Supporting information for:
QUILD: QUantum-regions Interconnected by Local Descriptions
Marcel Swarta,b,c,* and F. Matthias Bickelhaupta,*
a) Theoretische Chemie, Vrije Universiteit Amsterdam, De Boelelaan 1083, NL-1081 HV Amsterdam, The Netherlands
b) Institut de Química Computacional, Universitat de Girona, Campus Montilivi, E-17071 Girona, Spain
c) Institució Catalana de Recerca i Estudis Avançats (ICREA), E-08010 Barcelona, Spain
E-mail: [email protected], [email protected]
Setup of adapted delocalized coordinates
p. S2-S7
Starting coordinates of molecules in Baker test-set (Bohr)
p. S8-S10
Starting coordinates of molecules in weakly-bound test set (Å)
Table S1.
Number of geometry cycles for optimizing molecules of the
(at RHF/STO-3G), compared for various geometry optimization schemes.
p. S11
Baker
test
set
p. S12
Table S2.
Number of geometry cycles for optimizing molecules of the Baker test set (at PW91/TZ2P),
compared for various geometry optimization schemes.
p. S13
Table S3.
Number of geometry cycles for optimizing molecules of the weak-coordinates test set
(at PW91/TZ2P), compared for various geometry optimization schemes.
p. S14
Table S4.
Optimized geometry of H2O2 using numerical gradients with different finite difference steps
p. S15
Example QM/MM inputfile
p. S16
References
p. S17
S1
Setup of adapted delocalized coordinates
In this section, we describe how to setup the adapted delocalized coordinates as used in the paper.
Choice of coordinate system
The performance of geometry optimization techniques depends critically on the choice of coordinates to be used
in the optimization scheme. Although the use of 3N (N is the number of atoms) Cartesian coordinates is simple
and straightforward, it is not the best choice for the optimization, as the individual components are too strongly
coupled. For example, the Hessian, i.e., the matrix containing the second derivatives of the energy with respect to
a complete set of coordinates, is dense in the case of Cartesian coordinates. Better performances are obtained by
using Z-matrix coordinates.1 They represent a predefined set of bond distances, angles and dihedrals that as a
whole uniquely determines the atomic coordinates, and which usually has an almost diagonal Hessian matrix.
However, the performance of the Z-matrix depends critically on its definition, and can, when chosen unfavorably,
be even worse than that of Cartesian coordinates. Improved results may be obtained by using natural internal
coordinates,2-4 which in essence is a combined set of bonds, angles and dihedrals (the primitives), which are
chosen based on the actual geometry of the molecule to be studied. However, the choice of which primitives to
include, and with which weight factor, is still a matter of “hand work”. It is therefore quite complicated, and can
take several thousands of lines of computer code. An important step forward was made by the generation of
delocalized coordinates (vide infra), as formulated by Baker and co-workers.5-7
The delocalized coordinates setup works well for systems with strong coordinates but less satisfactory
performance is observed for weakly bound systems. This is easily understood as the weak coordinates are taken
into account with the same weight as the strong ones.8 Recently,9 we reported an adaptation of the delocalized
setup, which facilitates the use on weak coordinates as well (vide infra).
Delocalized coordinates setup
In the delocalized coordinates setup,5-7 the weight factors follow naturally from the actual geometry. Starting from
a set of primitives, containing all bonds, angles and dihedrals of connected atoms, the usual Wilson B matrix,10,11
relating the M primitives (!q) with the corresponding 3N Cartesian displacements (!x), is constructed:
"q = w p B"x = B p"x
(1)
In this formula, wp is a diagonal matrix containing a weight factor for each primitive. In the original delocalized
!
coordinates setup wp corresponds to the unit matrix. In the adapted delocalized coordinates each primitive has its
own associated weight (see eqs. 7-10 below).
Next, the M!M matrix G=BpBpT is formed and diagonalized, which results in two sets of eigenvectors (U and
R); the first set (U) of 3N–6 (for linear molecules 3N–5) non-redundant eigenvectors with eigenvalue !>0, and the
second set (R) of redundant eigenvectors with eigenvalue zero. The eigenvalue equation of G can thus be written
as:
# " 0&
G(UR) = (UR)%
(
$ 0 0'
!
(2)
S2
Only the first set of non-redundant eigenvectors (the set of active coordinates of the vectors contained in U) is
needed for the geometry optimization, and contains the weights for each of the primitives. The B matrix is then
transformed from primitive space to the active delocalized space (Bd =UTBp), and the “inverse” B matrix is formed
that transforms the Cartesian gradient to the delocalized gradient:
T "1
T "1
(B ) = (B B )
g = (B ) g
d
d
d
Bd
(3)
T "1
d
cart
Although the “inverse” B matrix could also be used to transform the Cartesian Hessian to delocalized space, it is
!
more advantageous to start from the Hessian matrix in primitive space (Hp). In this way, one can choose for each
primitive coordinate individually a force constant (see eq. 11 below) that is appropriate for that particular
coordinate. This matrix Hp is then transformed to our active optimization space according to H=UTHpU.
The optimization procedure is then carried out within the active delocalized space, using the coordinates s=UTq
within the delocalized space, together with the gradient g and the Hessian H. After a step has been taken in the
delocalized space, new Cartesian coordinates have to be formed involving a non-linear back-transformation,
which is solved in an iterative fashion:6
T "1
( )
X( k + 1) = X( k ) + B d
(k )[s " s( k )]
(4)
A more efficient way to solve this7 is by first choosing an intermediate Z-matrix as subset of the primitive
!
coordinates q. This intermediate Z-matrix is sufficient to completely determine the Cartesian coordinates and it is
used only in the back-transformation. Finally one iterates the primitive coordinates:
[
q( k + 1) = q( k ) + U sT " s( k )
T
]
(5)
From the current estimate of q, new Cartesian coordinates X are obtained through the Z-matrix back-
!
transformation, which lead to new values for the primitives q and subsequently to new values for the internal
coordinates s(k)=UTq. These are then compared with the known s, and the difference used to obtain an improved
estimate for q. In the current implementation, we construct the Z-matrix automatically in such a way as to include
all coordinates with the largest weights.
Strong versus weak coordinates: adapted delocalized coordinates
The delocalized coordinates setup works extremely well for strong coordinates, however, it was not yet designed
for weak, e.g. intermolecular, coordinates. In our attempts to remedy this, we obtained good results by assigning a
weight to each primitive coordinate to separate them into strong and weak coordinates, as proposed by Lindh and
co-workers.8 They used a model function to generate the weight, which they took from an earlier paper12 that dealt
with Hessian matrices based on parameterized force constants. As the early paper formulated the force constants
S3
only for the first three rows, and contains 15 parameters that were deduced from STO-3G Hartree-Fock
calculations, we decided to explore a generalized form that depends also on the actual geometry of the molecule.
We use for each atom-pair in the primitive coordinates a screening function "ij, that estimates the strength of the
corresponding interaction according to eq 6:
(
)
" ij = exp #{( rij /Cij ) #1}
(6)
Here, rij is the distance between atoms i and j, and Cij the sum of their covalent radii. The screening function is
!
around one for atoms that are covalently bonded, and lower for weaker coordinates. All bonds with a screening
function of 0.7 or higher are considered to be strong bonds, the other bonds are weak. The weights for the
different primitive coordinates are then obtained as:
w = " ij
12
w = ( " ij " jk )
!
w = ( " ij " jk " kl
!
w = ( " ij " jk " jl
!
!
[ f + (1# f ) sin$ ]
) [ f + (1# f ) sin$ ][ f + (1# f ) sin$ ]
) [ f + (1# f ) sin$ ][ f + (1# f ) sin$ ]
ijk
13
bond i-j
(7)
angle i-j-k, #ijk
(8)
ijk
jkl
dihedral i-j-k-l
(9)
ijk
ijl
improper i-j-k-l
(10)
13
The parts involving the sine function have been included to disfavor (near-)linear angles, and include a damping
factor f (which has a value of 0.12 in the current implementation). All coordinates with a weight above a certain
threshold (currently 0.3) are included in the primitive space, all others do not contribute significantly and can be
left out without problems. Transition State Reaction Coordinates (TSRC) and constraint coordinates are given a
special weight of 0.5 and 1.0 respectively.
Initial Hessian
The number of geometry cycles needed for convergence depends largely on the start-up Hessian that is used.
Baker6 proposed a simple scheme, using values of 0.5, 0.2 and 0.1 for bonds, angles and dihedrals respectively,
which seems to work well for systems containing strong coordinates only. Lindh and co-workers12 proposed
reduced values of 0.45, 0.15 and 0.005, which are further scaled with factors depending on the actual geometry.
The ADF program uses force constant values by Fischer and Almlöf.13 All of these schemes have been
implemented and tested, but finally we decided to adapt the scheme by Lindh and co-workers12 and reuse the
screening factors from equation 6, which leads to improved performance with the following adapted force
constant values:
k bond = 0.40 " # ij
k angle = 0.20 " # ij # jk
(11)
k dihedral = 0.01" # ij # jk # kl
kimproper = 0.01" # ij # jk # jl
!
S4
The use of the Hessian in primitive space also allows us to generate an appropriate starting Hessian for
transition state searches, when starting from a reasonable guess structure and when the coordinates involved in the
reaction coordinate (designated “TSRC coordinates”, also known as “transition vector”) are known, which is
usually the case. Along the TSRC coordinates, the energy achieves a maximum and the corresponding force
constants should thus be negative. Moreover, along the TSRC coordinates, in proximity to the transition state, the
Hessian must have one (and only one) negative eigenvalue. This can be achieved by using a damped negative
force constant value in the primitive Hessian for the TSRC coordinates. The force constants are damped with a
factor 0.10 to result in a small negative Hessian eigenvalue. While in all other cases the primitive Hessian has all
off-diagonal elements zero, we assign off-diagonal elements (between the TSRC coordinates only!) in this case in
order to couple the TSRC coordinates, which is crucial for describing the reaction coordinate. The values for the
off-diagonal elements are chosen such that only one negative eigenvalue results (all others are positive). This can
be achieved by using Hp,ij=Hp,ji=–(2Hp,iiHp,jj)1/2 (the minus sign serves to obtain an eigenvector of the negative
eigenvalue in which the TSRC coordinates have equal sign).
During the optimization, the Hessian is updated using either the BFGS (for minima) or Bofill (for transition
states) update schemes (see ref. 11 for details).
Optimization steps
Optimization techniques rely on a Taylor expansion of the energy E about the atomic coordinates X,1 which is
usually cut off at second order (quadratic model):
E k +1 = E k + gT X + XT HX + ..
(12)
Close to a minimum the energy surface will be quadratic, and as a result, the best guess for the step to take is
!
given by the Newton-Raphson step !X= -H-1g. The success of the step depends critically on the accuracy of the
curvature of the energy surface, i.e. the Hessian matrix, which in terms of number of geometry cycles should best
be recalculated at every step. More cost-effective11 however is to use an approximate Hessian Ha, with the
corresponding quasi-Newton step !X= -Ha-1g. This will lead to an increase of the number of geometry cycles, but
as the Hessian does not have to be calculated, it will also result in a decrease in the actual time used, saving in
practice up to 84% of computer time.11
Only close to the minimum is the energy surface in good approximation quadratic, and can the Taylor
expansion up to second order be trusted to be valid. This region is called the trust region, with a radius " that has a
default value of 0.2 Bohr. However, for the Baker set and weakly-bound test-set results given below, we used a
larger initial value of 0.4 Bohr, which is furthermore dynamically updated based on how well the quadratic model
represents the energy surface.14 If the quasi-Newton (QN) or Newton-Raphson (NR) step is smaller than ", the
QN/NR step is taken, else the restricted second order (RSO,14 also called level-shifted trust-region Newton
method)11 model is used. In the RSO model,14 a step is taken on the hypersphere of radius ", using a Lagrange
multiplier to ensure that the step length equals ".
Although at every point the QN/NR step is the best option, the geometry optimization is enhanced by using
GDIIS.15 The original paper proposed using the step as error vector, but later studies showed that it is more
S5
effective to use the gradient as error vector.16 When the GDIIS-QN step is larger than the trust radius and needs to
be restricted, the change in coordinates through the GDIIS equations is explicitly taken into account in the
restriction process. Furthermore, Farkas and Schlegel17 have proposed a set of four rules that the GDIIS vectors
have to fulfill. We have implemented the option to use either the step, the gradient or the “energy” vector (i.e.
Bij=giTHk-1gj)16 as error vector, and either with or without the Farkas-Schlegel rules. We observed that the best
performance is achieved by using the gradient as error vector, with a maximum of four GDIIS vectors, and
imposing the Farkas/Schlegel rules.
Handling of constraints
Baker used an elegant and efficient setup5 for dealing with constraints by introducing a Lagrangian multiplier #
for each constraint Ci(x), which facilitates imposing constraints that are not yet satisfied in the initial geometry
using the Lagrangian function:
Nc
L( x, " ) = E ( x ) + # "Ci ( x )
(13)
i
Each multiplier # is an additional degree of freedom within the optimization scheme, and results in one additional
!
negative Hessian eigenvalue. A stationary point of the Lagrangian function is found when grad(L) = 0, i.e. when
dL(x,!)/dx = 0 but in particular dL(x,!)/d! = 0, which means that all constraints are satisfied. Note that the
gradients of the Lagrangian function are given by:
Nc
dL( x, " ) /dx j = G j + # "i dCi ( x ) /dx j
(14)
i
dL( x, " ) /d"i = Ci ( x )
and the second derivative matrix by:
!
Nc
d 2 L( x, " ) /dx j dx k = H jk + # "i d 2Ci ( x ) /dx j dx k
i
2
d L( x, " ) /dx j d"i = dCi ( x ) /dx j
(15)
d 2 L( x, ") /d"i d" j = 0
Transition State Reaction Coordinates
!
Transition-state (TS) searches pose a greater challenge than “simple” minimization calculations, as there exists
one degree of freedom along which the energy should be maximized (in addition to the extra degrees of freedom,
associated with the inclusion of constraints, along which the energy must also be maximized). For all other
degrees of freedom the energy should be minimized. The quality of the (initial) Hessian and the Hessian update
method is therefore of vital importance for TS searches. The preparation of a model Hessian suited for TS
searches (vide supra) enhances the convergence properties of the TS searches. The ingredients for a smooth TS
S6
search are the TSRC coordinates, which the user has to define on input. These TSRC coordinates are used to
construct the model Hessian as well as to select the appropriate Hessian eigenvalue along which to maximize if
during the optimization more (or less) than one negative Hessian eigenvalue is observed. The selection of the
appropriate Hessian eigenvalue is done by choosing the one with the largest weight of the TSRC coordinates, in
combination with a weight function that depends on the difference with the lowest Hessian eigenvalue:
RC
w i = e" # i "# 1
0.005
%%c
2
ki
$ U jk
2
(16)
j=1 k
When the NR step is larger than the trust radius, the RSO method is used to restrict the step to the hypersphere
!
of the trust region. This amounts to finding a $ value that is in between the lowest and the second-lowest Hessian
eigenvalues, and which leads to the desired step with a length of ". However, depending on the Hessian
eigensystem this may not be possible at all times. Therefore, the Hessian eigenvalues (and the corresponding
gradients) along which to maximize the energy are temporarily multiplied by a factor -1. This leads to the RSO
method for minimizations for which it is guaranteed that a solution exists.
S7
Starting coordinates of molecules in Baker test-set (Bohr)
ACANIL01
O
N
C
C
C
C
C
C
C
C
H
H
H
H
H
H
H
H
H
6.7433416735
2.7512539832
-3.7595891909
-1.1366014546
0.0042737106
-1.5398535344
-4.1629370414
-5.2681107800
4.5738961053
4.1320702048
-4.6230675381
-0.0037780548
-0.7650560637
-5.3404165100
-7.3026657686
3.5808250620
5.9521203249
3.0821474360
3.0821474360
benzaldehyde
0.0000000000
-0.9199668058
-3.6204681333
-3.3872098429
-1.0031836305
1.1538710523
0.9183196876
-1.4672427143
0.8051152233
3.6405491893
-5.4717643625
-5.0876539749
3.0332653362
2.5875823952
-1.6480257355
-2.6647920859
4.6617819125
4.2349112447
4.2349112447
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
1.7007622024
-1.7007622024
0.0000000000
0.0000000000
1.3203724275
-0.1896215680
1.4959779753
-0.5925941212
-2.1147913804
2.2793324404
2.6808542134
-1.1884267581
-1.6814266653
2.5191205839
2.9918829189
-1.0441055740
-3.9073600209
-2.5692144717
3.9373524883
-0.0953103417
-3.3181015602
-2.0569602569
-1.2077535663
2.4606757746
3.2793490589
-4.8408251783
-2.1214772182
-0.4105139335
-3.3534713265
-2.8584292012
0.1052492525
2.9475903382
2.2195598714
5.3130657990
3.4669575743
1.1403259418
-0.2954284067
-0.2954284067
1.0044065219
1.0044065219
-1.5063099366
-1.5063099366
-1.5063099366
-1.5063099366
0.0000000000
0.0000000000
2.5013817152
-2.5013817152
4.1375406882
-4.1375406882
2.6698480410
2.6698480410
-2.6698480410
-2.6698480410
0.0000000000
0.0000000000
0.0000000000
0.0000000000
1.1338359981
-1.1338359981
3.0235626616
-3.0235626616
0.0000000000
2.4941929550
-2.4941929550
-3.5150316644
-3.5150316644
3.5150316644
3.5150316644
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
1.7677201632
-1.7677201632
0.0000000000
0.8997943198
0.8997943198
-1.7995886391
0.4769024955
-0.1589674985
-0.1589674985
-0.1589674985
O
C
C
C
C
C
C
C
H
H
H
H
H
H
0.0000000000
1.7987593946
-4.4058951880
-2.4302163638
-0.2218540433
1.6972673035
3.9768554826
-3.6804337975
-5.1014433292
-3.2498539233
-1.7454741756
0.5535143001
-0.8807169472
5.7352967900
4.0856267999
3.8685676971
C
C
C
C
C
C
H
H
H
H
H
H
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
1.6936030390
-1.6936030390
1.6936030390
-1.6936030390
N
N
C
C
C
C
C
C
C
C
C
C
C
C
H
H
H
H
H
H
H
H
H
H
H
H
acetylene
C
C
H
H
0.0000000000
0.0000000000
0.0000000000
0.0000000000
allene
C
C
C
H
H
H
H
0.0000000000
0.0000000000
0.0000000000
1.7677201632
-1.7677201632
0.0000000000
0.0000000000
0.0000000000
1.5584894515
-1.5584894515
0.0000000000
0.0000000000
0.0000000000
2.2815677027
-2.2815677027
2.2815677027
-2.2815677027
0.0000000000
0.0000000000
4.0494408785
-4.0494408785
4.0494408785
-4.0494408785
2.6345274540
-2.6345274540
1.3172637275
1.3172637275
-1.3172637275
-1.3172637275
4.6758915630
-4.6758915630
2.3379457815
2.3379457815
-2.3379457815
-2.3379457815
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
-2.2038894222
2.2038894222
-2.2038894222
2.2038894222
-2.2070662228
2.2070662228
-2.2070662228
2.2070662228
0.0000000000
0.0000000000
0.0000000000
0.0000000000
-3.9302267260
3.9302267260
-3.9302267260
3.9302267260
-3.9557397915
3.9557397915
-3.9557397915
3.9557397915
1.6783725186
-1.6783725186
1.6783725186
-1.6783725186
0.0000000000
0.0000000000
0.5648822305
-0.5648822305
-0.5648822305
0.5648822305
0.5634923517
-0.5634923517
-0.5634923517
0.5634923517
0.0000000000
0.0000000000
0.0000000000
0.0000000000
1.0222725320
-1.0222725320
-1.0222725320
1.0222725320
1.0738495675
-1.0738495675
-1.0738495675
1.0738495675
-0.4303131375
0.4303131375
0.4303131375
-0.4303131375
9.1797303761
-9.1797303761
5.3695570250
5.3695570250
-5.3695570250
-5.3695570250
2.7391294475
2.7391294475
-2.7391294475
-2.7391294475
1.3294862952
-1.3294862952
6.6793197706
-6.6793197706
6.3628346720
6.3628346720
-6.3628346720
-6.3628346720
1.8159664348
1.8159664348
-1.8159664348
-1.8159664348
10.0448317563
10.0448317563
-10.0448317563
-10.0448317563
-4.5596834563
0.0000000000
0.4079086834
4.3470436600
2.2989125280
-2.3286161020
0.0080681896
2.2891344003
3.0317870051
-2.5126891879
-0.0101164705
-4.7164936865
4.7050704451
-1.3879913800
4.0834657258
-4.8291667345
-4.8291667345
-6.4582328756
4.8360989641
4.8360989641
6.4098342016
-0.3802377304
-2.6205142045
-2.6205142045
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
1.6814572952
-1.6814572952
0.0000000000
1.6833237934
-1.6833237934
0.0000000000
0.0000000000
1.6878277887
-1.6878277887
caffeine
O
O
N
N
N
N
C
C
C
C
C
C
C
C
H
H
H
H
H
H
H
H
H
H
ammonia
N
H
H
H
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
benzidine
acetone
O
C
C
C
H
H
H
H
H
H
0.0000000000
-2.2595362151
-1.4347871225
1.1411808224
2.8972276388
2.0785184368
-0.5103443405
-1.4696981757
-3.5243404323
-4.2673340819
-2.7942683921
1.7780840801
4.8949599174
3.4703378645
benzene
ACHTAR10
O
O
N
C
C
C
C
H
H
H
H
H
H
H
H
H
6.1169594363
-0.4281183799
-2.9286935224
-3.4656164013
-1.5061149145
0.9961412279
1.5529020669
4.3100239391
4.6927731310
-0.0483891239
-4.4516781974
-5.4051670195
-1.9265366341
2.4915143918
S8
-1.3579649523
6.0035946545
-4.3469953034
-2.0214786850
2.4016649460
2.3896310664
-1.7351410012
-0.3965665155
-4.2862828595
-0.2338059695
3.6663062569
3.9123342656
3.8689942697
-6.5087149738
-6.0414687329
5.1502626147
5.1502626147
2.7518228859
5.1037416005
5.1037416005
2.6587883618
-8.3400356437
-6.4463452502
-6.4463452502
1,3-difluorobenzene
F
F
C
C
C
C
C
C
H
H
H
H
4.4509862937
-4.4509862937
2.2745931466
-2.2745931466
2.2746510932
-2.2746510932
0.0000000000
0.0000000000
4.0423269425
-4.0423269425
0.0000000000
0.0000000000
ethane
2.5307545457
2.5307545457
-1.3528497889
-1.3528497889
1.2738564047
1.2738564047
2.5872794109
-2.6664191923
-2.3725618150
-2.3725618150
4.6280488237
-4.7073077354
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
C
C
H
H
H
H
H
H
5.7744280988
-5.7744280988
0.7278545710
-0.7278545710
3.1106217391
-3.1106217391
3.3847993117
-3.3847993117
1.2377685123
-1.2377685123
1.4301426796
-1.4301426796
0.5520400805
-0.5520400805
4.7644595237
-4.7644595237
3.2499984362
-3.2499984362
0.0000000000
0.0000000000
-4.7025451180
4.7025451180
-3.6024924323
3.6024924323
-0.9879928692
0.9879928692
0.5712405534
-0.5712405534
3.2090770124
-3.2090770124
-6.7364640579
6.7364640579
-4.8006902051
4.8006902051
4.1394852226
-4.1394852226
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
O
C
C
H
H
H
H
H
H
5.2404816245
-5.2404816245
1.1570537551
-1.1570537551
1.5359683428
-1.5359683428
2.6570364766
-2.6570364766
5.6218667017
-5.6218667017
3.5488135300
-3.5488135300
7.5269778778
-7.5269778778
3.3453709195
-3.3453709195
0.0000000000
0.0000000000
-2.5515060844
2.5515060844
2.1516031685
-2.1516031685
-0.2747177008
0.2747177008
2.6220149256
-2.6220149256
4.0775601865
-4.0775601865
3.4123912749
-3.4123912749
6.1265701044
-6.1265701044
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
1.7998921359
1.1700814891
-0.4427300673
1.0709265512
0.0000000000
0.0000000000
-3.0339645013
2.7602612162
0.0865661776
3.0739195697
2.9984753577
-0.8406363794
1.2556106071
3.0367645867
-1.8973730406
1.2264794354
-0.1596897888
-1.7952491725
1.2812478780
-0.4061103776
-4.2039212942
-2.8220133604
-4.1585370224
-3.1281916081
-1.9274496235
0.5533222244
2.7951166661
5.4407882957
-3.9684871288
0.2440170158
-1.7750955024
-3.7917451573
-4.7732778874
-1.4218602449
1.0697292165
2.5463302073
2.7469591076
5.7186517345
6.9423075183
5.7840833923
-4.7146797589
-5.6007170921
-3.2278505048
1.9662426106
-0.1732023681
-1.3049911128
-0.0657104822
-0.0657104822
-0.8881734552
-2.1956541196
1.3527256647
1.3527256647
-2.1956541196
1.3527256647
1.3527256647
3.0363618918
-3.0363618918
0.0000000000
4.5475683923
3.5847502252
3.5847502252
-4.5475683923
-3.5847502252
-3.5847502252
O
C
C
C
C
H
H
H
H
-1.9030214241
0.6809819138
0.5734775872
-0.5753685961
0.0000000000
2.4013011875
-0.7574044511
-3.1706997287
-2.8981269221
-1.7083553458
1.5712715376
2.5648465740
-2.6448478161
0.1399745691
-0.8288785236
-0.7796921897
2.0556606244
1.6358456863
2.6515931879
4.3123566219
-0.6028216443
-2.7932914871
0.0751986389
O
O
N
N
N
C
C
C
C
C
C
H
H
H
H
H
H
H
H
H
0.0000000000
0.0000000000
0.0000000000
0.0000000000
2.1229004882
-2.1229004882
0.0000000000
2.1229004882
-2.1229004882
0.0000000000
0.8907097242
-1.2261270653
1.5045806421
2.0683434872
2.0683434872
-2.4603567438
-2.4603567438
-0.3851367880
0.0000000000
0.0000000000
0.0000000000
0.0000000000
1.6930689890
-1.6930689890
1.6719900859
-1.6719900859
0.0000000000
0.0000000000
1.3040964455
-1.3040964455
2.0768090816
-2.0768090816
2.5163904951
-2.5163904951
3.9939987516
-3.9939987516
-2.7115570299
1.3560027694
1.3560027694
-1.1487031091
-1.1487031091
2.9978275455
2.9978275455
-1.8493486908
-1.8493486908
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
3.9368391087
0.0000000000
-1.6271400548
-1.5552588229
-0.0651904381
0.0031311243
0.0439405564
-2.4430608128
1.6171293783
0.2591530667
1.6360598096
1.0906100804
-1.9165346596
-3.7791758444
2.2441717240
3.3993498640
-1.6379863218
-1.0929702054
-1.0991937071
0.9861261544
0.0000000000
0.0000000000
-0.1706316879
2.7569158479
-3.4369907595
-0.9638267251
0.7652697098
2.0425576532
0.5501227953
-0.6807021653
-0.2045617203
-2.6896672067
4.3540732145
3.2172553071
2.4763126542
-0.4680431662
0.1851679859
-4.2425936069
-3.7400107031
0.2493525730
5.0285854490
6.7554857214
-6.3898114507
-2.9885873892
1.7815227957
-4.4920500399
-2.4786015586
-5.3162641251
-0.0710045459
2.2227234000
4.7729517685
-4.5529495876
-1.9539596136
-6.3281771811
0.4447955065
-0.4753714331
2.4019443428
3.2164371594
0.1681049732
8.2542258083
2-hydroxybicyclopentane
O
C
C
C
C
C
H
H
H
H
H
H
H
H
0.0000000000
0.6127561157
-1.2524060881
-1.8999179628
2.6476459228
-0.1073209932
-2.8560102561
0.1334891998
3.5736810211
3.8069811050
-1.3357920208
-3.9012299267
-0.9340577957
1.5121816791
0.0000000000
1.7178782800
0.7543036674
-1.4918158979
0.0000000000
-0.5314032767
2.0556101735
3.4909403652
-1.3455861528
0.7983337878
-3.3415978304
-1.5404926982
0.2600298298
-0.8262002475
3.9763054942
-0.2567416031
1.7299107443
0.0621831143
-1.3619084898
-1.9909267561
2.0835309928
-1.2610334165
-0.0593319882
-2.9061371142
0.8588889111
-0.5391531045
-3.7457916036
3.4102048192
0.0000000000
0.0000000000
-0.9888963428
0.9888963428
1.6434445385
-0.7841792431
-2.0469823281
1.1877170327
hydroxysulphane
S
O
H
H
disilyl-ether
Si
Si
O
H
H
H
H
H
H
2.9495126852
0.4286436110
-1.4727499063
4.0579576902
0.0701756209
0.0701756209
-1.3618474095
-1.3618474095
-3.3800205031
histidine
dimethylpentane
C
C
C
C
C
C
C
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
1.4547876282
-1.4547876282
2.1445545504
-2.1445545504
2.1445545504
-2.1445545504
2.1445545504
-2.1445545504
furan
difuropyrazine
O
O
N
N
C
C
C
C
C
C
C
C
H
H
H
H
0.0000000000
0.0000000000
0.9704360937
-0.9704360937
0.9704360937
-0.9704360937
-1.9408721894
1.9408721894
ethanol
1,5-difluoronaphtalene
F
F
C
C
C
C
C
C
C
C
C
C
H
H
H
H
H
H
0.0000000000
0.0000000000
1.6808445513
1.6808445513
-1.6808445513
-1.6808445513
0.0000000000
0.0000000000
S9
0.0000000000
1.5564378813
0.7087897669
-2.2652276483
menthone
O
C
C
C
C
C
C
C
C
C
C
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
0.0000000000
-5.0659721163
-3.6034879641
-1.1377997194
0.6933582752
-0.8187936818
-3.4175581220
0.0832713856
3.1697784862
5.2396793740
2.7482073704
-5.7353404529
-6.8013953459
-3.1541951012
-4.8610977720
-1.7146320815
1.3353028580
-4.4104926442
-3.1022731235
-1.2751506361
0.8397929716
1.6571196240
4.0131457363
4.6990880959
7.0347568939
5.6688764509
4.5227783358
1.9890068391
1.4040260639
neopentane
0.0000000000
-1.2759209097
-1.4911122852
0.1218224964
-0.5332484664
-0.7642018905
-2.0641331109
-0.1824795775
1.1178842527
0.0000000000
3.9164865882
0.6922390336
-2.4428926379
-3.5014033901
-0.9026408224
2.1264781053
-2.4892597645
-1.6489160089
-4.1276730300
0.1934017618
-2.1081053115
1.1553128464
1.1016773541
0.0299064950
1.0585968570
-1.9848698773
5.0167778642
4.1353100814
4.8509633466
4.8350295665
0.4988504927
-2.0199506584
-2.0240250793
0.2469914093
2.7944276610
2.7786874577
-4.6618476891
0.3378091560
2.0585121222
1.0369291406
0.7566080964
0.4304593021
-2.3971538805
-3.5860303977
-1.8254234826
-0.1094906764
4.5693816544
2.7714295791
-6.2062554367
-4.9615771850
-4.9619504893
-1.5747354153
4.0774705629
1.9011131139
1.5689828577
0.9513248658
2.9726456776
-0.2582123254
0.0000000000
-3.8262984260
-1.3275291709
-1.4933539788
0.9402323872
1.5873508800
4.3731533082
-4.0017434683
-4.0017434683
-5.4898017896
-0.5794406045
-0.5794406045
-3.4726445808
2.5961434903
5.3338963865
5.3338963865
4.6073232530
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
1.6771680976
-1.6771680976
0.0000000000
1.6805369416
-1.6805369416
0.0000000000
0.0000000000
1.6938411125
-1.6938411125
0.0000000000
0.0000000000
-0.0307371822
-1.6302003168
1.9316390571
1.6491159373
-0.9599087477
-0.9599087477
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
1.6969519068
-1.6969519068
C
C
C
C
C
H
H
H
H
H
H
H
H
H
H
H
H
4.3049245548
0.0502472120
-1.3508783386
-4.2083887213
-0.1992065783
2.6061876658
3.3153790101
1.2846112117
1.2846112117
-1.2128146457
-5.0469559196
-5.0469559196
-4.8791103342
-1.4266859322
2.5675883965
2.5675883965
5.3898587304
O
N
N
N
N
N
C
C
C
C
C
C
H
H
H
H
H
1.5916930881
-1.1078124728
2.6143261592
-1.8166631951
2.5780491273
-1.9297963534
-1.9297963534
1.3150099345
-1.3150099345
1.3150099345
-1.3150099345
2.6509541041
-2.6509541041
2.6509541041
-2.6509541041
1.3595784802
-1.3595784802
2.3271380660
-2.3271380660
2.3271380660
-2.3271380660
4.6944935141
-4.6944935141
4.6944935141
-4.6944935141
5.4006870954
1.6745046894
-3.2977881039
2.4143500339
-2.0786863903
-1.0594193125
1.0450647735
-1.5782549038
-0.1007895891
-2.6656839215
3.1317795772
-0.3374432800
0.4229624123
-4.1154821045
3.7020414107
-2.9660143784
0.4081719917
0.0000000000
-4.0122480903
-2.6629858601
2.9809395408
1.9657781562
6.3338302157
-1.5886952821
-0.8359572675
-5.7640104211
-5.0779459969
0.5243588431
3.6706594226
-7.7366322874
-6.5145288177
4.4304595058
6.6862170583
7.6007612807
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
2.8756270087
-2.8756270087
0.0000000000
0.0000000000
2.8718881844
-2.8718881844
0.0000000000
4.5501206960
-4.5501206960
0.0000000000
2.8892367312
-2.8892367312
5.1495325026
-5.1495325026
2.9111238472
-2.9111238472
0.0000000000
0.0000000000
1.6602440278
1.6602440278
-3.3204880547
3.3161708326
-1.6580854160
-1.6580854160
5.2540268178
-2.6270134094
-2.6270134094
3.3362032089
-1.6681016042
-1.6681016042
2.9730839771
2.9730839771
1.6807381371
1.6807381371
-3.3614762743
-5.9461679533
0.5000983316
0.5000983316
0.5000983316
-0.6564595188
-0.6564595188
-0.6564595188
0.0455078719
0.0455078719
0.0455078719
-2.7167608516
-2.7167608516
-2.7167608516
-0.4683799864
-0.4683799864
3.2959941533
3.2959941533
3.2959941533
-0.4683799864
1,3,5-trifluorobenzene
naphtalene
C
C
C
C
C
C
C
C
C
C
H
H
H
H
H
H
H
H
0.0000000000
0.8897000000
-0.8897000000
-0.8897000000
0.8897000000
1.5138000000
1.5138000000
0.2654000000
-1.5138000000
-0.2654000000
-1.5138000000
-1.5138000000
-0.2654000000
-1.5138000000
0.2654000000
1.5138000000
1.5138000000
1,3,5-trisilacyclohexane
Si
Si
Si
C
C
C
H
H
H
H
H
H
H
H
H
H
H
H
methylamine
N
C
H
H
H
H
H
0.0000000000
0.8897000000
0.8897000000
-0.8897000000
-0.8897000000
1.5138000000
0.2654000000
1.5138000000
0.2654000000
1.5138000000
1.5138000000
-0.2654000000
-1.5138000000
-1.5138000000
-1.5138000000
-1.5138000000
-0.2654000000
pterin
mesityl-oxide
O
C
C
C
C
C
C
H
H
H
H
H
H
H
H
H
H
0.0000000000
-0.8897000000
0.8897000000
-0.8897000000
0.8897000000
-0.2654000000
-1.5138000000
-1.5138000000
1.5138000000
1.5138000000
0.2654000000
-1.5138000000
-1.5138000000
-0.2654000000
1.5138000000
0.2654000000
1.5138000000
4.5662599301
4.5662599301
-4.5662599301
-4.5662599301
2.3012120990
2.3012120990
-2.3012120990
-2.3012120990
0.0000000000
0.0000000000
6.3391558961
6.3391558961
-6.3391558961
-6.3391558961
2.3637514068
2.3637514068
-2.3637514068
-2.3637514068
F
F
F
C
C
C
C
C
C
H
H
H
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
4.4512477126
-4.4512477126
0.0000000000
2.2750112183
-2.2750112183
0.0000000000
2.2744659308
-2.2744659308
0.0000000000
4.0417664625
-4.0417664625
0.0000000000
2.5699290652
2.5699290652
-5.1398581302
1.3134783392
1.3134783392
-2.6269566786
-1.3131635176
-1.3131635176
2.6263270339
-2.3335149555
-2.3335149555
4.6670299102
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
-0.6980139011
0.3490069505
0.3490069505
0.0000000000
0.0000000000
0.0000000000
water
O
H
H
S10
0.0000000000
1.4815001599
-1.4815001599
Starting coordinates of molecules in weakly-bound test set (Å)
H2O ··· HCC–
Ar ··· H2
Ar
H
H
0.0000000000
0.3687590000
-0.3687590000
0.0000000000
0.0000000000
0.0000000000
2.7945120000
-1.3972560000
-1.3972560000
0.5960700000
1.5127000000
-2.1087700000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
-2.0851820000
2.0851820000
0.0000000000
0.0000000000
0.0000000000
0.8898650000
-0.8898650000
0.0000000000
0.0000000000
0.0000000000
0.8898650000
-0.8898650000
1.9949820000
3.0851460000
1.6310530000
1.6310530000
1.6310530000
-1.9949820000
-3.0851460000
-1.6310530000
-1.6310530000
-1.6310530000
0.0000000000
0.0000000000
-0.8869270000
0.8869270000
0.0000000000
0.0000000000
0.0000000000
0.8144430000
-0.8144430000
1.7948300000
2.8855470000
1.4234140000
1.4234140000
1.4234140000
-1.9472110000
-2.3344690000
-2.3344690000
-2.3344690000
-0.2735483200
-1.9813209100
0.5955431100
0.4141232800
-0.4636475800
3.3845412800
2.8476266800
2.4548279300
2.9968902100
-0.1412416300
-0.9187997300
0.9510259000
-0.9550035600
0.8701983600
-1.3332959500
-0.6102432700
-1.7863965400
-1.5866935800
0.0000000000
0.0000000000
0.0000000000
1.1497050000
-0.0000240000
-1.1496810000
-0.3532200000
0.6096910000
-0.4260590000
-0.1069430000
0.2765300000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
-0.3891100000
0.4251780000
0.1383230000
0.1383230000
0.0104870000
-0.3232000000
0.0000000000
0.0000000000
0.7659710000
-0.7659710000
0.0000000000
0.0000000000
0.1281240000
0.1039830000
-0.7234290000
0.0423780000
0.5710690000
0.5573920000
-0.6795160000
-0.1643450000
0.1704030000
0.3902120000
-0.0000030000
-0.5323960000
0.5677240000
-0.4315940000
O
2.1665530000
H
2.2927980000
H
1.1492120000
H
-2.9967370000
C
-0.6803380000
C
-1.9314880000
Charge -1
Ar ··· HF
H
F
Ar
Ar2
Ar
Ar
H2O ··· NH3
O
H
H
N
H
H
H
CH4 ··· CH4
C
H
H
H
H
C
H
H
H
H
0.0000000000
0.0000000000
1.0275280000
-0.5137640000
-0.5137640000
0.0000000000
0.0000000000
1.0275280000
-0.5137640000
-0.5137640000
CH4 ··· NH 3
C
H
H
H
H
N
H
H
H
0.0000000000
0.0000000000
0.5120670000
0.5120670000
-1.0241350000
0.0000000000
0.9404380000
-0.4702190000
-0.4702190000
CHBr3 ··· H2O2
C
Br
Br
H
Br
O
H
O
H
-0.1239074100
0.4852767400
1.2951035000
-0.3220035800
-1.8140311200
0.5791897800
0.9601966600
-0.4625118300
-1.2496694100
H2O ··· NH4+
N
-1.1691990000
H
-1.5153270000
H
-1.5227460000
H
-1.5227460000
H
-0.1095670000
H
2.1422270000
H
2.1422270000
O
1.5551310000
Charge +1
H2O ··· OH–
O
-1.1638150000
O
1.1691450000
H
-1.5066060000
H
-0.0163070000
H
1.5175840000
Charge -1
F
H
H
H
O
-1.2624680000
1.5375300000
-1.5404820000
-1.5404820000
0.6549290000
2.1509740000
F
F
H
H
-0.0099470000
-0.0045370000
0.0117550000
0.0003340000
-0.9418590000
0.4700210000
0.4742330000
-0.0008020000
-0.9664580000
0.4806110000
0.4806110000
0.0033230000
0.0015420000
0.0015420000
-0.0003680000
0.0000000000
0.0000000000
0.8339220000
-0.8339220000
0.0000000000
-0.7746450000
0.7746450000
0.0000000000
-0.3844640000
0.4113030000
0.2626440000
0.0040810000
-0.2935640000
-0.2500550000
-0.2056270000
0.3871320000
-0.2962760000
0.3648260000
-0.2111950000
0.6057980000
-0.5802760000
0.1856730000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
-1.9124520000
-1.0053010000
1.0907640000
1.0907640000
0.7362250000
0.1441680000
-0.1324880000
0.1541210000
0.1541210000
-0.3199230000
0.0000000000
0.0000000000
0.7694420000
-0.7694420000
0.0000000000
1.7428120000
-0.9961040000
-1.6078510000
0.8611430000
-0.1514110000
0.3612010000
-0.3254390000
0.1156500000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.3112110000
-0.4919020000
0.2135730000
0.2135730000
-0.3112110000
0.4919020000
-0.2135730000
-0.2135730000
0.0000000000
0.0000000000
-0.8141350000
0.8141350000
0.0000000000
0.0000000000
-0.8141350000
0.8141350000
NH3 ··· NH3
N
H
H
H
N
H
H
H
H2O ··· H3O+
O
1.1952950000
O
-1.1962250000
H
-1.6697680000
H
0.0009870000
H
-1.6928460000
H
1.6838960000
H
1.6786610000
Charge + 1
0.4519750000
0.2340610000
-0.4118190000
-0.0733040000
-0.0634340000
-0.8847830000
0.7473040000
HF ··· HF
H2O ··· H2O
O
O
H
H
H
H
2.3406100000
-1.4887830000
-1.9727560000
1.1209290000
HF ··· H 2O
H2O ··· CN–
O
1.5276760000
H
1.6282040000
H
0.5332700000
N
-1.2766640000
C
-2.4124860000
Charge -1
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
HCl ··· HCl
Cl
Cl
H
H
F– ··· HF
F
0.0000000000
H
0.0000000000
F
0.0000000000
Charge -1
1.9085280000
0.9524560000
2.3439890000
-1.0139360000
-1.3970810000
-1.4087530000
-1.3852030000
-0.3415410000
0.6190220000
-0.3711390000
0.1343970000
-0.0541620000
0.0134240000
S11
-1.5361740000
-0.9097660000
-2.1382650000
-2.1382650000
1.5361740000
0.9097660000
2.1382650000
2.1382650000
Table S1. Number of geometry cycles for optimizing molecules of the Baker test set (at RHF/STO-3G), compared
for various geometry optimization schemes.
Baker ref. 18
Lindh ref. 12
Eckert ref. 16
Bakken ref. 11
this work
water
6
4
4
4
4
ammonia
6
5
6
5
5
ethane
5
4
4
3
3
acetylene
6
5
6
4
5
allene
5
5
4
4
4
hydroxysulphane
8
8
7
7
6
benzene
4
3
3
3
3
methylamine
6
5
5
4
4
ethanol
6
5
5
4
5
acetone
6
5
5
4
5
disilyl-ether
8
11
9
8
7
1,3,5-trisilacyclohexane
8
8
6
9
6
benzaldehyde
6
5
5
4
4
1,3-difluorobenzene
5
5
5
4
4
1,3,5-trifluorobenzene
5
4
4
4
4
neopentane
5
5
4
4
4
furan
8
7
6
5
5
naphtalene
5
6
6
5
5
1,5-difluoronaphtalene
6
6
6
5
5
2-hydroxybicyclopentane
15
10
9
9
7
ACHTAR10
12
8
9
8
6
ACANIL01
8
8
8
7
8
benzidine
9
10
7
9
6
10
9
9
8
7
difuropyrazine
9
7
7
6
6
mesityl-oxide
7
6
6
5
6
histidine
19
20
14
16
12
dimethylpentane
12
10
10
9
6
caffeine
12
7
7
6
6
menthone
13
14
10
12
9
240
215
196
185
167
8
7
7
6
6
molecule
pterin
sum
rounded average
S12
Table S2. Number of geometry cycles for optimizing molecules of the Baker test set (at PW91/TZ2P), compared
for various geometry optimization schemes.
ADFcarta
QUILDcart
QUILDcart,w=1
QUILDdeloc
QUILDdeloc,w=1
water
4
4
4
3
4
ammonia
4
5
5
5
5
ethane
4
4
4
4
4
acetylene
4
4
5
4
5
allene
5
4
5
4
5
molecule
hydroxysulphane
10
9
9
6
6
benzene
3
3
3
3
3
methylamine
4
4
5
4
4
ethanol
6
4
5
5
5
acetone
5
5
5
4
5
disilyl-ether
11
6
10
8
10
1,3,5-trisilacyclohexane
10
4
4
5
6
benzaldehyde
7
5
5
5
5
1,3-difluorobenzene
5
4
4
4
4
1,3,5-trifluorobenzene
5
4
4
4
4
neopentane
4
4
4
4
4
furan
6
6
6
5
6
naphtalene
6
5
5
5
5
1,5-difluoronaphtalene
6
5
5
5
5
14
11
11
8
7
ACHTAR10
6
7
7
7
6
ACANIL01
7
6
6
6
6
benzidine
6
6
6
6
6
pterin
8
8
8
7
7
difuropyrazine
7
8
7
6
7
mesityl-oxide
6
6
6
5
5
34
48
47
13
13
dimethylpentane
5
5
4
4
4
caffeine
8
7
7
7
7
12
13
12
8
8
222
214
218
164
171
7
7
7
5
6
2-hydroxybicyclopentane
histidine
menthone
sum
rounded average
a) using old-style optimizer in ADF
S13
Table S3. Number of geometry cycles for optimizing molecules of the weak-coordinates test set (at PW91/TZ2P),
compared for various geometry optimization schemes.
molecule
ADFcarta
QUILDcart
QUILDcart,w=1
QUILDdeloc
QUILDdeloc,w=1
Ar ··· H2
109
9
12
8
13
Ar ··· HF
21
12
14
8
13
Ar ··· Ar
29
5
5
6
3
CH4 ··· CH4
38
7
10
7
10
CH4 ··· NH3
115
8
12
7
12
CHBr3 ··· H2O2
202
31
40
14
30
4
3
4
4
4
20
20
23
10
11
20
14
18
10
14
34
14
16
15
16
29
19
19
12
14
17
9
16
12
15
10
10
10
8
9
18
18
17
15
14
HCl ··· HCl
39
26
26
12
15
HF ··· H2O
12
13
16
10
13
HF ··· HF
18
18
23
11
13
NH3 ··· NH3
13
6
12
6
11
748
242
293
175
230
42
14
16
10
13
F– ··· HF
H2O ··· CN
–
H2O ··· H2O
H2O ··· H3O
+
H2O ··· HCC
–
H2O ··· NH3
H2O ··· NH4
H2O ··· OH
+
–
sum
rounded average
a) using old-style optimizer in ADF
S14
Table S4. Optimized geometrya of H2O2 using numerical gradients with different finite-difference steps
difstep (a.u.)
initial geometry
O-H (Å)
O-O (Å)
H-O-O (°)
H-O-O-H (°)
0.97
1.46
100.0
50.0
2.0·10
-2
0.9756
1.4693
99.86
110.69
1.0·10
-2
0.9756
1.4692
99.87
110.70
1.0·10
-3
0.9756
1.4692
99.87
110.78
1.0·10
-4
0.9756
1.4692
99.87
110.76
1.0·10-5
0.9756
1.4692
99.87
110.75
a) obtained using PW91/TZ2P with Spin-Orbit ZORA Hamiltonian and numerical gradients
S15
Geometry optimization with QM/MM treatment of water dimer
$ADFBIN/quild << eor
TITLE QM/MM calculation setup by pdb2adf: M. Swart, 2007
GEOMETRY
END
ATOMS
O
0.0000
H
-0.5220
H
-0.5220
O
0.0000
H
0.0570
H
0.9110
END
0.0000
0.2660
0.2660
-3.2000
-2.2440
-3.4950
0.0000
-0.7570
0.7570
0.0000
0.0000
0.0000
QUILD
NR_REGIONS=2
REGION 1
1-3
SUBEND
REGION 2
4-6
SUBEND
INTERACTIONS
TOTAL
description 1
REPLACE region 1
description 3 for description 2
SUBEND
DESCRIPTION 1 NEWMM
QMMM
FORCE_FIELD_FILE $ADFRESOURCES/ForceFields/amber95.ff
QMMM_INFO
1 OW
QM
-0.8340 H2O 1
O
2
3
2 HW
QM
0.4170 H2O 1
H1
1
3 HW
QM
0.4170 H2O 1
H2
1
4 OW
MM
-0.8340 H2O 2
O
5
6
5 HW
MM
0.4170 H2O 2
H1
4
6 HW
MM
0.4170 H2O 2
H2
4
SUBEND
END
SUBEND
DESCRIPTION 2 NEWMM
QMMM
FORCE_FIELD_FILE $ADFRESOURCES/ForceFields/amber95.ff
QMMM_INFO
1 OW
QM
-0.8340 H2O 1
O
2
3
2 HW
QM
0.4170 H2O 1
H1
1
3 HW
QM
0.4170 H2O 1
H2
1
SUBEND
END
SUBEND
DESCRIPTION 3
XC
GGA Becke-Perdew
END
BASIS
type TZP
core small
END
SCF
Converge 1.0e-5 1.0e-5
Iterations 99
END
INTEGRATION 5.0 5.0 5.0
CHARGE
0.0
SUBEND
END
ENDINPUT
eor
S16
References
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
Jensen, F. Introduction to computational chemistry; Wiley & Sons, 1998.
Billeter, S. R.; Turner, A. J.; Thiel, W. Phys Chem Chem Phys 2000, 2, 2177-2186.
Pulay, P.; Fogarasi, G. J Chem Phys 1992, 96, 2856-2860.
von Arnim, M.; Ahlrichs, R. J Chem Phys 1999, 111, 9183-9190.
Baker, J. J Comput Chem 1997, 18, 1079-1095.
Baker, J.; Kessi, A.; Delley, B. J Chem Phys 1996, 105, 192-212.
Baker, J.; Kinghorn, D.; Pulay, P. J Chem Phys 1999, 110, 4986-4991.
Lindh, R.; Bernhardsson, A.; Schutz, M. Chem Phys Lett 1999, 303, 567-575.
Swart, M.; Bickelhaupt, F. M. Int J Quant Chem 2006, 106, 2536-2544.
Wilson Jr., E. B.; Decius, J. C.; Cross, P. C. Molecular Vibrations—The Theory of Infrared and Raman Vibrational
Spectra; McGraw-Hill, Inc.: New York, 1955.
Bakken, V.; Helgaker, T. J Chem Phys 2002, 117, 9160-9174.
Lindh, R.; Bernhardsson, A.; Karlstrom, G.; Malmqvist, P. A. Chem Phys Lett 1995, 241, 423-428.
Fischer, T. H.; Almlof, J. J Phys Chem 1992, 96, 9768-9774.
Yeager, D. L.; Jensen, H. J. A.; Jørgensen, P.; Helgaker, T. U. In Geometrical derivatives of energy surfaces and
molecular properties; Jørgensen, P.; Simons, J., Eds.; Reidel Publishing: Dordrecht, 1986, p 229-241.
Csaszar, P.; Pulay, P. J Molec Struct 1984, 114, 31-34.
Eckert, F.; Pulay, P.; Werner, H. J. J Comput Chem 1997, 18, 1473-1483.
Farkas, O.; Schlegel, H. B. Phys Chem Chem Phys 2002, 4, 11-15.
Baker, J. J Comput Chem 1993, 14, 1085-1100.
S17