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Surface hopping study of the photodissociation dynamics of ICN− and BrCN−
Accepted Manuscript Surface Hopping Study of the Photodissociation Dynamics of ICN- and BrCNBernice Opoku-Agyeman, Anne B. McCoy PII: DOI: Reference:
S0009-2614(18)30011-3 https://doi.org/10.1016/j.cplett.2018.01.011 CPLETT 35363
To appear in:
Chemical Physics Letters
Received Date: Accepted Date:
13 November 2017 3 January 2018
Please cite this article as: B. Opoku-Agyeman, A.B. McCoy, Surface Hopping Study of the Photodissociation Dynamics of ICN- and BrCN-, Chemical Physics Letters (2018), doi: https://doi.org/10.1016/j.cplett.2018.01.011
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Surface Hopping Study of the Photodissociation Dynamics of ICN− and BrCN− Bernice Opoku-Agyeman Department of Chemistry and Biochemistry, The Ohio State University, Columbus, Ohio 43210, United States
Anne B. McCoy Department of Chemistry, University of Washington, Seattle, WA 98195, United States
Abstract In this work the efficacy of semi-classical surface hopping approaches is investigated through studies of the photodissociation dynamics of BrCN− and ICN− . BrCN− provides a challenging situation for semi-classical approaches as excitation to the first bright state yields both Br− + CN and Br∗ + CN− products. Further, this branching is highly sensitive to the amount of rotational energy in the CN0/− fragment. The results of semi-classical and quantum mechanical descriptions of the dynamics are compared when the classical dynamics are propagated in an adiabatic and diabatic representation. The implications of the differences between the classical and quantum treatments of J = 0 are also explored. Keywords: surface hopping, photodissociation, ICN− , ICN−
Email address: [email protected] (Anne B. McCoy)
Preprint submitted to Chemical Physics Letters
January 4, 2018
1
1. Introduction
2
Since the introduction of the semi-classical surface hopping approach [1, 2],
3
this technique along with its extensions [3, 4, 5] have evolved into the stan-
4
dard methods for incorporating electronic quantum effects in classical sim-
5
ulations. This approach can be developed from the quantum/classical Li-
6
ouville equation [5]. Surface hopping has been used in a broad range of
7
applications due to the relative ease with which it can be combined with
8
electronic structure calculations to perform non-adiabatic dynamics calcu-
9
lations in an adiabatic representation of the potential, where the requisite
10
forces and derivative couplings can be obtained on-the-fly. The approach
11
suffers from several known drawbacks, in particular challenges in treating
12
coherence and decoherence properly, as in its original and simplest imple-
13
mentation the surface hopping dynamics is propagated through a swarm of
14
independent trajectories and each hop is treated without consideration of
15
previous transitions between electronic states [6, 7, 5].
16
To test the limitations of this approach and develop appropriate corrections,
17
a series of model systems have been developed for which comparisons can
18
be made between the surface hopping dynamics and the exact quantum dy-
19
namics. The original set of two-state models were provided in the work of
20
Tully, which originally described the approach [2]. More recently Subotnik
21
and co-workers described several three-state models [4]. All of these models
22
focused on branching in scattering processes, and have been performed in an
23
adiabatic representation of the electronic structure.
24
The preference for the adiabatic representation in most of the surface hop-
25
ping studies to date arises from several factors. The first is the fact that the 2
26
adiabatic electronic states and their couplings can be determined unambigu-
27
ously from electronic structure calculations. In the adiabatic representation,
28
the couplings between the electronic states arise from the kinetic energy, and
29
are expressed as sums of terms that are proportional to the dot product of
30
the velocity of the atom of interest and the gradient of the electronic wave
31
function with respect to the coordinates of that atom. In a diabatic rep-
32
resentation, the couplings between the electronic states are included in the
33
description of the potential surfaces by identifying a linear combination of
34
the adiabatic states for which the kinetic energy coupling terms vanish. As
35
the kinetic coupling depends on the gradient of the electronic wave function
36
with respect to the nuclear coordinates, in the diabatic representation the
37
electronic wave function will not depend strongly on the nuclear coordinates.
38
In contrast to the adiabatic treatment there is not a well-defined procedure
39
for developing diabatic potentials, although several have been proposed and
40
implemented [8, 9]. As pointed out by Tully [10], since the couplings in the
41
adiabatic treatment depend on the velocities of the atoms, for low-kinetic
42
energy events the dynamics will be dominated by trajectories that remain on
43
one of the adiabatic potential surfaces. Therefore, one should expect fewer
44
transitions between surfaces in this representation. While at higher kinetic
45
energies the dynamics would be better described by a diabatic representa-
46
tion of the potentials, one expects that at these energies the quantum effects
47
would also become less important [10].
48
In the present study, we focus on the photodissociation dynamics of ICN−
49
and BrCN− . Unlike scattering processes, there is considerable excess kinetic
50
energy in both the translational and rotational coordinates in the geome-
3
51
tries at which the couplings between the electronic states are large. Based
52
on this, one might expect the dynamics to be more diabatic. Additionally,
53
experimental and quantum dynamics studies on the BrCN− system have in-
54
dicated that the branching between the two energetically accessible products
55
is highly sensitive to the amount of rotational energy in the CN0/− fragment.
56
Further, studies of the photodissociation of these ions in the presence of ar-
57
gon atoms or CO2 molecules illustrated surprisingly complicated dynamics
58
[11, 12, 13, 14].
59
At first glance, one might anticipate that the electronic structure of these
60
ions would be similar to that of a dihalide system. On the other hand, while
61
the excitation of IBr− to either of the two lowest-energy optically accessible
62
excited states yields a single photoproduct, excitation of ICN− or BrCN− to
63
the corresponding electronic state yields photoproducts in which both X− or
64
CN− can carry the excess charge. In the case of 2.5 eV excitation of ICN− ,
65
the dominant product channel (> 95%) corresponds to I− + CN [12, 14],
66
and the corresponding transition in BrCN− , only 40% of the photoproducts
67
correspond to Br− + CN, while the remaining products consist of CN− + Br∗
68
[11]. These findings were understood using quantum dynamics simulations.
69
Focusing on the product branching, an important difference between the
70
XCN− systems and the dihalides is the relatively small energetic gap between
71
the X∗ + CN− and the X− + CN product channels, as illustrated in Figure
72
1. In addition, X− + CN∗ channel, which would correlate to the I− + Br∗
73
channel in IBr− , is significantly higher in energy compared to the other three
74
product states rather than between the energies of the two product channels
75
of interest. In Figure 1, the energy spacings are based on the difference in
4
76
the electron affinities of the fragments [15, 16, 17], the energy of the first
77
electronically excited state of CN [18], as well as the spin-orbit splittings of
78
iodine and bromine [19]. Most notably, in BrCN− , the difference between
79
the electron affinities of Br and CN is almost exactly equal to the spin-orbit
80
splitting of Br, and the two product channels are nearly degenerate. This
81
can also be seen through examination of collinear cuts through the potential
82
curves for ICN− and BrCN− , shown in Figure 2.
83
Analysis of the electronic wave function shows that following excitation by
84
the energy depicted by the grey solid lines in Figure 2 (a) and (c), the system
85
is in the 2 Π1/2 electronic states and the excess charge is localized on the CN
86
end of the ion. Higher energy excitation represented by the grey dashed lines
87
initiates the dynamics on the 2 Σ+ state, where the charge is more equally
88
distributed in the ion. As a result, as the ion dissociates, there is a significant
89
redistribution of the charge. This occurs at X-CN distances of 4 to 6 ˚ A in
90
both BrCN− and ICN− [11, 12] and corresponds to the region in the panels
91
in Figure 2 where the green and blue dashed lines cross. As is seen in the
92
potential curves in Figure 2, by this distance, the potential energy is near
93
its asymntotic value and most of the initial excess potential energy has been
94
converted to kinetic energy. Based on the large kinetic energy in the region
95
where the couplings between the electronic states is large, photodissociation
96
of ICN− and BrCN− provide systems in which a diabatic picture of the
97
dynamics may be relevant.
98
Quantum dynamics calculations based on these potential surfaces reproduced
99
the observed branching ratios, although in the case of BrCN− the results of
100
the calculations were surprisingly sensitive to details of the potential surface
5
101
[11]. This sensitivity reflected the fact that for this anion, there was a strong
102
sensitivity of the branching to the Br·CN angle, and collinear calculations
103
based on BrCN− and BrNC− configurations yielded different photoproducts,
104
each with unit efficiency. Moving to the full-dimensional system, the branch-
105
ing was found to be sensitive to the range of Br·CN angles sampled over
106
the range of Br-CN distances over which the coupling between the two elec-
107
tronic states was largest (e.g. the distances over which the excess charge
108
relocalizes).
109
Such sensitivity sets up a particularly challenging scenerio for surface hoping
110
approaches. In this work, we focus on the photodissociation dynamics of
111
BrCN− and ICN− as described using surface hopping in both an adiabatic
112
and diabatic representation. We also explore sensitivities of the final results,
113
as indicated by the branching ratios and amount of rotational energy in the
114
CN or CN− fragment, to the choice of initial conditions. In particular, we
115
explore how the representation of total J = 0 in the classical dynamics affects
116
these quantities. The accuracy of the approaches is calibrated against the
117
results of quantum dynamics simulations.
118
2. Theoretical methods
119
In this work, we compare the results of quantum and semi-classical calcula-
120
tions of the product branching and rotational energy distribution following
121
excitation of ICN− and BrCN− to the accessible excited states, depicted by
122
the blue and green curves in the three panels of Figure 2.
123
For the semi-classical calculations, the initial conditions were sampled from a
124
Wigner distribution [20, 21] based on a fit of the ground state wave function 6
125
for ICN− or BrCN− to a three-dimensional Gaussian, expressed in the Jacobi
126
coordinates shown in Figure 3. By representing the ground state wave func-
127
tion as a product of three one-dimensional Gaussian functions, the Wigner
128
distribution function takes on a simple analytical form
W (r, pr , θ, pθ , R, pR ) = wr (r, pr ) wθ (θ, pθ ) wR (R, pR ) " r 2 # σα2 α − αe 2 2 wα (α, pα ) = exp −pα σα − π σα
(1) (2)
129
The parameters used to define these Gaussian functions are provided in the
130
Supporting Information. For these calculations, we used ground state poten-
131
tial surfaces that are functions of R and θ for ICN− and BrCN− , VXCN− (R, θ),
132
to evaluate the ground state wave functions. These are the same as potential
133
surfaces we used in earlier studies [11, 22]. To these surfaces, we added a
134
harmonic potential to describe the CN stretch. In this way, 1 2 V (R, r, θ) = VXCN− (R, θ) + µCN ωCN ∆r2 2
(3)
135
where µCN and ωCN are the reduced mass and harmonic frequency of the CN
136
stretch. The potentials for the excited states were generated in an analogous
137
manner.
138
Using the Wigner distribution, we randomly selected 10 000 sets of coordi-
139
nates and momenta, which were converted from Jacobi coordinates to three
140
Cartesian coordinates and momenta for each atom [23]. Although this ap-
141
pears to introduce a substantial increase in the dimensionality of the sys-
142
tem compared to the two-dimensions used for the quantum dynamics, six of
143
the degrees of freedom in the classical studies represent overall rotation and 7
144
translation, which are separable from the other degrees of freedom. The con-
145
version from Jacobi to Cartesian coordinates was performed to ensure that
146
these degrees of freedom do not contribute to the total kinetic energy [23].
147
We have also introduced an additional internal coordinate, r. As is seen in
148
Eq. (3), there is no potential coupling between r and the other two Jacobi
149
coordinates. The only coupling between these coordinates comes from the
150
r-dependence of the effective mass associated with θ. Since the harmonic
151
potential for the CN stretch is stiff, this coupling is expected to be weak, and
152
we will confirm that there is very little energy transfer between R/θ and r
153
in the following section.
154
Once constructed the initial coordinates and momenta were then propagated
155
using the adiabatic or diabatic representation of the excited state potential
156
functions in the absence of coupling between the electronic states, and on the
157
pair of coupled states using both the adiabatic and diabatic treatments of
158
the dynamics. In cases where couplings between the two states were consid-
159
ered, the system was allowed to sample both states based on Tully’s surface
160
hopping algorithm [10].
161
The potentials used to provide VXCN− (R, θ) in Eq. (3) are the ones devel-
162
oped for earlier quantum dynamics studies. Specifically, the electronic ener-
163
gies of the states of interest were evaluated at the spin-orbit multi-referece
164
configuration interaction level of theory with single and double excitations
165
(SO-MRCISD) using aug-cc-pVTZ basis sets and effective core potentials in
166
place of the core electrons in iodine and bromine [11, 22]. In that work, a
167
diabatic representation of the dynamics was used, and these diabatic poten-
168
tials are represented by the potential curves plotted with the dashed blue and
8
169
green dashed curves in the three panels of Figure 2. These potentials along
170
with the coupling between them are used to describe the diabatic represen-
171
tation in the present study. When we use an adiabatic representation for the
172
semi-classical studies, we evaluate the adiabatic potentials by obtaining the
173
eigenvalues of the 2 × 2 representation of the potential at a specific geometry
174
of the ion. The derivative couplings between these two states are evaluated
175
using the eigenvectors of this 2 × 2 matrix.
176
When a hop between the two surfaces takes place, the kinetic energy is
177
adjusted to ensure conservation of energy. In the adiabatic representation,
178
the velocity parallel to the derivative coupling vector is scaled to ensure
179
energy is conserved. In the diabatic representation, we require that the
180
velocity along the CN bond is unaffected by projecting out the component
181
of the velocity associated with the CN stretch before rescaling the velocities
182
of the individual atoms by a constant scaling factor to conserve energy.
183
One complication that arises is related to differences between how quantum
184
and classical mechanics describe the large amplitude X·CN bend when J = 0.
185
At J = 0 the zero-point energy in the bend corresponds to the harmonic fre-
187
quency (rather than half of that frequency). To account for this, the sampled √ bend coordinate and momentum are each multiplied by a factor of 2. We
188
also perform planar calculations in which this factor is not included, and the
189
corresponding system in the quantum dynamics simulation is confined to a
190
plane.
191
For comparisons with the semi-classical studies, quantum simulations were
192
also performed using the same initial wave functions that were used to gener-
193
ate the Wigner distributions. For the J = 0 calculations, the same approach
186
9
194
was used as has been described previously [11]. While the approach is the
195
same, the final results are slightly different due to the change in the initial
196
wave function. For these calculations, the bend contribution to the wave
197
function is described using a Discrete Variable Representation (DVR) based
198
on Legendre Polynomials. For the planar calculations, the bend is described
199
using a DVR based on particle-on-a-ring eigenstates. Otherwise the quantum
200
dynamics calculations for J = 0 and planar propagations are identical.
201
Results for both representations of the bend are described for excitation to
202
both the 2 Π1/2 and 2 Σ+ states, shown with blue and green lines in the three
203
panels of Figure 2.
204
3. Results and Discussion
205
Before comparing the quantum and semi-classical results we consider dif-
206
ferences between the two types of calculations. The first difference is the
207
dimensionality. The quantum calculations only include R and θ in Figure 3,
208
while the classical dynamics is propagated in all nine Cartesian coordinates,
209
under the constraint that J = 0. As such while the CN bond is rigid in the
210
quantum dynamics, it is allowed to vibrate in the semi-classical simulations.
211
On the other hand, the only coupling between the CN stretch and the other
212
degrees of freedom is through the rotational constant for CN. Since the CN
213
bond is stiff, this coupling is weak and little energy transfer between the the
214
CN stretch and the other degrees of freedom is expected. Analysis of the
215
amount of energy gained by the CN stretch (see Tables S2 and S5) shows
216
that it is < 10 cm−1 following excitation to the 2 Π1/2 state and between 40
217
and 80 cm−1 following excitation to the 2 Σ+ state. When surface hopping 10
218
is introduced the amount of energy transferred to or from the CN stretch
219
increases but remains smaller than 1% of the available energy.
220
The second difference comes in how J = 0 is reflected in the quantum and
221
semi-classical descriptions of the dynamics. Classically when J = 0 a tri-
222
atomic system is constrained to move in a plane, while quantum mechani-
223
cally the system has cylindrical symmetry. As noted above, the zero-point
224
energy in the X·CN bend is doubled in the quantum description compared
225
to a typical bending motion. This is expected to affect on the dynamics. In
226
particular, it should be reflected in the calculated branching fractions for the
227
photoproducts following excitation of BrCN− . This is due to the large role
228
of CN rotation in this branching.
229
The third difference is the one we are exploring here and that is the fully
230
quantum description of the nonadibatic dynamics compared to the surface
231
hopping approach used in these simulations.
232
The results of exciting the ions to the 2 Π1/2 and 2 Σ+ states are provided in
233
Tables 1 and 2, respectively. These tables focus on the branching fractions
234
and the rotational energy distributions. A more complete tabulation of the
235
energy partitioning of the photoproducts can be found in the Supporting
236
Information.
237
To start our analysis of these results, we compare the results obtained with-
238
out consideration of surface hopping. These are denoted as “no SH” in Tables
239
1 and 2. In these cases, we run the trajectories on either the diabatic or adi-
240
abatic potential and, as no transitions between electronic states are allowed,
241
only one product is generated. Comparing the average final rotational ener-
242
gies obtained from the quantum and classical calculations, we note significant
11
243
differences in the results of the J = 0 calculations, with 14 to 20% less energy
244
ending up in rotation in the classical calculation compared to when quantum
245
dynamics simulations are performed. This is attributed to the smaller phase
246
space sampled in these classical calculations, where J = 0 constrains the
247
atoms to move in a plane as opposed to the three-dimensional cylindrically
248
symmetric representation of J = 0 in quantum mechanics. When both cal-
249
culations are performed in a planar configuration, the differences are reduced
250
by an order of magnitude to 0.05 to 2%. While these results are reported for
251
calculations that consider only one potential surface, the importance of CN
252
rotation in determining the branching in BrCN− leads us to expect that the
253
product branching will be strongly affected when non-adiabatic effects are
254
introduced.
255
As we look at the product distributions and branching when surface hop-
256
ping is included in the simulations for the planar calculations, we find that
257
when the semi-classical dynamics is based on a diabatic representations of
258
the potentials the results are in better agreement with the quantum results,
259
although both the diabatic and adiabatic representations of the potential pro-
260
vide reasonably accurate results. One exception is the rotational energy of
261
the CN− product following photoexcitation of ICN− to the 2 Π1/2 state. This
262
is likely due to the relatively small branching fraction (3%) in this channel
263
and the resulting poorer statistics.
264
For the J = 0 calculations on ICN− , the results of the surface hopping calcu-
265
lations in a diabatic representation remain more accurate, although both the
266
adiabatic and diabatic representation of the potentials do a reasonably good
267
job of reproducing the results of the quantum dynamics. The accuracy of the
12
268
J = 0 calculations for BrCN− is less good, and this is attributed to the criti-
269
cal role of CN rotation on the branching. In the case of excitation to the 2 Σ+
270
state, neither surface hopping calculation reproduces the branching obtained
271
from the quantum simulation. While the adiabatic treatment reproduces the
272
branching, the final average rotational energy is in poor agreement with the
273
quantum results. In contrast, calculations in the diabatic representation pro-
274
vide more accurate rotational energy distributions, but the branching ratios
275
are less accurate than those obtained in the adiabatic representation. The
276
fact that calculations using the diabatic representation of the potential sur-
277
faces provide more accurate representations of the dynamics likely reflects the
278
much higher kinetic energy in these systems particularly following excitation
279
to the 2 Σ+ state.
280
As we analyzed the trajectories following photoexcitation of ICN− to the
281
2
Π1/2 excited state, we found a small, but statistically significant subset of
282
trajectories that underwent a frustrated dissociation prior to forming I +
283
CN− products. As is seen in Figures S1, in these trajectories, the I-CN
284
distance first increases, then decreases before the ultimate dissociation. This
285
happens in spite of there being no barrier along the potential, and the system
286
having more than 0.848 eV of kinetic energy in the photoproducts. The origin
287
of the longer-lived trajectories is a centrifugal barrier in R. This arises from
288
the fact that at J = 0 a large rotational kinetic energy in the CN products
289
requires that there is also a large orbital angular momentum in the I· · · CN
290
complex. This result provides support for the proposed explanation for the
291
observation of photoproducts following photoexcitation of ICN− which have
292
a mass consistent with ICN− when the ICN− is initially complexed with as
13
293
few as one argon atom [12]. Still, the large difference between the available
294
0.848 eV of kinetic energy compared to the roughly 0.05 eV binding energy
295
of a single argon atom makes the fact that recombination is observed with
296
only one argon atom present surprising.
297
4. Conclusions
298
In summary, we have explored the efficacy of surface hopping approaches
299
for studying the photodissociation dynamics of ICN− and BrCN− . Overall
300
we found the treatment to be quite accurate, although, in contrast to ear-
301
lier studies, we found the diabatic representation of the coupled electronic
302
states to provide somewhat more accurate description of the dynamics. This
303
is attributed in part to the more diabatic nature of the dynamics given the
304
relatively large kinetic energy release in these systems. Where we find dif-
305
ferences between the quantum and semi-classical results these are attributed
306
primarily to differences in the way the two treatments account for bending
307
motions in a linear molecule, particularly at low rotational energy, and not
308
to deficiencies in the surface hopping algorithm. It would be interesting to
309
explore how the introduction of solvating atoms or molecules affects the dy-
310
namics of these processes and further explore the apparent single atom caging
311
in the dissociation of ICN− in the presence of a single argon atom.
312
Acknowledgments
313
We thank Prof. Andrew S. Petit (California State University Fullerton) for
314
many helpful conversations during the execution of this project. Support of
315
this work by the National Science Foundation through grant CHE-1619660 14
316
is gratefully acknowledged. We also thank the Ohio Supercomputer Center
317
for an allocation of computing resources.
15
318
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17
Table 1: Branching ratios and diatomic product rotational energy in cm−1 , when BrCN− and ICN− are initially excited to the 2 Π state (see Fig. 2). Quantum (CN)
Classical (CN− )
(CN− )
Approacha
% X− b
ICN− ; J = 0
Adiabatic (no SH)
100
1372
NA
100
1179
NA
Diabatic (no SH)
0
NA
1662
0
NA
1392
Adiabatic (w/SH)
−−
−−
−−
90
869
2646
Diabatic (w/SH)
96
1346
3871
96
1342
1790
Adiabatic (no SH)
100
742
NA
100
741
NA
Diabatic (no SH)
0
NA
1050
0
NA
1045
Adiabatic (w/SH)
−−
−−
−−
93
521
2599
Diabatic (w/SH)
97
726
3469
97
931
1707
Adiabatic (no SH)
100
2148
NA
100
1743
NA
Diabatic (no SH)
0
NA
2138
0
NA
1764
Adiabatic (w/SH)
−−
−−
−−
36
698
1850
Diabatic (w/SH)
34
973
2304
52
712
2451
Adiabatic (no SH)
100
1354
NA
100
1344
NA
Diabatic (no SH)
0
NA
1453
0
NA
1443
Adiabatic (w/SH)
−−
−−
−−
40
349
1572
Diabatic (w/SH)
55
363
2122
59
474
2225
ICN− ; plane
BrCN− ; J = 0
BrCN− ; plane
Erot
Erot
% X−
(CN)
System
Erot
Erot
a
a complete description of the product state energy distribution is provided in the supporting information
a
Fraction of X− (as opposed to CN− ) products, X = I or Br.
18
Table 2: Branching ratios and diatomic product rotational energy in cm−1 , when BrCN− and ICN− are initially excited to the 2 Σ+ state (see Fig. 2). Quantum % X− b
(CN)
Classical (CN− )
% X−
(CN)
(CN− )
System
Approacha
ICN− ; J = 0
Adiabatic (no SH)
0
NA
6377
0
NA
Diabatic (no SH)
100
6288
NA
100
5071
NA
Adiabatic (w/SH)
−−
−−
−−
24
8225
4580
Diabatic (w/SH)
21
8174
6041
19
8141
4668
Adiabatic (no SH)
0
NA
4591
0
NA
4525
ICN− ; plane
BrCN− ; J = 0
BrCN− ; plane
Erot
Erot
Erot
Erot
5247
Diabatic (no SH)
100
4307
NA
100
4227
NA
Adiabatic (w/SH)
−−
−−
−−
19
8102
3911
Diabatic (w/SH)
14
7643
4163
14
8007
3936
Adiabatic (no SH)
0
NA
7450
0
NA
6026
Diabatic (no SH)
100
7590
NA
100
6063
NA
Adiabatic (w/SH)
−−
−−
−−
52
8036
4723
Diabatic (w/SH)
70
8445
6328
52
8332
4146
Adiabatic (no SH)
0
NA
5573
0
NA
5481
Diabatic (no SH)
100
5555
NA
100
5451
NA
Adiabatic (w/SH)
−−
−−
−−
49
7645
4055
Diabatic (w/SH)
54
7849
3544
49
8340
3221
a
Representation of the potential with or without surface hopping between states
a
Fraction of X− (as opposed to CN− ) products, X = I or Br.
19
FIGURES:
357
1.0
I* + CN
I* + Br
0.9 0.8
I + CN
I + Br*
Energy, eV
0.7 0.6
Br + CN
0.5
Br* + CN
0.4 0.3
I + Br
0.2 0.1 0.0
I + Br
I + CN
IBr
ICN
Br + CN BrCN
Figure 1: Relative energies of the photoproducts following photodissociation of IBr− , ICN− and BrCN− , based on electron affinities of I [17], Br [15] and CN [16], the energy of CN∗ [18] as well as the spin-orbit splittings of I and Br [19]. The products that contain electronically excited CN lie outside the scale of the plot with I− +CN* having an energy of 1.9 eV and Br− +CN* has an energy of 1.6 eV.
20
Br C N R
(a)
3
4
Br N C R
(b)
3 2Π
3/2/
2Π
Br– + CN*
2Σ+
1
2Π
0
-1
2Π
1/2
3/2
V, eV
V, eV
2
2
4
6
R, Å
2Π
1
8
10
2Π
1/2
Br– + CN Br* + CN– Br + CN–
3/2
2Σ+
2
2Π
2
6
R, Å
2Π
1/2
8
10
2Σ+
I* + CN–
1 2Π
0
-1 4
3/2/
I– + CN* Br– + CN*
2Σ+
-1
I C N R
(c)
2 3/2/ Π1/2
2
Br* + CN– Br– + CN 0 Br + CN–
2Σ+
4
3 2Π
1/2
V, eV
4
2Π
I– + CN
1/2
I + CN–
3/2
2Σ+
2
4
6
8
10
R, Å
Figure 2: One-dimensional cuts through the collinear (a) BrCN− (b) BrNC− and (c) ICN− potential surfaces, with the geometries illustrated at the top of the panels. In all cases, the solid lines represent the adiabatic potential surfaces, while the dashed lines provide the corresponding diabatic states. For consistency, the colors of the lines correspond to the asymptotic photoproducts. The initial average excitation energies considered in this work are shown with solid lines for excitation to the 2 Π1/2 state and dashed lines for excitation to the 2 Σ+ state. These lines correspond to excitation energies of 2.7 and 4.2 eV, respectively.
21
q
C
X R
N
Figure 3: The Jacobi coordinates (R, θ) used to define the XCN− (X = Br or I) geometry.
22
Highlights: -
Exploration of classical treatments of the photodissociation of ICN- and BrCNRigorous test of surface hopping in an adiabatic and diabatic representation Effect of differences in classical and quantum treatments of J = 0 on product branching
1.1
1.0
1.0
0.9
X* + CN0.9 V(R), eV
V(R), eV
1.1 Graphical Abstract 2Σ+
0.8 2Π
0.7
1/2
0.6
0.8
2Σ+
X* + CN 2Π
1/2
0.7 0.6
X + CN
X + CN
0.5
0.5
3
4
5
6 R, Å
7
8 3
4
5
6 R, Å
7
8
S0009-2614(18)30011-3 https://doi.org/10.1016/j.cplett.2018.01.011 CPLETT 35363
To appear in:
Chemical Physics Letters
Received Date: Accepted Date:
13 November 2017 3 January 2018
Please cite this article as: B. Opoku-Agyeman, A.B. McCoy, Surface Hopping Study of the Photodissociation Dynamics of ICN- and BrCN-, Chemical Physics Letters (2018), doi: https://doi.org/10.1016/j.cplett.2018.01.011
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Surface Hopping Study of the Photodissociation Dynamics of ICN− and BrCN− Bernice Opoku-Agyeman Department of Chemistry and Biochemistry, The Ohio State University, Columbus, Ohio 43210, United States
Anne B. McCoy Department of Chemistry, University of Washington, Seattle, WA 98195, United States
Abstract In this work the efficacy of semi-classical surface hopping approaches is investigated through studies of the photodissociation dynamics of BrCN− and ICN− . BrCN− provides a challenging situation for semi-classical approaches as excitation to the first bright state yields both Br− + CN and Br∗ + CN− products. Further, this branching is highly sensitive to the amount of rotational energy in the CN0/− fragment. The results of semi-classical and quantum mechanical descriptions of the dynamics are compared when the classical dynamics are propagated in an adiabatic and diabatic representation. The implications of the differences between the classical and quantum treatments of J = 0 are also explored. Keywords: surface hopping, photodissociation, ICN− , ICN−
Email address: [email protected] (Anne B. McCoy)
Preprint submitted to Chemical Physics Letters
January 4, 2018
1
1. Introduction
2
Since the introduction of the semi-classical surface hopping approach [1, 2],
3
this technique along with its extensions [3, 4, 5] have evolved into the stan-
4
dard methods for incorporating electronic quantum effects in classical sim-
5
ulations. This approach can be developed from the quantum/classical Li-
6
ouville equation [5]. Surface hopping has been used in a broad range of
7
applications due to the relative ease with which it can be combined with
8
electronic structure calculations to perform non-adiabatic dynamics calcu-
9
lations in an adiabatic representation of the potential, where the requisite
10
forces and derivative couplings can be obtained on-the-fly. The approach
11
suffers from several known drawbacks, in particular challenges in treating
12
coherence and decoherence properly, as in its original and simplest imple-
13
mentation the surface hopping dynamics is propagated through a swarm of
14
independent trajectories and each hop is treated without consideration of
15
previous transitions between electronic states [6, 7, 5].
16
To test the limitations of this approach and develop appropriate corrections,
17
a series of model systems have been developed for which comparisons can
18
be made between the surface hopping dynamics and the exact quantum dy-
19
namics. The original set of two-state models were provided in the work of
20
Tully, which originally described the approach [2]. More recently Subotnik
21
and co-workers described several three-state models [4]. All of these models
22
focused on branching in scattering processes, and have been performed in an
23
adiabatic representation of the electronic structure.
24
The preference for the adiabatic representation in most of the surface hop-
25
ping studies to date arises from several factors. The first is the fact that the 2
26
adiabatic electronic states and their couplings can be determined unambigu-
27
ously from electronic structure calculations. In the adiabatic representation,
28
the couplings between the electronic states arise from the kinetic energy, and
29
are expressed as sums of terms that are proportional to the dot product of
30
the velocity of the atom of interest and the gradient of the electronic wave
31
function with respect to the coordinates of that atom. In a diabatic rep-
32
resentation, the couplings between the electronic states are included in the
33
description of the potential surfaces by identifying a linear combination of
34
the adiabatic states for which the kinetic energy coupling terms vanish. As
35
the kinetic coupling depends on the gradient of the electronic wave function
36
with respect to the nuclear coordinates, in the diabatic representation the
37
electronic wave function will not depend strongly on the nuclear coordinates.
38
In contrast to the adiabatic treatment there is not a well-defined procedure
39
for developing diabatic potentials, although several have been proposed and
40
implemented [8, 9]. As pointed out by Tully [10], since the couplings in the
41
adiabatic treatment depend on the velocities of the atoms, for low-kinetic
42
energy events the dynamics will be dominated by trajectories that remain on
43
one of the adiabatic potential surfaces. Therefore, one should expect fewer
44
transitions between surfaces in this representation. While at higher kinetic
45
energies the dynamics would be better described by a diabatic representa-
46
tion of the potentials, one expects that at these energies the quantum effects
47
would also become less important [10].
48
In the present study, we focus on the photodissociation dynamics of ICN−
49
and BrCN− . Unlike scattering processes, there is considerable excess kinetic
50
energy in both the translational and rotational coordinates in the geome-
3
51
tries at which the couplings between the electronic states are large. Based
52
on this, one might expect the dynamics to be more diabatic. Additionally,
53
experimental and quantum dynamics studies on the BrCN− system have in-
54
dicated that the branching between the two energetically accessible products
55
is highly sensitive to the amount of rotational energy in the CN0/− fragment.
56
Further, studies of the photodissociation of these ions in the presence of ar-
57
gon atoms or CO2 molecules illustrated surprisingly complicated dynamics
58
[11, 12, 13, 14].
59
At first glance, one might anticipate that the electronic structure of these
60
ions would be similar to that of a dihalide system. On the other hand, while
61
the excitation of IBr− to either of the two lowest-energy optically accessible
62
excited states yields a single photoproduct, excitation of ICN− or BrCN− to
63
the corresponding electronic state yields photoproducts in which both X− or
64
CN− can carry the excess charge. In the case of 2.5 eV excitation of ICN− ,
65
the dominant product channel (> 95%) corresponds to I− + CN [12, 14],
66
and the corresponding transition in BrCN− , only 40% of the photoproducts
67
correspond to Br− + CN, while the remaining products consist of CN− + Br∗
68
[11]. These findings were understood using quantum dynamics simulations.
69
Focusing on the product branching, an important difference between the
70
XCN− systems and the dihalides is the relatively small energetic gap between
71
the X∗ + CN− and the X− + CN product channels, as illustrated in Figure
72
1. In addition, X− + CN∗ channel, which would correlate to the I− + Br∗
73
channel in IBr− , is significantly higher in energy compared to the other three
74
product states rather than between the energies of the two product channels
75
of interest. In Figure 1, the energy spacings are based on the difference in
4
76
the electron affinities of the fragments [15, 16, 17], the energy of the first
77
electronically excited state of CN [18], as well as the spin-orbit splittings of
78
iodine and bromine [19]. Most notably, in BrCN− , the difference between
79
the electron affinities of Br and CN is almost exactly equal to the spin-orbit
80
splitting of Br, and the two product channels are nearly degenerate. This
81
can also be seen through examination of collinear cuts through the potential
82
curves for ICN− and BrCN− , shown in Figure 2.
83
Analysis of the electronic wave function shows that following excitation by
84
the energy depicted by the grey solid lines in Figure 2 (a) and (c), the system
85
is in the 2 Π1/2 electronic states and the excess charge is localized on the CN
86
end of the ion. Higher energy excitation represented by the grey dashed lines
87
initiates the dynamics on the 2 Σ+ state, where the charge is more equally
88
distributed in the ion. As a result, as the ion dissociates, there is a significant
89
redistribution of the charge. This occurs at X-CN distances of 4 to 6 ˚ A in
90
both BrCN− and ICN− [11, 12] and corresponds to the region in the panels
91
in Figure 2 where the green and blue dashed lines cross. As is seen in the
92
potential curves in Figure 2, by this distance, the potential energy is near
93
its asymntotic value and most of the initial excess potential energy has been
94
converted to kinetic energy. Based on the large kinetic energy in the region
95
where the couplings between the electronic states is large, photodissociation
96
of ICN− and BrCN− provide systems in which a diabatic picture of the
97
dynamics may be relevant.
98
Quantum dynamics calculations based on these potential surfaces reproduced
99
the observed branching ratios, although in the case of BrCN− the results of
100
the calculations were surprisingly sensitive to details of the potential surface
5
101
[11]. This sensitivity reflected the fact that for this anion, there was a strong
102
sensitivity of the branching to the Br·CN angle, and collinear calculations
103
based on BrCN− and BrNC− configurations yielded different photoproducts,
104
each with unit efficiency. Moving to the full-dimensional system, the branch-
105
ing was found to be sensitive to the range of Br·CN angles sampled over
106
the range of Br-CN distances over which the coupling between the two elec-
107
tronic states was largest (e.g. the distances over which the excess charge
108
relocalizes).
109
Such sensitivity sets up a particularly challenging scenerio for surface hoping
110
approaches. In this work, we focus on the photodissociation dynamics of
111
BrCN− and ICN− as described using surface hopping in both an adiabatic
112
and diabatic representation. We also explore sensitivities of the final results,
113
as indicated by the branching ratios and amount of rotational energy in the
114
CN or CN− fragment, to the choice of initial conditions. In particular, we
115
explore how the representation of total J = 0 in the classical dynamics affects
116
these quantities. The accuracy of the approaches is calibrated against the
117
results of quantum dynamics simulations.
118
2. Theoretical methods
119
In this work, we compare the results of quantum and semi-classical calcula-
120
tions of the product branching and rotational energy distribution following
121
excitation of ICN− and BrCN− to the accessible excited states, depicted by
122
the blue and green curves in the three panels of Figure 2.
123
For the semi-classical calculations, the initial conditions were sampled from a
124
Wigner distribution [20, 21] based on a fit of the ground state wave function 6
125
for ICN− or BrCN− to a three-dimensional Gaussian, expressed in the Jacobi
126
coordinates shown in Figure 3. By representing the ground state wave func-
127
tion as a product of three one-dimensional Gaussian functions, the Wigner
128
distribution function takes on a simple analytical form
W (r, pr , θ, pθ , R, pR ) = wr (r, pr ) wθ (θ, pθ ) wR (R, pR ) " r 2 # σα2 α − αe 2 2 wα (α, pα ) = exp −pα σα − π σα
(1) (2)
129
The parameters used to define these Gaussian functions are provided in the
130
Supporting Information. For these calculations, we used ground state poten-
131
tial surfaces that are functions of R and θ for ICN− and BrCN− , VXCN− (R, θ),
132
to evaluate the ground state wave functions. These are the same as potential
133
surfaces we used in earlier studies [11, 22]. To these surfaces, we added a
134
harmonic potential to describe the CN stretch. In this way, 1 2 V (R, r, θ) = VXCN− (R, θ) + µCN ωCN ∆r2 2
(3)
135
where µCN and ωCN are the reduced mass and harmonic frequency of the CN
136
stretch. The potentials for the excited states were generated in an analogous
137
manner.
138
Using the Wigner distribution, we randomly selected 10 000 sets of coordi-
139
nates and momenta, which were converted from Jacobi coordinates to three
140
Cartesian coordinates and momenta for each atom [23]. Although this ap-
141
pears to introduce a substantial increase in the dimensionality of the sys-
142
tem compared to the two-dimensions used for the quantum dynamics, six of
143
the degrees of freedom in the classical studies represent overall rotation and 7
144
translation, which are separable from the other degrees of freedom. The con-
145
version from Jacobi to Cartesian coordinates was performed to ensure that
146
these degrees of freedom do not contribute to the total kinetic energy [23].
147
We have also introduced an additional internal coordinate, r. As is seen in
148
Eq. (3), there is no potential coupling between r and the other two Jacobi
149
coordinates. The only coupling between these coordinates comes from the
150
r-dependence of the effective mass associated with θ. Since the harmonic
151
potential for the CN stretch is stiff, this coupling is expected to be weak, and
152
we will confirm that there is very little energy transfer between R/θ and r
153
in the following section.
154
Once constructed the initial coordinates and momenta were then propagated
155
using the adiabatic or diabatic representation of the excited state potential
156
functions in the absence of coupling between the electronic states, and on the
157
pair of coupled states using both the adiabatic and diabatic treatments of
158
the dynamics. In cases where couplings between the two states were consid-
159
ered, the system was allowed to sample both states based on Tully’s surface
160
hopping algorithm [10].
161
The potentials used to provide VXCN− (R, θ) in Eq. (3) are the ones devel-
162
oped for earlier quantum dynamics studies. Specifically, the electronic ener-
163
gies of the states of interest were evaluated at the spin-orbit multi-referece
164
configuration interaction level of theory with single and double excitations
165
(SO-MRCISD) using aug-cc-pVTZ basis sets and effective core potentials in
166
place of the core electrons in iodine and bromine [11, 22]. In that work, a
167
diabatic representation of the dynamics was used, and these diabatic poten-
168
tials are represented by the potential curves plotted with the dashed blue and
8
169
green dashed curves in the three panels of Figure 2. These potentials along
170
with the coupling between them are used to describe the diabatic represen-
171
tation in the present study. When we use an adiabatic representation for the
172
semi-classical studies, we evaluate the adiabatic potentials by obtaining the
173
eigenvalues of the 2 × 2 representation of the potential at a specific geometry
174
of the ion. The derivative couplings between these two states are evaluated
175
using the eigenvectors of this 2 × 2 matrix.
176
When a hop between the two surfaces takes place, the kinetic energy is
177
adjusted to ensure conservation of energy. In the adiabatic representation,
178
the velocity parallel to the derivative coupling vector is scaled to ensure
179
energy is conserved. In the diabatic representation, we require that the
180
velocity along the CN bond is unaffected by projecting out the component
181
of the velocity associated with the CN stretch before rescaling the velocities
182
of the individual atoms by a constant scaling factor to conserve energy.
183
One complication that arises is related to differences between how quantum
184
and classical mechanics describe the large amplitude X·CN bend when J = 0.
185
At J = 0 the zero-point energy in the bend corresponds to the harmonic fre-
187
quency (rather than half of that frequency). To account for this, the sampled √ bend coordinate and momentum are each multiplied by a factor of 2. We
188
also perform planar calculations in which this factor is not included, and the
189
corresponding system in the quantum dynamics simulation is confined to a
190
plane.
191
For comparisons with the semi-classical studies, quantum simulations were
192
also performed using the same initial wave functions that were used to gener-
193
ate the Wigner distributions. For the J = 0 calculations, the same approach
186
9
194
was used as has been described previously [11]. While the approach is the
195
same, the final results are slightly different due to the change in the initial
196
wave function. For these calculations, the bend contribution to the wave
197
function is described using a Discrete Variable Representation (DVR) based
198
on Legendre Polynomials. For the planar calculations, the bend is described
199
using a DVR based on particle-on-a-ring eigenstates. Otherwise the quantum
200
dynamics calculations for J = 0 and planar propagations are identical.
201
Results for both representations of the bend are described for excitation to
202
both the 2 Π1/2 and 2 Σ+ states, shown with blue and green lines in the three
203
panels of Figure 2.
204
3. Results and Discussion
205
Before comparing the quantum and semi-classical results we consider dif-
206
ferences between the two types of calculations. The first difference is the
207
dimensionality. The quantum calculations only include R and θ in Figure 3,
208
while the classical dynamics is propagated in all nine Cartesian coordinates,
209
under the constraint that J = 0. As such while the CN bond is rigid in the
210
quantum dynamics, it is allowed to vibrate in the semi-classical simulations.
211
On the other hand, the only coupling between the CN stretch and the other
212
degrees of freedom is through the rotational constant for CN. Since the CN
213
bond is stiff, this coupling is weak and little energy transfer between the the
214
CN stretch and the other degrees of freedom is expected. Analysis of the
215
amount of energy gained by the CN stretch (see Tables S2 and S5) shows
216
that it is < 10 cm−1 following excitation to the 2 Π1/2 state and between 40
217
and 80 cm−1 following excitation to the 2 Σ+ state. When surface hopping 10
218
is introduced the amount of energy transferred to or from the CN stretch
219
increases but remains smaller than 1% of the available energy.
220
The second difference comes in how J = 0 is reflected in the quantum and
221
semi-classical descriptions of the dynamics. Classically when J = 0 a tri-
222
atomic system is constrained to move in a plane, while quantum mechani-
223
cally the system has cylindrical symmetry. As noted above, the zero-point
224
energy in the X·CN bend is doubled in the quantum description compared
225
to a typical bending motion. This is expected to affect on the dynamics. In
226
particular, it should be reflected in the calculated branching fractions for the
227
photoproducts following excitation of BrCN− . This is due to the large role
228
of CN rotation in this branching.
229
The third difference is the one we are exploring here and that is the fully
230
quantum description of the nonadibatic dynamics compared to the surface
231
hopping approach used in these simulations.
232
The results of exciting the ions to the 2 Π1/2 and 2 Σ+ states are provided in
233
Tables 1 and 2, respectively. These tables focus on the branching fractions
234
and the rotational energy distributions. A more complete tabulation of the
235
energy partitioning of the photoproducts can be found in the Supporting
236
Information.
237
To start our analysis of these results, we compare the results obtained with-
238
out consideration of surface hopping. These are denoted as “no SH” in Tables
239
1 and 2. In these cases, we run the trajectories on either the diabatic or adi-
240
abatic potential and, as no transitions between electronic states are allowed,
241
only one product is generated. Comparing the average final rotational ener-
242
gies obtained from the quantum and classical calculations, we note significant
11
243
differences in the results of the J = 0 calculations, with 14 to 20% less energy
244
ending up in rotation in the classical calculation compared to when quantum
245
dynamics simulations are performed. This is attributed to the smaller phase
246
space sampled in these classical calculations, where J = 0 constrains the
247
atoms to move in a plane as opposed to the three-dimensional cylindrically
248
symmetric representation of J = 0 in quantum mechanics. When both cal-
249
culations are performed in a planar configuration, the differences are reduced
250
by an order of magnitude to 0.05 to 2%. While these results are reported for
251
calculations that consider only one potential surface, the importance of CN
252
rotation in determining the branching in BrCN− leads us to expect that the
253
product branching will be strongly affected when non-adiabatic effects are
254
introduced.
255
As we look at the product distributions and branching when surface hop-
256
ping is included in the simulations for the planar calculations, we find that
257
when the semi-classical dynamics is based on a diabatic representations of
258
the potentials the results are in better agreement with the quantum results,
259
although both the diabatic and adiabatic representations of the potential pro-
260
vide reasonably accurate results. One exception is the rotational energy of
261
the CN− product following photoexcitation of ICN− to the 2 Π1/2 state. This
262
is likely due to the relatively small branching fraction (3%) in this channel
263
and the resulting poorer statistics.
264
For the J = 0 calculations on ICN− , the results of the surface hopping calcu-
265
lations in a diabatic representation remain more accurate, although both the
266
adiabatic and diabatic representation of the potentials do a reasonably good
267
job of reproducing the results of the quantum dynamics. The accuracy of the
12
268
J = 0 calculations for BrCN− is less good, and this is attributed to the criti-
269
cal role of CN rotation on the branching. In the case of excitation to the 2 Σ+
270
state, neither surface hopping calculation reproduces the branching obtained
271
from the quantum simulation. While the adiabatic treatment reproduces the
272
branching, the final average rotational energy is in poor agreement with the
273
quantum results. In contrast, calculations in the diabatic representation pro-
274
vide more accurate rotational energy distributions, but the branching ratios
275
are less accurate than those obtained in the adiabatic representation. The
276
fact that calculations using the diabatic representation of the potential sur-
277
faces provide more accurate representations of the dynamics likely reflects the
278
much higher kinetic energy in these systems particularly following excitation
279
to the 2 Σ+ state.
280
As we analyzed the trajectories following photoexcitation of ICN− to the
281
2
Π1/2 excited state, we found a small, but statistically significant subset of
282
trajectories that underwent a frustrated dissociation prior to forming I +
283
CN− products. As is seen in Figures S1, in these trajectories, the I-CN
284
distance first increases, then decreases before the ultimate dissociation. This
285
happens in spite of there being no barrier along the potential, and the system
286
having more than 0.848 eV of kinetic energy in the photoproducts. The origin
287
of the longer-lived trajectories is a centrifugal barrier in R. This arises from
288
the fact that at J = 0 a large rotational kinetic energy in the CN products
289
requires that there is also a large orbital angular momentum in the I· · · CN
290
complex. This result provides support for the proposed explanation for the
291
observation of photoproducts following photoexcitation of ICN− which have
292
a mass consistent with ICN− when the ICN− is initially complexed with as
13
293
few as one argon atom [12]. Still, the large difference between the available
294
0.848 eV of kinetic energy compared to the roughly 0.05 eV binding energy
295
of a single argon atom makes the fact that recombination is observed with
296
only one argon atom present surprising.
297
4. Conclusions
298
In summary, we have explored the efficacy of surface hopping approaches
299
for studying the photodissociation dynamics of ICN− and BrCN− . Overall
300
we found the treatment to be quite accurate, although, in contrast to ear-
301
lier studies, we found the diabatic representation of the coupled electronic
302
states to provide somewhat more accurate description of the dynamics. This
303
is attributed in part to the more diabatic nature of the dynamics given the
304
relatively large kinetic energy release in these systems. Where we find dif-
305
ferences between the quantum and semi-classical results these are attributed
306
primarily to differences in the way the two treatments account for bending
307
motions in a linear molecule, particularly at low rotational energy, and not
308
to deficiencies in the surface hopping algorithm. It would be interesting to
309
explore how the introduction of solvating atoms or molecules affects the dy-
310
namics of these processes and further explore the apparent single atom caging
311
in the dissociation of ICN− in the presence of a single argon atom.
312
Acknowledgments
313
We thank Prof. Andrew S. Petit (California State University Fullerton) for
314
many helpful conversations during the execution of this project. Support of
315
this work by the National Science Foundation through grant CHE-1619660 14
316
is gratefully acknowledged. We also thank the Ohio Supercomputer Center
317
for an allocation of computing resources.
15
318
References
319
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[4] Subotnik, J. E. J. Phys. Chem. A 2011, 115, 12083–12096.
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McCoy, A. B. J. Chem. Phys. 2014, 141, 084305/1–10.
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Lineberger, W. C.; McCoy, A. B. Angewandte Chemie 2012, 51, 2651–
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[13] Case, A. S.; McCoy, A. B.; Carl Lineberger, W. J. Phys. Chem. A 2013, 117, 13310–13318. [14] Martin, J. P.; Case, A. S.; Gu, Q.; Darr, J. P.; McCoy, A. B.; Lineberger, W. C. J. Chem. Phys. 2013, 139, 06415/1–10. [15] Blondel, C.; Cacciani, P.; Delsart, C.; Trainham, R. Phys. Rev. A 1989, 40, 3698–3701. [16] Bradforth, S. E.; Kim, E. H.; Arnold, D. W.; Neumark, D. M. J. Chem. Phys. 1993, 98, 800–810. [17] Andersen, T.; Haugen, H. K.; Hotop, H. J. Phys. Chem. Ref. Data 1999, 28, 1511–1533. [18] Arnold, J.; Nicholls, R. Journal of Quantitative Spectroscopy and Radiative Transfer 1972, 12, 1435 – 1436. [19] Moore, C. E. Atomic Energy Levels, 2nd ed.; National Bureau of Standards: Washington, DC, 1971.
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355
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356
Molecular Anions. Ph.D. thesis, The Ohio State University, 2017.
17
Table 1: Branching ratios and diatomic product rotational energy in cm−1 , when BrCN− and ICN− are initially excited to the 2 Π state (see Fig. 2). Quantum (CN)
Classical (CN− )
(CN− )
Approacha
% X− b
ICN− ; J = 0
Adiabatic (no SH)
100
1372
NA
100
1179
NA
Diabatic (no SH)
0
NA
1662
0
NA
1392
Adiabatic (w/SH)
−−
−−
−−
90
869
2646
Diabatic (w/SH)
96
1346
3871
96
1342
1790
Adiabatic (no SH)
100
742
NA
100
741
NA
Diabatic (no SH)
0
NA
1050
0
NA
1045
Adiabatic (w/SH)
−−
−−
−−
93
521
2599
Diabatic (w/SH)
97
726
3469
97
931
1707
Adiabatic (no SH)
100
2148
NA
100
1743
NA
Diabatic (no SH)
0
NA
2138
0
NA
1764
Adiabatic (w/SH)
−−
−−
−−
36
698
1850
Diabatic (w/SH)
34
973
2304
52
712
2451
Adiabatic (no SH)
100
1354
NA
100
1344
NA
Diabatic (no SH)
0
NA
1453
0
NA
1443
Adiabatic (w/SH)
−−
−−
−−
40
349
1572
Diabatic (w/SH)
55
363
2122
59
474
2225
ICN− ; plane
BrCN− ; J = 0
BrCN− ; plane
Erot
Erot
% X−
(CN)
System
Erot
Erot
a
a complete description of the product state energy distribution is provided in the supporting information
a
Fraction of X− (as opposed to CN− ) products, X = I or Br.
18
Table 2: Branching ratios and diatomic product rotational energy in cm−1 , when BrCN− and ICN− are initially excited to the 2 Σ+ state (see Fig. 2). Quantum % X− b
(CN)
Classical (CN− )
% X−
(CN)
(CN− )
System
Approacha
ICN− ; J = 0
Adiabatic (no SH)
0
NA
6377
0
NA
Diabatic (no SH)
100
6288
NA
100
5071
NA
Adiabatic (w/SH)
−−
−−
−−
24
8225
4580
Diabatic (w/SH)
21
8174
6041
19
8141
4668
Adiabatic (no SH)
0
NA
4591
0
NA
4525
ICN− ; plane
BrCN− ; J = 0
BrCN− ; plane
Erot
Erot
Erot
Erot
5247
Diabatic (no SH)
100
4307
NA
100
4227
NA
Adiabatic (w/SH)
−−
−−
−−
19
8102
3911
Diabatic (w/SH)
14
7643
4163
14
8007
3936
Adiabatic (no SH)
0
NA
7450
0
NA
6026
Diabatic (no SH)
100
7590
NA
100
6063
NA
Adiabatic (w/SH)
−−
−−
−−
52
8036
4723
Diabatic (w/SH)
70
8445
6328
52
8332
4146
Adiabatic (no SH)
0
NA
5573
0
NA
5481
Diabatic (no SH)
100
5555
NA
100
5451
NA
Adiabatic (w/SH)
−−
−−
−−
49
7645
4055
Diabatic (w/SH)
54
7849
3544
49
8340
3221
a
Representation of the potential with or without surface hopping between states
a
Fraction of X− (as opposed to CN− ) products, X = I or Br.
19
FIGURES:
357
1.0
I* + CN
I* + Br
0.9 0.8
I + CN
I + Br*
Energy, eV
0.7 0.6
Br + CN
0.5
Br* + CN
0.4 0.3
I + Br
0.2 0.1 0.0
I + Br
I + CN
IBr
ICN
Br + CN BrCN
Figure 1: Relative energies of the photoproducts following photodissociation of IBr− , ICN− and BrCN− , based on electron affinities of I [17], Br [15] and CN [16], the energy of CN∗ [18] as well as the spin-orbit splittings of I and Br [19]. The products that contain electronically excited CN lie outside the scale of the plot with I− +CN* having an energy of 1.9 eV and Br− +CN* has an energy of 1.6 eV.
20
Br C N R
(a)
3
4
Br N C R
(b)
3 2Π
3/2/
2Π
Br– + CN*
2Σ+
1
2Π
0
-1
2Π
1/2
3/2
V, eV
V, eV
2
2
4
6
R, Å
2Π
1
8
10
2Π
1/2
Br– + CN Br* + CN– Br + CN–
3/2
2Σ+
2
2Π
2
6
R, Å
2Π
1/2
8
10
2Σ+
I* + CN–
1 2Π
0
-1 4
3/2/
I– + CN* Br– + CN*
2Σ+
-1
I C N R
(c)
2 3/2/ Π1/2
2
Br* + CN– Br– + CN 0 Br + CN–
2Σ+
4
3 2Π
1/2
V, eV
4
2Π
I– + CN
1/2
I + CN–
3/2
2Σ+
2
4
6
8
10
R, Å
Figure 2: One-dimensional cuts through the collinear (a) BrCN− (b) BrNC− and (c) ICN− potential surfaces, with the geometries illustrated at the top of the panels. In all cases, the solid lines represent the adiabatic potential surfaces, while the dashed lines provide the corresponding diabatic states. For consistency, the colors of the lines correspond to the asymptotic photoproducts. The initial average excitation energies considered in this work are shown with solid lines for excitation to the 2 Π1/2 state and dashed lines for excitation to the 2 Σ+ state. These lines correspond to excitation energies of 2.7 and 4.2 eV, respectively.
21
q
C
X R
N
Figure 3: The Jacobi coordinates (R, θ) used to define the XCN− (X = Br or I) geometry.
22
Highlights: -
Exploration of classical treatments of the photodissociation of ICN- and BrCNRigorous test of surface hopping in an adiabatic and diabatic representation Effect of differences in classical and quantum treatments of J = 0 on product branching
1.1
1.0
1.0
0.9
X* + CN0.9 V(R), eV
V(R), eV
1.1 Graphical Abstract 2Σ+
0.8 2Π
0.7
1/2
0.6
0.8
2Σ+
X* + CN 2Π
1/2
0.7 0.6
X + CN
X + CN
0.5
0.5
3
4
5
6 R, Å
7
8 3
4
5
6 R, Å
7
8