Hellmann-Feynman theory Hamiltonians

Care must be taken in using the expressions above for obtaining nonlinear optical properties, because the values obtained may not be the same as those obtained from Eq. [4]. The results will be equivalent only if the Hellmann-Feyn-man theorem is satisfied. For the case of the exact wavefunction or any fully variational approximation, the Hellmann-Feynman theorem equates derivatives of the energy to expectation values of derivatives of the Hamiltonian for a given parameter. If we consider the parameter to be the external electric field, F, then this gives dE/dP = dH/d ) = (p,). For nonvariational methods, such as perturbation theory or coupled cluster methods, additional terms must be considered. 

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. 

In this section, we want to derive expressions for the second derivatives of the energy with respect to two components Pq,... and P. .. of the general electromagnetic field without relying on perturbation theory. According to the Hellmann-Feynman theorem, Eq. (3.7), the second derivative of the energy is equal to the first derivative of the expectation value of the derivative of the Hamiltonian for a non-zero value of the field, P 0, i.e. 

Having defined the perturbation Hamiltonians and perturbation operators we can now derive expressions for the cartesian components of the electric moments as derivatives of the energy in the presence of perturbing fields according to Eqs. (4.19) and (4.21). We have now two possibilities either we make use of the perturbation theory expansion of the energy, Eq. (3.15), or of the Hellmann-Feynman theorem. Let us start with perturbation theory. Because the moments are first derivatives we only need to consider the first-order energy correction, Eq. (3.29),... 

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