Also, interaction potentials are the tools for the next level of study of this chemistry, namely, the dynamics of clusters and energetics of bulk systems. It is model interaction potentials that are the focus of this review, not the whole of intermolecular interaction phenomena. Intermolecular interaction potentials have application at the junction of molecule-building-block chemistry and atom-building-block chemistry. Studying the influence of surrounding molecules or solvent molecules on chemical reactions or other atom-building-block phenomena requires knowledge of the weak interaction of the surrounding molecules with the reactive species.
In the history of interaction potentials, and likely for the immediate period ahead, what is learned from working at the detailed level drives, influences, or provides the knowledge base for everything less detailed. It is a logical progression of simulation technology. For the purposes of a review, there is some organizational advantage in selecting for the focus one certain level of detail, such as one of the extremes of highest accuracy or of being simplest for computation.
Rather than that, the objective here is to look at interaction potentials from the perspective of differing levels of accuracy and the type of understanding that aids the progression from the detailed to the practical. Again, it is essentially intact molecules that are the building blocks. Hence, it is reasonable that the intrinsic properties of isolated, intact molecules offer physical insight to intermolecular interaction and also offer a means, not necessarily a complete means, for constructing interaction potentials.
Also at hand as a means for surface construction but usually more laborious is direct quantum mechanical determination. Perhaps best of all is to do both by drawing on ab initio calculations to study the isolated constituent molecules and also to explore the role of properties in subtle electronic structure responses due to weak interaction. An interesting outcome of starting with molecular properties to understand weak interaction energetics is that property changes often can be put on an equivalent footing. That is, property surfaces for clusters can often be constructed along with potential surfaces using some of the same elements.
In that sense, the construction of intermolecular interaction potentials can yield back property information akin to that which may have gone into the potentials. This is important for gathering insight into the physical basis and electronic structure features of weak interaction; and in turn, it is important for the advantageous extraction of coarse potentials from the best detailed potential information available. This review is meant to focus on those property connections and hopefully to offer, thereby, a perspective that augments other recent reviews of weak interaction, two of which give emphasis to potential construction [16,17].
The subject, especially in the context of considering the progression from detailed to coarse, could be extended to include continuum representations of surrounding molecules. Furthermore, it is distinct from the notion of analyzing interactions using intrinsic molecular properties as much as possible. The objectives, then, are to take a fundamental view of constructing potentials, one that allows for a range of accuracy and a wide range of applications and one that exploits property information. This requires integrating some topics that are easily review subjects on their own.
Consequently, certain major development areas in weak interaction do not have the in-depth discussion that usually goes with reviews of weak interaction and hydrogen bonding [1—17], instead going only as far as needed to connect with the area of potential construction. The ultimate applications of the sort of intermolecular interaction potentials that are considered here are i detailed quantum dynamical treatment of rotation, vibration, and tunneling of small clusters, ii aggregation energetics, structures, and properties for clusters with up to hundreds of molecules, and iii molecular dynamics simulation for solvents in biomolecular and reactive problems.
Certain points of discussion will reflect that experience or, unintentionally, that bias.
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Distinction from Chemical Bonding Weak intermolecular interaction refers to hydrogen bonding, van der Waals attraction, London forces, long-range forces, and so on, through a number of different concepts that have been introduced throughout the history of chemical science. The essential unifying idea to define intermolecular interaction is that molecules attract, repel, or both, in the absence of forming ionic or covalent bonds.
The formation of covalent bonds involves rather sharp changes in orbital character, and often more than one electron configuration dominates the wave function at such points. This is easy to distinguish from weak interaction where the largely intact nature of the interacting molecules means that the electronic structure does not show sharp orbital or configurational changes. Ionic bonding is a little different to distinguish in that orbital changes may take place over longer distances and therefore seem less abrupt.
As well, a single configuration may dominate throughout the process. Nonetheless, orbitals change significantly as a result of forming an ionic bond, and this is a distinction from intermolecular interaction. There are changes at the atomic building block level that are a part of ionic bonding. For this discussion, interaction potentials for a partner molecule that is charged will be excluded as being at the fringe with respect to the definition of weak intermolecular interaction. With that, the broad meaning of hydrogen bonding will be limited herein to that of all the constituents being neutral.
The language of chemical bonding is sometimes used to discuss weak intermolecular interaction. Interactions of occupied and empty orbitals, mixing of orbitals, and so on, seem to be ready concepts that can be used to account for structural features of small clusters. Chemical bonding language can say the same thing, though not as directly. For the two clusters, the positively charged proton end of HF is favorably located near the 2p s orbitals of N2, and the alignment is collinear for the sake of s—s overlap.
In this case, the classical multipole and the quantum arguments are really the same; however, as long as the orbitals are not changing too much, the introduction of orbital interactions can be extra baggage. Furthermore, it may not be useful to bring to mind concepts used for changes in electronic structure, when mostly that is something not central to weak interaction. Can long-range weak interaction evolve into chemical bonding? Yes, but apart from ionic bonding, we should anticipate a potential surface bump between a region of weak attractiveness and the region of the chemically bound system.
The sharp changes for covalent chemical bonding tend to develop closer than do the attractions of intermolecular interaction, and those attractions do not smoothly evolve into the sharp changes of bond formation. Types of Elements The language of molecule building block interactions is that of several types of effects whose juxtaposition yields the net interaction potential.
These effects or contributing elements include 1 the electrostatic interactions of the unrelaxed charge distributions, 2 dispersion or the interaction of instantaneous multipoles, 3 the electrostatic effects of polarization and hyperpolarization of the molecular charge distributions, 4 the effects of penetration or overlap of constituent charge distributions and intermolecular electron exchange, and 5 charge transfer if not already included in the other pieces. The pieces might be obtained from ab initio calculation, from experimentally measured properties for some of the pieces, or from outright modeling, even guessing, of the individual pieces.
An important limitation inherent in this phenomenological approach was noted by Coulson a half century ago . The pieces in Eq. Coulson did a simple evaluation of pieces for the water—water interaction and made it clear that the magnitude of V would tend to be comparable to that of the individual elements. That is, the juxtaposition of positive and negative elements yields a net interaction that tends to be small, and the magnitude of the net interaction may be comparable to the magnitudes of one or more contributing elements.
Consequently, accuracy of V demands considerable accuracy in each of the individual elements. Also, it is not meaningful to argue about one or another of these terms being dominant. Instead, it is the subtle result of their combining that determines the structure and energetics of a cluster. It is useful to recognize that certain of the elements of interaction can arise in the absence of any electronic structure change while the others develop only because of some slight change in electronic structure.
For instance, the multipole term is in the first category, whereas mutual polarization is in the second. This really amounts to using the language of perturbation theory and distinguishing first-order from higher-order energy effects. Buckingham  did this at a very early point in the theory of intermolecular interactions. Furthermore, this view suggests that a base-level or starting form for any sort of interaction potential is that wherein effects of electronic structure changes are ignored as being a higher-order contribution.
Improving detail and accuracy then comes with steps to include higher-order elements. Pairwise Additivity and Cooperativity Pairwise additivity—namely, that a potential consists of additive pieces for each pair of species—is used here mostly in the sense of the species being the largely intact molecules making up a cluster, not the atoms in the molecules. This reference to whole molecules is done even if there are multiple sites in a molecule used for expressing a model potential. Were we to refer to atoms in a molecule, that would allow for three-body or nonpairwise additive potential elements in the simple cluster Ar—HF, for example, even though the weak interaction is between only two largely intact units, Ar and HF.
For any fixed H—F distance, those three-body terms get reclassified not changed as two-body terms if we refer to molecular units, not atomic sites, as the interacting bodies. This holds even with there being multiple sites in an individual molecular unit so long as the geometrical structure of the molecule is unchanging. These would involve products of property or parameter values for more than two molecules, and these are often referred to as cooperative or nonpairwise additive elements. A simple illustration is in the electrical interaction contributions.
While the interaction of permanent moments is pairwise additive, involving products of moments of only two different molecules at a time, the polarization energy can have a cooperative part. For some cluster of the molecules A, B, and C, the dipole polarization energy of A will be the polarizability of A, aA, multiplied by the square of the field experienced at A, F. Mutual or back polarization can be shown to produce contributions up to N-body for a system of N species. If we are willing to consider even the smallest of contributions for the sake of completeness, then it is important to realize that contributors in the category that requires a change in electronic structure can affect other contributors and give rise to cooperative terms not explicit in Eq.
For instance, if molecule A in a cluster is polarized by molecule B, then its dispersion interaction with molecule C may be affected. Perturbation theory provides a means for organizing the pieces and for realizing that there are higher-order and mixed terms, in general [10,11,14]. The task is to identify which are important for some desired level of accuracy. Hence, without even considering all the types of contributions, it is clear that not only will the interaction energy at one point be a subtle consequence of juxtaposition of elements, but also the whole potential surface may involve positive and negative terms that can be grouped into sets exhibiting like distance dependence.
The similar distance dependence developing with different contributing elements raises the important issue of uniqueness in the potential terms. Likewise, do we even need to know such terms precisely? There could be a seemingly good potential where too much of one element might be okay because of too little of another that happens to have the same variation with geometry over the surface. In other words, the juxtaposition of small competing terms that complicates accurate determination of a surface also implies that a correct total might not have an accurate breakdown into its pieces.
Should we be concerned? Perhaps not as basic a separation as particle kinetic and potential energies that form the fundamental Hamiltonian for the complete quantum description of the system, they are certainly ways 1 to break down the interaction for interpretation, 2 to guide the representation functional form of a surface, or 3 to set up model potentials by combining elements. The very definition of some of the elements shows the difficulty of trying to go beyond taking them as tools.
For example, charge transfer can be defined as net flow of charge across some chosen dividing surface between two interacting molecules. The choice of the dividing surface, as well as distinguishing charge flow due to charge transfer from that due to polarization, can be done in different ways. Different definitions can produce different pieces. While a consistent set of definitions and even partitioning of ab initio energies is possible, there is probably a limit in how far one can or should go in pinning down all the individual pieces. The case of HCN n clusters, where cooperative effects are important , has shown the type of disagreement that can ensue.
Essentially, one interpretation of the electronic structure changes of molecules in HCN chains was primarily that of charge transfer , while another was that of polarization [21,22]. One can go through more thorough analysis  for the sake of separating the two types of electronic structure changes, and what typically remains as charge transfer after entirely accounting for polarization seems to be a relatively small close-in effect.
From the standpoint of constructing potentials, the interpretation is less important than the fact that polarization energetics are the more straightforward to represent, being based on the simple laws of classical electrostatics. The pieces are the tools to build a potential; and the potential, not the tools, is the objective. The simplest level of physics that can account for the features of the interaction is likely to be the best blueprint for building potentials.
Anything less simple that remains can be added subject to the accuracy demands of the application. Uniqueness of potentials is an issue in a different sense. Two surfaces might yield like predictions and yet have different parameters or even different functional forms. This happens if the set of predictions, such as geometries or vibrational frequencies, sample surface regions where there is not a significant difference between the potentials. Along this line, it is interesting to mention studies of Arn—HF clusters. In one of these , the measured rotational constants of Ar1,2,3,4HF clusters [25—28] and the measured HF vibrational red shifts arising from interaction with argon atoms [29,30] were used as target values.
A quite simple potential was built on fixed electrical properties with very few adjustable parameters for the other terms. Repeated dynamical calculations were performed to find the sensitivity to the parameters and select a set that gave the best match with the target values. It proved possible to get quite close to the target values, and that led to evaluations for like properties in larger clusters with many more argon atoms. There are more elaborate and more accurate potentials  that have been applied to the vibrational dynamics of ArnHF clusters [32—34], and the results of the most recent of these studies  showed considerably better agreement with experimental values than those with the simpler model  see Table I.
We may argue that unique determination of individual pieces becomes an objective only to the extent that there are phenomena or quantitative determinations that call for a correspondingly detailed form of the overall potential. In all model potentials, though much more so in simple potentials, there can be offsetting errors, errors that are not manifested in whatever is being studied, and nonuniqueness. This is to achieve the best descriptions when simple potential forms are required.
Assessments of effects from simplifying potentials for computational advantage are therefore important. While there are still only small numbers of weak interaction problems for which 1 ab initio surface calculations have been carried to the highest levels, 2 correspondingly high-level vibrational—rotational calculations have been performed, and 3 extensive comparisons have been made with rotational and vibrational spectroscopic data— HF n clusters being a leading example [35,36]—there is reason for good confidence in calculational capability.
Calculations have been pushed to a point of very small error through extensive treatment of electron correlation and huge basis sets. Even if such calculations are not commonplace, they provide some of the most definitive information on interesting features such as tunneling barriers that are only obtained indirectly from experiment. They can also provide highly useful information on systems that have yet to be fully studied in the laboratory. Furthermore, what is determined from the very-high-level and benchmark studies can be used to work the most effectively at lower levels of treatment so as to keep the errors as small as possible.
Specifically, the crucial concerns of electron correlation and the adequacy of the basis set are ultimately guided by the high-end studies. With proper attention to basis-set needs and electron correlation, along with attention specific to the physics of weak interaction, useful potential surfaces even in the form of a collection of grid points can be obtained from ab initio calculation. Basis-Set Completeness and Superposition Perhaps more challenging to resolve than the choice of electron correlation treatment for weak interaction problems is the basis-set selection.
Partly, this reflects the fact that basis sets have traditionally been devised for describing chemical bonding, not for the subtle juxtaposition of effects in weak interaction. They have to do both for weak interaction potential evaluations. This issue in basis-set selection, the adequacy of the basis to describe the electronic structure effects that comprise the interaction, can impose more stringent requirements than for describing an isolated molecule.
Consider polarization as a contributor to interaction. A second problem in basis-set selection in conventional supermolecule calculations is that of superposition error or BSSE basis-set superposition error. The elimination of this artificial interaction effect tends to go slowly with increasing basis set size, except in the case of explicitly correlated basis functions as in R12 methods [38— 41]. Hence, there is an extensive literature on ways to exclude BSSE. The original idea, that of Boys and Bernardi , is to evaluate the artificial attraction of a molecule for a set of ghost functions, those being the ones that would be found on an interacting molecule.
They are ghost functions because the nuclei and electrons of the approaching molecule are not included in the evaluation. This artificial attraction is subtracted from the normally evaluated interaction in the Boys—Bernardi or counterpoise correction scheme. When large, flexible basis sets are used, as they should be to carefully describe the interaction contributions, the Boys—Bernardi scheme is generally accepted as the proper correction.
BSSE is a difficulty that is manifested in quite subtle ways, and considerable discussion [43—46] and developments with corrections for BSSE have continued. An interesting idea that may prove to be related to basis-set needs is how the extra electron of an anion is bound to a molecule. From a recent review of electrostatically formed anions by Simons and Skurski , it is clear that there are interesting localizations of negative charge. The two issues in basis-set selection, describing the interaction contributors and minimizing BSSE, are intertwined.
For instance, we have found that small or modest basis sets augmented in regions between monomers i. Yet, rather good values seem possible from these types of calculations with bond functions [49—51].
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Hence, it may be that some capability to describe certain interaction contributions should be included at the expense of BSSE. In fact, an opinion we expressed , that atom-centered functions relative to bond functions may be the best improvement of a small basis for a given computational cost, has met with a solid counterargument relating, in part, to describing dispersion .
The deepening of our understanding of what needs to be described, such as electrostatic-like binding of small excess charge mimicking anions, dispersion, or something else, will likely continue to refine and improve ideas for basis set selection. Use of smaller bases goes along with less reliability; however, with bases smaller than double-zeta in the valence plus one well-chosen set of polarization functions on all centers, including hydrogens, the reliability is so limited that results are not likely to be meaningful for most contemporary problems of weak interaction.
Electron Correlation Effects Having taken weak interaction to refer to systems where molecular electronic structure is largely intact, there is little reason to expect nondynamical electron correlation to be of major importance. In fact, the inclusion of dynamical correlation effects via the low level of second-order perturbation theory i. This is not to say that MP2 is sufficient for the highest accuracy. It is more a consequence of MP2 being relatively more suitable for weak interaction potentials than for the potentials of breaking or forming chemical bonds.
More complete treatment of correlation through higher order perturbative treatment or coupled cluster CC approaches [55—59] assures greater reliability. A significant approach for all types of correlation treatments, that of R12 theory [38— 41], uses many-electron basis functions that depend explicitly on interelectronic distances to deal with the interelectronic cusp problem.
Basis-set requirements are thereby reduced, though at the expense of certain computational steps to use the R12 bases. R12 treatments have been applied to problems of weakly bound clusters [60,61] with very good results. Density functional theory DFT  incorporates electron correlation at a very small computational cost, but its suitability for weak interaction still seems a somewhat open question.
There have been comparisons of DFT and conventional methods [63—68]. Mostly, an improvement over SCF level treatment seems possible, but there is a clear dependence on the choice of functional and on basis-set size. A single functional choice for spectroscopic accuracy in treating weakly bound clusters does not yet seem at hand, but that alone does not preclude application of DFT for lower levels of accuracy. With the computational cost advantage of DFT, the capability exists for treating large, extended clusters. Perturbative Analysis As Buckingham  has done, the intermolecular interaction can be formulated as a perturbing Hamiltonian and then elements of the interaction are associated with specific terms and specific orders of perturbation theory.
However, this introduces the complication of ensuring full electron antisymmetrization i. Symmetry-adapted perturbation theory SAPT [11,14,69—73] adapts the perturbative treatment to incorporate the antisymmetrization requirement as opposed to imposing it on the zero-order wave function. SAPT avoids the subtraction of large energy values that is necessarily part of a supermolecule ab initio calculation. The interaction evaluated in SAPT is defined so as to be free of BSSE; however, the other requirements on basis-set quality and for correlation effects still hold.
SAPT has yielded highly accurate interaction data, first for rare gas atoms interacting with small molecules [72—74] and more recently with molecule—molecule clusters such as the CO2 dimer . Further examples are the very accurate results achieved for Ne—HCN  and a pair potential for water . Another example study of perturbative treatment of the interaction potential has been a study of rare gas—HCN clusters  which included vibrational analysis. Representing Ab Initio Surfaces In principle, ab initio calculational results can provide a conventional grid of surface points spanning the geometrical degrees of freedom for the molecules in a weakly interacting cluster, and conventional techniques of surface fitting might be used to find an interaction potential.
The 3N — 6 degrees of freedom for a cluster with N atoms, though, is often sizable enough to make evaluation of a full grid unapproachable. Of course, the largely intact nature of molecules in a cluster can be a help, because it means that sometimes it will be reasonable to hold the intramolecular structural parameters fixed. This amounts to a Born— Oppenheimer-like separation of the relatively fast intramolecular vibrations from the usually slower intermolecular weak mode vibrations.
Another possibility is to limit the grid to an equilibrium or tunneling region of the geometrical space. While that may be suitable for finding certain features of the interaction potential, dynamical treatments will usually require a full representation of the surface, not one limited to a small region. For instance, diffusion quantum Monte Carlo treatment of cluster vibrational dynamics requires repeated potential evaluations at structures that evolve through random stepping.
Special techniques do exist for effectively using grid points in QMC [79,80]. For general purposes, it is essential that the energy can be evaluated as needed. Hence, the degrees of freedom of an interaction surface and its ultimate use should enter the considerations for collecting grid data. If ab initio energies have been evaluated as a grid of points, property analysis of the interaction can establish at least certain key functional forms for fitting the points or representing the surface in an explicit functional form.
One valuable outcome of analyzing the contributors to intermolecular interaction potentials is finding the pieces that are important. This will tend to keep the number of fitting parameters to a small number and keep the potential function as concise as possible for the targeted level of accuracy. In turn, this probably will keep the number of ab initio grid points to the fewest possible. A good idea for improving surfaces based on ab initio calculations is to empirically adjust the parameters in the potential so as to make the surfaces agree with experimental observations, or with any other feature.
Klopper et al. Morphing is the contemporary jargon for this process, and it is a good idea because it can correct for systematic problems in a surface. If one fits ab initio grid data to a potential whose form is based on the interacting pieces of weak interaction, as in Eq. Interaction-Specific Versus Transferable Potentials Weak interaction potentials can be constructed specific to a single system, such as the dimer of two molecules of water or the trimer of hydrogen fluoride.
They may contain terms that correspond to the usual contributing elements and may incorporate cooperative effects. In any case, the objective is a potential that is expressed in terms of geometrical parameters of the specific system as a whole. The alternative approach is to construct potentials with parameters tied to the molecule building blocks. This alternate approach imposes a type of constraint in that functional forms used to represent a surface must reflect the types of parameters.
The pieces making up multiple moment interaction each have a product of moment values from any pair of interacting molecules, and this goes along with a specific functional form. It is possible that many of the contributing elements to weak interaction can be represented by forms that assign parameters to the building blocks.
This has long been recognized for the interaction of fluctuating dipoles, or dispersion, between rare gas atoms. This type of relation is sometimes designated as a combining rule. It amounts to assigning transferable parameters to rare gas atoms, and there are two important things to notice. First, the number of parameters is smaller with imposed transferability. Second, the transferable approach, compared to an interaction-specific approach, has less adjustability and cannot be expected to offer the same level of quality.
It is fair to say that imposing transferability sacrifices a certain share of accuracy for the sake of broader applicability. How much is sacrificed depends on the whether there is a genuine basis for a transferable form. Clearly, for certain interaction elements, such as multipole moment interaction, a genuine basis is definitely the case. In the case of dispersion, a strong basis for a transferable form via a combining rule [83,84] has been evaluated and tested by Thakkar . The combining rule uses C6 coefficients for a pair of identical species and their dipole polarizabilities, a, to find a C6 coefficient for interactions of unlike species.
A formal expression that can be given for C6 coefficients  is that of an integral over frequency of the product of the dynamic dipole polarizabilities of the two interacting species. For like species, as in C6A--A , this would be an integral over the square of the dynamic dipole polarizability.
Going from use of Eq. Hence, it seems possible that interaction elements that do not immediately seem to have a physical basis via intrinsic molecular property values might nonetheless be cast in a transferable form.
Ab Initio Determination of Potential Elements There are two ideas used for ab initio determination of potential elements. One is to partition the ab initio energy into contributions in line with, or similar to, those in Eq. Another idea is to use ab initio calculations to evaluate properties that enter the interaction expression.
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Ab Initio Evaluation of Properties The most immediate properties for ab initio evaluation are electrical response properties. The electrical response properties are all derivatives of the moleculer energy. Therefore, they can be evaluated by methods that determine energy derivatives gradients, etc. For finite field evaluations, the Romberg approach of Champagne and Mosley  is especially good at ensuring numerical reliability and is easily implemented . By finite field or by direct evaluation of energy derivatives, there are requirements for reliability. First, basis sets must be flexible enough to describe relatively slight polarization changes in electronic structure.
Also, there are very large, multiply polarized bases that have been used in very critical evaluations [e. Electron correlation plays a role in electrical response properties; and where nondynamical correlation is important for the potential surface, it is likely to be important for electrical properties. It is also the case that correlation tends to be more important for higher-order derivatives. However, a deficient basis can exaggerate the correlation effect. A still greater correlation effect is possible, if not typical, for third derivative properties hyperpolarizabilities.
Ionic bonding can exhibit a sizable correlation effect on hyperpolarizabilities. For instance, the dipole hyperpolarizability b of LiH at equilibrium is about half its size with the neglect of correlation effects . For the many cases in which dynamical correlation is not significant, the nondynamical correlation effect on properties is fairly well determined with MP2. To give an example of correlation effects and the differences in correlation treatments, several calculations were done for trans-1,3-butadiene.
The dipole polarizability and second hyperpolarizability were obtained by finite field evaluations, and Fig. The results for a and g based on these curves are given in Table II. MP2 does a rather good job of accounting for the correlation effect when compared with the highest level treatment used, a Brueckner orbital BO double substitution coupled cluster level .
A much more complete study of the dipole polarizability of butadiene has been reported by Maroulis et al. Part of the process of building a model potential using electrical properties of interacting molecules is representing the permanent charge field. The most direct ab initio approach is to evaluate the moments of the charge distribution to some desired order and use them as the representation. Electron correlation in butadiene as a function of the strength of an electric field applied along the longitudinal x axis relative to the correlation energy at zero field strength.
The bottom two curves are also nearly coincident. They correspond to MP2 calculations done without correlating the carbon 1s orbitals and with the inclusion of correlation from these core orbitals. All the other correlation treatments were done without including core correlation effects. The alternative is to distribute low-order moments to selected sites in a large molecule. It is possible to do this on the basis of simply reproducing the molecule-centered moments.
The approximate form [,], using herein the designation ACCSD or ACCD, has been shown to yield potential curves, potential surface slices, and properties very close to the corresponding CC results [—]. The C6 dispersion coefficients for dipole—dipole dispersion between pairs of interacting species, the coefficients for terms involving higher multipolar dispersion, and coefficients for three-body dispersion terms can be and have been evaluated by ab initio techniques [—] as well as through relations to experimental optical data based on moments of the dipole oscillator strength [—].
These are parameters of the interaction, not properties.
The basis set and correlation requirements for adequate evaluation show, in part, the same requirements for describing polarizabilities; however, there are further needs and other than atom-centered functions are seen as being suited [49—52]. At the most basic level, this serves as a means of interpreting ab initio energetics more so than a distinct means for obtaining the energetics.
Morokuma [—] and Kollman  devised the key computational strategies to extract from ab initio calculations the contributions that could be associated with the different elements of noncovalent weak interaction. Different partitionings have been developed ; and perturbation theory, especially SAPT [11,14,69—73], directly gives an extensive partitioning of the interaction energies. This may offer the type of information to construct system-specific, full potential surface models from a relatively small number of ab initio surface points. Models for Parameters Used in Interaction Potentials The use of properties intrinsic to molecules for model interaction potentials requires obtaining those properties through ab initio calculations or in some cases through models of the properties themselves.
Ideas for this have existed for a long time independent of their use in interaction potentials [—]. We have followed the idea of roughly additive atomic contributions  to accomplish transferability. That is, local ai tensors for a given type of atom in a specific bonding environment e. A whole set of values can be built forcing this transferability i.
It is even useful, though less accurate, for the second dipole hyperpolarizabilities, g. A rather novel scheme for modeling molecular polarizabilities as distributed dipole polarizabilities has recently been reported . In this approach, the overall quadrupole induced in a molecule by an external field, as calculated with ab initio methods, is decomposed into induced dipoles distributed to atomic sites.
In turn, this yields the dipole polarizability values at those sites. In effect, this relates the overall dipole—quadrupole polarizability to a distribution of dipole polarizabilities. Recently, an additive scheme for bond polarizabilities has been incorporated with the MM3 force field  to facilitate evaluation of induced dipoles and other features associated with polarization. It is likely that approaches for modeling of polarizabilities, apart from thier role in interaction potentials, will continue to be developed and explored. Interaction Potential Models There are a growing number of interaction schemes based on properties to generate full potentials or else potential surface information for specific regions or specific objectives.
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There have long been interaction potentials that are empirical or entirely system-specific. However, in the spirit of the essential value of properties in interaction potentials, the view of interaction models given here is limited to those that a are not specific to a single pair interaction, at least in their development, and b utilize electrical properties.
In , the Buckingham—Fowler model  for the geometries of van der Waals clusters was introduced. This scheme uses the distributed multipole analysis DMA [,] representation of the permanent charge fields to obtain the electrostatic interaction. The repulsive part of the potential was treated with hard spheres of assigned diameters.
The approach has worked very well in giving the preferred orientations of monomers in clusters [5,8]. The hard sphere form of the repulsive part of the potential is sufficient for finding potential minima but not for representing the potential any closer. Our notions of weak interaction led us to put electrical effects and polarization upfront in modeling. The result was a potential energy surface scheme designated molecular mechanics for clusters MMC  which uses ab initio information on molecular electrical properties permanent moments, multipole polarizabilities, hyperpolarizabilities in the evaluation of the classical electrical interaction of a cluster.
This naturally combines the permanent charge field interaction with polarization energies. The other MMC potential elements were treated empirically and not as fully. This imposed transferability of parameters from one cluster to another has shown modest to quite good success for a number of mixed trimers [—], for instance. MMC has provided useful quantitative information on stabilities and structural parameters, though not always to an accuracy that can answer every question, especially those related to detailed dynamical behavior.
Again, there is an unavoidable trade-off between simplicity and accuracy. Sorenson, Gregory, and Clary  reported a study of the cluster of benzene and two water molecules which used the MMC representation for benzene , and Kong and Ponder have reported constructing an MMC type of model for water, with some interesting variations, as part of a force field program . The effective fragment potential EFP is a more recent scheme that goes beyond interaction modeling.
For reactions for example, Ref. For the spectator segment of the approach, the permanent moment interactions are corrected by a screening function to account for charge penetration effects  and is a stand-alone model potential scheme for small, weakly bound clusters [,,]. A model potential with polarization has been reported for the formaldehyde dimer . It is an example of a carefully crafted potential, which is system-specific because of its application to pure liquid formaldehyde, but which has terms associated with properties and interaction elements as in the above models.
Stone and co-workers have developed interaction potentials for HF clusters , water , and the CO dimer , which involve monomer electrical properties and terms derived from intermolecular perturbation theory treatment. SAPT has been used for constructing potentials that have enabled simulations of molecules in supercritical carbon dioxide . A final point in this section relates to transferability. There is, though, a useful connection. A transferable scheme may provide an initial model potential that may then be adjusted or morphed to improve accuracy for a specific application.
We have done essentially this for pure acetylene clusters . We started with the MMC representation for acetylene, but changed it slightly, using ab initio calculations, to achieve a better match of the measured rotational constant of the acetylene dimer  with a DQMC calculational result for the ground vibrational state. Tested on the trimer, tetramer, and deuterated forms of the dimer, the potential showed very good agreement with experimental rotational constants. We even calculated structures of larger acetylene clusters and their relation to the structures of the dimer and tetramer Fig.
The equilibrium structure of HCCH 13 calculated  with a polarizable model potential. The central molecule is seen essentially end-on. The 12 surrounding molecules are in three layers. The upper and lower layers have three molecules and resemble the structure of the cyclic trimer of acetylene. The middle layer of six acetylene molecules has a pinwheel-like arrangement. Puckering of the rings in the layers yields T-shaped orientations between acetylenes in different layers, along with the essentially T-shaped arrangements for adjacent molecules within each layer. The number of favorable T-shaped quadrupole—quadrupole interactions among acetylenes is thereby enhanced.
The detail to which the electrical response can be treated is something that can be improved in steps; however, the calculational organization stays essentially the same. As already mentioned, many properties may be the basis for constructing potentials; but to a good extent, the pieces needed in the evaluation for electrical analysis are the most involved to calculate. Sometimes, those pieces provide the values for using other property terms.
Hence, it is the electrical analysis which deserves particular care and attention. Also, there are three key reasons for considering these calculational aspects. For purposes of surface fitting ab initio grid points, as mentioned earlier, it is advantageous to know the functional forms of major terms—that is, the electrical property terms.
A straightforward, formal analysis readily provides this information. In terms of number of surface points and possibly gradient evaluations, the most extensive use of potential surfaces for weak interaction will be in dynamical treatments, either classical or quantum mechanical. For these, the cost of evaluation can be important; and with property-based models, one may have to consider the trade-off between cost and accuracy associated with how extensive is the treatment of electrical interaction.
To use electrical response to determine property surface information as discussed in the next section, or simply how to evaluate property changes associated with polarization of charge distributions, full electrical analysis is crucial. The polytensor approach introduced by Applequist  is a terrific organization of the problem of electrical interaction for high-level calculation because it can be continued uniformly to any order of multiple moment, any distribution of moments, and any number of interacting species. Furthermore, it can incorporate multipole polarization and hyperpolarization .
As such, it provides a scheme that can be coded for computer application in an open-ended fashion while also providing the formal analysis needed to extract functional forms of different electrical interaction pieces. The polytensor organization casts the electrical interaction evaluations in a form that is independent of the orders of multipoles included.
The polytensor organization makes it very clear that the key computational step for electrical parts of interaction potentials is where the geometry information separation distance and orientations enters. Overall, the evaluation of the elements needs to be done efficiently, and means are available for that . This can be used with response properties such as shielding polarizabilities to find property changes dues to electrical influence.
The evaluation is analogous to Eq. A first point of discussing the polytensor organization for evaluating electrical interaction energies is to see how computational effort grows with multipole order. The numbers of elements in M associated with a multipole of order 0 charge 1 dipole , 2, and 3 are 1, 3, 9, and 27, respectively. Though these numbers can be reduced by converting from Cartesian moments to irreducible spherical forms, the size of T grows as the square of the sum of these numbers—that is, as the square of the total number of elements in M.
Hence, from a computational standpoint, every order of multipole is a big step from the one before. For simulations of liquids that might involve hundreds of molecules, this computational complexity can pose a limitation. This leads to the second point in this section—that is, that the computational effort is largely in the T tensor, whether it is found explicitly or not.
However, it is difficult to represent a molecular charge distribution with a distribution of only a few point charges. The charges have to be relatively large, and this yields abrupt changes at certain regions. A comparison has been presented for water that shows this . Using many point charges instead of a few adds cost even with the simple singleelement T tensors, and it seems that the step to distributing a few dipoles may be more advantageous from a computational standpoint.
Indeed, for neutral species, the best overall scheme is probably to distribute dipoles and even quadrupoles to represent the charge field of a molecule. Further improvement might then come from a small number of point charges, each being very small in size, and this is in line with using DMA [,]. It may be premature to say that this is an optimum modeling strategy. We should anticipate that there is a lot more experience to be gained and that there are comparisons to be made. However, at the least, we can expect that limiting model potentials to forms that include point charges but not local dipoles is not necessarily computationally advantageous.
Two-body dispersion yields an interaction term that is relatively simple to calculate. More complete descriptions of dispersion will include higher-order terms. For dispersion sites that are not treated as spherical, there is an angular dependence via Legendre polynomials. For a system of only three interacting species, there should be little difference in using Eq. However, with many interacting species, Eq. To avoid redundant steps, every time a distance between two sites, rij , is updated or changed, the following are computed and stored with r ij: rij2 ; rij3 ; rij4 , and rij6.
Higher-order three-body dispersion coefficients e. In a simulation of the vapor—liquid equilibrium of pure argon, Bukowski and Szalewicz  showed there are important three-body effects; and because of certain cancellations, these were primarily DDD dispersion. Finally, it should be noted that the dispersion interaction that is at work at long range does not continue close-in. It has been recognized that damping out the dispersion close-in yields the correct behavior [—] and computationally simple damping functions have been devised [,].
These can be used for terms associated with dispersion in potential models. Yet there are certainly slight changes occurring, and they have interesting and revealing manifestations. Polarization is one of the clearest, most direct types of electronic structure changes taking place in intermolecular interaction. It may be the dominant change in many cases. This is significant because we can account for polarization changes to the electronic structure via multipole polarizabilities and hyperpolarizabilities.
In principle, ab initio calculations of potential surfaces can be accompanied by ab initio evaluations of property surfaces. However, this is likely to be a cumbersome task. On the other hand, if many properties reflect polarization changes in the electronic structures of the interacting species, then property surfaces should be well-suited to modeling. Indeed, the potential surfaces and property surfaces can be put on an equal footing via evaluation of the electrical influence of surrounding species i.
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