The macroscopic preparation of C60 has raised interesting questions concerning the mechanism of formation of spheroidal carbon clusters. C20 is the smallest carbon cluster which can exist as a fullerene cage; however, it is not known whether this structure is stable. There is a big surprise regarding the ground state geometry of C20: different methods give vastly different results!
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Above are shown the three candidates for the ground state geometry: RING (``1-D''), BOWL (``2-D'') and CAGE (``3-D''). Below each geometry is an electron density plot calculated within the Hartree-Fock approximation.
The striking feature about this problem is the enormous discrepancy between the various theoretical methods. Below is a graph which shows the relative energies of these three clusters for 4 different methods.
Our DMC calculations which were done with HF geometries have been confirmed by others recently, see Murphy and Friesner, Chem. Phys. Lett. 288, 403 (1998) and J.B. Anderson, to be published. For our DMC calculations, we used the following HF optimized geometries (from Ref. 1 in the paper):
[all coordinates in atomic units]
C20 Ring
-4.58917 6.31644 .00000 -7.42543 2.41267 .00000 -7.42543 -2.41267 .00000 -4.58917 -6.31644 .00000 .00000 -7.80755 .00000 4.58917 -6.31644 .00000 7.42543 -2.41267 .00000 7.42543 2.41267 .00000 4.58917 6.31644 .00000 .00000 7.80755 .00000 -6.16793 4.69866 .00000 -7.75176 .17588 .00000 -6.37469 -4.41408 .00000 -2.56270 -7.31801 .00000 2.22815 -7.42671 .00000 6.16793 -4.69866 .00000 7.75176 -.17588 .00000 6.37469 4.41408 .00000 2.56270 7.31801 .00000 -2.22815 7.42671 .00000
C20 Bowl
-2.18472 .70986 .82723 -1.35023 -1.85843 .82723 1.35023 -1.85843 .82723 2.18472 .70986 .82723 .00000 2.29715 .82723 -4.57900 1.48781 .03468 -2.82998 -3.89513 .03468 2.82998 -3.89513 .03468 4.57900 1.48781 .03468 .00000 4.81464 .03468 -6.01115 -.74342 -.43095 -1.15051 -5.94668 -.43095 5.30010 -2.93182 -.43095 4.42615 4.13471 -.43095 -2.56459 5.48722 -.43095 -4.42615 4.13471 -.43095 -5.30010 -2.93182 -.43095 1.15051 -5.94668 -.43095 6.01115 -.74342 -.43095 2.56459 5.48722 -.43095
C20 Cage
-.54030 -4.14154 -.03308 .54030 4.14154 -.03308 .86970 -2.98710 -2.21294 -.86970 2.98710 -2.21294 -2.73294 -2.54094 -.02746 2.73294 2.54094 -.02746 .80571 -2.95165 2.03893 -.80571 2.95165 2.03893 -2.84810 -.88895 2.22837 2.84810 .88895 2.22837 3.10967 -1.73783 1.37210 -3.10967 1.73783 1.37210 -.61797 -1.25317 -3.48965 .61797 1.25317 -3.48965 -.68634 -1.23650 3.60084 .68634 1.23650 3.60084 3.00377 -1.58399 -1.29065 -3.00377 1.58399 -1.29065 -3.00377 -1.01534 -2.18647 3.00377 1.01534 -2.18647
We have used the SBK pseudopotentials (J. Chem. Phys. 81, 6026 (1984)) in all calculations. Our optimized basis sets, the pseudopotential equivalents of 6-311G*, are given below:
CARBON:
L 5
1 15.4 -0.009352 0.018200
2 3.486 -0.161452 0.106082
3 1.046 0.123026 0.312724
4 0.3447 0.685931 0.493046
5 0.1128 0.326789 0.307014
L 1
1 0.2718 1.0 1.0
L 1
1 0.1213 1.0 1.0
D 1
1 0.8582 1.0
HYDROGEN:
S 5
1 33.865000 0.006068
2 5.094790 0.045316
3 1.158790 0.202845
4 0.325840 0.503711
5 0.102741 0.383567
S 1
1 0.325840
S 1
1 0.102741
P 1
1 0.757000
Calculated total energies [a.u.] for the C20 isomers are listed here:
|
RING BOWL CAGE |
Hartree-Fock -109.4326 -109.3724 -109.2681 |
DMC -112.67(1) -112.71(1) -112.63(1) |
Total Energy of the Carbon Atom:
HF: -5.3160
DMC: -5.4074
© J.C. Grossman 1995