Equation of State for PdH by a New Tight Binding Approach
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(1)
where aj, bi, ci and Xi are parameters determined by fitting to first principles calculations and f(r) is a cutoff function designed to extend up to five shells of atoms. In our calculations for the monatomic materials we have set the ci parameters to zero but we have included them in our calculations for binary compounds. On the other hand, in our monatomic materials studies we included the overlap matrix while in the present study of PdH we performed the fit using an orthogonal Hamiltonian. We have also introduced a volume dependence on the on-site TB parameters he assuming the following form,
he = cie + 13ep2/3
where
P = nl
+ "yep
413
(2)
e-62r
and %,, 13,, -Yeand 5 are parameters determined by the fit. In our monatomic calculations we used f = s, p and d. In our present calculation for PdH we split the d's into t2g and eg symmetries, thus introducing a fourth he.
31 Mat. Res. Soc. Symp. Proc. Vol. 408 ©1996 Materials Research Society
Finally our formalism, unlike other works,5-7 does not rely on a pair potential for the fitting of the total energy. As described in Ref. 3, instead of using a pair potential, we have shifted the individual eigenvalues by a constant that makes the sum of the shifted eigenvalues exactly equal to the eigenvalue sum. In our application for PdH we have a l0xl0 Hamiltonian (9 orbitals for Pd plus the is hydrogen orbital), which was chosen to be orthogonal and is determined by a total of 73 parameters found in Eqs. (1) and (2). TOTAL ENERGY RESULTS We have varied the above 73 parameters to simultaneously fit the energy bands and total energies for fcc and bcc Pd and for the NaCl, CsC1 and fluorite structures of PdH. The equation of state for all these structures is shown in Fig. 1. 0.250
0.200
0.150 n0
0.100 2.) E
0.050
CJ L-
0.000
-0.050 80
100
120 140 Volume/Unit Cell (Bohr 3 )
160
180
Figure 1. Equation of state for the Pd-H system.
The quantity E that is plotted in Fig. 1 is the energy of formation defined as follows: E = E(PdH,,) - nE(Pd) - 1/2 mE(H 2)
32
(3)
where E(Pd,,H,,) is the total energy of each compound, E(Pd) is the equilibrium energy of fcc Pd and E(H 2) is the total energy of the hydrogen molecule. 8 It is clear that all these structures fit very well with an rms error of about 1 mRy. The next question is whether this parametrization fits well with structures that we have not fitted. We have tested the Cu 3Au structure by comparing APW calculations for Pd3H to the TB Pd 3H. As is shown in Table I, the TB value of E for Pd3H is 95 mRy while the APW value is 156 mRy. Clearly we do not have quantitative agreement but the energy is well above the stable structure. Similarly, we find PdH 3 and Pd7H well above the NaCl structure but we did not perform corresponding APW calculations for comparison. In Fig. 1 we also show the Energy volume graph of the Li. structure. This was fitted at only the point shown by the diamond symbol which was enough to give us a reasonable position for this structure. In Table I we also list a TB-APW comparison of the
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