basis_sets

# Basis Sets

## Construction

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: $R_i(r) = r^{l_i} \sum_{j=1}^N c_{ij}\cdot \exp(-\alpha_j\cdot r^2)$

## File Format

We explain the file-format using the following example from the file $CP2K_HOME/data/BASIS_SET:  1 # Silicon 2 # 3 # Z(nuc) = 14 4 # Z(eff) = 4 5 # E(ref) = -3.739451 a.u. (DZV) 6 # 7 Si DZVP-GTH-PBE 8 2 9 3 0 1 4 2 2 10 1.1815290892 0.3214648125 0.0000000000 0.0458533253 0.0000000000 11 0.4454622072 -0.2454343061 0.0000000000 -0.2633419994 0.0000000000 12 0.1674585747 -0.7952663455 0.0000000000 -0.5433222352 0.0000000000 13 0.0564288769 -0.1828967955 1.0000000000 -0.3560783416 1.0000000000 14 3 2 2 1 1 15 0.4500000000 1.0000000000 Lines 1-6 are comments, because they start with #. Line 7 specifies the element of the basis set (here: silicon) and the name of the basis-set (here: 'DZVP-GTH-PBE'). Line 8 specifies the number of sets this basis set contains (here: 2). Line 9 specifies the composition of the first set. • The first number specifies the principal quantum number (here: 3). Since this basis-set is meant to be used with pseudo-potentials, it does not contain the core electrons. CP2K ignores this number. It is merely present for compatibility reasons and documentation. • The second number specifies the minimal angular quantum number$l_\text{min}$(here: 0). • The third number specifies the maximal angular quantum number$l_\text{max}$(here: 1). • The fourth number specifies the number of exponents$N$(here: 4). The following numbers specify the number of contracted basis functions for each angular momentum value$n_l$. • The fifth number specifies the number of contractions for$l=0$or s-functions (here: 2). • The sixth number specifies the number of contractions for$l=1$or p-functions (here: 2). Line 10-13 specify the coefficients of the first set. Each line consists of an exponent$\alpha_j$, followed by contraction coefficients$c_{ij}$. For example, line 10 starts with the exponent (1.181), followed by the two contraction coefficients for s-functions (0.321 and 0.0), followed by the two contraction coefficients for p-functions (0.046 and 0.0). The entire set consists of$\sum_{l=l_\text{min}}^{l_\text{max}} n_l \cdot (l+1)$basis functions. Each basis function consists of$N\$ terms - one for every exponent.

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}\cdot r \left [0.046 \cdot e^{-1.181r^2} -0.263 \cdot e^{-0.445r^2} -0.543 \cdot e^{-0.164r^2} - 0.356 \cdot e^{-0.056 r^2} \right ] \\ \varphi_4(\vec r) &= Y_{1,0}\cdot r \left [0.046 \cdot e^{-1.181r^2} -0.263 \cdot e^{-0.445r^2} -0.543 \cdot e^{-0.164r^2} - 0.356 \cdot e^{-0.056 r^2} \right ] \\ \varphi_5(\vec r) &= Y_{1, +1}\cdot r \left [0.046 \cdot e^{-1.181r^2} -0.263 \cdot e^{-0.445r^2} -0.543 \cdot e^{-0.164r^2} - 0.356 \cdot e^{-0.056 r^2} \right ] \\ \varphi_6(\vec r) &= Y_{1,-1} \cdot r \cdot e^{-0.056 r^2} \\ \varphi_7(\vec r) &= Y_{1,0} \cdot r \cdot e^{-0.056 r^2} \\ \varphi_8(\vec r) &= Y_{1, +1} \cdot r \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 r^2 \cdot e^{-0.45 r^2} \\ \varphi_{10}(\vec r) &= Y_{2,-1} \cdot r^2 \cdot e^{-0.45 r^2} \\ \varphi_{11}(\vec r) &= Y_{2,0} \cdot r^2 \cdot e^{-0.45 r^2} \\ \varphi_{12}(\vec r) &= Y_{2, +1} \cdot r^2 \cdot e^{-0.45 r^2} \\ \varphi_{13}(\vec r) &= Y_{2, +2} \cdot r^2 \cdot e^{-0.45 r^2} \end{align} 