Second Quantization and Hellmann-Feynman Theorem

In the preceding chapters we have seen the advantages of the second quantized approach rather from the formalistic point of view. Here we take another standpoint and show an example where second quantization involves a somewhat different interpretation of the results. The example will be somewhat peculiar (Surjan et al. 1988) we shall study the significance of the Hellmann-Feynman theorem in evaluating the first derivatives of the energy, which is topical in the field of optimizing molecular geometries, exponents of basis orbitals, etc. 

As known from quantum mechanics, the Hellmann-Feynman theorem states  

R is any parameter of the Hamiltonian (Hellmann 1937, Feynman 1939). Equation (14.1) is often written in form of an arbitrary variation as  

It is of some interest to consider whether there is a possibility to find a theoretical formalism permitting one to extend the applicability of the Hellmann-Feynman theorem to such cases. In what follows we shall take advantage of the second quantized formalism to investigate this point. In this framework one has a model Hamiltonian for which quantum-chemical wave functions expanded in a finite orbital basis obey a formal Hellmann-Feynman theorem. 

This holds in all cases when the usual Hellmann-Feynman theorem is violated only just due to the finiteness of the basis. 

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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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