Quantum Stress Focusing in Descriptive Chemistry
aa r X i v : . [ c ond - m a t . m t r l - s c i ] J a n Quantum Stress Focusing in Descriptive Chemistry
Jianmin Tao , , Giovanni Vignale , and I. V. Tokatly , Theoretical Division and CNLS, Los Alamos National Laboratory, Los Alamos, New Mexico 87545 Department of Physics, University of Missouri-Columbia, Columbia, Missouri 65211 European Theoretical Spectroscopy Facility (ETSF),Dpto. Fisica Materiales, Universidad del Pais Vasco, 20018 Donostia, Spain Moscow Institute of Electronic Technology, Zelenograd, 124498 Russia (Dated: November 3, 2018)We show that several important concepts of descriptive chemistry, such as atomic shells, bondingelectron pairs and lone electron pairs, may be described in terms of quantum stress focusing , i.e.the spontaneous formation of high-pressure regions in an electron gas. This description subsumesprevious mathematical constructions, such as the Laplacian of the density and the electron local-ization function, and provides a new tool for visualizing chemical structure. We also show that thefull stress tensor, defined as the derivative of the energy with respect to a local deformation, can beeasily calculated from density functional theory.
PACS numbers: 71.15.Mb,31.15.Ew,71.45.Gm
Atomic shell structure, electron pair domains, π -electron subsystems, etc., are common concepts in de-scriptive chemistry and play a significant role in modernelectronic structure theory. These concepts help us tovisualize the bonding between atoms in terms of smallgroups of localized electrons (e.g. two electrons of oppo-site spin in a simple covalent bond) and therefore playan important role in predicting new molecular structuresand in describing structural changes due to chemical re-actions.A precise quantitative description of small groups oflocalized electrons has been sought for a long time [1, 2],but none of the solutions proposed so far is completelysatisfactory. The natural candidate – the electronic den-sity n ( r ) – has a clear physical meaning, but fails to re-veal quantum mechanical features such as atomic shellstructure [3] or the localization of electron pairs of oppo-site spin in a covalent bond. The Laplacian ∇ n of thedensity [1, 4] provides a better way to visualize molecu-lar geometry, but lacks a clear physical significance andfails [5] to reveal the atomic shell structure of heavyatoms. Recently, a very useful indicator of electron lo-calization has been constructed [5] from the curvatureof the (spherically averaged) conditional pair probabilityfunction P σσ ( r , r ′ ) (the probability of finding an electronof spin σ at r given that there is another electron of thesame spin orientation at r ′ ) evaluated at r = r ′ . In thesimplest approximation this curvature is proportional to D σ ( r ) = τ σ − |∇ n σ | n σ , (1)where τ σ ( r ) = P l |∇ ψ lσ ( r ) | is the non-interacting ki-netic energy density of σ -spin electrons (atomic units m = ~ = e = 1 are used throughout), ψ lσ ( r ) are theoccupied Kohn-Sham orbitals of spin σ , and n σ is the σ -spin electron density ( n = n ↑ + n ↓ ). From D σ one constructs the electron localization function (ELF)[5] η σ ( r ) = 11 + ( D σ /τ TF σ ) , (2)where τ TF σ = (3 / π ) / n / σ is the Thomas-Fermikinetic energy density of σ -spin electrons. The ELF pro-vides excellent visualization of atomic shells as well asa quantitative description of valence shell electron pairrepulsion theory [6] of bonding and it has been recentlyextended to excited and time-dependent states [7]. Nev-ertheless, it has two defects: (1) it remains a mathemat-ical construction of dubious physical significance [8] and(2) it is difficult to calculate ELF beyond [9] the lowest-order approximation [Eq. (1)] and for this reason it canhardly be applied to strongly interacting systems.In this paper we propose a more physical way of lookingat atomic shells and bonds based on the idea of quantumstress focusing , by which we mean the spontaneous for-mation of regions of high and low pressure in the electrongas. The existence of such regions is nontrivial in viewof the fact that the electrostatic potential created by thenuclei has no maxima or minima (Earnshaw’s theorem).To understand how pressure maxima can arise we mustrecall a few basic concepts from the mechanics of contin-uum media. First, we introduce the stress tensor [10, 11], p ij ( r ), – a symmetric rank-2 tensor with the propertythat p ij is the i-th component of the force per unit areaacting on an infinitesimal surface perpendicular to thej-th axis [12]. The divergence of this tensor, P j ∂ j p ij ( r )( ∂ j being the derivative with respect to r j , the j -th com-ponent of r ), is the i -th component of the force per unitvolume exerted on an infinitesimal element of the elec-tron gas by the surrounding electrons. For convenience,in the following we will exclude from the stress tensor theelectrostatic Hartree contribution, which is then treatedas an external field on equal footing with the field of thenuclei. L r (bohr) p S p Ar atomK L M 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.5 1 1.5 2 2.5 3 3.5 4 L r (bohr) p S pZn atomKL M N 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.5 1 1.5 2 2.5 3 3.5 4 L r (bohr) p S pXe atomKL M N O FIG. 1: Indicators of electron localization as functions ofthe radial distance r constructed respectively from the non-interacting and interacting pressures p S and p [Eq. (9)] forthe Ar, Zn, and Xe atoms. The XC part of the pressure isevaluated with the GGA functional of Ref. [16]. The maximaof the pressure reveal electronic shells, indicated as usual bythe capital letters K,L,M,... Next, we separate p ij into a hydrostatic pressure part,which is isotropic, and a shear part, which is traceless: p ij ( r ) = δ ij p ( r ) + π ij ( r ) , Tr π ij = 0 . (3)Here p ( r ) ≡ Tr p ij ( r ) is the quantum pressure, i.e. thederivative of the internal energy with respect to a localdeformation that changes the volume of a small elementof the electron liquid without changing its shape. Sim-ilarly, the shear stress π ij ( r ) is the derivative of the in-ternal energy with respect to a local deformation thatchanges the shape of a small element of the liquid, with-out changing its volume. In a quantum system, these lo-cal deformations are implemented through a local changein the metrics (see Eq. (11) below and Ref. 11 for details). In the ground state of an electronic system the forces aris-ing from the divergence of p ij must exactly balance theexternal force exerted by the nuclei. In other words, thestresses satisfy the equilibrium condition − n ( r ) ∂ i v ( r ) = ∂ i p ( r ) + X j ∂ j π ij ( r ) , (4)where v ( r ) is the external potential due to the nuclei plusthe Hartree potential.Eq. (4) is not very useful if we don’t have explicit ex-pressions for p and π ij . However, certain approxima-tions can be made. For example, replacing the exactquantum pressure by the quasi-classical Fermi pressure p TF ( r ) = (3 π ) / n / and neglecting the shear termyields the well-known Thomas-Fermi equation for theequilibrium density. Because the electrostatic potential v ( r ) in an atom does not admit maxima and minima, weimmediately see that this approximation (as any approx-imation that neglects π ) cannot yield local maxima orminima in the pressure. In the special case of the spheri-cal Thomas-Fermi atom this implies that the density andthe pressure are both monotonically decreasing functionsof the radial distance r : the shell structure is absent.The situation changes radically when the shear stressis included. Now it is possible for the hydrostatic force ∂ i p to vanish because the electrostatic force n∂ i v canbe balanced by the shear force P j ∂ j π ij . It is there-fore possible to have maxima or minima in the pressure.We may picture the regions of high pressure as regionswhich are hard to compress, because of the high en-ergy cost of bringing the particles closer together againstthe “exchange-correlation hole”. These “maximally com-pressed regions” quite naturally correspond to the shellsand bonds of descriptive chemistry.We now show that regions of maximum and minimumpressure do occur in spherical atoms, with peaks corre-sponding to the K,L,M... shells of the standard shellmodel. To see this we don’t need to go any furtherthan the lowest-order approximation for the stress tensor,which has the form: p S ij = 12 X lσ ( ∂ i ψ ∗ lσ ∂ j ψ lσ + ∂ j ψ ∗ lσ ∂ i ψ lσ ) − δ ij ∇ n, (5)where the sum runs over the occupied Kohn-Sham or-bitals. For an atom of spherical symmetry this furthersimplifies to p S ij = p S δ ij + π S (cid:18) r i r j r − δ ij (cid:19) , (6)where p S is the noninteracting pressure given by p S = 13 X lσ |∇ ψ lσ | − ∇ n, (7)and π S = X lσ (cid:0) | ∂ r ψ lσ | − r − | ∂ θ ψ lσ | (cid:1) . (8) -5 0 5 10 15 20 25 0 0.5 1 1.5 2 2.5 3 3.5 4r (bohr)KL MK-LL-M Ar atomrp S (r)/n(r) r p S (r)/n(r) FIG. 2: Non-interacting pressure ( p S ) and shear stress ( π S )pressure in Ar. The peaks in p S identify the position of theshells and the peaks in π S define the boundary between theshells. Inset: distribution of stresses in the atom. The greyscale indicates the pressure distribution with dark (light) re-gions corresponding to high (low) value of the pressure in-dicator L (Eq. 9). The arrows indicate the direction of theshear force (2 / ∇ π + (2 /r ) π . Observe how the shells are“squeezed” by shear forces pointing in opposite directions. To graphically represent the noninteracting pressure p S , we define the dimensionless quantity ˜ p S = p S /p TF .Since this ratio diverges both at a nucleus (with a positivesign) and in the density tail (with a negative sign) [13],we find it convenient to plot the function L ( r ) = 12 p S p p S ) ! (9)which has values in the range (0 , L = 1 cor-responding to ˜ p S = ∞ and L = 0 corresponding to˜ p S = −∞ . Figures 1a–1c show conclusively the exis-tence of peaks and troughs of the noninteracting pres-sure (solid curves), each of which defines a spherical sur-face on which the gradient of p S vanishes. Since p TF is amonotonically decreasing function, the oscillatory behav-ior of these graphs is entirely due to p S : in particular, themaxima and minima of L are very close to the maximaand minima of p S . As discussed above, we interpret thepeaks in pressure as quantum shells. Figure 2 shows thebehavior of p S and π S as functions of r , and the inset de-picts schematically the distribution of shear forces in theatom. Observe that the minima of the shear stress arein correspondence with the maxima of the pressure, andviceversa. This is natural, because the gradients of p and π , i.e., approximately, the bulk force and the shear forcemust have opposite signs and largely cancel in order tobalance the relatively weak nuclear attraction. The factthat the shell is located at the minimum of the shearstress implies that the shear force has opposite signs onthe two sides of the shell, thus “locking” the shell intoposition.Two more points should be made about Eq. (7) for thepressure. The first is that the Laplacian of the density appears in it prominently, as a universal component ofthe pressure: this explains a posteriori the partial suc-cess of the Laplacian as an indicator of shell structure.More interestingly, the ELF (Eq. 2) is also closely relatedto the pressure. Indeed, the non-interacting pressure in-cludes a “bosonic” contribution, p , which is obtainedfrom Eq. (7) by putting all the electrons in the lowestenergy orbital ψ σ : p σ = (1 / |∇ n σ | / (4 n σ ) − (3 / ∇ n σ ] . (10)The excess pressure p ex σ ≡ p S σ − p σ ( r ) is called “Paulipressure” (since it is generated by the Pauli exclusionprinciple, which forces the occupation of higher energyorbitals) and is easily seen to coincide with D σ , the cur-vature of the conditional probability function (Eq. (1))and main ingredient of the ELF. Physically, high elec-tronic pressure means that it is difficult to bring the par-ticles closer together, which implies that the XC holesurrounding the electron is very “deep”, i.e. the proba-bility of finding another electron of the same spin aroundthe reference electron is low. This leads to a high valueof the ELF.One of the attractive features of our proposal is therelative ease with which strong electron-electron inter-actions can be included in the calculation of the stresstensor. Already p S includes some interaction effectsthrough the Kohn-Sham orbitals [14]. To include addi-tional exchange-correlation (XC) effects we observe thatin density functional theory (DFT) the XC energy is afunctional not only of the density n , but also, tacitly, ofthe metrics g ij of the space[11]: E xc = E xc [ n, g ij ].In conventional applications of DFT, one works withfixed Euclidean metrics g ij = δ ij , so there is no need toemphasize the dependence on the metrics. Here, how-ever, we are interested in small deformations, which canbe described by the coordinate transformation [15] ξ ( r ) = r − u ( r ), where u is an infinitesimal nonlinear functionof r . The transformation from r to ξ changes the densityfrom n ( r ) to, up to first order in u , ˜ n ( ξ ) = n ( ξ )+ u i ∂ i n ( ξ )(notice that i here are cartesian indices with the conven-tion that repeated indices are summed over), and themetric tensor from the Euclidean value δ ij to g ij ≡ ( ∂r α /∂ξ i )( ∂r α /∂ξ j ) = δ ij + ∂ i u j + ∂ j u i . (11)Since the XC energy cannot depend upon the arbi-trary choice of coordinates we must have E xc [ n, δ ij ] = E xc [˜ n, g ij ], which for an arbitrary infinitesimal transfor-mation implies n∂ i δE xc δn = ∂ j (cid:26) δ ij n δE xc δn − δE xc δg ij (cid:27) , (12)where the derivative with respect to n is taken at con-stant g ij and viceversa. Since the left-hand side of thisequation is the XC force density, the right-hand side mustbe the divergence of the XC stress tensor. This leads tothe result p xc ij = δ ij nv xc − δE xc δg ij , (13)where v xc is the XC potential defined by v xc ≡ δE xc /δn ,and the functional derivative is evaluated at g ij = δ ij .This important result shows that we may calculate theXC contribution to the stress tensor from a knowledge ofthe XC energy as a functional of density and metrics. Itturns out that semi-local XC functionals [16, 17] are of aform that can be easily generalized to include a nontrivialmetrics.For example, consider the generalized gradient approx-imation (GGA) for a spin-unpolarized system. The stan-dard form of this functional is E xc [ n ] = R d r e xc ( n, s ),where s = |∇ n | / (2 k F n ) is the reduced density gradient.Going to curvilinear coordinates we immediately obtainthe generalized form E xc [ n, g ij ] = Z d ξ p g ( ξ ) e xc (˜ n, ˜ s ) , (14)where g ≡ det( g ij ) is the determinant of the metrictensor, ˜ n ( ξ ) ≡ n ( r ( ξ )) is the density in the new co-ordinates, and ˜ s = p g ij ∂ i ˜ n∂ j ˜ n/ (2˜ k F ˜ n ), where ˜ k F =(3 π ˜ n ) / and g ij = ( g − ) ij . Using the identity δg = gg ij δg ij = − gg ij δg ij , we evaluate the functional deriva-tive of E xc [˜ n, g ij ] with respect to g ij , and then substituteit into Eq. (13). We finally obtain p xc ij = δ ij ( nv xc − e xc ) − ∂ i n∂ j n k F n |∇ n | ∂e xc ∂s . (15)The trace of this yields the XC pressure p xc = nv xc − e xc − s ∂e xc ∂s , (16)which exactly recovers the uniform-gas limit [10].In Figs. 1a–1c we have also plotted the normal-ized pressure L ( r ) including the XC correction (dashedcurves), which is calculated by the GGA of Ref. [16]. Inthe core region of the atom, where the density is high, p xc is utterly negligible. In the valence region, the XC con-tribution becomes relatively important, so the differenceis noticeable. In the density tail, both p S and p xc decayexponentially, causing the difference to be small. It ispossible, however, and indeed quite likely, that XC con-tributions become more important in strongly correlatedsystems, where the zeroth-order description provided bythe Kohn-Sham orbitals begins to break down. This re-mains a subject for future investigations.In conclusion, we have found that the analysis of thequantum stress field reveals important features of theelectronic structure of atoms. Although we have con-sidered only atoms so far, there is every reason to expect that stress focusing will also occur in molecules in cor-respondence of electron pair domains, such as the onesthat are commonly associated with single and multiplecovalent bonds and with the so-called lone-pair regionsof the molecule. In general, the local pressure – the traceof the stress tensor – is expected to provide an excellentquantitative description of electron localization. Moreinformation may emerge from the complete study of thestress tensor (its eigenvalues and principal axes) and itstopology – a study that is just beginning to be under-taken.We acknowledge valuable discussions with C.A. Ullrichand P.L. de Boeij. This work was supported by DOEunder Grant No. DE-FG02-05ER46203, and by DOEunder Contract No. DE-AC52-06NA25396 and GrantNo. LDRD-PRD X9KU at LANL (J.T). [1] R.F. W. Bader, Atoms in Molecules - A Quantum Theory (Oxford University Press, Oxford, 1990).[2] R.J. Gillespie and I. 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