Calculation, ab initio

Reliable potential energy surfaces (PES s) are prerequisite to obtaining accurate rate constants for chemical reactions. With the significant progress made in the field of quantum chemical methods in recent years as alluded to in the introduction, scientists not only can reproduce experimental thermochemical data but also can make accurate predictions on rate constants for which experimental data are unknown or uncertain. The Gaussian-X (GX) (X = 1, 2, 3) series of methods and the most recent new family of G3 methods, referred to as G3X, developed Pople et al. [24] and 

From a technical viewpoint, the density function (DFT) method can reliably reproduce experimental moleuclar parameters when a sufficient basis set is employed [26-28]. To improve the accuracy for the calculated species, various modifications of GX (X = 2, 3) methods have been proposed [4, 5, 25]. In the modified G3 method (G3X) [25], the structure and ZPE obtained at the B3LYP/6-31G(2df, p) level are used. 

For the total energy prediction with G2M, two versions of the method scheme [5] were used in this work depending on the size of the molecules involved in the reactions. They are represented as follows and will be indicated for a specific system in the related sections  

Ab initio Hartree-Fock calculations have been used to study the stability of the 4H- and 9a//-tautorners (17 and 18, respectively) of a series of derivatives of tetramethyl quinolizine-l,2,3,4-tetracarboxylate. These calculations (Table 1) have confirmed that the 411-tautomer is the thermodynamically more stable form 2003JST719 . 

Ab initio calculations using the CHF-GIAO approach on the optimized geometrical configurations of the compounds have also allowed to predict the 111, 13C, and 1SN nuclear magnetic resonance (NMR) spectra of the quinolizidine series. The calculated spectra fit fairly well the experimental data, with the exception of some signals 

Jonsson et al. have continued their MCSCF studies and calculated the hypermagnetizability and its anisotropy to interpret the Cotton-Mouton effect in CO2, N2O, OCS and CS2. They also assess the importance of the vibrational effects and show that they may sometimes be more important than the electronic contributions. 

Cemusak et al have used the Coupled HF method to study ring currents in six cyclic isomers of (CH)2B2N2. Correlated calculations show that all six have planar structures and Tt-electron diamagnetic ring currents. The nuclear shielding produced by the ring current is compared with that of benzene. 

Ligabue et al have calculated jr-electron current density maps from Coupled 

Hartree-Fock theory using the CTOCD method (see Section 2.4) for anthracene, phenanthrene and triphenylene. The magnetizabihties and nuclear shieldings are also calculated in the same approximation and are found to agree well with experimental values. Ferraro and Caputo have used the RPA (TDHF) method in conjunction with the CTOCD gauge transformation and a polarization propagator formahsm to calculate magnetic properties of HF, H2O, NH3 and CH4. 

Wigglesworth et al. have used LAOs to calculate nuclear shielding surfaces for water and its isotopes at the MCSCF level. The correlated surface for the proton is found to be similar to that obtained in an earlier uncorrelated calculation, but the oxygen surface is considerably changed in the correlated MCSCF method. An accurate force field has been used to represent the nuclear motion and good agreement with experimental isotope shifts has been found. Similar calculations have been performed for acetylene.  

The term ab initio ( from the beginning ) is used to describe calculations in which no use is made of experimental data. In an ab initio variational method, all three steps listed above are explicitly performed. In this chapter we describe a certain kind of ab initio calculation called the self-consistent field (SCF) method. This is one of the most commonly encountered types of ab initio calculation for atoms or molecules. We also describe a few popular methods for proceeding beyond the SCF level of approximation. 

3 Polarizabilities and Hyperpolarizabilities of Larger Molecules. - Ab Initio Calculations. At the most highly correlated level Christiansen et al.157 have used the CCS, CC2 and CCSD models to calculate the static polarizability of furan. Dispersion effects are included to make an estimate of the frequency-dependent polarizability. 

Ruud et a/.164 have attempted to introduce an integral screening procedure into direct SCF calculations of the second hyperpolarizabilities of large molecules. The screening simulates the correlation effects that are implicit macroscopically in the introduction of a dielectric constant. The introduction 

Kamada et al.165 166 have studied the effect of heavy atoms on the second hyperpolarizability of furan, thiophene, selenophene and tetrahydrofuran by a HF calculation with augmented basis sets. The heavier atoms are found to enhance the hyperpolarizability but -conjugation no longer contributes to its value in the presence of the heavier heteroatoms. 

Kucharski et al.161 have calculated the static / -hyperpolarizability of new sulphonamide amphiphiles using finite field SCF and INDO/S methods. In the latter case a solvent correction (SCRF option) was also included. The ab initio and INDO/S results for the isolated molecule were similar while the inclusion of the solvent correction increased the values by about 55-65%. Kassimi and Lin 168 have calculated the dipole moment and static polarizability of aza-substituted thiophene derivatives within the Hartree-Fock approximation. For a representative sub-set, correlation up to the MP4(SDQ4) level has been included. The results are expected to be accurate to within a few percent. 

Swart et al.169 have reported RHF, TD-DFT and Direct Reaction Field (DRF) calculations on the polarizabilities of a set of 15 organic molecules. They find that the RHF results are inferior to those of the other two methods and that the DFT method with the LB94 functional gives the best results for the polarizability anisotropy in molecules with -bonds. Howard et al.,7° have calculated the static polarizabilities of alkylsiloxanate and methoxysiloxanate anions using DFT with the BLYP functional. 

Direct calculations of vibrational optical activity are younger, but once theoretical obstacles to them were solved, they turned out to be easier despite the fact that such a calculation is a many step procedure and its protocol is rather complicated. The use of this procedure for absolute configuration determination is described in Section 8.5.1. A fundamental advantage of VCD calculations consists in the fact that we are dealing with molecules usually in their well-defined electronic state -the ground state. For the calculation of vibrational optical activity, we must calculate atomic polar and axial tensors [16, 92]. 

The calculation involves optimization of the molecular structure, computation of vibrational modes (by far the most demanding part computationally), computation of atomic polar and axial tensors and of all the sums leading to dipole and rotational strengths. As the last step, the theoretical VCD curve is simulated by using the empirical values for bandwidths. The quantum chemical part of the calculation 

Toroidal forms of carbon were predicted to be stable on the basis of molecular dynamics simulations using a Stillinger-Weber-type potential.To test these predictions, the stability of a C120 torus was compared with that of Ceo using ab initio self-consistent field calculations and ionization potentials determined by Koopman s theorem (Table 4). The C120 structure investigated hadDs symmetry, and appears to 

Atom No. Coordinates (in A) of each atom relative to the torus center  

In the following we give only a brief and nonmathematical description of the development of the MO method to obtain geometries and wave functions. Since it is computationally both more simple and more economical to carry out ab initio calculations within the MO formalism (than working with the VB formalism), the MO method is the most commonly used. 

Molecules in their ground state are typically treated using the so-called Born-Oppen-heimer approximation. This approximation is also known as the clamped nuclei approximation because it views the electrons as moving in a field of fixed nuclei. In other words, the total wave function, which is a function of nuclear and electronic coordinates, can be separated into a nuclear wave function and an electronic wave function. This approximation can be justified on the basis that electrons move much faster than nuclei and follow them quasi-instantly. 

MNDO gives substantially improved results as compared to MINDO/3. In 1985 an improved version of MNDO, called AMI, was published (85JA3902, 88TH1). Later MNDO was reparameterized to give MNDO-PM3 (89JCC10, 90JCC543). 

The computer program package MOPAC contains - MINDO/3, MNDO, AMI and PM3 [J.J.P. Stewart, MOPAC - 7.0 A semi-empirical Molecular Orbital Program, Program No 455, Quantum Chemistry Program Exchange (QCPE), Indiana University, Bloomington, IN 47405 USA], 

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Figure Al.5.1 Potential energy curve for NeF based on ab initio calculations of Archibong et al...

The multipole moment of rank n is sometimes called the 2"-pole moment. The first non-zero multipole moment of a molecule is origin independent but the higher-order ones depend on the choice of origin. Quadnipole moments are difficult to measure and experimental data are scarce [17, 18 and 19]. The octopole and hexadecapole moments have been measured only for a few highly syimnetric molecules whose lower multipole moments vanish. Ab initio calculations are probably the most reliable way to obtain quadnipole and higher multipole moments [20, 21 and 22]. 

There are higher multipole polarizabilities tiiat describe higher-order multipole moments induced by non-imifonn fields. For example, the quadnipole polarizability is a fourth-rank tensor C that characterizes the lowest-order quadnipole moment induced by an applied field gradient. There are also mixed polarizabilities such as the third-rank dipole-quadnipole polarizability tensor A that describes the lowest-order response of the dipole moment to a field gradient and of the quadnipole moment to a dipolar field. All polarizabilities of order higher tlian dipole depend on the choice of origin. Experimental values are basically restricted to the dipole polarizability and hyperpolarizability [21, 24 and 21]. Ab initio calculations are an imponant source of both dipole and higher polarizabilities [20] some recent examples include [26, 22] ... 

Unfortunately, the supennolecule approach [81, 82] is full of teclmical diflSculties, which stem chiefly from the very small magnitude of the interaction energy relative to the energy of the supennolecule. Even today, a novice would be ill-advised to attempt such a computation using one of the black-box computer programs available for perfonning ab initio calculations. 

That said, the remarkable advances in computer hardware have made ab initio calculations feasible for small systems, provided that various technical details are carefiilly treated. A few examples of recent computations... 

A few ab initio calculations are the main source of our current, very meagre knowledge of non-additive contributions to the short-range energy [91], It is unclear whether the short-range non-additivity is more or less important than the long-range, dispersion non-additivity in the rare-gas solids [28, 92],... 

There are many large molecules whose mteractions we have little hope of detemiining in detail. In these cases we turn to models based on simple mathematical representations of the interaction potential with empirically detemiined parameters. Even for smaller molecules where a detailed interaction potential has been obtained by an ab initio calculation or by a numerical inversion of experimental data, it is usefid to fit the calculated points to a functional fomi which then serves as a computationally inexpensive interpolation and extrapolation tool for use in fiirtlier work such as molecular simulation studies or predictive scattering computations. There are a very large number of such models in use, and only a small sample is considered here. The most frequently used simple spherical models are described in section Al.5.5.1 and some of the more common elaborate models are discussed in section A 1.5.5.2. section Al.5.5.3 and section Al.5.5.4. 

Dykstra C E 1988 Ab initio Calculation of the Structures and Properties of Molecules (Amsterdam Elsevier)... 

Hu C H and Thakkar A J 1996 Potential energy surface for interactions between N2 and He ab initio calculations, analytic fits, and second virial coefficients J. Chem. Phys. 104 2541... 

Many potential energy surfaces have been proposed for the F + FI2 reaction. It is one of the first reactions for which a surface was generated by a high-level ab initio calculation including electron correlation [47]. The... 

At the time the experiments were perfomied (1984), this discrepancy between theory and experiment was attributed to quantum mechanical resonances drat led to enhanced reaction probability in the FlF(u = 3) chaimel for high impact parameter collisions. Flowever, since 1984, several new potential energy surfaces using a combination of ab initio calculations and empirical corrections were developed in which the bend potential near the barrier was found to be very flat or even non-collinear [49, M], in contrast to the Muckennan V surface. In 1988, Sato [ ] showed that classical trajectory calculations on a surface with a bent transition-state geometry produced angular distributions in which the FIF(u = 3) product was peaked at 0 = 0°, while the FIF(u = 2) product was predominantly scattered into the backward hemisphere (0 > 90°), thereby qualitatively reproducing the most important features in figure A3.7.5. 

Detailed analyses of the above experiments suggest that the apparent steps in k E) may not arise from quantized transition state energy levels [110.111]. Transition state models used to interpret the ketene and acetaldehyde dissociation experiments are not consistent with the results of high-level ab initio calculations [110.111]. The steps observed for NO2 dissociation may originate from the opening of electronically excited dissociation chaimels [107.108]. It is also of interest that RRKM-like steps in k E) are not found from detailed quantum dynamical calculations of unimolecular dissociation [91.101.102.112]. More studies are needed of unimolecular reactions near tln-eshold to detennine whether tiiere are actual quantized transition states and steps in k E) and, if not, what is the origin of the apparent steps in the above measurements of k E). 

Roth H D, Weng H and Herbertz T 1997 CIDNP study and ab initio calculations of rigid vinylcyclopropane systems evidence for delocalized ring-closed radical cations Tetrahedron 53 10 051-70... 

This method, introduced originally in an analysis of nuclear resonance reactions, has been extensively developed [H, 16 and F7] over the past 20 years as a powerful ab initio calculational tool. It partitions configuration space into two regions by a sphere of radius r = a, where r is the scattered electron coordinate. 

More recently, the Duiming group has focused on developing basis sets that are optimal not for use in SCF-level calculations on atoms and molecules, but that have been optimized for use in correlated calculations. These so-called correlation-consistent bases [43] are now widely used because more and more ab initio calculations are being perfonned at a correlated level. 

Alfe D, Gillan M J and Price G D 2000 Constraints on the composition of the Earth s core from ab initio calculations Nature 405 172-5... 

Wei C M, Gross A and Scheffler M 1998 Ab initio calculation of the potential energy surface for the dissociation of H2 on the sulfur-covered Pd(IOO) surface Phys. Rev. B 57 15 572... 

Fukunishi Y and Nakatsu] H 1992 Modifications for ab initio calculations of the moderately large-embedded-cluster model. Hydrogen adsorption on a lithium surface J. Chem. Phys. 97 6535-43... 

Pulay P 1969 Ab initio calculation of force constants and equilibrium geometries in polyatomic molecules. I. Theory/Wo/. Phys. 17 197... 

Pulay P and Meyer W 1971 Ab initio calculation of the force field of ethylene J. Moi. Spectrosc. 40 59... 

Tarazona A, Kreisig S, Koglin E and Schwuger M J 1997 Adsorption properties of two cationic surfactant classes on silver surfaces studied by means of SERS spectroscopy and ab initio calculations Prog. Colloid Polym. Sol. 103 181-92... 

A further model Hamiltonian that is tailored for the treatment of non-adiabatic systems is the vibronic coupling (VC) model of Koppel et al. [65]. This provides an analytic expression for PES coupled by non-adiabatic effects, which can be fitted to ab initio calculations using only a few data points. As a result, it is a useful tool in the description of photochemical systems. It is also very useful in the development of dynamics methods, as it provides realistic global surfaces that can be used both for exact quantum wavepacket dynamics and more approximate methods. 

The adiabatic picture is the standard one in quantum chemistry for the reason that, not only is it mathematically well defined, but it is also that used in ab initio calculations, which solve the electronic Hamiltonian at a particular nuclear geometry. To see the effects of vibronic coupling on the potential energy surfaces one must move to what is called a diabatic representation [1,65,180, 181]. 

Note that the relations (23) are valid also if (22) is questionable. Brown [19] refined the approximation (23) by introducing the gn factor, describing the deviation of the mean values for Lj and fi om integers. Validity of the approximation (23) has been checked by means of explicit ab initio calculations, for example, in [20,21]. 

Now, we discuss briefly the situation when one or both of the adiabatic electronic states has/have nonlinear equilibrium geometry. In Figures 6 and 7 we show two characteristic examples, the state of BH2 and NH2, respectively. The BH2 potential curves are the result of ab initio calculations of the present authors [33,34], and those for NH2 are taken from [25]. 

Figure 9. Energy difference (absolute value) between the components of the X II electronic State of HCCS as a function of coordinates p, P2, and y. Curves represent the square root of the second of functions given by Eq. (77) (with e, = —0.011, 2 = 0.013, 8,2 = 0.005325) for fixed values of coordinates p, and P2 (attached at each curve) and variable Y = 4>2 Here y = 0 corresponds to cis-planar geometry and y = 71 to trans-planar geometry. Symbols results of explicit ab initio calculations.

The next question asked is whether there are any indications, from ab initio calculations, to the fact that the non-adiabatic transfonnation angles have this feature. Indeed such a study, related to the H3 system, was reported a few years ago [64]. However, it was done for circular contours with exceptionally small radii (at most a few tenths of an atomic unit). Similar studies, for circular and noncircular contours of much larger radii (sometimes up to five atomic units and more) were done for several systems showing that this feature holds for much more general situations [11,12,74]. As a result of the numerous numerical studies on this subject [11,12,64-75] the quantization of a quasi-isolated two-state non-adiabatic coupling term can be considered as established for realistic systems. 

Equation (151) can be applied to obtain/(0). Ab initio calculation for small enough q values will yield 19(0, q 0) and these, as is seen from Eq. (151), can be directly related to/(0) ... 

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