All the calculations are operated under qusntum chemistry software Q-Chem [19] [20].
The fictitious temperature θ for TAO-LDA used in all calculations is set to be 7 mHatrees suggested by founder [11] which is the largest fictious temperature θ performs similiarly as KS-DFT in single-reference system.
The singlet and triplet state geometries have been previously optimized at spin-unrestricted TAO-LDA(θ = 7 mHartree)/6-31G* level with their repective multiplicity. The singlet-triplet energy gap (ST gap) is defined as
E
ST = ET− E
S.
(2.15)The ionization potential and electron affinity are calculated at the corresponding sin-glet geometry optimized by TAO-LDA(θ = 7 mHartree)/6-31G*,which are defined repec-tively:
IP = E(N − 1) − E(N),
(2.16)and
EA = E(N ) − E(N + 1),
(2.17)where the E(N) is the energy of N-electron system. The fundamental gap is the difference between IP and EA:
F G = IP − EA = E(N − 1) − 2E(N) + E(N + 1).
(2.18)The source of initial geometry coordinates are described in section 3.2. The geometry of 342 isomers has been visulized by Avigadro 1.0.3 in Appendix B.
Chapter 3 Fullerenes
In 1996, the Nobel Prize in chemistry was awarded to Curl, Kroto and Smalley to praise their discover of the first fullerene molecule C60, which is a brand-new carbon al-lotrope except for graphite and diamond. This new-finding alal-lotrope has distinct, unique and fantastic electronic structure, chemical property, and optical property, which makes fullerenes and their derivatives like hetero fullerenes and endohedrofullerene a hot topic to study. This novel material have wide application in many subjects and ways, for in-stances, solar cells, drug delivery, superconductors, tumor research, hydrogen storage ma-terials [21--27].
3.1 Fullerenes
Fullerenes may exist in nature, detected from Planetary Nebula [28], and in artifact , detected from ancient Chinese ink sticks and Japanes Sumi [1] for a long time. Neverthe-less, Fullerenes haven't been found until modern times. In 1985, Kroto, Heath, O'Brien, Curl, and Smalley found the first fullerene C60 by using laser on vaporized graphite [2].
This new molecule is in shape of truncated icosahedron, composed of twelve perfect pen-tagonal faces and perfect hexagonal faces, which can be yielded from cutting off the icosa-hedron's twelve vertices in the one-third of each sides. Kroto et al named this new molecule by Buckminsterfullerene in honor of architect Richard Buckminster Fuller who inspired them the structure of C60. In 1987, Kroto named all family of carbon-based molecule
with properties of closed shape and composed of twelve pentagonal faces and any number of hexagonal faces as fullerene [29]. In 1990, W. Krätschmer and Donald R. Huffman find a new method to synthesize pure isolatable solid-state C60, which have opened a door to make further investigation of physical and chemical properties of fullerenes in experi-ment [30].
In this thesis, we focus on fullerenes( named as classical fullerenes in some litera-tures [31] [32]) discribed as a polyhedral carbon cage, which consistes of three-fold net-work, and composed of only pentagonal and hexgonal faces [33] [32]. The fullerene ver-tices, edges and faces must obey the Euler's polyhedron formula:
V + F − E = 2,
(3.1)where V is number of vertices, and F is number of faces, and E is the number of edges.
By definition of fullerenes, number of faces is composed of solely pentagonal faces and hexgonal faces, and each vertices is connecting to another three vertices, in mathematical language are
F = N
5+ N6,
(3.2)and
3V = 2E, (3.3)
where N5is number of pentagonal faces and N6is number of hexgonal faces. Acording to eqution 3.1, 3.2 and 3.2, fullerenes Cncan form only with n be an arbitrary even number from 20 and the number of pentagonal faces and hexagonal faces are respectively:
N
5 = 12 (3.4)and
N
6 =V
2
− 10.
(3.5)Compared to the same sp2 hybridization carbon atoms in flat graphite, the cage-like fullerenes with nonzero curvature are not in the minimun energy shape. Hence it is a
com-mon wisdom that the isolated-pentagon rule (IPR) holds water for most fullerenes, which means fullerenes without neighboring pentagonal faces are more stable than fullerenes with neighboring pentagonal faces. It is also a good indication to [3] suggest the more stable fullerene structures by isolated pentagon rule.
The number of different isomers of specific Cncan be solved practically when n is not large by ring spiral algorithm mentioned in papers proposed by D. E. Manolopoulos and coworkers [34].And the number of Cnfullerene isomers are listed in P. W. Fowler and D.
E. Manolopoulos' publication [3] or the data is on website [35]. The table 3.1 list the enumeration of fullerene isomers extracted from ref [3].
Table 3.1: The number of fullerene isomers from C20to C46. n Fullerene
20 1
22 0
24 1
26 1
28 2
30 3
32 6
34 6
36 15
38 17
40 40
42 45
44 89
46 116
The first fullerene of no adjacent pentagonal faces are C60, and next is C70. With smaller the n of Cn, the greater the proportion of adjacent pentagonal faces is. Down to the C20, all the twelve faces are pentagons. And the highly curvature gives a hint of highly reactivity and unstablity of small fullerenes [3]. Small fullerenes ( Cn,n is even number between 20 and 46 except for 22. ) occure in the production of larger fullerenes [36], but still difficult to isolate due to its highly curvature and reactivity except for C20 and C36[4] [5]. Because there are only a few experimental data of small fullerenes, the hope to understand their electronic properties turn to computational studies.
However, the strongly multi-referece character of small fullerenes lead to the
difi-ciency of describing small fullerenes by single-reference methods which is not capable to capture strongly static correlation [6--10]. The cost of high level ab initio method are limited to single or imcomplete fullerene isomers. And methods employed are rarely good enough to capture the static correlation of small fullerenes such that the electronic prop-erties are doubted.