By Andrei L. Tchougréeff

Hybrid tools of Molecular Modeling is a self-contained complex evaluate delivering step-by-step derivation of the constant theoretical photograph of hybrid modeling tools and the thorough research of the strategies and present sensible tools of hybrid modeling according to this conception. Hybrid equipment of Molecular Modeling offers its fabric in a sequential manner being attentive either to the actual soundness of the approximations used and to the mathematical rigor precious for functional constructing of the powerful modeling code. ancient feedback are given while it will be important to place the present presentation in a extra basic context and to set up relation with different components of computational chemistry. The reader must have event with uncomplicated strategies of computational chemistry and/or molecular modeling. easy wisdom of operators, wave services, electron densities is necessary. 

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Assembling the quantities Hkl and Mkl , into M ×M matrices H and M representing the Hamiltonian and the metric, respectively, and the amplitudes ui into a column-vector u, we rewrite the system of linear equations (eq. 48)) in the form (H − EM)u = 0 or Hu = EMu which is known as a “generalized matrix eigenvalue problem”. If the functional basis {Φk } is taken to be orthonormalized, Mkl = δkl ; the metric matrix in this basis becomes the unity matrix, M = 1. 49) Hu = Eu Solving the eigenvalues problem for a Hermitian matrix is equivalent to diagonalizing it by performing the similarity transformation of the matrix H by the unitary matrix U composed of its eigenvectors.

133) ˆH ˆ Pˆ Pˆ Ψ = E Pˆ Ψ ˆ Pˆ + Pˆ H ˆ Q(E ˆ Q ˆ−Q ˆH ˆ Q) ˆ −1 Q Pˆ H ˆ eff (E) and itself depends on energy. Its The expression in the square brackets is an H most important characteristic is that it acts in the subspace defined by the projection operator Pˆ (Pˆ -block) Im Pˆ , but its eigenvalues by construction coincide with the eigenvalues of the exact Hamiltonian. The eq. 133) represents a pseudoeigenvalue ˆ eff (E), where “pseudo” indicates problem as the operator in the right hand part is H its own dependence on the sought energy eigenvalue.

Obtaining the corrections to the wave functions (eigenvectors) depends on the character of the spectrum of the eigenvalues of the unperturbed ˆ (0) . Two major cases are distinguished: when all eigenvalues of the Hamiltonian H zero order problem eq. 51) are different it is referred to as the nondegenerate case. When some of the eigenvalues of the unperturbed problem coincide, it is referred to as a degenerate case. These cases are generally considered separately. 1. Nondegenerate case The problem of finding a vector is usually solved by representing the required vector as an expansion with respect to some natural set of basis vectors.

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