MCSCF excited state

In the process of determining the expansion coefficients that define an MCSCF wave function along lines similar to Eq. (7.10), CI calculations are carried out in die space of the orbitals that are active in the MCSCF. Excited states that can be generated by electronic excitations within that active space then have a corresponding root in diat limited CI window and, if one chooses, one can variationally optimize the orbitals for a root other than the one of lowest energy, i.e., other than the ground state. 

To use direct dynamics for the study of non-adiabatic systems it is necessary to be able to efficiently and accurately calculate electronic wave functions for excited states. In recent years, density functional theory (DFT) has been gaining ground over traditional Hartree-Fock based SCF calculations for the treatment of the ground state of large molecules. Recent advances mean that so-called time-dependent DFT methods are now also being applied to excited states. Even so, at present, the best general methods for the treatment of the photochemistry of polyatomic organic molecules are MCSCF methods, of which the CASSCF method is particularly powerful. 

For conhguration interaction calculations of double excitations or higher, it is possible to solve the Cl super-matrix for the 2nd root, 3rd root, 4th root, and so on. This is a very reliable way to obtain a high-quality wave function for the hrst few excited states. For higher excited states, CPU times become very large since more iterations are generally needed to converge the Cl calculation. This can be done also with MCSCF calculations. 

The use of Cl methods has been declining in recent years, to the profit of MP and especially CC methods. It is now recognized that size extensivity is important for obtaining accurate results. Excited states, however, are somewhat difficult to treat by perturbation or coupled cluster methods, and Cl or MCSCF based methods have been the prefen ed methods here. More recently propagator or equation of motion (Section 10.9) methods have been developed for coupled cluster wave functions, which allows calculation of exited state properties. 

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. 

So far everything is exact. A complete manifold of excitation operators, however, means that all excited states are considered, i.e. a full Cl approach. Approximate versions of propagator methods may be generated by restricting the excitation level, i.e. tmncating h. A complete specification furthermore requires a selection of the reference, normally taken as either an HF or MCSCF wave function. 

The energy of a wave function containing variational parameters, like an HF (one Slater determinant) or MCSCF (many Slater determinants) wave function. Parameters are typically the MO and state coefficients, but may also be for example basis function exponents. Usually only minima are desired, although in some cases saddle points may also be of interest (excited states). 

Finally, in order to ensure an homogeneous treatment of all excited states at the variational level, the MCSCF calculation should be averaged on the states under investigation. The lowest eigenfunetions of the MCSCF Hamiltonian will provide the zeroth-order wavefunetions to build the perturbation on. 

The lowest A, excited states of C3H2are of Rydberg type, arising from the promotion of one electron from the carbene lone pair orbital to 3s and 3p Rydberg orbitals, are better represented by orbitals generated by a MCSCF/SD treatment (Table 9). 

SCF, SCF-MI and MCSCF-MI level. Frequencies co were calculated directly from the force constants K by applying the harmonic approximation or by means of the Noumerov method [45,46]. In the latter case, frequencies were evaluated from the energy difference AE between the fundamental and the first excited states by applying Plank s law. This technique is undoubtedly more accurate than the harmonic approximation. 

The operator (4 66) together with the definition (4 67) of the matrix f gives a surprisingly powerful approximation to the full super-CI Hamiltonian. It has been used with great success in a number of calculations of a variety of MCSCF wave functions both for ground and excited states. 

Different electronic states have in many cases veiy differently shaped orbitals and the error introduced by using a common set cannot always be fully recovered by the MR-CI treatment. A well optimized wave function is especially important for the calculation of transition properties like the transition moments and the oscillator strength. A state specific calculation of the orbitals is more important for obtaining accurate values of the transition moments than extensive inclusion of correlation. Since excited states commonly exhibit large near-degeneracy effects in the wave function an MCSCF treatment then becomes necessary. 

The major obstacle in using the MCSCF approach in studies of excited states has been the orthogonality problem. If MCSCF calculations are performed on two electronic states of the same symmetry, the resulting wave functions will not be orthogonal to each other. Independent of symmetry, the two sets of orbitals will also be different. This leads to difficulties in the calculation of the transition properties, and the interpretation is obscured by the non-orthogonality of the two wave functions. Ideally one would like to perform the calculation on the excited states including the condition that it remains orthogonal to the lower states of the same symmetry. Such a procedure, however, leads to a formidable computational problem, which has so far not been satisfactorily solved. 

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. 

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