Density matrices second-quantization form

In the above equations, hpv are the usual one-electron integrals while [juv Ao] and [juA vo] are the standard bare and antisymmetrized two-electron integrals, respectively. To derive these formulae, one has merely to substitute the second quantized form of the total Hamiltonian and apply the above rules for the density matrix elements. The analogy of Eq. (27) to the corresponding HF formula is obvious. 

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. 

While early work [16, 19] on the CSE assumed that Nakatsuji s theorem [37], proved in 1976 for the integrodifferential form of the CSE, remains valid for the second-quantized CSE, the author presented the first formal proof in 1998 [20]. Nakatsuji s theorem is the following if we assume that the density matrices are pure A-representable, then the CSE may be satisfied by and if and only if the preimage density matrix D satisfies the Schrodinger equation (SE). The above derivation clearly proves that the SE imphes the CSE. We only need to prove that the CSE implies the SE. The SE equation can be satisfied if and only if... 

To complete the definition of the renormalization step for the left block, we also need to construct the new matrix representations of the second-quantized operators. In the product basis Z <8> p, matrix representations can be formed by the product of operator matrices associated with left, p j and the partition orbital p separately. Then, given such a product representation of O say, the renormalized representation O in the reduced M-dimensional basis / of LEFIi. p is obtained by projecting with the density matrix eigenvectors L defined above,... 

Instead of using the transition densities explicitly, we revert to a matrix-element form of (14.4.2), noting that in second quantization (Problem 14.4)... 

The density matrix in the spin-orbital representation was introduced in second quantization for the evaluation of one-electron expectation values in the following form... 

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