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< h1 class = "title" > Gaussian Gaussian< / h1 >
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< div id = "table-of-contents" >
< h2 > Table of Contents< / h2 >
< div id = "text-table-of-contents" >
< ul >
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< li > < a href = "#org75b028d" > 1. Summary< / a > < / li >
< li > < a href = "#org0603510" > 2. Atomic shell< / a >
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< ul >
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< li > < a href = "#orgdf52365" > 2.1. Type< / a > < / li >
< li > < a href = "#orgc6225f5" > 2.2. Access< / a > < / li >
< li > < a href = "#org6b8dfa1" > 2.3. Creation< / a > < / li >
< li > < a href = "#orga9e61b8" > 2.4. Printers< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#orgd615a55" > 3. Atomic shell pair couple< / a >
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< ul >
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< li > < a href = "#orgda0a937" > 3.1. Type< / a > < / li >
< li > < a href = "#org4991d87" > 3.2. Access< / a > < / li >
< li > < a href = "#orgb976b3f" > 3.3. Creation< / a > < / li >
< li > < a href = "#orgc9cfb8d" > 3.4. Printers< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#orgc9b4c06" > 4. Atomic shell pair< / a >
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< ul >
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< li > < a href = "#org2cc8ae6" > 4.1. Type< / a > < / li >
< li > < a href = "#org758a97d" > 4.2. Access< / a > < / li >
< li > < a href = "#orgf49ab7d" > 4.3. Creation< / a > < / li >
< li > < a href = "#org3b5b666" > 4.4. Printers< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#orgfc88a58" > 5. Summary< / a > < / li >
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< / ul >
< / div >
< / div >
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< h2 id = "org75b028d" > < span class = "section-number-2" > 1< / span > Summary< / h2 >
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< / div >
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< div id = "outline-container-org0603510" class = "outline-2" >
< h2 id = "org0603510" > < span class = "section-number-2" > 2< / span > Atomic shell< / h2 >
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< div class = "outline-text-2" id = "text-2" >
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< p >
Set of contracted Gaussians differing only by the powers of \(x\), \(y\) and \(z\), with a
constant < code > Angular_momentum.t< / code > , all centered on the same point.
< / p >
< p >
In other words, it is the set of all contracted shells sharing the same center.
< / p >
\begin{align*}
\chi_{n_x,n_y,n_z}(r) & = f(n_x,n_y,n_z) \sum_{j=1}^{n} \sum_{i=1}^{m} \mathcal{N}_{ij}\, d_{ij}\, g_{ij\,n_x,n_y,n_z}(r) \\
& = (x-X_A)^{n_x} (y-Y_A)^{n_y} (z-Z_A)^{n_z} f(n_x,n_y,n_z) \sum_{j=1}^{n} \sum_{i=1}^{m} \mathcal{N}_{ij}\, d_{ij}\, \exp \left( -\alpha_{ij} |r-R_A|^2 \right)
\end{align*}
< p >
where:
< / p >
< ul class = "org-ul" >
< li > \(g_{ij\,n_x,n_y,n_z}(r)\) is the $i$-th < code > PrimitiveShell.t< / code > of the $j$-th < code > Contracted_shell.t< / code > < / li >
< li > \(n_x + n_y + n_z = l\), the total angular momentum< / li >
< li > \(\alpha_{ij}\) are the exponents (tabulated) of the $j$-th < code > Contracted_shell.t< / code > < / li >
< li > \(d_{ij}\) are the contraction coefficients of the $j$-th < code > Contracted_shell.t< / code > < / li >
< li > \(\mathcal{N}_{ij}\) is the normalization coefficient of the $i$-th primitive shell
(< code > PrimitiveShell.norm_coef< / code > ) of the $j$-th < code > Contracted_shell.t< / code > < / li >
< li > \(f(n_x,n_y,n_z)\) is a scaling factor adjusting the normalization coefficient for the
particular powers of \(x,y,z\) (< code > PrimitiveShell.norm_coef_scale< / code > )< / li >
< / ul >
< / div >
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< h3 id = "orgdf52365" > < span class = "section-number-3" > 2.1< / span > Type< / h3 >
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< pre class = "src src-ocaml" > type t
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open Common
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< / pre >
< / div >
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< div id = "outline-container-orgc6225f5" class = "outline-3" >
< h3 id = "orgc6225f5" > < span class = "section-number-3" > 2.2< / span > Access< / h3 >
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< pre class = "src src-ocaml" > val ang_mom : t -> Angular_momentum.t
val center : t -> Coordinate.t
val coefficients : t -> float array array
val contracted_shells : t -> Contracted_shell.t array
val exponents : t -> float array array
val index : t -> int
val normalizations : t -> float array array
val norm_scales : t -> float array
val size_of_shell : t -> int
val size : t -> int
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< / pre >
< / div >
< table border = "2" cellspacing = "0" cellpadding = "6" rules = "groups" frame = "hsides" >
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< col class = "org-left" / >
< col class = "org-left" / >
< / colgroup >
< tbody >
< tr >
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< td class = "org-left" > < code > ang_mom< / code > < / td >
< td class = "org-left" > Total angular momentum : \(l = n_x + n_y + n_z\).< / td >
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< / tr >
< tr >
< td class = "org-left" > < code > center< / code > < / td >
< td class = "org-left" > Coordinate of the center \(\mathbf{A} = (X_A,Y_A,Z_A)\).< / td >
< / tr >
< tr >
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< td class = "org-left" > < code > coefficients< / code > < / td >
< td class = "org-left" > Array of contraction coefficients \(d_{ij}\). The first index is the index of the contracted function, and the second index is the index of the primitive.< / td >
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< / tr >
< tr >
< td class = "org-left" > < code > contracted_shells:< / code > < / td >
< td class = "org-left" > Array of contracted gaussians< / td >
< / tr >
< tr >
< td class = "org-left" > < code > exponents< / code > < / td >
< td class = "org-left" > Array of exponents \(\alpha_{ij}\). The first index is the index of the contracted function, and the second index is the index of the primitive.< / td >
< / tr >
< tr >
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< td class = "org-left" > < code > index< / code > < / td >
< td class = "org-left" > Index in the basis set, represented as an array of contracted shells.< / td >
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< / tr >
< tr >
< td class = "org-left" > < code > normalizations< / code > < / td >
< td class = "org-left" > Normalization coefficients \(\mathcal{N}_{ij}\). The first index is the index of the contracted function, and the second index is the index of the primitive.< / td >
< / tr >
< tr >
< td class = "org-left" > < code > norm_scales< / code > < / td >
< td class = "org-left" > Scaling factors \(f(n_x,n_y,n_z)\), given in the same order as < code > Angular_momentum.zkey_array ang_mom< / code > .< / td >
< / tr >
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< tr >
< td class = "org-left" > < code > size< / code > < / td >
< td class = "org-left" > Number of contracted functions, \(n\) in the definition.< / td >
< / tr >
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< tr >
< td class = "org-left" > < code > size_of_shell< / code > < / td >
< td class = "org-left" > Number of contracted functions in the shell: length of < code > norm_coef_scale< / code > .< / td >
< / tr >
< / tbody >
< / table >
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< pre class = "example" >
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< / pre >
< / div >
< / div >
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< div id = "outline-container-org6b8dfa1" class = "outline-3" >
< h3 id = "org6b8dfa1" > < span class = "section-number-3" > 2.3< / span > Creation< / h3 >
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< div class = "outline-text-3" id = "text-2-3" >
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< div class = "org-src-container" >
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< pre class = "src src-ocaml" > val make : ?index:int -> Contracted_shell.t array -> t
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val with_index : t -> int -> t
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< / pre >
< / div >
< table border = "2" cellspacing = "0" cellpadding = "6" rules = "groups" frame = "hsides" >
< colgroup >
< col class = "org-left" / >
< col class = "org-left" / >
< / colgroup >
< tbody >
< tr >
< td class = "org-left" > < code > make< / code > < / td >
< td class = "org-left" > Creates a contracted shell from a list of coefficients and primitives.< / td >
< / tr >
< tr >
< td class = "org-left" > < code > with_index< / code > < / td >
< td class = "org-left" > Returns a copy of the contracted shell with a modified index.< / td >
< / tr >
< / tbody >
< / table >
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< / div >
< / div >
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< div id = "outline-container-orga9e61b8" class = "outline-3" >
< h3 id = "orga9e61b8" > < span class = "section-number-3" > 2.4< / span > Printers< / h3 >
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< div class = "outline-text-3" id = "text-2-4" >
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< div class = "org-src-container" >
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< pre class = "src src-ocaml" > val pp : Format.formatter -> t -> unit
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< / pre >
< / div >
< / div >
< / div >
< / div >
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< div id = "outline-container-orgd615a55" class = "outline-2" >
< h2 id = "orgd615a55" > < span class = "section-number-2" > 3< / span > Atomic shell pair couple< / h2 >
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< div class = "outline-text-2" id = "text-3" >
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< p >
An atomic shell pair couple is the cartesian product between two sets of functions, one
set over electron one and one set over electron two. Both sets are atomic shell
pairs.
< / p >
< p >
These are usually called < i > shell quartets< / i > in the literature, but we prefer to use
< i > pair< / i > for two functions with the same electron, and < i > couple< / i > for two functions
acting on different electrons, since they will be coupled by a two-electron operator.
< / p >
< / div >
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< div id = "outline-container-orgda0a937" class = "outline-3" >
< h3 id = "orgda0a937" > < span class = "section-number-3" > 3.1< / span > Type< / h3 >
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< div class = "outline-text-3" id = "text-3-1" >
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< div class = "org-src-container" >
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< pre class = "src src-ocaml" > type t
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open Common
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< / pre >
< / div >
< / div >
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< / div >
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< div id = "outline-container-org4991d87" class = "outline-3" >
< h3 id = "org4991d87" > < span class = "section-number-3" > 3.2< / span > Access< / h3 >
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< div class = "outline-text-3" id = "text-3-2" >
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< div class = "org-src-container" >
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< pre class = "src src-ocaml" > val ang_mom : t -> Angular_momentum.t
val atomic_shell_a : t -> Atomic_shell.t
val atomic_shell_b : t -> Atomic_shell.t
val atomic_shell_c : t -> Atomic_shell.t
val atomic_shell_d : t -> Atomic_shell.t
val atomic_shell_pair_p : t -> Atomic_shell_pair.t
val atomic_shell_pair_q : t -> Atomic_shell_pair.t
val contracted_shell_pair_couples : t -> Contracted_shell_pair_couple.t list
val monocentric : t -> bool
val norm_scales : t -> float array
val zkey_array : t -> Zkey.t array
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< / pre >
< / div >
< table border = "2" cellspacing = "0" cellpadding = "6" rules = "groups" frame = "hsides" >
< colgroup >
< col class = "org-left" / >
< col class = "org-left" / >
< / colgroup >
< tbody >
< tr >
< td class = "org-left" > < code > ang_mom< / code > < / td >
< td class = "org-left" > Total angular momentum of the shell pair couple: sum of the angular momenta of all the shells.< / td >
< / tr >
< tr >
< td class = "org-left" > < code > atomic_shell_a< / code > < / td >
< td class = "org-left" > Returns the first atomic shell of the first shell pair.< / td >
< / tr >
< tr >
< td class = "org-left" > < code > atomic_shell_b< / code > < / td >
< td class = "org-left" > Returns the second atomic shell of the first shell pair.< / td >
< / tr >
< tr >
< td class = "org-left" > < code > atomic_shell_c< / code > < / td >
< td class = "org-left" > Returns the first atomic shell of the second shell pair.< / td >
< / tr >
< tr >
< td class = "org-left" > < code > atomic_shell_d< / code > < / td >
< td class = "org-left" > Returns the second atomic shell of the second shell pair.< / td >
< / tr >
< tr >
< td class = "org-left" > < code > atomic_shell_pair_p< / code > < / td >
< td class = "org-left" > Returns the first atomic shell pair that was used to build the shell pair.< / td >
< / tr >
< tr >
< td class = "org-left" > < code > atomic_shell_pair_q< / code > < / td >
< td class = "org-left" > Returns the second atomic shell pair that was used to build the shell pair.< / td >
< / tr >
< tr >
< td class = "org-left" > < code > contracted_shell_pair_couples< / code > < / td >
< td class = "org-left" > Returns the list of significant contracted shell pair couples.< / td >
< / tr >
< tr >
< td class = "org-left" > < code > monocentric< / code > < / td >
< td class = "org-left" > True if all four atomic shells have the same center.< / td >
< / tr >
< tr >
< td class = "org-left" > < code > norm_scales< / code > < / td >
< td class = "org-left" > Scaling factors of normalization coefficients inside the shell. The ordering is the same as < code > zkey_array< / code > .< / td >
< / tr >
< tr >
< td class = "org-left" > < code > zkey_array< / code > < / td >
< td class = "org-left" > Returns the array of < code > Zkey.t< / code > relative to the four shells of the shell pair couple.< / td >
< / tr >
< / tbody >
< / table >
< / div >
< / div >
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< div id = "outline-container-orgb976b3f" class = "outline-3" >
< h3 id = "orgb976b3f" > < span class = "section-number-3" > 3.3< / span > Creation< / h3 >
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< div class = "outline-text-3" id = "text-3-3" >
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< div class = "org-src-container" >
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< pre class = "src src-ocaml" > val make : ?cutoff:float -> Atomic_shell_pair.t -> Atomic_shell_pair.t -> t option
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< / pre >
< / div >
< p >
Default cutoff is \(\epsilon\).
< / p >
< table border = "2" cellspacing = "0" cellpadding = "6" rules = "groups" frame = "hsides" >
< colgroup >
< col class = "org-left" / >
< col class = "org-left" / >
< / colgroup >
< tbody >
< tr >
< td class = "org-left" > < code > make< / code > < / td >
< td class = "org-left" > Creates an atomic shell pair couple using two atomic shell pairs.< / td >
< / tr >
< / tbody >
< / table >
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< pre class = "example" >
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< / pre >
< / div >
< / div >
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< div id = "outline-container-orgc9cfb8d" class = "outline-3" >
< h3 id = "orgc9cfb8d" > < span class = "section-number-3" > 3.4< / span > Printers< / h3 >
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< div class = "outline-text-3" id = "text-3-4" >
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< div class = "org-src-container" >
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< pre class = "src src-ocaml" > val pp : Format.formatter -> t -> unit
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< / pre >
< / div >
< / div >
< / div >
< / div >
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< div id = "outline-container-orgc9b4c06" class = "outline-2" >
< h2 id = "orgc9b4c06" > < span class = "section-number-2" > 4< / span > Atomic shell pair< / h2 >
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< div class = "outline-text-2" id = "text-4" >
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< p >
Data structure to represent pairs of atomic shells. The products of
functions in the shell pair are one-electron functions.
< / p >
< p >
An atomic shell pair is an array of pairs of contracted shells.
< / p >
< / div >
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< div id = "outline-container-org2cc8ae6" class = "outline-3" >
< h3 id = "org2cc8ae6" > < span class = "section-number-3" > 4.1< / span > Type< / h3 >
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< div class = "outline-text-3" id = "text-4-1" >
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< div class = "org-src-container" >
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< pre class = "src src-ocaml" > type t
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open Common
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< / pre >
< / div >
< / div >
< / div >
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< div id = "outline-container-org758a97d" class = "outline-3" >
< h3 id = "org758a97d" > < span class = "section-number-3" > 4.2< / span > Access< / h3 >
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< div class = "outline-text-3" id = "text-4-2" >
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< div class = "org-src-container" >
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< pre class = "src src-ocaml" > val atomic_shell_a : t -> Atomic_shell.t
val atomic_shell_b : t -> Atomic_shell.t
val contracted_shell_pairs : t -> Contracted_shell_pair.t list
val ang_mom : t -> Angular_momentum.t
val monocentric : t -> bool
val norm_scales : t -> float array
val a_minus_b : t -> Coordinate.t
val a_minus_b_sq : t -> float
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< / pre >
< / div >
< table border = "2" cellspacing = "0" cellpadding = "6" rules = "groups" frame = "hsides" >
< colgroup >
< col class = "org-left" / >
< col class = "org-left" / >
< / colgroup >
< tbody >
< tr >
< td class = "org-left" > < code > atomic_shell_a< / code > < / td >
< td class = "org-left" > Returns the first < code > Atomic_shell.t< / code > which was used to build the atomic shell pair.< / td >
< / tr >
< tr >
< td class = "org-left" > < code > atomic_shell_b< / code > < / td >
< td class = "org-left" > Returns the second < code > Atomic_shell.t< / code > which was used to build the atomic shell pair.< / td >
< / tr >
< tr >
< td class = "org-left" > < code > contracted_shell_pairs< / code > < / td >
< td class = "org-left" > Returns an array of < code > ContractedShellPair.t< / code > , containing all the pairs of contracted functions used to build the atomic shell pair.< / td >
< / tr >
< tr >
< td class = "org-left" > < code > norm_scales< / code > < / td >
< td class = "org-left" > norm< sub > coef< / sub > .(i) / norm< sub > coef< / sub > .(0)< / td >
< / tr >
< tr >
< td class = "org-left" > < code > ang_mom< / code > < / td >
< td class = "org-left" > Total angular Momentum< / td >
< / tr >
< tr >
< td class = "org-left" > < code > monocentric< / code > < / td >
< td class = "org-left" > If true, the two atomic shells have the same center.< / td >
< / tr >
< tr >
< td class = "org-left" > < code > a_minus_b< / code > < / td >
< td class = "org-left" > Returns \(A-B\)< / td >
< / tr >
< tr >
< td class = "org-left" > < code > a_minus_b_sq< / code > < / td >
< td class = "org-left" > Returns \(\vert A-B \vert^2\)< / td >
< / tr >
< / tbody >
< / table >
< / div >
< / div >
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< div id = "outline-container-orgf49ab7d" class = "outline-3" >
< h3 id = "orgf49ab7d" > < span class = "section-number-3" > 4.3< / span > Creation< / h3 >
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< div class = "outline-text-3" id = "text-4-3" >
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< div class = "org-src-container" >
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< pre class = "src src-ocaml" > val make : ?cutoff:float -> Atomic_shell.t -> Atomic_shell.t -> t option
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< / pre >
< / div >
< p >
Creates an atomic shell pair from two atomic shells.
< / p >
< p >
The contracted shell pairs contains the only pairs of primitives for which
the norm is greater than < code > cutoff< / code > .
< / p >
< p >
If all the contracted shell pairs are not significant, the function returns
< code > None< / code > .
< / p >
< div class = "org-src-container" >
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< pre class = "src src-ocaml" > val of_atomic_shell_array : ?cutoff:float -> Atomic_shell.t array -> t option array array
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< / pre >
< / div >
< p >
Creates all possible atomic shell pairs from an array of atomic shells.
If an atomic shell pair is not significant, sets the value to < code > None< / code > .
< / p >
< / div >
< / div >
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< div id = "outline-container-org3b5b666" class = "outline-3" >
< h3 id = "org3b5b666" > < span class = "section-number-3" > 4.4< / span > Printers< / h3 >
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< div class = "outline-text-3" id = "text-4-4" >
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< div class = "org-src-container" >
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< pre class = "src src-ocaml" > val pp : Format.formatter -> t -> unit
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< / pre >
< / div >
< / div >
< / div >
< / div >
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< div id = "outline-container-orgfc88a58" class = "outline-2" >
< h2 id = "orgfc88a58" > < span class = "section-number-2" > 5< / span > Summary< / h2 >
< / div >
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< / div >
< div id = "postamble" class = "status" >
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< p class = "author" > Author: scemama< / p >
< p class = "date" > Created: 2023-06-17 Sat 00:26< / p >
< p class = "validation" > < a href = "http://validator.w3.org/check?uri=referer" > Validate< / a > < / p >
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< / div >
< / body >
< / html >
