Since the original HF method does not (by definition) take into account electron
correlation effects, in order to get the results in better agreement with experiment, a
few extensions, the so-called post-HF methods, have been proposed in following
years, e.g., CI—Configuration interaction [30–32], MP2 [33, 34], MP3 [35, 36],
and MP4 [37] Møller–Plesset Perturbation Theory, CC—Coupled Cluster Theory
[38, 39], QCI—Quadratic Configuration Interaction [40], CASSCF—Complete
Active Space Self-Consistent Field calculation [41, 42], MRSDCI—
Multi-Reference Single and Double Configuration Interaction [43] or Quantum
Chemistry Composite methods, like Gaussian G1 [44, 45], G2 [46], G3 [47], and
G4 [48], FPD Feller–Peterson–Dixon approach [49, 50], ccCA—correlation consistent Composite Approach [51], or CBS—Complete Basis Set methods [52]. Such
extensions improved considerably the quality of calculations, but at the expense of
significantly higher demand for computer resources and calculation time (in the
worst-case scenario, calculation time increases fivefold with the size of the system,
comparing to original HF method).
1.2.1.4 Density Functional Theory
HF method with post-HF extensions is still commonly used (especially by chemists
interested in molecular compounds), but in recent years the Density Functional
Theory (DFT) approach became to dominate in computational physics and in great
part of computational chemistry. The origins of density functional theory go back to
the first years of the twentieth century and the works of Thomas [53] and Fermi [54],
describing the properties of multi-electron systems using total electron density,
extended in following years by Dirac [55] (exchange energy term) and Weizsäcker
[56] (first-order correction for kinetic energy functional). These attempts turned out,
however, to be unsatisfactory, due to, inter alia, approximate character of the
expression for kinetic and exchange energy and complete neglect of electron correlation resulting in fundamentally wrong description of chemical bonding: The calculated total energy of any molecule was higher than the sum of the energies of
isolated atoms, and therefore, Thomas–Fermi–Dirac model predicted molecules to be
unstable [57]. The problem of inaccurate description of kinetic energy in the Thomas–
Fermi–Dirac–Weizsäcker model was circumvented in Kohn–Sham’s Density
Functional Theory (with electron density as a key variable in the description of the
properties of non-uniform electron gas in multi-electron systems) for which firm
mathematical foundation was given in 1964 in two Hohenberg–Kohn theorems [58].
The starting point in density functional theory is the Born–Oppenheimer
approximation, resulting in constant external Coulomb potential V ext (r), generated
by motionless atomic nuclei, in which all electrons move:
V ext ðrÞ ¼ À
X
a
Z a e
2
r À r a
j
j
ð1:32Þ
14
A. Koleżyński
correlation effects, in order to get the results in better agreement with experiment, a
few extensions, the so-called post-HF methods, have been proposed in following
years, e.g., CI—Configuration interaction [30–32], MP2 [33, 34], MP3 [35, 36],
and MP4 [37] Møller–Plesset Perturbation Theory, CC—Coupled Cluster Theory
[38, 39], QCI—Quadratic Configuration Interaction [40], CASSCF—Complete
Active Space Self-Consistent Field calculation [41, 42], MRSDCI—
Multi-Reference Single and Double Configuration Interaction [43] or Quantum
Chemistry Composite methods, like Gaussian G1 [44, 45], G2 [46], G3 [47], and
G4 [48], FPD Feller–Peterson–Dixon approach [49, 50], ccCA—correlation consistent Composite Approach [51], or CBS—Complete Basis Set methods [52]. Such
extensions improved considerably the quality of calculations, but at the expense of
significantly higher demand for computer resources and calculation time (in the
worst-case scenario, calculation time increases fivefold with the size of the system,
comparing to original HF method).
1.2.1.4 Density Functional Theory
HF method with post-HF extensions is still commonly used (especially by chemists
interested in molecular compounds), but in recent years the Density Functional
Theory (DFT) approach became to dominate in computational physics and in great
part of computational chemistry. The origins of density functional theory go back to
the first years of the twentieth century and the works of Thomas [53] and Fermi [54],
describing the properties of multi-electron systems using total electron density,
extended in following years by Dirac [55] (exchange energy term) and Weizsäcker
[56] (first-order correction for kinetic energy functional). These attempts turned out,
however, to be unsatisfactory, due to, inter alia, approximate character of the
expression for kinetic and exchange energy and complete neglect of electron correlation resulting in fundamentally wrong description of chemical bonding: The calculated total energy of any molecule was higher than the sum of the energies of
isolated atoms, and therefore, Thomas–Fermi–Dirac model predicted molecules to be
unstable [57]. The problem of inaccurate description of kinetic energy in the Thomas–
Fermi–Dirac–Weizsäcker model was circumvented in Kohn–Sham’s Density
Functional Theory (with electron density as a key variable in the description of the
properties of non-uniform electron gas in multi-electron systems) for which firm
mathematical foundation was given in 1964 in two Hohenberg–Kohn theorems [58].
The starting point in density functional theory is the Born–Oppenheimer
approximation, resulting in constant external Coulomb potential V ext (r), generated
by motionless atomic nuclei, in which all electrons move:
V ext ðrÞ ¼ À
X
a
Z a e
2
r À r a
j
j
ð1:32Þ
14
A. Koleżyński