Measuring correlated electron motion in atoms with the momentum-balance density
MMeasuring correlated electron motion in atoms with the momentum-balancedensity
Lucy G. Todd and Joshua W. Hollett
1, 2, a) Department of Chemistry, University of Winnipeg, Winnipeg, Manitoba, R3B 2E9, Canada Department of Chemistry, University of Manitoba, Winnipeg, Manitoba, R3T 2N2, Canada (Dated: 4 December 2020)
Three new measures of relative electron motion are introduced: equimomentum, antimomentum, andmomentum-balance. The equimomentum is the probability that two electrons have the exact same momen-tum, whereas the antimomentum is the probability their momenta are the exact opposite. Momentum-balance(MB) is the difference between the equimomentum and antimomentum, and therefore indicates if equal oropposite momentum is more probably in a system of electrons. The equimomentum, antimomentum andMB densities are also introduced, which are the local contribution to each quantity. The MB and MB den-sity of the extrapolated-Full Configuration Interaction wave functions of atoms of the first three rows of theperiodic table are analyzed, with a particular focus on contrasting the correlated motion of electrons withopposite and parallel spin. Coulomb correlation between opposite-spin electrons leads to a higher probabilityof equimomentum, whereas Fermi correlation between parallel-spin electrons leads to a higher probability ofantimomentum. The local contribution to MB, given an electron is present, is a minimum at the nucleus andgenerally increases as the distance from the nucleus increases. There are also interesting similarities betweenthe effects of Fermi correlation and Coulomb correlation (of opposite-spin electrons) on MB.
I. INTRODUCTION
Most of the difficulties encountered in the study of elec-tronic structure, both experimentally and computation-ally, arise from the complexity of the many-electron wavefunction. Therefore, both stand to benefit substantiallyfrom the unravelling this complexity. The unravelling canbegin by understanding the relative motion of electrons.Not only does this help demystify the mesmerizing many-electron wave function, but it also lends itself to the de-velopment of more suitable models of electronic structureand a more complete picture of quantum mechanics.Much has been learned about the relative position andmomentum of electrons, and most is due to the analy-sis of intracules.
Intracules are two-electron proba-bility distribution functions. Constructed from the N -electron wave function (approximate or exact), these re-duced probability densities reveal information regardingthe relative position, momentum, or po-sition and momentum of electrons. The analyses of-ten involve the calculation of intracule holes, whichare the difference between the exact and the Hartree-Fock intracule. An intracule hole explicitly describesthe effects of Coulomb correlation on that particularintracule, and subsequent inferences are made regard-ing the correlated motion of the electrons. Such stud-ies have led to a better understanding of the relativemotion of electrons due to radial and angular correla-tion in atoms, the correlated motion of electronsduring bond dissociation, and their correlated motionin the presence of a static electric field. However, formost intracules the spatial information ( i.e. absolute lo- a) Corresponding author: [email protected] cation of the electrons) is lost, which impedes the com-pletion of a picture of relative electron motion in atomsand molecules.By using a more complex object, such as the two-electron density, correlated electron movement can beanalyzed in real space through the construction ofCoulomb, exchange, or exchange-correlation holes.In this context, the hole is the two-electron density asa function of the position of the second electron when atest position is chosen for the first. Such an approachcan be used to demonstrate Fermi correlation and theFermi hole, which occur between electrons of parallel-spin and is due to the antisymmetry of the electronicwave function. This approach is also used to design andtest models for density functional approximations. Bychoosing a position in the molecule for the test electron,the complexity of the two-electron density is reduced buta complete picture of the relative motion of the electronsmust be synthesized from the combination of densitiesproduced from a collection of test electron positions.Alternatively, there are intracule-like probability den-sities that are not reduced to a single (1D or 3D)variable. Such probability densities, like the intexdistribution, retain absolute position information,that is often lost with simpler intracules, and can provideinformation regarding the relative positions of electronsat different locations in an atom or molecule. Similardistributions also exist for momentum space. In an effort to unravel some of the complexity of thecorrelated motion of electrons this article presents mul-tiple new quantum mechanical properties. The proper-ties are the equi- and antimomentum and momentum-balance, along with the corresponding equi- and antimo-mentum density and momentum-balance density. Themomentum-balance is a measure of the correlated mo-tion in an electronic system, and the momentum-balance a r X i v : . [ phy s i c s . c h e m - ph ] D ec density describes the local contribution of the electronicwave function to the correlated electron motion. In Sec-tion II, expressions for the equimomentum, antimomen-tum, and momentum balance are derived, along with thecorresponding densities. This is followed by a descrip-tion of the basic algorithm for the calculation of theseproperties in atoms and molecules using Gaussian basissets. Section IV presents an analysis and discussion ofthe momentum-balance and momentum-balance densityin atoms of the first 17 many-electron elements of theperiodic table. Then finally, the results are summarizedand future applications of these properties are discussedin Section V. Atomic units are used throughout. II. THEORY
The connection between electron position and mo-mentum is through the Fourier transform. The N -electron momentum-space wave function, Φ, in terms ofthe position-space wave function, Ψ, is given asΦ( p , p , . . . , p N ) = 1(2 π ) N (cid:90) e − i (cid:80) Nk =1 p k · r k × Ψ( r , r , . . . , r N ) d r d r . . . d r N , (1)where p k and r k are the electron momenta and posi-tions, respectively. The wave functions are spinless, thatis, they have been previously integrated with respect tothe electron spin-coordinates. When addressing the rel-ative motion of electrons, it is simpler to focus on elec-tron pairs rather than all N -electrons at once. There-fore, the two-electron reduced density matrix (2-RDM),Γ( r , r , r (cid:48) , r (cid:48) ), is formed via integration over the N -electron wave function,Γ( r , r , r (cid:48) , r (cid:48) ) = (cid:90) Ψ ∗ ( r (cid:48) , r (cid:48) , . . . , r N ) × Ψ( r , r , . . . , r N ) d r . . . d r N , (2)where the diagonal of the 2-RDM is the two-electron den-sity, Γ( r , r ) = Γ( r , r , r , r ). Following Equation (1),the two-electron momentum density, Π( p , p ), can beformed from the position-space 2-RDM,Π( p , p ) = 1(2 π ) (cid:90) e i [ p · ( r (cid:48) − r )+ p · ( r (cid:48) − r ) ] × Γ( r , r , r (cid:48) , r (cid:48) ) d r d r d r (cid:48) d r (cid:48) . (3)The two-electron momentum density gives the simulta-neous probability that one electron has momentum p and another has p . By inserting a Dirac delta func-tion, δ ( x ), and using the identity, δ ( x ) = π (cid:82) e − ikx dk ,the probability that two electrons have the exact samemomentum can be calculated, λ + = (cid:90) Π( p , p ) δ ( p − p ) d p d p = 1(2 π ) (cid:90) Γ( r , r , r + q , r − q ) d q d r d r , (4) which will be referred to as the equimomentum . Similarly,the probability that two electrons have the exact oppositemomentum is given by, λ − = (cid:90) Π( p , p ) δ ( p + p ) d p d p = 1(2 π ) (cid:90) Γ( r , r , r + q , r + q ) d q d r d r , (5)which will be referred to as the antimomentum . Thedifference between the two, will be referred to as the momentum-balance (MB) , µ = λ + − λ − (6)The MB is positive for a system in which it is more prob-able that the electrons have the same momentum, neg-ative for a system in which opposite momenta are moreprobable, and zero when both scenarios are equally likely.If the integrand for the equi- or antimomentum [Equa-tion (4) or (5)] is integrated over only one of the electroncoordinates the result is the local contribution to eachquantity, λ ± ( r ) = 1(2 π ) (cid:90) Γ( r , r , r + q , r ∓ q ) d r d q , (7)the equimomentum density, λ + ( r ), and the antimomen-tum density, λ − ( r ). The difference between the two isthe MB density, µ ( r ) = λ + ( r ) − λ − ( r ) (8)Although λ + ( r ) and λ − ( r ) are not probability densities,the MB density can still be analyzed to reveal the regionsof a wave function that contribute to electrons havingequal or opposite momenta. Such analysis is analogousto that seen with various types of energy densities. The sign of the MB density reveals whether there is alocal contribution to equi- or antimomentum, µ ( r ) > ⇒ contributes to equal momenta µ ( r ) < ⇒ contributes to opposite momentaInspection of the definitions of λ + ( r ) and λ − ( r ) [Equa-tion (S11)] in terms of the 2-RDM reveals a strong de-pendence of those densities and the MB density on theelectron density at r . Therefore, a reduced MB density, m ( r ), is defined (similar to an energy potential) to elim-inate that dependence, m ( r ) = µ ( r ) ρ ( r ) , (9)where ρ ( r ) is the one-electron density. It is expected that m ( r ) will reveal more fine details of the relative momentaof electrons in a system in comparison to µ ( r ).For wave functions constructed from an orbital basis,the equi- and antimomentum can be calculated by con-tracting the 2-RDM in the given basis, Γ abcd , with theappropriate two-electron integrals, λ ± = (cid:88) abcd Γ abcd [ abcd ] λ ± . (10)The two-electron integrals are given by[ abcd ] λ ± = 1(2 π ) (cid:90) φ a ( r ) φ b ( r + q ) × φ c ( r ) φ d ( r ∓ q ) d q d r d r , (11)where the orbitals, φ a , are assumed to be real. The inte-grals exhibit the very practical identity,[ abcd ] λ − = [ abdc ] λ + . (12)The equi- and antimomentum densities can be calculatedin the same manner, except the four-orbital integral isreplaced by the product of an orbital and a three-orbitalintegral, λ ± ( r ) = (cid:88) abcd Γ abcd φ a ( r ) [ bcd ] λ ± . (13)The three-orbital integral is given by,[ bcd ] λ ± = 1(2 π ) (cid:90) φ b ( r + q ) × φ c ( r ) φ d ( r ∓ q ) d q d r (14)which also has a useful identity,[ bcd ] λ − = [ bdc ] λ + . (15)Recurrence relations for the practical calculation of themolecular integrals, [ abcd ] λ ± and [ bcd ] λ ± , over Gaussian-type basis functions of arbitrary angular momentum areprovided in the supplementary material.Analysis of the relative motion of electrons via MB andthe MB density, is aided by decomposing the 2-RDM intoits various spin-components,Γ( r , r , r (cid:48) , r (cid:48) ) =Γ αα ( r , r , r (cid:48) , r (cid:48) ) + Γ ββ ( r , r , r (cid:48) , r (cid:48) )Γ αβ ( r , r , r (cid:48) , r (cid:48) ) + Γ βα ( r , r , r (cid:48) , r (cid:48) ) , (16)where α denotes “spin-up” and β denotes ”spin-down”.To demonstrate the relationship between µ and corre-lated electron motion, consider the Hartree-Fock wavefunction. Inserting the spin-resolved HF 2-RDM intoEquation (S12) yields an expression for the HF equi- andantimomentum, λ ± HF = 12 (cid:88) ij n αi n αj ([ iijj ] ααλ ± − [ ijji ] ααλ ± )+ n βi n βj (cid:16) [ iijj ] ββλ ± − [ ijji ] ββλ ± (cid:17) + n αi n βj (cid:16) [ iijj ] αβλ ± + [ iijj ] βαλ ± (cid:17) , (17)where n σi is the occupancy of σ -spin orbital i , with σ = α or β . The integral identity relating equimomen-tum and antimomentum [Equation (S9)] implies that theCoulomb-type integral for both quantities are equivalent,[ iijj ] λ − = [ iijj ] λ + . (18) Considering this, and the decomposition of µ into spin-components, µ HF = µ αα HF + µ ββ HF + µ αβ HF + µ βα HF , (19)the following simplification can be made, µ σσ HF = 12 (cid:88) ij n σi n σj ([ ijji ] σσλ − − [ ijji ] σσλ + ) . (20)The HF MB between electrons of parallel-spin is the dif-ference between the equi- and antimomentum contribu-tions that result from the exchange interaction ( i.e. an-tisymmetry). Given that there is no exchange interac-tion between electrons of opposite-spin, the HF MB ofopposite-spin electrons is exactly zero, µ σσ (cid:48) HF = 0 . (21)An analogous result is found for the MB density, µ σσ HF ( r ) = 12 (cid:88) ij n σi n σj φ σi ( r ) ([ jji ] σσλ − − [ jji ] σσλ + ) , (22)and µ σσ (cid:48) HF ( r ) = 0 . (23)It is clear from this result that non-zero MB and MBdensity only exists when electron motion is correlated.In other words, if electron motion is uncorrelated thenequimomentum and antimomentum are equally proba-ble, λ + = λ − and λ + ( r ) = λ − ( r ). In the case of the HFwave function, correlation occurs only between parallel-spin electrons and is due to the antisymmetry of the wavefunction, often referred to as Fermi correlation. There-fore, the MB and MB density of correlated ( e.g. post-HF)wave functions reveal the presence of correlation betweenopposite-spin electrons and the change in correlation be-tween parallel-spin electrons.
III. METHOD
All wave functions (including 1 and 2-RDMs) wereobtained using a determinant-driven selected configura-tion interaction (sCI) method known as CIPSI (Config-uration Interaction using a Perturbative Selection madeIteratively) in which the energies are extrapolatedto the full configuration interaction (FCI) result usingmultireference perturbation theory.
The all-electronextrapolated-FCI (exFCI) calculations were performedusing Quantum Package 2.0. The wave functions andRDMs for He were obtained using the cc-pVTZ, aug-cc-pVTZ and aug-cc-pVQZ/f basis sets, and all cal-culations beyond He were performed using the aug-cc-pCVTZ/f basis set.
Using the RDMs provided by Quantum Package 2.0,the equimomentum, antimomentum, and MB, along with
FIG. 1. Comparison of momentum-balance density of Heatom for multiple Dunning correlation-consistent basis sets. their densities were calculated using MUNgauss. Thedensities were calculated on a two-dimensional mesh,with a grid-spacing of 0.05 bohr, for visualization. Thedensities were also numerically integrated on the SG-1grid for comparison to analytically calculated equimo-mentum, antimomentum and MB values. IV. RESULTSA. Opposite-spin correlation
As an initial assessment of basis set dependence, theMB density of the He atom obtained using three differentDunning basis sets is shown in Figure 1. It is seen thatthere is significant deviation between the MB density cal-culated using the cc-pVTZ basis set and the density cal-culated using the aug-cc-pVTZ basis set. The deviationis most pronounced at the nucleus and decays as the dis-tance from the nucleus increases. Therefore, it appearsthe addition of diffuse functions, or “augmenting” the ba-sis set, (cc-pVTZ to aug-cc-pVTZ) is more important toan accurate description of the MB density than addingan extra set of valence basis functions (aug-cc-pVTZ toaug-cc-pVQZ). Furthermore, the MB density is far moresensitive to basis set than the one-electron density (Fig-ure 2). This may not be surprising considering it is acomparison between a one-electron property and a two-electron property. All subsequent properties of He thatfollow are obtained using the aug-cc-pVQZ basis set.In Figure 3, the He atom MB density, µ ( r ), reducedMB density, m ( r ), and one-electron density, ρ ( r ) are plot-ted together. Evident from Figures 1 and 3, µ ( r ) for He ispositive everywhere, and results in a total MB of 0.00373.The MB is the difference between an equimomentum λ + = 0 . λ − = 0 . FIG. 2. Comparison of total one-electron density of He atomfor multiple Dunning correlation-consistent basis sets.
FIG. 3. Total momentum-balance density and reducedmomentum-balance density of He, with the one-electron den-sity. times, they would find the electrons have the exact samemomentum 2 481 times and the exact opposite momen-tum 2 107 times. The difference, which is the MB, is dueto Coulomb correlation between the α and β electron.Their correlated motion favours equimomentum over an-timomentum, which is consistent with earlier studies ofangular correlation in the He atom. From Figures 1 and 2 it is clear that µ ( r ) closely followsthe shape of the ρ ( r ), albeit with a much smaller mag-nitude and slower decay. As mentioned previously, thisis due to the dependence of the two-electron density onthe one-electron density ( i.e. an electron must be presentto interact). This dependency is removed in the reducedMB density, m ( r ), which can be viewed as a MB poten-tial. It is seen that m ( r ) is actually a minimum at thenucleus and the potential contribution to MB increasesas the distance from the nucleus increases. Of course, - - - - FIG. 4. Renormalized µ αβ ( r ) of He, Li, Be and Be(fc) [frozencore]. The difference between the all-electron and frozen core µ αβ ( r ) for Be is included. while m ( r ) continues to grow as r increases, ρ ( r ) decaysand eventually extinguishes the contribution.Figure 4 presents the αβ MB density of He, Li andBe divided by the number of αβ electron pairs. It alsoincludes µ αβ ( r ) of Be calculated using a frozen-core ap-proximation to ascertain the role of the valence electronsin µ αβ ( r ). The frozen-core µ αβ ( r ) closely resembles thefully correlated µ αβ ( r ) which, unlike He and Li, has anode at small r and a negative peak at the nucleus. Itis evident that the valence electrons are mainly respon-sible for the structure of µ αβ ( r ) for Be. The differencebetween the frozen-core and fully correlated µ αβ ( r ) re-veals the contribution from the core, and core-valence,correlation and it is very similar to the µ αβ ( r ) of He andLi.The reduced MB densities, divided by the number of β electrons, of Be, Li and He are presented in Figure 5.Note, that for the spin-decomposed reduced MB density, m σσ (cid:48) ( r ) = m σσ (cid:48) ( r ) ρ σ ( r ) . The m αβ ( r ) are similar, in that theyhave a minimum at the nucleus and are everywhere pos-itive. However, the m αβ ( r ) of Li reaches a maximum at ∼ ρ α ( r )decays much slower than ρ β ( r ), unlike He and Be. Theamplitude of the Be m αβ ( r ), which is dominated by thevalence electrons, is significantly larger than that of Heand Li. Near the nucleus m αβ ( r ) of Be is negative andrelatively flat. It reaches a minimum at 0.374 bohr andhas node at 0.549 bohr. After crossing the node, m αβ ( r )increases sharply with a shoulder just beyond r = 1 bohr.The m αβ ( r ) of He also exhibits a shoulder, although itis less pronounced and further from the nucleus ( ∼ µ αβ ( r ) for Be to Ne are presented in Figure 6.With the exception of N, all the atoms have a µ αβ ( r )with the the same shape as that of Be. Proceeding fromBe to C the amplitude per αβ electron pair decreasesand the node contracts toward the nucleus, in the samemanner as the node of a 2 s atomic orbital with increasing FIG. 5. Renormalized m αβ ( r ) of He, Li, and Be. nuclear charge. Interestingly, µ αβ ( r ) for the N atom hasa positive peak at the nucleus, a minimum near r = 0 . µ αβ ( r ),such as the relative motion of core, and core-valence elec-tron pairs. With the addition of β -spin p -electrons, the µ αβ ( r ) of O to Ne have a shape similar to the µ αβ ( r ) ofthose atoms before N. From O to Ne, the amplitude ofthe maximum per electron pair decreases, while the am-plitude of the minimum at the nucleus increases, and thelocation of the node remains relatively stationary.For a description of the correlated momemnta that isunbiased by ρ α ( r ), the reduced MB densities, divided bythe number of β electrons, of Be to Ne are presented inFigure 7. As expected from the analysis of µ αβ ( r ), all m αβ ( r ) have the same shape with the exception of thatfor N, which is positive everywhere. The amplitude of m αβ ( r ) /N β decreases substantially moving from Be toC. Far from the nucleus ( r > . m αβ ( r ) /N β also decreases from O to Ne. However, theamplitude of m αβ ( r ) /N β near the nucleus is less ordered.The µ αβ ( r ), per αβ electron pair, for the third periodatoms are presented in Figure 8. The shape of µ αβ ( r )for the Na atom is the same as that for Ne. This is due - - - - - - FIG. 6. Renormalized µ αβ ( r ) of the atoms Be to Ne (above)and N to Ne (below). to the fact that the β -electron configuration remains thesame. Once a β electron is added (Mg atom and beyond),the magnitude of µ αβ ( r ) per αβ electron pair increasessubstantially, similar to that seen for Be in the secondperiod. Unlike atoms of the second period, the µ αβ ( r )of Mg and beyond have two nodes (like the 3 s atomicorbital) and are positive at the nucleus. Yet again, likethe second period atoms, µ αβ ( r ) > αβ reduced MB densities, per β electron, for thethird period atoms are presented in Figure 9. As sug-gested by their µ αβ ( r ), the m αβ ( r ) of Na closely resem-bles that of Ne. The m αβ ( r ) of Mg, like Be of the sec-ond period, has the the largest amplitude per β electron.Close to the nucleus, the features of m αβ ( r ) are the sameas those for the second period atoms, except the sign isreversed. These features are nested inside the same fea-tures (a minimum and a node) repeated, but with thesame sign as those of the second period atoms. It is alsointeresting to note that, unlike N, the m αβ ( r ) for the Patom has the same number of nodes as the other atomsof the period. However, like N, the amplitude of m αβ ( r )for P is relatively minuscule.The trend in µ αβ for the second and third period atomsis illustrated in Figure 10. For all atoms µ αβ >
0, whichcorresponds to opposite-spin electrons preferring to havethe same momentum rather than opposite, and periodic- × - FIG. 7. Renormalized m αβ ( r ) of the atoms Be to Ne (above)and N to Ne (below). FIG. 8. Renormalized µ αβ ( r ) of the atoms Na to Ar. ity is followed quite closely. There is a sharp increase in µ αβ moving from group 1 to 2 ( i.e. one valence electronto two) which illustrates the importance of valence elec-tron motion to µ αβ . As atomic number increases, andmore electrons are added to the valence shell, the valueof µ αβ decreases, which suggests less correlated motion.For open-shell systems µ αβ ( r ) (cid:54) = µ βα ( r ), and subse-quently m αβ ( r ) (cid:54) = m βα ( r ). However, the features ob-served in αβ densities are very similar to those observed - - × - × - FIG. 9. Renormalized m αβ ( r ) of the atoms Na to Ar. in the βα densities. Therefore, plots of µ βα ( r ) and m βα ( r ) can be found in the supplementary material. B. Parallel-spin correlation
As mentioned previously, Fermi correlation contributesto the MB of parallel-spin electrons. Not surprisingly, thecontribution of Fermi correlation is orders of magnitudelarger than that of Coulomb correlation. Consider the Neatom, where µ σσ HF ( r ) = − . µ σσ ( r ) = − . ∼ µ σσ <
0, which is parallel-spin electrons ● ● ● ● ● ● ● ●■ ■ ■ ■ ■ ■ ■ ■●■
FIG. 10. Opposite-spin momentum-balance of 2nd and 3rdrow atoms. - - - - FIG. 11. Renormalized µ αα ( r ) of the atoms Li to Ne. having the exact opposite momentum.The αα MB density per electron pair, µ αα ( r ) N α ( N α − / , foratoms of the second period are presented in Figure 11. Itis seen that µ αα ( r ) is negative everywhere, meaning allregions contribute to electrons having the exact oppositemomentum. Similar to the opposite-spin MB density, theamplitude of µ αα ( r ) is relatively small for Li. For the Beatom, which has only one valence α -electron, the ampli-tude of µ αα ( r ) is much more appreciable. But, it is stillsignificantly smaller than that for atoms with multiplevalence α -electrons. The addition of p -electrons for B,and the proceeding atoms, creates a shoulder in µ αα ( r )further from the nucleus. Moving from B to Ne the shoul-der contracts to the nucleus and the amplitude of µ αα ( r )per electron pair decreases.Figure 12 presents the αα reduced MB density, di-vided by the number of α electrons, for the second periodatoms. The amplitude of m αα ( r ) /N α increases slightlyfrom Li to Be, and then substantially from Be to B. Thisis accompanied by a change in the shape of m αα ( r ) dueto the addition of a p -electron. The amplitude for B is - - - - - FIG. 12. Renormalized m αα ( r ) of the atoms Li to Ne. - - - FIG. 13. Renormalized µ αα ( r ) of the atoms Na to Ar. the largest amongst the atoms of the second period, andthe amplitude gradually decreases from B to Ne. As withopposite-spin MB density, the magnitude of the reducedMB is a minimum at the nucleus, and the m αα ( r ) /N α converge to a small range of values at the origin.The µ αα ( r ) per electron pair for the third period atomsare plotted in Figure 13. Similar to the µ αα ( r ) of the sec-ond period atoms, the µ αα ( r ) of the third period are neg-ative everywhere. The µ αα ( r ) of the third period atomshave extra features due to the presence of additional coreelectrons. The µ αα ( r ) of Na and Mg resemble that ofNe. The secondary minimum near the nucleus decreasesin magnitude and almost disappears as α p -electrons areadded from Al to P. For those atoms with α p -electrons,there is a less pronounced shoulder at larger r .The αα reduced MB density per α electron is presentedfor the third period atoms in Figure 14. The transi-tion from atoms without valence p -electrons to those withthem is obvious. The m αα ( r ) of both Na and Mg haveminima in their valence region (similar to Li and Be),whereas the atoms beyond Mg exhibit a shoulder andcontinue to increase in amplitude. Also similar to thesecond period, the maximum amplitude of m αα ( r ) /N α - - - - - - - - - FIG. 14. Renormalized m αα ( r ) of the atoms Na to Ar. is reached by the atom with only one p -electron and theamplitude decreases across the period. Furthermore, thevalues of m αα ( r ) /N α converge to a short range of near-zero values close to the nucleus.Finally, the trend in the total MB between parallel-spin electrons for atoms of the second and third period isillustrated in Figure 15. The actual values can be foundin Table I of the supplementary material. Fermi correla-tion, or exchange, is mainly responsible for the shape of µ σσ ( r ) and the subsequent value of µ σσ . In agreementwith the findings of Koga, concerning the the angularseparation of electron momenta in atoms, the magnitudeof MB is strongly dependent on the p -electrons. This isseen with the dramatic decrease in the value of µ αα fromgroup 2 to 13, and the decrease in µ ββ seen from group 15to 16. Interestingly, the minimum value of µ αα occurs atgroup 14, rather than group 15 which has the maximumnumber of unpaired p -electrons. Also, it is seen that µ ββ slightly decreases when α p -electrons are added (group 2to 15), and µ αα decreases significantly as β p -electronsare added (group 15 to 17).The decrease in correlated electron motion, as indi-cated by the decrease in the magnitude of µ αα , and theactual overall parallel-spin MB ( µ αα + µ ββ ), from group14 to 18, is similar to the decrease seen in opposite-spin MB as the number of valence electrons is increased(10). Due to the calculation of results short of the ● ● ● ● ● ● ● ●■ ■ ■ ■ ■ ■ ■ ■◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲●■◆▲ - - - - FIG. 15. Parallel-spin momentum balance of 2nd and 3rdrow atoms. complete basis set limit, it is somewhat precarious todraw firm conclusions regarding the nature of this de-crease for opposite-spin electrons alone. However, > β electrons are given in the supplementary material due tothe similarities in features and trends compared to the α electron densities. C. Comparing correlation
As previously mentioned, the effect of Fermi correla-tion on relative momenta is orders of magnitude largerthan that of Coulomb correlation between parallel-spinelectrons. The effect of Coulomb correlation on opposite-spin electrons is not as small as that on parallel-spin elec-trons, but there is a difference in magnitude compared toFermi correlation. There is also a difference in the rela-tive direction of the correlated momenta; Coulomb corre-lation of opposite-spin electrons favours equimomentum,whereas Fermi correlation favours antimomentum. Fig-ures 16 and 17 compare the reduced MB density for Neand Ar by scaling the opposite-spin density so that it issimilar in magnitude to that of the parallel-spin density,and by reversing the sign. It is seen for both Ne andAr, that while m αα ( r ) and m αβ ( r ) differ in sign, andby an order of magnitude, their shapes are quite similar.Both m αα ( r ) and m αβ ( r ) have minimum amplitude atthe nucleus and are relatively flat in the core region. Both - - - - - FIG. 16. Comparison between m αα ( r ) and scaled m αβ ( r ) ofNe. - - - - - FIG. 17. Comparison between m αα ( r ) and scaled m αβ ( r ) ofAr. increase in magnitude beyond the core and have a slightshoulder as r increases from the core region to the valenceregion. In the valence region, both increase in amplitudetowards the extent of appreciable electron density. Atlarge r , the m ( r ) begin to differ more. For instance,the m αβ ( r ) of Ne approaches a maximum near r = 3,while m αα ( r ) continues to increase in amplitude. How-ever, the striking similarities of these densities, which areintimately connected to how opposite-spin and parallel-spin electrons avoid each other, cannot be overlooked. V. CONCLUSIONS
A more thorough understanding of the correlated mo-tion of electrons ( i.e. how electrons avoid each other)is key to unravelling the complexity of the N -electronwave function. Such understanding not only allows forthe conception of new, more effective, models of elec-tronic structure, but it also increases our basic knowl-0edge of quantum mechanics. The equimomentum, an-timomentum and momentum-balance, along with theircorresponding densities, provide a means for acheivingsuch understanding. In addition to being a completelynew descriptor of the motion of electrons, MB also servesas a correlated motion detector.The MB density, µ ( r ), povides a measure of the localcontribution to momentum-balance in a system of elec-trons. Due to the dependence of µ ( r ) on the probabilityan electron is found at r , it resembles the one-electrondensity. The reduced MB density, m ( r ), removes thisdependence and amplifies the more subtle aspects of thelocal contributions to MB.The MB, MB density, and reduced MB density canbe decomposed into spin contributions. The analysisof the opposite-spin and parallel-spin MB, MB density,and reduced MB density of the atoms He to Ar has re-vealed significant differences between the correlated mo-tion of opposite-spin and parallel-spin electrons. Fermicorrelation, or exchange, causes parallel-spin electrons tomove with opposite momentum more than equal momen-tum. For electrons of opposite spin, Coulomb correla-tion causes them to move with equal momentum morethan opposite momentum. The amount of correlatedmotion between parallel-spin electrons is strongly depen-dent on the angular momentum of the electrons present.Whereas, for opposite-spin electrons, the amount of cor-related motion is dependent on simply the number ofvalence electrons present. Also, the effect of Fermi corre-lation on relative electron momenta is an order of mag-nitude larger than that of Coulomb correlation.These new tools, particularly the reduced MB density,have also revealed some subtle similarities between thecorrelated motion of parallel-spin and opposite-spin elec-trons. For both, correlated motion, given an electron ispresent, is a minimum near the nucleus and grows as thedistance from the nucleus is increased.The quantities presented, their derivations, and prac-tical algorithms for their calculation are generalized toabitrary systems of multiple nuclei. Therefore, a naturalnext step is the investigation of correlated electron mo-tion in molecules. In addition, because MB and the MBdensities reveal the presence of electron correlation, theyare useful tools for understanding the role of electron cor-relation in different chemical phenomena ( e.g. bond for-mation/breakage, excited states, dispersion). They alsoprovide a useful tool for assessing the quality of variousmodels of electronic structure. These are just a few ofthe potential applications of these unique quantum me-chanical properties. SUPPLEMENTARY MATERIAL
See supplementary material for recurrence relations forintegrals over Gaussian-type basis functions, and addi-tional equations related to the algorithm for the compu-tation of equimomentum, antimomentum, momentum- balance and their corresponding densities. Also, see sup-plementary material for plots of βα and ββ MB densities,as well as numerical values for total atomic MB.
ACKNOWLEDGMENTS
JWH thanks the Natural Sciences and EngineeringResearch Council of Canada (NSERC) for a DiscoveryGrant, Compute/Calcul Canada for computing resourcesand the Discovery Institute for Computation and Synthe-sis for useful consultations.
DATA AVAILABILITY
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Warburton, MUNgauss.Memorial University, Chemistry Department, St. John’s, NLA1B 3X7 (2015), with contributions from A. Alrawashdeh, J.-P. Becker, J. Besaw, S.D. Bungay, F. Colonna, A. El-Sherbiny,T. Gosse, D. Keefe, A. Kelly, D. Nippard, C.C. Pye, D. Reid, K.Saputantri, M. Shaw, M. Staveley, O. Stueker, Y. Wang, and J.Xidos. P. M. W. Gill, B. G. Johnson, and J. A. Pople, Chem. Phys.Lett. , 506 (1993). P. M. W. Gill, B. G. Johnson, and J. A. Pople, Int. J. QuantumChem. , 745 (1991). P. M. W. Gill, Adv. Quantum Chem. , 141 (1994). Supplementary Material: Measuring correlated electron motion in atoms with themomentum-balance density
S.I RECURRENCE RELATIONS FOR λ + The equi- and antimomentum, in terms of the spinless2-RDM, Γ, are given by, λ ± = 1(2 π ) (cid:90) Γ( r , r , r + q , r ∓ q ) d q d r d r . (S1)Expanding the 2-RDM over a general set of basis func-tions, { φ r } , leads to the alternative expression for theequi- and antimomentum, λ ± = (cid:88) rstu Γ rstu [ rstu ] λ ± , (S2)where the two-electron integral over four basis functionsis given by,[ rstu ] λ ± = 1(2 π ) (cid:90) φ r ( r ) φ s ( r + q ) × φ t ( r ) φ u ( r ∓ q ) d q d r d r . (S3)In the case of a Gaussian basis set, the fundamental λ + integral over s -type functions with exponents α , β , γ , δ and centers A , B , C , D is given by,[ ] λ + = π / e − σR αβγ + βγδ + αβδ + αγδ ) / , (S4)where σ = αβγδαβγ + βγδ + αβδ + αγδ (S5)and R = A + C − B − D . (S6)Integrals of higher angular momentum can be con-structed using a five-term recurrence relation (RR). The RRs for augmenting the angular momentum of theGaussians on each centre, A , B , C and D , in the i -direction (where i = x, y or z ) are given by[( a + i ) bcd ] λ + = − σα R i [ abcd ] λ + + a i α − σ α [( a − i ) bcd ] λ + + b i σ αβ [ a ( b − i ) cd ] λ + − c i σ αγ [ ab ( c − i ) d ] λ + + d i σ αδ [ abc ( d − i )] λ + , (S7a) [ a ( b + i ) cd ] λ + = σβ R i [ abcd ] λ + + a i σ αβ [( a − i ) bcd ] λ + + b i β − σ β [ a ( b − i ) cd ] λ + + c i σ βγ [ ab ( c − i ) d ] λ + − d i σ βδ [ abc ( d − i )] λ + , (S7b)[ ab ( c + i ) d ] λ + = − σγ R i [ abcd ] λ + − a i σ αγ [( a − i ) bcd ] λ + + b i σ βγ [ a ( b − i ) cd ] λ + + c i γ − σ γ [ ab ( c − i ) d ] λ + + d i σ γδ [ abc ( d − i )] λ + , (S7c)[ abc ( d + i )] λ + = σδ R i [ abcd ] λ + + a i σ αδ [( a − i ) bcd ] λ + − b i σ βδ [ a ( b − i ) cd ] λ + + c i σ γδ [ ab ( c − i ) d ] λ + + d i δ − σ δ [ abc ( d − i )] λ + , (S7d)where a = ( a x , a y , a z ) is a vector of angular momentumquantum numbers, and i is a unit vector with value 1in the i -direction [ e.g. x = (1 , , µ = λ + − λ − . (S8)The effort required to calculate MB is halved by makinguse of the following identity,[ rstu ] λ − = [ rsut ] λ + . (S9)Insertion of the 2-RDM expansion (Equation S12) intoEquation S8 and use of the identity above (Equation S9)leads to a practical equation for the calculation of MB, µ = (cid:88) rstu Γ rstu ([ rstu ] λ + − [ rsut ] λ + ) . (S10)The equi- and antimomentum densities are given bythe following equation, λ ± ( r ) = 1(2 π ) (cid:90) Γ( r , r , r + q , r ∓ q ) d q d r . (S11)Similar to equi- and antimomentum, their correspondingdensities can also be expanded over a basis set, λ ± ( r ) = (cid:88) rstu Γ rstu φ r ( r ) [ stu ] λ ± , (S12)where the three-function integral is given by,[ stu ] λ ± = 1(2 π ) (cid:90) φ s ( r + q ) × φ t ( r ) φ u ( r ∓ q ) d q d r . (S13)If the basis set is comprised of Gaussian-type functions,then the fundamental λ + ( r ) integral is given by,[ ] λ + = e − τP βγ + βδ + γδ ) / , (S14)where τ = βγδβγ + βδ + γδ , (S15)and P = r − B + C − D . (S16)The integrals over functions of higher angular momentumcan be calculated using the following 4-term recurrencerelations,[( b + i ) cd ] λ + = τβ P i [ bcd ] λ + + b i β − τ β [( b − i ) cd ] λ + + c i τ βγ [ b ( c − i ) d ] λ + − d i τ βδ [ bc ( d − i )] λ + , (S17a)[ b ( c + i ) d ] λ + = − τγ P i [ bcd ] λ + + b i τ βγ [( b − i ) cd ] λ + + c i γ − τ γ [ b ( c − i ) d ] λ + + d i τ γδ [ bc ( d − i )] λ + , (S17b)[ bc ( d + i )] λ + = τδ P i [ bcd ] λ + − b i τ βδ [( b − i ) cd ] λ + + c i τ γδ [ b ( c − i ) d ] λ + + d i δ − τ δ [ bc ( d − i )] λ + . (S17c) The MB density is calculated as the difference betweenthe equi- and antimomentum densities, µ ( r ) = λ + ( r ) − λ − ( r ) . (S18)To efficiently calculate µ ( r ), the following identity isused, [ stu ] λ − = [ sut ] λ + . (S19)If calculating µ ( r ) for numerous values of r , it is desirableto reduce the number of integral evaluations, which canbe done with the following expression, µ ( r ) = (cid:88) rstu (Γ rstu − Γ rsut ) φ i ( r )[ stu ] λ + . (S20) S.II ATOMIC MOMENTUM-BALANCE DENSITIES
The µ βα ( r ), m βα ( r ) and µ ββ ( r ), m ββ ( r ) of the atomsof the first three rows of the periodic table are presentedin Figures S1 to S8. - - - FIG. S1. Renormalized µ βα ( r ) of the atoms Be to Ne. - - - - - - FIG. S2. Renormalized m βα ( r ) of the atoms Be to Ne. FIG. S3. Renormalized µ βα ( r ) of the atoms Na to Ar. - - × - × - FIG. S4. Renormalized m βα ( r ) of the atoms Na to Ar. - - - - - - FIG. S5. Renormalized µ ββ ( r ) of the atoms Be to Ne. - - - - - - - FIG. S6. Renormalized m ββ ( r ) of the atoms Be to Ne. - - - - - - FIG. S7. Renormalized µ ββ ( r ) of the atoms Na to Ar. - - - - - - - - - - - FIG. S8. Renormalized m ββ ( r ) of the atoms Na to Ar. S.III ATOMIC MOMENTUM-BALANCE
The total opposite-spin and parallel-spin momentum-balance for the atoms He to Ar are reported in TableI.
TABLE I. Spin-resolved and total momentum-balance of theatoms helium to argonAtom µ αβ µ αα µ ββ µµ