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Metallic Hydrogen in a More Accurate Wigner-Seitz Approximation Pairot Moontragoon and Udomsilp Pinsook The purpose of this work is to calculate the ground state energy of solid metallic hydrogen as a function of s and to determine s in which the ground r r state energy is minimum. We use the Wigner-Seitz method to estimate the energy band dispersion up to and use the most accurate correlation potential available. Hence the ground state energy calculated in this work is more accurate than in some previous work. Under a simple Coulomb potential, the minimum ground state energy is –1.10 Ry/atom at s 0 which corresponds to a o( k 4 ) r 1.8a density of 0.44 g/cm . Under screening, the minimum ground state energy is –1.07 Ry/atom at s 0 for the uniform screening and -0.94 Ry/atom at 3 rs 1.8a0 for Thomas-Fermi screening. r 1.6a Key words: ground state energy, metallic hydrogen and Wigner-Seitz method. Department of Physics, Faculty of Science, Chulalongkorn University, Bangkok, 10330, Thailand. J. Sci. Res. Chula. Univ., Vol. 28, No. 1 (2003) 13 ..……………………………………………………..Pairot Moontragoon and Udomsilp Pinsook พลังงานสถานะพื้นของโลหะไฮโดรเจนโดยการประมาณแ บบวิกเนอร์-ไซทซ์ที่ละเอียด ไพโรจน์ มูลตระกูล และ อุดมศิลป์ ปิ่นสุข ( 2 5 4 6 ) วารสารวิจัยวิทยาศาสตร์ จุฬาล งกรณ์มหาวิทยาลัย 28(1) จุดมุ่งหมายของงานวิจัยนี้ เพื่อคานวณพลังงานสถานะพื้นของโลหะไฮโดรเจนแข็ง ซึ่งเป็นฟังก์ชนของ ั rs ที่พลังงานสถานะพื้นมีค่าต่าสุด ผู้วิจัยใช้วิธีของวิกเนอร์ และระบุค่า rs - ไซทซ์เพื่อประมาณการกระจายของแถบพลังงานให้ละเอียดถึงอันดับ k 4 และใช้พลังงา นสหสัมพันธ์ที่ละเอียดที่สุดเท่าที่หาได้ ฉะนั้นพลังงานสถานะพื้นคานวณในงานวิจัยนี้จึงแม่นยากว่าของงานที่มีผู้วิจัยอื่นทาไว้ แล้ว ภายใต้พลังงานศักย์แบบคูลอมบ์ พลังงานสถานะพื้นต่าสุดมีค่า -1.10 Ry/atom ที่ rs เป็น 1 . 8 a 0 ซึ่งจะสอดคล้องกับความหนาแน่นประมาณ 0 . 4 4 g / c m 3 ภายใต้พลังงานศักย์กาบัง พบว่าพลังงานสถานะพื้นต่าสุดมีค่า -1.07 Ry/atom ที่ rs เป็น 1.6a 0 สาหรับศักย์กาบังแบบสม่าเสมอและ -0.94 Ry/atom ที่ rs เป็น 1.8a 0 สาหรับศักย์กาบังแบบโธมัส-เฟร์มี คาสาคัญ พลังงานสถานะพื้น โลหะไฮโดรเจน วิธีของวิกเนอร์ไซทซ์ 14 J. Sci. Res. Chula. Univ., Vol. 28, No. 1 (2003) Metallic Hydrogen in a More Accurate Wigner-Seitz Approximation..…………………………….. INTRODUCTION Hydrogen is the lightest element in the periodic table. In 1935 soon after quantum mechanics was fi rmly established, Wi gner predicted that hydrogen could become a metal under enormously intense pressure.(1) This is of great scientific interest because metallic hydrogen would definitely be the lightest metal. Some calculations indicated that it can be in a superconducting state even at high temperature.(2) A possible technological application is to use metallic hydrogen as a very compact source for fuel cells. Furthermore, some evidence shows that the cores of Jupiter and Saturn are composed of metallic hydrogen.(3) In this work, the ground state energy of metallic hydrogen is calculated by using the Hartree-Fock approximation. HartreeFock theory(4) is a simplification of a full problem of many electrons moving in a potential field. The motion of electrons can be separated from those of nuclei according to the Born-Oppenheimer approximation and the motion of nuclei will be neglected. The electrons are treated as identical particles which are described by antisymmetric wave functions. The Pauli exclusion principle is incorporated via the Slater determinant. The final form of the Hartree-Fock approximation can be w r i t t e n a s 2 2 N N 2 V ( r ) d 1 e E r 1 i r r 1 1 i 2 i j ( i j ) Ri R j 2m N i 1 1 N N e2 2 i j dd r1 r2 r1 r2 2 i j r1 r2 1N N e2 i r1 j r2 i r2 j r1 d r1 d r2 2 i j ( spin//) r1 r2 where are the electrons’ wave functions. The first term is the interaction between the electrons and the nuclei under a general potential . The second term is the Coulombic interaction among the nuclei. The third term is the Coulombic interaction among the electrons. This term is different from the 2 … (1) (r ) V (r ) classical picture in that the electrons are no longer a point charge but rather spread out in space. The fourth term is the so-called exchange interaction which is purely a quantum effect and has no classical analogue. The exchange energy for interacting electrons is not difficult to evaluate once we know the electron density.(1,4) 15 J. Sci. Res. Chula. Univ., Vol. 28, No. 1 (2003) ..……………………………………………………..Pairot Moontragoon and Udomsilp Pinsook In t hi s wor k, we shall t r y t o calculate some physical properties of metallic hydrogen such as the ground state energy as a function of s and s in which r r atomic cells with equal volume. In this method, an atomic cell of volume is replaced by a sphere of radius s where N the ground state energy is minimum. In order to do so, some further approximations will be carried out. Firstly, the Wigner-Seitz approximation, which reduces the many-body problem into a one-body problem and gives the dispersion of the energy band near k = 0, will be explained in section II. Secondly, the interactions between a given proton a n d a n electron are modeled by the Coulomb potential, a uniform screening potential and the Thomas Fermi screening potential. The correlation effect is included by using the most accurate correlation potential available.(5) The detail of these potentials and some analytic evaluations of the ground state energy are described in section III. Conclusions are in section IV. CELLULAR METHOD In general, the wave functions of electrons in a metal are determined by the so-called cellular method which was first proposed by Wigner and Seitz. (6) In general a metal is composed of arrays of ion cores and moving electrons. Thus the whole metal can be divided into 4 3 rs V . 3 r V This sphere is called the Wigner-Seitz cell. The key physics of this method is that even though electrons are nearly free to move, on the average, one Wigner-Seitz cell contains only one electron. Consequently, the potential function V r between electrons and their corresponding ion cores in eq. (1) can b e w r i t t e n a s V r U i r , where N Ui r i 1 … (2) is the potential between an electron and its ion core within cell Ui r 0 i and outside the cell. These potentials are assumed to be spherically symmetric. The explicit potentials will be discussed in section III. By inserting eq. (2) into eq. (1), the problem is reduced from a one N-body problem to N one-body problems, which are much easier to be solved. Then, the ground state energy can be expressed as(7) E0 N N 2 e2 2 NE 2 k 2 k k r2 dd r1 r1 r2 k k k a a r r 1 2 0 e k k k k dd . r1 r2 r2 r1 r1 r2 k k r1 r2 This approximation is called the WignerSeitz approximation. Note that energy is expressed in rydberg (Ry) and length in Bohr units ( 0 ). In this unit, 2 … (3) 1 , e 2 and 2 a 2 1. 2m 16 J. Sci. Res. Chula. Univ., Vol. 28, No. 1 (2003) Metallic Hydrogen in a More Accurate Wigner-Seitz Approximation..…………………………….. Next, we determine the electron wave functions and the corresponding energy as a function of k . This process is essential in the electronic band structure theory. The exact wave functions are extremely complicated because they are expressed by functions of complex k r e ik r u0 r u1 r k u2 r k 2 ... , numbers and their true symmetries are not known. However, because hydrogen has only one electron, when it becomes a metal, the conduction band is only half filled. Thus the expansion of the wave function near the bottom of the band can b e e x p r e s s e d a s ( 8 ) … (4) and the energy band dispersion is k E0 E 2 k 2 E 4 k 4 ... , where … (5) E0 E is the ground state energy of one-electron problem, and the coefficient 2 and 4 are to be determined. Due to E the periodic nature of a crystal, the wave functions must satisfy Bloch conditions and the electrons are not truly localized in a Wigner-Seitz cell but, rather, wandering around and hence the wave functions must also be continuous at the cell boundary, E E d solving i.e. 0 . A fundamental dr rs 1 d r 2 d U r E u 0 , 2 0 0 r dr dr u 0 is the radial part of the one-body ground state wave function, and 1 d r 2 d U ( r ) 1 E f 0 , 2 0 r2 r dr dr where = 0, 1, 2,….. In atomic physics, is called the quantum number of orbital angular momentum and is the radial weakness of this approach is that the higher order terms in k and the directional dependence of the wave functions are discarded. The directional dependency can be neglected if the lattice has high symmetry such as in bcc. The energy dispersion is the most important relation because it leads to the ground state energy in eq. (3). Bardeen(9) and Silverman(8) showed that 2 and 4 are given by … (6) … (7) The complete relation between f was shown in ref. 8 and 9. 1 must satisfy part of the corresponding wave function. However, the physical interpretation of in eq. (7) is quite different. It is just a f d 0 and then f ( r ) r u ( r ) .(8,9) 1 s s 0 s dr rs f f and u dummy function related to u in eq. (4). Because of the Wigner-Sietz approximation, we can use the one-body potential rather than the many-body potential U (r ) V (r ) 17 J. Sci. Res. Chula. Univ., Vol. 28, No. 1 (2003) ..……………………………………………………..Pairot Moontragoon and Udomsilp Pinsook and hence the index i can be omitted. Silverman(8) showed that rf E2 1 , f 1 rs 2 2 2 1 … (8) 2 2 4 rs E rf2 E 2 2 f1 E 4 rs E 2 , 5 15 f2 r u0 rs rE rs , E0 s and then … (9) where 2 2.21 5.81 k E 0 E 2 2 E 4 4 , Nk rs rs 4 f rs3u02 ( rs ) and f . 3 r work, we shall evaluate … (10) So far, this section has not referred to a specific metal. In the next section, we will specify some potentials for metallic hydrogen. Once the explicit potential functions U r are applied, the functions f2 using eq. (7) and perform a more accurate calculation by adding a correction term of order (k ) . k4 to u 0 , f1 and f 2 can u be evaluated. The ground state energy in eq. (3) can then be calculated. Styer and Ashcroft(7) have already evaluated 0 and 1 , and used eq. f (3) to calculate the ground state energy of the metallic hydrogen. The exact calculation is far more complicated so they preformed some sensible estimation for all terms in eq. (3). Nevertheless, they evaluated the first term in eq. (3) by expanding MODEL POTENTIALS Coulomb potential In this section, we assume that the interaction between the electron and its positive core is purely Coulombic, i.e. U (r ) 2 r in the natural units. The (k ) upto k 2 only. b r In this potential is long range. This means that the nuclei interact with all other nuclei in the metal. By solving eq. (7), we find that f2 Ar 2 e 1 F1 3 ;6; 2 b r , 1 b … (11) 18 J. Sci. Res. Chula. Univ., Vol. 28, No. 1 (2003) Metallic Hydrogen in a More Accurate Wigner-Seitz Approximation..…………………………….. where 1 1 is called a confluent hypergeometric F b E0 and y 2 b r . function or Kummer function. is an arbitrary constant. Its value is not needed because it is canceled in eq. (9). A 2 3rs 2 r 2 . Ur 3 r rs The uniform screening is short range. The potential vanishes at the cell boundary; this means that the nuclei do not feel the Screening effects Due to electrons moving nearly freely in the metal, the screening effect is extremely efficient. Thus our calculations are more realistic if we replace the Coulomb potential by a screening potential. For uniform screening, the potential is … (12) presence of the others. By inserting this equation into eq. (7) and solving by Frobenius method, we find that … (13) f2 where and A i bi r i 0 r b0 0 , b1 0 , b2 0 , b3 1 , bi 1 3 b 2 bi 1 E 0 bi 2 i 34 , i 4 . i i 1 6 rs rs 2 2 U r 1 e kTF r , r r … (14) For Thomas-Fermi screening, the potential is where 12 k TF 1 3 1 rs . Thomas- Fermi screening has an intermediate range. The potential extends over a couple of neighboring cells. By inserting eq. (14) into eq. (7) and using Frobenius method, we get … (15) f2 where and A i bi r , r i 0 b0 0 , b1 0 , b2 1 , 19 J. Sci. Res. Chula. Univ., Vol. 28, No. 1 (2003) ..……………………………………………………..Pairot Moontragoon and Udomsilp Pinsook i 1 i 1 kTF m1 bi bm E0 bi 2 , i 3 . 2 i i 1 6 m0 i m 1! Again, A in eq. (13) and (15) is an arbitrary constant and canceled out in eq. (9). 4 3 2 E0/(Ry/atom) 1 0 -1 -2 0 1 2 3 4 5 6 r(a0) Figure 1. The ground state energy E0(rs) with various potentials: Coulomb potential (dotted line), the uniform screening potential (solid line) and Thomas F e r m i p o t e n t i a l ( d a s h e d l i n e ) . By substituting u 0 , f1 (7) and f2 into eq. (8) and (9), we can easily calculate the energy dispersion in eq. (5), its average over the conduction band in eq. (10) and the ground state energy in eq. (3). (k ) To further improve the accuracy of the ground state energy, an accurate correlation energy c is added to eq. (3). The explicit expression for the correlation energy per electron is(10) x x 0 2 ln , x 2 2 b 1 Q bx 0 X x c rs Aln tan 2 x b X x 0 2b 2 x 0 1 Q X x Q Q tan 2 x b 20 … (16) J. Sci. Res. Chula. Univ., Vol. 28, No. 1 (2003) Metallic Hydrogen in a More Accurate Wigner-Seitz Approximation..…………………………….. where X x x 2 bx c , Q = 4c-b2 , A = 0.0621814, x 0 = 0.10498, b = 3.72744 and c = 12.9352. x rs , This correlation energy was proposed by Ceperley and Alder(5) and revived by Vosko, Wilk and Nusair(10) and accepted as the most accurate correlation energy available.(5) Nevertheless, we find that in the metallic density range, i.e. , the correlation energy s 0 0 r 1.66a0 , -1.038 r 1.65a0 a n d - 1 . 0 5 2 Ry/atom at rs 1.61a 0 for Coulomb, the is –1.078 Ry/atom at s Ry/atom at s uniform screening and Thomas-Fermi potentials respectively. Normal metals have s between 2.0a0- 6.0a0. The density of lithium is 0.542 g/cm3 and the density of iron is 7.87 g/cm3. As mentioned earlier, the major improvement of the calculation of the ground state energy is the improvement in the accuracy of . Thus the major difference between Styer and Ashcroft’s work(7) and ours is the correction term, r r 1a 6a used in(7) is only slightly different from ours. Therefore the major improvement of the calculation comes from the more accurate . The remaining task is to evaluate the second and the third terms in eq. (3). The second term is called Hartree energy. It is dependent on an integral of k . (k ) (k ) Unfortunately, in eq. (4) is not k good enough for the evaluation of Hartree energy. Thus we follow the estimation of this term proposed by Styer and Ashcroft.(7) The third term is exchange energy which is easily calculated once we know s . The explicit form of the exchange (r ) (r ) 5.81 E4 4 rs in eq. (10). Notice that E 4 is depended solely on the solution of eq. (6) and (7) which are subjected to change with the different model potentials only. In the other words, the value of 4 is depended E r energy is 0.916 . rs The ground state energy per electron of the metallic hydrogen as a function of s is shown in figure 1. For the Coulomb potential (dotted line), the minimum energy is - 1 . 1 0 R y/ a t o m a t s 0 which corresponds to a density of 0.44 g/cm 3. For uniform screening (solid line), the minimum energy is –1.07 Ry/atom at s 0 . For Thomas-Fermi screening (dashed line), the minimum energy is -0.94 Ry/atom at s 0 . For comparison, the minimum ground state energy calculated by Styer and Ashcroft(7) r r 1.8a on the screening or the unscreening potential only. The physical picture is that the screening effects affect the interactions between the electrons and their own nuclei, and the interactions among the nuclei. In Styer and Ashcroft’s work,(7) the screening effects have no obvious impact on the physical properties of the metallic hydrogen. This is in contrast with our findings. We see that the minimum ground state energy is higher under the ThomasFermi screening potential than under the Coulomb potential, and s is smaller, i.e. r r 1.6a r 1.8a smaller volume, with the uniform screening potential with the Coulomb potential. The volume can be small because the interaction caused by the uniform screening potential has a very short-range effect, i.e. vanishes at the cell boundary. Thus the nuclei can U (r ) J. Sci. Res. Chula. Univ., Vol. 28, No. 1 (2003) 21 ..……………………………………………………..Pairot Moontragoon and Udomsilp Pinsook hardly see each other and tend to stay closer together. CONCLUSIONS We used the Wigner-Seitz approximation and the most accurate correlation potential to calculate the minimum ground state energy and its corresponding s for solid metallic highest pressure available in laboratory, i.e. about 340 GPa. Furthermore, solid molecular hydrogen has lower ground state energy which is –1.1648 Ry/atom at s = 3.12 0 . Thus it is more likely that the r a r hydrogen. Metallic hydrogen is thought to be arrays of N atoms, which occupy Wigner-Seitz spherical cells. Thus the problem reduces from a one N-body equation to N one-body equations. We expand the energy band dispersion up to the order of . This leads to a more a c c u r at e gr o u n d s t at e e n e r gy. B y including the correlation energy, we found that under the simple Coulomb potential, the minimum ground state energies is - 1 . 1 0 R y/ a t o m a t s 0 which k4 solid molecular hydrogen will be found under pressure instead of the solid metallic hydrogen because it will tend to its lowest energy state. Solid metallic hydrogen might exist as a metastable state. If it is stable, there must be some other effects, apart from those effects included in this work, which reduce the ground state energy. Recently there has been a report on collective excitations, which produce extra binding energy(12) and might cause the metallization. However, this is beyond the scope of the present work. REFERENCES Wigner, E. and Huntington, H.B. (1935) J . Chem. Phys. 3, 764. 2. Maksimov, E. G., and Savrasov, D. Y. (2001) Solid State Commun. 119, 569. 3. Nellis, W. J. (1999) Philos. Mag. B 79, 6 5 5 . 4. Raimes, S. (1970) “The wave mechanics of electrons in metals”, (North-Holland P u b l i s h i n g C o mp a n y, A m s t e r d a m. 5. Parr, R. G. and Yang, W. (1989) “DensityFunctional Theory of Atoms and Molecules”, Oxford University Press, New York. 6. Callawa y, J . (1 964 ) “Energy B and Theory”, Academic Press, New York. 7. Styer, D. F. and Ashcroft, N. W. (1984) Phys. Rev. B 29, 5562. 8. Silverman, A. (1952) Phys. Rev. 85, 227. 9. Bardeen, J. (1938) J. Chem. Phys. 6, 367. 10. Vosko, S. J., Wilk, L. and Nusair, M. (1980) Can. J. Phys. 58, 1200. 11. Narayana, C., Luo, H., Orloff, J. and Ruoff, A. L. (1998) Nature 393, 46. 12. Straus, D. and Ashcroft, N. W. (1998) J. Phys.: Condens. Matter 10, 11135. 1. r 1.8a corresponds to a density of 0.44 g/cm 3. Under screening by a uniform background charge and Thomas-Fermi screening, the minimum ground state energies are -1.07 Ry/atom at s 0 and -0.94 Ry/atom r 1.6a at rs 1.8 a 0 respectively. Our results are different from Styer and Ashcroft’s in t hat t he scr eeni ng pot ent ials have significant effects on the physical properties of the metallic hydrogen, i.e. Thomas-Fermi screening produces higher ground state energy and the uniform screening produces smaller volume. Our calculations confirm that there is a possibility that metallic hydrogen exists because the ground state energy is negative. However, the configuration of metallic hydrogen used in this work is quite restricted, not general. In fact, the complete determination of the phase transition point is very difficult. According to experiments,(11) hydrogen does not become a metallic solid under the 22 Received: May 31, 2002 Accepted: June 4, 2003 J. Sci. Res. Chula. Univ., Vol. 28, No. 1 (2003) Metallic Hydrogen in a More Accurate Wigner-Seitz Approximation..…………………………….. J. Sci. Res. Chula. Univ., Vol. 28, No. 1 (2003) 23