Internally folded densities

Chemical Physics 63 (1981) 175-183 North-HoliandPublishingConiPany

INTERNALLY Ajit

FOLDED

DENSITIES

J. THAKKAR*

Cemre for Graduare Work in Chemisrry. Waterloo, Onrario, Canada NZL’3GI

Gtrelph-Waterloo

of Waterloo.

Unicersify

and A!fredo Depnrfmenr

Received

M. SIMAS of Clzemisr?,

and Vedene H. SMITH Jr. Q~een’s Unicrrsiry, Kingsron, Ontario,

Canndn

K7L 3.&V

8 June 1981

The 3q-dimensional Fourier transforms of the q-electron charge density and momentum density respectively are defined to be the q-electron form factor and internally folded density. For q = 1, these quantities are just the usual X-ray form factor and the 5(r) function recenrly introduced for the analysis of Compton profiles. Numerous properties of the one-electron internally folded density B(r) are derived. 5 (r) functions for the atoms from He through Ne are calculated and examined for chemical content with the help of the concept of reduced Compton profiles.

I. Introduction The 3q-dimensional Fourier transforms of the q-electron charge and momentum density respectively are defined to be the q-electron form factor and internally folded density. These quantities are of interest because for 4 = 1 they reduce to the usual X-ray form factor and the B(r) function which is used for the analysis of Compton profiles_ Section 2 provides general definitions of the q-electron quantities and establishes notation_ Section 3 delves into the derivation of numerous properties of the one-electron internally folded density B(r) and includes known properties of B(r) for the sake of completeness. Since internally folded densities are not as familiar as Compton profiles, section 4 presents B(r) functions for the atoms from hydrogen through neon. The concept of reduced Compton profiles, momentum densities and internally folded densities is introduced to facilitate the * NSERCC

University

Research

Fellow.

0301-0104/81/0000-0000/S02.75

examination of Compton profiles for chemical content above and over gross scale and size effects.

2. Fourier transforms of the q-electron and momentum densities

charge

Reduced density matrices are very useful in the analysis of electronic wavefunctions for atoms and molecules [l]. Since the wavefunctions are expressed in the position-space representation in an overwhelming majority of cases, it is usual to consider the position space representation of the qth order reduced density operator (later referred to as the r-space qmatrix) defined by

rqq,

x2 , . . .

,x,

[xi, xs,... ,XL,

=( >Ivqx;, xi, ..-,x;, N

Xq+l,

...

XN)

9

x*(xt,

@ 1981 North-Holland

X2, . . .

, h’)

dr,+l

dr,i2

**. drhr,

(1)

where Y(x*, x2, __., xN) is the position space representation of the N electron wavefunction, and x, = (r,, Ci) is a combined space-spin coordinate. However, one could equally well choose to consider the momentum space representation of the qth order reduced density operator (later referred to as the p-space q-matrix) defined by

yz. ._. . J-X) d_v,_l dv,--

xQ(yl,

___dys,

(2)

position space wavefunction: @(y*,

y2,

_..

x exp

dc+l __.duq du; ___dcr;,

... 6(u, -a;)

(3)

y,xr) =

-_i g C

rk

(27;)-3”” -

k=I

pk

I

I wxr,

x2. .--

3

xh.1

drl drz . . . dr,+

(6)

Hartree atomic units are used above as in the rest of this paper. The physically meaningful quantities are the q-electron charge density l?‘(rt,

rl, ._. , rq)

=P’(rl,

where @(yl, yj, . . . , _v,v_) is the momentum space representation of the N electron wavefunction, and yi = (Pi, a;) is a combined momentum-spin coordinate. In dealing with problems where spin is irrelevant, it is convenient to work with the spinless r-space q-matrix defined by

,

rz, . . .

(7)

, r, (rlr r7, . . . , rq).

and the q-electron

momentum

density

l-I’@@,, p3 ... >P4)

= rP’(p1, pz. ... 1pq IPI, p2, _._7Pq)-

(81

As is apparent from eqs. (5)-(S), the q-electron charge and momentum densities are not related by a 3q-dimensional Fourier transform Thus it is natural to examine their Fourier transforms:

and the spinless P-space q-matrix defined by II?Pl.

=

._. , pq lpi, .._, p;,

I I-P’(y1,.*-,yqly;,

.._,y;Mirrl-(T;j

...

.. . 6(cr, -CT;, da1 ._. da, do; ___drr;.

(4)

(spinless) P-space q--matrix is simply the 6q-dimensional Fourier transform of the (spinless) r-space q-matrix:

X

The

x exp

-i

I !?)h, 5

... , r, 1ri,

4

C pe - SL dpl ..-dp,. 1 k=l

(10)

is simply the X-ray form factor [2,3], the Bg(sl, .__, _FS) functions arose in an earlier dis-

cussion [4] of the relationship between form factors and momentum densities, and B1(s) has ___,

been extensively discussed in connection with analysis of Compton profiles [5-g]. It is easy to

ri)

see that

drl...

(rk-pe-rl-pi)

k=l

-i

F,(p)

II’4 ‘(p I,..., pqlpi ,...> Pi)

=(2*)-J”

exp

3

___ dr, dr; . . . dr;, because the momentum space wavefunction is the 3Ndimensional Fourier transform of the

(5)

(11)

Al.

F,(Pi

=

-pr,

J

Thkkar

. . . , eG -p4)

II’~‘(P,, _...pq Ip:, .._.p;, dPt .. . dP,,

177

er al. / Inrenxaiiy folded densiries

(12)

electron form factor, and Bq the q-electron internally folded density. It is possible to proceed to derive general properties of P, and F,

such as

where Pk = (pi +pk)/2,

and similarly

B,(O,O ,...) O)=F,(O,O ,..., o)=pqy). B&l,

=

.-. , sq)

J

rCq’(rl,.._ ,rqJrl+st,

. . . . r,+s,)drl

...dr., (13)

B,(r; -rlr .__, ri -rs) =

J

T”‘(rl, .._ ,rqlr;,

. .. ,ri)dR1

.._ dR,,

(14)

where Rn = (ri irk)/2_ Eqs. (14) and (12) make it clear that Bq and F, respectively are the relative or intracular coordinate projections of the r-space and p-space q-matrices. It is, of course, important not to confuse them with the intracular projections of the r-space and p-space +&~sities ; in particular the intracular projection of the pair density [lo, 111 bears no relationship to any of the B, functions. The spectral resolutions of I“” and II’“’ can be used in eqs. (11) and (13) and, if the q-rank of the q-matrices is finite, the order of integration and summation can be interchanged to obtain ~,(sr,

.__, s,) = k$, AY J qz)*(rl

xq?‘(rl,

. . . , r4) drl . . . dr,,

and

Fq(p~, . . . , p.q)=

; I=1

Alf’

I

+sl, _._, rq +-ss)

However, the rest of this paper will be devoted to an examination of B(r) =Bl(r) which shall be referred to as simply the folded density.

3. Properties and Compton

of the internally profile

For the sake of completeness the general equations of section 1 specialized to the case of q = 1 are quoted below.

B(x,y,z)=B(r) =~~(r)

=

I

exp [-ip

- r] n(p) dp,

i18)

where II&, p?, p,) = II(p) = l-I”‘(p) is the oneelectron momentum density, and

B(r) = 1 T”‘(s Is +r) ds,

a

=kg, vk

(1%

J \Erz(r +s) *k(s) ds,

(20)

(15)

where the Y,(r)rqp’(r) with occupation numbers malization condition is ~1;9)‘(p1+CL*,---.pq+Ilq)

x&‘(P,, ... , pq) de1 ... de,,

folded density

(16)

where the r-space and p-space natural q-states Up’ and 42’ respectively are the eigenstates of the r-space and p-space q-matrices with corresponding eigenvalues or occupation numbers A$‘, and n4 is the q-rank of rcq’. Evidently Bq and Fq respectively are essentially convex combinations of the autocorrelation functions associated with the r-space and p-space natural q-states. It may be appropriate to call Fq the q-

are the natural orbitals vi, =hf’. The nor-

B(0) =N. The Compton J(q=, qy. q.)=J(4)=

(21) profile [12] is given by

J

mJ)dS-

(22)

S

where S is the surface p * g = q2. Aiternatively (22) can be written [13] as (23j

J(q) = [ IUP) WP - q/q -4) dp, _-

Af.

178

where 4 = 141. Thus the Compton line is given by

Tlrakkar ei al. / Internally folded densities

profile along a

it follows that s IL(r)

If the momentum density has inversion symmetry, then both the folded density and Compton profile have inversion symmetry and are real. That is, if II(p) = IX-p) then Blr) = B*(r) = B(-r) and J(q) =J(-q). The exact II(p) and most n(p) calculated from approximate wavefunctions do have inversion symmetry. However, there are some types of approximate wavefunctions that lead to momentum densities which are not inversion symmetric [14]. The momentum density, Compton profile and folded density may all be expressed in terms of the pertinent spherical polar coordinates and expanded in spherical harmonics n(p)

= ; i l&“,(P) Yr,,*(ep, bp), ,=I, In=-,

(25) (26) (27)

Note that if the momentum density is inversion symmetric, then all the odd I spherical harmonics drop cut of the above expansions, that is, n,~(p)=Brm(r)=Jr,(qj=O for all odd I_ The leading term of the above expansions is proportional to the spherica! average of the appropriate quantity. Thus l-I(p) dR, = &,ip)/(4~)*“,

ii(p) = (4x)-’ B(rj = (.4~)-’ J(q) = (4;r)-’

I

B(r) dR, = B&r)/(4~)“‘,

Moreover expansion exp [-ip =45r

(29) (30)

from eqs. i18), (25) and (25), and the [ 151

- r] z i (-i)‘jl@ I=,, m=-I

where J(pr) is a spherical l3essel function. been previously shown [16, 171 that r Jl,(q)

(32)

ILA~)jt(pr)p’dp,

5 0

= 2~ 1 &l(p)

9(4/p)

It has

p dp,

(33)

141 where P!(x) is a Legendre polynomial. In the case of I = m = 0 the relations between the various spherical averages are obtained: s B(r) = 47 1

n(p)

jo(prJ p’ dp,

(341

0 E J(q)=21;

I

i=I(p) P dp.

(35)

14;

The inverse of eq. (35) djjdq

= -27

ii(q)q.

(36)

may be substituted in eq. (34) and an integration by parts carried out to yield z B(r) = 2

J Jcq,costqr) dq.

(37)

0

From an inspection clear that

of eqs. (18) and (241, it is

B(O, 0,z)

=

J

exp C-izp=lWpx, pyvpz ) dpx dp, dpz JE

[ J(q) dR, =J&)/(4&‘“. J

(28)

=47(-i)’

.I Ylm(6, $5) YEI(&, d$P), (31)

=

J

exp [-izp,]

J(0, 0,

p=) dp,.

(38)

--I:

This observation that the folded density along a line is the one-dimensional Fourier transform of the Compton profile along a line is the basis for a method of reconstructing three-dimensional momentum densities from experimental Compton profiles [18].

179

A.J. Thakkar et al. / Intemaliy forded densities

It is not hard to show that m‘

It is convenient to introduce the even extensions of the spherically averaged quantities. Thus J.=(4) =.%?I),

--co
(39)

B,(r) = B(H),

--a3
(40)

-co
)X(P) = I=hlPl),

030) =J(O,

B,(r)=B(r,O, II,(p)

O)=B(O,

B 0) =J(O,

0,4),

r,O)=B(O,

0, r),

= IUP, 0,O) = IUO, p. 0) = NO, 0, P).

Moreover

eq. (37) can be rewritten

(43) (44)

as

0

(47r-*

(43)

33 -2 s n =G4,

(46)

--2

and exactly analogous identities apply to the other directions. One can compare these with the well-known relations [3,19,20] co 1 q*&q)dq,

OSkS4,

(47)

0 (p-‘)

(4~)~’

(48)

and [21] a3 (p-a) = 2 I [3(0)-&q)] 0

q-* dq.

[ B(r) dr = / s(r)

rz dr = 2+;’ n(O)

0 (52)

= -rr[d’J(q)/dq’],,,. A relation anaIogous to eq. (SO) hoIds for B(r) along a line m m B(0,

=

0, z) dz =

I --m

B (x, 0,O) dx

(49)

I

~(0,

>r,

0) dy =2-J(O)

= sr(p-‘>.

(53)

-m

It is also easy to see that [d’B(O, 0, z)/dz?],,c

= -(p:),

(54)

(pf),

(55)

[d%(O, 0, z)/dr’&o=

and exactly analogous relations hold for the x and p directions. Additionally one can show that [d’B(r)/dr’],,o

= -(p’)/3,

(56)

[d%(r)/drJ],,c

= (pJ}/5_

(57)

In fact, eqs. (21), (56) and (57) all follow from the expansion B(r)

= 2J(O)

(51)

m

I --z

which is a direct analog of eq. (38). Evidently the moments of the directional Compton profile yield directional moments of the momentum density

(p”>=2(k+I)

1 B(r) r-’ dr = l B(r) r dr = (p-‘), 0

-co

0, P;) dpz =(P:),

(50) (c

m

J,(q) exp I--iqrl dq,

I iCJ(O,

B(r) dr = rj(O)

= sr(p_‘)/2,

(42)

m

B,(r) = f

r-‘) dr = j

(41)

The utility of these extensions lies in the fact that, in the case of systems with spherically symmetric momentum densities, they are identical with the pertinent functions along a line. Thus for a system with a spherically symmetric momentum density J,(q) =J(q,

(471)-l I B(r)

=N-ip’)r~‘/3!i(p’)r4/5!+~(r’).

(58)

Note that, despite what one might hastily conclude from eq. (34), the terms of order r’, r7, . . . do not vanish from expansion (58). One may verify this in the case of the hydrogen atom for which Bjr)=B(r)=e-‘(I

+rir’/3).

(59)

and

It is possible to show that [d” &j/d&,, = (27;/3j[2Z(dp/drj,=,,-(dh/dcrj;,=,II,

(60)

where p(r) is the spherical average of the position space one-electron density p!r)

=

(4r)-’

I

(61)

I-“‘(r) do,

and h (II) is the spherical average of the intracular projection of the electron pair density [lo, Ill ’ -1 1Z(f1j= (45rc_) x

J

T”‘(r,,

r2)

6(rc

-jr:

-r&

drl dr2.

(62)

If p(r) and /I(M) satisfy the cusp conditions [IO ,23,24], then eq. (60) may be rewritten as [d’~(r)/drs],~r,=-(2-ir/3)[4Z’p(0)th(0j].

(63)

The exact p(r) and h(rc) do satisfy the cusp conditions. However, most approximate wavefunctions are of finite l-rank and hence do not satisfy the electron-electron cusp condition for 12(II) and incorrectly predict that (.dh/du), =,) = 0. A further discussion may be found elsewhere [lo]. Eq. (60) is the direct analog-of the known [2S-271 asymptotic behaviour of II(p): l=I(p) = -(2/-;rj[2Z(dp/drj,=o - -“tb(p-‘“). - (dlz/dzrj,=olp

(64).

In the course of utilizing the above expressicns, it is important to remember that, in general, [d%(r)/dr”],,o

# [dkB,(r)/drk],=o.

(65)

Using a method identical to that used to derive some sum rules for X-ray and high energy electron scattering cross-sections [22], one may show that m 1 [N-&r),r-‘dr=-

j

0

0

m = -

I 0

r-‘(d&dr)

=-

r-id&)

dr = n(p)/4

(66)

48 7

I II

r-J[B(r)-N+{p’)r”/3!]dr.

(67)

4. Internally folded densities for the first row atoms

Since internally folded densities are less familiar than Compton profiles and momentum densities, L?(r) functions for all the atoms from hydrogen through neon are presented and ana!yzed in this section. The most accurate Compton profiles previously calculated from correlated wavefunctions [28-321 were fitted to a linear combination of hydrogenic lorentzian functions by a recently developed method [21]. The resulting expansions for J(q) were then inserted into eq. (37) to produce B(r) functions for all the atoms from helium through neon. The exact B(r) for the hydrogen atom is given by eq. (59). In classical thermodynamics and the theory of intermolecular forces it is useful to work with dimensionless quantities which are found to be roughly similar for many systems. For example, the interatomic potential V(R) is expressed as 1331 V(R) = &Z((R/Pj, where P is a characteristic energy such as the well depth, and R* is a characteristic distance such as the position of the minimum. The reduced interatomic potential u(x) is found to be approximately transferable from system to system [34] and the small differences between the reduced potentials are those that reflect subtle effects over and above the gross scale and size effects. We feel it may be useful to adopt a similar approach to Compton profiles and related functions. In order to proceed it is necessary to

AL

Thakknrrr

al. / Internnlly

define size and range units. Since an experimental Compton profile is best determined near the origin, it seems reasonable to use I(O) as the scale unit, and to define the range unit or characteristic momentum as follows q* = [-~(0)/(d’Jjdq’)s=,I]“”

= lJ(O)/Zr

I=f(O)]“‘.

(68) Notice that the experimentally accessible 4” is constructed from information which weights the small momentum region and hence the valence electrons. Thus the Compton profile may be written as f(q) =I(@

J&/4*),

(69)

where fr(.\-) is the redrrced Con~pron profile. It is easy to verify that ali one-electron ions such as H, He+, Li”, . . . have the snme reduced profile J,(X) = (1 +x2/6)-s,

(70)

with the size u_nitf(0) = 813~2 and the range unit q” = Z/46, where Z is the nuclear charge.

F_‘g. l-shows the reduced profiies Ir,(.r) = J(s)/J(O) of the atoms from H through Ne as a function of the dimensionless quantity .r = q/q*. Clearly it is only at larger values of q/4* that the subtle differences between the atoms show up. At q/q* = 4, the reduced profiles for Li

folded

181

demities

through Ne are ordered in the same manner as the columns of the periodic table. A reduced momentum density fiI,(s) consistent with the above reduced Compton profile

can be generated d&s)/d.r

from the analog of eq. (36)

= -2srl=ii,(x)s.

(71)

Thus it is clear that the appropriate scale unit is 27rii(O), the range unit is p* = q*, and we may write lTI(p) = 25;ii(O)Q&/p”).

(72)

For the one-electron ions the reduced momentum density takes the form l=&(r) = (25;)-‘(1+1’/6)-’

(73)

and the scale unit is 2~fT(0) = 16/irZ’. Fig. 2 shows the dimensionless densities I%(X) = fi(p)/2~rfI(O) for the atoms H through Ne as a function of the reduced momentum s = p/p”. Despite the fact that the dimensionless densities are all equal to 1/2a at the origin, they quickly fan out, and in the vicinity of p/p* = 0.5 the curves for N, 0, F and Ne are rather well separ-

ated with the latter three exhibiting maxima for non-zero p_ Evidently the dimensionless densities display chemical information over and above size and scale effects. A reduced internally folded density B,(x) consistent with the above reduced Compton profile

0.8

0

Fig. 1. Reduced Compton profiles for the atoms from hydrogen to neon. The curves are identified by the respective atomicnumbers.

05

1.0 P/p'

1.5

LO

Fig. 2. Dimensionless momentum densities for the atoms from hydrogen to neon. The curves are identified by the respective atomic numbers.

A.3.

182

can

be

generated

B,(x) = 2

I I,

Thakkor

et oi. / Itztemally

from the analog of eq. (37)

_Qy) cos (x!.) by.

(74)

&r/P).

For the one-electron folded density is BJx)

= (3&6/S)

(75)

ions the reduced internally

e-‘GX(l t&r

+2x’),

I 0

rz B(r) dr

/

,; B(r) dr.

in Hartree

atomic units

Atom

T(O)

nco,

cl*

r*

H He Li Be B C N 0 F Ne

C.8488 1.069 2.574 2.952 2.879 2.833 2.791 2.771 2.749 2.724

0.811 0.447 8.31 4.77

0.408 0.6!7 0.222 0.314

2.45 1.62 4.50 3.19

2.24 1.27 0.813 0.514 0.350 0.247

0.453 0.596 0.739 0.926 1.12 1.32

2.21 1.68 1.35 1.08 0.895 0.755

(76)

with the scale unit 7(0)/r*_= 8/3&z and the characteristic length r* = v/6/Z. Fig. 3 shows I?,(X) = r’B(r)/J(O) for the atoms from H through Ne as a function of x = r/r*, where the characteristic length r*= l/p*. From eqs. (50), (52) and (68) it is clear that (r*)2 =

densities

Table 1 Size and scale parameters

Thus it follows that the appropriate scale unit is J(O(o,g* = J(O)/r’, the characteristic length r* = l/p*, and we may write B(r) = [J(O)/r’]

folded

(77)

0

As might be expected from eq. (74), the fanning out of the reduced profiles seen at large q/q* in fig. 1 shows up at small r/r* in fig. 3. Finally the size and scale effects may be studied in table 1 which lists values of J(O), n(O), LJ* and r* for all the atoms considered in

this paper. The characteristic lengths show the expected decrease as one goes across the periodic table from lithium to neon. However, these Iengths cannot be expected to be proportionai to atomic radii because r*(He) > r*(Ne) despite the fact that r* weights the valence electrons. The weighting is verified by the following results computed from a near Hartree-Fock quality wavefunction for neon [35]: p&,, = 1.33, PLnce = 1.29, P;“,,, = 3.75 and, reciprocally, and rz,,, =0.267. r&l = 0.755, rzIcnce =0.775,

Acknowledgement The support of the Natural Sciences and Engineering Research Council of Canada is gratefully acknowledged. One of us (A.M.!%) wishes to thank the Conselho National de Desenvolvimento Cientifico e TechnolBgicoCNPq (Brazil) for a graduate fellowship.

References Cl]

0

0.5

1.0

s/r

1.5

2.0

3. Dimension!ess internally folded densities for the atoms from hydrogen to neon. ibe curves are identified by the respective atomic n*xmbers. Fig.

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WI A.J. Thakkar,