Chemical shifts of
Applications of density functional theory (DFT) calculations to organic chemistry are shown, beginning with geometry optimization and the calculation of vibrational frequencies, infrared (IR) intensities, and thermodynamic properties. The isotropic chemical shielding values and anisotropies relevant to nuclear magnetic resonance (NMR) can be calculated using gauge-invariant atomic orbitals (GIAOs); the calculation of spin-spin couplings is possible but time-consuming. For free radicals, hyperfine couplings and g tensors pertaining to EPR can be obtained. Regarding UV/vis spectra, wavelengths and oscillator strengths can be calculated by using a time-dependent Hamiltonian. In addition to gas-phase acidities, approximate pKa values can be obtained, provided that solvation is taken into account. Several sets of substituent parameters have been calculated: Hammett σ and σ+ parameters and inductive and mesomeric effects. Regarding reaction mechanisms, geometries and energies of intermediates and transition structures have been calculated for pericyclic reactions, nucleophilic aliphatic substitutions, electrophilic aromatic substitutions, additions, and eliminations.
- density functional theory
- magnetic resonance
- Hammett parameters
- reaction mechanisms
- pericyclic reactions
Focusing on density functional theory (DFT) calculations with Gaussian 09  and the B3LYP/6-311G(d,p) method, several applications to organic chemistry will be shown. After geometry optimization, which yields the total energy, a frequency calculation can be done, yielding the infrared spectrum (wave numbers and intensities) and, if requested, the Raman intensities and the thermodynamic properties (enthalpy, entropy, and Gibbs free energy).
Using a time-dependent Hamiltonian, UV/vis spectra can be calculated (wave lengths and oscillator strengths). Nuclear magnetic resonance (NMR) spectra can be calculated, providing isotropic shielding values as well as tensor data (anisotropies) of all magnetic nuclei, using gauge-invariant atomic orbitals (GIAOs). The calculation of spin-spin coupling constants is also possible but requires much more computational time. For free radicals, EPR data can be calculated: isotropic hyperfine coupling constants, hyperfine tensors, and
Substituent effects such as the
Regarding organic reaction mechanisms, pericyclic reactions are particularly well amenable to DFT calculations. Usually, the transition structure can be obtained which is characterized by a single imaginary frequency, which belongs to the reaction coordinate. For many other reaction types (substitutions, additions, eliminations, and rearrangements), at least an approximation to the transition structure can be calculated. Moreover, starting with such a structure and performing an optimization, the approximate dynamics of the reaction can be followed.
2. Geometries, energies, and thermodynamic data
2.1. Geometry optimization
As a starting point, a reasonable approximation to the geometry of the target molecule is required. Preferably, the coordinate file should be given as Z matrix, and standard bond lengths and angles may be used. A convenient tool for the generation of Z matrices is molden : in the Z-mat editor, start with methane, substitute by phenyl and finally by vinyl, and save as Z matrix (GAMESS). Next, the input file for the quantum-chemical calculation has to be created by supplementing the Z matrix file with the necessary parameters (see Appendix A).
After a successful calculation, the log file contains the energy (in Hartree) and the coordinates of the optimized structure. Again, it is advantageous to use a tool such as molden for analyzing the log file.
2.2. Calculation of thermodynamic properties
For a determination of the thermodynamic properties, it is necessary to calculate the (vibrational) frequencies. In the Gaussian input file, the preliminary coordinates have to be replaced by the optimized ones and the task “Opt” by “Freq”. (Actually, the request for “Freq Prop Pop = Full” additionally provides useful information such as charges and dipole moment. By default, the calculation is done for 298 K and 1.000 atm, but a different temperature or pressure may be specified.)
For the example molecule (
|SCF Done: E(RB3LYP)||−349.054882687 AU|
|Zero-point correction||0.159885 (Hartree/particle)|
|Thermal correction to energy||0.167687|
|Thermal correction to enthalpy||0.168631|
|Thermal correction to Gibbs free energy||0.127033|
|Sum of electronic and zero-point energies||−348.894997|
|Sum of electronic and thermal energies||−348.887196|
|Sum of electronic and thermal enthalpies||−348.886252|
|Sum of electronic and thermal free energies||−348.927850|
|Total E (thermal)||105.225 kcal/mol|
|Total CV||30.240 cal/mol-K|
|Total S (entropy)||87.551 cal/mol-K|
Most data are given in Hartree (see Appendix A), they refer to the formation from atomic nuclei and electrons. It is fairly easy to calculate the energy of formation from the atoms by subtracting the energies obtained for respective calculations of free atoms. In order to obtain approximate values for standard enthalpies of formation, bond energies and possibly enthalpies of phase changes (to the gas phase) have to be taken into account. It should be mentioned that the accuracy of these data, i.e., the agreement with experimental data, is not very good. It is advisable to restrain to energy (or enthalpy) differences of similar structures. Alternatively, approximate enthalpies of formation can be obtained more easily from semiempirical calculations (such as MNDO, AM1, or PM3).
Energies of some important free atoms (UB3LYP/6-311G(d,p) in Hartree): H, −0.502155930031; C, −37.8559889346; N, −54.5985431427; O, −75.0853856058; F, −99.7538096003; P, −341.280503655; S, −398.132082447; and Cl, −460.166160487.
Hence, the following energy of formation from the atoms is obtained for
Enthalpies required to generate free atoms from the elements in the standard state (kJ/mol) : H, 218.00; C, 716.67; N, 472.68; O, 249.17; F, 78.4; P, 314.55; S, 276.98; and Cl, 121.29.
These values have to be added to the above-given atomic energy of formation (ignoring somewhat the difference between energy and enthalpy), yielding the following energy of formation for our example: −8301.14 + 9 × 716.67 + 10 × 218.00 = 328.89 kJ/mol.
The energy can be converted to the enthalpy by means of Eq. (1), assuming the validity of the ideal gas law; Δ
In the example, 1 mole of product molecules (in the gas phase) is formed from 19 moles of atoms; therefore, Δ
The following experimental value for the enthalpy of formation of liquid
By comparison, a semiempirical AM1 calculation yields an enthalpy of formation of Δ
3.1. Vibrational spectroscopy: infrared and Raman
Vibrational frequencies and hence infrared (IR) and Raman spectra can be calculated (Gaussian keyword “Freq”). In Gaussian 09, the infrared intensities are calculated by default, but the Raman intensities can also be obtained (keyword “Freq = Raman”). The calculated frequencies can be assigned to the respective molecular motions. The visualization of vibrations is easily achieved by tools such as molden.
As an example,
3.2. Nuclear magnetic resonance (NMR)
The calculation of isotropic chemical shielding values
Using the abovementioned reference value for protons, the calculated chemical shifts are generally too small by about 0.5 ppm. In a survey of 21 natural products, a better fit for 13C nuclei, on the average, was obtained by using a reference value of 177.0 ppm instead .
For the example molecule
|Pos.||C calc||C exp||H calc||H pred|
It is also possible to calculate NMR spin-spin coupling constants, i.e., J [Hz] (Gaussian keyword “NMR = SpinSpin”), but at the expense of computational time. The results obtained with the B3LYP hybrid functional are much better than those of HF ab initio calculations.
It should be mentioned that NMR data provide an excellent and sensitive test for the accuracy of quantum-chemical calculations.
3.3. Electron paramagnetic resonance (EPR)
In the case of free radicals, unrestricted calculations have to be performed in which different orbitals are assigned to α and β spins. Whereas unrestricted Hartree-Fock (UHF) calculations yield poor results for hyperfine couplings (HFC) because of serious problems due to spin contamination, calculations with the UB3LYP hybrid functional yield fairly acceptable results . The calculations yield Mulliken spin densities (better designated as spin populations), isotropic HFC (Fermi contact coupling constants), and anisotropic hyperfine tensors.
The method will be illustrated using the ubisemiquinone-Q1 radical anion as example, which serves as a model compound for coenzyme Q10.
Figure 3 shows the calculated Mulliken spin densities and the calculated proton HFC of this radical anion. The rotation of the long side chain is hindered; therefore, the two methylene protons are inequivalent. Comparison with experimental data (ethanol, 230 K, in parentheses) : 6.44 (5.84), methyl protons, and 3.56 (3.68) and 2.11 (2.17) MHz, methylene protons. The
3.4. Electron spectroscopy (UV/vis)
In Hückel molecular orbital (HMO) theory, electronic excitation may be viewed as excitation of an electron from an occupied to an unoccupied orbital. The transition with the lowest energy, i.e., the longest wavelength, involves the excitation from the highest occupied molecular orbital (HOMO) to the lowest unoccupied molecular orbital (LUMO), although this transition might be forbidden.
In DFT, however, the Kohn-Sham orbitals are not suitable for this procedure, and a time-dependent Hamiltonian has to be used in the calculation (Gaussian keyword “TD”). The calculation gives the energies and the wavelengths of the excitations, the oscillator strengths
In the case of the symmetrical crystal violet cation, the HOMO is represented by two degenerate orbitals, and two excitations have the same wavelength, calculated as 504.7 nm (
For further examples, see .
4. Substituent effects
4.2. Estimating inductive and mesomeric effects by virtual 19F NMR
The relative contributions of inductive
The geometries were optimized for the planar conformation, and the shielding values were calculated for this conformation (0°) and for the orthogonal conformation with a dihedral angle of 90° for the central single bond (see Figure 5). The relative shielding values
4.3. Acids and bases:
The calculation of gas-phase acidities is straightforward, but they do not correlate well with experimental
Thus, Gibbs free energies for benzoic acid and a series of substituted benzoic acids (15 substituents both in
For the calculation of
5. Reaction mechanisms
5.1. Pericyclic reactions
In a pericyclic reaction, σ or π bonds change concertedly (“simultaneously”) along a perimeter, i.e., a cycle. They have first been studied theoretically by Woodward and Hoffmann (“the conservation of orbital symmetry”) . Typical examples are sigmatropic reactions such as the Cope rearrangement, cycloadditions such as the Diels-Alder addition, or electrocyclic reactions (ring closures or openings).
Pericyclic reactions are particularly well amenable to DFT calculations; the transition structure can usually be obtained. The transition structure is a saddle point in the energy hyperspace, i.e., the energy has a maximum along the reaction path (the reaction coordinate), but is minimized with respect to all other coordinates. This can be checked by a frequency calculation. Exactly one frequency should be imaginary, namely, the one pertaining to the reaction coordinate. Thus, the reaction dynamics can be visualized by looking at that vibration.
5.1.1. Cope rearrangement
The Cope rearrangement is a [3,3] sigmatropic reaction. As an example, the degenerate Cope reaction of 1,5-hexadiene is shown (see Figures 8 and 9). The calculated activation energy is 129 kJ/mol (DFT).
5.1.2. Diels-Alder addition
The Diels-Alder addition is a [4 + 2] cycloaddition, a diene reacts with a dienophile to form a (substituted) cyclohexene. As an example, the Diels-Alder addition of acrylonitrile to cyclopentadiene leading to the
5.1.3. Electrocyclic reactions
In an electrocyclic reaction, an unsaturated cycloalkane is formed from a conjugated polyene, or the reverse reaction occurs. Here, only thermally allowed electrocyclic reactions will be considered. For instance, cyclobutene is opened in a conrotatory manner to form 1,3-butadiene. (The calculated activation energy is 149 kJ/mol, and the calculated reaction energy is −39 kJ/mol, assuming that the most stable conformation of 1,3-butadiene is formed.) The example shown here is the disrotatory ring closure of 1,3,5-hexatriene to form 1,3-cyclohexadiene (see Figures 12 and 13). Starting with the most stable conformer of 1,3,5-hexatriene, the calculated activation energy is 252 kJ/mol, and the calculated reaction energy is −64 kJ/mol (DFT).
5.2. Nucleophilic aliphatic substitutions
The most important mechanisms for nucleophilic aliphatic substitutions are the single-step SN2 mechanism with backside attack of the nucleophile and a trigonal-bipyramidal transition state (for primary or secondary substrates) and the two-step SN1 mechanism with a carbenium ion intermediate (for secondary or tertiary substrates). Since these are ionic reactions, the progress in the gas phase may differ considerably from that in a polar solvent.
Considering first the degenerate SN2 reaction of fluoromethane with fluoride anion in the gas phase, the most stable species is a cluster of these two particles, which is formed in an exothermic reaction and calculated reaction energy −106 kJ/mol. The formation of the symmetric trigonal-bipyramidal transition state from this cluster requires an activation energy of 29 kJ/mol.
In the gas-phase reaction of chloromethane with fluoride anion (see Figures 14 and 15), the calculated reaction energy for the formation of fluoromethane and chloride anion at infinite distance is −198 kJ/mol. There is no activation energy for the forward reaction; the energy of the trigonal-bipyramidal transition state is lower by 19 kJ/mol than that of the cluster of chloromethane with fluoride. The most stable species is the cluster of fluoromethane with chloride anion, and the activation energy of the reverse reaction, starting with this cluster, would be 132 kJ/mol.
A typical example for an SN1 reaction is the reaction between
A true SN1 mechanism requires an efficient solvation of the intermediate carbenium ion by polar solvent molecules.
5.3. Electrophilic aromatic substitutions
In gas-phase reactions of benzene with a reactive cationic electrophile such as H+, Br+, NO2+, or CH3CH2+ (cf. Section 4.1), the reaction proceeds without any energy barrier to the σ complex and stops there. Some kind of π complex is formed on the reaction path, but it is not a true intermediate because it is not characterized by a local energy minimum.
In a more realistic scenario, the electrophile is a less reactive complex, e.g., of a halogen, an alkyl, or an acyl chloride, with a Lewis acid such as aluminum chloride or iron(III) bromide. Now, the reaction will usually stop at the π complex stage. In order to force the reaction to proceed to the σ complex, a strongly activating substituent such as oxido (i.e., phenolate anion) was introduced. After removal or replacement of this substituent, the optimization procedure allowed the study of either the backward reaction or the forward reaction to the products, possibly after a modification of the arrangement of the reaction partners.
As an example, the chlorination of benzene catalyzed by aluminum chloride will be considered in detail (see Figures 18 and 19). The overall reaction in the gas phase, yielding chlorobenzene and hydrogen chloride, is exothermic with a calculated reaction energy of −131 kJ/mol and a Gibbs free reaction energy of Δ
5.4. Additions and eliminations
5.4.1. Electrophilic addition
As an example, the addition of bromine to cyclohexene will be considered, yielding
6. Conclusions and outlook
In the field of molecular chemistry, the use of DFT in combination with efficient software and modern computer equipment allows the development of “virtual chemistry,” i.e., the prediction of essentially all molecular properties and of reaction paths. To a certain extent, supramolecular chemistry is also accessible to this method; molecular clusters and microdroplets of solvents can be simulated. It stands to a reason, however, that computational time increases heavily with molecular size (or cluster size). In the case of ab initio calculations, the proportionality is to the fourth power of the size of the basis set; in DFT, the situation might be somewhat better; computational time is proportional to roughly the third power of the number of orbitals involved, judging from NMR calculations.
It should be pointed out that present-day DFT is only an
The following conversion factors have been used in this study: 1 Hartree (a.u.) = 2625.50 kJ/mol, 1 cal = 4.184 J, and
|%Chk = methane|
|h||1 hc2||2 hch3|
|h||1 hc2||3 hch3||2 dih4|
|h||1 hc2||4 hch3||3 dih5|
(The first line specifies the checkpoint file; the second the method and the task, in this case the geometry optimization; the fourth the title, the sixth the total charge, here 0; and the multiplicity, usually 1. Then, the Z matrix follows immediately.)