Blyp

B-Half-and-Half-LYP28

ACM29

B3LYP29b

PW86/P86

PW9141

B88/P86

B88/LAP1

PW86/LAP1

Local functionals (LDA) Xa S21

Nonlocal functionals (posy-LDA) B8842

PW8639

B8842 B8842 PW8639

none VWN23

LYP43

P8640

P8640 LAP 132 LAP 132

aThe description of the original "half-and-half" approach can be found in Ref.28. B-Half-and-Half-LYP was obtained in a similar way, but the LYP correlation functional was used instead of the SVWN.

bIn the original work by Becke,29 the three-parameter hybrid exchange-correlation functional involves a linear combination of the B88, SVWN, and PW91 functionals with the coefficients fitted to experimental data. The B3LYP functional was obtained in a similar way, but the LYP correlation functional was used instead of the P86 one.

nection method recently proposed by Becke28,29, so as to eliminate the self-interaction effects. These schemes, also known as hybrid HF-DFT methods, have been shown recently by several researchers to be remarkably accurate in predicting various molecular properties29,31. Most recently, successful attempts to incorporate terms depending on the electron density Laplacian into the exchange-correlation functionals were also reported32.

The formalism of Kohn and Sham was implemented in several programs. Among the most widely used are: ADF33, which uses Slater basis sets; DGauss34, deMon/KS35, and Gaussian92/DFT36, which use Gaussian basis sets; DMol37, which uses numerical basis sets; and the programs implementing the Car-Parrinello dynamics38, which use the plane waves.

4.1.1.2 Applications

The Kohn-Sham equations have been applied to study gas-phase properties of many systems of biological interest. Most of such studies have been made for relatively small mol ecular systems for which the conventional post-Hartree-Fock ab initio methods could also be applied. For such systems, the DFT calculations served mainly the purpose to determine the range of applicability of known approximations of the exchange-correlation functional in the Kohn-Sham equations. There is also a still-growing amount of DFT results concerning larger systems which are of biological interest and for which DFT is the only quantum-chemical method that includes correlation effects available. The forthcoming section presents applications of the KS formalism to study properties of isolated molecules, molecular complexes, and chemical reactions.

Organic Molecules of Biological Interest. The works by Andzelm and Wimmer19 and by Johnson et al.20 were instrumental in expanding the range of applicability of the Kohn-Sham formalism to enter the domain of molecular systems of biological relevance. For large sample of organic molecules (including some of biological interest—H20, formic acid, for-mamide, pyrimidine, for instance), molecular properties derived from the DFT calculations were compared to the ones stemming from the post-Hartree-Fock calculations and from experiment (when available). Geometries, atomization energies, dipole moments, and vibrational frequencies were analyzed. Andzelm and Wimmer19 performed the DFT(S VWN) and DFT(B88/P86) calculations. It was found that for small molecules containing C, N, O, H, and F atoms, the DFT equilibrium bond lengths agreed with experiment within 0.01-0.02 A, whereas the bond and dihedral angles within 1-2 degrees. The DFT(SVWN) vibrational frequencies were too low, but as close to the experiment as those obtained from the second-order M0ller-Plesset (MP2) calculations. The studies by Johnson et al.20, in which several local and nonlocal functional were applied (S/null, SVWN, S/LYP, B88/null, B88/VWN, and BLYP) led to similar conclusions.

Similar analyses to these by Andzelm and Wimmer and by Johnson et al. were performed specifically for molecules of biological interest by St.-Amant et al.44. Functional groups which are commonly present in molecules of biological interest (mainly proteins) were investigated. The largest molecules were analogs of glycine and alanine dimers. Geometries, conformational energies and dipole moments were obtained from the DFT(SVWN) and DFT(B88/P86) calculations. Equilibrium geometries, relative energies of different con-formers, dipole moments, and molecular electrostatic potentials were analyzed. For the conformational analysis, 175 conformers of 79 organic molecules were selected. For 21 molecules, an analysis of the dipole moments was performed. Based on 35 comparisons of conformational energies for which experimental results were available, the authors concluded that the DFT(B88/P86) results were significantly better than the DFT(SVWN) ones. The root mean square (RMS) of the corresponding energy differences amounted to 0.8 kcal/mol and 0.5 kcal/mol for SVWN and B88/P86, respectively. The analysis of molecular geometries showed that the DFT(B88/P86) bond lengths were systematically overestimated compared to experimentally measured values. The average overestimation of the bond lengths ranged from 0.007 A for CC double bonds to 0.06 A for SS bonds. The corresponding differences between the experimental and MP2 results were significantly lower. The experimental and DFT(B88/P86) bond angles agreed within 1 degree. The MP2 and DFT(B88/P86) results for bond angles were at the same level of accuracy. Compared to experimental results, both the DFT(SVWN) and the DFT(B88/P86) calculations overestimated dipole moments (the average deviation amounted to 11.7% and 9.1 %, respectively). Smaller deviations (to 6.6% and 5.5%, respectively) were obtained in calculations involving an augmented basis set which included an extra set of diffuse and p functions on heavy atoms and a diffuse s function on hydrogen. The authors concluded that the DFT(B88/P86)

led to results approaching the MP2 ones and that they were better than the ones obtained from the Hartree-Fock calculations.

Rashin et al.45 obtained the dipole moment of 32 molecules of biological relevance by means of the DFT(SVWN) and DFT(B88/P86) calculations. The results showed a rather weak dependence of calculated dipole moments on the functional form of the exchange-correlation functional but a strong dependence on the basis set.

Numerous studies of properties of individual organic molecules of biological interest were reported recently. They dealt with molecules which represented fragments of proteins, biological membranes, and nucleic acids, drugs, and drug analogs. Some results will be discussed below.

Adamo, Barone, and collaborators46-48 studied isolated glycine, the smallest building block of proteins. Results showed that the DFT(B3LYP) conformational analysis led to results comparable to the ones obtained from the post-Hartree-Fock calculations. Salahub and collaborators studied the conformational equilibria in glycine and malonaldehyde using B88/P86, B88/LYP, ACM, B88/LAP1, and PW86/LAP1 functionals49. Both molecules contain internal hydrogen bonds and the relative energy differences between the most stable conformers are within 1 kcal/mol. The DFT(B88/P86), DFT(PW86/P86), and DFT(PW91) calculations led to wrong order of stability for the two conformers of glycine. The DFT(BLYP) and DFT(ACM) predicted the right order of stability but the relative energies were underestimated. Very good results were obtained using the DFT(B88/LAP1) and DFT(PW86/LAP1) calculations for both the relative energies of the glycine conformers and the energy of the hydrogen bond in malonaldehyde.

Florian and Johnson50 calculated vibrational frequencies in isolated formamide using the DFT calculations at the LDA (SVWN) and post-LDA (B88/LYP) levels. The DFT frequencies were compared with the ones derived from the Hartree-Fock and MP2 calculations, and from experiment. The authors found that the DFT(B88/LYP) frequencies were more in line with experiment then the MP2 ones. The DFT(SVWN) calculations led to geometry, force constants, and infrared spectra fully comparable to the MP2 results. The equilibrium geometry and vibrational frequencies of formamide were also the subject of studies by Andzelm et al.51. It was found that the DFT(B88/P86) calculations led to frequencies in a better agreement with experiment than those obtained from the CISD calculations.

Oie et al.52 studied the potential energy surfaces of 11 small conjugated molecules relevant to conformational equilibria of large biomolecules. The DFT(B88/P86) barriers to the rotation around the conjugated bond were compared to the ones derived from the MP2 and MP4 calculations. The geometries of minima located by means of both methodologies were in an excellent agreement (the largest torsional angle difference was 2.7 degrees). The rotational barriers were in a satisfactory agreement. The differences between the MP4 and the DFT relative energies were within 0.13-1.39 kcal/mol for non-amide molecules, whereas they were larger (2.71^4.88 kcal/mol) for molecules containing the amide group.

The DFT calculations were used to predict NMR parameters of several molecules of biological importance54. Predicting NMR spectra represents a challenging problem because of their strong dependency on the electronic structure and geometry. Salahub and collaborators reported several papers on this subject55-57. Malkin et al.55 reported a very good agreement between spin-spin coupling constants Jcc JCH, and J//H derived from the post-Hartree-Fock and DFT calculations, and from experiment for several organic molecules. In the DFT calculations, the following approximate exchange correlation functionals were used: SVWN, B88/P86, PW86/P86, and PW91. Compared to the post-Hartree-Fock meth ods, the DFT calculations led to worse coupling constants for systems containing lone pairs. A strong dependence of the estimated Fermi contact contribution to the coupling constants on the form of the approximate exchange-correlation functional was pointed out. Malkin et al.56 combined the Sum-Over-States perturbation theory (SOS) with the DFT calculations to derive shielding constants. The shielding constants did not appear to be sensitive to the analytic form of the exchange-correlation functional. The method was applied for several organic molecules including a model dipeptide. The shielding constants obtained with the DFT and post-Hartree-Fock electron densities were in very good agreement. In few cases, the shielding constants obtained by means of the SOS calculations combined with the Hartree-Fock electron densities were qualitatively wrong (e.g., the ozone molecule). In such cases, the SOS calculations combined with the DFT electron densities led to shielding constants in good agreement with experiment. Malkin et al.57 also reported very good agreement between experimental and the DFT/SOS calculated shielding constants for glycine.

Case58 investigated the effect of ring currents on NMR shielding constants by means of the DFT calculations. The studied rings included the ones commonly found in proteins and nucleic acids. The shielding constants were calculated for methane molecule placed in several positions relative to the ring. The calculations provided data needed to derive structural parameters from measured chemical shifts in proteins and nucleic acids.

The DFT calculations have also been applied to investigate radicals of biological inter-eS(48,53,54,59,6o Eriksson et al.53 calculated the hyperfine structure of small radicals built of H, N, C, O, F, and CI atoms. It was found that the anisotropic hyperfine couplings are relatively insensitive to the basis-set effects and to the functional form of the exchange-correlation functional. The isotropic hyperfine couplings were, however, strongly dependent on the approximate form of the exchange-correlation functional. The best results, in an excellent agreement with experiment, were obtained using the PW86/P86 functional for all neutral and cationic radicals, whereas for halide-containing anions, hyperfine structures were less accurate. The disagreement between experiment and the DFT was attributed to the wrong DFT equilibrium geometries for these compounds. Barone et al. obtained good ESR features for glycine radical using the DFT calculations at the LDA (SVWN) level48. O'Malley and Collins studied semi-quinoine anions, which are formed in the electron transfer reactions of photosyntesis, by the DFT(B3LYP) calculations59. Very accurate hyperfine coupling constants were obtained, but a strong basis set influence on the results was observed. For ring 13C atoms, basis sets of at least full double zeta quality were required. The hyperfine coupling constants for 1H, 170, and other 13C atoms were less basis-set dependent. Jensen et al.60 derived the spin densities of 3-methylindole from the DFT(B3LYP) calculations. This molecule was considered a model of tryptophan-191 radical of the enzyme cytochrome-c-peroxidase. The agreement between the experimental spin densities of the tryptophan-191 of cytochrome-c-peroxidase and the DFT spin densities of the cation radical of 3-methylindole supported the conclusion that tryptophan-191 radical is a cation radical. The spin densities derived from the MP2 and DFT(B3LYP) calculations were in a qualitative disagreement.

The components of nucleic acids have been the subject of continuous DFT stud-jes6r-65,67-69 jasjen ancj Fitzgerald calculated dipole moments and polarizabilities for a series of molecules of biological interest including nucleic acid bases (adenine, thymine, cy-tosine, and guanine) and their pairs (adenine-thymine and cytosine-guanine)61. A good correlation between DFT(HL), experimental, and MP2 results was obtained for dipole moments and polarizabilities. More detailed analyses of DFT(SVWN) and DFT(B88/P86) results, which included vibrational frequencies, were reported for isolated bases and their pairs by Santamaria and Vasques63. Similar analyses were made by Estrin et al.62 who used the LDA(SVWN), B88/P86, and PW86/P86 functional.

Sponer et al.65 compared the MP2 and the DFT energies for cytosine dimer in the base staking conformation. Some features (e.g., twist displacement of bases) of the MP2 potential energy surface were well reproduced by means of the DFT calculations; however, qualitatively wrong results were obtained for others (vertical displacement). ([The reported DFT energies were obtained by combining energies derived from the Kohn-Sham calculations (DFT(BLYP) or DFT(B3LYP)) with empirical terms representing dispersion.]) Bakalarski et al.64 studied the properties of isolated N-methylated nucleic bases in their fundamental tautomeric forms by means of the DFT(B3LYP), Hartree-Fock, and MP2 calculations. The dipole moment, rotational constants, and molecular electrostatic potential were calculated. Molecular properties (rotational constants, dipole moments) agreed better with experiment than the corresponding results from the Hartree-Fock calculations. The DFT results were comparable to the MP2 ones. The electrostatic potential (as measured by the magnitudes of fitted atomic charges) steming from the DFT calculations was closer to the MP2 one than to the electrostatic potential given by the Hartree-Fock calculations.

The tautomerism of heterocyclic compounds has been the subject of much interest due to its biological implications for the base pair formation in nucleic acids. A proper assessment of the relative stability of different tautomeric forms of isolated heterocyclic compounds seems, therefore, to be a prerequisite for modeling biological processes involving nucleic acids. For such compounds, relative energies of the most stable tautomeric forms are typically small (in the range of 1-2 kcal/mol) which makes quantum-chemical studies difficult. The relative energies of the most stable tautomers are frequently of the same order of magnitude as the differences among energies obtained from various the post-Hartree-Fock calculations. Since the differences between the zero-point energies in the most stable tautomers are typically in the 1 kcal/mol range, significant computational efforts are required to accurately calculate vibrational frequencies. The lack of conclusive ab initio results makes it difficult to assess the quality of the DFT results. A number of DFT studies on this subject have appeared recently62,66-71. The structure, relative energetics of tautomeric forms, and vibrational frequencies in uracil and in cytosine were studied by Estrin et al.62 who applied LDA(GL), and post-LDA (B88/P86 and PW86/P86) functionals. Good structures and dipole moments were obtained for all functionals, including the LDA. The DFT calculations with different post-LDA exchange functionals (B88 and PW86) led to similar relative energies of tautomers. The DFT calculations, for all functionals, predicted that the dioxo form is the most stable, in agreement with experiment and the high-level post-Hartree-Fock calculations. For cytosine, it was found that the energies of the three most stable tautomers were within 1 kcal/mol. Two of the most stable tautomers predicted by the DFT calculations had been observed experimentally. Hall et al.70 studied the energies of tautomeric forms in the 2-hydroxypyridine and in cytosine by means of the DFT calculations using the B88/VWN, BLYP, and the corresponding hybrid functionals. For each considered tautomeric form, the DFT and post-Hartree-Fock (QCISD(T) and MP4) calculations were made at the Hartree-Fock optimized geometry. For both studied molecules, the experimental relative energies of the most stable tautomeric forms are very close to each other (within 1.0 kcal/mol). The calculations did not lead to conclusive results for such small energy differences. The DFT(B88/VWN) and DFT(BLYP) calculations predicted a larger stability of the keto form, which is in a disagreement with experiment but in line with the MP2 results. The relative energies obtained using hybrid functionals agreed better with the MP4 ones. Adamo and Lejl66 reported DFT(SVWN) and DFT(B3LYP) studies of relative energies of 2-pyridone and its alternative tautomeric form—2 hydroxypyridine. The calculations were aimed at studying solvent effects, which will be discussed later in this chapter. Turning to gas-phase results, the DFT calculations predicted that the 2-pyridone was less stable than the 2-hydroxypyridine, in agreement with results obtained from the post-Hartree-Fock calculations. The stabilization energy of this tautomer amounted to —1.0, -3.8, -0.56, -1.20, and -0.47 kcal/mol for the Hartree-Fock, MP2, CI, DFT(SVWN), and DFT(B3LYP), respectively. In a similar study, Barone and Adamo71 reported good agreement between the DFT(B3LYP) and the MP2 results for the relative energies of 2-pyridone, 2-hydroxypyridine, and the transition state.

Kwiatkowski, Leszczynski, and collaborators obtained equilibrium geometries, vibrational spectra, relative energies of tautomers, dipole moments, and rotational constants of adenine67, cytosine68, and 2(lH)-pyridone69 by means of the DFT(B3LYP) and MP2 calculations. From comparisons between the DFT and the MP2 results, a general conclusion was drawn that, except for relative energies of tautomers and intensities of the IR absorption bands, the DFT results matched the ones steming from the MP2 calculations and were in good agreement with the available experimental data. The authors concluded that to predict relative energies of different tautomers (especially if they are small) the conventional ab initio post-Hartree-Fock methods are more reliable.

The computational advantages of the Kohn-Sham formalism make it a method of choice for conformational studies of large and flexible molecules. Many molecules of biological interest belong to this class and there is a continuously growing interest in applying the DFT calculations as an alternative to the semi-empirical or the Hartree-Fock calculations72-75. Topol and Burt72 studies conformational equilibria of 1,2-difluoroethane and inositol. Molecules containing inositol are commonly found in biomembranes as one of the few head groups of phospholipides. The conformational analysis demonstrated that the LDA calculations (DFT(BH)) can properly describe the ordering of small (1 kcal/mol) differences between energies of local minima on the potential energy surface. The authors showed also that the unsealed vibrational frequencies were in a reasonable agreement with experiment. The conformational energies of inositols were also studied by Liang et al.73, who performed a conformational analysis by means of the Hartree-Fock, MP2, DFT(SVWN), and DFT(B88/P86) calculations. Eight conformers were considered. Theoretical studies on such compounds are very valuable because of the lack of experimental conformational data. Unfortunately, the results of this analysis were not conclusive because of the small differences between energies of different conformers. Some common trends in the DFT(B88/P86) and the MP2 results were observed. Oie et al.74 performed a conformational analysis of ethylene glycol. Ethylene glycol is a model compound for diols which are employed as the central chemical core of several HIV-1 protease inhibitors. Geometries and relative conformational energies of 10 conformers were obtained by means of the Hartree-Fock, MP2, MP4, DFT(SVWN), and DFT(B88/P86) calculations. All the studied conformations were found to lie within 4.5 kcal/mol indicating an important flexibility of the ethylene glycol molecule. This conclusion was supported by results obtained from all theoretical methods. Rabinovitz et al.75 studied conformational properties in cyclopentapolycyclic aromatic hydrocarbons (PAHs) by means of the semi-empirical, Hartree-Fock, and DFT(SVWN) methods. Polycyclic hydrocarbons are known for their carcinogenetic activity. Due to their relatively large size, theoretical studies of their reactivity remains a challenge to quantum chemistry. The energetics and geometry of carbocation pairs which arc formed upon protonation of PAHs were studied by means of the DFT calculations combined with other methods for geometry optimization.

Metallo-organic Systems. Metalloorganic complexes can be found in a variety of biological systems that range from metallo-organic drugs to metallo-enzymes76. Pioneering studies of metallo-organic systems of biological importance were made by Aizman and Case77 by means of the multiple scattering method (MSXa)16 to study the model of the active center of proteins containing the 4Fe-4S cluster. The same method was used to investigate Fe(SR)4 (R = H, CH3) clusters78 which mimic the active site of iron-sulfur proteins and also to investigate ferryl intermediates79. The reader is encouraged to see the review by Case80 for early applications of Xa theory to metallo-organic systems.

More recently, the SCF-Xa scattered wave method was employed by Salomon and collaborators81,82 to investigate a series of copper(II) peroxide structures, in an attempt to rationalize the properties of such complexes of relevance to homocyanin and tyrosinase. Indeed, the latter systems are examples of copper-containing metallo-proteins that re-versibly bind and react with dioxygen. Results of such calculations were used to derive Cu-0 and O-O bonding interactions, magnetic exchange interactions, charge and spin distributions and excited-state transition energies, which all compared favorably with the corresponding experimental data. In particular, these investigations showed that, even at the Xa level of approximation, the DFT methods were able to elucidate the main features of active sites of metallo-proteins and to describe in a coherent and reliable way their possible reaction mechanisms with dioxygen.

Gosh et al. reported LDA(SVWN) studies of oxo(porphyrinato)iron(IV) complexes.83 These compounds have been detected in various peroxidases and are believed to be involved in the reaction mechanisms of other heme enzymes, such as cytochromes P450. Very Good Fe-0 distance and values of unsealed stretching frequencies, which were in excellent agreement with CASSCF results, published elsewhere, were obtained.

Case and collaborators84,85 reported studies of spin coupling in (Fe4S4)3+ using the post-LDA calculations. Parameters of the spin Hamiltonian were estimated using the DFT energies of a high-spin state as well as two different broken symmetry states. The parameters were compared to the ones derived from experimental measurements on the temperature dependence of the magnetic susceptibility. Good overall agreement between theory and experiment was found.

Bray and Deeth applied the DFT(SVWN) method to investigate the active site models of xantine oxidase86. The authors addressed the issue of the presence of (OH)~ at the Mow active site. The geometry of several active site models differing in the number of ligands was obtained by means of the geometry optimization at the DFT level. Resulting metal-ligand distances were compared with the ones stemming from extended X-ray absorption fine structure (EXAFS) experiments. The best agreement between calculated and EXAFS distances were observed for five-coordinate models in which one (OH)" was one of the ligands. The results of the calculations supported the hypothesis which assumes that the oxidation of xanthine to uric acid involves the presence of (OH)- at the oxidized, five-coordinated active site.

Cluster of metal ions and water molecules represent a special case of metallo-organic systems of biological interest. Most of biochemical reactions take place in aqueous solution. Since metal ions are present in biological solvents, all thermodynamic considerations of metal ions bound to biological macromolecules must involve the thermodynamics of ions in water. Clusters of water molecules and monovalent ions were studied by Combariza and Kestner87. For these systems, the Hartree-Fock calculations lead to the underestimated interaction energies and reliable ab initio results can be obtained only with approaches including electron correlation. The authors found that the DFT(B3LYP) calculations led to excellent energies, comparable to the MP2 ones and were significantly better than the ones obtained from the Hartree-Fock calculations.

Parrinello and collaborators investigated several metallorganic systems of biological interest.88-91 A particular implementation of the Kohn-Sham formalism (see the section The Car-Parrinello Method and Its Applications), in which pseudopotentials are used for core electrons, the Kohn-Sham orbitals are expanded using plane waves, and the energy minimization is not constrained to the Born-Oppenheimer surface, was applied to obtain ground state properties at equilibrium geometry. The structure, vibrational frequencies, and electronic properties of cisplatin and other Pt" complexes were studied by Carloni et al.88 and by Tornaghi et al.89 and compared with the available experimental data. These compounds are known for their anti-tumor activity. The studies of their properties represent an initial step in modeling their biological activity that involves interaction with DNA. A good agreement was shown between the available experimental data and the results of the DFT(B88/P86) calculations. Differences between the biological activity of cisplatin and that of carboplatin were rationalized by pointing out qualitative differences between their electronic structures. Lamoen and Parrinello90 studied the electronic and the structural properties of porphyrin, phyrazine, and their magnesium and palladium derivatives. Electronic properties of these compounds are relevant to oxygen and electron transport processes in biological systems. Molecules similar to free-base porphyrine can be found in important biomolecules like hemoglobin and chlorophyll. Structural results from calculation at the LDA level (PZ) were very close to the ones stemming from experiment and from the MP2 calculations. In addition, the electronic structure compared well with ab initio CAS-SCF calculations.

Carloni et al.91 applied the DFT(PZ) calculations to investigate the electronic structure of various models of oxydized and reduced Cu, Zn superoxide dismutase. The first stage of the enzymatic reaction involves the electron transfer from Cui7ion to superoxide. The theoretical investigations provided a detailed description of the electronic structure of the molecules involved in the electron transfer process. The effect of charged groups, present in the active center, on the electron transfer process were analyzed and the Argl41 residue was shown to play a crucial role.

Thermochemistry. The DFT calculations with gradient-dependent functionals are very useful in thermochemistry. Contrary to earlier models, namely Xa, which gives erratic results, and LDA, which systematically overestimates binding energies, the post-LDA calculations yield good results with average errors of order 6 kcal/mol in standard thermochemi-cal tests.29'92'93 Exchange-correlation functionals based on the adiabatic connection formula for the exchange-correlation hole29 reduce this error further to roughly 2 kcal/mol.29,30 At this level of accuracy, the DFT calculations can be considered a promising tool for predicting thermochemical properties of molecules of biological interest. Several DFT studies of atomization energies, ionization potentials, electron and proton affinities, heats of reaction, and bond energies in systems of biological relevance were reported recently19.20,29,30,32,92-96,98

Clementi and Chakravorty94 calculated atomization energies of about 50 organic molecules ranging from the ones as small as HF to the ones as large as pyridine. Several approximate exchange-correlation functionals, both the LDA and the post-LDA, were used. None of the considered functionals led to atomization energies of the chemical accuracy. Atomization energies obtained by means of nonlocal functionals were systematically closer to the experiment than the corresponding results obtained from the LDA calculations.

Andzelm and Wimmer19 studied the energetics of several reactions by means of the DFT(SVWN) and DFT(B88/P86) calculations. The DFT(B88/P86) bond separation energy for a set of typical organic reactions involving H, C, N, and O bond atoms was within 7 kcal/mol from experimental values. The similar level of accuracy was seen for the dissociation of bonds formed by C, N, and O atoms leading to radical products, the DFT(B88/P86) energies were comparable to the ones derived from the MP2 and MP4 calculations. Larger errors were detected for breaking bonds involving F atom.

Fournier and DePristo96 calculated bond energies in several small compounds containing disulfide bonds which are known to stabilize the tertiary structure of proteins. Bond dissociation energies are generally overestimated when LDA(SVWN) is used whereas the PW86/P86 functional brings them to within 5 kcal/mol of experimental values.

Lee et al.97 investigated the HF, HC1, and H2S dissociation reactions in water clusters. A good agreement between the DFT(BLYP) and the MP2 results was reported for binding energies (within 2 kcal/mol) and geometries. The studies showed also that at least four water molecules are needed for ionic products of dissociation to coexist.

Chandra and Goursot98 reported the DFT(SVWN), DFT(PW86/P86), DFT(B88/P86), DFT(PW86/LAP1), and DFT(B88/LAP1) studies of the proton affinities of the following organic molecules: H2CO, CH3CHO, CH3OH, C2HsOH, HCOOH, and CH3COOH. The SVWN proton affinities were systematically overestimated by 6-10 kcal/mol compared to experimental results. Significantly better results were obtained using nonlocal functionals. The PW86/P86, B88/P86, and PW86/LAP1 led also to overestimated proton affinities by 4.3-6.16, 2.2-4.4, and 0.9-2.2 kcal/mol, respectively. The best results were derived by means of the DFT(B88/LAP1) calculations which led to proton affinities that differed from the experimental values by —0.9 to +0.36 kcal/mol. These results are consistent with original work by Proynov et al.32 where the LAP1 functional was introduced to calculate atom-ization energies of several organic molecules including pyridine and pyridazine and to derive heats of reaction of simple organic reactions.

The structure and the proton affinity of triazene, a molecule related to triazenium ions some of which are known as putative carcinogens was the subject of DFT investigations by Schmiedenkamp et al.99 The DFT(SVWN), DFT(PW86/P86), and DFT(B88/P86) results were compared with the ones obtained from the Hartree-Fock and MP2 calculations. Structural and energetic results obtained from the nonlocal DFT calculations compared well to the MP2 ones. In the subsequent paper100 the authors performed similar analyses for 19 small organic compounds. The protonation site were either at nitrogen or at oxygen atoms. The results were not sensitive to the analytical form of the approximate exchange-correlation functional. Good results were obtained for nitrogen proton affinities, whereas oxygen affinities were systematically underestimated.

Hydrogen-Bonded Molecular Complexes. Gas-phase hydrogen-bonded complexes frequently attract attention for their relevance to similar complexes in biomolecular systems. Hydrogen bonds contribute to the stabilization energy of large biomolecules: nucleic acids and proteins. The nucleoside bases (adenine, thymine, uracil, cytosine, and guanine), which are the building blocks of nucleic acids, form hydrogen-bonded complexes of several types. The two polynucleotide chains of B-DNA are held together by hydrogen-bonded purine-pirymidyne base pairs101, for instance. In proteins, hydrogen bonds are abundant since the polypeptide backbone is built of groups that can form hydrogen bonds (NH as an acceptor and CO as a donor) and because several amino-acid side chains may be involved in hydrogen bond formation as donors (tryptophan and arginine) or both as acceptors and as donors (asparagine, glutamine, serine, threonine), or either as donors and acceptors dependent on the pH value (lysine, glutamic and aspartic acid, tyrosine, histidine)102. Polar groups of proteins frequently form hydrogen bonds with the solvent molecules (mostly water) or with other macromolecules103,104. In addition, strong hydrogen bonds are known to play the central role in several enzymatic reactions105,106. Consequently, any nonempirical attempt to study the stability of common biomolecules or enzymatic activity requires methods that can accurately describe hydrogen bonds. Hydrogen bonds represent a great challenge to quantum chemistry as far as chemical accuracy is required, as was demonstrated in the case of the hydrogen fluoride dimer by Latajka and Bouteiller107. For this system, detailed studies of a broad family of density functional methods using large basis sets (triple-zeta with diffuse functions and multiple sets of polarization functions) showed the good performance of the DFT methods compared to the conventional ab initio ones: Hartree-Fock, MP2, and quadratic configuration interactions with single and double excitations (QCISD).

A water dimer, which bears more relevance to biological systems, was recently investigated by many researchers. Results obtained using various implementations of the Kohn-Sham formalism were reported.87,109"116,119,122,124,125,128 Table 4.2 collects selected results.

From Table 4.2, it can be seen that the LDA calculations, regardless of the functional form of the exchange-correlation functional, led to too-small intermolecular distances (by about 0.15 A) and binding energies overestimated by about 2-3 kcal/mol. Water dimer properties (energy, dipole moment, geometry, and vibrational frequencies) predicted by the DFT calculations at the post-LDA level were significantly closer to experiment than the corresponding LDA results. The nonlocal functionals led to oxygen-oxygen distances ranging from 2.886 A to 2.943 A, which, being still too small, were fairly close to the experimental distance equal to 2.967 A117. The interaction energies corrected for the basis set superposition error (BSSE) and obtained at the post-LDA level were affected to some extent by the functional form of the exchange-correlation functional. The comparison of results obtained with the same basis set shows that the interaction energy may vary by as much as 1.5 kcal/mol depending on the choice of the approximate functional. In most cases, the equilibrium intermolecular distance increased upon the improvement of basis set.

The dependence of the DFT results on the basis set used to expand the Kohn-Sham orbitals is illustrated in Table 4.3, which collects equilibrium geometry properties of water dimer obtained with the same exchange-correlation functional (B88/P86) but with different basis sets.

It can be seen from Table 4.3 that the basis set superposition error was significantly reduced in the calculations with the largest basis sets. Surprisingly, results obtained with the plane-waves expansion of the Kohn-Sham orbitals, which should be free from the BSSE, differ from the ones obtained with the largest Gaussian basis set. One of the reasons of the discrepancy might be the periodicity (the length of the cubic unit cell was set to 8.464 A) in the plane waves calculations, or the poor convergence of the plane waves expansion for systems with relatively flat potential energy surface. Differences between results reported in Ref.109 and in Ref.114 for the same orbital basis set are due to the differences in the numerical evaluation of the exchange-correlation potential, which was fitted using different sets of auxiliary functions. Finally, the calculated value of the dipole moment of water dimer is very sensitive to the basis set.

Besides water dimer, larger clusters of water molecules were extensively investigated by means of the DFT calculations87,111_114,127,128. Laasonen et al.113 studied the structure, the energies, and the vibrational frequencies of small water clusters (up to eight molecules)

Table 4.2. Calculated properties of the water dimer. The interaction energy (Eint) in kcal/mol, the intermolecular distance (RQO) in A.

Elnt

Eint(BSSE)

Roo

Exc

basis set

Ref.

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