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By Andrei L. Tchougréeff

Hybrid tools of Molecular Modeling is a self-contained complex evaluate providing step-by-step derivation of the constant theoretical photograph of hybrid modeling tools and the thorough research of the recommendations and present sensible tools of hybrid modeling in keeping with this idea. Hybrid equipment of Molecular Modeling provides its fabric in a sequential method being attentive either to the actual soundness of the approximations used and to the mathematical rigor valuable for useful constructing of the powerful modeling code. ancient feedback are given whilst it's important to place the present presentation in a extra common context and to set up relation with different parts of computational chemistry. The reader must have adventure with easy strategies of computational chemistry and/or molecular modeling. simple wisdom of operators, wave services, electron densities is necessary. 

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By Andrei L. Tchougréeff

Hybrid tools of Molecular Modeling is a self-contained complex evaluate providing step-by-step derivation of the constant theoretical photograph of hybrid modeling tools and the thorough research of the recommendations and present sensible tools of hybrid modeling in keeping with this idea. Hybrid equipment of Molecular Modeling provides its fabric in a sequential method being attentive either to the actual soundness of the approximations used and to the mathematical rigor valuable for useful constructing of the powerful modeling code. ancient feedback are given whilst it's important to place the present presentation in a extra common context and to set up relation with different parts of computational chemistry. The reader must have adventure with easy strategies of computational chemistry and/or molecular modeling. simple wisdom of operators, wave services, electron densities is necessary. 

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This construct is extended to any number of orthonormal vectors. } be the set of its eigenvectors corresponding to the eigenvalues Ei , respectively. 95) H whose solutions are all operators Pˆ projecting to different possible subspaces ˆ spanned by eigenvectors of the operator H. 2. Resolvent Let us return to the Schr¨odinger equation for the projection operators Pˆi eq. 96) i This is simply the completeness relation for the eigenvectors of an Hermitian operˆ We introduce formally the resolvent of the operator H ˆ as a function of a ator H.

Tchougr´eeff where on the left side there is an operator in the brackets although on the right side it is a (complex) number. 99) ˆ ˆ Pˆi = Pˆi = R(z) ˆ ˆ Pˆi R(z) z Iˆ − H (z − Ei ) Pˆi = (z − Ei ) R(z) so that Pˆi z − Ei Summing this over all i’s yields, because of eq. 100) R(z) ˆ it is a matrix which reveals the structure of the resolvent of an Hermitian operator H: (operator) function of the complex variable z; it has simple poles Ei on the real axis ˆ which are the eigenvalues of H. 102) R(z) z Iˆ − H which is the Schr¨odinger equation in terms of the resolvent.

36)); neither is it assumed that the functions of the basis set are orthogonal. A priori they do not have any relation to the Hamiltonian under study – only boundary conditions must be fulfilled. 41) Ψ(ξ) = u k Φk k=1 so that the expansion amplitudes themselves take the part of the variational variables ξ and must be subject to the normalization condition Ψ(ξ)|Ψ(ξ) = 1. 44) u∗k Hkl ul ˆ E({ul }) = Ψ|H|Ψ = k,l=1 18 Andrei L. 47) Mkl = Φk |Φl is introduced. 48) l=1 The above equality must hold for each k, so that finding extrema of the auxiliary quadratic function is equivalent to a system of m linear equations.

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