Semi Empirical Schemes

The disadvantages are, of course, that the derived results depend on the severity of the NDO scheme and the values of the parameters. The latter depend on the quality and availability of experimental data. One seeks to derive a single set of atomic parameters which is capable of handling any environment that the atom finds itself in. The models are invariably valence AO only which therefore provides only limited flexibility. For example, the + 3 and + 5 oxidation states of phosphorous are distinct and place quite different demands on a model which seeks to describe both cases with a single set of valence 3s and 3p orbitals. One must take care to include representative examples of molecules with both oxidation states in order to derive a good set of parameters which will treat both equally. 

This type of problem is even more pronounced in TM chemistry where most elements have more than just two oxidation states. Add to this a relative scarcity of thermodynamic data and the obvious requirement to include d-orbitals into the valence-only treatment and it is no surprise that only a few INDO-based semi-empirical schemes are parameterised for metals [59,60] although results from these methods are encouraging (see Sect. 4). 

The real power of these approximate MO schemes lies in their ability to describe the most important orbital interactions. This relies mainly on getting the orbital energies and overlaps basically right. The former can be derived from AO energies while the latter are implicit in the molecular geometry. If we employ 

However, one must also be cautious when interpreting the results of approximate MO models. It is all too easy to endow the approximate method with the same virtues as the more rigorous ab initio scheme from which they are derived. Fenske has described [23] some of the possible pitfalls. For example, in Hartree-Fock theory there is a rigorous relationship between the stability of a molecule and its computed energy. Taking two isomeric species as an example, the one with the lowest computed energy is predicted to be the most stable. This relationship is often employed with approximate MO methods but it is no longer strictly valid. 

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The principal semi-empirical schemes usually involve one of two approaches. The first uses an effective one-electron Hamiltonian, where the Hamiltonian matrix elements are given empirical or semi-empirical values to try to correlate the results of calculations with experiment, but no specified and clear mathematical derivation of the explicit form of this one-electron Hamiltonian is available beyond that given above. The extended Hiickel calculations are of this type. 

The success of earlier empirical schemes in predicting the sign and magnitude of the enthalpy of formation can in part be attributed to making predictions for AB compounds. This can hide asymmetrical effects which are more apparent when other stoichiometries are considered. The empirical and semi-empirical schemes do... 

We see that the simple rectangular d band model reproduces the behaviour found by experiment and predicted by Miedema s semi-empirical scheme. However, we must stress that the model does not give credence to any theory that bases the heat of formation of transition-metal alloys on ionic Madelung contributions that arise from electronegativity differences between the constituent atoms because in the metallic state the atoms are perfectly screened and, hence, locally charge neutral. Instead, the model supports... 

The authors of this work proposed a semi-empirical scheme for the calculation of 13C chemical shift tensors based on the bond polarization theory (5). This method can reproduce 13C chemical shift tensors with deviations from experiment comparable to the errors of the ab initio methods. One major advantage is that the calculations can be performed for large molecular systems with hundreds of atoms even on a PC computer. In contrast to the ab initio method a set of empirical parameters is needed for the calculations. In the case of the bond polarization theory these parameters can be estimated directly from experimental chemical shifts solving a set of linear equations. 

Current interest in high temperature chemistry and the closely related thermodynamics of the actinides will provide additional stimuli for determining precise thermodynamic data in cryogenic as well as in higher temperature regions. The utopian era in the chemical thermodynamics of the lanthanides is sufficiently far off to occasion extension of shrewdly devised schemes to other classes of compounds. Use of the semi-empirical schemes already discussed—or theoretically based ones— plus the key compound concept may prove as effective here (desipte magnetic and electronic complications) as it has for hydrocarbon thermodynamics. 

As the name implies, semi-empirical calculations simplify the computational problem presented by the Schrodinger equation. Semi-empirical approaches eliminate some of the direct number crunching involved in ab initio calculations and instead replace portions of the calculation with values that may be taken from experimental data or other calculations that are parameterized to agree with empirical data. There is a variety of semi-empirical schemes that differ in the types of parameterizations that are made. Three common semi-empirical methods that are included in Spartan02 are called AMI, PM3, and MNDO. Each has strengths and weaknesses depending on the specific molecular environment one wishes to model. 

The full implications of the fact that one-electron SCF eigenvalues are, in semi-empirical schemes independent-particle eigenvalues are not always realized. In SCF—... 

It is often assumed that Koopmans Theorem can be generally employed for any MO calculation including those using semi-empirical schemes. The latter is not the case and care should be taken in assigning PES energies on the basis of such approximate methods [9]. 

Nevertheless, the increasing demands for models which can do more in terms of exploring PE surfaces continues to provide an impetus for the inclusion of TM atoms into the more sophisticated semi-empirical schemes. While the general performance of the models (vide infra) is encouraging, it is equally apparent that a significant amount of work remains before such methods enjoy the same level of accuracy for TM systems as they currently achieve for molecules comprising lighter, non-TM atoms. 

Yet even DFT is currently limited to molecules of at most a few hundred atoms. Semi-empirical schemes can handle larger systems but their accuracy is... 

The ligand-field model is a semi-empirical scheme of calculation that deals Avith the interpretation of the properties of complexes of metal ions with a partly filled /-shell. In this model the experiments are described in terms of empirically determined parameters whose coefficients are exactly calculated. 

It should be noted here that there are more approximate ways to perform DFT calculations that correctly describe the vdW forces. For example, several groups have recently proposed semi-empirical schemes to correct the usual DFT functionals so that the dispersion forces are modelled properly. " Also, a non-local DFT has been developed that incorporates the dispersion forces in a truly ab initio way this theoretical breakthrough leads to a very... 

This description of quantum mechanical methods for computing (hyper)polarizabilities demonstrates why, nowada, the determination of hyperpolarizabilities of systems containing hundreds of atoms can, at best, be achieved by adopting, for obvious computational reasons, semi-empirical schemes. In this study, the evaluation of the static and dynamic polarizabilities and first hyperpolaiizabilities was carried out at die Time-Dependent Hartree-Fock (TDOT) [39] level with the AMI [50] Hamiltonian. The dipole moments were also evaluated using the AMI scheme. The reliability of the semi-empirical AMI calculations was addressed in two ways. For small and medium-size push-pull polyenes, the TDHF/AMl approach was compared to Hartree-Fock and post Hartree-Fock [51] calculations of die static and dynamic longitudinal first hyperpolarizability. Except near resonance, the TDHF/AMl scheme was shown to perform appreciably better than the ab initio TDHF scheme. Then, the static electronic first hyperpolaiizabilities of the MNA molecule and dimer have been calculated [15] with various ab initio schemes and compared to the AMI results. In particular, the inclusion of electron correlation at the MP2 level leads to an increase of Paaa by about 50% with respect to the CPHF approach, similar to the effect calculated by Sim et al. [52] for the longitudinal p tensor component of p-nitroaniline. The use of AMI Hamiltonian predicts a p aa value that is smaller than the correlated MP2/6-31G result but larger than any of the CPHF ones, which results fi-om the implicit treatment of correlation effects, characteristic of die semi-empirical methods. This comparison confirms that a part of die electron... 

Consistency in Total Energy Calculations Implications for Empirical and Semi-Empirical Schemes. 

Pacchioni has recently carried out calculations on the low-lying states of Sn2 and Pb2. This author gives the impression that he is the first to carry out a comparative ab initio Cl calculation on these systems. We would like to clarify this further. First, his calculation starts with the Hafner-Schwarz model potentials in comparison to our relativistic ab initio potentials derived from numerical Dirac-Fock solutions of the atoms. Pacchioni s calculations ignore spin-orbit interaction. Our calculations include spin-orbit interaction in a relativistic Cl scheme in comparison to the non-relativistic Cl of Pacchioni. Thus, he obtains a Z), approximately twice the experimental value which he corrects by a semi-empirical scheme to arrive at a value close to our calculated value with a relativistic Cl. Our calculations have clearly demonstrated the need to carry out an intermediate-coupling Cl calculation for Pbj as a result of large spin-orbit contamination. Calculations without spin-orbit, such as Pacchioni s, have little relationship to the real Pb2 molecule. 

Celestino and Ermler have carried out calculations on Hg2 and TlHg. Calculations on Hg2 were carried out with a full four-electron Cl within 16 of the 22 valence and virtual orbitals. The spin-orbit interaction is ignored at the Cl stage but introduced after Cl using a semi-empirical scheme. For the TIHg molecule. Cl calculations included full correlation of five outer electrons with some restrictions. These authors have carried out calculations on a large number of low-lying states. 

Relativistic calculations of a number of polyatomic molecules have been carried out by both semi-empirical and ab initio methods. Many of the earlier calculations have been reviewed by Pyykko" as well as by Christiansen, Ermler and Pitzer. The semi-empirical methods include the relativistic extended Hiickel method, relativistic Xa method, etc. Most of the ab initio calculations on polyatomics initially omit the spin-orbit interaction. This is introduced at a later stage as a perturbation or by a semi-empirical scheme. 

A semi-empirical scheme is reintroduced for the conformational analysis of organic nitro compounds. In this scheme, proton chemical shifts are regarded as functions of short-range electrostatic contributions, through-space effects, and ring-currents, all of which are dependent on distances between atoms. 

This approach was applied to predict the valence state for a large number of Eu (Miedema 1976) and Yb (de Boer et al. 1979) intermetallic systems. A semi-empirical scheme was then used to obtain the quantities AH " (II) and AH "" (III). The result of this analysis for Eu intermetallics is shown in fig. 15. In this figure the valence state of a large number of Eu intermetallic compounds is plotted against the calculated heat of formation difference, AH " " (II)— AH " (III), for the compounds. It is seen that there is a critical value for the valence transition around 21 kcal/mol Eu, which in fact just corresponds to the value of En.iii for Eu metal. Considering the accuracy of A n,m and the calculated AH°° values, this perfect agreement must be considered to be somewhat fortuitous. However, the analysis shows that the separation between divalent and trivalent compounds is well described by the Miedema semiempirical scheme. 

In practice, the localized orbital is extracted from a model small molecule computed with the same semi-empirical method and incorporated into the QM/MM scheme. This is the basis of the Local Self-Consistent Field [37] which consists in solving the Hartree-Fock equations relative to the quanffim subsystem in which the electronic interactions with the electrons of the localized bond as well as the classical point charges are included. The adaptation of the PM3 semi-empirical scheme [38] to the AMBER force-field [9] and the analytical expression of the energy derivatives and changes of coordinates led to the so-called classical-quantum force-field (CQFF) [39]. In this force-field, the first atom belonging to the classical part and linked to the quanffim part by means of the... 

The use of known diatomic potentials to estimate the three-atom potential function is at the heart of the so-called London-Eyring-Polanyi(-Sato) (LEP(S)) method. This is a semi-empirical scheme based on the London equation, originally intended to deal with four one-electron S-state atoms. In its most primitive form, we begin by writing the potential between two atoms as a stun of a coulomb (Q)... 

There are numerous semi-empirical schemes, e.g., CINDO, MINDO, MNDO, AMI, and PM3, which differ in the amount of the full ab initio computation that is left out, and consequently, in the extent of parameterization. Different schemes also differ in the manner of parameterization, those mentioned above being parameterized with reference to experimental data as distinct from parameterization so as to most closely reproduce full ab initio results. Unlike MM. parameterization is not specific to each chemical bond type and the problem of incomplete parameterization is not encountered with simple organic compounds. Like MM, semi-empirical calculations are much less demanding of computer resources than full ab initio calculations. 

E(A ]B) is the energy of solution of a metallic atom A in a matrix B, A+1 is the (Z+1) element relative to A. Fig,6 shows the result of a study of some forty dilute alloys AB where the concentration of the dilute component A is 10% or less. The experimental shifts are analysed in terms of alloy heat of formation data according to a semi-empirical scheme due to Miedena. Again, the agreement with the above formula is good. For details, see ref. 8. 

In fact almost all went the way that Mulliken proposed, using the molecular orbital model. In this approach it was possible to formulate the equations in a manner suitable for calculation and to develop consistent approximation schemes that at least allowed semi-empirical calculations to be made. A number of semi-empirical schemes were developed, particularly for aromatic and conjugated systems, which can be regarded as inspired by the initial efforts of Hiickel in 1931 to use molecular orbitals in this area. For such systems the idea of a delocalised electron distribution came immediately out of the calculations, so that there was no need to invoke the bond or the idea of resonance. However the parameterisation schemes within these semi-empirical approaches, were cast in terms of integrals between hybrid orbitals, so that aspect of Pauling s ideas remained alive both in chemistry and in quantum chemistry. 

A semi-empirical scheme of the type described here has been used to construct a potential for the interaction between CO2 and a Pt(lll) surface. In ref. [196] this potential was used to study oxidation of CO on a platinum surface. 

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