Vibrational spectra of model Chromophores P. Giannozzi - - PowerPoint PPT Presentation

Vibrational spectra of model Chromophores

• P. Giannozzi

Democritos-INFM, Scuola Normale Superiore di Pisa, p.giannozzi@sns.it Isfahan, 2005/04/26 Work done in collaboration with V. Tozzini (NEST-INFM, SNS Pisa)

– Typeset by FoilT EX –

Vibrational Spectroscopy: experiment

• Observation of vibrational modes: phonons in crystals, normal modes in molecules,

is a powerful tool in materials characterization

• Vibrational spectroscopy is a sensitive probe of the atomic structure and of the

chemical bonding and thus of the electronic structure

• Most frequently used experimental techniques: Infrared and Raman – both simple

and effective Theoretical calculation of vibrational mode frequencies and intensities from first principles is very helpful in analyzing vibrational spectra

Vibrational frequencies: calculation

Normal mode frequencies, ω, and displacement patterns, U α

I for cartesian component

α of atom I, are determined by the secular equation:

• J,β
• Cαβ

IJ − MIω2δIJδαβ

• U β

J = 0,

where Cαβ

IJ is the matrix of inter-atomic force constants (IFC), i.e. second derivatives

• f the energy with respect to atomic positions:

Cαβ

IJ ≡ ∂2E({R})

∂Rα

I ∂Rβ J

= −∂F α

I

∂Rβ

I

where FI is the force acting on atom I at atomic position RI: FI = −∂E({R}) ∂RI

Vibrational frequencies: calculation (2)

Common methods – based on Density-functional Theory (DFT) – to calculate IFC’s:

• Frozen-phonon technique: finite differentiation of forces
• Density-functional Perturbation Theory (DFPT): direct calculation of second-order

derivatives of the energy Alternative method:

• Extract spectra from Molecular Dynamics runs, via the Fourier Transform of the

velocity-velocity autocorrelation function

Calculation of Infrared and Raman Intensities

Infrared (IR) Intensity: IIR(ν) =

• α
• Iβ

Z⋆αβ

I U β I (ν)

• 2

. is proportional to the square of the induced dipole. The effective charges Z⋆I is the polarization induced by an atomic displacement Non-resonant Raman intensities (Placzek approximation): IStokes(ν) ∝ (ωL − ων)4 ων rαβ(ν), rαβ(ν) =

• ∂χαβ

∂u(ν)

• 2

where χ is the electric polarizability, u(ν) is the normal mode coordinate along mode ν. Valid when ωL (incident laser frequency) << ωel (lowest electronic excitation energy)

Calculation of Infrared and Raman Intensities (2)

The effective charge matrix is a second order derivative of the energy can be calculated using:

• straightforward DFPT
• the Berry’s phase approach

The Raman tensor is a third-order derivative of the energy can be calculated using:

• DFPT for χ + Frozen-phonon (finite differences)
• DFPT using the (2n + 1) theorem: the (2n + 1)−th derivative of energy depends
• nly on derivatives up to order n of the charge density.
• DFPT + Second-order response to electric field
• Finite electric fields + Frozen phonon

Density-Functional Theory

Energy as a functional of the density n(r): E = Ts[n(r)] + EH[n(r)] + Exc[n(r)] +

• n(r)V (r)dr

is minimized by the ground-state charge density. Kohn-Sham equations for one-electron orbitals: (HKS − ǫi) ψi(r) = 0, HKS = − ¯ h2 2m∇2 + VH(r) + Vxc(r) + V (r) are solved self-consistently and the charge density is given by n(r) =

• i

|ψi(r)|2 (the sum is over occupied states)

Density-Functional Theory (2)

Let us assume that the external potential depend on some parameter λ Vλ(r) ≃ V (r) + λ∂V (r) ∂λ + 1 2λ2∂2V (r) ∂λ2 + ... (all derivatives calculated at λ = 0) and expand the charge density nλ(r) ≃ n(r) + λ∂n(r) ∂λ + 1 2λ2∂2n(r) ∂λ2 + ... and the energy functional accordingly: Eλ ≃ E + λ∂E ∂λ + 1 2λ2∂2E ∂λ2 + ... The first-order derivative ∂E/∂λ does not depend on any derivative of n(r) (Hellmann- Feynman theorem): ∂E ∂λ =

• n(r)∂V (r)

∂λ dr

Density-Functional Perturbation Theory (2)

The second-order derivative ∂2E/∂λ2 can be written as a functional of ∂n/∂λ that must be minimized by the correct (ground-state) value for ∂n/∂λ . This results in a set of linear equations that determine ∂n/∂λ. At the minimum (ground state): ∂2E ∂λ2 = ∂V (r) ∂λ ∂n(r) ∂λ dr +

• n(r)∂2V (r)

∂λ2 dr The result can be generalized to mixed derivatives: ∂2E ∂λ∂µ = ∂V (r) ∂λ ∂n(r) ∂µ dr +

• n(r)∂2V (r)

∂λ∂µ dr (the order of derivatives can be exchanged)

Calculation of Linear Response

The minimization of the second-order functional yields a set of equations: (H − ǫi)∂ψi(r) ∂λ +

• j

Kij ∂ψj ∂λ = −Pc ∂V ∂λ |ψi where Pc is the projector over unoccupied states, Kij is a nonlocal operator:

• Kij

∂ψj ∂λ

• (r) = 4
• ψi(r)
• e2

|r − r′| + δvxc(r) δn(r′)

• ψ∗

j(r′)∂ψj

∂λ (r′)dr′ and the linear charge response: ∂n(r) ∂λ = 2Re

• i

ψ∗

i (r)∂ψi(r)

∂λ This scheme can be recast into the “traditional” self-consistent scheme

Application: Raman spectra of model chromophores

The problem: characterization of the chromophore of proteins GFP (Green Fluorescent Proteins) and of DsRed (Red Fluorescent Protein).

O N N O H H H O O O O H NH2 NH2 NH2 Arg 96 Tyr 203 Gln 94 O O O H H H NH Thr 65 Glu 222 O H O NH N His 148 OH O N N O H H H O O O O H NH2 NH2 NH2 Arg 96 Gln 94 O O O H H H NH Thr 65 Glu 222 O O NH N His 148 OH H (a) (b) Thr 203 Tyr 203 Thr 203 Asn 146 Asn 146 Ser 205 Ser 205

Scheme of the chromophore of GFP in the protein

Model chromophores

O OH N N CH3 CH3

Model chromophore of GFP: HBDI

O OH N N CH3 CH3

Model chromophore 1 of DsRed: HBMPI Model chromophore 2 of DsRed: HBMPDI

O OH N N CH3 CH3

Raman Spectra for various Model Chromophores

Surface-Enhanced Raman Spectroscopy Results (HBDI Model Chromophore)

Surface-Enhanced Raman Spectroscopy Results (GFP protein)

Calculated Raman Intensities (HBDI Model Chromophore)

1000 2000 3000 Frequency (cm−1) Raman Intensity (arb. units)

GFP chromophore

(all modes)

300 500 700 900 1100 Frequency (cm−1)

(300−1200 cm−1 region)

Placzek’s approximation (non resonant Raman), DFPT + Frozen-phonon

Some relevant normal mode patterns ν5 ∼ 606cm−1

Suppressed in the protein by geometrical constraints (protein backbone)

Some relevant normal mode patterns ν8 ∼ 720cm−1

Broadened and shifted in the protein

Some relevant normal mode patterns ν11 ∼ 850cm−1

Shifted at lower frequencies in the protein

Resonance Raman

Raman spectra are often obtained in pre- resonant or resonant conditions, i.e. ωL < ωel

• r ωL ∼ ωel: The Placzek approximation does

not hold. The expression for the Raman intensity is much more complicated, involving sums over electronic excited states: σαβ(i → f) ∝

• e

f|Mα|ee|Mβ|i Ee − Ei − ¯ hωL − iΓe + f|Mβ|ee|Mα|i Ee − Ef − ¯ hωL − iΓe

• (M = electric dipole moment operator).

Calculation of pre-resonance Raman intensities

In pre-resonance conditions, under various approximations: Born-Oppenheimer, harmonic ground state, Franck-Condon, the Raman intensity is proportional to the gradients of the excited-state potential energy surface(s): σ(i → f) ∝

• e

|Me|2 ∂Ee ∂u(ν) calculated at the ground-state equilibrium positions Forces on ions on an excited state can be calculated using Time-Dependent DFT: Fe = − ∂ ∂RI (E + ∆Ee) where ∆Ee is the vertical excitation energy.

Calculation of (pre-)Resonance Raman intensities (2)

Convenient algorithm

• start a Molecular Dynamics run on the ground state, with initial velocities

proportional to the forces calculated on the excited-state surface

• extract the (mass-weighted) velocity-velocity autocorrelation function:

f(ω) = 1 √ 2π ∞

−∞

e−iωt

I

mIvI(t) · vI(0)dt (1) The resulting spectra mimic the pre-resonance Raman spectra, i.e. the height of the peaks is proportional to the Raman intensity, assuming that the main contribution comes from a specific excited-state

Results (preliminary) for HBMPDI

Red: Raman intensity calculated for the S2 excited state Green: as above for S3 Blue lines: harmonic frequencies Inset: combined Raman intensity for S2 and S3

Conclusions

• Calculation of vibrational frequencies and intensities from first principles is a useful

tool in materials characterization

• Both off-resonance and pre-resonance Raman intensities can be calculated from

first principles with a manageable computational effort