Population analysis and charge representation

In classical chemistry, chemical bonding is classified into ionic or covalent bonds (sharing of electrons).

Various methods are available to quantify atomic charges [82]. The differences in these methods arise from the algorithm used to partition the electron density. A graphical representation of the different methods can be seen in Figure 2-7.

2.3.1 Mulliken charge separation

The Mulliken population analysis is the oldest known partitioning of electron density. This method utilizes the representation of molecular wave-functions from basis functions and the electrons are distributed amongst the atomic orbitals of the atom [82–85].

The wave-functions (𝝋(𝒓)_𝒌) are expanded in terms of the basis function (𝝌_𝝁(𝒓)):

𝜑(𝑟)_𝑘 = ∑ ∑ 𝑐_𝜇𝑖

𝜇∈𝑖 𝑖

𝜒_𝜇(𝑟) (2.10)

The total number of electrons is defined as:

𝑁_𝑒= ∑ 𝑛_𝑘∫|𝜑(𝑟)_𝑘|²𝑑𝑟

𝑜𝑐𝑐𝑢𝑝𝑖𝑒𝑑

𝑘

(2.11)

where 𝑛_𝑘 is the number of orbital occupation.

Substituting equation 2.10 in 2.11:

𝑁_𝑒= ∑ ∑ ∑ 𝑛_𝑘𝑐_𝜇𝑘𝑐_𝑣𝑘∫ 𝜒_𝜇∙ 𝜒_𝑣𝑑𝑟

𝑜𝑐𝑐𝑢𝑝𝑖𝑒𝑑

𝑘 𝜇∈𝑖,𝑣∈𝑗 𝑖,𝑗

(2.12)

Defining a Density matrix as:

23 𝐷_𝜇𝑣= ∑ 𝑛_𝑘𝑐_𝜇𝑘𝑐_𝑣𝑘

𝑜𝑐𝑐𝑢𝑝𝑖𝑒𝑑

𝑘

(2.13)

and an overlap integral between 𝜒_𝜇 and 𝜒_𝑣:

𝑆_𝜇,𝑣 = ∫ 𝜒_𝜇∙ 𝜒_𝑣𝑑𝑟 (2.14) The total number of electrons can be written as:

𝑁_𝑒 = ∑ ∑ 𝐷_𝜇,𝑣∙ 𝑆_𝜇,𝑣

𝜇∈𝑖 𝑣∈𝑗 𝑖,𝑗

(2.15)

Since different basis functions represent the wave functions describing a system (e.g. a molecule) differently, the analysis becomes strongly dependent on the basis set chosen. Segall et al. [85] show that the magnitude of the atomic charge on a C atom in CO in the gas phase can differ by as much 0.3e depending on the basis set. Fonseca Guerra et al. [82] believe that the atomic charges resulting from a Mulliken population analysis are not of much use. The Natural Population analysis by Reed et al. [86]

is an improvement on the Mulliken analysis as they consider orthonormal natural atomic orbitals instead of basis atomic orbitals.

2.3.2 Hirshfeld charge separation

The Hirshfeld analysis separates the electron density of the system in proportion to the atomic ground states. The algorithm defines an electron density of a “promolecule” system which is typically the sum of the electron densities of the atomic ground states (𝒑(𝒓)_𝒊^𝟎) [82,87,88].

𝜌(𝑟)𝑃𝑟𝑜𝑚𝑜𝑙𝑒𝑐𝑢𝑙𝑒 = ∑ 𝑝(𝑟)_𝑖⁰

𝑖

(2.16)

The atomic ground states are typically spherical in nature. The weight function is then defined as:

𝑤(𝑟)_𝑖 = 𝑝(𝑟)_𝑖⁰ 𝜌(𝑟)𝑃𝑟𝑜𝑚𝑜𝑙𝑒𝑐𝑢𝑙𝑒

(2.17)

Using the electron density of the system (𝒑(𝒓)) and the nuclear charge (𝒁𝒊) the Hirshfeld atomic charge is then:

𝑄_{𝑖𝐻𝑖𝑟𝑠ℎ𝑓𝑒𝑙𝑑} = 𝑍_𝑖− ∫ 𝑤(𝑟)_𝑖 𝑝(𝑟)𝑑𝑟 (2.18)

2.3.3 Bader charge separation

The Bader analysis looks purely at the topology of the electron density [82,88,89]. The regions which the electron density are divided into Bader volumes, which are separated by curves representing the minimum of the electron density is a minimum (𝛻𝝆 = 𝟎) [82,88,89].

Typical algorithms from the Bader analysis require a large amount computational effort in finding the critically points where the electron density is a minimum [90]. The algorithm proposed by Henkelman et al. [89] uses the FFT grids produced from DFT calculations and finds the maxima on the grid using a steepest accent procedure. The procedure is significantly faster than previous algorithms with minimal losses in accuracy. This method was recoded in MATLAB and uses a CASTEP formatted density file to conduct the analysis.

Fonseca Geurrara et al. [82] believe the Bader charges to be too large and resulting in a misrepresentation of the partial atomic charges. However, the Bader charge assignment does not

24 originate from a full and complete charge separation and the calculated charge does therefore not represent the classical “charge” on a specific atom. The Bader charge analysis is still a popular method of electron density decomposition

Figure 2-7: Charge assignment for a hypothetical 1-D charge distribution (per the Mulliken charge separation (b) and the Hirshfeld and Bader charge separation method (c).

While all three methods mentioned above are widely used, the empirical assignment of charges to atoms can lead to poor approximations of the actual charge distribution, particularly if electron delocalization is prevalent in the system.

2.3.4 Mapping the electron density

Instead of assigning approximate charges to the atoms in the system, the electron density could be mapped out and slices of specific planes could be analysed. Figure 2-8 shows the electron density of a free CO molecule in a vacuum cut at a plane running through the C-O bond. As to be expected, the electron density is highest around the oxygen atom.

25 Figure 2-8: Electron density of CO in the gas phase (right)

While representations of the gas phases are easily understood, for surface models, the raw density map appears not to yield any insight into changes induced by the adsorption of CO on the surface or by lateral interactions involved as the electron density around the Fe cores overshadowed the density of the CO. This can be circumvented by subtracting the electron density of the clean surface, as shown in Figure 2-9. The result is electron density of the adsorbed CO and the electron density changes it induces on the metal. It is expected that the electrons not involved in the CO-Fe interaction will cancel each other out and the picture that remains will show the change in the electrons that are involved in the CO-Fe interaction. The changes to the metal are seen as a change in colour from dark blue to teal, which indicates the repositioning of electrons relative to the clean surface.

Figure 2-9: Electron density of the system of CO adsorbed on Fe (100) (left), Fe (100) (middle) and the resulting change electron density upon adsorption of CO on Fe (100 (right)

The electron density maps appear to give the same insights as the population analysis with the advantage of being a product (i.e. output) of a DFT simulation. Furthermore, a visual representation can at times be more easily understood then numerical values.

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