ONIOM for excited states

Describe an excited state

Most studies that employ ONIOM involve electronic ground states, but the method can describe excited states as well. Clearly, the high-level method must be able to describe an excited state (such as CASSCF, as used in this study) and the excitation must be (approximately) localized in the model system.
However, the choice for low-level method and electronic state is less clear. Assuming that the excitation is completely localized in the high-level region, one can argue for a groundstate low-level method, because the excitation is the same in $E_{real}^{low}$ and $E_{model}^{low}$ and therefore cancels from $E_{real}^{low}-E_{model}^{low}$ .

However, in this case, the interaction between the high- and low-level regions would be based only on the ground-state electronic structure, which may or may not be correct, depending on the system and process. It therefore seems formally necessary to employ an excited-state method for the low level, when calculating excited states with ONIOM. The excited-state energy is denoted $E^{*}$, and the ground-state energy as E (we discuss ground and excited-state potentials, but these could in fact be any pair of excited-state potentials). The ONIOM excited-state energy is then written as

$E^{*,ONIOM}=E_{model}^{*,high}+E_{real}^{*,low}-E_{model}^{*,low}$

The excitation energy (at a particular molecular geometry) is expressed as

$\mathrm{\Delta }{E}_{}^{ONIOM}$	$=E_{}^{*,ONIOM}-E_{}^{ONIOM}$
	$=(E_{model}^{*,high}+E_{real}^{*,low}-E_{model}^{*,low})-(E_{model}^{high}+E_{real}^{low}-E_{model}^{low})$
	$=(E_{model}^{*,high}-E_{model}^{high})+(E_{real}^{*,low}-E_{real}^{low})-(E_{model}^{*,low}-E_{model}^{low})$
	$=\mathrm{\Delta }{E}_{model}^{high}+\mathrm{\Delta }{E}_{real}^{low}-\mathrm{\Delta }{E}_{model}^{low}$

As mentioned above, in some cases we can assume (or approximate) that the excitation is localized in the high-level region and therefore that the interaction between regions does not depend on the state.
In this case $E_{real}^{*,low}-E_{model}^{*,low}\approx E_{real}^{low}-E_{model}^{low}$, and we can replace the low-level excited-state calculations with ground-state calculations.
In the resulting equations we indicate this substitution with the constrained low-level state (CLS) label:

$E^{*,ONIOM}\approx E^{*,ONIOM-CLS}=E_{model}^{*,high}+E_{real}^{low}-E_{model}^{low}$

and hence,

$\mathrm{\Delta }{E}_{}^{ONIOM-CLS}$	$=E_{}^{*,ONIOM-CLS}-E_{}^{ONIOM}$
	$=(E_{model}^{*,high}+E_{real}^{low}-E_{model}^{low})-(E_{model}^{high}+E_{real}^{low}-E_{model}^{low})$
	$=E_{model}^{*,high}-E_{model}^{high}$
	$=\mathrm{\Delta }{E}_{model}^{high}$

This is an attractive approximation, because ground-state-only methods can be used in the low level (e.g., molecular mechanics),reducing the complexity and the computational time and giving us a far greater choice of low-level methods.

Since $E_{real}^{*,low}-E_{model}^{*,low}\approx E_{real}^{low}-E_{model}^{low}$ , we could invert the above approximation and constrain the low level to be the excited state, which we indicate with the CLS* label.
In this case, the excited-state energy is equal to

$E^{*,ONIOM}=E^{*,ONIOM-CLS*}=E_{model}^{*,high}+E_{real}^{*,low}-E_{model}^{*,low}$

However, the ground-state ONIOM energy is written as

$E^{ONIOM}\approx E^{ONIOM-CLS*}=E_{model}^{high}+E_{real}^{*,low}-E_{model}^{*,low}$

(with excited-state low-level terms) and

$\mathrm{\Delta }{E}_{}^{ONIOM-CLS*}$	$=E_{}^{*,ONIOM-CLS}-E_{}^{ONIOM-CLS*}$
	$=(E_{model}^{*,high}+E_{real}^{*,low}-E_{model}^{*,low})-(E_{model}^{high}+E_{real}^{*,low}-E_{model}^{*,low})$
	$=E_{model}^{*,high}-E_{model}^{high}$
	$=\mathrm{\Delta }{E}_{model}^{high}$

The resulting energy difference $\mathrm{\Delta }{E}_{}^{ONIOM-CLS*}$ is the same as $\mathrm{\Delta }{E}_{}^{ONIOM-CLS}$ , although the ONIOM potential energy surfaces will not be the same, and, e.g., a stationary point for one approximation need not be a stationary point for the other. In fact, this CLS* approximation is not as attractive as the CLS approximation, because using an excited state for the low level increases the complexity of the calculation and the computational time and gives us fewer choices of low-level method. Consequently, we focus on the CLS approximation in the results presented in the next section.

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