the course of research, however, this practice has fallen into disuse. This can be
problematic in the prediction of the structural and energetic properties of the systems under investigation, especially when determination of the correct electronic
state is essential to achieve results that are even qualitatively correct.
It is known that there is more than one solution to the Hartree-Fock equations
[1–3]. In fact, within a finite basis there can be O(3
N ) solutions for a closed-shell
system, where N is equal to the number of basis functions used [1]. Most HF
algorithms populate the initial orbitals based on the aufbau principle, wherein the
lowest energy orbitals subject to the initial guess of LCAO coefficients are populated
to determine the lowest energy solution and thus the ground state of the molecule
[4–6]. There are cases, however, when the algorithm can produce an excited state
determinant rather than the ground state as has been noted in the literature [3, 7, 8].
Optimization to an excited state can happen when there is a small HOMO-LUMO
gap, when there are nearly degenerate determinants, or in other cases where a
multireference treatment is more appropriate such as when bonds are broken or
formed. Even multireference calculations depend upon single-reference methods as
the source of the initial orbitals from which an active space is chosen, and this is
often reflected in the rate of convergence of the multireference wave function.
Optimization of an excited state determinant with Hartree-Fock orbitals forms
the basis of extended Hartree-Fock theory for excited states [9]. However, it has
been shown that in the case of a closed shell system, an electron in a virtual orbital
does not experience the full interaction of the 2N electrons, and so a modified Fock
operator should be employed to obtain a well-defined excited state in terms of a
single determinant of Hartree-Fock orbitals. When the excited states are obtained
unintentionally through population based on orbital energies, the Fock operator is
not modified to account for the missing electronic interactions. Therefore, while the
excited states obtained may be representative of the dominant configuration of a
multiconfigurational excited state wave function, they do not include the entire
mean field correlation.
It is imperative that the optimized Hartree-Fock wave function be scrutinized to
ensure that the correct state has been determined. While there are some systems for
which optimization to an excited state determinant is not detrimental to geometry
optimizations (e.g. situations in which the potential energy curves (PEC) are mostly
parallel, such as with Ln(III)-halide bonds), [10] this frequently is not the case. The
optimized orbitals need to be investigated to ensure that the appropriate orbitals are
being populated, a process that generally can rely on chemical intuition.
A condition of Hartree-Fock convergence is that the orbital gradient is zero, i.e.
∂E/∂C i = 0. However, as this condition can be met at several places on the orbital
potential energy surface, the stability of the solution may need to be tested through
the calculation of the orbital Hessian matrix [11]. Negative eigenvalues indicate that
there is still a lower energy solution that can be reached; essentially, this indicates
that a saddle point in the potential energy surface has been found, rather than a
minimum. Such a situation could be remedied in a black-box manner by performing
a Hartree-Fock instability test. However, this is only beneficial when the HartreeFock solution is a saddle point on the orbital potential energy surface.
4
R. Weber et al.
problematic in the prediction of the structural and energetic properties of the systems under investigation, especially when determination of the correct electronic
state is essential to achieve results that are even qualitatively correct.
It is known that there is more than one solution to the Hartree-Fock equations
[1–3]. In fact, within a finite basis there can be O(3
N ) solutions for a closed-shell
system, where N is equal to the number of basis functions used [1]. Most HF
algorithms populate the initial orbitals based on the aufbau principle, wherein the
lowest energy orbitals subject to the initial guess of LCAO coefficients are populated
to determine the lowest energy solution and thus the ground state of the molecule
[4–6]. There are cases, however, when the algorithm can produce an excited state
determinant rather than the ground state as has been noted in the literature [3, 7, 8].
Optimization to an excited state can happen when there is a small HOMO-LUMO
gap, when there are nearly degenerate determinants, or in other cases where a
multireference treatment is more appropriate such as when bonds are broken or
formed. Even multireference calculations depend upon single-reference methods as
the source of the initial orbitals from which an active space is chosen, and this is
often reflected in the rate of convergence of the multireference wave function.
Optimization of an excited state determinant with Hartree-Fock orbitals forms
the basis of extended Hartree-Fock theory for excited states [9]. However, it has
been shown that in the case of a closed shell system, an electron in a virtual orbital
does not experience the full interaction of the 2N electrons, and so a modified Fock
operator should be employed to obtain a well-defined excited state in terms of a
single determinant of Hartree-Fock orbitals. When the excited states are obtained
unintentionally through population based on orbital energies, the Fock operator is
not modified to account for the missing electronic interactions. Therefore, while the
excited states obtained may be representative of the dominant configuration of a
multiconfigurational excited state wave function, they do not include the entire
mean field correlation.
It is imperative that the optimized Hartree-Fock wave function be scrutinized to
ensure that the correct state has been determined. While there are some systems for
which optimization to an excited state determinant is not detrimental to geometry
optimizations (e.g. situations in which the potential energy curves (PEC) are mostly
parallel, such as with Ln(III)-halide bonds), [10] this frequently is not the case. The
optimized orbitals need to be investigated to ensure that the appropriate orbitals are
being populated, a process that generally can rely on chemical intuition.
A condition of Hartree-Fock convergence is that the orbital gradient is zero, i.e.
∂E/∂C i = 0. However, as this condition can be met at several places on the orbital
potential energy surface, the stability of the solution may need to be tested through
the calculation of the orbital Hessian matrix [11]. Negative eigenvalues indicate that
there is still a lower energy solution that can be reached; essentially, this indicates
that a saddle point in the potential energy surface has been found, rather than a
minimum. Such a situation could be remedied in a black-box manner by performing
a Hartree-Fock instability test. However, this is only beneficial when the HartreeFock solution is a saddle point on the orbital potential energy surface.
4
R. Weber et al.