Hellmann-Feynman theorem matrix

The adiabatic coupling matrix elements, Fy, can be evaluated using an off-diagonal form of the Hellmann-Feynman theorem... 

By the Hellmann-Feynman theorem [46], the derivative of an eigenvalue Ilgu (s) of a hermitian (here, real symmetric) matrix II11 (.y) with respect to a parameter h[ is given by the diagonal element of the matrix product,... 

The quantum mechanical many-body nature of the interatomic forces is taken into account naturally through the Hellmann-Feynman theorem. Since the scheme usually uses a minimal basis set for the electronic structure calculation and the Hamiltonian matrix elements are parametrized, large numbers of atoms can be tackled within the present computer capabilities. One of the distinctive features of this scheme in comparison with other empirical schemes is that all the parameters in the model can be obtained theoretically. It is therefore very useful for studying novel materials where experimental data are not readily available. The scheme has been demonstrated to be a powerful method for studying various structural, dynamical, and electronic properties of covalent systems. 

Note that Eq. (60) is not the Hellmann-Feynman theorem,to which it bears a formal resemblance, since in Eq. (60) it is not the Hamiltonian operator i e(x X) but rather the Hamiltonian matrix H (X) which is being differentiated. Indeed, from Eq. (49a)... 

As usual, the bracket notation ( ) implies integration over the electronic coordinates only, while F/j(R ) and G/j(R ) are the first- and second-order nonadiabatic matrix coupling elements, respectively. By using a derivation similar to that employed for the Hellmann-Feynman theorem, it may be shown that... 

Contents Introduction. - Concept of Creation and Annihilation Operators. -Particle Number Operators. - Second Quantized Representation of Quantum Mechanical Operators. - Evaluation of Matrix Elements. - Advantages of Second Quantization. - Illustrative Examples. - Density Matrices. -Connection to Bra and Ket Formalism. - Using Spatial Orbitals. - Some Model Hamiltonians in Second Quantized Form. - The Brillouin Theorem. -Many-Body Perturbation Theory. -Second Quantization for Nonorthogonal Orbitals. - Second Quantization and Hellmann-Feynman Theorem. - Inter-molecular Interactions. - Quasiparticle Transformations. Miscellaneous Topics Related to Second Quantization -Problem Solutions. - References -Index. 

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