O.H. Mobarek and A.Z. Alzahrani*

Transcription

1 Int. J. Mathematical Modelling and Numerical Optimisation, Vol., No. 1, A Z-dependent variational solution of Thomas-Fermi-Dirac equation O.H. Mobarek and A.Z. Alzahrani* Department of Physics, Faculty of Science, King Abdulaziz University, P.O. Box 800, Jeddah 1589, Saudi Arabia oghamdi@kau.edu.sa azalzahrani@kau.edu.sa *Corresponding author Abstract: Using Thomas-Fermi-Dirac equation within the exchange-correlation scheme and a Z-dependent trial solution, the ground state binding energy for neutral, positively, and negatively charged atoms are calculated. Comparing to the results obtained in earlier works, the values of the binding energy estimated here for both light and heavy atoms agree nicely with the Hartree-Fock values. We have also calculated the radial expectation values and diamagnetic susceptibilities for some neutral and single ionised atoms. Our results are compared with the available theoretical and experimental findings. Keywords: Thomas-Fermi-Dirac; variational principle; binding energy; exchange-correlation; diamagnetic susceptibility. Reference to this paper should be made as follows: Mobarek, O.H. and Alzahrani, A.Z. (011) A Z-dependent variational solution of Thomas-Fermi-Dirac equation, Int. J. Mathematical Modelling and Numerical Optimisation, Vol., No. 1, pp Biographical notes: O.H. Mobarek is an Associate Professor in the Department of Physics, at King Abdulaziz University. His interest includes atomic and molecular physics. A.Z. Alzahrani is an Assistant Professor in the Physics Department at King Abdulaziz University. The electronic and structural properties of semiconductor and atomic surfaces represent the major field of his researches. He is also interested in computational and theoretical physics with emphasis on the density functional theory. 1 Introduction The fundamental electronic properties of any interacting system can be determined by solving the Schrödinger equation for N-electrons. Hartree-Fock (HF) or self-consistent field (SCF) approximation is successfully and widely used for calculating the electronic structure of some molecular and atomic systems (March, 1975; Martin, 004). The large-scale computational process in HF theory, however, makes the problem of determining the electronic properties much more difficult. However, this difficulty has Copyright 011 Inderscience Enterprises Ltd.

2 70 O.H. Mobarek and A.Z. Alzahrani been removed by introducing the principle of the density functional theory (DFT) (March, 1975; Martin, 004). In such a method, the electronic properties of a system of many interacting particles can be fully understood by examining its ground state density function which is considered as the basic variable. Among various density functional theories, Thomas-Fermi (TF) and Thomas-Fermi-Dirac (TFD) schemes have attracted immense interest due to their simplicity and adequate estimations of the ground state properties of free atoms and solids (Thomas, 197; Hohenberg and Kohn, 1964). However, the main difference between these approaches (viz. TF and TFD) is that the TF method neglected the exchange and correlation among the electrons whereas the TFD method considered a local approximation for exchange. Although these approximations are far behind the available electronic structure calculations, they are still in use today. Furthermore, the solutions of the TF and TFD equations are considered as good approximations to those of the more accurate quantum mechanical HF equations. One of the most sufficient tools to solve the TF or TFD equations is to make use of the variational principle. Since the exact solution of the TFD equations is proportional to the inverse of the sixth power of the radial distance from the nucleus, many researchers have presented variational solutions of the TF and TFD differential equations involving trial exponential wave functions satisfying the previous feature as well as the rest of the essential boundary conditions. The ground state energy, therefore, can be evaluated by minimising the density functional subject to the constraint on the total number of electrons. As an example of such a method, Csavinszky (1968, 197, 1976) and, later, Mobarek (1987) presented approximate variational solutions of the TF equation using several-parameters exponential solution. Despite that the results obtained for lighter atoms in their works were much better than those of the original TF equation, lack of accuracy was observed for the heavy atoms. Recently, Mobarek and Alattas (010) have also used a similar trial solution within the variational technique but with including the electron exchange term, (i.e., TFD). Unlike the Csavinszky s work, their results for binding energies show much better agreement with the HF results for the heaviest atoms instead of lighter atoms. According to these findings, the inclusion of Z in the trial solution is expected to improve the electronic structure of solids within TFD. Despite much available works for TF and TFD methods (Tomishima and Tonei, 1966; Lieb, 1981; Agil et al., 1987; Alhendi et al., 1989; Sabirov, 199; Porras and Moya, 1999), no Z-dependent exponential solutions, to the best of our knowledge, have been reported for the TFD equation using the electron exchange-correlation scheme. In the present work, we have presented a numerical solution of the TFD equations using a Z-dependent exponential trial function. We have carried out our calculations within the electron exchange-correlation scheme. The numerical results of the binding energies, radial expectation values, and diamagnetic susceptibilities are compared with those obtained by HF and original TF methods. Theoretical method Throughout the present calculations, we have adopted a variational principle to obtain solution of the TFD equations for neutral, positively, and negatively charged atoms. The electron-electron interactions were treated by including the exchange-correlation term within the differential equation of TFD and hence, the electron density. We have chosen a multi-parameters exponential function, which depends on the atomic number Z and

3 A Z-dependent variational solution of Thomas-Fermi-Dirac equation 71 satisfies the boundary conditions of TFD equation as well as the normalisation condition, to be our trial solution of the TFD differential equation. Our trial function has the form Zx γzx ( ) Φ= ae + be, (1) where a, b, α, and γ are adjustable parameters which make the function satisfies the initial conditions Φ (0) = 1, Φ( ) =Φ ( ) = 0, normalisation condition, and the TFD differential equation of the form (Mobarek, 1987) d Φ dx Φ ( x β ), = x + () where β is the exchange-correlation term which is defined as β = 0.94 Z. () Having known that the electron density within the atom has the form (Mobarek, 1987) Z 4π b0 Φ ( β x ) ρ() r = +, (4) where the dimensionless parameter x relates to the radial distance r through the equation with r = b x (5) 0, 1 b0 = Z, (6) the normalisation condition, over the total number of electrons, will be = N ρ() r d r. (7) It is of particular importance to mention that equation (4) is valid for both neutral and positively charged atoms. However, the Fermi-Amaldi (FA) (Gombash, 1951) correction has been introduced to this equation to make it valid for negatively charged atoms as following: Z 4π s0 Φ ( β x ) ρ () r N = +, N 1 (8) where r = s 0 x with N / N 0 = 1 0 = N N 1 9 s b 0.90 Z. (9) To evaluate the unknown parameters in the chosen trial function we use the variational principle, which defined as (Goldstein, 1965) L( ΦΦ,, x) = F( ΦΦ,, x) dx. (10) The functional F( ΦΦ,, x ) is chosen as

4 7 O.H. Mobarek and A.Z. Alzahrani 5/ 1/ / dφ Φ βφ β x Φ ( dx ) ( 1/ x ) ( ) ( ) 1 6 β 5 F ( ΦΦ,, x ) = x Φ. (11) Note that equations (10) and () are identical since the substitution of equation (11) in the Euler-Lagrange equation, F F = 0, Φ x Φ results in the TFD equation. Having considered the boundary condition (Φ(0) = 1 a + b = 1) and normalisation condition equation (10) will be minimised with respect to the unknown parameters a, b, α and γ and then the ground state properties, such as binding energy and susceptibilities, can be achieved straightforwardly. The zero-order binding energy of a single ionised atom is defined as (Mobarek, 1987) e ( ) N ( ) 1/ π a0 Z x0 7/ EZN (, = Z± 1) = Φ (0) Z, (1) where a 0 is the Bohr atomic radius of the hydrogen atom, e is the electronic charge, and x 0 is the position of zero (i.e., Φ(x 0 ) = 0). The value Φ (0) is known as the initial slope. For neutral atoms, the previous equation tends to be ( ) 1 7 9π 1/ e a0 7/ EZN (, = Z) = Φ (0) Z. (1) The ionisation energy, hence, can be computed using the formula I = EZN (, = Z± 1) EZN (, = Z ) (14) The expectation values of the radial and squared radial distance are known as r = ρ() r rd r = 4 π r ρ() r dr, (15) 4 r = ρ() r r d r = 4 π r ρ() r dr. (16) The diamagnetic susceptibilities, hence, are determined from the expression (Mobarek, 1987) χ = ena ( 6mc ) 0 r, (17) where m 0 is the electronic mass, N A is the Avogadro s number, and c is the speed of light. Detailed integration results for each of the above physical quantities are given in the Appendix. Results and discussion Using the variational principle with the boundary and normalisation conditions we have obtained the unknown parameters of the trial function for atoms with Z ranging from small to large numbers. These values are summarised in Table A1 alongside the value of

5 A Z-dependent variational solution of Thomas-Fermi-Dirac equation 7 the initial slope of the function. It is clearly noticed that the value of the initial slope is smaller than the TF value of 1.58, suggesting good choice of the trial solution and hence, expecting good ground state results. Figure 1 shows the variation of the trial solutions as a function of the radial distance for different values of atomic numbers, Z, as obtained by TF (Csavinszky, 1968), previous TFD (Mobarek and Alattas, 010), and the present work. It is clearly indicated that the trial function of the present work behaves quite similar to that used in the previous work with little mismatch with the TF for the region of small Z. Unlike that, we have observed good agreement between TFD and TF methods for larger atoms (with large Z value). This clearly suggests that the exchange and/or correlation terms within TF theory play a dominant role in the results for lighter atoms rather than heaviest atoms. Moreover, the inclusion of Z in the trial solution will show much effect in the results of the lighter atoms with small correction to those of the larger atoms. More interesting than these functions is the distribution of the electronic charge within the atom. To give a quantitative idea about this, Figure shows the radial distribution of charge with respect to the radial distance corresponding to different values of Z. We have clearly noticed that, for small Z, the present solution gives slightly good agreement with the TF around the nucleus origin comparing to the previous TFD (see Mobarek and Alattas, 010). However, larger values of Z lead to overlap between the present and previously reported solutions, suggesting that the main contribution to the binding energy comes from the region around the nucleus, (r 0.). An inspection of Table A reveals that the Z-dependent solution of TFD equations modifies the binding energy values for both light and heavy atoms of neutral atoms comparing to those obtained by TF and previous Z-independent TFD. We notice that the calculated results show excellent agreement with those obtained by HF method. The expectation values of the radial and squared-radial distances have also been evaluated as shown in Table A and Table A4. The results obtained here are slightly similar to those obtained by Z-independent trial solution. Both Z-dependent and independent solutions give much better results compared to TF method. This indicates that, for homogeneous gases, the radial expectation values do not be contributed by the number of electrons but be averaged over the whole electronic cloud. Moreover, the diamagnetic susceptibilities for some neutral atoms have been estimated and summarised in Table A5. These values are in good agreement with the theoretical values and experimentally measured data. Figure shows the change of the binding energies and the diamagnetic susceptibilities of some neutral atoms as functions of the atomic number. For the purpose of comparison, we have also plotted the HF results and previously reported TFD data. For single-positively charged atoms, the values of the adjustable parameters are shown in Table B1. It is of interest to note that these parameters tend to have similar values of those of the neutral atoms when the value of N Z slightly approaches the unity (viz. very large atoms). The numerical results for the binding energies, ionisation energies, expectation values, and diamagnetic susceptibilities are summarised in Table B to Table B5 and compared to the results of previous TFD (Mobarek and Alattas, 010) and HF (Snow et al., 1964). Generally, the calculated results have been noticeably improved. This improvement is attributed not only to the inclusion of the correlation term but also to the Z dependency in the choice of the trial function. We also

6 74 O.H. Mobarek and A.Z. Alzahrani N notice that, for atoms of 1, Z the present calculations of their binding energies are quite similar to those obtained in Mobarek and Alattas (010). Nevertheless, these values N saturate the TF values as well as the previous TFD for small atoms, 0.9. Z However, our results are in good agreement with the HF values. Despite that the expectation values of the radial separations in the present and previous TFD works are in good agreement with each other, a noticeable disagreement is observed for the expectation values of the square atomic distance for the mediated atoms. Otherwise we have obtained similar results for light and heavy atoms. The calculated magnitudes of the diamagnetic susceptibilities of some positively charged atoms, on the other side, show excellent agreement with the experimentally measured values, especially for those of large atomic number. These observations indicate that the inclusion of Z in the solution is much more useful for ionised atoms due to the existence of the attractive interactions between nucleons comparing to the neutral atoms. Figure 1 The variation of the trial wave function ϕ(r) as a function of the radial distance corresponding to different atomic numbers (a) Z = (b) Z = 10 (c) Z = 40 (d) Z = 90 within TF (black), previous TFD (red), and present TFD (blue) (see online version for colours) (a) (b) (c) (d) Note: The atomic units are used for both axis labels.

7 A Z-dependent variational solution of Thomas-Fermi-Dirac equation 75 Figure The distribution of the electronic charge r ρ(r) versus the square-root of the radial distance from the nucleus r with different atomic numbers (a) Z = (b) Z = 10 (c) Z = 40 (d) Z = 90 within TF (black), previous TFD (red), and present TFD (blue) (see online version for colours) (a) (b) (c) (d) Notes: In both TF and previous TFD, the trial solutions are of the same type while present TFD has a similar form but with Z dependency. The atomic units are used for both axis labels. Furthermore, Table C1 Table C5 summarise the results obtained for single-negatively charged atoms. These results seem to be of correct magnitudes comparing with those obtained for neutral and single-positively charged atoms. It is noted that the trial solution deals only with the change of the atomic number, therefore, it is very valid for neutral atoms. Considering positively or negatively charged atoms leads to that the trial solution does not differentiate between them. However, this shortcoming is totally resolved within the normalisation condition where both Z and N are involved. As a suggestion for better modification of results for positively and negatively charged atoms, a trial function employing both Z and N could be of interest and may improve their electronic properties to the best. For an example, one should attempt replacing the Z dependency in the present Z solution by both Z and N dependency, such as ( 1 + N ) Z. We believe that the results of the Z( 1+ ) neutral atoms will not be affected since the N Z equals Z in the case of N = Z.

8 76 O.H. Mobarek and A.Z. Alzahrani Figure The variation of binding energy as a function of the atomic number (upper panel). The inset shows the variation of the binding energy of very light atoms with Z 10. The change of the diamagnetic susceptibility of some neutral atoms with respect the atomic number is depicted (lower panel) (see online version for colours) 4 Summary and conclusions Within the framework of a Z-dependent trial function and correlation scheme we have presented numerical solutions for the electronic structure of neutral and single ionised atoms using TFD equation. It is found that the inclusion of Z within the solution modifies the binding energies of the light atoms as well as the heavy atoms. Moreover, it is found that multi-parameters trial functions are considered as one of the best solutions of the TFD model. Our results show good agreement with the previously reported theoretical and available experimental data.

9 References A Z-dependent variational solution of Thomas-Fermi-Dirac equation 77 Agil, I., Alharkan, A., Alhendi, H., Alnaghmoosh, A. and Naturforsch, Z. (1987) Vol. 4a, p.94. Alhendi, H., Alharkan, A., Alnaghmoosh, A. and Alagil, I. (1989) J. King Saud Univ. Sci., Vol. 1, p.105. Clementi, E. and Roetti, C. (1974) At. Data Nucl. Data Tables, Vol. 14, p.177. Csavinszky, P. (1968) Phys. Rev., Vol. 166, p.5. Csavinszky, P. (197) Phys. Rev., Vol. A, No. 8, p Csavinszky, P. (1976) Phys. Rev., Vol. B, No. 14, p Desclaux, J.P. (197) At. Data Nucl. Data Tables, Vol. 1, p.11. Goldstein, H. (1965) Classical Mechanics, Addison-Wiley. Gombash, P. (1951) Statistical Theory of the Atom and its Application, Russian translation, Moscow. Hohenberg, P. and Kohn, W. (1964) Phys. Rev., Vol. 16, p.b864. Lieb, E.H. (1981) Rev. Mod. Phys., Vol. 5, p.60. March, N.H. (1975) Self-Consistent Fields in Atoms, Pergamon, Oxford. Martin, R.M. (004) Electronic Structure: Basic Theory and Practical Methods, Cambridge University Press. Mobarek, O.H. (1987) Arab Gulf J. Sci. Res., Vol. 9, No. 1, p.6. Mobarek, O.H. and Alattas, A.A. (010) Prepared for submission. Porras, I. and Moya, A. (1999) Phys. Rev., Vol. A, No. 59, p Sabirov, R.K. (199) Opt. Spect., Vol. 75, p.1. Snow, E.C., Canfield, J.M. and Waber, J.T. (1964) Phys. Rev., Vol. 15, p.a969. Thomas, L.H. (197) Proc. Cambridge Phil. Soc., Vol., No. 5, p.54. Tomishima, Y. and Tonei, K. (1966) J. Phys. Soc. Jpn., Vol. 1, p.14. Appendix 1 The present calculations have been carried out within the variational principle which is defined from equation (10) as L( ΦΦ,, z) = F( ΦΦ,, x) dx, where the functional F( ΦΦ,, x ) is chosen as 5/ 1/ / 1 dφ Φ βφ 6β x Φ F( ΦΦ,, x) = β Φ. 1/ x dx 5 x Using our Z-dependent trial solution of the form [equation (1)] Zx γzx ( ) Φ= ae + be,

10 78 O.H. Mobarek and A.Z. Alzahrani the result of the above integral will be the following ( α + αγ+γ ) Z a b a b α( α+γ) 8ab γ( α+γ) L( ΦΦ,, x) = a α+ b γ+ + + α+γ α+ γ α+γ π a b ab a b a b a b Z 5 / 5 / α γ α+ γ α+ γ α+ γ α+γ 4 4 β a b 4ab 4a b a b Z α 4γ α+ γ α+γ α+γ (A.1) + Z πβ / a b a b ab α γ ( α+γ) ( α+ γ) / / / / / / β a b ab Z 4α 4 γ ( α+γ). Minimising the above equation with respect to the unknown parameters a, b, α, and γ according to the boundary conditions and the normalisation condition will result in the values of these unknown parameters for each Z value, as summarised in Table A1, Table B1, and Table C1. The normalisation condition in which the above equation being minimised is determined from the equation N = ρ() r d r. Using the charge density function along with the proposed trial solution [equations (1) and (4), respectively], the integral has the following expression π a b a b ab N = Z α γ ( α+γ) ( α+ γ) 1/ / / / / / / β a b ab 9 πβ a b / 5/ 5/ Z 4α 4 γ ( α+γ) 4Z α γ (A.) The radial expectation value, on the other hand, is evaluated from equation (15) as following r ρ() r rd r = 4 π r ρ() r dr.

11 A Z-dependent variational solution of Thomas-Fermi-Dirac equation 79 Using the definition of charge density, equation (4), we straightforwardly obtain the following integral value r / 1 a π(z α) = b 0 4 7Z α 1/ / ( βγ + πβ γ + π γ ) b54bz 405 (Z) b (Z) + 4 γ β b πz + 54a Z + α (Z( α+γ)) 5/ 1/ 4 bzβ 15 β (Z α) 6b Z. + 7a + π ( ) + 4 (Z( )) 5/ α+γ α α+ γ (A.) Similarly, the expectation value of the square-radial distance is obtained from equation (16) as following 4 r = ρ() r r d r = 4 π r ρ() r dr. Working out the integral results in r π α = 4Z α / 1 10a (Z ) b / / ( βγ + πβ γ + π γ ) b 486bZ 8505 (Z ) 10b (Z ) + 5 γ 4 β 5b πz + 486a Z + α (Z( α+γ)) 4 7/ (A.4) 1/ 5 64bZβ 5 β (Z α) 10b Z + 4a + π / ( ) (Z( )) α+γ α α+ γ

12 80 O.H. Mobarek and A.Z. Alzahrani Appendix Part A Table A1 Neutral atoms The values of the adjustable parameters for some neutral atoms are summarised, as obtained by the variational principle Z a b γ Φ (0) Note: The initial slopes are also tabulated. Table A The binding energy for some neutral atoms evaluated by the present work and compared to previously reported results using HF, TF, and TFD methods Z E(present) E (Mobarek and Alattas, 010) E (TF) E [HF (Desclaux, 197)] , , ,444.,4.66, , , , ,7. 8, , ,4.9 1, , , , ,10 18, ,16. 0,07.5 1, ,51.9,74.6 4,59.6 Notes: The energies are measured in atomic units. 1 This value is empirically obtained by Slater rules, see Csavinszky (197).

13 A Z-dependent variational solution of Thomas-Fermi-Dirac equation 81 Table A The expectation value of the squared radial distance for some neutral atoms estimated by the present work and compared to previously reported results using HF and TFD methods Z r (present) r (Mobarek and Alattas, 010) r (Mobarek, 1987) Table A4 The radial expectation value for some neutral atoms estimated by the present work and compared to previously reported results using HF and TFD methods Z r (present) r (Mobarek and Alattas, 010) r [HF (Clementi and Roetti, 1974)] Table A5 The diamagnetic susceptibilities (cm ) for some neutral atoms estimated by the present work and compared to previously reported results using TF method and experimental values Z χ 10 6 (present) χ 10 6 (Ref.[5]) χ 10 6 (exp.)

14 8 O.H. Mobarek and A.Z. Alzahrani Part B Table B1 Positively charged atoms The values of the adjustable parameters for some single-positively charged atoms are summarised, as obtained by the variational principle Z, N a b γ Φ (0) 10, , , , , , Note: The initial slopes are also tabulated. Table B The binding energy for some positively ionised atoms estimated by the present work and compared to previously reported HF, TF, and TFD results Z, N E (present) E (Mobarek and Alattas, 010) E (TF) E [HF (Snow et al., 1964)] I (present) 10, , , , , , Note: The energies are measured in atomic units. Table B The expectation value of the squared radial distance for some positively ionised atoms estimated by the present work and compared to previously reported results Z, N r (present) r (Mobarek and Alattas, 010) 10, , , , , , Table B4 The radial expectation value for some positively ionised atoms estimated by the present work and compared to previously reported results Z, N r (present) r (Mobarek and Alattas, 010) 10, , , , , ,

15 A Z-dependent variational solution of Thomas-Fermi-Dirac equation 8 Table B5 The diamagnetic susceptibilities (cm ) for some positively ionised atoms estimated by the present work and compared with previously reported results obtained by TF method and also experimental values Z, N χ 10 6 (present) χ 10 6 (Csavinszky, 197) χ 10 6 (exp.) 10, , , , , , Part C Table C1 Negatively charged atoms The values of the adjustable parameters for some single-negatively charged atoms are summarised, as obtained by the variational principle Z, N a b γ Φ (0) 10, , , , , Note: The initial slopes are also tabulated. Table C The binding energy for some negatively charged atoms estimated by the present work and compared to previously reported HF, TF, and TFD results N, Z E (present) 10, , , , , Note: The energies are measured in atomic units. Table C The expectation value of the squared radial distance for some negatively charged atoms estimated by the present work and compared to previously reported results N, Z r (present) r (Mobarek, 1987) 10, , , , , ,

16 84 O.H. Mobarek and A.Z. Alzahrani Table C4 The radial expectation value for some negatively charged atoms estimated by the present work and compared to previously reported results N, Z r (present) 10, , , , , , Table C5 The diamagnetic susceptibilities (cm ) for some negatively ionised atoms estimated by the present work and compared with previously reported results obtained by TF method and also experimental values N, Z χ 10 6 (present) χ 10 6 (Mobarek and Alattas, 010) χ 10 6 (exp.) 10, , , , , ,