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Working on basis set
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trex.org
@ -158,8 +158,8 @@ written in C.
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* Basis set
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We consider here basis functions centered on nuclei. Hence, we enable
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the possibility to define \emph{dummy atoms} to place basis functions
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in random positions.
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the possibility to define /dummy atoms/ to place basis functions in
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random positions.
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The atomic basis set is defined as a list of shells. Each shell $s$ is
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centered on a center $A$, possesses a given angular momentum $l$ and a
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@ -185,25 +185,28 @@ written in C.
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Some codes assume that the contraction coefficients are for a linear
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combination of /normalized/ primitives. This implies that a normalization
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constant for the primitive $ks$ needs to be computed and stored. If
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this normalization factor is not required, set $f_{ks}$ to one.
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this normalization factor is not required, $f_{ks}=1$.
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Some codes assume that the basis function are normalized. This
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implies the computation of an extra normalization factor, $\mathcal{N}_s$.
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If the the basis function is not normalized, set $\mathcal{N}_s=1$.
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If the the basis function is not considered normalized, $\mathcal{N}_s=1$.
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All the basis set parameters are stored in one-dimensional arrays:
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#+NAME: basis
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| ~type~ | ~str~ | | Type of basis set: "Gaussian" or "Slater" |
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| ~shell_num~ | ~int~ | | Total Number of shells |
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| ~shell_center~ | ~int~ | ~(basis.shell_num)~ | Nucleus on which the shell is centered ($A$) |
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| ~shell_ang_mom~ | ~int~ | ~(basis.shell_num)~ | Angular momentum ~0:S, 1:P, 2:D, ...~ |
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| ~shell_prim_num~ | ~int~ | ~(basis.shell_num)~ | Number of primitives in the shell ($N_{\text{prim}}$) |
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| ~shell_factor~ | ~float~ | ~(basis.shell_num)~ | Normalization factor of the shell ($\mathcal{N}_s$) |
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| ~prim_index~ | ~int~ | ~(basis.shell_num)~ | Index of the first primitive in the complete list |
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| ~prim_num~ | ~int~ | | Total number of primitives |
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| ~exponent~ | ~float~ | ~(basis.prim_num)~ | Exponents of the primitives ($\gamma_{ks}) |
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| ~coefficient~ | ~float~ | ~(basis.prim_num)~ | Coefficients of the primitives ($a_{ks}$) |
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| ~prim_factor~ | ~float~ | ~(basis.prim_num)~ | Normalization coefficients for the primitives ($f_{ks}$) |
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| ~type~ | ~str~ | | Type of basis set: "Gaussian" or "Slater" |
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| ~num~ | ~int~ | | Total Number of shells |
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| ~prim_num~ | ~int~ | | Total number of primitives |
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| ~nucleus_index~ | ~int~ | ~(nucleus.num)~ | Index of the first shell of each nucleus ($A$) |
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| ~nucleus_shell_num~ | ~int~ | ~(nucleus.num)~ | Number of shells for each nucleus |
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| ~shell_ang_mom~ | ~int~ | ~(basis.num)~ | Angular momentum ~0:S, 1:P, 2:D, ...~ |
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| ~shell_prim_num~ | ~int~ | ~(basis.num)~ | Number of primitives in the shell ($N_{\text{prim}}$) |
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| ~shell_factor~ | ~float~ | ~(basis.num)~ | Normalization factor of the shell ($\mathcal{N}_s$) |
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| ~shell_prim_index~ | ~int~ | ~(basis.num)~ | Index of the first primitive in the complete list |
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| ~exponent~ | ~float~ | ~(basis.prim_num)~ | Exponents of the primitives ($\gamma_{ks}) |
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| ~coefficient~ | ~float~ | ~(basis.prim_num)~ | Coefficients of the primitives ($a_{ks}$) |
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| ~prim_factor~ | ~float~ | ~(basis.prim_num)~ | Normalization coefficients for the primitives ($f_{ks}$) |
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#+CALL: json(data=basis, title="basis")
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@ -211,21 +214,78 @@ written in C.
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:results:
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#+begin_src python :tangle trex.json
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"basis": {
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"type" : [ "str" , [] ]
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, "shell_num" : [ "int" , [] ]
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, "shell_center" : [ "int" , [ "basis.shell_num" ] ]
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, "shell_ang_mom" : [ "int" , [ "basis.shell_num" ] ]
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, "shell_prim_num" : [ "int" , [ "basis.shell_num" ] ]
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, "shell_factor" : [ "float", [ "basis.shell_num" ] ]
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, "prim_index" : [ "int" , [ "basis.shell_num" ] ]
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, "prim_num" : [ "int" , [] ]
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, "exponent" : [ "float", [ "basis.prim_num" ] ]
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, "coefficient" : [ "float", [ "basis.prim_num" ] ]
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, "prim_factor" : [ "float", [ "basis.prim_num" ] ]
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"type" : [ "str" , [] ]
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, "num" : [ "int" , [] ]
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, "prim_num" : [ "int" , [] ]
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, "nucleus_index" : [ "int" , [ "nucleus.num" ] ]
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, "nucleus_shell_num" : [ "int" , [ "nucleus.num" ] ]
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, "shell_ang_mom" : [ "int" , [ "basis.num" ] ]
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, "shell_prim_num" : [ "int" , [ "basis.num" ] ]
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, "shell_factor" : [ "float", [ "basis.num" ] ]
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, "shell_prim_index" : [ "int" , [ "basis.num" ] ]
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, "exponent" : [ "float", [ "basis.prim_num" ] ]
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, "coefficient" : [ "float", [ "basis.prim_num" ] ]
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, "prim_factor" : [ "float", [ "basis.prim_num" ] ]
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} ,
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#+end_src
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:end:
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For example, consider H_2 with the following basis set (in GAMESS
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format), where both the AOs and primitives are considered normalized:
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#+BEGIN_EXAMPLE
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HYDROGEN
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S 5
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1 3.387000E+01 6.068000E-03
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2 5.095000E+00 4.530800E-02
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3 1.159000E+00 2.028220E-01
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4 3.258000E-01 5.039030E-01
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5 1.027000E-01 3.834210E-01
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S 1
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1 3.258000E-01 1.000000E+00
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S 1
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1 1.027000E-01 1.000000E+00
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P 1
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1 1.407000E+00 1.000000E+00
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P 1
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1 3.880000E-01 1.000000E+00
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D 1
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1 1.057000E+00 1.0000000
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#+END_EXAMPLE
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we have:
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#+BEGIN_EXAMPLE
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type = "Gaussian"
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num = 12
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prim_num = 20
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nucleus_index = [0 , 6]
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shell_ang_mom = [0 , 0 , 0 , 1 , 1 , 2 , 0 , 0 , 0 , 1 , 1 , 2 ]
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shell_prim_num = [5 , 1 , 1 , 1 , 1 , 1 , 5 , 1 , 1 , 1 , 1 , 1 ]
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shell_prim_index = [0 , 5 , 6 , 7 , 8 , 9 , 10, 15, 16, 17, 18, 19]
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shell_factor = [1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1.]
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exponent =
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[ 33.87, 5.095, 1.159, 0.3258, 0.1027, 0.3258, 0.1027, 1.407,
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0.388, 1.057, 33.87, 5.095, 1.159, 0.3258, 0.1027, 0.3258, 0.1027, 1.407,
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0.388, 1.057]
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coefficient =
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[ 0.006068, 0.045308, 0.202822, 0.503903, 0.383421, 1.0, 1.0,
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1.0, 1.0, 1.0, 0.006068, 0.045308, 0.202822, 0.503903, 0.383421, 1.0, 1.0,
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1.0, 1.0, 1.0]
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prim_factor =
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[ 1.0006253235944540e+01, 2.4169531573445120e+00, 7.9610924849766440e-01
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3.0734305383061117e-01, 1.2929684417481876e-01, 3.0734305383061117e-01,
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1.2929684417481876e-01, 2.1842769845268308e+00, 4.3649547399719840e-01,
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1.8135965626177861e+00, 1.0006253235944540e+01, 2.4169531573445120e+00,
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7.9610924849766440e-01, 3.0734305383061117e-01, 1.2929684417481876e-01,
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3.0734305383061117e-01, 1.2929684417481876e-01, 2.1842769845268308e+00,
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4.3649547399719840e-01, 1.8135965626177861e+00 ]
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#+END_EXAMPLE
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* Atomic orbitals
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Going from the atomic basis set to AOs implies a systematic
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