Differences

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--- basis_sets [2014/03/31 13:55] – oschuett
+++ basis_sets [2020/11/07 12:57] (current) – oschuett
@@ Line 5: / Line 5: @@
 Where $R(r)$ denotes the radial part and $Y_{lm}(\theta, \phi)$ [[wp>Spherical_harmonics|spherical harmonics]] for the angular part. From a physical point of view the best choice for the radial part would be [[wp>Slater-type_orbital|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:
-\[ R_i(r) = \sum_{j=1}^N c_{ij}\cdot \exp(-\alpha_j\cdot r^2) \]
+\[ R_i(r) = r^{l_i} \sum_{j=1}^N c_{ij}\cdot \exp(-\alpha_j\cdot r^2) \]
 ===== File Format =====
-We explain the file-format using the following example from the file ''$CP2K_HOME/tests/QS/BASIS_SET'':
+We explain the file-format using the following example from the file ''$CP2K_HOME/data/BASIS_SET'':
 <code>
    # Silicon
@@ Line 40: / Line 40: @@
   * 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 fourth number specifies the number of exponents $N$ (here: 4).
-The following number specify the number of contracted basis functions for each angular momentum value.
+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 (1.182), 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).
+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:
@@ Line 57: / Line 59: @@
 \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_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}\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}\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}\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}\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 e^{-0.056 r^2} \\
+\varphi_6(\vec r) &=  Y_{1,-1} \cdot r \cdot e^{-0.056 r^2} \\
-\varphi_7(\vec r) &=  Y_{1,0} \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 e^{-0.056 r^2}
+\varphi_8(\vec r) &=  Y_{1, +1} \cdot r \cdot e^{-0.056 r^2}
 \end{align}
@@ Line 78: / Line 80: @@
 \begin{align}
-\varphi_9(\vec r) &=  Y_{2,-2} \cdot e^{-0.45 r^2}  \\
+\varphi_9(\vec r) &=  Y_{2,-2} \cdot r^2 \cdot e^{-0.45 r^2}  \\
-\varphi_{10}(\vec r) &=  Y_{2,-1} \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 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 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 e^{-0.45 r^2}
+\varphi_{13}(\vec r) &=  Y_{2, +2} \cdot r^2 \cdot e^{-0.45 r^2}
 \end{align}