Reactive Intermediate Chemistry

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Further improvements in the flexibility with which the AOs in Eq. 4 are described mathematically can be obtained by adding a third independent basis function to a split valence basis set. In an anion, electrons are likely to be spread over a greater volume than in a neutral molecule, so adding very diffuse basis functions to the basis set for a negatively charged molecule is usually important. A further improvement in the basis set for a molecule would be to use two or three independent basis functions to describe, not only the valence AOs, but also the core AOs. Such basis sets are called, respectively, double-zeta or triple-zeta basis sets.

Another very important way to improve the basis sets used for LCAO–MO calculations is to abandon the tacit assumption that the AOs in a molecule are centered exactly on the nuclei of the atoms. This improvement is most easily accomplished by incorporating polarization functions into the basis sets. Polarization functions have one more angular node than the basis functions that they are polarizing; so mixing in polarization functions desymmetrizes basis functions by turning them into hybrid orbitals. Thus, 2p basis functions serve as polarization functions for the 1s basis functions on hydrogen; and, as shown graphically in Figure 1ðbÞ, 3d basis functions serve as polarization functions for the 2p basis functions on firstrow elements.

A final point about basis functions concerns the way in which their radial parts are represented mathematically. The AOs, obtained from solutions of the Schro¨dinger equation for one-electron atoms, fall-off exponentially with distance. Unfortunately, if exponentials are used as basis functions, computing the integrals that are required for obtaining electron repulsion energies between electrons is mathematically very cumbersome. Perhaps the most important software development in wave function based calculations came from the realization by Frank Boys that it would be much easier and faster to compute electron repulsion integrals if Gaussian-type functions, rather than exponential functions, were used to represent AOs.

However, especially for core AOs, where an exponential function has a very sharp cusp at the nucleus, several Gaussian functions are required even to begin to represent an exponential function accurately. For example, a split-valence basis set, developed by John Pople’s group and widely employed in current calculations, uses six Gaussian functions to represent the 1s core orbitals on first row atoms; whereas, a set of only three Gaussians, plus one more independent Gaussian, is used to represent the valence 2s and 2p AOs. With 3d functions included, to serve as polarization functions for the valence orbitals on the heavy atoms, Pople denotes

this Gaussian basis set as 6-31G(d) or, more commonly, 6-31G*.4a

A better Gaussian basis set, created by Pople and co-workers4b,c for first-row atoms, is 6-311G. It is valence triple zeta, in that it uses three independent basis functions to represent each valence AO. With one 4f and two independent 3d polarization functions on non-hydrogen atoms and one 3d and two independent sets of 2p polarization functions on hydrogen, in Pople’s notation the resulting basis set is called 6-311G(2d1f,2p1d).

Not all the basis sets that are in common use were created by Pople’s group. Also frequently employed and built into many programs for performing electronic

structure calculations are Dunning’s correlation-consistent (cc)-basis sets.5 As their name implies, the cc-basis sets are especially useful for calculations that include electron correlation. These basis sets provide a means for systematically increasing the amount of correlation energy that is recovered by such calculations.

3.2.3. Inclusion of Electron Correlation. HF calculations, performed with basis sets so large that the calculations approach the HF limit for a particular molecule, still calculate total energies rather poorly. The reason is that, as already discussed, HF wave functions include no correlation between electrons of opposite spins. In order to include this type of correlation, multiconfigurational (MC) wave functions, like that in Eq. 5, must be used.

¼ c1jc21c22 . . . c2mc2ni þ c2jc21c22 . . . c2mc2v i þ c3jc21c22 . . . c2v c2ni

þ c4 jc21c22 . . . camcbn c2v i jc21c22 . . . cbmcan c2v i . . . ð5Þ

In an MC wave function, the HF wave function in Eq. 2 is augmented by additional configurations, only four of which are actually given in Eq. 5. As shown pictorially in Figure 22.2, in these four additional configurations different pairs of electrons have been excited from two of the orbitals (cm and cn) that are doubly occupied in the HF wave function. The electrons have been excited into one of the many virtual orbitals (cv), which, if empty, are the HF wave function. It is the

Figure 22.2. Pictorial depiction of the HF and the four excited configurations that are given in Eq. 5. Many more configurations can be generated by excitations of electrons from other orbitals that are filled in the HF configuration into the many virtual orbitals that are unoccupied in this configuration.

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mixing of the excited configurations into the HF configuration that provides correlation between electrons of opposite spin, which is absent from the HF wave function.

For example, mixing of the second configuration in Eq. 5 into the HF configuration with a minus sign results in the mixing of cv into cn with a plus sign for one

of the electrons in c and with a minus sign for the other. Provided that c and c n p p p p n v overlap, the resulting orbitals, c1cn þ c2cv and c1cn c2cv, no longer

span exactly the same region of space. Therefore, the Coulombic repulsion energy in the HF wave function in Eq. 2, between the electrons of opposite spin in cn, is reduced in the MC wave function in Eq. 5.

3.2.3.1.Configuration Interaction Calculations. For the ground state of a molecule, the optimal coefficients for the MC wave function in Eq. 5 are those that causeto satisfy the Schro¨dinger equation with the lowest energy. Other sets of coefficients that cause to satisfy Eq. 1 give the wave functions for excited states of the molecule.

The different sets of coefficients that cause to satisfy Eq. 1, and the energy that corresponds to each wave function can be obtained by computing the energies of the interactions between each pair of configurations in Eq. 5, due to the Hamiltonian operator, H. If these interaction energies are displayed as a matrix, the coefficients that result in the MC wave function in Eq. 5 satisfying Eq. 1 are those that diagonalize this Hamiltonian matrix.

If good basis sets are used, the Hamiltonian matrices that are generated in configuration interaction (CI) calculations on even comparatively small molecules are very large. Fortunately, efficient computer algorithms have been developed for

computing the elements of these matrices and finding the wave functions for the lowest energy states, even for MR wave functions that consist of 106 configurations. For such a wave function, the effect of the Hamiltonian operator on is represented by a matrix that contains 106 106 ¼ 1012 elements.

If all possible excitations are included in a CI wave function, except for very small molecules, many more than 106 configurations are generated. Thus, full CI

calculations are almost always impractical or impossible.

One of the ways to reduce the size of CI calculations is by limiting the number of electrons that are simultaneously excited. For example, including only single and double excitations of electrons from the HF configuration gives what is called an SD–CI (or CISD) calculation. Sometimes the effects of including higher levels of excitation (e.g., triples and/or quadruples) are estimated, rather than obtained by actually performing an SDTQ–CI calculation. An SD–CI calculation in which the effect of including quadruple excitations is estimated would be denoted SD(Q)–CI.

3.2.3.2.Calculations Based on Perturbation Theory. An alternative to performing a CI calculation is to use perturbation theory. As in the expansion of a mathematical function in a power series, in perturbation theory the coefficients for and the energy of an MC wave function are expanded in a series. The more

terms in the series that are actually included, the more closely the coefficients and energies that are obtained by perturbation theory approximate the exact values that would be computed by a CI calculation with the same configurations.

If only the leading terms in the perturbation theory expansions are included, the computer resources required for such calculations are much more modest than those that would be needed for a CI calculation with the same set of configurations. This is the biggest advantage of calculations that are based on perturbation theory over CI calculations.

The lowest level of perturbation theory that can be used to calculate energies which include the effects of electron correlation is called second-order Mo¨ller– Plesset (MP2) perturbation theory.6 MP2 calculations include the interactions of the excited configurations with just the HF configuration and ignore the effects of the interactions of the excited configurations with each other. For closed-shell molecules, Brillouin’s theorem shows that singly excited configurations do not interact directly with the HF configuration. Therefore, only doubly excited configurations are included in an MP2 calculation.

Higher level calculations, based on higher orders of Mo¨ller–Plesset perturbation theory, can also be performed, albeit with the consumption of much more computer time. For example, an MP4SDTQ calculation uses fourth-order Mo¨ller–Plesset perturbation theory, includes excitations through quadruples, and gives better energies than MP2 does.

3.2.3.3.Size Consistency in CI Calculations. Not only are MPn calculations less demanding of computer resources than CI calculations that include the same levels of excitations, but MPn calculations are size-consistent; whereas, CISD calculations are not. A computational method is size consistent if the energy, obtained in a calculation on two identical molecules at infinity, is exactly twice the energy that is obtained in a calculation on just one of these molecules. The reason why CISD calculations are not size consistent is easy to understand.

If the CISD wave functions for two identical molecules are multiplied, to give the wave function for the pair of molecules, there are terms in the resulting wave function in which both molecules are doubly excited. Since these terms represent quadruple excitations from the HF configuration, they are not included in the CISD wave function for two identical molecules at infinity. Consequently, the CISD energy for a pair of identical molecules is higher than twice the CISD energy for an individual molecule.

Clearly, the way to remedy the lack of size consistency in a CISD calculation is to include the missing terms that contain quadruple excitations. Because these terms are products of simultaneous double excitations, their sizes can be determined from information that is available from CISD. Two methods that include quadruple excitations in this manner and, hence, are size consistent are discussed in Section 3.2.3.4.

3.2.3.4.Coupled-Cluster and Quadratic CI Calculations. The most powerful method for including electron correlation that can be used in calculations on

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medium-sized molecules is based on coupled-cluster (CC) theory. This method, which was developed by Cizek and successsfully implemented by Bartlett et al.,7 expresses the effects of quadruple and higher excitations as products of the effects of double excitations.

The CCSD calculations are similar, both in methodology and in the accuracy of the results obtained, to calculations performed with Pople’s quadratic CI method.8 Like CCSD calculations, QCISD calculations also explicitly include single and double excitations; and the effects of quadruple excitation in QCISD are obtained from quadrature of the effects of double excitations. However, CCSD does contain terms for the effects of excitations beyond quadruples, which are absent from QCISD.

In both methods the effects of triple excitations can be estimated. These effects can be significant, if quantitative accuracy is the goal of the calculations. However, performing a calculation at either the CCSD(T) or QCISD(T) level of theory comes at the cost of substantially increasing the computer time required, beyond that consumed by a CCSD or QCISD calculation.

3.2.3.5.G2, G3, and CBS Calculations. The Gaussian-2 (G2)9a and Gaussian-3 (G3)9b methods, developed by Pople and co-workers, and the complete basis set (CBS)10 method, developed by Petersson and co-worker, all attempt to provide

very accurate thermochemical predictions. The energies computed by very large calculations with very large basis sets are extrapolated from the results of high-level calculations with moderate-sized basis sets and lower level calculations with very large basis sets. Like the MPn, CCSD(T), and QCISD(T) methods, the G2, G3, and

CBS methods are available in the current Gaussian package of programs for performing electronic structure calculations.11

3.2.3.6.CASSCF and CASPT2 Calculations. For many molecules and reactions, a relatively small number of electrons (e.g., those in p orbitals) can be identified as being important. For such molecules and such reactions, variational calculations can be performed with configurations that provide correlation between just those active electrons. For each bonding orbital that is occupied by active electrons, a virtual orbital is also usually included in the orbital active space; and all possible configurations that arise from distributing the active electrons among the active orbitals are included in the resulting MC wave function.

In an MCSCF calculation, not only the coefficients of the multiple configurations in the MC wave function, but also the orbitals in them, are simultaneously optimized. An (n/m)MCSCF calculation, in which the n active electrons and m

active orbitals are chosen in the manner described in the preceding paragraph, is called a complete active space (CAS)SCF calculation.12

It is imperative to use CASSCF wave functions for singlet diradicals and other

open-shell molecules for which a single configuration provides an inadequate description of the wave function.13 However, perhaps surprisingly, CASSCF calcu-

lations often perform rather poorly in calculations on molecules and TSs with closed shells of electrons, if the active electrons are delocalized. An example is

provided by (6/6)CASSCF calculations with the 6-31G* basis set on the chair Cope rearrangement of 1,5-hexadiene, which gave a calculated activation enthalpy that was >12 kcal/mol above the experimental value.14

In order to get good agreement between CASSCF and experimental results, it is often necessary to include correlation between the active electrons and the other valence electrons.15 This type of correlation, which is called dynamic correlation, can be added to a CASSCF wave function by using the equivalent of MP2 perturbation theory. However, in an MP2 calculation on an MC wave function, rather than including double excitations from a single HF reference configuration, excitations from all of the configurations must be included.

Several different versions of second-order perturbation theory for multireference wave functions have been implemented; but the one currently in widest use is probably the CASPT2 method.16 This method was developed by Roos and co-workers17 in Lund, Sweden, and it is available in their MOLCAS package of computer programs.

A ‘‘CASMP2’’ module is also available in the Gaussian package of programs. However, at least as implemented in Gaussian 98, CASMP2 gives much poorer results than CASPT2. For example, Dr. David Hrovat in my group found (6/6) CASMP2/6-31G* calculations gave an enthalpy of activation for the Cope rearrangement that is in even worse agreement with experiment than the (6/6)CASSCF/ 6-31G* value.18

On the other hand, when the (6/6)CASSCF geometry for the Cope TS was partially reoptimized at the (6/6)CASPT2 level, (6/6)CASPT2/6-31G* calculations gave an enthalpy of activation for the Cope rearrangement that was 3 kcal/mol lower than the experimental value.19 Moreover, (6/6)CASPT2 calculations with the 6-311G* basis set gave an enthalpy of activation that was only 1 kcal/mol lower than the measured value.

CASPT2 calculations give excellent results for not only the enthalpic barriers to pericyclic reactions, such as the Cope rearrangement, but also for the excitation energies in the UV–vis spectra of conjugated molecules. Examples of the use of CASPT2 calculations of UV–vis spectra for the identification of RIs are provided by singlet 1,2,4,5-tetramethylbenzeneene diradical20 and each of the rearrangement products, formed sequentially from the radical cation of the syn dimer of cyclobutadiene.21

3.3. Calculations Based on Density Functional Theory

As discussed in Section 3.2.3.6, the major difficulty in wave function based calculations is that correlation between electrons of opposite spin must somehow be introduced into a theory that starts with the physically unrealistic premise that electrons of opposite spin move independently of each other. However, this tacit assumption not only provides a mathematically tractable starting point (i.e., HF theory) for wave function based calculations, but this assumption also underpins the entire concept of orbitals (i.e., wave functions for single electrons). The existence of MOs may be a construct, but it is a construct that has proven to be very useful in interpreting the results of both calculations and experiments.

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An attractive way to maintain the conceptual simplicity that arises from positing the existence of MOs, while improving upon the accuracy of HF calculations, would be to augment the Coulomb and exchange operators in the Fock operator with an additional operator that somehow accounts for correlation between electrons of opposite spin. This is essentially what is done in calculations based on DFT. Moreover, the general strategy behind modifying the Fock operator, so that it includes the effects of both exchange between electrons of the same spin and correlation between electrons of opposite spin, can be rigorously justified by two theorems. These theorems were proved by Walter Kohn, who shared the 1998 Nobel Prize in Chemistry with John Pople.

3.3.1.The Hohenberg–Kohn and Kohn–Sham Theorems. The first of these two theorems is called the Hohenberg–Kohn theorem. It proves that the energy of a molecule can, in principle, be computed exactly from the exact electron density. Since the electron density in a molecule is a mathematical function of the three spatial coordinates, and since functions of functions are called functionals, the Hohenberg–Kohn theorem can be formulated as stating that a functional exists from which the energy of a molecule can be computed exactly from the exact electron density. Unfortunately, the Hohenberg–Kohn theorem does not provide any information on what that functional is or how to go about finding it.

The second theorem is called the Kohn–Sham theorem. It proves that, if the exact electron density in a molecule can be represented by a single configuration, the energy, computed from the density, can be minimized by variationally optimizing the orbitals in that configuration. Therefore, a one-electron, Fock-like, operator can be used in the DFT analogue of Eq. 3, to find the optimal Kohn–Sham (KS) orbitals with which to compute the density. The same type of iterative process, employed in HF theory, can be used to find the KS orbitals that give the electron density of lowest energy for a given functional.

3.3.2.Functionals. The difference between the Fock operator, F, in wave function based calculations and the analogous operator in DFT calculations is that the Coulomb and exchange operators in F are replaced in DFT by a functional of the electron density. In principle, this functional should provide an exact formula for computing the Coulombic interactions between an electron in a KS orbital and all the other electrons in a molecule. To be exact, this functional must include corrections to the Coulombic repulsion energy, computed directly from the electron density, for exchange between electrons of the same spin and correlation between electrons of opposite spin.

In addition, the functional must somehow cancel the fictitious repulsion energy between an electron and itself, which arises if the electron density, due to all the electrons, is used to compute the Coulombic energy of a single electron. As discussed in Section 3.2.1, in HF theory cancellation of the self-repulsion energy results from the presence of the exchange operator in F. If this effect of Kj, in the Fock operator is not mirrored exactly by the functional chosen, the cancellation of the self-repulsion energy will not occur.

During the last decade of the twentieth century, creation and testing of functionals provided gainful employment for the many theoretical chemists who were interested in the development of DFT as an accurate method for use in electronic structure calculations. The chapter on DFT in Cramer’s book3 provides not only descriptions of different types of functionals but also information on how well each performs in calculating energies, bond lengths, and dipole moments for several different test sets of molecules.

Here it is sufficient to note that the functional, currently in use, which usually provides the most accurate results for organic molecules, is B3LYP. It consists of a three-parameter functional that was developed by Becke,22 combined with a correlation functional due to Lee, Yang, and Parr23 ‘‘Be three lip’’ is not an injunction in pidgin English to effect an impossible change in human physiology, but the acronym for the functional that is currently the one most widely used by chemists doing DFT calculations on organic and organometallic compounds.

3.4. Comparisons of Calculations Based on Wave Functions and on DFT

Wave function based calculations are often called ‘‘ab initio’’ (literally, ‘‘from the beginning’’) calculations, because energies, geometries, and properties are all computed in terms of fundamental quantities (i.e., Planck’s constant and the charge and mass of an electron). In contrast, no one has yet figured out how to find the holy grail of DFT—the functional that computes energies exactly from exact densities— ‘‘from the beginning’’. The successes and failures of each functional have to be evaluated by comparisons of the energies, bond lengths, etc. computed using the functional, with experimental values.

Some functionals, in particular B3LYP, incorporate parameters that are optimized by comparisons between DFT and experimental results. Thus, B3LYP calculations are, like the MINDO, MNDO, and AM1 methods that were developed by the late Michael Dewar in the 1970s and 1980s, semiempirical in nature. In this sense, B3LYP could be called ‘‘the AM1 of the twenty-first century’’.

However, most wave function based calculations also contain a semiempirical component. For example, the primitive Gaussian functions in all commonly used basis sets (e.g., the six Gaussian functions used to represent a 1s orbital on each first row atom in the 6-31G* basis set) are contracted into sums of Gaussians with fixed coefficients; and each of these linear combinations of Gaussians is used to represent one of the independent basis functions that contribute to each AO. The sizes of the primitive Gaussians (compact versus diffuse) and the coefficient of each Gaussian in the contracted basis functions, are obtained by optimizing the basis set in calculations on free atoms or on small molecules.4d

If the optimizations are performed in calculations on atoms, the basis functions for at least some of the atoms are usually scaled, because AOs in molecules are generally more compact than AOs on atoms. Thus, a set of scaling factors must be chosen. If the basis set optimizations are performed in calculations on small molecules, the molecules to use in the optimizations must be chosen. These types

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of choices in basis set construction are usually made by assaying how well different versions of a basis set perform in test calculations.4d Since basis set performance is usually judged by comparing the results of these test calculations to the results of experiments, this comparison introduces a semiempirical component into most of the contracted basis sets that are commonly used for electronic structure calculations.

The incorporation of experimental results into wave function based methods can be much less subtle. For example, G2 and G3 are both very high-level ‘‘ab initio’’ methods that have been designed to provide results that accurately mirror the results of experiments.9 However, part of the way close agreement is achieved between the enthalpies computed by the G2 or G3 methods and those found by experiments, is by carrying out large electronic structure calculations, extrapolating the energies obtained from them, and then applying semiempirical corrections to these energies.

Even if most wave function based calculations cannot claim superiority to DFT by virtue of being purely ab initio, wave function based calculations do have an advantage over those based on DFT. Wave function based calculations can be systematically improved; whereas DFT calculations cannot. For example, increasing the size of the basis set used and the amount of electron correlation energy recovered is not only guaranteed to provide a lower total energy for a molecule; but also, performing a better calculation usually (though not always) results in energy differences between molecules being computed more accurately.

In contrast, increasing the size of the basis set in B3LYP calculations or using a different functional will certainly affect the energy difference computed between two stationary points on a PES, but such changes may or may not increase the accuracy with which the energy difference is computed. Since, B3LYP is known to give poor results for energy differences between certain types of isomers,24 there is no assurance that one can always depend on B3LYP to be the best functional to use; but there is usually no way of knowing a priori whether a different functional will give better results.

On the other hand, DFT calculations, especially with the B3LYP functional, frequently do give results that are as good as those obtained from CCSD(T). The latter is currently the ‘‘gold standard level of theory’’3b for wave function based calculations that employ a single reference configuration.

However, a DFT calculation consumes much less computer time than a CCSD(T) calculation on the same molecule. The time it takes to perform DFT calculations scales no worse than the cube of the number of basis functions used; and efforts are being made by computational chemists to reduce this scaling toward being more nearly linear. In contrast, the time it takes to perform CCSD(T) calculations scales as the number of basis functions to the seventh power.

Since DFT calculations take much less time to perform than CCSD(T) calculations, DFT calculations can be carried out on much larger molecules than CCSD(T) calculations. In addition, with the same amount of computer resources, many more problems in chemistry can be addressed by DFT calculations than by CCSD(T) calculations.

A particular advantage of B3LYP calculations is that the IR frequencies, computed with this functional, are usually more accurate than those obtained from wave

function based calculations. The frequencies obtained from wave function based calculations have to be adjusted, using semiempirical scaling factors; whereas B3LYP frequencies usually require little or no scaling. Therefore, B3LYP calculations, rather than wave function based methods, are usually used in identifying RIs in matrix isolation and in time-resolved IR studies, by comparing calculated and observed IR frequencies.

UV–vis spectra can also be computed, using time-dependent (TD)DFT;25 and it is now possible to perform such calculations with relative ease, using the Gaussian suite of electronic structure codes.11,26 Using CASPT2 to compute UV– vis spectra, usually requires more computational expertise than using TDDFT.

On the other hand, since DFT calculations are usually based on a single electronic configuration, DFT has problems in dealing properly with some open-shell RIs. For example, B3LYP works fairly well for small radicals but gives poor spin distributions for radicals with extended conjugation.27 Inadequate solution of the self-repulsion problem causes B3LYP to fail very badly for radical cations.28 Finally, since DFT is based on a single electronic configuration, it does not properly describe singlet diradicals, for which an adequate wave function must contain at least two configurations.13

For calculations on many, if not most RIs, there is no reason to choose between DFT (e.g., B3LYP) and wave function based methods, if both types of calculations can be carried out. For example, since DFT calculations can be used much more economically than CCSD(T) calculations to optimize geometries, locate TSs, and perform vibrational analyses at these stationary points, performing single-point CCSD(T) calculations at B3LYP optimized geometries (denoted CCSD(T)// B3LYP) is an attractive option and a way of checking the accuracy of energy differences computed by B3LYP. To the extent that the results from B3LYP and CCSD(T) calculations agree, one can be confident that the answers provided by these two types of calculations are correct.

If the results of DFT and wave function based calculations disagree, then one can try to carry out wave function based calculations with larger basis sets and/or with inclusion of more electron correlation. If wave function based calculations at several different levels of electronic structure theory give similar results, it is likely that these calculations are correct and that the DFT calculations are in error.

4. APPLICATIONS OF ELECTRONIC STRUCTURE CALCULATIONS TO EXPLAINING AND PREDICTING THE CHEMISTRY OF THREE REACTIVE INTERMEDIATES—PHENYLNITRENE, CUBYL CATION, AND PROPANE-1,3-DIYL

The three examples in this section are taken largely from the work of our research group. Other, perhaps better, examples of the application of electronic structure calculations to the explanation and prediction of the behavior of RIs could have been selected; and some of them are provided in the Suggested Reading Section.