Contents

# Charge-Transfer

See also: Polarization & Polarization Models

The definition of the charge-transfer (CT) energy is still not settled. Rather, while we can physically describe the CT effect, it has proven difficult to come up with a reasonable definition of the CT energy. Think of charge-transfer as a delocalization process that leads to stability. Bring two molecules/atoms together and their electronic states will mix. Electronic charge is shared between the two (or more) interacting systems leading to energy lowering; this process is called charge-transfer and the energy lowering is the charge-transfer energy.

The term charge-transfer is a bit of a misnomer as it is possible to have a charge-tranfer-type energy stabilization without any nett charge transfered between the systems. Consider a symmetric, doubly H-bonded system like the pyridine dimer in its $D_{2h}$ configuration. There are two H-bonds in this system and charge is transfered symmetrically, with both molecules remaining neutral. So while charge is delocalised, no nett charge is transfered, but the charge-transfer stabilization is enormous.

We usually define interaction energy components using symmetry-adapted perturbation theory (SAPT), but SAPT doesn't give us a charge-transfer energy, instead the CT is included in the induction energy together with the polarization energy, that is, at second-order in perturbation theory, we have: $$ E_{\rm IND}^{(2)} = {\rm CT}^{(2)} + {\rm POL}^{(2)},$$ where ${\rm CT}^{(2)}$ is the second-order CT and ${\rm POL}^{(2)}$ the second-order polarization. There will be contributions from higher-order terms too. Our goal is to obtain a well-motivated definition of the CT.

Why bother about the CT? Well, there are a few reasons:

- Interpretation: we interpret the nature of bonding using the energy breakup. CT is an important component, so we need to define it in a reasonable manner.
- Polarization models: We need a good definition of the CT (at all orders) in order to define the polarization energy which is needed to define the damping used in the polarization model. From the practical point of view, this is probably the most important reason for needing to define the CT. And we need to be able to define it not only at equilibrium, but also at short-range, where damping effects become more important.

It's the latter reason that most concerns us here. The Williams-Stone-Misquitta (WSM) procedure can be used to obtain very accurate distributed polarizabilities and the Distributed Multipole Analysis (DMA) technique of Stone (1985 and 2005) can be used to obtain very accurate multipole models. But the polarization models additionally require knowledge of the damping: that is, the manner in which the $1/R$ expansion is made to die away at short $R$. Without damping $1/R \rightarrow \infty$ as $R \rightarrow 0$. We assume that the damping functions are the Tang & Toennies incomplete Gamma functions (see Stone's book on Intermolecular Forces), but we need to know what the damping constants associated with these are.

Too much damping and we will underestimate the importance of the polarization. Too little and we can grossly overestimate the polarization. The polarization energy is strongly non-additive, and the damping can effect the many-body nature of the polarization a lot. So we do need to get this right. Hence we worry about the CT.

## So how do we define the CT energy?

Since the CT energy is the stabilization energy upon charge delocalisation, it can be defined as: $$ {\rm CT} = E_{\rm int} - E_{\rm int}({\rm loc}),$$ where $E_{\rm int}$ is the interaction energy calculated in the usual way and $E_{\rm int}({\rm loc})$ is the localised interaction energy calculated, that is, some kind of localization has been performed to suppress the charge-transfer-type delocalization. There are various ways of defining the localised interaction energy, each leads to a definition of the CT energy.

### CT-SM09: Stone & Misquitta

**Ref**: A .J. Stone and A. J. Misquitta, Chem. Phys. Lett. **324**, 201-205 (2009).

There are (broadly) two kinds of basis sets used for interaction energies: the monomer centered (MC) type of basis in which basis functions are located on the molecular nuclei only, and the dimer centered (DC) type in which basis functions are additionally located on the sites of the partner monomers. The DC type of basis is used to calculate basis-set superposition error (BSSE) corrections in supremolecular calculations of the interaction energy. When using SAPT, there is no BSSE, so the MC basis should lead to sensible interactions. Only it doesn't. One reason for this is the charge-transfer energy (another is that the dispersion energy is slowly convergent with basis set, but this is another issue). To describe the CT we must allow charge-density relaxation onto the partner molecule. Hence the need for the DC-type of basis.

Important

See H. L. Williams, E. M. Mas, K. Szalewicz and B. Jeziorski, J. Chem. Phys., **103**, 7374-7391 (1995) for more details.

The Stone-Misquitta definition of the CT uses the MC/DC type of basis as follows: $$ {\rm CT^{(2)}(SM09)} = E^{(2)}_{\rm IND}({\rm DC}) - E^{(2)}_{\rm IND}({\rm MC}),$$ where we have use the induction energy and not the interaction energy. Here only the second-order CT is defined.

The CT-SM09 definition results in reasonable values for the CT (at equilibrium) with a reasonably small variation with basis set. For example, SM09 showed that for the water dimer at its minimum energy configuration the CT varies from -3.5 kJ/mol through -2.6 kJ/mol to -1.7 kJ/mol for the aug-cc-pVDZ, aug-cc-pVTZ and aug-cc-pVQZ basis sets. It is in reasonably wide use and is easy to calculate. But it has some serious shortcomings:

- Higher than second order contributions not included. These can be quite large.
- There is a non-negligible variation in CT with basis. See above example. In fact, as the basis gets complete, the distinction between the MC and DC types of basis vanish and the SM09 CT tends to zero!
- The basis-set variation grows larger at shorter separations. And in the bulk, many-body effects make shorter separations important.

There are more, subtle, problems with this definition. But just the above, lead to all sorts of problems when we attempt to use this definition to define the polarization model:

- A consequence of the above is that the separation of the induction energy into CT and POL gets less and less reliable as molecular separation decreases.
- And hence the damping needed in the polarization expansion cannot be determined, i.e., it is strongly basis set dependent.
- Therefore the many-body polarization energy (which is more sensitive to the damping) is very poorly determined.

### CT-Reg : Misquitta 2013

This is a new technique that I am still writing up so I will be brief. Here the idea is to get around all basis set issues by working suppressing the CT in real-space. The CT can be (nearly) completely suppressed for all intermolecular separations and independently of basis set using a theory called regularised SAPT, or R-SAPT. Think of regularization as filling in the nuclear potentials of the partner molecule(s) and therefore suppressing all tunneling into those potentials. We can define the second-order CT as: $$ {\rm CT}^{(2)}({\rm Reg}) = E^{(2)}_{\rm IND} - E^{(2)}_{\rm IND,reg}.$$

To get the higher-order contributions to the CT we use the polarization model as follows:

${\rm CT}^{(2)}({\rm Reg})$ allows the second-order CT to be defined at all separations. Therefore determining the damping becomes relatively straightforward.

Once the damping has been obtained, the infinite-order polarization can be calculated (by iterating to convergence). This gives us ${\rm Pol}^{2-\infty}$, where ${\rm Pol}$ indicates the energy from the model as opposed to ${\rm POL}$ which indicates the energy as calculated from an electronic structure method like SAPT.

Now we need to calculate ${\rm POL}$. If we assume that the SAPT energy component $\delta^{\rm HF}$ is all (or mostly) higher-order induction energy, then we can define:

$E^{(2-\infty)}_{\rm ind} = E^{(2)}_{\rm ind} + \delta^{\rm HF}$

And, using the result of the polarization model we can define:

${\rm CT}^{(2-\infty)}({\rm Reg}) = E^{(2-\infty)}_{\rm ind} - {\rm Pol}^{2-\infty} $

This procedure does have one shortcoming: because we need to go via the polarization model it is both cumbersome and only as accurate as the polarization model. However, because we define the main part of the CT (the second-order CT) directly via R-SAPT, the small uncertainities introduced by the polarization model are introduced for the third- and higher-order contributions only.

- Charge-Transfer =

See also: Polarization & Polarization Models

The definition of the charge-transfer (CT) energy is still not settled. Rather, while we can physically describe the CT effect, it has proven difficult to come up with a reasonable definition of the CT energy. Think of charge-transfer as a delocalization process that leads to stability. Bring two molecules/atoms together and their electronic states will mix. Electronic charge is shared between the two (or more) interacting systems leading to energy lowering; this process is called charge-transfer and the energy lowering is the charge-transfer energy.

The term charge-transfer is a bit of a misnomer as it is possible to have a charge-tranfer-type energy stabilization without any nett charge transfered between the systems. Consider a symmetric, doubly H-bonded system like the pyridine dimer in its $D_{2h}$ configuration. There are two H-bonds in this system and charge is transfered symmetrically, with both molecules remaining neutral. So while charge is delocalised, no nett charge is transfered, but the charge-transfer stabilization is enormous.

We usually define interaction energy components using symmetry-adapted perturbation theory (SAPT), but SAPT doesn't give us a charge-transfer energy, instead the CT is included in the induction energy together with the polarization energy, that is, at second-order in perturbation theory, we have: $$ E_{\rm IND}^{(2)} = {\rm CT}^{(2)} + {\rm POL}^{(2)},$$ where ${\rm CT}^{(2)}$ is the second-order CT and ${\rm POL}^{(2)}$ the second-order polarization. There will be contributions from higher-order terms too. Our goal is to obtain a well-motivated definition of the CT.

Why bother about the CT? Well, there are a few reasons:

- Interpretation: we interpret the nature of bonding using the energy breakup. CT is an important component, so we need to define it in a reasonable manner.
- Polarization models: We need a good definition of the CT (at all orders) in order to define the polarization energy which is needed to define the damping used in the polarization model. From the practical point of view, this is probably the most important reason for needing to define the CT. And we need to be able to define it not only at equilibrium, but also at short-range, where damping effects become more important.

It's the latter reason that most concerns us here. The Williams-Stone-Misquitta (WSM) procedure can be used to obtain very accurate distributed polarizabilities and the Distributed Multipole Analysis (DMA) technique of Stone (1985 and 2005) can be used to obtain very accurate multipole models. But the polarization models additionally require knowledge of the damping: that is, the manner in which the $1/R$ expansion is made to die away at short $R$. Without damping $1/R \rightarrow \infty$ as $R \rightarrow 0$. We assume that the damping functions are the Tang & Toennies incomplete Gamma functions (see Stone's book on Intermolecular Forces), but we need to know what the damping constants associated with these are.

Too much damping and we will underestimate the importance of the polarization. Too little and we can grossly overestimate the polarization. The polarization energy is strongly non-additive, and the damping can effect the many-body nature of the polarization a lot. So we do need to get this right. Hence we worry about the CT. ==So how do we define the CT energy?== Since the CT energy is the stabilization energy upon charge delocalisation, it can be defined as: $$ {\rm CT} = E_{\rm int} - E_{\rm int}({\rm loc}),$$ where $E_{\rm int}$ is the interaction energy calculated in the usual way and $E_{\rm int}({\rm loc})$ is the localised interaction energy calculated, that is, some kind of localization has been performed to suppress the charge-transfer-type delocalization. There are various ways of defining the localised interaction energy, each leads to a definition of the CT energy.

### CT-SM09: Stone & Misquitta

**Ref**: A .J. Stone and A. J. Misquitta, Chem. Phys. Lett. **324**, 201-205 (2009).

There are (broadly) two kinds of basis sets used for interaction energies: the monomer centered (MC) type of basis in which basis functions are located on the molecular nuclei only, and the dimer centered (DC) type in which basis functions are additionally located on the sites of the partner monomers. The DC type of basis is used to calculate basis-set superposition error (BSSE) corrections in supremolecular calculations of the interaction energy. When using SAPT, there is no BSSE, so the MC basis should lead to sensible interactions. Only it doesn't. One reason for this is the charge-transfer energy (another is that the dispersion energy is slowly convergent with basis set, but this is another issue). To describe the CT we must allow charge-density relaxation onto the partner molecule. Hence the need for the DC-type of basis.

Important

See H. L. Williams, E. M. Mas, K. Szalewicz and B. Jeziorski, J. Chem. Phys., **103**, 7374-7391 (1995) for more details.

The Stone-Misquitta definition of the CT uses the MC/DC type of basis as follows: $$ {\rm CT^{(2)}(SM09)} = E^{(2)}_{\rm IND}({\rm DC}) - E^{(2)}_{\rm IND}({\rm MC}),$$ where we have use the induction energy and not the interaction energy. Here only the second-order CT is defined.

The CT-SM09 definition results in reasonable values for the CT (at equilibrium) with a reasonably small variation with basis set. For example, SM09 showed that for the water dimer at its minimum energy configuration the CT varies from -3.5 kJ/mol through -2.6 kJ/mol to -1.7 kJ/mol for the aug-cc-pVDZ, aug-cc-pVTZ and aug-cc-pVQZ basis sets. It is in reasonably wide use and is easy to calculate. But it has some serious shortcomings:

- Higher than second order contributions not included. These can be quite large.
- There is a non-negligible variation in CT with basis. See above example. In fact, as the basis gets complete, the distinction between the MC and DC types of basis vanish and the SM09 CT tends to zero!
- The basis-set variation grows larger at shorter separations. And in the bulk, many-body effects make shorter separations important.

There are more, subtle, problems with this definition. But just the above, lead to all sorts of problems when we attempt to use this definition to define the polarization model:

- A consequence of the above is that the separation of the induction energy into CT and POL gets less and less reliable as molecular separation decreases.
- And hence the damping needed in the polarization expansion cannot be determined, i.e., it is strongly basis set dependent.
- Therefore the many-body polarization energy (which is more sensitive to the damping) is very poorly determined.

### CT-Reg : Misquitta 2013

This is a new technique that I am still writing up so I will be brief. Here the idea is to get around all basis set issues by working suppressing the CT in real-space. The CT can be (nearly) completely suppressed for all intermolecular separations and independently of basis set using a theory called regularised SAPT, or R-SAPT. Think of regularization as filling in the nuclear potentials of the partner molecule(s) and therefore suppressing all tunneling into those potentials. We can define the second-order CT as: $$ {\rm CT}^{(2)}({\rm Reg}) = E^{(2)}_{\rm IND} - E^{(2)}_{\rm IND,reg}.$$

To get the higher-order contributions to the CT we use the polarization model as follows:

${\rm CT}^{(2)}({\rm Reg})$ allows the second-order CT to be defined at all separations. Therefore determining the damping becomes relatively straightforward.

Once the damping has been obtained, the infinite-order polarization can be calculated (by iterating to convergence). This gives us ${\rm Pol}^{2-\infty}$, where ${\rm Pol}$ indicates the energy from the model as opposed to ${\rm POL}$ which indicates the energy as calculated from an electronic structure method like SAPT.

Now we need to calculate ${\rm POL}$. If we assume that the SAPT energy component $\delta^{\rm HF}$ is all (or mostly) higher-order induction energy, then we can define:

$E^{(2-\infty)}_{\rm ind} = E^{(2)}_{\rm ind} + \delta^{\rm HF}$

And, using the result of the polarization model we can define:

${\rm CT}^{(2-\infty)}({\rm Reg}) = E^{(2-\infty)}_{\rm ind} - {\rm Pol}^{2-\infty} $

This procedure does have one shortcoming: because we need to go via the polarization model it is both cumbersome and only as accurate as the polarization model. However, because we define the main part of the CT (the second-order CT) directly via R-SAPT, the small uncertainities introduced by the polarization model are introduced for the third- and higher-order contributions only.