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

β electrons, then the final term on the r.h.s. will be equal to the total number of doubly occupied orbitals, since the overlap integrals can then be computed as the Kronecker¨ δ. If all of the electrons having the minority spin occupy such orbitals, then the third term on the r.h.s. will exactly cancel the second, and the wave function will be a high-spin eigenfunction of S2 – indeed, this describes exactly the nature of an ROHF high-spin wave function. If, on the other hand, the orbitals of the electrons of minority spin have high amplitude in regions of space occupied to a lesser extent by electrons of the opposite spin, then the sum of the overlap integrals will be less than the second term on the r.h.s., and S2 will be greater than the presumably desired eigenvalue corresponding to the first term on the r.h.s. The degree to which the expectation value exceeds this eigenvalue reflects the spin contamination. Since the expectation value is larger than the expected value, the deviation derives from higher spin states contaminating the UHF wave function. So-called ‘spin-projection’ techniques can be used to remove these contaminating states, as discussed in more detail in Appendix C.

In DFT, there is no formal way to evaluate spin contamination for the (unknown) interacting wave function. As has already been discussed in Sections 8.5.1 and 8.5.3, however, the expectation value of S2 computed from Eq. (9.30) over the KS determinant can nevertheless sometimes provide qualitative information about the likely utility of the DFT results with respect to their interpretation as corresponding to a pure spin state compared to a mixture of different spin states.

9.1.5Polarizability and Hyperpolarizability

In Section 9.1.1, we discussed the molecular dipole moment as a measure of the inhomogeneity of the charge distribution. The dipole moment for an isolated molecule in a vacuum, which corresponds to that which would be computed in a typical electronic structure calculation, is often referred to as the ‘permanent’ electric dipole, µ0. However, if an electric field E is applied to the molecule, since the charge distribution interacts with the electric field through a new term in the Hamiltonian, the dipole moment will change. The magnitude of that change per unit of electric field strength defines the electric polarizability α, i.e.,

Note that since both µ and E are vector quantities, α is a second-rank tensor. The elements of α can be computed through differentiation of Eqs. (9.1) and (9.2). The difference between the permanent electric dipole moment and that measured in the presence of an electric field is referred to as the ‘induced’ dipole moment.

Experimentally, the dipole moment is usually determined by measuring the change in energy for a molecule when an electric field is applied – the so-called Stark effect. At low electric-field strength, the energy change is linear in field strength, and the slope of the line is the permanent electric dipole moment. At larger field strengths, the energy change becomes quadratic because the dipole moment begins to increase proportional to the polarizability, and this permits measurement of that quantity. For still larger field strengths, a

cubic contribution to the energy change can be measured (although technically it becomes increasingly challenging to fit the data reliably) and this change can be used to define the first hyperpolarizability, β (now a third-rank tensor).

It is possible to generalize this discussion in a useful way. Spectral measurements invariably assess how a molecular system changes in energy in response to some sort of external perturbation. The example presently under discussion involves application of an external electric field. If we write the energy as a Taylor expansion in some generalized vector perturbation X, we have

Thus, Eq. (9.32) makes more clear the measurement of the Stark effect, for instance. At low electric field strengths, the only expansion term having significant magnitude involves the first derivative, and it defines the permanent dipole moment. At higher field strengths, the second derivative term begins to be noticeable, and it contributes to the energy quadratically and defines the polarizability. Finally, we see naturally how additional terms in the Taylor expansion can be used to define the first hyperpolarizability, the second hyperpolarizability γ , etc. (Note that conventions differ somewhat on whether the 1/n! term preceding the corresponding nth derivative term is included in the value of the physical constant or not, so that care should be exercised in comparing values reported from different sources to ensure consistency in this regard.)

Analogous quantities to the electric moments can be defined when the external perturbation takes the form of a magnetic field. In this instance the first derivative defines the permanent magnetic moment (always zero for non-degenerate electronic states), the second derivative the magnetizability or magnetic susceptibility, etc.

Equation (9.32) is also useful to the extent it suggests the general way in which various spectral properties may be computed. The energy of a system represented by a wave function is computed as the expectation value of the Hamiltonian operator. So, differentiation of the energy with respect to a perturbation is equivalent to differentiation of the expectation value of the Hamiltonian. In the case of first derivatives, if the energy of the system is minimized with respect to the coefficients defining the wave function, the Hellmann–Feynman theorem of quantum mechanics allows us to write

Note that H here is the complete Hamiltonian, that is, it presumably includes new terms dependent on the nature of X. It is occasionally the case that the integral on the r.h.s. of Eq. (9.33) can be readily evaluated. Indeed, it is choice of X = E that leads to the definition of the dipole moment operator presented in Eq. (9.1).

However, even when it is not convenient to solve the integral on the r.h.s. of Eq. (9.33) analytically, or when Eq. (9.33) does not hold because the wave function is not variationally optimized, it is certainly always possible to carry out the differentiation numerically. That

is, one can compute the energy in the absence of the perturbation, then modify the Hamiltonian to include the perturbation (e.g., introduce an electric-field term), then compute the property as

where H is again the complete Hamiltonian and H(0) is the perturbation-free Hamiltonian. This procedure is called the ‘finite-field’ approach. In practice, one must take some care to ensure that computed values are numerically converged (a balance must be struck between using a small enough value of the perturbation that the limit holds but a large enough value that the numerator does not suffer from numerical noise).

Note that Eq. (9.34) can be generalized for higher derivatives, but numerical stability now becomes harder to achieve. Moreover, the procedure can be rather tedious, since in practice one must carry out a separate computation for each component associated with properties that are typically tensors. It is computationally much more convenient when analytic expressions can be found that permit direct calculation of these higher-order derivatives in a fashion that generalizes the procedure by which Eq. (9.33) is derived (not shown here).

As for the utility of different levels of theory for computing the polarizability and hyperpolarizability, the lack of high-quality gas-phase experimental data available for all but the smallest of molecules makes comparison between theory and experiment rather limited. As a rough rule of thumb, ab initio HF theory seems to do better for these properties than for dipole moments – at least there does not appear to be any particular systematic error. Semiempirical levels of theory are less reliable. DFT and correlated levels of MO theory do well, but it is not obvious for the latter that the improvement over HF necessarily justifies the cost, at least for routine purposes.

9.1.6ESR Hyperfine Coupling Constants

When a molecule carries a net electronic spin, that spin interacts with the (non-zero) spins of the individual nuclei. The energy difference between the two possibilities of the electronic and nuclear spins being either aligned or opposed in the z direction can be measured by electron spin resonance (ESR) spectroscopy and defines the isotropic hyperfine splitting (h.f.s.) or hyperfine coupling constant. If we were to pursue computation of this quantity using the approach outlined in the last section, we would modify the Hamiltonian to introduce a spin magnetic dipole at a particular nuclear position. The integral that results when Eq. (9.33) is used to evaluate the necessary perturbation is known as a Fermi contact integral. Isotropic h.f.s. values are determined as

where Sz is the expectation value of the operator Sz (1/2 for a doublet, 1 for a triplet, etc.), g is the electronic g factor (typically taken to be 2.0, the approximate value for a free electron), β is the Bohr magneton, gX and βX are the corresponding values for nucleus X,

and ρ(X) is the Fermi contact integral which, when the wave function can be expressed as a Slater determinant, can be computed as

µν

where Pα−β is the one-electron spin-density-difference matrix (computed as the difference between the two separate density matrices for the α and β electrons), and evaluation of the overlap between basis functions ϕµ and ϕν is only at the nuclear position, rX .

We have previously defined the one-electron spin-density matrix in the context of standard HF methodology (Eq. (6.9)), which includes semiempirical methods and both the UHF and ROHF implementations of Hartree –Fock for open-shell systems. In addition, it is well defined at the MP2, CISD, and DFT levels of theory, which permits straightforward computation of h.f.s. values at many levels of theory. Note that if the one-electron density matrix is not readily calculable, the finite-field methodology outlined in the last section allows evaluation of the Fermi contact integral by an appropriate perturbation of the quantum mechanical Hamiltonian.

For Eq. (9.35) to be useful the density matrix employed must be accurate. In particular, localization of excess spin must be well predicted. ROHF methods leave something to be desired in this regard. Since all doubly occupied orbitals at the ROHF level are spatially identical, they make no contribution to Pα−β ; only singly occupied orbitals contribute. As discussed in Section 6.3.3, this can lead to the incorrect prediction of a zero h.f.s. for all atoms in the nodal plane(s) of the singly occupied orbital(s), since their interaction with the unpaired spin(s) arises from spin polarization. In metal complexes as well, the importance of spin polarization compared to the simple analysis of orbital amplitude for singly occupied molecular orbitals (SOMOs) has been emphasized (Braden and Tyler 1998).

UHF, on the other hand, does optimize the α and β orbitals so that they need not be spatially identical, and thus is able to account for both spin polarization and some small amount of configurational mixing. As a result, however, UHF wave functions are generally not eigenfunctions of the operator S2, but are contaminated by higher spin states.

The challenge with unrestricted methods is the simultaneous minimization of spin ‘contamination’ and accurate prediction of spin ‘polarization’. The projected UHF (PUHF, see Appendix C) spin density matrix can be employed in Eq. (9.36), usually with somewhat improved results.

A complicating factor is that each spin density matrix element is multiplied by the corresponding basis function overlap at the nuclear positions. The orbitals having maximal amplitude at the nuclear positions are the core s orbitals, which are usually described with less flexibility than valence orbitals in typical electronic structure calculations. Moreover, actual atomic s orbitals are characterized by a cusp at the nucleus, a feature accurately modeled by STOs, but only approximated by the more commonly used GTOs. As a result, there are basis sets in the literature that systematically improve the description of the core orbitals in order to improve prediction of h.f.s., e.g. IGLO-III (Eriksson et al. 1994) and EPR-III (Barone 1995).

Of these four levels, the computation of the MP2 spin-density matrix is considerably more time-consuming than the other three. It is thus of interest to examine the accuracy of DFT methods, which by construction include electron correlation directly into their easily computed spin-density matrices. For the same geometries, the mean unsigned errors for the BVWN, BLYP, B3P86, and B3LYP levels of theory were 32.6, 32.6, 29.7, and 28.9 G. Somewhat surprisingly, these errors increased in every case when geometries were optimized at the corresponding DFT level, to 60.5, 54.4, 30.9, and 34.3 G. For this particular data set, several of the radicals seem prone to the DFT overdelocalization problem noted in Section 8.5.6. Guerra (2000) has shown similarly poor performance of the B3LYP functional in the context of vinylacyl radicals, where the functional strongly overestimates the stability of π delocalized radicals relative to σ alternatives, in contravention of experimental data.

In cases where overdelocalization is not a problem, however, DFT methods have proven to be quite robust for computing h.f.s. constants. For instance, Adamo, Cossi, and Barone (1999) have reported results for h.f.s. constants in the methyl radical using PW, B3LYP, and PBE1PBE that are competitive with correlated MO methods (Chipman 1983; Cramer 1991; Barone et al. 1993). Moreover, if a given system suffers from heavy spin contamination at the UHF level of theory, DFT may be the only reasonable recourse.

In general, then, DFT methods provide the best combination of accuracy and efficiency so long as overdelocalization effects do not poison their performance. The MP2 level of theory also provides a reasonably efficient way of carrying out h.f.s. calculations at a correlated level of theory. More highly correlated levels of MO theory are generally more accurate, but can be prohibitively expensive in large systems.

As a final note, although we have focused here on the computation of isotropic h.f.s. values, it is also straightforward to compute anisotropic hyperfine couplings, although these cannot be observed experimentally unless the system can be prevented from random tumbling (e.g., by freezing in a matrix or single crystal). Similarly, it is possible to calculate the electronic g value. These subjects are beyond the scope of the text, however, and interested readers are referred to relevant titles in the bibliography.

9.2 Ionization Potentials and Electron Affinities

As the general utility of semiempirical, HF, and DFT methods for the computation of IPs and EAs has already been discussed in some detail in Sections 5.6.1, 6.4.1, and 8.6.1, this section is restricted to a very brief recapitulation of the most important points relative to these properties.

Koopmans’ theorem suggests that the ionization energies for any orbital (usually ‘IP’ refers specifically to the ionization potential associated with the HOMO) will be equal to the negative of the eigenvalue of that orbital in HF theory. This provides a particularly simple method for estimating IPs, and because of canceling errors in basis-set incompleteness and failure to adequately account for electron correlation, the approach works reasonably well for the occupied orbitals in the highest energy range in ab initio HF wave functions (with semiempirical methods, performance is spottier). However, as one ionizes from orbitals