Higher order multipole moments for molecular dynamics simulations

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<ul><li><p>ORIGINAL PAPER</p><p>Higher order multipole moments for moleculardynamics simulations</p><p>Nuria Plattner &amp; Markus Meuwly</p><p>Received: 16 September 2008 /Accepted: 6 January 2009 /Published online: 5 March 2009# Springer-Verlag 2009</p><p>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.</p><p>Keywords Multipole moments . Molecular dynamics .</p><p>Point charges . Electron density distribution .</p><p>Distributed multipole analysis .</p><p>Cumulative atomic multipole moments .</p><p>Theory of atoms in molecules</p><p>Introduction</p><p>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</p><p>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 (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 (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 (r) over a suitably defined spaceoccupied by a particular atom for all desired multipoleranks. The partitioning of (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].</p><p>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 [1416].</p><p>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</p><p>J Mol Model (2009) 15:687694DOI 10.1007/s00894-009-0465-6</p><p>N. Plattner :M. Meuwly (*)Department of Chemistry, University of Basel,4056 Basel, Switzerlande-mail: m.meuwly@unibas.ch</p></li><li><p>context of MD simulations are then analysed and the effectsdemonstrated for model systems.</p><p>Choice of atomic multipole parameters for MDsimulations</p><p>The electrostatic energy of a multipole expansion up toquadrupole on two atoms a and b in spherical tensornotation can be written as</p><p>Eelec E qaqb X3</p><p>k1E qam</p><p>kb</p><p> X3</p><p>l1E mlaqb </p><p>X3</p><p>k1</p><p>X3</p><p>l1E mlam</p><p>kb</p><p> X5</p><p>m1E qa</p><p>mb</p><p> X5</p><p>n1E naqb </p><p>X3</p><p>k1</p><p>X5</p><p>m1E mka</p><p>mb</p><p> X5</p><p>n1</p><p>X3</p><p>l1E nam</p><p>kb</p><p>X5</p><p>m1</p><p>X5</p><p>n1E na</p><p>mb</p><p> ;</p><p>1where qa is the charge of atom a, mkb is the direction k ofthe dipole moment vector on atom b, na is the element n ofthe quadrupole tensor on atom a [23]. As an example, theinteraction between a charge qa and the </p><p>20b quadrupole</p><p>tensor element is given by</p><p>E qaq20b</p><p> qa20b1</p><p>2</p><p>3 rbz 21</p><p>R3; 2</p><p>where rbz and is the z-element of a unit vector pointing fromb to a and R is the interatomic distance.</p><p>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-</p><p>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.</p><p>Convergence and accuracy of distributed multipolepotentials</p><p>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 for waterfor different levels of accuracy. At rank two, fr typicallydiffers by a few percent on average from , 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 101010 for water and151515 for alanine, with 101 points in each directionoutside the van der Waals radii1.</p><p>Fig. 1 Differences (absolute values) between fr (potential frommultipole expansion) and (potential from ab initio calculations) forwater, based on a Hartree-Fock calculation with a 6-31G* basis set.</p><p>From left to right: distributed multipoles up to rank 0, rank 1, rank 2 andrank 3. Colour coding: black &gt;10 kJ/mol, red 510 kJ/mol, green 2.55 kJ/mol, blue 12.5 kJ/mol, yellow 0.51 kJ/mol, white </p></li><li><p>Conformational dependence of distributed multipolepotentials</p><p>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 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</p><p>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.</p><p>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.</p><p>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.</p><p>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].</p><p>Implementation in MD simulations</p><p>For the implementation of atomic multipole moments intoatomistic simulations, additional interaction terms and their</p><p>Fig. 2 Alanine conformationsc1 and c2</p><p>Fig. 3 Glycine conformations c1 and c2</p><p>Table 1 Comparison for water and alanine. HF Hartree-Fock, DMAdistributed multipole analysis</p><p>HF 6-31G* B3LYP Aug-cc-pVTZ</p><p>Water Alanine Water Alanine</p><p>DMA, rank 0</p><p>E (kJ/mol) 3.27 2.87 2.70 2.52</p><p>E (%) 64 2383 68 1653</p><p>DMA, rank 1</p><p>E (kJ/mol) 0.63 0.333 0.71 0.36</p><p>E (%) 41 178 61 73</p><p>DMA, rank 2</p><p>E (kJ/mol) 0.05 0.05 0.17 0.06</p><p>E (%) 1 6 4 8</p><p>Mulliken charges</p><p>E (kJ/mol) 0.52 0.84 2.18 6.05</p><p>E (%) 23 687 60 2993</p><p>Merz-Kollman</p><p>E (kJ/mol) 0.28 0.08 0.44 0.08</p><p>E (%) 18 35 37 18</p><p>J Mol Model (2009) 15:687694 689</p></li><li><p>Table2</p><p>Coordinates</p><p>andmultipolemom</p><p>entsforglycineconformations</p><p>(HF6-31G*).The</p><p>multipoles</p><p>onthehydrogensaretransferredto</p><p>theheavyatom</p><p>s</p><p>Coordinates</p><p>()</p><p>Atom</p><p>name(highestrank)</p><p>Conform</p><p>ation1</p><p>Conform</p><p>ation2</p><p>HNH2(rank0)</p><p>x=1.993,</p><p>y=0.619,</p><p>z=0.802</p><p>x=1</p><p>.919,y=</p><p>0.695,</p><p>z=0</p><p>.804</p><p>qa0.218431</p><p>0.218202</p><p>HNH2(rank0)</p><p>x=1.993,</p><p>y=0.621,</p><p>z=0</p><p>.800</p><p>x=1</p><p>.919,y=</p><p>0.696,</p><p>z=0.804</p><p>qa0.218482</p><p>0.218224</p><p>N(rank2)</p><p>x=1.950,</p><p>y=0.020,</p><p>z=0.000</p><p>x=1</p><p>.906,y=</p><p>0.100,</p><p>z=0.000</p><p>qa0</p><p>.445558</p><p>0.447569</p><p>b</p><p>0.000311</p><p>0.568593</p><p>0.155322</p><p>0.000311</p><p>0.568593</p><p>0.155322</p><p>c</p><p>0.245474</p><p>0.001489</p><p>0.000954</p><p>0.008565</p><p>0.826711</p><p>0.251683</p><p>0.000695</p><p>0.000380</p><p>0.134308</p><p>0.815526</p><p>C(rank2)</p><p>x=0.722,</p><p>y=0</p><p>.723,z=</p><p>0.000</p><p>x=0</p><p>.743,y=</p><p>0.750,z=</p><p>0.000</p><p>qa0</p><p>.094932</p><p>0.092121</p><p>b</p><p>0.000089</p><p>0.056946</p><p>0.375556</p><p>0.000089</p><p>0.056946</p><p>0.375556</p><p>c</p><p>0.612798</p><p>0.000074</p><p>0.000156</p><p>0.697168</p><p>0.012662</p><p>0.594929</p><p>0.000021</p><p>0.000036</p><p>0.711272</p><p>0.136218</p><p>H(rank0)</p><p>x=0.685,</p><p>y=1</p><p>.381,z=</p><p>0.867</p><p>x=0</p><p>.767,y=</p><p>1.402,z=</p><p>0.866</p><p>qa0.158468</p><p>0.160195</p><p>H(rank0)</p><p>x=0.686,</p><p>y=1</p><p>.381,z=</p><p>0.866</p><p>x=0</p><p>.767,y=</p><p>1.403,z=</p><p>0.865</p><p>qa0.158459</p><p>0.160199</p><p>CCOOH(rank2)</p><p>x=0</p><p>.540,y=</p><p>0.108,</p><p>z=0.000</p><p>x=0.628,</p><p>y=0</p><p>.095,z=</p><p>0.000</p><p>qa0.304407</p><p>0.304592</p><p>b</p><p>0.000013</p><p>0.121391</p><p>0.059910</p><p>0.000013</p><p>0.121391</p><p>0.059910</p><p>c</p><p>0.402671</p><p>0.000040</p><p>0.000037</p><p>0.135468</p><p>0.113704</p><p>0.404430</p><p>0.000065</p><p>0.000025</p><p>0.044386</p><p>0.258713</p><p>O(rank2)</p><p>x=0</p><p>.578,y=</p><p>1.295,</p><p>z=0.000</p><p>x=0.591,</p><p>y=1.234,</p><p>z=0.000</p><p>qa0</p><p>.458233</p><p>0.405414</p><p>0.000023</p><p>0.022297</p><p>0.443599</p><p>0.000023</p><p>0.022297</p><p>0.443599</p><p>c</p><p>0.229504</p><p>0.000121</p><p>0.000066</p><p>1.030753</p><p>0.083547</p><p>0.734215</p><p>0.000204</p><p>0.000232</p><p>0.395608</p><p>0.341832</p><p>OOH(rank2)</p><p>x=1</p><p>.635,y=</p><p>0.648,z=</p><p>0.000</p><p>x=1.648,</p><p>y=0</p><p>.705,z=</p><p>0.000</p><p>qa0</p><p>.391520</p><p>0.449517</p><p>aCharge[e]</p><p>q00</p><p>bDipole[ea 0]q10q11c</p><p>q11s</p><p>cQuadrupole[ea2</p><p>]q20q21c</p><p>q21s</p><p>q22c</p><p>q22s</p><p>690 J Mol Model (2009) 15:687694</p></li><li>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&gt;10 kJ/mol, red 510 kJ/mol,green 2.55 kJ/mol, blue12.5 kJ/mol, yellow0.51 kJ/mol, white </li><li><p>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 (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 momen...</p></li></ul>

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