CP2K uses Gaussian type orbitals as basis functions. Every basis function has the following form: $$\varphi_i(\vec r) = R_i(r) \cdot Y_{l_i, m_i}(\theta, \phi)$$

Where $R(r)$ denotes the radial part and $Y_{lm}(\theta, \phi)$ spherical harmonics for the angular part. From a physical point of view the best choice for the radial part would be Slater-type orbitals. However, CP2K uses contracted Gaussians instead, because they have nicer analytic properties. Contracted Gaussians are simply a weighted sum of primitive Gaussians with different exponents: $$R_i(r) = \sum_{j=1}^N c_{ij}\cdot \exp(\alpha_j\cdot r^2)$$

We explain the file-format using the following example from the file $CP2K_HOME/tests/QS/BASIS_SET:

 1   # Silicon
 2   #
 3   # Z(nuc) = 14
 4   # Z(eff) = 4
 5   # E(ref) = -3.739451 a.u. (DZV)
 6   #
 7   Si DZVP-GTH-PBE
 8     2
 9     3  0  1  4  2  2
10           1.1815290892   0.3214648125   0.0000000000   0.0458533253   0.0000000000
11           0.4454622072  -0.2454343061   0.0000000000  -0.2633419994   0.0000000000
12           0.1674585747  -0.7952663455   0.0000000000  -0.5433222352   0.0000000000
13           0.0564288769  -0.1828967955   1.0000000000  -0.3560783416   1.0000000000
14     3  2  2  1  1
15           0.4500000000   1.0000000000

Lines 1-6 are comments, because they start with #.

Line 7 specifies the element of the basis set (here: silicon) and the name of the basis-set (here: 'DZVP-GTH-PBE').

Line 8 specifies the number of sets this basis set contains (here: 2).

Line 9 specifies the composition of the first set.

• The first number specifies the principal quantum number (here: 3). Since this basis-set is meant to be used with pseudo-potentials, it does not contain the core electrons. CP2K ignores this number. It is merely present for compatibility reasons and documentation.
• The second number specifies the minimal angular quantum number $l_\text{min}$ (here: 0).
• The third number specifies the maximal angular quantum number $l_\text{max}$ (here: 1).
• The fourth number specifies the number of exponents (here: 4).

The following number specify the number of contractions for each angular momentum value $M$.

• The fifth number specifies the number of contractions for $l=0$ or s-functions (here: $M_s=2$).
• The sixth number specifies the number of contractions for $l=1$ or p-functions (here: $M_p=2$).

Line 10-13 specify the coefficients of the first set. Each line consists of an exponent $\alpha_j$, followed by contraction coefficients $c_{ij}$. For example, line 10 starts with the exponent (1.182), followed by the two contraction coefficients for s-functions (0.321 and 0.0), followed by the two contraction coefficients for p-functions (0.046 and 0.0).

The entire first set consists of the following 8 basis functions:

$$\begin{align} \varphi_1(\vec r) &= Y_{0,\ 0}\left [0.321 \cdot e^{1.181r^2} -0.245 \cdot e^{0.445r^2} -0.795 \cdot e^{0.164r^2} - 0.182 \cdot e^{0.056 r^2} \right ] \\ \varphi_2(\vec r) &= Y_{0,0} \cdot e^{0.056 r^2} \\ \varphi_3(\vec r) &= Y_{1,-1}\left [0.046 \cdot e^{1.181r^2} -0.263 \cdot e^{0.445r^2} -0.543 \cdot e^{0.164r^2} - 0.543 \cdot e^{0.056 r^2} \right ] \\ \varphi_4(\vec r) &= Y_{1,0}\left [0.046 \cdot e^{1.181r^2} -0.263 \cdot e^{0.445r^2} -0.543 \cdot e^{0.164r^2} - 0.543 \cdot e^{0.056 r^2} \right ] \\ \varphi_5(\vec r) &= Y_{1, +1}\left [0.046 \cdot e^{1.181r^2} -0.263 \cdot e^{0.445r^2} -0.543 \cdot e^{0.164r^2} - 0.543 \cdot e^{0.056 r^2} \right ] \\ \varphi_6(\vec r) &= Y_{1,-1} \cdot e^{0.056 r^2} \\ \varphi_7(\vec r) &= Y_{1,0} \cdot e^{0.056 r^2} \\ \varphi_8(\vec r) &= Y_{1, +1} \cdot e^{0.056 r^2} \end{align}$$

Line 14 specifies the composition of the second set.

Line 15 specify the coefficients of the second set.

The second set contributes the following 5 basis functions:

$$\begin{align} \varphi_9(\vec r) &= Y_{2,-2} \cdot e^{0.45 r^2} \\ \varphi_{10}(\vec r) &= Y_{2,-1} \cdot e^{0.45 r^2} \\ \varphi_{11}(\vec r) &= Y_{2,0} \cdot e^{0.45 r^2} \\ \varphi_{12}(\vec r) &= Y_{2, +1} \cdot e^{0.45 r^2} \\ \varphi_{13}(\vec r) &= Y_{2, +2} \cdot e^{0.45 r^2} \end{align}$$