Computational Physics (Lecture 21)

1. Computational Physics(Lecture 21) PHY4370

2. where g(r) is the pair distribution and S(k) is the angular average of S(k).

3. G(r, t) can be interpreted as the probability of observing one of the particles at r at time t if a particle was observed at r = 0 at t = 0. • This leads to the numerical evaluation of G(r, t), • which is the angular average of G(r, t). • If we write G(r, t) in two parts,

4. where ΔΩ(r, Δr) ≈4πr ^2 Δr is the volume of a spherical shell with radius r and thickness Δ r , and Δ N(r, Δ r ; t) is the number of particles in the spherical shell at time t. • The position of each particle at t = 0 is chosen as the origin in the evaluation of G_d(r, t) and the average is taken over all the particles in the system. • Note that this is different from the evaluation of g(r), • in which we always select a particle position as the origin and take the average over time. • We can also take the average over all the particles in the evaluation of g(r ). • Here Gs(r, t) can be evaluated in a similar fashion.

5. Because Gs(r, t) represents the probability for a particle to be at a distance r at time t from its original position at t = 0, we can introduce

6. which has many features similar to those of the phonon spectrum of the system: for example, a broad peak for the glassy state and sharp features for a crystalline state.

7. Thermodynamical quantities can also be evaluated from molecular dynamics simulations. • For example, if a simulation is performed under the constant pressurecondition, we can obtain physical quantities such as the particle density, pair-distribution function, and so on, at different temperature. • The inverse of the particle density is called the specific volume, denoted V_P (T ). • The thermal expansion coefficient under the constant-pressure condition is then given by which is quite different when the system is in a liquid phase than it is in a solid phase.

8. we can calculate the temperature-dependent enthalpy • H = E + PΩ

9. which is also quite different for the liquid phase than for the solid phase. • Formore discussions on the molecular dynamics simulation of glass transition, see Yonezawa(1991).

10. Other aspects related to the structure and dynamics of a system can be studied through molecular dynamics simulations. • The advantage of molecular dynamics over a typical stochastic simulation is that molecular dynamics can give all the information on the time dependence of the system, • which is necessary for analyzing the structural and dynamical properties of the system. • Molecular dynamics is therefore the method of choice in computer simulations of many-particle systems. • However, stochastic simulations, such as Monte Carlo simulations, are sometimes easier to perform for some systems and are closely related to the simulations of quantum systems.

11. Ab initio molecular dynamics • combines the calculation of the electronic structure and the molecular dynamics simulation for a system. • ab initio molecular dynamics, • which was devised and put into practice by Car and Parrinello (1985). • Classical MD: • The maturity of molecular dynamics simulation schemes, • the great advances in computing capacity • => possible to perform molecular dynamics simulations for amorphous materials, biopolymers, and other complex systems.

12. However, in order to obtain an accurate description of a specific system  we have to know the precise behavior of the interactions among the particles • the ions in the system. • Electrons move much faster than ions because the electron mass is much smaller than that of an ion. • The position dependence of the interactions among the ions in a given system is therefore determined by the distribution of the electrons (electronic structure) at the specific moment. • A good approximation of the electronic structure in a calculation can be obtained with all the nuclei fixed in space for that moment. • Born–Oppenheimer approximation, which allows the degrees of freedom of the electrons to be treated separately from those of the ions.

13. the interactions among the ions were given in a parameterized formbasedon experimental data, quantum chemistry calculations, or the specific conditions of the system under study. • All these procedures are limited due to the complexity of the electronic structure of the actual materials. • We can easily obtain accurate parameterized interactions for the inert gases, such as Ar, • But would have a lot of difficulties in obtaining an accurate parameterized interaction that can produce the various structures of ice correctly in the classical molecular dynamics simulation.

14. Acombined scheme is highly desirable. • We can calculate the many-body interactions among the ions in the system from the electronic structure calculated at every molecular dynamics time step and then determine the next configuration from such ab initio interactions. • This can be achieved in principle, but in practice the scheme is restricted by the existing computing capacity. • Thecombined method devised by Car and Parrinello (1985) was the first in its class and has been applied to the simulation of real materials.

15. Density functional theory • The density functional theory (Hohenberg and Kohn, 1964; Kohn and Sham, 1965) was introduced as a practical scheme to cope with the many-electron effect in atoms, molecules, and solids. • The theorem proved by Hohenberg and Kohn (1964) states that the ground-state energy of an interacting system is the optimized value of an energy functional E[ρ(r)] of the electron density ρ(r) and that the corresponding density distribution of the optimization is the unique ground-state density distribution.

18. which cannot be obtained exactly.

19. A common practice is to approximate it by its homogeneous density equivalent, the so-called local approximation, • in which we assume that Vxc(r) is given by the same quantity of a uniform electron gas with density equal to ρ(r). • This is termed the local density approximation. • The local density approximation has been successfully applied to many physical systems, including atomic, molecular, and condensed-matter systems. • The unexpected success of the local density approximation in materials research has made it a standard technique for calculating electronic properties of new materials and systems.

20. The procedure for calculating the electronic structure with the local density approximation can be described in several steps. • We first construct the local approximation of Vxc(r) with a guessed density distribution. • Then the Kohn–Sham equation is solved, and a new density distribution is constructed from the solution. • With the new density distribution, we can improve Vxc(r) and then solve the Kohn–Sham equation again. • This procedure is repeated until convergence is reached.

21. The Car---Parrinello simulation scheme • The Hohenberg–Kohn energy functional forms the Born–Oppenheimer potential surface for the ions in the system. • The idea of ab initio molecular dynamics is similar to the relaxation scheme. • We introduced a functional for the one-dimensional Poisson equation. Note that ρ(x) here is the charge density instead of the particle density.

22. The physical meaning of this functional is the electrostatic energy of the system. • After applying the trapezoid rule to the integral and taking a partial derivative of U with respect to φi, we obtain the corresponding difference equation: which would optimize (minimize) the functional (the electrostatic energy) as k →∞.

23. The indices k and k + 1 are for iteration steps, and the index n is for the spatial points. • The iteration can be interpreted as a fictitious time step, since we can rewrite the above equation as: • with p acting like a fictitious time step. • The solution converges to the true solution of the Poisson equation as k goes to infinity if the functional U decreases during the iterations.

24. The ab initio molecular dynamics is devised by introducing a fictitious time dependent equation for the electron degrees of freedom: • where μ is an adjustable parameter introduced for convenience, Λ_ij is the Lagrange multiplier, introduced to ensure the orthonormal condition of the wave functions ψi(r, t), and the summation is over all the occupied states.

25. Note that the potential energy surface E is a functional of the electron density as well as a function of the ionic coordinates Rn for n = 1, 2, . . . , Nc, with a total of Ncions in the system. • In practice, we can also consider the first-order time derivative equation, with d^2ψi (r, t)/dt^2 replaced by the first-order derivative dψi(r, t)/dt, • because either the first-order or the second-order derivative will approach zero at the limit of convergence. • Second-order derivatives were used in the original work of Car and Parrinello and were later shown to yield a fast convergence if a special damping term is introduced (Tassone, Mauri, and Car, 1994). • The ionic degrees of freedom are then simulated from Newton’s equation where Mnand Rn are the mass and the position vector of the nth particle.

26. The advantage of ab initio molecular dynamics is that the electron degrees of freedom and the ionic degrees of freedom are simulated simultaneously by the above equations. • Since its introduction by Car and Parrinello (1985), the method has been applied to many systems, especially those without a crystalline structure, namely, liquids and amorphous materials. • Progress in ab initio molecular dynamics has also included mapping the Hamiltonian onto a tight-binding model in • which the evaluation of the electron degrees of freedom is drastically simplified (Wang, Chan, and Ho, 1989).

27. Density functional theory: foundations • DFT is a theory of correlated many-body systems. • In close association with independent-particle methods, • Because it has provided the key step that has made possible development of practical, useful independent-particle approaches • Incorporate effects of interactions and correlations among the particles.

28. DFT has become the primary tool for calculation of electronic structure in condensed matter • Increasingly important for quantitative studies of molecules and other finite systems. • Remarkable successes of the approximate local density and generalized-gradient approximation functionals within the Kohn-Sham approach • Led to wide spread interest in DFT as the most promising aproach for accurate, practical methods in the theory of materials.

29. History • The modern formulation of DFT: • Originated in a famous paper written by P. Hohenberg and W. Kohn in 1964. • A special role can be assigned to the density of particles in the ground state of a quantum many-body system: the density can be considered as a basic variable. • All properties of the system can be considered to be unique functionals of the ground state density！

30. In 1965, Mermin extended the Hohenberg-Kohn arguments to finite temperature canonical and grand canonical ensembles. • The finite temperature extension hasn’t been widely used • Generality of DFT and the difficulty of realizing the promise of exact DFT. • In 1965, A classic work by W. Kohn and L. J. Sham, whose formulation of DFT has become the basis of much of present day methods for treating electrons ion atoms, molecules and condensed matter.