From Electronic Structure Theory to Simulating Electronic Spectroscopy Marcel Nooijen Anirban Hazra Hannah Chang University of Waterloo Princeton University

Beyond vertical excitation energies for “sizeable” molecules: (A) Franck-Condon Approach: Single surface (Born-Oppenheimer). Decoupled in normal modes. Computationally efficient. (B) Non-adiabatic vibronic models: Multiple coupled surfaces. Coupling across normal modes. Accurate but expensive.

Accurate vertical energy. What is best expansion point for harmonic approximation? Second order Taylor series expansion around the equilibrium geometry of the excited state. Accurate 0-0 transition. Excited state PES Second Taylor series expansion around the equilibrium geometry of the absorbing state. Accurate vertical energy. Ground state PES

Time-dependent picture of spectroscopy The broad features of the spectrum are obtained in a short time T. Accurate PES desired at the equilibrium geometry of the absorbing state.

Ethylene excited state potential energy surfaces CH stretch C C C=C stretch C C H H H H H H H H CH2 scissors C C Torsion C C H H H H

Vertical Franck-Condon Assume that excited state potential is separable in the normal modes of the excited states, defined at reference geometry Anharmonic Harmonic

Electronic absorption spectrum of ethylene: Rydberg + Valence state Experiment Geiger and Wittmaack, Z. Naturforsch, 20A, 628, (1965) Calculation

Rydberg state spectrum Valence state spectrum Total spectrum

Beyond Born-Oppenheimer: Vibronic models Short-time dynamics picture: Requires accurate time-dependent wave-packet in Condon-region, for limited time. Use multiple-surface model Hamiltonian that is accurate near absorbing state geometry. E.g. Two-state Hamiltonian with symmetric and asymmetric mode

Methane Photo-electron spectrum Fully quadratic vibronic model

Calculating coupling constants in vibronic model Construct model potential energy matrix in diabatic basis: Calculate geometry and harmonic normal modes of absorbing state. Calculate "excited" states in set of displaced geometries along normal modes. From adiabatic states construct set of diabatic states : Minimize off-diagonal overlap: Obtain non-diagonal diabatatic Use finite differences to obtain linear and quadratic coupling constants. Impose Abelian symmetry.

Advantages of vibronic model in diabatic basis "Minimize" non-adiabatic off-diagonal couplings. Generate smooth Taylor series expansion for diabatic matrix. The adiabatic potential energy surfaces can be very complicated. No fitting required; No group theory. Fully automated routine procedure. Model Potential Energy Surfaces Franck-Condon models. Solve for vibronic eigenstates and spectra in second quantization. Lanczos Procedure: - Cederbaum, Domcke, Köppel, 1980's - Stanton, Sattelmeyer: coupling constants from EOMCC calculations. - Nooijen: Automated extraction of coupling constants in diabatic basis. Efficient Lanczos for many electronic states.

Vibronic calculation in Second Quantization 2 x 2 Vibronic Hamiltonian (linear coupling) Vibronic eigenstates Total number of basis states Dimension grows very rapidly (but a few million basis states can be handled easily). Efficient implementation: rapid calculation of HC ,

Ethylene second cationic state: PES slices. torsion CC=0.0 CH2 rock, CC=0 CC stretch CH2 rock, CC=-1.5 torsion CC=2.0

Vibronic simulation of second cation state of ethylene Vibronic simulation of second cation state of ethylene. Simulation includes 4 electronic states and 5 normal modes. Model includes up to quartic coupling constants. (Holland, Shaw, Hayes, Shpinkova, Rennie, Karlsson, Baltzer, Wannberg, Chem. Phys. 219, p91, 1997)

Issues with current vibronic model + Lanczos scheme Computational cost and memory requirements scale very rapidly with number of normal modes included in calculation. For efficiency in matrix-vector multiplications we keep basis states (no restriction to a maximum excitation level) Vibronic calculation requires judicious selection of important modes, which is error prone and time-consuming. What can be done?

Transformation of Vibronic Hamiltonian Hamiltonian in second quantization: Electronic states ; Normal mode creation and annihilation operators Unitary transformation operator Define Hermitian transformed Hamiltonian Find operator , such that is easier to diagonalize …

Details on transformation “Displacement operators” “Rotation operators” After solving non-linear algebraic equations: Adiabatic, Harmonic!?

# of parameters in scales with # of normal modes We are neglecting three-body and higher terms in Eigenstates and eigenvalues of : : original oscillator basis, harmonic Transition moments: Easy calculation of spectrum after transformation. Akin to Franck-Condon calculation. # of parameters in scales with # of normal modes

Additional approximations: Assume is a two-body operator, if is two-body. Reasonable if is small Exact if (coordinate transformation) Allows recursive expansion of infinite series Method is “exact” for adiabatic harmonic surfaces (including displacements & Duschinsky rotations). Unitary expansion does not terminate, but preserves norm of wave function. [ Transformation with pure excitation operators can yield unnormalizable wfn’s Similarity transform changes eigenvalues Hamiltonian!! ]

Example: Core-Ionization spectrum ethylene Exact / Transformed Spectrum Potential along asymmetric CH stretch Franck-Condon spectrum

Analysis core-ionization spectra Localized core-hole Basis l,r: Delocalized core-hole Basis g, u: In localized picture: Displaced harmonic oscillator basis states Diagonalize 2x2 Hamiltonian Same frequencies as in ground state (very different from FC) Transformation works in delocalized picture. Slightly different frequencies. Symmetric mode coupling is the same in the two states.

Core Ionization Spectrum acetylene Individual states: exact Lanczos Franck-Condon spectrum Exact Lanczos / Transformation Perfect agreement Exact /Transformed

Benzene Core ionization Transformation with 6 electronic states, 11 normal modes

‘Breakdown’ of transformation method for strongly coupled E1u mode at 1057 cm-1

Combined Transform – Lanczos approach Identify "active" normal modes that would lead to large / troublesome transformation amplitudes. Transform to zero all ‘off-diagonal’ operators that do not involve purely active modes. Solve for and obtain a transformed Hamiltonian: Obtain as starting vector Lanczos Perform Lanczos with the simplified Hamiltonian .

Example: strong Jahn-Teller coupling Transformation Active / Lanczos Full Lanczos vs Full Transform Full Lanczos vs Partial Transform

Summary Vibronic approach is a powerful tool to simulate electronic spectra. Coupling constants that define the vibronic model can routinely be obtained from electronic structure calculations & diabatization procedure. Full Lanczos diagonalizations can be very expensive. Hard to converge. Vibronic model defines electronic surfaces: Can be used in (vertical and adiabatic) Franck-Condon calculations. No geometry optimizations; No surface scans. Efficient. “Transform & Diagonalize” approach is interesting possibility to extend vibronic approach to larger systems.

Anirban Hazra Hannah Chang Alexander Auer NSERC University of Waterloo NSF