In this section the Mulliken and Natural charge analysis methods, used in this chapter, are discussed. Charge analysis techniques are a means of interpreting Schr¨odinger’s wave equa- tion; the MPA and NPA methods achieve this by partitioning the wavefunction with reference to the basis functions used to compute it. This then allows qualitative analysis of the chemi- cal processes underpinning the system e.g. determining sites for nucleophilic or electrophilic attacks [188].
The electron density functionρ(r)is defined as the probability of finding an electron in a
volume dr,
Z
ρ(r)dr=N (6.1)
where integration is carried out over all space and this results inNelectrons. For HF theory, ρ(r)= N
∑
µ N∑
ν Pµν φµ(r)φν(r) (6.2)integratingρ(r)over all space, Z ρ(r)dr = N
∑
µ N∑
ν Pµν Z φµ(r)φν(r)dr (6.3) = N∑
µ N∑
ν Pµν Sµν (6.4) = N (6.5) (6.6) whereSµν is the overlap matrix.6.2.1
Mulliken Population Analysis
Mulliken proposed the first population analysis technique [189]. The implicit assumption is that the electron density can be partitioned according to the basis functions used to describe it. If density is attributed to a product of two functions on different centers, then the density is split evenly.
The MPA scheme uses theDαβSαβ matrix to apportion charge i.e. the matrix arising from the product of the Density and overlap matrices. The number of electrons associated with AO αis the diagonal elementDααSαα, whereas half the value of an off-diagonal elementDαβSαβ
is the number of electrons shared between AOsα and β. By summing all the contributions from AOs of a given atomic center, the number of electrons attributed to that center can be obtained. Thus the electron population on a given atomic centerAis defined as,
ρA= AO
∑
α2A AO∑
β Dα βSα β (6.7)which is then used to define the gross chargeQAon the atomic center according to
QA=ZA ρA (6.8)
whereZA is the atomic number of atomA(i.e the number of positively charged protons in the
nucleus of Atom A).
6.2.2
Natural Population Analysis
Wienhold et al. [224,298] describes the use of Natural population analysis to obtain nett charge on atoms. The NPA techniques uses natural orbitals to apportion electron density onto atomic and molecular orbitals.
Natural orbitals are the eigenfunctions of a quantity called the ‘first-order reduced Density matrix’. We now discuss the first-order density matrix, the definition of a natural orbital and its subsequent use in the NPA procedure to obtain nett charge.
6.2.2.1 First-order reduced density matrix
The motion of electrons in a system are described by the wavefunctionψ. The electron density functionρ(r)is obtained from the wavefunction,
ρ(r)=jψ(r)j
2
(6.9) Given ‘N’ electrons in a system, the probabilityP, of finding electron 1 located atx1, simulta-
neously when electron 2 is located atx2and so on for ‘N’ electrons is given by,
P=ψ(x1;x2;:::;xN) ψ(x1;x2;:::;xN) (6.10)
Thus the probability of finding electron 1, regardless of the locations of the other electrons is defined as,
P(x1)=N
Z
dx1dx2:::dxNψ(x1;x2;:::;xN) ψ(x1;x2;:::;xN) (6.11)
where,Nis a normalization factor defined such
Z
dx1ρ(x1)=N (6.12)
The probability function P(x1)can be generalized to a density matrixγ(x1;x 0
1), which is
termed the first-order reduced density matrix, γ(x1;x 0 1)=N Z dx1dx2:::dxNψ(x1;x2;:::;xN) ψ(x 0 1;x2;:::;xN) (6.13)
For HF theory, this can be obtained by expandingγ(x1;x 0 1)using functionsχi, γHF(x1;x 0 1)=
∑
a χa(x1)χ a(x1) (6.14)6.2.2.2 Natural Atomic Orbitals and NPA
The first-order reduced density matrix (γ) is Hermitian. On diagonalizingγthe eigenfunctions are called natural atomic orbitals (NAO) and the eigenvalues are occupation numbers. NAOs can be used to determine nett atomic charge [83]. We now briefly cover the process of ob- taining charges using NPA. Assume the basis functions for the system of interest have been
arranged such that all orbitals for center A are before those on center B and so on i.e., χA 1;χ A 2; χ A 3;::: ;χ B k+1 ; χ B k+2 ; χ B k+3 ;:::; χ C l+1 ;χ C l+2 ; χ C l+3 ; (6.15)
A Density matrixDcan be written in terms of blocks of basis functions at a given center,
D = 0 B B B B DAA DAB DAC ::: DAB DBB DBC ::: DAC DBC DCC ::: ::::::::::::::::::::: 1 C C C C A (6.16)
The NAOs for atom A are defined as those which diagonalize theDAAblock and similarly for atom B and so on. The definition of the sub-blocks are constrained to ensure eigenfunctions are orthonormal within the sub-block and to all other eigenfunctions.
The process of obtaining NPA charges begins by partitioning both the density and overlap matrices into sub-blocks ordered by the Atom (A) that the sub-block represents, its angular momentum (l) and the symmetry element (m) for the givenl.
All 2l+1 symmetry elements in the sub-block are averaged and the sub-blocks are inde-
pendently diagonalized to give eigenfunctions which form what are called pre-NAOs.
These pre-NAOs are classified by their occupancy into two groups, the ones with the high- est occupancy are called the Natural Minimum Basis (NMBs) and all the rest are called the Natural Rydberg Basis (NRBs).
As pre-NAOs from one center overlap with other pre-NAOs on another center, two sets of diagonalizations are done to obtain the intermediate NAOs – the NRBs are Schmidt orthogo- nalized w.r.t the NMBs and the NRBs are separately diagonalized using a weighted occupancy symmetric orthogonalization scheme [224].
This process removes overlaps and localized density onto specific atomic centers. The intermediate NAOs from the two sets of orthogonalizations are re-blocked and diagonalized to give the final NAOs. The diagonal elements of the density matrix formed using NAOs give the atomic population of each NAO. If the atomic populations for all the NAOs centered on a given atom are summed, the resulting charge population is called the natural atomic population.
6.2.3
The Radial Distribution Function
The RDF is a means of describing the structure of systems like gases and liquids [9, 156]. It measures the correlation between particles in a system and denotes the average probability of finding some particle at a distancerfrom a reference particle i.e. it is a spatial measure for the packing of particles around some central point.
Figure 6.2: Radial distribution function determined from a 100 ps molecular dynamics sim- ulation of liquid argon at a temperature of 100 Kelvin and a density of 1.396 g/cm3. Taken from [156].
The RDF is computed relative to an ideal gas2and is a dimensionless quantity [115, 223]. In this section the pair-wise RDFg(r)is utilized.
Figure 6.2 is a typical RDF obtained from an MD simulation. The graph, taken from [156], is for a 100 ps MD simulation of liquid argon. From the graph, at short distancesg(r)is zero
owing to the large short-range repulsive force between atoms. At 3.7 ˚A g(r) peaks at 3 and
this indicates that it is three times more likely that a pair of Argon atoms are separated at this distance. The RDF reduces at 5 ˚A approaching a minimum, indicating that at this distance the chances of finding a pair of particles is diminished compared to the 3.7 ˚A . At distances greater than 7 ˚A the value ofg(r)approaches one indicating a distinct lack of long range structure.
The pair-wise RDF is generated by computing the distance between a particle of interest and other particles in the system. These distances are sorted by distance and binned to create a histogram. Each population is then normalized by the number of particles that would be present in an ideal gas.
A Python code for computing the RDF was written to process the test systems used in this Chapter, this has been reproduced in the Appendix, Section A.4.
2An ideal gas is a theoretical system which has a uniform distribution of its particles over all of