1.2 Generation of Many Electron Spin Functions
21
state, the up-down-up situation depicted on the right is not the doublet state but rather
a superposition of the doublet and quartet spin functions given in Eqs. 1.40 and 1.41.
1.11 Calculate the overlap of the quartet and doublet spin functions given in
Eqs. 1.40 and 1.41 with the |αβα| determinant.
1.3 Perturbation Theory
Many body perturbation theory is one of the fundamental tools in Quantum Chemistry. It takes a central place both in the calculation of accurate energies and wave
functions, and in the analysis of results for reaching a better understanding of the
sometimes complicated physics contained in the system. There are basically two
flavors of many-body perturbation theory. The first is what one calls the diagonalizeand-then-perturb method, and the second one inverts this order, it follows a perturband-then-diagonalize approach.
When one is only interested in a single state that is well separated from all the
others, for example a non-degenerate ground state, the distinction is not very relevant. Using a proper zeroth-order wave function such as the one provided by the
Hartree–Fock approach, the effect of electron correlation can be estimated with any
standard perturbation scheme, being the Møller–Plesset implementation the most
common one.
However, it becomes a little more subtle when one wants to describe a collection of states of a quantum system that are close in energy, or when states with a
marked multiconfigurational character have to be described. The reference space is
now spanned by several Slater determinants that define a collection of electronic
states. Most approaches first diagonalize the reference space and then introduce
the effect of the external determinants with perturbation theory. In contrast, quasidegenerate perturbation theory (QDPT), first addresses the external determinants for
all the matrix elements among the reference determinants and then diagonalizes the
reference space to obtain the energies and wave functions of the states of interest.
1.3.1 Rayleigh–Schrödinger Perturbation Theory
There are only few systems for which the Schrödinger equation can be solved exactly.
Therefore, many schemes have been developed to obtain as accurate as possible
approximate solutions. The perturbative treatment is based on the partition of the
full Hamiltonian of the system in two parts.
ˆ
H = ˆ
H
(0) + λ ˆ
V
(1.58)
21
state, the up-down-up situation depicted on the right is not the doublet state but rather
a superposition of the doublet and quartet spin functions given in Eqs. 1.40 and 1.41.
1.11 Calculate the overlap of the quartet and doublet spin functions given in
Eqs. 1.40 and 1.41 with the |αβα| determinant.
1.3 Perturbation Theory
Many body perturbation theory is one of the fundamental tools in Quantum Chemistry. It takes a central place both in the calculation of accurate energies and wave
functions, and in the analysis of results for reaching a better understanding of the
sometimes complicated physics contained in the system. There are basically two
flavors of many-body perturbation theory. The first is what one calls the diagonalizeand-then-perturb method, and the second one inverts this order, it follows a perturband-then-diagonalize approach.
When one is only interested in a single state that is well separated from all the
others, for example a non-degenerate ground state, the distinction is not very relevant. Using a proper zeroth-order wave function such as the one provided by the
Hartree–Fock approach, the effect of electron correlation can be estimated with any
standard perturbation scheme, being the Møller–Plesset implementation the most
common one.
However, it becomes a little more subtle when one wants to describe a collection of states of a quantum system that are close in energy, or when states with a
marked multiconfigurational character have to be described. The reference space is
now spanned by several Slater determinants that define a collection of electronic
states. Most approaches first diagonalize the reference space and then introduce
the effect of the external determinants with perturbation theory. In contrast, quasidegenerate perturbation theory (QDPT), first addresses the external determinants for
all the matrix elements among the reference determinants and then diagonalizes the
reference space to obtain the energies and wave functions of the states of interest.
1.3.1 Rayleigh–Schrödinger Perturbation Theory
There are only few systems for which the Schrödinger equation can be solved exactly.
Therefore, many schemes have been developed to obtain as accurate as possible
approximate solutions. The perturbative treatment is based on the partition of the
full Hamiltonian of the system in two parts.
ˆ
H = ˆ
H
(0) + λ ˆ
V
(1.58)