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In the first section of this chapter, we have understood how a given scalar

func-tion V : R

ⁿ

→ R can be used to divide the set of points defining R

ⁿ

into different

subsets. Which points are clustered together into the same subset depends on

the topology of V . At this point, it should be obvious that meaningful scalar

functions should induce meaningful partitions. On the other side, the previous

section has clarified that all the information of our system is contained into its

N-body density matrix. Interestingly, its “reduced” versions are normally enough

for most of our chemical purposes. Thence, gathering together these ideas, we

may infer that scalar fields constructed from RDMs must carry chemical and

physical information, as well as its induced topology.

Here, we will show how the 1-RDM and the 2-RDM can be used to split R

³

into entities with chemical significance. Whereas the 1-RDM plays its role in the wonderful world of the induced topology through the electron density, the 2-RDM does it by following the road paved by the electron localization function (ELF). These two well-known scalar fields are (probably) the most notorious in Quantum Chemical Topology (QCT [22]).

3.3.1 The electron density

The topology induced by the electron density (equation 3.6) is the cornerstone of Bader’s theory of atoms in molecules [1, 23–25]. The form assumed by the distribution of charge in the molecule is the physical manifestation of the forces acting within the system. Concretely, the attractive force exerted by the nuclei is responsible for the single most important topological property exhibited by a molecular charge distribution of a many-electron system: ρ(r) exhibits (in gen-eral) local maxima only at the positions of the nuclei. As an example, the elec-tron density of BH

₃

in its molecular plane is plotted in Figure 3.3a, where each of the four maxima is located at a nuclear position.

^‡

With all the ρ-attractors localized, the next step consists of obtaining the separatrices of our dynamical system. They are represented in Figure 3.3b, where we can confirm that each attraction basin contains but one (and only one) nucleus. In this fashion, recog-nizable atomic forms are created within the charge distribution of the molecule:

we are in the presence of “atomic basins”.

This elegant procedure to define the atoms-in-the-molecule is, in essence, simple and intuitive. Moreover, atoms defined in such a manner exhibit interesting properties, but we will return to this subject later (Section 3.4).

‡This is not exactly true: the attractor of each hydrogen basin is not exactly located at the corresponding nuclear position. This is due to the fact that hydrogen nucleus is a weak electron density attractor.

FIGURE3.3: Topology induced by ρ in the BH₃molecule. The electron density in the molecular plane is shown in (a) whereas the atomic basin separatrices (zero-flux surfaces) are shown in (b). Isosurfaces of ρ > ρ_bcp, ρ = ρbcp and

ρ < ρ_bcp(top to bottom) are shown in (c).

To conclude this subsection, let us discuss about the intuitiveness of this atomic

definition by analyzing the ρ isosurfaces plotted in Figure 3.3c. The upper

iso-surface, characterized by a large value of ρ, is divided into four independent

pseudo-spheres, each one containing one nucleus. It seems natural to associate

the electron density enclosed by each sphere to the corresponding trapped

nu-cleus. This atomic classification of the electron density holds until the four

spheres merge (middle case). The critical density for which a pair of

“atomic-spheres” first bond each other define a special point in R

³

. It is known as bond

critical point (bcp) and, if we are to define the atoms as clearly bounded

enti-ties, we have no alternative but to consider it as part of the separation between

the bonded atoms. As we move into larger ρ isosurfaces (bottom one in Figure

3.3c), more points would form part of this separation which delimits the

molec-ular atoms. This is, exactly, how atoms are defined when the topology induced

by ρ is invoked.

3.3.2 The electron localization function

The electron localization function (ELF) [4, 5] was introduced by Becke and Edgecombe as a “simple measure of electron localization in atomic and molec-ular systems” [4]. In order to understand this function, we first need to define the same-spin pair probability function:

D_σ

(r

1

, r

₂

) = Γ

N⁽²⁾

([r

1

, σ], [r

2

, σ]) (3.8) It corresponds to the probability of finding simultaneously one electron at r

₁

and another at r

₂

, both with the same spin σ. Thus, the same-spin conditional probability function

^§

can be written as:

P_σ

(r

2

|r

1

) =

D_σ

(r

1

, r

₂

)

ρ_σ

(r

1

) (3.9)

where ρ

_σ

(r

1

) = Γ

N⁽¹⁾

(r

1

, σ). Using it, we can obtain the conditional probability of finding a same-spin electron at a distance s of the electron at r

₁

. This involves taking a spherical average on a sphere of radius s around r

₁

, S(s, r

1

):

p_σ

(s|r

1

) = 1 4π

Z

S(s,r1)

dSP_σ

(r

2

|r

1

) (3.10)

where the integration is done for the r

₂

spatial coordinate. For small values of

s, the Taylor expansion of p_σ

(s|r) takes the form [4]:

p_σ

(s|r) = 1 3

– 1 2

 ∇

²_r0D_σ

(r, r

⁰

) 

r⁰=r

ρ_σ

(r)

™

| {z }

C_σ(r)

s²

+ O (s

³

) (3.11)

§ It is the probability of finding a σ-spin electron at r₂ knowing that there is one σ-spin electron at r₁.

The coefficient of s

²

(except for the one-third factor), C

_σ

(r), tells us how large the same-spin conditional probability function is at each point in space. There-fore, it is considered an inverse measure of localization: the smaller this mag-nitude is, the more likely that an electron avoids electrons of equal spin. This amount can be calibrated with C

_σ

(r) for the homogeneous electron gas of the same density, C

_σ⁰

(r), giving rise to the adimensional ELF kernel:

χ_{E LF}

(r) =

C_σ

(r)

C_σ⁰

(r) (3.12)

As a consequence of the “inverse relationship”, and also in order to get a rea-sonable scaled quantity, the ELF (η) is defined as the Lorentzian projection of the ELF kernel:

η(r) =

1

1 + χ

E LF²

(r) (3.13)

Interestingly, the ELF was originally introduced assuming a Hartree-Fock for-mulation. In that situation, C

_σ

(r) is given by:

[C

σ

(r)]

^{H F}

= τ

σ

(r) − 1 4

|∇ρ

σ

(r)|

²

ρ_σ

(r) (3.14)

where the positive-definite kinetic energy density, τ

_σ

, is just P

i|σ

|∇φ

i

|

²

, whose summation is restricted to the φ

_i

molecular orbitals of σ spin.

As an example, the ELF topology in the water molecule is depicted in Figure 3.4a. We can recognize five attractors in the ELF topology, each one giving rise to an ELF basin (3.4b). Its attraction basins remind the standard Lewis structure for the water molecule. Clearly, one of the basins represents the oxygen (1s) core electrons, whereas two of them accounts for the two oxygen lone pairs.

Finally, the remaining basins describe the electrons involved in the two O-H

bonds.

FIGURE 3.4: Topology induced by the electron localization function (ELF) in the H₂O molecule. On the one hand, ELF isoline maps in the two symmetry planes of the molecule and an ELF isosurface of 0.905 are shown in (a). On the other hand, the five ELF basins, limited by ρ = 0.05 au, are depicted in

(b).

In document UNIVERSIDAD NACIONAL AGRARIA LA MOLINA (página 46-54)