PECD: Origin of the Angular Asymmetry

Powis Group

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The PhotoElectron Angular Distribution

Photoionization of randomly oriented molecules with polarized (or un-polarized) light produces a photoelectron angular distribution (PAD) that takes a well known form:

I(θ)=1 + βP2(cos θ).
P2 is the 2nd Legendre Polynomial function and provides a non-isotropic part to the overall distribution. The parameter, β ( −1 ≤ β ≤ +2 ) specifies the magnitude of the anisotropy which ranges from a pure cos2θ to sin2θ form, but which therefore remains symmetrical around the direction θ = 90°.

Less widely recognised, is that this equation is just a special (though common) sub-case of the more general expression:

Ip(θ) = 1 + b1pP1(cosθ) + b2pP2(cosθ)
The coefficients bnp are determined by the photoionization dynamics and depend on the light polarization — p = 0 for linear, p = ±1 for left, right circular polarization (CPL) states. Coupling of the electron and photon angular momenta introduces certain symmetry properties and constraints. For the P2 terms b20 = − ½ × b2±1 ≡ β . More particularly, b10 = 0 while b1±1 also vanishes for achiral molecules; in such circumstances this general expression simply reduces to the well-known first form.

But, for the specific case of a chiral molecule ionized with CPL, these P1 (1st Legendre Polynomial) coefficients are found to be antisymmetric for exchange of handedness of either the molecular enantiomer or the light polarization: b1+1 = − b1−1 .

Asymmetry in the PAD

The additional P1 term in the general PAD expression,opposite, is a simple cosθ function. For a chiral molecule ionized by CPL it therefore introduces a forward-backward asymmetry into the PAD – θ measured from the light beam propagation direction – which reverses with exchange of either the enantiomer or CPL state. Using the symmetry properties of the bnp coefficients, a quantitative measure of this PAD asymmetry for a given enantiomer/CPL pair is written:

GAD = Ip(0°) −Ip(180°) = 2b1p

The alternative asymmetry factors GAD and Γ provide essentially identical measures, and either allow experemental measurements to be related to the value of the b1p coefficient which can be calculated using appropriate quantum treatments.

Angle Resolved Circular Dichroism

For a given enantiomer, one can define the photoelectron circular dichroism (PECD) as the difference seen, at a fixed detection angle θ, between left and right CPL:

Γ0 = I+1(0°) −I−1(0°) = [I+1(θ) − I−1(θ)] cosθ = 2b1+1
– where the above expression has been simplified making use of the antisymmetry of the b1±1 coefficients. Γ therefore measures the dichroism in the angle-resolved photoelectron spectrum. For the other enantiomer, b1+1 will have opposite sign, so that the PECD, Γ, can also be expected to change sign.

Giant Chiral Asymmetry

The PECD chiral asymmetry discussed on this page is obtained from pure electric dipole interactions, unlike normal CD that requires much weaker electric quadrupole and magnetic dipole terms. Typical GAD or Γ0 values (10-2– 2×10-1) consequently exceed more conventional CD by 2–3 orders of magnitude. Indeed PECD is the strongest chiroptical asymmetry that can be observed in randomly oriented, dilute systems in the absence of any cooperative enhancement effects.

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Recording PECD

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This picture shows a 3D slice, mathematically inverted from a raw experimental 2D image, of valence photoelectrons from pure R-enantiomers of camphor. The HOMO photoelectron band appears as the reddish outer ring structure and the forward–backward (here it appears up–down) chiral asymmetry can be seen in this false colour representation. Also visible is a similar asymmetry, but with the opposite direction, from the HOMO-1 orbital ionization (the inner ring structure).

For sensitivity, however, we prefer to record PECD as a difference image, switching between left and right CPL. The asymmetry in such difference images appears as a positive signal in one direction, a negative in the other — as can be seen in other illustrations on this page.

Notice that GAD and Γ are essentially equivalent measures of the dynamical parameter b1p