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

asymptotically correct potentials, and Adamo and Barone (1999) have demonstrated that the hybrid PBE1PBE functional, for reasons that are not entirely clear, seems to be significantly less affected by high-energy errors than other hybrid functionals. Such ongoing developments together with efficient schemes for computing excited-state analytic derivatives (Furche and Ahlrichs 2002) make TDDFT the method of choice for molecules whose size precludes the use of very large multireference ab initio schemes. Within the area of inorganic chemistry, the accuracy of newly developed methods for computing transition-metal circular dichroism (CD) spectra with TDDFT is another very promising development (see, for example, Autschbach, Jorge, and Ziegler 2003). CD spectroscopy, together with optical rotatory dispersion (ORD), which is also open to computation (Beratan, Kondru, and Wipf 2002 and Rinderspacher and Schreiner 2004), is particularly useful in assigning absolute configuration for chiral molecules.

Finally, Shao, Head-Gordon, and Krylov (2003) have described a modification of TDDFT that permits formally multideterminantal target states to be described as spin–flip excitations from a single determinant reference state of different spin (SF-TDDFT). Thus, for example, the difficult singlet state(s) of trimethylenemethane, discussed in Chapter 7, may be generated from the single-determinantal triplet state by single spin–flip excitations (Slipchenko and Krylov 2003). This development substantially expands the range of excited states that may be addressed with TDDFT.

14.4 Sum and Projection Methods

The application of HF and KS-DFT is fundamentally limited to wave functions that can be expressed as single Slater determinants. This restricts their utility in dealing with states like the 1A2 state of phenylnitrene, which is two-determinantal in character (Figure 14.3). One can apply HF or KS-DFT to the single determinant that, restricted to representation of the singly occupied orbitals and with normalization implicit, is written

But, as described in more detail in Appendix C, this wave function, which configurationally corresponds to an α electron in the b1 orbital and a β electron in the b2 orbital, is neither a singlet nor a triplet, but a 50:50 mixture of the two, and this point is emphasized by the left superscript on in Eq. (14.14). While the wave function does not represent a pure spin state, we may take advantage of the prevailing situation by noting that we may write

where 3 and 1 represent the pure spin states (note that matrix elements of the Hamiltonian between different spin states are zero, leading to the simplification on going from the second to the third equality in Eq. (14.15)). The desired value in this process is the expectation

value of the Hamiltonian for 1 , which does not have a single-determinantal description. However, this expectation value is the only unknown in Eq. (14.15), because the expectation value for the triplet may be computed from the alternative triplet wave function

which is the Sz = 1 single-determinantal triplet one usually works with in HF or KS-DFT electronic structure codes. Thus, rearrangement of Eq. (14.15) indicates that the energy of the open-shell singlet may be determined as

1 |H |1 = 2 50:50 |H |50:50 − 3 |H |3 (14.17)

where the expectation values on the right are readily computed from single-determinantal HF or KS-DFT formalisms. This approach is known as the ‘sum method’. Although presented here in the context of singlet and triplet states, it has been generalized to the construction of Heisenberg spin ladders in arbitrarily complex systems, such as the iron–sulfur clusters found in biological electron-transport systems (Noodleman et al. 1995).

To provide a specific example of the method, near UV experiments have led to assignments of the vertical and adiabatic excitation energies for the 11Bg ← 11Ag transition in E-diazene (HN=NH), where the 1Bg state is open-shell. Table 14.4 compares sum-method predictions at the UHF and BLYP levels of theory to these experimental values, and also to published results at the MRCI level of theory. For this system, the HF results are systematically too high, and the DFT too low (cf. the sum method prediction for 1A2 phenylnitrene in Table 14.1), but are competitive with the much more expensive MRCI results. Note that all three levels do quite well at predicting the difference in vertical and adiabatic excitation energies.

The sum method is simple and fast, and it is an SCF approach, which can make it an attractive option compared to other alternatives in some cases. However, it also has some serious possible drawbacks. First, geometry optimization is tedious, since, from Eq. (14.17), the gradients of the open-shell singlet will be the difference between twice the gradients of the 50:50 system and the gradients of the triplet system, all at the same geometry. Secondly, and more importantly, Eq. (14.15) is rigorously correct only when all three wave functions, singlet, triplet, and 50:50, are written using the same MOs, in practice those that are SCF

a Lim et al. 1996.

b Kim, Shavitt, and Del Bene 1992. c Back, Willis, and Ramsay 1978.

If we cancel the equivalent sums on the left and right we are left with

We now multiply on the left by m and integrate, where m indexes the stationary state for which we are interested in measuring the probability of transition. This gives

Note that the expectation value on the l.h.s. of Eq. (14.28) is simply δmk , because of the orthogonality of the stationary-state eigenfunctions. Thus, only the term k = m survives, and we may rearrange the equation to

k

If we assume that our perturbation was small, and applied for only a short time, we may further assume that the expansion coefficients on the r.h.s. of Eq. (14.29) have their initial (ground-state) values. This leads to the further simplification

We now ask the question, for what values of m is |cm|2 large? Given a particular frequency of radiation ω, the magnitude of cm will be large if ωm0 is close to ω, thereby making the denominator in the second term in brackets very small (note that even when ωm0 is equal to ω, the expansion coefficient is well behaved because of the way the numerator approaches zero, cf. Section 10.5.2). This result is consistent with the notion that a photon of energy hν

is absorbed in the transition between the two states, although it takes a more sophisticated theoretical treatment to demonstrate this. However, this term fails to differentiate any one state m from another, all states being predicted to undergo transitions with equal probability at their respective frequencies.

It is the last term that accounts for differences in absorption probabilities. This term is the expectation value of the dipole moment operator (see Section 9.1.1) evaluated over different determinants. Its expectation value is referred to as the transition dipole moment.

The matrix elements rm0 are quite straightforward to evaluate. Before leaving them, however, it is worthwhile to make some qualitative observations about them. First, the Condon–Slater rules dictate that for the one-electron operator r, the only matrix elements that survive are those between determinants differing by at most two electronic orbitals. Thus, only absorptions generating singly or doubly excited states are allowed.

In addition, group theory can be used to assess when transition dipole moments must be zero. The product of the irreducible representations of the two wave functions and the dipole moment operator within the molecular point group symmetry must contain the totally symmetric representation for the matrix element to be non-zero (note that, if the molecule does not contain an inversion center, the operator r does not belong to any single irrep, except for the trivial case of C1 symmetry; see Appendix B for more details). A consequence of this consideration is that, for instance, electronic transitions between states of the same symmetry are forbidden in molecules possessing inversion centers.

The derivation above may be generalized to wave functions other than electronic ones. By evaluation of transition dipole matrix elements for rigid-rotor and harmonic-oscillator rotational and vibrational wave functions, respectively, one arrives at the well-known selection rules in those systems that absorptions and emissions can only occur to adjacent levels, as previously noted in Chapter 9. Of course, simplifications in the derivations lead to many ‘forbidden’ transitions being observable in the laboratory as weakly allowed, both in the electronic case and in the rotational and vibrational cases.

As a final point, let us consider the transition not simply between electronic states, but between wave functions described as products of (decoupled) electronic and vibrational states. That is, we consider wave functions of the form

where is the electronic wave function of Eq. (14.22) and is a vibrational wave function, e.g., as defined by Eq. (9.40). If we carry out the same analysis as above, for radiation of wavelengths that are far from regions associated with vibrational transitions (as UV/Vis is from IR), then we find that Eq. (14.31) generalizes to

where n indexes the vibrational wave functions of electronic state m, and we have assumed that the ground electronic state is also in its ground vibrational state. We now ask the question, when is the overlap between the vibrational wave functions (the so-called Franck–Condon

overlap) large? The rough rule of thumb is that ground-state vibrational wave functions have their maximum amplitude near the equilibrium structure, while excited vibrational wave functions have their maximum amplitudes near their turning points, which is to say where the PES along a vibrational coordinate rises to roughly the vibrational energy (see also Figure 9.7).

This refines the situation depicted in Figure 14.2, which suggests that a vertical transition occurs from the ground PES to the excited PES. More realistically, it occurs from the ground vibrational level of the ground state (which does indeed usually have a wave function well centered about the equilibrium structure) to any number of vibrational levels of the excited state. However, the only vibrational wave function in the single coordinate depicted in Figure 14.2 that is likely to have significant amplitude at the ground-state’s equilibrium geometry is the one with its turning point at that geometry. As the excited electronic state in this excited vibrational state has roughly the energy of the excited-state PES at the ground state’s equilibrium geometry, the picture drawn in Figure 14.2 is quantitatively valid, if somewhat opaque in justification (similar arguments can be made even for dissociative excited states). The major remaining error in the figure is the one-half quantum of ZPVE that is being ignored in the ground state for every coordinate that changes significantly between the ground and excited states. However, two states rarely differ in more than a very small number of coordinates, and this remaining error is typically no worse than that associated with the computation of the state-energy difference. It can, in any case, be corrected for in cases where treatments of increased quantitative accuracy are desired.

14.6 Solvatochromism

A surrounding condensed phase can have enormous impacts on the electronic spectroscopy of a given molecule. Certain dye molecules are sufficiently sensitive to the nature of a surrounding solvent that the color of their solutions can vary across the entire visible spectrum depending on the particular solvent chosen. This solvent effect on spectroscopy is known as solvatochromism.

The influence of solvent on UV/Vis absorption spectra is in some ways analogous to its influence on reaction coordinates. In this case, it is not differential solvation of connected stationary points that is of interest, but rather differential solvation of the PESs for different electronic states (Figure 14.7). Solvatochromism, shown in Figure 14.7 as it affects vertical absorption, derives from the differential solvation of the groundand excited-state potentialenergy surfaces. The blue shift illustrated in this example results from the equilibrium free energy of solvation for the ground state being larger in magnitude than G for the excited state. Note that G is not the equilibrium free energy of solvation of the excited state (which cannot be determined from this diagram, since the PES for the excited state in equilibrium with solvent is not shown), nor even the non-equilibrium solvation free energy of the excited state at the ground-state geometry, since it also includes effects associated with the changing ground-state geometry at which the vertical excitation takes place.

The subtlety of the situation derives from the different timescales involved. If we restrict our discussion to absorption, for the moment, the timescale of the absorption has already been noted to be on the electronic scale – effectively infinitely fast from the point of view

Figure 14.7 The dependence of solvatochromism on differential solvation of the groundand excited-state PESs

of nuclear motion. Thus, when we speak of the solvation of the excited state, at the point of absorption the solvent is equilibrated to the charge distribution of the ground state, but its nuclear motion is frozen on the absorption timescale, so it cannot reorganize until after the fact. Thus, the solvation of the excited state is a non-equilibrium solvation.

Indeed, things are slightly more complicated, because the electrons of the solvent can respond on the timescale of the absorption. Thus, in discussing solvent effects, it is helpful to separate the bulk dielectric response of the solvent, which is a function of ε, into a fast component, depending on n2 where n is the solvent index of refraction, and a slow component, which is the remainder after the fast component is removed from the bulk. The initially formed excited state interacts with the fast component in an equilibrium fashion, but with the slow component frozen in its ground-state-equilibrium polarization. The fast component accounts for almost the entire bulk dielectric response in very non-polar solvents, like alkanes, and about one-half of the response in highly polar solvents.

Because the solvent is fully equilibrated to the ground state but not to the excited state, it is often, but not always, the case that the ground state is better solvated than the excited state, and thus most absorptions are blue-shifted (moved to higher energy) in polar solvents. Applying identical arguments to the emission of a long-lived excited state, where the solvent has equilibrated to the excited state and thus will not be equilibrated to an instantaneously produced ground state, suggests that most spectroscopic emissions in polar solvents will be red-shifted.

While the above discussion has focused primarily on electrostatic interactions between solutes and polar solvents, experiment indicates that many absorptions in non-polar solutions

are red-shifted. This appears not to be a polarization effect, but a manifestation of improved dispersion interactions between the excited state, which because of its more highly excited electron(s) tends to be more polarizable than the ground state, and the solvent. Differential hydrogen bonding interactions can also play an important role in situations where solute –solvent interactions of this type are manifest.

Given the disparate nature of the physical interactions between the different electronic states and the solvent, and the non-equilibrium nature of the solvation of at least one state in the vertical process, theoretical models require a fairly high degree of sophistication in their construction to be applicable to predicting spectroscopic properties in solution. This requirement, coupled with the rather poor utility of available experimental data (most solution spectra show very broad absorption peaks, making it difficult to assign vertical transitions accurately in the absence of a very complex dynamical analysis), has kept most theory in this area at the developers’ level. A full discussion is beyond the scope of an introductory text, but we will briefly touch on a few of the key issues.

Continuum solvation models enjoy their usual advantage of efficiency, but the proper computation of the reaction field for the excited state requires that first the slow component is determined based on the ground-state charge distribution, and then the fast component based on the excited state, the latter process being iterative in the usual SCRF sense (Aguilar, Olivares del Valle, and Tomasi 1993; Mennucci, Cammi, and Tomasi 1998; Cossi and Barone 2000). In the absence of a surrounding solvent shell, however, differential dispersion and hydrogen bonding interactions must be accounted for in an ad hoc fashion after this accounting for polarization (Rauhut, Clark, and Steinke, 1993; Li, Cramer, and Truhlar 2000).

QM/MM approaches where the solute is QM and the solvent MM are in principle useful for computing the effect of the slow reaction field (represented by the solute point charges) but require a polarizable solvent model if electronic equilibration to the excited state is to be included (Gao 1994). With an MM solvent shell, it is no more possible to compute differential dispersion effects directly than for a continuum model. An option is to make the first solvent shell QM too, but computational costs for MC or MD simulations quickly expand with such a model. Large QM simulations with explicit solvent have appeared using the fast semiempirical INDO/S model to evaluate solvatochromic effects, and the results have been promising (Coutinho, Canuto, and Zerner 1997; Coutinho and Canuto 2003). Such simulations offer the potential to model solvent broadening accurately, since they can compute absorptions for an ensemble of solvent configurations.

14.7 Case Study: Organic Light Emitting Diode Alq3

Synopsis of Halls and Schlegel (2001) ‘Molecular Orbital Study of the First Excited State of the OLED Material tris(8-hydroxyquinoline)aluminum(III)’.

Many modern display technologies make use of organic light emitting diodes. These devices typically include two layers, at least one of which is organic, through which electrons and holes propagate. When a hole meets an electron in a single layer or at an interface, the recombination leads to a singlet exciton that fluoresces in the light-generating event. One small organic molecule that has proven to be useful in this regard is tris-(8- hydroxyquinoline)aluminum(III), also called Alq3 (Figure 14.8).