Journal of Molecular Structure (Theochem), 179 (1988) 273-292 Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands
MODELS AND MODELING IN THEORETICAL
JACOPO TOMASI Dipartimento di Chimica, UnioersitG di Piss, Via Risorgimento, 35 - I56100, Pisa (Italy) (Received 30 October 1987; in final form 7 March 1988)
ABSTRACT General considerations about models in theoretical chemistry are addressed to formulate criteria for the classification and judgment of the models. It is emphasized that, in most cases, scientific investigation is not performed with the aid of a single model, but rather using sets of related models, addressed to study large classes of chemical phenomena. Each set, which may be in competition with other sets, must satisfy, on the whole, some general requirements and a judgment must be made on the examination of the whole set. As an application of these methodological criteria, some model-building activities performed by our group in Pisa are analyzed. For brevity, attention is focussedon bimolecular interaction, chemical substitution, solvent interaction and electronic excitation effects only.
The word “model”, it has been remarked, is the most commonly used noun in modern scientific literature. An influence of fashion on this remarkable recurrence of the word is possible: scientists are influenced by the scale of values, attitudes and concepts implicitly accepted by the civil as well as by the scientific community. The explicit use of models pervades our everyday life: analyses and prescriptions on the economical, sociological, psychological facets of our society, to name a few, are based on this approach, borrowed from technical and physical disciplines. The widespread use of this concept has given rise to a progressive shift in the meaning of the word, losing contact with the original one related to the definition of paradigms to be assumed as reference and to objects to be imitated. The new “dynamical” meaning of the concept is actually of great importance in science. Scientific enquiry extensively uses models, based on appropriate theories, and often the elaboration of a model is the qualifying point of research. Chemistry too is deeply involved in the use of models. This fact is often acknowledged by researchers, although because of the conservative attitude, *Dedicated to Professor Bernard Pullman.
0 1988 Elsevier Science Publishers B.V.
common to many scientists, there is an inclination not to express it explicitly, when the involved subdiscipline is classified as “experimental”. When the “experimentalist” leaves his experimental apparatus to elaborate on the collected data, he enters a world of models, and often the experimental activity itself is based on models, material analogical models, according to the definition below. These considerations, expressed here rather vaguely, seem to me to be of some importance in putting in the correct context the status of “theoretical” chemistry, a chemical subdiscipline working, by definition, on models. Although not professionally qualified to speak about models “in se” - there is now a specific branch of scientific discipline, the model&tic, which addresses the analysis and formalizes the process giving rise to the formation of models in science - I will offer some remarks on this theme in the hope that they will be of some use in the evaluation of specific activities in the field of models in theoretical chemistry and, in particular, to clarify the strategy of the investigations performed by our group in Pisa which shall be taken as an example of these remarks in the second part of the paper. The subject covered in this paper, although not corresponding to the usual scheme of contributions to Theochem, seems to me convenient to honor Professor Pullman, an outstanding model maker in theoretical chemistry and biochemistry.
Before treating models in molecular sciences, and in theoretical chemistry in particular, it is convenient to consider some general characteristics of models. A model, according to the meaning this word has in science, is, by definition, incomplete with respect to the referent, which is generally a complex system. Only some features of the referent, which shall afterwards be designated as the “object system” are present in the model. The occurrence of different models referring to the same object system is quite common; these models may select different features of the object, because there is a different evaluation of what characterizes the object, or because there are distinct aspects of the object which deserve modeling. The co-existence of different models, which agree in the selection of the characteristic features and on the finality of the model, is possible because there are different levels in the hierarchy, or classification, of models. This accepted, and encouraged, the co-existence of alternative and competing models makes the introduction of judgement criteria necessary, which provide at least a loose evaluation (unsatisfactory, poor, acceptable, very good). Some simple criteria are given below.
A model must be non-contradictory. In particular, models related to the realm of science cannot be in contradiction with the basic principles accepted by contemporary science. Models which connect features of the object (or of the model) in contrast with some basic criteria (e.g. dimensionality) should be considered with some suspicion, because good performance of models of this kind may be due to chance, the models may be limited in their application, or they may obscure more satisfactory explanations of the same features. Simplicity
The unessential aspects of the object make the model obscure and reduce its significance. Ad hoc assumptions must be avoided (Ockam’s razor). The use of this criterion is not easy: models which are too simple may lose important hidden features of the object, and consequently be of little use in the scientific research. The balance between simplicity and completeness is delicate, and it is this balance which, in the end, measures the merits of the model. Related to simplicity is the transparency of the model: a good model is characterized by the facility with which it can be described, understood and applied. Stability
It should be possible to introduce modifications or complements into the model without destroying its internal structure. It should be possible to use a good simple model as a starting point for a sequence of models of increasing complexity to obtain increasingly accurate descriptions of the object properties. Utility
The model should provide information (or predictions) on some characteristics of itself which are not explicitly stated during its elaboration, and this information should be congruent with the corresponding characteristics of the object. In other words, a model must provide “surprises”. Actually, many models are tautologic in character, as mathematics is, but a useful model brings to light aspects which could be otherwise overlooked, and makes understandable aspects and properties of the object which could otherwise remain confused or misinterpreted. Generality
A good model should permit the identification of connections among distinct objects not evident during the elaboration of the model. There are wide classes
of object systems which are very different in nature, but share some attributes and properties, and one of the main goals of the scientific enquiry is to make these connections clear. These’ general criteria apply to all exact sciences and, with minor modifications, to human sciences also. In passing to chemistry, and then to theoretical chemistry, other remarks and other criteria could be added. To be brief, I include here only a few remarks on the scope of models in chemistry and on the process of modeling in molecular sciences. The scope of a model in chemistry is the interpretation of chemical phenomena and the prediction of the behavior of chemical systems under specific conditions imposed by the surrounding environment. Obviously, the prediction is related to the interpretation: a model with good predictive properties but devoid of interpretation is a poor model in physical sciences. A model which is finalized to a few aspects of the objects arises from a process of reduction of the information available on the object, and this process assumes specific characteristics in chemistry and theoretical chemistry. The process of reduction of information regards not only the passage from object to model, but also the utilization of the model. This is an important aspect of model design and of the utilization of a model. This process of reduction does not necessarily coincide with a “reductionist” position in the meaning this word has in the philosophy of science. Chemistry, and all the other molecular sciences, are involved with very complex object systems: the process of model building is, therefore, complex and cyclic. Model building starts from a preliminary interpretation of the object characteristics, and one of the possible uses of the model is just to obtain a better definition of the object itself. An interactive cycle, where a preliminary model is used to obtain a definition of the object, should enable a satisfactory model to be reached by successive approximation (if there are no collapses or spectacular failure, as often happens). To give an example, a biochemical reaction occurring in vivo may be a feature of different object systems ranging from the whole living body to a cell, to a substructure in the cell, to a limited amount of molecules or atoms. In fact these intermediate definitions of material systems (cell, substructure, etc.) are also models of a particular kind. A classification, or taxonomy, of models is necessary. It is convenient to use the following scheme Scheme material
1 f- iconic
An iconic model is based on its similarity in form with the object. Examples in
molecular sciences are the lock-and-key model in enzymology (abstract) and the balls and rigid sticks molecular models (material). An analogic model preserves some aspects of the form of the object, but gives emphasis to the behavior, or functional aspects of the object. So, a masses-andstrings molecular model is an analogic material model for studying vibrations, and a chemical system composed by a substrate and an appropriate synthetic polymer may be a material model for studying some aspects of enzymology. Later in this paper I shall give support to the contention that many models in quantum chemistry have the status of abstract analogic models. A symbolic model neglects the analogy of form with the object and relies solely on the analogy of function with the object. This is prima facie the realm of mathematical models (from quantum mechanics to thermodynamics), but the importance of non-mathematical models should not be neglected. A nice example of a symbolic abstract model of non-mathematical nature is the periodic table. Symbolic models may also be of material composition: for example, symbolic models for the study of molecular vibrations are both a set of coupled differential equations (abstract) and a set of coupled electric oscillators (material). This schematic attempt to classify models may be subject to several criticisms. The selection of the terms may be at variance with the terminology currently used in modelistics (I am not an expert), although it is similar to that used by Bunge [ 11.It gives emphasis to the structural aspects of the question leaving apart other considerations which can be of noticeable importance in the classification of models. This last point deserves more attention. Our understanding of the world of object systems derives from inquiries performed on models and, during this process, the status of a model may change. An important example in chemistry is the molecule: in the middle of the last century it was an abstract symbolic model (non-mathematical), later it gained the status of an analogical model, and eventually jumped from the world of models to the world of real, material object systems. So the ball-and-stick structures may no longer be considered a model of a model, but iconical modelizations of real objects. The same is happening for the Schroedinger description of material systems: this abstract symbolic (and mathematical) model has assumed in recent times a flavor of an analogical model, and it is often translated via computer graphics into iconic models. It should be clear that models play an important role in chemical research: often the material systems under investigation are analogical material models, and the interpretation of the experimental results relies on the use of abstract models, supported by analogical and iconic models. COMPOSITION OF MODELS IN THEORETICAL CHEMISTRY
As remarked above the “abstraction” of the models used in quantum chemistry has somewhat reduced in recent years. It is convenient to introduce a
distinction between the components of the models. This partition takes in0 account the aspects of “analogy” which the current models have and can be applied to the majority of models used in theoretical chemistry. (a) The material part of the model (or the material model) isthe portion of matter described by the model. It may correspond to the effective portion of matter in which a given phenomenon is observed, ox:to a reducth or simplification of it. (b) The physical aspects of the model! (the physical model) explicitly (or implicitly) considers the physical interact&on8of the &ject. In some cases it is convenient to introduce a distinction bet-n interactions involving only elements of the material system and interactions of the system with theexterior. (c ) The mathematical aspent%of the model (or the mathematical model) are the methods and approximationsused to describe thepbysical interactions in the material model. The consideration of th% division. of models for theoretical. studies is of help when judging models. Let us takemodels addressed to the &ndy of atuhemical reaction as an example. The material model may be reduoedto two ix&racting molecules, thus relying ORthe assumption that a,bimolecular encoun&r is sufficient to describe the reaction, or even simple material” models (Lttcouple of molecules of smaller dimension, or’more complicated models regardkd as a set of coupled oscillators) assuming that the mo%zcularremainder camb omit&d from the model; conversely, the material composition:may be enlanged by including a third body, a variable amount of so#!nt molecules, etc. ‘J&! phystil model may be reduced to the electrostatic: &rces in the qtmnturmformal&m acting in the bimolecular systems, or reduced some camponenti of the- or enlarged to non-electrostatic forces, or imdbmding~interactionwi& a the-al bath and so on (of course non-qua&al desarjiption is also-possible& The n&hematical model assesses the level of quanlmm mechan&al cal&ions (;HF, post-HF, etc.), the basis set, the approximations in the evaluatGon of matrix elements, and so on. The choice between time-dependent and! time-in&pendent formulations affects the physical as well a the mathem&cal aspeets of the model. A systematic examination of these aspects, coupled with thecriteria exposed previously, represents a substantial step in the judgement ofa model. @‘or example, at a given material and physical compoeition, the “amelioration” of the mathematical aspects (calculations of higher complexity and precision) could improve the “utility” of the model, but probably decreases its “transparency” and “simplicity”. Finally, since the primary goal of models is to obtain an interpretation of physical interaction phenomena, it is important to examine what the basic material unit is, in order to make this interpretation. The basic unit in molec-
ular sciences is the molecule, but chemists are not satisfied by this level of discrimination. Basic units of the submolecular level (canonical MOs, chemical group descriptions, atoms in a molecule, atomic hybrids, etc.) are not supported by evidence arising from object systems as molecules are, and correspond to a higher level of abstraction in model building. This aspect deserves particular attention when examining models in theoretical chemistry. QUANTUM CHEMICAL MODELS BASED ON CHARGE DISTRIBUTION
To substantiate these general remarks with some examples I will make use of the direct experience of our group in Pisa. To minimise the bibliographical references I will make use of a limited number of review papers without mention of the original publications. Attention will be focussed on quantum mechanical models in the time-independent approximation using as basic units single molecules, chemical group descriptions, or larger aggregates of molecules, depending on the case examined. The mathematical model always refers to ab initio SCF or post-HF methods, with expansion basis sets of various sizes and supplemented by a large set of analytical tools which can extract from the computations the necessary information for interpretation, and, in a following step, for the elaboration of more rapid and efficient algorithms for prediction, only the utility of the model has been ensured. The physical model includes, prima facie, all the quantum-mechanical interactions inside the material model, but much attention is paid to the role played by external fields (interactions with the exterior). A distinguishing feature is the systematic attempt to build up models in which only the classical part of the electrostatic interactions are considered (semiclassical models). This attempt, which assumes different connotations in models addressed to different phenomena, meets the criteria of simplicity and stability (and satisfies the requirement of reduction of information) whilst preserving, in principle, the criterion of self-consistency. The utility of these models will be judged on the basis of the results. The process of reduction of information is easily done on the basis of current formalism for the time-independent Schroedinger formulation. From the wavefunction of the material model QM(r; R), defined in terms of electronic (r) as well as nuclear (R) coordinates, it is easy to extract the first-order density matrix diagonal element r(r;M)
(1) In more complex systems the density matrix formalism allows the same quan-
tity to be obtained without starting from the wavefunction, c#&,which collects both the electronic and the nuclear contributions, and is partitioned according to the prescriptions of a specific set up of the model. Interactions among subunits (and interactions with the exterior) are computed, taking into consideration only classical interactions, and compared with full ab initio results and experimental data, when available. It is worth noting that the most natural reference for models of this type are ab initio models and not experimental data. One of the outstanding characteristics of the theoretical approach - especially when it relies on a theory which is able, in principle, to give a proper account of the phenomenon - is its analytical power, i.e., its ability of looking inside the phenomenon, putting in evidence aspects which are very difficult, or impossible, to discover with experimental tools. However, the utility of the model is finally determined by its capability to reproduce and interpret experimental results.
A STRATEGY FOR SEMICLASSICAL MODELS
For the sake of simplicity and transparency it is not convenient to introduce into the model the classical description of the interactions at their full extent. It is advisable to examine first simpler models, wherein only a portion of the interactions are considered, to submit the model to falsification tests, and then to pass on to more complete models until the full classical’ interactions are considered. In this way it is possible to put in evidence true quantum effects not described by classical interactions and to check if there is a model in this sequence of approximations which gives a satisfactory interpretation of the phenomenon, and then to use it to elaborate a mathematical procedure to be used for predictive purposes. Interpretations given in classical terms are, in general, more transparent and easier to translate into appropriate algorithms. If the analysis has not given satisfactory results, the semiclassical approximation may be used without destroying the model, and non-classical terms may be included step by step. Experience taught us that hurried generalizations are to be avoided: for each investigation it is convenient to define at the very beginning the phenomena under scrutiny, the field of variability in the composition of the material model, etc. This rather pedantic approach ultimately is of advantage: the generality of the model comes out naturally and with more evidence. Simpler models, adequate for one class of phenomena, fail to describe another class: often it is sufficient to go one step ahead in the sequence of models to obtain a satisfactory description of both classes, with a deeper understanding why the first model was sufficient for the first class and not for the second.
A first example: energetics of simple bimolecular interactions
An example we have used in preceding reports, regarding the process of bimolecular interactions A+B, on activity clarifies the above point fairly well. The phenomenon under investigation is the energetics of the process, i.e., dEAB ( ItAB). The capability of theoretical methods to examine this phenomenon allows direct scrutiny of not only the final result, but also the channels of approach, the affinity of B for A and the equilibrium geometry. It is necessary, however, to proceed step by step and to examine a statistically meaningful set of cases A,+B, A,+B, A,+B .... where B is fixed in composition and the A,s are selected from a wide, but not too general, class of compounds for example molecules composed by first row atoms and provided with lone pairs. B is then changed and similar interactions of the A,s with Bi, B2, B3... are examined. The simplest example, which provides more immediate insight into the study system, is that of the interaction between neutral molecules (A,) and charged species of small size (H+, Li+, Na+, ... Be2+, .... H30+, CH,+ , F-, Cl-, ... OH-, ...). In this case the first model assumes as a hypothesis that it is possible to describe the interaction on the whole range of distances in terms of classical interactions between rigid charge distributions, r(r; A) and r(r; B) and, furthermore, that it is possible to replace the complete description of subsystem B with a point charge qn. In this case the interaction may be reduced to the simple expression &B(RAB)
where VA ( RAB) is the electrostatic potential of A (generated by r( r; A) ) computed at the point RAB where the charge qs is placed. It is convenient to remark that there are no intuitive arguments to support this hypothesis for small RAB values, because other interactions, also partly classical in origin, should make important contributions. Comparison with ab initio calculations shows that the hypothesis is verified at relatively large distances, and gives fairly good results for proton affinity and AH+ equilibrium geometry, while it fails for other cations and anions. The correlation between proton affinities obtained with ab initio (mainly SCF, with different basis sets) and rigid distribution Coulombic models has been shown in previous papers [ 2-41: the slope of the regression line is less than unity because other effects are active, and the spread of the values around this line partly exceeds the effects due to chemical substitutions in the A,, set. However, on the whole it seems that the model has picked up the essence of the phenomenon for protonation acts. The next step consists of the introduction of a further classical term into the interaction, i.e., polarization effects, still preserving the description of B as a point charge. The improvement in the description of protonation acts is remarkable (correlation curves are given [2-41): this model may be applied to
obtain the proton affinity with an acceptable error of less than 0.6%, but the results for other species are worse than before. The reason for this failure is non-classical effects. In fact, starting from Li+, the presence of an electronic cloud in B gives rise to additional repulsive forces related to quantum effects. It is sufficient to introduce a repulsive potential for B of appropriate shape (e.g., a suitable step potential) to obtain results for the affinity of A, towards B in very good agreement with ab initio calculations. For larger B ions, especially anions, the model may be profitably supplemented by energy contributions due to the polarization of B. Other fine detail is given in larger material models by dispersion contributions (another energy contribution of non-classical origin which can, however, be modeled with sufficient accuracy) but the essence of the phenomenon in these simple association processes has been caught by the model. The same model applies well to a wider class of non-covalent interactions; much work has been done on hydrogen bonds, but the results are substantially the same for charge-transfer complexes, acid-base couples of different types, etc. Coming back to the original rigid Coulombic model, it appears that the addition of a repulsive term alone gives good results even for Li+. In other words, polarization of A, also plays a minor role in this relatively strong interaction. The interpretation is simple: because the repulsive force produces a larger R, than for the proton, the cation electric field is not sufficient to induce a substantial amount of polarization of A. The same holds for other complexes, and this is the reason why in modern literature the emphasis for H bonding and other non-covalent interactions is placed on electrostatic effects. Improvement of models for bimolecular interactions
The example summarized here required noticeable efforts. Other factors, not mentioned before, had to be checked: deformation of the internal geometry, dependence of the results (for the models as well as for the ab initio calculations) upon the basis set and upon the method employed to obtain r( r; A) and r( r;B ) , and their modification by polarization effects; the correction of basis set superposition errors; the setting up of analytical tools to analyze AEAB(R ) values, etc. Some of the intermediate results have been published separately, but the majority of the numerous numerical checks which were made remains in our files. I am adding this remark in order to avoid giving the impression, perhaps given by the brief summary reported above, that modeling is an easy task which does not require much effort and self-criticism. The efforts of checking and ameliorating the mathematical aspects of the model involves at present a considerable number of researchers, not directly related to our small group in Pisa. A review will be attempted on another occasion, and I will limit myself here to a recapitulation of our recent work on the correction of the
dimeric interaction for errors due to the incompleteness ployed in the ab initio calculations.
of the basis set em-
A CP-corrected description of the analysis of the non-covalent interaction processes
It is well known that the interaction energy dEAB (R) of a bimolecular interaction process is affected, inter alia, by mathematical (and nonphysical) errors due to imbalance in the basis sets used to compute the energy of the dimer and the reference energies of the isolated monomers A and B. The counterpoise procedure (CP) proposed several years ago by Boys and Bernardi  introduced more consistency into the basis sets used in the calculations; the procedure is simple to apply and, consequently it is a useful device in implementing models in which the material part may reach a considerable size. The interaction energy is thus corrected by a term, depending on Rm, in the following way (3) where &gT is the correction obtained by replacing the reference energies Ei and Ei in the definition of E with corresponding values obtained by extending the monomer basis set to the dimer size. The introduction of this correction into our strategy of building up models of increasing complexity must, however, be accompanied by a decomposition of dxgT into physically meaningful terms. In fact, the most important tool in models addressed to the interpretation and reproduction of dEAB is the decomposition of this quantity into components having a clear mathematical de& nition and a reasonable physical interpretation. For the examples shown previously, as well as for many other cases, we used the Kitaura and Morokuma [ 61 decomposition, which is computationally simple and satisfies the other prerequisite we need. To introduce CP corrections into the separate elements of this decomposition, we introduced partial enlargements of the monomeric basis sets, an action justified by the physical nature of the specific component. So, with minor modifications of the original code for the decomposition, we redifine this precious analytical tool in the following way (4) The superscript CP evidences the terms where a CP correction has been introduced. Of some importance for semiclassical models is the fact that this new de% nition does not alter the classical components of the interaction energy, namely the rigid electrostatic, EEs, and the polarization, EpL, terms. A change in math-
ematical model requires a revision of all the preceding steps, and the constancy of the leading terms make this revision easier. Anyway, a revision must be made. After some tests, we started a systematic analysis taking the material model (i.e., the chemical composition) as the variable, and using the qualifying features of the mathematical model (i.e., the basis set ) . The results obtained thus far seemed sufficient for us to write a review [ 71 of neutral hydrogen bonded dimers, and it will probably be followed by similar papers regarding charged H-bonded systems, and other types of non-covalent complexes, for which a large amount of the material is already computed. So, we may try, with some confidence, to anticipate some general conclusions: the general interpretation for non-covalent interactions made before is confirmed, with some changes which give more emphasis to the classical description of the model. The introduction of CP corrections reduces the spread in the predictions of association energy and equilibrium distance due to differences in the basis set. More confidence is gained in the results obtained with small basis sets, especially for the rigid Coulombic contributions, which are confirmed as the leading terms in the orientational conformation of dimers. The semiclassical model seems to have passed a severe test, confirming “stability” and “utility”. SUBMOLECULAR MODELS
The dimeric interactions considered in the preceding sections can also be used as an example to develop a following point. It is evident that a satisfactory interpretation of a bimolecular interaction cannot be reached when the molecular partners are taken as the basic elements of the analysis. A chemist desires to know why, for example, a specific site of the molecule is preferred over another for chemical attack by an incoming molecule, or at what extent and in what direction the reactivity of that site is altered by changes in the molecular structure. Theoreticians have a great freedom in selecting basic units at the submolecular level. Physically, atoms may represent a reasonable choice, chemically, functional groups are well suited to translate chemical experience, and chemical models, into a theoretical framework. The use of quantum mechanical methods makes possible other definitions of the basic subunits. The importance of the one-electron approximation in quantum chemistry suggests the use of molecular orbitals (of canonic form or subjected to appropriate orthogonal transformations) as a starting point in the analysis; other theories on the chemical bond give emphasis to atomic subunits (atomic hybrids) as the basic units of the description of molecular properties; other partitions of the molecular space, and then of the molecular charge
distribution, are suggested by the analysis of the intrinsic properties of the system (e.g., partition in terms of zero-flux surfaces). It is not necessary, we think, to report here a bibliography on this subject. The historical evolution of quantum chemistry is characterized by contributions of many scientists in this field, and to give a summary of this topic would in practice, be to recapitulate the history of theoretical chemistry. It will be sufficient to recall what has been said above: models in competition are a positive feature in this branch of the scientific enquiry and they must be judged according to the general criteria expressed in a preceding section. Furthermore, it is quite easy, in this specific case, to combine different approaches, thereby defining new models, each of which has specific features, specific merits and specific defects. Following the scheme adopted in the preceding section, I shall discuss briefly here only one of the models used by our group. Again, I shall not attempt to give an overview of similar approaches adopted in many other laboratories. The starting point is the one-electron description of the wavefunction of the molecule A; corrections due to the use of many-determinant wavefunctions are introduced afterwards. The canonical orbitals are subjected to a localization transformation (generally we have used the Boys transformation [ 81, but other localization procedures give similar results), and the localized charge distribution 2 ATy (r; A) 1, (r; A), supplemented by an adequate portion of nuclear charges, is used to define the basic submolecular charge distributions JJ~(r; A). The necessary number of ya (r; A) is collected to describe a chemical group, or molecular fragment, g, inside A y&A)
= C y,(r;A) ffeg
The total charge distribution of A, r(A), is unaffected by this partition r(r;A) = C y&A) GA
The charge description of groups may be supplemented by other descriptors, the nature of which depends on the phenomena under examination. Considering again two-body interactions leading to the formation of noncovalent complexes, and restricting the analysis to the energetics of the interaction act (other aspects of the interaction are of interest, of course, and can be dealt with using other descriptors), the quantities of interest in the semiclassical model are the electrostatic energy predictor V(r;A) and the polarization energy predictor P(r;A). The first of these factors is related to the electrostatic potential of A, V( r; A) V(r;A) = 1 V,(r;A) B
which, in turn, is related to the electrostatic energy of interaction between M and N
When r(r;B) is reduced to a single point charge qn at position RAB, the electrostatic interaction is just given by &s(&B)=
c K#LB;A) ( g
When r(r;B) is described by a discrete set of point charges, Ezs is obtained by a summation of V, (r; A) values modulated by the values of the qnx charges. The same is true for the polarization energy descriptor. When A is reduced to a point charge, we have E PL = - 1 P(R,n;A)
with P(r;A) = C P,(r;A)
Extension of the method to more complex descriptions of r( r;B) is straightforward. The operational definitions of these quantities have been reported previously [ 2-41; the same papers also give some numerical examples, others are reported in a review [91. After a sizeable number of analyses have been performed over a wide range of material models, these simple analytical tools can give an interpretation of numerous details of non-covalent interactions, for example: the order of affinity of small species (cations, anions, dipolar reactants like HF, HzO, NH, etc.), of different chemical groups in the same molecule, or of the same group in different molecules (e.g., reversal of affinity for different reactants in a given set of related molecules, like ethers, amines, ketones etc.); the interpretation of linearity in H-bonded dimers and the general conformational properties of dimers; and the interpretation of some steroelectronic effects, such as the anomerit effect, etc. The stability of the model with respect to the introduction of a finer definition of basic units seems to have been demonstrated. Prototypes for group charge distributions
It is possible to take the analysis a step further. The analysis mentioned above relies on subunits derived from the molecular component A of the material model. In chemistry we are accustomed to think of chemical groups as entities having characteristic properties which are modified to some extent,
but only slightly, by the remainder of the molecule. To introduce this feature into our model we may define a prototype charge distribution ~~(0) which is no longer related to a specific molecule A, but is obtained as an average over a set of charge distributions belonging to different molecules. Their operational definition [3,4] makes possible their direct transfer from a library of group charge distributions to describe in an approximate manner the molecular charge distribution. g;A Y,(O)=r(O, A) -r(A)
This approximation, though crude, is sufficient to describe the essential aspects of many chemical phenomena [2-41. In particular, the results of the analyses of molecular interactions mentioned before are confirmed in their essential aspects when the descriptors V, (r;A) and P, (r;A) are replaced by their directly transferable counterparts V, (0) and Pp (0). The description in terms of yg(0) contributions gives in particular a fairly good description of oneelectron molecular observables and also passed the rather severe tests related to the evaluation of the total molecular energy, such as conformational energy changes, energy balance for isodesmic reactions (a more detailed report on this point is in preparation). In this way, the model has reduced its analogy with the object system (the description of the material model is less specific than that given in terms of y,(A) functions), with a gain in simplicity. Furthermore, the field of applicability is remarkably enlarged: in fact when the ~~(0) models are used the previous calculation of aM is no longer necessary. The calculation of the molecular wavefunction is a bottleneck when attempting to study material systems of large size, as, for example, systems of biological interest. To quote an example, we are presently studying material systems of complex compositions where a component is a DNA fragment which may reach the molecular weight of 106. Molecular modification of the prototype charge distributions
Starting from the y,(O) models we can now try to go a step further in the opposite direction, i.e., in the direction of submolecular models more similar to the original y,(A) ones. To do this we again use the semiclassical approach. The strategy is similar to that exposed in a preceding section: a restrictive hypothesis on the nature of forces modifying ~~(0) to get y,(A) is formulated, and then tested with the opportune calculations. A restrictive hypothesis, which has thus far achieved fairly good results, is given below. The effect of the molecular remainder is reduced to the classical polarization effects, measured by the field F( A/g), i.e., the field produced by all the groups of A with the exception of g. In addition, the localized orbitals describing the electronic part of y,(O) are considered to be composed by generalized atomic
hybrids, e.g., a localized orbital corresponding to a bond between atoms M and N is written as A”,=C&hR+C&h&
The hypothesis is that the effect of the remainder is reduced to a parametrical dependence of the coefficients C, and C, alone on the perpendicular component of F( A/g), measured at the middle of the M-N bond. Several tests, performed mainly on one-electron observables of different types, show that these new charge distributions, here called rp(A/g), are more similar to those directly derived from the molecular wavefunction, i.e., the yp(A) of eqn. (5)) than to the prototypes rg(0). In other words, the semiclassical model has shown a comfortable degree of stability in the following sequence of approximations y,(O)-+y,(Alg)+y,(A)
This hypothesis is probably too restrictive in character, and better descriptions related to less restrictive hypotheses should be examined. But this is, for the moment, the end of the story. We have not yet performed experiments on the model addressed to falsify this hypothesis, or to define the limits of applicability. EXTENDING THE RANGE OF APPLICABILITY
OF THE MODEL
The above discussion should be sufficient to show how general criteria about models may be applied to judge a model and even to plan extensions and modifications of a given model. However, the exposition of the semiclassical model we have taken as an example is not complete. A concise exposition of other extensions of the model may shed further light on the generality of this model. To this end we shall select solvent effects and electronic excitation effects. Solvent effects The interpretation of bi- and poly-molecular interactions prompted us to extend the model to systems in the liquid phase. The material composition of this family of models (there is in fact a family of related models [lo] ) is given by a “solute” M, i.e., by a limited amount of molecules treated with the usual methods of quantum chemistry. The interaction with the whole “solvent” is described by an effective interaction operator Vint, which relies on thermally averaged descriptions of the medium surrounding the “solute”. These averages are represented by continuous distributions. The simpler definition of Vintrelies on the semiclassical approximation. The interaction between solute and solvent is reduced to classical electrostatic effects. It is expedient to reduce volume integrations to integrations of surfaces on the wall of cavity encircling the solute. In this way the interaction operator
Vintis reduced to an electrostatic operator V, which relies on the definition of an apparent charge distribution a(s) on the cavity surface [lo]. This definition makes it possible to use cavities of general shape, i.e., not limited to a sphere or to an ellipsoid, and avoids the inconvenience related to the use of analytical expansions of the solute-solvent interaction potential. Analogous definitions of the mathematical model, relying on surface integrations, have been elaborated for more complex models including, for example, the dispersion terms, or the consideration of a “solvent” with different properties in different portions of the space. These last versions have so far been applied to biochemical problems, where biomolecules (enzymes, membranes, DNA) have been considered, in some cases, as components of the systems for which a detailed description was not necessary, while their physical effects on the “solute” were important [lo]. Applications in other fields of chemistry, however, seem promising. Electrostatic solvent effects For simplicity let us consider only the initial simpler version. The Schroedinger equation is written in the following way WR+WW)lI/k=G4y~
The interaction operator V,( M ) depends on the solution v/t of this equation [which is actually a function of the electric field of the solvent polarized charge distribution r (r,M) computed on the cavity surface]. The dependence on the original parameters [P’ (r,M) in vacua] is not linear. From r (M) and Ei, information on the solvent effects on the “solute” and on the energetics of the whole system may be derived [lo]. I shall not review here the numerous applications thus far attempted with satisfactory results, but will limit the discussion to an aspect related to the previous discussion. Starting from eqn. (15)) in which M, as hinted before, may be a complex molecular model where chemical interactions may be active, we may repeat the analyses mentioned before, e.g., on dimerization or on the characterization of chemical groups, with a new factor in action, the field F, produced by the apparent charge distribution o. So, for example, it is possible to extend the chain of models illustrated in eqn. (14) y,(O)-ty,(Mlg)-ty~(Mlg;o)-ty~(M;a)
where the two last elements of the chain are the model group modified by the molecular field and by the solvent field at the same time, and the group description derived from eqn. (15), respectively. Furthermore, the solvent field, F, may be decoupled into group contributions. This adds a new degree of freedom to the analysis, making more detailed interpretation possible [ 41.
The study of chemical interactions in solution must be accompanied by the appropriate tools. The discussion here will be limited to the field of non-covalent interactions; for brevity, real chemical interactions, in vacua as well as in solution, have not been included in this synopsis, although examining the elaboration of the pertinent models could substantiate better our methodological analysis. As stated above, for non-covalent interactions in vacua an important analytical tool is comprised by the decomposition of the interaction energy performed according to the Kitaura-Morokuma method with CP corrections [see eqn. (4) 1. In solution, a redefinition of the decomposition leads to the following expression
We have used the symbol G instead of E’ to emphasize that, in solution, the basic energetic quantity is a free energy, and not an internal energy. By comparing eqn. (17) with eqn. (4) it is evident that a new term is present, namely GPDY which we have called the “partial desolvation” contribution [lo]. It is outside the scope of the present paper to show why this term is necessary and to show what its effect is on the interpretation of AGAB.It is more interesting to note that there is a strict analogy between this term and analogous terms introduced (a) when the molecular system (in vacua) is subjected to internal geometry deformation, (b) when the system is subjected to an external field, (c) when the system is inserted in a crystal lattice, and (d) when the system undergoes electronic excitation. The analytical tools used in the models, as for the model itself, are subjected to judgement criteria, among which generality, stability and utility play prominent roles. Electronic excitations
So far in this paper no mention has been made of application of the models to phenomena in which electronic excitations are involved, although such models have been built. Again I will not review the problems of extending the basic models to these phenomena and of the solutions we have elaborated but will discuss only one aspect. Electronic excitations often regard a limited portion of the molecule, the chromophore, with limited effects on the molecular remainder. In such cases it is possible to view the electronic excitation as a special case of chemical substitution, with a group g in the ground state replaced by a group g* in some excited state Y,(M,CS)-+Y,(M,EX)
The semiclassical analysis of intergroup effects is thus enlarged. It seems possible to obtain a rationale of the effects of excitation of g on
y (M,EX) using very simple concepts (electronic excitation of an electron to a vacant localized orbital) starting from (0) descriptions. Some indications on this theme are reported in a recent overview [ 111, wherein attention is paid to the geometric and reactivity effects of the excitation on the target system M. Of particular interest is the case of electronic excitations in solution. In this case we may couple fields deriving from the excitation process with fields deriving from the solvent, and also, according to the case, with fields deriving from chemical substitution inside a molecule and with fields related to chemical interactions [ 111. These complex networks of relationships have not been completely explored, but we are confident, on the basis of the results thus far obtained, that the generality and utility of the model will be confirmed. CONCLUSIONS
The analysis of a set of related models, as described in this paper, makes evident that it is convenient to look at specific activities in the field of theoretical chemistry not only on the basis of results displayed in single papers, but also considering the overall strategy and its congruence with general criteria about modeling. This analysis is done more easily by people involved in the implementation of that specific set of models, but explicit statements about the limits of a specific investigation, about the connection a specific contribution has with respect to a general strategy, should be of considerable help for other people to put in the proper context, and to evaluate, scientific contributions. There is now a tendency to separate, as a specific discipline, computational chemistry from theoretical chemistry. There are arguments to support this separation, but it seems to me that, in the vague definition of computational chemistry, the use of models derived from theoretical chemistry plays a very important role. Definitions of limits, of congruence with general criteria, etc. (i.e., with the topics treated here) are of particular importance in this case; if a definition is not satisfactorily achieved, the consequences on the scientific status of computational chemistry could eventually be rather severe. ACKNOWLEDGMENTS
Although the responsibility for errors, omissions and misunderstandings in the general discussion about models is mine, the personal experience I have tried to summarize in this paper and the large amount of actual results I have mentioned, are heavily dependent on the activities of many other persons. First of all Professor E. Scrocco, who taught us all to always have a wide vision of the problems even when involved in a strictly defined and detailed investigation. Drs. R. Bonaccorsi, C. Ghio and G. Alagona, as permanent members of our small staff, have contributed to the evolution and clarification of ideas and,
at the same time, performed the largest part of the work used here as examples. Contributions from other members of the staff at Pisa and from researchers working in other institutions, either formalized in published work or left to the status of informal discussions, are not forgotten, and are here collectively acknowledged.
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