Core Repulsion Integrals

From simple electrostatics the core-core repulsion (as used, for example, in CNDO/INDO) is  

In addition to this term, account must be taken of the decreasing screening of the nucleus by the electrons as the interatomic distance becomes very small. At very small distances the core-core term should approach the classical form. To account for this, an additional term is added to the basic core-core repulsion integral in MINDO/3 to give  

Here a B is a diatomic parameter. In the case of N-H and 0-H interactions only, this expression is replaced by  

The MINDO/3 parameters appearing in the above equations are discussed later in the parameter section. 

The basic idea of mixed model in MINDO/3 is the same as that used for CNDO and INDO and corrects Y b, which appears in the core Hamiltonian. Because the algorithm in calculating the Coulomb interaction in MINDO/3 is different from that used in CNDO and INDO, the procedure to correct Y b is also different from that in CNDO and INDO. 

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The two-center two-eleciroii. one-center two-electron, iwo-center one-electron, one-center one-electron, and core-core repulsion integrals involved in the above equations are discussed below. 

The core-core repulsion integrals are different for O-II and N-II interactions. They arc expressed as ... 

III fact, while this correction gives the desired behaviour at relatively long separations, it doLS not account for the fact that as two nuclei approach each other the screening by the core electrons decreases. As the separation approaches zero the core-core repulsion iimild be described by Coulomb s law. In MINDO/3 this is achieved by making the cure-core interaction a function of the electron-electron repulsion integrals as follows ... 

Electron-electron repulsion integrals, 28 Electrons bonding, 14, 18-19 electron-electron repulsion, 8 inner-shell core, 4 ionization energy of, 10 localization of, 16 polarization of, 75 Schroedinger equation for, 2 triplet spin states, 15-16 valence, core-valence separation, 4 wave functions of, 4,15-16 Electrostatic fields, of proteins, 122 Electrostatic interactions, 13, 87 in enzymatic reactions, 209-211,225-228 in lysozyme, 158-161,167-169 in metalloenzymes, 200-207 in proteins ... 

Core matrix elements, H, will be specified with individual methods. Indices k and m refer to closed and open shells, respectively c and y have their usual meaning of expansion coefficients and repulsion integrals, respectively. Numerical values of constants f, a, and b depend on the electronic configuration under study e.g., for a system having an unpaired electron in a nondegenerate... 

Thus, the current semi-empirical methods (MNDO, AMI and PM3) differ in the way in which core-repulsions are treated. Within the MNDO formalism the corerepulsion ( asmndo) is expressed in terms of two-centre, two-electron integrals (Eq. 5-4), where Za and Zb correspond to the core charges, Rab is the internuclear separation, and a a and aB are adjustable parameters in the exponential term [19]. 

Despite these modifications there remain a number of well-documented problems with the AM1/PM3 core-repulsion function [37] which has resulted in further refinements. For example, Jorgensen and co-workers have developed the PDDG (pair-wise distance directed Gaussian) PM3 and MNDO methods which display improved accuracy over standard NDDO parameterisations [38], However, for methods which include d-orbitals (e.g. MNDO/d [23,24], AMl/d [25] and AMI [39,40]) it has been found that to obtain the correct balance between attractive and repulsive Coulomb interactions requires an additional adjustable parameter p (previously evaluated using the one-centre two-electron integral Gss, Eq. 5-7), which is used in the evaluation of the two-centre two-electron integrals (Eq. 5-8). 

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],... 

The valence state ionization potential —the resonance integrals and the one-center electron repulsion integrals can be considered as basic parameters of the semiempirical method and can be adjusted to give optimal agreement. The core charges Z, indicate the number of 71 electrons the center M contributes to the n system, and the two-center electron repulsion integrals are obtained from an empirical relationship such as the Mataga-Nishimoto formula ... 

The initial Hiickel calculations can be employed to obtain preliminary values for the electron densities and bond orders, from which the self-consistent field matrix elements can be evaluated by introduction of the chosen core potentials and electron repulsion integrals.11 Table I lists the ionization potentials, electron affinities and nuclear charges employed in the present calculations. 

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