The Relative Alignment of Electron Momenta in Atoms and Molecules

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The Relative Alignment of Electron Momenta in Atoms and Molecules and the Effect of a Static Electric Field Joshua W. Hollett, and Wen Li J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.7b09439 • Publication Date (Web): 28 Sep 2017 Downloaded from http://pubs.acs.org on October 2, 2017

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The Journal of Physical Chemistry

The Relative Alignment of Electron Momenta in Atoms and Molecules and the Effect of a Static Electric Field Joshua W. Hollett∗,†,‡ and Wen Li∗,¶ †Department of Chemistry, University of Winnipeg, Winnipeg, Manitoba, R3B 2G3, Canada ‡Department of Chemistry, University of Manitoba, Winnipeg, Manitoba, R3T 2N2, Canada ¶Department of Chemistry, Wayne State University, Detroit, Michigan, 48202, USA E-mail: [email protected]; [email protected]

Abstract The relative momentum of electron pairs in atoms and small molecules is examined through calculation of the p1 · p2 probability distribution. The likelihood of aligned or anti-aligned momenta between paired electrons is determined from the calculated distributions. Coulomb correlation aligns the momenta of electron pairs, and the amount of alignment varies when considering momenta in specific directions in three-dimensional space. A static electric field is found to have competing effects on momentum alignment parallel and perpendicular to the electric field. However, the net effect of the electric field on alignment is significantly smaller than the effect of Coulomb correlation. Recent experimental advances suggest that such a correlation of electron momenta can now be measured directly using attosecond spectroscopic tools.

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Introduction Electron correlation underpins many fundamentally important many-body phenomena, from superconductivity to chemical bond formation. Rapid advances in attosecond spectroscopy have led to new tools for dissecting electron dynamics previously not directly accessible. 1–4 In particular, recent development of the strong-field based attosecond angular streaking technique, for the double ionization of atoms and molecules, allows for the measurement of the correlated momenta of two electrons within 500 as. 5 Such a method measures both the relative times of the electron ejections and also the relative direction of ejection. The interpretation of the observed two-electron momentum distributions requires extensive knowledge of the trajectory of the electrons and the influence of the laser pulse beyond ionization, but also knowledge of the initial momenta of the electrons 6 and the influence of the laser before ionization. 7 It has been shown that an electric field (typically 0.01-0.1 a.u.) present in strong-field approaches may influence the momentum distribution of a single active electron and needs to be considered. 8 However, whether and how the electric field affect the relative momenta between two electrons has not yet been investigated. Theoretically, information regarding the relative momenta of electrons pre-ionization can be obtained from intracules of the ground state electronic wave function. 9–17 Intracules are two-electron probability distribution functions. 10,13,16 The most common of which are the Position, 18 P (u), and Momentum intracules, 19 M (v), N (N − 1) hΨ|δ(u − r12 )|Ψi 2 N (N − 1) M (v) = hΦ|δ(v − p12 )|Φi 2 P (u) =

(1a) (1b)

where r12 = |r1 − r2 |, p12 = |p1 − p2 |, δ is the Dirac delta function, and Ψ and Φ are the N -electron wave functions in position and momentum space, respectively. P (u) and M (v) give the probability distributions of the relative distance u and momentum v between pairs of electrons of the wave function. The primary use of intracules at their inception was the 2 ACS Paragon Plus Environment

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measurement of the effects of Coulomb correlation. 18–25 The difference between an intracule for an exact wave function and an intracule for the Hartree-Fock (HF) wave function (single Slater determinant of one-electron spin-orbitals) is the intracule hole, 18

∆I(i) = I(i) − IHF (i)

(2)

where I(i) and IHF (i) represent generic two-electron probability distributions of the exact and HF wave functions. Commonly, the error in the HF approximation is simply referred to as electron correlation. 26 However, because the Slater determinant is antisymmetric it partially correlates the motion of parallel-spin electrons via Fermi correlation. The remainder of the correlation of the motion between parallel-spin electrons, and all of the correlation between opposite-spin electrons, is due to Coulomb correlation. 27 A few years ago, an intracule corresponding to the probability distribution of p1 · p2 was derived to measure the relative orientation of the momenta of electrons in atoms and molecules. 17 Herein, this intracule and the corresponding intracule hole are used to examine the correlated momentum of electrons in atoms and molecules. Also, as a preliminary query into the influence of a laser pulse on correlated electron trajectories, the effect of a static electric field on their relative motion is examined. First, the p1 · p2 intracule is introduced, and the manner in which information concerning specific electron pairs is extracted from the N -electron wave function is described. This is followed by the details of the calculations and the results for atoms and molecules. Conclusions are then drawn regarding the correlated momenta of electrons in atoms and molecules and the effect of a static electric field. Atomic units are used throughout.



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Theory The p1 · p2 intracule is given by,

B(β) =

N (N − 1) hΦ|δ(β − p1 · p2 )|Φi 2

(3)

Given that p1 · p2 = p1 p2 cos θ12 , where p1 = |p1 | and p2 = |p2 |, and θ12 is the angle between the two momentum vectors, B(β) provides information concerning the relative alignment of p1 and p2 and their simultaneous magnitudes (see Figure 1). Information regarding the

Figure 1: Value of β for different p1 and p2 orientations.

alignment of electron momenta along specified Cartesian axes or planes (e.g. specified by an electric field) can be obtained from intracules corresponding to 1D and 2D dot-products, Bi (β) and Bij (β), respectively. N (N − 1) hΦ|δ(β − pi,1 pi,2 )|Φi 2

(4)

N (N − 1) hΦ|δ (β − [pi,1 pi,2 + pj,1 pj,2 ]) |Φi 2

(5)

Bi (β) =

Bij (β) =

where i and j specify the Cartesian axes of interest (i.e. x, y or z). The fundamental integrals over Gaussian-type functions that may be used to construct these intracules for atoms and molecules, in conjuction with an appropriate recurrence relation, can be found in the Supporting Information (SI). From the 1D, 2D and 3D dot-product probability distributions, the likelihood that a pair of electrons have aligned momenta (θ12 < π/2), P + , or anti-aligned momenta (θ12 > π/2), 4 ACS Paragon Plus Environment

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P − , is measured by integration. Expressions for these probabilities are given by

P

+

∞

Z =

B(β)dβ

(6a)

B(β)dβ

(6b)

0

P

−

Z

0

= −∞

and are defined in an analogous matter for Bi (β) and Bij (β). For a given wavefunction, P + gives the likelihood that two electrons have aligned momenta in all of 3D space, whereas Pz+ gives the likelihood that the z-components of the momenta of the two electrons are aligned. + Pxy concerns the alignment of momenta projected onto the xy-plane.

The HF two-electron reduced density matrix (2-RDM) can be separated into Coulomb and exchange components and, as a consequence, so can any two-electron property. The B(β) intracule may be decomposed into Coulomb and exchange components as follows (for the closed-shell case) BHF (β) =

N/2 N/2 X X

2Baabb (β) − Babba (β)

(7)

a=1 b=1

where Baabb (β) are the Coulomb-type integrals and Babba (β) are the exchange-type integrals. 1 Baabb (β) = (2π)4

Z

1 Babba (β) = (2π)4

Z

∞

−∞

eitβ Z ∗ ∗ −ip·q × φa (r1 )φa (r1 + q)φb (r2 )φb (r2 − tp)e dr1 dr2 dqdp dt (8)

∞

eitβ −∞ Z ∗ −ip·q ∗ × φa (r1 )φb (r1 + q)φb (r2 )φa (r2 − tp)e dr1 dr2 dqdp dt (9)

where φa (r) and φb (r) are HF molecular orbitals. It can be shown (see SI) that the Coulomb components (present for each pair of electrons) of the HF 2-RDM do not contribute to momentum alignment or anti-alignment. Exchange, or Fermi correlation, which only occurs 5 ACS Paragon Plus Environment


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between parallel-spin electrons, does contribute to momentum alignment/anti-alignment. Therefore, under the HF approximation, the momenta of opposite-spin electrons are unaligned (neither aligned nor anti-aligned), and the alignment (or anti-alignment) of the momenta of parallel-spin electrons is due to Fermi correlation. Beyond wave functions of only two electrons, the extraction of information about specific1 pairs of electrons from an intracule is not straightforward. However, information regarding the correlated motion of electrons of the same orbital can be obtained via two simplifying assumptions: 1. Assume that the difference in the relative motion between two paired (same orbital) electrons, correlated in the field of N − 2 correlated electrons and correlated in the field of N − 2 HF electrons, is negligible. That is, the correlation of the two paired electrons with the remaining N − 2 electrons has little effect on their own relative motion. 2. Assume that the correlation of an electron pair does not significantly affect their relative motion with respect to the other N − 2 electrons. This is a reasonable assumption provided the paired electrons do not have significant diradical character (i.e. reasonable as long as the electrons are actually paired). Correlating a single pair of electrons in the field of N − 2 HF electrons is achieved by performing a Full Configuration Interaction (FCI) calculation in which just two electrons, in the same orbital, are active. The corresponding wave function is given by

ΨFCI = c0 ΨHF +

X

rs crs aa Ψaa

(10)

rs

where ΨHF is the HF wave function (ground state electron configuration) and Ψrs aa is a Slater determinant describing an electron configuration in which the electrons of occupied orbital a, φa , have been excited to the unoccupied (virtual) orbitals r and s. The presence of an overbar 1

Electrons of the atomic or molecular wave function are indistinguishable. However, this changes upon double-ionization in which the electrons may be identified by their orbitals of origin (within the Hartree-Fock approximation)


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rs are the CI expansion indicates β-spin, whereas the absence indicates α-spin. The c0 and caa

coefficients. The effect of this pair correlation on B(β) is given by the corresponding intracule hole, ∆B(β) = BFCI (β) − BHF (β)

(11)

The effect on alignment of the momenta of the electron pair is calculated either from the separate FCI and HF intracules or from the intracule hole itself,

∆P

+

Z

∞

=

∆B(β)dβ

(12a)

∆B(β)dβ

(12b)

0

∆P

−

Z

0

= −∞

The effect of correlating the electron pair in φa on momentum alignment may be decomposed into three components,

∆P ± = ∆P ± (a-a) + ∆P ± (a-R) + ∆P ± (R-R)

(13)

the effect of correlation on the alignment of the electron pair itself ∆P ± (a-a), the effect on the alignment of electrons of the pair relative to the remaining N − 2 electrons ∆P ± (a-R), and the effect on the alignment of the remaining N − 2 electrons with each other ∆P ± (R-R). Because the other N − 2 electrons are not correlated in ΨFCI , the alignment between these electrons is the same for ΨHF and ΨFCI , and consequently ∆P ± (R-R) = 0. Also, assumption 2 (above) implies that ∆P ± (a-R) = 0. This leads to an expression in which the effects of correlating the motion of the electrons in φa on P ± is solely due to the change in the relative motion between the electrons in φa , ∆P ± = ∆P ± (a-a)


(14)


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where ± ± ∆P ± (a-a) = PFCI (a-a) − PHF (a-a)

(15)

As mentioned above, opposite-spin electrons are neither aligned nor anti-aligned in the HF ± description and hence, PHF (a-a) =

1 . 2

± Assumption 1 implies that P ± (a-a) = PFCI (a-a).

Therefore, under the two assumptions above, the probability that the two electrons of φa are aligned/anti-aligned is given by

P ± (a-a) = ∆P ± +

1 2

(16)

Furthermore, the effect of a static electric field on an intracule, and consequently the alignment of electron momenta, may be determined by computing the difference in quantities from the electronic wave function in the presence and absence of an electric field,

∆E P ± = PE± − P0±

(17)

where P0± and PE± are the probability of alignment/anti-alignment in the absence and in the presence of an electric field of strength E, respectively.

Method Calculation of the HF and FCI wave functions was performed using the GAMESS quantum chemistry program. 28 Properties for He, Ne and H2 were determined with the aug-ccpV6Z(/fg) basis set, 29 and calculations on C2 H4 were performed with the aug-cc-pVTZ(/fg) basis set. 30,31 Calculations showed that intracule properties converge with respect to basis set size up to a certain electric field strength, which was system dependent (i.e. polarizability, ionization energy). Values reported reflect the convergence with the basis set. The B(β) intracules were calculated in Fourier space using in-house code ported to GAMESS and reverse Fourier transformed 13 and integrated using Mathematica. 32 The intracules in Fourier space 8 ACS Paragon Plus Environment

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were calculated on a 500 point Mura-Knowles grid 33,34 with the grid parameter R = 400. The static electric field was applied using the EFIELD option of GAMESS. For electric field strength, 1 a.u. = 5.1422082(15) × 1011 V m−1 .

Results and Discussion Atoms The p1 · p2 intracule hole for He in the absence of an electric field is shown in Figure 2. The hole, ∆B(β), is decomposed into radial and angular correlation effects. 17,19,20,35 Radial ΔB( β ) 0.02 0.01 -4

-2

0

2

4

β

ΔB ΔBrad ΔBang

-0.01 -0.02

Figure 2: The p1 · p2 intracule hole for He in the absence of an electric field, decomposed into angular and radial correlation components.

correlation concerns the correlated motion of the electrons in terms of their distance from the nucleus, when one is close the other is further away. Angular correlation involves the electrons avoiding each other by staying on opposite sides of the nucleus. As first noted by Banyard and Reed, 19 radial and angular correlation have two distinct effects on the relative momentum of electrons. It is seen in Figure 2, that the radial correlation effect is symmetric about the origin, meaning it only affects the relative magnitude of p1 and p2 , and has no influence on their relative orientation. There is an increased probability of moderate |β| values due to one electron being close to the nucleus (large p1 or p2 ) and the other being far away (small p2 or p1 ). Consequently, the probability of small |β| (both 9 ACS Paragon Plus Environment


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electrons far from the nucleus) and large |β| (both electrons close to the nucleus) is depleted. Angular correlation increases the probability of aligned momenta (β > 0) at the expense of anti-aligned momenta (β < 0). This suggests that angular correlation causes the electrons to oscillate with aligned momenta on opposite sides of the nucleus. 17,19 Angular correlation of the electrons of He leads to P + = 0.54083 [Equation 6a], which is approximately a 54% probability that the electrons have aligned momenta. The effect of the electric field on the relative orientation of the electron momenta of He is quantified by ∆E P + . Values of ∆E P + at varying field strengths E are presented in Figure 3 and Table 1.

It is seen that the overall probability (all) of the electrons having aligned ΔPE+ 0.0020 0.0015 0.0010 0.0005 0.0000◆●■ -0.0005 -0.0010

■

all ■ ∥ ◆⊥ ●

■ ●

■ ● ◆

■ ● ◆

●

◆ ◆

0.05

0.10

0.15

E 0.20

Figure 3: Change in alignment of electron momenta of He due to a static electric field.

Table 1: Alignment of momenta of electron pairs in He and Ne and change in alignment due to a static electric field. electron pair He

Ne (p-orbital)

direction all k ⊥ all k ⊥

P+ 0.54083 0.52027 0.53197 0.51380 0.50572 0.51788

∆E P + E = 0.05 E = 0.1 E = 0.15 0.00004 0.00016 0.00034 0.00009 0.00041 0.00100 -0.00002 -0.00014 -0.00037 0.00001 0.00005 0.00024 0.00001 0.00008 0.00042 -0.00003 -0.00014 -0.00042

momentum is increased with increased field strength, albeit not drastically. Recall that for He in the absence of an electric field P + = 0.54083, and from Table 1 it is deduced that + P0.15 = 0.54117.


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The probability of alignment of the electron momenta of He in specific directions relative to the static electric field is also presented in Figure 3 and Table 1. The effect on alignment along the axis parallel to the field, ∆Pk+ , is determined from the 1D Bi (β) [Equation (4)] where i is the direction of the field (e.g. z-direction). Whereas, the probabilty of alignment in the plane perpendicular to the field, ∆P⊥+ , is determined from the 2D Bij (β) [Equation (5)] where the ij-plane is perpendicular to the field direction (e.g. xy-plane). The overall effect on the alignment of the electron momenta is a combination of competing effects. Parallel to the + electric field direction the alignment of electron momenta is slightly enhanced, ∆PE,k > 0,

whereas perpendicular to the field direction the alignment of electron momenta is slightly + decreased, ∆PE,⊥ < 0.

For many-electron systems, information concerning the relative momentum of a pair of electrons can be extracted from the intracule hole of ΨFCI [Equations 11, 12a and 16], where a Full Configuration Interaction calculation is performed on only the electron pair of interest. The effect of a static electric field on the alignment of the electrons of a p-orbital of Ne is presented in Table 1. In the presence of an electric field, the highest energy orbital of Ne is a p-orbital aligned with the field direction (Figure 4). The probability of alignment of the momenta of the

Figure 4: HOMO of Ne in a static electric field of 0.15 a.u.

electrons in the p-orbital of Ne is 50.6 % in the direction parallel to the electric field, 51.8% in the plane perpendicular to the field, and 51.4% in all directions. This degree of alignment is significantly less than that seen for He. The general effect of an electric field, seen for He, is also observed for the p-orbital electrons of Ne. The static electric field causes a slight increase in the alignment of momenta 11 ACS Paragon Plus Environment


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in the direction parallel to the field, and a slight decrease in the plane perpendicular to it. For both He and Ne the effect of the electric field is at least an order of magnitude smaller than the effect of Coulomb correlation.

Molecules In the case of molecules, the particular orientation of the molecule with respect to the electric field has a significant influence on how the electron momenta respond to it. For H2 , two possible orientations are; H−H bond perpendicular to the field (orientation 1), and H−H bond parallel to the field (orientation 2), see Figure 5. Of course, there are an infinite

Figure 5: σ-orbital of H2 in two possible orientations with respect to a static electric field of 0.05 a.u.

number of orientations between these two. The alignment of the momenta of the electrons of H2 relative to the electric field direction, and the effect of the electric field on their alignment, are presented in Table 2. The overall probabilty that the two electrons of H2 have aligned momenta is 56.3%, which is about 2% more likely than that of He. When the H−H bond is perpendicular to the electric field direction (orientation 1), there is more alignment in the plane perpendicular to the field, 55.4%, compared to along the direction of the field, 52.6%. When the H−H bond is parallel to the electric field direction the amount of alignment parallel to the field and perpendicular to it are very similar, 54.5% and 54.0%, respectively. Upon application of an electric field to H2 , the general response of P + , Pk+ , and P⊥+ is the same as that for the electron pairs of He and the p-orbital of Ne. There is a slight increase


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Table 2: Alignment of momenta of electron pairs in H2 and C2 H4 and change in alignment due to a static electric field. electron pair H2

orientation 1

2

C2 H4 (HOMO)

1

2

C2 H4 (HOMO-1)

1

2

direction all k ⊥ all k ⊥ all k ⊥ all k ⊥ all k ⊥ all k ⊥

+

P 0.56267 0.52553 0.55434 0.56267 0.54538 0.54025 0.53184 0.51879 0.55156 0.53184 0.56378 0.52229 0.52165 0.50741 0.51933 0.52165 0.51643 0.51471

∆E P + E = 0.025 E = 0.05 0.00005 0.00019 0.00015 0.00065 -0.00009 -0.00038 -0.00004 -0.00019 0.00003 0.00014 -0.00005 -0.00024 0.00027 0.00010 -0.00031 -0.00051 -0.00205 -0.00003 -0.00001 0.00001 -0.00001 0.00032 0.00029 0.00024



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in alignment of the electron momenta in the direction parallel to the electric field, and there is a slight decrease perpendicular to the field. The slight increase in alignment is noticeably larger for orientation 1 compared to orientation 2. For C2 H4 , the two orientations chosen with respect to the static electric field and shapes of the HOMO and HOMO-1 are shown in Figures 6 and 7, respectively.

The HOMO of

Figure 6: HOMO of C2 H4 in two possible orientations with respect to a static electric field of 0.025 a.u.

Figure 7: HOMO-1 of C2 H4 in two possible orientations with respect to a static electric field of 0.025 a.u.

C2 H4 is the π-bonding orbital. In orientation 1, the electric field is perpendicular to the C−C bond and the nodal plane of the HOMO. While in orientation 2, the electric field is aligned with the C−C bond. In Table 2, it is seen that for orientation 1 the HOMO electrons are more likely to have their momenta aligned perpendicular to the electric field, P⊥+ = 0.552, compared to parallel, Pk+ = 0.519. The opposite is true for orientation 2, in which P⊥+ = 0.522 and Pk+ = 0.564. This suggests, that just like H2 there is more momentum alignment along the bond axis compared to perpendicular to the bond. For the HOMO of C2 H4 , the difference in the effect of the electric field on P⊥+ and Pk+ for 14 ACS Paragon Plus Environment

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orientations 1 and 2 is not suprising due to the significant difference in alignment of electron momenta in the plane of the molecule and perpendicular to it. In orientation 1, the effect of the electric field is typical of that seen for the pairs of electrons in He, Ne and H2 . That is, a slight increase in alignment parallel to the electric field and slight decrease perpendicular to it. However, in orientation 2, the behaviour is atypical. There is a slight decrease in alignment in all directions with a comparatively strong decrease in the direction parallel to the field, which is also parallel to the C−C bond. In the case of the HOMO-1 of C2 H4 (Figure 7), there is less alignment of the electrons in both the parallel and perpendicular directions, and overall. The effects of the electric field on alignment for orientation 1 are especially weak, but the effects on the HOMO-1 electrons in orientation 2 are significantly stronger than in orientation 1.

Conclusions New spectroscopic techniques have the ability to measure the relative momenta of electrons during double ionization events. In order to relate such measurements to molecular electronic structure, it is important to have a fundamental understanding of the relative momenta of electrons pre-ionization and the effect of an electric field. The B(β), Bi (β), and Bij (β) intracules can be used to measure the probability that the momenta of pairs of electrons are aligned or anti-aligned. Such a measurement is exact for systems of only two electrons, such as He and H2 . But under two simplifying assumptions, the probability of alignment (or anti-alignment) of the momenta of paired electrons (i.e. same orbital) in many-electron systems can also be measured. The key condition of the assumptions is that the electron pair in question does not have signifcant diradical character. Overall, paired electrons in atoms and molecules are more likely to have aligned momenta rather than anti-aligned (1% to 12% more likely, for the systems studied). The alignment is due to Coulomb correlation. Unless the orbital is spherically symmetric, the probability



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of alignment varies depending on the direction of the momenta in 3D space. In the case of both σ and π-bonds, there is more momentum alignment parallel to the bond axis than perpendicular to it. The presence of an electric field does not significantly alter the alignment of electron momenta. For the systems studied, the effect of a static electric field on momentum alignment is an order of magnitude smaller than that of Coulomb correlation. Therefore when considering the relative motion of electrons pre-ionization, the alignment of the electron momenta in the absence of an electric field is a sufficient description up until the electric field begins to drastically alter the electronic structure of the system. Furthermore, the weak effect of the electric field suggests that it has become possible to experimentally measure such momentum correlation, which is an important manifestation of electron dynamics in multi-electron systems. Future work will include extension to the final state of two continuum electrons, which will allow for direct comparison to experimental results and thus offer a new method for investigating the correlated motion of electrons.

Acknowledgement JWH thanks the Natural Sciences and Engineering Research Council of Canada (NSERC) for a Discovery Grant, Compute/Calcul Canada for computing resources, and the Discovery Institute for Computation and Synthesis for useful consultations. W. L. thanks the U.S. Department of Energy (DOE), Office of Science, Basic Energy Sciences (BES), under Award # DE-SC0012628 for financial support.

Supporting Information Available The following files are available free of charge. • pdp_Supporting_Info.pdf: Derivations and formulae


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This material is available free of charge via the Internet at http://pubs.acs.org/.

References (1) Hentschel, M.; Kienberger, R.; Spielmann, C.; Reider, G. A.; Milosevic, N.; Brabec, T.; Corkum, P.; Heinzmann, U.; Drescher, M.; Krausz, F. Attosecond Metrology. Nature 2001, 414, 509. (2) Feist, J.; Nagele, S.; Pazourek, R.; Persson, E.; Schneider, B. I.; Collins, L. A.; Burgdorfer, J. Probing Electron Correlation Via Attosecond Xuv Pulses in the Two Photon Double Ionization of Helium. Phys. Rev. Lett. 2009, 103, 063002. (3) Goulielmakis, E.; Loh, Z. H.; Wirth, A.; Santra, R.; Rohringer, N.; Yakovlev, V. S.; Zherebtsov, Z.; Pfeifer, T.; Azzeer, A. M.; Kling, M. F.; Leone, S. R.; Krausz, F. Real-Time Observation of Valence Electron Motion. Nature 2010, 466, 739. (4) Ott, C.; Kaldun, A.; Argenti, L.; Raith, P.; Meyer, K.; Laux, M.; Zhang, Y.; Blatterman, A.; Hagstotz, S.; Ding, T.; Heck, R.; Madronero, J.; Martin, F.; Pfeifer, T. Reconstruction and Control of a Time-Dependent Two-Electron Wave Packet. Nature 2014, 516, 374. (5) Winney, A.; Lee, S. K.; Lin, Y. F.; Liao, Q.; Adhikari, P.; Basnayake, G.; Schlegel, H. B.; Li, W. Attosecond Electron Correlation Dynamics in Double Ionization of Benzene Probed with Two-Electron Angular Streaking. Phys. Rev. Lett. 2017, 119, 123201. (6) MacDougall, P. J.; Levit, M. C. In A Matter of Density, 1st ed.; Sukumar, N., Ed.; John Wiley & Sons, Inc.: Hoboken, NJ, USA, 2013; Chapter 10, pp 257–270. (7) Ivanov, M. Y.; Spanner, M.; Smirnova, O. Anatomy of Strong Field Ionization. Journal of Modern Optics 2005, 52, 165.



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

(8) Dreissigacker, I.; Lein, M. Quantitative Theory for the Lateral Momentum Distribution after Strong-Field Ionization. Chem. Phys. 2013, 414, 69. (9) Gill, P. M. W.; O’Neill, D. P.; Besley, N. A. Two-Electron Distribution Functions and Intracules. Theor. Chem. Acc. 2003, 109, 241–250. (10) Gill, P. M. W.; Crittenden, D. L.; O’Neill, D. P.; Besley, N. A. A Family of Intracules, a Conjecture and the Electron Correlation Problem. Phys. Chem. Chem. Phys. 2006, 8, 15–25. (11) Crittenden, D. L.; Dumont, E. E.; Gill, P. M. W. Intracule Functional Models. II. Analytically Integrable Kernels. J. Chem. Phys. 2007, 127, 141103. (12) Dumont, E. E.; Crittenden, D. L.; Gill, P. M. W. Intracule Functional Models. I. AngleCorrected Correlation Kernels. Phys. Chem. Chem. Phys. 2007, 9, 5340–5343. (13) Bernard, Y. A.; Crittenden, D. L.; Gill, P. M. W. Intracule Functional Models. III. The Dot Intracule and its Fourier Transform. Phys. Chem. Chem. Phys. 2008, 10, 3447–3453. (14) Pearson, J. K.; Crittenden, D. L.; Gill, P. M. W. Intracule Functional Models. IV. Basis Set Effects. J. Chem. Phys. 2009, 130, 164110. (15) Hollett, J. W.; Gill, P. M. W. Intracule Functional Models. V. Recurrence Relations for Two-Electron Integrals in Position and Momentum Space. Phys. Chem. Chem. Phys. 2011, 13, 2972–2978. (16) Bernard, Y. A.; Loos, P. F.; Gill, P. M. W. The Distribution of r12.p12 in Quantum Systems. Mol. Phys. 2013, 111, 2414–2426. (17) Hollett, J. W.; Gill, P. M. W. Distributions of r1.r2 and p1.p2 in Atoms. J. Chem. Theory Comput. 2012, 8, 1657–1662.


Page 18 of 21

Page 19 of 21

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(18) Coulson, C. A.; Neilson, A. H. Electron Correlation in the Ground State of Helium. Proc. Phys. Soc. 1961, 78, 831–837. (19) Banyard, K. E.; Reed, C. E. Momentum Distributions of Two-Electron Systems: Electron Correlation and the Coulomb Hole. J. Phys. B 1978, 11, 2957–2967. (20) Banyard, K. E.; Ellis, D. J. A Distribution Function for Angular Correlation in He- and Be-like Ions. Mol. Phys. 1972, 24, 1291–1296. (21) Boyd, R. J.; Sarasola, C.; Ugalde, J. M. Intracule Densities and Electron Correlation in the Hydrogen Molecule. J. Phys. B 1988, 21, 2555–2561. (22) Wang, J.; Tripathi, A. N.; Smith Jr, V. H. The Influence of Electron Correlation on Anisotropic Electron Intracule and Extracule Densities. J. Chem. Phys. 1992, 97, 9188– 9194. (23) Cann, N. M.; Boyd, R. J.; Thakkar, A. J. Electron 33D Correlation Effects in the Rydberg-like and 31D States of Helium-like Ions. Int. J. Quant. Chem. 1993, 48, 1–14. (24) Pearson, J. K.; Gill, P. M. W.; Ugalde, J. M.; Boyd, R. J. Can Correlation Bring Electrons Closer Together? Mol. Phys. 2009, 107, 1089–1093. (25) Per, M. C.; Russo, S. P.; Snook, I. K. Anisotropic Intracule Densities and Electron Correlation in H2: A Quantum Monte Carlo Study. J. Chem. Phys. 2009, 130, 134103. (26) Löwdin, P.-O. Correlation Problem in Many-Electron Quantum Mechanics. I. Review of Different Approaches and Discussion of Some Current Ideas. Adv. Chem. Phys. 1959, 2, 207–322. (27) Tew, D. P.; Klopper, W.; Helgaker, T. Electron Correlation: The Many-Body Problem at the Heart of Chemistry. J. Comput. Chem. 2007, 28, 1307–1320.



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

(28) Schmidt, M. W.; Baldridge, K. K.; Boatz, J. A.; Elbert, S. T.; Gordon, M. S.; Jensen, J. H.; Koseki, S.; Matsunaga, N.; Nguyen, K. A.; Su, S.; et al., General Atomic and Molecular Electronic Structure System. J. Comput. Chem. 1993, 14, 1347–1363. (29) Wilson, A. K.; van Mourik, T.; Dunning Jr., T. H. Gaussian Basis Sets for Use in Correlated Molecular Calculations. VI. Sextuple Zeta Correlation Consistent Basis Sets for Boron Through Neon. J. Mol. Struct. 1996, 388, 339–349. (30) Dunning Jr., T. H. Gaussian Basis Sets for Use in Correlated Molecular Calculations. I. The Atoms Boron Through Neon and Hydrogen. J. Chem. Phys. 1989, 90, 1007. (31) Kendall, R. A.; Dunning Jr., T. H. Electron Affinities of the First-Row Atoms Revisited. Systematic Basis Sets and Wave Functions. J. Chem. Phys. 1992, 96, 6796–6806. (32) Wolfram Research, Inc., Mathematica, Version 11.2. Champaign, IL, 2017. (33) Mura, M. E.; Knowles, P. J. Improved Radial Grids for Quadrature in Molecular DensityâĂŘFunctional Calculations. J. Chem. Phys. 1996, 104, 9848. (34) Gill, P. M. W.; Chien, S. Radial Quadrature for Multiexponential Integrands. J. Comput. Chem. 2003, 24, 732–740. (35) Hollett, J. W.; McKemmish, L. K.; Gill, P. M. W. The Nature of Electron Correlation in a Dissociating Bond. J. Chem. Phys. 2011, 134, 224103.


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Graphical TOC Entry


The Relative Alignment of Electron Momenta in Atoms and Molecules

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