Second Quantization for Fermions

Second Quantization for Fermions - 29 Chapter 2 Second...

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29 Chapter 2 Second Quantization for Fermions Mario Piris Instituto Superior de Ciencias y Tecnología Nucleares, Ave Salvador Allende y Luaces, Quinta de los Molinos, La Habana 10600, Cuba . The object of quantum chemistry consists of interacting many particle systems of electrons and nuclei. An accurate description of such systems requires the solution of the many-particle Schrödinger equation. In principle, the n-body wave function in configuration space contains all possible information, but a direct solution of the Schrödinger equation is impractical. It is therefore necessary to resort to approximate techniques, and to work within the framework of more convenient representation of the quantum mechanic operators and wave functions: the Second Quantization. In a relativistic theory, the concept of Second Quantization is essential to describe the creation and destruction of particles 1 . However, even in a non-relativistic theory, Second Quantization greatly simplifies the discussion of many identical interacting particles 22 . This formalism has several distinct advantages: the second-quantized operators incorporate the statistics (Fermi in our case) at each step, which contrasts with the more cumbersome . L.A. Montero, L.A. Díaz and R. Bader (eds.), Introduction to Advanced Topics of Computational Chemistry , 29 - 39, 2003, © 2003 Editorial de la Universidad de La Habana, Havana.
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30 approach of using antisymmetrized products of single-particle wave functions. It also allows us to concentrate on the few matrix elements of interest, thus avoiding the need for dialing directly with the many-particle wave functions and the coordinates of all the remaining particles. Finally, the Green’s functions and density matrices, which contain the most important physical information concerning to the ground-state energy, the energy and lifetime of excited states, and other molecular properties, are easily expressed in this formalism. The Schrödinger equation Any problem on non-relativistic electronic structure requires the solution of the Schrödinger equation, Ψ = Ψ E H ˆ ( 1 ) where H ˆ and Ψ represent the Hamiltonian and the wave function of the system respectively. In the Born-Oppenheimer approximation, the electronic Hamiltonian for atoms and molecules (the systems of our interest) takes the form ∑∑ =+ = = = == = + = + = 1 n 1 i n 1 i j ij n 1 i i 1 n 1 i n 1 i j ij n 1 i N 1 I iI I n 1 i 2 i elec r 1 h ˆ r 1 r Z ˆ 2 1 H ˆ (2) where i h ˆ is the h-core operator, which contains the kinetic energy and the potential energy of interaction between nuclei and electrons. We should observe that this term is composed by the sum of operators involving the coordinates of the particles one at a time hence, it belongs to the group of the symmetric one-particle operators: = = n 1 i i n h ˆ A ˆ ( 3 ) The second term represents the potential energy of interaction between every pair of particles therefore; it belongs to the so-called symmetric two-particle operators, = = 1 n 1 i n 1 i j n 1 j . i 1 ij 1 ij n r ' 2 1 r B
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This note was uploaded on 10/29/2010 for the course MCMASTER U 1958 taught by Professor Richardbader during the Fall '03 term at McMaster University.

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Second Quantization for Fermions - 29 Chapter 2 Second...

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