Vibrational Anharmonic Calculations in Electronic Structure
Packages for Applications in Intramolecular Dynamics
Steven T. Shipman, Gordon G. Brown, and Brooks H. Pate
Department of Chemistry
University of Virginia
Motivation / Outline
With the anharmonic capabilities in G03, we can calculate a
number of quantities useful for high-resolution spectroscopy.
New quantities are:
Xij values
kijk values (3rd order matrix elements)
kiijk values (4th order matrix elements)
 values (vibration-rotation interaction)
But these calculations are somewhat resource intensive…
Are they worth it?
Going Beyond the Normal Mode Approximation
Hvib = 1/2 ∑ wr (pr2 + qr2) + 1/6 ∑ rst qr qs qt
r
rst
+ 1/24 ∑ rstu qr qs qt qu + ∑ Be j2

rstu
j = ∑ zij (qi pj – qj pi)
i<j
rst and rstu are 3rd and 4th derivatives of the potential energy with
respect to the normal modes. These are calculated numerically.
Only quartic elements with at most three distinct normal modes
are calculated by G03 (i.e. rrst, rrss, rrrs).
Barone, V., J. Chem. Phys. 122, 041408 (2005).
The Molecules Under Study
1-propyne
trifluoropropyne (tfp)
cyclopropylacetylene (cpa)
1-butyne
phenylacetylene (pa)
Experimental and Calculated Rotational Constants
Experimental
Calculated*
A
B
C
A
B
C
propyne
159139.0
8545.88
8545.88
159812.66
8514.42
8514.42
tfp
5718.84
2877.95
2877.95
5590.82
2848.33
2848.14
butyne
27147.76
4546.47
4086.87
27542.36
4476.46
4043.62
cpa
15756
3358.2
3191.8
15905.51
3308.65
3139.16
pa
5680.33
1529.74
1204.96
5667.47
1522.05
1199.82
Average Errors: A - 0.57%, B - 0.49%, C - 0.46%
* B3LYP/6-31+G(d,p) Opt=tight Int=ultrafine Nosymm
Vibrational Assignments and State Densities
First goal is to get rid of scaling factors and
to include the effects of major perturbations
to get better vibrational assignments.
Anharmonic corrections are primarily from
the diagonal elements of the Xij matrix.
State densities are the second goal; these are determined largely
from the full Xij matrix.
Assignments can be tested by literature values and
IR-FTMW DR.
State densities can be tested by mean-level spacings
from high-resolution IVR or by IR-FTMW DR.
Frequencies and State Densities
Mode Symm
n1
n2
n3
n4
n5
A1
A1
A1
A1
A1
Freq (e)
Freq (c)
3329.9
2165.4
1253.2
811.7
536.0
3343.2
2218.5
1223.8
792.7
524.4
Mode Symm
n6
n7
n8
n9
n10
E
E
E
E
E
Freq (e)
Freq (c)
1179.0
686.0
612.0
453.0
170.0
1120.4
690.5
596.2
436.0
175.7
Avg. error: 2.5%
Std. dev.: 1.3%
(v = 1)
(v = 2)
Butyne
CPA
PA
TFP
15.92 states/cm-1 (18.08 states/cm-1)
27.36 states/cm-1
177.6 states/cm-1 (200 states / cm-1)
1457 states/cm-1 (1250 states / cm-1)
State densities are reasonable; frequencies could use improvement…
Intramolecular Vibrational Redistribution (IVR)
From Ab Initio Results
Methodology
Use state count and matrix elements to generate and diagonalize H
Calculate bright state survival probability and Neff
P(t) = |< Ψ(0) | Ψ(t) >|2
Neff = (∑ Ii)2 / (∑ Ii2)
i
i
Difficulties: Relies on accurate energies (Xij) and accurate
matrix elements (kijk, kiijk).
IVR on Multiple Time Scales
Yoo, H.S.; Ph.D. Dissertation, University of Virginia, 2003.
Problems and Solutions for IVR Calculations
Anharmonically corrected frequencies are in error.
Acetylenic C-H stretch and C≡C stretches are significantly off.
Energy differences between pairs of states are not correct.
Overall effectiveness depends on matrix element and DE.
Large matrix elements seem to be more or less correct; these
should be used to identify important pathways which can be
explored in a hierarchical approach.
It would be interesting if these results could be used to place
bounds on random matrix calculations…
Vibration-Rotation Interactions for DRS
Methodology
Geometry optimization
Rotate to principal axis system
Calculate α’s and frequencies
Direct state count, prune list as necessary
Generate MW stick spectrum
Bvib = B0 –
3N-6
Σ αi (vi + di / 2)
i=0
For small molecules: test versus known values
For big molecules:
test versus DRS
Vibration-Rotation Interactions in Propyne
Mode Sym we (expt)
A
B
we (calc)
B (expt)
B (calc) A (expt) A (calc)
n1
n2
n3
n4
n5
A1
A1
A1
A1
A1
3467
3058
2138
1429
930
3484
3038
2227
1420
944
19.936
-2.518
45.269
11.992
37.774
19.205
1.278
40.872
55.580
40.173
…
1100
200
…
227.0
5.085
1676.663
74.775
-709.802
186.277
n6
n7
n8
n9
n10
E
E
E
E
E
3038
1492
1044
658
328
3104
1483
1058
630
353
0.779
-7.795
4.227
-5.396
-23.384
0.591
-30.555
4.594
-8.179
-20.359
510
1286
-1850
40.56
65.05
1084.442
1181.932
-895.030
42.421
82.143
159139.0 MHz
8545.88 MHz
| DB | = 8.40 MHz | DA | = 299 MHz
| DB |
= 9.83 x 10-4
B
| DA |
= 1.88 x 10-3
A
Pate, B.H.; Ph.D. Dissertation, Princeton, 1992.
Excited State Rotational Spectra of TFP
Anharmonic
17434.3 MHz
247.9 MHz
Ground state 202-303:
17267.58 MHz
Experimental lineshape arises
from IVR (motional narrowing).
Simulated with Expt’l ’s
17541.8 MHz
252.2 MHz
Experiment
17460 MHz
43.2 MHz
Excited State Rotational Spectra of TFP
Anharmonic
17434.3 MHz
247.9 MHz
Harmonic
17321.6 MHz
69.4 MHz
Simulated with Expt’l ’s
17541.8 MHz
252.2 MHz
Experiment
17460 MHz
43.2 MHz
Excited State Rotational Spectra of PA
322-423
303-404
313-414
321-422
312-413
Excited State Rotational Spectra of PA
322-423
303-404
313-414
321-422
312-413
IVR Exchange Narrowing
Experimental upper state spectra have a narrower distribution
than the simulated spectra.
This is a consequence of motional narrowing. The degree of
narrowing compared to the ab initio results allows us to
estimate the IVR timescale.
GIVR = (GROT)2 / Dnobs
TFP
PA (K=0)
Dnobs
43 MHz
18 MHz
GROT
248 MHz
65 MHz
GIVR
1430 MHz
235 MHz
tIVR
110 ps
670 ps
Pate, B.H., J. Chem. Phys. 109, 4396 (1998); Douglass, K.O. et al., J. Chem. Phys. 121, 6845 (2004).
Summary
The elimination of scaling factors is an important step forward
and should help to clarify issues of state identity.
The state densities seem to be good; more testing on a wider
range of systems is necessary to confirm this.
IVR simulations are difficult to get right; large matrix elements
at least seem correct. May be useful for modelling if not for
“real” calculations.
Vibration-rotation interactions are excellent! Shift of first
moment is correct and second moments are very plausible.
Acknowledgements
Current and former Pate Lab members
NSF-Chemistry
The Weibull Distribution
Mode occupancy is based on energy; rotational distributions
are skewed by the low-frequency modes.
The data are well-fit by a Weibull distribution with a shape
factor (k) of 2.
f (x; k, l, q) = (k / l) ((x – q) / l)k – 1 exp (- (x – q) / l)k
313-414
1st moment:
2nd moment:
m = l G (1 + 1 / k)
s2 = l2 G (1 + 2 / k) – m2
http://en.wikipedia.org/wiki/Weibull_distribution
Excited State Rotational Spectra of PA
303-404
10799.6 MHz
64.7 MHz
GS: 10759.8
312-413
11608.4 MHz
81.6 MHz
GS: 11551.2
313-414
10304.1 MHz
62.7 MHz
GS: 10253.5
Excited State Rotational Spectra of PA
321-422
11164.6 MHz
85.8 MHz
GS: 11105.5
Simulated:
303-404
312-413
313-414
322-423
321-422
10799.6 MHz
11608.4 MHz
10304.1 MHz
10981.4 MHz
11164.6 MHz
322-423
10981.4 MHz
77.0 MHz
GS: 10925.9
64.7 MHz
81.6 MHz
62.7 MHz
77.0 MHz
85.8 MHz
10759.8 MHz
11551.2 MHz
10253.5 MHz
10925.9 MHz
11105.5 MHz
Coriolis Coupling in Phenylacetylene
K2’s have coalesced to a single broad feature!
This is due to Coriolis coupling in addition to IVR.
Analysis of Coriolis times gives 2.2 ns for K1’s and 540 ps for K2’s.
Vibration-Rotation Interactions in Phenylacetylene
Mode Sym we (expt)
A1
3332
n1
A1
3078
n2
A1
3067
n3
A1
3047
n4
A1
2120
n5
A1
1601
n6
A1
1488
n7
A1
1192
n8
A1
1175
n9
A1
1028
n10
A1
998
n11
A1
760
n12
A1
465
n13
we (calc)
3353.8
3056.8
3076.2
3028.3
2169.5
1607.7
1493.0
1201.9
1184.2
1030.2
998.7
767.9
466.6
A (calc)
-0.01984
2.91460
2.96175
2.70491
1.15236
7.99449
7.03911
1.41715
2.85048
4.81951
-2.75985
1.91710
-0.96236
B (calc)
1.23932
0.40349
0.31789
0.26442
3.49554
1.64069
0.18442
2.54766
-0.68406
-0.33322
0.03965
665.75977
-0.08910
C (calc)
0.76859
0.36305
0.31680
0.27570
2.22070
-0.68204
0.22390
2.33401
0.28389
1.52035
0.81405
0.68657
0.63428
Narayanan, K. et al., Spectrochim. Acta Part A 52, 1703 (1996).
Vibration-Rotation Interactions in Phenylacetylene
Mode Sym we (expt)
n14
A2
968
n15
A2
842
n16
A2
418
n17
n18
n19
n20
n21
n22
n23
n24
B1
B1
B1
B1
B1
B1
B1
B1
985
915
756
689
613
530
349
140
we (calc)
967.7
841.7
402.9
983.2
914.4
749.0
680.6
594.6
524.4
355.9
157.4
A (calc)
8.10495
-0.26638
5.41654
B (calc)
0.87645
0.55779
-0.24506
C (calc)
0.00970
-0.00657
-0.66382
9.13881
7.14711
4.54851
-4.21976
83.73197
-50.33605
13.40457
-185.44936
0.84890
0.36767
-664.83298
0.38035
-0.48939
-0.23028
-0.27098
-2.60141
0.05945
-0.11219
-0.24832
-0.18773
-0.41874
-0.55570
-0.76070
-0.73606
Narayanan, K. et al., Spectrochim. Acta Part A 52, 1703 (1996).
Vibration-Rotation Interactions in Phenylacetylene
Mode Sym we (expt)
B2
3096
n25
B2
3059
n26
B2
1573
n27
B2
1447
n28
B2
1330
n29
B2
1282
n30
B2
1157
n31
B2
1070
n32
B2
649
n33
B2
613
n34
B2
513
n35
B2
152
n36
we (calc)
3078.6
3052.4
1578.4
1448.0
1336.3
1296.6
1163.7
1071.0
644.6
628.9
524.2
138.9
A (calc)
2.72440
2.65778
8.68141
-1.73869
3.12441
8.67497
-6.02185
-4.85601
-79.37246
2.35279
38.11380
187.14367
B (calc)
0.46571
0.34441
1.16494
1.13879
0.67025
1.32079
0.00641
0.63121
-0.59523
-0.15400
-0.44332
-1.45076
C (calc)
0.38697
0.32039
3.55127
1.38908
0.87899
0.92430
0.31289
-0.22052
-0.16895
0.30265
-0.26399
-1.62009
Narayanan, K. et al., Spectrochim. Acta Part A 52, 1703 (1996).
Vibration-Rotation Interactions in TFP
Mode Symm.
Freq. (expt) Freq. (calc)
B (expt)
B (calc) D (MHz)
n1
n2
n3
n4
n5
A1
A1
A1
A1
A1
3329.9
2165.4
1253.2
811.7
536.0
3343.2
2218.5
1223.8
792.7
524.4
4.27
…
…
– 0.549
– 4.149
3.449
9.249
0.050
2.994
0.150
+ 3.543
+ 4.299
n6
n7
n8
n9
n10
E
E
E
E
E
1179.0
686.0
612.0
453.0
170.0
1120.4
690.5
596.2
436.0
175.7
…
2.710
– 1.601
0.959
– 5.480
8.062
– 0.789
– 1.434
0.534
– 4.164
– 3.499
+ 0.167
– 0.425
+ 1.316
B = 2877.95 MHz
– 0.821
| D B | = 2.01 MHz
| DB |
= 6.98 x 10-4
B
Mills, I.M., Mol. Phys. 16, 345 (1969); Douglass, K.O. et al., J. Chem. Phys. 121, 6845 (2004).
IVR in Cyclopropylacetylene: Experimental
0.8
0.8
P0( t )
 t 
 12 
 1.5 10  0.6
exp 
0.6
 t 
exp 
 12 
 7 10 
0.4
 t 
 12 
 70 10 
0.4
exp 
0.2
0.2
0.0
0
 0.1
0
50
0
100
50
0
t
10
 12
150
100
150
150
1.1
1.0
1
0.8
0.8
P0( t )
 t 
 12 
 1.5 10  0.6
exp
0.6


 12 
 7 10 
exp
t
0.4
 t 
 12 
 70 10 
0.4
exp
0.2
0.2
0.0
0
Normalized Absorption Change
Bright State Survival Probability
t1 = 1.5 ps
t2 = 7.0 ps
t3 = 70 ps
1
t1 = 0.6 ps
t2 = 7.0 ps
t3 = 42 ps
1.0
1.1
1.0
0.8
0.6
0.4
0.2
0.0
0
50
100
150
1.0
0.8
0.6
0.4
0.2
0.0
0
 0.1
0
0
5
5
10
10
t
10
 12
15
15
Time (ps)
20
20
20
0
5
10
15
20
Time (ps)
Douglass, P.C.; Ph.D. Dissertation, Univ. of Virginia, 2007.