Monte carlo simulation of molecules and ions in liquid water

Thesolute molecule was `morphed' from a non-interacting `ghost' molecule inside a boxcontaining TIP3P/TIP4P water molecules under the periodic boundary conditions to its full potential f

MOLECULES AND IONS IN LIQUID WATER

MICHAEL YUDISTIRA PATUWO

NATIONAL UNIVERSITY OF SINGAPORE

2011

MOLECULES AND IONS IN LIQUID WATER

MICHAEL YUDISTIRA PATUWO

(B.Sc.(Hons), NUS)

A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF

PHILOSOPHY IN SCIENCE DEPARTMENT OF CHEMISTRY

NATIONAL UNIVERSITY OF SINGAPORE

2011

Derivation of reaction free energy of chemical processes in aqueous environmentscan be aided by the knowledge of hydration free energy Monte Carlo simula-tion can be done in conjunction with the thermodynamic perturbation method,

by means of Bennett's acceptance ratio, in order to obtain the free energy Thesolute molecule was `morphed' from a non-interacting `ghost' molecule inside a boxcontaining TIP3P/TIP4P water molecules under the periodic boundary conditions

to its full potential functions, by subjecting the two endpoint systems and mediate systems with soft-core solute-solvent interaction potentials to separate MCsimulations All MC simulations were performed using a homegrown Fortran90 pro-gram that allowed choice of solvent models and custom interaction functions, andwas well-tailored for morphing oriented works A good deal of attention was put onthe potential functions used for solute-solvent interaction While empirical functionswere used, they are largely consistent with established theoretical arguments andconstructs

inter-First and foremost, I would like to thank my supervisor and mentor, Assoc Prof.Ryan P A Bettens, for his guidance and counsel in both my undergraduate andgraduate years, for nurturing my interest in the eld of physical and computationalchemistry, and for being a constant source of inspiration for me I also thank everylecturer I've had the honour to be taught by in the past seven years, who had brought

me where I am right now, Dr Adrian M Lee for making me an acquaintance tothe eld of quantum chemistry and for his academic guidance, and Assoc Prof.Fan Wai Yip and Assoc Prof Kang Hway Chuan, who were the examiners for thequalication exam for my Ph.D candidature, for giving me a wake-up slap and put

me back on track Without all of you I would not have loved Chemistry the way I

do right now, and I would not have come this far

Many, many thanks to my junior and friend Le Hai Anh Ryo, whose ness and determination were invaluable to me; God bless her in her future endeav-ours My undergraduate and graduate friends who had been here in the same labtogether with me through thick and thin: Krishnan, Clara, Sandra, Hui Jia, Amelia,

resourceful-it would have been so dull wresourceful-ithout all of you I would also extend my thanks toLow Jia En, my acquaintance since secondary school and a great friend throughout

my years in NUS Being a chemistry major would not have been the same withoutyou

I also thank everyone who had helped me during my Ph.D candidature years

in various ways, Miss Suriawati Binte Sa'ad for her help in non-academic matters,Assoc Prof Thorsten Wohland for his ideas and initiative concerning the graduatematters, Yung Shing Gene and the CSE, for allowing me to use their machines to

do a signicant part of my work, and everyone else who had helped me in ways that

I may not be aware of or have forgotten

I am honoured to be a student of NUS, and I am forever grateful for the graduateresearch scholarship that I was oered Without it, it would have been impossible

Thank you my parents, for encouraging me in the academic path that I havechosen, and Stey and Priscy for always believing in their big brother Mami andPapi, I love you so much Thank you Benny, for sticking with me after we've movedout of NUS residential halls Thank you Qiu Xun, for sitting those physics lectureswith me, Kenny, my hall friend and gaming mate, all the cool people who were underASEAN scholarship with me back in the days, my NJC classmates, my friends in

NJ Chorale and Resonance old and young These four years would have been sodierent without your friendship

Looking back, I do not and will never regret the decision of coming to Singaporeand doing my Ph.D work in NUS The treasured memories will forever stay with

me, whatever the future may be

Summary i

1.1 Intermolecular ForcesA Preamble 1

1.2 Electrostatic interactions 7

1.2.1 Multipole moments and molecule in an electric eld 8

1.2.2 Electrostatic interactions between molecules 13

1.2.3 Distributed multipoles 19

1.3 Non-electrostatic interactions 25

1.3.1 Induction energy 26

1.3.2 Dispersion energy 31

1.3.3 Exchange-repulsion energy 39

1.4 Preliminary Calculations 40

2 Molecular Monte Carlo Simulation 55 2.1 Computer Simulation Methods 55

2.1.1 Metropolis Algorithm 62

2.1.2 Periodic Boundary Conditions 66

2.1.3 Ewald Summation Method 70

2.1.4 NpT ensemble 78

2.1.5 Free Energy 81

2.1.6 Acceptance Ratio Method 84

2.1.7 Morphing 89

2.1.8 Soft-core potential 94

2.2 The MC Program 99

2.2.1 General ow of the program 104

2.2.2 Preparative steps 106

3 Simulation of Neutral Molecules 111 3.1 Nitrogen 111

3.2 Methane 121

3.3 Methanol 128

3.4 Carbon dioxide 137

3.5 Butane 140

3.6 Benzene 146

3.7 Ethanoic acid 152

3.8 Ethanamide 158

3.9 Conclusion 163

4.2 Alanine 168

4.3 Asparagine 175

4.4 Neuraminidase inhibitors 183

4.4.1 Zanamivir 185

4.4.2 Peramivir 189

5 Conclusion 193 Bibliography 197 A Spherical harmonics 209 B Multipole moments 213 B.1 Operators 213

B.2 Geometry conversions for multipole moments 214

B.3 Changing the origin of multipole moments 215

C Interaction functions 219 D Program and Auxiliary Files 223 D.1 MC 223

D.2 Bennett_1000 226

D.3 shell 227

This thesis explores the theoretical framework behind intermolecular interaction,its adaptation into the practice of molecular simulation, and the validity of variouspotential models and other simulation parameters in determining the solvation Gibbsfree energy of molecules and ions We aim to arrive at a reliable simulation programthat can uncover crucial thermodynamic parameters of common molecules and ions

so as to understand better the chemistry of solvated compounds in aqueous solutions.Ultimately, our results can be used in conjunction with modern techniques, such

as fragmentation, to seal the gap between theoretical chemistry and the study ofmacromolecules and other complex systems

The rst chapter of this thesis deals primarily with the factors that contributetowards forces that act upon interacting molecules and ions as dictated by pertur-bation theory We separate them into two parts: electrostatic interaction, which isthe change in energy of the combined unperturbed system with respect to the sum

of the independent systems; and non-electrostatic interaction, which consists of thesecond order and higher interaction terms (induction and dispersion energies) andthe short-ranged exchange-repulsion term In the second half of this chapter, weprovide simplied approximations of the above interaction terms so that they can

be used as a set of dynamics in the simulation of molecules dissolved in water.Chapter 2 outlines the necessary elements of a Monte Carlo simulation formolecules The algorithms and framework for the computer simulation are dis-cussed here, and we select options best suited to our purposes The output of thesimulation needs to be parsed appropriately so useful data can be acquired We alsooutline the context and limitations of the homegrown Fortran90 program, MC, whichwas used to perform all our simulation works

The third and fourth chapter give the results of the simulation for neutral

molecules and zwitterions respectively We analyse the simulation data by ing at the average solute-solvent interaction energy, changes in Gibbs free energyfor every morphing step between intermediate systems, and the solvation structure

look-of key groups in the solute particle

Finally, at the end of this thesis are the appendices, which provide auxiliaryinformation on some of the more tedious elements in theoretical physics/chemistrythat is used in this work The last appendix gives a list of the data les contained

in the CD included at the back of this thesis

List of Tables

1.1 Distributed multipoles of the CO2molecule computed via nearest-site

algorithm using various basis sets 24

1.2 Distributed and central multipole moments of the methanol molecule 42 1.3 Geometry, parameters, and exhibited properties of the TIP3P and TIP4P water models 43

1.4 Fit parameters for Figure 1.5 51

2.1 Solvation energy of several solute molecules calculated via continuum models 58

2.2 Possible choices of morphing parameters in an MC run 108

3.1 Distributed multipoles of the N2 molecule for up to l = 2 112

3.2 Simulation results for dissolved nitrogen molecule 113

3.3 Simulation results for dissolved nitrogen molecule using C8 potentials 117 3.4 Optimised geometry for the methane molecule at B3LYP/cc-pVTZ 121 3.5 Multipole moments to be used in the simulation of solvated methane 122 3.6 Non-electrostatic parameters of CH4 123

3.7 Simulation results for dissolved methane molecule 124

3.8 Simulation results for dissolved methane molecule using modied AMBER parameters 127

3.9 Optimised geometry for the methanol molecule at B3LYP/cc-pVTZ 128 3.10 Multipole moments to be used in the simulation of solvated methane 129 3.11 Lennard-Jones parameters for the CH3OH molecule 131

3.12 Simulation results for dissolved CH3OH molecule 132

3.13 Distributed multipoles of the CO2 molecule for up to l = 4 138

3.14 Two Lennard-Jones models for CO2 138

3.15 Simulation results for two models of CO2 139

3.16 Potential models for solvated butane 141

3.17 Simulation results for dissolved butane molecule 142

3.18 Potential models for solvated benzene 147

3.19 Simulation results for dierent models of dissolved benzene molecule 148 3.20 Optimised geometry for the ethanoic acid molecule at B3LYP/aug-cc-pVTZ 153

3.21 Non-electrostatic parameters for solvated ethanoic acid 153

3.22 Simulation results for ethanoic acid 154

3.23 Optimised geometry for the ethanamide molecule at B3LYP/aug-cc-pVTZ 158

3.24 Lennard-Jones parameters for the nitrogen atom 159

3.25 Simulation results for ethanamide, parsed via dierent algorithms 159

4.1 Electric charges on alanine charge centres 169

4.2 Non-electrostatic parameters for solvated alanine 170

4.3 Simulation results for zwitterionic alanine 171

4.4 Electric charges on asparagine charge centres 175

4.5 C1 potential parameters for model C asparagine 177

4.6 Simulation results for zwitterionic asparagine 177

4.7 Simulation results for zwitterionic zanamivir 186

4.8 Simulation results for zwitterionic peramivir 190

A.1 List of spherical harmonics and regular spherical harmonics 211

B.1 Conversions between spherical and Cartesian geometry up to rank 3 214 C.1 List of interaction functions up to l1, l2 = 2 220

D.1 Files in MC directory, listed alphabetically Unless specied otherwise, all source les are compatible with the Fortran90 language (.f90 extension). 226

D.2 Files in Bennett_1000 directory, listed alphabetically. 227

D.3 Files in Shell directory, listed alphabetically. 228

List of Figures

1.1 Vectors convention on two interacting molecules 14

1.2 Non-additivity of induction energy 28

1.3 Preparation of solute-solvent interaction parameters 45

1.4 Calculated residual energy and tted non-electrostatic potential be-tween CO2 and a water molecule 49

1.5 Non-electrostatic contribution and t potentials to the ethanoate ion-water interaction energy 51

2.1 Periodic boundary conditions and minimum image convention 68

2.2 Ewald summation 73

2.3 Hess' cycle for free energy of reaction in aqueous medium 81

2.4 Intermediate state in sampling 90

2.5 Illustration of solute morphing for the calculation of free energy 93

2.6 Soft-core Lennard-Jones potential functions 95

2.7 Flow chart of the main top-level subroutine of the program MC 105

2.8 Plot of total solution energy and solute-solvent energy for an equili-bration simulation 107

2.9 Thermodynamic cycle for the hydration free energy of gaseous solutes.109 3.1 Geometry of nitrogen-water molecule pair for tting of non-electrostatic interaction parameters 111

3.2 Fitting for the non-electrostatic interaction potential between nitro-gen and water molecules 114

3.3 Electrostatic potential map for the N2 molecule 116

3.4 Estimated ∆G at various λ of solvated N2 117

3.5 Radial distribution function for the solvated N2 molecule 120

3.6 Fitting for the non-electrostatic interaction potential between

methane and water molecules 123

3.7 Estimated ∆G at various λ of solvated CH4 124

3.8 Estimated ∆G of solvated CH4 with smaller ∆λ 125

3.9 Estimated ∆G of solvated CH3OH 132

3.10 Electrostatic potential map for the CH3OH molecule 134

3.11 Radial distribution function for the solvated CH3OH molecule 136

3.12 Estimated ∆G at various λ of solvated butane 143

3.13 Radial distribution functions for water to terminal carbon of alkanes 145 3.14 Estimated ∆G at various λ of solvated benzene 149

3.15 Map of modied Lennard-Jones potential for the benzene molecule 150 3.16 Comparison between separate and integrated hydrogen atom potential 152 3.17 Estimated ∆G at various λ of solvated ethanoic acid 154

3.18 Radial distribution functions for water to carboxylic group of CH3COOH 157

3.19 Estimated ∆G at various λ of solvated ethanamide 161

3.20 Radial distribution functions for water to amide group of CH3CONH2 162 4.1 Optimised geometry of alanine zwitterion 169

4.2 Estimated ∆G at various λ of solvated zwitterionic alanine 171

4.3 Radial distribution functions for water to zwitterionic alanine 173

4.4 Optimised geometry of asparagine zwitterion 175

4.5 Estimated ∆G at various λ of solvated zwitterionic asparagine 178

4.6 Radial distribution functions for water to backbone of zwitterionic asparagine 180

4.7 Radial distribution functions for water to amide group of zwitterionic asparagine 182

4.8 Neutral and zwitterionic zanamivir 185

4.9 Radial distribution functions for water to zanamivir 187

4.10 Neutral and zwitterionic peramivir 189

4.11 Radial distribution functions for water to peramivir 190

B.1 Change of origin 217

C.1 Demonstration of local axes system 220

List of Symbols

In order of appearance

U Interaction energy .2

r Distance; scalar 2

r Distance/position; vector 9

ˆ M Multipole moment operator 9

ρ(r) Charge density 9

Rlm Regular spherical harmonics 9

l Rank of spherical harmonics 9

m Component of spherical harmonics 9

q Electric charge 9

κ Component of spherical harmonics (real form) 9

V Potential 10

α, β, γ Geometrical axes 10

Eα Electric eld strength in the α-axis 10

H Hamiltonian 10

H0 Interaction Hamiltonian 11

ˆ M0 Cartesian multipole moment operator; not traceless 11

ˆ q Monopole moment operator 11

ˆ Dipole moment operator 11

ˆ Q Quadrupole moment operator 11

ˆ O Octopole moment operator 12

W Total internal energy 13

U(n0) n-th order term to the interaction energy 13

U0, UEL First order term (electrostatic) to interaction energy 13

A, B, C Labels for molecules 13

a, b, c Labels for atoms or sites 14

ε0 Permittivity of space 14

TAB Interaction function 15

Ilm Irregular spherical harmonics 16

C(θ, ϕ) Renormalised spherical harmonics 18

χ(r) Gaussian basis function 21

a0 Bohr radius 23

Uind Induction energy 26

Udisp Dispersion energy 26

ααα 0 Dipole-dipole polarisability tensor element 27

Aα,α0 β 0 Dipole-quadrupole polarisability tensor element 27

Cαβ,α0 β 0 Quadrupole-quadrupole polarisability tensor element 27

Cn Constant associated with dependence on R−n 30

O(xn) Remaining terms with orders n and higher/lower 31

¯ α Mean polarisability 32

ν Frequency 36

ULJ A compound term containing all non-electrostatic interaction energies 44

ε Lennard-Jones potential, potential well depth 46

σ Lennard-Jones potential, collision diameter 46

N Number of particles 59

p Momentum; vector 59

ρ(x) Probability density 60

V Volume 60

T Temperature 60

E Internal energy 60

kB Boltzmann's constant (1.38066 × 10−23 JK-1) 60

Q Partition function 60

T Kinetic energy operator 63

V Potential energy operator 63

ξ Random number 64

L Box length 69

n Box vector (periodic boundary conditions) 69

rc Cut-o radius for short-range interaction potentials 69

ρ Number density 69

εr Relative dielectric constant 70

δ(x) Dirac delta function 72

k k-vector for reciprocal space electrostatic energy summation 74

p Pressure 78

V0 Standard volume 78

s Scaled position coordinates 78

G Gibbs free energy 81

A Helmholtz free energy 82

n Total number of time steps or statistically signicant congurations 85

λ Morphing parameter 88

g(r) Radial distribution function 119

Chapter 1 The Dynamics of Molecular

Systems

1.1 Intermolecular ForcesA Preamble

In order to obtain accurate, reliable results for any kind of molecular simulationprocedure, it is important that we rst build a structured model of intermolecularinteraction forces that is both realistic and practical Molecular Dynamics proce-dures, especially in the eld of biochemistry, are well known to utilize force eldmodels as done by CHARMM [23], AMBER [104, 38, 105], GROMOS [100, 35],MM2 [4]/MM3 [6]/MM4 [5], and many others These models have served us ex-ceedingly well in the eld of molecular and ionic simulations, especially those done

in the gas, and often the condensed, phaseand even at the present time theyare still routinely used and improved upon for their spectacular contributions inmolecular science

While the force-eld approach to calculate intermolecular interactions is nient and useful, a persistent problem with its application in molecular simulationslies in the fact that the parameters used to dene the molecular potential are usuallydetermined wholly from experimental data and sometimes empirically tted withinsucient regard to the principles that underpin it In other words, it often provesdicult to discern the individual components of this very potential that t in agree-

conve-ment with the established scientic model of atoms and molecules This is somethingthat warrants further inspection A working model that agrees well with the expec-tations of theory might be extrapolated to obtain parameters for other atoms andmolecular fragments without the need of their own experimental data Moreover, it

is often desirable to be able to express the dynamics of a molecular system via itscomponents (namely, electrostatic, induction, dispersion and exchange-repulsion),such as when interpretation and comparison of such gures are to be made, provid-ing a sense of meaning to bare numbers Simply tting the potential to arbitraryfunctions to obtain these contributions may not assign them correctly, as the number

of unknowns could be relatively large, and the subsequent interpretation of theseparameters would be misleading [133]

Being able to obtain ab initio parameters for atomic and molecular interactionswithout having to conduct and repeat an experimental procedure such as molecularbeam scattering [61]or at least a semi-empirical one, backed by quantum mechan-ical perturbation theory, would be advantageous when reliable and interpretableresults are of priority, and in this work we explore the practicality of such potentialfunctions when Monte Carlo simulations are performed

The interaction phenomena between two molecules, whether overall charged oruncharged, can be separated into the following contributions [127] (Interactions initalics are additive in nature):

U ∼ R−n `Long range' eects are contributions that vary with the inverse power of

distance, and can be represented in power series Note that despite beingcalled `long range', this does not mean that all interactions under this categorynecessarily transverse distances beyond molecular dimensions Electrostaticforce may extend slightly further away than the other interactions under thiscategory, especially in ions and highly polar molecules

 ElectrostaticProbably the most basic and easily analysable of all molecular forces, this energy arises from the classical Coulombic interaction between the

static ground-state charge distributions of the two molecules By `ground state',

we refer to the lowest-energy non-degenerate state of the individual molecules

as if they exist independently of each other By its nature this term can be either attractive or repulsive, and potentially dominates at very long range, as the rst-order term (interaction between two point charges) varies with R −1 , the largest possible power of distance for this type of interaction.

 InductionInduction energy arises from the modication of the electronic wavefunction of a particular molecule due to the combined inuence of its ground-state neighbours The largest term involves the summation of elec- tric eld strengths of surrounding particles by vector addition As a result, the magnitude of potential contributed by induction alone is not a simply ad- ditive property Regardless, it is always attractive in nature, as the electronic wavefunction always responds to lower the system's energy.

 DispersionAlso always attractive in nature, dispersion energy arises from the correlation between the movement of electrons belonging in either molecule as charge distributionsand hence the electrical environmentof the molecules constantly shift Lower energy states become favoured, and the result is an overall attraction force between both molecules The dispersion energy is ap- proximately additive in nature.

 ResonanceResonance only applies to pairs of molecules that have an overall degenerate state, that is, when one molecule occupy a degenerate state, or when two identical molecules occupy dierent states Ordinary closed-shell molecules exhibit no resonance unless they participate in high-energy reactions.

 MagneticA pair of molecules exhibit magnetic properties when they possess unpaired electrons in their orbitals, which once again is not a characteristic

of closed-shell molecules Magnetic eects from nuclear spin can be neglected for its quantity would be several magnitudes smaller Regardless, magnetic interactions due to both electronic and nuclear spin are generally too small

to be considered in the context of intermolecular forces, and they are often safely and reasonably neglected Organic molecules are usually closed-shell and exhibit no magnetic eects.

U ∼ e−αR `Short range' eects decay exponentially with distance and most do not usually

extend beyond the typical van der Waals radius of an atom These eects arisefrom the overlap of the electronic wavefunctions of the individual molecules,which results in the reorganisation of those wavefunctions As such their eect

is greatest at very short intermolecular distances

 ExchangeThis term represents the attractive potential between two molecules as a result of electrons of one molecule `spilling over' the general vicin- ity of the other molecule and falling under the inuence of its nuclei This term

is often paired with `repulsion', under the collective term `exchange-repulsion', due to the fact that the two phenomena are closely related as they both arise from the overlap of orbitals of dierent molecules [127]

 RepulsionThis term represents the repulsive potential between two molecules due to crowding of electrons in the same region of space Subject to the Pauli exclusion principle, the electronic orbitals are forced to redistribute such that a higher energy state is obtained Both exchange and repulsion will occur at the same time given two closed-shell molecules, but for an atom or

a molecule with an unpaired electron the attractive term oft dominate, as the exclusion principle only applies to electrons of the same spin trying to occupy the same region of space This term is the primary reason why two atoms rarely get very close to each other, in spite of nuclei size being about ve orders

of magnitude smaller than an angstrom (one of the shortest bond lengths in existence is that of H2, which is 0.74 Å).

 Charge TransferNot quite a full transference of electronic charge from one region to another, but more than a simple distortion in the electronic wave- function, charge transfer behaves similarly to induction energy, but varies ex- ponentially with distance instead of with its inverse power Like the induction energy, this term is always attractive and non-additive.

 PenetrationPenetration is, in a way, an `artefact' of the classical treatment

of the molecule in terms of electric multipole expansion, which will be explained

in greater detail later in this chapter At very short ranges multipole expansion

of electrostatic energy converges very slowly, and when it does, signicant

de-viation from its true value may be observed For convenience in computation, atoms are treated as point sites, but the distributed nature of electron clouds (as opposed to the static point charge approximation) inevitably gives rise to inaccuracies at short distances A correction term used to compensate for this eect is described as `penetration'.

 DampingAnalogous to penetration, damping is a correction term used to deal with the error associated with dispersion interaction at short ranges, if dispersion is to be expressed with a power series involving descending powers

of intermolecular distance At short distances, where orbital eects become prevalent, calculated dispersion energy will be overestimated by the power se- ries, reaching innity at zero separation while the interaction should remain

nite Dispersion interaction is `damped' at short distances, usually with a distance-dependent factor, to ensure that it does not go singular as R → 0.

For terms printed in italics, pairwise interactions can be summed up to give theoverall energy of the assembly, and many-body correction terms do not exist This isimportant because in any computational method it is most convenient to deal withquantities that can simply be added to one anothercalculations of three-body,four-body energy corrections etc greatly inate computational time and make itunwieldy For non-additive properties, their calculated total must be modied bymany-body corrections, in which the `excess' potential involving three or more (usu-ally three) molecules chosen from all possible permutations of the number of particles

in the system must be subtracted from the sum of the pairwise interactions This is,

of course, a major concern that often plagues simulations of very large assemblies.Corrections of such scale would consume a large amount of computational time, and

it is always better to nd ways to approximate these quantities, or look for native ways to model non-additive behaviour using simpler functions For example,the bulk of the non-additivity of polar systems can be accounted for by using sim-pler polarisation models, as shown from studies involving water clusters [59, 90, 84].Non-additivity in exchange interaction remains signicant for small clusters [59, 90],

alter-but the structuring via additional coordination shells for any given molecule in thecondensed phase and large clusters increase the long-range electrostatic (and to anextent, dispersion) potential to the point where the short-range non-additive onesbecome insignicant [84].

The goal of this work is to gure out the required terms of intermolecular tions analogous to theoretical predictions as far as possible, using empirical functionswhen necessary such that computational time is optimised without sacricing thereliability of calculation Fortunately, for closed-shell molecules, magnetic interac-tion is insignicant relative to the other long-range terms Resonance realisticallyonly applies to solvent molecules and only if their excited states are metastable, or ifthe simulation is conducted at high temperaturesand even then the contributionwould be relatively small A reasonable approximation to the long-range interactionterms is to consider only the electrostatic, induction, and dispersion terms betweensolute and solvent molecules, and among the solvent molecules Solute-solute inter-action will have to be treated separately as in ordinary dilute solutions they rarelycome in close contact with one another, and in simulations where periodic boundaryconditions [3] are used they are always kept at a sizeable distance

interac-In a simulation, molecules inevitably approach one another, and among thethree long-range interactions taken into consideration, two are always attractive,and one (electrostatic) could be either attractive or repulsive in nature Consideringthese interactions alone molecules will inevitably collapse into one another unless arepulsive energy function is added that can keep them apart Hence the exchange-repulsion potential, the most important of potential terms that decay exponentiallywith distance, must always be calculated at close distances in order to keep moleculesseparate It is noted that this is the primary goal of adding a repulsive potentialinto the sum of intermolecular interactions; at room temperatures molecules rarelyget too close to one another in reality, and these congurations make very littlestatistical contribution to the state of a system in equilibrium Thus, we postulate

that the exact magnitude of the repulsive potential may not often be as important

as the shape of the potential energy surface where the interaction energy is negative.One may aord to be a little more liberal in the assignment of the repulsive function,and hence we utilise a single exponential term to shape this `potential wall' thatarises from exchange-repulsion and charge-transfer mechanisms Penetration anddamping can also be moulded into this term, but alternatively these eects can also

be emulated by adding a separate potential into the interaction function, as it will

be demonstrated later on

Now that the interaction potential has been established, it is only appropriate

to look into them in greater depth Atoms in a given molecule are approximatelyspherical, and a practical strategy is to separate terms that depend strongly on theorientation of the interacting molecules from that which do not The electrostaticterm, in particular, has a signicant dependence on molecular orientation, and will

be treated separately Consequently, non-electrostatic terms are grouped togetherwhenever possible in a single overlying function for each particular atomic pair Theseparation between electrostatic and non-electrostatic terms (the latter sometimesreferred to as the van der Waals term) is known to be a common strategy adopted

in existing force eld models [73, 101]

to do so Solving the integral by the most direct numerical method, i.e dividingthe space around the molecules into grids and placing values for averaged chargedensity over those discrete, small regions of space would have to utilise a large mag-nitude of data arrays if this were to be repeated for every occupied basis function.This problem is aggravated for larger, more complex systems, and such treatmentcannot extend too far beyond the dimensions of the molecules themselves withoutcompromising accuracy and making the approximation unrealistic It is generallycommon practice to instead approximate the electrostatic energy of a molecule inthe presence of its neighbours via manipulation of molecular multipole moments.

As they are but analogues to the power series of a vector eld in 3-dimensionalspace, multipole moments are capable of reproducing any sort of potential pro-

le provided there exists a sucient number of terms to make the approximationreasonable Even the complicated potential prole of a complex molecule can berepresented in terms of multipole expansion, which converges at a quicker rate whenmultiple sites are used for this approximationthis will be discussed in greater de-tail in the rest of this chapter With this knowledge, all that's left is to be able todescribe the potential of a molecule in its multipole expansion such that a realisticand intuitive model can be obtained, one that could be represented in meaningfulnumbers and quantities First, we visit the foundations behind the construction ofmultipole moments

1.2.1 Multipole moments and molecule in an electric eld

The multipole moment representation of a distance-dierentiable, well-behaved tric eld in a spherical coordinate system can be expressed as

elec-ˆ

Mlm =

Z

with l and m to denote the `rank' (or `degree') and the `component' of the multipolemoment respectively The rank of a multipole controls the total angular momen-tum of the multipole, while its component gives information on the length of itsprojection along the direction of +z In this equation, Rlm is the regular spher-ical harmonics which denition and derivation can be found in Appendix A Theappendix also contains a more detailed explanation on constructing the basis ofmultipole moments.

We make the approximation that electrons and nuclei behave as point charges.This is, for all practical reasons, a reasonable approximation as the dimensions ofsubatomic particle are negligible when compared with the size of intervening space.The expression for charge density can then be expressed as distinct charge-carryingsites each representing a particle a associated with position vector r(a):

A molecule basically consists of discrete point charges in the form of protons

and electrons, and the electric eld generated by the presence of a molecule in aninstant of time can be approximated in a series of multipole expansions, much like

a power series Now, the potential of any arbitrary electric eld (assumed to bewell-behaved) can be expressed in the Taylor series:

Xα,β

3!

Xα,β,γ

Here, α, β, and γ represents any choice axis in a coordinate system, so for example,

in a 3-D Cartesian coordinates the summation sign sums up all combinations of x,

y, and z for each of the Greek letter The zeroth order term (the rst term in theequation, V (0)) is a scalar constant, and each subsequent term provides a `correction'

to the eld function using an electric eld term Eα, electric eld gradient term E0

αβand so forth For convenience, we shall write these terms using the capital letter Vinstead of the general convention of said quantities, as the physical quantities theyrepresent are easily deducible from the number of subscripts they have

Here, and at other instances for the rest of this work, we will also employ theEinstein summation convention to avoid the bulky summation signs (each term isnow instead summed independently over all three axes replacing the Greek lettersubscripts; multiple subscripts have all possible permutations summed up in thesame way) We have:

V (r) = V (0) + rαVα(0) + 1

2!rαrβVαβ(0) +

13!rαrβrγVαβγ(0) + (1.5)

As we have established earlier, a molecule is represented as an assembly of a pointcharges with magnitudes qa and position vectors r(a), which unperturbed internalenergy is associated with a certain Hamiltonian operator H Here, we are onlyinterested in the `interaction' Hamiltonian, that is, the excess energy contributed

by the presence of the electric eld `perturbation' This interaction Hamiltonian isgiven by:

H0 =Xa

2!Vαβ(0)

Xa

qarα(a)rβ(a) +1

3!Vαβγ(0)

Xa

qarα(a)rβ(a)rγ(a) + (1.6)

We then rewrite this equation as follows, with respect to a given origin:

qarα(a)rβ(a) rκ(a) (1.8)

We notice that the zeroth and rst rank of ˆM0 are equivalent to the multipoleexpressions of the respective ranks, ˆq and ˆpα (see Appendix B) It can be shownthat by subtracting the trace of ˆMαβ0 the expression of the quadrupole moment ˆQαβ,multiplied by a numerical factor, is retrieved [127] Similarly, subtracting the trace

of the third and subsequent orders of ˆM0 all yield the multipole moments of thecorresponding rank, each multiplied by a dierent numerical factor

(Note: κ in this equation is part of the Greek letter sequencing, and does NOT represent the quantum number of multipole component in the real form expression, which uses the same symbol)

In fact, this is exactly the reason why multipole moments are dened as they are.The trace of ˆM0 provides no contribution to the electrostatic energy, and multipolemoment operators are made traceless for this reason The expression in equation(1.9) indicates that multipoles of rank k only interacts with the k-th derivatives ofpotential, and each interaction term of a given rank is independent of the others.This is important, because it means that if we apply the spherical coordinate systemand use the real-only forms of multipole moments, we can rewrite the expression for

H0 as follows:

H0 =Xlκ

clMˆlκVlκ0where V0

lκ = Rlκ(∇)V |r=0 and cl is a constant associated with the multipole term

of rank l The summed term is the only scalar term that can be constructed fromˆ

(2k−1)!!Mˆ(k)

zz zVzz z = ckMˆzz z(k) Vzz z *from equation (1.9)

And here we see that the constant ck = (2k−1)!!1 Since ˆMlκ and V0

lκ are linearcombinations of ˆMαβ κ and V0

components of multipoles of the same rank We roll this constant into the potential

term such that Vlκ = clVlκ0 to obtain:

By perturbation theory, a non-uniform electric eld can be treated as a turbing potential that acts on the molecule, where the total internal energy of thesystem, W , is expressed as its unperturbed energy plus the `correction' terms inincreasing order, i e., W = W0+ U = W0 + (U0 + U00+ ) The rst-orderenergy of a molecule at its ground state |0i in an electric eld, also known as itspure electrostatic energy, can be expressed as:

per-U0 = h0|H0|0i

= ˆqV + ˆpαVα+1

3QˆαβVαβ+

1(3)(5)OˆαβγVαβγ +

lκ

ˆ

1.2.2 Electrostatic interactions between molecules

Consider two molecules in the vicinity of each other, as in Figure 1.1 Molecules

A and B are each located at origins r(A) and r(B), chosen arbitrarily, usually atpoints of interest such as the centre of nuclear charge or centre of mass, in relation

Figure 1.1: Vectors convention on two interacting molecules

to the their geometrical coordinates and atomic composition Particles in atom A,labelled a, are located at positions r(a) relative to its own origin, and similarly for

B The potential due to molecule A at origin B (henceforth referred to as `site' B)is:

#

a

qa4πε0

1

R −

#(1.13)

+ ˆMα0∇α1

R +

12!

+ ˆpα∇α1

R +

13ˆ

αβ κ+ (1.14)

In the above equation we dene the interaction function between molecules,

−5× (3RαRβ− R2δαβ) (1.15c)

TαβγAB = 1

4πε0∇α∇β∇γ

1R

inter-multipole term belongs to):

H0 = ˆqBVA(r(B)) + ˆpBαVαA(r(B)) +1

3ˆ

QBαβVαβA(r(B)) +

= TABqˆAqˆB+ TαAB qˆAˆBα − ˆpAαqˆB
+ TαβAB 1

The electrostatic energy between two charges varies with the inverse of distance,i.e UEL = q1q2

|r12| It has been demonstrated by Arfken [9] and Zare [148] that theexpansion in terms of spherical harmonics of |r1

12| takes the following form:

1

|r1− r2|=

Xlm

rl<

r>l+1(−1)

mCl,−m(θ1, ϕ1)Clm(θ2, ϕ2) (1.17)

where r< and r> are respectively the smaller and the larger of r1 and r2 We take

r1 = R = r(B) − r(A) and r2 = r(b) − r(a) Assuming R > |r(a) − r(b)| andsubstituting equation (A.4) (Appendix A), we obtain:

1

|R − (r(a) − r(b)) | =

∞Xl=0

lXm=−l(−1)mRl,−m(r(a) − r(b)) Ilm(R) (1.18)

The addition theorem in equation (B.5) can be put to use here Also note that

Rlm(−r) = (−1)lRlm(r) and l1+ l2 = l, and we get:

Rl,−m(r(a) − r(b)) =

∞X

#1 2

where C(θ, ϕ) are the renormalised spherical harmonics (equation (A.3)).

We see the electrostatic interaction energy dependence on R−l 1 −l 2 −1is from thisequation (and later on reected in Table C.1), in relation to the multipole ranks

of the interacting molecules that result in that particular term which is specied

by l1, l2, m1, m2, m The multipole terms ˆMlA

l 1 κ 1 ,l 2 κ 2, as all of them are simply constants that can each be evaluated andtabulated for convenience The list of interaction functions up to the quadrupole-quadrupole terms can be found in Appendix C (a more complete list of interactionfunctions can be found in sources such as Hattig and Hess [55] and Stone [128])

The electrostatic Hamiltonian of a pair of interacting molecules can then bewritten as