Excited states, approximate methods

The purpose of the present paper is to review the most essential elements of the excited-state MMCC theory and various approximate methods that result from it, including the aforementioned CR-EOMCCSD(T) [49,51,52,59] and externally corrected MMCC ]47-50, 52] approaches. In the discussion of approximate methods, we focus on the MMCC corrections to EOMCCSD energies due to triple excitations, since these corrections lead to the most practical computational schemes. Although some of the excited-state MMCC methods have already been described in our earlier reviews [49, 50, 52], it is important that we update our earlier work by the highly promising new developments that have not been mentioned before. For example, since the last review ]52], we have successfully extended the CR-EOMCCSD(T) methods to excited states of radicals and other open-shell systems ]59]. We have also developed a new type of the externally cor-... 

In a molecule with electrons in n orbitals, such as formaldehyde, ethylene, buta-1,3-diene and benzene, if we are concerned only with the ground state, or excited states obtained by electron promotion within 7i-type MOs, an approximate MO method due to Hiickel may be useM. 

The parameterization of MNDO/AM1/PM3 is performed by adjusting the constants involved in the different methods so that the results of HF calculations fit experimental data as closely as possible. This is in a sense wrong. We know that the HF method cannot give the correct result, even in the limit of an infinite basis set and without approximations. The HF results lack electron correlation, as will be discussed in Chapter 4, but the experimental data of course include such effects. This may be viewed as an advantage, the electron correlation effects are implicitly taken into account in the parameterization, and we need not perform complicated calculations to improve deficiencies in fhe HF procedure. However, it becomes problematic when the HF wave function cannot describe the system even qualitatively correctly, as for example with biradicals and excited states. Additional flexibility can be introduced in the trial wave function by adding more Slater determinants, for example by means of a Cl procedure (see Chapter 4 for details). But electron cori elation is then taken into account twice, once in the parameterization at the HF level, and once explicitly by the Cl calculation. 

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. 

Figure 3. Molecular-orbital diagrams as obtained by the ROHF method. Dashed lines indicate MOs dominated by the metal d-orbitals, the solid lines stand for doubly occupied or virtual ligand orbitals. Orbitals which are close in energy are presented as degenerate the average deviation from degeneracy is approximately 0.01 a.u. In the case of a septet state (S=3), the singly occupied open-shell orbitals come from a separate Fock operator and their orbital energies do not relate to ionization potentials as do the doubly occupied MOs (i.e. Koopmann s approximation). For these reasons, the open-shell orbitals appear well below the doubly occupied metal orbitals. Doubly occupying these gives rise to excited states, see text.

If the full molecular symmetry is assumed, the ground states of the cation radical of fulvalene and the anion radical of heptafulvalene are both predicted to be of symmetry by using the semiempirical open-shell SCF MO method The lowest excited states of both radicals are of symmetry and are predicted to be very close to the ground states in the framework of the Hiickel approximation these states are degenerate in both cases (Fig. 4). Therefore, it is expected that in both these radicals the ground state interacts strongly with the lowest excited state through the nuclear deformation of symmetry ( — with the result that the initially assumed molecular... 

In addition, since the HPHF wavefunction exhibits a two-determinantal form, this model can be used to describe singlet excited states or triplet excited states in which the projection of the spin momentum Ms=0. The HPHF approximation appears thus as a simple method for the direct determination of excited states (with Afs=0)such as the usual Unrestricted Hartree Fock model does for determining triplet excited states with Ms = 1. 

Having these severe approximations in mind, SCC-DFTB performs surprisingly well for many systems of interest, as discussed above. However, it has a lower overall accuracy than DFT or post HF methods. Therefore, applying it to new classes of systems should be only done after careful examination of its performance. This can be done e.g. by conducting reference calculations on smaller model systems with DFT or ab initio methods. A second source of errors is related to some intrinsic problems with the GGA functionals also used in popular DFT methods (SCC-DFTB uses the PBE functional), which are inherited in SCC-DFTB. This concerns the well known GGA problems in describing van der Waals interactions [32], extended conjugate n systems [45,46] or charge transfer excited states [47, 48],... 

The CC2 method [74] is an approximation to coupled cluster with singles and doubles (CCSD), and the excited state energies calculated have MP2 quality. An implementation that employs the resolution of identity (RI) approximation for two-electron integrals to reduce the CPU time is also available, RI-CC2 [75], which is suitable for large scale integral-direct calculations. This method has been implemented in TURBOMOLE [76],... 

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