In the HuLiS application, the characteristic polynomial P n ðxÞ of any molecule is
computed using the Balasubramanian algorithm [38]:
P n ðxÞ ¼
X n
k¼0
d k x
nÀk
ð13:25Þ
d k ¼
1
k
trðB kÀ1 Þ; d 0 ¼ À1
ð13:26Þ
B k ¼ BAðB kÀ1 À d k IÞ; B 0 ¼ A
ð13:27Þ
with A the topological matrix of the molecule (13.14) and I the identity matrix both
of n  n dimension. The n  n B k matrices and the d k coefficients are completely
defined in the recursive algorithm. The roots of P n ðxÞ are the energies of the
molecular orbitals of the molecule.
The topological resonance energy can be easily computed for one ring species:
the acyclic polynomial is the average of the Hückel and the Möbius system. As an
example, let us compare the topological resonance energies and the Breslow resonance energies, defined as the difference in energy between the cyclic and the open
molecule [39], Fig. 13.8. The agreement between both definitions is excellent.
Discussion about the status of different resonance energies is beyond the scope
of this work. The interested reader will find more information in [39, 40].
For polycyclic systems, one follows Ref. [41] and as an application, we can
compute the TRE of 1,4-Biphenylenedione. The decomposition shown in
Table 13.5 is easily done in HuLiS as a polynomial calculator is available in the
software. This calculator allows any linear combination of characteristic polynomials and searches for their roots. The calculation of TRE is thus at hand for any
molecule which HuLiS is able to treat.
References
1. Carissan Y, Hagebaum-Reignier D, Goudard N, Humbel S (2008) J Phys Chem A 112
(50):13256
2. Shaik SS, Hiberty PC (2008) A chemist’s guide to valence bond theory. Wiley-Interscience,
Hoboken
3. Cooper DL (ed) (2002) Valence bond theory. Elsevier Science
4. Chirgwin BH, Coulson CA (1950) Proc R Soc Lond Math Phys Eng Sci 201(1065):196
5. Jensen F (2006) Introduction to computational chemistry. Wiley
6. Deglmann P, Schäfer A, Lennartz C (2015) Int J Quantum Chem 115(3):107
7. Hirao K, Nakano H, Nakayama K, Dupuis M (1996) J Chem Phys 105(20):9227
8. Thorsteinsson T, Cooper D, Gerratt J, Karadakov P, Raimondi M (1996) Theor Chim Acta 93
(6):343
9. Glendening ED, Weinhold F (1998) J Comput Chem 19(6):593
10. Bader RF (1994) Atoms in molecules: a quantum theory. Oxford University Press, Oxford
11. Shaik S, Danovich D, Silvi B, Lauvergnat D, Hiberty P (2005) Chem Eur J 11(21):6358
13 Localized Structures at the Hückel Level …
359
computed using the Balasubramanian algorithm [38]:
P n ðxÞ ¼
X n
k¼0
d k x
nÀk
ð13:25Þ
d k ¼
1
k
trðB kÀ1 Þ; d 0 ¼ À1
ð13:26Þ
B k ¼ BAðB kÀ1 À d k IÞ; B 0 ¼ A
ð13:27Þ
with A the topological matrix of the molecule (13.14) and I the identity matrix both
of n  n dimension. The n  n B k matrices and the d k coefficients are completely
defined in the recursive algorithm. The roots of P n ðxÞ are the energies of the
molecular orbitals of the molecule.
The topological resonance energy can be easily computed for one ring species:
the acyclic polynomial is the average of the Hückel and the Möbius system. As an
example, let us compare the topological resonance energies and the Breslow resonance energies, defined as the difference in energy between the cyclic and the open
molecule [39], Fig. 13.8. The agreement between both definitions is excellent.
Discussion about the status of different resonance energies is beyond the scope
of this work. The interested reader will find more information in [39, 40].
For polycyclic systems, one follows Ref. [41] and as an application, we can
compute the TRE of 1,4-Biphenylenedione. The decomposition shown in
Table 13.5 is easily done in HuLiS as a polynomial calculator is available in the
software. This calculator allows any linear combination of characteristic polynomials and searches for their roots. The calculation of TRE is thus at hand for any
molecule which HuLiS is able to treat.
References
1. Carissan Y, Hagebaum-Reignier D, Goudard N, Humbel S (2008) J Phys Chem A 112
(50):13256
2. Shaik SS, Hiberty PC (2008) A chemist’s guide to valence bond theory. Wiley-Interscience,
Hoboken
3. Cooper DL (ed) (2002) Valence bond theory. Elsevier Science
4. Chirgwin BH, Coulson CA (1950) Proc R Soc Lond Math Phys Eng Sci 201(1065):196
5. Jensen F (2006) Introduction to computational chemistry. Wiley
6. Deglmann P, Schäfer A, Lennartz C (2015) Int J Quantum Chem 115(3):107
7. Hirao K, Nakano H, Nakayama K, Dupuis M (1996) J Chem Phys 105(20):9227
8. Thorsteinsson T, Cooper D, Gerratt J, Karadakov P, Raimondi M (1996) Theor Chim Acta 93
(6):343
9. Glendening ED, Weinhold F (1998) J Comput Chem 19(6):593
10. Bader RF (1994) Atoms in molecules: a quantum theory. Oxford University Press, Oxford
11. Shaik S, Danovich D, Silvi B, Lauvergnat D, Hiberty P (2005) Chem Eur J 11(21):6358
13 Localized Structures at the Hückel Level …
359