Article pubs.acs.org/JCTC
Description of Polar Chemical Bonds from the Quantum Mechanical Interference Perspective Felipe Fantuzzi and Marco Antonio Chaer Nascimento* Instituto de Química, Universidade Federal do Rio de Janeiro, Rio de Janeiro, RJ 21941-909, Brazil
ABSTRACT: The Generalized Product Function Energy Partitioning (GPF-EP) method has been applied to a set of molecules, AH (A = Li, Be, B, C, N, O, F), CO and LiF with quite different dipole moments, in order to investigate the role played by the quantum interference effect in the formation of polar chemical bonds. The calculations were carried out with GPF wave functions treating all the core electrons as a single Hartree−Fock group and the bonding electrons at the Generalized Valence Bond Perfect-Pairing (GVB-PP) level, with the cc-pVTZ basis set. The results of the energy partitioning into interference and quasiclassical contributions along the respective Potential Energy Surfaces (PES) show that the main contribution to the depth of the potential wells comes from the interference term, which is an indication that all the molecules mentioned above form typical covalent bonds. In all cases, the stabilization promoted by the interference term comes from the kinetic contribution, in agreement with previous results. The analysis of the effect of quantum interference on the electron density reveals that while polarization effects (quasi-classical) tend to displace electronic density from the most polarizable atom toward the less polarizable one, interference (quantum effects) counteracts by displacing electronic density to the bond region, giving rise to the right electronic density and dipole moment.
1. INTRODUCTION Chemical bond is usually considered to be a very well established concept in chemistry. Nevertheless, its origin and nature are still subject to much discussion and scientific inquiry.1−25 In spite of the fact that the minimum in the potential energy surface (PES) responsible for the bond in a stable system is followed by a decrease of potential energy and a rise of kinetic energy, as required by the virial theorem, this type of analysis does not provide a model for explaining bond formation. Since the total kinetic and potential energies are a direct consequence of the form of the total electronic density, the question of why the electronic density of a bonded molecule changes in such a way that results in an energy drop is unanswered by these quantities.1−3 It is possible to understand the reason why chemical bonds are formed by means of an alternative energy partitioning, derived from a density partitioning, in quasi-classical and interference contributions.1,4,5 This approach shows that the quantum mechanical interference effect promotes a change in the electron density which is responsible for the energy reduction that leads to the formation of covalent bonds. The results obtained for a variety of diatomic and polyatomic molecules based on this kind of energy © XXXX American Chemical Society
partitioning attest that such bonds are formed only in the presence of quantum interference.1−10 In previous papers, we developed an energy partition scheme, known as the Generalized Product Function Energy Partitioning Method (GPF-EP), and used it to investigate the chemical bond in different classes of molecules,5−10 including a range of homonuclear (AA) bonds5,6 in diatomic and polyatomic molecules.7−10 According to this approach, detailed in section 2, the total energy of the molecule is partitioned into a contribution due to the quantum interference effect and a contribution due to all the other effects, collectively referred as to quasi-classical or reference contribution. The molecules previously studied2−10 are all made of nonpolar or slightly polar bonds, and for these molecules the central role played by the interference energy can be easily verified to be determinant for the formation of each one of the bonds. Normally, nonpolar or slightly polar bonds are considered to be “covalent” bonds but this sort of association can lead to the unjustified conclusion that quantum Received: February 8, 2014
A
dx.doi.org/10.1021/ct500334f | J. Chem. Theory Comput. XXXX, XXX, XXX−XXX
Journal of Chemical Theory and Computation
Article η
interference only plays a dominant role in the formation of nonpolar or slightly polar bonds and, even worse, that polar bonds are predominantly noncovalent and that highly polar bonds are essentially “ionic”. In order to show that there is no direct relation between the quantum mechanical interference contribution to the formation of a chemical bond and its polarity, we propose to examine the interference energy in diatomic molecules, AB, presenting different polarities. This can be accomplished by selecting different pairs of distinct A and B atoms or by fixing one of them and varying the other, as to scan a reasonable range of values of dipole moments. For a comparative analysis the second choice is more convenient and the series of molecules A-H, with A = Li, Be, B, C, N, O, and F provides an excellent test, inasmuch as the dipole moment not only varies gradually in magnitude but also changes its sign as one moves from lithium hydride (μ = −5.88 D) to hydrogen fluoride (μ =1.82 D). In addition, from all AB (A, B ≠ H) molecules that can be formed with the first row elements, we also considered LiF and CO because they represent two excellent tests for the theory. According to any simple qualitative or semiquantitative analysis based on pure electrostatic models, LiF, being formed from atoms occupying the two extreme positions in the periodic table, should be the most “ionic” of the bonds, while CO should be less ionic but exhibiting a dipole moment with the negative end at the O atom, contrary to the experimental observations.
(2)
μ=1 η
π( r1⃗ r2⃗ , r1⃗′ r2⃗ ′) =
∑ π μ( r1⃗ r2⃗ , r1⃗′ r2⃗ ′) μ=1
+
1 2
η
η
⎡
∑ ∑ ⎣⎢ρ μ ( r1⃗ , r1⃗′)ρν ( r2⃗ , r2⃗ ′) − μ=1 ν>μ
⎤ 1 μ ρ ( r2⃗ , r1⃗′)ρν ( r1⃗ , r2⃗ ′)⎥ ⎦ 2 (3)
where η is the number of groups in the GPF function. The total electronic energy expression for a GPF wave function in terms of the reduced density matrices is ⎧ ⎪ ⎪ = ∑⎨ μ=1 ⎪ ⎪ ⎩ η
EGPF
∫
[hρ̂ μ ( r1⃗ , r1⃗′)]r1⃗ = r1⃗ ′ dr1⃗
⎫ ⎪ ⎡ π μ( r1⃗ r2⃗ , r1⃗′ r2⃗ ′) ⎤ ⎪ dr1⃗dr2⃗ ⎬ ⎢ ⎥ r r = r12 ⎣ ⎦ 1⃗ 1⃗ ′ ⎪ ⎪ r2⃗ = r2⃗ ⎭ ′ ⎧ ⎫ η η ⎪ ⎪ ⎡ ρ μ ( r1⃗ , r1⃗′)ρν ( r2⃗ , r2⃗ ′) ⎤ ⎪ ⎪1 ∑∑⎨ dr1⃗dr2⃗ ⎬ ⎢ ⎥ 2 r12 ⎣ ⎦ r1⃗ = r1⃗ ′ ⎪ μ=1 ν≠μ ⎪ ⎪ ⎪ r2⃗ = r2⃗ ⎭ ⎩ ′ ⎧ ⎫ η η ⎪ ⎪ ⎡ ρ μ ( r2⃗ , r1⃗′)ρν ( r1⃗ , r2⃗ ′) ⎤ ⎪1 ⎪ ∑∑⎨ dr1⃗dr2⃗ ⎬ ⎢ ⎥ 4 r12 ⎣ ⎦ r1⃗ = r1⃗ ′ ⎪ μ=1 ν≠μ ⎪ ⎪ ⎪ r2⃗ = r2⃗ ⎩ ⎭ ′
1 + 2
2. PARTITIONING OF THE INTERFERENCE ENERGY AND DENSITY The method for energy partitioning used herein is called Generalized Product Function Energy Partitioning (GPF-EP)5 and was derived by applying Ruedenberg’s4 original partitioning scheme to GPF wave functions constructed with Modern Valence Bond group functions. The GPF wave functions have been defined by McWeeny, and their general form is shown in eq 1:
∫
+
1 2
−
1 2
+
∑∑
∫
∫
M
M
A=1 B>A
ZAZB rAB
(4)
where ĥ is the one-electron operator, which includes both kinetic energy and electron-nuclei potential energy and r12 is the interelectronic distance. The expression in the first brackets stands for the intragroup energy, and the remaining part of the equation stands for the coulomb and exchange intergroup contributions, respectively. The GPF-EP method separates both ρ and π into interference and reference densities, the latter meaning the sum of quasi-classical densities. For instance, the interference and quasi-classical densities for a single group are
̂ Ψ1( r1ω ΨGPF = A{ ⃗ 1⃗ , r2⃗ ω⃗ 2 , ..., rN⃗ 1ω⃗ N1)Ψ2 × ( rN⃗ 1+ 1ω⃗ N1+ 1 , rN⃗ 1+ 2ω⃗ N1+ 2 , , ..., rN⃗ 2ω⃗ N2)...}
∑ ρ μ ( r1⃗ , r1′⃗ )
ρ( r1⃗ , r1′⃗ ) =
(1)
where  is the antisymmetrizer operator, the indexes (1) and (2) represent different orthogonal wave functions (groups), and ri and ωi are the electron spatial and spin coordinates, respectively. Each group is set to be orthogonal to all others, in the same way that Generalized Valence Bond (GVB) pairs are used in the GVB-PP (Perfect Pairing) wave function. However, in a GPF groups can be treated with different methods, and more than two electrons can be put in a single group. For saturated systems, the most convenient way to divide such groups is to define one group for each electron pair involved in a chemical bond and a single group for all the electrons belonging to the core. Since core electrons play a smaller role in most chemical bonds, treating them as a single Restricted Hartree−Fock (RHF) group is a reasonable approximation. In more complex systems, such as aromatic rings,10 it is necessary to define a group with more than two electrons, and carry out the calculations using a full GVB or, equivalently, a Spin-Coupled wave function. Only a brief discussion concerning the main aspects of the method will be presented. Detailed information about the method can be found elsewhere in the literature.5 The reduced density matrices of first and second order (RDM-1 and RDM-2) for GPF wave functions are as follows:
Nμ
ρIμ ( r1⃗)
=
∑ ′⟨r , s⟩μ p(r|s) (5)
r ,s
N μ ρQC ( r1⃗) =
μ
∑ (ϕrμ( r1⃗))2
(6)
r=1 μ
where μ is the selected group index, N is the number of electrons in this group, p(r|s) are the density matrix elements expressed in the orbital basis set, the prime character (′) on eq 5 indicates that diagonal elements (r = s) are not included in the sum and ⟨r,s⟩μ is the interference density for orbitals ϕr and ϕs: ⟨r , s⟩μ = ϕrμ( r ⃗)ϕsμ( r ⃗) −
1 ξ(r , s)[(ϕrμ( r ⃗))2 + (ϕsμ( r ⃗))2 ] 2 (7)
where ξ(r,s) is the overlap integral between ϕμr and ϕμs . Thus, the total energy of the system can be separated as follows: E[tot] = E[ref] + E[I] + E[II] + E[x] B
(8)
dx.doi.org/10.1021/ct500334f | J. Chem. Theory Comput. XXXX, XXX, XXX−XXX
Journal of Chemical Theory and Computation
Article
Table 1. Some Experimental and Calculated Properties of the Studied Moleculesa LiH ( Σ ) BeH (2Σ+) BH (1Σ+) CH (2Π) NH (3Σ−) OH (2Π) HF (1Σ+) CO (1Σ+) LiF (1Σ+) 1 +
|ΔXAB|
(μexp/D)b
(μcal/D)c
(Re/Å)d
(De/eV)e
(αA/au)f
1.1 0.6 0.2 0.4 0.9 1.4 1.9 1.0 3.0
−5.882 −0.228 1.270 1.460 1.539 1.660 −1.820 0.120 −6.325
−5.660 −0.212 1.712 1.459 1.649 1.684 −1.825 0.169 −6.478
1.5954 1.343 1.236 1.124 1.045 0.9706 0.9168 1.128 1.564
2.516 2.429 3.525 3.649 3.893 4.624 6.112 11.244 5.995
164.0 ± 3.4 (exp.) 37.31 20.53 11.26 7.6 ± 0.4 (exp.) 5.24 4.50
A positive dipole moment indicates A−B+ polarity. bLiH: ref 30. BeH, NH: ref 31. BH: ref 32. CH: ref 33. OH: ref 34. HF: ref 35. CO: refs 28 and 29. LiF: ref 36. cThis work. dRef 37. eCalculated from the experimental values (ref 38 for LiH and ref 37 for the other molecules) of ωe and D0. fRef 39. a
related to the σ-bond between the atoms was treated with the GVB-PP method, as well as the π bonds for the CO molecule. The lone pairs were also treated with the GVB-PP method in all cases where they could change qualitatively (and quantitatively in a reasonable degree) the characteristics of the chemical bonds. For the OH, HF, and LiF molecules, the description of the bond is pretty much independent of how the lone pairs are treated, and therefore, these electrons were included in the RHF group. Since the interference energy is only slightly affected by the size of the basis set and the use of basis sets of moderate sizes containing polarization functions correctly describes the energy partitioning,5,6 the cc-pVTZ basis set26 was used in all the present calculations. A version of the VB200027 code modified by our group was used in order to perform the GPF-EP partition. The energy partitioning was first performed for the diatomic molecules at their experimental equilibrium geometry. From this point, geometry scans were performed in order to obtain the partitioning profile from 0.60 Å to 5.00 Å, with a step size of 0.05 Å. The optimized orbitals from a given point were used as a guess for the subsequent calculation. To get a clearer picture of the effect of interference on the electron density, interference density contour diagrams, representing the change in the electron density promoted by the interference effect were constructed. The quasi-classical electron density associated with the bond formation was also obtained. In order to do that, the quasi-classical densities of the atoms were subtracted from the quasi-classical density of the molecule. The program DensPlot, developed by our group, was used to obtain these densities.
where E[ref] is the total reference energy, E[I] and E[II] are the first and second-order interference energies respectively, and E[x] is the total intergroup exchange energy, which arises from the antisymmetrization of the GPF wave function. It is important to emphasize that the E[x] term would not exist if the GPF wave function consisted of a single group. Consequently, the exchange term represents merely a symmetry correction to the reference energy and only arises because of the separation of the total wave function into different groups. Thus, the choice of the number of groups and the number of electrons in each group must be done with caution, since the results in terms of energy partition and its interpretation will strongly rely on the quality of this selection. In eq 8, the sum E[ref] + E[x] represents the quasi-classical contribution, E[ref+x] and E[I] + E[II] corresponds to the total interference contribution, E[I+II]. The reference and first-order interference energies can be further separated into kinetic (T[ref] and T[I]), electron− electron potential (Vee[ref] and Vee[I]) and electron-nuclei (Ven[ref] and Ven[I]) energies. The second-order interference energy E[II] consists exclusively of electron−electron repulsion terms and can be equivalently referred to as Vee[II]. These contributions can also be divided into intragroup and intergroup contributions as follows: η
E[ref] =
η
M
M
∑ Eμ[ref] + ∑ Eμ,ν[ref] + ∑ ∑ μ=1
μA
ZAZB rAB
(9)
η
E[I] =
∑ Eμ[I] (10)
μ=1
η
E[II] =
η
∑ Eμ[II] + ∑ Eμ,ν[II ] μ=1
μ
