Wave functions CASSCF

Table 4.3 Number of configurations generated in a [ , ]-CASSCF wave function...

The full Cl expansion within the active space severely restricts the number of orbitals and electrons that can be treated by CASSCF methods. Table 4.3 shows how many singlet CSFs are generated for an [n, n]-CASSCF wave function (eq. (4.13)), without any reductions arising from symmetry. 

For each occupied orbital, there will typically be one corresponding virtual orbital. This leads naturally to [n, m]-CASSCF wave functions where n and m are identical or nearly so. 

The structure on the left is biradical, while the two others are ionic, corresponding to both electrons being at the same carbon. The simplest CASSCF wave function which qualitatively can describe this system has two electrons in two orbitals, giving the three configurations shown above. The dynamical correlation between die two active electrons will tend to keep them as far apart as possible, i.e. favouring the biradical structure. Now... 

Ph CASSCF wave function resulting from all arrangements of the 3s and 3p electrons in the 3s and 3p orbitals. ... 

When the symmetry breaking of the wave function represents a biased procedure to decrease the weights of high energy VB stmctures which were fixed to umealistic values the tymmetry and single determinant constraints, one may expect that the valence CASSCF wave function will be symmetry-adapted, since this function optimizes the coefficients of all VB forms (the valence CASSCF is variational determination of the best valence space and of the best valence function, i.e. an optimal valence VB picture). In most problems the symmetry breaking should disappear when going to the appropriate MC SCF level. This is not always the case, as shown below. 

The extensive calculations of Serrano-Andres et al [31] have shown a spurious valence-Rydberg mixing in the CASSCF wave functions when valence (7t,7r )and Rydberg orbitals are optimized all together in a state average calculation it was shown that these orbitals loose their diffuse character and instead tend to provide an extra correlation to valence orbitals. To avoid such interaction, the orbitals used for the Cl treatment of the electronic spectrum were obtained by a two step procedure ... 

CASSCF wave function includes only the static correlation only a small number of electrons spanning frontier orbitals are correlated between them, while... 

Cl methods [21] add a certain number of excited Slater determinants, usually selected by the excitation type (e.g. single, double, triple excitations), which were initially not present in the CASSCF wave function, and treat them in a non-perturbative way. Inclusion of additional configurations allows for more degrees of freedom in the total wave function, thus improving its overall description. These methods are extremely costly and therefore, are only applicable to small systems. Among this class of methods, DDCI (difference-dedicated configuration interaction) [22] and CISD (single- and double excitations) [21] are the most popular. 

If the active space has been adequately chosen, the CASSCF wave function will include the most important CFs in the full Cl wave function. In this... 

The reference (zeroth-order) function in the CASPT2 method is a predetermined CASSCF wave function. The coefficients in the CAS function are thus fixed and are not affected by the perturbation operator. This choice of the reference function often works well when the other solutions to the CAS Hamiltonian are well separated in energy, but there may be a problem when two or more electronic states of the same symmetry are close in energy. Such situations are common for excited states. One can then expect the dynamic correlation to also affect the reference function. This problem can be handled by extending the perturbation treatment to include electronic states that are close in energy. This extension, called the Multi-State CASPT2 method, has been implemented by Finley and coworkers.24 We will briefly summarize the main aspects of the Multi-State CASPT2 method. 

Assume several CASSCF wave functions, T,-, i= l,N, obtained in a state average calculation. The corresponding (single state) CASPT2 functions are %i,i=l,N. The functions T, + x,- are used as basis functions in a variational calculation where all terms higher than second order are neglected. The corresponding effective Hamiltonian has the elements ... 

The (infinite) latitude for choosing different representations of the CASSCF wave-function may be exploited in order to bring it into a VB-like form. We adopt here the partitioning... 

It is imperative to use CASSCF wave functions for singlet diradicals and other open-shell molecules for which a single configuration provides an inadequate description of the wave function. However, perhaps surprisingly, CASSCF calculations often perform rather poorly in calculations on molecules and TSs with closed shells of electrons, if the active electrons are delocalized. An example is... 

In order to get good agreement between CASSCF and experimental results, it is often necessary to include correlation between the active electrons and the other valence electrons.This type of correlation, which is called dynamic correlation, can be added to a CASSCF wave function by using the equivalent of MP2 perturbation theory. However, in an MP2 calculation on an MC wave function, rather than including double excitations from a single HF reference configuration, excitations from all of the configurations must be included. 

The inactive and active orbitals are occupied in the wave function, while the external (also called secondary or virtual) orbitals span the rest of the orbital space, defined from the basis set used to build the molecular orbitals. The inactive orbitals are kept doubly occupied in all configurations that are used to build the CASSCF wave function. The number of electrons occupying these orbitals is thus twice the number of inactive orbitals. The remaining electrons (called active electrons) occupy the active orbitals. 

There exists today an alternative approach which has made MCSCF calculations on excited states feasible, also for rather large systems. A method has been developed which makes it easy to obtain orthogonal wave functions and transition densities from CASSCF wave functions optimized independently for a number of excited states of different or the same symmetry as the ground state. The method has been called the CAS State Interaction (CASSI) method. It will be briefly described below. 

If we transform the MO s such that condition (5 11) is fulfilled, the resulting transition density matrix will be obtained in a mixed basis, and can subsequently be transformed to any preferred basis The generators Epq of course have to be redefined in terms of the bi-orthonormal basis, but this is a technical detail which we do not have to worry about as long as we understand the relation between (5 9) and the Slater rules. How can a transformation to a bi-orthonormal basis be carried out We assume that the two sets of MO s are expanded in the same AO basis set. We also assume that the two CASSCF wave functions have been obtained with the same number of inactive and active orbitals, that is, the same configurational space is used. Let us call the two matrices that transform the original non-orthonormal MO s [Pg.242]

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