Chapter 1
Basic Concepts
Abstract In this chapter we examine some basic concepts of quantum chemistry to
give a solid foundation for the other chapters. We do not pretend to review all the
basics of quantum mechanics but rather focus on some specific topics that are central
in the theoretical description of magnetic phenomena in molecules and extended
systems. First, we will shortly review the Slater–Condon rules for the matrix elements
between Slater determinants, then we will extensively discuss the generation of spin
functions. Perturbation theory and effective Hamiltonians are fundamental tools for
understanding and to capture the complex physics of open shell systems in simpler
concepts. Therefore, the last three sections of this introductory chapter are dedicated
to standard Rayleigh–Schrödinger perturbation theory, quasi-degenerate perturbation
theory and the construction of effective Hamiltonians.
1.1 Slater Determinants and Slater–Condon Rules
The Slater determinant is the central entity in molecular orbital theory. The exact
N -electron wave function of a stationary molecule in the Born-Oppenheimer approximation is a 4N -dimensional object that depends on the three spatial coordinates and
a spin coordinate of the N electrons in the system. This object is of course too
complicated for any practical application and is, in first approximation, replaced
by a product of N orthonormal 4-dimensional functions that each depend on the
coordinates of only one of the electrons in the system.
Ψ(x 1 , y 1 , z 1 ,σ 1 , x 2 , y 2 , z 2 ,σ 2 ,...,x N , y N , z N ,σ N )
= φ a (x 1 , y 1 , z 1 ,σ 1 )φ b (x 2 , y 2 , z 2 ,σ 2 )...φ ω (x N , y N , z N ,σ N )
(1.1)
These one-electron functions are commonly referred to as spin orbitals and the product is known as the Hartree product Π . Obviously, the product suffers from important
deficiencies with respect to the foundations of Quantum Mechanics. The wave function is not antisymmetric with respect to the permutation of any two electrons, and
© Springer International Publishing Switzerland 2016
C. Graaf and R. Broer, Magnetic Interactions in Molecules and Solids,
Theoretical Chemistry and Computational Modelling,
DOI 10.1007/978-3-319-22951-5_1
1
Basic Concepts
Abstract In this chapter we examine some basic concepts of quantum chemistry to
give a solid foundation for the other chapters. We do not pretend to review all the
basics of quantum mechanics but rather focus on some specific topics that are central
in the theoretical description of magnetic phenomena in molecules and extended
systems. First, we will shortly review the Slater–Condon rules for the matrix elements
between Slater determinants, then we will extensively discuss the generation of spin
functions. Perturbation theory and effective Hamiltonians are fundamental tools for
understanding and to capture the complex physics of open shell systems in simpler
concepts. Therefore, the last three sections of this introductory chapter are dedicated
to standard Rayleigh–Schrödinger perturbation theory, quasi-degenerate perturbation
theory and the construction of effective Hamiltonians.
1.1 Slater Determinants and Slater–Condon Rules
The Slater determinant is the central entity in molecular orbital theory. The exact
N -electron wave function of a stationary molecule in the Born-Oppenheimer approximation is a 4N -dimensional object that depends on the three spatial coordinates and
a spin coordinate of the N electrons in the system. This object is of course too
complicated for any practical application and is, in first approximation, replaced
by a product of N orthonormal 4-dimensional functions that each depend on the
coordinates of only one of the electrons in the system.
Ψ(x 1 , y 1 , z 1 ,σ 1 , x 2 , y 2 , z 2 ,σ 2 ,...,x N , y N , z N ,σ N )
= φ a (x 1 , y 1 , z 1 ,σ 1 )φ b (x 2 , y 2 , z 2 ,σ 2 )...φ ω (x N , y N , z N ,σ N )
(1.1)
These one-electron functions are commonly referred to as spin orbitals and the product is known as the Hartree product Π . Obviously, the product suffers from important
deficiencies with respect to the foundations of Quantum Mechanics. The wave function is not antisymmetric with respect to the permutation of any two electrons, and
© Springer International Publishing Switzerland 2016
C. Graaf and R. Broer, Magnetic Interactions in Molecules and Solids,
Theoretical Chemistry and Computational Modelling,
DOI 10.1007/978-3-319-22951-5_1
1