Post-Hartree Fock methods

No matter how good the basis set is made by extension toward an infinite set, one encounters the Hartree-Fock limit on the accuracy of molecular energy, because the influence of one electron upon the others has not been fully accounted for in the SCF averaging procedure. The difference between a Hartree-Fock energy and the experimental energy is called the correlation energy. To remedy this fault, correlated models are made up which consist of a linear combination of the Hartree-Fock solution plus singly, doubly, etc. substituted wave functions 

Another method of progressing beyond the Hartree-Fock limit is by inclusion of many body perturbation terms (Atkins and Friedman, 1997) 

It is an attractive feature of ab initio wave function theory that there is a clear hierarchy of methods leading from Hartree-Fock to the exact solution of the Schrodinger equation. Post-Hartree-Fock methods can be divided into three main categories [88]. The first is based on (Mqller-Plesset) perturbation theory [89] and referred to as MPn where n is the order of the perturbation. MPn is excellent when Hartree-Fock already is giving a reasonable description, as is often the case for complexes involving Ad and 5d elements. Otherwise, it fails or might only converge slowly with the order n. MP2 can be used for medium size systems of 100-200 atoms. 

The second category is based on configuration interaction (Cl) in which the HF determinant To is augmented by a number of determinants T/ constructed from To by replacing one or more of the occupied HF orbitals with virtual HF orbitals 

In simple Cl, the expansion coefficients Dk are optimized in such a way that Tq has the minimum energy. The expansion in (22) can be slowly converging requiring millions of terms. The number of needed terms can be reduced in the multiconfiguration SCF procedure (MCSCF), where both Dk and the orbital expansion 

The third and last category is the couple cluster (CC) method [88]. In this scheme, one writes the wave function Pec as 

the Ti operator when working on Pq affords the ith excited Slater determinants. In practical, CC calculations T of (25) is truncated. Thus keeping T + T2 gives rise to CCSD whereas the addition of 7) and subsequently T4 leads to CCSDT and CCSDTQ, respectively. The CCSD scheme which scales as (nef is used routinely for up to 100 electrons. It is considered as the most accurate method for metal complexes in those cases where the reference HF determinant Po affords a reasonable description. CCSDT and CCSDTQ scales as (nef and (ne)l( , respectively, they can only be used for very small systems. 

So far, we have only discussed the four-component basis-set approach in connection with the simplest ab initio wave-function model, namely for a single Slater determinant provided by Dirac-Hartree-Fock theory. We know, however, from chapter 8 how to improve on this model and shall now discuss some papers with a specific focus on correlated four-component basis-set methods. 

An efficient approach to improve on the Hartree-Fock Slater determinant is to employ Moller-Plesset perturbation theory, which works satisfactorily well for all molecules in which the Dirac-Hartree-Fock model provides a good approximation (i.e., in typical closed-shell single-determinantal cases). The four-component Moller-Plesset perturbation theory has been implemented by various groups [519,584,595]. A major bottleneck for these calculations is the fact that the molecular spinor optimization in the SCF procedure is carried out in the atomic-orbital basis set, while the perturbation expressions are given in terms of molecular spinors. Hence, all two-electron integrals required for the second-order Moller-Plesset energy expression must be calculated from the integrals over atomic-orbital basis functions like 

This transition to the molecular spinor basis is called four-index transformation for obvious reasons and has been discussed for the four-component case by Esser et al. [596] (see also Ref. [525]). 

More appropriate than perturbation approaches for improving on the energy are variational approaches (under the specific caveats discussed in chapter 8 with respect to the negative-energy states), because the total electronic energies obtained are much better controlled, an essential property since the exact reference is not known for any interesting many-electron molecule. In particular, we shall address the second-generation MCSCF methods mentioned in chapter 8. For reference to molecular Cl and CC theory, please consult sections 8.5.2 and 8.9, respectively. 

By contrast to the numerical MCSCF method discussed in the last chapter, the basis-set approach has the convenient advantage that the virtual orbitals come for free by solution of the Roothaan equation. While the fully numerical approaches of chapter 8 do not produce virtual orbitals, as the SCF equations are solved directly for occupied orbitals only and smart b)q)asses must be devised, this problem does not show up in basis-set approaches. Out of the m basis functions, only N with N C WJ are occupied, while the diagonaliza-tion of the matrix Fock operator produces a full set of m orthogonal molecular spinor vectors that can be efficiently employed in the excitation process of any Cl-like method. 

Although HF theory is useful in its own right for many kinds of investigations, there are some applications for which the neglect of electron correlation or the assumption that the error is constant (and so will cancel) is not warranted. Post-Hartree-Fock methods seek to improve the description of the electron-electron interactions using HF theory as a reference point. Improvements to HF theory can be made in a variety of ways, including the method of configuration interaction (CI) and by use of many-body perturbation theory (MBPT). It is beyond the scope of this text to treat CI and MBPT methods in any but the most cursory manner. However, both methods can be introduced from aspects of the theory already discussed. 

Earlier it was argued that the many-electron wave function (the true solution to the electronic Schrodinger equation) could be expanded in terms of an infinite series of single determinantal wave functions [Equation (A. 13)]  

The variational method is used to find the optimum expansion in terms of the configurations that is, the energy is expressed as an expectation value as was done in equation (A.9), 

The set of ci linear equations must then be solved for the energies and coefficients. This is accomplished by diagonalization of the Hamiltonian matrix H, whose elements are defined by 

Excited-state energies and wave functions are automatically obtained from Cl calculations. However, the quality of the wave functions is more difficult to achieve. The equivalent of the HF description for the ground state requires an all-singles Cl (SCI). Singly excited configurations do not mix with the HF determinant, that is, 

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The application of density functional theory to isolated, organic molecules is still in relative infancy compared with the use of Hartree-Fock methods. There continues to be a steady stream of publications designed to assess the performance of the various approaches to DFT. As we have discussed there is a plethora of ways in which density functional theory can be implemented with different functional forms for the basis set (Gaussians, Slater type orbitals, or numerical), different expressions for the exchange and correlation contributions within the local density approximation, different expressions for the gradient corrections and different ways to solve the Kohn-Sham equations to achieve self-consistency. This contrasts with the situation for Hartree-Fock calculations, wlrich mostly use one of a series of tried and tested Gaussian basis sets and where there is a substantial body of literature to help choose the most appropriate method for incorporating post-Hartree-Fock methods, should that be desired. 

Trioxane 210 has been used as a model system by Gu and coworkers to study the antimalarial drug artemisinin 211 (Scheme 137) [97CPL234, 99JST103]. It is the boat/twist form rather than the chair conformer of 210 that describes the subunit in 211. Moreover, geometric parameters and vibrational frequencies can only reliably be computed at the DFT level and by post-Hartree-Fock methods. B3-LYP/6-31G calculations on the conformers of 3,3,6,6-tetramethyl-1,2,4,5-tetroxane show that the chair conformer is stabilized with respect to the twisted conformer by about -2.8 kcal/mol [00JST85]. No corresponding boat conformer was found. 

One of the simplest chemical reactions involving a barrier, H2 + H —> [H—H—H] —> II + H2, has been investigated in some detail in a number of publications. The theoretical description of this hydrogen abstraction sequence turns out to be quite involved for post-Hartree-Fock methods and is anything but a trivial task for density functional theory approaches. Table 13-7 shows results reported by Johnson et al., 1994, and Csonka and Johnson, 1998, for computed classical barrier heights (without consideration of zero-point vibrational corrections or tunneling effects) obtained with various methods. The CCSD(T) result of 9.9 kcal/mol is probably very accurate and serves as a reference (the experimental barrier, which of course includes zero-point energy contributions, amounts to 9.7 kcal/mol). 

Bartlett, R. J., Stanton, J. F., 1995, Applications of Post-Hartree-Fock Methods A Tutorial , Rev. Comput. Chem., 5, 65. 

Sodupe, M., Bertran, J., Rodriguez-Santiago, L., Baerends, E. J., 1999, Ground State of the (H20)2 Radical Cation DFT versus Post-Hartree-Fock Methods , J. Phys. Chem. A, 103, 166. 

Bartlett RJ, Stanton JF (1994) Applications of post-hartree-fock methods a tutorial. In Lipkowitz KB Boyd DB (eds) Reviews in computational chemistry, vol. 5. Wiley-VCH, New York, pp 65—169... 

Schmiedenkamp, A. M., I. A. Topol, S. K. Burt, H. Razafinjanahary, H. Chermette, T. Pfatzgraff, and C. J. Michejda. 1994. Triazene Proton Affinities A comparison between Density Functional, Hartree-Fock, and Post-Hartree-Fock Methods. J. Comp. Chem. 875, 875. 

Rodney J. Bartlett and John F. Stanton, Applications of Post-Hartree-Fock Methods A Tutorial. 

Up to now, only a few organocatalytic reactions of the above types have been investigated with post-Hartree-Fock methods. Potential reasons are computational costs, spatial and conformational flexibility (ab initio methods do not necessarily find... 

Keywords strongly correlated electrons nondynamic correlation density matrix renormalization group post Hartree-Fock methods many-body basis matrix product states complete active space self-consistent field electron correlation... 

Ground State of the (H20)2 Radical Cation DFT Versus Post-Hartree-Fock Methods. 

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