2.1 Computational Methods
37
(e.g. Møller-Plesset Perturbation theory) have been developed, alongside other theories (e.g. Density Functional Theory) in an attempt to better model the complex
electron behaviour.
2.1.3 Multi-Reference Methods
A particular drawback to the neglect of correlation in HF comes in the form of
non-dynamic (or static) correlation—the fact that in some cases a single-reference
wavefunction cannot describe a given electronic state [6]. For example, for the H 2
molecule, HF provides a reasonable estimate for the geometry around the equilibrium
geometry. However, if the bond is elongated, the Slater determinant generates a
wavefunction which can be represented as
H
α
. . . H
β
+
H
β
. . . H
α
+
H
−αβ
. . . H
+
+
H
+
. . . H
−αβ
(2.13)
Thus the HF scheme builds a wavefunction including equal weightings of two
ionic states. This leads to considerable issues in reproducing the asymptotic limit of
H 2 dissociation [5].
These problems, amongst others, have been approached using configurational
interaction (CI) methods [7]. The full theory of these methods is extensive and can
be found in many textbooks, including References [5] and [8]. In CI methods, the
ansatz wavefunction is taken as a linear combination of determinants (derived from
HF), and are weighted by an appropriate coefficient,
C I = c 1 φ 1 + c 2 φ 2 . . .
(2.14)
To expand beyond the HF method, CI methods introduce excited determinants,
in which an electron is permitted to occupy a valence HF orbital. This introduces
extra flexibility to the electrons within the calculation, and reflects the fact that
excited state orbital structures become important at large perturbations away from
equilibrium, and when considering electronic excitations. This addition therefore
offers a means to treat non-dynamic correlation. Hence, Eq. 2.14 can be re-written
in terms of the number of excited determinants used and their excitation level, here
denoted S (single) and D (double) excitations (higher order excitations can be used),
C I = c 1 φ H F +
S
c s φ s +
D
c D φ D . . . =
i=0
c i φ i
(2.15)
To understand how non-dynamical correlation improves the description of H 2
dissociation, the nature of the frontier orbitals must first be considered, Fig. 2.1.
The HF method constricts both electrons to the bonding (φ 1 ) orbital, whereas
inclusion of the φ 2 antibonding orbital in the CI-based approach permits these electrons to occupy different sides of the nodal plane. Hence, on dissociation, CI methods
37
(e.g. Møller-Plesset Perturbation theory) have been developed, alongside other theories (e.g. Density Functional Theory) in an attempt to better model the complex
electron behaviour.
2.1.3 Multi-Reference Methods
A particular drawback to the neglect of correlation in HF comes in the form of
non-dynamic (or static) correlation—the fact that in some cases a single-reference
wavefunction cannot describe a given electronic state [6]. For example, for the H 2
molecule, HF provides a reasonable estimate for the geometry around the equilibrium
geometry. However, if the bond is elongated, the Slater determinant generates a
wavefunction which can be represented as
H
α
. . . H
β
+
H
β
. . . H
α
+
H
−αβ
. . . H
+
+
H
+
. . . H
−αβ
(2.13)
Thus the HF scheme builds a wavefunction including equal weightings of two
ionic states. This leads to considerable issues in reproducing the asymptotic limit of
H 2 dissociation [5].
These problems, amongst others, have been approached using configurational
interaction (CI) methods [7]. The full theory of these methods is extensive and can
be found in many textbooks, including References [5] and [8]. In CI methods, the
ansatz wavefunction is taken as a linear combination of determinants (derived from
HF), and are weighted by an appropriate coefficient,
C I = c 1 φ 1 + c 2 φ 2 . . .
(2.14)
To expand beyond the HF method, CI methods introduce excited determinants,
in which an electron is permitted to occupy a valence HF orbital. This introduces
extra flexibility to the electrons within the calculation, and reflects the fact that
excited state orbital structures become important at large perturbations away from
equilibrium, and when considering electronic excitations. This addition therefore
offers a means to treat non-dynamic correlation. Hence, Eq. 2.14 can be re-written
in terms of the number of excited determinants used and their excitation level, here
denoted S (single) and D (double) excitations (higher order excitations can be used),
C I = c 1 φ H F +
S
c s φ s +
D
c D φ D . . . =
i=0
c i φ i
(2.15)
To understand how non-dynamical correlation improves the description of H 2
dissociation, the nature of the frontier orbitals must first be considered, Fig. 2.1.
The HF method constricts both electrons to the bonding (φ 1 ) orbital, whereas
inclusion of the φ 2 antibonding orbital in the CI-based approach permits these electrons to occupy different sides of the nodal plane. Hence, on dissociation, CI methods