Cn (n=2-4): current status

Abstract

The major aspects of the C₂, C₃ and C₄ elemental carbon clusters are surveyed. For C₂, a brief analysis of its current status is presented. Regarding C₃, the most recent results obtained in our group are reviewed with emphasis on modelling its potential energy surface which is particularly complicated due to the presence of multiple conical intersections. As for C₄, the most stable isomeric forms of both triplet and singlet spin states and their possible interconversion pathways are examined afresh by means of accurate ab initio calculations. The main strategies for modelling the ground triplet C₄ potential are also discussed. Starting from a truncated cluster expansion and a previously reported DMBE form for C₃, an approximate four-body term is calibrated from the ab initio energies. The final six-dimensional global DMBE form so obtained reproduces all known topographical aspects while providing an accurate description of the C₄ linear-rhombic isomerization pathway. It is therefore commended for both spectroscopic and reaction dynamics studies.This article is part of the theme issue 'Modern theoretical chemistry'.

Keywords: C2; C3; C4; carbon clusters; double many-body expansion; many-body expansion; potential energy surfaces.

Figure 1.

Optimized bending potentials for C₃ at the MRCI/AVTZ//FVCAS/AVTZ level of theory as a function of the angle. Shown in the key are the irreducible representations in C_2v symmetry for each state, and correlations within the C_s point group. Also shown are the associated correlations for linear geometries in symmetry, and corresponding dissociation limits. Cis are indicated by the cross symbol. Energies are given with respect to the global minimum of C₃ of symmetry. (Online version in colour.)

Figure 2.

MO diagram and valence electronic configuration of C₃ at equilateral triangular geometries. The ¹A₁ (in C_2v) component of the ¹E′ term (in D_3h) is schematically shown. MOs are labelled according to D_3h irreps with the corresponding correlations with C_2v point group in parentheses. Up and down arrows denote as usual α and β spin states. Natural orbitals obtained from a state-averaged FVCAS/AVTZ wave function are also shown. (Online version in colour.)

Figure 3.

Slater determinants arising from excitations in the e′² valence configuration of C₃ (figure 2). All MOs are labelled according to D_3h irreps with the corresponding C_2v correlations given in parentheses. Up and down arrows denote α and β spin states. (Online version in colour.)

Figure 4.

Optimized FVCAS/AVTZ bending potential for the four electronic states of C₃ arising from excitations in the e′² configuration (see equations (2.1)–(2.5)). Shown in the key are the irreps, in C_2v symmetry, for each state. Cis are indicated by the symbol ×. (Online version in colour.)

Figure 5.

PES cuts along the bending coordinate Q₃ (for Q₂=0 and Q₁=4.548a₀) for the three lowest ¹A′ electronic states of C₃ over the range of −0.020a₀≤ Q₃≥0.020a₀. Symbols indicate MRCI/AVTZ points, and the solid lines the associated eigenvalues of equation (2.6) as obtained from the least-squares fitting procedure. (Online version in colour.)

Figure 6.

Contour plot of the lowest eigenvalues of the JT+PJT vibronic Hamiltonian of C₃ (equation (2.6)) for Q₁=4.548a₀; (a) 1 ¹A′; (b) 2 ¹A′; (c) 3 ¹A′. Contours in (a) are equally spaced by −0.0300 mE_h starting at 0.0000 mE_h, with values of 0.0168 mE_h and −0.2000 mE_h for (b), and 0.0500 mE_h and 11.7224 mE_h for (c). Cis are indicated by crosses (×). (Online version in colour.)

Figure 7.

Adiabatic evolution of the 1 ¹A′, 2 ¹A′ and 3 ¹A′ wave functions of C₃ along a closed circuit C in the two-dimensional branching plane encircling one or more Cis; c₀ is the coefficient of the leading determinant (CSF) of the FVCAS/AVTZ wave function for each state. Four closed loops are considered that encircle: (a) the central (D_3h) Ci; (b) one C_2v Ci; (c) two Cis; (d) four Cis. See also text. (Online version in colour.)

Figure 8.

(a) Relaxed triangular plot in hyperspherical coordinates [171] depicting the location and symmetry of all stationary points of the DMBE I PES for ground-state C₃. Solid black lines are equally spaced by 0.005 E_h, starting at −0.2904 E_h. Dashed red and blue lines are equally spaced by 0.001 E_h, starting at 0.00067 E_h and −0.0206E_h, respectively. (b) Optimized reaction path along C_2v arrangements. The plot starts at , which corresponds to the C+C₂ limit and leads to the linear global minimum at via the Ci region. The insert shows an amplified view of the nuclear configuration space illustrating the correct cusped behaviour at the D_3h and C_2v crossing seams. The dots indicate fitted points, with the lowest curve shifted by −0.0025 E_h for visibility. Indicated by DMBE JT is the PES without the V _I,EHF′′⁽³⁾(R) contributions in table 3. (Online version in colour.)

Figure 9.

Evolution of D_3h and C_2v crossing seams versus Q₁ as predicted from the DMBE II PES for ground-state C₃: (a) Q₁=4.000a₀; (b) Q₁= 4.520a₀; (c) Q₁=4.997a₀; (d) Q₁=5.750a₀. Contours are spaced as follows (from (a) to (d)): 0.025 mE_h, starting at −0.1710 E_h; 0.015 mE_h, starting at −0.2392 E_h; 0.012 mE_h, starting at −0.2081 E_h and 0.022 mE_h, starting at − 0.1048 E_h. Cis are indicated by crosses. (Online version in colour.)

Figure 10.

Relaxed triangular plot in hyperspherical coordinates [171] for C₃: (a) DMBE/ES/ABW [117]; (b) ABW [34]; (c) DMBE/ES/SS [117]; (d) SS [115]. Key for all global forms are as in figure 8. For local forms, contours are equally spaced by 0.005 E_h, starting at −0.2904 E_h. Navy dashed lines in panels (a) and (c) define cut-off energies (E₀) of equation (2.15). (Online version in colour.)

Figure 11.

Minimum energy paths for interconversion of linear and rhombic C₄ in both the triplet (blue online) and singlet (red) ground-state PESs. Energies are given with respect to the l- form and have been obtained at the ve-CASDC/CBS level of theory. Structural parameters and vibrational frequencies for each structure are in tables 6 and 7. (Online version in colour.)

Figure 12.

Molecular orbital diagrams and valence electronic configurations for (a). l- and (b). r-C₄(¹A_g). Molecular orbitals are labelled according to and D_2h irreps, respectively, with the corresponding correlations with D_2h and C_2v point groups given in parentheses. Up and down arrows denote α and β spin states, respectively. Natural orbitals obtained from FVCAS/AVTZ wave functions are also depicted. (Online version in colour.)

Figure 13.

ORC paths for interconversion of l- and r-C₄(³B_2g) via lr-C₄(³A′′) transition state. The inactive coordinate corresponds to the peripheral bond length of the r-C₄(³B_2g) structure (R), with all other degrees of freedom optimized at each grid point. Stationary structures at the CAS(8,12)/AVTZ, ve-CASDC/CBS and ve-CASDC/CBS//CAS(8,12)/AVTZ levels are shown by dots, diamonds and triangles, respectively. Energies are relative to l-. The MRCI(Q)-8-S and curves are discussed in §4b. (Online version in colour.)

Figure 14.

ORC paths for the r-C₄(¹A_g) and l-C₄(¹Δ_g) isomerization. (a) Interconversion of r-C₄(¹A_g) and d-C₄(¹A′). The inactive coordinate corresponds to the angle formed between peripheral and cross-ring bond lengths (α). (b) Interconversion of d-C₄(¹A′) and l-C₄(¹Δ_g) via the dl-C₄(¹A′) transition state. The inactive coordinate corresponds to the ring-opening angle β. Stationary structures at the CAS(8,12)/AVTZ level are indicated by solid dots. Energies are relative to l-. (Online version in colour.)

Figure 15.

(a) Energetics of the various asymptotic channels of l- shown in equation (3.5a). The atomization energy (at 0 K) for C₄ is taken from [42] with the zero-point vibrational energy retrieved from [184]. The corresponding experimental data for and C₂(a³Π_u) are from [35]. (b) Inter-particle coordinate system employed in the construction of the DMBE-(2+3) and DMBE-(2+3+4) PESs. (Online version in colour.)

Figure 16.

Partially relaxed (2.2≤R_e/a₀≤2.6) contour plot for insertion of a C atom into a C₃ triatomic molecule. (a) DMBE/ES/SS-(2+3) PES. (b) DMBE/ES/SS-(2+3+4) PES. Contours in panel (a) are equally spaced by 0.03 E_h, starting at −0.8370 E_h. In (b), they are 0.03 E_h, and −0.7127 E_h in the same order.

Figure 17.

As in figure 16, but for a partially relaxed (2.2≤R_e/a₀≤2.6) contour plot for the C atom moving around C₃ which lies along the x-axis with the origin fixed at the central carbon atom. Solid black lines are in panel (a) equally spaced by 0.01 E_h, starting at −0.8864 E_h. In panel (b), the corresponding values are 0.015 E_h and −0.7127 E_h. Dotted grey and dashed black lines are equally spaced by 0.005 and 0.0002 E_h, starting at −0.5624 and −0.5326 E_h, respectively. (Online version in colour.)

Figure 18.

Relaxed 1D cuts for dissociation of l- into and as obtained from DMBE/ES/SS-(2+3) and DMBE/ES/SS-(2+3+4). Also shown are the approximate four-body interaction energies obtained at the ve-CASDC/CBS level and the additional term . (Online version in colour.)

Figure 19.

Partially relaxed contour plot (2.2≤R_e/a₀≤2.6) of the DMBE/ES/SS-(2+3+4) PES for a C₂ fragment moving around another C₂ molecule which lies along the x-axis with the origin fixed at the centre of the C−C bond length. Contours are equally spaced by 0.015 E_h, starting at −0.7127 E_h. (Online version in colour.)

Figure 20.

Partially relaxed contour plot (2.4≤R_e/a₀≤3.4) for C_2v insertion of a C₂ fragment into another C₂ diatomic molecule as obtained from the DMBE/ES/SS-(2+3+4) PES. Contours are equally spaced by 0.015 E_h, starting at −0.6770 E_h. (Online version in colour.)