CaF: All Spectra and All Dynamics R. W. Field, J. J. Kay, S. L. Coy, V. S. Petrovic, S. N. Altunata, B. M. Wong, and Ch. Jungen NSF.

CaF: All Spectra and All Dynamics

R. W. Field, J. J. Kay, S. L. Coy, V. S. Petrovic, S. N. Altunata, B. M. Wong, and Ch. Jungen

NSF

Multichannel Quantum Defect Theory• MQDT builds a complete description of something

complicated, CaF, by combining complete (all states) descriptions of two isolated parts, CaF+ and e-, at infinite separation.

• Space is divided into an infinite simple region and a compact region of complex interactions.

• The Heff matrix would be of infinite dimension.• An e-, incident with definite ε,ℓ, scatters inelastically off of

the ion-core in a definite v+,N+ state.• The building blocks are channels rather than individual

electronic states. A channel consists of infinite number of states, with the e- in a particular mixed-ℓ form converging to a v+,N+ state of the ion-core.

MQDT• The scattering of the e- off of the ion-core is described by

the “Reaction Matrix,” K, which is expressed in terms of quantum defect matrix elements: Kij = tan(πμij).

• .

• The μij are expanded as power series in internuclear distance, R, and energy. μij, dμij/dR, and dμij/dε are the fitted quantities.

• The MQDT equations yield eigenquantum defects and eigenvectors, which may be expressed in either the Hund’s case (b) or (d) basis sets.

• There are many useful, a priori known transformations between basis sets.

ψi

(N ) E,r,Ωe( ) = fl ε i ,r( )δ i, j − K i, j(rv) ε( ) gl ' ε j ,r( )

j∑

⎡

⎣⎢

⎤

⎦⎥ΦlN+

N( ) Ωe( )

i = l ,v+ ,N+{ } , j = l ',v+ ',N+ '{ } ,ε i =εv+N+ =E −E

v+N+ ,∞.

Quantum defect matrix element values and derivatives obtained from fits to CaF Σ, Π, Δ, and Φ states. Uncertainties are indicated in parentheses. If no numerical value is given, the parameter has been held fixed at zero.

Quality of fit for vibrationally-excited levels with low-n*

Quality of fit in the vicinity of n* = 7.0. Vibronic states at this energy that belong to different vibrational quantum numbers are interleaved. Here, the classical period of electronic motion [proportional to (n*)3] is approximately equal to the classical period of vibrational motion. Vibronic perturbations are frequent.

Example of a strong vibronic (homogeneous) perturbation. In the absence of the perturbation, the 7.36 ‘p’ Π v=0 and 6.36 ‘p’ Π v=1 levels are nearly degenerate. The perturbation causes a ~45 cm-1 splitting of the levels and complete mixing of the two zero-order wavefunctions.

Quality of fit in the n* = 16.5 – 17.5 region. Above n*16, rotational interactions are ubiquitous and quite strong, causing the destruction of regular rotational patterns, which is evident here.

Nonpenetrating States: Stacked Plot for N, N+ Assignment

Nonpenetrating States Live Near Integer n*

High Resolution Detective Work Required

Vibrational Autoionization: Single Channel (HLB-RWF)

Case (b) to Case (d) Transformation

case (b) case (d)even N f Σ,Π,Δ,Φ N+=N-3,N-1,N+1,N+3+ parity d Σ,Π,Δ N+=N-2,N, N+2

[10] p Σ,Π N+=N-1,N+1 s Σ N+=N

- parity f -,Π,Δ,Φ N+=N-2,N,N+2 [6] d -,Π,Δ N+=N-1,N+1

p -,Π N+=N s - N+=-

Multichannel Autoionization• Want total autoionization rate from every

n*,N,v+=1 level and fractional yields into different N+ levels of CaF+ v+=0.

• Transform the μ and dμ/dR matrices from case (b) to case (d).

• Solve the MQDT equations to get eigenvectors expressed in case (d).

• For the optically selected eigenstate, compute expectation value of [dμ/dR(d)]Tdμ/dR(d). This can be broken down into individual N+ contributions

You Can Get Everything You Want

Even if you do not need it!

ψi

(N ) E,r,Ωe( ) = fl ε i ,r( )δ i, j − K i, j(rv) ε( ) gl ' ε j ,r( )

j∑

⎡

⎣⎢

⎤

⎦⎥ΦlN+

N( ) Ωe( )

i = l ,v+ ,N+{ } , j = l ',v+ ',N+ '{ } ,ε i =εv+N+ =E−E

v+N+ ,∞.

Ψr>rc

(N ) = BlN+v+(N ) ψ

lN+v+(N )

lN+v+∑

Ki, j(rv ) ε( ) = Λ N+ N( )

Λ∑ χ

v+N+

R( )K l , ′l(el )(Λ) R,ε( )χv+ '

N+ ' R( )dR∫⎡⎣ ⎤⎦ N+ ' Λ

N( )

i = l ,v+ ,N+{ } , j = l ',v+ ',N+ '{ }

Λ N+ N( )= −1( )

N−Λ 1+ −1( )p−N++l

2

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

21+δΛ0( )

⎡

⎣⎢⎢

⎤

⎦⎥⎥

12

2N+ +1( )12 N+ l N

0 Λ −Λ

⎛

⎝⎜

⎞

⎠⎟

K (rv ) ε( )B = -P(E)B, i.e.,

P(E) +K (rv) ε( )⎡⎣ ⎤⎦B =0

Pi, j( N , p) E( ) =tanπν

v+N+ E( )δ i, j

i = l ,v+ ,N+{ } , j = l ',v+ ',N+ '{ }