Semiempirical approximations

In theory, once the activity of an electrolyte in solution is known, the activity of the solvent can be determined by the Gibbs-Duhem integration (see section 2.11). In practice, the calculation is prohibitive, because of the chemical complexity of most aqueous solutions of geochemical interest. Semiempirical approximations are therefore preferred, such as that proposed by Helgeson (1969), consisting of a simulation of the properties of the H20-NaCl system up to a solute... 

In the first place the Hiickel approximation for the tt electrons has been replaced by a self-consistent field (SCF) procedure, generally in a semiempirical approximation of the Pariser-Parr-Pople type completed with some limited configuration interaction (PPP-CI method).59 00 Second, the a skeleton of the molecules has been treated by the Del Re procedure61 for saturated systems (which is the counterpart for the a electrons of the Hiickel method for 77 electrons) as refined for the a skeletons of conjugated heterocycles by... 

The Hamiltonian for a molecular system in a general semiempirical approximation can be represented as a sum of one- and two-center contributions ... 

Theoretical studies of the basicity of pyrazoles, using the semiempirical approximations as well as the STO-3G and 4-31G methods have enhanced the understanding of the differences in basicity between the gas phase and the aqueous solution. To rationalize the relative gas-phase and solution basicity and acidity of pyrazole, it is necessary to take into account the lone pair/lone pair repulsion in the pyrazolate anion (6.5 kcal mol1), the adjacent NH/lone pair attraction in pyrazole (l.Okcal mol1) and the NH+/NH+ repulsion in the pyrazolium cation (6.5 kcal mol1). Solvation by water, and to a lesser extent by DMSO, modifies these values to the point that the position of the equilibria can be reversed. 

Good, rigorous SCF calculations on polyatomic molecules are long, difficult, and tedious to program and inevitably expensive in computer time. What we needed was a simple semiempirical approximate method for three-dimensional molecular orbital calculations. 

The reason we employ two rather distinct methods of inquiry is that neither, by itself, is free of open methodological issues. The method of molecular dynamics has been extensively applied, inter alia, to cluster impact. However, there are two problems. One is that the results are only as reliable as the potential energy function that is used as input. For a problem containing many open shell reactive atoms, one does not have well tested semiempirical approximations for the potential. We used the many body potential which we used for the reactive system in our earlier studies on rare gas clusters containing several N2/O2 molecules (see Sec. 3.4). The other limitation of the MD simulation is that it fails to incorporate the possibility of electronic excitation. This will be discussed fmther below. The second method that we used is, in many ways, complementary to MD. It does not require the potential as an input and it can readily allow for electronically excited as well as for charged products. It seeks to compute that distribution of products which is of maximal entropy subject to the constraints on the system (conservation of chemical elements, charge and... 

Quantum mechanical calculations of molcular orbitals have been performed on five examples (8-azapurine, -hypoxanthine, -guanine, -adenine, and -xanthine) by two methods (a) a semiempirical approximation, which included contributions from the a electrons of the skeleton, and (b) the CNDO approximation, which included contributions from all the valence electrons of the molecule. The results were tabulated in parallel for each of the three possible positions of the triazole proton. In all 15 entries, the highest occupied and the lowest unoccupied molecular orbitals were calculated and also the dipole moment, the molecular energy, and the UV absorption maxima (the last-named showed only a modest agreement with experimental results). It was concluded that both types of calculation indicated that relative stabilities for the three tautomers (in each of the five sets) should decrease in the order HN-9, HN-7, and HN-8, and that the HN-8 tautomers should be 85 to 125 kJ (20-30 kcal) per mol less stable than the other two. However, it had to be admitted that, in all sets of three isomers examined experimentally, the HN-8 member has never been found inferior in stability. ... 

This semiempirical approximation neglects two-center integrals such as... 

Based on the idea of orbital electronegativity introduced by Hinze-Whitehead-Jaffe, in 1981 Ponec had introduced the orbital electronegativity based on the semiempirical approximation CNDO ( complete neglecting differential overlapping , the complete neglect of the differential-orbital overlap even for the orbitals belonging to the same atom) (Ponec, 1981). 

Similarly, the three-body dynamical correlation is semiempirically approximated by the form[15,17]... 

As in all molecular orbital theories, the fundamental computation in SEMOT is to determine the electronic wavefunction for the molecule of interest. Semiempirical approximations were developed of necessity, when computers were too slow for ab initio calculations to be chemically useful [53]. Although many of the approximations are severe, the use of empirical parameters was successful in restoring accuracy. In many cases, the resulting accuracy even exceeds that of simple ab initio calculations, since higher order effects are incorporated into the parameterization. Thus, semiempirical methods remain popular today despite the widespread availability of inexpensive, powerful computers. 

Ab Initio and Semiempirical Approximations and Modeis. The general formulas of ab initio methods can be used for calculations directly and without approximation, only if the set of wave functions and eigenenergies is known with sufficient accuracy. This is true only in the case of the very simplest atoms and molecules. The from first principles approach is to calculate the wave functions by solving the Schrodinger equation exactly. For large molecules, such as polymer fragments and chains, it is currently impossible to produce a set of exact wave functions or to obtain sufficient spectroscopic data. Therefore various assumptions are introduced to alleviate the complexity of the calculations semiempirical models contain such assumptions—which are beyond the scope of this chapter. However one may rely on semiempirical quantum-mechanical methods to predict (at least qualitatively) properties of polymer chain fragments. 

Semiempirical Approximation. The method that adopts a semiempirical approximation to a uses the Andersen-Hinthome local thermodynamic equilibrium (LTE) model [185] for estimating the degree of ionization. The model does not take account of any details of individual ionization processes but assumes that the region at and near the surface involved in sputtering can be approximated by a dense plasma in local thermodynamic equilibrium. The plasma has an associated temperature T. The ratio of the concentrations of two elements X and Y in a matrix B, if sufficiently dilute, can then be written as... 

Unfortunately, the van der Waals equation of state is quantitatively inaccurate (see Chapter 8) and was essentially abandoned by physicists and chemists, who considered it only a clever collage of insightful, but semiempirical, approximations. Its status was considerably enhanced, at least among theoreticians, by the work of Kac et al. (1963), who showed in what sense it could be considered an exact theoretical result for a definite, although physically unrealistic, model of hard spheres with weaklong tmge attractive forces. But this theoretical insight did not improve its accuracy. 

Lennard-Jones potential phys chem A semiempirical approximation to the potential... 

DFT is exact in principle, provided that Exc [p] is known, in which case (see O Eq. 7.53) is an exact representation of the ground state energy Eo (see O Eq. 7.8). In practice, however, Exc [p] is not - and presumably never will be - known exactly therefore (semiempirical) approximations are used. 

Use a computer program such as CAChe or Spartan to find the semiempirical approximations to the SCF orbitals for benzene, using several methods, such as AMI, PM3, and extended Htickel. Compare your orbitals with those obtained using the Htickel method. Calculate the wavelength of the lowest-energy transition (from the highest occupied MO to the lowest unoccupied MO) for each method. Compare the values with the experimental value 180 nm and with each other. 

Certain trends in the development and application of hybrid potentials are evident. Up to now most studies have been done with potentials that use semiempirical methods as the approximation. It is likely that these potentials will remain the most widely used in the future as they are relatively inexpensive to apply, although they will be improved, either by reparametrizing the semiempirical method for each new problem of interest or by using newer semiempirical approximations. Wider application of potentials with ab initio methods is also likely. Particularly promising are hybrid DFT/MM potentials as DFT methods are one of the most inexpensive types of ab initio methods yet they can provide results of an accuracy equivalent to the more expensive correlated molecular orbital methods. 

Early on Higgins et al, found that the HE method based on the AMI semiempirical approximation gives the best overall results for calculation of lEs of radicals. To evaluate... 

We have used the outer valence Green s function (OVGF) method coupled with semiempirical approximations to calculate the IPs of a variety of molecules ranging from benzene to fullerenes. We also extended the method to calculate IPs... 

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