44 Chapter 3. First-principles based modeling of inter-atomic force constants in conventional alloy theory, a cluster expansion [108] has to be fitted to the ground state ener- gies of a large number of ordered states. While the calculation for a given ordered structure is a relatively routine task with modern first-principles electronic structure codes, this procedure has to be repeated for many configurations in order to properly fit a cluster expansion which makes the whole procedure computationally demanding. Similar is the case for the supercell technique. For calculation of phonon excitations, the computational cost rises even further.
Thus, in spite of having a suitable self-consistent, analytical technique to perform the required averaging over various configurations in the disordered systems, the calculations of phonon spectrum in random alloys were rather limited because of these practical difficulties.
In this chapter, we present a new approach to calculate the inter-atomic force constants in random alloys. The formalism is a combination of an ab initio method which computes inter- atomic interactions based upon the detailed electronic structures and the intuitive argument about dependence of stretching and bending force constants on bond lengths. In combination with the ICPA, which does the desired averaging over various configurations in the disordered alloy, addressing both mass as well as force constant disorder, we employ our approach to investigate the interrelations between the inter-atomic force constants and the lattice dynamics of Pd0.96Fe0.04 and Pd0.9Fe0.1 [109]. We present results on phonon frequencies and elastic constants for these two alloys. Significant insight about the inter-atomic interactions between various pairs of chemical specie is obtained in the course of our investigations.
In the next two sections, we discuss the Density functional theory (DFT) [110, 111] and the Density functional perturbation theory (DFPT) [112], which we use for ab initio calcu- lations of inter-atomic force constants. The modeling strategy to calculate force constants in random alloys and the corresponding results on phonon frequencies obtained by the ICPA method are discussed in the following section which also illustrates limitations of the present approach. At the very end, we put forward the formulation of a more appropriate modeling strategy which does away with the limitations of the present approach.
3.2. Density functional theory (DFT) 45 electronic problem onto a non-interacting one. The formulation of the DFT originated in a famous article written by P. Hohenberg and W. Kohn in 1964 [110]. They established two theorems which constitute the theoretical foundation of DFT:
Theorem I: For any system of interacting particles in an external potential Vext(r), the potential Vext(r) is determined uniquely, except for a constant, by the ground state particle density n0(r).
Theorem II: A universal functional for the energy E(n) in terms of the density n(r) can be defined, valid for any external potential Vext(r). For any particular Vext(r), the exact ground state energy of the system is the global minimum value of this functional and the density n(r) that minimizes the functional is the exact ground state density n0(r).
The meaning of the first theorem is that the ground state density n0(r) completely de- termines all the properties of a given many body system. It allows an enormous conceptual simplification of the quantum mechanical problem of searching for the ground state properties of a system of interacting electrons, because it replaces the traditional description based on wave functions (which depend on 3N independent variables, N being the number of electrons) with a much more tractable description in terms of the single body charge density, which de- pends on only three variables. Hohenberg and Kohn also demonstrated that there exists a universal functional F[n(r)] of the electron charge density such that total energy functional becomes
E[n(r)]=F[n(r)]+ Z
Vext(r)n(r)dr (3.1)
where F[n(r)] contains the kinetic energy and the mutual Coulomb interaction of the elec- trons and Vext(r) represents the external potential acting on the electrons. The minimization of this functional with the condition that the total number of particles, N, is preserved:
Z
n(r)dr= N, (3.2)
directly gives the ground state energy and charge density, from which all the other physical properties can be extracted.
46 Chapter 3. First-principles based modeling of inter-atomic force constants However, a major difficulty in the direct application of this remarkably simple result is that the form of the functional, F[n] is unknown. In order to solve the problem, Kohn and Sham [111] introduced a further development to transform DFT into a practically useful tool.
The Hohenberg and Kohn theorems hold independently of the precise form of the electron- electron interaction. In particular, when the strength of the electron-electron interaction van- ishes, F [n] defines the ground state kinetic energy of a system of non-interacting electrons as a functional of its ground state charge density distribution T0[n]. This fact was used by Kohn and Sham (1965) to map the problem of a system of interacting electrons onto an equivalent non-interacting problem. To this end, the unknown functional F [n] is cast in the form
F [n]= T0[n]+ e2 2
Z n (r) n r′
|r−r′| drdr′+Exc[n] (3.3) where the second term is the classical electrostatic self-interaction of the electron charge density distribution and the so called exchange-correlation energy Excis defined by Eq. (3.3).
Variation of the energy functional with respect to n (r) with the constraint that the number of electrons be kept fixed leads formally to the same equation as would hold for a system of non-interacting electrons subject to an effective potential, also called the self-consistent field (SCF) potential, whose form is,
VS CF(r)= V (r)+e2
Z n r′
|r−r′|dr′ +vxc(r), (3.4) where
vxc(r)≡ δExc
δn (r) (3.5)
is the functional derivative of the exchange-correlation energy, also called the exchange- correlation potential.
The power of this trick is that, if one knew the effective potential VS CF(r), the problem for non-interacting electrons could be trivially solved without knowing the form of the non- interacting kinetic energy functional T0. To this end, one should simply solve the one-electron Schrodinger equation: