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Part 2 Atomic Orbital Based Response Theory 2.6 Conclusion The atomic orbital (AO) based response equations have been derived using the second quantization framework. In particular, the proof of pairing is considered. Since the diagonal elements in κ are not redundant in the AO basis, the proof given in the MO basis cannot be directly applied. However, it is shown that there is also pairing in the AO basis. An AO response solver has been implemented similar to the solver in the MO basis with a few exceptions. The lack of diagonal dominance in the electronic Hessian in the AO basis makes preconditioning a difficult task. Optimally, the AO solver should be implemented in a linear scaling manner with only matrix multiplications and additions, and without reference to the MO basis. However, currently a transformation is made to the MO basis where the preconditioning is carried out followed by a transformation back to the AO basis. The redundant orbital rotations, which are simply left out of the MO equations, are removed in the AO formulation using projection operators. The response equations and molecular property expressions are simpler in the AO formulation than in the MO formulation. To demonstrate how expressions for properties can easily be derived in the AO response framework, the expression for the geometrical gradient of the singlet excited state has been derived. To illustrate the possibilities of the AO optimization methods presented in Part 1, joined with the AO response solver presented in this part of the thesis, test calculations are given for cases where DIIS diverged when optimizing the reference state. The averaged polarizability and the lowest excitation energy are given as well as the excited state dipole for one of the examples. The derivation and implementation of the various molecular properties is straightforward in the AO formulation compared to the MO formulation as exemplified by the excited state geometrical gradient. Especially the derivation of higher derivatives of molecular properties is simplified, and it will thus be natural to expand our response program in this direction. However, before calculations of molecular properties of large and complex molecules can be carried out in a truly linear scaling framework, the problems related to preconditioning of the AO solver must be solved. 76
Part 3 Benchmarking for Radicals 3.1 Introduction To corroborate the reliability of ab initio quantum chemical predictions of molecular properties, it is important to investigate and describe strengths and weaknesses of the many-electron models through systematic benchmark studies on different kinds of molecules. Regarding open-shell molecules, benchmarks have been reported comparing open- and closed-shell molecules examining the accuracy of molecular properties computed by various many-electron models. In a study of the atomization energies of 11 small molecules 67 no significant difference in the performance for closed- and open-shell molecules was found for the CCSDT model. However, in another study 68 it was found that even though the CCSD(T) model performs convincingly for closed-shell molecules, the performance for open-shell molecules is less impressive. In this part of the thesis full configuration interaction (FCI) benchmarks of molecular properties for the small open-shell molecules CN and CCH are presented. In the FCI model, all Slater determinants arising from distributing the electrons in the given one-electron basis with correct symmetry and spin-projection are included. Errors due to truncation of the many-electron basis are thus eliminated in an FCI calculation and it provides important benchmarks for other many-electron models. For open-shell molecules, the number of FCI benchmarks is limited and the work presented in this part of the thesis is an attempt to improve on this situation. We thus hope our results will serve as valuable benchmarks for further analysis of open-shell methods. 3.2 Computational Methods All calculations have been carried out with the quantum chemical program package LUCIA 69 , using integrals and Hartree-Fock (HF) orbitals obtained from the DALTON 70 program. The calculations 77
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PhD Thesis Optimization of Densitie
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Contents Preface ..................
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Summary............................
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List of Publications This thesis in
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