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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.
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