ajusted to give the best agreement with the ab
initio molecular electrostatic potential phi^AI, computed on a set of
gridpoints {r_g} around the molecule.

Although partial charges of a nonpolarizable model can be extracted
from a single potential map, adjusting the polarizabilities requires a
series of /perturbed/ potential maps {phi^AI_p}, each one representing
the molecule in the presence of a small point charge at a given
position r_p. The molecular mechanics model for the molecule under
the influence of perturbation p is a collection of point charges {q_i
- delta_i} at atomic positions {r_i} and Drude charges {delta_i} at
positions {r_i + d_pi}. The model electrostatic potential for the
p-th perturbation, at the g-th gridpoint, is

phi_pg({q}) = sum_i [ (q_i-\delta_i)/(|r_i-r_g|)
+ (delta_i)/(|r_i+d_pi-r_g|} ]

The optimal displacements {d_pi} depend on the position r_p of the
perturbating charge, as well as on the atomic and Drude charges. All
charges are adjusted to minimize the discrepancy between the ab initio
and model potential maps, i.e., we find the charges that minimize the
following chi^2 function:

chi^2 = chi^2_\phi({q}) = sum_pg [ phi^AI_pg - phi_pg({q}) ]^2

Because of the implicit charge-dependence of the displacements
{d_pi}, the system of equations

(@ chi^2)/(@ q) = 0

where q designates either {q_i} or {delta_i}, has to be solved
iteratively. We use the Levenberg-Marquardt algorithm, specially
designed to minimize chi^2-type functions.

Restrained charge fitting:
Solving equations for chi^2 = chi^2_phi, one usually ends up
with partial charges and polarizabilities having poor chemical significance
(e.g. charges on carbon > 1). For nonpolarizable models, it was shown
that fitted charges of neighboring atoms were highly correlated, and,
more generally, that the atomic point charges model of the potential
was largely overparametrized. It is therefore desirable to either
remove charge contributions that have a negligible effect on the
potential, or to penalize any deviation from some intuitive'' (or
conservative'') reference charge, given that the restraint doesn't
significantly deteriorate the quality of the fit.

The original RESP scheme of Bayly et al. minimizes chi^2 =
chi^2_phi + chi^2_r, with either

chi^2_r = A sum_i (q_i - qbar_i)^2

or

chi^2_r = A sum_i [ sqrt(q_i^2 + b^2) - b ]

The first restraint is forcing the charges q_i to their reference''
values qbar_i, and the second restraint is favoring smaller
charges. The force constant A is chosen so that undesirable charge
deviations are penalized while chi^2_phi stays close to its
unrestrained value. It assumes a uniform restraint force A, independent
of the atom type. A more flexible scheme would allow various A's, but
this has not been implemented.

Although the RESP scheme was formulated for nonpolarizable,
partial charges models, it is generalizable to models with Drude
polarizabilities. Equation may be written

chi^2_r = sum_i [ A ( sqrt(q_i^2 + b^2) - b )
+ A'( sqrt(delta_i^2 + b'^2) - b' ) ]

where distinct force constants A and A' are used for the atomic and
Drude charges, along with distinct hyperbolic stiffnesses b and b'. We
thus separately penalize the net atomic charges {q_i} and the Drude
charges {delta_i = sqrt(alpha_i / k_D)}.

The restraint function has the form

chi^2_r = N_p N_g sum_i [
w_i S(q_i - qbar_i) + w'_i S(delta_i - deltabar_i) ],

where N_p is the number of perturbations and N_g is the number of
gridpoints.

The restraints are not applied directly to the charges of the
particles, but to the net charges {q_i} and dipoles
{-delta_i,delta_i}. The weights {w_i} and {w'_i} are read from the
WMAIN array and the initial atomic charges are taken as reference
charges {qbar_i} and {deltabar_i}.

The function S(q) describes the shape of the penalty as the deviation
increases. Two basic shapes are available:

PARA Parabolic shape, S(q) = q^2

HYPE Hyperbolic shape, S(q) = sqrt(q^2+b^2)-b

Parameter b (keyword BHYP) is 0.1 electron by default. The additional
keyword DHYP, with default value B, is used for Drude charges. To
produce S(q) = |q|, set b=0.

The FLAT keyword modifies the shape:

S(q+FLAT) if q < -FLAT,
S'(q) = 0 if -FLAT < q < FLAT,
S(q-FLAT) if FLAT < q.

The default value is FLAT=0. The additional keyword DFLAt, with
default value FLAT, is used for Drude charges.

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Purpose of the commands

EQUIvalent atom-selection

This block allows explicit equivalences between atoms to be stated.
Default value is no equivalences, i.e. each atom is unique in the fitting
procedure. Multiple EQUIvalence keywords are allowed. For each EQUI keyword,
the selected atoms are made equivalent.

[ RESTraint [PARAbolic|HYPErbolic]
[BHYP real] [DHYP real] [FLAT real] [DFLAt real] ]

RESTraint keyword invokes RESP restrained fitting. Not specifying
the RESTraint keyword causes unrestrained fitting to be performed. The
charges and polarizabilities are restrainted to their initial values
(for parabolic penalty function, invoked by keyword PARA, which is also
default) or to zero (in the case of the hyperbolic restraint, HYPE keyword).
The restraint forces (penalty weight) are taken from WMAIN array. They can
be assigned to individual atoms but in practice a uniform stiffness
parameter works well for the whole system (see example below).
A choice between PARAbolic or HYPERbolic function can be made for the
penalty function in the case of the restrained fitting. The PARAbolic shape
introduces the penalty function in the form S(q) = q^2 where q is charge
deviation from the restrained value. The HYPErbolic penalty function is
S(q) = sqrt(q_^2 + B^2) - B, where B is the parabola stiffness parameter.
BHYP keyword sets the stiffness for atomic charges.
DHYP keyword penalizes the atomic polarizability (i.e. the Drude charges).
FLAT keyword introduces a flat well potential,i zeroing the penalty for
the charge deviation in the range from -FLAT to +FLAT.
DFLAT keywords has simular effect for atomic polarizabilites (i.e. Drude
charges).

atom-selection-1 atom-selection-2

SELEct ... END The first atom selection specifies the atoms to fit.
This is an obligatory keyword.
SELEct ... END The second atom selection specifies the atoms
contributing to the electrostatic potential. This is an obligatory keyword.
In most common cases both selections should be pointing to all atoms of the
system excluding the perturbation ion. All other (non-selected) atoms are
contributing to the potential energy, and are considered as a perturbation
(this is how the CALcium perturbation atom is handled).

[NITEr int] [TOLErance real] [COUNter int]

NITEr - maximum number of Levenberg-Marquardt (least square) iterations.
Default value NITE=50. If the program does not converge in 50 iterations most
likely something is wrong with the input data.
TOLErance - relative tolerance on the convergence of the minimized
function (chi^2 corresponding to ESP deviation and penalty contribution) for
Levenberg-Marquardt algorithm. Default value TOLE=1.0E-4. Setting bigger
value is not advised. Smaller values may cause convergence problems.
COUNter is number of iterations under the tolerance it needs to
converge. Default value COUN=2. In most cases setting COUN=1 will result in
the fitting requiring less number of LM steps but the results may be highly
questionable. COUN=2 is proven to be safe. Greater values can be used to test
convergence to assue that the real minimum is identified though this is not
necessary. Inspection of "lambda" variable (an equivalent to level shifting
in QM) from the program output having values 0.05 and below is usually a good
indication of convergence. Smaller final value for "lambda" indicates better
result of the fitting.

NCONf int UPOT int UOUT int NPERt int [int]

NCONf specifies the number of conformations to be used in the
electrostatic fit. Typically 1 conformation is used.
UPOT is the file unit number from which to read the unperturbed ESP
map. The format of this file is: Number of lines: ngrid(iconf) Format:
Xgrid Ygrid Zgrid Potential (4f15.6) For NCONf > 1, units UPOT+1,
UPOT+2, ..., UPOT+NCONf-1 will also be read. These files should have
been open before FITCharge execution.
UOUT is the scratch file unit. The file is used for temporary storage of
NPERt is the number of perturbations for each conformation, e.g. NPERT 40
indicates that 40 perturbation ESP maps are calculated in QM jobs and
provided for charge fitting. NPERT 40 42 indicates that 40 perturbed ESP maps
are available for the first conformation and 42 maps are available for th
second one.

TEST

This is a test case to compare CHARMM Drude and QM electrostatic potentials
generated in the position of perturbation ions. This requires perturbation
ions and grid points being placed at the same locations, giving equal number
of perturbation ions and grid points. No fitting will be performed in this
case. CHARMM and QM potential along with differences in static and perturbed
potential will be printed out on the unit specified by UOUT keyword. The order
of columns is the following: perturbation ion numer, QM static ESP in the
position of the specified perturbation ion, CHARMM static ESP, QM perturbed
ESP, CHARMM perturbed ESP, QM polarization component, CHARMM polarization
component.

UPPOt int UCOOrd int [ALTInput]

UPPOt is input unit for the perturbed ESP maps. The file format is
Number of lines: npert(iconf)*NGRID(iconf) Format: Potential (1f15.6)
For NCONF > 1 (multiple conformation fitting), units UPPOT+1, UPPOT+2, ...,
UCOOrd is the unit number of the first file with model compound coordinates
and a perturbation ion. Coordinates are in CHARMM format. NPERt number of such
files has to be provided. All files have to be opened before invoking
FITCharge.
ALTInput switches on the alternative input for coordinates. In this mode,
the coordinates of the atoms of the second selection are read from UCOOrd, for
each conformation and perturbation.

[UPRT int] [ASCAle real] [RDIPole real]

UPRT is file unit for final printout of the FITCharge results. The data are
printed in the form of a CHARMM stream file.
ASCAle is the polarizability (alpha) scaling factor. Useful to scale
gas-phase polarizabilities. The scaling keeps atomic charges intact.
RDIPole is reference dipole for charge scaling. The charges will be
scaled to reproduce the reference dipole.
VTHOLe allows the fitting of chemical type dependent thole parameters
in addition to the charges. If this flag is not included a constant value of
a_i = 1.3 for all chemical types will be used to fit the charges.
This corresponds to a parameter a = a_i + a_j = 2.6 which is the THOLE parameter
in the old Drude command syntax.

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Example

set residue cyt

! potential for unperturbed system will be read from this file
open read card unit 11 name @residue.pot0

! potential for perturbed systems
open read card unit 21 name @residue.pot

! ESP calculated by CHARMM; a scratch file
open write card unit 30 name @residue.scratch

! fitcharge results will be stored here
open write card unit 90 name @residue.charge.optimized

! all the positions of the 0.5 charge; for alternative input
open read card unit 31 name @residue.all

! set weighting factor for restraints
scalar wmain set 1.0d-5 select segid @residue end

FITCHARGE -
equivalent select type H4* end - ! make atoms H41 and H42 equivalent
select segid @residue end - ! atoms to fit
select segid @residue end - ! ESP contributing atoms
restraint para - ! invoke restrained fitting
flat 0.0 dflat 0.1 - ! use flat well potential for polarizability
upot 11 uout 30 -
NITE 50 - ! look for input errors if job does not converge in 50 steps
NCONF 1 - ! 1 conformation will be used in fitting
NPERT 57 - ! 57 perturbed QM ESP maps were given on input
uppot 21 -
ucoord 31 altinput - ! use alternative input
ascale 0.742 - ! Scale polarizability in analogy with SWM4P water model
rdipole 6.72 - ! Cytosize B3LYP/aug-cc-pVDZ gas-phase dipole moment
uprt 90 ! results will be saved in the form of a CHARMM script

Victor Anisimov at victor@outerbanks.umaryland.edu

References:
1) Bayly et al, JPC 97 (40), 10269, 1993
2) V.M.Anisimov, G.Lamoureux, I.V.Vorobyov, N.Huang, B.Roux,
A.D.MacKerell,Jr. JCTC, 2004, Vol.1, No.1

Task required for charges/polarizabilility fitting not yet included in CHARMM:
- ion placement and grid generation around the model compound
- QM ESP calculation
- extraction of ESP data from the QM output files

Scripts to perform these functions may be requested on the CHARMM forum.

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Limitations

1. Unperturbed QM ESP map (static) is not included into charge and
polarizability fitting when Drude model is employed.
2. In the Lone-Pair case "altinput" keyword is mandatory.