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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
manyparticle Schrödinger equation. In principle, the nbody 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 nonrelativistic theory, Second
Quantization greatly simplifies the discussion of many identical interacting particles
22
. This
formalism has several distinct advantages: the secondquantized 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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approach of using antisymmetrized products of singleparticle wave functions. It also allows
us to concentrate on the few matrix elements of interest, thus avoiding the need for dialing
directly with the manyparticle 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 groundstate energy, the energy and lifetime
of excited states, and other molecular properties, are easily expressed in this formalism.
The Schrödinger equation
Any problem on nonrelativistic 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 BornOppenheimer 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 hcore 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 oneparticle 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 socalled symmetric twoparticle 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.
 Fall '03
 RichardBader

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