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Part 2
Atomic Orbital Based Response Theory
2.1 Introduction
The first part of this thesis was concerned with the optimization of the one electron density matrix
for Hartree-Fock (HF) and density-functional theory (DFT). From such an optimized density,
information about excited states and how the system reacts to a perturbation (e.g. an external
electric field) may be obtained using response theory. Response theory and the derivation of
molecular properties will be the subject of this part of the thesis.
Response theory provides a rigorous approach for calculating molecular properties. As for the SCF
optimization algorithms, the theory has usually been formulated in the molecular orbital (MO) basis
which is inherently delocal, making the implicated matrices non-sparse. A reformulation in the local
atomic orbital (AO) basis is thus necessary to obtain linear scaling algorithms and permit
calculations of properties for large systems. Such a reformulation, in which an exponential
parameterization of the density matrix is employed, is given in a paper by Larsen et al. 61 .
The AO formulation of the response functions has a number of advantages compared to the MO
formulation, besides locality. The response equations and molecular property expressions are
simpler in the AO basis as the involved matrices (e.g. the Fock and property matrices) enter the
equations in the basis they are evaluated in originally. No transformation between bases is necessary
in the AO formulation as it is in the MO formulation. The AO formulation is particular convenient
for perturbation dependent basis sets. In the MO formulation a set of perturbation dependent
orthonormal molecular orbitals must be introduced. These orbitals have no physical content and
thus add artificial complexity to the problem. To exemplify the benefits of the AO formulation, the
expression for the excited state geometrical gradient is derived in Section 2.4.
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