Topology of conical/surface intersections among five low-lying electronic states of
CO2: Multireference configuration interaction calculations
Bo Zhou, Chaoyuan Zhu, Zhenyi Wen, Zhenyi Jiang, Jianguo Yu, Yuan-Pern Lee, and Sheng Hsien Lin
Citation: The Journal of Chemical Physics 139, 154302 (2013); doi: 10.1063/1.4824483
View online: http://dx.doi.org/10.1063/1.4824483
View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/139/15?ver=pdfcov Published by the AIP Publishing
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Topology of conical/surface intersections among five low-lying
electronic states of CO
2: Multireference configuration interaction
calculations
Bo Zhou,1,2Chaoyuan Zhu,1,a)Zhenyi Wen,2Zhenyi Jiang,2Jianguo Yu,3Yuan-Pern Lee,1 and Sheng Hsien Lin1
1Department of Applied Chemistry, Institute of Molecular Science and Center for Interdisciplinary
Molecular Science, National Chiao-Tung University, Hsinchu 30050, Taiwan
2Institute of Modern Physics, Northwest University, Xi’an 710069, People’s Republic of China 3Department of Chemistry, Beijing Normal University, Beijing 100875, People’s Republic of China (Received 16 April 2013; accepted 25 September 2013; published online 14 October 2013)
Multi-reference configuration interaction with single and double excitation method has been utilized to calculate the potential energy surfaces of the five low-lying electronic states1A1,1A2,3A2,1B2, and3B
2 of carbon dioxide molecule. Topology of intersections among these five states has been fully analyzed and is associated with double-well potential energy structure for every electronic state. The analytical potential energy surfaces based on the reproducing kernel Hilbert space method have been utilized for illustrating topology of surface crossings. Double surface seam lines between 1A
1 and3B2states have been found inside which the3B2 state is always lower in potential energy than the 1A
1 state, and thus it leads to an angle bias collision dynamics. Several conical/surface intersections among these five low-lying states have been found to enrich dissociation pathways, and predissociation can even prefer bent-geometry channels. Especially, the dissociation of O(3P)+ CO can take place through the intersection between3B
2and1B2states, and the intersection between3A2 and1B
2states. © 2013 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4824483]
I. INTRODUCTION
The collisions of O(3P, 1D) with CO(1+) have been widely studied both experimentally and theoretically.1–6 A
large amount of electronic energy can be efficiently trans-ferred through a spin-orbit induced surface/conical intersec-tion involving a long-lived (several vibraintersec-tions) collision com-plex intermediates. The vibrational relaxation of CO(1+) by O(3P) is several orders of magnitude faster than predicted by the conventional theory of translational to vibrational energy transfer.7 In CO
2 photodissociation experiments,8 the disso-ciation at 185 nm is considered as a spin forbidden process which involves two steps: an electronic transition from the ground state (11+) to an upper bounded single state 1B2 and then transition to an unbounded triplet 3B2 state. The transition from the 3B2 state back to the ground state can take place only in the presence of some perturbation such as spin-orbit couplings. However, the spin-orbital induced sur-face crossing between 3B
2 and1A1 states cannot explain an unusual enrichment of17O isotope.9 Therefore, the topology
of surface/conical intersections between the lowest electronic singlet and triplet states must play an important role for the dissociation mechanism of carbon dioxide molecule.
The conical/surface intersection problem of carbon dioxide molecule has attracted attention for high-level ab
initio quantum chemistry calculations. Julienne et al.10
have reported self-consistent-field and small configuration-interaction calculation of the potential energy curves along
a)Electronic mail: [email protected]
the bending coordinate for the four lowest bent singlet states. Simkin et al.11 have calculated the potential energy surfaces
(PESs) of the ground state (1+
g) and excited electronic states (3B2,1B2), and their calculations indicated that no crossing point is found between1B
2 and3B2 states. However, by us-ing the complete active space self consistent field (CASSCF) and multireference configuration interaction (MRCI) meth-ods, Spielfiedel et al.12 found that there are surface
cross-ing regions between 1B
2 and 3B2 states at CO bond length
RCO= 1.243 Å and OCO bond angle in 90◦< αOCO<100◦. Spielfiedel et al.12 also pointed out that the predissociation
of 1B
2 state can occur through a triplet state3A2 intersec-tion with the singlet state1B
2. Therefore, it is still not very clear whether or not the 1B
2 state has intersection with3A2 state or/and3B
2 state. The purpose of the present work is to study this problem and we try to carry out extensive high-level ab initio calculations for potential energy surfaces of the ground and excited states in order to understand the dis-sociation mechanism of carbon dioxide molecule. We have calculated PESs of the low-lying states 1A
1,1B2,1A2,3B2, and3A
2using the un-contracted MRCI method, and then we have fitted the calculated PESs into the analytical form to ex-plore topology of intersections among these five low-lying electronic states.
The paper is organized as follows: in Sec.IIwe present a brief description of the high-level ab initio calculation meth-ods and algorithm of analytical fitting to PES. Then we report detailed analysis and discussion of potential energy surfaces and related surface/conical intersections in Sec.III. Conclud-ing remarks are given in Sec.IV.
154302-2 Zhouet al. J. Chem. Phys. 139, 154302 (2013)
II. COMPUTATIONAL METHODS A. High-levelab initio calculations
The ground and several low-lying excited electronic state PESs of CO2 are calculated at the un-contracted multi-reference configuration interaction with single and double excitation (MRCISD) level using the Xi’an-CI code.13 The
un-contracted MRCI(SD) method is based on the graphical unitary group approach (GUGA) and the hole-particle cor-respondence. The details of the method have been given in the original publications.13–16 We utilized the state averaged
CASSCF (SA-CASSCF) method with cc-pVTZ(6d, 10f) ba-sis sets to generate the reference space by the GAMESS pro-gram package.17 The multiconfiguration self-consistent field
wave function consists of a complete configuration expansion for 12 electrons distributed in the 10 active valence orbitals. The active space CAS(12, 10) is chosen to have over 3000 configuration state functions (CSFs) under C2vgroup symme-try and 7000 CSFs under Csgroup symmetry. If all CAS con-figurations are chosen as the reference states in the MRCI(SD) calculation with the cc-pVTZ basis set, the number of CSFs can exceed 3.4× 107and 6.1× 108under C
2vand Csgroup symmetry, respectively. It is extremely time-consuming for constructing potential energy surfaces under such high level calculation. In order to reduce computational cost, we only keep those CSFs with absolute coefficients larger than a pres-elected threshold in the reference space.
For instance, the number of CSFs in CI space is 34 964 540 without threshold selection, and it is reduced to 7 202 161 with the threshold set up to be 0.01 for ground state 1A1 with CO bond length Rco = 1.164 Å and OCO angle αoco = 180◦. We found that the electronic energy of the ground state is −5119.4911 eV (−5120.0092 eV with Davidson correction) without the threshold selection, while it is−5119.4710 eV (−5120.0258 eV with Davidson correc-tion) with 0.01 threshold selection. There is the correlation energy loss of 0.0201 eV with 0.01 threshold selection, but computation cost is reduced by 80%. For electronic triplet3B
2 state, the number of CSFs in CI space is reduced to 7 804 228 from 64 665 634 by 0.01 threshold selection and the correla-tion energy loss is 0.0633 eV. We think that such small cor-relation energy loss is quite acceptable for potential energy surface calculations. On the other hand, we found that the electronic energy of the ground1A
1 state is−5119.3967 eV calculated by internally contracted MRCI method imple-mented by Molpro software,18,19 and this is 0.0955 eV higher than −5119.4911 eV obtained by the present un-contracted MRCI method. In general, the electronic energy calculated from internally contracted MRCI (ic-MRCI) is higher than that calculated from un-contracted MRCI method. This difference can become bigger while the basis set is smaller. In the following, all the calculations in the present work are based on un-contracted MRCI with the threshold selection 0.01.
One of the authors (Z.W.) has recently developed an improved version of the configuration-based multi-reference second-order perturbation approach (CB-MRPT2) according to the formulation of Lindgren on perturbation theory of a de-generate model space.20 The
diagonalize-then-perturb-then-diagonalize (DPD) model has been implemented and thus the DPD-MRPT2 method is considered to be an approximate MRCISD. It is most capable to rectify the shortcomings of the CASPT2 method.21 Therefore, we also carried out the DPD-MRPT2 calculations in the present study.
B. Analytical fitting of potential energy surface In order to explore topology of surface/conical intersec-tions among the five low-lying electronic states, we fit the calculated PESs into the analytical functions. We apply the reproducing kernel Hilbert space (RKHS) method22 to con-struct the analytical PESs. Two-dimensional (2D) analytical potential energy surface is constructed as a function V(R, θ ) in which R is the CO bond length and θ is the OCO bond an-gle. Furthermore, we introduce the new variables x= αR (α is the scaling factor, and 0.5 is used in our calculations) and y = (1 + cos θ)/2, so that 0 ≤ x < ∞ and 0 ≤ y ≤ 1. We adopt the reciprocal power (RP) and Taylor spline (TS) reproducing kernel as follows:23 Qn1,m1;n2(x, y, x , y)= qRP n1,m1(x, x )qT S n2 (y, y ), (1) where qRP n1,m1(x, x ) is defined as qn,mRP(x, x)= n2x−(m+1)> B(m+ 1, n)2 ×F1 −n + 1, m + 1; n + m + 1,x< x> (2)
in which x> and x< represent the larger and the smaller, re-spectively, for x and x. B(a, b) is the beta function and2F1(a,
b; c; z) is the Gauss’ hypergeometric function. The TS
repro-ducing kernel is given by
qnT S(x, x)= n−1 i=0 xixi+ ξnT(x, x) (3) and ξnT(x, x)= nx<nx>n2−1F1 1,−n + 1; n + 1;x< x> . (4)
We use Q3, 5; 7(x, y, x, y) with n1= 3, m1= 5, and n2= 7 in Eq.(1)to construct the 2D PESs of CO2. This means that cal-culations are performed under the C2vgroup symmetry. Each of the PESs is obtained from the completely filled 13× 60 2D grids in which there are 780 ab initio data points for CO bond length in the region of [0.6, 2.37] Å with step interval 0.03 Å and the bond angle in the region of [60◦, 180◦] with step in-terval 10◦. The overall root-mean-square (rms) error for the analytical PES is estimated as 0.0438 eV for1A
1, 0.0846 eV for1A
2, 0.0494 eV for1B2, 0.0403 eV for3B2, and 0.0503 eV for3A
2.
III. RESULTS AND DISCUSSION A. Vertical excitation energies
The ground state of CO2 is in the linear
ge-ometry 1+
TABLE I. Orbital space and main configurations of CO2 for the ground
state and the low-lying excited states in D∞h and C2vsymmetry,
respec-tively. Core orbitals (1σ2
g1σu22σg22σu23σg2) that are all doubly occupied are
not shown here.
Orbital 6 7 8 9 10 11 12 13 14 15 C2v 4a1 3b2 1b1 5a1 1a2 4b2 6a1 2b1 7a1 5b2
D∞h 4σg 3σu 1πu 1πu 1πg 1πg 2πu 2πu 5σg 4σu 1+ g(1A1) 2 2 2 2 2 2 0 0 0 0 1− u(1-1A2) 2 2 2 2 1 2 1 0 0 0 1u(2-1A 2) 2 2 2 2 2 1 0 1 0 0 1u(1-1B 2) 2 2 2 2 1 2 0 1 0 0 2 2 2 2 2 1 1 0 0 0 1 g(3-1A 2) 2 2 2 2 1 2 0 0 1 0 1 g(2-1B 2) 2 2 2 2 2 1 0 0 1 0 3+ u(1-3B2) 2 2 2 2 2 1 1 0 0 0 3u(2-3B 2) 2 2 2 2 1 2 0 1 0 0 3u(1-3A 2) 2 2 2 2 1 2 1 0 0 0 3− u(2-3A2) 2 2 2 2 2 1 0 1 0 0 3 g(3-3B 2) 2 2 2 2 2 1 0 0 1 0 3 g(3-3A 2) 2 2 2 2 1 2 0 0 1 0
shown in Table I. The electronic configurations of the lowest excited states of CO2 at linear geometry are (1σ2
g1σu22σg22σu23σg23σu24σg21πu21πu21πg21πg12πu1) for
u−, u+, u states and (· · · 1πu21πu21πg21πg15σg1) for the g states. The 5σg and 2πu (D∞h symmetry) are molecular or-bitals corresponding to 6a1, 2b1, and 7a1, respectively, in C2v group symmetry. The 5σgorbital has mixed valence-Rydberg character so that large augmented polarized basis set should be used. As pointed out by Spielfiedel,12 in any electronic
configuration in which either of these two a1 orbital (6a1, 7a1) are occupied, the energetically lower a1 orbital always has more 2πu character. The same conclusion is true for the triplet states. However, the behavior of these two a1states is opposite in singlet states1A2and1B2. The orbital in 6a1and 7a1 with higher energy has more 2πu character. Thus, 5σg (6a1) molecular orbital has lower energy than 2πu(2b1, 7a1) orbital at the D∞hgroup symmetry.
The ground state of CO2 in C2v group symmetry is the 1A
1 state with electronic configurations (see Table I) (1a2
11b222a122b223a123b224a215a121b211a224b22). The B2 states are correlated with (4b1
21a226a11), (4b221a122b11), and (4b121a227a11) configurations. It is found that the mixing of the 1B
2 con-figurations designated as 4b1
21a226a11 (11B2) and 4b221a122b11 (21B
2) is required to ensure that the final 1B2 states ap-proach correctly into 1
g and 1u states at linear geom-etry. In the calculation of triplet 3B
2 states, we optimize 3B
2 states by a state-average of all the three states. The A2 states are correlated with (4b221a216a11), (4b221a217a11), and (4b211a222b11) configurations. Similarly, we optimize3A2states in a state-averaged procedure with the same weight. The 3A
2 states and 3B2 states succeed to converge to the 3u and 3
g states, respectively, as shown in Fig. 1. For the 1A
2 states, the computed 1A2 state approaches to 1u and 1−
u states correctly if the configurations are designated as the mixing of 11A
2 and 21A2 states. But if three-state av-erage procedure is used during the optimization, the results
FIG. 1. The one-dimensional potential energy curves for the low-lying ex-cited states of CO2along the bond angle in the region of 190◦> αoco>170◦
at fixed bond length Rco= 1.111 Å and 1.164 Å, respectively. (a) At Rco
= 1.111 Å for singlet states, the solid lines correspond to1A
2and dashed
lines correspond to1B2. (b) The same as (a) except at 1.164 Å. (c) At Rco
= 1.111 Å for triplet states, the solid lines correspond to3B
2 and dashed
lines correspond to3A
2. (d) The same as (c) except at Rco= 1.164 Å.
converge to ˜ X 4a2 13b225a211b124b21a222b1(0.6358) 4a123b225a211b124b221a27a1(0.6802) , ˜ A 4a213b225a121b214b21a222b1(−0.6766) 4a2 13b225a121b124b221a27a1(0.6362) , and ˜ B4a213b225a121b214b221a26a1(0.9146) . The ˜X, ˜A, and ˜Bstates correspond to the1−
u,1u, and1 g states, respectively. While the two1B
2 states succeed to con-verge to the ˜X(1 uat 180◦) and ˜A(1 gat 180◦) as follows: ˜ X 4a213b225a121b211a224b27a1(0.6598) 4a2 13b225a121b121a24b222b1(0.6526) and ˜ A4a123b225a121b211a224b26a1(−0.9212) . This causes that energy of the 1
u state is about 0.04 eV higher estimated from 1A
2 than from1B2 state. The energy of1
154302-4 Zhouet al. J. Chem. Phys. 139, 154302 (2013)
TABLE II. Vertical excitation energies of CO2for Rco= 1.164 Å (2.20 Bohr) and αoco= 180◦, and all energies
are relative to the energy of the electronic ground state E(1A1)= −5119.4704 eV. (−5120.0255 eV with Davidson
correction) at linear geometry. States
D∞h C2v MRCI (0.00) MRCI+ Qa MRCI(0.01)b MRCI+ Q DPD-MRPT2 3+ u 3B2 8.27 8.19 8.38(8.35c) 8.15 8.23 3u 3A 2 8.77 8.72 8.81(8.83) 8.56 8.77 3B 2 8.79 8.73 8.84(8.83) 8.80 8.77 3− u 3A2 9.16 9.14 9.22(9.21) 9.18 9.20 3 g 3A 2 9.71 9.55 9.77(8.61) 9.54 10.05 3B 2 9.71 9.52 9.82(8.61) 9.58 10.09 1− u 1A2 9.14 9.11 9.17(9.19) 8.90 9.19 1u 1A 2 9.25 9.17 9.28(9.28) 9.26 9.30 1B 2 9.48 9.38 9.48(9.28) 9.34 9.45 1 g 1A 2 10.00 10.25 10.10(9.00) 9.89 10.42 1B 2 9.92 9.84 10.09(9.00) 9.88 10.17
aUn-contracted MRCI method with Davidson correction (+Q).
bUn-contracted MRCI method with the reference configuration threshold selection set up to be 0.01. cThe theoretical results from Spielfiedel’s work12with internally contracted MRCI method.
from1B2state. We also notice that the equilibrium geometry of 1
g state (31Aand 31A) is linear with nonsymmetrical CO bond lengths under Cssymmetry.
The vertical excitation energies are summarized in TableIIwith the comparison to theoretical work of Spielfiedel
et al.12 The present results are inconsistent with the recent
work by Grebenshchikov,24 in which the vertical excitation
energies are in the order of E(1−
u) < E(1u) < E(1 g) based on larger basis set and internally contracted MRCI method. This is different from Knowles25 and Spielfiedel et al.’s12
work which gives the order of E(1
g) < E(1u−) < E(1u). For the triplet states, vertical excitation energies are in the order of E(3u+) < E(3u) < E(3 g) < E(3u−) from the present calculation and it is different from Spielfiedel et al.’s12 work in the order of E(3u+) < E(3u) < E(3u−) < E(3 g). For the adiabatic excitation energies, both the present and Spielfiedel et al.’s12 methods turn out to be 5.53 eV and
5.80 eV (5.62 eV and 5.88 eV with configuration selection) for1A
2and1B2, respectively. This is in agreement with Clyne and Thrush’s suggestion26that the first excited singlet states
of CO2are bent1A2and1B2states which are fairly similar in adiabatic excitation energy. The equilibrium bond length of 1A
1state in the present calculation is 1.162 Å that is slightly longer than the experimental value 1.160 Å.27 The
equilib-rium geometries of the lowest 1,3A
2 and 1,3B2 states calcu-lated from available MRCISD methods are summarized in Table III. It is only 1B2 state that can be directly compared with the experimental values. In Dixon’s classical work,28the rotational structures of1B2 state have been detected, and the equilibrium geometry of1B2state is estimated as bond length
Rco= 1.246 ± 0.008 Å and bond angle αOCO = 122◦ ± 2◦. The present calculation predicted Rco = 1.255 Å and αOCO = 118.0◦for1B
2state and it is in reasonable agreement with the experimental value. As shown in Table III, the excita-tion energy from un-contracted MRCISD method is closest to the experimental value. The differences of vertical excita-tion energies between un-contracted and internally contracted
MRCISD methods are in the range of 0.01 to 0.22 eV. The energy differences induced by the configuration selection are in the range of 0.05 to 0.23 eV. The present un-contracted MRCISD method without configuration selection gives the lower excitation energy as compared with the internally con-tracted MRCISD methods.
Furthermore, we carried out the DPD-MRPT2 cal-culations to conform that both un-contracted MRCI and
TABLE III. Optimized geometries of the low-lying bent excited states of CO2with available MRCISD methods.
Geometry
Dimension of Emrci
States Methods Rco(Å) αoco(◦) CI spaces (eV) 1B 2 MRCISDa 1.255 118.0 34909004 5.80 MRCISD(0.01)b 1.265 118.5 6201153 5.88 ic-MRCISDc 1.254 118.0 666920 5.83 ic-MRCISD(0.01)d 1.258 117.1 374383 6.06 Expt.e 1.246 122.0 5.70 1A 2 MRCISD 1.252 127.0 34657216 5.53 MRCISD(0.01) 1.255 127.5 11583986 5.62 ic-MRCISD 1.254 127.4 737290 5.54 ic-MRCISD(0.01) 1.253 127.4 562053 5.53 3B 2 MRCISD 1.245 118.5 64665634 4.66 MRCISD(0.01) 1.251 114.0 13362429 4.83 ic-MRCISD 1.247 118.2 1110878 4.88 ic-MRCISD(0.01) 1.242 119.0 658302 4.72 3A 2 MRCISD 1.254 127.3 64259728 5.32 MRCISD(0.01) 1.250 129.0 22097203 5.37 ic-MRCISD 1.254 127.7 1111126 5.35 ic-MRCISD(0.01) 1.253 127.7 825967 5.33
aUn-contracted MRCISD method from the Xi’an CI program.
bUn-contracted MRCISD method with configuration selection (threshold 0.01) from the
Xi’an CI program.
cInternally contracted MRCISD method from the MOLPRO program suite.18 dInternally contracted MRCISD method with configuration selection (threshold 0.01)
from the MOLPRO program suite.18 eExperimental values from Dixon’s work.28
DPD-MRPT2 methods give the same order for excitation en-ergies as shown in Table II and Table S1 in supplementary material.34 We believe that the present un-contracted MRCI method predicted correct order of vertical excitation energies for the lower excited states.
B. 1D potential energy curves for bent valence excited states of CO2
We first carried out calculations for the one-dimensional potential curves in the region of the bond angle 190◦> αoco
>170◦at fixed bond length Rco= 1.111 and Rco= 1.164 Å, respectively, for the low-lying excited states. 1D potential curves are plotted in Fig.1. We found that the lowest1B
2state (1
gat 180◦) forms a conical intersection with the second1B2 state (1
uat 180◦) at Rco= 1.111 Å. For three singlet 1A
2 states, the dominating configura-tions are (4b2
21a216a11), (4b211a222b11), and (4b221a217a11) in en-ergy sequence, the 3-1A
2 (1-1A2) has local minimum (max-imum) at OCO angle around 180◦ where these two states form an avoided crossing, while the 2-1A
2 state is almost unchanged in bond angle 190◦ > αoco >170◦ as shown in Figs.1(a)and1(b).
For the triplet states of CO2, the conical intersection is formed between the 2-3A2and 3-3A2 at Rco= 1.111 Å and
αoco= 180◦as shown in Fig.1(c), and three triplet3B2states have similar behavior to their counterpart of three singlet1A
2 states. Especially, the dominating configurations are also the same. Both 3-3A
2 (1-3A2) and 3-3B2(2-3B2) states correctly approach to 3
g (3u) state at linear geometry. An avoided crossing is formed between the 3-3B
2and 2-3B2states at Rco slightly short than 1.111 Å and αoco = 180◦ as shown in Fig.1(c).
We again carried out DPD-MRPT2 calculations for con-ical intersections and avoided crossings for both low-lying singlet and triplet states, and detailed information is shown in Figs. S1, S2, S3, S4, and S5 in supplementary material.34
DPD-MRPT2 calculations basically conform what we have obtained from un-contracted MRCI calculations discussed above. Besides, the 2-1A2, 2-3A2, and 2-3B2states, which are predominantly described by the electronic configurations in-volving the b1component of the 2πumolecular orbital, have only very small barriers at linear geometry. This complicates photodissociation of CO2by the fact that the valence singlet and triplet states1,3−
u and1,3u form conical intersections with the diffuse1,3
g states in the Franck-Condon region of the absorption spectrum.
C. 2D potential energy surfaces for bent valence excited states of CO2
We next carried out calculations for the two-dimensional potential contours in terms of bond lengths Rcoand the OCO angle αocounder C2v group symmetry. Then we utilized ana-lytical fitting method introduced in Sec.II Bto obtain analyti-cal potential energy surfaces for the ground state1A
1, and the low-lying excited states1A
2,3A2,1B2, and3B2as plotted in Fig.2. * 6.33 (a) 0.8 1 1.2 1.4 1.6 1.8 2 2.2 1A 1 60 80 100 120 140 160 180 5.62 (b) 0.8 1 1.2 1.4 1.6 1.8 2 2.2 1A 2 60 80 100 120 140 160 180 5.37 (c) 0.8 1 1.2 1.4 1.6 1.8 2 2.2 3A 2 60 80 100 120 140 160 180 5.88 (d) 0.8 1 1.2 1.4 1.6 1.8 2 2.2 1B 2 60 80 100 120 140 160 180 4.83 (e) 0.8 1 1.2 1.4 1.6 1.8 2 2.2 3 B2 60 80 100 120 140 160 180
FIG. 2. Contour plot of the fitted potential energy surfaces for the low-lying electronic states (a)1A
1, (b)1A2, (c)3A2, (d)1B2, and (e)3B2. The two
axes are the bond lengths Rco in Å, and the OCO angle αoco in degree.
The minimal values of the potential energy surfaces have been plotted in eV with respect to the ground-state1A
1 minimum. The contour increment
is 0.05 eV.
We found that the ground state potential energy surface has a double-well structure with the global minimum at bond length Rco= 1.162 Å (linear geometry) and local bent mini-mum at Rco= 1.32 Å and α = 73.4◦. There is an energy gap of 6.33 eV between the two minima. The C2vconstrained saddle point that was found at Rco= 1.30 Å and α = 91◦connects these two minima with potential barrier of 0.433 eV above the bent minimum. The transition state of the ground state is very close to the local bend minimum. The present calculation is basically in agreement with the result obtained by Xanth-eas and Ruedenberg,29which estimate the saddle point at R
co = 1.32 Å and α = 94.2◦with a potential barrier of 0.607 eV above the local bend minimum. All other four excited states 1A
2,3A2,1B2, and 3B2are quite similar to the ground state 1A
1, and they all have double-well structures. The local min-ima of1A
2 and1B2 states are located at smaller bond angle (αOCO<60◦) which are out of the plotted region in Fig.2.
154302-6 Zhouet al. J. Chem. Phys. 139, 154302 (2013)
D. The topology of surface/conical intersections among1A
1,1A2,3A2,1B2, and3B2states
We finally exhibited the topology of intersections among five electronic states1A
1,1A2,3A2,1B2, and3B2, and we an-alyzed the reaction pathways via those surface intersections. It should be noted that the3B
2state is the lowest triplet state of CO2and it is correlated to the lowest3Aand3u+states in
Csand D∞hgroup symmetry, respectively. The surface crossing between1A
1and3B2states has the great influence on the dynamical behavior of O–CO reaction. We found that this singlet-triplet surface crossing of1A
1and 3B
2 states occurs at CO bond length Rco >1.6 Å when the OCO angle is larger than 120◦, as shown in Fig.3(a). There is an interesting area for surface crossing from bond angle 85◦ to 105◦, within which the3B2 surface is always below the 1A1 surface for any bond length, and this special area has energy about 5.4 eV above the ground state. It should be noted that3B2 state at the equilibrium geometry has energy just 4.85 eV above the ground state. This special area is very close to the photodissociation threshold that has energy 5.45 eV above the ground state.30 It is well known that the
singlet and triplet potential energy surface crossing happens as the elongation of CO bond length (Rco > 1.6 Å).4 The present calculation shows that the bending vibration mode at
high quantum number can also lead to the singlet and triplet surfaces crossing, and it results in another pathway for the re-laxation of CO+ O(3P) system.
The crossing between 1B2 and 3B2 states also has the great importance in understanding the photodissociation pro-cess of CO2. From Simkin et al.’s MRCI calculation,11 no crossing point is found between 1B
2 and3B2 states. On the other hand, from Spielfiedel et al.’s MRCI calculation,12the
crossing area is found at the bond length 1.243 Å and bond angle from 90◦ to 100◦. The present calculation does show crossing region around bond length 1.0 Å and bond angle 180◦ as plotted in Fig.3(b), and otherwise the1B
2 state has potential energy higher than the 3B
2 state. This area is very close to the conical intersection point of the first and second 1B
2 state reported in Sec. III B, and thus it can play an im-portant role in the photodissociation process of CO2. How-ever, the area of surface intersection between 1B2 and 3B2 has energy about 13 eV above the 1A1 ground state. From the present calculations, we notice that the energy difference between the1B
2and3B2states is small (smaller than 1.0 eV) when the OCO angle is smaller than 140◦. Vibrational excita-tion is very important when the interacexcita-tion between these two states is taken into account. This makes the interaction of1B
2 and3B
2 surfaces complicated. At about Aoco = 120◦ which is close to the minimum of3B
2 states, the energy level is in
FIG. 3. The topology of conical intersections between (a)1A
1and3B2states, (b)3B2and1B2states, (c)1B2and3A2states, (d)1A1and3A2states, (e)1A2
and1B
the region of 5.0–6.0 eV, which is consistent with the surface intersection area (5.72 eV) suggested by Lin’s work31 and Kinnersley’s work.32 From this point of view, the present re-sults agree with the experiments and thus we present an alter-native interaction mechanism. The detailed analysis is shown in Fig. S6 of supplementary material.34
The surface crossing between1B
2and3A2states is found to occur in two regions as shown in Fig.3(c); one is near the linear structure and bond length Rco < 1.05 Å and another is in the region at bond angle αOCO <140◦. Therefore, we think that the predissociation of1B
2state may occur through surface crossing between triplet state3A
2and the singlet state 1B
2.
We found that the 3A
2 state has energy always higher than the 3B2 states, and there is no crossing between these two states. The3A2 and3B2 states became degenerate at CO bond length Rco>2.3 Å. However, the3A2surface has cross-ing with the1A1 surface in the region at the bond angle αoco
>118◦and bond length Rco>1.85 Å as shown in Fig.3(d). The present calculation shows complicated topology pat-tern of surface crossing between1A
2and1B2states, and these two potential surfaces are quite close in energy as shown in Fig.3(e). We do not find crossing point between1A
1and1A2 states. On the other hand, we did find conical intersection be-tween the1A
1and1B2states, which takes place in the region of Rco<1.2 Å and 92◦< αoco<100◦as shown in Fig.3(f). This crossing area is overlapped with the crossing area be-tween3B
2and1A1states.
In order to demonstrate important reaction pathways, we plot the crossing seams in Fig. 4 between the1A1 and3B2 states, the1B2and3A2states, and the1B2and1A2states. We found there are double crossing seams between the1A1 and 3B
2 states as shown in Fig.4(a), the lower seam has energy 4.89 eV above the ground state at local minimum of the geom-etry Rco= 1.28 Å and αoco= 108◦, while the upper seam has energy 6.63 eV above ground state at local minimum corre-sponding to the geometry Rco= 1.33 Å, αoco= 85.6◦. There is only one crossing seam between1B
2and3A2states as shown in Fig.4(b), and it has energy 5.95 eV above the ground state at local minimum Rco= 1.25 Å and αoco= 109◦. The crossing seam of1A
2and1B2have three parts, one is bond length Rco
>1.61 Å and αoco>150◦with the energy 11.0 eV above the global minimum. The second is Rco<1.0 Å and αoco>150◦ with the energy 15.9 eV above the global minimum. The third is the continuous seam as bond angle smaller than 140◦ as shown in Fig.4(c), and it has energy 5.92 eV above the ground state at local minimum of Rco= 1.25 Å and αoco= 113◦. This means that the crossing points for these two pairs (one is1B2 and3A
2, and another is1A2and1B2) are quite close in both energy and geometry. We found there is energy jump at Rco = 1.64690 Å as shown in attached window of Fig.4(c)and we have analyzed that this is due to projection from two di-mensions (see Fig.3(e)) to one dimension.
We conclude from the present study that the predissoci-ation of1B
2 state can occur through the surface intersection between the triplet 3B
2 state and the singlet1B2 state, and through the intersection between the triplet3A
2state and the singlet1B
2 state. It has been known for a long time that all these states can predissociate in linear structures.33 However,
FIG. 4. Conical intersection seam lines between (a) the1A1and3B2states,
(b) the1B2and3A2states, and (c) the1B2and1A2states, which are
corre-sponding to Figs.3(a),3(c), and3(e), respectively. The seam plotted in (c) represents the intersection of1B
2and1A2states when the OCO bond angle
less than 140◦. All energies are relative to the global minimal of the ground state (1A1 state). The bond length is given in Angstrom, and the energy is
in eV.
we found that there are even more predissociation pathways from bent geometry as shown in Figs.3and4.
IV. CONCLUDING REMARKS
Un-contracted MRCISD method has been used to calcu-late the potential energy surfaces of the five low-lying elec-tronic states1A
1,1A2,3A2,1B2, and3B2of CO2. By applying the reproducing kernel Hilbert space method to fit potential energy surfaces of these five low-lying states in the analytical forms, we demonstrated topology of the conical/surface in-tersections near the equilibrium geometry of these five states. We first confirmed that the vertical excitation energies from the present level calculations agree with the recent high-level MRCI calculations, and we also noticed the deviations from the early internally contracted MRCI works. Then we extended the present un-contracted MRCI method to calcu-late conical/surface intersections in both linear geometry and bend geometry of CO2. We found that conical intersection be-tween singlet1uand1 gstates in the case of linear geom-etry corresponds to conical intersection between the first and second1B
2states in the case of bend geometry at bond length
Rco= 1.111 Å. Actually, we found that the conical intersec-tion between triplet 3
u and 3 g states occurs at the same region of the corresponding two singlet states. In the case of bend geometry, we found that the dissociation of O(3P) + CO can occur through the surface intersection between3B
154302-8 Zhouet al. J. Chem. Phys. 139, 154302 (2013)
and1B
2 states, and the surface intersection between3A2and 1B
2 states. The ground state 1A1 has a large area of inter-section with the lowest triplet 3B
2 state around bond angle at 95◦, and this provides a potentially wide-region relaxation pathways. The1B2 state also has a small intersection region with the1A1 state around Rco<1.2 Å and αoco = 95◦. This makes the relaxation mechanism of 1B2 state more compli-cated than the previous works. We have used the newly devel-oped DPD-MRPT2 method to confirm the calculations from the un-contracted MRCI method and this makes the present high-level calculations in solid base for conical/surface in-tersections. In the near future, we need to do un-contracted MRCI calculations for potential energy surfaces in Cs sym-metry and trajectory surface hopping must be performed for quantitatively studying dissociation reaction of CO2.
ACKNOWLEDGMENTS
B. Zhou would like to thank Postdoctoral Fellowship sup-ported by National Science Council of the Republic of China under Grant No. 100-2811-M-009-056. This work is sup-ported by National Science Council of the Republic of China under Grant No. 100-2113-M-009-005-MY3. C. Zhu would like to thank the MOE-ATU project of the National Chiao Tung University for support.
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