Supplementary Material for: A Supervised Fitting Approach to Force Field
Parametrization with Application to the SIBFA Polarizable Force Field
Mike Devereux, Nohad Gresh, Jean-Philip Piquemal and Markus Meuwly
1.1 Fitting Protocol using I-NoLLS
During I-NoLLS fitting, the general strategy after each evaluation of the Jacobian matrix was
to first examine the results of SVD. Often, a large reduction in the total error was predicted
using only the first few singular directions. The distance in parameter space associated with
adding each additional singular direction was balanced against the predicted reduction in the
total error it would provide, to try to find the maximum possible improvement for the minimum
possible parameter change. Early on in the fitting process, a large Levenberg-Marquardt
constant was also used to help to reduce the step-size away from the current parameter
values, and the constraints outlined in Eq. 13 of the main text were used to ensure values did
not drift too far from their initial SBK or hand-fitted values either. A simple reduction factor
could also be applied after selecting SVD and Levenberg-Marquardt values to scale the
chosen parameter step down before it was submitted for testing. The chosen parameter step
was then trialed, and if it led to a reduction in the total error it was accepted and the next
evaluation of J began. If it failed or led to an increase in the total error, the singular
components, Levenberg-Marquardt parameters and reduction factor were made more
conservative and the new step was tested until reduction in the total error was achieved.
Without these measures an erroneous step that either greatly increased the total error or
caused a crash during evaluation of the energy in SIBFA was highly likely, highlighting the
advantage of a supervised approach over a fully automated procedure for the current
application. Rejecting steps that led to an increase in the total error also yielded significant
time-savings by reducing the total number of evaluations of J necessary.
Towards the end of the process the fit typically becomes more linear, the predicted
improvement when including all singular components after SVD becomes increasingly small
and little user supervision is required. Convergence of the fit is achieved when no further
significant reduction in the total error is possible, as evaluated using the variance (σ2)
reported by I-NoLLS.
2.1 (H2O)n and [Mg(H2O)n]2+ validation complex energies
Fig. S1 SIBFA vs RVS energies for the series of (H2O)n (top) and [Mg(H2O)n]2+ (bottom) complexes used in the
validation set. Parameters used in SIBFA were P1 (black circles), P2 (green diamonds), P3 (red squares) and P4
(blue triangles). All energy components are included (electrostatic, repulsion, polarization, charge transfer and
total binding energy). The black dashed line is included at y=x to guide the eye.
2.2 Parameters adjusted during I-NoLLS fitting
Below is a table showing parameters adjusted by I-NoLLS during fitting and final values for
models P1, P2, P3 and P4. Standard parameter names are given as they are used in the
SIBFA code, available upon request from the authors. Colored values relate specifically to
H2O (green), formamide (red) or imidazole (blue).
The definition of the parameters is detailed below regarding each energy contribution.
The effective radii (six-lettered code) are denoted with a 'w' in the second
position. Letters 'r', 's' or 'p', and 't' are the radii used for Erep, Epol and Ect,
respectively. The effective radii used for the penetration term of EMTP* have two
'p's as the first two letters.
The involvement of these radii can be found in Refs. S1 for Erep, Epol and Ect,
and in Refs. S2 and S3 for EMTP* .
EMTP*. Cnumpe, dnumpe, cdipnu, and paramc are the parameters used for Epen.
See eg, Ref. S2. The first two ones correspond to parameters gamma and delta of
equation 3. Cdipnu corresponds to khi of equations 7 and 8.
Erep. Cofrea and cofreb are the multiplicative constants of the S2/R and S2/R2
components of Erep, respectively, and alfrea and alfreb are their corresponding
Gaussian exponents. Details on the expressions of Erep are in Ref. S3 and S4.
Epol. Rampol and vampol are the multiplicative factor and the exponent of the
screening Gaussian function used to screen the electrostatic field exerted on a
given ligand. They correspond to parameters E and F of equation 13 of Ref. S1.
They are ligand-specific and are thus listed for water first, and for then
formamide and imidazole.
Ect. Cwhydr is the effective radius for polar H atoms acting as electron-acceptors
for the charge-transfer contribution. It corresponds to parameter UM* of equation
18 of Ref. S1. Prop and alphf are the multiplicative constant and the exponential
of the charge-transfer contribution in the case of non-metal cation complexes.
They intervene in equation 16 of Ref. S4. Procat and proro3cat are cation-specific
constants. The first is the multiplicative factor of Ect when the cation acts as the
electron-acceptor. The second is the 'self-potential' of the cation. They correspond
to parameters SM and FM of equations 15 and 17 of Ref. S1.
For both
Erep and Ect,
the
“dwlpi”
values
correspond
to
small
increments/decrements of the effective radius of a heavy atom along the direction
of its pi lone pairs in the case of formamide and imidazole.
Regarding Mg(II). The parameters eg, ppen, pw, and egg, are the Mg(II) effective
radii used for Erep, EMTP, Epol and Ect, respectively. Pkm1, pkm8, pkm612 and
pkm712 are the multiplicative constants which are used for the pairs Mg-H, Mg-O,
Mg-C and Mg-N, respectively.
References.
S1. Gresh, N. J. Comput. Chem. 1995, 16, 856.
S2. Piquemal, J.-P., Gresh, N., Giessner-Prettre, C. J. Phys. Chem. A., 2003, 107,
10353.
S3. Piquemal, J.-P., Chevreau, H., Gresh, N. J. Chem. Theory Comput. 2007 3,
824.
S4. Gresh, N. J. Phys. Chem. A 1997, 101, 8680.
Emtp
Erep
Epol
Ect
Parameter
ppoxyg
pphydr
ppcjug
ppnypr
ppocar
ppen
cnumpe
dnumpe
paramc
cdipnu
coefpe
rwoxyg
rwhydr
rwcjug
rwnpyr
rwocar
eg
pkm1
pkm8
pkm612
pkm712
cofrea (*104)
cofreb (*104)
alfrea
alfreb
dwlpi
dwlpi
dwlpi2
dwlpi
pwoxyg
swoxyg
pwhydr
swhydr
pwcara
swcara
pwnitr
swnitr
pwocar
swocar
pw
rampol
vampol
rampol
vampol
rampol
vampol
twcarb
twnitr
twocar
twoxyg
cwhydr
prop
alphf
proro3cat
procat
egg
P1
1.410
1.100
1.605
1.450
1.440
0.755
2.440
2.250
1.420
2.450
1.000
1.448
1.240
1.550
1.650
1.480
1.265
11.700
14.000
11.600
12.900
4.400
4.500
9.440
14.000
0.000
0.000
0.000
0.000
1.448
1.500
1.300
1.200
1.900
1.900
1.650
1.600
1.480
1.425
1.265
0.680
1.400
0.680
1.400
1.050
1.650
1.700
1.650
1.450
1.500
1.700
0.660
9.500
2.501
0.950
2.009
P2
1.410
1.100
1.605
1.500
1.440
0.755
2.440
2.250
1.420
2.450
1.000
1.448
1.240
1.550
1.650
1.450
1.285
7.500
9.400
8.020
8.250
3.244
3.244
9.140
14.000
0.000
0.000
0.000
0.000
1.448
1.500
1.300
1.200
1.900
1.900
1.650
1.600
1.480
1.425
1.285
0.630
1.450
0.654
1.420
1.050
1.650
1.700
1.650
1.450
1.500
1.700
0.660
9.500
3.751
0.591
2.509
P3
1.303
1.179
1.499
1.439
1.382
0.702
2.437
2.208
1.420
2.568
0.932
1.486
1.299
1.423
1.730
1.461
1.197
11.658
14.571
11.892
11.564
4.107
4.185
9.781
14.898
0.002
0.028
-0.072
-0.014
1.463
1.473
1.267
1.217
1.899
1.988
1.684
1.526
1.444
1.337
1.331
0.689
1.631
0.667
1.564
0.729
1.602
1.758
1.622
1.541
1.597
1.813
0.771
9.414
2.472
0.934
2.029
P4
1.306
1.179
1.499
1.460
1.381
0.702
2.443
2.214
1.420
2.567
0.933
1.476
1.339
1.422
1.688
1.488
1.368
7.419
9.766
8.199
7.743
3.043
3.151
9.484
14.159
0.023
0.077
-0.092
-0.071
1.465
1.476
1.269
1.217
1.900
1.987
1.683
1.529
1.445
1.337
1.347
0.694
1.649
0.667
1.572
0.727
1.606
1.701
1.556
1.370
1.578
1.821
0.776
9.572
3.637
0.451
2.389