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Spin-polarized electronic structure calculations for nickel and NiH
Journal
of the Less-Common
Metals,
123 (1986)
SPIN-POLARIZED ELECTRONIC NICKEL AND NiH B. SZPUNAR* Department
37
37
- 43
STRUCTURE
CALCULATIONS
FOR
and W. E. WALLACE of Chemistry,
University
of Pittsburgh,
Pittsburgh,
PA 15260
(U.S.A.)
P. STRANGE Department
(Received
of Physics,
November
University
of Bristol,
Tyndall
Avenue,
Bristol
858
ITL
(U.K.)
4, 1985)
Summary Self-consistent density functional calculations have been performed with a view to examining the effect of the hydrogen impurity on the magnetic, cohesive and bonding properties of nickel. These calculations show that pure nickel is ferromagnetic (m = 0.57 ~(a per atom) and its hydrides are paramagnets.
1. Introduction The transition metal hydrides have been studied systematically, experimentally and theoretically [ 1 - 41. Previous band structure calculations were non-spin-polarized calculations and gave us an accurate picture of the electronic properties of paramagnetic binary hydrides, but self-consistent spinpolarized calculations for hydrides and accurate total energy estimation are still open questions [ 5, 61. The one-electron results for the heat formation of transition-metal hydrides obtained by Gelatt et al. [l] demonstrated that the hydrogen Coulomb energy plays an important role, and Coulomb energy corrections must be included to obtain reasonable agreement with these experiments. The suggestions about the importance of electrostatic factors due to electron transfer between the hydrogen and metal atoms in site occupations have been made by Shoemaker and Shoemaker [ 71. The LMTOASA method used is based on density-functional [8] theory and the total Hamiltonian includes the electrostatic interaction of electrons with nuclei, electron-electron Coulomb repulsion and, as well, nuclei-nuclei interaction [ 91. Therefore, the method is very useful for our purpose. Nickel is a very interesting material as it crystallizes in the simple f.c.c. structure and under high hydrogen pressure forms hydrides. Pure nickel is a *Present Canada.
address:
0022-5088/86/$3.50
Department
of Physics,
Queen’s
@ Elsevier
University,
Sequoia/Printed
Kingston,
K7L 3N6,
in The Netherlands
38
ferromagnet but during hydrogenation becomes paramagnetic. There are no self-consistent spin-polarized band structure calculations which can describe this behavior.
2. Method We have performed self-consistent density functional calculations for nickel and its hydrides with a view to examining the effect of the hydrogen impurity on the magnetic, cohesive and bonding properties of nickel, A frozen core approximation has been used. The argon core in the nickel was frozen and the outer ten nickel electrons plus the single hydrogen electron were included in the iterations to self-consistency. The core charge density for hydrogen is assumed to be zero, and all contributions to the charge density are from the valence electron. The exchange-correlation energy was described using the Von Barth-Hedin approximation [lo]. All calculations were performed using the standard LMTO method [ 9, 111. Spin-orbit coupling was neglected but all other relativistic effects were included. The calculations were all self-consistent. We sampled 95 K points in the irreducible wedge of the brillouin zone, and the density of states was integrated with an energy step not exceeding 1 mRy. The calculations for pure nickel were performed on an f.c.c. lattice with the experimental value of the lattice constant equal to 6.64 a.u. This corresponds to an atomic sphere radius of 2.595 a.u. In the hydrides we used the same atomic sphere volume around the nickel ion as in the elemental case. In stoichiometric NiH, hydrogen occupies the octahedral sites (the NaCl structure) [ 121 with corresponding expansion of lattice parameters. We assume that changes of lattice parameters to 7.032 a.u. [ 131 are totally due to the metallic hydrogen sphere, and it gives the hydrogen sphere radius equal to 1.486 au. for the octahedral site. Our sphere radius of hydrogen is higher than that used by Switendick (Rn = 1.33 au.) [14] due to the atomic sphere approximation where overlapping muffin-tin (MT) spheres are used instead of the non-overlapping MT spheres applied by Switendick.
3. Results The local densities of states decomposed by 1 for s, p, d electrons for nickel and hydrogen sites are shown in Figs. 1 and 2. For pure nickel the results are very similar to those of Moruzzi et al. [15] but differ slightly due to our use of the experimental lattice constant which is different from the equilibrium theoretical lattice constant. Our calculations have an extra degree of freedom over and above those of Moruzzi et al. [ 151, in that we have allowed 1 = 3 character on the nickel site. Figure 3 presents band structure results for pure nickel spin-up and spin-down electrons, and Fig. 4, for comparison, presents the band structure for paramagnetic NiH.
39
,. cc
3.2
r
-
4ou - I.0
8 -0.8
(a)
-0.6
-0.4
ENERGY
-0.2 (Ryl
0.0
- 1.0
0.2 @I
-0.8
-0.6 ENERGY
-0.4
-0.2
0.0
0.2
(Ry)
Fig. 1. (a) The d-electron density of states in nickel (full line); the broken line shows the integrated total density of states. (b) The s-electron density of states in nickel (full line); the broken line shows the p-electron density of states.
In the ASA the assumption is made that each nucleus is screened spherically by the electronic charge cloud in the surrounding cell. The core and atomic charge density on nickel and hydrogen sites in NM are presented in Figs. 5 and 6. The core charge density for hydrogen is assumed to be zero and all contributions to hydrogen charge density in NiH are from valence electrons. 4. Discussion 4.1. Magnetism and electronic specific heat The zero temperature spin magnetic moment of nickel is well known [15]. We find a value of 0.57 pg atom -‘. This is due to the d-band splitting contribution and a small negative contribution of -0.03 pg atom-’ from sp and f electrons. The d-band occupancy in this case is 4.56 electrons majority spin and 3.96 electrons in the minority spin band. The results for metallic nickel presented in Table 1 are almost identical with LMTO-ASA results published by Anderson et aZ. [ 161. Now we examine what happens when hydrogen is introduced into nickel. The hydrogen s-band center is well below any of the nickel valence bands. Inclusion of hydrogen does not introduce any new bands, though the s-band width is increased. To fill up the bands to the correct number of electrons, we obviously have to include an extra electron of hydrogen. The majority spin d-states are virtually filled, so it is energetically most favorable
-1.0 -0,8 ENERGY
-0.6
!Hyl
-0.4
-0,2
ENERGY
0.0
0.2
04
(Ry)
(b) 35r
-1.0 ENERGY (c)
CRyI
0.8
-0.6
-04
-0.2
ENERGY
00
0.2
0.4
IRyl
(d)
Fig. 2. (a) The d-electron density of states on the nickel site in NiH (full line); the broken line shows the integrated total local density of states on the nickel site. (b) The s-electron density of states on the nickel site in NiH (full line); the broken line shows the p-electron density of states. The results are for spin-up and -down electrons as marked by arrows. (c) The selectron density of states on the hydrogen site in NiH (full line); the broken line shows the p-electron density of states. (d) The d-electron density of states on the hydrogen site in NiH (full line); the broken line shows the integrated total, local density of states on the hydrogen site.
to put the extra electron con~ibution in a spin-down d-band. This destroys the magnetic moment and to first order, we have a simple rigid band picture with the effect of raising ep from -0.132 rydbergs for pure nickel to -0.084 rydbergs for NiH. However, as we can see from Table 1, the total charge transfer from hydrogen to nickel is only 0.122 electrons per atom for the assumed atomic spheres radii. The hydrogen modifies the d-electron’s density on nickel and raises the height of a peak close to the Fermi energy. In Fig. 7 we present experimental results of the specific heat measurements of nickel hydride [17], These results are for the non-stoichiometric nickel hydride. One does not observe a linear dependence of the specific heat on the concentration of hydrogen, which is what is expected for two phases in
41
Fig. 3. The band structure of pure nickel for (a) spin-up and (b) spin-down electrons (all energies in rydbergs, atomic sphere radius equals 2.595 a.u.).
-I 0
L._....._l L. W
\I/ r
20 /
x
v,
K
L-1
-10
r
i -8
-6
-4 LOGN
-2
I.__/ 0
2
4
IR)
Fig. 4. The band structure of NiH paramagnet (all energies in rydbergs, atomic sphere radius of nickel equals 2.595 a.u., atomic sphere radius of hydrogen equals 1.486 a.u.). Fig. 5. Atomic (core + valence) charge density in NiH on nickel site (full line) and core electron charge density (broken line).
equilibrium. On the right side of the diagram we tried to extrapolate our results for stoichiometric Nil3 to nonstoichiometric by using the rigid bandfilling model. We can see that if not all octahedral sites are occupied by hydrogen, the electronic specific heat increases when the concentration of hydrogen decreases. From Fig. Z(a) note that nonstoichiometric nickel hydrides with concentrations of hydrogen lower than 50% are more energetically favorable, due to the flat tail above the very high peak and below the Fermi energy in the d-electron density on the nickel site. There is a po~ib~ty that these hydrogen vacancies play a crucial role in pressurecomposition hysteresis effects.
42
0.8 -10
I -8
I -6
I -4
I
t
I
-2
0
2
0.7
1
I 0.0
4
I
I
0.5
1.0
X(H)
LOGN (R)
Fig. 6. Atomic (core + valence) charge density in NiH on hydrogen site. The core electron charge density is zero. Fig. 7. The electronic specific heat measurements of nonstoichiometric nickel hydrides as functions of hydrogen concentration by Wolf and Baranowski [16] (0). The full line shows calculated values from the band-filling model.
TABLE
1
Band occupancy Element
Band
for metallic
nickel and NiH Occupation Metallic
Ni
0.329
0.335 0.357 0.382 4.559 3.959 0.039 0.040 H
S
P d
number
nickel
NiH 0.273 0.273 0.367 0.367 4.383 4.383 0.039 0.039 0.710 0.142 0.023
4.2. Energy In the density functional theory we minimize the total energy with respect to changes in charge density. Thus the charge density is calculated self-consistently. The cakulation is initiated using stand’ard relativistic atomic charge densities. The results of spin-polarized LETS-ASA calculations are summarized in Table 2. In all calculations the total energy con-
Nickel atom Hydrogen atom Metallic nickel Nickel hydride Nickel hydride Nickel hydride
Material
energy
-3037.1776 -0.9236 -3037.7964 -3039.003 -3038.9875 -3039.0135
(rydbergs)
Total
0.0004 1.101 2.013 0.3714
(rydbergs)
(rydbergs)
-0.619 -0.902 -0.887 -0.913
Madelung energy
Cohesive energy
Results of spin-polarised LMTO-ASA calculations
TABLE 2
2.595 2.595 2.595 2.49
Ni
1.486 1.3 1.486
H
Atomic sphere radius (au.)
-0.2685 -0.2145 -0.1001 -0.004
(Mbar)
Pressure
transfer
0.1215 0.2309 0.0398
(electrons)
Charge H-+Ni
-0.132 -0.084 -0.067 0.005
(rydbergs)
Fermi energy
-0.813 -0.863 -0.892 -0.878
(rydbergs)
Muffin-tin zero energy
44
verged to better than 1 millirydbergs, but the uncertainties inherent in the local density functional and in the LMTO-ASA method make the second and third figures in the total-energy numbers quoted in Table 2 unreliable. Our cohesive energy of nickel metal is of greater magnitude than in ref. 15. This is due to the difference in lattice constant and our inclusion of I = 3 in the calculation. Table 2 shows that the expected behavior of nickel and its hydrides is reproduced by density functional theory. Solid nickel is energetically more favorable than atomic nickel and the hydride is more stable than metallic nickel and hydrogen separately. The heat of formation of NiH can be easily calculated from the values of the cohesive energy of NiH and metallic nickel if we assume the value of the energy of hydrogen molecule (H,) as equal to -2.266 rydbergs [l]. We obtain the heat of formation per hydrogen atom as -0.074 rydbergs. According to Gelatt et al. [ 11, the experimental value of the heat of formation for NiH is about 0.025 rydbergs and the theoretical value obtained by them is approximately 0.07 rydbergs. This is in contradiction to our results and recent experimental results obtained by Baranowski [ 181, where they obtained a small negative value for the heat of formation of NiH and AhHem = -2.100 + 0.140 kcal mall ‘, which corresponds to AS = -0.0034 rydbergs H-‘. In Table 2 we present results for various atomic sphere radii. The atomic sphere radius corresponding to the experimental value of the lattice constant of pure nickel has been used. This value is not equal to the calculated, equilibrium value and therefore we obtained a negative value of the pressure. The atomic sphere radius for hydrogen equal to 1.486 a.u. has been calculated using an experimental value of the lattice expansion during hydrogenation [ 181. This value is slightly higher than the value used by Switendick [ 141 because in the LMTO-ASA method crossing atomic spheres are used. The heat of formation decreases very much as the atomic sphere radius of hydrogen decreases (AE = -0.0585 rydbergs H-’ for RH = 1.3 a.u.) and the Madelung energy increases due to higher charge transfer. This is the well-known “size effect” observed experimentally, where the hydrogen capacity of a material increases when the size of the hole where it is absorbed increases. The interesting result from Table 2 is the evidence of the importance of the Coulomb energy in NiH. The Madelung energy in NiH increases to 1.1 rydbergs for RH = 1.486 a.u. 4.3. Bonding In pure nickel the bonding is dominated by the minority spin of d electrons with smaller bonding contributions compared with the majority spin d. We expect the strength of the nickel bonds to be roughly proportional to nd (5 - nd), where nd is the number of occupied d-states per spin direction [19]. So this picture is compatible with the magnetic properties of nickel. As in most transition metals the valence s-states are antibonding. If we add hydrogen to nickel in the octahedral site, we see that the hydrogen s electron hybridizes with the nickel s band so the bonding is s-s like. Our
45
calculations support previous calculations that no new bands are introduced during hydrogenation, but a metal-hydrogen bonding state appears [ 3 - 51. The minority spin d bands are now nearly filled and their contribution to the bonding is lowered to that of the majority spin. The electronic structure of the hydride shows that the center of the hydrogen p-band is over 3.3 Ry above EF. Therefore, the hydrogen p character given in Table 1 is unlikely to be pure p like, but is due to overlapping nickel d orbitals being expanded in spherical harmonics about the hydrogen lattice site.
Acknowledgements This work was supported by a grant from the National Science Foundation. One of us (B.S.) would like to thank Dr. F. Pourarian and Dr. H. K. Smith for much useful discussion. We are grateful to the Physics Department, Queen’s University, for technical assistance in preparation of this pub~cation.
References 1 C. D. Gelatt, Jr., H. Ehrenreich and J. A. Weiss, Phys. Rev. B, 17 (1978) 1940. 2 C. D. Gelatt, in L. D. Bennet (ed.), Theory of AEloy Phase Formation, Metallurgical Sot., AIME, New York, 1980. 3 A. C. Switendick, Theoretical studies of hydrogen in metals, Rep. Sand-78-0250 (Sandia National Laboratory, Albuquerque, NM). 4 P. C. P. Bouten and A. R. Miedema, J. Less-Common Met., 71 (1980) 147. 5 M. Gupta, J. Less-Common Met., 201 (1984) 36. 6 A. C. Switendick, Ber. Bunsenges. Phys. Chem., 76 (1972) 535. 7 D. P. Shoemaker and C. B. Shoemaker, J. Less-Common Met., 68 (1979) 43. 8 P. Hohenberg and W. Kohn, Phys. Rev. B, 136 (1964) 864. 9 H. L. Skriver (ed.), The LMTO Method, Springer, Berlin, 1984, p. 41. 10 U. Von Barth and L. Hedin, J. Phys. C, 5 (1972) 1629. 11 0. K. Andersen, Phys. Rev. B, 12 (1975) 3060. 12 B. Stalinski, Phys. Chem., 76 (1972) 724. 13 G. G. Libowitz, in W. A. Benjamin (ed.), The Solid State C~ernist~ of Binary Metat Hydrides, New York, 1965. 14 A. C. Switendick, Solid State Commun., 8 (1970) 1463. 15 V. L. Moruzzi, J. F. Janak and A. R. Williams, Calculated Electronic Properties of Metals, Pergamon, Oxford, 1978. 16 0. K. Andersen, 0. Jepsen and D. Glotrel, Highlights of condensed matter theory, Enrico Fermi School LXXXIX, 1985 (Sot. Ital. Fisica, Bologna, Italy). 17 G. Wolf and B. Baranowski, J. Phys. Chem. Solids, 32 (1971) 1649. 18 B. Baranowski, Ber. Bunsenges. Phys. Chem., 76 (1972) 714. 19 D. G. Pettifor, J. Phys. F, 7 (1977) 613.
of the Less-Common
Metals,
123 (1986)
SPIN-POLARIZED ELECTRONIC NICKEL AND NiH B. SZPUNAR* Department
37
37
- 43
STRUCTURE
CALCULATIONS
FOR
and W. E. WALLACE of Chemistry,
University
of Pittsburgh,
Pittsburgh,
PA 15260
(U.S.A.)
P. STRANGE Department
(Received
of Physics,
November
University
of Bristol,
Tyndall
Avenue,
Bristol
858
ITL
(U.K.)
4, 1985)
Summary Self-consistent density functional calculations have been performed with a view to examining the effect of the hydrogen impurity on the magnetic, cohesive and bonding properties of nickel. These calculations show that pure nickel is ferromagnetic (m = 0.57 ~(a per atom) and its hydrides are paramagnets.
1. Introduction The transition metal hydrides have been studied systematically, experimentally and theoretically [ 1 - 41. Previous band structure calculations were non-spin-polarized calculations and gave us an accurate picture of the electronic properties of paramagnetic binary hydrides, but self-consistent spinpolarized calculations for hydrides and accurate total energy estimation are still open questions [ 5, 61. The one-electron results for the heat formation of transition-metal hydrides obtained by Gelatt et al. [l] demonstrated that the hydrogen Coulomb energy plays an important role, and Coulomb energy corrections must be included to obtain reasonable agreement with these experiments. The suggestions about the importance of electrostatic factors due to electron transfer between the hydrogen and metal atoms in site occupations have been made by Shoemaker and Shoemaker [ 71. The LMTOASA method used is based on density-functional [8] theory and the total Hamiltonian includes the electrostatic interaction of electrons with nuclei, electron-electron Coulomb repulsion and, as well, nuclei-nuclei interaction [ 91. Therefore, the method is very useful for our purpose. Nickel is a very interesting material as it crystallizes in the simple f.c.c. structure and under high hydrogen pressure forms hydrides. Pure nickel is a *Present Canada.
address:
0022-5088/86/$3.50
Department
of Physics,
Queen’s
@ Elsevier
University,
Sequoia/Printed
Kingston,
K7L 3N6,
in The Netherlands
38
ferromagnet but during hydrogenation becomes paramagnetic. There are no self-consistent spin-polarized band structure calculations which can describe this behavior.
2. Method We have performed self-consistent density functional calculations for nickel and its hydrides with a view to examining the effect of the hydrogen impurity on the magnetic, cohesive and bonding properties of nickel, A frozen core approximation has been used. The argon core in the nickel was frozen and the outer ten nickel electrons plus the single hydrogen electron were included in the iterations to self-consistency. The core charge density for hydrogen is assumed to be zero, and all contributions to the charge density are from the valence electron. The exchange-correlation energy was described using the Von Barth-Hedin approximation [lo]. All calculations were performed using the standard LMTO method [ 9, 111. Spin-orbit coupling was neglected but all other relativistic effects were included. The calculations were all self-consistent. We sampled 95 K points in the irreducible wedge of the brillouin zone, and the density of states was integrated with an energy step not exceeding 1 mRy. The calculations for pure nickel were performed on an f.c.c. lattice with the experimental value of the lattice constant equal to 6.64 a.u. This corresponds to an atomic sphere radius of 2.595 a.u. In the hydrides we used the same atomic sphere volume around the nickel ion as in the elemental case. In stoichiometric NiH, hydrogen occupies the octahedral sites (the NaCl structure) [ 121 with corresponding expansion of lattice parameters. We assume that changes of lattice parameters to 7.032 a.u. [ 131 are totally due to the metallic hydrogen sphere, and it gives the hydrogen sphere radius equal to 1.486 au. for the octahedral site. Our sphere radius of hydrogen is higher than that used by Switendick (Rn = 1.33 au.) [14] due to the atomic sphere approximation where overlapping muffin-tin (MT) spheres are used instead of the non-overlapping MT spheres applied by Switendick.
3. Results The local densities of states decomposed by 1 for s, p, d electrons for nickel and hydrogen sites are shown in Figs. 1 and 2. For pure nickel the results are very similar to those of Moruzzi et al. [15] but differ slightly due to our use of the experimental lattice constant which is different from the equilibrium theoretical lattice constant. Our calculations have an extra degree of freedom over and above those of Moruzzi et al. [ 151, in that we have allowed 1 = 3 character on the nickel site. Figure 3 presents band structure results for pure nickel spin-up and spin-down electrons, and Fig. 4, for comparison, presents the band structure for paramagnetic NiH.
39
,. cc
3.2
r
-
4ou - I.0
8 -0.8
(a)
-0.6
-0.4
ENERGY
-0.2 (Ryl
0.0
- 1.0
0.2 @I
-0.8
-0.6 ENERGY
-0.4
-0.2
0.0
0.2
(Ry)
Fig. 1. (a) The d-electron density of states in nickel (full line); the broken line shows the integrated total density of states. (b) The s-electron density of states in nickel (full line); the broken line shows the p-electron density of states.
In the ASA the assumption is made that each nucleus is screened spherically by the electronic charge cloud in the surrounding cell. The core and atomic charge density on nickel and hydrogen sites in NM are presented in Figs. 5 and 6. The core charge density for hydrogen is assumed to be zero and all contributions to hydrogen charge density in NiH are from valence electrons. 4. Discussion 4.1. Magnetism and electronic specific heat The zero temperature spin magnetic moment of nickel is well known [15]. We find a value of 0.57 pg atom -‘. This is due to the d-band splitting contribution and a small negative contribution of -0.03 pg atom-’ from sp and f electrons. The d-band occupancy in this case is 4.56 electrons majority spin and 3.96 electrons in the minority spin band. The results for metallic nickel presented in Table 1 are almost identical with LMTO-ASA results published by Anderson et aZ. [ 161. Now we examine what happens when hydrogen is introduced into nickel. The hydrogen s-band center is well below any of the nickel valence bands. Inclusion of hydrogen does not introduce any new bands, though the s-band width is increased. To fill up the bands to the correct number of electrons, we obviously have to include an extra electron of hydrogen. The majority spin d-states are virtually filled, so it is energetically most favorable
-1.0 -0,8 ENERGY
-0.6
!Hyl
-0.4
-0,2
ENERGY
0.0
0.2
04
(Ry)
(b) 35r
-1.0 ENERGY (c)
CRyI
0.8
-0.6
-04
-0.2
ENERGY
00
0.2
0.4
IRyl
(d)
Fig. 2. (a) The d-electron density of states on the nickel site in NiH (full line); the broken line shows the integrated total local density of states on the nickel site. (b) The s-electron density of states on the nickel site in NiH (full line); the broken line shows the p-electron density of states. The results are for spin-up and -down electrons as marked by arrows. (c) The selectron density of states on the hydrogen site in NiH (full line); the broken line shows the p-electron density of states. (d) The d-electron density of states on the hydrogen site in NiH (full line); the broken line shows the integrated total, local density of states on the hydrogen site.
to put the extra electron con~ibution in a spin-down d-band. This destroys the magnetic moment and to first order, we have a simple rigid band picture with the effect of raising ep from -0.132 rydbergs for pure nickel to -0.084 rydbergs for NiH. However, as we can see from Table 1, the total charge transfer from hydrogen to nickel is only 0.122 electrons per atom for the assumed atomic spheres radii. The hydrogen modifies the d-electron’s density on nickel and raises the height of a peak close to the Fermi energy. In Fig. 7 we present experimental results of the specific heat measurements of nickel hydride [17], These results are for the non-stoichiometric nickel hydride. One does not observe a linear dependence of the specific heat on the concentration of hydrogen, which is what is expected for two phases in
41
Fig. 3. The band structure of pure nickel for (a) spin-up and (b) spin-down electrons (all energies in rydbergs, atomic sphere radius equals 2.595 a.u.).
-I 0
L._....._l L. W
\I/ r
20 /
x
v,
K
L-1
-10
r
i -8
-6
-4 LOGN
-2
I.__/ 0
2
4
IR)
Fig. 4. The band structure of NiH paramagnet (all energies in rydbergs, atomic sphere radius of nickel equals 2.595 a.u., atomic sphere radius of hydrogen equals 1.486 a.u.). Fig. 5. Atomic (core + valence) charge density in NiH on nickel site (full line) and core electron charge density (broken line).
equilibrium. On the right side of the diagram we tried to extrapolate our results for stoichiometric Nil3 to nonstoichiometric by using the rigid bandfilling model. We can see that if not all octahedral sites are occupied by hydrogen, the electronic specific heat increases when the concentration of hydrogen decreases. From Fig. Z(a) note that nonstoichiometric nickel hydrides with concentrations of hydrogen lower than 50% are more energetically favorable, due to the flat tail above the very high peak and below the Fermi energy in the d-electron density on the nickel site. There is a po~ib~ty that these hydrogen vacancies play a crucial role in pressurecomposition hysteresis effects.
42
0.8 -10
I -8
I -6
I -4
I
t
I
-2
0
2
0.7
1
I 0.0
4
I
I
0.5
1.0
X(H)
LOGN (R)
Fig. 6. Atomic (core + valence) charge density in NiH on hydrogen site. The core electron charge density is zero. Fig. 7. The electronic specific heat measurements of nonstoichiometric nickel hydrides as functions of hydrogen concentration by Wolf and Baranowski [16] (0). The full line shows calculated values from the band-filling model.
TABLE
1
Band occupancy Element
Band
for metallic
nickel and NiH Occupation Metallic
Ni
0.329
0.335 0.357 0.382 4.559 3.959 0.039 0.040 H
S
P d
number
nickel
NiH 0.273 0.273 0.367 0.367 4.383 4.383 0.039 0.039 0.710 0.142 0.023
4.2. Energy In the density functional theory we minimize the total energy with respect to changes in charge density. Thus the charge density is calculated self-consistently. The cakulation is initiated using stand’ard relativistic atomic charge densities. The results of spin-polarized LETS-ASA calculations are summarized in Table 2. In all calculations the total energy con-
Nickel atom Hydrogen atom Metallic nickel Nickel hydride Nickel hydride Nickel hydride
Material
energy
-3037.1776 -0.9236 -3037.7964 -3039.003 -3038.9875 -3039.0135
(rydbergs)
Total
0.0004 1.101 2.013 0.3714
(rydbergs)
(rydbergs)
-0.619 -0.902 -0.887 -0.913
Madelung energy
Cohesive energy
Results of spin-polarised LMTO-ASA calculations
TABLE 2
2.595 2.595 2.595 2.49
Ni
1.486 1.3 1.486
H
Atomic sphere radius (au.)
-0.2685 -0.2145 -0.1001 -0.004
(Mbar)
Pressure
transfer
0.1215 0.2309 0.0398
(electrons)
Charge H-+Ni
-0.132 -0.084 -0.067 0.005
(rydbergs)
Fermi energy
-0.813 -0.863 -0.892 -0.878
(rydbergs)
Muffin-tin zero energy
44
verged to better than 1 millirydbergs, but the uncertainties inherent in the local density functional and in the LMTO-ASA method make the second and third figures in the total-energy numbers quoted in Table 2 unreliable. Our cohesive energy of nickel metal is of greater magnitude than in ref. 15. This is due to the difference in lattice constant and our inclusion of I = 3 in the calculation. Table 2 shows that the expected behavior of nickel and its hydrides is reproduced by density functional theory. Solid nickel is energetically more favorable than atomic nickel and the hydride is more stable than metallic nickel and hydrogen separately. The heat of formation of NiH can be easily calculated from the values of the cohesive energy of NiH and metallic nickel if we assume the value of the energy of hydrogen molecule (H,) as equal to -2.266 rydbergs [l]. We obtain the heat of formation per hydrogen atom as -0.074 rydbergs. According to Gelatt et al. [ 11, the experimental value of the heat of formation for NiH is about 0.025 rydbergs and the theoretical value obtained by them is approximately 0.07 rydbergs. This is in contradiction to our results and recent experimental results obtained by Baranowski [ 181, where they obtained a small negative value for the heat of formation of NiH and AhHem = -2.100 + 0.140 kcal mall ‘, which corresponds to AS = -0.0034 rydbergs H-‘. In Table 2 we present results for various atomic sphere radii. The atomic sphere radius corresponding to the experimental value of the lattice constant of pure nickel has been used. This value is not equal to the calculated, equilibrium value and therefore we obtained a negative value of the pressure. The atomic sphere radius for hydrogen equal to 1.486 a.u. has been calculated using an experimental value of the lattice expansion during hydrogenation [ 181. This value is slightly higher than the value used by Switendick [ 141 because in the LMTO-ASA method crossing atomic spheres are used. The heat of formation decreases very much as the atomic sphere radius of hydrogen decreases (AE = -0.0585 rydbergs H-’ for RH = 1.3 a.u.) and the Madelung energy increases due to higher charge transfer. This is the well-known “size effect” observed experimentally, where the hydrogen capacity of a material increases when the size of the hole where it is absorbed increases. The interesting result from Table 2 is the evidence of the importance of the Coulomb energy in NiH. The Madelung energy in NiH increases to 1.1 rydbergs for RH = 1.486 a.u. 4.3. Bonding In pure nickel the bonding is dominated by the minority spin of d electrons with smaller bonding contributions compared with the majority spin d. We expect the strength of the nickel bonds to be roughly proportional to nd (5 - nd), where nd is the number of occupied d-states per spin direction [19]. So this picture is compatible with the magnetic properties of nickel. As in most transition metals the valence s-states are antibonding. If we add hydrogen to nickel in the octahedral site, we see that the hydrogen s electron hybridizes with the nickel s band so the bonding is s-s like. Our
45
calculations support previous calculations that no new bands are introduced during hydrogenation, but a metal-hydrogen bonding state appears [ 3 - 51. The minority spin d bands are now nearly filled and their contribution to the bonding is lowered to that of the majority spin. The electronic structure of the hydride shows that the center of the hydrogen p-band is over 3.3 Ry above EF. Therefore, the hydrogen p character given in Table 1 is unlikely to be pure p like, but is due to overlapping nickel d orbitals being expanded in spherical harmonics about the hydrogen lattice site.
Acknowledgements This work was supported by a grant from the National Science Foundation. One of us (B.S.) would like to thank Dr. F. Pourarian and Dr. H. K. Smith for much useful discussion. We are grateful to the Physics Department, Queen’s University, for technical assistance in preparation of this pub~cation.
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