Modeling electronic and photoelectron spectra: Theory and ... · 3. Electronic structure methods for spectra calculations (EOM-CC approaches). 4. Examples: Photoelectron spectroscopy

Modeling electronic and photoelectron spectra:Theory and examples

Anna I. KrylovUniversity of Southern California

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Helsinki,December 2013

Thursday, December 19, 2013

OutlinePart I: Fundamentals. 1. Basics of photoelectron spectroscopy. What do we see and what do we learn? 2. Franck-Condon factors: double-harmonic approximation, Duschinsky rotations, and beyond. 3. Electronic structure methods for spectra calculations (EOM-CC approaches). 4. Examples: Photoelectron spectroscopy of diradicals.

Part II. Examples from GFP studies. 1. Photoelectron spectra of HBDI and phenolate (a lesson about VDE).2. Non-Condon effects in 2PA spectra of GFP.

Part III. Spectroscopy in condensed phase. 1. Methodology (QM/MM, QM/EF, sampling).2. Example: Photoelectron spectra of aqueous phenol and phenolate.

Conclusions and outlook.


< ξin|ξfn� >

< φd|µ|Ψel >

17

VI. FIGURES

S0

VDE

ADE

D0

0-0

0-1

0-2

0-3

0-4

0-5

0-6

E

FIG. 1: Model ground (S0) and electron-detached (D0) states potential energy surfaces (left) and the

corresponding photoelectron spectrum (right) for a diatomic molecule (or, more generally, a single

Franck-Condon active mode). The most intense vibronic band correponds to VDE.

FIG. 2: Model GFP chromophore: p-hydroxybenzylidene-2,3-dimethylimidazolinone (HBDI). The

deprotonated (anionic) form is shown.

Photoelectron spectra: What do we see and what do we learn

Energy balance: Ek=hv - ADE - Evib

Peak intensities (within same band): Cross-sections/angular distributions:

RNN

Energy balance: orbital energies and vibrational levelsPeak intensities: structuralchanges, vibrational structureCross-sections/angular distributions: orbital shape

Vertical

Adiabatic


Photoelectron spectra: Different electronic statesof the target

Lone pairs: weak vibrational progression

Water orbital energies (hartree)

# Symmetry Energy, au

1 1a1 -20.562 2a1 -1.353 1b1 -0.714 3a1 -0.575 1b2 -0.50

: broadvibrational progression

σOH


Photoelectron spectra: Important terms

1. Ionizaion of neutrals: Ionization energy (IE)

2. Ionization of anions: Detachment energy (DE)

3. Vertical and adiabatic values, Tee versus T00

4. Experimental: Onset and

5. Intensities (cross-sections), angular distributions, anisotropies

λmax


17

VI. FIGURES

S0

VDE

ADE

D0

0-0

0-1

0-2

0-3

0-4

0-5

0-6

E

FIG. 1: Model ground (S0) and electron-detached (D0) states potential energy surfaces (left) and the

corresponding photoelectron spectrum (right) for a diatomic molecule (or, more generally, a single

Franck-Condon active mode). The most intense vibronic band correponds to VDE.

FIG. 2: Model GFP chromophore: p-hydroxybenzylidene-2,3-dimethylimidazolinone (HBDI). The

deprotonated (anionic) form is shown.

Photoelectron spectra: Caveats

Theory: VDE(VIE), ADE(AIE), and FCFsIn general, VDE is not the same as and onset does not always equal ADE.

Experiment: Onset (origin) and λmax

λmax


Ψi(r,R) = Φi(r;R)ξin(R)

Ψf (r,R) = Φf (r;R)ξfn�(R)

Φf (r;R) = ΦN−1(r;R)Ψel

Pif ∼ | < Ψi(r,R)|µ|Ψf (r,R) > ·E|2

Pif ∼ |�

dRξinξfn�

�dr(Φi(r;R)µΦf (r;R))|2 =

|�

dRξinµ(R)ξfn� |2

Electronic and photoelectron spectra: TheoryTime-dependent PT + dipole approximation:

Initial and final states (in the Born-Oppenheimer picture):

In photoionization/photodetachment:

Thus:


�ξin(R)µ(R)ξfn�(R)dR ≈< Φi|µ|Φf > |R=R0 < ξin|ξ

fn� >

I ∼< ξin|ξfn� >2 |µif (R0)E|2

Electronic and photoelectron spectra: Theory cont-dAssuming (Condon approximation):

Franck-Condon approximation:

µ(R) ≈ µ(R0)

Important points: Cross-section/selection rules: Polarization/angular distributions: Shape (vibronic progression): Franck-Condon factors

< ξin|ξfn� >

µif =< Φi|µ|Φf >cos(µif , E)


Calculating photoelectron spectra

Franck-Condon factors: |<χi|χf>|2

One-dimensional case & harmonic approximation: Need geometry change or change in frequency to have non-zero FCF

final state

initial state


Calculating Franck-Condon factors: |<χi|χf>|2

1. Compute many-dimensional VCI wave functions... Very expensive, but could be done (e.g., using Multimode).

2. Double-harmonic approximation: Assume both initial and final states are described by harmonic potentials; account for difference in normal modes by Duschinsky rotation.

3. Parallel mode double-harmonic approximation: Use same normal coordinates for both states.

ezSpectrum: calculating FCFs using double-harmonic appr-nInput: Equilibrium geometries of the two states; Normal modes and frequencies (one or two states).


When Duschinsky rotations are important

Parallel modes

even in the low-energy part of the spectrum (below 2000 cm-1),where the VCI method is expected to have the highest accuracy.Our previous work, which calculated infrared spectra of theHCOH isomers, achieved excellent agreement with experiment;we expect that current results will be of use in an experimentaldiscrimination of the photoelectron spectra of HCOH.

Acknowledgment. This work is conducted under the auspicesof the iOpenShell Center for Computational Studies of ElectronicStructure and Spectroscopy of Open-Shell and ElectronicallyExcited Species supported by the National Science Foundationthrough the CRIF:CRF CHE-0625419 + 0624602 + 0625237grant. A.I.K. and J.M.B. also acknowledge support of theDepartment of Energy (DE-FG02-05ER15685 and DE-FG02-97ER14782, respectively).

References and Notes

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74, 617.(24) Koopmans, T. Physica 1933, 1, 104.(25) Oana, M.; Krylov, A. I. J. Chem. Phys. 2007, 127, 234106.(26) Kamarchik, E.; Braams, B.; Krylov, A. I.; Bowman, J. M. ezPES;

http://iopenshell.usc.edu/downloads/.(27) Raghavachari, K.; Trucks, G. W.; Pople, J. A.; Head-Gordon, M.

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Soc. 2003, 126, 5042.(33) Brown, A.; McCoy, A. B.; Braams, B. J.; Jin, Z.; Bowman, J. M.

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122, 044308.(36) Huang, X.; Braams, B. J.; Bowman, J. M. J. Phys. Chem. A 2006,

110, 445.(37) Xie, Z.; Braams, B. J.; Bowman, J. M. J. Chem. Phys. 2005, 122,

224307.(38) Huang, X.; Braams, B. J.; Bowman, J. M. J. Phys. Chem. 2005,

122, 044308.(39) Kendall, R. A., Jr.; Dunning, T. H.; Harrison, R. J. J. Chem. Phys.

1992, 96, 6796.(40) Watson, J. K. G. Mol. Phys. 1968, 15, 479.(41) Bowman, J. M. Acc. Chem. Res. 1986, 19, 202.(42) Carter, S.; Culik, S. J.; Bowman, J. M. J. Chem. Phys. 1997, 107,

10458.(43) Stanton, J. F.; Gauss, J.; Watts, J. D.; Lauderdale, W. J.; Bartlett,

R. J. ACES II, 1993. The package also contains modified versions of theMOLECULE Gaussian integral program of J. Almlof and P.R. Taylor, theABACUS integral derivative program written by T. U. Helgaker, H. J. Aa.Jensen, P. Jørgensen, and P. R. Taylor and the PROPS property evaluationintegral code of P. R. Taylor.

(44) Shao, Y.; Molnar, L. F.; Jung, Y.; Kussmann, J.; Ochsenfeld, C.;Brown, S.; Gilbert, A. T. B.; Slipchenko, L. V.; Levchenko, S. V.; O’Neil,D. P.; Distasio, R. A., Jr.; Lochan, R. C.; Wang, T.; Beran, G. J. O.; Besley,N. A.; Herbert, J. M.; Lin, C. Y.; Van Voorhis, T.; Chien, S. H.; Sodt, A.;Steele, R. P.; Rassolov, V. A.; Maslen, P.; Korambath, P. P.; Adamson,R. D.; Austin, B.; Baker, J.; Bird, E. F. C.; Daschel, H.; Doerksen, R. J.;Drew, A.; Dunietz, B. D.; Dutoi, A. D.; Furlani, T. R.; Gwaltney, S. R.;Heyden, A.; Hirata, S.; Hsu, C.-P.; Kedziora, G. S.; Khalliulin, R. Z.;Klunziger, P.; Lee, A. M.; Liang, W. Z.; Lotan, I.; Nair, N.; Peters, B.;Proynov, E. I.; Pieniazek, P. A.; Rhee, Y. M.; Ritchie, J.; Rosta, E.; Sherrill,C. D.; Simmonett, A. C.; Subotnik, J. E.; Woodcock, H. L., III; Zhang,W.; Bell, A. T.; Chakraborty, A. K.; Chipman, D. M.; Keil, F. J.; Warshel,A.; Herhe, W. J.; Schaefer, H. F., III; Kong, J.; Krylov, A. I.; Gill, P. M. W.;Head-Gordon, M. Phys. Chem. Chem. Phys. 2006, 8, 3172.

(45) Gauss, J.; Stanton, J. F.; Bartlett, R. J. J. Chem. Phys. 1991, 95,2623.

(46) Koziol L. Krylov, A. I. ezVibe; http://iopenshell.usc.edu/downloads/.(47) Bowman, J. M.; Carter, S.; Huang, X. Int. ReV. Phys. Chem. 2003,

22, 533.(48) Jensen, P.; Bunker, P. R. J. Chem. Phys. 1988, 89, 1327.(49) Handy, N. C., Jr.; Knowles, P. J.; Carter, S. J. Chem. Phys. 1991,

94, 118.(50) Salem, L.; Rowland, C. Angew. Chem., Int. Ed. Engl. 1972, 11,

92.(51) Diradicals; Borden, W. T., Ed.; Wiley: New York, 1982.(52) Bonacic-Koutecky, V.; Koutecky, J.; Michl, J. Angew. Chem., Int.

Ed. Engl. 1987, 26, 170.(53) Walsh, A. D. J. Chem. Soc. 1953, 2260.(54) Hoffmann, R.; Zeiss, G. D.; Van Dine, G. W. J. Am. Chem. Soc.

1968, 90, 1485.(55) Condon, E. Phys. ReV. 1926, 28, 1182.(56) Duschinsky, F. Acta Physicochim. USSR 1937, 7, 551.(57) Mozhayskiy V. A. Krylov, A. I. ezSpectrum; http://iopenshell.

usc.edu/downloads/.

JP903476W

Figure 9. The effect of rotations of normal coordinates on Franck-Condon factors within the parallel-mode approximation. (a) The correctoverlap between wave functions on lower (q!!) and upper (q!) surfaces.(b) The overlap when lower normal coordinates are rotated to coincidewith upper coordinates. (c) The overlap when upper normal coordinatesare rotated to coincide with lower coordinates.

HCOH Diradical Photoelectron Spectra Calculation J. Phys. Chem. A, Vol. 113, No. 27, 2009 7809

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76w even in the low-energy part of the spectrum (below 2000 cm-1),

where the VCI method is expected to have the highest accuracy.Our previous work, which calculated infrared spectra of theHCOH isomers, achieved excellent agreement with experiment;we expect that current results will be of use in an experimentaldiscrimination of the photoelectron spectra of HCOH.










































usc.edu/downloads/.

JP903476W



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Modes are rotated


Electronic structure methods for reliable geometries/energy differences

Mind open-shell character of the states involved....

Spin-contamination, symmetry breaking, etc....

EOM-CC: a versatile set of methods for various open-shell states.

“Equation-of-motion coupled-cluster methods for electronically excited and open-shell species: The Hitchhikers guide to Fock space”

A.I. Krylov , Ann. Rev. Phys. Chem. 59, 433 2008

See also webinar on EOM-CC (link from Q-Chem site):http://www.youtube.com/watch?v=Adf_F6IatrU&feature=youtu.be


http://www.youtube.com/watch?v=Adf_F6IatrU&feature=youtu.be

http://www.youtube.com/watch?v=Adf_F6IatrU&feature=youtu.be

Equation-of-Motion Coupled-Cluster Theory

T: satisfies CCSD equations for the reference

Specific EOM-CC model: -choice of excitation level in T & R; -choice of the reference and type of R.


EOM-IP: Ψ(N) =R(-1)Ψ0(N+1)

Ψ0 Ψi Ψija

EOM-EE: Ψ(Ms=0) =R(Ms=0)Ψ0(Ms=0)

Ψ0 Ψia Ψij

ab

EOM-EA: Ψ(N) =R(+1)Ψ0(N-1)

Ψ0 Ψa Ψiab

Ψ0 Ψia

EOM-SF: Ψ(Ms=0)=R(Ms=-1)Ψ0(Ms=1)

EOM MODELS: CHOICE OF R and Φ0


Photoelectron spectra: Examples from diradical studies1. Para-benzyne diradical: Success of parallel mode double-harmonic approx-n

2. HCOH diradical: when Duschinsky rotations are important.


Para-benzyne diradical: two nearly degenerate electronic states (singlet and triplet).Photodetachment spectrum from C6H6- (Wenthold et. al):probes singlet-triplet gap and provides structural information.

Vanovschi, Krylov, Wenthold, Theor. Chem. Acc. 120, 45 (2008)

+

+λ

Photoelectron spectra: Para-benzyne diradical states

b1u

a1g 3Bu

1Ag

R(Ms=-1) 0.14 eV


Calculated photoelectron spectrum

CCSD/6-311+G** frequencies and geometries for the anion and the triplet neutral, SF-5050/6-31G** frequencies and geometries for the singlet neutral


Calculated and experimental spectra


Photoelectron spectrum of cis-HCOH diradical

Parallel mode (using neutral’s modes)versus VCI

Parallel mode (using cation’s modes)versus VCI

the nodal structure (column c in Figure 9). Therefore, for largerelative rotations of normal coordinates, it can be more accurateto use the normal coordinates of the cation within the parallel-mode approximation, especially if the active modes have similarfrequencies on the neutral state.

VII. Conclusions

We report accurate configuration interaction calculations ofvibrational levels of the cis and trans isomers of HCOH+ andHCOD+. The photoelectron spectra from the ground vibrationalwave functions of the two neutral isomers are also presented.

HCOH+ is derived by removing an electron from a doublyoccupied lone pair orbital on the carbon atom (Figure 3), withantibonding contribution along CO. This leads to large structuralchanges upon ionization, including shortening of the CO bondand an increase in the H-C-O angle due to increased shybridization on C. Changes in harmonic frequency are due tostructural changes and in the reduced repulsion betweenelectrons on O and the C center in the cation.

VCI fundamental excitations are within 35 cm-1 of theharmonic ones for the lowest four normal modes, whereas the

CH and OH stretches show anharmonicities over 150 cm-1. Dueto the large difference in equilibrium structures on the neutraland cation surfaces, nonzero Franck-Condon factors arecalculated for energies up to 7000 cm-1. The progressions arelocalized into select frequencies; namely, two in-plane bendsand the CO stretch. This is rationalized in terms of thegeometrical differences. Photoelectron spectra for the HCODisotopes are significantly different from those for HCOH; thisis due to the suppression of D motion in the normal modevibrations.

The photoelectron spectra in the parallel-mode harmonicapproximation were also calculated and compared with the VCIspectra. This approximation was fairly accurate for the low-energy part of the spectrum, especially in duplicating intensitiesof the three active fundamental excitations in both isomers. Forcombinations and overtones, the harmonic intensities for thestrong peaks are accurate only to within a factor of 2 for cis-HCOH. However, the parallel-mode harmonic approximationis slightly more accurate for trans-HCOH than for cis-.

The calculated photoelectron spectra for cis- and trans-HCOHare qualitatively different, which should make an experimentalidentification possible. Moreover, these differences are present

Figure 7. Comparison between VCI (black lines) and parallel-modeharmonic oscillator approximation (red lines) using normal coordinatesof the neutral (top) and cation (bottom) for the Franck-Condon factorsof cis-HCOH. Harmonic intensities are not scaled to match VCI.

Figure 8. Comparison between VCI (black lines) and parallel-modeharmonic oscillator approximation (red lines) using normal coordinatesof the neutral (top) and cation (bottom) for the Franck-Condon factorsof trans-HCOH. Harmonic intensities are not scaled to match VCI.

7808 J. Phys. Chem. A, Vol. 113, No. 27, 2009 Koziol et al.

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type lone pairs (N, O atoms) lead to singlet ground statesbecause these lone pairs can mix with carbon’s ! orbital. Thiscan raise it enough so that pairing the electrons in " becomesenergetically favorable. For example, in HCOH, the singlet stateis about 1 eV below the triplet.

The vertical (adiabatic) IEs of HCOH are 9.45 (8.76) and9.44 (8.79) eV for the cis and trans isomers, respectively, ascomputed at the CCSD(T)/cc-pVTZ level (ZPE excluded). Thehighest occupied molecular orbital (HOMO) on HCOH is a lonepair on carbon with a minor contribution on OH, which providesantibonding character along the CO bond. (Figure 3). The firstionization removes an electron from the HOMO, with largegeometrical changes in the equilibrium structure. The CO bondis shortened by 0.097 and 0.093 Å in the cis and trans isomers,respectively. Ionization from an sp2 orbital on carbon increasesthe carbon’s overall s character; the HCO angle increases by25.5 and 22.6°. The HOC angle also increases, by 6.1 and 9.9°.

The displaced equilibrium structures strongly affect the shapeof the PES, and harmonic frequencies show strong differencesupon ionization (Table 3). The largest change is in the COstretch, which increases by 370 cm-1 in both isomers uponionization. This is due to the shortening of the CO bond. TheCH stretching frequency increases upon ionization, by 286 and216 cm-1. The remaining four frequencies decrease. The OHstretch decreases by 213 and 255 cm-1; this follows from thelonger OH bond in the cation, due to increased donation intothe electron-depleted carbon. The remaining three are bendingmodes involving the OH group; the oxygen lone pairs encounterless steric hindrance with a single electron on C in these motions.

Two barriers on the HCOH+ PES, which separate the cis andtrans wells, are 6190 and 7085 cm-1 above the trans minimum(Figure 1). The respective transition states represent in-planeand out-of-plane rotation of H around the oxygen, respectively.These transition states are lower in energy relative to the neutral(by 6717 cm-1 for the linear, and by 3578 cm-1 for the out-of-plane). This also is due to decreased repulsion between theelectrons on O and C: in out-of-plane rotation, the HOC angleremains essentially constant. In in-plane rotation, this anglechanges, and the oxygen’s electron density is brought closer tothe carbon center. The ionized carbon atom presents a muchsmaller barrier for this interaction; hence, the disproportionateeffect of ionization on the two barriers.

It should be noted that Figure 1 in our previous paper9 had atypographical error: energies of the two barriers connecting cis-and trans-HCOH were incorrectly labeled relative to the cisminimum, rather than to the trans, as indicated. In addition, theneutral PES was optimized to replicate harmonic frequencies;we have since created a similar PES that replicates barrierheights accurately with only a moderate decline in the accuracyof the harmonic frequencies. Both PESs are available fordownload from the iOpenShell Web site.

IV. Photoelectron Spectra of HCOH

Vibrational levels of HCOH+ up to 3600 cm-1 are listed inTable 4. Considering the fundamental excitations, the first fourlevels (up to 1700 cm-1) are accurately described by theharmonic approximation, with an average deviation betweenharmonic and VCI excitation energies of 35 cm-1. The higherstretches show large deviations from the harmonic approxima-tion; VCI decreases the CH and OH stretch fundamentalfrequencies by approximately 150 and 190 cm-1, respectively.

The photoelectron spectra for the two isomers are shown inFigure 5, and positions and intensities are tabulated in Tables 5and 6. The intensities are unitless; an intensity of 1 correspondsto full overlap between the neutral and cation wave functions.

The cis-HCOH photoelectron spectrum is given in Figure 5and Table 5. In the low-energy region (0-2000 cm-1), thelowest-frequency mode, #6, has no intensity. This is the onlymode that is not fully symmetric. If this normal mode wasseparable in the PES, then transitions to odd levels would beforbidden by symmetry. The other four fundamentals in thisrange have appreciable intensity, with #5 and 2#5 being thestrongest. In cis-HCOH+, #5 is a scissoring of OH and CH,which moves the molecule toward linearity. From Figure 4,displacement along this mode brings the cation into theFranck-Condon region. #4 increases one angle and decreasesthe other one. It is active because the difference in H-C-Oangles in neutral and cation structures is much larger than the

Figure 3. The highest occupied molecular orbital of cis- (left) andtrans-HCOH (right).

TABLE 3: Comparison of Harmonic Frequencies (cm-1)between the Neutral and the Cation PESs

opwag

ipbend

ipbend

COstretch

CHstretch

OHstretch

cis-HCOH 1014 1238 1476 1335 2768 3655cis-HCOH+ 931 996 1159 1711 3054 3442trans-HCOH 1098 1214 1508 1326 2853 3754trans-HCOH+ 970 997 1254 1694 3069 3499

TABLE 4: The HCOH+ VCI Vibrational Levels below 3600cm-1, and Corresponding Levels for HCOD+ (cm-1)

no.statelabel cis-HCOH+ cis-HCOD+ trans-HCOH+ trans-HCOD+

1 #6 905 746 935 7812 #5 949 822 967 8223 #4 1126 1099 1211 11434 #3 1684 1671 1664 16555 2#6 1811 1475 1858 15476 #5 + #6 1874 1575 1915 16047 2#5 1885 1641 1933 16448 #4 + #6 2045 1835 2149 19129 #4 + #5 2052 1947 2141 198210 2#4 2224 2167 2405 225911 #3 + #6 2582 2436 2597 241712 #3 + #5 2624 2488 2626 247713 3#6 2711 2193 2779 230114 #5 + 2#6 2782 2306 2826 236515 #3 + #4 2801 2753 2863 278516 3#5 2823 2456 2908 246317 2#5 + #6 2825 2395 2903 243018 #2 2896 2384 2933 247819 #4 + 2#5 2954 2784 3054 281820 #4 + 2#6 2962 2557 3100 267121 #4 + #5 + #6 2990 2691 3090 275122 2#4 + #5 3122 3038 3298 310923 2#4 + #6 3156 2877 3340 301624 #1 3248 2900 3328 292725 3#4 3300 3191 3578 334726 2#3 3347 3321 3306 328727 #3 + 2#6 3481 3176 3519 317728 #3 + #5 + #6 3542 3253 3574 328529 #3 + 2#5 3560 3304 3589 3301


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9034

76w

Koziol, Mozhayskiy, Braams, Bowman, Krylov;JPCA 113, 7802 (2009)


even in the low-energy part of the spectrum (below 2000 cm-1),where the VCI method is expected to have the highest accuracy.Our previous work, which calculated infrared spectra of theHCOH isomers, achieved excellent agreement with experiment;we expect that current results will be of use in an experimentaldiscrimination of the photoelectron spectra of HCOH.










































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When Duschinsky rotations are important

Difference between using initial and target state normal modes in parallel-mode approximation


Part I: Conclusions

Modeling spectra: - get electronic structure right. - FCFs: can use simple models or more sophisticated vibrational structure calc-s. - angular distributions, cross sections - Dyson orbitals (another talk....)

Can learn about target states of the ionized/detached system that are not accessible by electronic spectroscopy.







Photoelectron/electronic spectra: Examples from GFP studies

1. GFP: brief introduction.

2. Electron detachment from GFP chromophore and phenolate: Be careful about vertical energies!

3. Non-Condon effects in GFP absorption.


GFP: Green Fluorescent ProteinLight-emitter in bioluminescence of the jellyfish Aequorea Victoria (also found in corals, copepods, lancelets, and combjellies).

The chromophore is formed auto-catalytically, upon folding.Can be cloned and expressed in living cells and animals ->

used to tag proteins -> genetic biomarker

Heim et al., PNAS 91 12501 (1994);Tsien, Annu. Rev. Biochem. 67 509 (1998). When lit up in the dark, the pigs glow green

In daylight, their eyes and skin are green-tinged

Taiwan breeds green-glowing pigs


HBDI (model GFP chromophore), phenol, and phenolate

HBDI: model GFP chromophore

closest electronic state is a dark state, 1A! derived from theHOMO-3!HOMO transition !in terms of the anionic orbit-als".

Figure 6 shows an overall energy diagram for the threeforms of HBDI. Adiabatically, the doubly oxidized form is9.98 eV above the ground state of the anion. The respectivevalue of VDE !corresponding to removing two electrons" is10.37 eV. The adiabatic IE from the ground state of the dou-blet radical is 7.59 eV !computed using the anion’s VDEvalue of 2.54 and 0.15 eV relaxation energy of the neutralradical". Another relevant value is the energy gap betweenthe excited states !D1 and D2" of the doublet radical and thecation. Using the same values of vertical detachment andrelaxation energies, and 1.52 and 3.37 eV for the verticalD0!D1,2 excitation energies, we estimate the ionization en-ergy of the electronically excited doublet radical as 6.07 and4.22 eV, respectively. Finally, the energy gap between theexcited singlet state of the anion and the two excited states ofthe radical are 1.29 and 3.14 eV, respectively.

To evaluate whether two-electron oxidation of the anionis possible using the electron acceptors employed in Ref. 13,it is necessary to estimate standard reduction potentials cor-responding to the detachment and ionization transitions de-scribed above. Rigorous calculations of the oxidation/reduction potentials require free energy calculations,however, a simpler approach based on the correlation be-tween detachment/ionization energies and the standard re-duction potentials can be employed:54

E ° = !! 2.59 ! 0.26" + !0.56 ! 0.03" · VIE, !1"

where E° corresponds to the following half reaction:

A+ + e = A , !2"

and VIE is the ionization energy of A. This equation wasderived by comparing experimental values of the reversibleoxidation potentials !in DMF or ACN solvent" for a set of 14

organic molecules with their gas-phase ionization energies.Applying this equation, we arrive at the values of E° sum-marized in Table III. Equation !1" was derived using IEs inthe range of 5.6–9.0 eV and its application to lower values!such as the S1!D1,2 transitions from Table III" assumesthat the trend can be extrapolated. Thus, estimations for thelower IEs are of a semiquantitative value only.

Oxidation of the ground-state doublet radical !D0" cor-responds to E° =1.66 V, which is much larger than E° of thestrongest oxidizing agent employed in Ref. 13!K3Fe!CN"6 ,E° =0.42 V". However, the oxidation of theelectronically excited neutral radical corresponds to E°=0.81 V and E° =!0.12 V for the D1 and D2 states, respec-tively. Thus, electronically excited doublet radical !in the D2state" can be oxidized even by the weakest oxidizing agentfrom Ref. 13 !E° =!0.32 V", whereas the oxidation of D1 isless likely. Finally, the electronically excited anion can beeasily oxidized producing excited states of the doublet !E°=!1.87 V and E° =!0.83 V for D1 and D2 states, respec-tively". Note that the S1!D0 transition is spontaneous in thegas phase and does not require the presence of oxidizingagents due to the autoionizing nature of S1. By comparingthe electronic configurations of the doublet states !see Fig. 4"and the S1 state !Fig. 3", we note that these transitions in-clude significant one-electron character. Thus, a possiblemechanism of the oxidative redding emerging from thepresent study is presented in Scheme 1.

The first step involves photoexcitation, and the blue lightis sufficient to generate this transition. The second and thirdsteps are one-electron oxidation steps with E° =!0.88 V,which is within the reach of the oxidizing agents employedin Ref. 13. The two-electron oxidation is consistent with thestoichiometric results from Ref. 13. Moreover, the closed-shell character of the cation is consistent with the relativelychemically stable nature of the red form of GFP. Althoughthe chemical structure of the cation suggests high reactivitywith respect to nucleophilic agents !e.g., water", the proteinmay be sufficiently effective in shielding this fragile species.Finally, the absorption of the cationic form is redshifted by

FIG. 6. Energy diagram of the relevant electronic states of deprotonatedHBDI.

TABLE III. Standard oxidation potentials corresponding to different one-electron oxidation transitions computed using gas-phase detachment andionization energies !see text".

TransitionIE

!eV"E°!V"

D0!cation 7.59 1.66!0.49D1!cation 6.07 0.81!0.44D2!cation 4.22 !0.23!0.39S1!D1 1.29 !1.87!0.30S1!D2 3.14 !0.83!0.35

SCHEME 1. Possible mechanism of one-photon two-electron oxidation ofthe deprotonated HBDI anion.

115104-5 Oxidation of GFP chromophore J. Chem. Phys. 132, 115104 !2010"

Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp

Anionic chromophore (~phenolate):responsible for fluorescence


Redox properties of GFP and its chromophores

1. Motivation: new experiments, redox sensors.

2. Need to understand energetics of oxidation/reduction.

3. Gas-phase studies of model chromophores: intrinsic electron donating ability.


Gas-phase IEs and photoelectron spectra of HBDI1. First prediction of VIE and AIE for HBDI: Epifanovsky, 2009. VIE=2.54 eV. Suggested resonance character of the bright state (Eex=2.6 eV).2. Experiments of Jocksuch, Verlet, Fielding, Andersen: observation of photoelectrons.

closest electronic state is a dark state, 1A! derived from theHOMO-3!HOMO transition !in terms of the anionic orbit-als".

Figure 6 shows an overall energy diagram for the threeforms of HBDI. Adiabatically, the doubly oxidized form is9.98 eV above the ground state of the anion. The respectivevalue of VDE !corresponding to removing two electrons" is10.37 eV. The adiabatic IE from the ground state of the dou-blet radical is 7.59 eV !computed using the anion’s VDEvalue of 2.54 and 0.15 eV relaxation energy of the neutralradical". Another relevant value is the energy gap betweenthe excited states !D1 and D2" of the doublet radical and thecation. Using the same values of vertical detachment andrelaxation energies, and 1.52 and 3.37 eV for the verticalD0!D1,2 excitation energies, we estimate the ionization en-ergy of the electronically excited doublet radical as 6.07 and4.22 eV, respectively. Finally, the energy gap between theexcited singlet state of the anion and the two excited states ofthe radical are 1.29 and 3.14 eV, respectively.

To evaluate whether two-electron oxidation of the anionis possible using the electron acceptors employed in Ref. 13,it is necessary to estimate standard reduction potentials cor-responding to the detachment and ionization transitions de-scribed above. Rigorous calculations of the oxidation/reduction potentials require free energy calculations,however, a simpler approach based on the correlation be-tween detachment/ionization energies and the standard re-duction potentials can be employed:54

E ° = !! 2.59 ! 0.26" + !0.56 ! 0.03" · VIE, !1"

where E° corresponds to the following half reaction:

A+ + e = A , !2"

and VIE is the ionization energy of A. This equation wasderived by comparing experimental values of the reversibleoxidation potentials !in DMF or ACN solvent" for a set of 14

organic molecules with their gas-phase ionization energies.Applying this equation, we arrive at the values of E° sum-marized in Table III. Equation !1" was derived using IEs inthe range of 5.6–9.0 eV and its application to lower values!such as the S1!D1,2 transitions from Table III" assumesthat the trend can be extrapolated. Thus, estimations for thelower IEs are of a semiquantitative value only.

Oxidation of the ground-state doublet radical !D0" cor-responds to E° =1.66 V, which is much larger than E° of thestrongest oxidizing agent employed in Ref. 13!K3Fe!CN"6 ,E° =0.42 V". However, the oxidation of theelectronically excited neutral radical corresponds to E°=0.81 V and E° =!0.12 V for the D1 and D2 states, respec-tively. Thus, electronically excited doublet radical !in the D2state" can be oxidized even by the weakest oxidizing agentfrom Ref. 13 !E° =!0.32 V", whereas the oxidation of D1 isless likely. Finally, the electronically excited anion can beeasily oxidized producing excited states of the doublet !E°=!1.87 V and E° =!0.83 V for D1 and D2 states, respec-tively". Note that the S1!D0 transition is spontaneous in thegas phase and does not require the presence of oxidizingagents due to the autoionizing nature of S1. By comparingthe electronic configurations of the doublet states !see Fig. 4"and the S1 state !Fig. 3", we note that these transitions in-clude significant one-electron character. Thus, a possiblemechanism of the oxidative redding emerging from thepresent study is presented in Scheme 1.

The first step involves photoexcitation, and the blue lightis sufficient to generate this transition. The second and thirdsteps are one-electron oxidation steps with E° =!0.88 V,which is within the reach of the oxidizing agents employedin Ref. 13. The two-electron oxidation is consistent with thestoichiometric results from Ref. 13. Moreover, the closed-shell character of the cation is consistent with the relativelychemically stable nature of the red form of GFP. Althoughthe chemical structure of the cation suggests high reactivitywith respect to nucleophilic agents !e.g., water", the proteinmay be sufficiently effective in shielding this fragile species.Finally, the absorption of the cationic form is redshifted by

FIG. 6. Energy diagram of the relevant electronic states of deprotonatedHBDI.

TABLE III. Standard oxidation potentials corresponding to different one-electron oxidation transitions computed using gas-phase detachment andionization energies !see text".

TransitionIE

!eV"E°!V"

D0!cation 7.59 1.66!0.49D1!cation 6.07 0.81!0.44D2!cation 4.22 !0.23!0.39S1!D1 1.29 !1.87!0.30S1!D2 3.14 !0.83!0.35

SCHEME 1. Possible mechanism of one-photon two-electron oxidation ofthe deprotonated HBDI anion.

115104-5 Oxidation of GFP chromophore J. Chem. Phys. 132, 115104 !2010"

Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp

Epifanovsky, Polyakov, Grigorenko, Nemukhin, Krylov, JCTC 5, 1895 (2009); Horke, Verlet, PCCP 14, 8511 (2012); Mooney, Sanz, McKay, Fitzmaurice, Aliev, Caddick, Fielding, JPCA, 7943 (2012); Forbes, Nagyand.Jockusch, Int. J. Mass. Spectrom., 308,155 (2011); Bochenkova, Andresen, Faraday Discussion, in press (2013).

Source VIE AIEKrylov (2009) 2.54 2.45

Verlet (2012) 2.8+/-0.1 2.6+/-0.2

Fielding (2012) 2.85+/-0.1

Bochenkova (2013) 2.62 2.52

Andersen (2013) 2.68


Calibration of theory: Phenolate anionExpt-l VDE = 2.18 eV (Fielding 2010).

0.1 Phenolate anion

Experiment.ADE=2.253 eV (Lineberger 1992)

ADE=2.15±0.15eV (Fielding 2010)

VDE = 2.18 eV (Fielding 2010)

Table 1: Vertical detachment energies of the phenolate anion

Method Geometry VDE, eV

ωB97X-D/cc-pVTZ ωB97X-D/cc-pVTZ 2.03

ωB97X-D/aug-cc-pVTZ ωB97X-D/cc-pVTZ 2.20

EOM-IP-CCSD/6-311+,+G(d,p) RIMP2/cc-pVTZ 2.01

EOM-IP-CCSD/cc-pVTZ RIMP2/cc-pVTZ 2.00

EOM-IP-CCSD/aug-cc-pVTZ RIMP2/cc-pVTZ 2.25

basis. lin. dep. thresh. 3

orth. at. orb 393

CCSD/cc-pVTZ RIMP2/cc-pVTZ 1.97

CCSD(T)/cc-pVTZ RIMP2/cc-pVTZ 2.01

CCSD/aug-cc-pVTZ RIMP2/cc-pVTZ 2.20


orth. at. orb 393

CCSD(T)/aug-cc-pVTZ RIMP2/cc-pVTZ 2.27


orth. at. orb 393

CCSD/aug-cc-pVTZ ωB97X-D/cc-pVTZ 2.18


orth. at. orb 393

CCSD(T)/aug-cc-pVTZ ωB97X-D/cc-pVTZ 2.24


orth. at. orb 393

1

1. EOM-IP-CCSD, CCSD(T), and DFT: within 0.05 eV.2. Theory: within 0.06 eV from exp-t.3. aug-cc-pVTZ is essential!


Phenolate spectrum

2.1 2.2 2.3 2.4 2.5 2.6Energy, eV

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Inte

nsi

ty, a.

u.

unsclaed, unshifted FWHF=0.01

experiment, Lineberger (PES)

unscaled, shifted

scaled (0.94), shifted

Figure 1: Computed S0-D0 photoelectron spectrum for the phenolate anion (T=300 K). Ex-perimental spectrum by Lineberger et al. [1], vibrational temperature 300-400 K.

2

Expt: Gunion, Gilles, Polak, Lineberger, Int. J. Mass Spectrom. Ion Proc. 117, 601 (1992).Theo: Bravaya, Krylov, JPCA in press (2013).

AIE Expt. 2.25 eV Theo. 2.19 eV (-0.06 eV difference)

00 line: most intense

00

19

2.2 2.3 2.4 2.5 2.6Energy, eV

0

0.1

0.2

0.3

Inte

nsity, a.u

.2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8

Energy, eV

0

0.1

0.2

0.3

0.4

Inte

nsity, a.u

.

!11

0-0

0-!111

0-!112

0-!113

!111 -0

0-!114

FIG. 4: Photoelectron stick spectrum of the phenolate anion computed in double-harmonic parallelnormal mode approximation at T=300K (top panel). Atoms displacement vectors corresponding tothe active normal mode are also shown. Comparison of the computed and experimental photoelectronspectrum [35] (T∼300K) (bottom panel). The computed spectrum is shifted by 0.12 eV so that themaxima of the most intense bands coincide.


Back to HBDI puzzle...

Revised theoretical estimates (wB97X-D/aug-cc-pVTZ): VDE=2.76 eV & ADE=2.65 eV.

ADE is in excellent agreement with expt (2.6+/-0.2 eV). VDE seems to be in agreement too....

But what is VDE? Expt: Maximum in photoelectron spectrum;Calcs: Vertical energy difference between the anion and neural.

Need to compute spectrum and see if these two coincide.


HBDI spectrum

Horke, Verlet, PCCP 14, 8511 (2012); Mooney, Sanz, McKay, Fitzmaurice, Aliev, Caddick, Fielding, JPCA, 7943 (2012); Forbes, Nagyand.Jockusch, Int. J. Mass. Spectrom., 308,155 (2011).

18

2.50 2.75 3.00 3.25 3.50

0.0

0.2

0.4

0.6

0.8

1.0

1.2

Toker et al. (355 nm)

Mooney et al. (330 nm)

Horke and Verlet (355 nm)

Forbes and Jockusch

Inte

nsity, a.u

.

Energy, eV

FIG. 3: Photoelectron spectra measured at comparable excitation wavelengths as reported by Horke

and Verlet [11], by Mooney et al. [10], and by Tokeret al. [12]. The difference in the shape of the PES

by Mooney et al. [10] in comparison to two other PESs can be possibly explained by different excitationwavelength (λ=330 nm), which may result in population of anion metastable states. Photodetachment

signal from the action spectroscopy study by Forbes and Jockusch et al. [9] is also shown. Note that

the photodetachment signal from Ref. [9] cannot be directly compared to PES because photoelectrons

with all possible kinetic energies contribute to the former.


20

2.65 2.7 2.75 2.8 2.85 2.9 2.95Energy, eV

0

0.05

0.1

0.15

0.2

0.25

Inte

nsity,

a.u

.

!50

!44

!49 !48

!43

0-0

!491 -0

!501 -0

0-!431

0-!191

0-!4410-

!491

0-!481

0-!501

!29

2.4 2.6 2.8 3 3.2 3.4Energy, eV

0

0.2

0.4

0.6

0.8

1

Inte

nsity,

a.u

.

experiment

shifted PES, HWHM=0.09 eV

PES, HWHM=0.09 eV

PES, HWHM=0.05 eV

FC stick spectrum

FIG. 5: Photoelectron stick spectrum of the HBDI anion computed in double-harmonic parallel nor-

mal mode approximation at T=300K (top panel). Atoms displacement vectors corresponding to the

active normal modes are also shown. Comparison of the computed and experimental photoelectron

spectra [12] (T∼300K) (bottom panel).

HBDI spectrum

Bravaya, Krylov, JPCA in press (2013)


20

2.65 2.7 2.75 2.8 2.85 2.9 2.95Energy, eV

0

0.05

0.1

0.15

0.2

0.25

Inte

nsity, a.u

.

!50

!44

!49 !48

!43

0-0

!491 -0

!501 -0

0-!431

0-!191

0-!4410-

!491

0-!481

0-!501

!29

2.4 2.6 2.8 3 3.2 3.4Energy, eV

0

0.2

0.4

0.6

0.8

1

Inte

nsity,

a.u

.

experiment

shifted PES, HWHM=0.09 eV

PES, HWHM=0.09 eV

PES, HWHM=0.05 eV

FC stick spectrum

FIG. 5: Photoelectron stick spectrum of the HBDI anion computed in double-harmonic parallel nor-

mal mode approximation at T=300K (top panel). Atoms displacement vectors corresponding to the

active normal modes are also shown. Comparison of the computed and experimental photoelectron

spectra [12] (T∼300K) (bottom panel).

HBDI spectrum

Expt: Mooney, Sanz, McKay, Fitzmaurice, Aliev, Caddick, Fielding, JPCA, 7943 (2012).Theo: Bravaya, Krylov, JPCA in press (2013)

Computed spectra:- Max is not at VDE but at 00 transition(~ADE). - True theo VDE: ~2.65 eV and not 2.76 eV.

Exp. spectrum is reproduced when peaks are broadened by 0.09 eV. Exp-theo: 0.04 eV


Conclusions

1. Exp. and theoretical ADE agree (~0.04 eV), but only when theo. spectrum is broadened (0.09 eV).

2. Computed VDE is meaningless because spectral maximum corresponds to 00 transition (ADE). This seems to be quite often the case in polyatomic molecules.


Non-Condon effects in two-photon absorption of GFP

1. Motivation for 2-photon (2PA) spectroscopies.2. 2PA versus 1PA lineshapes.3. GFP example.

!a1

!a2

!a3

!n1

!n2

Figure 2: Relevant molecular orbitals of deprotonated HBDI (left) and HBDI (right), HF/6-

311G* and the corresponding solutions of the Huckel model. In the ground state of the anion,

both φa1 and φa

2 are doubly occupied, and the bright state is derived by the φa2 → φa

3 excitation.

In the protonated form (neutral HBDI), the MOs are no longer of allylic character, and the

bright excited state resembles a simple ethylene-like φn1 → φn

2 transition.

Figure 3: The effect of increasing the number of diffuse functions in the basis set on the

density of states and the convergence of the lowest excited and the bright ππ∗ state. The

calculations were performed with CIS and the basis set was varied from 6-311G(2pd,2df) to

6-311(4+,2+)G(2pd,2df).

23


Non-Condon effects in one- and two-photon absorption of GFP chromophore

Two-photon absorption (2PA): - Quadratic dependence of laser power, better spatial resolution; - Employs lower energies (e.g., 1.3 eV instead of 2.6 eV):better penetration, less photo-damage (reduced phototoxicity and photobleaching).

2PA: predicted by Maria Goeppert-Mayer in 1931|0>

|f>


Calculation of the 1PA and 2PA spectra

In the Franck-Condon approximation,spectral shape is Pof=|µ(R0) <χi|χf>|2

- Overlap of vibrational wave functions (Franck-Condon factors); - Electronic transition dipole moment or 2PA electronic factors: neglect the dependence of electronic transition matrix elements on nuclear coordinates.2PA and 1PA spectra have the same shape (but selection rules may be different).

Spe

ctru

m final state

initial state

Pif ∼ |�

dRξinξfn�

�dr(Φi(r;R)µΦf (r;R))|2


Non-Condon effects in 1PA and 2PA of GFP chromophore

TPA measurements of GFP: Hosoi, Ramaguchi, Mizuno, Miyawaki, Tahara, J. Phys. Chem. B. 112, 2761 (2008). The 2PA spectrum is blue-shifted by 700 cm-1 relative to 1PA. “Dark” state was postulated.

Our calculations of Franck-Condon and electronic transition factors: Vibronic effects change the shape of 2PA spectra resulting in the blue shift (no dark state is necessary).

1

HBDI/methanol

EGFP


Pif ∼ |�

dRξinµ(R)ξfn� |2

µ(R) =< Φi|µ|Φf >

Methodology:1. Ground-state structure and frequencies:RI-MP2/cc-pVTZ using Q-CHEM.2. Excited state structure: SOS-CIS(D)/cc-pVTZ using Q-CHEM.3. 1PA transition dipoles: CIS/cc-pVDZ (Q-CHEM).4. FCFs: parallel mode double-harmonic approximation usingezSpectrum.5. 2PA cross sections: TD-DFT/B3LYP and CIS with 6-31G*using Dalton. 6. Including non-Condon effects:


5

TABLE I: The displacement of the most active normal coordinates (calculated at the S0 equilibrium

geometry) for the S1 state of the HBDI anion. The frequencies and the calculated one- and two-

photon cross-sections are also shown for each displaced geometry. The displacements are given as

dimensionless scale factors along mass-weighted, normalized normal modes.

Mode Assignment Frequency (cm!1) !q !OPA !TPA(GM)

Reference geometry - - 1.5959 2.8003

3 in-plane bend 79 -0.646 1.5902 2.5043

9 rock 203 0.183 1.6045 2.9221

11 stretch 246 0.298 1.5986 2.6691

12 stretch 273 -0.297 1.6133 2.3382

13 stretch 333 -0.221 1.5946 2.5552

19 out-of-plane bend 584 0.117 1.5867 2.6776

24 NCC bend 734 0.101 1.5885 2.2879

30 breathing 854 0.138 1.5614 3.0841

61 C=C stretch 1628 -0.066 1.5783 2.9221

necessary to use a small basis set to avoid mixing of the bright states with the electron-

detached continuum[11], which would normally be shifted to higher energies in a protein

environment.

The full results of these calculations are shown in Table I. Images showing the nor-

mal modes and more details of the calculation are available as supplementary information.

Absolute TPA cross sections computed using B3LYP and CIS are also provided in SM.

The gas-phase one-photon absorption spectrum of HBDI was calculated both with the

Condon approximation and including non-Condon e!ects and the two spectra are shown

in Figure 1. The FCFs reveal several important vibrational progressions. The low fre-

quency modes, !3, !9, !11, !12, and !13 contribute significantly to the overall broadening

since their 0-1 transitions have intensity both by themselves and in combination with any

other Franck-Condon active mode. Although the 0-0 transition has the largest individual

FCF, the higher density of peaks corresponding to combinations of low frequency modes in

the region between 300-400 cm!1 serves to shift the peak of the simulated spectrum. The

full spectrum (i.e., including non-Condon e!ects) is also compared with the gas-phase ex-

Franck-Condon active modes and cross sectionsdependence on nuclear geometry


Calculated FCFs and the 1PA/2PA spectra

0

0.2

0.4

0.6

0.8

1

0 1000 2000 3000 4000 5000

Rela

tive

Inte

nsity

Energy (cm-1)

0-0

0-1 3

0-1 9

0-1 11

0-1 12

0-1 13

0-1 190-1 24

0-1 30

0-1 61

non-Condon OPACondon OPA

FCFs

- Several modes are FC-active, mostly those involving the bridge moiety; - Explicit dependence of µ(R) in 1PA is not very important;- Including explicit dependence of 2PA cross section results in 500 cm-1 blue shift relative to the 1PA spectrum.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 1000 2000 3000 4000 5000

Rela

tive

Inte

nsity

Energy (cm-1)

OPATPA


Why 2PA cross section depends so strongly on nuclear displacements?

3

into electronic and nuclear parts such that

!(!re,!rN) ! "e(!re;!rN)#vib(!rN), (2)

where #vib(!rN) is the vibrational wavefunction and the dependence of "e(!re;!rN ) on the

nuclear coordinates is parametric. Substituting this back into Eq. (1) gives

P0f " [

!"e0(!re;!rN)#

vib0 (!rN)!µ"

ef (!re;!rN )#

vibf (!rN)d!red!rN ]

2. (3)

Introducing the electronic transition moment,

!µ0fe (!rN) =

!#e0(!re;!rN )!µ(!re)#

ef (!re;!rN)d!re, (4)

the final expression used for the transition probability is obtained as

P0f " [

!#vib0 (!rN)!µ

0fe (!rN)#

vibf (!rN)d!rN ]

2. (5)

This expression takes into account both the spatial overlap of initial and final vibrational

wavefunctions as well as the dependence of the transition moment on the nuclear geometry.

In the Condon approximation, the dependence of µ0f on nuclear coordinates is neglected

giving rise to familiar Franck-Condon factors (FCFs) defined as overlaps of vibrational

wavefunctions of the initial and final states, ["#vib0 (!rN)#vib

f (!rN)d!rN ]2. As demonstrated in

this letter, the inclusion of a transition moment with explicit dependence on the nuclear

coordinates, i.e., non-Condon e"ects, can lead to both qualitative and quantitative changes

in calculated spectra.

TPA is a non-linear process, and the electronic factors that govern these transitions can

be computed as second-order response properties [15, 16]. The total probability of TPA

can be described by an expression similar to Eq. (5) in which !µ0fe is replaced by the TPA

transition moment, $0fTPA

:

$0fTPA

=1

15

0f#ij

S0fii S

!

jj + 2S0fij (S

0f )!ij, (6)

where the indices i and j run over Cartesian coordinates and S0fij is the two-photon transition

matrix element. The two-photon transition matrix element can itself be expressed as a sum

over transitions between ground, |0#, excited, |n#, and final, |f#, states:

S0fij =

#n

[$0|µi|n#$n|µj|f#

%0,n % %+

$0|µj|n#$n|µi|f#

%0,n % %]. (7)

3

into electronic and nuclear parts such that

!(!re,!rN) ! "e(!re;!rN)#vib(!rN), (2)

where #vib(!rN) is the vibrational wavefunction and the dependence of "e(!re;!rN ) on the

nuclear coordinates is parametric. Substituting this back into Eq. (1) gives

P0f " [

!"e0(!re;!rN)#

vib0 (!rN)!µ"

ef (!re;!rN )#

vibf (!rN)d!red!rN ]

2. (3)

Introducing the electronic transition moment,

!µ0fe (!rN) =

!#e0(!re;!rN )!µ(!re)#

ef (!re;!rN)d!re, (4)

the final expression used for the transition probability is obtained as

P0f " [

!#vib0 (!rN)!µ

0fe (!rN)#

vibf (!rN)d!rN ]

2. (5)

This expression takes into account both the spatial overlap of initial and final vibrational

wavefunctions as well as the dependence of the transition moment on the nuclear geometry.

In the Condon approximation, the dependence of µ0f on nuclear coordinates is neglected

giving rise to familiar Franck-Condon factors (FCFs) defined as overlaps of vibrational

wavefunctions of the initial and final states, ["#vib0 (!rN)#vib

f (!rN)d!rN ]2. As demonstrated in

this letter, the inclusion of a transition moment with explicit dependence on the nuclear

coordinates, i.e., non-Condon e"ects, can lead to both qualitative and quantitative changes

in calculated spectra.

TPA is a non-linear process, and the electronic factors that govern these transitions can

be computed as second-order response properties [15, 16]. The total probability of TPA

can be described by an expression similar to Eq. (5) in which !µ0fe is replaced by the TPA

transition moment, $0fTPA

:

$0fTPA

=1

15

0f#ij

S0fii S

!

jj + 2S0fij (S

0f )!ij, (6)

where the indices i and j run over Cartesian coordinates and S0fij is the two-photon transition

matrix element. The two-photon transition matrix element can itself be expressed as a sum

over transitions between ground, |0#, excited, |n#, and final, |f#, states:

S0fij =

#n

[$0|µi|n#$n|µj|f#

%0,n % %+

$0|µj|n#$n|µi|f#

%0,n % %]. (7)

Note quadratic dependence on µtr.

Why does µtr depend on geometry? Huckel model:

µtr~ x0 (CC bond length at the bridge).

!a1

!a2

!a3

!n1

!n2

Figure 2: Relevant molecular orbitals of deprotonated HBDI (left) and HBDI (right), HF/6-

311G* and the corresponding solutions of the Huckel model. In the ground state of the anion,

both φa1 and φa

2 are doubly occupied, and the bright state is derived by the φa2 → φa

3 excitation.

In the protonated form (neutral HBDI), the MOs are no longer of allylic character, and the

bright excited state resembles a simple ethylene-like φn1 → φn

2 transition.

Figure 3: The effect of increasing the number of diffuse functions in the basis set on the

density of states and the convergence of the lowest excited and the bright ππ∗ state. The

calculations were performed with CIS and the basis set was varied from 6-311G(2pd,2df) to

6-311(4+,2+)G(2pd,2df).

23

and the eigenstates are:

φn1 =

1√2(pB + pI) (11)

φn2 =

1√2(pI − pB) (12)

The resulting HOMO-LUMO excitation energy is 2|α|.

Thus, the model predicts lower excitation energies in the deprotonated form, which is

indeed supported by the calculations and the experiment. For example, vertical excitation

energies of the anionic and protonated forms are 2.62 and 3.83 eV, respectively [computed

by SOS-CIS(D)/cc-pVTZ, see Ref. 16]. Likewise, detuned resonance in meta-form of anionic

HBDI and pCA results in higher excitation energies.20,21 A similar trend is observed in the

two isomers of anionic pCA — the excitation energy of the carboxylate form is almost 1 eV

higher than that of the phenolate form.19

This model can also be used to analyze the trends in oscillator strengths (similar to the

analysis of Dewar and Longuet-Higgins27). For a linear arrangement of the atoms, such that

pB is located at x = 0, pP at x = -x0, pI at x = x0, i.e., �pB|x|pB� = 0, �pP |x|pP � = −x0,

�pI |x|pI� = x0, and assuming zero overlap between the orbitals centered on different atoms

(�pP |x|pB� = �pP |x|pI� = �pB|x|pI� = 0), the transition dipole moment matrix element for the

anionic form corresponding to the first excitation (HOMO →LUMO) is:

�φa2|x|φa

3� =1

2√2

�−pP + pI |x|pP −

√2pB + pI

�≈ − 1

2√2�pP |x|pP �+

1

2√2�pI |x|pI� =

1√2x0

(13)

and in the neutral (protonated) form:

�φn1 |x|φn

2 � =1

2�pB + pI |x|pI − pB� ≈

1

2�pI |x|pI� −

1

2�pB|x|pB� =

1

2x0 (14)

Thus, the first excited state (π → π∗) in the two forms corresponds to the φa2 → φa

3 and

φn1 → φn

2 transitions giving rise to the following values of the transition dipole moments:

| �φa2|x|φa

3� |2 =x202 and | �φn

1 |x|φn2 � |2 =

x204 .

Indeed, computed transition dipole moments of the anionic and neutral HBDI are 4.05 and

3.18 a.u., respectively.16 pCA− exhibit a similar trend, although in this case the difference in

the computed oscillator strength is much higher 30 times).19

5

and the eigenstates are:

φn1 =

1√2(pB + pI) (11)

φn2 =

1√2(pI − pB) (12)

The resulting HOMO-LUMO excitation energy is 2|α|.

Thus, the model predicts lower excitation energies in the deprotonated form, which is

indeed supported by the calculations and the experiment. For example, vertical excitation

energies of the anionic and protonated forms are 2.62 and 3.83 eV, respectively [computed

by SOS-CIS(D)/cc-pVTZ, see Ref. 16]. Likewise, detuned resonance in meta-form of anionic

HBDI and pCA results in higher excitation energies.20,21 A similar trend is observed in the

two isomers of anionic pCA — the excitation energy of the carboxylate form is almost 1 eV

higher than that of the phenolate form.19

This model can also be used to analyze the trends in oscillator strengths (similar to the

analysis of Dewar and Longuet-Higgins27). For a linear arrangement of the atoms, such that

pB is located at x = 0, pP at x = -x0, pI at x = x0, i.e., �pB|x|pB� = 0, �pP |x|pP � = −x0,

�pI |x|pI� = x0, and assuming zero overlap between the orbitals centered on different atoms

(�pP |x|pB� = �pP |x|pI� = �pB|x|pI� = 0), the transition dipole moment matrix element for the

anionic form corresponding to the first excitation (HOMO →LUMO) is:

�φa2|x|φa

3� =1

2√2

�−pP + pI |x|pP −

√2pB + pI

�≈ − 1

2√2�pP |x|pP �+

1

2√2�pI |x|pI� =

1√2x0

(13)

and in the neutral (protonated) form:

�φn1 |x|φn

2 � =1

2�pB + pI |x|pI − pB� ≈

1

2�pI |x|pI� −

1

2�pB|x|pB� =

1

2x0 (14)

Thus, the first excited state (π → π∗) in the two forms corresponds to the φa2 → φa

3 and

φn1 → φn

2 transitions giving rise to the following values of the transition dipole moments:

| �φa2|x|φa

3� |2 =x202 and | �φn

1 |x|φn2 � |2 =

x204 .

Indeed, computed transition dipole moments of the anionic and neutral HBDI are 4.05 and

3.18 a.u., respectively.16 pCA− exhibit a similar trend, although in this case the difference in

the computed oscillator strength is much higher 30 times).19

5


Conclusions:

1. Blue shift in the 2PA spectrum: due to non-Condon effects and not due to a “dark state”.

2. Similar results obtained for the PYP chromophore explaining recent experiments of Prof. Delmar Larsen (UC Davis). Also, observed in other FPs.

3. 2PA cross section depends on nuclear geometry more than that of 1PA because of its quadratic nature. Huckel model explains the trend.







Spectroscopy in condensed phase

1. How to include environment: Explicit solvent models (QM/MM and QM/EFP).

2. Sampling.

3. Solvent effects on spectra: Shift + inhomogeneous broadening.

4. Phenol and phenolate example.


Treating the environment

Environment (solvent, protein, etc)

Implicit Explicit (QM/MM)

PCM – Tomasi, MennucciCOSMO – Barone, CossiC-PCM – Barone, Cossi, KlamtD-PCM – Tomasi, Barone, CossiIEF-PCM – Tomasi, Barone, CossiSMx - Truhlar SSVPE – ChipmanSWIG - Herbert

Point charge models: OPLS force field CHARMM force field Amber force fieldAMEOBA – q, µ, Θ & polarizablePolarizable model – Berne, Friesner, WarshelPolarizable embedding:- Effective fragment potential (Gordon, Slipchenko, Krylov) - polarizable embedding by Kongsted, Christiansen, and co-workers

Semi-explicitLangevin dipole – polarizable and rotatable dipoles on a grid - Warshel


QM/MM: Hybrid approach

• QM part – treats the most important part of the system where all the chemistry happens.

• MM part – the environment.

• Point charge models for MM, e.g., OPLS, CHARMM, AMBER force-fields.

• Our approach: EFP

Warshel, Levitt, J. Mol. Biol. 103, 227 (1976) Singh, Kollman, JCC 7, 718 (1986)

!"#$%&'""()&*%&(

Hybrid QM/MM

QM

MM

•  Large number of degrees of freedom in real systems.

•  Cannot be treated with ab initio method alone.

•  Hybrid quantum mechanical molecular mechanical (QM/MM) method

•  QM part – Treats the most important part of the system where all the chemistry happens.

•  MM part – The environment.

Warshel, Levitt, J. Mol. Biol., 103, 227 (1976)

Singh, Kollman, J. Comput. Chem, 7, 718 (1986)

QM MM QM MMH H H H != + +


Effective Fragment Potential

EFP

EFP

QM

1. Discrete QM/MM. Treats each of the molecules as separate entities.2. Electronic embedding. Explicitly accounts for the effect of the MM part by including the perturbative potential due to the MM part in the QM Hamiltonian.3. Polarizable. Includes response of the environment to electronic excitations of a solute.4. Interaction energy can be divided into 4 components – electrostatics, polarization, exchange-repulsion, and dispersion energy.5. Fully ab initio: no empirical parameters. Originally developed by Mark Gordon

and coworkers. EOM-EFP: Lyudmila Slipchenko (Purdue)

Extension to proteins: Laurent, Ghosh

Q-Chem implementation:Ghosh, Kosenkov, Vanovschi, Williams, Herbert, Gordon, Schmidt, Slipchenko, Krylov, JPCA 114, 12739 (2010).


Benchmark on microsolvated phenol and phenolate

Solvent induced shifts in VIE: -0.53 - +1.95 eV

Errors against full EOM-IP: below 0.05 eV

Ghosh, Isayev, Slipchenko, Krylov, JPCA 115, 6028 (2011);Ghosh et al, JPCB 116 7269 (2012).

Table 1: ∆VIEs (eV) of microsolvated phenol and phenolate computed using the EOM/EFP

and full EOM-IP-CCSD with the 6-31+G(d) basis set. Error in ∆VIE, i.e., ∆VIE(EOM/EFP)-

∆VIE(EOM) is tabulated.

System Full IP-CCSD IP-CCSD/EFP Error in ∆VIE

Phenol(H2O) (PH1) -0.53 -0.55 -0.02

Phenol(H2O)2 (PH2) -0.07 -0.10 -0.03

Phenol(H2O)3 (PH3) -0.24 -0.29 -0.05

Phenol(H2O)4 (PH4) -0.23 -0.26 -0.03

Phenolate(H2O) (P1) +0.63 +0.60 -0.03

Phenolate(H2O)2 (P2) +1.16 +1.13 -0.03

Phenolate(H2O)3 (P3) +1.57 +1.61 +0.04

Phenolate(H2O)4 (P4) +1.95 +1.99 +0.04

(for the lowest ionization) of microsolvated (1-4 water molecules) phenol and phenolate

computed by EOM/EFP and full EOM-IP-CCSD in the 6-31+G(d) basis set. ∆VIE is

defined as the solvent-induced shift in the lowest VIE:

∆VIE = VIEmicrosolvated −VIEgas−phase. (11)

As in our previous studies on thymine,22 the errors in IEs due to EFP approximation are

less than 0.05 eV for the microsolvated phenol structures and 0.04 eV for microsolvated

phenolate. Moreover, the correct ordering of the IEs in the different structures is preserved

in the EFP calculations. Another important point is that the errors in EOM/EFP VIEs

remain almost constant when going from one to four water molecules, which is expected

because the polarization of the solvent is treated self-consistently in the EFP scheme.

Fig. 5 presents several lowest VIEs for the P1, P2, P3, and P4 structures. We note that

the lowest 3 VIEs are accurately determined by the EOM/EFP calculations. However, as

we consider higher VIEs, the errors increase (≈0.15-0.3 eV) and there are few VIEs that

are missing with EOM/EFP as compared to the full EOM calculations. This is because

higher ionized states can be delocalized over the phenolate and water moieties (causing

large errors) or even fully localized on water (causing missing IEs when the QM part

consists only of phenolate). This is illustrated in Fig. 6, which shows the leading MOs

describing the ionized states in the P1 structure. The missing IE (6.63 eV) has a leading

12


Equilibrium sampling (MD) for neutral and charged systems

• Neutral systems (phenol, thymine): can use periodic boundary condition (PBC)

• PBC with charged system causes interaction between the charges

• Spherical boundary condition is used

• Langevin dipole and continuum model used to remove the effect of surface

Florian, Warshel, JPC, 101, 5584 (1997).


VIE of thymine in bulk water: Convergence with system size

Effect of shells of water around thymine is not monotonic!Requires about 5 shells of water for the IE to converge.The computed shift agrees with experiment

Ghosh et al, JPCA 115, 6028 (2011).


VIEs of phenol/phenolate in bulk water

Spectral lineshape is reproduced very well.Absolute values for lowest VIE: Phenol: 7.9 eV (exp: 7.8 eV). Gas phase: 8.6 eV Phenolate: 7.7 eV (exp. 7.1 eV). Gas phase: 2.0 eV.

Ghosh, Roy, Seidel, Winter, Bradforth, Krylov, JPCB 116 7269 (2012).

Phenol (water): Solvent shift: -0.7 eV

Phenolate (water):Solvent shift: +5.7 eV


Conclusions:1. EFP is a 1st principle parameter-free polarizable force field.2. Can be used in hybrid QM/EFP scheme.3. Benchmark results for ionization energy shows remarkable agreement with full QM methods such as EOM-IP-CCSD.4. Bulk solvation effects on VIE of thymine, phenol, and phenolate agree with experiment.


Conclusions and outlook:

Spectroscopy: way to interrogate molecular structure.

“The spectrum is the message from the molecule”

But need theory to decode the message.

New spectroscopies - new challenges for theory


THANKS:


Documents

Modeling electronic and photoelectron spectra: Theory and ... · 3. Electronic structure methods for spectra calculations (EOM-CC approaches). 4. Examples: Photoelectron spectroscopy