Integration over the $k$ space

Consider a function $f$ defined in the $k$-space. The sum of $f$ over ${\bf k}$ is

$$I=\sum_{\bf k} f({\bf k})$$ (1)

(1) can be written as

$$\sum_{\bf k} f({\bf k})=\frac{L^3}{(2\pi)^3}\sum_{\bf k} f({\bf k}) \frac{(2\pi)^3}{L^3},$$ (2)

where $L$ is the sample size along each direction, and $\frac{(2\pi)^3}{L^3}$ is the volume occupied by one state in the $k$-space. If the available states are continuous rather than discrete, the quantity $\frac{(2\pi)^3}{L^3}$ can be seen as an infinitesimal volume $(\Delta k)^3$. As a consequence, the sum becomes an integral, and (1) can be expressed as

$$\sum_{\bf k} f({\bf k}) = \frac{L^3}{(2\pi)^3}\sum_{\bf k} f... ...ta k)^3 \approx \frac{L^3}{(2\pi)^3} \int f({\bf k}) d{\bf k}.$$ (3)

In general, we can generalize to a generic dimensionality $d$ and make the following substitution

$$\sum_k f(k)\rightarrow 2 \times \frac{L^d}{(2\pi)^d} \int f(k) (d k)^d$$ (4)

where we have also considered the spin degeneracy. This relation will result to be very useful in the derivation of the electron density whose general expression at the equilibrium is  [1]

$$n({\bf r})=\sum_i f_0(E_i)\vert\Psi_i({\bf r})\vert^2$$ (5)

where the sum is performed over the single energy levels $E_i$, and $f_0$ is the Fermi-Dirac distribution function.