Open-shell implementation

EOMCCSD(T) calculations, while facilitating the open-shell implementation of the CR-EOMCCSD(T) method employing the restricted open-shell Hartree-Fock (ROHE) orbitals [59]. Indeed, the use of spin-orbital energy differences (ca -f -f Cc — — ej — e ) instead of the complete form of the diagonal matrix elements of involving triply excited determinants to... 

Another way of constructing wave functions for open-shell molecules is the restricted open shell Hartree-Fock method (ROHF). In this method, the paired electrons share the same spatial orbital thus, there is no spin contamination. The ROHF technique is more difficult to implement than UHF and may require slightly more CPU time to execute. ROHF is primarily used for cases where spin contamination is large using UHF. 

We continue this chapter with a presentation of the basic concepts and notations relevant to D-functional theory (Section 111). We then review the fundaments of the NOF theory (Section IV) and derive the corresponding Euler equations (Section V). The Gilbert [15] and Pernal [81] formulations, as well as the relation of Euler equations with the EKT, are considered here. The following sections are devoted to presenting our NOF theory. The cumulant of the 2-RDM is discussed in detail in Section VI. The spin-restricted formulations for closed and open-shells are analyzed in Sections Vll and VIII, respectively. Section IX is dedicated to our further simplification in order to achieve a practical functional. In Section X, we briefly describe the implementation the NOF theory for numerical calculations. We end with some results for selected molecules (Section XI). 

With few exceptions, all investigations of matrix-isolated reactive intermediates are done by absorption spectroscopy, in the UV-vis and/or in the IR spectral range, or, in the case of open-shell species, by ESR. Occasionally, one also finds studies where emission or Raman scattering of reactive intermediates is probed in matrices, but these studies are few and far between, so we will focus in this section on the first group of techniques that can be easily implemented with commercially available equipment. 

All electron calculations were carried out with the DFT program suite Turbomole (152,153). The clusters were treated as open-shell systems in the unrestricted Kohn-Sham framework. For the calculations we used the Becke-Perdew exchange-correlation functional dubbed BP86 (154,155) and the hybrid B3LYP functional (156,157). For BP86 we invoked the resolution-of-the-iden-tity (RI) approximation as implemented in Turbomole. For all atoms included in our models we employed Ahlrichs valence triple-C TZVP basis set with polarization functions on all atoms (158). If not noted otherwise, initial guess orbitals were obtained by extended Hiickel theory. Local spin analyses were performed with our local Turbomole version, where either Lowdin (131) or Mulliken (132) pseudo-projection operators were employed. Broken-symmetry determinants were obtained with our restrained optimization tool (136). Pictures of molecular structures were created with Pymol (159). 

We have previously defined the one-electron spin-density matrix in the context of standard HF methodology (Eq. (6.9)), which includes semiempirical methods and both the UHF and ROHF implementations of Hartree-Fock for open-shell systems. In addition, it is well defined at the MP2, CISD, and DFT levels of theory, which permits straightforward computation of h.f.s. values at many levels of theory. Note that if the one-electron density matrix is not readily calculable, the finite-field methodology outlined in the last section allows evaluation of the Fermi contact integral by an appropriate perturbation of the quantum mechanical Hamiltonian. 

In this section, we briefly discuss some of the electronic structure methods which have been used in the calculations of the PE functions which are discussed in the following sections. There are variety of ab initio electronic structure methods which can be used for the calculation of the PE surface of the electronic ground state. Most widely used are Hartree-Fock (HF) based methods. In this approach, the electronic wavefunction of a closed-shell system is described by a determinant composed of restricted one-electron spin orbitals. The unrestricted HF (UHF) method can handle also open-shell electronic systems. The limitation of HF based methods is that they do not account for electron correlation effects. For the electronic ground state of closed-shell systems, electron correlation effects can be accounted for relatively easily by second-order Mpller-Plesset perturbation theory (MP2). In modern implementations of MP2, linear scaling with the size of the system has been achieved. It is thus possible to treat quite large molecules and clusters at this level of theory. 

Wang, F. and Ziegler T., Excitation energies of some dl systems calculated using time-dependent density functional theory an implementation of open-shell TDDFT theory for doublet-doublet excitations. Mol.Phys (2004) 102 2585 -2595. 

In the present chapter, we will focus on the simulation of the dynamics of photoexcited nucleobases, in particular on the investigation of radiationless decay dynamics and the determination of associated characteristic time constants. We use a nonadiabatic extension of ab initio molecular dynamics (AIMD) [15, 18, 21, 22] which is formulated entirely within the framework of density functional theory. This approach couples the restricted open-shell Kohn-Sham (ROKS) [26-28] first singlet excited state, Su to the Kohn-Sham ground state, S0, by means of the surface hopping method [15, 18, 94-97], The current implementation employs a plane-wave basis set in combination with periodic boundary conditions and is therefore ideally suited to condensed phase applications. Hence, in addition to gas phase reference simulations, we will also present nonadiabatic AIMD (na-AIMD) simulations of nucleobases and base pairs in aqueous solution. 

The HF method treats electron-electron interactions at a mean field level, with the Hartree and exchange interactions exactly written. The method can be implemented either in its spin restricted form (RHF), for closed shell systems, or in the unrestricted form (UHF) for open-shell or strongly correlated systems. In the first case, the one-electron orbitals are identical for electrons of both spin directions, while UHF can account for a non-uniform spin density. The one electron orbitals, which are determined in the course of the self-consistent resolution of the HF equations, are expanded on an over-complete basis set of optimized variational functions. 

G2(MP2)-RAD, which implements restricted-open-shell versions of both coupled cluster and perturbation theories. The latter method has been shown to generally yield reliable results when applied to open-shell systems [66]. 

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