Integrals semiempirical estimate

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 calculations are set up with the same general structure as a HF calculation in that they have a Hamiltonian and a wave function. Within this framework, certain pieces of information are approximated or completely omitted. Usually, the core electrons are not included in the calculation and only a minimal basis set is used. Also, some of the two-electron integrals are omitted. In order to correct for the errors introduced by omitting part of the calculation, the method is parameterized. Parameters to estimate the omitted values are obtained by fitting the results to experimental data or ah initio calculations. Often, these parameters replace some of the integrals that are excluded. 

For simple Hiickel calculations, the integrals aUu and fiuv are estimated by the semiempirical relations... 

Abstract. We have calculated the scalar and tensor dipole polarizabilities (/3) and hyperpolarizabilities (7) of excited ls2p Po, ls2p P2- states of helium. Our theory includes fine structure of triplet sublevels. Semiempirical and accurate electron-correlated wave functions have been used to determine the static values of j3 and 7. Numerical calculations are carried out using sums of oscillator strengths and, alternatively, with the Green function for the excited valence electron. Specifically, we present results for the integral over the continuum, for second- and fourth-order matrix elements. The corresponding estimations indicate that these corrections are of the order of 23% for the scalar part of polarizability and only of the order of 3% for the tensor part... 

In semiempirical calculations, yfJlj 10.8 eV is commonly used for two electrons occupying a p-orbital on the same carbon atom p and the integrals yfJJ, for pffv are usually estimated by a point charge approximation, y e2l(rllv + a), where a = < 2/ =133 pm, that is, y, 1440/(fy1 /pm + 133) eV. 

The integrals and yAB are calculated from semiempirical formulas that contain adjustable parameters. These are optimized in order to reproduce properties of a given class of molecules in an optimal manner (see Sec. 4). The intra-atomic electron repulsion integral yAA is estimated from experimentally measured ionization energies and electron affinities (see, for example, Ref. [23]). 

It is a trivial observation that the bond energy depends on the value of bond length. This fact was used in semiempirical models of quantum chemistry to estimate the value of resonance integral [13]. Recently the empirical model of estimating bond energy from bond length was presented for the systems built up of CC bonds [14]. Pauling proposed [15] a fractional bond number, n, defined as ... 

Valence-state ionization potentials are used to estimate integrals in semiempirical calculations (Sections 17.2 and 17.3). The valence-state ionization potential for a 2p electron in an sp -hybt vAvLoA carbon atom is the energy difference between the valence stale of jp -hybridized C and the valence state of ip. jiybridized C. ... 

Used in the calculation of the Hamiltonian parameters the value of the hopping integral t = 2.7 eV, hybridization potential F= -1.43 eV estimated from quantum chemical calculations of the electronic structure of CNTs within the semiempirical MNDO [14]. Electron energy impurity Sj =-5.72 eV was assessed using the method described in Refs. [6, 7]. 

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