Two-electron integral derivative

These specifications define choices (a) and (b) in the MNDO formalism and thus constitute the MNDO model. The original implementation of the model may be summarized as follows [19]. Conceptually the one-center terms are taken from atomic spectroscopic data, with the refinement that slight adjustments of the parameters are allowed in the optimization to account for possible differences between free atoms and atoms in a molecule. Any such adjustments should be minor to ensure that the one-center parameters remain close to their spectroscopic values and thus retain their physical significance. The one-center two-electron integrals derived from atomic spectroscopic data are considerably smaller than their analytically calculated values, which is (at least partly) attributed to an average incorporation of electron correlation effects. For reasons of internal consistency, these integrals provide the one-center limit (/ AB = 0) of the two-center two-electron integrals A o- ),... 

The one-center two-electron integrals in the MNDO method are derived from experimental data on isolated atoms. Most were obtained from Oleari s work L. Oleari, L. DiSipio, and G. DeMich-ells. Mol. Phys., 10, 97( 1977)1, but a few were obtained by IDewar using fits to molecular properties. 

Using the above asymptotic forms of the two-center two-electron integrals, the parameters and Ag can be derived. Certainly, parameter A is different for different orbitals even though they reside on the same atom. Dewar used AM to represent the parameter A obtain ed via G s, AD to represen t th e param eter A obtain ed via Hj,p, and AQ to represent the parameter A obtained from Hpp. 

The first and second derivatives of the energy with respect to the X variables ( 0) and "(O)) can be written in term of Fock matrix elements and two-electron integrals in the MO basis. For an RHF type wave function these are given as... 

The gradient of the energy is an off-diagonal element of the molecular Fock matrix, which is easily calculated from the atomic Fock matrix. The second derivative, however, involves two-electron integrals which require an AO to MO transformation (see Section 4.2.1), and is therefore computationally expensive. 

Consider now the case where the perturbation A is a specific nuclear displacement, A"i Xk + AX t. The derivatives of the one- and two-electron integrals are of two types, those involving derivatives of the basis functions, and those involving derivatives of the operators. The latter are given as... 

The A(0) and B(0) matrices as defined in Eqs. (41 and 42) depend on the orbital energies and several two-electron integrals collected into the elements defined in Eq. (43). The orbitals are chosen to be real. In this basis the derivatives of orbital energies with respect to a magnetic perturbation are zero. Therefore, only the derivatives of the elements are needed to evaluate A(1) and B(1). 

This energy expression forms the basis for the derivation of the MCSCF optimization methods. Note that the information about the molecular orbitals (the MO coefficients) is contained completely within the one- and two-electron integrals. The density matrices D and P contain the information about the Cl coefficients. 

Semiempirical molecular orbital methods23-25 incorporate parameters derived from experimental data into molecular orbital theory to reduce the time-consuming calculation of two-electron integrals and correlation effects. Examples of semiempirical molecular orbital methods include Dewar s AMI, MNDO, and MINDO/3. Of the three quantum chemical types, the semiempirical molecular orbital methods are the least sophisticated and thus require the least amount of computational resources. However, these methods can be reasonably accurate for molecules with standard bond types. 

Calculate the integrals Trs, Vrs for each nucleus, and the two-electron integrals (ru ts) etc. needed for Grs, as well as the overlap integrals Srs for the orthogonalizing matrix derived from S (see step 3). Note in the direct SCF method (Section 5.3) the two-electron integrals are calculated as needed, rather than all at once. 

The two-electron integrals pq kl] are < p(l)0fc(2) e2/ri2 0,(l)0j(2) > and may involve as many as four orbitals. The models of interest are restricted to one and two-center terms. Two electrons in the same orbital, [pp pp], is 7 in Pariser-Parr-Pople (PPP) theory[4] or U in Hubbard models[5], while pp qq are the two-center integrals kept in PPP. The zero-differential-overlap (ZDO) approximation[3] can be invoked to rationalize such simplification. In modern applications, however, and especially in the solid state, models are introduced phenomenologically. Particularly successful models are apt to be derived subsequently and their parameters computed separately. 

Let us summarize. The calculation of Cl first anharmonicities requires no storage or transformation of second and third derivative two-electron integrals, but the full set of first derivative MO integrals is needed. One must construct and transform one set of effective density elements for third derivative integrals and 3M — 6 sets of effective densities for second derivative integrals. In addition to the 3N — 6 MCSCF orbital responses k(1) and the Handy-Schaefer vector Cm needed for the Hessian, the first anharmonicity requires the solution of 3JV — 6 response equations to obtain (1). 

I. Panas, Chem. Phys. Lett., 184, 86 (1991). Two-Electron Integrals and Integral Derivatives Revisited. 

MO coefficients. CPHF equations involve (first) derivatives of the one- and two-electron integrals with ... 

The erivative of a p-function can thus be written in terms of an s- and a d-type Gaussian function, t he one- and two-electron integrals involving derivatives of basis functions are... 

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