ab initio spin-orbit coupling in spectroscopy and...

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


01/02 WSU-PChem

William R. Wiley


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


01/02 WSU-PChem

William R. Wiley


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”


01/02 WSU-PChem

William R. Wiley


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


01/02 WSU-PChem

William R. Wiley


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)


01/02 WSU-PChem

William R. Wiley


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 )


01/02 WSU-PChem

William R. Wiley


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+


01/02 WSU-PChem

William R. Wiley


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.)


01/02 WSU-PChem

William R. Wiley


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.)


01/02 WSU-PChem

William R. Wiley


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


01/02 WSU-PChem

William R. Wiley


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


01/02 WSU-PChem

William R. Wiley


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+)


01/02 WSU-PChem

William R. Wiley


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


01/02 WSU-PChem

William R. Wiley


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


01/02 WSU-PChem

William R. Wiley


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


01/02 WSU-PChem

William R. Wiley


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')


01/02 WSU-PChem

William R. Wiley


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)$$$$

ab initio spin-orbit coupling in spectroscopy and...

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