S1
Supporting information
Calculation of Converged Rovibrational Energies and Partition Function for Methane using VibrationalRotational Configuration Interaction
Arindam Chakraborty, Donald G. Truhlar
Department of Chemistry and Supercomputing Institute, University of Minnesota, Minneapolis, MN 55455-0431
Joel M. Bowman, and Stuart Carter
Cherry L. Emerson Center of Scientific Computation and Department of Chemistry, Emory University, Atlanta, GA 30322
Date of preparation of this supplementary information: April 18, 2004
Contents of the supporting information:
page
1. Comparison of the rovibrational energies of selected vibrational states
S2
2. Density of states at selective energies for J = 0 vibrational calculation
~
3. Convergence of Q with respect to J
S5
S7
~
4. Convergence of Q with respect to the change in rovibrational basis
S8
5. Symmetry labels for vibrational and rotational states of methane
S10
S2
1. Comparison of the rovibrational energies of selected vibrational states
Table S-1. Comparison of the minimum and the maximum rovibrational energy (in cm1) of selected vibrational states, computed
using various VCI bases at J = 15.a,b
715

a
4165
4390
5650
1 2 3 4
d0 c
min
0000
1
1248(2)
1248(9)
1248(2)
1248(8)
1248(3)
1248(8)
1248(3)
1248(8)
0100
2
2696(4)
2732(5)
2693(3)
2729(3)
2693(2)
2729(3)
2693(2)
2729(3)
0001
3
2437(6)
2588(1)
2433(6)
2585(3)
2433(7)
2585(5)
2433(7)
2585(5)
0002
6
3667(9)
4007(11)
3621(9)
3991(11)
3621(9)
3980(12)
3621(9)
3976(12)
0101
6
3905(5)
4243(13)
3867(5)
4200(12)
3867(8)
4200(12)
3866(8)
4200(12)
0003
10
4875(11)
5395(7)
4819(11)
5352(7)
4818(10)
5352(9)
4813(10)
5302(9)
0102
12
5087(9)
5512(9)
5062(10)
5465(9)
5050(10)
5465(9)
5048(10)
5464(9)
max
min
max
min
max
min
max
Calculations were performed using 3-mode representation with 15 Gauss-Hermite integration points and 12 harmonic oscillator
functions per mode.
b
c
The values of K are provided in the parentheses. They were obtained from the state with the largest CI coefficient.
The minimum and maximum energy levels were selected from 31 d 0 states (with M = 0) that have the indicated values of the four
vibrational quantum numbers.
S3
Table S-2. Comparison of the minimum and the maximum rovibrational energy (in cm1) of selected vibrational states, computed
using various VCI bases at J = 20.a,b
1 2 3 4

a
715
4165
4390
5650
d0 c
min
max
0000
1
2183(–5)
2183(7)
2182(–1)
2183(8)
2182(–2)
2183(9)
2182(–2)
2183(9)
0100
2
3633(1)
3695(–4)
3628(4)
3690(–7)
3628(2)
3690(–10)
3628(2)
3689(–10)
0001
3
3342(7)
3546(12)
3335(9)
3537(8)
3335(8)
3537(11)
3335(8)
3537(11)
0002
6
4534(–5)
4987(11)
4488(1)
4948(12)
4488(–8)
4948(12)
4488(–8)
4948(12)
0101
6
4640(8)
5222(12)
4737(7)
5186(7)
4737(9)
5186(15)
4728(9)
5181(15)
0003
10
5735(8)
6389(11)
5654(8)
6317(18)
5652(9)
6316(17)
5648(9)
6314(17)
0102
12
5952(10)
6513(12)
5885(10)
6530(15)
5884(10)
6462(17)
5791(10)
6459(17)
min
max
min
max
min
max
Calculations were performed using 3-mode representation with 15 Gauss-Hermite integration points and 12 harmonic oscillator
functions per mode.
b
c
The values of K are provided in the parentheses. They were obtained from the state with the largest CI coefficient.
The minimum and maximum energy levels were selected from 41 d 0 states (with M = 0) that have the indicated values of the four
vibrational quantum numbers.
S4
Table S-3. Comparison of the minimum and the maximum rovibrational energy (in cm1) of selected vibrational states, computed
using various VCI bases at J = 25.a,b
1 2 3 4

a
715
4165
4390
5650
d0 c
min
max
0000
1
3375(1)
3377(–6)
3375(2)
3376(–6)
3375(2)
3376(–7)
3375(2)
3376(–7)
0100
2
4827(6)
4917(19)
4651(6)
4911(17)
4651(7)
4911(19)
4651(7)
4911(19)
0001
3
4497(–4) 4741(14)
4489(–5)
4733(15)
4489(–6)
4733(15)
4489(–6)
4732(15)
0002
6
5656(4)
6248(16)
5611(7)
6175(17)
5611(6)
6175(19)
5610(6)
6175(19)
0101
6
5780(11) 6465(17)
5740(9)
6426(18)
5740(10)
6426(19)
5739(10)
6425(19)
0003
10
6833(12) 7498(19)
6742(10)
7542(22)
6741(11)
7537(22)
6737(11)
7518(22)
0102
12
6943(17) 7498(21)
6817(18)
7545(21)
6918(17)
7545(22)
6917(17)
7544(22)
min
max
min
max
min
max
Calculations were performed using 3-mode representation with 15 Gauss-Hermite integration points and 12 harmonic oscillator
functions per mode.
b
c
The values of K are provided in the parentheses. They were obtained from the state with the largest CI coefficient.
The minimum and maximum energy levels were selected from 51 d 0 states (with M = 0) that have the indicated values of the four
vibrational quantum numbers.
S5
2. Density of states at selective energies for J = 0 vibrational calculation
The density of state at energy E was calculated by dividing the total number of states in the range ( E   , E   ) by 2 . The results
shown below were calculated using two different values of  for J = 0 vibrational calculation with a VCI basis of 5650.
Table S-4. Density of states at specific energies for J = 0 vibrational calculations
Density of states ((cm1) 1)
E (cm1)
 = 30 cm1
 = 60 cm1
2552
0.05
0.05
2713
0.10
0.06
3816
0.03
0.03
3982
0.13
0.09
5430
0.35
0.26
5642
0.32
0.26
8423
0.75
0.77
S6
Figure caption
Fig. S-1. Number of states that are below a cutoff energy E cut for the J = 0 vibrational calculation with a VCI basis of 5650.
Number of energy levels
1200
1000
800
600
400
200
0
0
2000
4000
6000
Ecut(cm-1)
8000
10000
S7
~
3. Convergence of Q with respect to J
~
Table S-5. Convergence of Q with respect to J max .a
a
T (K)
J max  20
J max  30
J max  40
J max  50
100
117.4
117.4
117.4
117.4
200
329.3
329.3
329.3
329.3
300
608.9
609.0
609.2
609.0
400
967.4
968.2
968.2
968.2
500
1442
1448
1448
1448
600
2084
2109
2109
2109
700
2958
3032
3032
3032
800
4142
4319
4321
4321
900
5727
6097
6103
6103
1000
7895
8598
8616
8616
Calculations were performed with a VCI basis of 5650, and using 3-mode representation with 15 Gauss-Hermite integration points
and 12 harmonic oscillator functions per mode.
S8
~
4. Convergence of Q with respect to the change in rovibrational basis
~
Table S-6. Convergence of QJ with respect to the change in rovibrational basis for J = 5.a
a
N Vib
100 K
200 K
300 K
400 K
500 K
600 K
700 K
800 K
900K
1000 K
250
17.09
52.53
76.99
95.95
114.87
137.07
164.78
199.95
244.45
300.09
500
17.09
52.53
76.99
95.95
114.88
137.08
164.81
200.09
244.98
301.64
Calculations were performed using 3-mode representation with 15 Gauss-Hermite integration points and 12 harmonic oscillator
functions per mode.
~
Table S-7. Convergence of QJ with respect to the change in rovibrational basis for J = 11.a
a
N Vib
100 K
200 K
300 K
400 K
500 K
600 K
700 K
800 K
900K
1000 K
200
3.63  102
5.06
26.46
62.33
109.39
168.49
243.10
338.27
460.01
615.00
300
3.63  102
5.06
26.46
62.33
109.40
168.50
243.12
338.33
460.22
615.51
Calculations were performed using 3-mode representation with 15 Gauss-Hermite integration points and 12 harmonic oscillator
functions per mode.
S9
~
Table S-8. Convergence of QJ with respect to the change in rovibrational basis for J = 16.a
a
N Vib
100 K
200 K
300 K
400 K
500 K
600 K
700 K
800 K
900K
1000 K
100
2.13  106
0.06
1.66
9.38
27.82
60.78
112.48
188.20
294.19
437.11
200
2.13  106
0.06
1.66
9.39
27.84
60.84
112.71
189.02
296.73
443.88
Calculations were performed using 3-mode representation with 15 Gauss-Hermite integration points and 12 harmonic oscillator
functions per mode.
~
Table S-9. Convergence of QJ with respect to the change in rovibrational basis for J = 21.a
a
N Vib
100 K
200 K
300 K
400 K
500 K
600 K
700 K
800 K
900K
1000 K
100
2.47  1012
0.000
0.02
0.44
2.77
9.69
25.15
54.30
103.08
178.72
200
2.47  1012
7.81  105
0.02
0.46
2.78
9.75
25.31
54.53
103.50
179.13
Calculations were performed using 3-mode representation with 15 Gauss-Hermite integration points and 12 harmonic oscillator
functions per mode.
S10
5. Symmetry labels for vibrational and rotational states of methane
Table S-10 gives the symmetries of the overtone levels of the degenerate modes for methane. Table S-11 gives the transformations
from irreducible representations of the group of all rotations and reflections to irreducible representations of the point group of methane,
and Table S-12 gives the symmetries of the rotational states that result from this transformation. The final symmetries of the states are
obtained by taking direct products of the vibrational symmetries, such as those in Table S-10, with the rotational symmetries, such as
those in Table S-12.
S11
Table S-10. Symmetry types of overtone levels of degenerate vibrations for Td point group.a
a
Vibrational level
Resulting statesb
(e)2
A1 + E
(e)3
A1 + A2 + E
(e)4
A1 + 2E
(e)5
A1 + A2 + 2E
(e)6
2A1 + A2 + 2E
(e)7
A1 + A2 + 3E
(f2)2
A1 + E + F2
(f2)3
A1 + F1 + 2F2
(f2)4
2A1 + 2E + F1 + 2F2
(f2)5
A1 + E + 2F1 + 4F2
(f2)6
3A1 + A2 + 3E + 2F1 + 4F2
(f2)7
2A1 + 2E + 4F1 + 6F2
G. Herzberg, Infrared and Raman Spectra of Polyatomic Molecules (Lancaster Press, Inc., New York, 1945), pp. 125127.
b
These states are degenerate for a harmonic oscillator but are split by the anharmonicity.
S12
Table S-11. Transformation of DJg and D Ju to Td point group.a
Class
E

0
DJg
D Ju
2J+1
8C2

2
3
1 2
sin( J  )
2 3
sin
2J+1
H. A. Jahn, Proc. Roy. Soc. A 171, 450 (1939).
3
1 2
sin( J  )
2 3
sin
a


3
3C2
6σd


1
sin( J  )
2
1
sin( J  )
2
sin

2
1
sin( J  )
2
sin

2
sin


2
1
 sin( J  )
2
sin
6S4

2

2
1 
sin( J  )
2 2
sin

4
1 
 sin( J  )
2 2
sin

4
S13
Table S-12. Representation of DJg and D Ju in irreducible representations of the Td point group.a
a
J
Reduction of DJg
Reduction of D Ju
0
A1
A2
1
F1
F2
2
E + F2
E + F1
3
A2 + F1+ F2
A1 + F1 + F2
4
A1 + E + F1+ F2
A2 + E + F1 + F2
5
E + 2F1 + F2
E + F1 + 2F2
6
A1 + A2 + E + F1 + 2F2
A1 + A2 + E + 2F1 + F2
7
A2 + E + 2F1 + 2F2
A1 + E + 2F1 + 2F2
8
A1 + 2E + 2F1 + 2F2
A2 + 2E + 2F1 + 2F2
9
A1 + A2 + E + 3F1 + 2F2
A1 + A2 + E + 2F1 + 3F2
10
A1 + A2 + 2E + 2F1 + 3F2
A1 + A2 + 2E + 3F1 + 2F2
11
A2 + 2E + 3F1 + 3F2
A1 + 2E + 3F1 + 3F2
12
2A1 + A2 + 2E + 3F1 + 3F2
A1 + 2A2 + 2E + 3F1 + 3F2
13
A1 + A2 + 2E + 4F1 + 3F2
A1 + A2 + 2E + 3F1 + 4F2
14
A1 + A2 + 3E + 3F1 + 4F2
A1 + A2 + 3E + 4F1 + 3F2
15
A1 + 2A2 + 2E + 4F1 + 4F2
2A1 + A2 + 2E + 4F1 + 4F2
H. A. Jahn, Proc. Roy. Soc. A 171, 450 (1939).