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Electron Localizability Indicators from Spinor Wavefunctions Alexey I. Baranov* For the fully relativistic 4-component many-electron wavefunction six flavors of electron localizability indicators (ELI) have been proposed. Their counterparts, suitable for the application to the 2-component wavefunctions, have been also derived. Six proposed indicators have been tested on Ar and Rn atoms and one of them, the ELI-D for spatially antisymmetrized electron pairs, has been found to reveal atomic shell structures at quantitative level. Shell structures of all the atoms of periods

4–7 of the periodic table have been obtained using this indicator and compared with these obtained from the nonrelativistic limit calculations as well as from scalar-relativistic (zeroC 2014 Wiley Periodorder regular approximation) calculations. V icals, Inc.

Introduction

or by nonrelativistic hamiltonians with noncollinear magnetic Zeeman term.[7,8] They can contain spin-nondiagonal terms which depend on space coordinates, for example, ~ a ~ p or ~ r~ bð~ rÞ (where ~ a are Dirac matrices, ~ p —momentum operator, ~ r —Pauli spin matrices, and ~ bð~ rÞ—magnetic field). Spin and space rotations are then coupled and no global spin quantization axis can be chosen. All the spin-resolved properties are then not unique but depend on the choice of coordinate system and no such system exists where these hamiltonians would commute with z-component spin operator. Therefore, their eigenfunctions are not pure spin states, the natural orbitals of corresponding density matrices are not pure spin state functions[9] and no unique partitioning of density matrices into spinresolved contributions is possible. This precludes any sensible use of same- or opposite-spin-based indicators. Alternatively, the 1- and 2-density matrices can be partitioned into singlet- and triplet-coupled electron pair contributions,[9–13] which are invariant under rotations in spin space. Using these quantities, ELI-q for singlet- and ELI-D for tripletcoupled electron pairs have been introduced recently[14] for nonrelativistic scalar wavefunctions. This article presents an extension of ELI for spinor wavefunctions and their applications to the results of fully relativistic numerical Dirac–Coulomb–Kohn–Sham atomic calculations. A comparison with the ELI evaluated from same level calculations in nonrelativistic limit and scalar-relativistic zero-order regular approximation (ZORA)[15] calculations has also been made.

Electron localizability indicators (ELI) are a family[1,2] of functionals, capable of reproducing the shell structure of atoms across the periodic table at quantitative level. Their ability to reconstitute the populations of the shells in close accordance with Aufbau principle makes them valuable instrument for chemical bonding analysis. All these functionals are constructed from two distributions, derived from the reduced density matrices, using the x-restricted population method.[3] First one, called control function, is used to partition the space into sufficiently small compact regions called microcells. The name of the control function distinguishes different flavors of ELI, for example, ELI-q is based on the number of electrons and ELI-D on the number of electron pairs. The partitioning must obey the condition that the integral of control function over each microcell delivers the same number x given as a fixed parameter defining the partitioning. Second distribution, called sampling function, is then integrated over each microcell. After factorizing out x-dependence, one obtains quasicontinuous distribution of appropriate indicator. Being defined on a firm theoretical ground without using any arbitrary chosen reference system, these indicators can be evaluated from any calculation able to deliver control and sampling functions and not only in coordinate but also in momentum space.[4] Various 1- and 2-particle densities are usually taken as control and sampling functions. Probably the most popular representative is the ELI-D for r-spin electrons where the control function is an electron pair density for r-spin pairs and the sampling function is an electron density for r-spin electrons. It delivers valuable partitioning of space into core, penultimate shells, and various valence basins already from the 1determinantal wavefunctions.[1] Many important systems have to be described by multicomponent hamiltonians having in general case spinor eigenfunctions, for example, by fully relativistic 4-component manyelectron hamiltonians or their reduced 2-component forms[5,6]

DOI: 10.1002/jcc.23524

A. I. Baranov Max Planck Institute for Chemical Physics of Solids, N€ othnitzer Strasse 40, 01187, Dresden, Germany E-mail: [email protected] Contract grant sponsor: Deutsche Forschungsgemeinschaft; Contract grant number: BA-4911/1-1 C 2014 Wiley Periodicals, Inc. V

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Methodology Single component wavefunctions Any scalar many-electron wavefunction must be antisymmetric with respect to the exchange of coordinates of two particles: r r P^12 Wð~ r 1 r1 ;~ r 2 r2 ; . . . ; ~ x N Þ5P^12 P^12 Wð~ r 1 r1 ;~ r 2 r2 ; . . . ; ~ x NÞ r r 5P^12 P^12 Wð~ r 1 r1 ;~ r 2 r2 ; . . . ; ~ x N Þ5Wð~ r 2 r2 ;~ r 1 r1 ; . . . ; ~ x NÞ 52Wð~ r 1 r1 ;~ r 2 r2 ; . . . ; ~ x NÞ

r1 5r1 ;r2 5r2

(1)

here ~ x 5ð~ r; rÞ are combined spatial and spin electron coordir nates. The actions of permutation operators for space (P^12 ) r and spin (P^12 ) coordinates on many-electron wavefunction are: r r 1 r1 ;~ r 2 r2 ; . . . ; ~ x N Þ5Wð~ r 2 r1 ;~ r 1 r2 ; . . . ; ~ x NÞ P^12 Wð~ r r 1 r1 ;~ r 2 r2 ; . . . ; ~ x N Þ5Wð~ r 1 r2 ;~ r 2 r1 ; . . . ; ~ x NÞ P^12 Wð~

(2)

ðs=tÞ

The projection operators P^12 for singlet-/triplet-coupled electron r pairs can be defined either via permutation of spin indices P^12 [12]: 1 ðs=tÞ r P^12 5 ð17P^12 Þ 2

(3)

r or of spatial coordinates P^12 of two electrons[11]:

1 ðs=tÞ r P^12 5 ð16P^12 Þ 2

(4)

Both definitions are equivalent due to eq. (1) and the fact that the square of any permutation operator is an identity operator. It is easy to verify that both projection operators are ðs=tÞ ðs=tÞ idempotent: ðP^12 Þ2 5P^12 . r ^s 1  ~ ^s 2 Þ and Using Dirac spin exchange identity[16] P^12 5 12 ð114~ 2 3 the fact that ^s i 5 4  1, one can rewrite projection operators as[13]:   1 ðsÞ ^2 22~ S 12 P^12 5 2 1^ ðtÞ S P^12 5 ~ 2 12

(5)

2

^ ^s i are spin operators for each electron and ~ ^s 1 1~ ^s 2 where ~ S 12 5~ is the total spin operator for two electrons. Total squared spin of electron pair corresponding to singlet or triplet-projected wavefunctions is then: 1 ^2 ~ ðtÞ ^ 2 ðsÞ ^2 ^2 ^2 ~ S 12 P^12 W5 ð2~ S 12 2S 12~ S 12 ÞW5ð~ S 12 22ðP^12 Þ2 ÞW5 2 ðtÞ ^2 ^2 ^2 ð~ S 12 22P^12 ÞW5ð~ S 12 2~ S 12 ÞW50

(6)

1 ^2 ~ ðtÞ ðtÞ ^ 2 ðtÞ ^2 ~ S 12 P^12 W5 ~ S 12 S 12 W52ðP^12 Þ2 W52P^12 W 2 Therefore, any singlet-projected wavefunction possesses the value of total squared spin for two electrons S12 ðS12 11Þ equal to 0, corresponding to singlet-coupled pair and any tripletprojected wavefunction possesses the value of S12 ðS12 11Þ52 corresponding to triplet-coupled pair. Taking the squared modulus of projected wavefunctions, summing over all spin indi566

Journal of Computational Chemistry 2014, 35, 565–585

r2 ces, and integrating over all spatial coordinates except ~ r 1 ;~ delivers the corresponding density matrices: ! N XXð ðs=tÞ 0 0 q2 ð~ d~ x3 . . . r 1 ;~ r 2 ;~ r 1 ;~ r 2 Þ5 2 r1 r2 (7) ð h i ðs=tÞ x N W ð~ d~ x 01 ; ~ x 02 ; . . . ; ~ x N ÞP^12 Wð~ x 1; ~ x 2; . . . ; ~ x NÞ 0 0 These density matrices are symmetric/antisymmetric with respect to the exchange of one pair of spatial coordinates of two electrons [eq. (4)]: ðsÞ

ðsÞ

q2 ð~ r 01 ;~ r 02 ;~ r 2 ;~ r 1 Þ5q2 ð~ r 01 ;~ r 02 ;~ r 1 ;~ r 2Þ ðtÞ

(8)

ðtÞ

q2 ð~ r 01 ;~ r 02 ;~ r 2 ;~ r 1 Þ52q2 ð~ r 01 ;~ r 02 ;~ r 1 ;~ r 2Þ ðs=tÞ

and their diagonal parts q2 ð~ r 1 ;~ r 2 ;~ r 1 ;~ r 2 Þ can be interpreted times probability densities that two electrons at ~ r1 as NðN21Þ 2 and ~ r 2 are coupled to a singlet or triplet pair.[11] Integrating out second electron coordinate results in the singlet/triplet one-electron density matrices, whose diagonal parts are the one-electron densities for electrons coupled to singlet/triðtÞ plet pairs.[9,10,13] ELI for singlet- !ðsÞ q and triplet-coupled !D electron pairs can then be obtained using corresponding one- and two-electron densities as control and sampling functions[14]:  ðsÞ N12 2 q2 ð~ r;~ rÞ 2 4 N21 ½qðsÞ ð~ rÞ   12 3=8 ðtÞ !D ð~ rÞ5qðtÞ ð~ rÞ ðtÞ rÞ g ð~

1 !ðsÞ rÞ5 q ð~



(9)

ðtÞ

r;~ r; ~ r; In case of triplet-coupled electrons the ontop part q2 ð~ ~ rÞ is identically zero as follows from eq. (8). The first leading term in the Taylor expansion of triplet pair density around the electron coalescence is of second order,[14,17] therefore, its curvature gðtÞ appears in the formula for triplet ELI-D. For singletcoupled electrons the ontop density does not necessarily vanishes and thus appears in the formula of !ðsÞ q . For single-determinantal closed-shell wavefunction, the ontop pair density and one-electron density for singletcoupled electrons are given by[14]: q2 ð~ rÞ 4 1 N12 qðsÞ ð~ qð~ rÞ rÞ5 4 N21 ðsÞ q2 ð~ r; ~ rÞ5

(10)

X where qð~ rÞ5 i ni jui ð~ rÞj2 is the total electron density. Therefore, in this case ELI-q for singlet-coupled electrons yields a constant distribution equal to unity everywhere.[14] The ingredients of triplet ELI-D in case of singledeterminantal closed-shell wavefunction are: " # 3 ½rqð~ rÞ2 g ð~ rÞsð~ rÞ2 rÞ5 qð~ 2 8 ðtÞ

3 N22 qð~ rÞ rÞ5 q ð~ 4 N21 ðtÞ

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(11)

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where sð~ rÞ5 12

X

n jrui ð~ rÞj i i

2

is the positive definite kinetic

energy density.* ELI-D for triplet-coupled electrons yields the nonuniform distribution with rich topology even for closedshell single-determinantal wavefunctions, which makes it a universal indicator for chemical bonding analysis.[14]

r k Þ  uck rk ð~ r k Þg these density matrices can be reprenors fuXk ð~ sented as: X 0 X 02 X1 X2

q2 1

0 X X0  0 X  0 ð~ r 01 ;~ r 02 ;~ r 1 ;~ r 2 Þ5 crspq ur 1 ð~ r 1 Þus 2 ð~ r 2 ÞuXp1 ð~ r 1 ÞuXq2 ð~ r 2Þ

rspq

(16) †

4-component wavefunctions Many-particle spinor wavefunction for N electrons can be represented as mN-component tensor WX1 X2 ...XN ð~ r 1 ;~ r 2 ; . . . ;~ r N Þ,[6,18] where each component depends on spatial coordinates of all particles. In 4-component case m 5 4 with Xk 2 fLa; Lb; Sa; Sbg. The index Xk can also be represented as combined Xk  ck rk , where ck 2 fL; Sg and rk 2 fa; bg. In 2-component case small components with ck 5S vanish and m 5 2 with Xk 5rk 2 fa; bg. Antisymmetry principle of N-electron spinor wavefunction can be expressed as[18]: r 1 ;~ r 2 ; . . . ;~ r N Þ5WX2 X1 ...XN ð~ r 2 ;~ r 1 ; . . . ;~ r NÞ P^12 WX1 X2 ...XN ð~ 52WX1 X2 ...XN ð~ r 1 ;~ r 2 ; . . . ;~ r NÞ

(12)

r X Permutation operator can also be represented as P^12 5P^12 P^12 : r P^12 WX1 X2 ...XN ð~ r 1 ;~ r 2 ; . . . ;~ r N Þ5WX1 X2 ...XN ð~ r 2 ;~ r 1 ; . . . ;~ r NÞ X r 1 ;~ r 2 ; . . . ;~ r N Þ5WX2 X1 ...XN ð~ r 1;~ r 2 ; . . . ;~ r NÞ P^12 WX1 X2 ...XN ð~

m 2 ða1br12 1Oðr12 ÞÞ Wðr12 ; . . .Þ  r12

(13)

In 4-components case permutation operator P^12 exchanges not only spin labels but also L and S labels and thus can be X LS r further represented[18] as P^12 5P^12 P^12 . Therefore, the operators constructed according to eqs. (3) and (4) are not equivalent as: X

r r r r LS r LS r r P^12 52P^12 P^12 52P^12 P^12 P^12 P^12 52P^12 P^12 6¼ 2P^12

(14)

Six independent indicators (symmetric/antisymmetric to the permutations of space, spin, and component) can be constructed, which are considered later. ELI are evaluated from the diagonal parts of various density matrices. For spinor many-electron wavefunction a general two-electron reduced density matrix is a m4-component tensor, obtained by taking partial trace over X3 . . . XN components of the squared wavefunction modulus tensor and integrating out ~ r 3 . . .~ r N spatial electron coordinates: X0 X0 X X r 01 ;~ r 02 ;~ r 1 ;~ r 2 Þ5 q2 1 2 1 2 ð~

N 2

!

X ð

ð 0 0 d~ r 3 . . . d~ r N WX 1 X 2 X3 ...XN 

X3 ...XN

ð~ r 01 ;~ r 02 ;~ r 3 ; . . . ;~ r N ÞWX1 X2 X3 ...XN ð~ r 1 ;~ r 2 ;~ r 3 ; . . . ;~ r NÞ N

!

(15)

where L€ owdin normalization to electron pairs[19] has 2 been used. For most consistent Fock space formulation[20] of relativistic quantum chemistry,[21] in the finite basis of one-particle spi-

*

†

a r a^s a^p a^q jWi. Here W is many-electron wavewhere crspq 5hWj^ † function and a^r ; a^p are usual creation and annihilation operators for one-electron states. For single-determinantal ansatz with orthonormal spinors crspq 5 12 nr ns ðdrp dsq 2drq dsp Þ. Local behavior of relativistic 4-component many-electron wavefunction at electron–electron coalescence point differs from that of nonrelativistic one.[18,22] This difference deserves a special attention here, as for the evaluation of ELI one needs to calculate integrals over microcells around electron coalescence point in the limit of microcell volume going to zero. Both nonrelativistic and relativistic wavefunctions near electron–electron coalescence point can be represented using following general expansion in powers of interelectronic distance 0  r12 5r2 2r1 < , where m is the lowest noninteger power of r12[18,22]:

Strictly speaking, only when multiplied with the proper dimensionality factor.

(17)

where a and b are coefficients independent of r12. In nonrelativistic case m50, whereas in case of 4-component hamiltonians, m can be slightly negative[18,22] with m  2c22 (c is the speed of light) which means that the wavefunction should be weakly singular at electron–electron coalescence point. In practical calculations, this singularity can be absent, that is, m50 and then the ontop value can be used for the integration as in nonrelativistic case. If the singularity is however present, it poses no practical problems for integral evaluation as it is then multiplied by volume element, for example, 2 4pr12 dr12 [18] and: ðR 0 ðR 0

2 dr12 r12 W ðr12   ÞWðr12   Þ 

r 312m 212m dr12 r12 5jaj2 12 jR0 5jaj2 312m

ðR

2 2m dr12 r12 jaj2 r12 5jaj2

0 312m

R R3  jaj2 1OðmÞ 312m 3

(18)

that is, ignoring the singularity delivers the result with the accuracy c22. Indicators derived from (anti)symmetrized spatial coordinate projections If projection operators for spinor wavefunctions are constructed via symmetrization/antisymmetrization (correspondingly—s/a) of spatial coordinates in analogy with eq. (4): 1 1 rðs=aÞ r r LS P^12 5 ð16P^12 Þ5 ð17P^12 P^12 Þ 2 2

(19)

LS then, due to the presence of P^12 operator, the projected wave2 functions will not have definitive values of ~ S 12 operator [contrary to eqs. (6)] and thus will not be describing electrons 1

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and 2 as being coupled to singlet or triplet pairs. Therefore, the words “singlet” or “triplet” should not be used for these projections. However, corresponding pair density matrices are symmetric/antisymmetric to the permutation of spatial coordinates of one-electron by construction and conditions analogous to given by eq. (8) will be fulfilled: q2 ð~ r 01 ;~ r 02 ;~ r 2 ;~ r 1 Þ5q2 ð~ r 01 ;~ r 02 ;~ r 1 ;~ r 2Þ rðsÞ

rðsÞ

(20)

rðaÞ 0 rðaÞ 0 q2 ð~ r 1 ;~ r 02 ;~ r 2 ;~ r 1 Þ52q2 ð~ r 1 ;~ r 02 ;~ r 1 ;~ r 2Þ

rðs=aÞ

ð~ r 1 ;~ r 2 Þ5

1XXX c ½uc1 r1  ð~ r 1 Þusc2 r2  ð~ r 2 Þucp1 r1 ð~ r 1Þ 2 c1 r1 c2 r2 rspq rspq r

r 2 Þ6urc1 r1  ð~ r 1 Þusc2 r2  ð~ r 2 Þucp1 r1 ð~ r 2 Þucq2 r2 ð~ r 1 Þ ucq2 r2 ð~ (21)

ð

2 rðs=aÞ r 1 Þ5 r 1 ;~ r 2Þ qrðs=aÞ ð~ d~ r 2 q2 ð~ N21 " 1 X X 5 c uc1 r1  ð~ r 1 Þucp1 r1 ð~ r 1Þ N21 c1 r1 rsp rsps r

(22) #

XX crspq Scsp2 r2 ;c1 r1 ucr 1 r1  ð~ r 1 Þucq2 r2 ð~ r 1Þ 6 c2 r2 rspq

The overlap integrals between spinor components introduced above are: ð Scsq1 r1 ;c2 r2 5 d~ r 2 usc1 r1  ð~ r 2 Þucq2 r2 ð~ r 2Þ

 rðsÞ N12 2 q2 ð~ r;~ rÞ 2 4 N21 ½qrðsÞ ð~ rÞ

2

568

"



# XX c1 r1 ;c2 r2 c2 r2 ;c1 r1 q ð~ rÞ2 q ð~ rÞq ð~ rÞ 2

c1 r1 c2 r2

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s

!#

nr ns Scsr2 r2 ;c1 r1 ucr 1 r1  ð~ rÞucs 2 r2 ð~ rÞ

rs

(26) where (27)

r

are the components of electron density tensor for which trace X qð~ rÞ5 c r qc1 r1 ;c1 r1 ð~ rÞ is the total electron density. 1 1

Pair density for spatially antisymmetrized electrons vanishes identically at electron coalescence point, as easily seen from, for example, eq. (21). An indicator for spatially antisymmetrized electrons should therefore be based on higher order terms of Taylor expansion of corresponding pair density around electron coalescence point. The spatially antisymmetrized projected wavefunction around electron–electron coalescence point has the following form: 1 r WrðaÞ ð~ r 1 ;~ r 2 ; . . .Þ5 ð12P^12 ÞWð~ r 1 ;~ r 2 ; . . .Þ 2   1 m 2 m 2 ða1br12 1Oðr12 ÞÞ2r21 ða1br21 1Oðr21 ÞÞ 5 r12 2  1 m 1 m m 2 2 5 r12 ða1br12 Þ2r12 ða2br12 1Oðr12 ÞÞ 5 r12 ð2br12 1Oðr12 ÞÞ 2 2 (28) so that diagonal part of appropriate density matrix is:

(24)

The prefactor is kept for compatibility with the original nonrelativistic formula [eq. (9)], so that applied to the nonrelativistic limit wavefunction, this indicator delivers the nonrelativistic ELI-q for singlet-coupled electrons. This indicator shows, how many spatially symmetrized electron pairs can be found in the small volume around evaluation point ~ r, when this volume contains fixed fraction of electron density of spatially symmetrized electrons. In case of 1-determinantal wavefunctions, the ontop pair density and one-electron densities become (Appendix A “Densities for space (anti)Symmetrized Projections”): 1 rðsÞ q2 ð~ r;~ rÞ5

X

ns Scss2 r2 ;c1 r1 qc1 r1 ;c2 r2 ð~ rÞ2

(23)

Ontop pair density for spatially symmetrized electron pairs is not necessarily zero and electron localizability indicator for spatially symmetrized electron pairs can be constructed using corresponding ontop pair density and the one-electron density in completely similar way, as nonrelativistic ELI-q for singletcoupled pairs[14]: 1 rÞ5 !qrðsÞ ð~

X

" XX XX 1 Nqð~ rÞ2 n2r jucr 1 r1 ð~ rÞj2 7 2ðN21Þ c1 r1 r c1 r1 c2 r2

X qc1 r1 ;c2 r2 ð~ rÞ5 nr urc1 r1  ð~ rÞucr 2 r2 ð~ rÞ

Pair and one-electron densities for spatially (anti)symmetrized electrons are (see Appendix A “Densities for space (anti)Symmetrized Projections” for details): q2

qrðs=aÞ ð~ rÞ5

(25)

rðaÞ

r 1 ;~ r 2 ;~ r 1 ;~ r 2 Þ  WrðaÞ ð~ r 1 ;~ r 2 ; . . .ÞWrðaÞ ð~ r 1 ;~ r 2 ; . . .Þ q2 ð~ 2m 2 3 2m 3  r12 ðjbj2 r12 1Oðr12 ÞÞ  jbj2 r12 2ð11mÞ 1r12 Oðr12 Þ

(29)

which necessarily means that the first derivatives also vanish identically at the coalescence point. The second derivatives, however, do not go to zero but have the same type of singu2m larity r12 as squared wavefunction and the value of curvature at coalescence can be used for the integration [cf. eq. (18)]. Applying D-restricted partitioning, following indicator can then be constructed in a way similar to ELI-D for triplet-coupled electrons[14]:   12 3=8 rðaÞ !D ð~ rÞ5qrðaÞ ð~ rÞ rðaÞ rÞ g ð~

(30)

Using traditional interpretation of ELI-D[1] one can state that this indicator is proportional to the charge of spatially antisymmetrized electrons in the microcell around the evaluation point ~ r, which is necessary to have certain small given fraction of spatially antisymmetrized electron pairs in that microcell. The curvature of spatially antisymmetrized pair density at coalescence grðaÞ ð~ rÞ is given by (see Appendix A “Densities for Space (anti)Symmetrized Projections” for details): WWW.CHEMISTRYVIEWS.COM

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h XXX grðaÞ ð~ rÞ5 crspq urc1 r1  ð~ rÞrusc2 r2  ð~ rÞucp1 r1 ð~ rÞrucq2 r2 ð~ rÞ c1 r1 c2 r2 rspq

2urc1 r1  ð~ rÞrusc2 r2  ð~ rÞrucp1 r1 ð~ rÞucq2 r2 ð~ rÞ

i (31)

In 1-determinantal case it becomes:    1 rqð~ rÞ ~ rqð~ rÞ ~ grðaÞ ð~ rÞ5qð~ rÞsð~ rÞ2 1ijð~ rÞ 2ijð~ rÞ 2 2 2  c1 r1 ;c2 r2  X  1 rq ð~ rÞ ~c1 r1 ;c2 r2 1ij rÞsc2 r2 ;c1 r1 ð~ rÞ2 ð~ rÞ qc1 r1 ;c2 r2 ð~ 1 2 2 c1 r1 ;c2 r2  c2 r2 ;c1 r1  rq ð~ rÞ ~c2 r2 ;c1 r1 2ij ð~ rÞ 2

evaluation point from given fraction of electron charge of singlet-coupled electrons. As above, the prefactor from nonrelativistic ELI-q for singlet-coupled electrons is kept here for the compatibility. ELI-D for triplet (i.e., spin-symmetrized) electrons can be constructed using D-restricted partitioning,[14] where the space is divided into microcells each enclosing a fixed fraction DrðsÞ of spin-symmetrized electron pair. Using Taylor expansion of spin-symmetrized pair density and taking the ontop part as leading term, the number of spin-symmetrized pairs in the microcell l is approximately given by: rðsÞ

DrðsÞ  q2 ð~ r l ;~ r l ÞVl2 l

(36)

(32) where overlap integrals Scrs1 r1 ;c2 r2 were defined in eq. (23), electron density tensor components qc1 r1 ;c2 r2 ð~ rÞ are given by eq. (27) and additionally following quantities have been introduced: 1X sc1 r1 ;c2 r2 ð~ rÞ5 nr rurc1 r1  ð~ rÞrucr 2 r2 ð~ rÞ 2 r  1 X  c1 r1  jc1 r1 ;c2 r2 ð~ rÞ5 nr ur ð~ rÞrucr 2 r2 ð~ rÞ2rucr 1 r1  ð~ rÞucr 2 r2 ð~ rÞ 2i r (33) Although these two quantities are labeled here like nonrelativistic kinetic energy and current densities, they should not be interpreted like relativistic kinetic energy and current densities, as in 4-component case, they are defined in completely different way.[6] Only when applied to the nonrelativistic limit wavefunctions, they turn to expressions formally equivalent to nonrelativistic current density and kinetic energy density, as discussed later. Indicators derived from (anti)symmetrized spin projections Alternatively, one can choose antisymmetrization/symmetrization of spin indices and construct the projection operators similarly to eq. (3). 1 1 rða=sÞ r r LS P^12 5 ð17P^12 Þ5 ð16P^12 P^12 Þ 2

2

  rðaÞ 1 N12 2 q2 ð~ r;~ rÞ 2 4 N21 ½qrðaÞ ð~ rÞ

Vl 

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi DrðsÞ rðsÞ

q2 ð~ r l ;~ r lÞ

(37)

Integration of the spin-symmetrized one-electron density rðsÞ over the microcell results in the charge 1D of spinsymmetrized electrons, which is necessary to have a given number of spin-symmetrized electron pairs DrðsÞ in the microcell. This can be calculated approximately as: rðsÞ

1D

 qrðsÞ ð~ r l ÞVl sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi DrðsÞ qrðsÞ ð~ r l Þ pffiffiffiffiffiffiffiffiffiffi rðsÞ 5 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi DrðsÞ  q ð~ r lÞ rðsÞ rðsÞ q2 ð~ r l ;~ r lÞ q2 ð~ r l ;~ r lÞ

(38)

For fixed infinitesimally small DrðsÞ , the ELI-D for spinsymmetrized electrons can then be defined as: qrðsÞ ð~ rÞ rðsÞ !D ð~ rÞ5 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rðsÞ q2 ð~ r;~ rÞ

(34)

Using relativistic extension[23] of Dirac spin exchange idenr ^ ^ tity P^12 5 12 ð11~ R1  ~ R 2 Þ, it is easy to show, that projected wavefunctions will then fulfill analogs of eqs. (6) and thus will describe the electron pair coupled to singlet (R12 50) or triplet (R12 51). However, corresponding density matrices will not fulLS fill eq. (8) due to the presence of P^12 operator and none of pair densities will necessarily vanish at electron coalescence point, resulting in nonzero ontop densities. ELI-q for singlet (i.e., spin-antisymmetrized) electrons can then be constructed using q-restricted partitioning just as it was done in original publication[14]: !rðaÞ rÞ5 q ð~

The volume of the microcell containing fixed fraction DrðsÞ of spin-symmetrized electron pair is then

(39)

Thus defined indicator shows, how large is the charge of triplet-coupled electrons in the small volume around evaluation point when it contains the fixed fraction of tripletcoupled electron pairs. Applying projection operators [eq. (34)] to the general multideterminantal wavefunction one obtains (see Appendix A “Densities for spin (anti)Symmetrized Projections”) for the spin (anti)symmetrized (correspondingly—a/s) pair densities: h 1XXX crspq urc1 r1  ð~ r 1 Þusc2 r2  ð~ r 2 Þucp1 r1 ð~ r 1Þ 2 c1 r1 c2 r2 rspq i ucq2 r2 ð~ r 2 Þ7ucr 1 r1  ð~ r 1 Þusc2 r2  ð~ r 2 Þucp1 r2 ð~ r 1 Þucq2 r1 ð~ r 2Þ rða=sÞ

q2

ð~ r 1 ;~ r 2 Þ5

(35)

(40)

Thus defined indicator shows, how many singlet (spin-antisymmetrized) pairs can be formed in a small volume around

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ð 2 rða=sÞ d~ r 2 q2 qrða=sÞ ð~ r 1 Þ5 ð~ r 1 ;~ r 2Þ N21 " 1 X X 5 c uc1 r1  ð~ r 1 Þucp1 r1 ð~ r 1Þ N21 c1 r1 rsp rsps r

(41) #

XX crspq Scsq2 r2 ;c2 r1 ucr 1 r1  ð~ r 1 Þucp1 r2 ð~ r 1Þ 7 c2 r2 rspq

where the overlap integrals are defined by eq. (23). In case of 1-determinantal wavefunctions, the ingredients of these indicators become: rða=sÞ

q2

ð~ r;~ rÞ5

" XX 1 2 q ð~ rÞ2 qc1 r1 ;c2 r2 ð~ rÞqc2 r2 ;c1 r1 ð~ rÞ 4 c1 r1 c2 r2

XX 7 ðqc1 r1 ;c1 r2 ð~ rÞqc2 r2 ;c2 r1 ð~ rÞ2qc1 r1 ;c2 r1 ð~ rÞqc2 r2 ;c1 r2 ð~ rÞÞ

#

r rðaÞ pairs due to eq. (44) as P^12 operator has no effect when ~ r5~ r 1 5~ r 2 . However, corresponding one-electron densities (e.g., for c(a) and rðsÞ electrons) will be different as one of the spatial coordinates is integrated out there and the final indicators (e.g., !cðaÞ and !qrðsÞ ) will also be different. These components q (anti)symmetrized indicators should be interpreted in the similar way as those from previous section.

Indicators for 2-component wavefunctions In 2-component case only large components of bispinors are LS not zero.[6] P^12 operator is then equivalent to the identity operator as any two electrons have the same (large) components. Therefore, this operator can be omitted from eqs. (13) and (14), and the projection operators become: rðs=aÞ2c P^12

c1 r1 c2 r2

qrða=sÞ ð~ rÞ5 X s

XX XX 1 Nqð~ rÞ2 n2r jucr 1 r1 ð~ rÞj2 7 2ðN21Þ c1 r1 r c1 r1 c2 r2 X

ns Scss2 r2 ;c2 r1 qc1 r1 ;c1 r2 ð~ rÞ2

cðs=aÞ2c P^12

!#

nr ns Scsr2 r2 ;c2 r1 ucr 1 r1  ð~ rÞucs 1 r2 ð~ rÞ

rs

(43) where the components of electron density tensor qc1 r1 ;c1 r2 were defined earlier. Indicators derived from (anti)symmetrized large and small component projections Formally, it is also possible to construct projections symmetric/ antisymmetric (correspondingly—s/a) to the permutation of large and small blocks of wavefunctions: 1 1 cðs=aÞ LS r r P^12 5 ð16P^12 Þ5 ð17P^12 P^12 Þ 2 2

(44)

Obviously, neither analogs of eq. (6) nor of eq. (8) will be then fulfilled and thus projected wavefunctions will describe neither singlet-/triplet-coupled electron pairs nor have vanishing values at coalescence. It seems to be difficult to give any chemically meaningful interpretation to these projections. One can choose following indicators using q-restriction for symmetrized and D-restriction for antisymmetrized projections: cðsÞ

!cðsÞ rÞ5 q ð~

q2 ð~ r;~ rÞ ½qcðsÞ ð~ rÞ

2

qcðaÞ ð~ rÞ cðaÞ rÞ5 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !D ð~ cðaÞ q2 ð~ r;~ rÞ

(45)

Journal of Computational Chemistry 2014, 35, 565–585

(47)

As one immediately sees, the symmetrization of space is then equivalent to the antisymmetrization of spin and, reverse, space antisymmetrization is equivalent to spin symmetrization. The former can again be called singlet and the latter the triplet projection, as eq. (6) will be fulfilled, as well as eq. (8). Projection operators, based on component permutation reduce in the 2-component case to the trivial unity operator, yielding original wavefunction (from symmetrized) or zero operator, yielding zero function (from antisymmetrized). Therefore, for 2-component wavefunctions there are only two independent nontrivial projections, for which ELI-q for singletcoupled electrons and ELI-D for triplet-coupled electrons[14] can be introduced. Singlet ELI-q for 2-component case can be calculated from either eqs. (35) or (24), taking ontop singlet pair density and singlet one-electron density from 2-component wavefunction. However, for 2-component triplet ELI-D one can not directly use eq. (39), as it was derived assuming nonzero ontop triplet pair density, which is identically zero in 2-component case. Instead one can use eq. (30), derived under assumption of zero ontop part. Expressions for ingredients of singlet ELI-q and triplet ELI-D for 2-component wavefunctions can be directly obtained from corresponding 4-component expressions for space (anti)symmetrized indicators by omitting the summation over components: ðs=tÞ

q2

ð~ r 1 ;~ r 2 Þ5

1XXX c 2 r1 r2 rspq rspq

h i urr1  ð~ r 1 Þurs 2  ð~ r 2 Þurp1 ð~ r 1 Þurq2 ð~ r 2 Þ6urr 1  ð~ r 1 Þurs 2  ð~ r 2 Þurp1 ð~ r 2 Þurq2 ð~ r 1Þ

(46)

Formulae for corresponding ontop pair densities and oneelectron densities for general CI ansatz and 1-determinantal case are similar to those from above section and are given in Appendix A “Densities for Large and Small Component (anti)Symmetrized Projections”. As shown there, the ontop pair densities are identical for c(a) and rðsÞ and, reverse, for c(s) and 570

rða=sÞ2c P^12

(42)

"

1 1 r r 5 ð16P^12 Þ5 ð17P^12 Þ 2 2 1 1 r r 5 ð17P^12 Þ5 ð16P^12 Þ 2 2 1 1 LS 5 ð16P^12 Þ5 ð161Þ 2 2

(48) r 1 Þ5 qðs=tÞ ð~ " X rsp

ð 2 1 X ðs=tÞ d~ r 2 q2 ð~ r 1;~ r 2 Þ5 N21 N21 r1 XX

crsps urr 1  ð~ r 1 Þurp1 ð~ r 1 Þ6

#

crspq Srsp2 ;r1 urr 1  ð~ r 1 Þurq2 ð~ r 1Þ

r2 rspq

(49)

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gðtÞ ð~ rÞ5

XX

Computational details

crspq

r1 r2 rspq

h i urr1  ð~ rÞrurs 2  ð~ rÞurp1 ð~ rÞrurq2 ð~ rÞ2urr 1  ð~ rÞrurs 2  ð~ rÞrurp1 ð~ rÞurq2 ð~ rÞ

(50) and in 1-determinantal case: ðsÞ q2 ð~ r; ~ rÞ5

ðs=tÞ

q

" # XX 1 2 rÞ2 qr1 ;r2 ð~ rÞqr2 ;r1 ð~ rÞ q ð~ 2 r1 r2

(51)

" XX 1 ð~ rÞ5 n2r jurr 1 ð~ rÞj2 Nqð~ rÞ2 2ðN21Þ r1 r

!# X XX X ns Srss2 ;r1 qr1 ;r2 ð~ rÞ2 nr ns Srsr2 ;r1 urr1  ð~ rÞurs 2 ð~ rÞ 7 r1

r2

s

rs





 1 rqð~ rÞ ~ rqð~ rÞ ~ 1ijð~ rÞ 2ijð~ rÞ rÞ5qð~ rÞsð~ rÞ2 gðtÞ ð~ 2 2 2   X 1 rqr1 ;r2 ð~ rÞ ~r1 ;r2 r1 ;r2 r2 ;r1 1 1ij q ð~ rÞs ð~ rÞ2 ð~ rÞ 2 2 r1 ;r2  r2 ;r1  rq ð~ rÞ ~r2 ;r1 2ij ð~ rÞ 2

(52)

(53)

where X qr1 ;r2 ð~ rÞ5 nr urr 1  ð~ rÞurr 2 ð~ rÞ

Results and Discussion

r

sr1 ;r2 ð~ rÞ5

1X 2

r1 ;r2 1 ~ ð~ rÞ5 j 2i

r

nr rurr 1  ð~ rÞrurr 2 ð~ rÞ

All the calculations have been made for neutral isolated atoms taken in ground electronic configurations.[24] In cases of partially occupied subshells fractional occupancies were used to accomplish a spherical symmetry. Numerical 4-component Dirac–Coulomb–Kohn–Sham LDA (Perdew–Wang 92[25]) restricted calculations have been performed using radial Dirac equation solver routines of Elk program.[26] Coupled Dirac equations have been solved on the logarithmic radial grid using predictor–corrector method. Total number of radial points used was about 1032104 and the maximal radius was 20–40 a.u. Spline interpolation has been used for the calculation of derivatives. Point nuclei model has been used. The speed of light for the calculations in the nonrelativistic limit was increased by a factor of 106. Indicators, introduced above for 4-component wavefunctions, have been evaluated from density matrices constructed from Kohn–Sham spinors in 1-determinantal ansatz.‡ Restricted scalar-relativistic ZORA[15] calculations have been performed using Slater basis set of QZ4P level with ADF code[27] using same Perdew–Wang LDA parameterization. ELID for same-spin electrons[1] has been evaluated from density matrices constructed in 1-determinantal ansatz.[3] The effect of picture change[15] for the densities has been ignored. Numerical evaluation of shell electron populations has been done with the program DGrid.[28]

(54)

X   nr urr1  ð~ rÞrurr 2 ð~ rÞ2rurr1  ð~ rÞurr 2 ð~ rÞ r

are the components of nonrelativistic electron charge, kinetic energy, and current density tensors.† In the collinear case a global spin quantization axis exists and all the spinors can be chosen as eigenfunctions of S^z operator. Then the pair densities [eq. (48)] can be further reduced to those reported in Refs. [13] and [14] (see Appendix B). If one would straightforwardly evaluate 4-component ELI from the 4-component nonrelativistic limit wavefunction, !qrðaÞ rðaÞ and !rðsÞ will give singlet ELI-q !ðsÞ will yield triplet ELI-D q q . !D rðsÞ and !D will yield infinity, because of zero ontop density for spin-symmetrized, that is, triplet-coupled electrons in nonrelativistic case. !qcðsÞ will yield the ratio between spinless ontop pair density and squared total density which is when being multiplied by four is identical to spinless ELI-q (eq. (53) in Ref. cðaÞ 14]). !D will yield indefinite result as both quantities in numerator and denominator will be zero.

Ar atom Figure 1 shows all six localizability indicators, evaluated from fully relativistic 4-component calculation for Ar atom. In addirðaÞ tion, !D , evaluated from nonrelativistic limit calculation, is also shown on the inset. The most well pronounced shell structure is delivered by ELI-D rðaÞ for spatially antisymmetrized pairs !D , where the shell boundaries are clearly marked by radial minima. The integration of the electron density inside thus defined shells gives the following populations: K—2.207, L—7.868, and M—7.923. In case of outermost shell, the integration region extends to 6 a.u from the nucleus and the total charge of three shells was 17.998. These populations are in close agreement with the same-spin ELI-D shell populations from Hartee–Fock (HF) and highly correlated wavefunction calculations[29] as well as electron localization function (ELF) shell populations, obtained from Clementi–Roetti HF calculation.[30] For the nonrelativistic limit calculation the diagram rðaÞ for !D , equivalent to ELI-D for triplet-coupled electrons, looks at the chosen scale completely identical to the relativistic one, as expected for light element. The only essential difference is observed near the nucleus, where the nonrelativistic ELI-D demonstrates cusp with nonzero value at nuclear position, while the

‡

†

Strictly speaking, only when multiplied with the proper dimensionality factors.

It should be noted that thus constructed matrices are not exact but approximated even for closed-shell atoms, since they are evaluated from noninteracting Kohn–Sham states. Journal of Computational Chemistry 2014, 35, 565–585

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Figure 1. ELI from fully relativistic calculation for Ar atom (black). The inset at top left diagram shows the behavior close to nucleus for fully relativistic rðaÞ (black) as well as for nonrelativistic limit (blue) calculation. Vertical dashed lines mark shell boundaries given by minima of the !D .

relativistic one has its limit at zero value. As shown in Appendix C, this difference can be explained by different behavior of relativistic and nonrelativistic wavefunctions close to nucleus.[22] ELI-q for spatially symmetrized electron pairs !rðsÞ q also shows certain structure and its maxima are relatively close to the minrðaÞ ima of !D indicator, marked by vertical dashed lines. This could seem somewhat unexpected, recalling that ELI-q for spatially symmetrized electron pairs in nonrelativistic case is equivalent to ELI-q for singlet-coupled electrons, showing in general case shells as maxima and their boundaries as minima.[14] However, in this 4-component case spatially symmetrized electron pairs are not equivalent to singlet-coupled electrons. !rðsÞ valq ues are very close to one, that is, to the value obtained from nonrelativistic limit calculation and the largest deviations from this value are observed close to the nucleus. rðsÞ ELI-D for spin-symmetrized (triplet) pairs !D has shallow local minima near shell boundaries. Its values increase with the distance from the nucleus, which can be rationalized as decrease of triplet 572

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ontop density in the denominator of eq. (39). For the nonrelativistic wavefunctions, this ontop density is identically zero and visually, rðaÞ this increase of !D values can be imagined, as if electrons become more and more nonrelativistic with the ontop density running to zero as the distance from nucleus increases. For the nonrelativistic limit wavefunction, this indicator should turn to the infinity. ELI-q for spin-antisymmetrized (singlet) pairs !rðaÞ shows q only one clear maximum, shifted considerably from K/L shell rðaÞ boundary given by !D . Its diagram resembles that of !rðsÞ q and also has the values very close to one, which is the value for the nonrelativistic limit. cðaÞ Component-antisymmetrized pair ELI-D !D shows shallow rðaÞ minima near shell boundaries given by !D . In the nonrelativistic limit, there are no component-antisymmetrized pairs and in this case the indicator looses its meaning. ELI-q for component-symmetrized pairs !qcðsÞ shows shallow rðaÞ maxima close to the shell boundaries given by !D . Its values are very close to its nonrelativistic limit value 0.25, equal to WWW.CHEMISTRYVIEWS.COM

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rðaÞ

Figure 2. ELI from fully relativistic calculation for Rn atom (black). At top left diagram also shown !D rðaÞ Vertical dashed lines mark shell boundaries given by minima of the !D .

the one quarter of the value of spinless ELI-q for nonrelativistic single-determinantal wavefunction.[14] Rn atom Figure 2 shows all six indicators, evaluated for Rn atom from fully relativistic calculation. Additionally shown is ELI-D for spatially antisymmetrized electron pairs from nonrelativistic limit calculation. Other indicators are not shown for that limit, as for the single-determinantal case, they either show uniform distributions or become meaningless. Again, clear shell structure can be recognized only from ELI-D for spatially antisymmetrized electron pairs. The shell populations obtained from this partitioning (Table 1) are reasonably close to the integer populations, resulting from Aufbau principle and to the populations obtained from scalar-relativistic ZORA calculations. The largest deviation from ideal occupation number is observed for N-shell

indicator from nonrelativistic limit calculation (blue).

(more than 2 e) and is probably due to large ideal shell population itself (32 e). For comparison, the corresponding deviation of M-shell (18 e), obtained from ELF shell structure[30] can be close to one. rðaÞ In contrast to Ar, !D diagram for nonrelativistic limit calculation looks quantitatively different to the relativistic one here. Quantitative characteristics of shells for both fully relativistic and nonrelativistic limit calculations are given in Table 1. Well known relativistic contraction is observed for all shells and is most pronounced for outer shells. From the orbital-based analysis[6,31,32] it is known, that the s- and p-shells are usually contracted, while d- and f-shells can experience an expansion. In ELI the contributions from all the shells are summed for each one- and two-electron ingredients and the total effect is, therefore, the contraction of all the shells. Another interesting relativistic effect is that shells in the nonrelativistic limit calculation are much strongly pronounced (e.g., difference between shell radial maxima and intershell radial minima is larger) than Journal of Computational Chemistry 2014, 35, 565–585

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rðaÞ

rðaÞ

Table 1. Shell Structure of Rn atom from the topology of !D obtained from fully relativistic (rel), scalar-relativistic ZORA [15, 27] (scZORA) and nonrelativistic limit (nrl) calculations.

qK rK qL rL qM rM qN rN qO rO qP

rel

scZORA

nrl

2.238 0.0215 8.765 0.0822 18.288 0.2230 29.412 0.5923 18.595 1.4614 8.685

2.198 0.0212 8.799 0.0823 18.339 0.2237 29.349 0.5924 18.6189 1.4647 8.700

2.273 0.0260 9.177 0.0912 18.378 0.2350 29.268 0.6130 18.519 1.5342 8.388

Shell radii are in a.u.

in fully relativistic case. Different behaviour near nucleus of rðaÞ !D indicators obtained from relativistic and nonrelativistic calculations is also observed here. Other five ELI flavors also show certain features, which are, rðaÞ however, less pronounced, than for !D . For instance, rðsÞ cðaÞ rðsÞ !q ; !D , and !D mark shell boundaries with maxima or minima, but only for few inner shells. ELI-q for componentrðaÞ symmetrized electrons shows weak maxima close to the !D shell boundaries for all the shells, but the shell populations are too far from those expected from Aufbau principle.

Thus !D is the best choice for the analysis of the shell structures from fully relativistic calculations at singledeterminantal level. Complete tables with the atomic shell rðaÞ structure parameters evaluated from relativistic !D indicator for all the atoms of periods 4 to 7 from the fully relativistic and nonrelativistic limit calculations are given in the Appendix D, as well as shell structure parameters from scalar-relativistic ZORA calculations. Elements of period 4 Figure 3 summarizes some selected shell parameters for the two outermost shells of the elements of fourth period, obtained from 4-component fully relativistic, its nonrelativistic limit and scalar-relativistic ZORA calculations. One observes that for outer shells the difference between fully relativistic, nonrelativistic, and scalar-relativistic results is quite small. Well known trends in the shell occupations and the contraction of shell radii along the period are easily recognizable. The comparison with Clementi–Roetti HF results[30] shows a bit larger discrepancies. For the two inner shells the differences between fully relativistic, nonrelativistic, and scalar-relativistic calculations are even smaller in the absolute scale (Table 2). One can observe K;L K;L K;L that rnrl > rrel  rscZORA and qKnrl > qKrel > qKscZORA whereas L L L qnrl  qrel  qscZORA . Along the period the radii of the first two shells slightly contract monotonically. The population of the Kshell stays nearly constant along the period, whereas the population of L-shell slightly increases monotonically. Elements of period 5

Figure 3. Shell radii (in a.u.) for third (M-) shell and shell populations for third (M-) and fourth (N-) shells for the elements of fourth period from the topology of ELI-D, calculated from fully relativistic Dirac–Kohn–Sham LDA (black), its nonrelativistic limit (blue) and scZORA (red). [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

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Figure 4 summarizes some selected shell parameters for the two outermost shells of the elements of fifth period. In contrast to the previous period, there is a clear difference between relativistic and nonrelativistic results, concerning the appearance of the fourth (O-) shell. In the nonrelativistic calculation, the O-shell is revealed for all the elements, except for Pd (electron configuration ½Kr4d10 ), where it is empty and the outer boundary of fourth (N-) shell does not exist. In relativistic calculations, both fully- and scalar-relativistic, this shell disappears also for its neighbors from Ru to Ag. This disappearance at first sight seems to be a relativistic effect due to contraction of the outer s-shell and relativistic expansion of the penultimate shell containing d-orbitals.[6,31,32] However, similar effect is observed for the valence shell in the nonrelativistic results for late d-metals of sixth period (see below), which points out that there could also be other reasons for that. For scalar-relativistic ZORA calculations, the results can be sensitive to the basis set used. For instance, using TZ2P basis instead of QZ4P causes the O-shell to appear for Ag. It is, however, clearly an artifact, as it is very weakly pronounced, its inner radial boundary lies quite far away from nucleus (at 3.69 a.u.) and, therefore, its population is small ( rrel rscZORA . For most of the atoms of sixth period, the nonrelativistic shell populations are a bit larger than relativistic. For L-shell, the nonrelativistic populations increase monotonically along the period, while the relativistic show weak maximum in the middle. Elements of period 7 Figure 6 summarizes some selected shell parameters for the three outermost shells of the elements of seventh period. For these elements the difference between nonrelativistic and relativistic calculations is remarkable except for the populations of outermost Q-shell, where it is less pronounced. The difference between fully relativistic and scalar-relativistic ZORA results also becomes visible. For inner shells the differences between fully relativistic, nonrelativistic, and scalar-relativistic calculations are smaller in the absolute scale (Table 5). One can observe for the radii that K;L;M;N;O K;L;M;N;O K;L;M;N;O rnrl > rrel  rscZORA . For the populations of the L;M shells one sees that qKnrl > qKrel > qKscZORA , qL;M nrl > qscZORA L;M N N N > qrel , whereas qrel > qscZORA > qnrl . Figure 4. Shell radii (in a.u.) for fourth (N-) shell and shell populations for fourth (N-) and fifth (O-) shells for the elements of fifth period from the topology of ELI-D calculated from fully relativistic Dirac–Kohn–Sham LDA (black), its nonrelativistic limit (blue) and scZORA (red). [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.] K;L;M K;L;M absolute scale (Table 3). One can observe that rnrl > rrel K;L;M K;L K;L K;L M M M rscZORA ; qnrl > qrel qscZORA but qscZORA > qrel > qnrl . The radii of the inner shells also decrease monotonically as for the elements of fourth period. The populations of the Kshell stay nearly constant along the period, of L-shell slightly increase and of M-shell show weak maximum for Pd.

Elements of period 6 Figure 5 summarizes some selected shell parameters for the three outermost shells of the elements of sixth period. Usual trends in the occupations of the last three shells and penultimate shell radii contraction can be easily recognized. As for fourth period, for late d-metals the outermost shell does not show up. This holds for both relativistic and nonrelativistic results with the exception of Hg for which the nonrelativistic calculation shows weakly pronounced O-shell occupied by 2.1 e. The absence of the valence shells in nonrelativistic results for Pt and Au means thus that this shell disappearance is not necessarily caused by only relativistic effects. In general, relativistic and nonrelativistic results differ slightly for almost all elements. Fully relativistic results and scalar-relativistic ZORA results are very close. Scalar-relativistic ZORA results are sensitive to the basis set used. Reducing basis set level from QZ4P to TZ2P causes the valence shell of Tl atom to disappear in contrast with the fully relativistic result.

Figure 5. Shell radii (in a.u.) for fifth (O-) shell and shell populations for fourth (N-), fifth (O-) and sixth (P-) shells for the elements of sixth period from the topology of ELI-D, calculated from fully relativistic Dirac–Kohn– Sham LDA (black), its nonrelativistic limit (blue) and scZORA (red). [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

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rðaÞ

!D indicator has been used to evaluate quantitative atomic shell structure for all the atoms of periods 4–7 of the periodic table of elements from Dirac–Kohn–Sham LDA results. The results have been compared with the shell structures obtained from same level calculations in nonrelativistic limit as well as with the shell structures revealed by nonrelativistic ELI-D from scalarrelativistic ZORA Kohn–Sham results and ELF shell structure from HF Clementi–Roetti wavefunctions (for the elements of fourth period). Close agreement has been found between shell structures from fully relativistic and scalar-relativistic ZORA calculations for almost all elements. Minor differences between them have been observed almost only for the elements of seventh period. Nonrelativistic shell structures differ noticeably from relativistic ones for the elements of fifth to seventh periods. The most remarkable is the difference in the number of shells revealed for late transition metals of fifth to sixth periods.

Acknowledgment The author thanks Dr. M. Kohout and Dr. F. R. Wagner for helpful discussions.

Figure 6. Shell radii (in a.u.) for sixth (P-) shell and shell populations for fifth (O-), sixth (P-) and seventh (Q-) shells for the elements of seventh period from the topology of ELI-D, calculated from fully relativistic Dirac–Kohn–Sham LDA (black), its nonrelativistic limit (blue) and scZORA (red). [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

Conclusions For the fully relativistic many-electron 4-component wave functions six independent ELI flavors constructed from (anti)symmetrized over (1) space, (2) spin, and (3) component coordinate electron densities have been proposed. Their 2-component counterparts have also been introduced and analyzed. Indicators for fully relativistic wavefunctions have been tested on Ar and Rn Dirac–Kohn–Sham LDA results. All six indicators show on their radial dependence certain features pointing on the presence of atomic shells but only an indicator for spatially antirðaÞ symmetrized electron pairs !D is capable to deliver the shell structure at quantitative level with the shell populations similar ðtÞ to those from nonrelativistic ELI-D !D and !r2 D .

APPENDIX A: Derivation of 1- and 2-Electron Densities Densities for Space (anti)Symmetrized Projections Using eqs. (16) and (19) one obtains space (anti)symmetrized 2 e density matrix as: q2 ð~ r 01 ;~ r 02 ;~ r 1 ;~ r 2Þ  X X 1 r c0 r0 c0 r0 c r c r r 01 ;~ r 02 ;~ r 1 ;~ r 2Þ ð16P^12 Þq21 1 2 2 1 1 2 2 ð~ 5 2 c0i 5ci ;r0i 5ri c1 r1 c2 r2  1 X X  c1 r1 c2 r2 c1 r1 c2 r2 0 0 c1 r1 c2 r2 c1 r1 c2 r2 0 5 q ð~ r 1 ;~ r 2 ;~ r 1 ;~ r 2 Þ6q2 ð~ r 1 ;~ r 02 ;~ r 2 ;~ r 1Þ 2 c1 r1 c2 r2 2 h 1XXX crspq urc1 r1  ð~ r 01 Þusc2 r2  ð~ r 02 Þucp1 r1 ð~ r 1 Þucq2 r2 ð~ r 2Þ 5 2 c1 r1 c2 r2 rspq i r 01 Þusc2 r2  ð~ r 02 Þucp1 r1 ð~ r 2 Þucq2 r2 ð~ r 1Þ 6urc1 r1  ð~ rðs=aÞ

(55) The pair densities are then immediately obtained from final r 1 and ~ r 02 5~ r 2 . Consequent integraexpression by setting ~ r 01 5~ ~ tion over r 2 and rescaling yields one-electron densities:

 ð ð 2 1 XXX rðs=aÞ d~ r 2 q2 ð~ r 1 ;~ r 2 ;~ r 1 ;~ r 2 Þ5 crspq urc1 r1  ð~ r 1 Þucp1 r1 ð~ r 1 Þ d~ r 2 usc2 r2  ð~ r 2 Þucq2 r2 ð~ r 2 Þ6urc1 r1  ð~ r 1 Þucq2 r2 ð~ r 1Þ N21 N21 c1 r1 c2 r2 rspq   ð X X 1 XX d~ r 2 usc2 r2  ð~ uc1 r1  ð~ r 2 Þucp1 r1 ð~ r 2 Þ5 c r 1 Þucp1 r1 ð~ r 1Þ Scsq2 r2 ;c2 r2 6 urc1 r1  ð~ r 1 Þucq2 r2 ð~ r 1 ÞScsp2 r2 ;c1 r1 N21 c1 r1 rspq rspq r c2 r2 c2 r2 |fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl}

qrðs=aÞ ð~ r 1 Þ5

5dsq

" # XX 1 X X c1 r1  c1 r1 c2 r2 ;c1 r1 c1 r1  c2 r2 ~ ~ ~ ~ 5 c u ðr 1 Þup ðr 1 Þ6 crspq Ssp ur ðr 1 Þuq ðr 1 Þ N21 c1 r1 rsp rsps r c2 r2 rspq (56)

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XXX rðaÞ rÞ5r~2r 2 q2 ð~ r 1 ;~ r 2 Þj~r 5~r 1 5~r 2 5 crspq grðaÞ ð~

rðaÞ

The pair density for spatially antisymmetrized electrons q2 rðaÞ ð~ r 1 ;~ r 2 Þ5q2 ð~ r 1 ;~ r 2 ;~ r 1 ;~ r 2 Þ is necessarily zero at electron coalescence point (~ r 1 5~ r 2 ). As shown in the section “Indicators derived from (anti)symmetrized spatial coordinate projections”, the first leading term in Taylor expansion is of second order, that is, spherically averaged curvature: rðaÞ r~r 2 q2 ð~ r 1 ;~ r 2 Þ5

1XXX 2 c1 r1

crspq ½urc1 r1  ð~ r 1 Þrucs 2 r2  ð~ r 2Þ

c2 r2 rspq

ucp1 r1 ð~ r 1 Þucq2 r2 ð~ r 2 Þ1urc1 r1  ð~ r 1 Þusc2 r2  ð~ r 2 Þucp1 r1 ð~ r 1 Þrucq2 r2 ð~ r 2 Þ (57) c1 r1  c2 r2  c1 r1 c2 r2 r 1 Þrus ð~ r 2 Þup ð~ r 2 Þuq ð~ r 1Þ 2ur ð~ 2urc1 r1  ð~ r 1 Þusc2 r2  ð~ r 2 Þrucp1 r1 ð~ r 2 Þucq2 r2 ð~ r 1 Þ rðaÞ r~2r 2 q2 ð~ r 1;~ r 2 Þ5

1XXX c ½uc1 r1  ð~ r 1 Þr2 usc2 r2  ð~ r 2Þ 2 c1 r1 c2 r2 rspq rspq r

c1 r1 c2 r2 rspq

(59)

½urc1 r1  ð~ rÞrusc2 r2  ð~ rÞucp1 r1 ð~ rÞrucq2 r2 ð~ rÞ 2urc1 r1  ð~ rÞrusc2 r2  ð~ rÞrucp1 r1 ð~ rÞucq2 r2 ð~ rÞ

as terms containing second derivatives mutually cancel. In 1-determinantal case, when crspq 5 12 nr ns ðdrp dsq 2drq dsp Þ one obtains for ontop density: rðsÞ

q2 ð~ r;~ rÞ5

1XXX nr ns ðdrp dsq 2drq dsp Þ 4 c1 r1 c2 r2 rspq

rÞusc2 r2  ð~ rÞucp1 r1 ð~ rÞucq2 r2 ð~ rÞ1urc1 r1  ð~ rÞucs 2 r2  ð~ rÞucp1 r1 ð~ rÞucq2 r2 ð~ rÞ ½urc1 r1  ð~  X 1XX X 5 nr ucr 1 r1  ð~ rÞucr 1 r1 ð~ rÞ ns ucs 2 r2  ð~ rÞucs 2 r2 ð~ rÞ 4 c1 r1 c2 r2 r s X X nr ucr 1 r1  ð~ rÞucr 1 r1 ð~ rÞ ns ucs 2 r2  ð~ rÞucs 2 r2 ð~ rÞ 1 r s X X nr ucr 1 r1  ð~ rÞucr 2 r2 ð~ rÞ ns ucs 2 r2  ð~ rÞucs 1 r1 ð~ rÞ 2

ucp1 r1 ð~ r 1 Þucq2 r2 ð~ r 2 Þ12urc1 r1  ð~ r 1 Þrusc2 r2  ð~ r 2 Þucp1 r1 ð~ r 1 Þrucq2 r2 ð~ r 2Þ r 1 Þusc2 r2  ð~ r 2 Þucp1 r1 ð~ r 1 Þr2 ucq2 r2 ð~ r 2Þ 1urc1 r1  ð~

r

s

r

s

 X X 2 nr ucr 1 r1  ð~ rÞucr 2 r2 ð~ rÞ ns ucs 2 r2  ð~ rÞucs 1 r1 ð~ rÞ

r 1 Þr2 usc2 r2  ð~ r 2 Þucp1 r1 ð~ r 2 Þucq2 r2 ð~ r 1Þ 2urc1 r1  ð~

" # XX 1 2 c1 r1 ;c2 r2 c2 r2 ;c1 r1 5 q ð~ rÞ2 q ð~ rÞq ð~ rÞ 2 c1 r1 c2 r2

22urc1 r1  ð~ r 1 Þrusc2 r2  ð~ r 2 Þrucp1 r1 ð~ r 2 Þucq2 r2 ð~ r 1Þ r 1 Þusc2 r2  ð~ r 2 Þr2 ucp1 r1 ð~ r 2 Þucq2 r2 ð~ r 1 Þ 2urc1 r1  ð~ (58)

(60) c1 r1 ;c2 r2

ð~ rÞ is the electron density tensor [eq. (27)]. Onewhere q electron densities are:

which at coalescence point becomes:

 X X XX 1 nr ns ðdrp dss 2drs dsp Þucr 1 r1  ð~ r 1 Þucp1 r1 ð~ r 1 Þ6 nr ns ðdrp dsq 2drq dsp Þucr 1 r1  ð~ r 1 Þucq2 r2 ð~ r 1 ÞScsp2 r2 ;c1 r1 2ðN21Þ c1 r1 rsp c2 r2 rspq X XX X X 1 nr ns drp dss urc1 r1  ð~ r 1 Þucp1 r1 ð~ r 1 Þ2 nr ns drs dsp urc1 r1  ð~ r 1 Þucp1 r1 ð~ r 1 Þ6 nr ns drp dsq urc1 r1  ð~ r 1 Þucq2 r2 ð~ r 1 ÞScsp2 r2 ;c1 r1 5 2ðN21Þ c1 r1 rsp rsp c2 r2 rspq "  X XX XX XX 1 c1 r1  c2 r2 c2 r2 ;c1 r1 5 7 nr ns drq dsp ur ð~ r 1 Þuq ð~ r 1 ÞSsp dss ns nr jucr 1 r1 ð~ r 1 Þj2 2 n2r jucr 1 r1 ð~ r 1 Þj2 2ðN21Þ c2 r2 rspq s c1 r1 r c1 r1 r |fflfflfflfflffl{zfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} 5N 5qð~ r 1Þ # XXX XXX X c1 r1  c2 r2 c2 r2 ;c1 r1 c1 r1  c2 r2 c2 r2 ;c1 r1 6 nr ns ur ð~ r 1 Þus ð~ r 1 ÞSsr 7 nr ur ð~ r 1 Þur ð~ r 1Þ ns Sss qrðs=aÞ ð~ r 1 Þ5

c1 r1 c2 r2

c1 r1 c2 r2

rs

r |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

s

5qc1 r1 ;c2 r2 ð~ r 1Þ

5

  XX X X X X 1 Nqð~ r 1 Þ2 n2r jucr 1 r1 ð~ r 1 Þj2 7 ns Scss2 r2 ;c1 r1 qc1 r1 ;c2 r2 ð~ r 1 Þ2 nr ns Scsr2 r2 ;c1 r1 urc1 r1  ð~ r 1 Þucs 2 r2 ð~ r 1Þ 2ðN21Þ c1 r1 r c1 r1 c2 r2 s rs (61)

The curvature of the spatially antisymmetrized pair density at coalescence is then:

grðaÞ ð~ rÞ5

XX 1X nr ns ½urc1 r1  ð~ rÞrucs 2 r2  ð~ rÞucr 1 r1 ð~ rÞrucs 2 r2 ð~ rÞ2ucr 1 r1  ð~ rÞrusc2 r2  ð~ rÞrucr 1 r1 ð~ rÞucs 2 r2 ð~ rÞ 2 rs c1 r1 c2 r2

rÞrucs 2 r2  ð~ rÞucs 1 r1 ð~ rÞrucr 2 r2 ð~ rÞ1urc1 r1  ð~ rÞrucs 2 r2  ð~ rÞrucs 1 r1 ð~ rÞucr 2 r2 ð~ rÞ 2urc1 r1  ð~  X X X 1XX X 5 nr jucr 1 r1 ð~ rÞj2 ns jrucs 2 r2 ð~ rÞj2 2 nr ucr 1 r1  ð~ rÞrucr 1 r1 ð~ rÞ ns rusc2 r2  ð~ rÞucs 2 r2 ð~ rÞ 2 c1 r1 c2 r2 r s r s 2

X

 X X X nr urc1 r1  ð~ rÞrucr 2 r2 ð~ rÞ ns rusc2 r2  ð~ rÞucs 1 r1 ð~ rÞ1 nr urc1 r1  ð~ rÞucr 2 r2 ð~ rÞ ns rusc2 r2  ð~ rÞrucs 1 r1 ð~ rÞ

(62)

s r s r XXX X XX 1 XX 2 2 c1 r1 c2 r2 c1 r1  c2 r2 nr jur ð~ rÞj ns jrus ð~ rÞj 1 nr ur ð~ rÞur ð~ rÞ ns rusc2 r2  ð~ rÞrucs 1 r1 ð~ rÞ 5 2 c1 r1 r c2 r2 s c1 r1 c2 r2 r s

 XX XXX X XX nr urc1 r1  ð~ rÞrucr 1 r1 ð~ rÞ ns rusc2 r2  ð~ rÞucs 2 r2 ð~ rÞ2 nr ucr 1 r1  ð~ rÞrucr 2 r2 ð~ rÞ ns rusc2 r2  ð~ rÞucs 1 r1 ð~ rÞ 2 c1 r1

r

c2 r2

s

c1 r1 c2 r2

r

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Introducing following one-electron quantities: 1X sc1 r1 ;c2 r2 ð~ rÞ5 nr rurc1 r1  ð~ rÞrucr 2 r2 ð~ rÞ 2 r c1 r1 ;c2 r2 1 ~ j ð~ rÞ5 2i

X

  nr urc1 r1  ð~ rÞrucr 2 r2 ð~ rÞ2rurc1 r1  ð~ rÞucr 2 r2 ð~ rÞ

of one-electron spinors. As discussed in the section “Indicators for 2-component wavefunctions”, only in nonrelativistic limit they turn to the expressions formally equivalent to nonrelativistic kinetic energy and current densities.

r

(63) one can compactly write the curvature for 1-determinantal case as:   1 rqð~ rÞ ~ rqð~ rÞ ~ 1ijð~ rÞÞð 2ijð~ rÞ grðaÞ ð~ rÞ5qð~ rÞsð~ rÞ2 2 2 2 " X 1 qc1 r1 ;c2 r2 ð~ rÞsc2 r2 ;c1 r1 ð~ rÞ

Densities for spin (anti)symmetrized projections Using projection operators based on spin permutation to the two-electron density matrix, one obtains: rða=sÞ

(64)

q

rða=sÞ

5

1XXX c ½uc1 r1  ð~ r 1 Þusc2 r2  ð~ r 2 Þucp1 r1 ð~ r 1 Þucq2 r2 ð~ r 2Þ 2 c1 r1 c2 r2 rspq rspq r 7ucr 1 r1  ð~ r 1 Þusc2 r2  ð~ r 2 Þucp1 r2 ð~ r 1 Þucq2 r1 ð~ r 2 Þ

(65) Integration over ~ r 2 and multiplication by

2 N21

5

5dsq

XX 1 X X c uc1 r1  ð~ r 1 Þucp1 r1 ð~ r 1 Þ7 crspq Scsq2 r2 ;c2 r1 urc1 r1  ð~ r 1 Þucp1 r2 ð~ r 1Þ N21 c1 r1 rsp rsps r c2 r2 rspq

Inserting 1-determinantal density matrix crspq 5 12 nr ns ðdrp dsq 2 drq dsp Þ into eq. (65) one obtains for ontop pair densities (~ r5~ r 1 5~ r 2 ): rða=sÞ

q2

qrða=sÞ ð~ r 1 Þ5

ð~ rÞ 7

XX 1X nr ns ðdrp dsq 2drq dsp Þ rÞusc2 r2  ð~ rÞucp1 r1 ð~ rÞucq2 r2 ð~ rÞ urc1 r1  ð~ 4 rspq c1 r1 c2 r2

# nr ns ðdrp dsq 2drq dsp ÞScsq2 r2 ;c2 r1 urc1 r1  ð~ r 1 Þucp1 r2 ð~ r 1Þ

c2 r2 rspq

5

X X X 1 N nr jucr 1 r1 ð~ r 1 Þj2 2 n2r jucr 1 r1 ð~ r 1 Þj2 2ðN21Þ c1 r1 r r 7

X X c 2 r2

rÞucs 2 r2  ð~ rÞucr 1 r2 ð~ rÞucs 2 r1 ð~ rÞ 7urc1 r1  ð~

2

s

X

ns Scss2 r2 ;c2 r1

X

nr urc1 r1  ð~ r 1 Þucr 1 r2 ð~ r 1Þ

r

nr ns Scsr2 r2 ;c2 r1 urc1 r1  ð~ r 1 Þucs 1 r2 ð~ r 1Þ

5

r 1 Þusc2 r2  ð~ rÞucs 1 r2 ð~ r 1 Þucr 2 r1 ð~ rÞ 6urc1 r1  ð~

7

X X X c1 r1 c2 r2

#

s

 XX 1 Nqð~ r 1 Þ2 n2r jucr 1 r1 ð~ r 1 Þj2 2ðN21Þ c1 r1 r

ns Scss2 r2 ;c2 r1 qc1 r1 c1 r2 ð~ r 1 Þ2

X

nr ns Scsr2 r2 c2 r1 urc1 r1  ð~ r 1 Þucs 1 r2 ð~ r 1Þ



rs

(68)

c1 r1 c2 r2

(67)

Journal of Computational Chemistry 2014, 35, 565–585



rs

rÞucs 2 r2  ð~ rÞucs 1 r1 ð~ rÞucr 2 r2 ð~ rÞ 2urc1 r1  ð~

XX ðqc1 r1 c1 r2 ð~ rÞqc2 r2 c2 r1 ð~ rÞ2qc1 r1 c2 r1 ð~ rÞqc2 r2 c1 r2 ð~ rÞÞ 7

(66)

" X X 1 nr ns ðdrp dss 2drs dsp Þurc1 r1  ð~ r 1 Þucp1 r1 ð~ r 1Þ 2ðN21Þ c1 r1 rsp

XX

1XXX 5 nr ns ½urc1 r1  ð~ rÞucs 2 r2  ð~ rÞucr 1 r1 ð~ rÞucs 2 r2 ð~ rÞ 4 c1 r1 c2 r2 rs

" X X c1 r1 c2 r2 1 2 rÞ2 q ð~ rÞqc2 r2 c1 r1 ð~ rÞ 5 q ð~ 4 c1 r1 c2 r2

#

and from eq. (66):

 rÞusc2 r2  ð~ rÞucp1 r2 ð~ rÞucq2 r1 ð~ rÞ 7urc1 r1  ð~

578

yields:

" # ð X X X 2 1 X rða=sÞ c1 r1  c1 r1 c2 r2 c2 r2 c1 r1  c1 r2 c2 r2 ;c2 r1 d~ r 2 q2 ur ð~ ð~ r 1 Þ5 ð~ r 1 ;~ r 2 Þ5 c r 1 Þup ð~ r 1Þ Ssq 7 ur ð~ r 1 Þup ð~ r 1 ÞSsq N21 N21 rspq rspq c1 r1 c2 r2 c2 r2 |fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl} "

5

c0i 5ci ;r0i 5ri

1 X X c1 r1 c2 r2 c1 r1 c2 r2 5 ½q ð~ r 1 ;~ r 2 ;~ r 1 ;~ r 2 Þ7qc21 r1 c2 r2 c1 r2 c2 r1 ð~ r 1 ;~ r 2 ;~ r 1 ;~ r 2 Þ 2 c1 r1 c2 r2 2

c1 r1 ;c2 r2

j are labeled here Although quantities sc1 r1 ;c2 r2 and ~ like nonrelativistic kinetic energy s and current density ~ j, they should not be interpreted like those, as in 4-component case these quantities are defined in completely different way.[6] Rather one should consider them as sums of mixed derivatives

ð~ r 1 ;~ r 2 ;~ r 1 ;~ r 2Þ

# XX 1 r c01 r01 c02 r02 c1 r1 c2 r2 ^ ð17P 12 Þq2 5 ð~ r 1 ;~ r 2 ;~ r 1 ;~ r 2Þ 2 c1 r1 c2 r2

c1 r1 ;c2 r2

  c2 r2 ;c1 r1 # 1 rqc1 r1 ;c2 r2 ð~ rÞ ~ c1 r1 ;c2 r2 rq ð~ rÞ ~ c2 r2 ;c1 r1 1ij 2ij ð~ rÞ ð~ rÞ 2 2 2 2

q2

"

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Densities for large and small component (anti)symmetrized projections The derivation is completely analogous to that given in the previous section but instead of permuting spin indices r1 and r2 , one permutes component indices c1 and c2 . The final results for the pair density from general CI ansatz are:

either a or b component necessarily vanishes and thus any state r, s, p, or q has certain spin index. Thus, for pair densities one obtains from eq. (48): ðs=tÞ

q2

ð~ r 1 ;~ r 2 Þ5

 X crspq urr1  ð~ r 1 Þusr2  ð~ r 2 Þurp1 ð~ r 2 Þurq2 ð~ r 1Þ 6

1XXX cðs=aÞ q2 ð~ r 1 ;~ r 2 Þ5 c ½uc1 r1  ð~ r 1 Þusc2 r2  ð~ r 2 Þucp1 r1 ð~ r 1 Þucq2 r2 ð~ r 2Þ 2 c1 r1 c2 r2 rspq rspq r r 1 Þucs 2 r2  ð~ r 2 Þucp2 r1 ð~ r 1 Þucq1 r2 ð~ r 2 Þ 6ucr 1 r1  ð~

rspq

5 (69)

Using identity crspq 52crsqp which follows from the anticommutation rule for annihilation operators, one can rewrite ontop density as: 1XXX cðs=aÞ q2 ð~ r;~ rÞ5 c ½uc1 r1  ð~ rÞusc2 r2  ð~ rÞucp1 r1 ð~ rÞucq2 r2 ð~ rÞ 2 c1 r1 c2 r2 rspq rspq r rÞusc2 r2  ð~ rÞucp1 r2 ð~ rÞucq2 r1 ð~ rÞ 7urc1 r1  ð~

which is identical to the ontop part of eq. (65). The one-electron density is obtained by integration of pair 2 density over ~ r 2 and multiplying by N21 : ð 2 cðs=aÞ d~ r 2 q2 ð~ r 1 ;~ r 2Þ N21 " X X 1 X urc1 r1  ð~ 5 c r 1 Þucp1 r1 ð~ r 1Þ Scsq2 r2 c2 r2 N21 rspq rspq c1 r1 c2 r2 |fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl} qcðs=aÞ ð~ r 1 Þ5

#

 1XX X c ur1  ð~ r 1 Þusr2  ð~ r 2 Þurp1 ð~ r 1 Þurq2 ð~ r 2Þ 2 r1 r2 rspq rspq r  X 7 crsqp urr1  ð~ r 1 Þusr2  ð~ r 2 Þurq2 ð~ r 1 Þurp1 ð~ r 2Þ

1XX 5 2 r1 r2 7

rspq



X rr1 sr2 pr1 qr2

X

rr1 sr2 pr1 qr2

(70)

X 6 urc1 r1  ð~ r 1 Þucp2 r1 ð~ r 1 ÞScsq2 r2 ;c1 r2

 1XX X c ur1  ð~ r 1 Þusr2  ð~ r 2 Þurp1 ð~ r 1 Þurq2 ð~ r 2Þ 2 r1 r2 rspq rspq r

5

crr1 sr2 pr1 qr2 urr ð~ r 1 Þusr ð~ r 2 Þupr1 ð~ r 1 Þuqr2 ð~ r 2Þ 1

2

 crr1 sr2 qr2 pr1 urr ð~ r 1 Þusr ð~ r 2 Þuqr2 ð~ r 1 Þupr1 ð~ r 2Þ 1

2

1 X X r1 r2 r1 r2 ½q ð~ r 1 ;~ r 2 Þ7qr21 r2 r2 r1 ð~ r 1 ;~ r 2 Þ 2 r1 r2 2 (73)

For singlet pairs same-spin (r1 5r2 ) parts cancel mutually and one is left just with opposite-spin parts: 1 ðsÞ q2 ð~ r 1 ;~ r 2 Þ5 ½qabab ð~ r 1 ;~ r 2 Þ2qabba ð~ r 1 ;~ r 2 Þ1qbaba ð~ r 1 ;~ r 2 Þ2qbaab ð~ r 1 ;~ r 2 Þ 2 2 2 2 2

(74)

5dsq

(71)

c2 r2

" 1 X X 5 c uc1 r1  ð~ r 1 Þucp1 r1 ð~ r 1Þ N21 c1 r1 rsp rsps r # XX c2 r2 ;c1 r2 c1 r1  c2 r1 6 crspq Ssq ur ð~ r 1 Þup ð~ r 1Þ

which is equivalent to eq. (48) in Ref. [13], assuming L€ owdin normalization of the density matrices.[19] In case of triplet-coupled pairs, both same-spin and opposite-spin contributions will be present in the pair density and one obtains: ðtÞ q2 ð~ r 1 ;~ r 2 Þ5qaaaa ð~ r 1 ;~ r 2 Þ1qbbbb ð~ r 1 ;~ r 2Þ 2 2

1 1 ½qabab ð~ r 1 ;~ r 2 Þ1qabba ð~ r 1 ;~ r 2 Þ1qbaba ð~ r 1 ;~ r 2 Þ1qbaab ð~ r 1 ;~ r 2 Þ 2 2 2 2 2 (75)

c2 r2 rspq

which is equivalent to eq. (49) in Ref. [13], assuming L€ owdin normalization of the density matrices.

which for 1-determinantal ansatz becomes: ð 2 cðs=aÞ d~ r 2 q2 ð~ r 1 ;~ r 2Þ N21  XX n2r jucr 1 r1 ð~ r 1 Þj2 Nqð~ r 1 Þ2

qcðs=aÞ ð~ r 1 Þ5 5

1 2ðN21Þ

c1 r1

APPENDIX C: Nuclear Cusp Condition for ELI-D

r

 X X X X 6 nr Scrr2 r2 ;c1 r2 qc1 r1 c2 r1 ð~ r 1 Þ2 nr ns Scsr2 r2 c1 r2 ucr 1 r1  ð~ r 1 Þucs 2 r1 ð~ r 1Þ c1 r1 c2 r2

r

General ansatz for the radial dependence of the electronic wavefunction near the nucleus is[22]:

rs

(72)

APPENDIX B: Pair Densities for Singlet/Triplet Pairs in Spin-Collinear Case In the spin-collinear case each one-electron spinor can be chosen as an eigenfunction of S^z operator which means, that

W  r m ½11ar1br 2 1Oðr 3 Þ

(76)

here r is the distance between an electron and nucleus, r ! 0. In the nonrelativistic case m50 and in relativistic case m  2c22 .[22] In the nonrelativistic case, then W  11ar1br2 1Oðr 3 Þ and for the electron density one obtains: Journal of Computational Chemistry 2014, 35, 565–585

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q  W2  ½11ar1br 2 1Oðr 3 Þ2 5112ar1Oðr 2 Þ

(77)

The curvature of Fermi hole near nucleus position is ( 0 means derivative over r):

W0 mr m21 ½11ar1Oðr 2 Þ1r m ½a12br1Oðr 2 Þ ar 1 5mr m21 ½11ar1 1Oðr 2 Þr 21 ½11að11 Þr1Oðr 2 Þ1OðmÞ m m (81)

0

gW2 W 2 5½112ar1Oðr 2 Þ½a12br1Oðr 2 Þ2 5½112ar1Oðr2 Þ ½a2 12abr1Oðr 2 Þ5a2 12a3 r12abr1Oðr 2 Þ  112ða1b=aÞr1Oðr 2 Þ

(78) and then for ELI-D one obtains: !qg23=8  ½112ar1Oðr 2 Þ½112ða1b=aÞr1Oðr 2 Þ23=8 5½112ar1Oðr 2 Þ½12 5112ar2

3ða2 1bÞr 1Oðr 2 Þ 4a

(79)

3ða2 1bÞr 5a2 23b 1Oðr 2 Þ511 r1Oðr 2 Þ 4a 4a

(80)

The derivative of the wavefunction over r is:

1 0 gW2 W 2  r 2m ½112ar1Oðr 2 Þr 22 ½11að11 Þr1Oðr 2 Þ2 m 1 2m22 2 5r ½112ar1Oðr Þ½112að11 Þr1Oðr 2 Þ m 1 2 22 5r ½112að21 Þr1Oðr Þ1OðmÞ m

(82)

and then for ELI-D one obtains:

which means, that ELI-D should demonstrate Kato-like cusp near the nucleus, what indeed is observed (cf. inset on Fig. 1). In the relativistic case m 6¼ 0 and then for the electron density one has: q  W2 5r 2m ½112ar1Oðr 2 Þ

The curvature of spatially antisymmetrized pair density at electron coalescence near nucleus position is then:

1 !qg23=8  r 2m ½112ar1Oðr 2 Þ½r 22 ð112að21 Þr1Oðr 2 ÞÞ23=8 m 3a 1 2m13=4 2 5r ½112ar1Oðr Þ½12 ð21 Þr1Oðr 2 Þ 4 m 2am23a 2 3=4 5r ½11 r1Oðr Þ1OðmÞ 4m (83) which means, that ELI-D for fully relativistic case should behave near nucleus as power function, what indeed is observed (cf. inset on Fig. 1).

APPENDIX D: ELI-D Shell Radii and Electron Numbers for the Elements of 4–7 Periods of Periodic Table Table 2. ELI-D shell structure parameters for the atoms of fourth period from fully relativistic (top line), scalar-relativistic ZORA (middle line), and nonrelativistic limit (bottom line) calculations.

K

Ca

Sc

Ti

V

Cr

Mn

Fe

Co

580

qK

rK

qL

rL

qM

rM

2.210 2.210 2.212 2.211 2.212 2.214 2.213 2.214 2.215 2.214 2.215 2.217 2.215 2.217 2.219 2.217 2.219 2.222 2.218 2.219 2.222 2.220 2.220 2.223 2.220 2.221 2.225

0.1342 0.1342 0.1351 0.1262 0.1261 0.1271 0.1192 0.1191 0.1202 0.1129 0.1128 0.1139 0.1072 0.1072 0.1084 0.1022 0.1022 0.1034 0.0975 0.0974 0.0987 0.0933 0.0932 0.0945 0.0893 0.0892 0.0906

7.865 7.863 7.864 7.862 7.860 7.860 7.887 7.889 7.887 7.922 7.926 7.922 7.964 7.970 7.965 8.027 8.033 8.029 8.061 8.070 8.064 8.113 8.121 8.120 8.168 8.175 8.177

0.6696 0.6692 0.6716 0.6148 0.6145 0.6168 0.5699 0.5703 0.5721 0.5311 0.5316 0.5333 0.4970 0.4979 0.4994 0.4681 0.4689 0.4706 0.4403 0.4412 0.4428 0.4164 0.4171 0.4190 0.3948 0.3954 0.3976

8.012 8.022 8.010 8.009 8.009 8.007 8.729 8.725 8.736 9.549 9.541 9.561 10.416 10.403 10.433 12.043 12.068 12.069 12.218 12.191 12.242 13.134 13.109 13.160 14.058 14.045 14.085

3.2072 3.2301 3.2125 2.4962 2.4950 2.5025 2.3086 2.3079 2.3180 2.1806 2.1787 2.1922 2.0828 2.0805 2.0962 2.2719 2.2969 2.2907 1.9345 1.9269 1.9503 1.8732 1.8661 1.8900 1.8184 1.8162 1.8362

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qN 0.897 0.889 0.898 1.914 1.915 1.914 2.169 2.170 2.159 2.314 2.317 2.298 2.404 2.410 2.382 1.712 1.670 1.679 2.503 2.520 2.471 2.533 2.550 2.497 2.554 2.558 2.512 (Continued)

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Table 2. (Continued)

Ni

Cu

Zn

Ga

Ge

As

Se

Br

Kr

qK

rK

qL

rL

qM

rM

qN

2.220 2.222 2.226 2.222 2.223 2.227 2.222 2.224 2.228 2.225 2.223 2.230 2.226 2.224 2.232 2.225 2.224 2.232 2.225 2.225 2.232 2.225 2.225 2.233 2.225 2.226 2.234

0.0857 0.0856 0.0871 0.0824 0.0823 0.0838 0.0793 0.0792 0.0806 0.0764 0.0763 0.0779 0.0737 0.0735 0.0752 0.0711 0.0710 0.0727 0.0687 0.0686 0.0703 0.0664 0.0663 0.0681 0.0642 0.0641 0.0660

8.223 8.229 8.235 8.292 8.294 8.310 8.333 8.333 8.353 8.370 8.369 8.394 8.402 8.401 8.428 8.432 8.427 8.459 8.457 8.451 8.488 8.479 8.472 8.511 8.498 8.491 8.532

0.3753 0.3757 0.3781 0.3582 0.3583 0.3611 0.3412 0.3412 0.3442 0.3256 0.3254 0.3287 0.3111 0.3110 0.3144 0.2978 0.2976 0.3011 0.2854 0.2852 0.2888 0.2740 0.2737 0.2774 0.2633 0.2631 0.2668

14.988 14.976 15.017 16.984 17.061 16.991 16.864 16.869 16.892 17.008 17.037 17.023 17.058 17.110 17.065 17.069 17.105 17.070 17.066 17.096 17.063 17.058 17.080 17.051 17.049 17.056 17.039

1.7690 1.7671 1.7878 2.0963 2.1485 2.1125 1.6831 1.6853 1.7036 1.5284 1.5375 1.5451 1.3856 1.3989 1.4001 1.2647 1.2718 1.2768 1.1628 1.1675 1.1733 1.0765 1.0790 1.0860 1.0027 1.0028 1.0115

2.568 2.573 2.522 1.502 1.422 1.472 2.581 2.574 2.527 3.396 3.368 3.352 4.315 4.265 4.275 5.274 5.244 5.238 6.252 6.228 6.218 7.238 7.223 7.205 8.228 8.227 8.195

Shell radii are in a.u.

Table 3. ELI-D shell structure parameters for the atoms of fifth period from fully relativistic (top line), scalar-relativistic ZORA (middle line), and nonrelativistic limit (bottom line), calculations.

Rb

Sr

Y

Zr

Nb

Mo

Tc

Ru

Rh

Pd

qK

rK

qL

rL

qM

rM

qN

rN

2.232 2.226 2.238 2.226 2.226 2.240 2.228 2.227 2.242 2.229 2.227 2.240 2.232 2.227 2.243 2.229 2.228 2.244 2.230 2.228 2.243 2.231 2.228 2.242 2.230 2.228 2.246 2.229 2.228 2.243

0.0624 0.0621 0.0641 0.0603 0.0602 0.0623 0.0585 0.0584 0.0605 0.0568 0.0567 0.0588 0.0553 0.0551 0.0573 0.0537 0.0536 0.0558 0.0522 0.0521 0.0544 0.0509 0.0507 0.0530 0.0495 0.0494 0.0518 0.0482 0.0481 0.0505

8.510 8.506 8.550 8.529 8.521 8.564 8.545 8.543 8.584 8.561 8.561 8.607 8.578 8.583 8.629 8.601 8.599 8.651 8.615 8.614 8.670 8.634 8.635 8.697 8.653 8.653 8.717 8.672 8.680 8.746

0.2534 0.2531 0.2570 0.2441 0.2439 0.2478 0.2355 0.2354 0.2393 0.2275 0.2274 0.2314 0.2200 0.2199 0.2240 0.2130 0.2129 0.2170 0.2063 0.2062 0.2104 0.2001 0.2000 0.2043 0.1942 0.1941 0.1985 0.1886 0.1887 0.1930

17.030 17.042 17.019 17.014 17.020 17.000 17.022 17.027 17.005 17.035 17.043 17.015 17.068 17.076 17.041 17.078 17.088 17.048 17.072 17.085 17.039 17.100 17.110 17.060 17.110 17.118 17.066 17.134 17.126 17.082

0.9379 0.9382 0.9466 0.8816 0.8815 0.8901 0.8355 0.8357 0.8440 0.7946 0.7950 0.8029 0.7597 0.7603 0.7678 0.7254 0.7260 0.7333 0.6921 0.6929 0.7000 0.6647 0.6653 0.6723 0.6377 0.6382 0.6452 0.6138 0.6138 0.6211

8.351 8.349 8.311 8.332 8.334 8.295 8.877 8.878 8.875 9.592 9.594 9.621 11.217 11.336 11.299 12.418 12.464 12.481 12.354 12.323 12.443 16.036 16.027 14.948 17.008 17.001 16.137 17.966 17.966 17.930

3.6144 3.6203 3.6379 2.8567 2.8571 2.8840 2.6242 2.6295 2.6607 2.4785 2.4867 2.5231 2.6821 2.7745 2.7546 2.7562 2.7983 2.8204 2.2844 2.2795 2.3421 – – 3.0610 – – 3.1944 – – –

qO 0.859 0.857 0.863 1.893 1.892 1.893 2.326 2.321 2.291 2.582 2.573 2.516 1.903 1.777 1.786 1.674 1.619 1.576 2.731 2.749 2.605 – – 1.052 – – 0.835 – – – (Continued)

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Table 3. (Continued)

Ag

Cd

In

Sn

Sb

Te

I

Xe

qK

rK

qL

rL

qM

rM

qN

rN

qO

2.230 2.227 2.247 2.230 2.228 2.243 2.239 2.227 2.246 2.230 2.227 2.251 2.233 2.227 2.250 2.227 2.227 2.249 2.230 2.227 2.248 2.225 2.228 2.245

0.0470 0.0469 0.0494 0.0459 0.0457 0.0482 0.0449 0.0446 0.0472 0.0437 0.0435 0.0462 0.0427 0.0425 0.0452 0.0416 0.0415 0.0443 0.0407 0.0405 0.0433 0.0397 0.0396 0.0424

8.688 8.696 8.762 8.700 8.708 8.786 8.706 8.721 8.801 8.727 8.732 8.814 8.733 8.743 8.832 8.749 8.754 8.847 8.753 8.762 8.862 8.763 8.768 8.877

0.1832 0.1833 0.1878 0.1781 0.1782 0.1828 0.1733 0.1734 0.1781 0.1687 0.1688 0.1736 0.1643 0.1644 0.1693 0.1601 0.1602 0.1651 0.1561 0.1561 0.1612 0.1522 0.1523 0.1575

17.131 17.125 17.076 17.129 17.122 17.068 17.126 17.119 17.063 17.122 17.116 17.055 17.119 17.112 17.045 17.113 17.108 17.039 17.110 17.104 17.030 17.107 17.100 17.022

0.5894 0.5895 0.5967 0.5667 0.5668 0.5740 0.5457 0.5456 0.5530 0.5260 0.5260 0.5333 0.5075 0.5076 0.5150 0.4903 0.4904 0.4977 0.4741 0.4742 0.4816 0.4588 0.4589 0.4664

18.952 18.952 18.201 17.673 17.668 17.696 17.723 17.767 17.745 17.692 17.749 17.726 17.663 17.706 17.694 17.625 17.657 17.658 17.587 17.604 17.623 17.552 17.551 17.592

– – 3.1559 2.1843 2.1827 2.2227 1.9720 1.9874 2.0073 1.7856 1.8033 1.8206 1.6410 1.6516 1.6698 1.5195 1.5266 1.5455 1.4176 1.4211 1.4414 1.3306 1.3309 1.3528

– – 0.714 2.270 2.274 2.208 3.206 3.164 3.144 4.231 4.174 4.155 5.254 5.211 5.179 6.287 6.254 6.209 7.322 7.303 7.238 8.353 8.353 8.264

Shell radii are in a.u.

Table 4. ELI-D shell structure parameters for the atoms of sixth period from fully relativistic (top line), scalar-relativistic ZORA (middle line), and nonrelativistic limit (bottom line), calculations.

Cs

Ba

La

Ce

Pr

Nd

Pm

Sm

Eu

Gd

Tb

582

qK

rK

qL

rL

qM

rM

qN

rN

qO

rO

2.254 2.226 2.263 2.236 2.227 2.259 2.241 2.224 2.251 2.256 2.226 2.262 2.243 2.225 2.264 2.238 2.224 2.270 2.236 2.225 2.266 2.245 2.223 2.254 2.240 2.222 2.267 2.245 2.223 2.260 2.241 2.221 2.257

0.0392 0.0387 0.0418 0.0381 0.0379 0.0409 0.0374 0.0370 0.0400 0.0367 0.0363 0.0394 0.0358 0.0355 0.0388 0.0350 0.0347 0.0382 0.0343 0.0341 0.0374 0.0337 0.0333 0.0366 0.0330 0.0327 0.0362 0.0323 0.0320 0.0355 0.0317 0.0314 0.0348

8.756 8.777 8.884 8.772 8.783 8.894 8.771 8.792 8.915 8.770 8.796 8.922 8.781 8.801 8.927 8.794 8.806 8.931 8.793 8.809 8.948 8.790 8.815 8.972 8.798 8.819 8.966 8.797 8.820 8.986 8.804 8.825 8.999

0.1486 0.1486 0.1540 0.1450 0.1451 0.1505 0.1416 0.1417 0.1472 0.1385 0.1386 0.1442 0.1355 0.1356 0.1414 0.1327 0.1327 0.1385 0.1298 0.1299 0.1359 0.1271 0.1272 0.1333 0.1246 0.1247 0.1308 0.1220 0.1221 0.1284 0.1197 0.1198 0.1262

17.098 17.094 17.009 17.095 17.089 17.001 17.094 17.086 16.994 17.127 17.123 17.033 17.195 17.191 17.112 17.238 17.237 17.163 17.289 17.285 17.214 17.340 17.338 17.269 17.392 17.392 17.334 17.429 17.429 17.368 17.507 17.507 17.466

0.4445 0.4446 0.4522 0.4309 0.4310 0.4387 0.4183 0.4184 0.4261 0.4070 0.4071 0.4150 0.3968 0.3970 0.4051 0.3866 0.3869 0.3951 0.3769 0.3772 0.3855 0.3677 0.3680 0.3764 0.3590 0.3592 0.3678 0.3500 0.3505 0.3591 0.3424 0.3428 0.3517

17.521 17.522 17.565 17.498 17.499 17.544 17.535 17.533 17.577 18.141 18.141 18.231 19.205 19.207 19.417 19.864 19.866 20.113 20.546 20.547 20.829 21.242 21.247 21.561 21.958 21.953 22.307 22.295 22.288 22.590 23.415 23.404 23.822

1.2550 1.2553 1.2767 1.1890 1.1894 1.2103 1.1373 1.1375 1.1580 1.1089 1.1094 1.1315 1.0967 1.0977 1.1245 1.0722 1.0734 1.1014 1.0495 1.0509 1.0801 1.0283 1.0301 1.0603 1.0092 1.0102 1.0421 0.9800 0.9809 1.0110 0.9735 0.9740 1.0083

8.529 8.536 8.419 8.520 8.522 8.418 9.095 9.091 9.065 9.437 9.432 9.353 9.626 9.625 9.370 9.919 9.915 9.613 10.187 10.182 9.833 10.435 10.429 10.033 10.659 10.662 10.213 10.890 10.887 10.539 11.077 11.086 10.541

4.1476 4.1636 4.2048 3.3201 3.3221 3.3887 3.0914 3.0915 3.1813 3.0313 3.0326 3.1257 3.1111 3.1130 3.2055 3.0591 3.0592 3.1563 3.0102 3.0106 3.1102 2.9637 2.9665 3.0667 2.9178 2.9213 3.0249 2.7350 2.7396 2.8508 2.8324 2.8356 2.9475

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qP 0.822 0.814 0.829 1.870 1.867 1.869 2.261 2.267 2.190 2.272 2.277 2.197 1.946 1.943 1.900 1.946 1.946 1.902 1.947 1.945 1.903 1.948 1.942 1.905 1.952 1.946 1.907 2.348 2.349 2.254 1.959 1.953 1.911 (Continued)

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Table 4. (Continued)

Dy

Ho

Er

Tm

Yb

Lu

Hf

Ta

W

Re

Os

Ir

Pt

Au

Hg

Tl

Pb

Bi

Po

At

Rn

qK

rK

qL

rL

qM

rM

qN

rN

qO

rO

qP

2.242 2.220 2.257 2.242 2.222 2.261 2.231 2.220 2.262 2.238 2.216 2.268 2.244 2.218 2.272 2.257 2.219 2.286 2.243 2.217 2.271 2.238 2.213 2.268 2.240 2.215 2.270 2.241 2.216 2.270 2.236 2.213 2.260 2.232 2.208 2.260 2.248 2.211 2.276 2.228 2.212 2.270 2.236 2.211 2.257 2.250 2.206 2.274 2.253 2.204 2.280 2.254 2.207 2.281 2.234 2.206 2.279 2.246 2.204 2.276 2.238 2.199 2.273

0.0311 0.0308 0.0343 0.0305 0.0302 0.0338 0.0298 0.0296 0.0333 0.0293 0.0290 0.0328 0.0288 0.0285 0.0324 0.0284 0.0280 0.0320 0.0277 0.0274 0.0314 0.0272 0.0269 0.0309 0.0267 0.0264 0.0305 0.0262 0.0260 0.0301 0.0257 0.0255 0.0295 0.0252 0.0250 0.0291 0.0249 0.0245 0.0289 0.0243 0.0241 0.0284 0.0239 0.0237 0.0279 0.0235 0.0232 0.0277 0.0231 0.0228 0.0274 0.0228 0.0224 0.0271 0.0222 0.0220 0.0267 0.0219 0.0216 0.0264 0.0215 0.0212 0.0260

8.804 8.827 9.010 8.802 8.828 9.015 8.814 8.830 9.024 8.812 8.835 9.026 8.803 8.834 9.032 8.796 8.825 9.033 8.802 8.826 9.050 8.806 8.829 9.062 8.800 8.828 9.069 8.800 8.827 9.080 8.801 8.829 9.096 8.800 8.834 9.108 8.789 8.830 9.103 8.805 8.828 9.121 8.793 8.826 9.141 8.791 8.830 9.144 8.782 8.829 9.142 8.770 8.822 9.147 8.783 8.818 9.157 8.765 8.817 9.164 8.765 8.818 9.177

0.1173 0.1174 0.1240 0.1151 0.1152 0.1219 0.1129 0.1130 0.1198 0.1108 0.1109 0.1178 0.1088 0.1089 0.1159 0.1068 0.1068 0.1140 0.1048 0.1049 0.1121 0.1029 0.1030 0.1104 0.1010 0.1012 0.1086 0.0993 0.0994 0.1069 0.0975 0.0976 0.1053 0.0958 0.0960 0.1037 0.0941 0.0943 0.1022 0.0925 0.0927 0.1007 0.0909 0.0911 0.0992 0.0894 0.0896 0.0978 0.0879 0.0881 0.0964 0.0864 0.0866 0.0950 0.0850 0.0852 0.0937 0.0836 0.0838 0.0925 0.0822 0.0825 0.0912

17.568 17.568 17.535 17.630 17.629 17.607 17.693 17.691 17.681 17.753 17.754 17.757 17.819 17.817 17.834 17.864 17.872 17.885 17.912 17.916 17.939 17.950 17.958 17.986 17.992 18.003 18.031 18.025 18.038 18.071 18.059 18.070 18.111 18.093 18.100 18.145 18.121 18.132 18.180 18.150 18.158 18.209 18.171 18.183 18.239 18.193 18.202 18.262 18.216 18.224 18.292 18.234 18.246 18.317 18.253 18.266 18.338 18.271 18.281 18.360 18.288 18.298 18.378

0.3347 0.3350 0.3442 0.3273 0.3276 0.3369 0.3202 0.3205 0.3300 0.3134 0.3136 0.3233 0.3068 0.3071 0.3169 0.3002 0.3005 0.3104 0.2938 0.2941 0.3041 0.2876 0.2879 0.2980 0.2816 0.2820 0.2922 0.2758 0.2762 0.2865 0.2702 0.2706 0.2810 0.2648 0.2652 0.2757 0.2596 0.2600 0.2706 0.2545 0.2549 0.2657 0.2496 0.2500 0.2608 0.2448 0.2452 0.2562 0.2402 0.2406 0.2517 0.2357 0.2361 0.2473 0.2313 0.2318 0.2431 0.2271 0.2276 0.2390 0.2230 0.2235 0.2350

24.153 24.137 24.589 24.898 24.888 25.362 25.646 25.639 26.140 26.400 26.393 26.922 27.158 27.153 27.708 27.569 27.558 28.048 27.902 27.904 28.318 28.179 28.191 28.535 28.409 28.423 28.708 28.607 28.607 28.846 28.775 28.766 28.959 28.918 28.908 29.049 29.073 29.040 29.150 29.177 29.139 29.209 29.240 29.204 29.235 29.290 29.253 29.253 29.332 29.290 29.265 29.361 29.318 29.272 29.384 29.337 29.273 29.400 29.351 29.272 29.412 29.359 29.268

0.9569 0.9570 0.9926 0.9410 0.9413 0.9776 0.9258 0.9261 0.9633 0.9111 0.9114 0.9496 0.8971 0.8974 0.9364 0.8753 0.8754 0.9126 0.8534 0.8540 0.8889 0.8317 0.8327 0.8654 0.8102 0.8117 0.8423 0.7893 0.7904 0.8196 0.7690 0.7697 0.7975 0.7492 0.7498 0.7761 0.7316 0.7315 0.7568 0.7128 0.7126 0.7368 0.6934 0.6933 0.7164 0.6748 0.6747 0.6971 0.6569 0.6569 0.6787 0.6397 0.6397 0.6611 0.6232 0.6232 0.6443 0.6074 0.6074 0.6282 0.5923 0.5922 0.6130

11.273 11.291 10.692 11.463 11.475 10.836 11.646 11.660 10.974 11.825 11.836 11.106 11.998 12.013 11.232 12.060 12.062 11.411 12.298 12.292 11.800 12.688 12.688 12.358 13.222 13.237 13.070 13.961 13.947 13.936 14.825 14.791 14.879 15.830 15.798 15.882 19.778 19.789 19.295 20.648 20.665 20.194 21.565 21.579 19.025 19.403 19.849 18.996 19.025 19.435 18.890 18.928 19.176 18.790 18.811 18.969 18.692 18.697 18.798 18.601 18.595 18.651 18.519

2.7922 2.7984 2.9114 2.7534 2.7592 2.8767 2.7159 2.7233 2.8434 2.6795 2.6862 2.8114 2.6442 2.6552 2.7805 2.4648 2.4733 2.6049 2.3309 2.3442 2.4912 2.2341 2.2550 2.4204 2.1681 2.1978 2.3837 2.1539 2.1641 2.3812 2.1549 2.1527 2.3868 2.1785 2.1739 2.3996 – – – – – – – – 2.4444 2.1495 2.2924 2.2158 1.8994 1.9963 2.0163 1.7632 1.8110 1.8596 1.6462 1.6711 1.7315 1.5467 1.5597 1.6248 1.4614 1.4669 1.5342

1.964 1.953 1.913 1.968 1.956 1.915 1.973 1.958 1.916 1.979 1.964 1.918 1.985 1.964 1.920 2.464 2.462 2.341 2.850 2.844 2.624 3.146 3.122 2.794 3.344 3.295 2.855 3.373 3.367 2.800 3.309 3.333 2.697 3.132 3.154 2.558 – – – – – – – – 2.106 3.090 2.659 3.076 4.412 4.019 4.138 5.464 5.233 5.200 6.546 6.408 6.267 7.631 7.551 7.330 8.712 8.678 8.388

Shell radii are in a.u.

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Table 5. ELI-D shell structure parameters for the atoms of seventh period from fully relativistic (top line), scalar-relativistic ZORA (middle line), and nonrelativistic limit (bottom line), calculations.

Fr

Ra

Ac

Th

Pa

U

Np

Pu

Am

Cm

Bk

Cf

Es

Fm

Md

No

Lr

Rf

qK

rK

qL

rL

qM

rM

qN

rN

qO

rO

qP

rP

qQ

2.295 2.197 2.333 2.315 2.200 2.340 2.265 2.200 2.299 2.282 2.199 2.304 2.265 2.196 2.305 2.276 2.188 2.301 2.264 2.190 2.290 2.273 2.192 2.287 2.259 2.192 2.306 2.308 2.191 2.332 2.259 2.186 2.311 2.273 2.178 2.303 2.270 2.178 2.309 2.289 2.180 2.286 2.287 2.181 2.298 2.302 2.182 2.298 2.330 2.179 2.339 2.306 2.172 2.309

0.0215 0.0208 0.0261 0.0212 0.0205 0.0258 0.0205 0.0202 0.0252 0.0203 0.0198 0.0250 0.0198 0.0194 0.0247 0.0195 0.0190 0.0243 0.0191 0.0187 0.0240 0.0188 0.0184 0.0237 0.0184 0.0181 0.0236 0.0184 0.0178 0.0235 0.0177 0.0175 0.0231 0.0175 0.0171 0.0228 0.0172 0.0168 0.0226 0.0170 0.0166 0.0222 0.0167 0.0163 0.0220 0.0165 0.0160 0.0218 0.0163 0.0157 0.0218 0.0158 0.0154 0.0214

8.736 8.813 9.154 8.714 8.806 9.160 8.734 8.801 9.191 8.709 8.796 9.191 8.717 8.794 9.204 8.701 8.793 9.216 8.709 8.786 9.230 8.693 8.775 9.244 8.690 8.769 9.229 8.635 8.761 9.217 8.664 8.758 9.237 8.649 8.756 9.246 8.638 8.747 9.253 8.601 8.737 9.284 8.587 8.725 9.271 8.571 8.714 9.284 8.546 8.707 9.277 8.560 8.701 9.297

0.0809 0.0811 0.0901 0.0796 0.0798 0.0889 0.0783 0.0786 0.0877 0.0770 0.0773 0.0866 0.0758 0.0761 0.0855 0.0746 0.0749 0.0844 0.0735 0.0738 0.0834 0.0723 0.0727 0.0824 0.0712 0.0716 0.0814 0.0701 0.0705 0.0805 0.0690 0.0694 0.0795 0.0680 0.0684 0.0786 0.0669 0.0674 0.0777 0.0659 0.0664 0.0769 0.0649 0.0654 0.0760 0.0639 0.0644 0.0752 0.0629 0.0635 0.0744 0.0620 0.0626 0.0735

18.295 18.316 18.393 18.307 18.329 18.409 18.332 18.343 18.433 18.340 18.357 18.444 18.373 18.385 18.488 18.395 18.411 18.521 18.411 18.432 18.561 18.439 18.463 18.601 18.463 18.486 18.639 18.484 18.505 18.664 18.513 18.536 18.713 18.526 18.561 18.758 18.552 18.587 18.787 18.582 18.610 18.829 18.604 18.635 18.877 18.615 18.663 18.913 18.642 18.682 18.938 18.644 18.702 18.974

0.2191 0.2195 0.2312 0.2152 0.2157 0.2275 0.2114 0.2120 0.2239 0.2078 0.2083 0.2203 0.2043 0.2049 0.2171 0.2009 0.2016 0.2139 0.1977 0.1983 0.2108 0.1945 0.1952 0.2079 0.1914 0.1921 0.2049 0.1883 0.1891 0.2020 0.1854 0.1862 0.1993 0.1825 0.1833 0.1967 0.1797 0.1806 0.1940 0.1770 0.1778 0.1915 0.1743 0.1752 0.1891 0.1717 0.1726 0.1866 0.1691 0.1701 0.1843 0.1666 0.1676 0.1819

29.424 29.371 29.261 29.441 29.383 29.252 29.458 29.396 29.251 29.474 29.409 29.248 29.568 29.497 29.304 29.626 29.550 29.330 29.684 29.604 29.349 29.768 29.682 29.392 29.829 29.735 29.409 29.860 29.764 29.407 29.944 29.834 29.439 30.002 29.885 29.451 30.056 29.930 29.464 30.113 29.976 29.471 30.172 30.020 29.477 30.227 30.063 29.484 30.261 30.090 29.468 30.299 30.115 29.463

0.5780 0.5779 0.5984 0.5643 0.5642 0.5846 0.5514 0.5513 0.5715 0.5390 0.5389 0.5590 0.5292 0.5291 0.5492 0.5189 0.5188 0.5388 0.5090 0.5089 0.5287 0.5001 0.5000 0.5196 0.4909 0.4907 0.5102 0.4813 0.4812 0.5003 0.4733 0.4731 0.4921 0.4649 0.4647 0.4835 0.4568 0.4566 0.4752 0.4489 0.4486 0.4671 0.4413 0.4409 0.4592 0.4339 0.4334 0.4516 0.4262 0.4258 0.4437 0.4188 0.4183 0.4361

18.510 18.569 18.458 18.442 18.506 18.406 18.498 18.555 18.439 18.581 18.637 18.478 19.169 19.250 19.411 19.586 19.684 19.982 20.047 20.175 20.602 20.750 20.935 21.680 21.366 21.524 22.422 21.736 21.891 22.724 22.713 22.830 23.996 23.448 23.542 24.823 24.228 24.298 25.675 25.059 25.098 26.549 25.949 25.946 27.443 26.910 26.835 28.357 27.411 27.260 28.628 28.076 27.850 28.991

1.3875 1.3926 1.4578 1.3233 1.3282 1.3910 1.2769 1.2811 1.3409 1.2365 1.2405 1.2955 1.2116 1.2172 1.2814 1.1836 1.1901 1.2566 1.1580 1.1661 1.2345 1.1398 1.1508 1.2271 1.1227 1.1314 1.2108 1.0983 1.1063 1.1815 1.0930 1.0978 1.1817 1.0804 1.0836 1.1686 1.0694 1.0709 1.1566 1.0601 1.0597 1.1453 1.0527 1.0503 1.1348 1.0478 1.0421 1.1251 1.0287 1.0204 1.0961 1.0198 1.0088 1.0764

8.970 8.956 8.573 8.923 8.921 8.581 9.318 9.316 9.193 9.802 9.813 9.933 10.468 10.450 10.097 10.984 10.947 10.462 11.455 11.391 10.779 12.086 11.979 10.918 12.410 12.342 11.115 12.544 12.463 11.458 12.940 12.937 11.424 13.143 13.167 11.539 13.301 13.361 11.631 13.407 13.514 11.701 13.454 13.610 11.752 13.429 13.668 11.784 12.874 13.293 11.487 13.077 13.471 12.464

4.3527 4.3809 4.5027 3.4415 3.4639 3.6575 3.1826 3.2062 3.4448 2.9975 3.0299 3.3129 3.0572 3.0809 3.3336 3.0063 3.0266 3.2835 2.9579 2.9767 3.2360 3.0590 3.0749 3.3172 3.0124 3.0334 3.2750 2.8187 2.8387 3.1060 2.9246 2.9527 3.1961 2.8832 2.9110 3.1598 2.8431 2.8746 3.1252 2.8043 2.8400 3.0923 2.7666 2.8017 3.0609 2.7299 2.7659 3.0308 2.4568 2.5488 2.7354 2.3747 2.4177 2.7570

0.778 0.757 0.807 1.882 1.846 1.854 2.414 2.385 2.198 2.830 2.788 2.404 2.460 2.427 2.197 2.456 2.428 2.197 2.451 2.424 2.199 2.014 1.976 1.884 2.004 1.954 1.884 2.459 2.427 2.215 1.987 1.922 1.885 1.981 1.915 1.886 1.976 1.903 1.887 1.972 1.890 1.889 1.969 1.887 1.890 1.967 1.882 1.892 2.973 2.794 2.882 3.078 2.997 2.529

Shell radii are in a.u.

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Keywords: electron localizability indicator  atomic shell structure  Dirac  Coulomb hamiltonian  spinor wavefunctions  zero-order regular approximation calculations How to cite this article: A. I. Baranov. J. Comput. Chem. 2014, 35, 565–585. DOI: 10.1002/jcc.23524 [1] M. Kohout, Faraday Discuss. 2007, 135, 43. [2] C. Gatti, P. Macchi, Eds., Modern Charge-Density Analysis, Chapter 9; Springer: Dordrecht, 2012. [3] M. Kohout, Int. J. Quantum Chem. 2004, 97, 651. [4] M. Kohout, F. R. Wagner, Y. Grin, Int. J. Quantum Chem. 2006, 106, 1499. [5] P. Pyykk€ o, Annu. Rev. Phys. Chem. 2012, 63, 45. [6] M. Reiher, A. Wolf, Relativistic Quantum Chemistry; Wiley-VCH Verlag GmbH: Weinheim, 2009. [7] J. K€ ubler, K.-H. H€ ock, J. Stich, A. R. Williams, J. Phys. F: Met. Phys. 1988, 18, 469. [8] S. Sharma, J. K. Dewhurst, C. Ambrosch-Draxl, S. Kurth, N. Helbig, S. Pittalis, S. Shallcross, L. Nordstr€ om, E. K. U. Gross, Phys. Rev. B. 2007, 98, 196405. [9] R. McWeeny, W. Kutzelnigg, Int. J. Quantum Chem. 1968, 2, 187. [10] W. Kutzelnigg, Z. Naturforschg. 1963, 18, 1058. [11] W. Kutzelnigg, In Explicitly Correlated Wave Functions in Chemistry and Physics: Theory and Applications; J. Rychlewski, Ed.; Kluwer: The Netherlands, 2002; pp. 14–17. [12] D. L. Cooper, R. Ponec, T. Thorsteinssohn, G. Raos, Int. J. Quantum Chem. 1996, 57, 501. [13] V. N Staroverov, E. R. Davidson, Int. J. Quantum Chem. 2000, 77, 651. [14] M. Kohout, F. R. Wagner, Y. Grin, Theor. Chem. Acc. 2008, 119, 413. [15] E. van Lenthe, E. J. Baerends, J. G. Snijders, J. Chem. Phys. 1993, 99, 4597.

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Received: 18 September 2013 Revised: 12 December 2013 Accepted: 17 December 2013 Published online on 17 January 2014

Journal of Computational Chemistry 2014, 35, 565–585

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