(2) JOURNAL OF MATHEMATICAL PHYSICS 49, 012302 共2008兲. Stability of atoms and molecules in an ultrarelativistic Thomas-Fermi-Weizsäcker model Rafael D. Benguriaa兲 Department of Physics, Pontificia Universidad Católica de Chile, Casilla 306, Correo 22 Santiago, Chile. Michael Lossb兲 School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332, USA. Heinz Siedentopc兲 Mathematisches Institut, Ludwig-Maximilians-Universität München, Theresienstrasse 39, 80333 München, Germany 共Received 24 October 2007; accepted 18 December 2007; published online 31 January 2008兲. We consider the zero mass limit of a relativistic Thomas-Fermi-Weizsäcker model of atoms and molecules. We find bounds for the critical nuclear charges that ensure stability. © 2008 American Institute of Physics. 关DOI: 10.1063/1.2832620兴. I. INTRODUCTION. The zero mass limit of the relativistic Thomas-Fermi-Weizsäcker 共henceforth ultrarelativistic TFW兲 energy functional for nuclei of charges zi ⬎ 0 共which need not be integral兲 located at Ri, i = 1 , . . . , K, is defined by2,3. 共兲 = a2. 冕. 共ⵜ1/3兲2dx + b2. 冕. 4/3dx −. 冕. V共x兲共x兲dx + D共, 兲.. 共1兲. Here, 共x兲 艌 0 is the electron density, K. V共x兲 = ␣ 兺 i=1. zi , 兩x − Ri兩. 共2兲. the electrostatic potential created by the nuclei, D共, 兲 =. ␣ 2. 冕. 共x兲共y兲 dxdy, 兩x − y兩. 共3兲. the electronic repulsion, and ␣ = e2 / បc ⬇ 1 / 137 is the fine structure constant. In units in which ប = c = 1, the constants a2 and b2 in 共1兲 are given, respectively, by a2 =. 3 共32兲2/3 82. 共4兲. and. a兲. Electronic mail: email@example.com. Electronic mail: firstname.lastname@example.org. Electronic mail: email@example.com.. b兲 c兲. 0022-2488/2008/49共1兲/012302/7/$23.00. 49, 012302-1. © 2008 American Institute of Physics.
(3) 012302-2. J. Math. Phys. 49, 012302 共2008兲. Benguria, Loss, and Siedentop. 3 b2 = 共32兲1/3 . 4. 共5兲. In the nonrelativistic case, some emphasis has been placed on the question of an appropriate choice of the coefficient of the gradient term connected to the kinetic energy. The nonrelativistic gradient correction 共ⵜ兲2 / was initially derived by Weizsäcker14 with a value = 1, whereas the systematic gradient expansion by Kirznits6,5 leads to = 91 . In the derivation of Tomishima and Yonei,13 = 51 . Lieb8,7 showed that the Weizsäcker term introduces a z2 correction to the leading z7/3 term of the nonrelativistic ground state energy. Adapting the coefficient in front of the Weizsäcker term such that the correction agrees with the leading z2 correction of the nonrelativistic quantum mechanics, the Scott correction, leads Lieb to propose = 0.185. In relativistic quantum mechanics, the energy remains unchanged to leading order, whereas the Scott correction is smaller than in nonrelativistic quantum mechanics,4 i.e., we cannot expect to have the same value as in the TFW functional. However, we are not yet in a position to proceed with Lieb’s strategy to infer the coefficient of the gradient correction from this. In particular, that would require showing that the massive equivalent of the function leaves indeed the leading energy contribution unchanged and the gradient term yields again a z2 correction. Let us first consider the atomic case, i.e., the case K = 1, z1 = z, R1 = 0. Because of simple scaling considerations, if we minimize the energy functional 共1兲 over all functions for which each of the terms in 共1兲 makes sense, the infimum of the energy functional is either zero or minus infinity. In the first case, we say that the atom is stable. Otherwise, we say that the atom is unstable. Our purpose here is to determine the range of values of the zi’s for which the atom or molecule is stable. The following result holds in the atomic case 共i.e., for K = 1, z1 = z, R1 = 0兲.1 Theorem 1.1: Let. 共兲 = a2. 冕. 共ⵜ1/3兲2dx + b2. 冕. 4/3dx −. 冕. z␣. dx + D共, 兲, 兩x兩. 共6兲. with D共 , 兲 given by (3). Then,. inf 共兲 =. 再. − ⬁ for z ⬎ 4ab/3␣ + 7a3/6b3 0. for z ⬍ 4ab/3␣. 冎. 共7兲. where the infimum is taken over all non-negative functions 共x兲, such that 苸 L4/3共R3兲, ⵜ1/3 苸 L2共R3兲, and D共 , 兲 ⬍ ⬁. Remarks: 共i兲. 共ii兲. 共iii兲. It follows from 共7兲 that if z ⬍ 4ab / 共3␣兲, the atom is stable, whereas if z ⬎ 4ab / 共3␣兲 + 7a3 / 共6b3兲, the atom is unstable. The exact critical value of z 共zc, say兲 dividing the region of stability from the region of unstability is not known. However, it turns out that for the physical values of the constants, the gap between the upper and lower bounds on zc is less than 1 and therefore negligible 共see the following remarks兲. For the physical values of a and b given by 共4兲 and 共5兲, the atom will be stable if z ⬍ 冑3 / 2 / ␣ ⬇ 167.8冑. Thus, if = 91 共i.e., the value used by Kirznits,6,5兲, the atom is stable if z ⬍ 56. If = 51 共i.e., the value used by Tomishima and Yonei13兲, the atom is stable if z ⬍ 75. Finally, using the value of Lieb,8,7 the atom is stable if z ⬍ 73. As for the value of the gap, using the physical values of the constants, one gets 7a3 / 共3b6兲 = 共7 / 12兲冑33 / 2 ⬍ 0.021⬍ 1 for all the values of considered above. Thus, the gap is negligible from the physical point of view.. For the molecular case, i.e., when K ⬎ 1 and V is given by 共2兲, the following result was proven in Ref. 1..
(4) 012302-3. J. Math. Phys. 49, 012302 共2008兲. Ultrarelativistic Thomas-Fermi-Weizsäcker. Theorem 1.2: Let. 共兲 = a2. 冕. 共ⵜ1/3兲2dx + b2. 冕. 4/3dx −. 冕. Vdx + D共, 兲,. 共8兲. with V given by (2) and D共 , 兲 given by (3). Then, K. inf 共兲 = 0. if Z = 兺 zi 艋 i=1. 4ab , 3␣. 共9兲. where the infimum is taken over all non-negative functions , such that 苸 L4/3共R3兲, ⵜ1/3 苸 L2共R3兲, and D共 , 兲 ⬍ ⬁. The above result is just a trivial extension of the atomic to the molecular case and in some sense is the best possible when the interaction between the nuclei is not taken into account. In fact, if we neglect the nuclear interaction, one can always think of the possibility of putting all the nuclear charges at the same point and reducing the molecular case to the atomic case, which K zi. The deficiencies of the above result are obvious. The goal explains the condition 共9兲 on Z ⬅ 兺i=1 of this paper is to have a result that yields stability for reasonable values of the nuclear charges. For that purpose, the nucleus-nucleus interaction, U⬅␣. zz. i j , 兺 1艋i⬍j艋K 兩Ri − R j兩. 共10兲. plays a key role because it prevents the possibility of putting the nuclear charges on top of each other. Our main result is the following theorem for the molecular case. Theorem 1.3: Let 共兲 be given by (8) for functions as in Theorem 1.2, with V given by (2). Then, we have stability, i.e., inf 共兲 + U 艌 0, provided that 0 艋 zi 艋 where x 苸 共0 , 1兲 is the root of. 4a b冑1 − x, 3␣. 冉冊. 1 − x b4 4 = 2 x3 a 3. 2. 1 . 2␣共4 + 9␣4兲. 共11兲. 共12兲. Remark: For the physical values of a and b given by 共4兲 and 共5兲 and the physical value of the 1 兲 in 共12兲, Theorem 1.3 says that the molecule will be fine structure constant taken 共i.e., ␣ = 137 1 stable if each zi 艋 55, if = 9 共i.e., the value used by Kirznits7,6兲. If = 51 共i.e., the value used by Tomishima and Yonei13兲 the molecule is stable if each zi 艋 74. Finally, using the value of Lieb,8,7 the molecule is stable if each zi 艋 71. These bounds on each individual nuclear charge are almost the same as those embodied in the atomic case 共i.e., the ones given in Theorem 1.1 above兲. In the next section, we give the proof of Theorem 1.3. II. IMPROVED RESULTS ON THE STABILITY OF MOLECULES. In this section, we prove Theorem 1.3. Our proof relies in a modified uncertainty principle 共see Theorem 2.1 below兲 which is of independent interest. As we mentioned in the Introduction, the nucleus-nucleus interaction plays a key role in the stability of molecules. As in Ref. 9, we may use the fact that the energy is separately concave in the nuclear charges and each charge zi varies between 0 and z. The minimum of a concave function is always on the boundary and hence the.
(5) 012302-4. J. Math. Phys. 49, 012302 共2008兲. Benguria, Loss, and Siedentop. value of zi wants to be either 0 or z. If it is zero, we have one nucleus less, and if it is z, then we are in the case we are considering. First, we need some notation. We introduce the nearest neighbor, or Voronoi, cells15 兵⌫ j其Kj=1 defined by ⌫ j = 兵x兩兩x − R j兩 艋 兩x − Rk兩其.. 共13兲. The boundary of ⌫ j, ⌫ j, consists of a finite number of planes. We also define the distance D j = dist共R j, ⌫ j兲 = 21 min兵兩Rk − R j兩兩j ⫽ k其.. 共14兲. One of the key ingredients we need in the sequel is an electrostatic inequality of Lieb and Yau.11,12 Define a function ⌽ on R3 with the aid of the Voronoi cells mentioned above. In the cell ⌫ j, ⌽ equals the electrostatic potential generated by all the nuclei except for the nucleus situated in ⌫ j itself, i.e., for x 苸 ⌫ j, K. ⌽共x兲 ⬅ z 兺 兩x − Ri兩−1 .. 共15兲. i=1. i⫽j. If is any bounded Borel measure on R 共not necessarily positive兲, then 3. 1 2. 冕冕 R3. 冕. K. 1 兩x − y兩 d共x兲d共y兲 − ⌽共x兲d共x兲 + U 艌 z2 兺 D−1 j . 3 3 8 j=1 R R −1. 共16兲. We will also need a localization result for the kinetic energy which will allow us to control the Coulomb potential near each nuclei. This localization result for the UTFW model is given by Theorem 2.1 below. Theorem 2.1: 共Modified uncertainty principle兲 For any smooth non-negative function f on the closed ball BR of radius R, we have the estimate a2. 冕. 兩ⵜf共x兲兩2dx + b2. BR. 冕. f共x兲4dx 艌 ab. 冕冋 BR. BR. 册. 2 4 − f共x兲3dx. 3兩x兩 R. Remark: Notice that the factor 34 is best possible and it agrees with the sharp value given in Theorem 1.1. To prove this theorem, we need the following preliminary result. Lemma 2.1: Let u共r兲 be any smooth function with u共R兲 = 0. Then, the following uncertainty principle holds. 冏冕. 冏 冉冕. 共3u共兩x兩兲 + u⬘共兩x兩兲兩x兩兲f共x兲3dx 艋 3. BR. BR. BR. There is equality if and only if f= . 冕. 冊 冉冕 1/2. 兩ⵜf共x兲兩2dx. 1 r. 关su共s兲兴ds + C. 0. for some constants C and . Proof: Set gi共x兲 = u共兩x兩兲xi , where u is a smooth function with u共R兲 = 0 and note that. u共兩x兩兲2兩x兩2 f共x兲4dx. 冊. 1/2. ..
(6) 012302-5. 冕. BR. J. Math. Phys. 49, 012302 共2008兲. Ultrarelativistic Thomas-Fermi-Weizsäcker. 冕 兺冕. 共3u共兩x兩兲 + u⬘共兩x兩兲兩x兩兲f共x兲3dx = 兺 =. f共x兲关 jg j共x兲兴f共x兲2dx. j. BR. j. BR. f共x兲 j共g j f 2兲共x兲dx − 2 兺 j. 冕. f共x兲2g j共x兲 j f共x兲dx.. BR. Integrating the first term by parts yields. 兺j. 冕. BR. f共x兲 j共g j f 2兲共x兲dx = − 兺 j. 冕. j f共x兲g j共x兲f共x兲2dx +. BR. 冕. f共x兲3u共兩x兩兲兩x兩dS共x兲,. BR. where the boundary term vanishes since u共R兲 = 0. Thus,. 冕. BR. 共3u共兩x兩兲 + u⬘共兩x兩兲兩x兩兲f共x兲3dx = − 3 兺 j. 冕. f共x兲2g j共x兲 j f共x兲dx.. BR. Using Schwarz’ inequality on the last term yields. 冏冕. 冏 冉冕. BR. 冊 冉冕 1/2. 共3u共兩x兩兲 + u⬘共兩x兩兲兩x兩兲f共x兲3dx 艋 3. 兩ⵜf共x兲兩2dx. BR. u共兩x兩兲2兩x兩2 f共x兲4dx. BR. 冊. 1/2. .. Schwarz’ inequality is an equality if and only if. j f = − g j共x兲f共x兲2 , 䊐. which can easily be integrated and yields the stated function. Proof of Theorem 2.1: To prove the theorem, pick. u共r兲 =. 冉 冊. 1 1 1 − 2 r R. in the lemma which leads to the inequality. 冕冋 BR. 册. 3 3 1 − f共x兲3dx 艋 2 兩x兩 2R. 冉冕. 冊 冉冕 冉 冊 1/2. 兩ⵜf共x兲兩2dx. BR. 1−. BR. 兩x兩 R. 2. f共x兲4dx. 冊. 1/2. .. Next, applying the inequality between the arithmetic and geometric mean yields a2. 冕. 兩ⵜf共x兲兩2dx + b2. BR. 冕冉 冊 1−. BR. 兩x兩 R. 2. f共x兲4dx 艌 ab. 冕冋 BR. 册. 2 4 f共x兲3dx, − 3兩x兩 R. from which the theorem follows. 䊐 In order to prove our main result, we consider the total energy, 共兲 + U, and we split the 兰4/3 term into two parts, i.e., 共b21 + b22兲 兰 4/3. For later discussions, it is important to remark that the parameter b2 can be chosen arbitrarily in the interval 共0 , b兲. Then, we use Theorem 2.1 with f 3 = and BR = B j, the largest ball inscribed in the corresponding Voronoi cell ⌫ j, to get. 共兲 + U 艌 b21. 冕. R3. 4/3dx −. 冕. R3. K. V共x兲共x兲 + ab2 兺 j=1. 冕冉 Bj. 冊. 2 4 共x兲dx + D共, 兲 + U. − 3兩x − R j兩 D j 共17兲. In order to cancel the Coulomb singularity inside B j, we choose the parameter.
(7) 012302-6. J. Math. Phys. 49, 012302 共2008兲. Benguria, Loss, and Siedentop. 3 ␣z . 4 a. b2 =. 共18兲. The restrictions on b2 will give restrictions on z to ensure stability. With the help of the Voronoi cells, we now define W共x兲 ⬅ ⌽共x兲 +. z 兩x − R j兩. 共19兲. W共x兲 ⬅ ⌽共x兲 +. 2ab2 , Dj. 共20兲. if x 苸 ⌫ j and 兩x − R j兩 艌 D j, whereas. if x 苸 ⌫ j and 兩x − R j兩 艋 D j. Now, if we restrict to values of z such that z 艋 4ab2 / 共3␣兲, we can finally write. 共兲 + U 艌 1共兲 + 2共兲,. 共21兲. with. 1共兲 = b21. 冕. R3. 4/3dx − ␣. 冕. R3. W共x兲共x兲 + ␣. 冕. R3. ⌽共x兲dx. 共22兲. and. 2共兲 = D共, 兲 − ␣. 冕. R3. ⌽共x兲共x兲dx + U.. 共23兲. Using the Lieb-Yau electrostatic inequality 共16兲, we have K. 2共 兲 艌 ␣. 1 z2 . 兺 8 j=1 D j. 共24兲. On the other hand, it is simple to estimate 1共兲 from below since it is simple to solve the variational principle inf 1共兲. Thus, we get. 冉 冊冕. 1 3 1共 兲 艌 − ␣ 4 4 4b21. 3. R3. 共W − ⌽兲+4dx.. 共25兲. From 共21兲, 共24兲, and 共25兲, we get. 冉 冊冕. 1 3 共兲 + U 艌 − ␣4 4 4b21. K. 3. R3. 共W −. ⌽兲+4dx. 1 z2 +␣ 兺 . 8 j=1 D j. 共26兲. Now, using 共20兲, we compute. 冕. 共W − ⌽兲+4dx =. Bj. 冉 冊 2ab2 Dj. 4. 64a4b42 4 D3j = . 3 3D j. 共27兲. Since any Voronoi cell is contained in a half space, one calculates as in Ref. 10 that. 冕. ⌫ j∖B j. 共W − ⌽兲+4dx =. 冕 冉 ⌫ j∖B j. z 兩x − R j兩. 冊. 4. Finally, using the estimates 共27兲 and 共28兲 in 共26兲, we get. dx 艋. 3z4 . Dj. 共28兲.
(8) 012302-7. J. Math. Phys. 49, 012302 共2008兲. Ultrarelativistic Thomas-Fermi-Weizsäcker K. 共兲 + U 艌 M 兺 j=1. 1 , Dj. 共29兲. where. 冉 冊冉. 3 1 M = − ␣4 4 4b21. 3. 3z4 +. 冊. 64a4b42 ␣z2 + . 3 8. 共30兲. If M ⬎ 0, then 共兲 + U ⬎ 0 and the molecule is stable. Using the fact that b22 = b2 − b21 together with 共18兲, the condition M 艌 0 can be written solely in terms of b1 as. a2. b2 − b21 b61. 艋. 冉冊 4 3. 2. 1 . 2共4␣ + 9␣5兲. 共31兲. Finally, in order to allow for the largest possible value of z 共equivalently the largest possible value of b2兲, b1 must be chosen so that we have equality in 共31兲. Since the left side of 共31兲 is decreasing as a function of b1 in the allowed interval 共0 , b兲, we conclude the proof of our Theorem 1.3. Remark: Note that we do not need a bound on the fine structure constant to ensure stability in this model. This is due to the absence of the exchange term.. ACKNOWLEDGMENTS. R.D.B. was supported by FONDECYT 共Chile兲 Project Nos. 706-0200 and 106-0651 and CONICYT 共Chile兲 PBCT Proyecto Anillo de Investigación en Ciencia y Tecnología ACT30/2006. M.L. was supported by NSF Grant No. DMS-0600037. H.S. was supported by the Deutsche Forschungsgemeinschaft Grant No. SI 348/13-1. 1. Benguria, R. D. and Pérez-Oyarzún, S., “The ultrarelativistic Thomas-Fermi-von Weizsäcker model,” J. Phys. A 35, 3409–3414 共2002兲. Engel, E. and Dreizler, R. M., “Field-theoretical Approach to a Relativistic Thomas-Fermi-Weizsäcker Model,” Phys. Rev. A 35, 3607–3618 共1987兲. 3 Engel, E. and Dreizler, R. M., “Solution of the relativistic Thomas-Fermi-Dirac-Weizsäcker Model for the Case of Neutral Atoms and Positive Ions,” Phys. Rev. A 38, 3909–3917 共1988兲. 4 Frank, R. L., Siedentop, H., and Warzel, S., “The Ground State Energy of Heavy Atoms: Relativistic Lowering of the Leading Energy Correction,” Commun. Math. Phys. 共in press兲. 5 Hodges, C. H., “Quantum Corrections to the Thomas-Fermi Approximation—The Kirzhnits Method,” Can. J. Phys. 51, 1428–1437 共1973兲. 6 Kirznits, D. A., Field Theoretical Methods in Many Body Systems 共Pergamon, New York, 1967兲. 7 Lieb, E. H., “Analysis of the Thomas-Fermi-von Weizsäcker Equation for an Infinite Atom without Electron Repulsion,” Commun. Math. Phys. 85, 15–25 共1982兲. 8 Lieb, E. H., “Thomas-Fermi and Related Theories of Atoms and Molecules,” Rev. Mod. Phys. 53, 603–641 共1981兲. 9 Lieb, E. H. and Daubechies, I., “One electron relativistic molecules with Coulomb interaction,” Commun. Math. Phys. 90, 497–510 共1983兲. 10 Lieb, E. H., Loss, M., and Siedentop, H., “Stability of Relativistic Matter via Thomas-Fermi Theory,” Helv. Phys. Acta 69, 974–984 共1996兲. 11 Lieb, E. H. and Yau, H., “Many-Body Stability Implies a Bound on the Fine-Structure Constant,” Phys. Rev. Lett. 61, 1695–1697 共1988兲. 12 Lieb, E. H. and Yau, H., “The Stability and Instability of Relativistic Matter,” Commun. Math. Phys. 118, 177–213 共1988兲. 13 Tomishima, Y. and Yonei, K., “Solution of the Thomas-Fermi-Dirac Equation with a Modified Weizsäcker Correction,” J. Phys. Soc. Jpn. 21, 142–153 共1966兲. 14 von Weizsäcker, C. F., “Zur Theorie de Kernmassen,” Z. Phys. 96, 431–458 共1935兲. 15 Voronoi, G., “Nouvelles applications des paramètres continus à la théorie des formes quadratiques,” J. Reine Angew. Math. 133, 97–178 共1907兲. 2.