All-valence semiempirical methods

In contrast to the pseudopotential methods where the Hartree-Fock method is used to construct the subset of orbitals spanning the core and valence carrier subspaces, whereas the calculation in the valence subspace can be performed at any level of correlation accounting, for the overwhelming majority of the semi-empirical methods, the electronic structure of the valence shell is described by a single determinant (HFR) wave function eq. (1.142). 

Fock operator in the basis of AOs were set equal to ionization potential characteristic for the AO at hand  

The self-consistent nature of the Fock operator is sometimes modeled in the methods without interaction by the schemes relating the ionization potentials of each AO with the overall electronic population (or effective charge) of a given atom and sometimes with the orbital populations of the AOs centered on the considered atom. This generally leads to the expressions of the form [55]  

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From all two-electron integrals (pv per), only the diagonal terms of type (pp pp) remain. This reduces the size dependence of semiempirical methods from formally N4 (as is the case for HF methods) to only N2. Here lies the main reason why semiempirical methods are able to treat very large systems with up to N= 10000 electrons. (It has to be kept in mind that for the PPP method, the number of electrons and the number of atomic basis functions are identical. This is, in general, not the case for all-valence semiempirical methods and certainly not for ab initio methods using extended basis sets.)... 

Semiempirical MO methods neglect the explicit consideration of atomic inner shells and usually use only a minimal basis set for the valence electrons (all-valence semiempirical methods). Sometimes this basis is enhanced by addition of polarization, Rydberd, and ghost orbitals 18-49-53,83,177-179,215,228-232... 

The description of configuration interaction given for rr-electron methods is also valid for all-valence-electron methods. Recently, two papers were published in which the half-electron method was combined with a modified CNDO method (69) and the MINDO/2 method was combined with the Roothaan method (70). Appropriate semiempirical parameters and applications of all-valence-electron methods are most probably the same as those reviewed for closed-shell systems (71). 

Until now, applications of semiempirical all-valence-electron methods have been rare, although the experimental data for a series of alkyl radicals are available (108,109). In Figure 9, we present the theoretical values of ionization potentials calculated (68) for formyl radical by the CNDO version of Del Bene and Jaffe (110), which is superior to the standard CNDO/2 method in estimation of ionization potentials of closed-shell systems (111). The first ionization potential is seen, in Figure 9, to agree fairly well with the experimental value. Similarly, good results were also obtained (113) with some other radicals (Table VII). 

A nonempirical approach to the chemical reactivity may of course be made along the same lines as has been practised for years in treatments by semiempirical all-valence electron methods. Typically, the results of such treatments provide qualitative explanation of the observed facts and give guidance for further experiments. Here we shall deal only with what may be taken as the ultimate goal of ab initio calculations in the field of chemical reactivity - the predictions of absolute values of equilibrium and rate constants. 

In order to improve the model further we are currently taking quantum effects in the lattice into account, i.e. treating the CH units not classically but on quantum mechanical basis. To this end we use an ansatz state similar to Davydov s so-called ID,> state [96] developed for the description of solitons in proteins. However, there vibrations are coupled to lattice phonons, while in tPA fermions (electrons) are coupled to the lattice phonons. The results of this study will be the subject of a forthcoming paper. Further we want to improve the description of the electrons by going to semiempirical all valence electron methods or even to density functional theories. Further we introduce temperature effects into the theory which can be done with the help of a Langevin equation (random force and dissipation terms) or by a thermal population of the lattice phonons. Starting then the simulations with an optimized soliton geometry in the center of the chain (equilibrium position) one can study the soliton mobility as function of temperature. Further in the same way the mobility of polarons can be... 

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. 

In this review we summarized our experience with the development and applications of semiempirical Pariser-Parr-Pople (PPP)-type and all-valence-electron methods to electronic spectra of radicals. After the era of PPP calculations on closed-shell molecules and the advent of semiempirical all-valence-electron methods, the electronic spectra of radicals represented a new challenge for molecular orbital (MO) theory. It was a time when progress in experimental techniques resulted in accumulation of a vast amount of data on the electronic spectra of radicals of various structural types. Compared to closed-shell molecules, the electronic spectra of some radicals exhibited peculiar features bands in the near infrared, many transitions in the whole UV/vis region, and some bands of extraordinary intensity. Clearly, without the help of MO theory, their interpretation seemed even harder than with closed-shell molecules. 

We decided to undertake this with the aim of formulating a generally applicable computational scheme for radicals which would be a natural extension of the PPP and semiempirical all-valence-electron methods for closed-shell molecules. We used for this purpose the self-consistent-field open-shell methods of Longuet-Higgins and Pople [2] and of Roothaan [3], we derived all expressions necessary for CI-S calculations [4, 5], and we tested the semiempirical open-shell PPP-type and INDO/S calculations systematically for various classes of radicals. 

Since the development of semiempirical MO methods has been repeatedly reviewed (for example, for all-valence-electron methods and n electron methods ), we shall concentrate here only on the essentials. 

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-... 

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. 

The semiempirical methods represent a real alternative for this research. Aside from the limitation to the treatment of only special groups of electrons (e.g. n- or valence electrons), the neglect of numerous integrals above all leads to a drastic reduction of computer time in comparison with ab initio calculations. In an attempt to compensate for the inaccuracies by the neglects, parametrization of the methods is used. Meaning that values of special integrals are estimated or calibrated semiempirically with the help of experimental results. The usefulness of a set of parameters can be estimated by the theoretical reproduction of special properties of reference molecules obtained experimentally. Each of the numerous semiempirical methods has its own set of parameters because there is not an universial set to calculate all properties of molecules with exact precision. The parametrization of a method is always conformed to a special problem. This explains the multiplicity of semiempirical methods. 

Semiempirical methods of calculation with consideration of all valence electrons have been used only recently but already have given results on the reactivities of some aromatic and heteroaromatic com-pounds. " Thus, to analyze the reactivities of thiophene and the isomeric thienothiophenes 1-3 to electrophilic substitution, the semiempirical SCF LCAO MO method CNDO/2 was used, taking into account all valence electrons.The 3s, 3p, and 3d orbitals have been taken into account for the sulfur atom. Tlie reactivities were estimated from the difference between bond energies of the initial and the protonated molecule (in a complex). ... 

The reactivities of isomeric thienothiophenes calculated in n -electron approximation by the PPP method, and those calculated considering all valence electrons, show reasonable agreement. It should be noted, however, that the choice of parameters in PPP calculations is somewhat arbitrary, especially for heavy atoms (e.g., sulfur). This may lead to a discrepancy between theoretical (in 7r-electron approximation) and experimental estimation of reactivities. For example, Clark applied the semiempirical method PPP SCF MO to calculate the reactivities of different positions in thienothiophenes 1—3, thiophene, and naphthalene from the localization energy values and found the following order of decreasing reactivity for electrophilic substitution thieno[3,4-b]-thiophene (3) > thieno[2,3-Z>]thiophene (I) > thieno [3,2-b]thiophene... 

We now consider the PPP, CNDO, INDO, and MINDO two-electron semiempirical methods. These are all SCF methods which iteratively solve the Hartree-Fock-Roothaan equations (1.296) and (1.298) until self-consistent MOs are obtained. However, instead of the true Hartree-Fock operator (1.291), they use a Hartree-Fock operator in which the sum in (1.291) goes over only the valence MOs. Thus, besides the terms in (1.292), f/corc(l) m these methods also includes the potential energy of interaction of valence electron 1 with the field of the inner-shell electrons rather than attempting a direct calculation of this interaction, the integrals of //corc(/) are given by various semiempirical schemes that make use of experimental data furthermore, many of the electron repulsion integrals are neglected, so as to simplify the calculation. 

The ground-state wave function of cytosine has been calculated by practically all the semiempirical as well as nonempirical methods. Here, we shall discuss the application of these methods to interpret the experimental quantities that can. be calculated from the molecular orbitals of cytosines and are related to the distribution of electron densities in the molecules. The simplest v-HMO method yielded a great mass of useful information concerning the structure and the properties of biological molecules including cytosines. The reader is referred to the book1 Quantum Biochemistry for the application of this method to interpret the physicochemical properties of biomolecules. Here we will restrict our attention to the results of the v-SCF MO and the all-valence or all-electron treatments of cytosines. 

The electronic structure of the pyrrolizine anion has been investigated by the HMO-,115 the ASMO-SCF-MI-,117 and a CNDO/2-SCF semiempirical method of Pople-Santry-Segal, taking into account all valence electrons... 

During the early years at McGill University, Whitehead s group concentrated on experimental nuclear quadrupole resonance spectroscopy123 and a variety of n- and all-valence electron semiempirical molecular orbital methods.124 His recent interests have included topics as diverse as density functional theory125 and related topics,126 and molecular models of surfactants. 

The quantum mechanical methods described in this book are all molecular orbital (MO) methods, or oriented toward the molecular orbital approach ab initio and semiempirical methods use the MO method, and density functional methods are oriented toward the MO approach. There is another approach to applying the Schrodinger equation to chemistry, namely the valence bond method. Basically the MO method allows atomic orbitals to interact to create the molecular orbitals of a molecule, and does not focus on individual bonds as shown in conventional structural formulas. The VB method, on the other hand, takes the molecule, mathematically, as a sum (linear combination) of structures each of which corresponds to a structural formula with a certain pairing of electrons [16]. The MO method explains in a relatively simple way phenomena that can be understood only with difficulty using the VB method, like the triplet nature of dioxygen or the fact that benzene is aromatic but cyclobutadiene is not [17]. With the application of computers to quantum chemistry the MO method almost eclipsed the VB approach, but the latter has in recent years made a limited comeback [18],... 

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