called electron paramagnetic resonance) [10].

serves as a model compound for coenzyme Q10.

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 timedependent 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 f, and reports the orbitals which are involved. The vibrational fine structure and the consequences of the Franck-Condon principle are not taken into account.

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 (f = 0.806); experimental data for the absorption maxima: 591.0 and 540.5 nm.

For further examples, see [13].

#### 4. Substituent effects

#### 4.1. Hammett σ parameters

The Hammett σ parameters refer to the acidities of substituted benzoic acids which will be considered in Section 4.3. Regarding electrophilic aromatic substitution (see Section 5.3), a modified set has to be used, at least for the para positions (σ+ parameters). The Hammett equation is

$$\log \frac{k\_i}{k\_0} = \sigma \rho \tag{4}$$

4.2. Estimating inductive and mesomeric effects by virtual 19F NMR

Table 2. Calculated (DFT) and literature data [14] for Hammett σ/σ+ parameters.

fluoro-1,3-butadienes were chosen, in two conformations (Figure 5).

shielding values at 0 and at 90, σrel(

(e.g., alkyl groups) (see Table 3).

The relative contributions of inductive (I) and mesomeric (M) effects might be inferred from a comparison of the Hammett σ/σ+ parameters (see Section 4.1) for the para (I + M) and meta (I + M/3) positions. Yet, a different approach is taken here. Using DFT, the obvious target to look for should be the charge density distribution. However, it turned out that the Mulliken charges, at least, did not yield satisfying results. Therefore, calculated isotropic 19F shielding values were used as a probe of local charge density. As a suitable system, 4-substituted (E,E)-1-

Substituent σ meta DFT σ meta lit. σ+ para DFT σ+ para lit. H 0.00 0.00 0.00 0.00 Methyl 0.11 0.10 0.37 0.31 t-Butyl 0.23 0.10 0.47 0.31 Phenyl 0.17 0.00 0.67 0.18 Hydroxy 0.09 0.13 0.64 0.92 Methoxy 0.06 0.05 0.87 0.78 Amino 0.29 0.16 1.38 1.30 Dimethylamino 0.47 0.10 1.65 1.70 Fluoro 0.39 0.35 0.00 0.07 Chloro 0.39 0.40 0.06 0.11 Bromo 0.36 0.41 0.03 0.15 Nitro 0.78 0.73 0.88 0.79 Cyano 0.72 0.56 0.62 0.66 Trifluoromethyl 0.57 0.46 0.58 0.53 Acetyl 0.24 0.36 0.33 0.47 Carboxy 0.29 0.32 0.35 0.42 Sulfonyl 0.58 0.64 0.47 0.73

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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 σrel(

90 should be proportional to the inductive (I) effect, and the differences of the relative

mesomeric (M) effect. The reference compound is, of course, the unsubstituted compound (R = H). These data were calibrated against the Hammett σ/σ+ parameters, yielding a slope of 20.447, i.e., the data have to be divided by this factor. According to the sign convention of the Hammett σ/σ+ parameters, electron-withdrawing groups (EWG, I, M) have a positive sign (e.g., nitro and cyano), whereas electron-releasing groups (ERG, +I, +M) have a negative sign

19F)0σrel(

19F) at

19F)90, should be proportional to the

where ρ is the reaction parameter and k0 and ki are the rate constants for the unsubstituted and substituted compounds, respectively. The σ/σ+ parameters for electrophilic aromatic substitution have been determined from the relative stabilities of the σ complexes as averages for the following four reactions: protonation, bromination, nitration, and alkylation (by ethyl groups). A linear fit of σ/σ+ (literature data [14]) versus calculated σ (DFT) was determined for 17 substituents both in meta and in para positions, yielding a squared correlation coefficient of r 2 = 0.932 (see Figure 4 and Table 2).

Figure 4. Plot of literature data for Hammett σ/σ+ parameters versus calculated values (DFT).

Table 2. Calculated (DFT) and literature data [14] for Hammett σ/σ+ parameters.

4. Substituent effects

86 Density Functional Calculations - Recent Progresses of Theory and Application

4.1. Hammett σ parameters

2 = 0.932 (see Figure 4 and Table 2).

r

The Hammett σ parameters refer to the acidities of substituted benzoic acids which will be considered in Section 4.3. Regarding electrophilic aromatic substitution (see Section 5.3), a modified set has to be used, at least for the para positions (σ+ parameters). The Hammett equation is

where ρ is the reaction parameter and k0 and ki are the rate constants for the unsubstituted and substituted compounds, respectively. The σ/σ+ parameters for electrophilic aromatic substitution have been determined from the relative stabilities of the σ complexes as averages for the following four reactions: protonation, bromination, nitration, and alkylation (by ethyl groups). A linear fit of σ/σ+ (literature data [14]) versus calculated σ (DFT) was determined for 17 substituents both in meta and in para positions, yielding a squared correlation coefficient of

¼ σρ (4)

log ki k0

Figure 4. Plot of literature data for Hammett σ/σ+ parameters versus calculated values (DFT).

#### 4.2. Estimating inductive and mesomeric effects by virtual 19F NMR

The relative contributions of inductive (I) and mesomeric (M) effects might be inferred from a comparison of the Hammett σ/σ+ parameters (see Section 4.1) for the para (I + M) and meta (I + M/3) positions. Yet, a different approach is taken here. Using DFT, the obvious target to look for should be the charge density distribution. However, it turned out that the Mulliken charges, at least, did not yield satisfying results. Therefore, calculated isotropic 19F shielding values were used as a probe of local charge density. As a suitable system, 4-substituted (E,E)-1 fluoro-1,3-butadienes were chosen, in two conformations (Figure 5).

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 σrel( 19F) at 90 should be proportional to the inductive (I) effect, and the differences of the relative shielding values at 0 and at 90, σrel( 19F)0σrel( 19F)90, should be proportional to the mesomeric (M) effect. The reference compound is, of course, the unsubstituted compound (R = H). These data were calibrated against the Hammett σ/σ+ parameters, yielding a slope of 20.447, i.e., the data have to be divided by this factor. According to the sign convention of the Hammett σ/σ+ parameters, electron-withdrawing groups (EWG, I, M) have a positive sign (e.g., nitro and cyano), whereas electron-releasing groups (ERG, +I, +M) have a negative sign (e.g., alkyl groups) (see Table 3).

4.3. Acids and bases: pKa values

Ka was calculated according to

The calculation of gas-phase acidities is straightforward, but they do not correlate well with experimental pKa values [13]. This situation is only partially improved by taking the bulk properties of the solvent (water) into account (Gaussian keyword "SCRF = (Solvent = Water)"). A much better approximation is obtained when additionally a few water molecules are taken into account explicitly, e.g., two water molecules in the case of carboxylic acids (see Figure 6). Thus, Gibbs free energies for benzoic acid and a series of substituted benzoic acids (15 substituents both in meta and in para positions) as well as the respective anions were calculated. ΔG� = G�(anion) � G�(acid) was converted from Hartree to kJ/mol (see Appendix A), and log

The relative pKa values were found to be quite reasonable but have to be scaled. Since the difference log Ka (substituted benzoic acid)-log Ka (benzoic acid) should be equal to the Hammett σ parameter, a linear fit of σ (literature data [14]) versus calculated Δ(log Ka) (DFT)

was determined, yielding a slope of 0.3437 and a squared correlation coefficient of r

¼ �2:3026 RT log Ka (5)

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pKa ¼ � logKa (6)

2 = 0.967

89

ΔG�

Figure 6. 3D models of dihydrates of benzoic acid and benzoate anion.

Figure 5. 3D models of two conformations of 1-fluoro-5,5-dimethyl-1,3-hexadiene.

Table 3. Calculated inductive (σI) and mesomeric (σM) effects (DFT) (see text).

#### 4.3. Acids and bases: pKa values

Figure 5. 3D models of two conformations of 1-fluoro-5,5-dimethyl-1,3-hexadiene.

88 Density Functional Calculations - Recent Progresses of Theory and Application

Table 3. Calculated inductive (σI) and mesomeric (σM) effects (DFT) (see text).

Substituent σI σM H 0.00 0.00 Nitro 0.47 0.99 Cyano 0.35 0.57 Acetyl 0.16 0.69 Carboxy 0.16 0.75 Methoxycarbonyl 0.12 0.69 Trifluoromethyl 1.02 �0.29 Fluoro 0.39 �0.43 Chloro 0.33 �0.23 Bromo 0.30 �0.21 Methyl �0.04 �0.25 Hydroxy 0.18 �0.74 Methoxy 0.16 �0.74 Amino 0.01 �1.21 Dimethylamino �0.02 �1.17 The calculation of gas-phase acidities is straightforward, but they do not correlate well with experimental pKa values [13]. This situation is only partially improved by taking the bulk properties of the solvent (water) into account (Gaussian keyword "SCRF = (Solvent = Water)"). A much better approximation is obtained when additionally a few water molecules are taken into account explicitly, e.g., two water molecules in the case of carboxylic acids (see Figure 6).

Thus, Gibbs free energies for benzoic acid and a series of substituted benzoic acids (15 substituents both in meta and in para positions) as well as the respective anions were calculated. ΔG� = G�(anion) � G�(acid) was converted from Hartree to kJ/mol (see Appendix A), and log Ka was calculated according to

$$
\Delta G^{\circ} = -2.3026 \, RT \log K\_{\circ} \tag{5}
$$

$$pK\_a = -\log K\_a \tag{6}$$

The relative pKa values were found to be quite reasonable but have to be scaled. Since the difference log Ka (substituted benzoic acid)-log Ka (benzoic acid) should be equal to the Hammett σ parameter, a linear fit of σ (literature data [14]) versus calculated Δ(log Ka) (DFT) was determined, yielding a slope of 0.3437 and a squared correlation coefficient of r 2 = 0.967

Figure 6. 3D models of dihydrates of benzoic acid and benzoate anion.

conservation of orbital symmetry") [15]. Typical examples are sigmatropic reactions such as the Cope rearrangement, cycloadditions such as the Diels-Alder addition, or electrocyclic

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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

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

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 endo-product is shown (see Figures 10 and 11). The calculated

N

N

activation energy is 40 kJ/mol, and the calculated reaction energy is �132 kJ/mol (DFT).

reactions (ring closures or openings).

5.1.1. Cope rearrangement

5.1.2. Diels-Alder addition

129 kJ/mol (DFT).

reaction dynamics can be visualized by looking at that vibration.

Figure 8. Scheme of the degenerate cope reaction of 1,5-hexadiene.

Figure 9. Cope reaction: 3D models of reactant, transition structure, and product.

N

Figure 10. Scheme of the Diels-Alder reaction between cyclopentadiene and acrylonitrile.

+

Figure 7. Plot of Hammett σ parameters versus calculated relative log Ka values (DFT).

(Figure 7). That means, the substituent effect on the calculated pKa value is overestimated by about a factor of 3.

For the calculation of absolute pKa values, the considerable Gibbs free solvation energy of the proton has to be taken into account (ΔG� = �1120.39 kJ/mol). The calculated Gibbs free energy of the dissociation of benzoic acid in the gas phase is ΔG� = 1446.04 kJ/mol. For the hydration (dihydrate model, vide supra), ΔG� = �47.35 kJ/mol and ΔG� = �302.76 kJ/mol are obtained for benzoic acid and benzoate anion, respectively; in total, ΔG� = 1375.80 kJ/mol. Hence, the estimate for the Gibbs free energy of the dissociation of benzoic acid in aqueous solution is ΔG� = 1446.04 � 1375.80 = 70.24 kJ/mol corresponding to a pKa value of 12.31 (exp. 4.19). To put it differently, the model applied here accounts for about 96.7% of the true Gibbs free energy of solvation.

#### 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") [15]. 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).

Figure 8. Scheme of the degenerate cope reaction of 1,5-hexadiene.

Figure 9. Cope reaction: 3D models of reactant, transition structure, and product.

#### 5.1.2. Diels-Alder addition

(Figure 7). That means, the substituent effect on the calculated pKa value is overestimated by

Figure 7. Plot of Hammett σ parameters versus calculated relative log Ka values (DFT).

90 Density Functional Calculations - Recent Progresses of Theory and Application

For the calculation of absolute pKa values, the considerable Gibbs free solvation energy of the proton has to be taken into account (ΔG� = �1120.39 kJ/mol). The calculated Gibbs free energy of the dissociation of benzoic acid in the gas phase is ΔG� = 1446.04 kJ/mol. For the hydration (dihydrate model, vide supra), ΔG� = �47.35 kJ/mol and ΔG� = �302.76 kJ/mol are obtained for benzoic acid and benzoate anion, respectively; in total, ΔG� = 1375.80 kJ/mol. Hence, the estimate for the Gibbs free energy of the dissociation of benzoic acid in aqueous solution is ΔG� = 1446.04 � 1375.80 = 70.24 kJ/mol corresponding to a pKa value of 12.31 (exp. 4.19). To put it differently, the model applied here accounts for about 96.7% of the true Gibbs free energy of solvation.

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

about a factor of 3.

5. Reaction mechanisms

5.1. Pericyclic reactions

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 endo-product is shown (see Figures 10 and 11). The calculated activation energy is 40 kJ/mol, and the calculated reaction energy is �132 kJ/mol (DFT).

Figure 10. Scheme of the Diels-Alder reaction between cyclopentadiene and acrylonitrile.

5.2. Nucleophilic aliphatic substitutions

29 kJ/mol.

would be 132 kJ/mol.

H

Figure 14. Scheme of an SN2 reaction.

H

H

Cl

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

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

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,

F Cl

+ +

F Cl F

H

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H H

H H

H

A typical example for an SN1 reaction is the reaction between tert-butanol and hydrogen chloride, yielding tert-butyl chloride and water (or the reverse reaction; see Figures 16 and 17). In the gas phase, this reaction is slightly exothermic with a calculated reaction energy of �8 kJ/mol (DFT). The concurrent elimination reaction (E1), yielding isobutene, water, and hydrogen chloride, is endothermic with a calculated reaction energy of 47 kJ/mol. For the reaction to proceed, it is necessary to form the protonated alcohol. Only the formation of an ion pair of chloride and protonated tert-butanol is conceivable, requiring an estimated activation energy of about 165 kJ/ mol. The formation of a free tert-butyl carbenium ion is not feasible, because the energy required would be about 682 kJ/mol. The reaction should rather proceed by a backside attack similar to

Figure 15. SN2 reaction: 3D models of reactants, transition structure, and products.

in the gas phase may differ considerably from that in a polar solvent.

Figure 11. Diels-Alder reaction: 3D models of reactants, transition structure, and product.

#### 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).

Figure 12. Scheme of the electrocyclic ring closure of 1,3,5-hexatriene.

Figure 13. Electrocyclic ring closure of 1,3,5-hexatriene: 3D models of reactant (two conformations), transition structure, and product.

#### 5.2. Nucleophilic aliphatic substitutions

5.1.3. Electrocyclic reactions

(DFT).

and product.

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

Figure 13. Electrocyclic ring closure of 1,3,5-hexatriene: 3D models of reactant (two conformations), transition structure,

Figure 11. Diels-Alder reaction: 3D models of reactants, transition structure, and product.

92 Density Functional Calculations - Recent Progresses of Theory and Application

Figure 12. Scheme of the electrocyclic ring closure of 1,3,5-hexatriene.

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.

Figure 15. SN2 reaction: 3D models of reactants, transition structure, and products.

A typical example for an SN1 reaction is the reaction between tert-butanol and hydrogen chloride, yielding tert-butyl chloride and water (or the reverse reaction; see Figures 16 and 17). In the gas phase, this reaction is slightly exothermic with a calculated reaction energy of �8 kJ/mol (DFT). The concurrent elimination reaction (E1), yielding isobutene, water, and hydrogen chloride, is endothermic with a calculated reaction energy of 47 kJ/mol. For the reaction to proceed, it is necessary to form the protonated alcohol. Only the formation of an ion pair of chloride and protonated tert-butanol is conceivable, requiring an estimated activation energy of about 165 kJ/ mol. The formation of a free tert-butyl carbenium ion is not feasible, because the energy required would be about 682 kJ/mol. The reaction should rather proceed by a backside attack similar to

Figure 17. SN1 reaction: 3D models of reactants, intermediate, and products.

the SN2 reaction and stop at a cluster of tert-butyl chloride and water (reaction energy 19 kJ/ mol).

A true SN1 mechanism requires an efficient solvation of the intermediate carbenium ion by polar solvent molecules.

H C l +

Figure 19. Electrophilic chlorination of benzene. Top row: first π complex, approximate first transition structure, and σ

complex. Bottom row: approximate second transition structure and second π complex (of products).

Figure 18. Scheme of the AlCl3-catalyzed electrophilic substitution of benzene by chlorine.

Cl

Cl

Al Cl Cl

Cl H

+

Cl

Al Cl Cl

Cl

Cl

+ Cl Cl + Cl

Al Cl Cl Cl Cl

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Cl

Al Cl Cl Cl H

Cl

Al Cl Cl

#### 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 ΔrG = 143 kJ/mol (DFT). The reaction involves three intermediates, π complex 1, σ complex, and π complex 2, which were calculated as local minima and two transition structures, which could not yet be identified unambiguously. The crucial energy barrier is most likely the transition state leading to the σ complex. Taking the energies of the separated reactants as reference, the relative energies are as follows: 45 kJ/mol for π complex 1, approximately 208 kJ/mol for transition state 1, 80 kJ/mol for the σ complex, approximately

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Figure 18. Scheme of the AlCl3-catalyzed electrophilic substitution of benzene by chlorine.

the SN2 reaction and stop at a cluster of tert-butyl chloride and water (reaction energy 19 kJ/

CH3

Cl OH2 Cl

H3C CH3

H Cl + O

A true SN1 mechanism requires an efficient solvation of the intermediate carbenium ion by

and stops there. Some kind of π complex is formed on the reaction path, but it is not a true

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,

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 ΔrG = 143 kJ/mol (DFT). The reaction involves three intermediates, π complex 1, σ complex, and π complex 2, which were calculated as local minima and two transition structures, which could not yet be identified unambiguously. The crucial energy barrier is most likely the transition state leading to the σ complex. Taking the energies of the separated reactants as reference, the relative energies are as follows: 45 kJ/mol for π complex 1, approximately 208 kJ/mol for transition state 1, 80 kJ/mol for the σ complex, approximately

+ (cf. Section 4.1), the reaction proceeds without any energy barrier to the σ complex

, Br+

CH3

H

H

CH3 CH3 +

> , NO2 + , or

In gas-phase reactions of benzene with a reactive cationic electrophile such as H+

intermediate because it is not characterized by a local energy minimum.

possibly after a modification of the arrangement of the reaction partners.

mol).

CH3CH2

polar solvent molecules.

5.3. Electrophilic aromatic substitutions

OH

94 Density Functional Calculations - Recent Progresses of Theory and Application

Figure 17. SN1 reaction: 3D models of reactants, intermediate, and products.

H3C

Figure 16. Scheme of an SN1 reaction.

H3C

H3C

Figure 19. Electrophilic chlorination of benzene. Top row: first π complex, approximate first transition structure, and σ complex. Bottom row: approximate second transition structure and second π complex (of products).

27 kJ/mol for transition state 2, �169 kJ/mol for π complex 2 (global minimum), and �131 kJ/ mol for the separated products.

#### 5.4. Additions and eliminations

#### 5.4.1. Electrophilic addition

As an example, the addition of bromine to cyclohexene will be considered, yielding trans-1,2 dibromocyclohexane (see Figures 20 and 21). First, a π complex is formed (relative energy �22 kJ/mol with respect to the separated reactants) and, next, a bicyclic bromonium ion, which is more stable than the respective carbenium ion (by roughly 100 kJ/mol). Finally, the diaxial conformer of trans-1,2-dibromocyclohexane is formed, which is actually more stable than the diequatorial conformer by 7 kJ/mol. This finding is somewhat surprising and stands in contrast to previous assumptions. Apparently, electrostatic repulsion favors the diaxial form, whereas in monosubstituted cyclohexanes, the substituent prefers the equatorial position. The reaction energy is �108 kJ/mol, and the Gibbs free reaction energy is �50 kJ/mol; this is due to the unfavorable reaction entropy.

6. Conclusions and outlook

products.

H3C

H3C

H3C

A. Appendix

of orbitals involved, judging from NMR calculations.

OH

Figure 22. Scheme for the E1 elimination of water from tert-butanol.

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

Figure 23. Elimination of water from tert-butanol: 3D models of reactants, approximate transition state, and π complex of

H3C CH3

H H H

+ H Cl + O

Cl

OH2

H

97

H

H3C CH3

Cl H

http://dx.doi.org/10.5772/intechopen.70751

Spectroscopy, Substituent Effects, and Reaction Mechanisms

It should be pointed out that present-day DFT is only an approximate theory. Therefore, it is

The following conversion factors have been used in this study: 1 Hartree (a.u.) = 2625.50 kJ/

Computational details. For all computations in Chapters 2 to 4, Gaussian 09 was used B3LYP/ 6–311(d,p) [1]. For most computations in Chapter 5, deMon2k was used [16]. For the

necessary to check the quality of the computational results against experimental data.

mol, 1 cal = 4.184 J, and pK = ΔG [kJ/mol] /5.708008 (at T = 298.15 K).

Figure 20. Scheme for the electrophilic addition of bromine to cyclohexene.

Figure 21. Electrophilic addition of bromine to cyclohexene: 3D models of reactants, π complex, bromonium ion, and product.

#### 5.4.2. Elimination

In Section 5.2, it was already briefly mentioned that the reaction between tert-butanol and hydrogen chloride might proceed as an elimination instead of a substitution. The mechanism is E1, and isobutene (2-methylpropene) is formed as product (see Figures 22 and 23).

Figure 22. Scheme for the E1 elimination of water from tert-butanol.

Figure 23. Elimination of water from tert-butanol: 3D models of reactants, approximate transition state, and π complex of products.

#### 6. Conclusions and outlook

27 kJ/mol for transition state 2, �169 kJ/mol for π complex 2 (global minimum), and �131 kJ/

As an example, the addition of bromine to cyclohexene will be considered, yielding trans-1,2 dibromocyclohexane (see Figures 20 and 21). First, a π complex is formed (relative energy �22 kJ/mol with respect to the separated reactants) and, next, a bicyclic bromonium ion, which is more stable than the respective carbenium ion (by roughly 100 kJ/mol). Finally, the diaxial conformer of trans-1,2-dibromocyclohexane is formed, which is actually more stable than the diequatorial conformer by 7 kJ/mol. This finding is somewhat surprising and stands in contrast to previous assumptions. Apparently, electrostatic repulsion favors the diaxial form, whereas in monosubstituted cyclohexanes, the substituent prefers the equatorial position. The reaction energy is �108 kJ/mol, and the Gibbs free reaction energy is �50 kJ/mol; this is due to the

In Section 5.2, it was already briefly mentioned that the reaction between tert-butanol and hydrogen chloride might proceed as an elimination instead of a substitution. The mechanism

Figure 21. Electrophilic addition of bromine to cyclohexene: 3D models of reactants, π complex, bromonium ion, and

+

Br

Br

Br Br

is E1, and isobutene (2-methylpropene) is formed as product (see Figures 22 and 23).

Br

Br

Figure 20. Scheme for the electrophilic addition of bromine to cyclohexene.

mol for the separated products.

96 Density Functional Calculations - Recent Progresses of Theory and Application

5.4. Additions and eliminations

5.4.1. Electrophilic addition

unfavorable reaction entropy.

+

Br

Br

5.4.2. Elimination

product.

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 approximate theory. Therefore, it is necessary to check the quality of the computational results against experimental data.

#### A. Appendix

The following conversion factors have been used in this study: 1 Hartree (a.u.) = 2625.50 kJ/ mol, 1 cal = 4.184 J, and pK = ΔG [kJ/mol] /5.708008 (at T = 298.15 K).

Computational details. For all computations in Chapters 2 to 4, Gaussian 09 was used B3LYP/ 6–311(d,p) [1]. For most computations in Chapter 5, deMon2k was used [16]. For the preparation of input files, i.e., the generation of Z matrices, and the visualization of the results, molden [2] was of great help.

[3] Weast RC, editor. Handbook of Chemistry and Physics. 57th ed. Cleveland, Ohio: CRC

Spectroscopy, Substituent Effects, and Reaction Mechanisms

http://dx.doi.org/10.5772/intechopen.70751

99

[6] Kaupp M, Bühl M, Malkin VG, editors. Calculation of NMR and EPR Parameters. Weinheim:

[7] Kirste B. 3D Structure Determination of Natural Products by 13C–NMR-Controlled DFT Calculations [Internet]. 2016. Available from: http://kirste.userpage.fu-berlin.de/chemis-

[9] Kirste B. DFT calculations of hyperfine coupling constants of organic π radicals and comparison with empirical equations and experiment. Magnetic Resonance in Chemistry.

[10] Gerson F, Huber W. Electron Spin Resonance Spectroscopy of Organic Radicals. Weinheim:

[11] Kirste B. Untersuchung paramagnetischer organischer Verbindungen in flüssigen Kristallen und in Festkörpern mit der magnetischen Resonanz (EPR/ENDOR) [habilitation thesis]. Berlin; 1985. Available from: http://kirste.userpage.fu-berlin.de/ag/kirste/pdf/

[12] Möbius K, Savitsky A. High-Field EPR Spectroscopy on Proteins and their Model Sys-

[13] Kirste B. Applications of density functional theory to theoretical organic chemistry.

[14] Smith MB, March J. March's Advanced Organic Chemistry. 6th ed. Hoboken, New Jersey:

[15] Woodward RB, Hoffmann R. The conservation of orbital symmetry. Angewandte Chemie

[16] Koster AM, Calaminici P, Casida ME, Dominguez VD, Flores-Moreno, et al. deMon2k, Version 2. Cinvestav, Mexico City: The deMon Developers; 2006 http://www.demon-

Chemical Sciences Journal. 2016;7:127. DOI: 10.4172/2150-3494.1000127

(International Ed. in English). 1969;8:781-853. DOI: 10.1002/anie.196907811

[4] NIST Chemistry WebBook. http://webbook.nist.gov/chemistry/

try/nmr/nmrdft/index.html [Accessed: 2017-07-03]

[8] NMRShiftDB. https://nmrshiftdb.nmr.uni-koeln.de/

2016;54:835-841. DOI: 10.1002/mrc.4467

bkihabil.pdf Accessed: 2017–07-03

tems. Cambridge: RSC Publishing; 2009

[5] Spectral Database for Organic Compounds, SDBS. http://sdbs.db.aist.go.jp/

Press; 1976

Wiley-VCH; 2004

Wiley-VCH; 2003

Wiley; 2007

software.com/

Sample Gaussian input file: methane

%Chk = methane #B3LYP/6–311G(d,p) Opt Methane 0 1 c h 1 hc2 h 1 hc2 2 hch3 h 1 hc2 3 hch3 2 dih4 h 1 hc2 4 hch3 3 dih5 hc2 1.089 hch3 109.47 dih4 �120.0 dih5 120.0

(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.)

#### Author details

Burkhard Kirste

Address all correspondence to<EMAIL_ADDRESS>
Institute of Chemistry and Biochemistry, Freie Universität Berlin, Germany

#### References

preparation of input files, i.e., the generation of Z matrices, and the visualization of the results,

(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 multiplic-

[1] Frisch MJ, Trucks GW, Schlegel HB, Scuseria GE, Robb MA, et al. Gaussian 09. Walling-

[2] Schaftenaar G. CMBI. The Netherlands: Molden http://www.cmbi.ru.nl/molden/

ity, usually 1. Then, the Z matrix follows immediately.)

Address all correspondence to<EMAIL_ADDRESS>
Institute of Chemistry and Biochemistry, Freie Universität Berlin, Germany

ford CT: Gaussian Inc.; 2013 http://www.gaussian.com/

Author details

Burkhard Kirste

References

molden [2] was of great help.

%Chk = methane

Methane

c

0 1

h 1 hc2

hc2 1.089 hch3 109.47 dih4 �120.0 dih5 120.0

#B3LYP/6–311G(d,p) Opt

h 1 hc2 2 hch3

h 1 hc2 3 hch3 2 dih4 h 1 hc2 4 hch3 3 dih5

Sample Gaussian input file: methane

98 Density Functional Calculations - Recent Progresses of Theory and Application

**Chapter 5**

Provisional chapter

**Spectral Calculations with DFT**

Spectral Calculations with DFT

Ataf Ali Altaf, Samia Kausar and Amin Badshah

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

Spectra calculations are an important branch of theoretical modeling, and due to the significant improvements of high-level computational methods, the calculated spectra can be used directly and sometimes help to correct the errors of experimental observations. On the other hand, theoretical computations assist the experimental assignments. The authors discuss three spectral calculations (UV-Vis, IR and NMR) that are the most widely used. UV-Visible spectrum can be carried out employing time-dependent density functional theory (TDDFT) with B3LYP/631G(d,p) and CAM-B3LYP functional method to illustrate the characteristics of vertical electronic excitations. The vibrational spectra can be generated from a list of frequencies and intensities using a Gaussian broadening function method. NMR chemical shifts can be calculated by density functional theory individual gauge for localized orbitals (DFTIGLO) method and by gauge including

DOI: 10.5772/intechopen.71080

Keywords: spectral calculation, vibrational spectra, DFT NMR calculation, TDDFT for

Early days of quantum chemistry go back to Thomas-Fermi and Thomas-Fermi-Dirac models of the electronic structure of atoms, which gave the concept of articulating some parts or all of the molecular energy as a functional of the electron density, and then comes the traditional Hartree-Fock (HF) theory [1]. HF theory was a simplest ab initio technique and among the first principles of quantum-chemical theories, being attained directly from the Schrodingerwave equation that did not incorporate any pragmatic contemplations. Although such theories and techniques proved beneficial, it was density functional theory (DFT) that laid a demanding theoretical foundation in 1964 by an outstanding result established by Hohenberg

> © The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

distribution, and reproduction in any medium, provided the original work is properly cited.

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

http://dx.doi.org/10.5772/intechopen.71080

atomic orbitals (GIAO) approach.

UV-visible spectra

1. Introduction

and Kohn [2].

Ataf Ali Altaf, Samia Kausar and

Amin Badshah

Abstract

**Chapter 5** Provisional chapter

#### **Spectral Calculations with DFT** Spectral Calculations with DFT

Ataf Ali Altaf, Samia Kausar and Amin Badshah Ataf Ali Altaf, Samia Kausar and

Additional information is available at the end of the chapter Amin Badshah

http://dx.doi.org/10.5772/intechopen.71080 Additional information is available at the end of the chapter

#### Abstract

Spectra calculations are an important branch of theoretical modeling, and due to the significant improvements of high-level computational methods, the calculated spectra can be used directly and sometimes help to correct the errors of experimental observations. On the other hand, theoretical computations assist the experimental assignments. The authors discuss three spectral calculations (UV-Vis, IR and NMR) that are the most widely used. UV-Visible spectrum can be carried out employing time-dependent density functional theory (TDDFT) with B3LYP/631G(d,p) and CAM-B3LYP functional method to illustrate the characteristics of vertical electronic excitations. The vibrational spectra can be generated from a list of frequencies and intensities using a Gaussian broadening function method. NMR chemical shifts can be calculated by density functional theory individual gauge for localized orbitals (DFTIGLO) method and by gauge including atomic orbitals (GIAO) approach.

DOI: 10.5772/intechopen.71080

Keywords: spectral calculation, vibrational spectra, DFT NMR calculation, TDDFT for UV-visible spectra

#### 1. Introduction

Early days of quantum chemistry go back to Thomas-Fermi and Thomas-Fermi-Dirac models of the electronic structure of atoms, which gave the concept of articulating some parts or all of the molecular energy as a functional of the electron density, and then comes the traditional Hartree-Fock (HF) theory [1]. HF theory was a simplest ab initio technique and among the first principles of quantum-chemical theories, being attained directly from the Schrodingerwave equation that did not incorporate any pragmatic contemplations. Although such theories and techniques proved beneficial, it was density functional theory (DFT) that laid a demanding theoretical foundation in 1964 by an outstanding result established by Hohenberg and Kohn [2].

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

The Hohenberg-Kohn theorems state that the exact ground state energy of the system is produced by a unique functional of the electron density r. The first theorem demonstrates a many-particle system n(x,y,z) for which one-to-one plotting happens between the ground state electric density and the ground state wave function. The second theorem verifies that total electric energy of the system E[n(x,y,z)] is minimalized by the ground state density [3]. The work of Kohn and Sham instantly surveyed and paved the way for hands-on computational applicability of DFT to physical systems, by linking a reference-state comprising of a set of noninteracting one-particle orbitals with a particular functional. The reference orbitals determined by a set of operative one-particle Schrodinger-wave equations are called Kohn-Sham (KS) equations [4]; one of them is depicted in Eq. (1).

$$\left(-\frac{1}{2}\nabla^2\left(\overrightarrow{r}\right) + V\_{KS}\right)\Psi\_i = \varepsilon\_i \Psi\_i \tag{1}$$

and biochemical systems. It is obvious that local geometrical and electronic structures influence magnetic resonance parameters, i.e., shielding tensors, nuclear spin-spin coupling parameters, hyperfine tensors and g-tensors. For speculations of NMR parameters, a practical approximation of the real system is depicted by a model system in which 50–100 atoms are treated with ab initio techniques. Over the last years, DFT advanced by the capability to include effects of electron correlation in a very effectual means to the forefront of field of

Spectral Calculations with DFT

103

http://dx.doi.org/10.5772/intechopen.71080

Competent calculations of excited states properties are emerging field of interest for quantum chemists; hence, they are developing interesting solutions for these properties. Because of advancement of computations based on the time-dependent density functional theory (TD-DFT) [10] in recent years, calculations of electronic structures in the excited states have become a motivation of interest. A commanding method of carrying out exact quantum mechanical calculations of the intervalence absorption spectrum is provided by the time-dependent concept of electronic spectroscopy. A corporeal picture of the effects of the coupling of electronic and nuclear motions became available by employing calculations in the time realm due to reason that the time development of the wave-packet can be tracked and inferred as well [11]. Plentiful successes of this technique have been recently revised as electronic spectra are often calculated with TD-DFT. Vertical excitation energies, absorption wavelength and oscillator

The influence of vibrational spectroscopy as an analytical tool in several fields is obvious. Qualitative association of bands and specific structures or chemical groups is basic interpretation of vibrational spectra. In contrast to nuclear magnetic resonance where nuclear spin is associated with one peak or multiplet is proved advantageous, united motion of all of the nuclei in sample is reason for observed bands in vibrational spectra. For N nuclei, there are at most 3 N-6 experiential fundamental bands and so far the matrix of internuclear force interactions, i.e., the second derivative matrix in harmonic approximation, has (3 N-6)(3 N -5)/2 exclusive terms. The extraction of force constants from vibrational frequencies is yet an undetermined mathematical problem as many bands are not actually observed; e.g., there are overtones, combination bands and nonconformities from the harmonic approximation; hence,

For the ground state's potential energy surface computation, discovery of effectual codes delivers a possible answer to the problem. A technique that will precisely determine the bonding, as well as intermolecular interactions, will be helpful for calculating vibrational spectra. The ground state properties and potential energies are accurately calculated by DFT approaches; therefore, DFT excellently and proficiently calculates the vibrational spectra from first principles [7]. The absolute values of the frequencies are high in contrast to experimental values while using DFT models as they seem to characterize the bonding pretty well and comparison with experimental trends is possible. DFT methods give sufficiently high accuracy of normal mode calculations as the restrictions of the harmonic approximation are often a foremost cause of divergence between theory and experiment.

calculating NMR parameters [8, 9].

strength calculations can be performed by this technique [12].

2.1. Theoretical vibrational spectra analysis

the problem seems obstinate [13].

where VKS is a local one-body potential defined as total density of the noninteracting system and is the same as the density of a real system.

#### 2. Spectral implementations of DFT

The density functional theory (DFT) has become a powerful tool in computational chemistry owing to its usefulness. Spectroscopic analysis of chemical entities by this technique emerged as commanding implementation. Prediction of frequencies and spectral intensities by DFT calculations are indispensable nowadays for interpreting the experimental spectra of complex molecules. Much advancement has been made in the past decade to design new DFT approaches that can be employed into available quantum-chemical computational programs. For calculation of the molecular and electronic structures of ground-state systems and various spectral parameters related to NMR, ESR, UV-Vis and IR, various density functional practices are available now. Functionals available today can strive with best previous ab initio methods [5].

For experimental spectroscopists, theoretical computation of vibrational frequencies has become practically essential these days as specifically in problematic and uncertain cases it assists to assign and interpret experimental infrared/Raman spectra. Previously, the HF method was used in many studies to calculate vibrational frequencies, but it has long been identified that this approach miscalculates these frequencies even occasionally to a disturbing degree. Inadequate handling of electron correlation and anharmonicity of the vibrations are found to be the main contributions to error [6]. DFT largely overcome the errors as theoretical computations helped to interpret experimental conclusions depicting that using DFT these theoretical data can nearly be attained at the harmonic level. Calculation of optimized geometry, IR intensities, vibrational frequencies and Raman scattering activities can be done by employing different density functional approaches [7].

Taking NMR spectroscopy into consideration, DFT-based new methods are developed that are appropriate for the satisfactory hypothetical interpretation of NMR spectra of various chemical and biochemical systems. It is obvious that local geometrical and electronic structures influence magnetic resonance parameters, i.e., shielding tensors, nuclear spin-spin coupling parameters, hyperfine tensors and g-tensors. For speculations of NMR parameters, a practical approximation of the real system is depicted by a model system in which 50–100 atoms are treated with ab initio techniques. Over the last years, DFT advanced by the capability to include effects of electron correlation in a very effectual means to the forefront of field of calculating NMR parameters [8, 9].

Competent calculations of excited states properties are emerging field of interest for quantum chemists; hence, they are developing interesting solutions for these properties. Because of advancement of computations based on the time-dependent density functional theory (TD-DFT) [10] in recent years, calculations of electronic structures in the excited states have become a motivation of interest. A commanding method of carrying out exact quantum mechanical calculations of the intervalence absorption spectrum is provided by the time-dependent concept of electronic spectroscopy. A corporeal picture of the effects of the coupling of electronic and nuclear motions became available by employing calculations in the time realm due to reason that the time development of the wave-packet can be tracked and inferred as well [11]. Plentiful successes of this technique have been recently revised as electronic spectra are often calculated with TD-DFT. Vertical excitation energies, absorption wavelength and oscillator strength calculations can be performed by this technique [12].

#### 2.1. Theoretical vibrational spectra analysis

The Hohenberg-Kohn theorems state that the exact ground state energy of the system is produced by a unique functional of the electron density r. The first theorem demonstrates a many-particle system n(x,y,z) for which one-to-one plotting happens between the ground state electric density and the ground state wave function. The second theorem verifies that total electric energy of the system E[n(x,y,z)] is minimalized by the ground state density [3]. The work of Kohn and Sham instantly surveyed and paved the way for hands-on computational applicability of DFT to physical systems, by linking a reference-state comprising of a set of noninteracting one-particle orbitals with a particular functional. The reference orbitals determined by a set of operative one-particle Schrodinger-wave equations are called Kohn-Sham

þ VKS

where VKS is a local one-body potential defined as total density of the noninteracting system

The density functional theory (DFT) has become a powerful tool in computational chemistry owing to its usefulness. Spectroscopic analysis of chemical entities by this technique emerged as commanding implementation. Prediction of frequencies and spectral intensities by DFT calculations are indispensable nowadays for interpreting the experimental spectra of complex molecules. Much advancement has been made in the past decade to design new DFT approaches that can be employed into available quantum-chemical computational programs. For calculation of the molecular and electronic structures of ground-state systems and various spectral parameters related to NMR, ESR, UV-Vis and IR, various density functional practices are available now. Functionals available today can strive with best previous ab

For experimental spectroscopists, theoretical computation of vibrational frequencies has become practically essential these days as specifically in problematic and uncertain cases it assists to assign and interpret experimental infrared/Raman spectra. Previously, the HF method was used in many studies to calculate vibrational frequencies, but it has long been identified that this approach miscalculates these frequencies even occasionally to a disturbing degree. Inadequate handling of electron correlation and anharmonicity of the vibrations are found to be the main contributions to error [6]. DFT largely overcome the errors as theoretical computations helped to interpret experimental conclusions depicting that using DFT these theoretical data can nearly be attained at the harmonic level. Calculation of optimized geometry, IR intensities, vibrational frequencies and Raman scattering activities can be done by

Taking NMR spectroscopy into consideration, DFT-based new methods are developed that are appropriate for the satisfactory hypothetical interpretation of NMR spectra of various chemical

Ψi ¼ εiΨi (1)

(KS) equations [4]; one of them is depicted in Eq. (1).

102 Density Functional Calculations - Recent Progresses of Theory and Application

and is the same as the density of a real system.

2. Spectral implementations of DFT

employing different density functional approaches [7].

initio methods [5].

� 1 2 ∇2 r !

> The influence of vibrational spectroscopy as an analytical tool in several fields is obvious. Qualitative association of bands and specific structures or chemical groups is basic interpretation of vibrational spectra. In contrast to nuclear magnetic resonance where nuclear spin is associated with one peak or multiplet is proved advantageous, united motion of all of the nuclei in sample is reason for observed bands in vibrational spectra. For N nuclei, there are at most 3 N-6 experiential fundamental bands and so far the matrix of internuclear force interactions, i.e., the second derivative matrix in harmonic approximation, has (3 N-6)(3 N -5)/2 exclusive terms. The extraction of force constants from vibrational frequencies is yet an undetermined mathematical problem as many bands are not actually observed; e.g., there are overtones, combination bands and nonconformities from the harmonic approximation; hence, the problem seems obstinate [13].

> For the ground state's potential energy surface computation, discovery of effectual codes delivers a possible answer to the problem. A technique that will precisely determine the bonding, as well as intermolecular interactions, will be helpful for calculating vibrational spectra. The ground state properties and potential energies are accurately calculated by DFT approaches; therefore, DFT excellently and proficiently calculates the vibrational spectra from first principles [7]. The absolute values of the frequencies are high in contrast to experimental values while using DFT models as they seem to characterize the bonding pretty well and comparison with experimental trends is possible. DFT methods give sufficiently high accuracy of normal mode calculations as the restrictions of the harmonic approximation are often a foremost cause of divergence between theory and experiment.

It is reasonable to outspread the approach by including molecular interactions after given achievement of the method employed to small molecules. There are two aspects that can be reasons for inconsistency, i.e., anharmonicity and hydrogen bonding, both result in a lower frequency for a specific normal mode than projected by harmonic approximation [14].

#### 2.1.1. Spectral generation via Gaussian broadening function

Computation of vibrational spectra of molecules in their ground and excited states can be done by using the Gaussian program. The program can also designate the dislocations of the molecule as it undergoes normal modes of vibrations along with prediction of spectral frequencies and intensities. The spectra can be produced from a list of frequencies and intensities employing a Gaussian broadening function as depicted in Eq. (2) [13]:

$$I(\nu) = \sum\_{k}^{N} \frac{A\_k}{\sqrt{2\pi\sigma}} \exp\left\{-\frac{(\upsilon - \upsilon\_0)^2}{2\sigma^2}\right\} \tag{2}$$

Hb el þ VNN � �Ψel ¼ UΨel (4)

Spectral Calculations with DFT

105

http://dx.doi.org/10.5772/intechopen.71080

i, j ¼ 1, 2,…::3N (5)

(6)

where Hb el is the one-electron Hamiltonian and VNN is the operator of potential energy between the nuclei of molecular system intended for numerous molecular geometries to discover

The second derivative of energy regarding the position of nuclei is factor on which molecular frequencies hinge on. Hartree-Fock theory, DFT-B3LYP and other approaches make available the analytic second derivative of energy. Input for frequency calculation is given by the optimized energy [16]. These derivatives are estimated at the equilibrium geometry with origin at the middle of mass by manipulating a set of second derivatives of molecular energy U respecting the 3 N nuclear Cartesian coordinates of a coordinate system as in Eq. (5) [15].

equilibrium geometry of the molecule, i.e., geometry optimization.

∂2U ∂Xi∂Xj � �

> Fy ¼ 1 mimj � �1=2

Then, set of 3 N linear equations in 3 N unknowns in Eq. (7) are being solved.

If coefficient of determinant vanishes, this set of consistent equations has a nontrivial explana-

This determinant is of order 3 N and on extension gives a polynomial whose highest power of λk

vk ¼ λ1=2 k

By solving Eq. (9), six of the λk values originated will be zero giving six frequencies with zero value, in correspondence to the three translational and three rotational degrees of freedom of

k , so the determinantal equation will yield 3 N roots (some of which may be the same) for λk.

det Fij � δijλk

Calculation of molecular harmonic vibrational frequencies is now done as follows [15]:

Fij � δijλk

where mi is mass of the nucleus in correspondence to coordinate Xi.

X 3N

j¼1

<IP_ADDRESS>. Calculating vibrational frequencies

tion as follows:

is λ3N

Mass-weighted force constant matrix elements give:

eq

∂2U ∂Xi∂Xj � �

eq

� �Ijk ¼ 0 i, j ¼ 1, 2, …::3N (7)

� � ¼ 0 (8)

2π (9)

For each of the N vibrational modes calculated, intensity of each band is Ak in km/mol.

#### 2.1.2. Calculating the vibrational frequencies

Study of a molecular system behind calculating the vibrational frequencies [7]. If springs are considered Hookean, for example, equations of motion can readily be solved when force is proportional to the displacement and we can find that vibrational frequencies are associated with force constants and masses of atoms. For instance, in a simple molecule like CO where there is only one spring, the frequency is calculated as in Eq. (3).

$$\nu = \frac{1}{2\pi} \sqrt{k/\mu} \tag{3}$$

where 1 μ ¼ 1 mc þ 1 ma and k is the spring constant. The value of k can be computed from DFT calculations at equilibrium state, bond length k ¼ ∂2E ∂x2 . It is essential to perform geometry optimization prior to frequency calculations as these calculations are effective only at fixed points on the potential energy surface.

#### <IP_ADDRESS>. Calculating second derivative of energy

Before calculating the second derivative of energy, the first step in quantum chemical calculations is optimization of molecular geometry. Vibrational frequency calculations are done after optimizing geometry by using the same method and basis set as used for calculating vibrational frequency. Model's development is constructed on spontaneously rational values for interbond angles, bond distances and dihedral angles in the absence of experimental data. It is usually done by solving the following Schrodinger wave equation [15]:

$$\left(\hat{H}\_{el} + V\_{\text{NN}}\right)\Psi\_{el} = \mathcal{U}\Psi\_{el} \tag{4}$$

where Hb el is the one-electron Hamiltonian and VNN is the operator of potential energy between the nuclei of molecular system intended for numerous molecular geometries to discover equilibrium geometry of the molecule, i.e., geometry optimization.

The second derivative of energy regarding the position of nuclei is factor on which molecular frequencies hinge on. Hartree-Fock theory, DFT-B3LYP and other approaches make available the analytic second derivative of energy. Input for frequency calculation is given by the optimized energy [16]. These derivatives are estimated at the equilibrium geometry with origin at the middle of mass by manipulating a set of second derivatives of molecular energy U respecting the 3 N nuclear Cartesian coordinates of a coordinate system as in Eq. (5) [15].

$$\left(\frac{\partial^2 U}{\partial \mathbf{X}\_i \partial \mathbf{X}\_j}\right)\_{eq} \text{ i, j = 1, 2, ..., 3N} \tag{5}$$

Mass-weighted force constant matrix elements give:

It is reasonable to outspread the approach by including molecular interactions after given achievement of the method employed to small molecules. There are two aspects that can be reasons for inconsistency, i.e., anharmonicity and hydrogen bonding, both result in a lower frequency for a specific normal mode than projected by harmonic approxima-

Computation of vibrational spectra of molecules in their ground and excited states can be done by using the Gaussian program. The program can also designate the dislocations of the molecule as it undergoes normal modes of vibrations along with prediction of spectral frequencies and intensities. The spectra can be produced from a list of frequencies and intensities employing a Gaussian broadening function as depicted in

2πσ p exp � ð Þ v � v0

2

(2)

(3)

2σ2 ( )

2.1.1. Spectral generation via Gaussian broadening function

104 Density Functional Calculations - Recent Progresses of Theory and Application

2.1.2. Calculating the vibrational frequencies

points on the potential energy surface.

<IP_ADDRESS>. Calculating second derivative of energy

Ið Þ¼ ν X N

there is only one spring, the frequency is calculated as in Eq. (3).

usually done by solving the following Schrodinger wave equation [15]:

calculations at equilibrium state, bond length k ¼ ∂2E

k

Ak ffiffiffiffiffiffiffiffi

For each of the N vibrational modes calculated, intensity of each band is Ak in km/mol.

ν ¼ 1 2π

optimization prior to frequency calculations as these calculations are effective only at fixed

Before calculating the second derivative of energy, the first step in quantum chemical calculations is optimization of molecular geometry. Vibrational frequency calculations are done after optimizing geometry by using the same method and basis set as used for calculating vibrational frequency. Model's development is constructed on spontaneously rational values for interbond angles, bond distances and dihedral angles in the absence of experimental data. It is

Study of a molecular system behind calculating the vibrational frequencies [7]. If springs are considered Hookean, for example, equations of motion can readily be solved when force is proportional to the displacement and we can find that vibrational frequencies are associated with force constants and masses of atoms. For instance, in a simple molecule like CO where

> ffiffiffiffiffiffiffiffi k=μ q

ma and k is the spring constant. The value of k can be computed from DFT

∂x2 . It is essential to perform geometry

tion [14].

Eq. (2) [13]:

where 1

μ ¼ 1 mc þ 1

$$F\_{\mathcal{Y}} = \frac{1}{\left(m\_i m\_j\right)^{1/2}} \left(\frac{\partial^2 U}{\partial X\_i \partial X\_j}\right)\_{eq} \tag{6}$$

where mi is mass of the nucleus in correspondence to coordinate Xi.

Then, set of 3 N linear equations in 3 N unknowns in Eq. (7) are being solved.

$$\sum\_{j=1}^{3N} \left( F\_{i\bar{j}} - \delta\_{i\bar{j}} \lambda\_k \right) I\_{\bar{j}k} = 0 \quad i, j = 1, 2, \dots \\ \text{3N} \tag{7}$$

If coefficient of determinant vanishes, this set of consistent equations has a nontrivial explanation as follows:

$$\det\left(F\_{\vec{\eta}} - \delta\_{\vec{\eta}}\lambda\_k\right) = 0 \tag{8}$$

This determinant is of order 3 N and on extension gives a polynomial whose highest power of λk is λ3N k , so the determinantal equation will yield 3 N roots (some of which may be the same) for λk.

#### <IP_ADDRESS>. Calculating vibrational frequencies

Calculation of molecular harmonic vibrational frequencies is now done as follows [15]:

$$
\sigma\_k = \frac{\lambda\_k^{1/2}}{2\pi} \tag{9}
$$

By solving Eq. (9), six of the λk values originated will be zero giving six frequencies with zero value, in correspondence to the three translational and three rotational degrees of freedom of molecule. In practice, one may find six vibrational frequencies with values almost zero because equilibrium geometry never originates with infinite precision. The remaining 36-N vibrational frequencies depict molecular-harmonic vibrational frequencies.

#### <IP_ADDRESS>. Zero-point energy calculations for maximum vibrations

Zero-point energy is defined as the sum over all the vibrational modes for a molecule with maximum number of vibrational modes and calculated as:

$$E\_{\rm ZPE} = \sum\_{i} \frac{1}{2} \hbar v\_i \tag{10}$$

ii. In Method section, select ground state (HF, Restricted or DFT Restricted and B3LYP according to required calculation) and select a basis set, Charge-0 and Spin-Singlet. Insert SCF = Tight in the Additional Keywords section and if your calculation is in vacuo, select

Spectral Calculations with DFT

107

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iii. Open the output file of the optimized structure in Gauss View. Verify that the optimization calculation has converged by checking that the maximum force, RMS force, maxi-

iv. Go to Results and select the 'Vibrations' option. View the IR spectrum by pressing the 'Spectrum' button and check out whether the number of modes are in accordance with theoretical calculations or not. Each vibrational mode can be visualized by highlighting it

NMR is based on the principle that the energy of a system containing nuclear or electron magnetic moments arising from the spin of a particle, in the existence of an external stationary magnetic field, depends upon the direction of the magnetic moment with respect to the external field. One can thus measure the energy difference for different directions of the electronic magnetic moment (electronic Zeeman effect) or of nuclear magnetic moment (nuclear Zeeman effect) by employing a suitable oscillating external magnetic field as a

NMR parameters are considered as quantities which are determined shifts, indirect spin-spin coupling constant and direct dipole-dipole coupling constants. This electronic structure in turn confidentially interrelated to local and global geometry and hence internal flexibility and intramolecular interactions influences NMR parameters. Chemical shifts along with spin-spin coupling constants found an average experimentally in comparison to those values belonging to all geometrical arrangements arising throughout the sequence of NMR experiment. Unfortunately, chemical shifts are generally found to be dependent on the internal dynamics or on the intermolecular interactions in various descriptions. Consequently, for obtaining structural information, most experimental NMR techniques imply coupling constants or the nuclear Overhauser effect instead of chemical shifts. However, ab initio calculations provide the understanding to structure-chemical shifts or spin-spin coupling constant associations, which can make the experimental data interpretation much easier in this sense. That is the reason that highly capable and steadfast computational calculations

As commonly noted that, when results of sufficient quality are attainable by employing various strategies based on wave function properties. These procedures are restricted to small- and medium-sized systems unfortunately. With the advancements of density functional methods, it is possible to acquire pertinent results even for larger molecules, such as fragments of proteins and nucleic acids [22] where the electron correlation effects are

mum displacement and RMS displacement parameters are all converged.

'None' in the Solvation dialog box. Submit the calculation.

in the 'Display Vibrations' table and pressing the 'Start' button.

3. DFT methods for nuclear magnetic resonance (NMR)

probe [20].

are in high demand [21].

#### 2.1.3. Normal coordinate analysis

Normal coordinate analysis gives comprehensive explanation of vibrational modes. It is termed a practice that calculates the vibrational frequencies involving observed frequencies of more preferably infrared and Raman harmonic frequencies to equilibrium geometry, force constants and atomic masses of an oscillating system. In assigning vibrational spectra, normal coordinate analysis proved beneficial, but reliable intramolecular force constants influence its predictive ability [17].

#### 2.1.4. B3LYP density functional method

DFT syndicates accuracy with computational rapidity and user-friendliness for investigating the ground state characteristics in sturdily bound systems. Consistent highly reliable hybrid DFT methods (discussed in Section 3.1) make them more proficient and endearing. B3LYP functional [18] (discussed in hybrid methods of Section 3.1) is the most extensively used among all hybrid density functional methods as it is considered to give most exact vibrational frequencies of compounds only if calculated frequencies are scaled by a uniform scaling factor. Scaling is done to compensate for all probable causes of inaccuracy produced owing to electronic structure method–related inaccuracies, e.g., basis set insufficiencies and estimated handling of electron-correlation and nuclear motion treatment inaccuracies [19].

#### 2.1.5. IR calculations in Gaussian software

Gaussian is a most employed computational chemistry software program, whereas Gauss View is an inexpensive full featured graphical user interface for Gaussian. One can submit inputs to Gaussian and can observe output graphically, which is usually produced by Gaussian software via using Gauss View. For IR spectrum calculations in Gaussian, the following are the steps:

i. After predicting the number of vibrational modes and expected regions for frequencies for molecule by theoretical calculations, build molecule in Gauss View. Go to Calculate from Gauss View toolbar and select Gaussian. In the Job Type dialog box, select Opt + Freq and optimize to a 'Minimum' Calculate Force Constants-'Never,' Compute Raman- 'Default,' deselect any other option.

#### 3. DFT methods for nuclear magnetic resonance (NMR)

molecule. In practice, one may find six vibrational frequencies with values almost zero because equilibrium geometry never originates with infinite precision. The remaining 36-N vibrational

Zero-point energy is defined as the sum over all the vibrational modes for a molecule with

Normal coordinate analysis gives comprehensive explanation of vibrational modes. It is termed a practice that calculates the vibrational frequencies involving observed frequencies of more preferably infrared and Raman harmonic frequencies to equilibrium geometry, force constants and atomic masses of an oscillating system. In assigning vibrational spectra, normal coordinate analysis proved beneficial, but reliable intramolecular force constants influence its

DFT syndicates accuracy with computational rapidity and user-friendliness for investigating the ground state characteristics in sturdily bound systems. Consistent highly reliable hybrid DFT methods (discussed in Section 3.1) make them more proficient and endearing. B3LYP functional [18] (discussed in hybrid methods of Section 3.1) is the most extensively used among all hybrid density functional methods as it is considered to give most exact vibrational frequencies of compounds only if calculated frequencies are scaled by a uniform scaling factor. Scaling is done to compensate for all probable causes of inaccuracy produced owing to electronic structure method–related inaccuracies, e.g., basis set insufficiencies and estimated

Gaussian is a most employed computational chemistry software program, whereas Gauss View is an inexpensive full featured graphical user interface for Gaussian. One can submit inputs to Gaussian and can observe output graphically, which is usually produced by Gaussian software via using Gauss View. For IR spectrum calculations in Gaussian, the following are

i. After predicting the number of vibrational modes and expected regions for frequencies for molecule by theoretical calculations, build molecule in Gauss View. Go to Calculate from Gauss View toolbar and select Gaussian. In the Job Type dialog box, select Opt + Freq and optimize to a 'Minimum' Calculate Force Constants-'Never,' Compute

handling of electron-correlation and nuclear motion treatment inaccuracies [19].

i 1 2

hvi (10)

EZPE ¼ X

frequencies depict molecular-harmonic vibrational frequencies.

<IP_ADDRESS>. Zero-point energy calculations for maximum vibrations

106 Density Functional Calculations - Recent Progresses of Theory and Application

maximum number of vibrational modes and calculated as:

2.1.3. Normal coordinate analysis

2.1.4. B3LYP density functional method

2.1.5. IR calculations in Gaussian software

Raman- 'Default,' deselect any other option.

the steps:

predictive ability [17].

NMR is based on the principle that the energy of a system containing nuclear or electron magnetic moments arising from the spin of a particle, in the existence of an external stationary magnetic field, depends upon the direction of the magnetic moment with respect to the external field. One can thus measure the energy difference for different directions of the electronic magnetic moment (electronic Zeeman effect) or of nuclear magnetic moment (nuclear Zeeman effect) by employing a suitable oscillating external magnetic field as a probe [20].

NMR parameters are considered as quantities which are determined shifts, indirect spin-spin coupling constant and direct dipole-dipole coupling constants. This electronic structure in turn confidentially interrelated to local and global geometry and hence internal flexibility and intramolecular interactions influences NMR parameters. Chemical shifts along with spin-spin coupling constants found an average experimentally in comparison to those values belonging to all geometrical arrangements arising throughout the sequence of NMR experiment. Unfortunately, chemical shifts are generally found to be dependent on the internal dynamics or on the intermolecular interactions in various descriptions. Consequently, for obtaining structural information, most experimental NMR techniques imply coupling constants or the nuclear Overhauser effect instead of chemical shifts. However, ab initio calculations provide the understanding to structure-chemical shifts or spin-spin coupling constant associations, which can make the experimental data interpretation much easier in this sense. That is the reason that highly capable and steadfast computational calculations are in high demand [21].

As commonly noted that, when results of sufficient quality are attainable by employing various strategies based on wave function properties. These procedures are restricted to small- and medium-sized systems unfortunately. With the advancements of density functional methods, it is possible to acquire pertinent results even for larger molecules, such as fragments of proteins and nucleic acids [22] where the electron correlation effects are indirectly accounted through the exchange-correlation functional. DFT-based NMR calculations have seen a rapid expansion during the last 10 years—that is perhaps best designated with the word explosion. Since the publication of these calculations, methods have already entered the standard repertoire of quantum chemistry in short time length. More comprehensive and technical as well as more general reviews are available as theoretical portrayal of NMR chemical shifts based on the more traditional ab initio procedures has seen a marvelous progress as well [23]. For NMR calculations, density functional theory (DFT) is recently proved to substitute the traditional Hartree-Fock (HF) and post-HF methods [24]. Inclusion of electron correlation effects in a very efficient way to calculate NMR parameters is a rather new field of application in DFT over the last years.

#### 3.1. Calculations of NMR parameters

Total energy E of an n-electron system is expressed exactly on the basis of DFT approach as in Eq. (11) [2].

$$E = \sum\_{i}^{n} \int d\vec{r} \Psi\_{i}^{\*} \left(\frac{p^{2}}{2} + V\_{N}\right) \Psi\_{i} \frac{1}{2} \int d\vec{r}^{\dagger} \vec{r}^{\dagger} \frac{\rho\left(\vec{r}^{\dagger}\right) \rho\left(\vec{r}^{\dagger}\right)}{|\vec{r}^{\dagger} - \vec{r}^{\dagger}|} + E\_{\text{XC}} \tag{11}$$

In Eq. (11), the f g Ψi is a set of n orthonormal one-electron functions, r = Pn i Ψ∗ i Ψiis electronic

density of the system, VN is the external (nuclear) potential, and p ! is momentum operator. Hence, the kinetic and potential energy of a model system with the same density is signified by the first integral but without incorporating electron-electron communication. The second term represents itself as coulomb interaction of electron density. The XC energy and E proper (EXC) are functionals of the density although exact functional arrangement for EXC is not known (defined through Eq. (11)). Any application of DFT is mainly concerned with the assessment of numerous approximations; hence, Kohn-Sham (KS) equations are typically derived from Eq. (11) [4, 25].

$$
\hbar\_{\rm KS} \Psi\_i = \varepsilon\_i \Psi\_i \tag{12}
$$

where A !

3.1.1. Exchange-correlation functional

approximations used for Exc are as follows:

<IP_ADDRESS>. The local density approximation (LDA)

<IP_ADDRESS>. The generalized gradient approximations (GGA)

split into its exchange and correlation parts as:

EGGA XC rα; rβ h i ¼

Exc in turn written as:

Eq. (15). The ELDA

be generically inscribed as:

is the vector potential of the field. Though this is not the entire division in DFT. Rather,

h i !

!

rð Þr Exc½ � r dr (16)

� �dr (18)

C (19)

XC is also typically

C (17)

depicted in Eq. (14) [27].

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http://dx.doi.org/10.5772/intechopen.71080

(15)

109

the XC energy (EXC) becomes a relativistic for current density functional, which interprets in

EXC½ � p ����! substitute EXC p; j

Quality of the description centers exclusively on precision of the approximation to Exc as mentioned earlier that obvious form of the exchange-correlation functional (Exc) is not identified yet. Unfortunately, there is no systematic mode of improving exchange-correlation functionals although the pursuit for improved and better functionals is at the very core of DFT. Prevailing XC functionals can be roughly categorized into three distinct groups [28] as several

The idea of a nonvariating electron gas laid the foundation of this model, which approximates

ð

Exc½ � r is the exchange-correlation energy per particle of a uniform electron gas of density r(r) in

XC ½ � r can be correctly split into exchange and correlation parts as:

The generalized gradient approximations (GGA) for Exc are not lone functions of local density r(r) but also functions of the gradient of charge density ∇r(r) in comparison to LDA. They can

X þ ELDA

f rα; rβ; Δrα; Δrβ

X þ EGGA

XC ½ �¼ r ELDA

ð

XC ½ �¼ r EGGA

The GGA exchange functional's representatives are functionals by Becke, 1988 (B or B88) [18] and by Perdew and Wang, 1986 (P or PW86) [29] and are the most familiar ones. Perdew and Wang also established the PW91 exchange-correlation functional [30] comprising exchange

ELDA XC ½ �¼ r

ELDA

α and β denote to "up" and "down" spin, respectively, in Eq. (17). Now, EGGA

EGGA

nonrelativistic system to the electron density r and the current density j

$$h\_{KS} = \frac{p^2}{2} + V\_{KS} = \frac{p^2}{2} + V\_N + \int d\vec{r} \frac{\rho\left(\vec{r'}\right)}{|\vec{r'} - \vec{r'}|} + V\_{XC} \tag{13}$$

The XC potential VXC is the functional derivative of the XC energy EXC with respect to the density, r. The inclusion of magnetic fields is another extension, which can be introduced. Obviously, this is essential to all the properties to be introduced. The magnetic field B ! is most accessibly acquaint with supposed minimal coupling as in Eq. (13) [26].

$$
\overrightarrow{p'} = \overrightarrow{p'} + \overrightarrow{A} \,/\circledast \tag{14}
$$

where A ! is the vector potential of the field. Though this is not the entire division in DFT. Rather, the XC energy (EXC) becomes a relativistic for current density functional, which interprets in nonrelativistic system to the electron density r and the current density j ! depicted in Eq. (14) [27].

$$E\_{\rm XC}[p] \xrightarrow{substrate} E\_{\rm XC}\left[p, \overrightarrow{j}^{\cdot}\right] \tag{15}$$

#### 3.1.1. Exchange-correlation functional

indirectly accounted through the exchange-correlation functional. DFT-based NMR calculations have seen a rapid expansion during the last 10 years—that is perhaps best designated with the word explosion. Since the publication of these calculations, methods have already entered the standard repertoire of quantum chemistry in short time length. More comprehensive and technical as well as more general reviews are available as theoretical portrayal of NMR chemical shifts based on the more traditional ab initio procedures has seen a marvelous progress as well [23]. For NMR calculations, density functional theory (DFT) is recently proved to substitute the traditional Hartree-Fock (HF) and post-HF methods [24]. Inclusion of electron correlation effects in a very efficient way to calculate NMR parameters is a rather

Total energy E of an n-electron system is expressed exactly on the basis of DFT approach as in

Ψi 1 2 ð dr1 ! r 2 ! r r 1 !� �

Hence, the kinetic and potential energy of a model system with the same density is signified by the first integral but without incorporating electron-electron communication. The second term represents itself as coulomb interaction of electron density. The XC energy and E proper (EXC) are functionals of the density although exact functional arrangement for EXC is not known (defined through Eq. (11)). Any application of DFT is mainly concerned with the assessment of numerous approximations; hence, Kohn-Sham (KS) equations are typically derived from

2 þ VN þ

The XC potential VXC is the functional derivative of the XC energy EXC with respect to the density, r. The inclusion of magnetic fields is another extension, which can be introduced.

> ! þ A !

Obviously, this is essential to all the properties to be introduced.  (13) [26].

r r 2 !� �

hKSΨi ¼ εiΨi (12)

þ EXC (11)

! is momentum operator.

þ VXC (13)

=c (14)

!

is most

i Ψiis electronic

i Ψ∗

∣r 1 ! � r 2 ! ∣

new field of application in DFT over the last years.

108 Density Functional Calculations - Recent Progresses of Theory and Application

3.1. Calculations of NMR parameters

E ¼ Xn i

ð d r !Ψ∗ i p2 2 þ VN � �

hKS ¼ p2

Eq. (11) [2].

Eq. (11) [4, 25].

Quality of the description centers exclusively on precision of the approximation to Exc as mentioned earlier that obvious form of the exchange-correlation functional (Exc) is not identified yet. Unfortunately, there is no systematic mode of improving exchange-correlation functionals although the pursuit for improved and better functionals is at the very core of DFT. Prevailing XC functionals can be roughly categorized into three distinct groups [28] as several approximations used for Exc are as follows:

#### <IP_ADDRESS>. The local density approximation (LDA)

The idea of a nonvariating electron gas laid the foundation of this model, which approximates Exc in turn written as:

$$E\_{\rm XC}^{LDA}[\rho] = \int \rho(r) \epsilon\_{\rm xc}[\rho] dr \tag{16}$$

Exc½ � r is the exchange-correlation energy per particle of a uniform electron gas of density r(r) in Eq. (15). The ELDA XC ½ � r can be correctly split into exchange and correlation parts as:

$$E\_{X\mathbb{C}}^{LDA}[\rho] = E\_X^{LDA} + E\_{\mathbb{C}}^{LDA} \tag{17}$$

#### <IP_ADDRESS>. The generalized gradient approximations (GGA)

The generalized gradient approximations (GGA) for Exc are not lone functions of local density r(r) but also functions of the gradient of charge density ∇r(r) in comparison to LDA. They can be generically inscribed as:

$$E\_{\rm XC}^{\rm GGA}\left[\rho\_a, \rho\_\beta\right] = \int \left(\rho\_a, \rho\_\beta, \Delta\rho\_a, \Delta\rho\_\beta\right) dr\tag{18}$$

α and β denote to "up" and "down" spin, respectively, in Eq. (17). Now, EGGA XC is also typically split into its exchange and correlation parts as:

$$E\_{\rm XC}^{\rm GGA}[\rho] = E\_{\rm X}^{\rm GGA} + E\_{\rm \mathcal{C}}^{\rm GGA} \tag{19}$$

The GGA exchange functional's representatives are functionals by Becke, 1988 (B or B88) [18] and by Perdew and Wang, 1986 (P or PW86) [29] and are the most familiar ones. Perdew and Wang also established the PW91 exchange-correlation functional [30] comprising exchange and correlation contributions; if used separately, both functionals are symbolized by PW91. The exchange fragment of PW91 is analogous to B88 and the correlation part is a modified form of P86. Currently, possibly the most prevalent correlation functional accounts to contributions of Lee et al. (LYP) [31]. Because of presence of some indigenous components, it differs from the other GGA functionals.

#### <IP_ADDRESS>. The hybrid methods

Exchange-correlation functionals in which exchange part is composed of particular Hartree-Fock exchange and pure density functionals for exchange are called hybrid functionals. These functionals consist of a mixture of Hartree-Fock exchange with DFT exchange and correlation in other words as depicted in Eq. (19).

$$E\_{\rm XC}^{hydrid} = \mathcal{c}^{HF} E\_{\rm X}^{HF} + \mathcal{c}^{DFT} E\_{\rm C}^{DFT} \tag{20}$$

for localized orbitals (DFTIGLO) method [9] and gauge including atomic orbitals (GIAO)

One appraises matrix fundamentals of the Hamiltonian in terms of a basis of fielddependent atomic orbitals (AO) in the GIAO method. By insertion of an intricate phase factor denoting the position of basis function, which is usually nucleus, the basic functions are made obviously dependent on the magnetic field. Such orbitals are termed as London atomic orbitals (LAO) or the gauge including atomic orbitals (GIAO). Matrix elements involved in GIAOs only differentiate in the vector potentials is the essence of idea, in that way entirely eradicating the reference to a complete gauge origin. Employment of field-dependent GIAOs as basic functions [33] is possibly the best solution to the gauge

> i 2c B ! xR ! a :r ! χa r

function that is adjusted at position. The field-dependent prefactor Eq. (21) guarantees that only differences of position vectors give the impression in expectation values. This eradicates any origin dependence, even for estimated MOs and finite basis sets. For showing the complete basis set limit, correspondence of the GIAO and the simpler common gauge methods to the NMR shielding are depicted. Nucleus-attached basis functions in geometry optimization processes have been equated with specific AOs (Eq. (21)) by assigning a field-dependent phase

> ! !

tion of Eq. (21). The absolute shielding σ is interrelated to the more acquainted chemical shift δ

where σref is the absolute shielding of reference compound (e.g., tetramethylsilane [TMS] for

H, 13C and 29Si NMR). Opposite sign between δ and σ can be noted, whereas GIAO shielding

division is not exceptional as only the total shielding is an apparent quantity, but for the GIAO

has been divided into its paramagnetic and diamagnetic fragments.  (22)

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111

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! is customary field-free basis

) can be derived from the GIAO formula-

(24)

δ ¼ σref � σ (23)

χa B ! ; r ! ¼ exp

Working equations for the NMR shielding tensor (σ

approach [32].

problem.

where R !

factor to them.

as follows [33]:

tensor is specified in DFT as [34]:

method, it has been defined exclusively.

1

where σ ! !

a and χa B

! ; r

3.2.2. The GIAO method

#### <IP_ADDRESS>. Becke's three-parameter hybrid functional (B3LYP) method

An illustrative example of the above-mentioned hybrid representations is Becke's threeparameter hybrid functional (B3), depicted in Eq. (20). [18], and exchange energy is being calculated by using this (B3) functional. If correlation functional LYP or PW91 is assumed for EGGA C , B3LYP or B3PW91 will be obtained, respectively. The earlier mentioned is most widely used exchange-correlation functional nowadays, and in many research domains throughout, it was also rigorously active. Since GGAs are better in comparison to LSDA, they could also employ a functional form like this (B3) as follows:

$$E\_{\rm XC}^{\rm B3} = (1 - a)E\_X^{\rm LSDA} + aE\_X^{\rm HF} + b\Delta E\_X^{\rm B88} + (1 - c)E\_\mathcal{C}^{\rm LSDA} + cE\_\mathcal{c}^{\rm GGA} \tag{21}$$

B3LYP uses this formulation, with LYP for ΔEGGA c .

#### 3.2. Shielding tensor calculations

The most basic DFT calculations of NMR chemical shifts were done years ago in which very small basis sets were employed and inappropriate approximation to the exchange-correlation (XC) functional was done due to which point of practical applicability was lost. Also in any handling of magnetic fields, another central exertion that surfaces is the so-called gauge problem [28] be it in DFT or otherwise.

#### 3.2.1. Solving gauge problem

The gauge requirement should disappear just as any anticipation value together with NMR possessions that can only hinge on the values of observable quantities. Certainly, this is the case for particular solutions of, for example, the KS equations, Eq. (12a, b) where large (immeasurable) basis sets are employed. A strong dependence on the choice of gauge is of main concern for approximate solutions with smaller (infinite) basis sets. Current applications of DFT for calculating NMR chemical shifts have used density functional theory individual gauge for localized orbitals (DFTIGLO) method [9] and gauge including atomic orbitals (GIAO) approach [32].

#### 3.2.2. The GIAO method

and correlation contributions; if used separately, both functionals are symbolized by PW91. The exchange fragment of PW91 is analogous to B88 and the correlation part is a modified form of P86. Currently, possibly the most prevalent correlation functional accounts to contributions of Lee et al. (LYP) [31]. Because of presence of some indigenous components, it differs

Exchange-correlation functionals in which exchange part is composed of particular Hartree-Fock exchange and pure density functionals for exchange are called hybrid functionals. These functionals consist of a mixture of Hartree-Fock exchange with DFT exchange and correlation

> HFEHF X þ c

An illustrative example of the above-mentioned hybrid representations is Becke's threeparameter hybrid functional (B3), depicted in Eq. (20). [18], and exchange energy is being calculated by using this (B3) functional. If correlation functional LYP or PW91 is assumed for

C , B3LYP or B3PW91 will be obtained, respectively. The earlier mentioned is most widely used exchange-correlation functional nowadays, and in many research domains throughout, it was also rigorously active. Since GGAs are better in comparison to LSDA, they could also

X þ bΔEB88

c .

The most basic DFT calculations of NMR chemical shifts were done years ago in which very small basis sets were employed and inappropriate approximation to the exchange-correlation (XC) functional was done due to which point of practical applicability was lost. Also in any handling of magnetic fields, another central exertion that surfaces is the so-called gauge

The gauge requirement should disappear just as any anticipation value together with NMR possessions that can only hinge on the values of observable quantities. Certainly, this is the case for particular solutions of, for example, the KS equations, Eq. (12a, b) where large (immeasurable) basis sets are employed. A strong dependence on the choice of gauge is of main concern for approximate solutions with smaller (infinite) basis sets. Current applications of DFT for calculating NMR chemical shifts have used density functional theory individual gauge

DFTEDFT

X þ ð Þ 1 � c ELSDA

C þ cEGGA

c (21)

C (20)

Ehybrid XC ¼ c

X þ aEHF

<IP_ADDRESS>. Becke's three-parameter hybrid functional (B3LYP) method

110 Density Functional Calculations - Recent Progresses of Theory and Application

employ a functional form like this (B3) as follows:

B3LYP uses this formulation, with LYP for ΔEGGA

XC ¼ ð Þ 1 � a ELSDA

EB3

3.2. Shielding tensor calculations

problem [28] be it in DFT or otherwise.

3.2.1. Solving gauge problem

from the other GGA functionals.

in other words as depicted in Eq. (19).

<IP_ADDRESS>. The hybrid methods

EGGA

One appraises matrix fundamentals of the Hamiltonian in terms of a basis of fielddependent atomic orbitals (AO) in the GIAO method. By insertion of an intricate phase factor denoting the position of basis function, which is usually nucleus, the basic functions are made obviously dependent on the magnetic field. Such orbitals are termed as London atomic orbitals (LAO) or the gauge including atomic orbitals (GIAO). Matrix elements involved in GIAOs only differentiate in the vector potentials is the essence of idea, in that way entirely eradicating the reference to a complete gauge origin. Employment of field-dependent GIAOs as basic functions [33] is possibly the best solution to the gauge problem.

$$\chi\_a \left( \overrightarrow{B} \; , \overrightarrow{r} \right) = \exp \left[ \frac{i}{2c} \left( \overrightarrow{B} \mathbf{x} \overrightarrow{R}\_a \right) . \overrightarrow{r} \right] \chi\_a \left( \overrightarrow{r} \right) \tag{22}$$

where R ! a and χa B ! ; r ! are the conforming GIAO and χa r ! is customary field-free basis function that is adjusted at position. The field-dependent prefactor Eq. (21) guarantees that only differences of position vectors give the impression in expectation values. This eradicates any origin dependence, even for estimated MOs and finite basis sets. For showing the complete basis set limit, correspondence of the GIAO and the simpler common gauge methods to the NMR shielding are depicted. Nucleus-attached basis functions in geometry optimization processes have been equated with specific AOs (Eq. (21)) by assigning a field-dependent phase factor to them.

Working equations for the NMR shielding tensor (σ ! ! ) can be derived from the GIAO formulation of Eq. (21). The absolute shielding σ is interrelated to the more acquainted chemical shift δ as follows [33]:

$$
\delta = \sigma\_{\text{ref}} - \sigma \tag{23}
$$

where σref is the absolute shielding of reference compound (e.g., tetramethylsilane [TMS] for 1 H, 13C and 29Si NMR). Opposite sign between δ and σ can be noted, whereas GIAO shielding tensor is specified in DFT as [34]:

$$
\stackrel{\rightarrow}{\vec{\sigma}}\_{} = \stackrel{\rightarrow}{\sigma^p}\_{} + \stackrel{\rightarrow}{\sigma^p}\_{} \tag{24}
$$

where σ ! ! has been divided into its paramagnetic and diamagnetic fragments. Generally, this division is not exceptional as only the total shielding is an apparent quantity, but for the GIAO method, it has been defined exclusively.

#### 3.2.3. The IGLO method

The use of disseminated gauge origins was depicted by the individual gauge for localized orbitals method (IGLO) [9], which was the first to empower the organized learning of nuclear shielding in larger systems. Evaluation of shielding tensor in terms of localized molecular orbitals (MO) is the idea of approach. To minimalize the absolute value of paramagnetic involvement, individual gauge origins of shielding tensor are selected. In the IGLO method, local phase factors are involved to molecular orbital in comparison to the GIAO approach where a phase factor is associated to each atomic orbital. In computational analysis point of view, IGLO approach is less challenging in contrast to the GIAO method since the use of distinct approximations of derivative two-electron integrals by this approach is most probable. By using localized orbitals, previously mentioned approximations are empowered; hence, it is important to select an appropriate basis set in an IGLO calculation [35]. According to IGLO approach, it is possible to allot comparable exponential prefactors to other objects, e.g., to localized MOs as a replacement for GIAO method. In IGLO method, specific integrals are easier to estimate analytically. Gauge difficulty has been resolute recently by retaining numerical integration or by techniques that were used in geometry optimization measures.

The portrayal of calculations follows Ref. [37]. By meaning, the nuclear spin-spin coupling tensor JMNuv is the second derivative of the total energy of the system with respect to the spins

IMu¼INv¼0

The orientationally averaged value of the nuclear spin-spin coupling tensor (JMNuv ) of nuclear

Since calculations for TMS are encompassed in many software, DFT-B3LYP-GIAO is the 6-311 + G(2d.p) [38], a suitable basis set, which easily determines the chemical shifts relative to TMS. The use of smaller 6-31 + G(d,p) basis set for calculations is most suitable for students compared to larger basis sets such as the 6-311 + G(2d,p) or even the very large 6-311++G (3df,3pd) basis sets that are employed for research purposes on small molecules. Many of the movements necessitate much less than 3 hours fixed for a standard research laboratory session even while using the larger basic sets. First of all, in this method, the structure is optimized in all the cases, and then to establish that either optimized structure is at least a local minimal or not, vibrational frequencies are determined. After the complete optimization of the structure, shielding constants are calculated using the GIAO method. Calculations for chemical shifts are carried out by subtracting the value of the screening constant (σ), which is calculated from the

value of the screening constant calculated for TMS using the same level of theory [38].

Basic considerations in NMR calculations are as follows: first, how many types of protons are there in molecule and which are they and second, how many NMR signals are expected to see and in which regions. In the case of an NMR spectrum calculation, geometry optimization calculation is needed first of all and then optimized structure is used to perform NMR spec-

i. Follow the calculations in Section 2.1.5 till step i for geometry optimization calculations and then follow steps used to perform an HNMR spectrum calculation of the optimized

ii. Press the Calculate button in the Gauss View toolbar and select Gaussian. In the Job Type dialog box, select 'NMR' and 'GIAO method. Submit the calculation after checking all

parameters. Open the output file and select the NMR option from Results.

u, v ¼ f g x; y; z (25)

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113

3 JMNxx þ JMNyy þ JMNzz (26)

δ ¼ σTMS\_σCalculated (27)

of the nuclei M and N as depicted in Eq. (24).

JMNuv ¼ ∂2E Ið Þ Mu:INv ð Þ δIMu:δINv

spin-spin coupling constant ð Þ JMN is the focus of interest in many cases.

JMN ¼ 1

3.4. Chemical shift calculation from DFT-B3LYP-GIAO method

3.5. NMR calculation steps using Gaussian software

trum calculation using the same method and basis set.

structure with the same method and basis set.

#### 3.3. Nuclear spin-spin coupling constant calculations

Highly appropriate ab initio calculations of spin-spin coupling constants are still very infrequent and usually deal with the simplest molecules only in comparison to excessive advancement in concept and computer codes for shielding tensor scheming. Shielding tensors are the reasons behind sensitivity of nuclear spin spin coupling constants to the correlation effects and basis set quality [36]. There are four significant contributions which causes other problems to the nuclear spin-spin coupling constants, are enlisted as follows:

HF calculations of the spin-spin coupling constants of large molecules lead to deprived contract with experimental numbers and further empirical scaling has to be done. On the other hand, the use of post–Hartree-Fock methods and prolonged basis sets for demanding handlings of all the four contributions is very inefficient and time-consuming.

In DFT, only a single paper by Fukui [23] published years ago was worthy initially, where the blend of finite perturbation theory (FPT) and the LCAO-Xa method was employed and only calculations of the FC contribution to the spin-spin coupling constant were carried out, which gave somewhat poor results likely as too poor basis set in combination with the Xa potential was used. As in Ref. [37], a new method to calculate the nuclear spin-spin coupling tensor using DFT procedure is developed sideways to the shielding tensor calculations in accordance. The portrayal of calculations follows Ref. [37]. By meaning, the nuclear spin-spin coupling tensor JMNuv is the second derivative of the total energy of the system with respect to the spins of the nuclei M and N as depicted in Eq. (24).

$$J\_{\rm MN\_{uv}} = \left[\frac{\partial^2 E(I\_{\rm Mu}, I\_{\rm Nv})}{(\delta I\_{\rm Mu}, \delta I\_{\rm Nv})}\right]\_{I\_{\rm Mu} = I\_{\rm Nv} = 0} \quad \text{u/v} = \{\mathbf{x}, y, \mathbf{z}\} \tag{25}$$

The orientationally averaged value of the nuclear spin-spin coupling tensor (JMNuv ) of nuclear spin-spin coupling constant ð Þ JMN is the focus of interest in many cases.

$$J\_{\rm MN} = \frac{1}{3} \left( J\_{\rm MN\_{xx}} + J\_{\rm MN\_{yy}} + J\_{\rm MN\_{zz}} \right) \tag{26}$$

#### 3.4. Chemical shift calculation from DFT-B3LYP-GIAO method

3.2.3. The IGLO method

optimization measures.

a. Fermi contact (FC),

c. Spin dipolar (SD)

b. Paramagnetic spin orbit (PSO)

d. Diamagnetic spin orbit (DSO)

3.3. Nuclear spin-spin coupling constant calculations

112 Density Functional Calculations - Recent Progresses of Theory and Application

the nuclear spin-spin coupling constants, are enlisted as follows:

dlings of all the four contributions is very inefficient and time-consuming.

The use of disseminated gauge origins was depicted by the individual gauge for localized orbitals method (IGLO) [9], which was the first to empower the organized learning of nuclear shielding in larger systems. Evaluation of shielding tensor in terms of localized molecular orbitals (MO) is the idea of approach. To minimalize the absolute value of paramagnetic involvement, individual gauge origins of shielding tensor are selected. In the IGLO method, local phase factors are involved to molecular orbital in comparison to the GIAO approach where a phase factor is associated to each atomic orbital. In computational analysis point of view, IGLO approach is less challenging in contrast to the GIAO method since the use of distinct approximations of derivative two-electron integrals by this approach is most probable. By using localized orbitals, previously mentioned approximations are empowered; hence, it is important to select an appropriate basis set in an IGLO calculation [35]. According to IGLO approach, it is possible to allot comparable exponential prefactors to other objects, e.g., to localized MOs as a replacement for GIAO method. In IGLO method, specific integrals are easier to estimate analytically. Gauge difficulty has been resolute recently by retaining numerical integration or by techniques that were used in geometry

Highly appropriate ab initio calculations of spin-spin coupling constants are still very infrequent and usually deal with the simplest molecules only in comparison to excessive advancement in concept and computer codes for shielding tensor scheming. Shielding tensors are the reasons behind sensitivity of nuclear spin spin coupling constants to the correlation effects and basis set quality [36]. There are four significant contributions which causes other problems to

HF calculations of the spin-spin coupling constants of large molecules lead to deprived contract with experimental numbers and further empirical scaling has to be done. On the other hand, the use of post–Hartree-Fock methods and prolonged basis sets for demanding han-

In DFT, only a single paper by Fukui [23] published years ago was worthy initially, where the blend of finite perturbation theory (FPT) and the LCAO-Xa method was employed and only calculations of the FC contribution to the spin-spin coupling constant were carried out, which gave somewhat poor results likely as too poor basis set in combination with the Xa potential was used. As in Ref. [37], a new method to calculate the nuclear spin-spin coupling tensor using DFT procedure is developed sideways to the shielding tensor calculations in accordance. Since calculations for TMS are encompassed in many software, DFT-B3LYP-GIAO is the 6-311 + G(2d.p) [38], a suitable basis set, which easily determines the chemical shifts relative to TMS. The use of smaller 6-31 + G(d,p) basis set for calculations is most suitable for students compared to larger basis sets such as the 6-311 + G(2d,p) or even the very large 6-311++G (3df,3pd) basis sets that are employed for research purposes on small molecules. Many of the movements necessitate much less than 3 hours fixed for a standard research laboratory session even while using the larger basic sets. First of all, in this method, the structure is optimized in all the cases, and then to establish that either optimized structure is at least a local minimal or not, vibrational frequencies are determined. After the complete optimization of the structure, shielding constants are calculated using the GIAO method. Calculations for chemical shifts are carried out by subtracting the value of the screening constant (σ), which is calculated from the value of the screening constant calculated for TMS using the same level of theory [38].

$$
\delta = \sigma\_{\text{TMS\\_off\\_calculated}} \tag{27}
$$

#### 3.5. NMR calculation steps using Gaussian software

Basic considerations in NMR calculations are as follows: first, how many types of protons are there in molecule and which are they and second, how many NMR signals are expected to see and in which regions. In the case of an NMR spectrum calculation, geometry optimization calculation is needed first of all and then optimized structure is used to perform NMR spectrum calculation using the same method and basis set.

iii. In the 'SCF GIAO magnetic shielding dialog box, select 'H' option in Element drop-down list, and in the Reference list, select the calculation method earlier used. The number of peaks of magnetic shielding observed is noted. Infer calculated spectrum and compare it with theoretical calculations. With this, multiplicity of the peaks, temperature at which NMR experiment is simulated and width of the NMR peaks can be discussed.

determination of electronic excitation spectrum in typical dipole approximation [41]. The poles' strengths are provided by oscillator strengths (fi) or by transition dipole moment (μi)

Electronic absorption spectra calculation is foremost applicability of TD-DFT after the geometry got optimized by using linear response theory. First, the Hohenberg-Kohn theorem statement that dynamic linear response of a system of N-electrons is determined by the groundstate charge density was discussed earlier but no applied mean was found to build this response. A productive procedure was provided by TD-DFT from which absorption spectra

Step 1: The calculations comprised of promulgating Kohn-Sham orbital incidence of a trivial

Step 2: Then induced dipole moment, μinduced(t), is calculated by Eq. (28), which depicts the

calculations of dynamic polarizability in Eq. (29) are carried out after applying Fourier trans-

ð

f I ω2

α t � t

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vappliedð Þ¼ r; t εð Þt :r (28)

α ωð Þ¼ μ ωð Þ=Eð Þ ω (30)

rn rð Þ :t rd (29)

I � ω2 (31)

Imα ωð Þ þ iη (32)

0 ð Þεð Þt :Final

dynamic perturbation, e.g., considering system being perturbed by Eq. (27).

difference among time-dependent and permanent dipole moments of a system.

Step 3: Dynamic polarizability, α, is expressed by linear term, μinducedðÞ¼ t Ð

Step 4: Sum-over-states (SOS) form of dynamic polarizability has the form:

μinducedðÞ¼� t e

α ωð Þ¼ X

Sð Þ¼ ω

I6¼0

where ωI and fI are corresponding vertical excitation energies and conforming oscillator strength. Finally, spectrum was obtained employing Eq. (31) also termed as Lorentzian broadened stick

> 2ω π

Such a procedure has been executed in many computer codes, e.g., in Octopus code [43], and has benefited that absorption spectrum of a very huge molecule over a wide range of energies can be calculated even though using only reasonable spectral resolution. This function parallels to optical absorption spectrum, e.g., calculating spectra of benzene [44] by TD-DFT

consistently also termed as intensity of the optical transitions.

4.1. Absorption spectra calculation

can be calculated by following steps [42]:

form convolution theorem.

spectrum:

approach depicted in Figure 1.

iv. Open the log file and search for or scroll to the line "SCF GIAO Magnetic Shielding tensor (ppm)." For simple NMR experiments, the signals are found at the chemical shifts, which appear after the heading "isotropic." All other numbers relate to the directionality of the NMR signals.

### 4. Time-dependent density functional theory (TD-DFT) for UV-vis spectra

As classic DFT is concerned with ground stationary state and applications, i.e., UV-vis spectroscopy, photochemistry, NLO and others, comprise either electronic excited states or timedependent electronic characteristics, which are in agreement with time-dependent density functional theory (TD-DFT). From classic paper of Runge and Gross [39], formal TD-DFT is traced back, which strained to firm up former efforts on the same topic. Therefore, Runge-Gross TD-DFT is two decades younger than Hohenberg Kohn-Sham theory [2], which is about stationary ground state as mentioned earlier. Four theorems proposed in the Runge-Gross paper. First, Runge-Gross theorem states that external potential up to an additive function of time is regulated by time-dependent charge density, r(r,t), in cooperation with preliminary wave function (Ψ0).

Various chemical and physical molecular properties provide basis for electronic spectra. Different chemical and physical effects can easily be computationally investigated by modifying the spectral characteristics of molecules as many stimulating chemical problems are included in both ground and excited states of molecules. As for ground state properties of atoms, molecules and solids, DFT gave an effective explanation; hence, in order to designate photochemical and photophysical procedures, DFT formalism has to be expanded to excited states. So, for excited state calculations, time-dependent DFT (TD-DFT) is established as an operative tool [40] as it provides first principle technique for calculating excitation energies and various response-related characteristics in density functional outline.

There are classically one or more low energy excited states for a molecule that can be designated as valence-MO-valence-MO single electronic excitations and replicated in spectra. Energies of excited determinants are essential for multiplet energy calculation such as employing spin-orbital number (i) to the empty spin-orbital number (j). Accordingly, specific states are denoted as π ! π\* , n ! π\* transitions. Relative easiness of character sometimes smoothes the process of concluding wave functions for these states. The calculation of the dynamic answer of charge density proposes a rigorous direction to time-dependent simplification of DFT formalism. The poles of dynamic polarizability regulate excitation energies permitting the determination of electronic excitation spectrum in typical dipole approximation [41]. The poles' strengths are provided by oscillator strengths (fi) or by transition dipole moment (μi) consistently also termed as intensity of the optical transitions.

#### 4.1. Absorption spectra calculation

iii. In the 'SCF GIAO magnetic shielding dialog box, select 'H' option in Element drop-down list, and in the Reference list, select the calculation method earlier used. The number of peaks of magnetic shielding observed is noted. Infer calculated spectrum and compare it with theoretical calculations. With this, multiplicity of the peaks, temperature at which

NMR experiment is simulated and width of the NMR peaks can be discussed.

4. Time-dependent density functional theory (TD-DFT)

114 Density Functional Calculations - Recent Progresses of Theory and Application

response-related characteristics in density functional outline.

of the NMR signals.

for UV-vis spectra

wave function (Ψ0).

denoted as π ! π\*

iv. Open the log file and search for or scroll to the line "SCF GIAO Magnetic Shielding tensor (ppm)." For simple NMR experiments, the signals are found at the chemical shifts, which appear after the heading "isotropic." All other numbers relate to the directionality

As classic DFT is concerned with ground stationary state and applications, i.e., UV-vis spectroscopy, photochemistry, NLO and others, comprise either electronic excited states or timedependent electronic characteristics, which are in agreement with time-dependent density functional theory (TD-DFT). From classic paper of Runge and Gross [39], formal TD-DFT is traced back, which strained to firm up former efforts on the same topic. Therefore, Runge-Gross TD-DFT is two decades younger than Hohenberg Kohn-Sham theory [2], which is about stationary ground state as mentioned earlier. Four theorems proposed in the Runge-Gross paper. First, Runge-Gross theorem states that external potential up to an additive function of time is regulated by time-dependent charge density, r(r,t), in cooperation with preliminary

Various chemical and physical molecular properties provide basis for electronic spectra. Different chemical and physical effects can easily be computationally investigated by modifying the spectral characteristics of molecules as many stimulating chemical problems are included in both ground and excited states of molecules. As for ground state properties of atoms, molecules and solids, DFT gave an effective explanation; hence, in order to designate photochemical and photophysical procedures, DFT formalism has to be expanded to excited states. So, for excited state calculations, time-dependent DFT (TD-DFT) is established as an operative tool [40] as it provides first principle technique for calculating excitation energies and various

There are classically one or more low energy excited states for a molecule that can be designated as valence-MO-valence-MO single electronic excitations and replicated in spectra. Energies of excited determinants are essential for multiplet energy calculation such as employing spin-orbital number (i) to the empty spin-orbital number (j). Accordingly, specific states are

process of concluding wave functions for these states. The calculation of the dynamic answer of charge density proposes a rigorous direction to time-dependent simplification of DFT formalism. The poles of dynamic polarizability regulate excitation energies permitting the

, n ! π\* transitions. Relative easiness of character sometimes smoothes the

Electronic absorption spectra calculation is foremost applicability of TD-DFT after the geometry got optimized by using linear response theory. First, the Hohenberg-Kohn theorem statement that dynamic linear response of a system of N-electrons is determined by the groundstate charge density was discussed earlier but no applied mean was found to build this response. A productive procedure was provided by TD-DFT from which absorption spectra can be calculated by following steps [42]:

Step 1: The calculations comprised of promulgating Kohn-Sham orbital incidence of a trivial dynamic perturbation, e.g., considering system being perturbed by Eq. (27).

Step 2: Then induced dipole moment, μinduced(t), is calculated by Eq. (28), which depicts the difference among time-dependent and permanent dipole moments of a system.

Step 3: Dynamic polarizability, α, is expressed by linear term, μinducedðÞ¼ t Ð α t � t 0 ð Þεð Þt :Final calculations of dynamic polarizability in Eq. (29) are carried out after applying Fourier transform convolution theorem.

Step 4: Sum-over-states (SOS) form of dynamic polarizability has the form:

$$
\sigma\_{\text{applied}}(r, t) = \varepsilon(t). \mathbf{r} \tag{28}
$$

$$
\mu\_{induced}(t) = -e \int \mathbf{m}(r.t) \mathbf{r} d\mathbf{r} \tag{29}
$$

$$
\alpha(\omega) = \mu(\omega) / \mathcal{E}(\omega) \tag{30}
$$

$$a(\omega) = \sum\_{I \neq 0} \frac{f\_I}{\omega\_I^2 - \omega^2} \tag{31}$$

where ωI and fI are corresponding vertical excitation energies and conforming oscillator strength.

Finally, spectrum was obtained employing Eq. (31) also termed as Lorentzian broadened stick spectrum:

$$S(\omega) = \frac{2\omega}{\pi} \text{Im}\alpha(\omega + i\eta) \tag{32}$$

Such a procedure has been executed in many computer codes, e.g., in Octopus code [43], and has benefited that absorption spectrum of a very huge molecule over a wide range of energies can be calculated even though using only reasonable spectral resolution. This function parallels to optical absorption spectrum, e.g., calculating spectra of benzene [44] by TD-DFT approach depicted in Figure 1.

Figure 1. Absorption spectra of benzene.

#### 4.1.1. Calculating the oscillator strength

It is an important point to note that TD-DFT calculations not only give excitation energies (ω) but also provide respective oscillator strengths. Oscillator strengths are actually pure numbers in a complete basis set [42].

$$\,\_{I}f\_{I} = \frac{2m\_{e}}{3h} \omega\_{I} |\langle 0|r|I\rangle|2\tag{33}$$

on oscillator strength carrying subspace so that absorption spectrum is possibly calculated with less excitations in a fixed energy space. A common problem considered is that a large number of excitations have to be estimated with the purpose of covering energy space of concentration, while figure of absorption spectrum is determined by only some of them

Another detailed DFT method is presented, proposed in ref. [46], to calculate UV absorption

where ℏω is the input running radiation energy and ℏωjk is energy difference between ground (j) and excited (k) state. Γjk is termed as line broadening, which is characterized by electron

As high harmonic generation is possible with TD-DFT methods, emission spectra [48] are

ð

Absorption spectrum is the fundamental property of a system. One working on absorption spectra calculations is mainly concerned with the lowest excited states commonly. Though, excitations over a broader energy range may be essential, consequentially requiring very challenging calculations while working for large molecular complexes and high density of states (DOS) materials [49]. Hence, for large systems approximate, precise and efficient computational methods have to be looked for. LR-TD-DFT is extensively employed to compute absorption spectra of larger systems as well [50]. To estimate the absorption spectrum in full LR-TD-DFT context, a two-sided Lanczos process is proposed in Ref. [51]. To acquire an accurate estimation of absorption spectrum, a more standard Lanczos algorithm with an

Density response of a system by applying an external time-dependent perturbation in linear

� � � � dteiωt d2

dt2 d tð Þ

� � � �

2

μjk � �2 ħωjk � ħω � iΓjk

Ijk ≈ ωjk

vibration broadening and inversely proportional to the period of k state.

σemisionð Þ¼ ω

4.2. Linear response TD-DFT(LR-TD-DFT) and absorption spectrum

response formulation of TD-DFT (LR-TD-DFT) is usually studied as [52]:

appropriately selected inner product was employed.

4.2.1. LR-TD-DFT formulation

! space. Intensity of absorption is in direct

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(35)

(36)

! from ground jth to kth excited state; hence, it can be

because of their greater oscillator strength (fI) [45].

spectra. The integration is completed over real r

4.1.2. Calculating the intensity of absorption

proportion to transition dipole moment μjk

estimated by formula [47]:

4.1.3. Emission spectra calculation