Nondynamical correlation

Furthermore, one must consider where to include spin-orbit effects in relation to both dynamical and nondynamical correlation. The most widely used approach is to construct an effective Hamiltonian for spin-orbit coupling from MRCI wavefunctions that are built from a common orbital set. The small set of wavefunctions is often sufficient to describe spin-orbit coupling between the nonrelativistic surfaces, and was used to good effect by Werner in the reaction dynamics calculations presented in this symposium. Another approach is to obtain a set of natural orbitals from MRCI calculations and use these in a Cl calculation that includes spin-orbit interaction. In this way dynamical correlation, nondynamical correlation and spin-orbit interaction are treated on the same footing, albeit with some compromise on the first of these three. 

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... 

Two different correlation effects can be distinguished. The first one, called dynamical electron correlation, comes from the fact that in the Hartree-Fock approximation the instantaneous electron repulsion is not taken into account. The nondynamical electron correlation arises when several electron configurations are nearly degenerate and are strongly mixed in the wave function. 

When the HF wave function gives a very poor description of the system, i.e. when nondynamical electron correlation is important, the multiconfigurational SCF (MCSCF) method is used. This method is based on a Cl expansion of the wave function in which both the coefficients of the Cl and those of the molecular orbitals are variationally determined. The most common approach is the Complete Active Space SCF (CASSCF) scheme, where the user selects the chemically important molecular orbitals (active space), within which a full Cl is done. 

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. 

However, in a large number of closed shell molecules, a single Slater determinant describes the ground state wave function fairly accurately. Even in such cases inclusion of excited state configuration results in substantial lowering of total electronic energy, and this is referred to as nondynamic 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. 

We have proposed a canonical transformation (CT) theory to describe dynamic correlation in bonding situations where there is also significant nondynamic... 

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. 

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