05_Electron localization function - Yiwei666/08_computional-chemistry-learning-materials- GitHub Wiki
1. A simple measure of electron localization in atomic and molecular systems （1990，first propose ELF）
ABSTRACT
We introduce in this work a new approach to the identification of localized electronic groupsin atomic and molecular systems. Our approach is based on local behavior of the Hartree-Fockparallel-spin pair probability and is completely independent of unitary orbital transformations.We derive a simple electron localization function”(ELF) which easily reveals atomic shellstructure and core, binding, and lone electron pairs in simple molecular systems as well.
INTRODUCTION
Of major importance in descriptive chemistry is the concept of localized groups of electrons, encompassing such notions as atomic shells, binding and lone electron pairs, pi-electron subsystems, etc. However, despite its undeniable utility, the concept of spatially localized electrons is theoretically and mathematically elusive. The canonical orbitals of Hartree-Fock theory are delocalized throughout the space of a molecule or crystal and do not suggest localized electronic groups. On the other hand, it is well known that equivalent localized orbitals can be generated by unitary transformations of canonical orbitals, leaving Hartree-Fock total energy unchanged, according to various prescriptions. Unfortunately, such transformations are not unique and may even result in qualitatively different views of certain bonding situations. The dichotomy between "σ-" and "π-bent" multiple bonds is a classic example. Ultimately, neither of such equivalent views can be given preference at the Hartree-Fock theoretical level. Hartree-Fock theory is defined entirely by its one-body density matrix (see Sec. II), which is completely invariant with respect to unitary orbital transformations. Theoretically meaningful definitions of electron localization must therefore be sought in the density matrix itself (or related functions) and not in the orbitals.
Alternative, orbital-independent descriptions of electron localization have received some attention in recent literature. Bader and co-workers have noted that the total electronic density alone reveals atomic shell structure, electron pairs, etc., through the topography of its Laplacian ∇²ρ. This approach fully invokes the spirit of the so-called "density-functional theory of many-electron systems, which rigorously asserts that the total electronic density is the fundamental independent variable of many-electron theory. The Laplacian of the density does not, however, completely reveal the expected shell structure of heavy atomic systems. Also, its wide variability, ranging from a negatively infinite value at atomic nuclei to an unbounded positive value elsewhere, is somewhat inconvenient for purposes of graphical representation.
Moreover, Bader and co-workers have emphasized that electron localization is fundamentally related to parallel-spin pair probability and its associated Fermi hole function, through which the effects of Pauli exchange repulsion are directly reflected. In the present work, a new orbital-independent measure of electron localization based on consideration of the Hartree-Fock pair probability is developed. This work is a natural extension of previous investigations by one of the authors (A.D.B.) of the short-range behavior of the Fermi hole function in inhomogeneous systems and is similar in some respects to the work of Luken on localized orbitals and Fermi hole mobility. In Sec. II of this paper, we introduce a new "electron localization function" (ELF) which depends on total electronic density, its gradient, and also the kinetic energy density. Application of this ELF to the noble gas atoms Ne through Rn in Sec. III easily and completely reveals their expected shell structure. In Sec. IV, ELF is applied to some classic freshman chemistry problems to illustrate its usefulness in identifying core, binding, and lone-pair regions in molecular systems, and, finally, concluding remarks are offered in Sec. V.
V. CONCLUSIONS
Essentially, we have introduced a novel and simple method for the mapping of electron pair probability (Hartree-Fock parallel spin) in multi-electron systems. Pair probability is properly, of course, a six-dimensional function, so we focus on its spherically averaged local behavior as a function of a reference point. Nevertheless, we find that the local pair probability, as conveyed by our proposed Electron Localization Function (ELF), generates interesting pictures of atomic shell, core, binding, and lone-pair regions in atomic and molecular systems. Such information is not contained in the total electronic density itself, though its Laplacian (∇²) is very useful in this regard. Our ELF does not require the calculation of localized molecular orbitals and, indeed, is invariant with respect to unitary orbital transformations.
ELF provides a faithful visualization of the Valence Shell Electron Pair Repulsion Theory (VSEPR) in action. After all, the source of "electron pair repulsion" in VSEPR theory is predominantly exchange (i.e., not dynamical) correlation, which, through its origins in the Hartree-Fock parallel-spin pair probability, is precisely what ELF represents. We feel that the present electron localization function is, therefore, a valuable addition to the quantum chemical arsenal of descriptive and interpretative theoretical tools.
2. ELF: The Electron Localization Function (1997,review)
2.1. Covalent versus Ionic
The division of chemical bonds into homopolar or primarily ionic bonds has proven to be very useful in the language and comprehension of chemistry. Naturally, the ideal covalent bond is simply defined in element-element bonds, but strictly speaking, only when the surroundings of both bonding partners are identical. The ideal ionic bond with complete charge transfer does not exist. In general, a bonding interaction intermediate to ionic and covalent bonds is observed: a polar bond. [46]
The definition of the range of influence for ELF attractors permits a new formulation of ionicity in which the delimitation and mutual significance of covalent and ionic parts are simplified. [47] First, a qualitative impression is gained upon analyzing the form of the attractors. If they are more spherically distributed around the cores, either a more ionic or a van der Waals interaction is present. If the covalency of a bond increases, the migration of the attractor becomes more distinctive between the centers until a totally symmetric topology is achieved in the ideal covalent case. The position of the attractor between the centers can quantitatively be used to define the extent of polarity of a chemical bond. [1481]
As long as the attractor lies on the line connecting the cores and can be separated from the cores themselves by a trajectory, and the attractor does not circumscribe the core, a situation is reached that is usually described as polar covalency. An ionic formulation is suitable if the attractor is close to the core region of one of the atoms and no longer on the connecting line. One must take into account that no clear separation is possible.
The combination of the division of the electron density according to Bader [izl] and ELF in domains (WBs) is a second way for a complete quantitative description of the chemical bond. The actual charges of the atoms can be determined by the partition of the density in ranges of atomic influence. The covalent contribution of the bond is obtained by the number of electrons in the mutually shared electronic range of influence and attractor (partition of ELF). This is illustrated in Section 42 with the results of calculations on several intermetallic phases.
3. First-principles study on microstructure of CaO-Al2O3-B2O3 slag
The electron localization function (ELF) was widely used in the description of chemical bonds between atoms 42(https://www.sciencedirect.com/science/article/pii/S0167732222022772#b0210), 43(https://www.sciencedirect.com/science/article/pii/S0167732222022772#b0215), 44(https://www.sciencedirect.com/science/article/pii/S0167732222022772#b0220). In the fourth stage of No.3, the ELF of each atom at 1447 fs was shown in Fig. 6. Three main states of O acted in high temperature slag, namely Onb (O1 and O2 shown in Fig. 6 (a)-(b)), Ob (O7 and O13 shown in Fig. 6(c)-(d)) and Ot (O5 and O10 shown in Fig. 6(e)-(h)). Among them, the electron of Ca tightly localized around the atomic nucleus, forming a regular spherical, indicated that Ca and O existed in the form of ionic bond. Nevertheless, electrons around B and Al nuclei disappeared, but all were localized around O. Moreover, the ELF of O atom formed a unregular sphere, but extended from O to B or Al, formed a small synapse. This indicated that there was neither ionic bond nor covalent bond between B or Al and O, but a charge-transfer bond 45(https://www.sciencedirect.com/science/article/pii/S0167732222022772#b0225), 46(https://www.sciencedirect.com/science/article/pii/S0167732222022772#b0230), 47(https://www.sciencedirect.com/science/article/pii/S0167732222022772#b0235).
In order to further explain the evolution of BIII to BIV, the ELFs (Fig. 10) of B2, O5 and O8 and the change of Bader charge of related atoms (Table 4) at 1437 fs, 1447 fs, 1457 fs, 1467 fs and 1477 fs were analyzed. It can be seen that, with O5 approached B2, O5 first interacted with the BIII electron cloud and then formed BIV with stable electron cloud. The charge transfer process was that O5 distributed the charge to O6, O7 and O8 which was originally coordinated with B2. The charge redistribution occurred after atoms bonded stably. At 1477 fs, O5 was Ot, so the Bader charge of O5 was larger. In addition, a stable electronic cloud connection was formed between the B-O bonds, which was beneficial to the stability of the B-O network structure.
In order to further explain the evolution of AlⅣ, AlⅤ and AlⅥ, the ELFs (Fig. 13) and the change of Bader charge (Table 5) were analyzed. As shown in Fig. 13, there were few electron cloud connections between each O atom bonded with Al (Fig. 13, 855 fs, 925 fs, 969 fs and 1009 fs), indicated that the structural stability of Al-O was worse than that of B-O, which would be the essential reason for Al2O3 being amphoteric oxide. In the bonded process of Al3-O8 and Al3-O11, respectively, as shown in Fig. 13 at 865 fs and 979 fs, O8 and O11 exchanged charge with the original O in AlIV and AlV, respectively, so that the concentration of electron cloud between O atoms increases, and the electron cloud connection between O atoms bonded with Al was formed dynamically, and disappeared after stable bonded. In addition, from the Bader charge shown in Table 5, O8 obtained charge with change from AlIV to AlV, while O11 lost charge as transformation from AlV to AlVI, and other O atomic charge were in a dynamic equilibrium state. Overall, the structural stability of AlV and AlVI was poor, resulting that the structural transformation was easy occurrence. Therefore, the existence of AlV and AlVI in slag was beneficial to the depolymerization of slag, and the basic depolymerization of amphoteric oxide Al2O3 was better.
4. Structural Diversity and Electron Confinement in Li4N: Potential for 0-D, 2-D, and 3-D Electrides
Determining whether a material is an electride or not is not simple. One needs to look for both valence electron density off the nuclei, and for high values of the electron localization function (ELF). (98-100) These criteria have also been suggested by Martinez-Canales et al. (31) Let us look for both in the Li4N structures we have found. The ELF plots for structure 1 are illustrated in Figure 7. The degree of electron localization is well gauged by ELF. The value of ELF (usually denoted by η) is normalized in the range from 0 to 1. η = 0.5 corresponds to the homogeneous electron gas, while regions where η is close to 1 correspond to well-localized electrons, such as cores, bonds, and lone pairs. (101)Figure 7a shows the isosurface of ELF with η = 0.7. There are two different regions where electrons are localized. One is the region around the nitride ions, namely the center of NLi6 octahedra—these are the nitride core regions. High ELF values are also found in the interstitial region, in the elemental bcc Li layer.
We scrutinize the ELF contours in the elemental Li layer in Figure 7b,c, showing two different plane cuts through the layers. In both contours we can see distinct ELF attractors (non-nuclear maxima of the electron density) located in the interstitial regions. They certainly look like a signature of an electride. Some of ELF plots for the other structures we found are shown later, but others are shown in the SI. Briefly, we found that by the ELF criterion there is localization of electrons off the atoms in all the predicted Li4N structures, including the segregated Li4N and amorphous Li4N structures. In the SI we discuss the ELF plots for S3, the most stable Li4N structure found in this study. It shows features similar to those exhibited in Figure 7—at the boundary between the Li layer and c-Li3N layer, there are signs of localization in the form of ELF attractors.
5. First-principles study of the surface of silica and sodium silicate glasses
In this section, we discuss the nature of the chemical bonding in the glasses using the electron localization function (ELF) [100]. The ELF is related to the probability distribution η(r) of electron pairs, divided by the corresponding distribution for a uniform electron gas. By definition, η takes at any point of space a value that lies between 0 and 1. A value of 1 corresponds to a perfect localization of the electron pairs, while a value of 0.5 corresponds to that of a uniform electron gas. Details of the calculation can be found in Ref. [101].
In Fig. 13, we illustrate some of the properties of the ELF for the case of the silica glass surface. Figure 13(a) shows the isosurface of the distribution evaluated at the value η = 0.83. The region we consider includes a SiO4 tetrahedron with one NBO (marked as O1) and three BO (O2–O4). For each BO, we observe a hemispherical domain along each Si-O bond [see, for example, the bridge Si1-O2-Si2 in Fig. 13(a)], and this domain can be assigned to a pair of bonding electrons. One also finds a banana-shaped domain at the reflex side of the Si-BO-Si bridge, which is orthogonal to the Si-BO-Si plane. This domain is assigned to two lone pairs of electrons, i.e., the four valence electrons that are not involved in bonding. These nonbonding domains are substantially larger than the bonded hemispherical domains along the Si-O bonds, in agreement with the ELF mapping of the SiOSi linkage in silicate minerals [102]. For the NBO atoms, as for example the atom labeled O1, we observe that, aside from the bond pair domain, a concave hemispherical-shaped domain can be found and it seems to have a rotational symmetry along the Si-NBO direction. This domain can be ascribed to the nonbonding electrons, and it appears to have a larger volume than the nonbonding domain electron domain for BO. This observation is reasonable since presumably there are five nonbonding electrons for the NBO, while only four for the BO.
Figure 13(b) shows the two-dimensional contour plot of the ELF in a plane spanned by Si1, O2, and Si2, i.e., for a BO, and Fig. 13(c), for the plane given by O2, Si1, and O1, i.e., for a NBO. The aforementioned bonding and nonbonding domains are clearly visible from the contour plots. In addition, one recognizes from Fig. 13(c) that the probability distribution of electron pairs around the NBO is more spread out than that of the BO. This observation can be rationalized by the fact that the NBO has more free volume on the side opposite to the Si-O bond than the BO atoms.
A further important structural unit, namely a 2M ring, is depicted in Fig. 13(d). One notices that the O atoms in the 2M ring, O7 and O8, have electron pair domains that are similar to the ones of ordinary BO atoms, e.g., O2 in Fig. 13(a). Figures 13(e) and 13(f) show the ELF contour plots corresponding to two Si-O-Si linkages associated with the 2M ring. [Note that the Si-O-Si linkage in Fig. 13(e) involves an edge-sharing Si, Si4.] One sees that the angle Si3-O5-Si4 is much larger than the one in Fig. 13(b), demonstrating that the strong angular constraint in the 2M ring also affects the linkages of its neighbors. Consequently, the bond and lone pair domains around the BO in Fig. 13(e) are not as well structured as the ones in Fig. 13(b). Figure 13(f) shows the ELF contour plots of the 2M ring. One observes that the bond and lone pair domains are well structured and can be clearly distinguished. Another noticeable feature is that the bond paths, i.e., the lines connecting neighboring atoms, are no longer axes of symmetry for the bond pair domains. This is likely due to the strong repulsion of the electrons from the two opposing esBO atoms.
To describe the ELF in a more quantitative manner we show in Figs. 13(g) and 13(h) the line profile of the ELF along the bond paths starting from the oxygen atom (r = 0). Note that all BO in Figs. 13(a) and 13(g) are ordinary corner-sharing BO, i.e., csBO. Figure 13(g) shows that the ELF of the NBO-Si bond is smaller than the one of the BO-Si bond, implying that the ELF around the NBO is more spread out, in agreement with the contour plot in Fig. 13(c). In addition, we note that the BO-Si bond peaks at a larger r that the Si-NBO bond (see the values in the parentheses of the legend), in agreement with the observation that for the NBO the ELF is extended in the direction opposite to the Si-O bond. Also included in Fig. 13(g) is the ELF profile corresponding to a Si-BO-Si linkage in the interior of the sample and which has an angle close to the Si1-O2-Si2 linkage shown in Fig. 13(a). The presence of the surface does not seem to affect in a significant manner the ELF profile of the Si-BO bonds, although the BO-Si bond length (indicated by the vertical arrows) in the interior is slightly smaller than the surface BO-Si (see also Table II). Figure 13(h) compares the ELF line profiles of the esBO-esSi and csBO-esSi bonds, and one notices that the ELF of the esBO-esSi bonds shifts to a larger r relative to the csBO-esSi bonds but seems to have the same maximum height. However, since for the esBO-esSi bond, the bond path does not pass through the maximum of the ELF [see Fig. 13(f)], the real maximum value of the ELF for this bond is in fact higher than the one for the csBO-esSi bond, i.e., the electrons are more localized.
Figure 14 shows the ELF results for the NS3 glass surface. We note that, in addition to the structural modification discussed in the previous sections, the presence of Na induces also changes in the bonding. For example, Fig. 14(a), the bond pair domain for the NBO-Si bond O1-Si1 is much smaller that the corresponding domain in silica [Fig. 13(a)]. Figure 14(b) shows that the presence of Na also leads to an asymmetry of the lone pair domain of the NBO (i.e., O1). This effect is also seen from the two-dimensional (2D) contour plot in the plane defined by Na1-O1-Si1 [Fig. 14(c)]. For the NBO, O1, we note that the domains in the directions of the Na atoms can be ascribed to the Na-O bond pair interaction superimposed on the lone pair domains [Fig. 14(c)]. Similar results were found for earth materials containing alkali metals [102].
Figure 14(d) shows a 2M ring with one of the Si atoms connected to a NBO and its nearby Na atoms. Figure 14(e) shows that, for the 2M rings, the distribution is no longer symmetric around the O7(esBO)-Si3 connection, an observation that is coherent with the finding for the 2M rings in silica (see Fig. 13). For the NBO, O6, we find that the ELF contour plot is quite similar to the one for O1 shown in Fig. 14(b), in spite of the presence of the neighboring 2M ring. Figure 14(f) clearly shows that the ELF for the esBO (O7) bonded to the Na is less spread out than the distribution for the other esBO (O8) in the 2M ring, demonstrating that O7 is indeed bonded to the Na atom.
Figure 14(g) shows the average ELF line profiles of various types of O-Si bonds. (Note that the NBO atom connected to an esSi atom is denoted as NBO2M.) One observes that the ELF profile of the NBO-csSi bond is very similar to the one of the NBO2M-esSi bond, indicating that the NBO-Si bond character is basically independent of the Si type. Furthermore, we find that the ELF values of the NBO-Si bonds are smaller than that of the esBO-esSi bond, in accordance with the fact that the distribution of the electron pairs around the NBO is more spread out than the one for the esBO-esSi bond. (Also here we recall that the ELF for the esBO-esSi is not symmetric with respect to the connecting axis [see Fig. 14(c)], and hence the maximum value is even higher). For all the three NBO-Si bonds, the maximum of the ELF is located at r ≈ 0.68 Å, independent of the bond type. Figure 14(h) shows the profiles for the O-Na pairs and one sees that the maxima of the curves are located at r ≈ 0.61, 0.63, and 0.67 Å for the NBO2M-Na, NBO-Na, and esBO-Na bonds, respectively. These results indicate that the character of the O-Na bond is more sensitive to the changes in the local environment than the NBO-Si bond. We also note that the maxima of the ELF for the O-Na bonds are closer to the oxygen atoms (at r = 0) than the ones of the O-Si bonds. This result demonstrates that the O-Na is less covalent (i.e., more ionic) than the O-Si bonds. In addition, based on the locations of the ELF maxima, it can be deduced that the esBO-Na bond is more covalent than the NBO-Na bonds. Finally, we note that the locations of the maxima of the ELF profiles for the NBO-Si and esBO-esSi bonds are very close to the corresponding values found for the silica glass. This similarity indicates that the presence of Na affects the position of the bond pair domains of the O-Si bonds only weakly.
6. Effect of alkaline oxides on aluminate slag structure by first principles calculation
In the diagram of PDOS, if a certain orbit of the two atoms had a strong overlap, a strong chemical bond was formed between them 33(https://www.sciencedirect.com/science/article/pii/S0167732223018949#b0165). There was a weak overlap between the p orbital of the alkaline cations and the p orbital of O at −6 ∼ 21 eV, and the d orbital of Ca and Ba also overlapped with the p orbital of O (Fig. 6), demonstrating that the chemical bond formed between the alkaline cations and O was weak. The ELF was widely used in the description of chemical bonds between atoms 34(https://www.sciencedirect.com/science/article/pii/S0167732223018949#b0170), 35(https://www.sciencedirect.com/science/article/pii/S0167732223018949#b0175), 36(https://www.sciencedirect.com/science/article/pii/S0167732223018949#b0180). The value of ELF was between 0 and 1, which indicated that the electron was completely delocalized and fully localized or lone pair electrons, respectively 37(https://www.sciencedirect.com/science/article/pii/S0167732223018949#b0185). The electrons of Ca, Na, Li, Ba and Mg were closely around the nucleus to form a regular sphere, and there was neither electronic interaction nor small synapses with O (Fig. 7), indicating that ionic bonds were formed between the alkaline cations and O.
3.4. Structure of aluminates and mechanism of alkaline oxides It has been concluded in CaO-Al2O3-B2O3 ternary slag that, besides the anion of the electron to the cation to produce a small synapse, there was an electron distribution between the anions as show in the ELF of [AlO4]5-, [AlO5]7-, [AlO6]9-, [BO3]3- and [BO4]5- and moreover the electron distribution between the anions around B more than that of Al 26(https://www.sciencedirect.com/science/article/pii/S0167732223018949#b0130). In order to better understand the charge distribution of Al-O network structure, the ELF isosurface of 0.1 and 0.8 in CaO-Al2O3 binary slag was presented in three dimensions showing in Fig. 8(a) and (b), respectively. Fig. 8(a) clearly showed that there were a small number of localized electrons between O and O, which surround Al to form a closed Al-O tetrahedron structure, and it can be seen from Fig. 8(b) that localized electrons around O form small synapses pointing to Al. To sum up, in the tetrahedral structure of Al-O, in addition to forming charge-transfer bonds between Al-O, chemical bonds were also formed between O and O, so it can be considered that the structural unit of Al-O was a whole formed by O wrapped around Al.