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Documentation
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@ -1,10 +1,7 @@
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open Util
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open Lacaml.D
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type t =
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| Lowdin of Mat.t
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| Canonical of Mat.t
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| Svd of Mat.t
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type t = Mat.t
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module Am = AngularMomentum
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module Bs = Basis
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@ -12,38 +9,34 @@ module Cs = ContractedShell
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let make_canonical_spherical ~thresh ~overlap basis =
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let ao_num = Bs.size basis in
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let cart_sphe = Mat.make ao_num ao_num 0.
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and i = ref 0
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and n = ref 0 in
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Array.iter (fun shell ->
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let submatrix =
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SphericalToCartesian.matrix (Cs.ang_mom shell)
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in
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ignore @@ lacpy ~b:cart_sphe ~br:(!i+1) ~bc:(!n+1) submatrix;
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i := !i + Mat.dim1 submatrix;
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n := !n + Mat.dim2 submatrix;
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) (Bs.contracted_shells basis);
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let s = gemm ~transa:`T ~m:!n cart_sphe overlap in
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let overlap = gemm s ~n:!n cart_sphe in
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let s = canonical_ortho ~thresh ~overlap (Mat.identity !n) in
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gemm cart_sphe ~k:!n s
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let make_canonical ~thresh ~basis ~cartesian ~overlap =
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let make_canonical ~cartesian ~thresh ~basis ~overlap =
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let result =
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if cartesian then
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canonical_ortho ~thresh ~overlap (Mat.identity @@ Mat.dim1 overlap)
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else
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match basis with
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| None -> invalid_arg
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"Basis.t is required when cartesian=false in make_canonical"
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| Some basis ->
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make_canonical_spherical ~thresh ~overlap basis
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let make_canonical_spherical basis =
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let ao_num = Bs.size basis in
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let cart_sphe = Mat.make ao_num ao_num 0.
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and i = ref 0
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and n = ref 0 in
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Array.iter (fun shell ->
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let submatrix =
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SphericalToCartesian.matrix (Cs.ang_mom shell)
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in
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ignore @@ lacpy ~b:cart_sphe ~br:(!i+1) ~bc:(!n+1) submatrix;
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i := !i + Mat.dim1 submatrix;
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n := !n + Mat.dim2 submatrix;
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) (Bs.contracted_shells basis);
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let s = gemm ~transa:`T ~m:!n cart_sphe overlap in
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let overlap = gemm s ~n:!n cart_sphe in
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let s = canonical_ortho ~thresh ~overlap (Mat.identity !n) in
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gemm cart_sphe ~k:!n s
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in
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Canonical result
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if cartesian then
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canonical_ortho ~thresh ~overlap (Mat.identity @@ Mat.dim1 overlap)
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else
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match basis with
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| None -> invalid_arg
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"Basis.t is required when cartesian=false in make_canonical"
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| Some basis -> make_canonical_spherical basis
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@ -59,16 +52,12 @@ let make_lowdin ~thresh ~overlap =
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let u_vec' =
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Mat.init_cols (Mat.dim1 u_vec) (Mat.dim2 u_vec) (fun i j -> u_vec.{i,j} *. u_val.{j})
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in
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let result =
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gemm u_vec' ~transb:`T u_vec
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in
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Lowdin result
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gemm u_vec' ~transb:`T u_vec
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let make ~cartesian ?(thresh=1.e-12) ?basis overlap =
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let make ?(thresh=1.e-12) ?basis ~cartesian overlap =
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(*
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make_lowdin ~thresh ~overlap
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*)
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make_canonical ~cartesian ~basis ~thresh ~overlap
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make_canonical ~thresh ~basis ~cartesian ~overlap
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12
Basis/Orthonormalization.mli
Normal file
12
Basis/Orthonormalization.mli
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@ -0,0 +1,12 @@
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(** Orthonormalization of the basis. *)
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type t = Lacaml.D.Mat.t
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val make: ?thresh:float -> ?basis:Basis.t -> cartesian:bool -> Overlap.t -> t
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(** Returns a matrix or orthonormal vectors in the basis. The vectors are
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linearly independent up to a threshold [thresh]. If [cartesian] is
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[false], the [basis] argument needs to be given and the space spanned by
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the vectors is the same as the space spanned by the basis in spherical
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coordinates (5d,7f,...).
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*)
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(* Orthogonalization matrix *)
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let m_X =
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match Lazy.force simulation.overlap_ortho with
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| Lowdin x -> x
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| Canonical x -> x
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| Svd x -> x
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Lazy.force simulation.overlap_ortho
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in
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@ -169,13 +169,13 @@ let array_product a =
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Array.fold_left ( *. ) 0. a
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let diagonalize_symm h =
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let v = lacpy h in
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let w = Vec.create (Mat.dim1 h) in
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let diagonalize_symm m_H =
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let m_V = lacpy m_H in
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let m_W = Vec.create (Mat.dim1 m_H) in
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let result =
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syevd ~vectors:true ~w v
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syevd ~vectors:true ~w:m_W m_V
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in
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v, result
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m_V, result
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let xt_o_x ~o ~x =
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gemm o x
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@ -184,16 +184,19 @@ let xt_o_x ~o ~x =
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let canonical_ortho ?thresh:(thresh=1.e-6) ~overlap c =
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let d, u, _ = gesvd (lacpy overlap) in
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let d0 = Vec.sqrt d in
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let n = Vec.fold (fun accu x -> if x > thresh then accu + 1 else accu) 0 d in
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let d = Vec.map (fun x ->
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if x >= thresh then 1. /. x
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else 0. ) ~y:d d0
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let d_sqrt = Vec.sqrt d in
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let n = (* Number of non-negligible singular vectors *)
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Vec.fold (fun accu x -> if x > thresh then accu + 1 else accu) 0 d
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in
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if n < Vec.dim d0 then
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let d_inv_sq = (* D^{-1/2} *)
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Vec.map (fun x ->
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if x >= thresh then 1. /. x
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else 0. ) ~y:d d_sqrt
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in
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if n < Vec.dim d_sqrt then
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Printf.printf "Removed linear dependencies below %f\n" (1. /. d.{n})
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;
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Mat.scal_cols u d ;
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Mat.scal_cols u d_inv_sq ;
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gemm c u
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@ -67,11 +67,19 @@ val diagonalize_symm : Lacaml.D.mat -> Lacaml.D.mat * Lacaml.D.vec
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(** Diagonalize a symmetric matrix. Returns the eigenvectors and the eigenvalues. *)
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val xt_o_x : o:Lacaml.D.mat -> x:Lacaml.D.mat -> Lacaml.D.mat
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(** Computes X{^T}.O.X *)
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(** Computes {% $\mathbf{X^\dag\, O\, X}$ %} *)
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val canonical_ortho: ?thresh:float -> overlap:Lacaml.D.mat -> Lacaml.D.mat -> Lacaml.D.mat
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(** Canonical orthogonalization. [overlap] is the overlap matrix, and the last argument
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contains the vectors to orthogonalize. *)
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(** Canonical orthogonalization. [overlap] is the overlap matrix {% $\mathbf{S}$ %},
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and the last argument contains the vectors {% $\mathbf{C}$ %} to orthogonalize.
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{%
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\begin{align}
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\mathbf{U\, D\, V^\dag} & = \mathbf{S} \\
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\mathbf{C_\bot} & = \mathbf{C\, U\, D^{-1/2}}
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\end{align}
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%}
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*)
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val debug_matrix: string -> Lacaml.D.mat -> unit
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(** Prints a matrix in stdout for debug *)
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