1Garth J. Simpson

Objectives:1. Discuss the benefits of coordinate-

independent visualization of molecular tensors.

2. Describe sagittary representations for the resonant molecular tensor.

3. Introduce space-filling hyper-ellipsoid representations for both the molecular and ensemble (surface) tensors.

4. Introduce vector-sphere (unit-sphere) representations for the molecular and ensemble tensors.

Garth J. Simpson

Department of Chemistry Purdue University

3Garth J. Simpson

Computational chemistry typically produces outputs as tables of 18 or 27 numbers.

1. It can be difficult to connect these numbers to particular molecular structural motifs.

2. For a particular vibrational transition, the coordinate system of the molecule will generally be different that the local mode of the transition of interest.

3. Very different sets of tensor elements are recovered depending on the particular choice of the internal molecular coordinates selected.

Visual representations

Numerical representation

4Garth J. Simpson

Computational chemistry typically produces outputs as tables of 18 or 27 numbers.

1. It can be difficult to connect these numbers to particular molecular structural motifs.

2. For a particular vibrational transition, the coordinate system of the molecule will generally be different that the local mode of the transition of interest.

3. Very different sets of tensor elements are recovered depending on the particular choice of the internal molecular coordinates selected.

Visual representations

Numerical representation

5

Linear polarizability. The one-photon resonances associated with the linear polarizability are described by the transition moment, which is a vector property in the molecule.

The linear polarizability matrix is constructed by the Kronecker (outer) product of the transition moment with its transposed conjugate.

0nµ

Garth J. Simpson

Perry, J.M.; Moad, A.J.; Begue, N.J.; Wampler, R.D.; Simpson, G.J. J. Phys. Chem. B, 109, 2005,20009-20026.

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Raman polarizability.The Raman tensor is a 3×3 matrix. As long as the virtual states in the Raman process are far from resonance (i.e., adiabatic limit), the tensor is symmetric Just like the moment of inertia matrix, the symmetric Raman tensor can be described by three principal eigenvectors and their corresponding eigenvalues. In this reference frame, only the three diagonal elements within the Raman matrix are nonzero.

0nα

Garth J. Simpson

Perry, J.M.; Moad, A.J.; Begue, N.J.; Wampler, R.D.; Simpson, G.J. J. Phys. Chem. B, 109, 2005,20009-20026.

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Raman polarizability.Red, dashed lines indicate principal Raman elements with negative sign.

Solid, blue lines indicate positive sign.

0nα

Garth J. Simpson

Perry, J.M.; Moad, A.J.; Begue, N.J.; Wampler, R.D.; Simpson, G.J. J. Phys. Chem. B, 109, 2005,20009-20026.

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VibrationallyResonant SFG.

( )(2)0 0res n ir n nSβ ω α µ= ⊗

The combined representations for the transition moment and Raman tensor together with the assumed complex-valued lineshape function recover the resonant contributions to the molecular tensor in any arbitrary coordinate system.

0nµ

0nα

Garth J. Simpson

Perry, J.M.; Moad, A.J.; Begue, N.J.; Wampler, R.D.; Simpson, G.J. J. Phys. Chem. B, 109, 2005,20009-20026.

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ElectronicallyResonant SHG.

( )(2)0 02res n n nSβ ω µ α= ⊗

The combined representations for the transition moment and TPA tensor together with the assumed complex-valued lineshape function recover the resonant contributions to the molecular tensor in any arbitrary coordinate system.

0nµ

0nα

Garth J. SimpsonOrder flips relative

to vib. SFG

Lineshaperesonance about 2ω instead of ωir.

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Raman polarizability.The Raman tensor is a 3×3 matrix. As long as the virtual states in the Raman process are far from resonance (i.e., adiabatic limit), the tensor is symmetric Just like the moment of inertia matrix, the symmetric Raman tensor can be described by three principal eigenvectors and their corresponding eigenvalues. In this reference frame, only the three diagonal elements within the Raman matrix are nonzero.

0nα

Garth J. Simpson

Perry, J.M.; Moad, A.J.; Begue, N.J.; Wampler, R.D.; Simpson, G.J. J. Phys. Chem. B, 109, 2005,20009-20026.

11Garth J. Simpson

Worked example. What are the nonzero tensor elements for this transition?

x'

y'

68o

39o

First, the transition moment ( )( )

0

0 00

sin 39 0.080.13

0.101cos 39ea eaµ

− − = =

To generate α in the reference (x',y') coordinate system, first define it in the principal coordinate system (x0,y0) then rotate.

0.1 e⋅a0

0.2 e2⋅a02⋅Eh

-1

( ) ( ) ( )2 2 1 0 0 2 2 10 0

2.40 0.090 0.229

' 0.2 68 680 0.229

0.90 0.387

h he a E R R e a Eα − −

− = ⊗ = −

Now combine.

( ) ( ) 3 3 10

0.0070.0190.0190.0320.009

0.0230.0230.039

ir ir hS S e a Eβ ω α µ ω −

− − − = ⊗ = −

βx'x'x'

βy'y'y'

Perry, J.M.; Moad, A.J.; Begue, N.J.; Wampler, R.D.; Simpson, G.J. J. Phys. Chem. B, 109, 2005,20009-20026.

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Limitations of the sagittary representations.

Resonant SFG.

( )(2)0 0res n ir n nSβ ω α µ= ⊗

This representation ONLY works well for individual chromophores and individual exciton transition, but cannot be trivially generalized to represent the NLO properties of weakly coupled ensembles.

0nµ

0nα

Garth J. Simpson

13Garth J. Simpson

Hyperellipsoid representations can be generated both for single transitions and for ensembles. The amplitude of the hyperellipsoid surface is proportional to the magnitude of βz'z'z' (or χZZZ) as the z'-axis (or Z-axis for the ensemble) is redefined at each point across the surface of a sphere. The sign information is carried in the color, with red indicating negative and blue positive.

Wampler, R.D.; Moad, A.J.; Moad, C.W.; Heiland, R.; Simpson, G.J. Acc. Chem. Res., 40, 2007,953-960.

14Garth J. Simpson

Hyperellipsoids can be constructed for surfaces, crystals, or any other second-order active structure.

Individual structural motifs (collagen helix)

Complete proteins (SIV protease)

Crystals (P212121lattice)

Wampler, R.D.; Moad, A.J.; Moad, C.W.; Heiland, R.; Simpson, G.J. Acc. Chem. Res., 40, 2007,953-960.

15Garth J. Simpson

1. Hyperellipsoids do not incorporate the complete information content of the tensor, recovering only the coparallel polarized component at each position. Additional contributions from cross-polarizations are lost in this representation.

2. Inversion to recover the numerical values for the tensor elements is nontrivial and is not generally complete. The primary utility of the hyperellipsoids lies in qualitative characterization for building of intuition.

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Originally developed by Adam Tuer and Virgis Barzda, University of Toronto.Each vector on the surface of the sphere represents the polarization induced in the molecule by driving fields polarized along the axis connecting the origin to that point in the sphere.

The hyper-ellipsoids recover only the projection normal to the sphere surface.

Tuer, A.; Krouglov, S.; Cisek, R. Tokarz, D.; Barzda, V. J. Comput. Chem. 2011, 32,1128-34. Garth J. Simpson

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Example: D4 point group (422 crystal class), in which only orthogonally polarized SHG is allowed by symmetry.

Only one unique tensor element is allowed in a D4(422) crystal:

Tuer, A.; Krouglov, S.; Cisek, R. Tokarz, D.; Barzda, V. J. Comput. Chem. 2011, 32,1128-34. Garth J. Simpson

XYZ XZY YXZ YZXχ χ χ χ= = − = −

The crystals exhibit SHG activity by symmetry, but cannot produce coparallelSHG in any crystal orientation. Therefore, the hyperellipsoid is identically zero everywhere, but the vector-sphere is not.

C4

axis

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Moad, A.J.; Moad, C.W.; Perry, J.M.; Wampler, R.D.; Goeken, G.S.; Begue, N.J.; Shen, T.; Heiland, R.; Simpson, G.J. J. Comput. Chem. 2007, 28,1996-2002. Garth J. Simpson

http://nlopredict.sourceforge.net/

Tensor visualization, rotation, and manipulation software package based on the mathematical framework described in this meeting.

• Performs evaluation of the local-frame and ensemble-frame responses as a function of orientation.

• Incorporates all the visualization techniques described in this presentation for both single molecules and ensembles (where appropriate).

• Open-source software available through Sourceforge.net.• Piggybacks as an add-in to the protein visualization package UCSF

Chimera (http://www.cgl.ucsf.edu/chimera/ ).

19Garth J. Simpson

SummaryI. Sagittary representations

Pros: i) intuitive and rigorously complete when coupled with the linshapefunction, ii) invertible to numerically recover the complete molecular tensor in any reference frame.Cons: only applicable to individual transitions and not ensembles.

II. Hyperellipsoid representations.Pros: i) applicable both to individual transitions and ensembles, ii) visually intuitive.Cons: i) only recovers coparallel polarized component and therefore has the potential to be misleading, ii) not invertible to recover complete molecular tensor.

III. Vector sphere / unit sphere representationsPros: i) applicable both to individual transitions and ensembles, ii) invertible to recover complete ensemble tensor in any reference frame (in principle).Cons: visually more complex than the alternative representations.