NDDO approach

The first (1967) of the Dewar-type methods was PNDDO [35], partial NDDO), but because further development of the NDDO approach turned out to be unexpectedly formidable [33], Dewar s group temporarily turned to INDO, creating MINDO/1 [36] (modified INDO, version 1). The third version of this method, MINDO/3, was said [33] [to have] so far survived every test without serious failure , and it became the first widely-used Dewar-type method. Keeping their promise to return to NDDO the group soon came up with MNDO (modified NDDO). MINDO/3 was made essentially obsolete by MNDO, except perhaps for the study of carbocations (Clark has summarized the strengths and weaknesses of MINDO/3, and the early work on MNDO [37]). MNDO (and MNDOC and MNDO/d) and its descendants, the very popular AMI and PM3, are discussed below. Briefly mentioned are a modification of AMI called SAMI and an... 

There is also work with the RPH based on semiempirical methods. However, semiempirical methods are parametrized for equilibrium geometries and, accordingly, do not necessarily represent all parts of a reaction valley in a consistent way. Because of this, Truhlar and co-workers have proposed modifications of known semiempirical methods so that all their adjustable parameters are varied to reproduce experimental or ab initio data for specific reactions. Since most of the semiempirical methods presently in use are based on the NDDO approach, the term specific reaction parameter NDDO model (NDDO-SRP) has been coined." The SRP parametrization changes the NDDO model from being qualitatively incorrect to semiquantitatively accurate and, accordingly, provides a much cheaper basis to apply the RPH. 

In order to overcome these weaknesses, Pople and co-workers reverted to a more complete approach that they first proposed in 1965 [14], neglect of diatomic differential overlap (NDDO). In NDDO, all four-center integrals (pv are considered in which p and v are on one center, as are 2 and cr (but not necessarily on the same one as and v). Furthermore, integrals for which the two atomic centers are diEFer-ent are treated in an analogous way to the one-center integrals in INDO, resulting... 

The term "semi-empirical" has been reserved commonly for electronic-based calculations which also starts with the Schrodinger equation.9-31 Due to the mathematical complexity, which involve the calculation of many integrals, certain families of integrals have been eliminated or approximated. Unlike ab initio methods, the semi-empirical approach adds terms and parameters to fit experimental data (e.g., heats of formation). The level of approximations define the different semi-empirical methods. The original semi-empirical methods can be traced back to the CNDO,12 13 NDDO, and INDO.15 The success of the MINDO,16 MINDO/3,17-21 and MNDO22-27 level of theory ultimately led to the development of AMI28 and a reparameterized variant known as PM3.29 30 In 1993, Dewar et al. introduced SAMI.31 Semi-empirical calculations have provided a wealth of information for practical applications. 

The semiempirical molecular orbital (MO) methods of quantum chemistry [1-12] are widely used in computational studies of large molecules. A number of such methods are available for calculating thermochemical properties of ground state molecules in the gas phase, including MNDO [13], MNDOC [14], MNDO/d [15-18], AMI [19], PM3 [20], SAMI [21,22], OM1 [23], OM2 [24,25] MINDO/3 [26], SINDOl [27,28], and MSINDO [29-31]. MNDO, AMI, and PM3 are widely distributed in a number of software packages, and they are probably the most popular semiempirical methods for thermochemical calculations. We shall therefore concentrate on these methods, but shall also address other NDDO-based approaches with orthogonalization corrections [23-25],... 

The first reported approach along these lines was the localized self-consistent-field (LSCF) method of Ferenczy et al. (1992), originally described for the NDDO level of theory. In this case, the auxiliary region consists of a single frozen orbital on each QM boundary atom. 

The third approach is the complete neglect of differential overlap approximation (CNDO), in which only the one- and two-center, two-electron integrals remain. The direct application of these methods (NDDO, INDO, or CNDO) is not useful because of the approximations, so it is necessary to include parameters in place of all or some of the integrals. These parameters are based on atomic or molecular experimental data. 

In this section, we consider a family of semiempirical implementations of the antisymmetrized product of the strictly local geminals (SLG). Quite naturally, this approach applies only to compounds (largely organic) with well localized two-center two-electron bonds. It had been originally developed for an old-fashioned MINDO/3 type of parametrization of the molecular Hamiltonian and then extended to the more contemporary NDDO family of parametrizations. First, the description of the wave function is given in detail and then the energy functional is described and analyzed. Its variation provides the equilibrium values of the electronic structure variables (ESVs) relevant for this method. 

As can be seen from Equation 6.33, the one-electron part of the Fock operator (cf. Equation 6.25 and Equation 6.26) is set to be proportional to the overlap between atoms A and B with respect to the two specified AOs. Note that this approach violates the NDDO formalism consequently, the approach was called MNDO, and the successor models AMI, PM3, and PM5 are in fact MNDO-type methods. The one-electron matrix element of Equation 6.33 represents the kinetic and potential energy of the electron in the field of the two nuclei (represented by their core charges), and the P terms are the respective single-atom resonance integrals. 

It has been customary to classify methods by the nature of the approximations made. In this sense CNDO, INDO (or MINDO), and NDDO (Neglect of Diatomic Differential Overlap) form a natural progression in which the neglect of differential overlap is applied less and less fully. It is now clearer that there is a deeper division between methods, related to their objectives. On the one hand are approximate methods which set out to mimic the ab initio molecular orbital results. The objective here is simply to find a more economical method. On the other hand, some workers, recognizing the defects of the MO scheme, aim to produce more accurate results by the extensive use of parameters obtained from experimental data. This latter approach appears to be theoretically unsound since the formalism of the single-determinant wavefunction and the Hartree-Fock equations is retained. It can be argued that the use of the single-determinant wavefunction prevents the consistent achievement of predictions better than those obtained by the ab initio scheme where no further... 

The range of zero differential overlap methods such as CNDO, INDO, and NDDO originally suggested by Pople et al.7z continues to have wide use in spite of the fact that they are less accurate than the methods just discussed. Of these NDDO might be expected to be the most accurate but there has been surprisingly little use of this approach. NDDO neglects integrals by the use of... 

With the NDDO methods, tautomeric equilibria,22o especially in heterocycles,216-219,223,224,227,232,233 have been a favorite topic for study using the BKO approach. The tautomeric equilibria of many heterocyclic systems are exquisitely sensitive to solvation,i i3>2i4 making them interesting test cases for the validation of any solvation model. A detailed comparison is presented later in the section on relative free energies in heterocyclic equilibria. A comprehensive study of the stabilization of a wide variety of carbon radical and ionic centers has also been reported.21 ... 

The CDS parameters, on the other hand, are expected neither to be solvent-independent nor to be clearly related to any particular solvent bulk observable, especially insofar as they correct for errors in the NDDO wavefunc-tion and its impact on the ENP terms. The CDS parameters also make up empirically for the errors that inevitably occur when a continuous charge distribution is modeled by a set of atom-centered nuclear charges and for the approximate nature of the generalized Born approach to solving the Poisson equation. Hence, the CDS parameters must be parameterized separately against available experimental data for every solvent. This requirement presents an initial barrier to developing new solvent parameter sets, and at present, published SMx models are available for water only (although a hexadecane parameter seH will be available soon). 

The SMx aqueous solvation models, of which the most successful are called AM1-SM2,27 AMl-SMla, and PM3-SM3, °- adopt this quantum statistical approach, which takes account of the ENP and CDS terms on a consistent footing. The NDDO models employed are specified as the first element (AMI or PM3) of each identifier. It is worth emphasizing that the SMx models specifically calculate the absolute free energy of solvation—a quantity not easily obtained with other approaches. We have reviewed the development and performance of the models elsewhere.203 We anticipate our further observations later in this chapter by noting that the mean unsigned error in predicted free energies of solvation is about 0.6-0.9 kcal/mol for the SMx models for a data set of 150 neutral solutes that spans a wide variety of functionalities. A number of examples are provided later in this chapter. 

Quantum chemical computations of potential energies surface sections along the reaction pathway (PEES) for interaction of typical electrophiles (halogensilanes HaSiX (X = F, Cl, Br, I), trimethylchlorosilane [48,49], acetyl chloride [51]) and nucleophiles (hydrogen halides HX (X = F, Cl, Br, I), water, aliphatic amines, aliphatic alcohols [52], amino acids [53] and substituted phenols [54]) with the silica OH group in a cluster approach using semiempirical AMI, NDDO, MNDO and MNDO/H methods were performed. Representative PEES is shown in Fig.l. 

Conventionally, there are three levels of integral approaches CNDO (complete neglect of differential overlap), INDO (intermediate neglect of differential overlap), and NDDO (neglect of diatomic differential overlap). The last is the best of the approximations since it keeps the superior multipoles of charge distributions in the two-center interactions (unlike the others that shorten after the monopole). 

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