E. Lieb, "Thomas--Fermi and Related Theories of Atoms and Molecules", Reviews of Modern Physics, 1981, Vol. 53, No. 4, pp. 603--641.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....of an atom is then defined as E(Z) min N1 E(N;Z) and our problem is to compute E(Z) asymptotically for large Z. Building on the previous work of Thomas [43] Fermi [22] Dirac [6] and Scott [37] see the In this survey we neglect electron spin to simplify notation. survey article of Lieb [30]) Schwinger [36] proposed the refined formula 8 (1:1) for some explicitly defined positive constants c 0 , c 1 . After the early work of Lieb and Simon [31] on molecules, Hughes and Siedentop Weikard [24, 40, 41, 42] gave a rigorous proof of the Scott conjecture , namely E(Z) Gammac ....

E. Lieb, "Thomas--Fermi and Related Theories of Atoms and Molecules", Reviews of Modern Physics, 1981, Vol. 53, No. 4, pp. 603--641.

....only the first, third, and fourth terms in Eq. 2.3 need be considered. There have been several attempts to solve the electronic problem using approximate methods based on non interacting single particle wavefunctions (Hartree and Hartree Fock[61] or on the charge density (Thomas Fermi Dirac[62]) More recently, solutions based upon Density Functional Theory[22, 23] also based upon the charge density) have become the preferred method of solution due to their accuracy and computational efficiency. 2.2 Density Functional Theory The most crucial aspect of an electronic structure ....

E. H. Lieb, Thomas-fermi and related theories of atoms and molecules, Rev. Mod. Phys. 53, 603 (1981).

....the full 3N dimensional equation (1. 2) by simpler (usually non linear) ones in 3 dimensions (Thomas Fermi theory and Hartree Fock theory) The relation of these approximations with the N electron Schr odinger equation has been analysed in detail for big atoms, see e.g. Lieb and Simon [10] Lieb [9], and Lieb and Simon [11] One important observation and motivation for the development of these and other approximation schemes was the insight that in order to calculate the energy E or one and two electron operator expectation values there is no need for the full wave function (x 1 ; x 2 ; ....

Lieb, E. H. Thomas-fermi and related theories of atoms and molecules. Rev. Modern Phys. 53, 4 (1981), 603-641.

....electrons by (x) dx = x) dx = We shall discuss Hartree Fock theory in greater detail in Sect. 3 and ThomasFermi theory in greater detail in Sect. 4. For a complete discussion of TF theory we refer the reader to the original paper by Lieb and Simon [14] or the review by Lieb [8]. Here we shall just mention that the functions are the unique solutions to the set of equations (x) 4 ae (x) Gamma 4Z ffi(x) 5) 6) N : 7) Here is a non negative parameter called the chemical potential, which is also uniquely determined from the ....

....we refer the reader to Sect. 8. 1.3. REMARK. The total energy of the atom in Thomas Fermi theory is dx Gamma Z (x)jxj (x)jx Gamma yj (y)dx dy Gammae 0 Z (8) where e 0 is the total binding energy of a neutral TF atom of unit nuclear charge. Numerically [8], e 0 = 2(3 Delta 3:67874 = 0:7687: 9) For a neutral atom, where N = Z, the above inequality is an equality. The inequality states that in Thomas Fermi theory the energy is smallest for a neutral atom. We can now state two of the main results in this paper. 1.4. THEOREM (Potential ....

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E. H. Lieb,Thomas-Fermi and related theories of atoms and molecules. Rev. Modern Phys., 53 no. 4, 603--641, (1981).

....density minimizing the functional (11) corresponds to a neutral system. For future use we shall denote by OE(R; Z; x) the unique solution OE of (14) and OE at (R; Z; x) the one corresponding to a single atom of charge Z located at R 2 R 3 . We refer to the original paper [19] or the review [15] for the proofs of these statements and for further results on Thomas Fermi theory. We consider the following function: f fi (R; Z) E fi TF;R;Z C fi TF M X J=1 Z 7=3 j (15) 8 Using the scaling properties of Thomas Fermi theory we have f fi (R; Z) Z 7=3 f fi i Z 1=3 R; ....

....prove the lemma for Z j 1 and fi = fi phys and in this case we omit the superscript. Then we have the following (a Feynman Hellman type result) Z i f (R; Z) lim x7 R i [OE(R; Z; x) Gamma OE at (R i ; Z i ; x) 18) The right side of (18) is non negative because of Teller s Lemma (see [15] Theorem 3.4) which implies that f (R; Z) is non decreasing in any of the Z i arguments. Therefore we have that f (R; Z) f (R; 1 : 1) 19) Let H( be defined by H( f (R; 20) Applying Feynman Hellman s formula again we obtain d d H( M X i=1 lim x7 R i [OE(R; ....

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E.H. Lieb, Thomas-Fermi and related theories of atoms and molecules. Rev.Mod.Phys., 53, 603--641 (1981)

....The story began twenty years ago from a rather theoretical standpoint with the fundamental contributions of E.H. Lieb, B. Simon, H. Brezis and coworkers, and continued with the works of P L. Lions. A rapid list of the most significant articles in this field should include at least the following [22, 23, 24] (For a complete list, we refer to [5, 11, 12] To this day, it sounds reasonable to claim that most of the molecular models of Quantum Chemistry are now well understood mathematically and have been carefully analyzed 1 .The focus has now turned either towards the side of the study of the ....

E. H. Lieb, Thomas-Fermi and related theories of atoms and molecules,Rev. Mod. Phys., 53, 4, 1981, pp 603-641.

....to the proof of this theorem. For the proof we will be interested only in the case Z N 2Z. Lieb ( 6] and [7] has given a simple argument to handle the case N 2Z. Thomas Fermi theory will play a central role in the proof. We recall a few fundamental facts. For a nice detailed discussion see [5]. 1. E(Z) C TF Z 7 = 3 as Z 1. 2. Let ae TF be Thomas Fermi density. Then: a. R R 3 ae TF (x) dx = Z b. For jxj Z Gamma 1 = 3 , ae TF (x) Cjxj Gamma6 . c. R jxj R ae TF (x) dx C ae TF R Gamma3 for R Z Gamma 1 = 3 , for some constant C ae TF . 1. Key Estimate ....

Lieb, E. H. (1981) "Thomas--Fermi and Related Theories of Atoms and Molecules" Reviews of Modern Physics Vol. 53, no. 4.

....atoms (although not entirely satisfactory) and not so good for small ones, and with the exception of item 5, the other properties look obvious in nature. But a rigorous analysis had to wait until 1977, when it was proved that atoms behave as Thomas Fermi theory predicts in the limit Z 1 (see [Li] and [LS] Other people also considered the problem of atomic energies for large Z (in the limit as Z 1) Scott predicted a correction for Thomas Fermi energy that recently has been proved correct (see [Hu] and [SW] and other atomic models due to Dirac, von Weizsacker and others were created, ....

....Corollary: Let E(Z) inf 2H jj jj 2 =1 hHLB ; i Then E(Z) E(Z) CEZ 5 = 3 Proof: Let Z be a normalized ground state for HZ . By the virial theorem, we know that h Gamma Delta Z ; Z i = GammaE(Z) where Gamma Delta means the laplacian in R 3Z . From TF theory, we know (see [Li] or [LS] that E(Z) C TF Z 7 = 3 o i Z 7 = 3 j which implies that Z satisfies the hypotheses of Theorem 7.1; therefore, E(Z) hHZ Z ; Z i = hHLB Z ; Z i hE Z ; Z i E(Z) CEZ 5 = 3 69 In order to prove Theorem 7.1, first note that E = 1 2 Z R 0 V (R) dR R 5 ....

Lieb, E. "Thomas--Fermi and Related Theories of Atoms and Molecules" Reviews of Modern Physics Vol 53 no. 4, (1981)

....2 , and u (n) x) c n x Gamma2 Gamman . 3. u reaches a single maximum Omega 2 c at the point r c , and the equation u(x) Omega 2 has exactly two solutions r 1 r 2 ( Omega Gamma for Omega 2 (0 ; Omega c ) In particular, max Z 1 = 3 Omega c . We refer the reader to [L1] for an excellent account of Thomas Fermi theory, to [Hi] for basic details on the Thomas Fermi function, and to [Hug] SW2] and [FS6] for many of the other properties needed for atoms. The only number theoretic ingredient in our proof is the following estimate well known to ....

Lieb, E. (1981) "Thomas--Fermi and Related Theories of Atoms and Molecules" Reviews of Modern Physics Vol 53 no. 4, 603--641.

....2 , and u (n) x) c n x Gamma2 Gamman . 3. u reaches a single maximum Omega 2 c at the point r c , and the equation u(x) Omega 2 has exactly two solutions r 1 r 2 ( Omega Gamma for Omega 2 (0 ; Omega c ) In particular, max Z 1 = 3 Omega c . We refer the reader to [L1] for an excellent account of Thomas Fermi theory, to [Hi] for basic details on the Thomas Fermi function, and to [Hug] SW2] and [FS6] for many of the other properties needed for atoms. The only number theoretic ingredient in our proof is the following estimate well known to ....

Lieb, E. (1981) "Thomas--Fermi and Related Theories of Atoms and Molecules" Reviews of Modern Physics Vol 53 no. 4, 603--641.

....conditions on V and a, the interval ( Gamma1; 1) is a gap in the essential spectrum of T V (a) and hence the above trace coincides with the number of eigenvalues in this gap. Our basic tools are the methods of coherent states used previously in the work of Berezin [4] Li and Yau [13] Lieb [14], and Thirring [22] This paper follows closely earlier work of Evans, Lewis, Siedentop, and Solovej [7] Throughout this paper ( Delta; Delta) and k Delta k are used to denote the usual inner product and norm in L 2 (R 3 ) 4 ; the norms of L p (R 3 ) and L p (R 3 ) 4 are both ....

....follows from Lemma 5 and (14) 4.2. An upper bound for Gamma tr(T V (a) 2 Gamma 1 Gamma Phi) Gamma . The main idea here is to use (14) with a good approximating d in order to produce an upper bound for the operator T V (a) 2 Gamma 1 Gamma Phi. For a similar treatment see Lieb [14], p.621. Let O(p; q) p; q 2 R 3 , be a 4 Theta 4 matrix with eigenvectors o , 1; 2; 3; 4, orthonormal in C 4 . Define the normalized coherent state F O fl (x) e ip Deltax= g ae (x Gamma q)o (p; q) 22) cf. F U fl (x) above) For = 1; 2; 3; 4; let M (p; q) be a function ....

Lieb, Elliott H.: Thomas-Fermi and related theories of atoms and molecules. Rev. Mod. Phys., 53(4):603--641, October 1981.

....(0; 1) y(0) 1 ; y(1) 0 : From (16) 18) one can read off the basic properties of V TF . For instance, the order of magnitude of V TF is given by (19) GammaV TF (x) min Phi Zjxj Gamma1 ; jxj Gamma4 g (x 2 R 3 ) A detailed discussion of Thomas Fermi theory is given by Lieb [L]. We close this introduction by mentioning several open problems on the density (1) The first natural problem is to estimate ae Gamma ae sc for the Thomas Fermi potential of a molecule. The most we can hope for here is probably (20) Z R 3 Z R 3 Gamma ae(x) Gamma ae sc (x) ....

E. Lieb, Thomas-Fermi and Related Theories of Atoms and Molecules, Rev. Mod. Phys. 53; no. 4; part I (1981), 603--641.

....the models arising in Density Functional Theory: we refer the reader to [14, 41] for an introduction to the general features and the physical foundations of such models. Mathematically, it is a well known fact that the problem (1) 2) has a unique minimizing density, denoted by ae (see E.H. Lieb [29], R. Benguria, H. Br ezis E.H. Lieb [5] or P. L. Lions [35] and that, denoting u = p ae , u is a solution to Gamma4u [ 5 3 ae 2=3 Gamma Phi ]u = Gamma u ; 3) where we denote by Phi = X k2 1 jx Gamma kj Gamma ae 1 jxj ; the effective potential the ....

E. H. Lieb, Thomas-Fermi and related theories of atoms and molecules, Rev. Mod. Phys., 53, 4, 1981, pp. 603-641.

....j j Z d 3 y jx i Gamma yj ae(y) 1 A Gamma 1 2 D(ae; ae) 4) and is defined on DN . In the case of atoms and molecules with N electrons a natural candidate for ae is the corresponding Thomas Fermi (TF) density ae TF , and the Mean Field potential becomes the TF potential OE TF (see [11]) We introduce h TF : Gamma Delta Gamma K X j=1 Z j jx Gamma R j j Z d 3 y jx Gamma yj ae TF = Gamma Delta Gamma OE TF (x) 5) self adjointly realized on D. By general arguments oe ess (h TF ) 0; 1) and oe disc (h TF ) Gamma1; 0) Moreover, if N Z then the chemical ....

....h TF : Gamma Delta Gamma K X j=1 Z j jx Gamma R j j Z d 3 y jx Gamma yj ae TF = Gamma Delta Gamma OE TF (x) 5) self adjointly realized on D. By general arguments oe ess (h TF ) 0; 1) and oe disc (h TF ) Gamma1; 0) Moreover, if N Z then the chemical potential = 0 [11] and joe disc (h TF )j 1, whereas 0 and joe disc (h TF )j = 1 for N Z. We set PN : Gamma1;0) h TF ] in case tr 1 f ( Gamma1;0) h TF ]g N . Otherwise PN : N X i=1 j i ih i j ; 6) is a spectral projection onto N eigenfunctions i of h TF with lowest possible negative ....

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E.H. Lieb. Thomas-Fermi and related theories of atoms and molecules. Rev. Mod. Phys., 53:603--604, 1981.

....R(Z; Z) O(1) 6) this is part of a much stronger conjecture saying that as Z 1 the atomic structure shows a universal behavior, which is to say that the quantities in (6) actually converge to non zero values as Z 1. In Thomas Fermi theory this universality has been known for some time (see [5]) In the present paper we will indeed compare with TF theory. In the Thomas Fermi von Weizsacker theory universality was recently proved in [8] It is not hard to prove that Q c (Z) CZ; I(Z; Z) CZ 4 = 3 and R(Z; Z) CZ Gamma 1 = 3 : 7) In [6] it was proved that Q c (Z) o(Z) This ....

.... 2 dxdy; The Normalized 2 point Correlation Outside R, C ffi (R) 1 Q ffi (R) 2 Z ae (2) x; y) R (x) 2 R (y) 2 dxdy: We will prove an upper bound to the ionization energy in terms of these quantities by using a very simple trick which in fact goes back to Benguria (see [5]) and was used in [4] to prove N c 2Z 1. The idea here is to use the trick on the outside problem (jxj R) The same method was used in [8] Theorem 4 For all ffi 0 and R 0 I h ffi (R) Gamma 1 2 C ffi (R)Q ffi (R) i R Gamma1 X ffi ; 10) where the error term is bounded by ....

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Elliott H. Lieb. Thomas-Fermi and related theories of atoms and molecules. Rev. Mod. Phys., 53:603--604, 1981.

....(plus a constant) and the number of eigenvalues, respectively. More specifically, we consider the Schrodinger operator H tf : Gamma Delta Gamma OE tf (x) 28) acting on a dense domain in L 2 (IR 3 ) Omega C 2 . Here, OE tf (x) is the neutral Thomas Fermi potential (see e.g. [19]) which, implicitly, depends on Z and R. We define the projection d onto its negative eigenvalues (counting multiplicities) e 1 = e 2 e 3 = e 4 Delta Delta Delta e 2M 0, with corresponding orthonormal eigenvectors f 1 ; f 2M , so d : Gamma1;0) h H tf i = 2M X j=1 jf ....

E.H. Lieb. Thomas-Fermi and related theories of atoms and molecules. Rev. Mod. Phys., 53:603--604, 1981.

....content ourselves knowing that Thomas Fermi theory is simpler than the Schrodinger equation. We will postpone the derivation of the Thomas Fermi equations until later. The problem to understand Thomas Fermi theory mathematically was tackled in 1973 with the work of Lieb and Simon (see [LS] and [Li]) which is now a central piece in modern mathematical physics. In their setting, large atoms were viewed as a limit Z 1. Since then, large Z asymptotics have become the mathematical paradigm of large atoms. In particular, the work of Lieb and Simon proves that (1) is the leading expression as ....

.... paper lies in the asymptotic formula, as Z goes to infinity, for the atomic energy E(Z) Gammac TFZ 7 = 3 1 8 Z 2 Gamma c s Z 5 = 3 O i Z 5 3 Gammaa j ; a 0: 5) The first term above was introduced by Thomas and Fermi in [Th] Fe] and proved rigorously in [LS] See also [Li] for a review of Thomas Fermi theory) The Z 2 term was discovered by Scott in [Sc] and proved to be true in a series of papers by Hughes Siedentop Weikard, in [Hu] SW1] SW2] and [SW3] Its generalization to molecules was obtained by Ivrii Sigal ( IS] The Z 5 = 3 term was obtained ....

Lieb, E. (1981) "Thomas--Fermi and Related Theories of Atoms and Molecules" Reviews of Modern Physics Vol 53 no. 4, 603--641.

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E. H. Lieb, Thomas Fermi and related theories of atoms and molecules, Rev. Mod. Phys. 53 (1981) 603--640.

....GP ( x)j 4 d 3 x (1.9) is the mean density. The minimizer Phi GP of (1.6) with the condition (1.3) depends on N and a, of course, and when this is important we denote it by Phi GP N;a . Mathematically, the GP equation is quite similar to the Thomas Fermi von Weizs acker equation [8] and its basic properties can be established by similar means. See [7] Sect. 2 and Appendix A. The idea is now that for dilute gases one should have E GP E QM and ae QM ( x) fi fi Phi GP ( x) fi fi 2 j ae GP ( x) 1.10) where the quantum mechanical particle density in the ....

....(N; a) E GP R (N; a) 1 Gamma (const. N Gamma1=10 ) 4.11) The constant may depend on R, but this is of no harm, since we first take N 1 and then R 1, using E GP R E GP and (1.13) This proves (1.15) Convergence of the densities, 1. 16) is obtained in a standard way (see, e.g. [8]) by variation with respect to the external potential. 5 Concluding remarks Bose Einstein condensation in the ground state is a concept that involves the full one particle density matrix fl N ( x; x 0 ) N Z Psi 0 ( x; x 2 ; x N ) Psi 0 ( x 0 ; x 2 ; x N )d x 2 ....

E.H. Lieb, Thomas-fermi and related theories of atoms and molecules, Rev. Mod. Phys. 53, 603--641 (1981).

No context found.

E. Lieb, "Thomas--Fermi and Related Theories of Atoms and Molecules", Reviews of Modern Physics, 1981, Vol. 53, No. 4, pp. 603--641.

No context found.

Lieb, Elliott H.: Thomas-Fermi and related theories of atoms and molecules. Rev. Mod. Phys., 53(4):603--641, October 1981.

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Lieb, E. H., Thomas-Fermi and related theories of atoms and molecules. Rev. Modern Phys. 53, no. 4 (1981), 603--641.

No context found.

Lieb, E. H., Thomas-Fermi and related theories of atoms and molecules. Rev. Modern Phys. 53, no. 4 (1981), 603--641.

No context found.

E.H. Lieb, Thomas-Fermi and related theories of atoms and molecules. Rev.Mod.Phys., 53, 603-641 (1981)

No context found.

Lieb, E. "Thomas--Fermi and Related Theories of Atoms and Molecules" Reviews of Modern Physics Vol 53 no. 4. (1981)

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