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Ratings and Reviews 0 0 star ratings 0 reviews. 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 [86] is especially good at ensuring numerical reliability and is easily implemented [87]. 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. The correlation consistent basis sets of Dunning and co-workers [88] are among available bases that include a number of E-Book Center , Phone: 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. As molecule size E-Book Center , Phone: 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 more E-Book Center , Phone: 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. One immediate outcome was the confirmation that electrical E-Book Center , Phone: 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.
There have long been interaction potentials that are empirical or entirely system-specific. Indeed, there are hundreds of potentials that have been used for the specific interaction of a water molecule with another E-Book Center , Phone: 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.
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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. It incorporates electrical response and is targeted E-Book Center , Phone: 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 moment polytensor, M, is the single index array of the usual moment elements in canonical order from the zeroth moment charge to the first dipole , second quadrupole , and so on, to any desired level of termination: Hyperpolarizabilities E-Book Center , Phone: 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 []. In polyensor form, the interaction energy between moments at site A and moments at site B is a matrix inner product: 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 [37]. 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. The first term is a dipole—dipole—dipole DDD [] term: Via substitution with the law of cosines, this can be expressed in terms of the distances: 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: 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. How well does this work? For one, we have shown that for carbon monoxide in a series of carbonmonoxyheme proteins, the experimentally observed correlation of 13C chemical shifts, 17O chemical shifts, 17O nuclear quadrupole coupling, and CO vibrational frequency shifts arose because each property change had to due with polarization of the CO by particular distal ligands in the proteins [].
We have shown that polarization accounts for the evolution of the cluster dipole hyperpolarizability of HF 2 as the monomers approach [], how the ranges of structural nonequivalences in the chemical shifts of proteins are matched by the representative sizes of the shielding polarization [] first derivative of the nmr shielding tensor with respect to an external electric field , and we have developed a generalized picture of Sternheimer shielding [].
In the study of a subtle property, the nuclear quadrupole coupling constants of weakly bound clusters [], a hybrid calculation helped reveal the polarization nature of the effect. The charge field of a perturbing molecule was built into a one-electron operator that was used in the evaluation of the nuclear quadrupole coupling constant of the perturbed molecule, but the perturbing molecule was not otherwise included no nuclei, electrons, or basis functions.
The ab initio calculations showed a correlation between interaction strength and effect on nuclear quadrupole coupling through a series of clusters, and this was in very good agreement with experimentally determined values. An earlier and much more extensive series of ab initio studies by Cummins, Bacskay, and Hush [— ] had provided a strong picture of nuclear quadrupole coupling being E-Book Center , Phone: These studies [—] are a very good indication that the primary way in which the property change takes place is via the polarizing, charge field influence of a perturbing molecule.
Hence, the basic of idea of obtaining property changes along with energy changes seems to hold in a scheme that takes good account of the electrical part of the interaction. That is, in evaluating the electrical contribution to an interaction potential, the external electrical potential at a molecule or site in a molecule has to be evaluated, and it is this information which can feed the computational process for calculating a property change.
Bridging Quantum Mechanical Treatment to Models of Surroundings Interaction modeling based on properties of molecules, properties that of course are determined by quantum mechanics, amounts to a bridge from the quantum to the classical picture. But the reverse is also important, and we can see this from the example just discussed [] involving work done in collaboration with the late H. Gutowsky and his group. These are a measure of the electric field gradient at the 14N nucleus, and such evaluation usually requires careful quantum mechanical treatment because of sensitivity to subtle and closein features of the electronic wavefunction.
The objective, accounting for the trend in eQq with the interaction strength of the three partner molecules, was realized by an ab initio calculation done on HCN alone through representing the partner species by a one-electron operator. This is a classical to quantum bridge, one that precludes intermolecular quantum effects. It suggests that weak interaction effects of surrounding molecules on a species being described quantum mechanically can, to a good extent, be incorporated as an electrical influence, and this is much simpler than a full quantum mechanical treatment of surrounding species.
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In complex systems, there is an advantage in partitioning physical space into regions described with different levels of treatment, starting with the highest levels for regions of greatest interest. Electrical analysis, if used in these methods, can go both directions at the interface. The effective potentials used to represent the non-quantum mechanical spectator molecules in the ab initio treatment quantum region include Coulomb interactions, polarization, and exchange repulsion.
Permanent charge fields offer an influence bridge between quantum and nonquantum regions in complex systems. If the polarization response in the nonquantum region can be represented more fully, then it may be possible to bridge mutual polarization effects, too. This more physically complete interface might mean that the boundary between quantum and non-quantum regions could be pulled closer to the active region—a computational benefit. That is, the external electrical environment arising from nearby molecules acts at a site or at distributed sites within a molecule.
The basis for following this in a potential model is twofold. First, the electronic structure changes due to weak interaction are slight, which can be easily seen by examining ab initio electron densities of weakly perturbed molecules. Second, it seems clear from analyzing property changes due to weak interaction that, primarily, the source of the slight electronic structure changes is polarization.
The future in model construction, particularly that which seeks to follow properties along with energies, will likely involve more detailed treatment of polarization response. Within the framework of the response being local, the detail is increased by the straightforward act of distributing more sites within a molecule. However, going beyond a local response is another approach to adding detail, though of a different sort; and possibly, there are ways to provide models with the capability to represent somewhat more substantial electronic structure changes than has been considered so far.
The interaction of long-chain molecules such as polymers is a problem area where the nature of polarization response can be a significant concern on its own.
An example is from a study of parallel hexatriene molecules carried out to represent a truncated form of solid-state polyacetylene []. This study included both ab initio calculations and an electrostatic model using polarizability, a, and second hyperpolarizability, g, tensors distributed to the carbon centers. The ab initio calculations on a single hexatriene molecule were used to find the distributed tensors for the electrical analysis.
The ab initio evaluations benchmarked the electrostatic model calculations both for E-Book Center , Phone: Of course, the model treatment could be extended to much more than seven hexatrienes; hence, values were obtained for the asymptotic limit of an infinite number of parallel chains. As chain length increases, there is the possibility that the polarization response may be more than can be described by a sequence of point dipoles being induced at the atomic sites used in this local repsonse treatment.
For longchain molecules or otherwise extended species, polarization might result in a net flow of charge from one end of a molecule to the other. Stone [] and Munn [] have considered this with respect to charge. The key is to have a charge—charge polarizability or susceptibility at sites in a molecule whereby a potential at one site acts to change the net amount of charge at another site, a nonlocal interaction.
Stone connects the dipole induced in a region with charge—charge, charge—dipole, and dipole—dipole polarizabilities, the first two corresponding to a flow of charge into or out of the region in response to an external potential. The analysis offers a good physical picture by breaking down a molecule into regions and allowing for charge to flow from region to region in addition to being shifted within the region.
The charge—charge, charge—dipole, and dipole—dipole polarizabilities, though, are more complicated than the usual molecular dipole polarizabilities because they are defined for pairs of sites. Right now, the evaluation of charge susceptibilities and the related values in interaction models is challenging. There have been signficant developments for practical approaches [,]. However, an important link emerges from the work of Hunt and co-workers [—], who have shown how geometrical derivatives are related to nonlocal charge susceptibilities.
Derivatives of a molecular electronic energy with respect to geometrical parameters are routinely calculated with ab initio techniques; hence, this may provide a useful route to the necessary properties. Particularly important is the connection of these response features with forces in molecules and geometrical derivatives as also discussed by Fowler and Buckingham [] E-Book Center , Phone: There is another difficulty, that of achieving transferability; however, it may be that the need for this more elaborate treatment of polarization will tend to be quite specific, making transferability less important.
As expressed at the outset of this review, it is important to realize that interaction analysis and model development is an industry serving several needs ranging from coarse forms to very detailed and precise forms and from systemspecific to generally applicable. That the molecules are not changed by interaction in the abrupt ways that go along with chemical bonding implies that a simpler level of physics should be at work, one that can be exploited by relying, as much as possible, on intrinsic properties of molecules.
The focus herein has been on constructing and building models tied to using properties, and specifically on doing so in a mostly fundamental way as opposed to working backwards to achieve known outcomes or making educated guesses for very simple types of potentials.
The latter type of approach is widespread, and this is perhaps because there is such a strong need for hydrogen-bonding potentials, particularly for the theoretical study of liquids [,]. Working directly from an analysis of the interaction, its elements, the full functional forms for those elements, and the parameters in those elements is an increasingly common route to potentials for dynamical simulation.
It is a route with much promise, and so that trend will continue. There have been many creative directions for developing potentials, and a considerable understanding has developed in the last two decades, especially. However, a statement by Buckingham well before that in a report [18] continues to hold: There are highquality ab initio approaches for electrical response information, for dispersion, and for properties and parameters, and there are fully developed calculational approaches for very extensive treatment of electrical interaction.
Within the last 5—10 years, there has been strong attention given to evaluating and dissecting short-range or close-in effects, anisotropy in dispersion, and other subtle aspects needed at some point in potential construction. These fundamental developments represent the high end of our understanding of weak interaction. An argument of this report is that basing model interaction potentials on intrinsic properties of molecules as much as possible ensures conciseness in the potential function and the greatest prospect for transferability as opposed to being system-specific.
Furthermore, property-based potentials are probably the E-Book Center , Phone: The future of this type of technology is in increasing the range of tunability to greater and greater accuracy, and a number of possible steps have been mentioned. The slight changes that do take place in the electronic structure of weakly interacting species, excluding those being influenced only by rare gas atoms, seem to be largely polarization induction changes. This argument is made on the basis of the variety of property changes that can be accounted for by evaluating only a polarization response.
This implies getting back other property information, or, in other words, obtaining property surfaces along with potential energy surfaces via the modeling that is done. We can speculate that polarization being the primary electronic structure change has another interesting outcome: It is likely to be a strong contributor to cooperative effects, perhaps a very dominant effect among cooperative elements in certain clusters. Our own evaluations on many clusters have shown four-body polarization effects to be very much smaller than three-body polarization effects.
It is possible that with adequate treatment of intermolecular polarization response, maybe only through three-body effects, the crucial cooperativity needed to connect gas-phase potentials with condensed phase behavior may be realized. Three-body dispersion might be a further improvement for this connection. Various contemporary efforts at constructing model potentials e. It is probably at a stage like that of the technology of ab initio calculations when small basis sets, small molecules, and limited treatments of electron correlation were typical.
That would be a time about 30 years ago or so. The comparison, though, does not hold for accuracy, which is already quite high in many types of models, and it does not mean that model potentials will be getting very much more complex and extensive. Rather, the comparison with the development of ab initio electronic structure technology is to suggest that this is an area of computational chemistry that is likely to be very significant on its own.
There is growing consensus on means for constructing potentials, and there are more and more critical comparisons and tests. There are likely to be fundamental improvements in the technology, things that would compare to the successive developments of high-level electron correlation treatments of ab initio technology. Also, there are issues unlike those of the ab initio methodologies, such as how far to go in devising potentials that are transferable. In many ways, this suggestion of an emerging computational area follows the idea that weak interaction is its own chemistry with molecules, not atoms, as the building blocks.
Whereas ab initio electronic structure treats atomic building blocks, E-Book Center , Phone: This gives even more impetus to model construction that puts property changes on an equal or nearly equal footing with the energetics. Acknowledgments The support of the National Science Foundation for investigations related to weak interaction over a considerable period and recently via Grant CHE is gratefully acknowledged. Buckingham, in Intermolecular Interactions: From Diatomics to Biopolymers, B.
A 51, A , Scripta 21, A 5, A 20, A , ; , B 3, C Solid State Phys. Le Seur and A.
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Ionic and Covalent Crystals IV. Crystal Orbital Methods A. Density Functional Theory C. Other Approaches Acknowledgments References I. This curiosity has been spurred in recent decades by the increasing importance of the communications industry in the world and the parallel need for materials having specific properties for electronic, optical, and other devices.
With few exceptions, a useful nonlinear optical material will be in the solid phase—for example, a single crystal or a poled polymer embedded in a film. Ironically, the quantum chemical calculations of nonlinear optical properties have for the most part been concerned with a single microscopic species.
Advances in Chemical Physics (Volume 126)
Much has been learned in this way about appropriate molecular construction, but the ultimate goal must be to investigate the nonlinear optical NLO properties in the solid phase. In the physics arena the theoretical determination of NLO properties of solids has been more advanced, though not to the degree that has been achieved for simple gas-phase molecules using modern quantum chemical practices.
For example, density functional theory in its crudest form has been frequently adopted to find some NLO properties for semiconductors. A glaring example of lack of progress is the third-order susceptibility of quartz. There is, as yet, no rigorous calculation of this quantity, even though it is the reference point for nearly all NLO measurements. It is our opinion that in the next few decades this situation is going to change—that is, the field of single molecule calculations will be saturated and attention will turn to the more practically relevant solid phase.
This makes it an opportune time to review what computational strategies have been already developed. This, in turn, will indicate the more profitable lines of research to be pursued in the coming years. As the situation currently stands, there are two extreme approaches: For i , in its simplest disguise, single molecules lie side-by-side and the properties of the solid are just an appropriate combination of the molecular ones. For ii , the whole solid is considered as a giant molecule and the computational approach is that of standard quantum chemical calculations.
An important variation of ii is to make use of the translational symmetry that exists in a crystal or a periodic polymer, and this leads to so-called crystal orbital methods. Here is an important break from the conventional molecular calculations and one that is likely to see a great deal more use in the NLO field in the near future. Refinement of the oriented-gas model, including more and more precise accounting for interspecies interactions, is another likely avenue for progress. The macroscopic optical responses of a medium are given by its linear and nonlinear susceptibilities, which are the expansion coefficients of the material polarization, P, in terms of the Maxwell fields, E [1—3].
For the sake of clarity, the tensor nature and frequency-dependence of the susceptibilities have been omitted. For an isolated molecule, the electric dipole moment, m, is expanded in a Taylor series of the external field: As we will see, these microscopic responses are, in many schemes, used as a starting point to evaluate the macroscopic responses, and the effects of the surroundings are introduced subsequently.
In such cases, the effects of the surroundings can be decomposed into two contributions. The second effect is the difference between the local electric field that hyper polarizes the entities and the macroscopic field that defines the bulk susceptibility. The inhomogeneity and the nonlocality of the responses to the applied fields are consequently crucial aspects to be included in these determinations.
In tune with the above introductory remarks, we have arranged this review in the following way: Section II deals with the oriented gas model that employs simple local field factors to relate the microscopic to the macroscopic nonlinear optical responses. The supermolecule and cluster methods are presented in Section III as a means of incorporating the various types of specific interactions between the entities forming the crystals.
The field-induced and permanent mutual hyper polarization of the different entities then account for the differences between the macroscopic and local fields as well as for part of the effects of the surroundings. Other methods for their inclusion into the nonlinear susceptibility calculations are reviewed in Section IV. In Section V, the specifics of successive generations of crystal orbital approaches for determining the nonlinear responses of periodic infinite systems are presented.
Finally, E-Book Center , Phone: In each section, the pros and cons of the different approaches are noted and illustrated by giving selected references, which focus on particular aspects of the methodology, or the calculations, or the characterization of materials. In this model, molecular hyperpolarizabilities are assumed to be additive and the macroscopic crystal susceptibilities are obtained by performing a tensor sum of the microscopic hyperpolarizabilities of the molecules that constitute the unit cell.
The effects of the surroundings are approximated by using simple local field factors. Equation 3 assumes that the crystal symmetry is high enough that the fLon tensors are diagonal in the crystallographic frame. The validity of this model requires that the intermolecular forces most often van der Waals be much weaker than their intramolecular analogs, which are associated with the chemical bonds. Conversely, using the hyperpolarizabilities of the isolated molecules, Eq.
Of course, the gap between the model and the true values depends upon how large the van der Waals forces, the intermolecular charge transfer effects, the hydrogen bonds, and so on, are. Its E-Book Center , Phone: The third related to the generated wave at os originates from the fact that the nonlinear contribution to the displacement vector is the product of the nonlinear polarization and the local field factor [5].
The local fields, and therefore the local field factors, should account, in principle, for the microscopic nature of the unit cell; but, in practice, this is far too complicated [6] and such an approach would negate the simplicity of the model. It is important to recall that by using this expression for fJo1 and fKo2 , one assumes that the nonlinear contribution to the local field is negligible. This relationship can be improved by considering an ellipsoidal cavity for which analytical expressions can be found from solving the Maxwell equations with boundary conditions [7].
Further improvements to these local field factor expressions take account of the nondiagonality of the local-field tensor in the crystallographic frame [7], the self-polarization correction, which is a type of dressing of the isolated gas species [6], and retardation effects. Just consider the change from a sphere to an ellipsoid where the large axis is twice as large as the other two. The aim was to assess the validity of an additivity scheme for substituted benzenes.
A similar approach was also later adopted by Oudar and Zyss [8] to study methyl- 2,4-dinitrophenyl -aminopropanoate MAP crystals. The combination of crystal SHG data, treated using the oriented gas approximation, with EFISH electric-field-induced second harmonic generation measurements on MAP in solution enabled them to demonstrate the two-dimensional character of its first hyperpolarizability tensor.
The differences between the theoretically and experimentally deduced b-tensor components were as much as one order of magnitude. Subsequently, this model was used to define the MAP crystal structure, which exhibits the highest phase-matched crystal nonlinearity. In their next paper, these authors [9] addressed the effect of the crystal point group on the relation between the microscopic and macroscopic nonlinear responses for both second harmonic generation SHG and the electro-optic dcPockels dc-P effect.
Adopting the oriented gas approximation and assuming a one-dimensional character for the chromophores, they considered the influence of phase-matching conditions for SHG and determined the optimal orientation of the molecular conjugation axis within the crystalline frame. They also pointed out the limits of this additivity-based scheme for the electro-optic dc-Pockels effect due to the vibrational contributions associated with lowfrequency intermolecular vibrations.
In the same period, Zyss and Berthier [10] applied the additivity scheme in a slightly different way in order to determine the macroscopic second-order nonlinear response of urea crystals. A Coulomb point-charge potential was located around the urea molecules to account for the hydrogen bonds and the electrostatic interactions. This potential modified the electronic properties of the molecule in the crystal with respect to its isolated state and was found in a self-consistent way using the molecular wave functions.
During the next 20 years, the oriented gas approximation was frequently applied to connect microscopic and solid-state macroscopic NLO responses of organic systems. Table I gives, for both second- and third-order processes, information on typical investigations that are discussed in more detail in the following paragraphs. In this way, they demonstrated the efficiency of combining the quantum chemical calculations of the hyper polarizabilities with electron crystallography in order to characterize compounds for which it is difficult to grow sufficiently large crystals for the purpose of measuring their NLO properties.
Subsequently, Voigt-Martin et al. Because in the crystals the MHBA molecules form pairs linked by hydrogen bonds, the first hyperpolarizability of MHBA dimers was evaluated and combined with the usual Lorenz—Lorentz local field factor for spherical entities. This, however, led to little improvement, probably because the dimer is less spherical than the monomer. Typically crystal unit cells may contain 2; 4;. This approach is related to the earlier work of Zyss and Berthier [10] as well as to that of Lin and Wu [16] and can be seen as a hybrid between the oriented gas model and the supermolecule approach see Section III.
It enables one to take into account the interactions between the molecules and, particularly, the effects that originate from their large dipole moment. It has been used by Zhu et al. Due to their large third-order nonlinear responses, conjugated materials have received the attention of several groups, the most studied systems being the prototypical p-conjugated polyacetylene PA , polydiacetylene PDA and its mesomeric polybutatriene PBT form, and the s-conjugated polysilane PSi. Due to p-electron delocalization, the longitudinal second hyperpolarizability per unit cell of the p-conjugated systems increases with chain length until a linear region is reached where the number of unit cells is between 20 and As a E-Book Center , Phone: Consequently, the surrounding effects have only to be included for the two directions perpendicular to the chain axis.
However, as a result of experimental difficulties three-dimensional disorder, chain defects, etc. In this regard, using local field factor corrections has always reduced the gap between theory and experiment. The same approach was later employed for substituted PSi chains of different conformations [21]. Another study is due to Luo et al. The amorphous nature of the system justifies the choice of the SCRF method, the removal of the sums in Eq.
The oriented gas model can also, in principle, be used to evaluate the nonlinear susceptibilities of inorganic crystals with or without accounting for the Coulomb effects in the hyperpolarizability determinations see Section VI. Without these Coulomb effects, which can be large, Cheng et al. The microscopic-to-macroscopic transformation and the agreement with experiment have, however, been criticized [28]. There is no doubt that such approaches deserve to be pursued further.
The results can be analyzed by association with the different types of bonds present, and these range from the weak van der Waals and hydrogen bonds of molecular crystals to the strong bonds of ionic and covalent crystals. The quality of the calculated macroscopic NLO responses depends on the level of the theoretical treatment as well as on the size, shape, and boundaries of the composite species. Both aspects are equally important.
When dealing with molecular crystals, Langmuir—Blodgett films, or chromophores dispersed in polymer matrices, besides possible strong ion-pair intermolecular interactions, there are two types of bond: Therefore the distinction between the field aspect and the molecular aspect is clear. In this case, one can refer to the method as either the supermolecule or the cluster approach, but, for the sake of clarity, we will only use the former. On the other hand, in the case of covalent or ionic crystals, the two types of effect are intertwined and one refers to the cluster approach only because the system is no longer a supermolecule.
Many theoretical investigations have addressed the impact of intermolecular interactions upon molecular properties by adopting the supermolecule approach, although the number of interacting entities included is often small. There are fewer studies on covalent and ionic clusters.
The specifics of these two kinds of investigation as well as a brief survey of the most representative ones are addressed in Sections III. This convergence is related to the evolution of the response as a function of the size and the shape of the aggregate and to the extrapolation procedures if used to determine the properties of the infinite system. There, since the chain length dependence is very strong, several extrapolation schemes have been employed [29—32]. However, in contradistinction to polymers, the systems under study here spread in all spatial directions and the number of atoms in the system can be very large before reaching convergence or even before being able to extrapolate.
Indeed the former converges faster because the end-chain effects are mostly removed [33]. In this framework the search for a simple scheme that could reduce the threedimensional problem to a problem of lower dimensionality is of great value.