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Parameter Fitting Procedure

# 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 of charge fitting commands¶

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 *note select:(chmdoc/select.doc)

## 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 adjusted 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

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 grid points.

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.

## 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 CHARMM calculated ESP.
• 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, ..., UPPOT+NCONf-1 will also be read.
• 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.

## 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

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.

## 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.