90
K. B. Be´ c et al.
that needs to be fulfilled within the iterative procedure of solving the Hartree equations. The most far-reaching approximation assumed in the HF approach is neglecting
the Coulomb correlation, which is often described as the mean-field charge distribution approximation of the electron correlation, since the HF method effectively
averages the electron–electron interactions. This causes an inherent inability of the
method to properly describe London dispersion. In order to step beyond a mean-field
approximation of independent particles, so-called post-HF methods have been developed. The HF theory has a critical historical importance, being the first developed
quantum theory with practical implementation. Nowadays, pure HF calculations are
rarely used. However, the method is still widely adopted for calculations of the
initial wavefunction of a quantum system, thus representing the preliminary step for
calculations at higher levels of theory. On the other hand, a hierarchy of increasingly accurate methods based on the HF results exist, in which more than one Slater
determinant is employed.
5.3.2.2 Post-HF Approaches
Different populations of atomic orbitals by electrons or electron configurations in
a quantum system are possible. When any given electron changes its configuration,
which can be described as an excitation into another orbital, the distribution of the
other electrons in the molecule adjusts to minimize the total energy of the system.
Thus, the motion of electrons is not independent but correlated, which lowers the
total energy of the system. However, in the HF approach, any given electron only
interacts with the average potential of all the other electrons in the system. To amend
this shortcoming, post-HF methods aiming at a more accurate treatment of electron
correlation effects were introduced. This may be accomplished in different ways, but
unequivocally increases the computational complexity of the method by orders of
magnitude.
In the simplest case, the electron correlation energy can be treated as a perturbation of the electronic state described in the HF formalism. As long as electron
correlation has a relatively small contribution to the total energy, it can be expressed
via a perturbing Hamiltonian corresponding to a correction added to the HF Hamiltonian. Since the unperturbed HF state is known, the perturbative correction is solvable
using approximate methods, e.g., via an asymptotic series. The practical formulation of this approach is based on Møller–Plesset theory (MP) of a given order k.
Zeroth-order wavefunction corresponds to an unperturbed HF state, and the firstorder perturbation correction (MP1) to the HF energy can be shown to be equal
to zero, which implies that only second- (MP2) and higher-order MP expressions
are practically meaningful. Among those, MP2 bears the highest practical usefulness and finds broad applications. In most cases, higher-order perturbations (such
as MP3, MP4, and MP5) do not improve the accuracy by an acceptable margin and
display a huge computational demand. The MP2 method has become particularly
widely applied since it is the most efficient approach to take electron correlation
K. B. Be´ c et al.
that needs to be fulfilled within the iterative procedure of solving the Hartree equations. The most far-reaching approximation assumed in the HF approach is neglecting
the Coulomb correlation, which is often described as the mean-field charge distribution approximation of the electron correlation, since the HF method effectively
averages the electron–electron interactions. This causes an inherent inability of the
method to properly describe London dispersion. In order to step beyond a mean-field
approximation of independent particles, so-called post-HF methods have been developed. The HF theory has a critical historical importance, being the first developed
quantum theory with practical implementation. Nowadays, pure HF calculations are
rarely used. However, the method is still widely adopted for calculations of the
initial wavefunction of a quantum system, thus representing the preliminary step for
calculations at higher levels of theory. On the other hand, a hierarchy of increasingly accurate methods based on the HF results exist, in which more than one Slater
determinant is employed.
5.3.2.2 Post-HF Approaches
Different populations of atomic orbitals by electrons or electron configurations in
a quantum system are possible. When any given electron changes its configuration,
which can be described as an excitation into another orbital, the distribution of the
other electrons in the molecule adjusts to minimize the total energy of the system.
Thus, the motion of electrons is not independent but correlated, which lowers the
total energy of the system. However, in the HF approach, any given electron only
interacts with the average potential of all the other electrons in the system. To amend
this shortcoming, post-HF methods aiming at a more accurate treatment of electron
correlation effects were introduced. This may be accomplished in different ways, but
unequivocally increases the computational complexity of the method by orders of
magnitude.
In the simplest case, the electron correlation energy can be treated as a perturbation of the electronic state described in the HF formalism. As long as electron
correlation has a relatively small contribution to the total energy, it can be expressed
via a perturbing Hamiltonian corresponding to a correction added to the HF Hamiltonian. Since the unperturbed HF state is known, the perturbative correction is solvable
using approximate methods, e.g., via an asymptotic series. The practical formulation of this approach is based on Møller–Plesset theory (MP) of a given order k.
Zeroth-order wavefunction corresponds to an unperturbed HF state, and the firstorder perturbation correction (MP1) to the HF energy can be shown to be equal
to zero, which implies that only second- (MP2) and higher-order MP expressions
are practically meaningful. Among those, MP2 bears the highest practical usefulness and finds broad applications. In most cases, higher-order perturbations (such
as MP3, MP4, and MP5) do not improve the accuracy by an acceptable margin and
display a huge computational demand. The MP2 method has become particularly
widely applied since it is the most efficient approach to take electron correlation