Department of Chemistry - Washington State University
01/02 WSU-PChem
William R. Wiley
Environmental Molecular Sciences Laboratory
Kirk A. PetersonDepartment of Chemistry, Washington State Universityand the Environmental Molecular Sciences LaboratoryPacific Northwest National LaboratoryRichland, WA
Ab Initio Spin-Orbit Coupling in Spectroscopy and Dynamics
1.5
2.0
2.5
3.0
3.5
4.0
3 4 5 6 7 8
X1Σ+
21Σ+
B3Π
23Π
13Σ–
Ener
gy (
eV)
R (a.u.)
AC1 AC2
Department of Chemistry - Washington State University
01/02 WSU-PChem
William R. Wiley
Environmental Molecular Sciences Laboratory
Outline of Talk
• Methods of computing spin-orbit effects
• Basis sets and electron correlation
All-electron benchmark calculations: Atoms and light diatomics
Effective 1-electron operators: Pseudopotentials vs. all-electron
• Applications
BrO : low-lying electronic states: predissociation of A2Π3/2
HOBr: Singlet-triplet interactions in the UV/Vis absorption spectrum
BrCl: preliminary results for the B3Π(0+) ← X1Σ+ system
Department of Chemistry - Washington State University
01/02 WSU-PChem
William R. Wiley
Environmental Molecular Sciences Laboratory
Spin-Orbit Coupling: It’s Not Just for Heavy Atoms
• Predissociation of excited electronic states by states of different spin multiplicity
• Intersystem crossing and phosphorescence of excited triplet states in organic molecules
• Altering the shape of potential energy surfaces in exit and/or entrance channels
• Fine structure in high resolution spectroscopy
• Altering ground state chemical reactions by inducing transitions between different potential energy surfaces
• Thermochemistry to within “chemical accuracy”
Department of Chemistry - Washington State University
01/02 WSU-PChem
William R. Wiley
Environmental Molecular Sciences Laboratory
Options for Computing Spin-orbit Effects ab Initio
• 4-component methods based on the Dirac equation
– computationally very expensive; few programs available
• 2-component spin-orbit schemes
– incorporates SO effects into the orbitals
– requires significant work to implement into standard ab initio codes
• Perturbation treatments
– include SO when setting up the CI matrix
– calculate SO matrix elements between small number of spin-free states
operators:
1- and 2-electron Breit-Pauli; Douglas-Kroll-Hess; effective 1-electron
Department of Chemistry - Washington State University
01/02 WSU-PChem
William R. Wiley
Environmental Molecular Sciences Laboratory
HSO = A L . S^
A B
J→
L→ S
→
R→
Λ Σ
Ω
Angular momenta in a diatomic molecule
J (total) = L (orbital) + S (spin) + R (rotational)
Department of Chemistry - Washington State University
01/02 WSU-PChem
William R. Wiley
Environmental Molecular Sciences Laboratory
Operators Used in the Present Work
HSO
Z
ri i i
i rij i i j
i ji ij
= ×( ) ⋅
∑ − ×( ) ⋅ +( )
∑≠
12
2 12
2 13 3
2α αλ
λλ
λr p s r p s s
1) The Breit-Pauli spin-orbit operator
123 1231-electron
Zλ is the actual nuclear charge2-electron
spin-same-orbit &spin-other-orbit
2) Effective 1-electron operator via quasi-relativistic pseudopotentials
Contains the difference between 2-component relativistic pseudopotentials
* Includes scalar relativistic & some 2-electron effects
HV r
lP i P iSO
l il i i l
l
L
i=
+∑
∑− 2
2 1
1 ∆ λλ λ
λλ
λ
( )( ) ( )l s
Calculation of Spin-Orbit Coupled Eigenstates
Diagonalize Hel + HSO in a basis of spin-free (Λ-S) eigenfunctions
use a basis of the lowest 5 valence states: X1A’, 21A’, 11A”, 13A’, 13A”(labeled by S and Ms)
Example: HOBr
J
J
JJ
H
HH
H
B
BB
B
F
FF
F
SCF 2p +2s +1s385
390
395
400
405
J
J
J
JJ J
H
H
H
HH H
B
B
B
B
B B
F
F
F
F
F F
SCF
3p
+3s
+2p
+2s
+1s
780
800
820
840
860
880
900
J
J
J
J
JJ
J J J
H
H
H
H
HH
H H H
B
B
B
B
BB
B B B
F
F
F
F
FF
F F F
SCF
4p
+4s
+3d
+3p
+3s
+2p
+2s
+1s
3100
3200
3300
3400
3500
3600
3700
cc-pCVDZ
cc-pCVTZ
cc-pCVQZ
cc-pCV5ZExpt
The all-electron Breit-Pauli operator: Basis Set and Electron Correlation Effects for the Spin-Orbit Splittings of F, Cl, Br
Splittings in cm-1 , CISD wavefunctions
F Cl Br
SCF
+2p
+2s
+3s
3p
+1s
+3p
+3d
+4s
4p
+2p
+2s
+3s
+1s
SCF
180
200
220
240
CASSCF Valence Val+2p
Basis Set and Electron Correlation Effects for the Spin-Orbit Splittings of Small (Light) Molecules
cc-pCVDZ
cc-pCVDZ
cc-pCVTZ
cc-pCVTZ
cc-pCVQZ
cc-pCVQZExpt’l
Expt’l
X2Πr NS X2Π i ClO
(Splittings in cm-1)
Effects ofValence-state Spin-Orbit CI:
240
260
280
300
320
340
CASSCF Valence Val+2p
+0.01 cm-1 +10.4 cm-1
500
600
700
800
900
1000
CASSCF Valence +3d +2p3s3p3d
All-electron Breit-Pauli vs. pseudopotentials: The X ΠΠΠΠ state of BrO2Sp
littin
g (c
m-1)
Expt’l
cc-pCVDZ
cc-pCVTZ
cc-pCVQZ
Effects ofValence-state + 60–70 cm-1Spin-Orbit CI:
Atomic Br(2P) results:
“best” all-electron*: 3583 cm-1
Rel. Pseudopotential: 3670 cm-1
Expt’l: 3685 cm-1
* cc-pCV5Z, all electrons corr.
1-e- pseudopotential results (cc-pVnZ)
Bromine Monoxide: low-lying valence electronic states
Previous Experimental Work:
(i) Numerous high-res. studies on the X2Π3/2 state (JPL, NOAA, Ottawa)
→ equil. geom., IR freq., SO splitting, etc.
(ii) near-UV region dominated by the A2Π3/2 ← X2Π3/2 transition
emission, absorption → UV cross sections for atmospheric monitoring
• with high res.: Barnett et al. (Ottawa), Wheeler et al. (Bristol), and
Wilmouth et al. (Harvard)
Previous Theoretical Work:
Nothing on the excited states of BrO. Recent calculations on ClO by
Orr-Ewing and co-workers (Bristol) and Toniolo et al (Milan).
( >50% of all stratospheric bromine is in the form of BrO )
Department of Chemistry - Washington State University
01/02 WSU-PChem
William R. Wiley
Environmental Molecular Sciences Laboratory
Valence States of BrO
12
31
23
Λ-S State Ω = |Λ+Σ|
2 x 2Σ- 1/2
1 x 2Σ+ 1/22 x 2Π 3/2, 1/2
Br(2Pu) + O(3Pg) 1 x 2∆ 5/2, 3/22 x 4Σ- 3/2, 1/21 x 4Σ+ 3/2, 1/22 x 4Π 5/2, 3/2, 1/2, 1/21 x 4∆ 7/2, 5/2, 3/2, 1/2
(27 total)1 x 2Σ- 1/2
2 x 2Σ+ 1/2Br(2Pu) + O(1Dg) 3 x 2Π 3/2, 1/2
2 x 2∆ 5/2, 3/21 x 2Φ 7/2, 5/2
(42 total)
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
2.5 3 3.5 4 4.5 5 5.5 6 6.5
The Doublet States (ΛΛΛΛ-S) of BrO
(obtained via MRCI+Q/aug-cc-pVQZ calculations)
X2Π
A2Π
32Π 42Π
12∆
22∆
12Σ–
22Σ –
12Σ+
22Σ+
Br(2P) + O(3P)
Br(2P) + O(1D)
Ener
gy (
eV)
R (a.u.)
+ Quartet States
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
2.5 3 3.5 4 4.5 5 5.5 6 6.5
14Σ–
24Σ–14Π 24Π
4∆ 4Σ+
Department of Chemistry - Washington State University
01/02 WSU-PChem
William R. Wiley
Environmental Molecular Sciences Laboratory
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
2.5 3 3.5 4 4.5 5 5.5 6
ΩΩΩΩ=1/2 States [ Case (c) coupling throughout ]
Br(2P) + O(3P)
Br(2P) + O(1D)
X2Π
a4Σ–
A2Π
Ener
gy (
eV)
R (a.u.)
Department of Chemistry - Washington State University
01/02 WSU-PChem
William R. Wiley
Environmental Molecular Sciences Laboratory
ΩΩΩΩ=3/2 States [ Case (c) coupling throughout ]
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
2.5 3 3.5 4 4.5 5 5.5 6
Br(2P) + O(3P)
Br(2P) + O(1D)
X2Π
a4Σ–
A2Π
Ener
gy (
eV)
R (a.u.)
Department of Chemistry - Washington State University
01/02 WSU-PChem
William R. Wiley
Environmental Molecular Sciences Laboratory
Spectroscopic Constants of the X ΠΠΠΠ3/2 and A ΠΠΠΠ3/2 states2 2
Te re ωe ωexe(eV) (Å) (cm-1) (cm-1)
X2Π 0 1.726 729 4.9
X2Π3/2 0 1.724 734 4.9(1.717) (733)
∆(1/2–3/2) 848 cm-1 +0.007 –13.0 +0.2(968) (+0.007) (-15)
A2Π 3.42 1.941 533 5.6
A2Π3/2 3.28(3.27)
All values at the MRCI+Q/aug-cc-pV5Z level of theory
Department of Chemistry - Washington State University
01/02 WSU-PChem
William R. Wiley
Environmental Molecular Sciences Laboratory
Predissociation of the BrO A ΠΠΠΠ3333////2222 State2
The 3 high resolution studies performed to date indicate that:
• The only bands showing rotational structure are the v’,v”=7,0 & 12,0 and perhaps higher v’,0
• Bands with v’=0 & 1 are very diffuse; v’=1 is strongly perturbed
• With increasing J, the 7,0 band tunes towards a crossing while the 12,0 band first tunes away and then into another crossing (linewidth minimum at 3.887 eV) ;
12,0 has a slightly shorter lifetime than the 7,0 (2 vs. 2.5 ps)
• D0(A) = 1.107±0.017 eV ; D0(X) = 2.394±0.017 eV (Wilmouth et al.)
3.0
3.5
4.0
4.5
3 3.5 4 4.5 5
The A ΠΠΠΠ1/2 State with possible ΩΩΩΩ=1/2 perturbers2
Ener
gy (
eV)
R (a.u.)
a4Σ–
A2Π1/2
2Σ –
2Σ –
4Σ –
2Σ+
4Σ+
4∆
2Σ+
4Π
4Π
2Π
3.0
3.5
4.0
4.5
3 3.5 4 4.5 5
The A ΠΠΠΠ3/2 State with possible ΩΩΩΩ=3/2 perturbers2
Ener
gy (
eV)
R (a.u.)
a4Σ–
4Σ –
4Σ+
2∆
4Π2Π
4∆
4Π
2∆
v=1
7
12
A2Π3/2
At the crossing (r=4.34 bohr) θ=52º
and H12 = cosθ sinθ [E(32Π) – E(A2Π)] = 200 cm-1(d)
Interaction of the A2ΠΠΠΠ and 32ΠΠΠΠ states: a weakly avoided crossing
• non-adiabatic coupling matrix elements (NACMEs) were calculated as a function of R by numerical differentiation of the MRCI wavefunctions with an aug-cc-pV5Z basis set
• These were integrated to yield the mixing angles θ(R), i.e., the transformation between the adiabatic and diabatic basis.
NA
CM
E
Mixing A
ngle, θ
R (Bohr)
0
5
10
15
0
15
30
45
60
75
90
3 3.5 4 4.5 5 5.5 6
∂∂
= ∂∂
θR R
ad adΨ Ψ2 1
ΨΨ
ΨΨ
1
2
1
2
d
d
ad
ad
=
−
cos sin
sin cos
θ θθ θ
θ θ( )RR
dRRad ad
R
R= + ∂
∂ ′∫ ′0
0
2 1Ψ Ψ
3.0
3.5
4.0
4.5
3 3.5 4 4.5 5
400
200
80
40
650220
70
The A ΠΠΠΠ3/2 State with possible ΩΩΩΩ=3/2 perturbers & coupling ME's2
Ener
gy (
eV)
R (a.u.)
a4Σ–
4Σ –
4Σ+
2∆
4Π2Π
4∆
4Π
2∆
A2Π3/2
3.0
3.5
4.0
4.5
3 3.5 4 4.5 5
Department of Chemistry - Washington State University
01/02 WSU-PChem
William R. Wiley
Environmental Molecular Sciences Laboratory
The Low-Lying Electronic States of BrCl: preliminary results
3-4 excited electronic states are involved in the UV and near-UV spectrum
– the A3Π(1), B3Π(0+), C1Π(1) & D(0+) states (~550 - 235 nm)
Several non-adiabatic interactions have been observed
Recent Experimental Work
Cao et al. (1994)
Cooper et al. (1998)
Park et al. (2000)
At λ~235 nm: D(0+) absorption 3 product channels observed:
Br*+Cl (0.6), Br+Cl*(0.2), Br+Cl(.2) (||) (⊥ ) (||)
λ~310-410 nm: C1Π(1) absorption; Br+Cl formed
λ>410 nm: absorption via B3Π(0+) with a ⊥ contribution; Cl*/Cl branching ratio increases
Expected Low-Lying Electronic States of BrClFor the Homonuclear Halogens, e.g., Br2:
p5 p5
↑
↑σg
σu*
πu
πg*↑
↑
↑↑
↑↑
↑
↑
2440 X1Σ+g
2440
2431
2341
2422
1Σ+g
3Πu
3Πg
1Πu
1Πg
3Σg–
1∆g
1Σ+g
1g
1u2u
2g
0g+
0g–
0u–
0u+
1u
0g+
1g
0g+
1g2g
0g+
2P1/2 + 2P1/2
2P1/2 + 2P3/2
2P3/2 + 2P3/2
X
A
B
1u, 0g, 0u
2g, 2u, 1g, 1u, 1g,1u, 0g, 0u, 0g, 0u
3u, 2g, 2u, 1u, 1g,1u, 0g, 0u, 0g, 0u
+ –
+ –+ –
+ –+ –
23 total Ω states
Mulliken label
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
3 4 5 6 7 8
The Singlet States (ΛΛΛΛ-S) of BrCl(MRCI+Q/aug-cc-pVQZ)
X1Σ+
21Σ+
11Σ–
11Π
21Π
11∆
Br(2P) + Cl(2P)
Ener
gy (
eV)
R (a.u.)
Triplet States
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
3 4 5 6 7 8
The ΩΩΩΩ=0+ states of BrCl
Br + ClBr + Cl*
Br* + ClBr* + Cl*
1.5
2.0
2.5
3.0
3.5
4.0
3 4 5 6 7 8
X1Σ+
21Σ+
B3Π
23Π
13Σ–
Ener
gy (
eV)
R (a.u.)
AC1 AC2
1.5
2.0
2.5
3.0
3.5
4.0
3 4 5 6 7 8
The ΩΩΩΩ=1 states of BrCl
13Π
11Π
23Π
21Π
13Σ+
23Σ+
13Σ–
13∆
Ener
gy (
eV)
R (a.u.)
B3Π(0+)
Department of Chemistry - Washington State University
01/02 WSU-PChem
William R. Wiley
Environmental Molecular Sciences Laboratory
Near Re the 13ΠΠΠΠ state has large SO matrix elements withlow-lying singlets:
At AC2 the composition of the two 0+ states are:
II 0+ : 62% 13Σ- 1% 13Π 32% 21Σ+ 2% 23Π
III 0+ : 54% 13Π 2% 13Σ- 27% 21Σ+ 16% 23Π
<13Π | HSO | X1Σ+> = 601 cm-1
<13Π | HSO | 11Π > = 937 cm-1
<13Π | HSO | 21Π > = 453 cm-1
<13Π | HSO | 21Σ+> = 371 cm-1
<X1Σ+ | µ | 13Σ-(0+)> = 0.017 Debye
<X1Σ+ | µ | A3Π(1)> = 0.024 Debye
<X1Σ+ | µ | B3Π(0+)> = 0.071 Debye
<X1Σ+ | µ | C1Π(1)> = 0.148 Debye
Department of Chemistry - Washington State University
01/02 WSU-PChem
William R. Wiley
Environmental Molecular Sciences Laboratory
Hypobromous Acid (HOBr)UV/Vis Absorption Spectrum and Photodissociation Dynamics
0.0000
0.0005
0.0010
0.0015
0.0020
0.0025
0.0030
0.0035
0 100 200 300 400 500 600
Orlando & Burkholder
Crowley et al.
λ (nm)
Important contributor to both homogeneous and heterogeneous removalprocesses of stratospheric ozone
• Several studies of UV-Vis absorption X-sections
• First observation of the lowest triplet state of HOBr by Sinha and co-workers
• Photodissociation study of OH+Br (product distributions, vector correlation, etc.) by Sinha and co-workers at ~500 nm
a3A” state borrows intensity from B1A’
Dissociation is rapid →
B1A’
a3A”
A1A”
HOBr Computational Details
Correlation treatment : full valence CAS-reference multireference CI with Davidson correction
• 246 reference configurations, all single and double excitations wrt to these
• Davidson correction added for approximate treatment of higher excitations
• Calculate a total of 5 electronic states: X1A′, a3A′′ , B1A′′ , b3A′, and A1A′
• Relativistic effective core potential on Br
Basis set(s) : series of 3 correlation consistent basis sets:
cc-pVDZ + diffuse + spd/sp : 54 contracted functions
cc-pVTZ + diffuse + spd/sp : 95 contracted functions
cc-pVQZ + diffuse + spd/sp : 161 contracted functions (~2 hrs CPU per point)
!! pointwise extrapolate to complete basis set (CBS) limit
Grid: ~1000 points calculated with each of the 3 basis sets (~3000 calculations)
ROH (ao) = 1.4 – 3.0; RBrO (ao) = 2.6 – 10.0; θHOBr = 0 – 180º
+ near-equilibrium data for HOBr, HBrO, and the HOBr → HBrO TS
+ additional ROH for θ ≤ 80º
Department of Chemistry - Washington State University
01/02 WSU-PChem
William R. Wiley
Environmental Molecular Sciences Laboratory
0.0
2.0
4.0
6.0
8.0
4 0 6 0 8 0 100 120 140 160 180
0.0
2.0
4.0
6.0
8.0
2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5
The low-lying excited states of HOBr (ΛΛΛΛ-S states)
r(OH)=1.83 ao
θ = 103.2o
R(BrO), ao θ (deg.)
X1A'
a3A"
B1A'
A1A"
b3A'
X1A'
A1A"
b3A'
a3A"
B1A'
1Σ+
3Π
1Π
r(OH)=1.83 ao
r(BrO)=3.474 ao
(MRCI+Q/CBS, energies in eV)
1Σ–
3Σ–
3Σ+
1,3∆
Br(2P) +
OH(2Π)
3.0 4.0 5.0 6.0 7.0 8.0
180
160
140
120
100
80
60
40
20
0
0 20 40 60 80 100120140160180200220240
50
45 65
80
5
ROH = 1.82 ao The X1A’ state of HOBr
RBrO, ao
θ, d
eg.
1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
8.0
7.0
6.0
5.0
4.0
3.0
5
55
105
155
RBr
O, a o
ROH, ao
θ = 102.8o
2.0 3.0 4.0 5.0 6.0 7.0
180
160
140
120
100
80
60
40
20
0
0 25 50 75 100 125 150 175 200 225
90
15
65
110
105
RBrO = 3.10 ao
ROH, ao
θ, d
eg.
HOBr: 708 bound statesHBrO: 74 localized states
2.40
3.40
3.00
4.00
3.80
3.75 5.00 6.25 7.50 8.75 10.00
175
150
125
100
75
50
2.53.03.54.04.55.05.56.06.57.07.58.08.59.09.510.0r1t_ev_fit__1__3_rs
2.20
2.40
2.40
3.00
3.00
3.40
3.75 5.00 6.25 7.50 8.75 10.00
175
150
125
100
75
50
2.53.03.54.04.55.05.56.06.57.07.58.08.59.09.510.010.5r1t_ev_fit__1__3_rs
2.40
3.40
3.40
4.20
2.40
3.75 5.00 6.25 7.50 8.75 10.00
175
150
125
100
75
50
2.53.03.54.04.55.05.56.06.57.07.58.08.59.09.510.0r1t ev fit 1 3 rs
5.20
4.80
4.00
2.802.40
3.75 5.00 6.25 7.50 8.75 10.00
175
150
125
100
75
50
2.53.03.54.04.55.05.56.06.57.07.58.08.59.09.510.010.5r1t ev fit 1 3 rs
a3A" b3A'
A1A" B1A'
r(BrO), bohr
The
ta, d
egs.
The
ta, d
egs.
The
ta, d
egs.
The
ta, d
egs.
r(BrO), bohr
r(BrO), bohrr(BrO), bohr
3.20
3.20
4.80
3.60
4.00
2.60
2.20
3.75 5.00 6.25 7.50 8.75 10.00
3.00
2.75
2.50
2.25
2.00
1.75
1.50
2 2 2 3 3 3 4 4 4 4 4 5 5 5 6 6 6 6 6 7r1r2_ev_fit__1__3_rs
a3A"
3.20
5.00
5.40
4.60
3.40
3.402.40
3.75 5.00 6.25 7.50 8.75 10.00
3.00
2.75
2.50
2.25
2.00
1.75
1.50
2 3 4 4 4 5 6 6 6 7 8 8 8 9 10r1r2_ev_fit__1__3_rs
b3A'
3.40
4.004.80
4.60
3.002.40
3.75 5.00 6.25 7.50 8.75 10.00
3.00
2.75
2.50
2.25
2.00
1.75
1.50
2 3 4 4 4 5 6 6 6 7 8 8 8 9 10 1010r1r2 ev fit 1 3 rs
B1A'4.60
5.20
5.00
3.00
3.80
3.20
2.40
3.75 5.00 6.25 7.50 8.75 10.00
3.00
2.75
2.50
2.25
2.00
1.75
1.50
3 4 5 6 7 8 9 10r1r2 ev fit 1 3 rs
A1A"
r(BrO), bohr
r(O
H),
boh
r
r(O
H),
boh
rr(
OH
), b
ohr
r(O
H),
boh
r
r(BrO), bohr
r(BrO), bohrr(BrO), bohr
-0.05
0.00
0.05
0.10
0.15
0.20
0.25
2 3 4 5 6 7 8-0.20
-0.15
-0.10
-0.05
0.00
0.05
0.10
2 3 4 5 6 7 8
Transition dipole moments
The oscillator strengths for both the singlet-singlet and singlet-triplettransitions are governed at least in part by the transition dipole momentfunctions
For HOBr, these turn out to be strongly dependent on the level of theory
X A A Ax
1 1′ ′′µ
X A B Ay z1 1′ ′µ ,
y
z
re
–– ACPF--- MRCI.... CAS
H O
Br
z
y
R(BrO), ao
re
r(OH)=1.83 ao
θ = 103.2º
–– ACPF--- MRCI.... CAS
x
-1000
-800
-600
-400
-200
0
200
60 80 100 120 140 160 180
Representative Spin-Orbit Matrix Elements (cm-1)
-600
-400
-200
0
200
400
600
800
60 80 100 120 140 160 180-1000
-500
0
500
1000
60 80 100 120 140 160 180
x-component y-component z-component
Theta (deg.)
<X1A’|HSO|b3A’>
<X1A’|HSO|a3A”>
<21A’|HSO|b3A’>
<21A’|HSO|a3A”>
<11A”|HSO|a3A”>
r(OH)=1.83 ao
r(BrO)=3.474 ao
<11A”|HSO|b3A’>
<21A’|HSO|a3A”>
<X1A’|HSO|a3A”>
<11A”|HSO|b3A’>
3.0
3.5
4.0
4.5
5.0
80 100 120 140 160 180
Influence of Spin-Orbit Coupling on the Potential Energy Surfaces
Theta (deg.)
Ener
gy (
eV)
13A”
21A’
11A”13A’
Spin-free state
SO-coupled state
3Π
1Π
r(OH)=1.83 ao
r(BrO)=3.474 ao
0.0
0.0050
0.010
0.015
0.020
100 200 300 400 500 600
totalB1A′ ← X1A′
A1A″ ← X1A′
wavelength (nm)
H O
BrR
• CASSCF transition dipoles and SO matrix elements• Effective 1-D potentials:
0.0
0.0020
0.0040
0.0060
0.0080
0.010
100 200 300 400 500 600
A1A″ ← X1A′
B1A′ ← X1A′
b3A′ ← X1A′
a3A″ ← X1A′
(x 5)
Approximate Spectra with and without Spin-Orbit Effects
Cross sections obtained from 1-d wavepacket propagations (Å2)
w/o SO w/ SO
σ ω ω
tot ( ) ( )∝ ∫−∞
+∞dt S t ei t
S t tf f( ) ( ) ( )= Ψ Ψ0
Ψ Ψf fi i iE( ) ( )0 = µ
Preliminary absorption cross sections from 3-dimensional calculations
wavelength (nm)
0
5
10
15
20
25
200 250 300 350 400 450 500 550 600
xyztotal
0
5
10
15
20
25
30
35
200 250 300 350 400 450 500 550 600
Crowley & co-workers
Burkholder & Orlando
Theory: no spin-orbit couplingExperimental spectrum
• in collaboration with Dr. Dimitris Skouteris and Prof. Hans-Joachim Werner at Univ. Stuttgart
• wavepacket propagations carried out on a total of 8 excited states constructed from 4 spin-free
(diabatic) states with spin-orbit off-diagonal couplings (ACPF transition dipoles and MRCI SO)
• diagonalization of Hel + Hso currently does not include the ground state
Inclusion of spin-orbit coupling
0
5
10
15
20
25
-1.5
-1
-0.5
0
0.5
1
200 250 300 350 400 450 500
Total w/ SO
Total w/o SO
diff(SO-noSO)
0.0
0.2
0.4
0.6
0.8
1.0
350 400 450 500 550
xyz
Enlarged region near 450 nm
SO coupling between A1A" and b3A' states broaden the 2nd peak
The intensity of the X1A' → a3A" transition is strongly underestimated
Department of Chemistry - Washington State University
01/02 WSU-PChem
William R. Wiley
Environmental Molecular Sciences Laboratory
Calculations in progress
• Include the 41A' state to provide a source for more intensity
borrowing by the a3A" state
– the 41A' state lies at 9 eV, but its transition moment with
the ground state is ~1 a.u. (10x greater than the 21A' state)
a3A" oscillator strength:
w/o 41A' or X1A' : 4.3 x 10-6
w/ 41A' & X1A' : 1.7 x 10-5 (factor of 4)
• Use a partially adiabatic representation, with dynamics run on the same number of states (8) as before
(i.e., block diagonalize X1A', 21A', 41A' and a3A')
Department of Chemistry - Washington State University
01/02 WSU-PChem
William R. Wiley
Environmental Molecular Sciences Laboratory
Dr. Andreas Nicklass (halogen atoms, BrCl)
Prof. Joe Francisco, Purdue Univ. (BrO)
Dr. Dimitris Skouteris and Prof. H.-J. Werner, Univ. Stuttgart (HOBr)
Acknowledgments
National Science Foundation (Career program)
U.S. Dept of Energy (Basic Energy Sciences)$$$$
