Nondynamic correlation

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... 

It is possible to divide electron correlation as dynamic and nondynamic correlations. Dynamic correlation is associated with instant correlation between electrons occupying the same spatial orbitals and the nondynamic correlation is associated with the electrons avoiding each other by occupying different spatial orbitals. Thus, the ground state electronic wave function cannot be described with a single Slater determinant (Figure 3.3) and multiconfiguration self-consistent field (MCSCF) procedures are necessary to include dynamic electron correlation. 

Formulas 21.1 through 21.3 are designed for Hartree-Fock wave functions. There are some attempts to define similar indices using wave functions obtained via methods including electron correlation [19]. Similarly, to the situation with respect to basis set improvement, the results based on correlated wave functions do not necessarily make the qualitative picture of bonding easier to understand. An exception is when there is a significant nondynamical correlation in the system,... 

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. 

Results using W1 and W2 theories are shown in Table 2.1. For W2 theory we find a mean absolute deviation (MAD) of 0.23 kcal/mol, which further drops to 0.18 kcal/mol when the NO, O2, and F2 molecules are deleted (all of which have mild nondynamical correlation in common). Our largest deviation is 0.70 kcal/mol. We can hence state that W2 meets our design goals. 

While CCSD and especially CCSD(T) are known [36] to be less sensitive to nondynamical correlation effects than low-order perturbation theoretical methods, some sensitivity remains, and deterioration of W1 and W2 results is to be expected for systems that exhibit severe nondynamical correlation character. A number of indicators exist for this, such as the T diagnostic of Lee and Taylor [64], the size of the largest amplitudes in the converged CCSD wavefunction, and natural orbital occupations of the frontier orbitals. 

One pragmatic criterion which we have found to be very useful is the percentage of the TAE that gets recovered at the SCF level. For systems that are wholly dominated by dynamical correlation, like CH4 and H2, this proportion exceeds 80 %, while it drops to 50 % for the N2 molecule, 02 is only barely bound at the SCF level, and F2 is even metastable. In the W1 /W2 validation paper [26], we invariably found that large deviations from what appeared to be reliable experimental data tend to be associated with strong nondynamical correlation, and a small SCF component of TAE (e.g. 27 % for NO2, 32 %for F2O, and 15 % for CIO). 

Further improvement of accuracy, as well as applicability to systems exhibiting nondynamical correlation, will almost certainly require some level of treatment of connected quadruple excitations. 

On the other hand, the PAs at the para and ortho positions are manifestly more sensitive to the electron correlation treatment than the PA at the nitrogen. From the point of view of the % diagnostic (a measure for the importance of nondynamical correlation), aniline (7 =0.0113) and N-anilinium (7 =0.0096) are very similar (and basically purely single-reference), while the p- and o-protonated species exhibit very mild multireference character (0.0149 and 0.0157, respectively). Since protonation at these sites thus involves a noticeable change in 7i, the PA is expected to be more sensitive to the correlation method. 

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... 

We will not attempt to survey the vast existing literature for the problem of constructing dynamic correlations when nondynamic correlations are present. Instead, we refer the reader to a number of excellent recent reviews [17, 26]. The many different theories in the literature all share common elements but typically employ different approximations or adopt different points of view. It is common to stress either a wavefunction ansatz picture or an effective... 

The generic chemical problem involving both dynamic and nondynamic correlation is illustrated in Fig. 1. The orbitals are divided into two sets the active orbitals, usually the valence orbitals, which display partial occupancies (assuming spin orbitals) very different from 0 or 1 for the state of interest, and the external orbitals, which are divided into the core (largely occupied in the target state) or virtual (largely unoccupied in the target state) orbitals. The asymmetry between... 

Figure 1. Multireference problems involve both dynamical and nondynamical correlation. The nondynamical correlation is accounted for by the CASCI/CASSCF/DMRG wavefunction, which is made of multiple configurations generated in the active space with a fixed number of active electrons. The dynamical correlation is recovered on top of the multiconfigurational reference by correlating the active orbitals with orbitals in the external space (i.e., core and virtual orbitals.)...

Nondynamic correlation is associated with active-active space correlations, while dynamic correlation is associated with correlations between the active-external and external-external spaces. 

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. 

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... 

Multireference There is no division into occupied and virtual orbitals, all orbitals appear on an equal footing in the ansatz (Equation 8). In particular, the Hartree-Fock reference has no special significance here. For this reason, we expect (and observe) the ansatz to be very well balanced for describing nondynamic correlation in multireference problems (see e.g., refs. 10-12). Conversely, the ansatz is inefficient for describing dynamic correlation, since to treat dynamic correlation one would benefit from the knowledge of which orbitals are in the occupied and virtual spaces. 

Let us recall, finally, that ec CCSD approaches exploit the complementarity of the Cl and CC methods in their handling of the dynamic and nondynamic correlations. While we use the Cl as an external source of higher than pair clusters, Meissner et al. [10,72-74] exploit the CC method to correct the Cl results (thus designing the CC-based Davidson-type corrections). This aspect will also be addressed below. 

Let us consider the 5s, 5p, 5d orbitals of lead and Is orbital of oxygen as the outercore and the ai, a2, os, tti, tt2 orbitals of PbO (consisting mainly of 6s, 6p orbitals of Pb and 2s, 2p orbitals of O) as valence. Although in the Cl calculations we take into account only the correlation between valence electrons, the accuracy attained in the Cl calculation of Ay is much better than in the RCC-SD calculation. The main problem with the RCC calculation was that the Fock-space RCC-SD version used there was not optimal in accounting for nondynamic correlations (see [136] for details of RCC-SD and Cl calculations of the Pb atom). Nevertheless, the potential of the RCC approach for electronic structure calculations is very high, especially in the framework of the intermediate Hamiltonian formulation [102, 131]. 

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... 

---