It is shown that the electronic contribution to the dipole moment, calculated from a solution to the Thomas—Fermi—Dirac or Thomas—Fermi equations, should be equal and opposite to the nuclear contribution. Thus, the Thomas—Fermi—Dirac and Thomas—Fermi theories predict vanishing dipole moments for all molecular systems

Goodisman, Jerry

English

Syracuse University Research Facility and Collaborative Environment

Syracuse University SURFACE Chemistry - Faculty Scholarship College of Arts and Sciences 9-21-1972 Dipole Moments in Thomas-Fermi-Dirac and Thomas-Fermi Theories. Jerry Goodisman Syracuse University Follow this and additional works at: https://surface.syr.edu/che  Part of the Chemistry Commons Recommended Citation Goodisman, J. (1972). Dipole moments in thomas-fermi-dirac and thomas-fermi theories. Chemical Physics Letters, 17(4), 471-473. This Article is brought to you for free and open access by the College of Arts and Sciences at SURFACE. It has been accepted for inclusion in Chemistry - Faculty Scholarship by an authorized administrator of SURFACE. For more information, please contact surface@syr.edu. Volume 17, number 4 CHEMICAL PHYSICS LETTERS 15 December 1972 DIFULE MOMENTS IN THOMAS-FERMI-DIRAC AND TMOMAS-FERIMI TREORIES It is shown that the electronic contributiori to the dipok moment, calculated from 8 solution to the Thomas- Fermi-Dirac or Thomas-Fermi equations, should be equal and opposite to the nuclear contribution. Thus, the Thomas-Fermi-Dirac and Thomas-Fermi theories predict vanishing: dipole moments for at1 mokculrrr systems. Recently, we have been performing calcdations of molecular electronic charge distributions according to the Thomas- Fermi-Dirac theory. Our method [I] permits treatment of anjjt cylindrically symmetric molecule; only for homopolar diatomic systems [I& 3) have accurate solutions to the TFD equation been presented so far. For the heteropolar diatomic moIe- cules we treated, we found an electric dipole moment (nuclear minus electronic contributions) of zero, within the estimated accuracy of our calculations. In the present article, it is shown that the moment should be exactly zero for the TFD density of a neutral system. Brinkman and Peperxak [41, after fmding zero moments from Thomas-Femli calculations on H,O, presented a proof that the Thomas-Fermi (TF) densi- ty always gives zero moments. The proof has been criticized [S] , since it assumes that the potential goes as r-4 when r, the distance from the nuclei, is large. This assumption permits one to state that a certain in- tegraf over a sphere of radius r approaches zero as r approaches infinity. However, assuming that the po- tential goes as rW4 is equivalent o assuming the non- existence of a dipoie moment. The Thomas-Fermi- Dirac (TFD) electron density is finite in extent, so that integrals over infinitely Iarge surfaces will not oc- cur in our proof. Having proved the theorem for TFD, we will show how it may be extended to cover TF. in our proof, we will require some results given by Sheldon [25. Some of his proofs will also be repeated in a notation more convenient for our use. In the TED theory, the ciectran density p is chosen to minimize with the normalization constraint N=Jdr/J. (2) Here, VN is the potential due to the nuclei, ok = $j’(~Zi2)2’3&Z~, KG = g(3/;l)“3e2, and N is the num- ber of electrons. Let nucleus A, with charge Z,, be located at rA, with cz, =N, hf for a neutral system. By making W - tiV stationary, with h a Lagrange multiplier, one finds +kp2’3 - +JpJ + p* = h , (4 where Sheidon [ZI shows one must consider the possibility of a discontinuous soiution, i.e., p = 0 outside a boundary surface C. The boundary is dhos,on to m&e the ener= a Volume 17, number 4 CHEMiCAL. PHYSICS LETTERS‘ 15 December 1972 minimum, by considering another. solution, differing molecule is zero. According to (4), QI is constant on slightly from fhe first, and being bounded by a sur- the boundary, i.e., the boundary is kequipotential, face c. Its properties are distlkguished by primes. so that WI is normal to the boundary, and the nor- Letting subscripts C and u refer to the voIumes within mal component of V@ is of the same sign at all points C and between C and c respectively, Sheldon noted on the boundary. Since the surface integral of VcP*ds that vanishes, the normal component of VQ, is zero on IV’ - IV = W& - W + rut = h(N-&) + iv; (6) the boundary, and because of the stationary principle obeyed by the v@l = 0 on the boundary of the molecule . (12) first solution within C. Since u is small, Outside the boundary, p vanishes. Since there are no lines of force extending out of the molecule, one can integrate a1 to infinity, starting at the boundary, to ” show cPl(-) = al(boundary). Then, using (lo), ~N:(KATpz’3-K=P”3f~,), (7) Qpl = 0 on the boundary of the mo!ecule . iW where Ni = N - A& Now, using (4) and (7) in (6), Inserting (8) and (13) in (4), we have W’ - ~~=~~(-~~lip2’“+~KiI~“3). X = ---&K~/K~ . (14) This will be negative if l11’3 is greater than ~K~/K~. The electronic contribution to the dipole moment in Therefore, the boundary surface is defined by the x direction is PC = (+Ka/Kk)3 . (8) The TFD equations, thkn are = -( 1/4ne)J [V*(xck, ) - 2dQI /ti] dr , (15) V2G1 =o, P < ~K~IKX’ 3 (9b) where ‘P, is given in terms of p by eq. (4). The boundary conditions are, for a neutral system, where rhI is the distance to nucleus M. Eqs. (9) are ac- tually satisfied everywhere that the nuclear charge density vanishes. If eq. (9a) is used in cq. (2), with the volume of in- tegration the volume of the molecule, excluding small spheres centered at the nuclei, Green’s theorem may be used to give a surface integral: N= -(1/477ej JVQ,*dS. (11) S The contribution of the sphere centered on nucleus M is Z,, _ This means, because of the neu traiity condition (2), that the surface integral over the boundary of the 472. where we integrate over the region where ,o is non- zero, except for small volumes around the nuclei. If these vo!umes are taken as srnail cylinders of radius R with axes in the x direction, da1 /dx may be integrated immediately over X, withy and z fEed, to give @t at the limits of the integration, which may be the boundary of the molecule, or a face of a small cylinder. In the former case, a1 = 0: the contributions of the faces give zero as R + 0. Converting the remainder of (15) to a surface integral, we have (16) S The contribution of the outer boundary surface vanishes because of (12) and (13 j. The contribution of the cylinder about nucleus M may be calculated exact- ly as follows. Let the cylinder have radius R and extend from x = {I (rl < xbI) to x = 52 (cl > xhi). Then the surface integral of V(x+,)*dS is Volume Z 7, number 4 CHEMICAL PHYSICS LETTERS 15 December 1972 7 -1/z - [Xl - X&f I K, - XhJ K5-, Q-l 13 after some algebra. Note that 10-I -“& l’* = I& -XhiI = Xhl - {I and [(fZ-x,,)2 ] “* = c2 - xhf. Letting R approach 0, and cl, and c2 approach xhi from the left and the right, the first two terms in the above integral vanish, and one obtains 27iz [(25,-x ) + (?.$ -x hf _ hl 1 hi )] + 4iT.z x hi M e - Now the total dipole moment is -eC jxpdr=o. hI ZhI Xhl + e (17) Therefore, the net dipole moment for the Thomas- Fermi-Dirac electron density vanishes. To extend this result to the Thomas-Fermi electron density, we have only to note the well-known fact that the TFD model goes over to the TF model when electronic exchange energy is neglected. This corresponds to neglecting the third term in (l), or to putting K~ = 0. The surface C is at infmity, and pc = 0 [eq. @)I. Now di the re- sults above are independent of the value of Kg. There- fore, the result (17) hoids for the TF model as well. It- also holds for modifications like that of Comb& [6, section 1 I], which include some of thk effect of correlation, but involve a change in the value of K,. It is interesting to note that the chemical potential [6, pp_ 38, 581 of the Thomas-Fermi atom is zero for all neutral atoms. In the TFD theory,*, chemical potential is which also has the same value (-&i/gk) for all neutral systems. Thus, there is no tendency for charge transfer from one atom to another: all atoms have the same electronegativity. This helps to explain the ab- sence of dipole moments. In our calculations, as in calculations performed by other methods, the errors in the electron density are likely to be largest on the periphery. The density at the periphery has a great effect on the expectation value of x, because of the weighting by the operator. Thus, large errors are likely to occur in calculations of dipole moments. The deviation of the electronic con- tribution to the dipole moment from the nuclear con- tribution should be interpreted as a measure of the numerical error in the calculaticn, since we have shown that the exact TFD or TF density must give a net dipole moment of zero. References [l] J. Slczak and J. Goodismnn, to be submitted for publica- tion_ 12) J.W. Sheldon, Phys. Rev. 99 (1955) 1291. [3] J. Coodismnn. Phys. Rev. 3 (1971) 1819. (41 H.C. Brinkmnn and B. Pepemk, Physica 21 (1955) 48. [S] N.H. March, Advan. Phys. 6 (1957) 1, seep. 44. [6] P. Combis, Die stcttistische Theorie des Atoms und ihre Anwendungen (Springer, Berlin, 1949). 473 

Dipole Moments in Thomas-Fermi-Dirac and Thomas-Fermi Theories.

https://surface.syr.edu/cgi/viewcontent.cgi?article=1078&amp;context=che

Dipole Moments in Thomas-Fermi-Dirac and Thomas-Fermi Theories.

Abstract

Similar works

Full text

Available Versions

Syracuse University Research Facility and Collaborative Environment