Modern Ab-initio Calculations Based on Tomas-Fermi-Dirac Theory with High-Pressure Environment

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1 American Journal o Quantum Chemistry and Molecular Spectroscopy 016; 1(1): doi: /j.ajqcms Modern Ab-initio Calculations Based on Tomas-Fermi-Dirac Theory with High-Pressure Environment Sergey Seriy Material Science Department, Institute o Dynamic Systems, Komsomolsk-on-Amur City, Russian Federation address: serggray@mail.ru To cite this article: Sergey Seriy. Modern Ab-initio Calculations Based on Tomas-Fermi-Dirac Theory with High-Pressure Environment. American Journal o Quantum Chemistry and Molecular Spectroscopy. Vol. 1, No. 1, 016, pp doi: /j.ajqcms Received: October 1, 016; Accepted: November 7, 016; Published: December 1, 016 Abstract: Thomas-Fermi theory is an approimate method, which is widely used to describe the properties o matter at various hierarchical levels (atomic nucleus, atom, molecule, solid, etc.). Special development achieved using Thomas-Fermi theory to the theory o treme states o matter appearing under high pressures, high temperatures or strong ternal ields. Relevant sections o physics and related sciences (astrophysics, quantum chemistry, a number o applied sciences) determine the scope o Thomas-Fermi theory. Popularity Thomas-Fermi theory is related to its relative simplicity, clarity and versatility. The latter means that result o the calculation by Thomas-Fermi theory applies immediately to all chemical elements: the transition rom element to element is as simple scale transormation. These eatures make it highly convenient tool or qualitative and, in many cases, quantitative analysis. Keywords: Quantum Mechanics, Ab-initio Calculations, Thomas-Fermi Theory, Material Science, Temperature and Pressure Environment, Supercomputer Calculations, Nanostructures, Ab-initio Molecular Dynamics, Numerical Methods 1. Introduction Thomas-Fermi theory was originally proposed by Thomas and Fermi [1-4] to describe the electron shell o a heavy atom, which is characterized by a relatively uniorm distribution o the electron density. Thomas-Fermi theory is the semi-classical (or Wentzel-Kramers-Brillouin) limit in relation to sel-consistent Hartree ield equations, and thereore modiication o this model are associated with a more detailed account o the correlation, change, quantum and multi-shell eects. Initial approimate nature o Thomas-Fermi theory has a dual nature. Remain outside the model, irstly, correlation eects, relecting the inaccuracy o the Hartree method and the associated sel-consistent dierence (average) true interaction rom the true physical interaction. Secondly, in Thomas-Fermi theory, quantum eects are not considered responsible approimate nature o the semi-classical description o the atom. The report amines the theory o these eects, allowing to ind the limits o applicability o Thomas-Fermi theory in its original orm and generalize the model beyond the scope o its applicability. The presence o correlation corrections are caused by dierence o sel-consistent ield o the actual ield inside the atom. These corrections are the result o the anti-symmetry o the electron wave unctions and are interpreted as the change correlation eects. Additionally, appear also the eects o the power correlation. We begin by considering the eects o correlation, which in turn are divided into two classes. This is primarily the eects o statistical correlation (change eects), describing the eect o the Pauli principle on the interaction o particles. Electrons with parallel spins are hold at a greater distance rom each other than in the singlet state, and the radius o such a correlation coincides with the de Broglie wavelength o an electron. Correlation eects o the second class (called correlation eects o power or simply correlation eects) relect the inaccuracy o the principle o independent particles, i.e., the inability to talk about the state o a single electron in the eective average ield o the other particles because o their mutual inluence, beyond the sel-consistent description. Being the power eects, this kind o correlation eects are characterized by a dimensionless parameter equal to the ratio o perturbation theory o the average energy o the coulomb

2 8 Sergey Seriy: Modern Ab-initio Calculations Based on Tomas-Fermi-Dirac Theory with High-Pressure Environment interaction between pairs o particles to their average kinetic energy. In Thomas-Fermi [3, 4] theory introduced accounting Dirac s correlation eects [], this pansion has been called the theory o Thomas-Fermi-Dirac. The report shows that the total energy o the electrons can be pressed in terms o the spatial dependence o the electron density according to the Thomas-Fermi-Dirac theory. In this calculations, the energy o the atom is based on nuclear-electron and electron-electron interactions (which can also be represented as a unction o the electron density). Quantum corrections arise rom the use o the semi-classical ormalism and relect the presence o non-local electron density communication to the potential in consequence o the uncertainty principle.. Theoretical Procedures A variational technique can be used to derive the Thomas-Fermi equation, and an tension o this method provides an oten-used and simple means o adding corrections to the statistical model [5, 6]. Thus, we can write the Fermi kinetic energy density o a gas o ree electrons at a temperature o zero degrees absolute in Eq. (1). U 5 3 = c ρ (1) Where, c = (3/10)(3π ) /3. The electrostatic potential energy density is the sum o the electron-nuclear and the electron-electron terms. We can write this as Eq. (). e n e n v Ep = Up + Up = ( v + ) ρ () Where, v n is the potential due to the nucleus o charge Z, v e is the potential due to the electrons, and the actor o 1/ is included in the electron-electron term to avoid counting each pair o electrons twice. With denoting distance rom the nucleus, the total energy o the spherical distribution is given by Eq. (3). v (3) 5 e 3 n E = ( c ρ ( v + ) ρ)4π d The pression or density on the Thomas-Fermi model: ρ = σ 0 (E - V) 3/ (4) σ 0 = (3/5 c ) 3/ (5) Eqs. (4) and (5) are obtained by minimizing Eq. (3) subject to the auiliary condition that the total number o particles N, remains constant. The potential energy V is a unction o position in the electron distribution. E is the Fermi energy, or chemical potential, and is constant throughout a given distribution. The Thomas-Fermi equation ollows rom Eq. (4) and Poisson s equation. The tendency or electrons o like spin to stay apart because o clusion principle is accounted or by the inclusion in Eq. (3) o change energy, the volume density o which is given by Eq. (6). U 4 3 = c ρ (6) Where, c = (3/4)(3/π) 1/3. Minimization o the total energy now leads to Eq. (7) c ρ cρ ( E' V) = 0 (7) 3 3 Eq. (7) is quadratic in ρ 1/3. From Eq. (7), we get Eq. (8). Where, ρ = σ ( τ + ( E' V + τ ) j ) (8) c τ 0 = (9) 15c Now Poisson equation with the density given by Eq. (9) leads to the TFD (Thomas-Fermi-Dirac) equation. In the ollowing two slides, we propose additional energy terms to be included in Eq. (3). The incorporation o these terms leads to a simple quantum- and correlation-corrected TFD equation. The quantum-correction energy density ollows rom a slight change in the derivation due to March and Plaskett [6]. March and Plaskett have demonstrated that the TF (Thomas-Fermi) approimation to the sum o one-electron eigenvalues in a spherically symmetric potential is given by the integral: I = (l + 1) E( nr, l) dnrdl (10) Where, the number o states over which the sum is carried is written as Eq. (11). N = (l + 1) dnrdl (11) where, E(nr, l) is the pression or the semi-classic eigen-values considered as unctions o continuous variables [1, 6]; n r is the radial quantum number; l is the orbital quantum number; the region o integration is bounded by n r = -1/, l = -1/, and E(n r, l) = E. We have included a actor o two in these equations to account or the spin degeneracy o the electronic states. The Fermi energy E is chosen so that Eq. (11) gives the total number o states being considered, the N electrons occupying the N lowest states. 3 P P (1) 5 3π 3 I = ( + V) 4π d With considerable manipulation, Eq. (10) becomes TF energy Eq. (1) and Eq. (11) reveals the TF density through Eq. (13).

3 American Journal o Quantum Chemistry and Molecular Spectroscopy 016; 1(1): N P 3π 3 = 4 π d (13) Both integrals being taken between the roots o E = V(). We have written these results in atomic units, so that P, the Fermi momentum, is deined by Eq. (14). P = ( E' V) (14) It is pertinent to amine the error in the TF sum o eigen-values, as given by Eq. (1), or case o pure Coulomb ield. The WKB eigen-values in Coulomb ield is given by Eq. (15). z En, ( ) r l = ( n + l + 1) r (15) And let us consider the levels illed rom n = 1 to n = v, where n is the total quantum number deined by n = n r + l + 1. Then, or any value o v, we can evaluate the error in the TF approimation to the sum o eigen-values, comparing always with the correct value, -Z v. Scott correction to the total binding energy is obtained by letting v become very large. Although the sum o one-electron eigen-values is not the total energy o the statistical atom because o the electron-electron interaction being counted twice, we might pect to improve the calculated binding energy greatly by correcting this sum in some manner, since the chie cause o the discrepancy is certainly the large error in the electron-nuclear potential energy. This correction can be perormed by imposing a new lower limit on l in the integrations above. When we introduce a new lower limit l min and a related quantity which we call the modiication actor, we obtain, ater more manipulation, slightly dierent pressions corresponding to Eqs. (1) and (13). a = l 1 min + (16) From these revised pressions, we can identiy a quantum-corrected TF density pression as Eq. (17), and corrected kinetic energy density as Eq. (18). 3 a ρ = σ0( E' V ) (17) 5 3 a Uk = c ρ + ρ (18) Revised lower limit on the volume integrals, say 1, is the lower root o: E V - a / = 0 (19) For < l, ρ must vanish (stay zero), and we have thus termed 1 the inner density cuto distance. We can call the second term on the right-hand side o Eq. (18) the quantum-correction energy density and write it in the more consistent orm as Eq. (0). c U = ρ (0) q q Where, c q is deining as Eq. (1): c a q = (1) The modiication actor a, is determined by the initial slope o the potential unction. For interpreting these results, it is helpul to consider just what we have done in changing the lower limit o the orbital quantum number. Since the lower limit l = -1/ must correspond to an orbital angular momentum o zero, we have, clearly, eliminated states with angular momentum o magnitude between zero and a cuto value L c = aħ. Corresponding to L c at every radial distance is now a linear cuto momentum: P c = aħ/, and we can rewrite Eq. (17) in terms o the Fermi momentum and cuto momentum: σ 3 0 ρ = ( P P ) 3 c () At radial distances less than l, momenta are prohibited over the entire range rom zero to P, so the electron density vanishes. This interpretation must be modiied somewhat when change and correlation eects are included; or then the Fermi momentum is no longer simply given by Eq. (14), cept very near the nucleus. We can deine 1 as in the absence o interactions, i.e., as the lower o the roots o Eq. (19), but it is not correct to demand that the density vanish at the upper root. Instead, we require only that the density is real. Correlation correction [1, 5, 6], the original TF equation describes a system o independent particles, while the introduction o change energy, which leads to the TFD equation, represents a correction or the correlated motion o electrons o like spin. The remainder o the energy o the electron gas is termed the correlation energy, by its inclusion we are recognizing that electrons, regardless o spin orientation, tend to avoid one another. In tensions o the statistical model, there have been suggested at least two dierent pressions, or the correlation energy that approach, in the appropriate limits, Wigner s low-density ormula and the pression due to Gell-Mann and Brueckner at high densities. In addition to these, Gombas and Tomishima [6] have utilized pansions o the correlation energy per particle in powers o ρ 1/3 about the particle density encountered at the outer boundary o the atom or ion. In this pansion, the term o irst-order can be considered as a correction to the change energy, and it ollows that the TFD solutions or a given Z then correspond to correlation-corrected solutions or a modiied value o Z. Aside rom rather poor approimation o the correlation energy, a drawback to this procedure is that the TFD solutions must be at hand. I solutions representing speciied degrees o

4 10 Sergey Seriy: Modern Ab-initio Calculations Based on Tomas-Fermi-Dirac Theory with High-Pressure Environment compression are desired, the method would appear to be impractical. It is interesting and ortunate that over density range o interest it is apparently possible to approimate the correlation energy per particle quite closely by an pression o orm: u c c 1 6 = c ρ (3) where, we have set c c = 0.084, and compared this approimation with the values due to Carr and Maradudin [5]. So, Thomas-Fermi and the Thomas-Fermi-Dirac equations are nonlinear dierential equations which must be solved numerically. The procedure is to assume a solution in the orm o a power series, and to start the numerical integration rom the series solution. The series coeicients o the solution may be evaluated in terms o the initial slope o the potential unction. Each value o the initial slope then gives rise to a particular potential curve. The TFD equation possesses a solution which becomes tangent to the ais at ininity. This solution has an initial slope which has been determined or He by Latter. Application o the TFD method ollows by surrounding each nucleus in by a cell which contains just Z electrons. The boundaries o the cell are planes which perpendicularly bisect the lines joining each nucleus with the nearest or nt nearest neighbors. Because o the high order o symmetry o the charge distribution, the electric ield at points outside the cell should then be very small and drop o rapidly with distance. distribution in the orm: e v cq = ρ ρ ρ ( + ) ρ + ρ (4) n U c c cc v Where all quantities appearing in the equation have been previously deined. By minimizing the integral o U over the volume occupied by the charge, while requiring that the total number o electrons be ied, we obtain Eq. (5). 4 c Where, τ 1 = 5 c 1 1 R ρ τ1ρ υ0ρ = 0 (5) 4 7, υ = c σ c 0 The electron density is ound as a unction o R by solving Eq. (3), a quartic in ρ 1/6. To accomplish this we write a resolvent cubic equation in terms o another variable, say y:, y 3 + τ 1 y + R y + (τ 1 R-υ 0 ) = 0 (6) Let us use the same symbol y, to denote any real root o this cubic equation. We can then press the our roots o the quartic, and hence our pressions or the electron density, in terms o y. One o these pressions possesses the proper behavior in reducing to previously obtained results in the neglect o correlation and change eects, namely: ρ 1 ( τ ψ ) 8 3 = y + R (7) Where, ψ = τ + τ + + (8) 1 y( 1 y y R) We note that ψ vanishes when correlation is neglected, since y = -τ 1 is then root o Eq. (6). In the amiliar manner we now deine a modiied TFD potential unction θ by Eq. (9), and rom Poisson equation and Eq. (7) we obtain Eq. (30). Zθ = (E - V + τ 0 ) (9) Fig. 1. Various TFD solutions. The approimation is then made that the ield outside the cell can be neglected, and the polyhedron is replaced by a spherically symmetric distribution o charge having the same volume as that o the polyhedron. The solutions o the Thomas-Fermi equation or inite border are thus capable o describing the element in various states o compression (pressure), ig Derivation and Discussion From the results o the preceding slides, we can now press the total energy per unit volume o the charge In terms o θ: π 3 ( τ1 + ψ + y + R) ; 1, θ = Z 0; < 1. (30) Zθ a 3 R = 4 σ0 ( τ 0) (31) Eqs. (6), (8), (30) and (31) constitute the dierential relationship to be satisied at each step in the integration. We could, o course, write immediately the solutions o Eq. (6) in analytic orm, but it proves convenient in the numerical treatment to obtain a root by the Newton-Raphson method, since a good irst guess in the iteration is available rom the

5 American Journal o Quantum Chemistry and Molecular Spectroscopy 016; 1(1): previous integration step. The boundary conditions on Eq. (30) are: 1 As nucleus approached the potential must become that nucleus alone, or θ(0) = 1; At outer boundary, o distribution o N electrons: (3) N = ρ4π d = Z θ d 1 1 Integration by parts yields: N θ( 1) = 1 + 1θ ( 1), ( θ θ) 1 =, and since we have Z the usual condition: θ( ) = θ ( ) + (33) Z In addition to potential and density distributions, total binding energies o atoms are o special interest to us here. For the proper evaluation o energies, the arbitrary constant that is present originally in both the electrostatic potential energy and the Fermi energy must be speciied. The state o ininite separation o the constituent particles is normally taken to have zero energy. We thereore ollow the usual convention and i the potential at the edge o the neutral atom at zero or all values o. For an ion the potential energy o an electron at the boundary is taken as: V = (34) The deining relation, Eq. (34), now gives at the boundary: Or solving or the Fermi energy, Zθ( ) = ( E + + τ ) (35) 0 θ( ) E Z = τ0 (36) The total electron-nuclear potential energy given by E n = Z p 4 d ρ π (37) 1 While or the electron-electron potential energy we have: e 1 e Ep v ρ4π d 1 = (38) From Eq. (9) and the relation V = - (v n + v e ), this becomes: Zθ 0 ρ π 1 e 1 ( n Ep = Ep + τ N + EN 4 d ) (39) Other energy integrals are, with an obvious notation: 5 3 E c ρ 4π d = (40) 4 q = q d (41) E c ρ π 4 3 E c ρ 4π d = (4) 7 6 Ec cc ρ 4π d 4. Results and Conclusions = (43) It was pointed out in the introduction that the quantum-corrected TFD equation yields atomic binding energies in good agreement with perimental values and with the results o DFT (density unctional theory) calculations [5]. Multi-shells eects relect irregularities physical quantities due to the discrete energy spectrum, but in the case o the continuous spectrum o these eects may occur as a result o intererence o de Broglie waves and allow the model to take into account the shells structure o the atom. Multi-shells eects associated with the discrete spectrum o bound electrons in atomic systems (atom, ion, atomic cell, etc.). Multi-shells eects, unlike quantum-change eects, aect the chemical potential E', but practically have no eect on the value o sel-consistent potential V [7]. Thereore, when they accounting or the calculation o a corresponding shells correction, is not necessary to solve the Poisson s equation. Just use the normalization condition with the same sel-consistent potential in TFD model. For shell corrections E' sh primary role o shell eects reduces to a shit o the chemical potential E' [8]. E ' sh 1 sin( πkv( λ)) = 1 k (44) π k k = 1 Where, v = n r + 1/, λ = l + 1/. Sotware implementation o this modiied Thomas-Fermi theory and calculations (or ample, rare gas atoms, Table 1 and Fig. ) taking into account quantum, change and correlation corrections showed that the corrections really lead to a rapprochement and converge results with perimental data, and also the results obtained by the DFT approimation [5]. Total energy calculations by Thomas-Fermi, DFT and perimental data are shown in the summary Table 1. Nt step o this work include program realization or multi-shell and gradient corrections also [1, 6]. Table 1. Results comparisons. Z E TFD (a.u.) E DFT (a.u.) E p (a.u.)

6 1 Sergey Seriy: Modern Ab-initio Calculations Based on Tomas-Fermi-Dirac Theory with High-Pressure Environment Reerences [1] Kirzhnits, D. A.; Lozovik, Y. E.; Shpatakovskaya, G. V. Sov. Phys. Usp. 1975, 18, 649. [] Dirac, P. Proc. Cambr. Philos. Soc. 1930, 6, 376. [3] Thomas, L. Proc. Cambr. Philos. Soc. 197, 3, 54. [4] Fermi, E. Rend. Accad. Naz. Lincei 197, 6, 60. [5] Barnes, J. F. Quantum- and correlation-corrected Thomas-Fermi-Dirac equation. Phys. Rev. 140, (1965). [6] Barnes, J. F. Los Alamos Scientiic Laboratory o the University o Caliornia, Fig.. The electron density(ternal shells) o the rare atoms (a) helium (He), (b) argon (Ar), (c) neon (Ne) and (d) krypton (Kr), computed on the present model agree closely with their crystal radii. Acknowledgements The author would like to thank the Keldysh Institute o Applied Mathematic, Moscow, Russian Federation; Tohocu University, Sendai city, Japan; Los Alamos scientiic laboratory, Oakland, USA. [7] Karpov, V. Y.; Shpatakovskaya, G. V. Journal o Eperimental and Theoretical Physics Letters 013, 98, [8] Shpatakovskaya, G. V. Journal Physics-Uspekhi. 01, 55, [9] H. J. Monkhorst and J. D. Pack, Phys. Rev. B 13, 5188, [10] F. Birch, J. Geophys. Res. 83, 157, 1978.