Quantum Stress Focusing in Descriptive Chemistry

0.26 MB·English

by  Jianmin Tao

Quantum Stress Focusing in Descriptive Chemistry
Jianmin Tao1,2, Giovanni Vignale2, and I. V. Tokatly3,4
1Theoretical Division and CNLS, Los Alamos National Laboratory, Los Alamos, New Mexico 87545
2Department of Physics, University of Missouri-Columbia, Columbia, Missouri 65211
3European Theoretical Spectroscopy Facility (ETSF),
Dpto. Fisica Materiales, Universidad del Pais Vasco, 20018 Donostia, Spain
4Moscow Institute of Electronic Technology, Zelenograd, 124498 Russia
(Dated: February 2, 2008)
8 Weshowthatseveralimportantconceptsofdescriptivechemistry,suchasatomicshells, bonding
0 electron pairs and lone electron pairs, may be described in terms of quantum stress focusing, i.e.
0 the spontaneous formation of high-pressure regions in an electron gas. This description subsumes
2 previous mathematical constructions, such as the Laplacian of the density and the electron local-
ization function, and providesa new tool for visualizing chemical structure. Wealso show that the
n
fullstresstensor,definedasthederivativeoftheenergywithrespect toalocaldeformation, canbe
a
J easily calculated from density functional theory.

4
PACSnumbers: 71.15.Mb,31.15.Ew,71.45.Gm
1

]
i Atomic shell structure, electron pair domains, π- constructs the electron localization function (ELF)[5]
c
electron subsystems, etc., are common concepts in de-
s
rl- seclerciptrtoivneicchsetrmuicstturyreatnhdeoprlay.y aTshiegsneificcoanncterpotlsehinelpmoudsertno ησ(r)= 1+(Dσ1/τσTF)2, (2)
t
m visualize the bonding between atoms in terms of small
. groups of localized electrons (e.g. two electrons of oppo- where τσTF = (3/10)(6π2)2/3n5σ/3 is the Thomas-Fermi
at site spin in a simple covalent bond) and therefore play kinetic energydensity of σ-spinelectrons. The ELFpro-
m animportantroleinpredictingnewmolecularstructures vides excellent visualization of atomic shells as well as
- and in describing structural changes due to chemical re- a quantitative description of valence shell electron pair
d actions. repulsion theory [6] of bonding and it has been recently
n
extended to excited and time-dependent states [7]. Nev-
o A precise quantitative description of small groups of
ertheless, it has two defects: (1) it remains a mathemat-
c localizedelectronshas been soughtfora long time [1, 2],
[ but none of the solutions proposed so far is completely ical construction of dubious physical significance [8] and

(2) it is difficult to calculate ELF beyond [9] the lowest-
1 satisfactory. The naturalcandidate – the electronic den-
v sity n(r) – has a clear physical meaning, but fails to re- order approximation [Eq. (1)] and for this reason it can
7 veal quantum mechanical features such as atomic shell hardly be applied to strongly interacting systems.
1 Inthispaperweproposeamorephysicalwayoflooking
structure[3]orthe localizationofelectronpairsofoppo-
1 site spin in a covalent bond. The Laplacian ∇2n of the atatomic shellsandbondsbasedonthe idea ofquantum
2
stress focusing, by which we mean the spontaneous for-
. density [1, 4] provides a better way to visualize molecu-
1 mationofregionsofhighandlowpressureintheelectron
lar geometry, but lacks a clear physical significance and
0
gas. The existence of such regions is nontrivial in view
fails [5] to reveal the atomic shell structure of heavy
8
ofthe factthat the electrostaticpotentialcreatedby the
0 atoms. Recently, a very useful indicator of electron lo-
: calization has been constructed [5] from the curvature nuclei has no maxima or minima (Earnshaw’s theorem).
v
i ofthe (sphericallyaveraged)conditionalpairprobability Tounderstandhowpressuremaximacanarisewemust
X functionP (r,r′)(theprobabilityoffindinganelectron recallafew basicconcepts fromthe mechanicsofcontin-
σσ
r of spin σ at r given that there is another electron of the uummedia. First,weintroducethestresstensor[10,11],
a same spin orientation at r′) evaluated at r = r′. In the pij(r), – a symmetric rank-2 tensor with the property
simplest approximationthis curvature is proportional to that pij is the i-th component of the force per unit area
acting on an infinitesimal surface perpendicular to the
j-th axis [12]. The divergence of this tensor, ∂ p (r)
|∇n |2 j j ij
Dσ(r)=τσ − 8nσσ , (1) (p∂ojnbenetinogftrh)e, idsetrhiveatii-vthewcoitmhproenspenecttotfothrje,ftohrPecej-pthercuonmit-
volume exerted on an infinitesimal element of the elec-
where τ (r) = 1|∇ψ (r)|2 is the non-interacting ki- tron gas by the surrounding electrons. For convenience,
σ l 2 lσ
netic energy density of σ-spin electrons (atomic units inthefollowingwewillexcludefromthestresstensorthe
m = ~ = e2 =P1 are used throughout), ψ (r) are the electrostatic Hartree contribution, which is then treated
lσ
occupied Kohn-Sham orbitals of spin σ, and n is the as an externalfield on equal footing with the field of the
σ
σ-spin electron density (n = n + n ). From D one nuclei.
↑ ↓ σ
2
1 Inthegroundstateofanelectronicsystemtheforcesaris-
K L M
0.9 ing from the divergence of p must exactly balance the
ij
0.8 pS Ar atom external force exerted by the nuclei. In other words, the
0.7
stresses satisfy the equilibrium condition
0.6 p
L 0.5 −n(r)∂ v(r)=∂ p(r)+ ∂ π (r), (4)
i i j ij
0.4
j
0.3 X
0.2 wherev(r)istheexternalpotentialduetothenucleiplus
0.1 the Hartree potential.
0 0 0.5 1 1.5 2 2.5 3 Eq. (4) is not very useful if we don’t have explicit ex-
r (bohr) pressions for p and π . However, certain approxima-
ij
1 K tions can be made. For example, replacing the exact
0.9 L M
quantum pressure by the quasi-classical Fermi pressure
0.8 pTF(r) = 2(3π2)2/3n5/3 and neglecting the shear term
0.7 Zn atom 5
0.6 yields the well-known Thomas-Fermi equation for the
L 0.5 equilibrium density. Because the electrostatic potential
0.4 v(r) inanatomdoes notadmitmaximaandminima, we
pS
0.3 N p immediatelyseethatthisapproximation(asanyapprox-
0.2
imation that neglects π) cannot yield local maxima or
0.1
minimainthe pressure. Inthe specialcaseofthe spheri-
0
0 0.5 1 1.5 2 2.5 3 3.5 4 calThomas-Fermiatomthisimpliesthatthedensityand
r (bohr)
thepressurearebothmonotonicallydecreasingfunctions
of the radial distance r: the shell structure is absent.
1
0.9 LKM N The situation changes radically when the shear stress
O
0.8 Xe atom is included. Now it is possible for the hydrostatic force
0.7 pS ∂ip to vanish because the electrostatic force n∂iv can
0.6
be balanced by the shear force ∂ π . It is there-
L 0.5 p fore possible to have maxima or minjimjaiijn the pressure.
0.4 P
We may picture the regions of high pressure as regions
0.3
0.2 which are hard to compress, because of the high en-
0.1 ergycostofbringingthe particlesclosertogetheragainst
0 the“exchange-correlationhole”. These“maximallycom-
0 0.5 1 1.5 2 2.5 3 3.5 4
r (bohr) pressed regions” quite naturally correspond to the shells
and bonds of descriptive chemistry.
FIG. 1: Indicators of electron localization as functions of
We now show that regionsof maximum and minimum
the radial distance r constructed respectively from the non-
interacting and interacting pressures pS and p [Eq. (9)] for pressure do occur in spherical atoms, with peaks corre-
sponding to the K,L,M... shells of the standard shell
the Ar, Zn, and Xe atoms. The XC part of the pressure is
evaluatedwiththeGGAfunctionalofRef.[16]. Themaxima model. To see this we don’t need to go any further
of the pressure reveal electronic shells, indicated as usual by thanthelowest-orderapproximationforthestresstensor,
thecapital letters K,L,M,... which has the form:
1 1
pS = (∂ ψ∗ ∂ ψ +∂ ψ∗ ∂ ψ )− δ ∇2n, (5)
ij 2 i lσ j lσ j lσ i lσ 4 ij
Next, we separatepij into a hydrostaticpressurepart, Xlσ
which is isotropic, and a shear part, which is traceless:
where the sum runs over the occupied Kohn-Sham or-
bitals. For an atom of spherical symmetry this further
p (r)=δ p(r)+π (r), Tr π =0. (3)
ij ij ij ij simplifies to
Here p(r)≡ 13Tr pij(r) is the quantum pressure, i.e. the S S S rirj δij
p =p δ +π − , (6)
derivative of the internal energy with respect to a local ij ij r2 3
(cid:18) (cid:19)
deformation that changes the volume of a small element
where pS is the noninteracting pressure given by
of the electron liquid without changing its shape. Sim-
ilarly, the shear stress πij(r) is the derivative of the in- pS = 1 |∇ψ |2− 1∇2n, (7)
ternal energy with respect to a local deformation that 3 lσ 4
lσ
changes the shape of a small element of the liquid, with- X
out changingits volume. In a quantum system, these lo- and
caldeformationsareimplementedthroughalocalchange πS = |∂ ψ |2−r−2|∂ ψ |2 . (8)
r lσ θ lσ
inthemetrics(seeEq.(11)belowandRef.11fordetails).
lσ
X(cid:0) (cid:1)
3
25 appears in it prominently, as a universal component of
K-L Ar atom the pressure: this explains a posteriori the partial suc-
20
K cess of the Laplacian as an indicator of shell structure.
15 L-M Moreinterestingly,theELF(Eq.2)isalsocloselyrelated
to the pressure. Indeed, the non-interacting pressure in-
10
cludes a “bosonic” contribution, p0, which is obtained
5 L from Eq. (7) by putting all the electrons in the lowest
0 M rp S(r)/n(r) energy orbital ψ0σ:
-5 rpS(r)/n(r) p0σ =(1/3)[|∇nσ|2/(4nσ)−(3/4)∇2nσ]. (10)
0 0.5 1 1.5 2 2.5 3 3.5 4
r (bohr) The excess pressure pexσ ≡ pSσ −p0σ(r) is called “Pauli
pressure” (since it is generated by the Pauli exclusion
FIG. 2: Non-interacting pressure (pS) and shear stress (πS)
pressure in Ar. The peaks in pS identify the position of the principle, which forces the occupation of higher energy
shells and the peaks in πS define the boundary between the orbitals) and is easily seen to coincide with Dσ, the cur-
shells. Inset: distribution of stresses in the atom. The grey vature of the conditional probability function (Eq. (1))
scale indicates the pressure distribution with dark (light) re- and main ingredient of the ELF. Physically, high elec-
gions corresponding to high (low) value of the pressure in- tronicpressuremeansthatitisdifficulttobringthepar-
dicator L (Eq. 9). The arrows indicate the direction of the ticles closer together, which implies that the XC hole
shear force (2/3)∇π +(2/r)π. Observe how the shells are
surrounding the electron is very “deep”, i.e. the proba-
“squeezed” by shear forces pointing in opposite directions.
bilityoffindinganotherelectronofthesamespinaround
the reference electron is low. This leads to a high value
of the ELF.
To graphically represent the noninteracting pressure
pS, we define the dimensionless quantity p˜S = pS/pTF. One of the attractive features of our proposal is the
relative ease with which strong electron-electron inter-
Sincethisratiodivergesbothatanucleus(withapositive
actions can be included in the calculation of the stress
sign) and in the density tail (with a negative sign) [13],
S
tensor. Already p includes some interaction effects
we find it convenient to plot the function
through the Kohn-Sham orbitals [14]. To include addi-
1 p˜S tional exchange-correlation(XC) effects we observe that
L(r)= 1+ (9) in density functional theory (DFT) the XC energy is a
2 1+(p˜S)2!
functional not only of the density n, but also, tacitly, of
which has values in the ranpge (0,1), with L = 1 cor- the metrics gij of the space[11]: Exc =Exc[n,gij].
S In conventional applications of DFT, one works with
responding to p˜ = ∞ and L = 0 corresponding to
p˜S = −∞. Figures 1a–1c show conclusively the exis- fixed Euclidean metrics gij = δij, so there is no need to
emphasize the dependence on the metrics. Here, how-
tence of peaks and troughs of the noninteracting pres-
ever, we are interested in small deformations, which can
sure (solid curves),eachof whichdefines a sphericalsur-
face on which the gradientof pS vanishes. Since pTF is a bedescribedbythecoordinatetransformation[15]ξ(r)=
r−u(r), where u is an infinitesimal nonlinear function
monotonicallydecreasingfunction,theoscillatorybehav-
iorofthesegraphsisentirelyduetopS: inparticular,the ofr. Thetransformationfromrtoξ changesthedensity
fromn(r)to,uptofirstorderinu,n˜(ξ)=n(ξ)+u ∂ n(ξ)
maxima and minima of L are very close to the maxima i i
and minima of pS. As discussed above, we interpret the (notice that i here arecartesianindices with the conven-
tion that repeated indices are summed over), and the
peaks in pressure as quantum shells. Figure 2 shows the
behaviorofpS andπS asfunctions ofr, andtheinsetde- metric tensor from the Euclidean value δij to
picts schematicallythe distributionofshearforcesinthe
g ≡(∂r /∂ξ )(∂r /∂ξ )=δ +∂ u +∂ u . (11)
atom. Observe that the minima of the shear stress are ij α i α j ij i j j i
in correspondence with the maxima of the pressure, and
Since the XC energy cannot depend upon the arbi-
viceversa. Thisisnatural,becausethegradientsofpand
trary choice of coordinates we must have Exc[n,δij] =
π, i.e., approximately, the bulk force and the shear force
Exc[n˜,gij], which for an arbitrary infinitesimal transfor-
must have opposite signs and largely cancel in order to
mation implies
balance the relatively weak nuclear attraction. The fact
that the shell is located at the minimum of the shear δExc δExc δExc
n∂ =∂ δ n −2 , (12)
stress implies that the shear force has opposite signs on i δn j ij δn δg
(cid:26) ij (cid:27)
the two sides of the shell, thus “locking” the shell into
position. where the derivative with respect to n is taken at con-
TwomorepointsshouldbemadeaboutEq.(7)forthe stant g and viceversa. Since the left-hand side of this
ij
pressure. The first is that the Laplacian of the density equationistheXCforcedensity,theright-handsidemust
4
be the divergence of the XC stress tensor. This leads to that stress focusing will also occur in molecules in cor-
the result respondence of electron pair domains, such as the ones
that are commonly associated with single and multiple
pxijc =δijnvxc−2δδEgxc, (13) covalent bonds and with the so-called lone-pair regions
ij ofthemolecule. Ingeneral,thelocalpressure–thetrace
of the stress tensor – is expected to provide an excellent
where vxc is the XC potential defined by vxc ≡δExc/δn,
quantitative description of electron localization. More
and the functional derivative is evaluated at g = δ .
ij ij information may emerge from the complete study of the
This important result shows that we may calculate the
stress tensor (its eigenvalues and principal axes) and its
XCcontributiontothestresstensorfromaknowledgeof
topology – a study that is just beginning to be under-
the XC energy as a functional of density and metrics. It
taken.
turns outthat semi-localXC functionals [16, 17]are ofa
formthatcanbeeasilygeneralizedtoincludeanontrivial WeacknowledgevaluablediscussionswithC.A.Ullrich
metrics. and P.L. de Boeij. This work was supported by DOE
Forexample,considerthegeneralizedgradientapprox- under Grant No. DE-FG02-05ER46203, and by DOE
imation(GGA)foraspin-unpolarizedsystem. Thestan- under Contract No. DE-AC52-06NA25396 and Grant
dard form of this functional is Exc[n] = drexc(n,s), No. LDRD-PRD X9KU at LANL (J.T).
where s = |∇n|/(2k n) is the reduced density gradient.
F
R
Going to curvilinear coordinates we immediately obtain
the generalized form
[1] R.F.W.Bader,AtomsinMolecules-AQuantumTheory
Exc[n,gij]= dξ g(ξ)exc(n˜,s˜), (14) (Oxford UniversityPress, Oxford, 1990).
Z [2] R.J. Gillespie and I. Hargittai, The Vsepr Model of
p
where g ≡ det(g ) is the determinant of the metric Molecular Geometry (Allyn & Bacon, 1992).
tensor, n˜(ξ) ≡ ni(jr(ξ)) is the density in the new co- [3] P.W. Ayers and R.G. Parr, Int. J. Quantum Chem. 95,
ordinates, and s˜ = gij∂ n˜∂ n˜/(2k˜ n˜), where k˜ = 877 (2003).
i j F F [4] R.F.W. Bader and H. Essen, J. Chem. Phys. 80, 1943
(3π2n˜)1/3 and gij =p(g−1)ij. Using the identity δg = (1984).
ggijδgij = −ggijδgij, we evaluate the functional deriva- [5] A.D. Becke and K.E. Edgecombe, J. Chem. Phys. 92,
tiveofExc[n˜,gij]withrespecttogij,andthensubstitute 5397 (1990).
it into Eq. (13). We finally obtain [6] R.J.Gillespie, Molecular Geometry(VanNostrandRein-
hold, London, 1972).
pxijc =δij(nvxc−exc)− 2k∂inn∂|∇jnn|∂∂esxc . (15) [7] TR.evB.uArn7u1s,,0M10.A50.L1(.RM)a(r1q9u8e5s),. and E.K.U. Gross, Phys.
F [8] A. Savin,J. Mol. Struc: (Theochem) 727, 127 (2005).
[9] E. Matito, B. Silvi, M. Duran, and M. Sol´a, J. Chem.
The trace of this yields the XC pressure
Phys. 125, 024301 (2006).
[10] O.H. Nielsen and R.M. Martin, Phys. Rev. B 32, 3780
xc s∂exc
p =nvxc−exc− , (16) (1985).
3 ∂s [11] I.V.Tokatly,Phys.Rev.B71,165104(2005);71,165105
(2005); 75, 125105 (2007).
which exactly recoversthe uniform-gas limit [10].
[12] pij(r)isformallydefinedasthederivativeoftheinternal
In Figs. 1a–1c we have also plotted the normal- energy with respect to a local strain uij(r), arising from
ized pressure L(r) including the XC correction (dashed a deformation in which each particle is displaced from
curves), which is calculated by the GGA of Ref. [16]. In rn to rn +u(rn), with uij ≡ (∂iuj +∂jui)/2. This de-
thecoreregionoftheatom,wherethedensityishigh,pxc formation corresponds to a local change in metrics [see
is utterly negligible. In the valence region, the XC con- Eq. (11)] which modifies the Hamiltonian according to
the formulas given in Ref. [11].
tribution becomes relatively important, so the difference
is noticeable. In the density tail, both pS and pxc decay [13] The positive divergence at a nucleus is caused by the
Laplacianofthedensity,whichdivergesas1/rforr→0.
exponentially, causing the difference to be small. It is
The negative divergence in the density tail (r → ∞)
possible, however, and indeed quite likely, that XC con- reflects the negative value of pS in the non-classical
tributions becomemoreimportantinstronglycorrelated regions where the wave function decays exponentially:
systems, where the zeroth-order description provided by pS r→→∞−4n|ǫKmSax|/3, where ǫKmSax is thehighest occupied
the Kohn-Sham orbitals begins to break down. This re- Kohn-Sham orbital energy of the atom.
[14] W.KohnandL.J.Sham,Phys.Rev.140,A1133(1965).
mains a subject for future investigations.
[15] J. Tao, G. Vignale, and I.V. Tokatly, Phys. Rev. B 76,
In conclusion, we have found that the analysis of the
195126 (2007).
quantum stress field reveals important features of the
[16] J.P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev.
electronic structure of atoms. Although we have con- Lett. 77, 3865 (1996).
sideredonlyatomssofar,thereiseveryreasontoexpect [17] J.Tao,J.P.Perdew,V.N.Staroverov,andG.E.Scuseria,
5
Phys.Rev.Lett. 91, 146401 (2003).