Nondynamical correlation energy

Figure 11. Schematic representation of the partitioning of the potential energy into extended Hartree-Fock (EHF), including the nondynamical correlation energy, and dynamical correlation (corr). See text.

Wang, S. G., Schwarz, W. H. E., 1996, Simulation of Nondynamical Correlation in Density Functional Calculations by the Optimized Fractional Occupation Approach Application to the Potential Energy Surfaces of 03 and... 

For systems devoid of nondynamical correlation effects, this is the largest individual contribution to the molecular binding energy. Its basis set convergence is relatively rapid, yet our discussion will be disproportionately long because a number of the dramatis personae that reappear in the remainder of the story need to be introduced here. 

The valence correlation component of TAE is the only one that can rival the SCF component in importance. As is well known by now (and is a logical consequence of the structure of the exact nonrelativistic Bom-Oppenheimer Hamiltonian on one hand, and the use of a Hartree-Fock reference wavefunction on the other hand), molecular correlation energies tend to be dominated by double excitations and disconnected products thereof. Single excitation energies become important only in systems with appreciable nondynamical correlation. Nonetheless, since the number of single-excitation amplitudes is so small compared to the double-excitation amplitudes, there is no point in treating them separately. 

In modest sized systems, we can treat the nondynamic correlation in an active space. For systems with up to 14 orbitals, the complete-active-space self-consistent field (CASSCF) theory provides a very satisfactory description [2, 3]. More recently, the ab initio density matrix renormalization group (DMRG) theory has allowed us to obtain a balanced description of nondynamic correlation for up to 40 active orbitals and more [4-13]. CASSCF and DMRG potential energy... 

The major advantage of a 1-RDM formulation is that the kinetic energy is explicitly defined and does not require the construction of a functional. The unknown functional in a D-based theory only needs to incorporate electron correlation. It does not rely on the concept of a fictitious noninteracting system. Consequently, the scheme is not expected to suffer from the above mentioned limitations of KS methods. In fact, the correlation energy in 1-RDM theory scales homogeneously in contrast to the scaling properties of the correlation term in DPT [14]. Moreover, the 1-RDM completely determines the natural orbitals (NOs) and their occupation numbers (ONs). Accordingly, the functional incorporates fractional ONs in a natural way, which should provide a correct description of both dynamical and nondynamical correlation. 

We also performed extensive DFT studies on both the full target system and the model for calibration purposes. For details of one-electron basis sets used please consult Ref. (55). We used the B3LYP functional but found the ground-state potential energy surface to be relatively insensitive to the chosen functional (note though that this does not mean that DFT gives the correct surfaces, as important nondynamical correlation effects are... 

The second approach to treating nondynamical correlation has an air of the ostrich about it ignore the spin symmetry of the wave function and use unrestricted Haxtree-Fock (UHF) theory as the single configuration description [7]. Since the UHF wave function comprises one spin-orbital for each electron, a molecular UHF wave function should dissociate to atomic UHF wave functions, for example. This is certainly not the case for spin-restricted Hartree-Fock (RHF) molecules and atoms in general. And there is an attractive simplicity about UHF — no active orbitals to identify, and so forth. However, where nondynamical correlation would be important in an RHF-based treatment, the UHF method will suffer from severe spin-contamination, while where nondynamical correlation is not important the RHF solution may be lower in energy than any broken-symmetry UHF solution, so potential curves and surfaces may have steps or kinks where the spin symmetry is broken in the UHF treatment. 

It is essential to use selection of reference configurations in the MRCI procedure with the larger basis set, as the MRCI(CAS) calculation would involve several million CSFs. The result labelled Valence(0.025) is obtained with only the valence configurations — that is, it is the MRCI(0.025) wave function with all excitations to the virtual orbitals omitted. This wave function will be considerably smaller than the CASSCF wave function, but if we are to describe the nondynamical correlation acceptably it should produce similar results. This is one reason for not using a selection threshold of (0.05) the Valence(0.05) binding energy was found to be quite different from the CASSCF value [74]. 

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