ORIGINAL PAPER
Higher order multipole moments for moleculardynamics simulations
Nuria Plattner & Markus Meuwly
Received: 16 September 2008 /Accepted: 6 January 2009 /Published online: 5 March 2009# Springer-Verlag 2009
Abstract In conventional force fields, the electrostaticpotential is represented by atom-centred point charges. Thischoice is in principle arbitrary, but technically convenient.Point charges can be understood as the first term ofmultipole expansions, which converge with an increasingnumber of terms towards the accurate representation of themolecular potential given by the electron density distribu-tion. The use of multipole expansions can thereforeimprove the force field accuracy. Technically, the imple-mentation of atomic multipoles is more involved than theuse of point charges. Important points to consider are theorientation of the multipole moments during the trajectory,conformational dependence of the atomic moments andstability of the simulations which are discussed here.
Keywords Multipole moments . Molecular dynamics .
Point charges . Electron density distribution .
Distributed multipole analysis .
Cumulative atomic multipole moments .
Theory of atoms in molecules
Introduction
In atomistic force fields such as CHARMM or AMBER[12, 26], the electrostatic potential is usually represented bypoint charges. Charges centred on an atom are not anintrinsic physical property but rather a convenient concept,and are typically obtained from a population analysis of theHartree-Fock wave function from electronic structure
calculations [13], or by fitting atomic charges that optimallyreproduce the electrostatic potential or interaction energies[2, 20]. The physical property from which charges arederived is the molecular electron density ρ(r), where r is apoint in three-dimensional (3D) space. It has been demon-strated in the past that point charges can be understood as afirst term in a multipole expansion [21, 22]. The potentialthat corresponds to the multipole expansion convergestowards the potential derived from ρ(r) on which theanalysis is based. Several algorithms have been developedfor obtaining atom centred multipole expansions. The mostwidely used are distributed multipole analysis (DMA) [25],cumulative atomic multipole moments (CAMM) [21] andthe theory of atoms in molecules (AIM) [1]. All thesemethods include a partitioning of the electron density andan integration of ρ(r) over a suitably defined spaceoccupied by a particular atom for all desired multipoleranks. The partitioning of ρ(r), the allocation of atomicmoments to particular atoms, and therefore the resultingmultipole expansion, its convergence, basis set dependence,conformational dependence and transferability differ be-tween the three methods. A direct comparison of DMA (inits previous form [22]) and AIM has been published [7].
Atomic multipole moments have been found to beimportant for modelling molecular interactions in com-plexes, solids and proteins [17]. In particular, they havebeen applied successfully to organic crystal structureprediction [3], and they have also been shown to improveresults of molecular dynamics (MD) simulations for severalsystems [14–16].
In the following, practical issues in using atomicmultipole moments in MD simulations are considered.The properties of electrostatic multipoles are compared todifferent point charge models and specific problemsconcerning the implementation of atomic multipoles in the
J Mol Model (2009) 15:687–694DOI 10.1007/s00894-009-0465-6
N. Plattner :M. Meuwly (*)Department of Chemistry, University of Basel,4056 Basel, Switzerlande-mail: [email protected]
context of MD simulations are then analysed and the effectsdemonstrated for model systems.
Choice of atomic multipole parameters for MDsimulations
The electrostatic energy of a multipole expansion up toquadrupole on two atoms a and b in spherical tensornotation can be written as
Eelec ¼ E qaqbð Þ þX3
k¼1
E qamkb
� �þX3
l¼1
E mlaqb
� �
þX3
k¼1
X3
l¼1
E mlam
kb
� �þX5
m¼1
E qaΘmb
� �þX5
n¼1
E Θnaqb
� �
þX3
k¼1
X5
m¼1
E mkaΘ
mb
� �þX5
n¼1
X3
l¼1
E Θnam
kb
� �
þX5
m¼1
X5
n¼1
E ΘnaΘ
mb
� �;
ð1Þwhere qa is the charge of atom a, mk
b is the direction k ofthe dipole moment vector on atom b, Θn
a is the element n ofthe quadrupole tensor on atom a [23]. As an example, theinteraction between a charge qa and the Θ20
b quadrupoletensor element is given by
E qaq20b
� � ¼ qaΘ20b
1
2
3 rbz� �2�1
� �
R3; ð2Þ
where rbz and is the z-element of a unit vector pointing fromb to a and R is the interatomic distance.
The practical use of atomic multipole moments for MDsimulations should take a number of considerations intoaccount. First, to justify the additional computational cost,the electrostatic potential derived from the multipoleexpansion has to be significantly more accurate than acorresponding point charge potential. Second, the compu-
tational expense depends sensitively on the multipole rankat which the expansion can be truncated with only smallremaining differences between the exact electrostaticpotential and the potential from the multipole expansion.Third, the accuracy of the electrostatics inside the van derWaals radii is of less importance since at very short rangethe interaction is dominated by the repulsive part of theLennard-Jones (LJ) potential. Fourth, for the parametrisa-tion of larger molecules, the atomic multipole momentsshould be transferable between smaller fragments withoutlarge errors (e.g. transferability between amino acids indifferent peptides). Finally, the change of the multipolemoments with molecular conformation should either besmall or should be described by an analytical function. Inthe following, the properties of atomic moments obtainedfrom DMA [24] are analysed, using alanine and glycine asexamples, covering the points mentioned above.
Convergence and accuracy of distributed multipolepotentials
For all the molecules and ab initio methods studied, thedistributed multipole expansion converges to within 10% atrank two (i.e. quadrupole). To illustrate this, Fig. 1 showsthe difference between potential fr (derived from themultipole expansion) and the ab initio potential φ for waterfor different levels of accuracy. At rank two, fr typicallydiffers by a few percent on average from φ, and issignificantly more accurate than potentials from pointcharges for the same molecule, using Mulliken charges orcharges fitted according to the Merz-Kollman scheme [2].Quantitative evaluations for water and alanine are shown inTable 1. The evaluation is carried out with a probe chargeof 1, based on a cubic grid of 10×10×10 Å for water and15×15×15 Å for alanine, with 101 points in each directionoutside the van der Waals radii1.
Fig. 1 Differences (absolute values) between fr (potential frommultipole expansion) and φ (potential from ab initio calculations) forwater, based on a Hartree-Fock calculation with a 6-31G* basis set.
From left to right: distributed multipoles up to rank 0, rank 1, rank 2 andrank 3. Colour coding: black >10 kJ/mol, red 5–10 kJ/mol, green 2.5–5 kJ/mol, blue 1–2.5 kJ/mol, yellow 0.5–1 kJ/mol, white <0.5 kJ/mol
1 Multipole parameters identical to those from Table 2 for allmolecules considered can be obtained from the authors on request.
688 J Mol Model (2009) 15:687–694
Conformational dependence of distributed multipolepotentials
To illustrate the conformational dependence of multipole-derived potentials, two conformations (c1 and c2) ofalanine and glycine, are considered (see Figs. 2, 3). Table 2provides coordinates and multipole moments for glycinefrom which the potentials in Fig. 4 were calculated. Thisexample shows that the differences between � and fr areconsiderably larger if the multipoles are transferred toanother conformation. A quantitative analysis of the differ-ences between fr and φ is provided in Table 3, wheredifferences between the electrostatic potentials from abinitio calculations and different charge models are reported.In order to calculate the data for another conformation, themultipoles are rotated according to the geometry of theatoms in the new conformation. The tables also show datafor the conformational dependence for point charges, whichare calculated according to the Merz-Kollmann scheme [2].Also for point charges, the deviations in another conforma-tion are considerably larger. The data show that theconformational dependence of fr has to be taken intoaccount. Otherwise the accuracy gained by using multipoleexpansions instead of point charges is likely to be
compromised. The conformational dependence of fr hasalready been investigated in earlier studies for the DMAand for the CAMM algorithm [9, 18]. There are differentpossible solutions to this problem, not all of which areapplicable in general.
For some molecules, the change of the atomic multipolemoments with conformation can be described explicitly byan analytical function. This has been done successfully forCO [16]. Unlike this particularly simple case, moreelaborate methods need to be used for molecules such asglycine and alanine. For the conformational dependencearound torsion angles, the use of short Fourier series hasbeen shown to significantly improve the electrostaticpotential [10]. A combined solution for atomic multipolestogether with inter- and intramolecular polarisation hasbeen suggested [19], but no explicit dynamics has beencarried out with this procedure for larger molecules such asamino acids. Furthermore, it is unclear how well thisapproach is able to reproduce atomic multipole moments ofanother conformation, since only the combined electrostaticplus polarisation energies have been evaluated and theindividual contribution of each term to different conforma-tions is not available.
Another possible solution is to use conformationallyaveraged multipole moments to better reproduce � onaverage. This has been done here for glycine and alanine byshifting multipole moments with ranks higher than zero tofewer sites. Furthermore, the relative atomic radii forLebedev integration in the distributed multipole analysisbased on the self-consistent field (SCF) density from thecorresponding calculation can be varied to minimiseconformational dependence. The data are shown in Table 3.This is not a general solution, but the data presented hereshows that, at least for the cases considered here, such aprocedure reduces the error due to conformational changes.
Finally, it may also be possible to use an iterativescheme in which the multipole moments are recalculatedperiodically. Such an approach has already been proposedand tested for energy minimisations of crystal structures [8].
Implementation in MD simulations
For the implementation of atomic multipole moments intoatomistic simulations, additional interaction terms and their
Fig. 2 Alanine conformationsc1 and c2
Fig. 3 Glycine conformations c1 and c2
Table 1 Comparison for water and alanine. HF Hartree-Fock, DMAdistributed multipole analysis
HF 6-31G* B3LYP Aug-cc-pVTZ
Water Alanine Water Alanine
DMA, rank 0
ΔE (kJ/mol) 3.27 2.87 2.70 2.52
ΔE (%) 64 2383 68 1653
DMA, rank 1
ΔE (kJ/mol) 0.63 0.333 0.71 0.36
ΔE (%) 41 178 61 73
DMA, rank 2
ΔE (kJ/mol) 0.05 0.05 0.17 0.06
ΔE (%) 1 6 4 8
Mulliken charges
ΔE (kJ/mol) 0.52 0.84 2.18 6.05
ΔE (%) 23 687 60 2993
Merz-Kollman
ΔE (kJ/mol) 0.28 0.08 0.44 0.08
ΔE (%) 18 35 37 18
J Mol Model (2009) 15:687–694 689
Tab
le2
Coo
rdinates
andmultip
olemom
entsforglycineconformations
(HF6-31
G*).The
multip
oles
onthehy
drog
ensaretransferredto
theheavyatom
s
Coo
rdinates
(Å)
Atom
name(highestrank
)Con
form
ation1
Con
form
ation2
HNH2(rank0)
x=1.99
3,y=
0.61
9,z=
0.80
2x=
−1.919
,y=
0.69
5,z=
−0.804
qa0.21
8431
0.21
8202
HNH2(rank0)
x=1.99
3,y=
0.62
1,z=
−0.800
x=−1
.919
,y=
0.69
6,z=
0.80
4
qa0.21
8482
0.21
8224
N(rank2)
x=1.95
0,y=
0.02
0,z=
0.00
0x=
−1.906
,y=
0.10
0,z=
0.00
0
qa−0
.445
558
−0.447
569
μb
0.00
0311
−0.568
593
0.15
5322
0.00
0311
−0.568
593
0.15
5322
Θc
0.24
5474
0.00
1489
−0.000
954
0.00
8565
0.82
6711
0.25
1683
0.00
0695
0.00
0380
−0.134
308
−0.815
526
C(rank2)
x=0.72
2,y=
−0.723
,z=
0.00
0x=
−0.743
,y=
−0.750
,z=
0.00
0
qa−0
.094
932
−0.092
121
μb
0.00
0089
−0.056
946
0.37
5556
0.00
0089
−0.056
946
0.37
5556
Θc
−0.612
798
0.00
0074
−0.000
156
0.69
7168
0.01
2662
−0.594
929
−0.000
021
−0.000
036
0.7112
72−0
.136
218
H(rank0)
x=0.68
5,y=
−1.381
,z=
−0.867
x=−0
.767
,y=
−1.402
,z=
0.86
6
qa0.15
8468
0.16
0195
H(rank0)
x=0.68
6,y=
−1.381
,z=
0.86
6x=
−0.767
,y=
−1.403
,z=
−0.865
qa0.15
8459
0.16
0199
CCOOH(rank2)
x=−0
.540
,y=
0.10
8,z=
0.00
0x=
0.62
8,y=
−0.095
,z=
0.00
0
qa0.30
4407
0.30
4592
μb
0.00
0013
−0.121
391
0.05
9910
0.00
0013
−0.121
391
0.05
9910
Θc
−0.402
671
−0.000
040
−0.000
037
−0.135
468
0.1137
04−0
.404
430
−0.000
065
−0.000
025
0.04
4386
−0.258
713
O(rank2)
x=−0
.578
,y=
1.29
5,z=
0.00
0x=
0.59
1,y=
1.23
4,z=
0.00
0
qa−0
.458
233
−0.405
414
Μ0.00
0023
−0.022
297
−0.443
599
0.00
0023
−0.022
297
−0.443
599
Θc
−0.229
504
0.00
0121
−0.000
066
−1.030
753
−0.083
547
−0.734
215
−0.000
204
−0.000
232
−0.395
608
0.34
1832
OOH(rank2)
x=−1
.635
,y=
−0.648
,z=
0.00
0x=
1.64
8,y=
−0.705
,z=
0.00
0
qa−0
.391
520
−0.449
517
aCharge[e]
q00
bDipole[ea 0]q1
0q11c
q11s
cQuadrup
ole[ea2
]q2
0q2
1cq2
1sq2
2cq2
2s
690 J Mol Model (2009) 15:687–694
Fig. 4 Differences (absolutevalues) between multipoleexpansion and ab initio potentialfor glycine in two differentconformations based on HF6-31G* calculations. Upper rowMultipole moments calculatedfrom conformation 1 of glycine,lower row multipole momentscalculated from conformation 2of glycine. Left column Com-parison with ab initio potentialof conformation 1, right columncomparison with ab initiopotential of conformation 2. Thex-axis of the second conforma-tion is inverted in this represen-tation in order to allow bettercomparison of the two confor-mations. Colour coding: Black>10 kJ/mol, red 5–10 kJ/mol,green 2.5–5 kJ/mol, blue1–2.5 kJ/mol, yellow0.5–1 kJ/mol, white <0.5 kJ/mol
Glycine Alanine Glycine Alanine
c1 c2 c1 c2 c1 c2 c1 c2HF 6-31G** HF 6-31G** B3LYP cc-pVQZ B3LYP cc-pVTZ
Charges of c1
ΔE (kJ/mol) 0.09 0.53 0.07 1.03 0.11 0.44 0.08 1.05
ΔE (%) 21 60 18 330 29 39 35 1317
Charges of c2
ΔE (kJ/mol) 0.17 0.15 0.75 0.08 0.17 0.15 0.47 0.10
ΔE (%) 55 34 418 13 47 30 613 73
DMA of c1
ΔE (kJ/mol) 0.07 0.26 0.10 1.27 0.07 0.19 0.11 1.11
ΔE (%) 4 30 11 295 5 25 15 829
DMA of c2
ΔE (kJ/mol) 0.16 0.07 0.63 0.10 0.14 0.11 0.54 0.11
ΔE (%) 40 4 250 5 36 9 423 7
Setup with three multipole sites for glycine, four sites for alanine
DMA of c1
ΔE (kJ/mol) 0.10 0.12 0.10 0.32 0.10 0.13 0.10 0.46
ΔE (%) 6 9 7 77 8 8 11 206
Table 3 Conformational depen-dence of glycine and alanine;charges are calculated accordingto the Merz-Kollman scheme
J Mol Model (2009) 15:687–694 691
first derivatives have to be provided. This can be donebased on published interaction terms [23]. For the actualimplementation it is important to take into account thathigher order multipoles (κ≥1, i.e. dipole) depend on thecoordinate system in which they were calculated. This canbe handled through molecular reference axes systems and isconsidered in some more detail here. For distributedmultipoles, moments obtained from GDMA [24] aredefined with respect to the standard orientation of themolecule, obtained from the underlying GAUSSIAN [4]calculation. In a MD simulation the definitions of themultipole moments have to agree with the overall orienta-tion of the corresponding molecule. Therefore, molecularreference axis systems have to be defined. In the simplestcase (linear molecule) one reference axis, defined by thetwo atoms, is sufficient. For water, all three axes have to bedefined. This can be done by aligning the z-axis along theaxis pointing from the oxygen to the centre of the twohydrogens, the y-axis from one hydrogen to the otherhydrogen, and the x-axis being then defined by orthogo-nality (see Fig. 5). For larger molecules, such as glycineand alanine, several reference axis systems have to bedefined, since the different parts of the molecule can rotatewith respect to each other. Finally, the multipole momentsor the multipole interaction between molecules have to berotated at each step of the simulation to the correspondingmolecular reference axis system. This procedure ensuresthat the orientation of the atomic multipole moments,defined with respect to the equilibrium structure, agreeswith the orientation of the molecule in space.
For molecular moments, the torques on the molecule canbe calculated from the multipole interactions. This is not
possible for atom-centred moments, as torques on pointparticles are not meaningful. For rigid molecules, the torqueon the atoms can be transformed to torques on themolecules, which is not possible for flexible molecules.Therefore, molecular reference axis systems have to beused. Assigning a reference axis system is in principle atechnical issue, since it corresponds to a limited number ofsymmetry groups from which all molecular geometries canbe assembled. Nevertheless, an additional issue arises as therotation of the atomic multipole moments slightly changesthe interaction energy at each timestep because thereference axis system, in which the atomic multipoles aredefined, rotates by a small amount between two consecu-tive time steps. The description of this rotation in terms ofatom centred forces is difficult as, for each interacting pairof atoms, the forces depend not only on the relative positionof these two atoms (as for all other force field terms), butalso on the position of the other atoms belonging to thereference axis systems of that atom pair. Thus, in a rigorousanalytical approach, additional forces will appear for allatoms that define the reference axis system. To the best ofour knowledge no complete treatment is as yet available. Ifthe overall orientational change at each step is small, theseadditional forces are small and can be neglected.
However, for cases where the energy contribution arisingfrom the atomic multipoles is larger, the effect may becomesignificant. The problem is different when the moleculesare treated as rigid units, as done in the DL MULTI forcefield [5] where the forces are separated into a translationalpart (applied to the multipole sites) and a rotational part(applied to the molecular centre of mass) [11]. However, inthis case the level of energy conservation depends on thechoice of the electrostatic cutoffs. Furthermore, it has to benoted that this approach can be applied rigorously only torigid molecules or rigid molecular fragments. It remains tobe seen whether a generalisation of separating the rotationaland translational part, or a more heuristic correction ispreferable. In any case, for an approach that conservesenergy, a combination of atomistic and molecular terms inthe forces is required. One other way to circumvent thisproblem is to “absorb” the effect by using an NVT ensemblefor which, however, careful testing of the simulation resultsis necessary to establish that no artifacts occur. It isparticularly important that total energy is correctly parti-tioned between translational and rotational degrees offreedom according to the equipartition theorem.
Practical applications in atomistic simulations
As a test system to highlight the use of higher multipolemoments in a realistic simulation, a mixed CO/watersystem is considered. The water molecules are described
Fig. 5 Local reference axis systems for two water molecules in aglobal Cartesian coordinate system
692 J Mol Model (2009) 15:687–694
by the TIP3P [6] potential, the CO molecules by differentdistributed multipole expansions that have already beenused for CO in myoglobin and have been described indetail [16]. In the following, they will be referred to asModels B (multipoles up to rank 1 on carbon and up to rank2 on oxygen), C (multipoles up to rank 2 on carbon and upto rank 3 on oxygen) and D (up to rank 3 on both atoms).Differences in the degree of energy conservation are foundbetween rigid and flexible molecules, and between differentparametrisations.
The first test system is a cluster of four water moleculesand two CO molecules (see Fig. 6). The energy conservationis evaluated during 100 ps at 5 K, first for one CO in thestatic field of the other molecules, second for one CO andone water free to move, and third for one CO and two waters
free to move (Fig. 7). For models B and D, these changesleave the energy conservation nearly unaffected. For modelC, the changes destabilise the trajectory. These examplesshow that seemingly small changes in the simulationconditions can affect the stability of the trajectories. Theenergy change between two conformations of the system andfor each atom may be small. Nevertheless, it can become
Fig. 8 CO clathrate system to illustrate energy conservation in a2×2×2 unit cell of a CO clathrate hydrate
Fig. 6 Water/CO cluster structure. Molecules that are fixed during thetrajectory are shown with smaller atom spheres, molecules that arefree to move are shown with large atom spheres
-0.6
-0.3
0
0.3
0.6
-0.6
-0.3
0
0.3
0.6
δ E
[kc
al/m
ol]
20 30 40 50 60 70 80 90 100 110 120
time [ps]
-0.6
-0.3
0
0.3
0.6
Fig. 7 Energy conservation dE ¼ E � E� �
of water/CO clusterduring 100 ps at 5 K. Top Only one CO free to move, centre oneCO and one water free to move, bottom one CO and two waters free tomove. The different colours represent different multipole models onthe CO molecules: B (black), C (red), D (green)
0 20 40 60 80 100
time [ps]
0
100
200
300
400
500
δ E
[kc
al/m
ol]
Fig. 9 Energy conservation of CO clathrate during 100 ps at 100 K.Black CO positions fixed, waters free to move, no energy correctionterm. Red All molecules free to move, no correction term. The energyvalues are absolute values of the total energy at each step minus theaverage total energy during trajectory two [in the case of trajectorytwo (red), the average would be meaningless, therefore the average oftrajectory three (green) is subtracted for comparison]
J Mol Model (2009) 15:687–694 693
very important due to self-amplification. This is explainedthrough the fact that energy changes affect only the rotationalenergy, which increases due to the rotation of the atomicmultipole moments. If the rotational energy increases, theorientational change of the multipole moments alsoincreases. Thus, once the error in energy exceeds a certainthreshold, it increases exponentially.
To demonstrate this effect, trajectories are shown for aCO clathrate system (Fig. 8), which consists of watermolecules arranged in an ice-like structure around COmolecules [14]. For the technical evaluation here, water isdescribed by the TIP3P water model and for CO model C isused. If the CO atoms are fixed and the waters move in thefield of the atomic multipole expansions, the total energy isconserved because no reorientation of the atomic multipolestakes place. If the CO molecules are free to move, theenergy starts to drift and the self-amplifying effectmentioned above appears (see Fig. 9).
Conclusions
In conclusion, the present work has shown that atom centredmultipole moments are a useful way forward to generaliseand improve the description of electrostatic interactions forMD simulations. Technically, the implementation is moreinvolved than for point charges, since the orientation of themultipole moments has to be taken into account. Theorientation of the multipole moments can be describedthrough reference axis systems. For rigid molecules, asatisfactory solution that conserves the total energy has beenrecently presented in the literature [11]. However, for fullyflexible molecules, no general solution is as yet available. Ageneralised implementation for fully flexible moleculesrequires particular care in order to ensure that total energyis conserved in the simulations. For cases in which theperformance of conformationally dependent multipoles withflexible molecules has been carefully examined, clearadvantages over point charges in the calculated observableshave been found [15, 16]. Overall, the accuracy of the forcefield is improved by including higher multipole moments.For larger molecules, the conformational dependence ofmultipole moments has to be considered and becomespotentially important to maintain the accuracy highermultipole moments can provide.
Acknowledgement The authors gratefully acknowledge financialsupport from the Swiss National Science Foundation.
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