Visualization of electron correlation in autoionizing states above

the 3p threshold in magnesiumw

Arthur Reber,a Christopher Gordonb and R. Stephen Berry*a

aDepartment of Chemistry and The James Franck Institute, The University of Chicago,Chicago, Illinois 60637, USA

bLaboratory School, The University of Chicago, Chicago, Illinois, USA

Received 9th June 2005, Accepted 20th July 2005First published as an Advance Article on the web 2nd August 2005

The conditional probability density has been calculated for a number of autoionizing states (AIS) in Mg abovethe 3p threshold. The correlation in such high energy AIS has not been extensively studied and provides insightinto the rovibrator behavior of two-electron atoms. The calculations have been done by configuration interaction(CI) with a B-spline basis. This allows for the simultaneous study of the effects of electron correlation and thewidths and angular distribution of photoelectrons in multiphoton ionization. The states have been assignedapproximate vibrational quantum numbers, and a correlation between the approximate quantum numbers andthe photoelectron distribution is observed. Probability distribution for one electron when the other, representedby the small spike, is at its most probable distance from the nucleus. This is a distribution for the doubly-excited1Se state commonly labeled as the 4s2 state.

1. Introduction

A great deal of interest has been shown in the exact energiesand properties of the autoionizing states of themagnum atom.1–6

Furthermore, the nature of electron correlation in doubly-excited two-electron atoms has also generated interest.7–12

Previous studies of correlation in alkaline earth atoms havefocused primarily on their ground and low-energy excitedstates. With the advent of methods based on the B-spline basis,and with expanded computational capacities, the calculation ofwave functions and properties of diffuse doubly-excited statesand doubly-excited states with large angular momenta hasbecome straightforward.

This work is in part motivated by interest in the rovibra-tional behavior of doubly-excited states in two-electron atoms.A primary goal of previous work has been to attempt toidentify approximate constants of motion which provide fora description of the observed states. The molecular model oftwo-electron atoms has offered profound insight into electroncorrelation. Examination of the correlated behavior of thevalence electrons in alkaline earth atoms (and doubly-excitedhelium as well) are primarily located on opposite sides of thenucleus and undergo large-amplitude bending vibrationscoupled with collective rotations. In other words, the correla-tion is so strong that the appropriate zero-order description ofthe quantization of the valence electrons in these atoms is interms of collective, rotor–vibrator behavior, rather than per-turbed states of single-electron angular momenta. This simpleview, substantiated by scrutiny of the wave functions, hasprovided insight into low-lying doubly-excited states. Will itdo the same for high energy and high angular momentumdoubly-excited states?

The connection between photoionization and collisionalionization properties and the collective motions of the elec-trons in the populated AIS is also of considerable interest. Theuse of the L2 integrable B-spline basis has allowed for theaccurate calculation of photoionization and multiphoton ioni-zation of simple atoms near doubly-excited states.13 A com-parison of the approximate constants of motion in the doubly-excited states and their photoelectron distribution providesadditional insight and utility to the interpretation of dynamicsof doubly-excited states.Considerable work has gone into studying the conditional

probability distributions in two-electron atoms. The break-through by Herrick and Kellman showed that the energylevel patterns of doubly-excited helium revealed collect-ive rotations and vibrations similar to that of a triatomicmolecule.14,15 The wave functions and correlated proba-bility distributions of doubly-excited states in the alkaline-earth atoms have been studied by Krause and Berry.10,11

Previous work includes the development of overlap-optimizedrotor–vibrator wave functions by Hunter and Berry,16 andBatka and Berry have compared a variety of methods usingconditional probability distributions, oscillator strengths, andangular distributions.17 Morushita and Lin studied conditionalprobabilities and approximate quantum numbers in three-electron atoms.18–20 Ellis has developed an operator whichrepresents the spatial correlation in two-electron atoms.21

O’Neil and Gill have studied correlation in a Hooke’s lawatom.22 Osenda et al. have further demonstrated the impor-tance of electron correlation in the dynamics of entangle-ment.12 Also, Luc-Koenig et al.29 have calculated multi-photon ionization cross sections including two-photon ATIusing R-matrix methods.Considerable experimental work has been done connecting

electron correlation to the angular distribution of electrons.Hunter et al.23 and Mullins et al.24 have studied photoelectronangular distributions from excited states in barium in order toprobe configuration mixing in the states, and this group hasalso studied effects in strontium and calcium.25,26 Ford et al.27

and El-Marji et al.28 have studied electron-impact double

w Electronic supplementary information (ESI) available: Animationsshowing the effect of varying a in the conditional probability distribu-tion diagrams for Mg. See http://dx.doi.org/10.1039/b508158k

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ionization differential cross sections in magnum at variableenergies and find profound changes in the differential crosssection which provided information on electron correlation inthe atom.

Here, we present the conditional probability densities of avariety of doubly-excited states lying above the 3p thres-hold in magnum, for which L ¼ 0–3. The wave functionsare calculated from a B-spline basis in an configuration inter-action calculation (CI); for this light atom, LS couplingis assumed. The conditional probability distributions arestudied and tentatively assigned rotational and vibrationalquantum numbers. The exact positions and widths of theAIS are calculated, together with the two- and three-photonangular distributions from the decay of the AIS. The cal-culations show that the conditional probability distribut-ions are useful for identifying states with similar angulardistributions.

2. Theory

The calculation of energy levels and wave functions for two-electron atoms is a straightforward process. The general formof the CI wave functions is given in eqn. (1).

CCI ¼ AX

n1 l1n2 l2

Cn1l1n2l2fn1l1ðr1Þfn2l2

ðr2ÞYLMl1l2ðr1; r2Þ ð1Þ

A is the antisymmetrization operator; the radial wave functionsare

fnl ¼XNi¼1

cnli BiðrÞ; ð2Þ

where Bi(r) is the ith B-spline and the cinl are the appropriate

expansion coefficients. The angular functions Yl1l2

LM(r1, r2) arethe coupled spherical harmonics,

YLMl1l2ðr1; r2Þ ¼

Xm1 ;m2

Yl1m1ðy1;f1ÞYl2m2

ðy2;f2Þhl1;m1; l2;m2jLMi:

ð3Þ

The wave functions are optimized by diagonalizing as de-scribed by Bachau et al.13

The conditional probability density is derived from thesolution of the two-electron wave functions by integrating overthe Euler angles to give the rotational trace of the diagonaltwo-electron density matrix in terms of the variables r1, r2, and

y12. This gives eqn. (4).

rðr1; r2; y12Þ ¼X

l1;n1 ;l2;n2;l02 ;n02 ;

c�l01 n01 l02 n02c�l1n1l2n2

f�n01

� ðr1Þf�n02ðr2Þfn1ðr1Þfn2

ðr2Þ

� 1

2½ð2l 01 þ 1Þð2l1 þ 1Þð2l 02 þ 1Þð2l 02 þ 1Þ�1=2

�Xminðl1þl01;l2þl02Þ

k

�1l�kð2kþ 1Þ

�l1 l01 k

0 0 0

!l2 l02 k

0 0 0

!l1 k l01

l02 l l2

( )

� Pkðcos y12Þ

ð4Þ

The one-particle density is then found by integrating over r1and y12 to give eqn. (5).

SðrÞ ¼Z inf

0

r21dr1

Z p

0

sin y12dy12rðr1; r2 ¼ r; y12Þ ð5Þ

Finally, the two-variable conditional probability density ofprincipal interest here is given by eqn. (6).

r(r1,y12 |r2 ¼ a) ¼ r(r1,r2 ¼ a,y12)/S(r ¼ a) (6)

The conditional probability density is then calculated for a gridof points for a fixed value of a and then plotted. This allows for

Table 1 Configurations and amplitudes for AIS with L ¼ 0, S ¼ 0

Amplitude for L ¼ 0 AIS states

1Se 4s2 0.916 3Se 4s5s 0.905

4p2 0.280 4s6s �0.3534s5s �0.24 4p5p 0.209

4d2 0.115 3d4d �0.0931Se 3d2 0.800 3Se 3d4d 0.784

3d4d �0.529 3d5d �0.5614p2 0.406 4p5p 0.199

4s5s 0.157 4s6s �0.1291Se 4s5s 0.768 3Se 4s6s 0.700

4s6s �0.533 4s7s �0.6404p5p 0.217 4s5s 0.243

4s2 0.179 3d5d �0.1141Se 4s7s 0.715 3Se 4s8s 0.805

4s6s �0.554 4s7s �0.3824s5s �0.271 4s6s �0.3413d5d 0.176 4s9s �0.216

Table 2 Configurations and amplitudes for AIS with L ¼ 1, S ¼ 1

Amplitude for L ¼ 1 AIS states

1Po 4s4p 0.786 3Po 4s4p 0.939

4p3d �0.465 4s5p �0.1824s5p �0.299 4p5s 0.169

4p4d 0.256 4p3d �0.1671Po 4s5p �0.706 3Po 4p3d �0.892

3d4p 0.541 3d4f 0.258

4s6p 0.267 4s5p 0.217

3d4f 0.260 4p4d 0.1901Po 4s6p �0.628 3Po 4s5p 0.786

3d4f 0.445 4s6p �0.5333d4p 0.370 4s4p 0.175

4s4p 0.322 3d4p 0.1391Po 4s7p �0.591 3Po 4s7p 0.611

3d5f �0.435 3d5f 0.398

4s6p 0.410 3d4f �0.3343d4f 0.360 4s5p �0.250

Table 3 Configurations and amplitudes for AIS with L ¼ 2, S ¼ 0

Amplitude for L ¼ 2 AIS states

1De 4s3d 0.858 3De 4s3d 0.929

3d5s �0.258 3d5s �0.2383d2 �0.254 4s4d �0.2224s6d �0.213 4p4f �0.149

1De 3d2 0.775 3De 4s4d 0.724

3d4d �0.460 3d5s �0.4374p2 0.352 4s5d �0.4134s4d �0.147 3d5s �0.177

1De 4s4d 0.695 3De 3d7s 0.686

5s3d �0.479 3d6s �0.4064s5d �0.316 4s8d �0.3194p62 �0.294 4s7d 0.295

1De 3d5s 0.608 3De 3d4d 0.811

3d6s �0.541 3d5d �0.5374s5d �0.300 4p5p 0.144

3d4s 0.250 3d5s �0.098

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the study of the two-electron density as a function of geometry,a particularly powerful tool to examine radial and angularcorrelation.

The wave functions are represented in a basis of two-electronstates constructed from a B-spline basis.13,30 B-splines are anL2 integrable basis; for each angular momentum, a basis of 190B-splines of order 10 was constructed in a spherical cavity witha radius of 75 au. The order of the basis refers to the order ofthe polynomial and the number of nonzero basis functions at a

given point. At the edge of the cavity, a basis function isremoved if it does not conform to the boundary conditions.The Mg21 core is represented by an analytical model potentialthat reproduces the valence-core potential resulting from self-consistent calculations. Also, we included a phenomenologicalpotential that represents polarization of the core.31 The one-electron states are calculated by diagonalizing the Mg1 Ha-miltonian with the B-spline basis; the diagonalization is per-formed while imposing orthogonality with the Mg21 core. Thetwo-electron wave functions were evaluated in a basis ofconfigurations built from the one-electron wave orbitals. Thenumbers of configurations used for each calculation were: 326for L ¼ 0, 113 for L ¼ 1, 477 for L ¼ 2, and 259 for L ¼ 3. Forthe angular distribution calculations, we used a basis of 650B-splines in a box of 250 au radius.The multiphoton cross sections were evaluated in the dipole

approximation for linearly polarized light. For the two-photoncase, the transition amplitude is found by

Mð2Þgm ¼

Xn

hgjD � ejnihnjD � ejmiEg þ o� En

þ limZ!0�

ZdEnhgjD � ejnihnjD � ejmiEg þ o� En � iZ

ð7Þ

Table 4 Configurations and amplitudes for AIS with L ¼ 3, S ¼ 1

Amplitude for 1Fo AIS states

1Fo 4p3d 0.813 3Fo 4p3d 0.895

3d5p �0.400 3d5p �0.2974s4f 0.381 4p4d �0.2393d4f 0.124 4s4f 0.199

1Fo 4s4f 0.645 3Fo 4s4f 0.723

4s5f �0.504 4s5f �0.5395p3d 0.434 3d5p 0.327

4p4d �0.228 4p4d �0.1521Fo 3d6p 0.568 3Fo 3d5p �0.607

5p3d �0.517 3d6p �0.5414s5f �0.433 3d4f �0.3583d4f �0.262 3d5f 0.276

1Fo 3d5f �0.665 3Fo 3d4f 0.669

3d4f 0.549 3d5f �0.5304s6f 0.301 3d6p �0.3014s5f �0.274 3d5p �0.237

Table 5 The configurations, symmetries, molecular mode quantum

numbers, positions with respect to the Mg21 ion and widths of the AIS

lying above the 3p threshold

Position width and assignment

Conf. Sym. Quant. Position Width/eV

4s2 1Se [1,1,0] �0.354 74 1.65[�1]3d2 1Se [0,2,0] �0.298 36 1.09[�1]4s5s 1Se [0,0,1] �0.283 99 7.46[�2]4s7s 1Se [1,1,0] �0.263 19 7.00[�4]4s5s 3Se [0,0,1] �0.290 59 3.43[�3]3d4d 3Se [0,2,0] �0.268 46 4.29[�4]4s6s 3Se [0,0,1] �0.264 93 2.09[�3]4s8s 3Se [1,1,0] �0.253 93 9.11[�4]4s4p 1Po [1,0,0] �0.312 05 5.26[�3]4s5p 1Po [0,0,1] �0.288 50 5.40[�3]4s6p 1Po [0,2,1] �0.272 13 1.11[�3]4s7p 1Po [0,2,1] �0.263 03 2.46 [�3]4s4p 3Po [0,0,1] �0.325 48 7.91[�2]4p3d 3Po [1,0,0] �0.279 62 1.13[�1]4s5p 3Po [1,1,0] �0.275 47 1.75[�2]4s7p 3Po [0,1,1] �0.258 75 9.89[�3]4s3d 1De [1,1,0] �0.331 29 3.56[�1]3d2 1De [0,2,0] �0.303 14 2.59[�1]4s4d 1De [2,1,0] �0.284 17 2.83[�2]3d5s 1De [2,1,0] �0.272 58 1.43[�1]4s3d 3De [0,0,1] �0.333 91 9.26[�2]4s4d 3De [1,1,0] �0.287 44 2.02[�3]3d7s 3De [0,1,2] �0.273 85 2.50[�2]3d4d 3De [0,0,1] �0.270 63 9.40[�3]4p3d 1Fo [0,0,1] �0.299 70 1.06[�2]4s4f 1Fo [2,0,0] �0.271 49 4.15[�4]3d6p 1Fo [0,0,2] �0.264 13 2.91[�3]3d5f 1Fo [0,1,1] �0.259 09 6.47[�5]4p3d 3Fo [0,0,1] �0.303 81 3.08[�2]4s4f 3Fo [0,1,2] �0.272 42 8.99[�3]3d5p 3Fo [2,2,0] �0.264 11 1.43[�2]3d4f 3Fo [0,2,2] �0.261 55 2.29[�3]

Fig. 1 Conditional probability distributions for the 1Se AIS. A, 4s2,a ¼ 5.87, B 3d2, a ¼ 4.34, C 4s5s, a ¼ 5.61, D 4s7s, a ¼ 6.12. The peakindicates the position of the fixed electron, and the plot is centeredaround the nucleus in a 25 au box unless otherwise noted.

3278 P h y s . C h e m . C h e m . P h y s . , 2 0 0 5 , 7 , 3 2 7 6 – 3 2 8 3 T h i s j o u r n a l i s & T h e O w n e r S o c i e t i e s 2 0 0 5

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Here, n represents all possible intermediate (bound and con-tinuum) states, g represents the ground state, D is the dipoleoperator and e is the polarization vector. The velocity gauge ofthe dipole operator is used throughout this calculation. Thecorresponding three-photon amplitudes are given by eqn. (8).

Mð3Þgm ¼ lim

Z!0�

Xn1;n2

ZZdEn1dEn2

� hgjD � ejn1ihn1jD � ejn2ihn2jD � ejmiðEg þ o� En1 � iZÞðEg þ 2o� En2 � iZÞ

ð8Þ

The discretization is then performed by varying the box size toensure that the energy for each true continuum state associatedwith a pole and its discrete representation match precisely. Theintermediate-state wave functions are calculated with the L2

integrable close-coupling method developed by Cortes andMartın.32 The multichannel continuum is transformed into asum of orthogonal uncoupled continuum states. The un-coupled channels are found by diagonalizing the Hamiltonianin a basis of two-electron configurations. Green’s functionsintroduce the coupling of the channels.

The method developed by Sanchez and Martın34 providesthe treatment of the final continuum states in the Feshbachformalism.33 We computed the resonant and nonresonantcontributions to the wave functions separately because findingthe uncoupled channels first, and then doing the channelcouplings, is simpler than calculating the both resonant andnonresonant contributions together. The P projection operatorselects the nonresonant configurations and the Q operatorselects the doubly-excited configurations. This way, one canevaluate the widths and positions of the doubly excited statesin a single calculation. It also clarifies the role of correlation inthe spectra by identifying the Feshbach state, and the weightsof the contributing configurations. The angular distributionsare calculated from the transition amplitudes using the equa-tions given in ref. 1 and 2. This model assumes that the recoil ofthe atom is negligible, and that the photons are linearlypolarized, with parallel polarizations.

3. Results

We have calculated the conditional probability distributions ofthe four lowest-energy doubly-excited states in magnesiumabove the 3p threshold for L ¼ 0–3, with natural parity. The

Fig. 3 Conditional probability distributions for 1Po AIS. A, 4s4p, a ¼5.61, B 4s5p, a ¼ 6.64, C 4s6p, a ¼ 6.38, D 4s7p, a ¼ 5.36. The peakindicates the position of the fixed electron, and the plot is centeredaround the nucleus in a 25 au box unless otherwise noted.

Fig. 2 Conditional probability distributions for 3Se AIS. A, 4s5s, a ¼5.61, B 3d4d, a ¼ 4.59, C 4s6s, a ¼ 5.61, D 4s8s, a ¼ 5.36. The peakindicates the position of the fixed electron, and the plot is centeredaround the nucleus in a 25 au box unless otherwise noted.

P h y s . C h e m . C h e m . P h y s . , 2 0 0 5 , 7 , 3 2 7 6 – 3 2 8 3 3279T h i s j o u r n a l i s & T h e O w n e r S o c i e t i e s 2 0 0 5

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two-electron wave functions have been calculated, and theamplitudes of the major configurations have been tabulatedin Tables 1–4. The energies from these calculations are reason-ably converged. For most of the states, one can characterizethem in terms of symmetric stretches, bends, asymmetricstretches, and rigid-rotor excitations. The approximate quan-tum numbers, the positions of the fully coupled doubly-excitedstates in the continuum and their widths are shown in Table 5.The best way we have found to assign symmetric and antisym-metric stretches is to study short movies of the conditionalprobability distribution as a is varied. To see these movies, visithttp://berrygroup.uchicago.edu/Bart/Corrmovie.html or seethe electronic supplementary information (ESI).w

Fig. 1 shows the conditional probability densities of the AISin the 1Se states above the 3p threshold. The 4s2 state behavesas a bending mode, as shown by its high electron densityperpendicular to the radial vector from the atomic core to‘‘electron 2’’, and the strong radial node indicates a symmetricstretch. The 3d2 state shows two quanta of bend, with electrondensity both perpendicular to the core and across the core. The4s5s state shows an antisymmetric stretch. The 4s7s statereveals a pattern similar to that of the 4s2 state, with onequantum of bend and symmetric stretch.

Fig. 2 plots the conditional probability distribution of the 3Se

states above the 3p threshold 4s5s has an antisymmetricstretch. 3d4d shows two quanta of bend, with two nodes inthe angular distribution. 4s6s has an antisymmetric stretch,and 4s8s reveals a quantum of bend and one of symmetricstretch, respectively.Figs. 3 and 4 show the L ¼ 1 state conditional probability

distributions. The 4s4p 1Po state has one quantum of sym-metric stretch, and the 4s4p 1Po state has one quantum ofantisymmetric stretch. The 4s6p state has two quanta of bend,with electron density both perpendicular and opposite to thevector from the core to ‘‘electron 2’’, and a quantum ofantisymmetric stretch. The 4s7p plot shows two quanta ofbend and one of symmetric stretch. In Fig. 4, the 4s4p 3Po stateshows an antisymmetric stretch, and the 4p3d 3Po state shows asymmetric stretch. The 4s5p 3Po state demonstrates a bend anda symmetric stretch. The 4s7p3Po state has a quantum ofsymmetric stretch and one quantum of bend.Figs. 5 and 6 show the L ¼ 2 conditional probability

distributions. 4s3d 1De has a symmetric stretch and bend,and 3d2 1De has two quanta of bend. 4s4d 1De has onequantum of bend and two quanta of symmetric stretch, asdoes 3d5s 1De. The 4s3d 3De state shows an antisymmetric

Fig. 4 Conditional probability distributions for 3Po AIS. A, 4s4p, a ¼6.63, B 4p3d, a¼ 7.40, C 4s5p, a¼ , D 4s7p, a¼ 6.38, box¼ 35 au. Thepeak indicates the position of the fixed electron, and the plot is centeredaround the nucleus in a 25 au box unless otherwise noted.

Fig. 5 Conditional probability distributions for 1De AIS. A, 4s3d, a ¼5.61, B 3d2, a ¼ 4.08, C 4s4d, a ¼ 7.09, D 3d5s, a ¼ 6.07, box ¼ 35 au.The peak indicates the position of the fixed electron, and the plot iscentered around the nucleus in a 25 au box unless otherwise noted.

3280 P h y s . C h e m . C h e m . P h y s . , 2 0 0 5 , 7 , 3 2 7 6 – 3 2 8 3 T h i s j o u r n a l i s & T h e O w n e r S o c i e t i e s 2 0 0 5

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stretch. 4s4d 3Do has one quantum of symmetric stretch andone of bend. 3d7s 3De has one quantum of bend and two ofantisymmetric stretch, and 3d4d 3De has one quantum ofantisymmetric stretch.

Figs. 7 and 8 plot the L ¼ 3 conditional probabilitydistributions. The 4p3d 1Fo state reveals an antisymmetricstretch. The 4s4f 1Fo state shows two quanta of symmetricstretch, and the 3d6p 1Fo state shows two quanta of antisym-metric stretch. The 3d5f 1Fo state reveals one quantum ofantisymmetric stretch. The 4p3d 3Fo AIS has a quantum ofantisymmetric stretch, and the 4s3d 3Fo state has one quantumof bend and two quanta of antisymmetric stretch. The 3d5p 3Fo

state reveals two quanta of bend, and two quanta of symmetricstretch. The 3d4f 3Fo state has two quanta of bend and two ofantisymmetric stretch.

Fig. 9 shows the energies, symmetries and approximatequantum numbers of the lowest-energy doubly-excited statesabove the 3p threshold. A fourth quantum number, the rota-tional quantum number might also be applied using energy andangular momentum conservation. Due to excessive mixing inthe states, vibrational and rotational constants may not befound. For example, in the 1Se symmetry, both 4s2 and 4s7s

AIS have probability distributions showing one quantum ofsymmetric stretch and one of bend. The total angular momen-tum is zero in each, so they both must have the same rotationalquantum numbers. On the other hand, the magnitudes of theapproximate quantum numbers are clearly rising with energy.So while an accurate estimate of the energy may not be made,there is clearly a loose correlation between the assignment andthe doubly-excited states energy.Figs. 10–13 show the angular distributions of photoelectrons

for two-photon ATI1 from the ground state for Figs. 10 and 12,and three-photon ATI from the ground state2 for Figs. 11 and13. The three-photon process requires two photons to reach thecontinuum and the third photon induces a continuum–con-tinuum transition. The angular distributions for the case inwhich the ion is left in the 3s state generally reveal a minimaldependence on energy. However, when the ion is left in the 3pstate, we can see some correlation between the conditionalprobability distributions and the angular distributions ofphotoelectrons. The [1,1,0] distribution in the 1Se 4s2, 4s7sAIS and the 1De 4s3d demonstrate a p-wave distribution,although the 4s7s 1Se and 4s3d 1De are slightly more spreadout. The antisymmetric modes of the 1Se 4s5s, the 1Po 4s5p, and

Fig. 6 Conditional probability distributions for 3De AIS. A, 4s3d, a ¼6.38, B 4s4d, a ¼, C 3d7s, a ¼ 9.64, D 3d4d, a ¼ 6.43, box ¼ 35 au. Thepeak indicates the position of the fixed electron, and the plot is centeredaround the nucleus in a 25 au box unless otherwise noted.

Fig. 7 Conditional probability distributions for 1Fo AIS. A, 4p3d, a ¼7.50, B 4s4f, a ¼ 6.27, box ¼ 30 au, C 3d6p, a ¼ 5.09 (Box ¼ 35 au, D3d5f, a¼ 4.59, box¼ 35 au. The peak indicates the position of the fixedelectron, and the plot is centered around the nucleus in a 25 au boxunless otherwise noted.

P h y s . C h e m . C h e m . P h y s . , 2 0 0 5 , 7 , 3 2 7 6 – 3 2 8 3 3281T h i s j o u r n a l i s & T h e O w n e r S o c i e t i e s 2 0 0 5

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the 1Fo 4s3d both show a similar perpendicular feature in thephotoelectron distribution. The 1De 4s4d and 3d7s both have[2,1,0], and the photoelectron distributions shown in Figs. 12G

and 12H are also quite similar. They are also qualitativelysimilar to the distributions from states with a [1,1,0] descrip-tion. It should be noted that the 1Se 3d2 and 1Fo 4s4f AIS resultin similar and striking photoelectron distributions despite theirdifferent conditional probability distributions. The photoelec-tron distributions and the conditional probability distributionsshow that useful information from the approximate quantumnumbers are correlated with an observable, the photoelectrondistribution.

Fig. 8 Conditional probability distributions for 3Fo AIS. A, 4p3d, a ¼8.35 30 au, B 4s4f, a ¼ 6.20 35 au, C 3d5p, a ¼ 7.5, D 3d4f, a ¼ 5.98,box ¼ 35 au. The peak indicates the position of the fixed electron, andthe plot is centered around the nucleus in a 25 au box unless otherwisenoted.

Fig. 9 The energies in au with respect to the Mg21 ion symmetry, andapproximate quantum numbers [symmetric stretch, bend, antisym-metric stretch] for AIS above the 3p threshold.

Fig. 10 Photoelectron angular distributions for two-photon ATIpopulating the 1Se AIS, for A and E the 4s2 AIS left in the 3s and 3pstates; B and F, 3d2 states, C and G in 4s5s and D and H in 4s7s.

Fig. 11 Photoelectron angular distributions for three-photon processpopulating the 1Po AIS: for A and E the 4s4p AIS left in the 3s and 3pstate; B and F, in 4s5p, C and G in 4s6p and D and H 4s7p.

3282 P h y s . C h e m . C h e m . P h y s . , 2 0 0 5 , 7 , 3 2 7 6 – 3 2 8 3 T h i s j o u r n a l i s & T h e O w n e r S o c i e t i e s 2 0 0 5

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4. Conclusions

We have studied the conditional probability distributions for aseries of autoionizing states (AIS) of Mg above the 3p ioniza-

tion threshold, and assigned the states with approximatequantum numbers based on molecular modes. We have com-pared these distributions and assignments with two observa-bles, the energy and the photoelectron distribution based ontwo- and three-photon ionization. The angular distributionshave been calculated with an L2 integrable B-spline basis in theFeshbach formalism. The assignments and conditional prob-ability distributions appear to be correlated with the angulardistributions, indicating that this is a situation in which theseassignments are useful. Excessive mixing of the states, however,limits the applicability of these approximate quantum numbersin the determination of the energies of the states. Furthermore,these results indicate that the usefulness of conditional prob-ability distributions and interpretation in terms of collectivequantization is likely to be even more striking in comparingconditional probability distributions to two-electron ioni-zation.

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Fig. 12 Photoelectron angular distributions for two-photon processpopulating the 1De AIS, for A and E the 4s3d AIS left in the 3s and 3pstate B and F, 3d2, C and G 4s4d and D and H 3d5s.

Fig. 13 Photoelectron angular distributions for three-photon processpopulating the 1Fo AIS, for A and E the 4p3d AIS left in the 3s and 3pstate B and F, 4s4f, C and G 3d6p and D and H 3d5f.

P h y s . C h e m . C h e m . P h y s . , 2 0 0 5 , 7 , 3 2 7 6 – 3 2 8 3 3283T h i s j o u r n a l i s & T h e O w n e r S o c i e t i e s 2 0 0 5

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