Dynamical correlation methods

For these reasons, in the MCSCF method the number of CSFs is usually kept to a small to moderate number (e.g. a few to several thousand) chosen to describe essential correlations (i.e. configuration crossings, near degeneracies, proper dissociation, etc, all of which are often tenned non-dynamicaI correlations) and important dynamical correlations (those electron-pair correlations of angular, radial, left-right, etc nature that are important when low-lying virtual orbitals are present). 

As larger atomic basis sets are employed, the size of the CSF list used to treat a dynamic correlation increases rapidly. For example, many of the above methods use singly- and doubly-excited CSFs for this purpose. For large basis sets, the number of such CSFs (N ) scales as the number of electrons squared uptimes the number... 

RHF to UHF, or to a TCSCF, is almost pure static correlation. Increasing the number of configurations in an MCSCF will recover more and more of the dynamical correlation, until at the full Cl limit, the correlation treatment is exact. As mentioned above, MCSCF methods are mainly used for generating a qualitatively correct wave function, i.e. recovering the static part of the correlation. 

A more balanced description requires MCSCF based methods where the orbitals are optimized for each particular state, or optimized for a suitable average of the desired states (state averaged MCSCF). It should be noted that such excited state MCSCF solutions correspond to saddle points in the parameter space for the wave function, and second-order optimization techniques are therefore almost mandatory. In order to obtain accurate excitation energies it is normally necessarily to also include dynamical Correlation, for example by using the CASPT2 method. 

Luminescence lifetime spectroscopy. In addition to the nanosecond lifetime measurements that are now rather routine, lifetime measurements on a femtosecond time scale are being attained with the intensity correlation method (124), which is an indirect technique for investigating the dynamics of excited states in the time frame of the laser pulse itself. The sample is excited with two laser pulse trains of equal amplitude and frequencies nl and n2 and the time-integrated luminescence at the difference frequency (nl - n2 ) is measured as a function of the relative pulse delay. Hochstrasser (125) has measured inertial motions of rotating molecules in condensed phases on time scales shorter than the collision time, allowing insight into relaxation processes following molecular collisions. 

The electronic structure method used to provide the energies and gradients of the states is crucial in photochemistry and photophysics. Ab initio electronic structure methods have been used for many years. Treating closed shell systems in their ground state is a problem that, in many cases, can now be solved routinely by chemists using standardized methods and computer packages. In order to obtain quantitative results, electron correlation (also referred to as dynamical correlation) should be included in the model and there are many methods available for doing this based on either variational or perturbation principles [41],... 

Perturbative methods (CASPT2 [17], NEVPT2 [18]) add the dynamical correlation in an effective way, using multiconfigurational second-order perturbation theory on the CASSCF input states. These methods have proved to be suitable for studying problems in spectroscopy, photochemistry, and so on [19, 20]. 

It It has become customary in quantum chemistry to subdivide electron correlation effects into two classes according to the methods used to account for them. The first are termed non-dynamic and they can be recovered by the above type of small Cl. On the other hand, all deficiencies which remain once these non-dynamic effects are taken care of are attributed to dynamic correlation. 

In ab initio methods (which, by definiton, should not contain empirical parameters), the dynamic correlation energy must be recovered by a true extension of the (single configuration or small Cl) model. This can be done by using a very large basis of configurations, but there are more economical methods based on many-body perturbation theory which allow one to circumvent the expensive (and often impracticable) large variational Cl calculation. Due to their importance in calculations of polyene radical ion excited states, these will be briefly described in Section 4. 

Due to the size of the variational problem, a large Cl is usually not a practicable method for recovering dynamic correlation. Instead, one usually resorts to some form of treatment based on many-body perturbation theory where an explicit calculation of all off-diagonal Cl matrix elements (and the diagonalization of the matrix) are avoided. For a detailed description of such methods, which is beyond the scope of this review, the reader is referred to appropriate textbooks295. For the present purpose, it suffices to mention two important aspects. 

However, if this is not the case, the perturbations are large and perturbation theory is no longer appropriate. In other words, perturbation methods based on single-determinant wavefunctions cannot be used to recover non-dynamic correlation effects in cases where more than one configuration is needed to obtain a reasonable approximation to the true many-electron wavefunction. This represents a serious impediment to the calculation of well-correlated wavefunctions for excited states which is only possible by means of cumbersome and computationally expensive multi-reference Cl methods. 

Luckily, this impasse was removed through the recent introduction of the CASPT2 model, which combines a powerful procedure for treating cases of strong non-dynamic correlation (CASSCF) with a very economical one for treating dynamic correlation296. As will be shown below, the CASPT2 method works very well for polyene radical cations. 

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