Chapter 5. Quantum chemistry in Molecular Modeling


5.4 Limitations of the HF method; Electron correlation.

Restricted Hartree-Fock SCF theory has some painful shortcomings. Consider for example the dissociation of the H2 molecule :
H+    +    H-   <------ H-H -------->  H.    +    H.

A "dissociation catastrophe" occurs because the separated hydrogen atoms cannot be described using doubly occupied orbitals, so that H2 tends to dissociate in H+ and H-, which can be described with a doubly occupied orbital on H-. This problem does not occur in the UHF method, but this method has the disadvantage that it does not give pure spin states.

An additional limitation of the HF method in general is that due to the use of the independent particle approximation the instantaneous correlation of the motions of electrons is neglected, even in the Hartree-Fock limit.
The difference between the exact energy (determined by the Hamiltonian) and the HF energy is known as the correlation energy: Ecorrelation = Eexact - EHF < 0

Even though EHF is approximately 99% of Esub>exact the difference may be chemically important.

Several approaches are known that try to calculate the correlation energy after Hartree-Fock calculations (post-HF methods). We will very briefly discuss

HF theory gives a wavefunction which is represented as a Slater determinant. In the conceptually simple Configuration Interaction (CI) method a linear combination of Slater determinants is constructed, using the unoccupied "virtual" orbitals from the SCF-calculation :

The total wavefunction is written as :

In principle, the exact correlation energy can be obtained from a full CI calculation in which all configurations are taken into consideration.
Unfortunately this is not possible for all but the smallest systems. Moreover, the problem is aggrevated when the size of the basis set is increased, on the way towards the Hartree-Fock limit. Thus, the theoretical limit of the exact (time-independent, non-relativistic) Schrödinger equation cannot be reached.

Even for small systems the number of excited configurations is enormously large. A popular way to truncate the CI expansion is to consider only singly and doubly excited configurations (CI-SD). The energy, calculated as the expectation value of the Hamiltonian for CISD is :

To perform the calculation one needs the two-electron integrals over Molecular Orbitals. The computation of these is very time-consuming, even when the integrals over AO's are available :

In general, CI is not the practical method of choice for the calculation of correlation energy because full CI is not possible, convergence of the CI expansion is slow, and the integral transformation time-consuming.
Moreover truncated CI is not size-consistent, which means that the calculation of two species at large separation does not give the same energy as the sum of the calculations on separate species. This is because a different selection of excited configurations is made in the two calculations. An advantage of the CI method is that it is variational, so the calculated energy is always greater than the exact energy.

Although CI is not recommendable as a method for ground states CI-singles (CIS) has been advocated as an approach to computation of excited state potential energy surfaces [10].

A different approach to electron correlation has become very popular in recent years : Møller-Plesset Perturbation Theory.
The basic idea is that the difference between the Fock operator and the exact Hamiltonian can be considered as a perturbation :

Corrections can be made to any order of the energy and the wavefunction :

The most popular method is the lowest level of correction, MP2.

An enormous practical advantage is that MP2 is fast (of the same order of magnitude as SCF), while it is rather reliable in its behavior, and size consistent. A disadvantage is that it is not variational, so the estimate of the correlation energy can be too large. In practice MP2 must be used with a reasonable basis set (6-31G* or better). Subsequent MP-levels MP3, MP4 (usually MP4 SDQ) are more complicated and much more time-consuming.
For example, for pentane (C5H12) with the 6-31G(d) basis set (99 basis functions) an MP2 energy calculation took about 4 times the amount of time needed for SCF, while MP4 took almost 90 times that time [7].

Multiconfiguration SCF (MCSCF) or Complete Active Space SCF (CASSCF) is a special method in which HF-orbitals are optimized simultaneously with a "small" CI.
This can be used to study problems where the Hartree-Fock method is inappropriate (e.g. when there are low-lying excited states), or to generate a good starting wavefunction for a subsequent CI calculation.

The MCSCF method requires considerable care in the selection of the basis set and especially the active space, and should not be considered for routine use.
In contrast to the HF, MPn and CI methods, MCSCF does not provide a "model chemistry" because each problem requires different choices.
MCSCF methods are essential for the study of processes in which transitions between potential energy surfaces occur, such as in photochemical reactions [11, 12]. A combination of MP2 with MCSCF has recently been explored by Roos et al. [13]. This seems to be a very promising method for excited states.

Other methods to determine the correlation energy are under development.
At this point it is useful to note another promising development, that of density functional theory. This is a method in which the two-electron integrals are not computed in the conventional way. Application of this approach to molecular systems is still in its infancy, but rapid developments are to be expected in the next few years, in particular driven by the desire to be able to compute larger systems, e.g. metal complexes and organometallic compounds.

References:

[7] Foresman, J.B.; Frisch, A.
Exploring Chemistry with Electronic Structure Methods: A Guide to Using Gaussian, Gaussian Inc., 1993.
[8] D. Feller and E.R Davidson,
in Reviews in Computational Chemistry, K.B. Lipkowitz and D.B. Boyd, eds., VCH, 1990, pp. 1 - 43.
[9] Brouwer, A.M.; Bezemer, L.; Jacobs, H.J.C.,
Recl. Trav. Chim. Pays-Bas 1992, 111, 138-143
[10] Foresman, J.B.; Head-Gordon, M.; Pople, J.A.; Frisch, M.J.,
J. Phys. Chem., 1992, 96, 135 - 149
[11] Palmer, I.J.; Ragazos, I.N.; Bernardi, F.; Olivucci, M.; Robb, M.A.,
J. Am. Chem. Soc. 1993,115, 673 - 682
[12] Olivucci, M.; Ragazos, I.N.; Bernardi, F.; Robb, M.A.,
J. Am. Chem. Soc. 1993, 115, 3710 - 3721
[13] Roos, B.O.; Andersson, K.; Fülscher, M.P.,
Chem. Phys. Lett., 1992, 192, 5 - 13

Next paragraph, 5.5 Energy calculations
Previous paragraph 5.3 Solving the electronic Schrödinger equation : Hartree-Fock Self-Consistent Field theory.
Chapter 5 MM Syllabus 1995 MODIFIED November 8, 1995
Fred Brouwer, Lab. of Organic Chemistry, University of Amsterdam.