Department of Chemistry - Washington State University 01/02 WSU-PChem William R. Wiley Environmental Molecular Sciences Laboratory Kirk A. Peterson Department of Chemistry, Washington State University and the Environmental Molecular Sciences Laboratory Pacific Northwest National Laboratory Richland, 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 X 1 Σ + 2 1 Σ + B 3 Π 2 3 Π 1 3 Σ – Energy (eV) R (a.u.) AC1 AC2
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Ab Initio Spin-Orbit Coupling in Spectroscopy and Dynamicstyr0.chem.wsu.edu/~kipeters/Chem537/pdfs/SO.pdf · Options for Computing Spin-orbit Effects ab Initio ¥ 4-component methods
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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
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
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:
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)