NDDO methods MNDO approximation

Recently, the MNDO type methods (MNDO [32], AMI [33] and PM3 [34]) have been tested for their ability to produce reliable MEP maps. These semi-empirical methods, just as the CNDO and INDO methods, are ZDO methods, and are based on the more sophisticated NDDO approximation [35]. 

NDDO [21] goes beyond INDO in that the ZDO approximation (Section 6.2.1, point (3)) is not applied to orbitals on the same atom, i.e. ZDO is used only for atomic orbitals on different atoms. NDDO is the basis of the currently popular semiempirical methods developed by M. J. S. Dewar and by coworkers who took up the torch MNDO, AMI and PM3 (as well as SAMI, PM5, and PM6). NDDO methods are the gold standard in general-purpose semiempirical methods, and the rest of this chapter concentrates on them. 

The popular semiempirical methods, MNDO (Dewar and Thiel, 1977), Austin Model 1 AMI Dewar et al., 1985), Parameterized Model 3 (PM3 Stewart 1989a 1989b), and Parameterized Model 5 (PM5 Stewart, 2002), are all confined to treating only valence electrons explicitly, and employ a minimum basis set (one 5 orbital for hydrogen, and one 5 and three p orbitals for all heavy atoms). Most importantly, they are based on the NDDO approximation (Stewart, 1990a, 1990b Thiel, 1988, 1996 Zemer, 1991) ... 

Many approximate molecular orbital theories have been devised. Most of these methods are not in widespread use today in their original form. Nevertheless, the more widely used methods of today are derived from earlier formalisms, which we will therefore consider where appropriate. We will concentrate on the semi-empirical methods developed in the research groups of Pople and Dewar. The former pioneered the CNDO, INDO and NDDO methods, which are now relatively little used in their original form but provided the basis for subsequent work by the Dewar group, whose research resulted in the popular MINDO/3, MNDO and AMI methods. Our aim will be to show how the theory can be applied in a practical way, not only to highlight their successes but also to show where problems were encountered and how these problems were overcome. We will also consider the Hiickel molecular orbital approach and the extended Hiickel method Our discussion of the underlying theoretical background of the approximate molecular orbital methods will be based on the Roothaan-Hall framework we have already developed. This will help us to establish the similarities and the differences with the ab initio approach. 

Two of these approximations (INDO and NDDO) have received considerable attention in the past 20 years. The most widely used software package that incorporates these approximations is known as MOPAC, which is available from QCPE. " The program was created by J. J. P. Stewart. A related program is AMPAC, which is also available from QCPE. These programs incorporate the MINDO/S and MNDO implementations of the INDO and NDDO methods, respectively. Both programs also include a more recent semiempirical NDDO implementation called AMl, and MOPAC has PM3. ... 

The neglect of diatomic differential overlap (NDDO) method [236] is an improvement over the INDO approximation, since the ZDO approximation is applied only for orbital pairs centered at different atoms. Thus, all integrals pv Xa) are retained provided p and v are on the same atomic center and A and a are on the same atomic center, but not necessarily the center hosting p and v. In principle, the NDDO approximation should describe long-range electrostatic interactions more accurately than INDO. Most modern semiempirical models (MNDO, AMI, PM3) are NDDO models. 

The purpose of this article is to describe the two-electron approximations used in the NDDO series of semiempirical methods, MNDO (see MNDO), AMI (sec AMI), and PM3 (see PM3), and to discuss the implications of these approximations on the ability of the resulting methods to model systems of chemical interest. 

The NDDO (Neglect of Diatomic Differential Overlap) approximation is the basis for the MNDO, AMI, and PM3 methods. In addition to the integralsused in the INDO methods, they have an additional class of electron repulsion integrals. This class includes the overlap density between two orbitals centered on the same atom interacting with the overlap density between two orbitals also centered on a single (but possibly different) atom. This is a significant step toward calculatin g th e effects of electron -electron in teraction s on different atoms. 

The MNDO, AMI and PM3 methods are parameterizations of the NDDO model, where the parameterization is in terms of atomic variables, i.e. referring only to the nature of a single atom. MNDO, AMI and PM3 are derived from the same basic approximations (NDDO), and differ only in the way the core-core repulsion is treated, and how the parameters are assigned. Each method considers only the valence s- and p-functions, which are taken as Slater type orbitals with corresponding exponents, (s and... 

Due to the integral approximations used in the MNDO model, closed-shell Pauli exchange repulsions are not represented in the Hamiltonian, but are only included indirectly, e.g., through the effective atom-pair correction terms to the core-core repulsions [12], To account for Pauli repulsions more properly, the NDDO-based OM1 and OM2 methods [23-25] incorporate orthogonalization terms into the one-center or the one- and two-center one-electron matrix elements, respectively. Similar correction terms have also been used at the INDO level [27-31] and probably contribute to the success of methods such as MSINDO [29-31],... 

Numerous other semiempirical methods have been proposed. The MNDO method has been extended to d functions by Theil and coworkers and is referred to as MNDO/d.155>156 For second-row and heavier elements, this method does significantly better than other methods. The semi-ab initio method 1 (SAM1)157>158 is based on the NDDO approximation and calculates some one- and two-center two-electron integrals directly from atomic orbitals. 

Because of convention, the symbols for the chemical potential, used in Equation 6.44 and Equation 6.45, and the dipole moment are the same. Further evaluation of Equation 6.48 proceeds through introduction of the LCAO-MO expansion (Equation 6.18) and, dependent on the level of theory, consideration of relevant approximations such as the NDDO formalism (Equation 6.31) in the case of semiempirical MNDO-type methods. Because the calculation of the dipole moment is usually considered a somewhat demanding test of the quality of the wavefunctions employed in the quantum chemical model, this property is included in the comparative statistical analysis of various methods to calculate molecular descriptors as presented in Section V. 

The term "semi-empirical" has been reserved commonly for electronic-based calculations which also starts with the Schrbdinger equation. " 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, NDDO, and INDO. The success of the MINDO, MINDO/3, " and MNDO " level of theory ultimately led to the development of AMl and a reparameterized variant known as PM3. In 1993, Dewar et al. introduced SAMI. Semi-empirical calculations have provided a wealth of information for practical applications. 

Theoretical studies at varying levels of approximation (CNDO/2, NDDO, MIN-DO/2, MINDO/3, MNDO, and STO-3G methods) have been carried out in order to predict the tautomerization energy AH0 (80CPL(69)537, 83JA3568). In all the cases the stability of the hydroxy form is strongly overestimated. 

In the case of transition metal complexes, the CNDO theory was first applied by Dahl and Ballhausen [24] to MnO,. Their scheme was later extended to INDO by Ziegler [25] and implemented into the general package ODIN [26], Better known is the INDO program ZINDO [27] by M. Zemer and the NDDO implementation due to D.S. Marynick [28]. Both have been applied with some success in transition metal chemistry for structure determination and studies of excited states. Attempts have also been made to extend AMI, PM3 and MNDO to transition metals. All in all it must be said that the methods based on integral approximations have been more prolific in studies of main group compounds than transition metal complexes. The reason for that is likely the considerable extra complexity added by the -orbitals combined with the fact that other attractive schemes are available for d-block compounds. 

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