Applying the Constant Number of Electrons In Dft Density Optimisation

Applying the Constant Number of Electrons In Dft Density Optimization

In Density Functional Theory (DFT), optimizing the electron density is a crucial step in calculating the energy of a system. However, understanding the process of density optimization can be a bit confusing.

The Derivative of Energy in Density Optimization

The derivative of the energy in DFT density optimization is given by the following equation:

Here, v(r) is the external potential, and v_i(r) is the Hartree potential, which represents the electron-electron repulsion. The last term in the equation represents the exchange-correlation potential, which is usually treated using an approximate functional.

Chemical Potential in Density Optimization

The chemical potential is a critical quantity in DFT. It is defined as the derivative of the energy with respect to the number of particles:

where V is the volume of the system, and the integral is taken over the entire domain of the system.

Density Optimization Procedure

The density optimization procedure aims to find the electron density that minimizes the energy of the system. The optimization is usually done iteratively by updating the density at each step. One of the most widely used methods is the self-consistent field (SCF) procedure.

The next step density is obtained using the following equation:

where t is a step size to help convergence, and k denotes the iteration number.

Constant Number of Electrons in Density Optimization

One thing that might confuse people is the fact that the number of electrons is kept constant during the density optimization procedure. This means that the total charge density of the system is fixed, and the optimization only involves redistributing the charge density among the atoms.

It might seem that this approach oversimplifies the problem, and that it cannot account for the complexity of molecular orbitals beyond the s orbital. However, it turns out that this is not the case. In fact, for a given number of electrons, the optimal density can be found by minimizing the energy with respect to the density subject to the constraint of a fixed number of electrons.

Implementing the Density Optimization Procedure

Implementing the density optimization procedure can be challenging, especially for complex systems. However, starting with simple systems can be a good way to learn how it works. For example, in the case of the hydrogen atom, we can write down the exact expression for the energy and optimize the density using the SCF procedure.

import numpy as np

def v_ext(r):
    """External potential for hydrogen atom."""
    return -1/r

def v_hartree(n, r):
    """Hartree potential for hydrogen atom."""
    return np.trapz(n*(r<1e-10)/r, r)

def energy(n, r):
    """Total energy of hydrogen atom."""
    e_kin = 3/10*(3*np.pi**2)**(2/3)*np.trapz(n**(5/3), r)
    e_hartree = 0.5*v_hartree(n, r)*np.trapz(n, r)
    e_ext = np.trapz(v_ext(r)*n, r)
    return e_kin + e_hartree + e_ext

def density_optimization():
    """Optimize electron density of hydrogen atom."""
    r = np.linspace(0.1, 30, 10000)
    n = np.random.rand(len(r))*np.exp(-r)
    # SCF procedure
    for _ in range(100):
        mu = energy(n, r)
        n = n - 0.1*(v_ext(r) + v_hartree(n, r) + (3*np.pi**2)**(2/3)*n**(2/3) - mu)
    return n

density_optimization()

Here, we define functions for the external potential, the Hartree potential, and the total energy of the hydrogen atom. We then use the SCF procedure to optimize the density. Note that we use a random initial density as the starting point.

Conclusion

Density optimization is a fundamental step in DFT calculations. The constant number of electrons constraint might seem oversimplified, but it turns out to be a valid approach for finding the optimal electron density. Implementing the density optimization procedure can be challenging, but starting with simple systems can help you learn how it works.