Differences

This shows you the differences between two versions of the page.

--- basis_sets [2014/03/31 13:22] – oschuett
+++ basis_sets [2020/08/21 10:15] – external edit 127.0.0.1
@@ 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 42: / Line 42: @@
   * 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 following numbers specify the number of contracted basis functions for each angular momentum value.
-  * The fifth number specifies the number of contractions for $l=0$ or s-functions (here: $M_s=2$).
+  * 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: $M_p=2$).
+  * 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.
@@ Line 53: / Line 53: @@
 \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_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_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 78: @@
 \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}