write-up: running bsj calculations (LA-CC 9869)

This discusses how to use t12 programs to solve the Poisson equation of dielectic solvation models for electrostatic hydration free energies of ions, molecules, and complexes. This write-up is structured in four sections:

The physical and algorithm issues underlying this work are discussed for the scientific public in the references: L. R. Pratt, G. J. Tawa, G. Hummer, A. E. García, and S. A. Corcelli, Int. J. Quant. Chem. 64, 121 (1997) `` Boundary integral methods for the Poisson equation of continuum dielectric solvation models ;'' and S. A. Corcelli, J. D. Kress, L. R. Pratt, and G. J. Tawa, Pacific Symposium on Biocomputing `96, edited by L. Hunter and T. E. Klein (World Scientific, Singapore, 1995), 142. `` Mixed direct-iterative methods for boundary integral formulations of continuum dielectric solvation models .'' This latter paper is available as a postscript file. ["bsj" originally meant "boundary sampling jazz."]

Algorithm comments

The burden of the calculation is to set-up and solve simultaneous linear equations A.x = b. The solution x is the electrostatic potential on the molecular surface; b is the bare (no solvation) electrostatic potential on the molecular surface. In many cases we have studied we required 10K (or more) sampling (or tesselation) points on the molecular surface. So A.x = b typically represents several thousand simultaneous linear equations.

These linear equations are solved with a "mixed direct-iterative" strategy suggested by multi-grid techniques. First, a coarsened problem is solved by a direct method to provide the spatially slowly varying features of the solution x. This coarse solution (as an initial estimate) is then embedded on a finer scale spatial lattice. The higher resolution answer is then sought for the problem on the fine lattice which has many more points. The higher resolution answer is obtained by "nested iteration:"

  1. Gauss-Seidel iteration on the fine lattice until either:
  2. When a pending "stall" is anticipated, the calculation addresses the error in the residual A.xa - b where xais the current approximate solution. With the view that this error should be spatially slowly varying, the calculation solves for the error on the coarse lattice using a direct method and intermediate results (factorized matrices) from the initial coarse solution.
It is very typical for the calculation to take three Gauss-Seidel passes, then break to upgrade the error with a coarse calculation, and then resume iterative refinement on the fine lattice.

Note from these algorithm comments:

Two codes are used in sequence

(back to the top?) Since the number of linear equations can be large, the memory requirements for these calculations can be large. In order to be specific and thus economical with memory requirements the calculation is carried-out with two programs that are run in sequence:

  1. scope - sets up ("scopes") the lattices on the molecular surface, prunes and notes-down necessary lattice info, and writes that to the temporary local file tlat.dat. The executable is /n/home4/lrp/source/ellpde/JCHOL.06/scope.
  2. jchol - picks up the lattice info established by scope and placed in the local temporary file tlat.dat, then finishes the calculation. The executable is /n/home4/lrp/source/ellpde/JCHOL.06/jchol.

I typically run these calculations with a mini-script, e.g. rjc: 

rm tlat.dat
/n/home4/lrp/source/ellpde/JCHOL.06/scope < $1 > $2
/n/home4/lrp/source/ellpde/JCHOL.06/jchol < $1 >> $2
rm tlat.dat
Then I do something like 
%rjc input.go output.out &
It is discussed below what the input and output should be.

Note the inelegant treatment of the temporary, local lattice data file tlat.dat. It is written by scope, read by jchol, then not used further. If you get in a situation where you run scope a second time before jchol is finished reading tlat.dat from a earlier execution of scope, you might "step-on" the data that jchol needs and things might get screwed-up. jchol does this input promptly on start-up, once and for all. As you can see from the script above, I've been (mostly) careful to clean-up tlat.dat so these confusions don't do serious damage.

Example input-output: imidazole

back to the top? Here I give a full example of input and output. The application molecule is imidazole with the molecular data obtained from Greg Tawa.

The input procedure is slightly indirect because the program permits an interactive use. That 'interactive feature' is not very special (or that useful) for serious calculations after the input data required is settled.

(I needn't discuss that interactive use here, except to say that if you merely run scope as, e.g.: 

%scope
you will get prompts for input to be typed at the terminal, and then terminal output. That helps for code development, debugging, etc. where you run really simple special cases and immediately get results that you don't need to save.)

The necessary molecular information is in a file that I refer to as the "setup" file, e.g., for imidazole im.b.liq.set:

(jump to output?) 

 im.b.liq.set - check version 6
 77.4
 1.0
 324
 100
 4000
 9
  7    .00000E+00    .11050E+01    .00000E+00
  6   -.10910E+01    .28200E+00    .00000E+00
  7   -.74100E+00   -.98300E+00    .00000E+00
  6    .63600E+00   -.98400E+00    .00000E+00
  6    .11200E+01    .29800E+00    .00000E+00
  1    .21190E+01    .70000E+00    .00000E+00
  1   -.90000E-02    .21120E+01    .00000E+00
  1   -.21020E+01    .66100E+00    .00000E+00
  1    .11970E+01   -.19050E+01    .00000E+00
  N    -.90285E-01    .15000E+01
  C     .23237E+00    .22300E+01
  N    -.71590E+00    .15000E+01
  C     .21736E+00    .22300E+01
  C    -.37469E+00    .22300E+01
  H     .22838E+00    .17100E+01
  H     .31803E+00    .17100E+01
  H     .10239E+00    .11600E+01
  H     .82346E-01    .17100E+01
 n
 1 186
 2 36
 3 186
 4 36
 5 36
 6 36
 7 36
 8 186
 9 36

The description of this input is as follows:

  1. ``im.b.liq.set - check version 6''; a title or caption of your chosing to help identify the output; no more than 72 characters.
  2. 77.4; the solvent dielectric constant, here water at p=1 atm. and 300K; READ(9,*) esol.
  3. 1.0; the "dielectric constant" for the molecule interior; lots of fancy stuff can happen if you make it not 1.0. BUT the physical meaning of that fancy stuff is fuzzy and I advise always putting 1.0 here; READ(9,*) emol.
  4. 774; maximum number of coarse lattice points (or plaquettes or plax); I almost always explicitly set the number of plax on each sphere (see below) and then put here the sum of those numbers of plax; actual number of surviving (nonburied) lattice points will be less than this; READ(9,*) nplax. With the currently compiled scope this must be no greater than 10 000.
  5. 100; "number of (MC) points per plaque;" certain integrations in the program are done by Monte Carlo and this parameter allows you to scale the number of integration points; typically integrals over a sphere use this number times the number of plax on the sphere; READ(9,*) nppp.
  6. 75000; maximum number of fine lattice points; you have to over-estimate this because scope will end-up with much fewer due to intricacies of the lattice generation and the fact that lots of lattice points initially generated will be buried; e.g. the results here "requested" 75000 fine lattice points but the calculation ended-up with 11792(see below); READ(9,*) nfl
    7. . With the currently compiled scope this must be no greater than 100 000.
  7. 9; the number of partial charges (sources) here; READ(9,*) nsrcs
  8. the next 9 (nsrcs) lines are READ(9,*) nz(m), x(m), y(m), z(m); the atomic number, cartesian positions, in Å, of each atom.
  9. the next 9 (ncnts) lines are READ(9,*) as(m), q(m), rad(m); as(m): `atomic symbol,' a CHARACTER*2 identifier of your choosing for this atom, e.g., Cl for chlorine; q(m) the partial charge (in e) located at this atom center; and rad(m): a radius parameter (in Å) for this atom center. There are some new tricks possible in this input list.
    1. It is permissible to include an `atomic center' with zero partial charge in order to inquire about the electrostatic potential at some observation point of interest. This was the function of the rarely used `observation points' of the previous implementation. In such a case you would probably put the radius for that `atomic center' equal to zero also. Then you get the moral equivalant of the old `observation points.'
    2. It is permissible to have an atomic center with a non-zero partial charge but a radius of value zero. This corresponds to an atomic center with no exclusion sphere located there. Presumably this center is buried as far as the description of the molecular volume is concerned, e.g., H-atoms that are enveloped in `extended atom' spheres of methyl and methylene groups. In the previous version this was handled by having a list of centers (with their radii) of exclusion that was fully separate from the list of positions with partial charges.
    3. Finally, the program will supply a default value for an atomic radius with the following device: you enter this line as, e.g.:
      " H .82346E-01 , , "
      I.e. the ",," indicates an empty slot for the radius and the program provides one.
  10. n; "do you want output data for subsequent graphics?" I have a Mathematica notebook that can produce color-coded, three-dimensional graphics for the computed electrostatic potential on the molecular surface; if you say "y" then the following line must be a filename for a file that will be written locally that will contain that graphics data, e.g., im.b.liq.gg.
  11. Notice that 774 = 3*186 + 6*36. The subsequent lines establish target plaque numbers for each sphere but are optional. If no subsequent lines appear, all spheres are given "36" as the target plaque numbers. [unless the radius set for that sphere is zero; then the target plaque number is zero too -- evidently the intention was no exclusion sphere there at all.] Any subsequent line that appears it should have a form "ij" where "i" is the atom number (in the list of centers) and "j" is the target plaque number for sphere "i". I imagine that a conventional way of running such calculations would be to first accept the default "36" target for all spheres. Then take a look and decided where it is worthwhile to boost the target plaque numbers on particular spheres. A typical case that calls for higher target plaque number is the case of the hexa-aquo-ferric ion. The central ferric ion is nearly buried but not quite, and is highly charged so that it is imprudent to neglect the surface area associated with the sphere on the central ion. Then you would want to jack-up the target plaque number for that ion, so that the pruning of buried plaque-points leaves a few on that surface. Otherwise, no surviving plaque-points on a particular sphere means no (zero) fine lattice points either. So, check that there are surviving points on the spheres that you think are important. If not, then boost the target plaque points for those spheres.

    If no subsequent lines are included then the program, when rewriting the input data for archival purposes, will write-out one such line for each atom. You can edit these lines if you want to specify these target plaque numbers in subsequent calculations.

    Re-emphasis: if the radius of a particular center is zero, you will get zero for the target plaque number there too.

    Otherwise, the smallest reasonable number is "6" (that is roughly: tesselating a sphere with the points of the embedded octahedron) but you might choose from: 3(5j-1)/2 = 6, 36, 186, 936, 4686, 23436, ... (those numbers are already too large to use). This formula is historical but it conveys the horse-sense that to make a two dimensional lattice significantly more dense the density should increase asymptotically by about a factor of 5. You are best off trying to use a minimal coarse lattice: The final accuracy is just limited by the density of the fine lattice. So you want a rough-and-ready coarse lattice that doesn't miss spheres that are important to you. In cases where an important sphere is almost buried but not quite, I jack-up the requested plaque numbers for that sphere so that some coarse lattice points will survive unburied on that exposed surface.

Now to run with this input I construct another input file, e.g. im.b.liq.go: 

n
im.b.liq.set
im.b.liq.set
These three lines say
  1. n; I don't want to do an interactive set-up.
  2. im.b.liq.set; this is the name of the local file with the input data - the data/file that we just plowed through.
  3. im.b.liq.set; this is the name of the local file (can be different from above though I almost never use that) to which the input data can be written - in case you want a separate record of what was the input for this run.

Then as indicated above, I run 

 %rjc im.b.liq.go im.b.liq.out &

The results (im.b.liq.out) I get are shown below. I annotate this typical output with a few comments set in bold type.

(jump to input?) 


 character input: 
                          interactive set-up? (y/n) =
                                                  n

 character input: 
              filename (char*20) for the input file =
                               im.b.liq.set        

  the input file is im.b.liq.set        

  date: Tue Jun 30 15:46:49 1998

 @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@
 Copyright Notice for JCHOL
 Copyright (1996), The Regents of the University of California.
 This software was produced under U.S. Government contract
 W-7405-ENG-36 for Los Alamos National Laboratory, which is operated
 by the University of California for the U.S. Department of Energy.
 The U.S. Government is licensed to use, reproduce, and distribute
 this software in accordance with the terms of the contract. Neither
 the Government nor the University makes any warranty, express or
 implied, or assumes any liability or responsibility for the use of
 this software.
 @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@


  im.b.liq.set - check version 6                                          
  setting-up ...

  SET-UP FOR JCHOL.06 RUN -- NOW SCOPING THE LATTICE

    the value of the solution dielectric constant is 
  			esol =      77.4000

    the value of the interior dielectric constant is 
  			emol =       1.0000

    target (max) number of plax for this calculation is
  			nplax =       324

    target (max) number of fine lattice points is
                        nfl =     75000

    target (max) total MC points for this calculation is
    (fine-plax)*(points/plaq) =     75000 *      100 =   7500000

    there are   9 sources
    i         x(A)         y(A)         z(A)         q/e
    1/N       .000        1.105         .000        -.090
    2/C     -1.091         .282         .000         .232
    3/N      -.741        -.983         .000        -.716
    4/C       .636        -.984         .000         .217
    5/C      1.120         .298         .000        -.375
    6/H      2.119         .700         .000         .228
    7/H      -.009        2.112         .000         .318
    8/H     -2.102         .661         .000         .102
    9/H      1.197       -1.905         .000         .082

                                  TOTAL:             .000
You can check here that the input data for the positions and partial charges were correct. Note that the sum of the partial charges is zero - imidazole is a neutral molecule. 

    INTEGER SEED USED WITH RNUM =      -899243209

    the molecular volume is the union of spherical cavities
    there are   9 cavity spheres

    i      xcnt(A)     ycnt(A)     zcnt(A)     R(A)    
       ---->     a(A^2)      v(A^3)    x-fraction-area

    1      .000       1.105        .000       1.500
       ---->   .000E+00     .000E+00   .000E+00
    2    -1.091        .282        .000       2.230
       ---->   .341E+02     .337E+02   .546E+00
    3     -.741       -.983        .000       1.500
       ---->   .296E+01     .100E+01   .105E+00
    4      .636       -.984        .000       2.230
       ---->   .200E+02     .251E+02   .320E+00
    5     1.120        .298        .000       2.230
       ---->   .175E+02     .243E+02   .280E+00
    6     2.119        .700        .000       1.710
       ---->   .139E+02     .674E+01   .379E+00
    7     -.009       2.112        .000       1.710
       ---->   .187E+02     .127E+02   .508E+00
    8    -2.102        .661        .000       1.160
       ---->   .146E+00     .152E-02   .865E-02
    9     1.197      -1.905        .000       1.710
       ---->   .139E+02     .678E+01   .379E+00

 TOTALS:       .121E+03     .110E+03
You can check here that the input data for the position and radii of each sphere was correct. The --> line gives exposed areas and volumes for each sphere where the volume of each sphere is the faceted volume bounded by spherical surfaces and pair cutting planes. 

  the will be no graphics output this time

  give the filename (char*20) for the setup data file

  setup data file is im.b.liq.set        

  done with set-up, here we go ...

    SOBOL SAMPLING FOR SPHERICAL LATTICE

    distribution of points over exposed surface of cavities
    there are   9 cavity spheres
    i       xcnt(A)    ycnt(A)    zcnt(A)    points
    1       .000      1.105       .000         0
    2     -1.091       .282       .000        18
    3      -.741      -.983       .000        20
    4       .636      -.984       .000        12
    5      1.120       .298       .000        13
    6      2.119       .700       .000        14
    7      -.009      2.112       .000        18
    8     -2.102       .661       .000         3
    9      1.197     -1.905       .000        13

    THERE ARE   111 PLAQUETTES ON THE EXPOSED SURFACE
We anticipated 774 plaques and got 111. Note that our surface area estimate (above) was that sphere #1 had no (ZERO) exposed surface. Sphere # 8 had only a small exposed surface area; only about 0.9% of its total surface was exposed. Only 3 of the 186 plaquettes (3/186=0.016) on that sphere survived, unburied on the exposed surface. Notice that the program doesn't permit fine lattice points on spheres that have no coarse lattice points - the fine lattice points would have no coarse plaquettes to call their home. 

    no coarse points on center     1, no fine points permitted either

    SOBOL SAMPLING FOR SPHERICAL LATTICE

    distribution of points over exposed surface of cavities
    there are   9 cavity spheres
    i       xcnt(A)    ycnt(A)    zcnt(A)    points
    1       .000      1.105       .000         0
    2     -1.091       .282       .000      2563
    3      -.741      -.983       .000       490
    4       .636      -.984       .000      1508
    5      1.120       .298       .000      1309
    6      2.119       .700       .000      1775
    7      -.009      2.112       .000      2386
    8     -2.102       .661       .000         9
    9      1.197     -1.905       .000      1780

    THERE ARE     11820 FINE LATTICE POINTS EXPOSED
We anticipated 75000 fine lattice points and got 11820. 

    (min,max) plaque occupancy in HOME = (      2,    298)
Occupancy is: fine lattice points on (coarse) plaquettes. You should never get 0 (zero) for the minimum occupancy - something is wrong in that case. You would like substantial occupancy in all plaquettes.

scope ends here and jchol starts-up. 


  date: Tue Jun 30 15:52:30 1998

 @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@
 Copyright Notice for JCHOL
 Copyright (1996), The Regents of the University of California.
 This software was produced under U.S. Government contract
 W-7405-ENG-36 for Los Alamos National Laboratory, which is operated
 by the University of California for the U.S. Department of Energy.
 The U.S. Government is licensed to use, reproduce, and distribute
 this software in accordance with the terms of the contract. Neither
 the Government nor the University makes any warranty, express or
 implied, or assumes any liability or responsibility for the use of
 this software.
 @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@


  picking-up lattice, resuming ...

  im.b.liq.set - check version 6                                          

    INTEGER SEED USED WITH RNUM =      -899243550


    SAMPLING ESTIMATE OF FRACTIONAL SOLID ANGLES

    (min,max) sample size in FSA = (       93481,      468580)


  ------------------------------
  results for the solvated case:
  ------------------------------

      0-th coarse: rms-residual/rms-right-side =   .10000E+01
    ___                                          ____________

    close boundary element pairs that required MC:   138
                                                  ______

    row-sum with largest absolute value    .4265E-02
                                        ____________

    BOTH MATRIX AND VECTOR ARE NOW READY -- SOLVE !

    RESULTS FOR INTERIOR POINTS
    there are   9 sources
    i         x(A)         y(A)         z(A)        phi(VOLTS)
    1         .000        1.105         .000       -.31203E+00
    2       -1.091         .282         .000        .31064E+00
    3        -.741        -.983         .000        .12840E+01
    4         .636        -.984         .000        .65070E+00
    5        1.120         .298         .000       -.26628E-01
    6        2.119         .700         .000       -.38881E+00
    7        -.009        2.112         .000       -.92401E+00
    8       -2.102         .661         .000       -.72575E-01
    9        1.197       -1.905         .000        .57990E+00

  EXCESS CHEMICAL POTENTIAL (DMU) = -.50489323E+00 eV
  ===============================
Results after the initial coarse calculations. "DMU = -.50489323E+00 eV" is the electrostatic hydration free energy. 1 eV = 23.0609 kcal/mol. 


  V-PASS GAUSS-SEIDEL ITERATIVE REFINEMENT, TOLERANCE =   .10000E-03
    iteration   rms-residual / rms-right-side    factor
       1              .20848E+00              .00000E+00
       2              .15694E-01              .13284E+02
       3              .23146E-02              .67806E+01

      1-th coarse: rms-residual/rms-right-side =   .37483E-03
    ___                                          ____________

  EXCESS CHEMICAL POTENTIAL (DMU) = -.55549344E+00 eV
  ===============================
Results after the first batch of Gauss-Seidel relaxations plus the first course update. "factor" (".13284E+02" after the second iteration) is the current the rate of reduction of the error. The calculation will stay with the Gauss-Seidel relaxation as long as this factor doesn't decrease or for three passes. After updating the error on the coarse lattice "1-th coarse:" the error ".37483E-03" is still larger than the tolerance "0.10000E-03." The free energy has decreased by about 0.051eV from the initial coarse result. 

  V-PASS GAUSS-SEIDEL ITERATIVE REFINEMENT, TOLERANCE =   .10000E-03
    iteration   rms-residual / rms-right-side    factor
       1              .25883E-03              .00000E+00
       2              .43802E-04              .59090E+01
The tolerance is achieved after the second GS iteration here. The calculation terminates. 

    RESULTS FOR INTERIOR POINTS
    there are   9 sources
    i         x(A)         y(A)         z(A)        phi(VOLTS)
    1         .000        1.105         .000       -.32088E+00
    2       -1.091         .282         .000        .33508E+00
    3        -.741        -.983         .000        .14162E+01
    4         .636        -.984         .000        .67641E+00
    5        1.120         .298         .000       -.24269E-01
    6        2.119         .700         .000       -.40223E+00
    7        -.009        2.112         .000       -.95097E+00
    8       -2.102         .661         .000       -.11687E+00
    9        1.197       -1.905         .000        .56070E+00

  EXCESS CHEMICAL POTENTIAL (DMU) = -.55549656E+00 eV
  ===============================
Notice that the last change in the free energy is 0.000003 eV - impressively small. This isn't the accuracy of the solution to the physical problem but it just says that you've achieved the accuracy of this particular lattice. 

  date: Tue Jun 30 21:59:35 1998
This calculation took about 6 hours and 13 minutes of the wall clock time and produced a value -0.555 eV (-12.8 kcal/mol) for the electrostatic part of the hydration free energy.

Comparison with Greg's published results: Greg reports -12.6 kcal/mol from his calculation that is not trivially identical. The present answer is slightly lowered, I prefer it for now, and in any case the results are not vastly different. I believe the uncertainty in the numerical solution here is about 0.1 kcal/mol. My experience is that this accuracy at the last digit is sensitive to the density of the lattice on the H(-N) proton here (atom #8). A calculation with the `next coarser' lattice (no plaques or fine points on this proton -- is it nearly buried! -- 93 plaques and 11792 fine points overall) produces -12.7 kcal/mol.

One source of confusion is such comparisons is associated with "charge leakage" and the fitting that produces partial charges for the classical electrostatic calculation. Of course, all classical electrostatic calculations should produce the same answer for the same defined electrostatic problem (charges and boundaries).

However, different calculations rely on the data differently. This can induce slight differences between two calculations that start from the same electronic structure results. The electronic structure results naturally have some charge outside the molecular volume. The partial charges fitted to the electrostatic potential on an exterior surface are all inside. My calculation directly treats the electrostatic potential and uses precisely the electrostatic potential on the molecular surface. Thus, if the partial charges reproduce that potential, I don't care where other charges might be and I argue that the whole procedure is correct and consistent.

However, most of the alternative calculations (Greg's too) directly treat the electric field normal to the molecular surface. Thus, if the charges are fitted to the electrostatic potential on the surface but used to calculate the normal electric field, then an error is introduced to the extent that the charges don't adequately described the electrostatic potential in a larger neighborhood of the molecular surface. This is my interpretation of the "charge leakage" problem. Not a problem for me. My more general advice to practioneers is that the partial charges should be fit to the electrostatic potential in a layer of non-zero width containing the molecular surface. A satisfactory fit would then permit accurate and consistent evaluation of both the electrostatic potential and the normal electric field on the molecular surface.


Pictures can be worth thousands of words. Here are pictures of imidazole and imidazolium showing the high resolution solutions that have been obtained. These solutions use about 11K fine points on the molecular surfaces. Pointilists can obtain an impression of the graininess of the tesselation of the molecular surface. The coloring is a device to show contours of electrostatic potential on the molecular surface. It doesn't indicate standard values of the electrostatic potential but more red is more positive and more blue is more negative.

Radii collections

back to the top? Here I copy some values for atomic radii that have been reported and used. This is no endorsement. Although it is scarcely ever acknowledged, such radii depend on temperature, pressure, and composition of the solvent. Thus, although the authors here don't address that important issue, these radii are explicitly relevant to aqueous solutions at high dilution with (T,p) near standard conditions.

A. A. Rashin and K. Namboodiri, J. Phys. Chem.91, 6003 (1987)
atom types H C(sp3) C(sp<3) N(hb) N(not) O F P S Cl
radii (Å) 1.16 2.46 2.46 1.5 2.3 1.5 1.423 1.97 1.937

A. A. Rashin, L. Young, and I. A. Topol, Biophys. Chem.51, 359 (1994)
atom types H(-O or -N) H(-C) C N O SH-
radii (Å) 1.16 1.71 2.46 1.5 1.5 1.97

T. N. Truong and E. V. Stephanovich, Chem. Phys. Letts.240, 253 (1995)
atom types H C(sp3) C(sp<3) N(hb) N(not) O F P S Cl
radii (Å) 1.16 2.3 1.7 1.5 2.2 1.4 1.423 2.35 1.97 1.937

E. V. Stephanovich and T. N. Truong, Chem. Phys. Letts.244, 65 (1995)
atom types H C(sp3) C(sp<3) N(hb) N(not) O F P S Cl
radii (Å) 1.172 2.096 1.635 1.738 2.126 1.576 1.28 2.279 2.023 1.75

Radii for many inorganic ions treated as spheres can found in A. A. Rashin and B. Honig, J. Phys. Chem.89, 5588 (1985). Example results for important molecular ions are R(OH-)=1.48 Å and R(NH4+)=2.13 Å. For peptides, you can also check-out ``Atomic radii for continuum electrostatic calculations based on molecular dynamics free energy simulations,'' M. Nina, D. Beglov, and B. Roux, J. Phys. Chem. B 101, 5239 (1997).

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