The Quantum Interference Contribution to the Dipole Moment of

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The Quantum Interference Contribution to the Dipole Moment of Diatomic Molecules David Wilian Oliveira de Sousa, and Marco Antonio Chaer Nascimento J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.7b11760 • Publication Date (Web): 16 Jan 2018 Downloaded from http://pubs.acs.org on January 17, 2018

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The Journal of Physical Chemistry

The Quantum Interference Contribution to the Dipole Moment of Diatomic Molecules David Wilian Oliveira de Sousa, Marco Antonio Chaer Nascimento* Instituto de Química, Universidade Federal do Rio de Janeiro, Rio de Janeiro, Brazil

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ABSTRACT

The Interference Energy Partitioning Analysis method developed by our group and used to study the nature of the chemical bond was extended to partition the electric dipole moment in quasiclassical and interference contributions. Our results show that interference participates in charge displacement in polar molecules, providing, directly or indirectly, a relevant contribution for the total dipole moment. A linear correlation was found between the interference contribution of the dipole moment from the bond electron group, µINT(bond), and the difference of electronegativity of the atoms which form the bond, ∆XAB. This interesting result reinforces the fact that electronegativity is not a property of an atom alone, but rather a property of the atom in the molecule and that ∆XAB can only be associated with that part of the total charge displacement resulting from the formation of the chemical bond. The partitioning of the total dipole moment into quasi-classical and interference contributions provides new insights about the reasons for the failure of the ∆XAB criterion in predicting the correct orientation of the dipole moment in several molecules. The results of the present work also bring additional evidences for the previously proposed mechanism of formation of polar bonds.


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1. INTRODUCTION The concept of chemical bond is in the core of chemistry and understanding its nature is perhaps the most important problem of theoretical chemistry. From the early theories of the chemical bond, the most successful was the one by Lewis,1 who proposed that chemical bonds would result from atoms sharing pairs of “valence” electrons and the polarity of a bond would result from an unequal sharing of the pair of electrons by the atoms. Pauling2 translated Lewis’ ideas to the framework of quantum mechanics, using this valence bond approach, and analyzed the polarity of a bond in terms of the difference of electronegativity (∆XAB) of the atoms A and B involved in the bond. Pauling also considered the pure covalent and ionic bonds as the two extremes of a scale and that a given A-B bond could be somewhat ionic or covalent depending on the difference of electronegativity (∆XAB) of the atoms A and B. Thus, the value of ∆XAB could be used to predict the polarity of a bond. This criterion became the standard one and, traditionally, chemical bonds are classified as covalent, polar and ionic, depending on the corresponding value of ∆XAB. However, we have recently shown3 that this criterion can be very deceiving and does not resist to a simple analysis using diatomic molecules formed by the first-row atoms, besides inducing chemist to believe that there is something fundamentally different among these “types” of bonds. For example, from the diatomic hydrides of the second-period elements, FH should be the most polar and BH the least polar, which is not the case. Besides, FH and LiH molecules should be considered ionic, which was a somewhat controversial subject,4–9 but most of the studies agree that it is also not the case. The simple ∆XAB model would also predict that CO molecule is more polar than CH. It not only fails in that prediction, but also fails in attributing the right orientation of the dipole, which is Cδ–Oδ+. Other well-known examples of dipole “inversion” are the CF,10 BF, BCl, AlF, and AlCl molecules.11 In a recent work, our group have analyzed the chemical bond in a series of polar diatomic molecules3 using modern valence bond wave functions and our Interference Energy Analysis (IEA) method.12 This powerful approach is rooted on the fact that chemical bonds result from the quantum



interference between electronic states.13 By using this method, we were able to show that the traditional way of classifying bonds as pure covalent, polar and ionic is meaningless because all bonds have the same origin, quantum interference, and as it takes one orbital from each atom to build up interference, all chemical bonds are covalent.14 A simple mechanism of the formation of polar bonds (Figure 1) can be envisaged as follows: first, consider two non-interacting atoms A and B, whose electronic densities associated to the valence electrons are and . The total density for the non-interacting atoms is simply the sum of the two densities, and , which will be denoted as quasi-classical density,

, . As the atoms approach and start to feel the presence of each other, the electronic clouds will distort due to electrostatic effects which cause some electronic density displacement to the less polarizable atom. Since this distortion results from pure electrostatic effects, the total density remains the sum of the individual ones. At some point, the electronic densities overlap giving rise to quantum interference and the total density is not anymore the sum of two atomic densities. The individual orbitals which interact must be summed and then squared in order to get the correct density, yielding 2 , where the last term accounts for the quantum interference effect. This term is also responsible for displacing some of the electron density from regions next to the nuclei to the bonding region.


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Figure 1. Scheme of the mechanism of formation of polar chemical bonds. (1) Isolated atoms; the total density is the sum of the squares of the wave functions of each atom. (2) As the atoms approach, but do not interact directly, the electronic cloud distorts towards the less polarizable atom. This is a pure polarization effect, which does not account for the formation of the bond. (3) The bonding orbitals overlap and the total density is not anymore the sum of the two separated parts. The interference effect displaces some electronic density from the regions close to the nuclei to the bonding region. In this work, we elaborate a very simple and straightforward method to analyze separately the roles of interference and quasi-classical effects in the charge displacement when a polar bond is formed. Such analysis works as a direct verification of the previously discussed mechanism,3 as well as it provides additional insight about the reasons for the inversion of the electric dipole orientation in several molecules.

2. THEORY

The electric dipole moment operator of a molecule with N electrons and M nuclei, ^, is defined as:

^ − ^ ^

(1)

where ^ is the position operator of each electron, ^ of the nuclei, and ZA is the nuclear charges.

The average dipole moment for the molecule in the state | is given by:

|^| − ^ ^ ! " # ^

(2)

where ρ is the electronic density. Our partitioning method12 is based on generalized product wave functions (GPFs)15 and on the reduced density matrices (RDMs) formalism. Both the total density



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and energy can be partitioned in quasi-classical (i.e., non-interference) and interference contributions:

INT

(3a)

$ $ $INT

(3b)

Making use of equation (3a) one can also partition any density-based property, such as the dipole moment, in quasi-classical and interference contributions. By using a one-electron orbital basis we can rewrite the density as follows:

%

% &%

(4)

where Pij is an element of the first-order RDM. The total dipole moment is, therefore:

%

− '% &% ^

(5)

where '% ( ||% ) are the dipole moment integrals in the orbital basis. In the same basis, the forms of the quasi-classical and interference densities can be expressed as:12

(6a)

1 INT * % − ,% - % ./ &% 2

(6b)

%0

where ,% ( 1% ) are the overlap integrals. Hence, one can obtain the final expressions of the quasi-classical and interference contributions to the dipole moment µ:

− ' ^

%0

1 INT − *'% − ,% -' '%% ./ &% 2

(7a)

(7b)

where the nuclear charge part was included in the quasi-classical contribution. Furthermore, since GPF wave functions can be factored groups of electrons and the one-electron density is additive by ACS Paragon Plus Environment

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groups, we can determine, separately, the contributions from the core electrons, lone pairs and bonding pairs to the quasi-classical and interference components of the dipole moment, as: 3

2

QC,INT QC,INT 2

(8)

where O is the number of groups.

3. COMPUTATIONAL DETAILS

In this work we have used the VB2000 2.716 / GAMESS17 (December, 2014) programs modified by our group to include the GPF energy partitioning program. All calculations were performed at the experimental equilibrium geometries, using both the cc-pVTZ and aug-cc-pVTZ basis sets. Systematic studies18,19 have shown that augmented triple-zeta quality basis sets are usually sufficient for accuracy in electric dipole moment calculations. Generalized Valence Bond wave functions20 were used at the perfect-pairing approximation (GVB-PP), since the result dipole moments obtained at this level are quite reasonable for most of the studied molecules. The GVB wave function can be treated as a special case of the GPF. The electron group splitting was made as follows: the core electrons (all 1s electrons, 2s from N, O, F, and Cl atoms, and 2p and 3s from Cl atom) are kept doubly occupied and treated at Hartree-Fock level. The remaining valence electrons, including the lone pairs, are treated in the perfect pairing approximation. GAMESS stores dipole moment integrals in atomic orbital basis, (4 |5|4% ), (4 |6|4% ), and

(4 |7|4% ). Each one-electron orbital is given by ∑2 92 42 , so we can easily write the X, Y and Z components of the dipole moment as:



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:% 92 9%; 42 |5|4; 2

;

2

;

2

;

The Quantum Interference Contribution to the Dipole Moment of

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