Semiempirical method

Semiempirical methods, of whieh there are quite a few, differ in the proportion of caleulations from first prineiples and the relianee on empirieal substitutions. Different methods of parameterization also lead to different semiempirieal methods. Huekel and extended Huekel ealeulations are among the simplest of the semiempirieal methods. In the next two seetions, we shall treat a semiempirieal method, the self eonsistent field method, developed by Paiiser and Parr (1953) and by Pople (1953), whieh usually goes under the name of the PPP method. 

One example of the use of semiempirical methodology is provided in an article detailing a molecular-dynamics simulation of the beta domain of metallothionein with a semiempirical treatment of the metal core.73 The beta domain of rat liver metallothionein-2 contains three-metal centers. In this study, three molecular variants with different metal contents—(1) three cadmium ions, (2) three zinc ions, and (3) one cadmium ion and two zinc ions—were investigated using a conventional molecular dynamics simulation, as well as a simulation with a semiempirical quantum chemical description (MNDO and MNDO/d) of the metal core embedded in a classical environment. For the purely classical simulations, the standard GROMOS96 force-field parameters were used, and parameters were estimated for cadmium. The results of both kinds of simulations were compared to each other 

Numerous semiempirical methods have been used to calculate dipole moments (e g, PPP, CNDO/2, CNDO/S, other CNDO variations, INDO, INDO/S, MNDO, MINDO/3, AMI, HAM3, etc ). They can be divided into 7t-electron and all-valence-electron methods. In u-electron methods such as, e g., the PPP (LCI-SCF-MO) method, only the 7i-component of the dipole moment is obtained and the o-component has to be computed separately. As in the case of empirical methods, one possibility is to calculate the o-component as a vector sum of the individual o-bond and group moments. These values are readily available from several sources [4-6,11,18,19,85] The resulting total (overall) dipole moment is then computed as a vector sum of the TT-moment and the o-moment. 

The o-components can also be obtained by calculation of all o-moments according to the approach suggested by Mulliken and Coulson [86,87], with the inclusion of homopolar and atomic dipoles and using Slater atomic orbitals [88-90]. Other possibilities are to calculate the o-electronic charges on the individual atoms [91-93] or the Del Re approach [94,95]. Some of the eariy papers devoted to dipole moments calculated by the PPP method were later subjected to a criticism by Exner [18], In one of the older studies, satisfactory dipole moments were obtained for pyridine and pyrrole using the variable electronegativity SCF method [96], 

The use of the various all-valence-electron methods (CNDO/2, CNDO/S, INDO, and the other above-mentioned methods) gives values of total dipole moments which are generally in good agreement with experimental values [97-99]. 

The CNDO/2 method [100,101] and the related methods have been successfully used to compute dipole moments [97,102] and there are numerous publications devoted to the use of the CNDO and INDO methods for calculations of dipole moments of heterocyclic compounds. However, in certain cases, some of these methods tend to lead to somewhat higher dipole moments than the actual experimental values. 

A number of authors have compared the validity of various semiempirical methods and optimized the parameters [103-106] It does not seem practical to give references to all calculations on this topic. Selected recent calculations of dipole moments of pyrimidine and purine bases and thdr derivatives and analogs will be mentioned here as practical examples [107-114]. Numerous references on semiempirical calculations of dipole moments of different types of heterocycles including pyrimidines and purines can be found in our publications [115-133]. 

In contrast to molecular mechanics force fields, modern semiempirical methods are classified as an SCF electron-structure theory (wave function-based) method [8]. Older (pre-HF) 

In its widely used form, the MNDO approximation and its variants (AM 1 and PM3) are suitable for the computation of properties of compounds of first row elements. The computational effort (often referred to by computational chemists as cost ) scales approximately with N2 (N = number of particles - nuclei and electrons). These methods serve as efficient tools for searching large conformational spaces, e.g. as a preliminary to subsequent higher level computations however, errors in the computations are less systematic than in ab initio methods. This is particularly evident when an error cannot be related to a physically measurable quantity. 

A key advantage of semiempirical methods is that they give heats of formation directly. Small cyclic hydrocarbons are typically computed to be too stable, and sterically crowded structures are predicted to be too unstable. This is because semiempirical methods do not describe weak interactions well, e.g. those arising from London dispersion forces thus, they would not be suitable to describe, for instance, molecular structures that rely heavily on hydrogen bonding interactions. 

In contrast to molecular mechanics force fields, modern semiempirical methods are classified as an SCF electron-structure theory (wavefunction-based) method [12]. Older (pre-HF) semiempirical approaches such as extended Hvickel theory, which can be classified as a one-electron effective Hamiltonian approach, involve drastic approximations but rely on the researcher s intuition and ability to extrapolate from simple computations to meaningful chemistry. This method is not used much these days but still plays a role in determining the band structures of organic polymers, most of which are carbon-rich by definition [13]. 

Nucleus-independent chemical shifts, introduced by Schleyer et al. as a measure for aromaticity [20], can also be computed at the GIAO-MNDO level of theory, making this useful tool also available to much larger aromatic structures such as graphenes and fullerenes [21]. The ab initio reference data are generally reproduced well, but four-membered rings are still problematic. The aromatic properties of Hiickel-type hydrocarbons, Mobius aromatics, and three-dimensional cage com- 

Our calculations with several established semiempirical schemes (INDO/S [66], MNDO [67], AMI [68], PM3 [69], and MNDO/d [70]) show that all these methods significantly underestimate the electronic coupling between r-stacked base pairs as compared with HF results. Typically, the matrix elements derived from semiempirical calculations are three to six times smaller ( ) than the corresponding HF values. 

This limitation has been overcome with a special NDDO-HT parameterization for calculating hole coupling matrix elements in DNA-related systems [72]. As reference data, coupling matrix elements were calculated for a set of 130 structures of WCP dimers with different step parameters at the HF/6-31G level. As discussed below in more detail, electronic couplings between neighboring pairs are extremely sensitive to conformational fluctuations of the DNA structure. For instance, the matrix element between base pairs in 

Molecular Modeling for the Design of Novel Performance Chemicals and Materials 

In the ab initio methods, MOs are usually expressed as the linear combinations of a finite number of basis functions  

The total electronic wave function is expressed as a linear combination of Slater determinants 

Hartree and Fock (Fock 1930 Hartree 1928) formalism uses the single SD form of the total wave function, which is solved under self-consistent field (SCF) approximation. This involves an iterative process in which the orbitals are improved cycle to cycle until the electronic energy reaches a constant minimum and the orbitals no longer change. Upon convergence of the SCF method, the minimum-energy MOs produce an electric field that generates the same orbitals and hence the self-consistency. 

The quantities are the energies of, respectively, the singlet and the triplet transition of the electron from the i- to the j-orbital, and K j is the exchange integral of Eq. (2.13). 

Equations (2.26) and (2.27) mean that in terms of the method under discussion there occur, respectively, a singlet or a triplet excited states lying close to the ground state. 

The triplet instability of the RHF solutions is a necessary, but insufficient, condition for the conclusion as to the biradical character or the triplet ground state of a given system, which would be important for an analysis of the internal mechanism of a number of reactions (see Sect. 5.1). Usually, a reliable result may be achieved in such cases by passing to the UHF approximation. 

Generally, instability of the HF solutions signifies drawing together of the PES s of different electron states and shows the need for the inclusion of correlation effects in order that a more exact description of the molecular system may be achieved. Since the effects of instability of the HF solutions are easy to diagnose, their detection should be viewed as procedure valuable for the prediction of structural features and reaction mechanisms. 

In nonempirical methods of calculation, usually about 70% of the computer time is spent on computing the integrals of the interelectron interaction (/iv A(t) in Eq. (2.5). As the size of a molecular system is increased, the number of such integrals grows roughly proportionally to (N is the size of the AO basis set of 

---

A highly readable account of early efforts to apply the independent-particle approximation to problems of organic chemistry. Although more accurate computational methods have since been developed for treating all of the problems discussed in the text, its discussion of approximate Hartree-Fock (semiempirical) methods and their accuracy is still useful. Moreover, the view supplied about what was understood and what was not understood in physical organic chemistry three decades ago is... 

Jones et al. [144,214] used direct dynamics with semiempirical electronic wave functions to study electron transfer in cyclic polyene radical cations. Semiempirical methods have the advantage that they are cheap, and so a number of trajectories can be run for up to 50 atoms. Accuracy is of course sacrificed in comparison to CASSCF techniques, but for many organic molecules semiempirical methods are known to perform adequately. 

In formulating a mathematical representation of molecules, it is necessary to define a reference system that is defined as having zero energy. This zero of energy is different from one approximation to the next. For ah initio or density functional theory (DFT) methods, which model all the electrons in a system, zero energy corresponds to having all nuclei and electrons at an infinite distance from one another. Most semiempirical methods use a valence energy that cor-... 

Semiempirical methods are parameterized to reproduce various results. Most often, geometry and energy (usually the heat of formation) are used. Some researchers have extended this by including dipole moments, heats of reaction, and ionization potentials in the parameterization set. A few methods have been parameterized to reproduce a specific property, such as electronic spectra or NMR chemical shifts. Semiempirical calculations can be used to compute properties other than those in the parameterization set. 

Many semiempirical methods compute energies as heats of formation. The researcher should not add zero-point corrections to these energies because the thermodynamic corrections are implicit in the parameterization. 

Semiempirical calculations have been very successful in the description of organic chemistry, where there are only a few elements used extensively and the molecules are of moderate size. Some semiempirical methods have been devised specifically for the description of inorganic chemistry as well. The following are some of the most commonly used semiempirical methods. 

The Huckel method and is one of the earliest and simplest semiempirical methods. A Huckel calculation models only the 7t valence electrons in a planar conjugated hydrocarbon. A parameter is used to describe the interaction between bonded atoms. There are no second atom affects. Huckel calculations do reflect orbital symmetry and qualitatively predict orbital coefficients. Huckel calculations can give crude quantitative information or qualitative insight into conjugated compounds, but are seldom used today. The primary use of Huckel calculations now is as a class exercise because it is a calculation that can be done by hand. 

PM3/TM is an extension of the PM3 method to include d orbitals for use with transition metals. Unlike the case with many other semiempirical methods, PM3/TM s parameterization is based solely on reproducing geometries from X-ray diffraction results. Results with PM3/TM can be either reasonable or not depending on the coordination of the metal center. Certain transition metals tend to prefer a specific hybridization for which it works well. 

The typed neglect of differential overlap (TNDO) method is a semiempirical method parameterized specifically to reproduce NMR chemical shifts. This... 

Semiempirical Methods of Electronic Structure Calculation G. A. Segal, Ed., Plenum, New York (1977). 

In order to obtain the best accuracy results as quickly as possible, it is often advantageous to do two geometry optimizations. The first geometry optimization should be done with a faster level of theory, such as molecular mechanics or a semiempirical method. Once a geometry close to the correct geometry has been obtained with this lower level of theory, it is used as the starting geometry for a second optimization at the final, more accurate level of theory. 

A basis set is a set of functions used to describe the shape of the orbitals in an atom. Molecular orbitals and entire wave functions are created by taking linear combinations of basis functions and angular functions. Most semiempirical methods use a predehned basis set. When ah initio or density functional theory calculations are done, a basis set must be specihed. Although it is possible to create a basis set from scratch, most calculations are done using existing basis sets. The type of calculation performed and basis set chosen are the two biggest factors in determining the accuracy of results. This chapter discusses these standard basis sets and how to choose an appropriate one. 

Unlike semiempirical methods that are formulated to completely neglect the core electrons, ah initio methods must represent all the electrons in some manner. However, for heavy atoms it is desirable to reduce the amount of computation necessary. This is done by replacing the core electrons and their basis functions in the wave function by a potential term in the Hamiltonian. These are called core potentials, elfective core potentials (ECP), or relativistic effective core potentials (RECP). Core potentials must be used along with a valence basis set that was created to accompany them. As well as reducing the computation time, core potentials can include the effects of the relativistic mass defect and spin coupling terms that are significant near the nuclei of heavy atoms. This is often the method of choice for heavy atoms, Rb and up. 

Some density functional theory methods occasionally yield frequencies with a bit of erratic behavior, but with a smaller deviation from the experimental results than semiempirical methods give. Overall systematic error with the better DFT functionals is less than with HF. 

Another related issue is the computation of the intensities of the peaks in the spectrum. Peak intensities depend on the probability that a particular wavelength photon will be absorbed or Raman-scattered. These probabilities can be computed from the wave function by computing the transition dipole moments. This gives relative peak intensities since the calculation does not include the density of the substance. Some types of transitions turn out to have a zero probability due to the molecules symmetry or the spin of the electrons. This is where spectroscopic selection rules come from. Ah initio methods are the preferred way of computing intensities. Although intensities can be computed using semiempirical methods, they tend to give rather poor accuracy results for many chemical systems. 

Transition structures are more dihicult to describe than equilibrium geometries. As such, lower levels of theory such as semiempirical methods, DFT using a local density approximation (LDA), and ah initio methods with small basis sets do not generally describe transition structures as accurately as they describe equilibrium geometries. There are, of course, exceptions to this, but they must be identihed on a case-by-case basis. As a general rule of thumb, methods that are empirically dehned, such as semiempirical methods or the G1 and G2 methods, describe transition structures more poorly than completely ah initio methods do. 

Ah initio trajectory calculations have now been performed. However, these calculations require such an enormous amount of computer time that they have only been done on the simplest systems. At the present time, these calculations are too expensive to be used for computing rate constants, which require many trajectories to be computed. Semiempirical methods have been designed specifically for dynamics calculations, which have given insight into vibrational motion, but they have not been the methods of choice for computing rate constants since they are generally inferior to analytic potential energy surfaces fitted from ah initio results. 

The most commonly used semiempirical for describing PES s is the diatomics-in-molecules (DIM) method. This method uses a Hamiltonian with parameters for describing atomic and diatomic fragments within a molecule. The functional form, which is covered in detail by Tully, allows it to be parameterized from either ah initio calculations or spectroscopic results. The parameters must be fitted carefully in order for the method to give a reasonable description of the entire PES. Most cases where DIM yielded completely unreasonable results can be attributed to a poor fitting of parameters. Other semiempirical methods for describing the PES, which are discussed in the reviews below, are LEPS, hyperbolic map functions, the method of Agmon and Levine, and the mole-cules-in-molecules (MIM) method. 

The simplest and most quickly computed models are those based solely on steric hindrance. Unfortunately, these are often too inaccurate to be trusted. Molecular mechanics methods are often the method of choice due to the large amount of computation time necessary. Semiempirical methods are sometimes used when molecular mechanics does not properly represent the molecule. Ah initio methods are only viable for the very smallest molecules. These are discussed in more detail in the applicable chapters and the sources mentioned in the bibliography. 

A few of the earliest methods did truncate the atom on the dividing line between regions. Leaving this unfilled valence is reasonable only for a few of the very approximate semiempirical methods that were used at that time. 

SMx semiempirical methods for very modest computational demands. 

Most of the semiempirical methods are not designed to correctly predict the electronic excited state. Although excited-state calculations are possible, particularly using a CIS formulation, the energetics are not very accurate. However, the HOMO-LUMO gap is reasonably reproduced by some of the methods. 

There is one semiempirical program, called HyperNMR, that computes NMR chemical shifts. This program goes one step further than other semiempiricals by defining different parameters for the various hybridizations, such as sp carbon vs. sp carbon. This method is called the typed neglect of differential overlap method (TNDO/1 and TNDO/2). As with any semiempirical method, the results are better for species with functional groups similar to those in the set of molecules used to parameterize the method. 

Another semiempirical method, incorporated in the VAMP program, combines a semiempirical calculation with a neural network for predicting the chemical shifts. Semiempirical calculations are useful for large molecules, but are not generally as accurate as ah initio calculations. 

Semiempirical calculations tend to be qualitative. In some cases, the correct trends have been predicted. In other cases, semiempirical methods give incorrect signs as well as unreasonable magnitudes. 

Semiempirical methods where they have been shown to reproduce the correct trends... 

Many semiempirical methods have been created for modeling organic compounds. These methods correctly predict many aspects of electronic structure, such as aromaticity. Furthermore, these orbital-based methods give additional information about the compounds, such as population analysis. There are also good techniques for including solvation elfects in some semiempirical calculations. Semiempirical methods are discussed further in Chapter 4. 

---