Repulsion integrals

IlyperChem uses 16 bytes (two double-precision words) of storage for each electron repulsion integral. The first 8 bytes save thecom-pressed four indices and the second S bytes store the value of the integral. Each index lakes 16 bits. Thus the maximum number of basis fiinctions is 65,535. This should satisfy all users of IlyperChem for the foreseeable future. 

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. 

In addition to this term, account must be taken of the decreasing screen in g of then iicleus by th e electron s as the in teratom ic dis-tance 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 MlXnO/3 to give ... 

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

Che two-centre repulsion integrals 7ab in MINDO/3 are calculated using the following niaction ... 

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 ... 

Using Program SCF for ethylene and 1,3,5-hexatriene, list the electron repulsion integrals in the foiiii Yjj, Yj2, and so on. Take the coordinates from Figure 8-6. Try small variations in the atomic coordinates to see what their influence is on Yy. 

There is considerable variation in the values assigned to the election repulsion integrals in Exercise 8.9.1. Salem (1966) points out that calculation using Slater orbitals leads to... 

The calculation of the two-electron repulsion integrals in ab initio method is inevitable and time-consuming. The computational time is mainly dominated by the performance of the two-electron integral calculation. The following items can control the performance of the two-electron integrals. 

Set this threshold to a small positive constant (the default value is 10" ° Hartree). This threshold is used by HyperChem to ignore all two-electron repulsion integrals with an absolute value less than this value. This option controls the performance of the SCF iterations and the accuracy of the wave function and energies since it can decrease the number of calculated two-electron integrals. 

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. 

Because of the use of two double-precision words for each integral, HyperChem needs, for example, about 44 MBytes of computer main memory and/or disk space to store the electron repulsion integrals for benzene with a double-zeta 6-3IG basis set. 

The two-center two-electron repulsion integrals ( AV Arr) represents the energy of interaction between the charge distributions at atom Aand at atom B. Classically, they are equal to the sum over all interactions between the multipole moments of the two charge contributions, where the subscripts I and m specify the order and orientation of the multipole. MNDO uses the classical model in calculating these two-center two-electron interactions. 

The term ( iv X.o) in Equation 32 signifies the two-electron repulsion integrals. Under the Hartree-Fock treatment, each electron sees all of the other electrons as an average distribution there is no instantaneous electron-electron interaction included. Higher level methods attempt to remedy this neglect of electron correlation in various ways, as we shall see. 

Many simple sehemes have been put forward for these repulsion integrals, which are usually written. They are taken to depend on the type of atoms that basis funetion x, and Xj are centred on, and on the distance between the atomic centres. Pariser and Parr made use of the uniformly charged sphere representation illustrated in Figure 8.1. 

The repulsion integrals can be calculated from standard formulae, and they come to the values shown in Table 8.5. 

Craig, D. P., Proc. Roy. Soc. [London) A202, 498, Electronic levels in simple conjugated systems. I. Configuration interaction in cyclobutadiene. (ii) All the interelectron repulsion integrals, three- and four-centered atomic integrals, are included. 

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 ... 

The first summation requires electron repulsion integrals with four virtuaJ indices. Efficient algorithms that avoid the storage of these integrals have been discussed in detail [20]. For every orbital index, p, this OV contraction must be repeated for each energy considered in the pole search it is usually the computational bottleneck. 

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