A Distance-Dependent Parameterization of the Extended Hubbard Model for
Conjugated and Aromatic Hydrocarbons Derived From Stretched Ethene
Thomas G. Schmalza,*, Luis Serrano-Andrésb, Vicenta Saurib, Manuela Merchánb,
Josep M. Olivac,*
a
Texas A&M University at Galveston, P. O. Box 1675, Galveston, TX 77553, USA
b
Instituto de Ciencia Molecular, Universidad de Valencia,
P.O. Box 22085, ES-46071 Valencia, Spain
c
Instituto de Química-Física “Rocasolano”, CSIC, Serrano 119, ES-28006, Madrid,
Spain
Supplementary Information
* To whom correspondence should be addressed: schmaltz@tamug.edu, J.M.Oliva@iqfr.csic.es
1
Table 1: Vertical transition energies (eV) and other properties of the singlet, triplet and
Rydberg states of ethene (D2h), obtained at the gas-phase equilibrium geometry.1
a
b
c
e
-
0.96
11.6
11.6
f
8.03
0.95
29.7
20.7
0.30
g
4.44
0.95
11.7
11.7
-
f
9.33
0.96
93.0
102.0
0.02
-
11B1u (V)
7.98
13B1u (T)
4.39
21B1u (3d)
9.40
MS-RASPT2
x2 
d
x2 
11A1g (N)
a
a

CASPT2
f
MS-RASPT2(12,2,2;4,6,8)(SD), using the notation from Ref. 2. Twelve electrons correlated into four  MOs in
RAS1, (*) valence MOs, (*) CC bond MOs and two more b2g MOs, being one of them the 3dxz Rydberg
orbital, in RAS2, and four * plus four additional MOs to avoid intruder states in RAS3. Up to two holes/particles
allowed in RAS1/RAS3.
b
The weight of the RASSCF reference function in the first order wave function.
c
RASSCF expectation values of x2 (in au2), where x is the coordinate perpendicular to the molecular plane.
d
MS-RASPT2 expectation values of x2 (in au2), where x is the coordinate perpendicular to the molecular plane.
e
MS-RASPT2 oscillator strengths.
f
MS-CASPT2(2,3)/ANO 5s4p2d/3s2p plus a set of 1s1p1d diffuse functions in the center of the molecule at the gas-
phase equilibrium geometry. See revision of data in Ref. 3.
g
CASPT2(2,11)/ANO 5s4p2d/3s2p plus a set of even-tempered 2s2p1d functions on C atoms. See therein Ref. 4.
2
Table 2: Vertical transition energies (eV) and other properties of the singlet, triplet and
Rydberg states of ethene (D2h), obtained at the MP2/cc-pVTZ optimized ground-state
equilibrium geometry.
CASPT2
11A1g (N)
f
7.98
13B1u (T)
4.39
21B1u (3d)
g
8.62
h
h
4.60
9.40
f
a
b
c
e
x2 
-
0.96
11.5
11.5
i
8.07
0.95
30.0
20.8
0.30
i
4.51
0.95
11.7
11.7
-
9.37
0.96
92.9
102.1
0.02
7.84
4.48
x2 
d

MS-RASPT2
-
11B1u (V)
a
a
f
MS-RASPT2(12,2,2;4,6,8)(SD), using the notation from Ref. 2. Twelve electrons correlated into four  MOs in
RAS1, (*) valence MOs, (*) CC bond MOs and two more b2g MOs, being one of them the 3dxz Rydberg
orbital, in RAS2, and four * plus four additional MOs to avoid intruder states in RAS3. Up to two holes/particles
allowed in RAS1/RAS3.
b
The weight of the RASSCF reference function in the first order of perturbation wave function.
c
RASSCF expectation values of x2 (in au2), where x is the coordinate perpendicular to the molecular plane.
d
MS-RASPT2 expectation values of x2 (in au2), where x is the coordinate perpendicular to the molecular plane.
e
MS-RASPT2 oscillator strengths.
f
MS-CASPT2(2,3)/ANO 5s4p2d/3s2p plus a set of 1s1p1d diffuse functions in the center of the molecule at the gas-
phase equilibrium geometry. See revision of data in Ref. 3.
g
h
i
CASPT2(2,11)/ANO 5s4p2d/3s2p plus a set of even-tempered 2s2p1d functions on C atoms. See therein Ref. 4.
CASPT2(2,2)/TZVP at the MP2/6-31G* optimized ground-state equilibrium geometry. See Ref. 5.
CASPT2(8,10)/aug-cc-pTZV at the MP2/6-31G* optimized ground-state equilibrium geometry. See Ref. 6.
3
Table 3: Vertical transition energies (eV) and other properties of the singlet, triplet and
Rydberg states of trans-1,3-butadiene (C2h), obtained at the MP2/cc-pVTZ optimized
ground-state equilibrium geometry.
CASPT2
11Ag
a
f
g
f
g
f
g
f
g
6.23 6.47
21Ag (V)
6.27 6.63
13Bu
3.20 3.34
13Ag
4.89 5.16
c
21.1
21.0
6.38
i
5.97
0.80
38.7
26.1
0.67
6.43
i
6.57
0.82
38.2
27.5
-
3.30
i
3.33
0.82
21.5
21.5
-
i
5.12
0.83
22.1
21.9
-
f
f
h
7.67
0.83
57.4
67.8
-
f
h
7.32
0.83
79.1
91.7
0.03
7.47 7.48
21Bu (3p)
6.70 7.07
MS-RASPT2(4,0,1;0,4,9)(S), using the notation from Ref. 2. RAS1 empty, RAS2 * valence MOs, and RAS3
including nine (n=3) Rydberg MOs. A single particle (S) allowed in RAS3.
b
The weight of the RASSCF reference function in the first order of perturbation wave function.
c
RASSCF expectation values of z2 (in au2), where z is the coordinate perpendicular to the molecular plane.
d
MS-RASPT2 expectation values of z2 (in au2), where z is the coordinate perpendicular to the molecular plane.
e
MS-RASPT2 oscillator strengths.
f
CASPT2(4,8)/ANO 6s3p1d/2s1p plus a set of even-tempered 2p functions on C atoms. See therein Ref. 4.
g
e
0.83
5.08
x2 
d
-
31Ag (3d)
a
b
x2 
-
11Bu (V)
a

MS-RASPT2
MS-CASPT2(4,4)/TZVP at the MP2/6-31G* optimized ground-state equilibrium geometry. See Ref. 5.
h
Experimental data. See therein Ref. 5.
i
CASPT2(4,8)/aug-cc-pTZV at the MP2/6-31G* optimized ground-state equilibrium geometry. See Ref. 6.
4
Table 4: Vertical transition energies (eV) and other properties of the singlet, triplet and
Rydberg states of trans-1,3,5-hexatriene (C2h), obtained at the MP2/cc-pVTZ optimized
ground-state equilibrium geometry.
CASPT2
11Ag
f
g
f
g
f
g
f
g
21Ag (V)
5.20 5.42
13Bu
2.55 2.71
13Ag
4.12 4.31
23Bu
a
f
5.01 (5.68)
0.74
46.7
34.7
0.96
h
5.37
0.76
31.8
31.5
-
h
2.68
0.77
31.4
31.4
-
h
4.31
0.77
31.3
31.3
-
5.27
0.77
32.0
32.0
-
6.51
0.77
94.7
95.0
-
5.33
2.67
4.22
i
e
31.2
j
x2 
d
31.2
h
6.24 6.22
c
0.77
5.18
-
b
x2 
-
5.01 5.31
a

MS-RASPT2
-
11Bu (V)
31Ag (3p)
a
f
MS-RASPT2(6,0,1;0,4,9)(S), using the notation from Ref. 2. RAS1 empty, RAS2 * valence MOs, and RAS3
including nine (n=3) Rydberg MOs. A single particle (S) allowed in RAS3.
b
The weight of the RASSCF reference function in the first order of perturbation wave function.
c
RASSCF expectation values of z2 (in au2), where z is the coordinate perpendicular to the molecular plane.
d
MS-RASPT2 expectation values of z2 (in au2), where z is the coordinate perpendicular to the molecular plane.
e
MS-RASPT2 oscillator strengths.
f
CASPT2(6,8)/ANO 6s3p1d/2s1p plus a set of even-tempered 2p functions on C atoms. See therein Ref. 4.
g
MS-CASPT2(6,6)/TZVP at the MP2/6-31G* optimized ground-state equilibrium geometry. See Ref. 5.
h
CASPT2(6,11)/aug-cc-pTZV at the MP2/6-31G* optimized ground-state equilibrium geometry. See Ref. 6.
i
Experimental data. See therein Ref. 5.
j
Present RASPT2 result.
5
Table 5: Vertical transition energies (eV) and other properties of the singlet, triplet and
Rydberg states of benzene (D6h), obtained at the MP2/ cc-pVTZ optimized ground-state
equilibrium geometry.
CASPT2
11Ag
f
g
f
g
f
g
f
g
f
g
f
g
4.84 5.05
11B1u
6.30 6.45
11E1u
7.03 7.07
13B1u
3.89 4.17
13E1u
4.49 4.90
13B2u
5.49 5.76
a
f
c
e
0.78
29.7
29.9
h
4.91
0.77
29.8
29.8
0.00005
h
6.15
0.75
30.9
30.9
0.0008
h
6.86
0.72
31.5
32.0
0.89
h
4.09
0.78
29.4
29.4
-
h
4.70
0.77
29.7
29.6
-
h
5.40
0.75
30.2
30.0
-
7.43
0.78
92.0
92.0
0.00012
7.36
4.13
4.76
5.72
z2
d
-
6.57
i
b
z2
4.96
7.16 7.41
a

MS-RASPT2
-
11B2u
21E1u (3p)
a
f
MS-RASPT2(6,0,1;0,4,9)(S), using the notation from Ref. 2. RAS1 empty, RAS2 * valence MOs, and RAS3
including nine (n=3) Rydberg MOs. A single particle (S) allowed in RAS3.
b
The weight of the RASSCF reference function in the first order of perturbation wave function.
c
RASSCF expectation values of z2 (in au2), where z is the coordinate perpendicular to the molecular plane.
d
MS-RASPT2 expectation values of z2 (in au2), where z is the coordinate perpendicular to the molecular plane.
e
MS-RASPT2 oscillator strengths.
f
CASPT2(6,13) for singlet and CASPT2(6,12) for triplet states/ANO 4s3p2d/3s2p plus a set of 1s1p1d diffuse
functions in the center of the molecule. See therein Ref. 7.
g
MS-CASPT2(6,6)/TZVP at the MP2/6-31G* optimized ground-state equilibrium geometry. See Ref. 5.
h
CASPT2(6,13)/aug-cc-pTZV at the MP2/6-31G* optimized ground-state equilibrium geometry. See Ref. 6.
i
Experimental data. See therein Ref. 5.
6
Additional notes:
(Ethene) As discussed in section III, the V state is well-described by MS-RASPT2, solving the
valence-Rydberg mixing in the multi-configurational reference wave function pointed out in
previous works.2,3
On the other hand, the present multi-state result based on large scale
RASPT2 wave functions does not give support to the recent estimate of the vertical transition
energy for the V state obtained at 8.62 eV.8 Most probably the one-electron basis set employed
was not diffuse enough to describe properly the 3dxz Rydberg-type orbital at the CASPT2 level,
and the computation was performed just with the two  MOs active.8 Indeed, in the same study,
the 11B1u state was estimated at 7.83 eV when the active space is enlarged and the one-electron
basis set is extended, which is approximately equal to the 7.8 eV found by Davidson.9
(Hexatriene and Butadiene) It is worth noticing that the valence-Rydberg mixing for hexatriene
was not found in the early CASPT2 study4 unlike in the present work. For the 21Ag state of
butadiene, the MS-RASPT2 transition energy is estimated at 6.57 eV, near 6.55 eV from an
extensive MR-CI study10 but overestimated regarding the early CASPT2 result of 6.27 eV.4 In
hexatriene, the 21Ag state is found at 5.37 eV at MS-RASPT2 level which seems to be
acceptable when comparing to the early CASPT2 value of 5.20 eV.4 But the values given by
Schreiber et al.8 seem to be inaccurate at MS-CASPT2 because of a lack of diffuseness in the
one-electron basis set and missing Rydberg orbitals in the active space. In a recent work,6 the
singlet-state energy values come closer to the present results both in butadiene and hexatriene.
7
REFERENCES
1. G. Herzberg, Molecular Spectra and Molecular Structure III. Electronic Spectra
and Electronic Structure of Polyatomic Molecules (D. Van Nostrand Company, Inc.,
Princeton, New Jersey, 1967).
2. V. Sauri, L. Serrano-Andrés, A. R. M. Shahi, L. Gagliardi, S. Vancoillie, K. Pierloot,
J. Chem. Theory Comput. 7, 153 (2011).
3. J. Finley, P. A. Malmqvist, B. O. Roos, L. Serrano-Andres, Chem. Phys. Lett. 288,
299 (1998).
4. L. Serrano-Andrés, M. Merchán, I. Nebotgil, R. Lindh, B. O. Roos, J. Chem. Phys.
98, 3151 (1993).
5. D. F. Evans, J. Chem. Soc. 1735 (1960).
6. M. R. Silva, M. Schreiber, S. P. A. Sauer, W. Thiel, J. Chem. Phys. 133, 174318
(2010).
7. J. Lorentzon, P. A. Malmqvist, M. Fulscher, B. O. Roos, Theor. Chim. Acta, 91, 91
(1995).
8. M. Schreiber, M. R. Silva-Junior, S. P. A. Sauer, and W. Thiel, J. Chem. Phys. 128, 134110
(2008)
9. E. R. Davidson and A. A. Jarzecki, Chem. Phys. Lett. 285, 155 (1998).
10. M. Dallos and H. Lischka, Theor. Chem. Acc. 112, 16 (2004).
8