Theoretical Exploration of the Potential and Force Acting on One

Article pubs.acs.org/JPCA

Theoretical Exploration of the Potential and Force Acting on One Electron within a Molecule Dong-Xia Zhao* and Zhong-Zhi Yang* School of Chemistry and Chemical Engineering, Liaoning Normal University, Dalian 116029, China ABSTRACT: The potential and force acting on one electron within a molecule (PAEM and FAEM) have been investigated and analyzed. The PAEM, defined as the interaction energy on one electron provided by all the nuclei and the remaining electrons in a molecule, can be precisely expressed and calculated by ab initio method and our in-house program. Although the analysis of the scalar function PAEM is similar to that of the molecular electron density in the Bader’s AIM theory, the former is distinct from the latter mainly in three points: (a) The minus gradient of the PAEM is the force acting on one electron within a molecule (FAEM). (b) The bond center is defined in terms of the features of FAEM and PAEM between two bonded atoms, and it is a two-dimensional attractive center whereas a nucleus is a threedimensional attractive source for electrons. We have calculated the physical quantities of one electron at the bond center, such as Dpb, the Hessian matrix, and its eigenvalues. Interestingly, it is found that the force constant and frequency of the electron interflow around the bond center are well correlated with those corresponding quantities for the nuclear vibration which relate to the bond strength, for some series of diatomic molecules. (c) The bond center locates at a different point from that of the critical point of the electron density in the Bader’s AIM theory, which will lead to different partitioning of the molecular space into the atomic regions. topological analysis of electron density.5,6,13,14,17 In recent decades, this approach has become a standard method to explore the nature of the chemical bond in molecules and extended systems.18−28 In 1990, Becke and Edgecombe7,8 proposed a local scalar function, the electron localization function (ELF) denoted by η(r), which is related to the Fermi hole curvature. As shown by Savin29 and co-workers, the ELF measures the excess kinetic energy density due to the Pauli repulsion. In the region of space where the Pauli repulsion is weak (single electron or opposite spin-pair behavior) the ELF is close to unity, whereas where the probability to find the same-spin electrons close together is high the ELF tends to zero. As the ELF is a scalar function, the analysis of its gradient field can be carried out to locate its attractors (the local maxima) and the corresponding basins.29,30 The picture of the molecule provided by the ELF analysis8 is consistent with the Lewis valence theory,31 and therefore, it is possible to assign a chemical meaning to the attractors and to their basins.30 The topological analysis of the ELF gradient field also provides a tool for the study of the evolution of the bonding along a reaction pathway.32 The topological analysis of the electron localization function has been applied to a lot of molecules and complexes representative of the weak, medium and strong hydrogen bond.30,33−35

1. INTRODUCTION A scalar field related to a molecule contains abundant information on molecular structures and properties. Even a simple nuclear potential generated by the atomic nuclei of a molecule may provide some important information, just as Parr, Gadre, and Bartolotti,1 as well as Tal, Bader, and Erkku2 had traced the fundamental role of the nuclear potential in determining the topological properties of charge distribution and had studied the structural homeomorphism between the electronic charge density and the nuclear potential of a molecular system in terms of their topological properties. Many three-dimensional (3D) atomic and molecular scalar fields have become subject of extensive studies in physical and chemical sciences, such as electron density,3,4 the Laplacian of electron density,5,6 electron localization function (ELF),7,8 and molecular electrostatic potential (MEP). Particularly, the electron density is a scalar field that can be experimentally accessed9 in principle and contains all necessary information for the ground state of the molecular system, according to the Hohenberg−Kohn10,11 theorems of density functional theory (DFT).12 In the 1970s, Bader proposed a theory of atoms in molecules (AIM) in terms of the topological analysis of electron density.5,6,13 In this theory, the molecular space6,13,14 is divided into adjacent nonoverlapping regions, the atomic basins, which have additive properties and enable the atomic charges to be rigorously defined. Furthermore, a bond is defined by a bond path (BP)15 and a bond critical point (BCP).16 The BP is a line in space linking the nuclei of bonded atoms where the electron density is a maximum with respect to any neighboring line. The various molecular structures and graphs were defined, studied, and discussed according to the © XXXX American Chemical Society

Special Issue: International Conference on Theoretical and High Performance Computational Chemistry Symposium Received: February 27, 2014 Revised: August 9, 2014

A

dx.doi.org/10.1021/jp5020466 | J. Phys. Chem. A XXXX, XXX, XXX−XXX

The Journal of Physical Chemistry A

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Molecular electrostatic potential (MEP), a scalar field, is familiar to chemists. MEP is a well-established tool for exploring molecular reactivity, intermolecular interactions, and a variety of other chemical phenomena. Klarner and Kahlert36 studied and explored the relationship between the topology of the MEP and molecular reactivities. Gadre and his collaborators showed that a detailed investigation of the topology of the MEP is able to reveal subtle changes of the spatial electronic distribution due to changes in the molecular framework.37−39 Leboeuf, Köster, Jug, and Salahub studied the topology of the MEP of 18 molecules, which was calculated in the framework of Kohn−Sham density functional theory.40 The topology of the MEP offers a satisfactory explanation to various reactivity features of the molecular systems.41 Suresh, Koga, and Gadre have revisited Markovnikov addition to alkenes via MEP.42 Suresh and co-workers have investigated Hammett constants, reactivity, Clar’s theory, and so on, as well as defined the lone pair by using MEP.42−46 In recent years, our group proposed the potential acting on one electron within a molecule (PAEM), a molecular scalar function, which is defined as the interaction energy of a local electron with the rest of particles, namely, all the nuclei and remaining electrons. The PAEM has been employed to define the atomic radius for an atom47−49 and an ion,50,51 and the molecular intrinsic characteristic contour for a molecule,52−55 which have been used to demonstrate the dynamic pictures of both molecular shape56,57 and frontier electron density for an organic chemical reaction about F + C2H4.58 The PAEM has been developed further to display the molecular face (MF)59 and to explain the mechanism of the Markovnikov reactions of alkenes60 and SN2(C) and SN2(Si) reactions61 and to show its relation to the force constant and bond length62 and as well as to evaluate the molecular surface and volume.63 Particularly, we have constructed a PAEM-MO diagram from which a clear distinction between van der Waals interaction and chemical bonding is intuitively demonstrated.64 As we know, exploration of the electron density of an electronic system has led to the birth of the density functional theory (DFT) of atoms and molecules (Hohenberg,10 Kohn,11 Parr and Yang12). The analysis of the electron density of a molecule has given rise to the Bader’s theory of Atoms in Molecules.6 What can we get from the one-electron potential PAEM? The PAEM is a scalar function in a three-dimensional (3D) space, like the electron density, ELF, Fukui Function, MEP, and so on mentioned above. A particular feature of the PAEM is that the minus of the gradient of PAEM is the force acting on one electron in a molecule. Force is the origin of movement. Therefore, the analysis of the PAEM and its gradient field will provide new insight into the molecular properties and structures. In this paper, we will investigate and analyze 3D PAEM graph and its topological characteristics, propose a descriptor of strength of a chemical bond, and thereby define the bond center that is a 2D electron attractor. The theoretical framework of the PAEM and FAEM is given in section 2, the details of calculation results and analysis are presented in section 3, and at last the summary is given.

of one electron at position r ⃗ with the remaining electrons and all the nuclei in a molecule and expressed as V ( r ⃗) = − ∑ A

ZA | r ⃗ − RA⃗ |

+

1 ρ( r ⃗)

∫

ρ2 ( r ⃗ , r ′⃗ ) | r ⃗ − r ′⃗ |

= Vne( r ⃗) + Vee( r ⃗)

d r ′⃗ (1)

where the first term, Vne(r)⃗ , is the attractive potential provided by all the nuclei, and the second term, Vee(r)⃗ , is the interaction potential given by all the remaining electrons of the molecule. In eq 1, ZA, R⃗ A, and r ⃗ are the nuclear charge and positions of nucleus A and the electron concerned, ρ(r)⃗ is the electron density and ρ2(r,⃗ r′⃗ ) is the two-electron density function, i.e., the probability function of finding one electron at r ⃗ and another electron at r′⃗ simultaneously. In the treatment of the PAEM, the electron at position r ⃗ belongs to the molecular system and hence has the exchange interaction with the remaining electrons. Therefore, the PAEM considers one internal electron, whereas the molecular electrostatic potential (MEP) considers one external test positive charge. Their distinction is discussed in detail in ref 62. 2.2. Force Acting on One Electron within a Molecule (FAEM) and the Bond Center. The force acting on one electron within a molecule (FAEM) is defined as the force exerted on one electron at r ⃗ by the remaining electrons and all the nuclei; i.e., it is equal to the minus of the gradient of the potential acting on one electron within a molecule (PAEM), −∇V(r)⃗ , as expressed in eq 2. F (⃗ r ⃗) = −∇V ( r ⃗) ⎛ ∂V ( r ⃗) ∂V ( r ⃗) ∂V ( r ⃗) ⃗⎞ = −⎜ i⃗ + j⃗ + k⎟ ∂y ∂z ⎠ ⎝ ∂x = Fx( r ⃗) i ⃗ + Fy( r ⃗)j ⃗ + Fz( r ⃗)k ⃗

(2)

where i,⃗ j,⃗ k⃗ denote the unit vectors in the x, y, and z directions, respectively. Fx(r)⃗ denotes the x component of the FAEM, i.e., the value along the x direction, and Fx(r)⃗ = −∂V(r)⃗ /∂x, and Fy(r)⃗ and Fz(r)⃗ have expressions similar to that for the Fx(r)⃗ . It should be pointed out that the FAEM force F(r)⃗ is formally related to the Ehrenfest force15,65 Fe(r)⃗ through Fx(r)⃗ = −Fe(r)⃗ /ρ(r)⃗ where ρ(r)⃗ is the electron density. 2.3. Topological Characteristics of the PAEM. The topological characteristics of PAEM are provided by the analysis of its associated gradient field (its minus is the FAEM). Although the topological analysis of the PAEM is very similar to that of the electron density as in Bader’s AIM theory, the former is distinct from the latter mainly in three points: (a) The minus gradient of the PAEM is the force acting on one electron within a molecule (FAEM). (b) The bond center is defined in terms of the features of FAEM and PAEM between two bonded atoms, and it is a two-dimensional attractive center located on the potential bond path that is the minimum potential line with respect to any neighboring line between the two boned atoms, whereas a nucleus is a three-dimensional attractive source for electrons. We have calculated the physical quantities of one electron at the bond center, such as Dpb, the Hessian matrix and its eigenvalues. Interestingly, it is found that the force constant, and frequency of the electron interflow around the bond center are well correlated with those corresponding quantities for the nuclear vibration which relate to the bond strength, for some series of diatomic molecules. (c)

2. THEORETICAL FRAMEWORK 2.1. Formulism of the PAEM. The potential acting on one electron within a molecule (PAEM) has been expressed by ab initio method at CI62 and RHF52 levels. Here, we give a brief description of the PAEM. It is defined as the interaction energy B



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expression of the FAEM. Thus, we calculated Fx(r)⃗ at r ⃗ using the usual approximate method: Fx(r)⃗ ≈ −[V(x2,y,z) − V(x1,y,z)]/(x2 − x1), when the difference between x2 and x1 is very small. Using the similar method, Fy(r)⃗ and Fz(r)⃗ were obtained and then the FAEM at every point of the chosen space can be determined. In terms of eq 3, the various pcp’s of some molecules were searched by the numerical interpolation. If the element of the Hessian matrix at rp⃗ cp was known, its corresponding Hessian matrix was constructed according to eq 4. Then its diagonalization yielded three eigenvectors and three real eigenvalues using the Jacobi method. Each pcp point was then classified by the signature S which is the algebraic sum of sign of the eigenvalues of the Hessian matrix of the PAEM.

The bond center or the PAEM bond critical point locates at a different point from that of the critical point in the Bader’s AIM theory. A PAEM critical point (pcp) is a point where the gradient of the scalar field V(r)⃗ vanishes, as expressed in eq 3.

∇V ( r ⃗)|rpcp ⃗ = 0

(3)

Thus, a pcp is a point where the corresponding scalar field, V(r)⃗ , has an extremum. To characterize the critical points of the PAEM, as usual, we assume that, for a sufficiently well-behaved scalar field, the Hessian matrix of the field, as shown in eq 4 exists at a pcp point, ⎛ ∂ 2V ∂ 2V ∂ 2V ⎞ ⎜ 2 ⎟ ∂x ∂y ∂x ∂z ⎟ ⎜ ∂x ⎜ 2 ⎟ 2 2 ⎜∂ V ∂ V ∂ V ⎟ H[V ( rpcp ⃗ )] = ⎜ ∂y∂x ∂y 2 ∂y∂z ⎟ ⎜ ⎟ ⎜ ∂ 2V ∂ 2V ∂ 2V ⎟ ⎜ ⎟ 2 ⎝ ∂z ∂x ∂z ∂y ∂z ⎠r = r

pcp

3. RESULTS AND ANALYSIS Using the formalism and computational methods mentioned above, we have calculated and investigated the PAEMs for several series of diatomic molecules, such as HX, LiX, NaX, KX, and X2 molecules (X = F, Cl, Br, and I) and AH (M = Li, Na, K, Rb) as well as H2, F2, and Li2, and polyatomic molecules H2O, NH3, CH4, C2H4, and (NH2)2CO, cyclic molecules C4H4 and C6H6, as well as heterocyclic molecules C4OH4 and C4SH4. 3.1. Graph and Analysis of the Potential Acting on an Electron within a Molecule (PAEM). 3.1.1. 3D Graph Representation of PAEM. For the better understanding of the character of the PAEM for a molecule, it is worthy to investigate the graph representation of PAEM. Actually, the PAEM V(r)⃗ spans a fourth dimension because the electron coordinate r ⃗ runs in a three dimensions. When we represent the PAEM in a 3-dimensional graph, we may take a plane on which the PAEM is calculated. Of course, we can take as many planes as wanted to do this kind of graph representations. Here take a water molecule as an example. A water molecule contains three nuclei and two O−H chemical bonds. The experimental equilibrium O−H bond length is 0.9575 Å and ∠HOH bond angle is 104.51°.68 H2O molecule is planar and has C2v symmetry. The O atom is placed at the origin of the Cartesian coordinate, the bisector of angle H−O−H could be set as the y axis and a line perpendicular to y axis in the molecular plane is chosen as the x axis. The calculation details of PAEM implemented by an accurate ab initio method and our own program has been described in section 2.4. When the position of one electron r ⃗ runs on the xy plane of H2O, the calculated values of V(r)⃗ is displayed in the z-axis. A three-dimensional representation of the PAEM, V(r)⃗ , is formed and drawn in Figure 1a. For the PAEM of H2O, there are three deep potential wells that originate from the nuclear attraction: a wider potential well is around the O atom with the larger nuclear charge whereas two narrower potential wells are around two H atoms with the smaller nuclear charges. A significant feature of the PAEM surface is that there is a potential saddle point (psp) relating each O−H bond region where the PAEM value is a local maximum along the O−H chemical bond (bold line) and a local minimum in the direction perpendicular to the O−H bond (bold line). As displayed in Figure 1a, the crossing point between the two bold solid lines is the PAEM saddle point (psp), which is marked by the black solid uptriangle. The psp corresponds to the electron coordinate that is marked by the magenta solid star on the O−H bond and called the bond center (bc) and is also a PAEM critical point. There is no such feature between two H atoms. The vast remaining part is the

(4)

which is a real and symmetric matrix. Consequently, at any given pcp point, the Hessian matrix can be diagonalized and then yields three real eigenvalues, λi (i = 1, 2, 3). The corresponding eigenvectors are the principal axes. The characterization of a pcp depends on its rank and signature. The rank, R, of a pcp is the number of eigenvalues that are different from zero, and the signature, S (as eq 5), is the algebraic sum of the signs of the three eigenvalues: 3

S=

∑ sign(λi) i=1

(5)

Thus, a pcp is classified by the ordered pair (R, S). A (3, +3) is a local minimum, a (3, +1) is a minimum in two directions and a maximum in the other direction, a (3, −1) is maximum in two directions and a minimum in the other direction, and a (3, −3) is a local maximum. 2.4. Calculation of PAEM and FAEM. In the configuration interaction (CI) method, ρ(r)⃗ 62 and ρ2(r,⃗ r′⃗ )62 in eq 1 can be specifically expressed as a combination of molecular integrals obtained by an ab initio method.66 We used the MELD66 package developed by Davidson to perform the ab initio calculation for obtaining the molecular integrals, then employed an in-house code to calculate the electron density, second reduced density matrix, and then PAEM expressed in eq 1. Thus, the PAEM was calculated at each point of a grid covering the molecule, with certain spacing between the grid points. Ab initio calculations were implemented at the configuration interaction single double (CISD) level with the near Hartree− Fock quality Gaussian type orbital basis sets67 for some diatomic molecules and in conjunction with the 6-31+G(d,p) basis set for polyatomic molecules. All calculations were carried out on an SGI O300 server (16 × 500 MHz MIPS R14000 CPU) and an SGI Octane2 workstation (2 × 400 MHz MIPS R12000 CPU). The FAEM, F(r)⃗ , is a vector that has a length and a direction. To obtain the FAEM, we calculate Fx(r)⃗ , Fy(r)⃗ , and Fz(r)⃗ . Because Fx(r)⃗ = −∂V(r)⃗ /∂x, it is obtained if the expression of the first derivative of the PAEM was analytically known. However, it is not easy to write out the concrete analytical C



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region, the potential gradually ascends with increasing the distance between the electron and nuclei and then approaches to zero at the infinite, which is a correct asymptotic behavior. It is noted that in Figure 1a the detailed mark is only done for one O−H bond region and it has the same feature for another O− H bond region. As shown in Figure 1b, each contour represents an iso-value line of PAEM. The value of the PAEM with the bold solid line is equal to −2.3704 au, which is equal to PAEM at the psp. The crossing point between the solid line and each O−H bond is the bond center (bc) marked by the magenta star. There is no bond center point between the two H atoms, which corresponds to the fact that there is no chemical bond between them. It is interesting to briefly discuss the features of PAEM in the section parallel to the molecule plane. We have investigated the PAEM graphs on a series of plane which parallel to the molecule planes of H2O molecule. Take a plane as an example, the distance between this plane and the molecular plane is 0.5 au. The PAEM graph is displayed in Figure 1c. On this plane, as we see, the PAEM has three pits and two saddle points. There are two chemical bonds in H2O molecule. For the section line of every OH chemical bond, the bottoms of the pits and the saddle point of the PAEM correspond to 3 electron coordinates that locate on a straight line that parallel to the bond axis of H2O molecule. The pit that is just above the H nucleus is narrow, and the pit that is just above the O nucleus is wide. The PAEM at the saddle point of in this case is −1.8295 hartree, which is higher than that of the saddle point (−2.3849 hartree) in the PAEM picture as the electron runs on the molecular plane as shown in Figure 1a. Actually, we have carried out a series of calculations as the plane parallel to the molecular plane take a series of values of the distances between it and the molecular plane. When the electron moves along the line of an OH chemical bond axis and its parallel lines, the corresponding PAEM curves are collected in Figure 1d. Giving a bird’s view of the PAEM values, when the bird flies from the inside to the outside in the real 3dimensional space, the depths of the PAEM pits or wells look shallower and shallower, and the saddle points or the barriers between the two atoms get higher and higher; in other words, the total PAEMs get flatter and flatter. 3.1.2. Topological Analysis of PAEM. The further information on PAEM is provided by the curvature of PAEM at its critical points. Each topological feature of PAEM, whether it is a maximum, a minimum, or a saddle, is associated with a PAEM critical point (abbreviated to pcp), where the minus gradient of the PAEM, that is, the force acting on one electron F(rp⃗ cp) = −∇V(rp⃗ cp) = 0. For H2O, a local minimum is obtained at the position of each nucleus; thus there are three (3, +3) pcp’s. Further, a (3, +1) pcp is a minimum in the two directions and a maximum in the third direction, so a (3, +1) pcp can be obtained at a position between the two bonded atoms. This position is called the bond center. However, there is no (3, +1) pcp point between two H atoms, which corresponds to no chemical bond between them. The topological structure of H2O from the PAEM is shown in Figure 2. The points are clearly displayed in the relief maps of the PAEM in Figure 1 as the electron runs on the molecular plane of H2O. The critical points of the PAEM were also calculated and explored using the method similar to that for H2O for other molecules, such as HF, H2, F2, Li2, LiF, and some polyatomic molecules, such as NH3, CH4, C2H4, and

Figure 1. (a) 3D graph representation of the PAEM, electron running on the molecular plane of H2O, in which the red solid circle denotes the O atom and the two black circles denote H atoms, the blue solid up-triangle denotes the psp, and the magenta solid star denotes the bc (only marked one bc here; the other bc on the other O−H bond is omitted). The separation between psp and bc points denotes the gap of the PAEM. (b) Display of the iso-PAEM of the water molecule. Values of FAEM above an arbitrarily chosen value are not shown. (c) 3D graph representation of the PAEM on the plane parallel to the molecular plane and being 0.5 au far from the molecular plane. (d) Collection of the PAEM curves along an OH bond axis and its parallel lines, the distances between it and those lines whose separations are 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, and 0.7 au, respectively.

out-bond region of the PAEM, which is the region except the near-nuclei and bond regions from the whole PAEM. In this D



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n+3, n−3, n+1, and n−1 denote the number of (3, +3), (3, −3), (3, +1), and (3, −1) pcp’s in the molecule, respectively. The number of the particular pcp’s for the diatomic, polyatomic and cyclic molecules is collected in Table 1. The value of n+3 − n−3 − n+1 + n−1 = 1 satisfies the Poincaré−Hopf relationship of the number of the various pcp’s of the PAEM for molecules. Table 1. Number of Various Types of PAEM Critical Points (pcp) for Molecules of Interesta no. of various pcp molecule

n+3●

n+1★

H2O HF H2 F2 Li2 LiF C2H4 urea ((NH2)2CO) NH3 CH4 C4H4 C6H6 C4H4O C4H4S C4H5N

3 2 2 2 2 2 6 8 4 5 8 12 9 9 10

2 1 1 1 1 1 5 7 3 4 8 12 9 9 10

n−1☆

1 1 1 1 1

n−3

n+3 − n+1 + n−1 − n−3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

a

For the C6H6 molecule, the critical points are classified into three types: (3, +3) denotes the position of every atom, (3, +1) denotes the position of the bond pcp, and (3, −1) denotes the position of the ring pcp. In the C6H6 molecule, the number of (3, +3) is 12, the number of (3, +1) is 12, the number of the ring pcp is 1. n+3 − n+1 + n−1 = 1. The number of these critical points satisfies the Poincare−Hopf relationship.

Among the critical points of the PAEM, the (3, +1) pcp is particularly important in the topological analysis of the PAEM because this point is associated with formation of a bond. A (3, +3) pcp is a local minimum and practically coincides with a nucleus (although rigorously there is a cusp in PAEM at the nucleus). A (3, −1) pcp is associated with the formation of the ring surface for cyclic molecules. Generally speaking, the topological characteristic properties from PAEM analysis is homomorphic to those of the electron density analysis provided by Bader’s AIM theory, but there is something distinct as shown below. 3.2. Force Acting on One Electron in a Molecule (FAEM)Analysis around the Bond Center. 3.2.1. Bond Center in Diatomic AB moleculeA 2D Electron Attractor. For describing the FAEM, we first take the HF molecule as an example. The F atom is placed at the origin of the Cartesian coordinate, and H atom is located on the x-axis. The experimental equilibrium bond length for HF is 1.7327 au. PAEM, electron density (Den) and FAEM (including Fx(r)⃗ and Fy(r)⃗ ) at some points (the electron coordinates) along the H− F bond were calculated and also listed in Table 2. The curves of the PAEM (Figure 3a) are displayed on two lines which are along and perpendicular to the H−F bond, respectively. The electron coordinates of the PAEM at the saddle point is marked by the magenta star on the x axis. This electron position is called the bond center (bc) of the H−F bond. At this point, the force acting on one electron F(rp⃗ cp) = −∇V(rp⃗ cp) = 0.0. On the other hand, according to the PAEM-MO diagram,64 the main

Figure 2. Graphs of the critical points for molecules, including HF, H2O, H2, F2, Li2, LiF, C2H4, (NH2)2CO, NH3, and CH4, as well as C4H4, C6H6, C4H4S, and C4H4O. The solid circle denotes the pcp (3, +3), i.e., the nuclei position. The solid magenta star denotes the pbcp (3, +1). The open magenta star denotes the bcp (3, −1).

(NH2)2CO. Their topological characters and structures are also displayed in Figure 2. For several cyclic molecules, such as C4H4, C6H6, C4OH4, and C4SH4, there are three types of the PAEM critical points, which are (3, +3) and (3, +1) and (3, −1) pcp’s. A (3, −1) pcp is a maximum in the two directions and a minimum in the other direction for the cyclic molecules. A pcp can be located at the circle center of these molecules. The topological structures of these molecules are also displayed in Figure 2. The representing ways of (3, +3) and (3, +1) are same as those of the diatomic molecules. The solid magenta star denotes the position of the (3,+1) pcp, whereas the open magenta star denotes the position of the (3, −1) pcp. E



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Table 2. For the HF Molecule, the x and y Coordinates of the Electrons and the Nuclei, PAEM, Den, and the x and y Components of the FAEM, Fx(r⃗), Fy(r⃗) as Well as the Absolute Values of FAEM, |F(r⃗)|a positions

F nucleus

point g point h

bond center

point l point m

H nucleus

x/au

y/au

PAEM

Den

Fx(r⃗)

Fy(r⃗)

|F(r⃗)|

−0.60 −0.50 −0.40 −0.30 −0.20 −0.10 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.10 1.14 1.20 1.30 1.40 1.50 1.60 1.70 1.7327 1.80 1.90 2.00 2.10

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

−5.500 −7.552 −10.992 −17.597 −31.515 −74.366 -8982.291 −74.419 −31.658 −17.844 −11.225 −7.794 −5.792 −4.513 −3.670 −3.120 −2.784 −2.625 -2.609 −2.639 −2.858 −3.390 −4.560 −7.709 −30.708 -999.210 −14.920 −6.015 −3.762 −2.734

1.221 1.509 1.941 3.630 13.680 74.356 433.867 75.171 13.935 3.526 1.731 1.323 1.107 0.925 0.767 0.640 0.543 0.473 0.455 0.425 0.395 0.3812 0.3814 0.394 0.417 0.432 0.344 0.249 0.181 0.132

0.170 0.275 0.502 1.026 2.838 447.539 0.00 −447.532 −2.829 −1.022 −0.502 −0.272 −0.164 −0.106 −0.070 −0.044 −0.025 −0.007 0.000 0.012 0.038 0.085 0.216 1.307 0.361 0.000 −1.235 −0.558 −0.164 −0.081

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

0.170 0.275 0.502 1.026 2.838 447.539 0.00 447.532 2.829 1.022 0.502 0.272 0.164 0.106 0.070 0.044 0.025 0.007 0.000 0.012 0.038 0.085 0.216 1.307 0.361 0.000 1.235 0.558 0.164 0.081

a

x and y denote the electronic coordinates. PAEM denotes the potential acting on an electron within a molecule. Den denotes the electron density. Fx(r)⃗ and Fy(r)⃗ denote the x and y partials of the force acting on an electron within a molecule (FAEM), respectively. The units of the properties are atomic units (au).

at h moves toward the H nucleus, the F(xh) decreases and points to the F nucleus. When the electron at l moves toward the F nucleus, on the other hand, the F(xl) decreases and points to the H nucleus. When points h and l reach the same point, | F(xh)| is equal to |F(xl)|, and they both are equal to zero. This point is the bond center (bc). For HF molecule, an isovalue line of PAEM overlaid with the force lines of −∇V(r)⃗ , FAEM, has been displayed in Figure 3b. The solid balls denote the F and H nuclei. The magenta star denotes the bond center of the H−F bond. Red and green lines denote the force lines. The two green lines originate at infinity and terminate at the bond center (marked by magenta star). The red lines originate at infinity and terminate at the F and H nuclei. The other two red lines originate at the bond center and terminate at the F and H nuclei. The arrow of every force line denotes the direction of the FAEM at the point and the size of arrow schematically denotes the size of FAEM. In the case of HF, the FAEM may be drawn on the plane of HF, because of its linear structure. But the FAEM defined here, −∇V(r)⃗ , actually distributes in three-dimensional space. The 3D manifold of the FAEM can be generated by just rotating the plane shown in Figure 3b around the symmetry axis of HF. Two important aspects of this 3D manyfold are related to the bond center (bc) shown in Figure 3c and to a nucleus shown in Figure 3d. The center of Figure 3d is a nucleus at which all the

valence bonding Molecular Orbital (MVBMO) (3σ MO) in energy is higher than the PAEM value at this point. From Table 2, it is noted that the bc position (0.5927 au from the H nucleus) is different from the position of the electron density minimum (0.330 au from the H nucleus) which is the critical point in Bader’s AIM theory. The value of 0.5927 au of bc is more close to the covalent radius of H atom (0.6992 au) than 0.330 au of the CP point in the AIM theory.6 Let us see the force FAEM in more detail. As shown in Table 2, at the bond center (bc), between F and H atoms, the force acting on one electron within a molecule is equal to zero. Suppose that xg and xh represent two electron coordinates, which are located between the F nucleus and bc point. F(xg) and F(xh) represent the forces acting on one electron at the points xg and xh, respectively. When xg = 0.4 au, F(xg) = −1.022 au; when xh = 0.5 au, F(xh) = −0.502 au. The positive direction of the FAEM is chosen as F atom pointing to H atom. The values of F(xg) and F(xh) are negative. Their directions point to F atom and |F(xg)| > |F(xh)|. On the other side, xl and xm represent two electron coordinates for any two points that are located between bc point and H nucleus. When xl = 1.40 au, F(xl)=0.085 au; when xm = 1.5 au, F(xm) = 0.216 au. Thus, | F(xl)| < |F(xm)|, the directions of the FAEMs at points l and m are all pointing to the H nucleus. Let us see how the FAEM change with changing the location. When the electron locating F



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Figure 4. (a) Display of the FAEM lines of an H2O molecule overlaid with iso-PAEM. (b) Lines of the force acting on an electron within the water molecule on the molecular plane. The solid red circle denotes the O atom, and two solid green circles denote the H atoms. Two solid magenta stars denote the bond centers. The red and green lines denote the FAEM lines, and black lines denote the iso-PAEM lines. The arrows denote the direction of the FAEM.

Figure 3. (a) Curves of the PAEM along and perpendicular to H−F bond. (b) Isovalue line of PAEM of an HF molecule overlaid with FAEM lines. (c) Portraits for the force lines that terminate and originate at the bond center. (d) Portraits for the force lines that terminate at the nucleus.

PAEM contour lines, as shown in Figure 4a. (2) Every force line originates or terminates at a point where −∇V(r)⃗ vanishes, i.e., at a critical point of PAEM. (3) The force lines cannot cross each other because −∇V(r)⃗ defines only one direction at each point r ⃗ except at the critical points. (4) The bond center is a characteristic of a chemical bond. 3.3. Characteristic Properties around the Bond Center. 3.3.1. Relationship between PAEM and Nuclear Vibrational Energy. The Dpb is the minus of the PAEM value, i.e., −V(rbc), at the bond center for a chemical bond. The Dpb for some types of diatomic molecules of interest are listed in Table 3. We have shown the physical quantity, Dpb, and its relationship with bond length.62 In our present manuscript, we mainly discuss some correspondence between the electron interflow and nuclear vibration around the bond center through the three aspects, as well as the relationship between the Dpb and bond dissociation energy. The nuclear vibrational energy closely relates to the strength of the chemical bond. The larger the nuclear vibrational energy is, the stronger the chemical bond is. The vibrational energy, Evib, is calculated through the expression derived from68

force lines terminate. Therefore, the nucleus is regarded as a 3D electron attractive source. In the middle of the diagram shown in Figure 3b, there are two force lines originating from the bond center, and all the other force lines direct to it or terminate at it. Noteworthy, the bond center is a 2D electron attractor, although it does not possess any positive charge. 3.2.2. H2O Molecule. The isovalue lines of the PAEM of an H2O molecule overlaid with the FAEM force lines are shown in Figure 4 on the molecular plane. The positions where the FAEMs are equal to zero are found to be three nuclei and the two bond centers located at two O−H bonds, as marked by the magenta stars in Figure 4. As shown in Figure 4a,b, all the red lines (force line) originate at infinity and terminate at three nuclei (nuclei O and two H), and four green lines originate at the bond centers located between the two bonded atoms and terminate at one of the three nuclei. They divide the molecular plane into three regions, which are one O and two H atomic regions. Every nucleus is an attractive source in the three dimensions. The bond center is an attractive source in the two dimensions. As described above, the graphs of negative of the gradient vector field of the PAEM (FAEM) have illustrated the following general properties. (1) Because the gradient vector of a scalar filed is in the direction of the steepest increase in the scalar field, the force lines of −∇V(r)⃗ are perpendicular to the iso-

Evib /hc = ωe(v + 1/2) − ωeχe (v + 1/2)2 + ···

(6)

where ωe is the spectroscopic constant for a diatomic molecule at the ground electronic state, v is the vibrational quantum number, h is Planck’s constant, and c is the speed of light. In this customary formulation, the constant ωe has dimension of inverse length; in ref 68 it is given in units of cm−1. ωe and ωeχe G



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Table 3. Experimental Bond Length, the Minus Value of PAEM at the Bond Center, −V(rbc), (Dpb), the Force Constants of Electron Interflow (N/cm), and the Energy and Force Constants of the Nuclear Vibration from Experimental Data for Some Diatomic Molecules force constant of electron interflowa (N/cm)

a

molecule name

bond length68 (Å)

−V(rbc), Dpb, (hartree)

a

b

nuclear vibration energy × 103 (hartree)

force constant of the nuclear vibration68 (N/cm)

HF HCl HBr HI LiH NaH KH RbH LiF LiCl LiBr LiI NaF NaCl NaBr NaI KF KCl KBr KI F2 Cl2 Br2 I2

0.9169 1.2746 1.4145 1.6090 1.5949 1.8873 2.2440 2.3670 1.5639 2.0207 2.1704 2.3919 1.9260 2.3609 2.5020 2.7115 2.1716 2.6667 2.8208 3.0478 1.4119 1.9878 2.2811 2.6663

2.6140 1.9342 1.7600 1.5747 0.8277 0.6714 0.5883 0.6054 1.2099 1.0003 0.9040 0.8486 0.9114 0.7912 0.7611 0.7214 0.8537 0.7211 0.6896 0.6449 2.0166 1.6267 1.3740 1.1855

−274.25 −120.31 −95.39 −72.06 −25.88 −24.39 −18.25 −16.88 −69.24 −32.30 −26.60 −20.66 −52.27 −28.50 −24.89 −19.99 −42.58 −22.52 −19.58 −15.96 −9.41 −3.71 −2.68 −1.82

105.33 44.02 34.47 25.15 8.11 5.63 3.99 4.08 15.54 7.78 6.53 5.18 8.21 4.85 4.25 3.36 5.26 3.87 3.36 2.67 2.62 1.13 0.77 0.50

9.3251 6.7535 5.9831 5.2150 3.1758 2.6479 2.2245 2.1182 2.0650 1.4596 1.2790 1.1286 1.2162 0.8314 0.6863 0.5865 0.9683 0.6387 0.4843 0.4243 2.0754 1.2720 0.7399 0.4880

9.66 5.16 4.12 3.14 1.03 0.78 0.56 0.52 2.50 1.43 1.20 0.97 1.76 1.09 0.94 0.76

4.7 3.23 2.46 1.72

The a column denotes the force constant along the chemical bond; the b column denotes the force constant perpendicular to the chemical bond.

and A denotes Li, Na, K). When the value of Dpb is large, the corresponding nuclear vibration energy is large. The Dpb is also the characteristic quantity of representing the strength of chemical bond. The potential of one electron at this point of bond center closely relates to the nuclear vibrational energy. Thus, the Dpb of a chemical bond is a useful parameter for a description of chemical bond strength. 3.3.2. Force Constant of the Electron Interflow around the Bond Center. We can construct the corresponding Hessian matrix of the PAEM at the bond center rb⃗ c where the gradient of the PAEM is equal to zero. The three eigenvalues of the Hessian matrix at rb⃗ c can be obtained by performing the diagonalization of the matrix, out of which one eigenvalue λ1 characterizes the direction along the chemical bond, and the other two eigenvalues characterize the directions perpendicular to the chemical bond, λ2, and λ3, respectively. For the HF molecule, as shown in Figure 3a, the PAEM curve (bold line) along the F−H bond looks partly like a downward parabolic curve between the F and H nuclei. The change of the PAEM around the rb⃗ c is approximately represented as 1/2kX2 (X denotes the canonical coordinates shift from the rb⃗ c and k is the force constant). The force constant is equal to the eigenvalue of Hessian matrix along the chemical bond, λ1 = −17.6151 hartree/(bohr)2, according to the harmonic oscillator. The value is negative due to the downward parabola of the PAEM around rb⃗ c. The PAEM curve perpendicular to the F−H bond at rbc (1.14, 0.0, 0.0) (in the y direction) looks partly like an upward parabola around the bc point, which results in a positive eigenvalue λ2 = 6.7651

are listed in the ref 68. v in eq 6 is set to zero. The nuclear vibrational energies calculated by the first two terms of eq 6 are also included in Table 3 for better comparison. Through Figure 5, we find that −V(rbc) values, the Dpb at the rb⃗ c, have an excellent linear correlation with the experimental nuclear vibration energies for the three types of diatomic molecules (AH, HX, AX, and X2, in which X is F, Cl, Br, and I,

Figure 5. Relationships of the Dpb and the nuclear vibrational energy for the AH and HX (A = Li, Na, K, Rb; X = F, Cl, Br, I), AX (A not including Rb), and X2. H



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hartree/(bohr)2 of the PAEM Hessian matrix at the bond center, bc point. Because the HF molecule has axial symmetry, the third eigenvalue is equal to the second, λ3 = λ2 = 6.7651 hartree/(bohr)2 in the another direction also perpendicular to the HF bond axis through the bond center (1.14, 0.0, 0.0). Accordingly, the three force constants characterize their respective directions, respectively. A unit of the force constant in hartree/(bohr)2 is equal to 15.5690 N cm−1. The eigenvalues, λ1, λ2, and λ3, can be converted to −274.2497, +105.3255, +105.3255 N cm−1, respectively. Now we consider the force acting one electron (FAEM) and the electron motion around the bond center. In one case of the FAEM pointing to the F or H nuclei along the chemical bond, when one electron moves toward F from the bond center, we assume that the electron belongs to F rather than to H. The eigenvalue λ1 characterizes the motion of exchanging one electron around the bond center between two bonded atoms in the bond direction. In the other case of the FAEM pointing to the bond center along the direction perpendicular to the chemical bond, if one electron at rb⃗ c moves upward or downward, the electron is pulled toward the bond center by the force FAEM. In either case, the electron frequently interflowing (or exchanging) through the bond center makes the H atom and the F atom attract each other or form the H−F bond. The movement of the electron around the bond center is called the electron interflow here. The value of the force constant λ1 along the chemical bond is much bigger than those of λ2 and λ3, in the directions perpendicular to the H−F chemical bond. Using the method described above, the force constants of the electron interflow at rb⃗ c have been calculated for a series of diatomic molecules, AH and HX (X = F, Cl, Br, I; A = Li, Na, K, Rb), as well as AX and X2, as listed in Table 3. For a comparison, the representative force constants68 for the bond stretching are also listed in Table 3, which are derived from the values of the nuclear harmonic vibration frequencies, ωe. The force constants for the bond stretching represent the strength of the chemical bonds. The bigger the force constant is, the stronger the chemical bond is. The fair linear correlations between the electronic interflow at the bond center and nuclear vibration are found as well. Figure 6 displays the relationship of force constants between the electron interflow around the bond center on one side and

nuclear vibration at the nuclear equilibrium on the other side for AH and HX (X = F, Cl, Br, I; A = Li, Na, K, Rb) and AX and X2 (A = Li and Na). A good linear relation of the force constants between them has been found and y = 28.23x − 7.066, R = 0.9880, where y corresponds to the absolute value of force constant of the electronic interflow at the bond center along the chemical bond and x to the force constant of the nuclear vibration at the equilibrium. For the directions perpendicular to the chemical bond, y = 11.48x − 8.630, R = 0.9860, where y corresponds to the force constant of the electronic interflow at the bond center perpendicular to the chemical bond and x to the nuclear vibration at the equilibrium. The greater the value of the force constant for the bond stretch is, the bigger the force constants of the electronic interflows of both along the chemical bond and perpendicular to the chemical bond are. The force constants of the electronic interflow have a strong correlation with the physical quantities for describing the chemical bond strength. 3.3.3. Relationship of the Frequencies of the Electron Interflow and the Nuclear Vibration. The harmonic vibrational frequency ωe is expressed as

ωe = 1/2πc k /μ

(7)

where c, k, and μ denote the velocity of the light, the force constant and the reduced mass, respectively. We have described above that the electron movement around the bond center rb⃗ c is called the electron interflow. Thus, we may assume that the frequency of the electron interflow has a similar expression as shown in eq 7. The frequency of the electron interflow at rb⃗ c was also calculated according to eq 7, but here μ is the electronic mass and k denotes the force constants λ1, λ2, and λ3 of the electron interflow around the bond center in the three directions. For HF molecule, the force constant of the electronic interflow at the bond center along the F−H chemical bond is equal to −274.2497 N/cm, and its corresponding frequency is 9.2050i × 107 cm−1 (i is the imaginary unit), which is called the imaginary frequency. The force constant perpendicular to the F−H chemical bond is equal to 105.33 N/cm, so its frequency is 5.7045 × 107 cm−1. Along the other two perpendicular directions, the forces acting on the electron at rb⃗ c all direct to the bond center, which are the reverting forces. Here, we would point out the following interesting fact. There is a strong similarity between the molecular reaction dynamics and the electron interflow along the direction of the chemical bond. The famous intrinsic reaction coordinate (IRC) theory developed by Fukui and Morokuma69−71 in molecular reaction dynamics describes that the reactant passes through the transition state (a saddle point on the potential energy surface) and then reaches the product. Similarly, in the present description, the electron in one nuclear region may pass through the bond center (corresponding to a saddle point on the PAEM) and then enters into the other nuclear region, and vice versa. Very similar to the negative eigenvalue of the Hessian matrix at the transition state on the IRC route of the potential energy surface (PES), there is a negative eigenvalue (the force constant) of the Hessian matrix of the PAEM being constructed at the bond center. In the transition state theory, the negative eigenvalue at the transition state of the PES intimately relates to the reaction rate constant. By the similarity, here the negative eigenvalue and its related imaginary frequency of the electron interflow intimately relates to the rate of exchanging the electron between the two nuclear regions. This

Figure 6. Relationships of the force constant between the electron interflow and nuclear vibration for the AH, HX (A = Li, Na, K, Rb; X = F, Cl, Br, I), AX (A = Li, Na), and X2. I



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vibration, the stronger the corresponding chemical bond. Therefore, the greater the frequency of the electron interflow at the bond center of a chemical bond, the stronger the corresponding chemical bond. It is found that the value of the frequency of the electron interflow at the bond center is approximately 104 times as large as that of the nuclear vibration of two atoms bonded by the corresponding chemical bond. It would be expected that the correlation can be extended to predict the nuclear frequency in situ acting on two adjacent atoms in a polyatomic molecule, because the electron interflow frequency at the bond center of the PAEM can be calculated theoretically whereas the nuclear vibration frequency in situ for polyatomic molecules and its corresponding bond strength are very difficult to be experimentally measured. As discussed above, there is a corresponding relationship between the electron appearing at the bond center on one side and the nucleus lying at the equilibrium geometry on the other side, as shown in Table 5. The bond center is defined when the

strongly suggests that there exists a correlation between the electron interflow frequency and the nuclear vibration frequency. Using eq 7, the frequencies of the electron interflow at the bond center have been calculated for some series of diatomic molecules, AH, HX, AX, and X2. Figure 7 shows a correlation

Table 5. Correlation of Electron versus Nucleus in a Molecule

Figure 7. Relationships between the nuclear vibration frequencies and the electron interflow frequencies along and perpendicular to the chemical bond (cm−1).

graph between the nuclear vibration frequencies and the electron interflow frequencies along and perpendicular to the chemical bonds for HX and AH types of molecules. Along the chemical bond as marked by the open uptriangle, the correlation is a very good linearity as shown by a linear regression of an R of 0.9918, a slope of 2.0963, and the intercept of 104.60. As perpendicular to the chemical bond shown by the solid uptriangle, the correlation is also linear, as shown by a linear regression of an R of 0.9952, the slope of 1.3992, and the intercept of −322.18. For AX and X2 type molecules, their respective linear regression equations are listed in Table 4. Whether the

LiX NaX KX X2 LiX NaX KX X2

linear regression along the chemical bond

perpendicular to the chemical bond

y y y y y y y y

= = = = = = = =

5.0846x − 27.9470 5.45015x + 1068.89558 5.65566x + 1168.75634 5.23263x + 1740.00629 2.2182x + 162.44856 5.45015x + 1068.89558 5.65566x + 1168.75634 5.23263x + 1740.00629

⇔

nucleus

bond center (bc)

↔

equilibrium position

the force acting on the electron equals zero −[dV(r⃗)/dr]|r=rbc = 0

↔

the force acting on the nucleus equals zero −[dE(R)/dR]|R=Req = 0

↔

nuclear vibration the nuclear vibrational energy, Evib the force constant vibrational frequency

electron interflow −V(rbc), (Dpb) the force constant at rbc interflow frequency

minus gradient of the PAEM is equal to zero, i.e., the F(rb⃗ c) = −∇V(r)⃗ |r⃗bc = 0, as the nuclear equilibrium geometry is obtained when the gradient of the potential energy is equal to zero. For the electron, at the bond center, the negative of PAEM, −V(rbc) or Dpb, the force constant, the frequency on one side has an intimate relation with the nuclear vibration energy, the representative force constants for the bond stretch, the harmonic vibration frequency on the other side for a molecule, respectively. Therefore, with this relationship, the physical quantities at the bond center are able to represent the chemical bond strength. The bond center is a characteristic of the chemical bond. For these diatomic molecules, we have studied the correlation between D pb and the dissociation energy, respectively. For HX(X = F, Cl, Br, and I) molecules, the linear regression equation y = 253.37x − 82.399, and the correlation coefficient, R, is 0.9988. For MH (M = Li, Na, K, and Rb), the equation is y = 321.72x − 29.043, and R is 0.9718. For LiX molecules, the equation is y = 718.01x − 121.3, and R is 0.9740. For NaX molecules, the equation is y = 1098.8x − 477.27, and R is 0.9942. For KX molecules, the equation is y = 612.61x − 151.5, and R is 0.9802. y in these equations denotes the dissociation energy (kJ·mol−1) taken from the experimental value68 and x represents the Dpb whose unit is hartree. From these linear equations, it can be found the there is a good relationship between them. Table 6 lists some homogeneous sets of diatomic molecules (HX, LiX, NaX, and KX (X = F, Cl, Br, and I)), their related

Table 4. Linear Regression Equations and Their Correlation Coefficients between the Electron Interflow Frequencies and the Nuclear Vibrational Frequencies for Four Series of the Diatomic Molecules (X = F, Cl, Br, I) molecule

electron

correlation coefficient (R) 0.9973 0.9924 0.9846 0.9754 0.9997 0.9827 0.9824 0.9850

frequencies of the electron interflows at the bond centers are imaginary or real, they increase with the increasing of the experimental frequencies of the nuclear vibrations. The greater the frequency of nuclear vibration, the bigger the value of the frequency of the electron interflow for each diatomic molecule, as is well-known that the greater the frequency of the nuclear J



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Table 6. For Homogeneous Set of Molecules (HX, LiX, NaX, and KX (X = F, Cl, Br, and I)), the Distances of pcp with X and H(M), D(pcp−X) and D(pcp−H(M)) (Unit: Atomic Unit), and Dipole Moments (Debye Units) as Well as Their Linear Equations and Correlation Coefficient (R)

molecules

D(pcp− X)/au

D(pcp− H(M))/ au

dipole moment (DM)/ debye68

HF HCl HBr HI LiF LiCl LiBr LiI NaF NaCl NaBr NaI KF KCl KBr KI

1.139 1.639 1.847 2.132 1.629 2.259 2.461 2.782 1.810 2.420 2.620 2.930 1.840 2.480 2.700 3.000

0.593 0.770 0.826 0.908 1.326 1.560 1.641 1.738 1.830 2.041 2.108 2.194 2.264 2.559 2.631 2.759

1.826178 1.1086 0.8272 0.448 6.3274 7.12887 7.268 7.428 8.156 9.00117 9.1183 9.236 8.585 10.269 10.628 10.8

a

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AUTHOR INFORMATION

Corresponding Authors

*D.-X. Zhao. Tel: (86) 411-82159607. Fax: (86) 41182158977. E-mail:[email protected]. *Z.-Z. Yang. Tel: (86) 411-82159607. Fax: (86) 411-82158977. E-mail: [email protected].

linear equation and the correlation coefficient (R)a

Notes

The authors declare no competing financial interest.

y = −4.3793x + 4.4443 R = 0.9991

■

ACKNOWLEDGMENTS

■

REFERENCES

The authors are very grateful to Prof. E. R. Davidson for providing us the MELD package and other kind helps. The authors give great thanks to the editor and reviewers for their suggestions and comments that help us a lot for improving our manuscript. Thanks are also given to the support of the National Natural Science Foundation of China (Nos. 21133005 and 21073080), as well as program for Liaoning Excellent Talents in University LNET (LJQ2013111), and Natural Science Foundation of Liaoning (2014020150).

y = 2.7408x + 2.746 R = 0.9848

y = 3.0808x + 2.5831 R = 0.9765

y = 4.7203x − 1.9817 R = 0.9771

(1) Parr, R. G.; Gadre, S. R.; Bartolotti, L. J. Local Density Functional Theory of Atoms and Molecules. Proc. Natl. Acad. Sci. U. S. A. 1979, 76, 2522−2526. (2) Tal, Y.; Bader, R. F. W.; Erkku, J. Structural Homeomorphism Between the Electronic Charge Density and the Nuclear Potential of a Molecular System. Phys. Rev. A 1980, 21, 1−11. (3) Collard, K.; Hall, G. G. Orthogonal Trajectories of the Electron Density. Int. J. Quantum Chem. 1977, 12, 623−637. (4) Bader, R. F. W.; Stephens, M. E. Spatial Localization of the Electronic Pair and Number Distributions in Molecules. J. Am. Chem. Soc. 1975, 97, 7391−7399. (5) Bader, R. F. W. Molecular Fragments or Chemical Bonds. Acc. Chem. Res. 1975, 8, 34−40. (6) Bader, R. F. W. Atoms in Molecules, A Quantum Theory; Clarendon: Oxford, U.K., 1990. (7) Becke, A. D.; Edgecomb, K. A Simple Measure of Electron Localization in Atomic and Molecular Systems. J. Chem. Phys. 1990, 92, 5397−5403. (8) Silvi, B.; Savin, A. Classification of Chemical Bonds Based on Topological Analysis of Electron Localization Functions. Nature 1994, 371, 683−686. (9) Koritsanszky, T. S.; Coppens, P. Chemical Applications of X-Ray Charge-Density Analysis. Chem. Rev. 2001, 101, 1583−1628. (10) Hohenberg, P.; Kohn, W. Inhomogeneous Electron Gas. Phys. Rev. B 1964, 136, 864−871. (11) Kohn, W.; Sham, L. Self-Consistent Eqations Including Exchage and Correlation Effects. Phys. Rev. A 1965, 140, 1133−1138. (12) Parr, R. G.; Yang, W. Density Functional Theory of Atoms and Molecules; Oxford University Press: Oxford, U.K., 1989. (13) Bader, R. F. W. Atoms in Molecules. Acc. Chem. Res. 1985, 18, 9−15. (14) Bader, R. F. W. A Quantum Theory of Molecular Structure and Its Appllcatlons. Chem. Rev. 1991, 91, 893−928. (15) Bader, R. F. W. A Bond Path: A Universal Indicator of Bonded Interactions. J. Phys. Chem. A 1998, 102, 7314−7323. (16) Popelier, P. L. A. Quantum Molecular Similarity. 1. BCP Space. J. Phys. Chem. A 1999, 103, 2883−2890. (17) Bader, R. F. W.; Legare, D. A. Properties of Atoms in Molecules: Structure and Reactivities of Boranes and Carboranes. Can. J. Chem. 1992, 70, 657−676. (18) Wiberg, K. B.; Bader, R. F. W.; Lau, C. D. H. A Theoretical Analysis of Hydrocarbon Properties: I. Bonds, Structures, Charge Concentrations and Charge Relaxations. Can. J. Chem. 1987, 109, 985−1001.

y denotes dipole moment and x denotes D(pcp−H(M)).

quantities, i.e., dipole moment. When we correlate the positions of the bond center that are denoted by the distances between pcp and H (M = Li, Na, and K) atoms with the experimental dipole moments,68 it is found that the correlation is fairly linear for a homogeneous set, as listed in Table 6.

4. SUMMARY The potential acting on one electron in a molecule (PAEM), V(r)⃗ , is a scalar field. The 3D representations, theoretical analysis, and topology characters of the PAEM have been drawn and explored. A significant feature of the PAEM surface is that there are potential saddle points (psp). The negative of the potential, Dpb, that is the gap between the saddle point and the zero energy level of one electron, at this position is able to characterize how easily one electron moves from one nuclear region to another nuclear region through the bond region and it can also characterize the bond strength. The force acting on one electron in a molecule, −∇V(r)⃗ = 0, has been investigated. The bond center (rb⃗ c) between the two bonded atoms, where −∇V(rb⃗ c) = 0, has been defined, and it is found that a bond center is a 2D electron attractor, although it does not possess any positive charges. The movement of the electron around the bond center has been assumed as the electron interflow. Through the analysis of the Hessian matrix of the PAEM around the bond center, the force constant and frequency of the electron interflow around the bond center have been calculated and correlated with the nuclear vibration between the two bonded atoms. In summary, through analysis and investigation of PAEM and FAEM, it is shown that the quantities of the electron interflow around the bond center on one side correlate with the corresponding quantities of the nuclear vibration around the equilibrium geometry of the two atoms (Table 5) on the other side, which provides new insight into the nature of chemical bonds. K



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Theoretical Exploration of the Potential and Force Acting on One

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