Wiener index
The Wiener index - named after Harry Wiener - is the oldest topological descriptor that maps the structure of a molecule in a number.
formulation
- W: Wiener index
- N: number of non-hydrogen atoms in the structure
- d ij : Number of bonds on the shortest path between atoms i and j
The factor 1 / 2 states that every path is received only once in the index.
use
The Wiener index is used in the methods of the quantitative structure-activity relationship (QSPR) to correlate substance properties such as the saturation vapor pressure with the molecular structure . Since the Wiener index does not differentiate between different atoms, it can only be used meaningfully within a homologous series .
Sample calculation
The Wiener index for 3-ethylhexane is W = 72. It is the sum of the distances
- 1-2 (1), 1-3 (2), 1-4 (3), 1-5 (4), 1-6 (5), 1-7 (3), 1-8 (4),
- 2-3 (1), 2-4 (2), 2-5 (3), 2-6 (4), 2-7 (2), 2-8 (3),
- 3-4 (1), 3-5 (2), 3-6 (3), 3-7 (1), 3-8 (2),
- 4-5 (1), 4-6 (2), 4-7 (2), 4-8 (3),
- 5-6 (1), 5-7 (3), 5-8 (4),
- 6-7 (4), 6-8 (5),
- 7-8 (1)
- → 1 + 2 + 3 + 4 + 5 + 3 + 4 + 1 + 2 + 3 + 4 + 2 + 3 + 1 + 2 + 3 + 1 + 2 + 1 + 2 + 2 + 3 + 1 + 3 + 4 + 4 + 5 +1 = 72.
Sample values
material | Wiener index |
---|---|
n-hexane | 35 |
2-methylpentane | 32 |
3-methylpentane | 31 |
2,3-dimethylbutane | 29 |
These four examples with the identical empirical formula (C 6 H 14 ) make it clear that the Wiener index is highest without branching and that it becomes smaller with increasing branching in the molecular structure; with the same number of branches, it becomes smaller with increasing molecular symmetry .
Calculation of the maximum values for W
The maximum Wiener index applicable to the unbranched molecule can easily be calculated from the total number of "non-H" atoms (n) as follows:
- It is always a sum of square numbers:
- If n is even, the squares of the odd numbers between 0 and n are summed up.
- If n is odd, the squares of the even numbers between 0 and n are summed.
examples for
- n = 6: 1 2 + 3 2 + 5 2 = 35
- n = 13: 2 2 + 4 2 + 6 2 + 8 2 + 10 2 + 12 2 = 364
One can use as a simplified calculation formula.
Table of maximum values for W
up to n = 21:
n | 2 | 3 | 4th | 5 | 6th | 7th | 8th | 9 | 10 | 11 | 12 | 13 | 14th | 15th | 16 | 17th | 18th | 19th | 20th | 21st |
(n-1) n = odd | 1 | 3 | 5 | 7th | 9 | 11 | 13 | 15th | 17th | 19th | ||||||||||
(n-1) 2 | 1 | 9 | 25th | 49 | 81 | 121 | 169 | 225 | 289 | 361 | ||||||||||
Sum = W | 1 | 10 | 35 | 84 | 165 | 286 | 455 | 680 | 969 | 1330 | ||||||||||
(n-1) n = even | 2 | 4th | 6th | 8th | 10 | 12 | 14th | 16 | 18th | 20th | ||||||||||
(n-1) 2 | 4th | 16 | 36 | 64 | 100 | 144 | 196 | 256 | 324 | 400 | ||||||||||
Sum = W | 4th | 20th | 56 | 120 | 220 | 364 | 560 | 816 | 1140 | 1540 |
literature
- ↑ Harry Wiener (1947): Structural determination of paraffin boiling points . J. Am. Chem. Soc., 69, pp. 17-20. doi : 10.1021 / ja01193a005