Cramer C.J. Essentials of Computational Chemistry Theories and Models

NAlO

hn

N O

Figure 14.8 Structure of Alq3 with only one ligand drawn in full. The two states involved in the photoemission are computed to be highly localized to a single aryl ligand, and indeed to resemble the analogous HOMO and LUMO of 8-hydroxyquinoline (inset at right). How are issues of molecular geometry and absorption/emission spectroscopy related? What levels of theory would be expected to be most successful, and economical, for predicting geometry, or absorption/emission, or both? How does the inorganic aluminum atom complicate things if at all?

Halls and Schlegel approached Alq3 with an interest primarily in its excited-state properties. Prior studies on the ground state had provided some information about its molecular geometry, vibrational spectroscopy, and vertical absorption spectroscopy.

Geometry optimization of excited states can be tedious since analytic derivatives are available for a more limited range of theories than is true for ground states. When the excited state cannot be described at the single-configuration SCF level (i.e., it is not the lowest energy state of some irrep of the molecular point group), then the next simplest approach is CIS, for which analytic derivatives are indeed available. The size of Alq3 presents some basis set limitations, and Halls and Schlegel decided to employ 3-21+G . [Note that it is not entirely clear what is meant by this basis set name, since usually the first ‘*’ is really ‘( )’, and means polarization only on second row atoms (see Section 6.2.3), while the second ‘*’ presumably implies p functions on H atoms; however, it would be very unbalanced to have polarization functions only on Al and H, so it seems likely that the authors used first-row polarization functions as well, perhaps borrowing them from the 6-31G(d) basis set, where such functions have been defined.] To check the likely utility of this basis set, they compared it to a large polarized valence triple-ζ basis set for the S1 excited state of simple 8-hydroxyquinoline (which was small enough to permit geometry optimization to be carried out with the larger basis). Differences in the structures predicted using the two basis sets were sufficiently small that Halls and Schlegel were confident about using 3-21+G for Alq3.

However, while the CIS level has been shown in many instances to be fairly good for optimizing excited-state geometries, it is not particularly good in this system for excitation energies, as shown in Table 14.5. The ab initio CIS results drastically overestimate the absorption energies in every case, irrespective of basis-set quality. Note, however, that the CIS/INDO/S results do very well indeed, particularly with the largest window of 15 occupied and 15 virtual orbitals used in generating the singly excited states. This illustrates how much one can achieve with careful parameterization.

The TDDFT results, in this case using the B3LYP functional, are the most satisfactory of the non-empirical results. Comparison of 3-21+G results with previously published

Table 14.5 Measured and predicted absorption and emission energies (eV) for 8-hydroxyquinoline and Alq3

6-31G(d) results again would seem to indicate that the former basis set may be regarded as adequate.

With respect to emission, the geometry computed for the excited-state Alq3 structure at the CIS/3-21+G level was found to be significantly different from that for the ground state. This leads to a Stokes shift of 0.8 eV, which is not particularly well reproduced at the CIS level but is quite accurately predicted at the TD-B3LYP level.

Having obtained good agreement with experiment for the various spectroscopic data, Halls and Schlegel go on to analyze the MOs involved in the photoluminescence. They find that the orbitals involved are highly localized on a single one of the three aryl ligands in Alq3, and that these orbitals are quite similar to those involved in the S0 → S1 absorption/emission of 8-hydroxyquinoline itself. They also express the difference between the excited-state geometry and the ground-state geometry of Alq3 in terms of the normal modes (this procedure is in essence a multilinear regression involving displacement vectors). They find that a particular normal mode having high intensity in the vibrational spectrum makes a significant contribution in this analysis, thereby rationalizing observations of vibrational structure in the absorption and emission spectra of Alq3 under matrix isolation conditions. They carry out their vibrational analysis using BLYP/6-31G(d) frequencies, as these were found to be in very good agreement with the experimental ground-state IR spectrum.

The work of Halls and Schlegel illustrates particularly effectively how different levels of theory may be used for studying different aspects of a complex chemical problem. Furthermore, repeated comparisons of theoretical predictions to experimental measurements in order to validate the chosen levels of theory provides solid support for the quality of further predictions using those levels.

Bibliography and Suggested Additional Reading

Aguilar, M. A. 2001. ‘Separation of the Electric Polarization into Fast and Slow Components: A Comparison of Two Partition Schemes’ J. Phys. Chem. A, 105, 10 393.

Cave, R. J., Burke, K., and Castner, E. W., Jr. 2002. ‘Theoretical Investigation of the Ground and Excited States of Coumarin 151 and Coumarin 120’ J. Phys. Chem. A, 106, 9294.

516 14 EXCITED ELECTRONIC STATES

Ciofini, I. and Daul, C. A. 2003. ‘DFT Calculations of Molecular Magnetic Properties of Coordination Compounds’, Coord. Chem. Rev., 238, 187.

Foresman, J. B., Head-Gordon, M., Pople, J. A., and Frisch, M. 1992. ‘Toward a Systematic Molecular Orbital Theory for Excited States’ J. Phys. Chem., 96, 135.

Jensen, F. 1999. Introduction to Computational Chemistry , Wiley: Chichester.

Koch, W. and Holthausen, M. C. 2000. A Chemist’s Guide to Density Functional Theory , Wiley-VCH: Weinheim.

Krylov, A. I., Slipchenko, L. V., and Levchenko, S. V. ‘Breaking the Curse of the Non-dynamical Correlation Problem: the Spin – Flip Method’, ACS Symp. Ser., in press.

Levine, I. N. 2000. Quantum Chemistry , 5th Edn., Prentice Hall: New York.

Reichardt, C. 1990. Solvents and Solvent Effects in Organic Chemistry , VCH: New York.

Yarkony, D. R. 1998. ‘Conical Intersections: Diabolical and Often Misunderstood’, Acc. Chem. Res., 31, 511.

Zerner, M. C. 1996. ‘Intermediate Neglect of Differential Overlap Calculations on the Electronic Spectra of Transition Metal Complexes’, in Metal –Ligand Interactions , Russo, N. and Salahub, D. R., Eds., Kluwer: Dordrecht, 493.

Ziegler, T., Rauk, A., and Baerends, E. J. 1977. Theor. Chim. Acta, 43, 261.

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15

Adiabatic Reaction Dynamics

15.1 Reaction Kinetics and Rate Constants

Consider an arbitrary equilibrium system

where no particular stoichiometry is implied. When the system is displaced from equilibrium, by addition of more of a particular species, by a change in temperature and/or pressure, or by any other influence, empirical observation has shown that the rate at which equilibrium is reestablished may be expressed as

where kφ is a phenomenological rate constant (distinguished from an elementary rate constant as defined later on), [W] represents the concentration of species W (usually expressed in units of molarity or partial pressure), and each concentration term has associated with it an exponent that is sometimes referred to as the ‘molecularity’ of the species. Often, but not always, molecularities have integral values, including zero. Note that since we are measuring a return to equilibrium, all concentration terms are functions of time t, as are kφ and the rate itself.

The a priori prediction of all of the variables appearing on the r.h.s. of Eq. (15.2) is a challenging task, to say the least. This is particularly true because the equilibrium of Eq. (15.1) may involve the simultaneous operation of a large number of individual chemical reactions, with some possibly involving very low concentrations of reactive intermediates, the presence of which may be difficult to establish experimentally. In order to make progress, a critical simplification is to break the overall process down into so-called elementary steps. To simplify matters a bit, we will consider only adiabatic reaction steps, that is, reactions taking place on a single PES without any change in electronic state (the topic of non-adiabatic dynamics is discussed briefly in Section 15.5). For practical purposes, there are only two kinds of elementary reactions: unimolecular and bimolecular.

Essentials of Computational Chemistry, 2nd Edition Christopher J. Cramer

2004 John Wiley & Sons, Ltd ISBNs: 0-470-09181-9 (cased); 0-470-09182-7 (pbk)

15.1.1Unimolecular Reactions

The simplest unimolecular reaction may be expressed in equilibrium form as

k1

A −−−−−− B (15.3)

k−1

where A and B are isomeric, e.g., conformationally or constitutionally. The unimolecular rate constants above and below the equilibrium arrows are associated with the forward and reverse steps in the equilibrium process. Thus, the rate at which A is converted into B is k1[A] while the rate at which B is converted into A is k−1[B]. These rate constants truly are ‘constants’, i.e., they are independent of time.

Note that at equilibrium, the rate at which A is converted into B must be exactly equal to the rate at which B is converted into A, i.e., the system is stationary with respect to reactant and product concentrations. Thus

where Keq is the equilibrium constant for Eq. (15.3). So, it is a straightforward task to measure the ratio of the elementary rate constants, but how is any one measured individually?

If we consider the system perturbed from equilibrium – let us suppose that there is an excess of A – then the rate of return to equilibrium may be expressed either as the rate of disappearance of A, i.e., −d[A]/dt, or as the rate of appearance of B, i.e., d[B]/dt. Using

If the second term on the r.h.s. can be ignored, either because k−1 k1 (a so-called ‘irreversible’ reaction), or because we start with [A] [B] and only observe the system over a time frame where that relationship continues to hold, then we may rearrange Eq. (15.6) to give the first-order rate expression

where ‘first order’ implies that the sum of the exponents for concentration terms on the r.h.s. of the general rate expression written in the form of Eq. (15.2) is one. Integration of both

Thus, experimentally, one plots the logarithm of the concentration ratio against time under conditions where Eq. (15.7) holds in order to determine k1. The reverse rate constant k−1 may either be determined analogously, or from Eq. (15.5) once k1 is known.

Note that for a first-order reaction, the time required for the reactant concentration to drop by some constant factor is a simple function of the rate constant. Thus, for instance, the halflife τ1/2, which is the time required such that [A]τ1/2 = 12 [A]0 for any starting concentration, may be determined from Eq. (15.8) to be k1−1 ln 2.

Fragmentation is another possible unimolecular reaction. A fragmentation reaction may be expressed as

−

(in principle, fragmentations involving more than two products all of which are produced simultaneously are possible, but examples are very rare). The rate for disappearance of A when in excess of its equilibrium value is

The only difference from an experimental viewpoint between Eqs. (15.6) and (15.10) is that Eqs. (15.7) and (15.8) can now be made to apply by ensuring that either one (or both) of B and C have vanishingly small concentrations over the course of the rate measurement.

15.1.2Bimolecular Reactions

The opposite of a fragmentation reaction is a condensation reaction, i.e.,

k1

A + B −−−−−− C (15.11)

k−1

Note that in practice the abstract species A and B may themselves already be molecules or supermolecules formed from prior condensations, but simple probability arguments make condensation reactions simultaneously involving more than two species impossible under most sets of experimental conditions. The rate law associated with eq. 15.11 is

where k1 is a second-order rate constant because it multiplies a set of concentrations whose exponents sum to 2. The simplest evaluation of k1 proceeds by arranging for a vanishingly

small concentration of C over the course of the measurement (or choosing an irreversible reaction) and a vast excess of B. In that case, the rate expression becomes

which is identical to Eq. (15.7) except that k1 , the so-called pseudo-first-order rate constant which may be measured in exactly the fashion already described above for a normal firstorder rate constant, is the product of the second-order rate constant k1 and the effectively constant [B]0 when B is in excess.

Bimolecular reactions having more products than the single species produced from a condensation are also possible, and their rate laws are constructed and measured in a fashion analogous to Eqs. (15.12) and (15.13). Note that the special case of a bimolecular reaction involving two molecules of the same reactant has a rate law that is particularly simple to integrate and work with.

15.2 Reaction Paths and Transition States

Theory may play two particularly important roles in rationalizing and predicting chemical reaction dynamics. As noted in the last section, the first step to understanding the dynamical behavior of a complex chemical system is breaking down the overall system into its constituent elementary processes. From a theoretical standpoint, the likely importance of various processes may be qualitatively assessed from the potential energy surfaces of putative reactions. Reactions with very high barriers will be less likely to play an important role, while low-barrier reactions will be more likely to do so.

Moreover, the PES helps to define the scope of each elementary reaction. Thus, for instance, a bimolecular condensation that involves the formation of two new bonds between the reacting species may either proceed in a concerted fashion, with only a single predicted TS structure, or it may proceed as a stepwise process with two TS structures; the stepwise process is really two elementary reactions – first a condensation and then a unimolecular rearrangement.

To say that an overall process involves two different TS structures presupposes, however, some sort of trajectory that the reacting system maps out on the PES. In general, when chemists think of a system moving on a PES, they tend to think about a particular path called the minimum-energy path (MEP) or sometimes the intrinsic reaction coordinate (IRC). The MEP is the path downwards from a saddle point to a minimum that would be followed by a ball rolling on a surface if its velocity were infinitely damped at every point; an example of such a path is given in Figure 1.4. When the potential energy surface is expressed in mass-weighted coordinates, the MEP is also the path that follows the steepest gradient at every point. The mass-weighted Cartesian coordinates for an atom are simply the Cartesian coordinates scaled by the square root of the atomic mass. Mass-weighted internal coordinates can be generated by diagonalization of the mass-weighted Cartesian coordinate force constant matrix. This coordinate system is a very convenient one in which to work since the gradients for many electronic structure methods are available to facilitate the following of the MEP.

It is worth digressing for a moment to note that following an MEP is often crucial to understanding the nature of a TS structure. Sometimes, when a molecule has a single imaginary frequency, visualization of the corresponding normal mode does not necessarily make it obvious what the reaction coordinate is. It can often happen that the TS structure that has been located corresponds to some process other than the one of interest, e.g., a TS structure for the internal rotation of a methyl group may be found when the desired TS structure was for some bond-making or bond-breaking process. In such a case, following the MEP will lead, in each direction, to the ultimate minimum energy structures connected by the TS structure. On complex potential energy surfaces, such connections can be critical to understanding the overall topology of the PES (see, for instance, Gustafson and Cramer 1995).

Although the MEP and its connection to TS structure(s) is tremendously useful as a conceptual tool, it can also be somewhat misleading to the extent that it focuses analysis on the PES itself. It should always be kept in mind that the equilibria and kinetics of reacting systems are nearly always governed by the free energy of populations of molecules, and not the potential energy of single molecules. To the extent the free energy describes a thermal distribution of particles composing the reacting system, one may think of the system as a cloud hovering over the PES, with the density of the cloud thinning as it rises according to Boltzmann statistics. Within the cloud, individual molecules may be exchanging energy with one another to rise and fall relative to the PES, but the net distribution remains dictated by temperature. A reacting system may be thought of as a cloud over the PES headed towards a mountain pass whose saddle point is the TS structure. However, the passage of the cloud over the pass need by no means take place directly over the TS structure. Depending on how wide the pass is and how tall the cloud is, many cloud particles may be able to pass arbitrarily far to the left and right of the TS structure (when the pass is very narrow one says that the reaction has an entropic bottleneck, meaning that little variation in degrees of freedom other than the reaction coordinate is permitted).

So, while the TS structure, by virtue of being a stationary point on the PES, can be informative about the height of the pass, and local topology (by Taylor expansion of the surface about the stationary point), it is only one representative of the population of molecules passing from reactants to products. As such, one should be rather careful not to confuse the TS structure, which is the stationary point, with the transition state, which may be somewhat more rigorously defined for an N -atom system as a surface having 3N − 7 degrees of freedom (i.e., one less than the reactants) through which the reactive flux is maximized. That is, the ratio of the number of molecules crossing the surface in the direction reactants → products to the number crossing in the opposite direction in a given time interval is maximal. To make the distinction between the TS structure and the transition state more clear, it is helpful to return to a somewhat older term for the latter, namely, the ‘activated complex’. The remainder of this chapter will hew to this distinction as closely as possible.

Returning to kinetics, while theory can be advantageously used to decompose a complex system into its constituent series of elementary reactions, we have not yet described any relationship between a theoretical quantity associated with the individual elementary reactions and their forward and reverse rate constants. It is axiomatic that reactions with high-energy