Communication: Chemical bonding in carbon dimer isovalent series from the natural orbital functional theory perspective

The natural orbital functional theory admits two unique representations in the orbital space. On the one hand, we have the natural orbitals themselves that minimize the energy functional, and which afford for a diagonal one-particle reduced density matrix but not for a diagonal Lagrangian orbital energy multipliers matrix. On the other hand, since it is possible to reverse the situation but only once the energy minimization has been achieved, we have the so-called canonical representation, where the Lagrangian orbital energy multipliers matrix is diagonal but the one-particle reduced density matrix is not. Here it is shown that the former representation, the natural orbital representation, accounts nicely for the quadrupole bond character of the ground states of C₂, BN, CB⁻, and CN⁺, and for the double bond order character of the isovalent

$^{1}\Sigma _{g}^{+}$

$Σg+1$ state of Si₂. Similarly, the canonical orbital representation accounts correctly for the ionization spectra of all these species.

In spite of the vivid discussion that has recently sprouted in the literature about the existence of the atomic and molecular orbitals (MO)^1,2 and their measurability,^3–7 orbitals still constitute one of the most used tools by chemists to describe, understand, and predict chemical behavior.⁸ 

However, it is worth noting that the molecular orbitals themselves stem from approximate solutions to the Schrödinger equation, and, in addition of being non observable, in the strict quantum mechanical sense of the expression, they are normally subjected to some intrinsic large arbritariness. Thus, any unitary transformation of any set of optimized molecular orbitals computed with approximate methods that yield the one-particle reduced density matrix (1-RDM) idempotent^9,10 gives another equivalent set of molecular orbitals which keeps invariant all observables, including the total energy of the system. Consequently, the particular set of orbitals that a chemist uses constitutes no more than one choice among many equally eligible sets. Additionally, we have the so-called Valence Bond (VB) molecular orbitals¹¹ that result from optimizing the energy of the selected valence bond structures that expand the VB wave function.

The enormous interpretative power of molecular orbitals stems from their direct linkage with the central concept of chemistry, namely, the chemical bond. In spite of other more fundamental descriptions like Bader's quantum theory of atoms in molecules,¹² molecular orbitals give an appealingly attractive visual picture of the chemical bonding between the atoms in a molecule, and it is because of this direct relationship that they have been extensively used by chemists in almost all fields of chemistry.¹³ Furthermore, chemists have produced a number of related concepts like bond order, delocalized, localized, σ-type bonding orbitals, π-type bonding orbitals, etc.,² which constitute central pieces of chemists’ modern common language.

In this vein, the quantification of the strength of a chemical bond is customarily made by its bond order, which is estimated by the inspection of the molecular orbitals, as to decide whether they are bonding, antibonding, or non-bonding, and their occupation numbers, as to quantify how much each orbital contributes to the overall chemical bond order. This approach has produced a number of pleasant outcomes, like the recent characterization of the quintuple bond of the uranium dimer,¹⁴ but, as it has been stated above, it is subjected to some degree of arbritariness that for depending on which orbital set the inspected conclusion might be different. A nice example of the latter is the research by Shaik et al. on the bond order of carbon dimer and its isovalent eight-valence electron species,¹⁵ which has been recently enriched by an enlightening discussion with Rzepa and Hoffmann.¹⁶ Thus, for C₂, BN, CN⁺, and CB⁻, inspection of the orbitals computed from MO theory suggests a bond order of two. On the other hand, a deep inspection of the VB hybrid molecular orbitals pinpoints to a likely valence bond structure with a bond order of four, and it is claimed that the properties of C₂ can be predicted quite well using such quadruple bond valence bond structure.^15,17 Furthermore, for Si₂ and Ge₂, inspection of both sets of orbitals agrees in assigning a bond order of two.

This disparity between the bonding schemes produced by MO and the VB orbitals should not be regarded as a surprise given the differences between MO and VB orbitals, and it makes the analysis of molecular orbitals to be not an out-of-the-box like automatic operation. MO theory molecular orbitals correspond to the eigenfunctions of the Fock noninteracting quasiparticle operator. The MO theory molecular orbitals are orthogonal. VB molecular orbitals are constructed from hybrid atomic orbitals to represent the various valence bond configurations that expand the VB wave functions. The VB orbitals are non-orthogonal. Hence, two different pictures might arise from these two different methods.

Recently, we have reported on the advantages of Natural Orbital Functional (NOF) theory^18–20 molecular orbitals analysis.^21–24 In NOF theory, the energy functional of the 1-RDM is expressed by means of its spectral representation in terms of the natural orbitals and their occupation numbers. The solution of Euler equations for the natural orbitals and their occupation numbers gives the sought after minimum energy variational solution. Refer to the supplementary material for a detailed description of NOF theory.²⁵ 

In this paper, we have used the Piris natural orbital functional 5 (PNOF5) reconstruction of the 2-RDM,^26–30 based on a pairing scheme of the natural orbitals, which is allowed to vary along the energy minimization process. It has been recently reported²² that PNOF5 provides natural orbitals whose inspection yields chemical bonding pictures resembling those obtained from the empirical valence shell electron pair repulsion theory (VSEPR)³¹ and the Bent's rule,³² as well as with the one emerging from the VB theory. Natural orbitals provide a diagonal 1-RDM (Γ) and a non-diagonal Lagrangian orbital energy multipliers matrix (Λ). Therefore, orbital energies are ill-defined in the natural orbital representation. In order to obtain well-defined orbital energies, we must have the orbital energy multipliers matrix Λ diagonal, as it has been discussed in more detail in an earlier publication.²⁴ This can be achieved by a unitary transformation, but only at the energy minimum for it is only at the minimum where the matrix Λ is hermitian, ergo only at this point exists a unitary matrix that diagonalizes it. These two unique sets of orbitals, which diagonalize both Γ and Λ, separately, belong to the same solution and, therefore, they complement each other. Both sets represent orbitals in the coordinate space, but orbital occupations and orbital energies do not come together in neither of the two representations. Thus, orbital occupations should be read from the natural orbital representation and orbital energies from the canonical orbital representation.

The computed PNOF5 natural and canonical molecular orbitals of the carbon dimer are shown in Fig. 1. Note that the natural orbitals yield a representation with four bonding orbitals (quasi) doubly occupied, very supportive of a formal bond order of four, as claimed by Shaik et al.¹⁵ However, there is a number of differences between our description and that of the VB calculations of Shaik et al.¹⁵ which should be pointed out. Thus, the VB computations assume a sp hybridization scheme for each of the two carbon atoms and then four molecular bonding orbitals are set by combining the two (one on each carbon atom) σ-type orbitals inwardly pointing, the four (two on each carbon atom) π-type orbitals, and the fourth bond is made of the combination of the two σ-type outwardly pointing orbitals. Consequently, VB predicts three strong bonds and one weaker bond, the so-called putative bond. Indeed, their estimated bond energies are 4.35 eV (100.4 kcal/mol) for the σ(C–C) bond, 4.08 eV (94.2 kcal/mol) for the two degenerate π(C–C) bonds, and 0.62 eV (14.3 kcal/mol) for the fourth weak bond.¹⁵ It is worth noting that these orbital bond energies do not correspond to the ionization energies.¹¹ 

The natural orbitals found in the PNOF5 calculations are one of σ-type and three hybrid bent orbitals of pseudo-π symmetry, also referred as banana orbitals,³³ in the sense that their nodal plane is not a plane of symmetry, but they conserve an inversion point in such a way that bonding pseudo-π orbitals are of ungerade symmetry and antibonding pseudo-π orbitals are of gerade symmetry. The occupation numbers of these orbitals are 1.997 e and 1.841 e, for the σ and pseudo-π orbitals, respectively, while 0.003 e and 0.159 e for their corresponding antibonding orbitals. This leads to an effective bond order (EBO) of 3.52 (see Table I). The EBO is calculated according to the definition of Roos and co-workers,^34,35 namely, ∑(η_b − η_ab)/2, where η_b and η_ab state for the natural orbital occupation of the bonding and antibonding orbitals, respectively. Observe that this value corresponds to a formal bond order of four, in agreement with the VB prediction. Besides, all these orbitals arise from the combination of sp³ hybrid atomic orbitals (see Figure 2 for a schematic view of the formation of these orbitals), and not from hybrid sp and p atomic orbitals, as in VB.

Additionally to this arresting visual picture of the quadruple bond of C₂, we can regain the connection between orbital energies and ionization energies by changing over to the canonical representation.¹⁸ In this representation, the overall point-group symmetry adapted picture is recovered within NOF theory by the unique unitary transformation, which diagonalizes the matrix of the Lagrangian orbital energy multipliers and keeps the total energy invariant. This transformation yields the canonical orbitals shown on the right of Fig. 1. It should be mentioned that these canonical orbitals resemble those obtained at common molecular orbital calculations such as HF or Kohn-Sham density functional theory (KS-DFT), and, hence, no hybrid atomic orbitals are observed. Good approximate ionization energies can be extracted from this representation as shown in Table I.

Inspection of the data shown in Table I reveals that our calculations predict a principal ionization energy of 12.7 eV and a second ionization energy of 15.4 eV for C₂, estimated by using the Extended Koopmans Theorem (EKT).⁴¹ Their orbital energies (λ_ii) are 11.67 eV and 14.24 eV, which correspond to the canonical doubly degenerate π-type bonding orbital and to the antibonding σ* orbital composed mainly of the 2s atomic orbitals. Notice that our principal ionization energy estimation agrees satisfactorily with the experimental mark. Similar conclusions, regarding bond orders and ionization energies, can be drawn for BN, CB⁻, and CN⁺, as shown in Table I.

Chemical bonding in the related isovalent

$Σg+1$ state of the silicon dimer is known to be different.¹⁵ The

$^{1}\Sigma _{g}^{+}$

$Σg+1$ state of Si₂ is indeed an excited state lying significantly above the

$^{3}\Sigma _{u}^{-}$

$Σu−3$ ground state.⁴² Silicon dimer, in general, must have a very different electronic structure as compared to carbon dimer, as suggested by its very different chemical properties. The nature of the chemical bond of its

$^{1}\Sigma _{g}^{+}$

$Σg+1$ state is not an exception.

Figure 3 displays its computed PNOF5 natural and canonical molecular orbitals for silicon dimer. Inspection of the left panel of Fig. 3 reveals that bonding now is made by one σ-type orbital and one π-type orbital with populations of 1.990 and 1.836 e⁻ and two degenerate lone-pair non-bonding hybrid orbitals localized on each of the silicon atoms, with a population of 1.935 e⁻. Their corresponding antibonding orbitals have populations of 0.009, 0.164, and 0.064 e⁻, respectively. This renders a formal bond order of two. Notice that axis of the antibonding σ-type orbital (highest lying orbital) does not match with the molecular axis.

The canonical orbital representation of the Si₂ dimer, shown in the right panel of Fig. 3, features three filled σ-type molecular orbitals and one bonding π molecular orbital, in agreement with previous CI results.⁴² Furthermore, the EKT estimated principal ionization energy agrees satisfactorily with the experimental mark, as shown in Table I, which gives confidence on the reliability of our computed NOF orbitals.

In summary, we have put forward that the natural orbital representation and canonical orbital representation nicely account for the quadrupole bond character of ground states of C₂, BN, CB⁻, and CN⁺ and their ionization energy spectra. Additionally, they correctly predict that the

$Σg+1$ state of the isovalent Si₂ has a double bond character instead. Our analysis has demonstrated for this case that the two representations mentioned above merge into one unified picture. Namely, the coordinate space visualization of chemical bonding in terms of the localized natural orbitals, and the connection between ionization energies and the orbital energies in terms of the canonical orbitals, yield a unified molecular orbital picture of the chemical bonding of this interesting species.

Financial support comes from Basque Government (IT588-13) and the Spanish Office for Scientific Research (Grant No. CTQ2011-27374). The SGI/IZO-SGIker UPV/EHU (supported by the National Program for the Promotion of Human Resources within the National Plan of Scientific Research, Development and Innovation—Fondo Social Europeo and MCyT) is gratefully acknowledged for generous allocation of computational resources. J.M.M. would like to thank the Spanish Ministry of Science and Innovation for funding through a Ramón y Cajal fellow position (Grant No. RYC 2008-03216). Financial support from REA-FP7-IRSES TEMM1P (GA 295172) is also acknowledged.