Table of Contents
1. Todo [50%]
[-]create the unit tensor operator and check that it plays well[X]create an option to disallow contractions of the tensor with itself.[X]create an option for only allowing linked contractions[X]implement symmetry filter: Viajb should be the same as Vaibj and so on. So the symmetry is a node Vertauschungssymmetrie.[ ]compile with gcl[ ]test with clasp (it takes too long to compile in nixos)[ ]implement parity symmetry
2. Prolog
(defpackage :arponen (:use :cl)) (in-package :arponen)
;; todo define packages and all that
(in-package :arponen)
(defmacro assert-equal (left right)
`(assert (equal ,left ,right)))
(defmacro assert! (expr &rest args)
`(assert (not ,expr) ,@args))
(defmacro assert!-equal (left right)
`(assert (not (equal ,left ,right))))
(defmacro assert-condition (expr condition-type &rest body)
`(handler-case (progn ,expr
(assert nil))
(,condition-type ,@body)))
(defmacro assert-errmsg (expr condition-type error-message)
`(assert-condition ,expr ,condition-type
(m) (assert-equal (format nil "~a" m)
,error-message)))
3. Nomenclature
In order to read better the code and organize it better, we should begin by having a clear nomenclature of the objects we are dealing with.
A diagram is in its graph-theoretic sense a set of nodes with vertices. In the case of a diagram in chemistry a vertex is interpreted as a contraction and uncontracted vertices are legs of a diagram.
We will not make the distinction between a legged-node and an unlegged-node, as we do not need it for our purposes.
Therefore, a simple picture of a diagram can be akin to the following representation:
\ \ \ v------------- Node *========* / \ / \ <--------- Leg / \
3.1. Types
In Lisp, we will represent a short list of objects in the following manner:
A tensor
(tensor-name [node])
A contracted expression
((contraction (point-1 point-2)) (tensor-product))
These conceps are simply represented as simple lisp predicates as follows
(defun contraction? (expr)
(and (listp expr)
(listp (car expr))
(eq (caar expr) 'contraction)
;; body
(listp (cadr expr))))
(defun tensor? (expr)
(and (listp expr)
(listp (cadr expr))))
(assert (contraction? '((contraction ((a b) (c d))) nil))) (assert (contraction? '((contraction nil) ((t (a i) (b j)))))) (assert (tensor? '(t (a i) (b j)))) (assert (tensor? '(1 nil))) (assert (tensor? '(0 nil)))
4. Main variables
The following variables act as flags for the behaviour of the program, you can set them temporarily in a `let` block in order to use them or set them globally with `setq`.
4.1. Verbosity and logging
To control the verbosity of the program, set the *print-log*
variable.
(defvar *print-log* t
"Wether to print the log messages for the contractions and so on")
;; TODO: implement log levels
(defmacro logger (fmt &rest args)
`(when *print-log*
(eval (format t ,fmt ,@args))))
(defmacro logger! (&rest args)
`(let ((*print-log* t))
(logger ,@args)))
4.2. Contractions within the same tensor
In general, the contraction searcher will search for contractions according to the contraction rules between all nodes.
However, for most many-body diagrammatics, one is mostly interested in contractions between different tensors.
(defvar *allow-self-contractions* nil "Wether or not to allow a tensor to search for contractions with its nodes.")
If you want to allow self-contractions you can set this option as
(setq *allow-self-contractions* t)
4.3. Connected diagrams
As in coupled-cluster theories, you can also only look for connected contraction possibilities.
This settings is disables by default so you will have to explicitly set it to have connected diagrams, see the examples for coupled-cluster theories.
;; TODO: maybe this should be called linked, ;; check with lindgren paper.
(defvar *only-connected-diagrams* nil "Wether to look for contractions that create connected diagrams.")
4.4. Node symmetry
If you do not want the contractions to be filtered automatically
by node symmetry set this variable to nil.
(defvar *filter-node-symmetry* t)
4.5. Parity symmetry
(defvar *filter-parity-symmetry* nil "Wether to filter contractions according to parity symmetry.")
5. Combinatorics
This code relies a lot on combinatorics and set-theoretic functions, this section describes their implementation in order to maintain the package self-contained.
5.1. Cartesian product
We implement a cartesian product in the form of a macro that avoids recursion but however writes it with depending on the loop macro.
(defmacro cartesian-product (&rest lists)
(let* ((indices (loop for i from 1 to (length lists)
collect (gensym (format nil "~a-i-" i))))
(initial-value `(loop for ,(car (last indices)) in ',(car (last lists))
collect `(,,@indices))))
(reduce
(lambda (x y)
`(loop for ,(car x) in ',(cadr x)
nconc ,y))
(mapcar #'list (butlast indices) (butlast lists))
:from-end t
:initial-value initial-value)))
With tests:
(assert-equal (cartesian-product (H P) (a b c) (1 2 3 5))
'((H A 1) (H A 2) (H A 3) (H A 5)
(H B 1) (H B 2) (H B 3) (H B 5)
(H C 1) (H C 2) (H C 3) (H C 5)
(P A 1) (P A 2) (P A 3) (P A 5)
(P B 1) (P B 2) (P B 3) (P B 5)
(P C 1) (P C 2) (P C 3) (P C 5)))
(assert-equal (cartesian-product (H (P)) ((a)))
'((H (A)) ((P) (A))))
5.2. Permutations
(defun all-permutations (lst &optional (remain lst))
(cond ((null remain) nil)
((null (rest lst)) (list lst))
(t (append
(mapcar (lambda (l) (cons (first lst) l))
(all-permutations (rest lst)))
(all-permutations (append (rest lst) (list (first lst)))
(rest remain))))))
And tests
(assert-equal (all-permutations '(a b))
'((A B) (B A)))
(assert-equal (all-permutations '(a b c))
'((A B C) (A C B) (B C A) (B A C) (C A B) (C B A)))
(assert-equal (all-permutations '(a b c d))
'((A B C D) (A B D C) (A C D B) (A C B D) (A D B C) (A D C B)
(B C D A) (B C A D) (B D A C)
(B D C A) (B A C D) (B A D C) (C D A B) (C D B A) (C A B D)
(C A D B) (C B D A) (C B A D)
(D A B C) (D A C B) (D B C A) (D B A C) (D C A B) (D C B A)))
5.3. Node pairs building
In order to find contractions, we will be concerned with a product of tensors \((t_1, \ldots, t_n)\) and every tensor will have a series of nodes, let us denote the whole indices of the nodes from \(0\) to \(N - 1\) where
\[ N = \sum_i \#\mathrm{nodes}(t_i) \]
get-node-pairs will give us pairs of nodes to search
for contractions in.
We have to restrict the possible pairs according to:
- only one occurrence of a pair combination is allowed in order not to search for the same contractions twice, this means that only \((i, j)\) pairs are allowed where \(i \leq j\).
- in the case we want to search only for contractions
between different tensors, we can provide
a
group-lengthslist which will only allow for pairs of different groups.
(defun get-node-pairs (n &key (group-lengths nil))
;; check that group-lengths is well built
(when group-lengths (assert (eq n (apply #'+ group-lengths))))
(let ((successive-lengths
;; successive-lengths
;; should be simply (g0 (+ g0 g1) ... (+ g0 .. gn))
;; where gj \in group-lengths
(reverse (maplist (lambda (lst) (apply #'+ lst))
(reverse group-lengths)))))
(labels ((from-i (i)
(if group-lengths
;; find the first group where i
;; is smaller, this means the next group
;; starts there
(find i successive-lengths :test #'<)
i)))
(loop for i from 0 below n
nconcing (loop for j from (from-i i) below n
collect `(,i ,j))))))
;; trivial examples
(assert-equal (get-node-pairs 1) '((0 0)))
(assert-equal (get-node-pairs 2) '((0 0) (0 1) (1 1)))
(assert-equal (get-node-pairs 3) '((0 0) (0 1) (0 2) (1 1) (1 2) (2 2)))
;; 2 3
;; (0 1 || 2 3 4)
(assert-equal (get-node-pairs 5 :group-lengths '(2 3))
'((0 2) (0 3) (0 4)
(1 2) (1 3) (1 4)))
(assert-equal (get-node-pairs 5)
'((0 0) (0 1) (0 2) (0 3) (0 4)
(1 1) (1 2) (1 3) (1 4) (2 2)
(2 3) (2 4) (3 3) (3 4) (4 4)))
;; 2 3 1 3
;; (0 1 || 2 3 4 || 5 || 6 7 8)
(assert-equal (get-node-pairs 9 :group-lengths '(2 3 1 3))
'((0 2) (0 3) (0 4) (0 5) (0 6) (0 7) (0 8)
(1 2) (1 3) (1 4) (1 5) (1 6) (1 7) (1 8)
(2 5) (2 6) (2 7) (2 8)
(3 5) (3 6) (3 7) (3 8)
(4 5) (4 6) (4 7) (4 8)
(5 6) (5 7) (5 8)))
;; V T1 T2
;; (0 1 || 2 || 3 4)
(assert-equal (get-node-pairs 5 :group-lengths '(2 1 2))
'((0 2) (0 3) (0 4)
(1 2) (1 3) (1 4)
(2 3) (2 4)))
5.4. Pair combinations
Given a product of tensors, we will want to have
which pair of nodes can have contractions, this is given
by the get-node-pairs function.
But in general we will want to have \(n_c\) contractions,
taken from combinations of these node pairs (node-a node-b).
For example, if we are looking for 3 contractions in total,
and we have the pairs of nodes where we can find these contractions
(p1 ... pn), then we will want to look for instance
first three times in the pair of nodes p1 for 3 successful
contractions, in pair-index notation this would represent the
list
(0 0 0)
where 0 is the index of the position of p1 in the pair list.
We call these lists pair combinations.
A given pair combination describes the potential connections of the tensors and represent a whole class of diagrams. In particular, the linkedness and connectedness of diagrams are encoded in these lists and we use them to decide if a given diagram is linked or not. ;; TODO: check link or connected exactly
(defmacro ordered-subsets-with-repetition (n space-size)
(when (> n 0)
(let* ((vars (loop for i below (1+ n) collect (gensym))))
`(let ((,(car vars) 0))
,(reduce (lambda (x other-loop)
`(loop for ,(cdr x) from ,(car x) below ,space-size
,@(if (null other-loop)
`(collect `(,,@(cdr vars)))
(list 'nconcing other-loop))))
(mapcar #'cons vars (cdr vars))
:initial-value nil
:from-end t)))))
(assert-equal (ordered-subsets-with-repetition 1 2)
'((0) (1)))
(assert-equal (ordered-subsets-with-repetition 2 2)
'((0 0) (0 1) (1 1)))
(assert-equal (ordered-subsets-with-repetition 2 3)
'((0 0) (0 1) (0 2) (1 1) (1 2) (2 2)))
(assert-equal (ordered-subsets-with-repetition 2 5)
'((0 0) (0 1) (0 2) (0 3) (0 4) (1 1) (1 2) (1 3)
(1 4) (2 2) (2 3) (2 4) (3 3) (3 4) (4 4)))
(assert-equal (ordered-subsets-with-repetition 3 3)
'((0 0 0) (0 0 1) (0 0 2) (0 1 1) (0 1 2)
(0 2 2) (1 1 1) (1 1 2) (1 2 2) (2 2 2)))
;; here we would need 4 contractions between a set of
;; 4 pairs of nodes
(assert-equal (ordered-subsets-with-repetition 4 4)
'((0 0 0 0) (0 0 0 1) (0 0 0 2) (0 0 0 3) (0 0 1 1) (0 0 1 2)
(0 0 1 3) (0 0 2 2) (0 0 2 3) (0 0 3 3) (0 1 1 1) (0 1 1 2)
(0 1 1 3) (0 1 2 2) (0 1 2 3) (0 1 3 3) (0 2 2 2) (0 2 2 3)
(0 2 3 3) (0 3 3 3) (1 1 1 1) (1 1 1 2) (1 1 1 3) (1 1 2 2)
(1 1 2 3) (1 1 3 3) (1 2 2 2) (1 2 2 3) (1 2 3 3) (1 3 3 3)
(2 2 2 2) (2 2 2 3) (2 2 3 3) (2 3 3 3) (3 3 3 3)))
5.5. Utils
;; functions taken from uruk
(defun flatten-list (ls)
(cond
((and (consp ls)
(atom (car ls)))
`(,(car ls) ,@(flatten-list (cdr ls))))
((and (consp ls)
(consp (car ls)))
`(,@(flatten-list (car ls)) ,@(flatten-list (cdr ls))))
(t ls)))
(defmacro thread-first (var &rest forms)
(let ((init var))
(loop for f in forms
do (setf init (setf f (cons (car f)
(cons init (cdr f))))))
init))
(defmacro thread-last (var &rest forms)
(let ((init var))
(loop for f in forms
do (setf init (setf f (cons (car f)
(reverse (cons init
(reverse (cdr f))))))))
init))
(multiple-value-bind (expression _ )
(macroexpand '(thread-first x (+ 5) (* 8)))
(declare (ignorable _))
(assert-equal '(* (+ x 5) 8)
expression))
(multiple-value-bind (expression _ )
(macroexpand '(thread-last x (+ 5) (* 8)))
(declare (ignorable _))
(assert-equal '(* 8 (+ 5 x))
expression))
(defun symbols-repeated-p (lst)
(let ((symbols (flatten-list lst))
s)
(loop while (setq s (pop symbols))
if (> (count s symbols) 0)
do (return t))))
(let ((vals '(((a b c) . nil)
((a (a) b c) . t)
((((a)) ((b e f g)) ((((b))))) . t))))
(loop for (lst . val) in vals
do (assert (eq (symbols-repeated-p lst) val))))
6. Arithmetic expressions
(defun expr-to-lists (exp)
(case (if (atom exp) t (car exp))
(* (reduce (lambda (x y)
(reduce #'append
(loop for -x in x
collect (loop for -y in y
collect (append -x -y)))))
(mapcar #'expr-to-lists (cdr exp))
:initial-value '(nil)
:from-end t))
(+ (reduce #'append (mapcar #'expr-to-lists (cdr exp))))
(t (list (list exp)))))
(defun expr-power (n expr)
`(* ,@(mapcar (constantly expr) (loop for i below n collect nil))))
Some extensive tests..
(assert-equal (expr-to-lists '(* (a) (e))) '(((a) (e))))
(assert-equal (expr-to-lists '(* a b c (* d e (* e f e))))
'((a b c d e e f e)))
(assert-equal (expr-to-lists '(+ (+ (a) (e))
(b)
(c)))
'(((a)) ((e)) ((b)) ((c))))
(assert-equal (expr-to-lists '(+ (+ a e) (+ b c)))
'((a) (e) (b) (c)))
(assert-equal (expr-to-lists '(* a (+ b c) (+ d (* e l))))
'((A B D) (A B E L) (A C D) (A C E L)))
(assert-equal (expr-to-lists '(* (+ f v)
(+ 1 t1 t2 (* q1 q1) (* k1 k2) (* f2 f2))
(+ r1 r2)
|0>|))
'((F 1 R1 0>)
(F 1 R2 0>)
(F T1 R1 0>)
(F T1 R2 0>)
(F T2 R1 0>)
(F T2 R2 0>)
(F Q1 Q1 R1 0>)
(F Q1 Q1 R2 0>)
(F K1 K2 R1 0>)
(F K1 K2 R2 0>)
(F F2 F2 R1 0>)
(F F2 F2 R2 0>)
(V 1 R1 0>)
(V 1 R2 0>)
(V T1 R1 0>)
(V T1 R2 0>)
(V T2 R1 0>)
(V T2 R2 0>)
(V Q1 Q1 R1 0>)
(V Q1 Q1 R2 0>)
(V K1 K2 R1 0>)
(V K1 K2 R2 0>)
(V F2 F2 R1 0>)
(V F2 F2 R2 0>)))
(assert-equal (expr-to-lists (expr-power 2 '(+ a b)))
'((A A) (A B) (B A) (B B)))
(assert-equal (expr-to-lists (expr-power 3 '(+ a b)))
'((A A A) (A A B) (A B A) (A B B)
(B A A) (B A B) (B B A) (B B B)))
(assert-equal (expr-to-lists (expr-power 4 '(+ a b)))
'((A A A A) (A A A B) (A A B A) (A A B B) (A B A A)
(A B A B) (A B B A) (A B B B) (B A A A) (B A A B) (B A B A)
(B A B B) (B B A A) (B B A B) (B B B A) (B B B B)))
(assert-equal
(expr-to-lists
'(* (+ (fab) (fij) (fai) (fia) (vpqrs) (v...))
(+ (1) (t1) (t2) (* (t1) (t1)) (* (t1) (t2)) (* (t2) (t2)))
(+ (r1) (r2))))
'(((FAB) (1) (R1)) ((FAB) (1) (R2)) ((FAB) (T1) (R1)) ((FAB) (T1) (R2))
((FAB) (T2) (R1)) ((FAB) (T2) (R2)) ((FAB) (T1) (T1) (R1))
((FAB) (T1) (T1) (R2))
((FAB) (T1) (T2) (R1)) ((FAB) (T1) (T2) (R2))
((FAB) (T2) (T2) (R1)) ((FAB) (T2) (T2) (R2))
((FIJ) (1) (R1))
((FIJ) (1) (R2))
((FIJ) (T1) (R1))
((FIJ) (T1) (R2))
((FIJ) (T2) (R1))
((FIJ) (T2) (R2))
((FIJ) (T1) (T1) (R1))
((FIJ) (T1) (T1) (R2))
((FIJ) (T1) (T2) (R1))
((FIJ) (T1) (T2) (R2))
((FIJ) (T2) (T2) (R1))
((FIJ) (T2) (T2) (R2))
((FAI) (1) (R1))
((FAI) (1) (R2))
((FAI) (T1) (R1))
((FAI) (T1) (R2))
((FAI) (T2) (R1))
((FAI) (T2) (R2))
((FAI) (T1) (T1) (R1))
((FAI) (T1) (T1) (R2))
((FAI) (T1) (T2) (R1))
((FAI) (T1) (T2) (R2))
((FAI) (T2) (T2) (R1))
((FAI) (T2) (T2) (R2))
((FIA) (1) (R1))
((FIA) (1) (R2))
((FIA) (T1) (R1))
((FIA) (T1) (R2))
((FIA) (T2) (R1))
((FIA) (T2) (R2))
((FIA) (T1) (T1) (R1))
((FIA) (T1) (T1) (R2))
((FIA) (T1) (T2) (R1))
((FIA) (T1) (T2) (R2))
((FIA) (T2) (T2) (R1))
((FIA) (T2) (T2) (R2))
((VPQRS) (1) (R1))
((VPQRS) (1) (R2))
((VPQRS) (T1) (R1))
((VPQRS) (T1) (R2))
((VPQRS) (T2) (R1))
((VPQRS) (T2) (R2))
((VPQRS) (T1) (T1) (R1))
((VPQRS) (T1) (T1) (R2))
((VPQRS) (T1) (T2) (R1))
((VPQRS) (T1) (T2) (R2))
((VPQRS) (T2) (T2) (R1))
((VPQRS) (T2) (T2) (R2))
((V...) (1) (R1))
((V...) (1) (R2))
((V...) (T1) (R1))
((V...) (T1) (R2))
((V...) (T2) (R1))
((V...) (T2) (R2))
((V...) (T1) (T1) (R1))
((V...) (T1) (T1) (R2))
((V...) (T1) (T2) (R1))
((V...) (T1) (T2) (R2))
((V...) (T2) (T2) (R1))
((V...) (T2) (T2) (R2))))
(assert-equal (expr-to-lists '(* (+ (T1 (P6 H6))) (+ (T1 (P5 H5)))))
'(((T1 (P6 H6)) (T1 (P5 H5)))))
(assert-equal
(expr-to-lists '(+ 1 (T1 (P6 H6))
(T2 (P3 H3) (P4 H4))
(* (+ (T1 (P6 H6))) (+ (T1 (P5 H5))))
(* (+ (T1 (P6 H6))) (+ (T2 (P3 H3) (P4 H4))))
(* (+ (T2 (P3 H3) (P4 H4))) (+ (T2 (P1 H1) (P2 H2))))))
'((1)
((T1 (P6 H6)))
((T2 (P3 H3) (P4 H4)))
((T1 (P6 H6)) (T1 (P5 H5)))
((T1 (P6 H6)) (T2 (P3 H3) (P4 H4)))
((T2 (P3 H3) (P4 H4)) (T2 (P1 H1) (P2 H2)))))
7. Index spaces
(defun match-index-to-space (index orbital-space) (find index (cdr orbital-space)))
(progn (assert (match-index-to-space 'k '(H i j k l)))
(assert (not (match-index-to-space 'H '(H i j k l)))))
(defun find-space-by-leg (index orbital-spaces) (find index orbital-spaces :test #'match-index-to-space))
(progn (assert (equal (find-space-by-leg 'k '((P a b c) (H i j k l)))
'(H I J K L)))
(assert (not (find-space-by-leg 'a '((H i j k l))))))
(defun find-space-by-name (name orbital-spaces) (find name orbital-spaces :key #'car)) (defun find-space-name-by-leg (leg orbital-spaces) (car (find leg orbital-spaces :test #'match-index-to-space)))
(assert-equal (find-space-by-name 'p '((PQ p q r s) (p a b c))) '(p a b c))
(let ((spaces '((H k l i) (P a b c) (PQ p q r s)))
(vals '((i . h)
(p . pq)
(q . pq)
(b . p))))
(loop for (v . result) in vals
do (assert (eq (find-space-name-by-leg v spaces) result))))
TODO: Tests
(defun traverse-nodes (fn tensor)
(destructuring-bind (name . nodes) tensor
`(,name ,@(mapcar fn nodes))))
(defun traverse-legs (fn tensor)
(traverse-nodes (lambda (node) (mapcar fn node)) tensor))
(defun tensor-to-description (tensor &key orbital-spaces)
(traverse-legs (lambda (leg) (find-space-name-by-leg leg orbital-spaces))
tensor))
(assert-equal (tensor-to-description '(V (i k) (l a))
:orbital-spaces
'((H i j k l) (P a b c d)))
'(V (H H) (H P)))
7.1. Tensor sum
(defun tensor-sum (&rest expressions)
`(+ ,@(reduce (lambda (tsr rest)
(if (atom tsr)
(cons tsr rest)
(case (car tsr)
(+ (append (cdr tsr) rest))
(t (cons tsr rest)))))
expressions
:from-end t
:initial-value nil)))
(assert-equal (tensor-sum '(T (A b) (c d)))
'(+ (T (a b) (c d))))
(assert-equal (tensor-sum '(T (A b) (c d)) '(V (e i)))
'(+ (T (a b) (c d))
(V (e i))))
(assert-equal (tensor-sum '(* (t (a b) (c d)) (f (k l))) '(v (e i)))
'(+ (* (T (A B) (C D)) (F (K L)))
(V (E I))))
(assert-equal (tensor-sum '(+ a b c d (* e d)) '(h1 h2))
'(+ A B C D (* E D) (H1 H2)))
;; this one is very useful
(assert-equal (tensor-sum '(+ a b c d) '(+ e d) '(+ h1 h2))
'(+ A B C D E D H1 H2))
8. Tensor matching
(defun match-target-with-tensor-1 (target tensor &key orbital-spaces)
(unless (eq (length target) (length tensor))
(return-from match-target-with-tensor-1 nil))
(notany #'null
(loop for target-tensor in (mapcar #'list (cdr target) (cdr tensor))
collect
(let ((spaces (mapcar (lambda (i) (find i orbital-spaces :key #'car))
(car target-tensor))))
(assert (eq (length (car target-tensor)) (length (cadr target-tensor))))
(notany #'null (mapcar #'match-index-to-space
(cadr target-tensor)
spaces))))))
(assert (match-target-with-tensor-1 '(V (H P) (P))
'(t (i b) (a))
:orbital-spaces
'((H i)
(P b a))))
(assert (not (match-target-with-tensor-1 '(V (H P) (P))
'(t (i b) (c)) ;; here
:orbital-spaces
'((H i)
(P b a)))))
(assert (not (match-target-with-tensor-1 '(V (H P))
'(t (i b) (c)) ;; here
:orbital-spaces
'((H i)
(P b a)))))
(defun match-target-with-tensor (target tensor &key orbital-spaces)
"Here we check that Vaibj is equivalent to Viajb and so on always.
This is general to all tensors.
It works for any dimension thanks to permuting all the legs of
the tensor."
(let ((all-targets (mapcar (lambda (x) `(,(car target) ;; name
,@x)) ;; feet
(all-permutations (cdr target)))))
(loop for tt in all-targets
thereis (match-target-with-tensor-1
tt tensor
:orbital-spaces orbital-spaces))))
(progn
(assert (match-target-with-tensor '(V (H P) (P H))
'(t (a i) (j b))
:orbital-spaces
'((H i j)
(P b a))))
(assert (not (match-target-with-tensor '(V (H P) (P H))
'(t (i a) (j b))
:orbital-spaces
'((H i j)
(P b a))))))
9. Symmetries
This section discusses how to implement and encode symmetries of the diagrams.
9.1. Node symmetry
For instance, all operators in quantum chemistry have a node symmetry whereby exchanging the positions of the electrons the tensor remains unchanged. For instance, for the coulomb integrals \( V^{pq}_{rs} \) this is encoded in the relation
\begin{equation*} V^{pq}_{rs} = V^{qp}_{sr} \end{equation*}and from the second-quantization point of view this is also transposing a pair number of times the \(q\)-operators.
We can encode these properties in lisp by just saying by which replacements the tensors remain unchanged, for instance for
(V (p s) (q r))
we would write as symmetries
;; main two-body node symmetry ((p . q) (s . r))
and we can write a simple function to apply this symmetry to tensor nodes
(defun apply-symmetry-to-nodes (symmetry-equivalence object)
(let* ((temp-symbols (mapcar (lambda (x) (declare (ignorable x))
(gensym)) symmetry-equivalence))
(equiv-forward (mapcar (lambda (x y) (cons (cdr x) y))
symmetry-equivalence temp-symbols))
(equiv-backward (mapcar (lambda (x y) (cons y (car x)))
symmetry-equivalence temp-symbols)))
(sublis equiv-backward
(sublis symmetry-equivalence
(sublis equiv-forward object)))))
(defun apply-symmetries-to-nodes (symmetry-equivalences object)
(mapcar (lambda (x) (apply-symmetry-to-nodes x object)) symmetry-equivalences))
And in fact it is a very general function that works on every
tree thanks to sublis:
(assert-equal (apply-symmetry-to-nodes '((P . Q) (S . R))
'((P S) (Q R)))
;;
'((Q R) (P S)))
(assert-equal (apply-symmetry-to-nodes '((a . b) (i . j))
'(T (a i) (b j) (c k)))
;;
'(T (B J) (A I) (C K)))
(let ((contraction '((contraction (P2 P5) (H2 H3) (H1 H4) (P1 P3))
(V (h1 p1) (h2 p2))
(T (p3 h3) (p4 h4))
(T (p5 h5)))))
(destructuring-bind ((cts _ a b c) v tabij tai . nil) (apply-symmetry-to-nodes
'((p3 . p4) (h3 . h4))
contraction)
(assert-equal tabij '(T (p4 h4) (p3 h3)))
(assert-equal tai '(t (p5 h5)))
(assert-equal v '(V (h1 p1) (h2 p2)))
(assert-equal (list a b c) '((h2 h4) (h1 h3) (p1 p4)))))
(assert-equal (apply-symmetry-to-nodes '((p . q))
'(V (P s) (q r)))
;;
'(V (Q S) (P R)))
(assert-equal (apply-symmetries-to-nodes '(((p . q) (s . r)) ((p . s)) ((r . q)))
'(V (P s) (q r)))
'((V (Q R) (P S))
(V (S P) (Q R))
(V (P S) (R Q))))
Mostly however it is quite tedious to write these equivalences
by hand so we can use the make-node-symmetry function to
create a well-named symmetry equivalence.
(defun triangle-pairs (n)
(loop for fst below n
append (loop for snd from (1+ fst) below n
collect (list fst snd))))
(defun iota (n &key (start 0))
(declare (optimize (safety 0) (speed 3) (debug 0)) (fixnum n) (fixnum start))
(loop for i below n collect (+ i start)))
(defun all-transpositions (ls &key (test #'eq) )
(loop for perm in (all-permutations ls)
collect (remove-duplicates (loop for i in perm
for j in ls
if (not (funcall test i j))
collect (cons i j))
:test
(lambda (x y)
(and (eq (car x) (cdr y))
(eq (cdr x) (car y)))))))
(defun make-node-symmetry (nodes)
(flet ((idxs-to-syms (idxs)
(loop for idx in idxs
append (let ((node-a (nth (car idx) nodes))
(node-b (nth (cdr idx) nodes)))
(loop for a in node-a
for b in node-b
collect (cons a b))))))
(let* ((combinations (all-transpositions (iota (length nodes)))))
(remove-if #'null (mapcar #'idxs-to-syms combinations)))))
And it should of course work for higher dimensional tensors.
;; utility function
(assert-equal (triangle-pairs 1) nil)
(assert-equal (triangle-pairs 2) '((0 1)))
(assert-equal (triangle-pairs 3) '((0 1) (0 2) (1 2)))
;; fail gracefully for one dimensional diagrams
(assert! (make-node-symmetry '((p q))))
(assert-equal (make-node-symmetry '((p s) (q r)))
'(((P . Q) (S . R))))
(assert-equal (make-node-symmetry '((p0 h0) (p1 h1) (p2 h2)))
'(;; node 1 <> node 2
((P1 . P2) (H1 . H2))
;; node 1 <> 0 && 2 <> 1 && 0 <> 2
((P1 . P0) (H1 . H0) (P2 . P1) (H2 . H1) (P0 . P2) (H0 . H2))
;; node 0 <> node 1
((P0 . P1) (H0 . H1))
;; node 2 <> 0 && 0 <> 1 && 1 <> 2
((P2 . P0) (H2 . H0) (P0 . P1) (H0 . H1) (P1 . P2) (H1 . H2))
;; node 0 <> node 2
((P0 . P2) (H0 . H2))))
(let ((result '(((P2 . P3) (H2 . H3)) ;; 2 <> 3
((P2 . P1) (H2 . H1) (P3 . P2) (H3 . H2) (P1 . P3) (H1 . H3))
((P1 . P2) (H1 . H2)) ;; 1 <> 2
((P3 . P1) (H3 . H1) (P1 . P2) (H1 . H2) (P2 . P3) (H2 . H3))
((P1 . P3) (H1 . H3)) ;; 1 <> 3
((P1 . P0) (H1 . H0) (P2 . P1)
(H2 . H1) (P3 . P2) (H3 . H2)
(P0 . P3) (H0 . H3))
((P1 . P0) (H1 . H0) (P2 . P1)
(H2 . H1) (P0 . P2) (H0 . H2))
((P1 . P0) (H1 . H0) (P3 . P1)
(H3 . H1) (P0 . P2) (H0 . H2)
(P2 . P3) (H2 . H3))
((P1 . P0) (H1 . H0) (P3 . P1)
(H3 . H1) (P0 . P3) (H0 . H3))
((P0 . P1) (H0 . H1)) ;; 0 <> 1
((P0 . P1) (H0 . H1) (P2 . P3) (H2 . H3))
((P0 . P2) (H0 . H2) (P1 . P3) (H1 . H3))
((P2 . P0) (H2 . H0) (P3 . P1)
(H3 . H1) (P1 . P2) (H1 . H2)
(P0 . P3) (H0 . H3))
((P2 . P0) (H2 . H0) (P0 . P1)
(H0 . H1) (P1 . P2) (H1 . H2))
((P2 . P0) (H2 . H0) (P0 . P1)
(H0 . H1) (P3 . P2) (H3 . H2)
(P1 . P3) (H1 . H3))
((P2 . P0) (H2 . H0) (P3 . P2)
(H3 . H2) (P0 . P3) (H0 . H3))
((P0 . P2) (H0 . H2)) ;; 0 <> 2
((P3 . P0) (H3 . H0) (P0 . P1)
(H0 . H1) (P1 . P2) (H1 . H2)
(P2 . P3) (H2 . H3))
((P3 . P0) (H3 . H0) (P0 . P1)
(H0 . H1) (P1 . P3) (H1 . H3))
((P0 . P3) (H0 . H3)) ;; 0 <> 3
((P3 . P0) (H3 . H0) (P0 . P2)
(H0 . H2) (P2 . P3) (H2 . H3))
((P3 . P0) (H3 . H0) (P2 . P1)
(H2 . H1) (P0 . P2) (H0 . H2)
(P1 . P3) (H1 . H3))
((P1 . P2) (H1 . H2) (P0 . P3) (H0 . H3)))))
(assert-equal (make-node-symmetry '((p0 h0) (p1 h1) (p2 h2) (p3 h3)))
result)
(assert-equal (length result) (- (* 4 3 2) 1)))
In the case of a list of tensors we can define the following function
(defun find-effective-nodes-list (list-of-tensors)
(let* ((keys (remove-duplicates (mapcar #'car list-of-tensors) :test #'equal))
(result-alist (mapcar (lambda (k) (cons k '())) keys)))
(mapc (lambda (tsr)
(let ((current (assoc (car tsr) result-alist)))
(rplacd (assoc (car tsr) result-alist)
(append (cdr current) (cdr tsr)))))
list-of-tensors)
(mapcar #'cdr result-alist)))
(defun make-symmetries-in-node-list (list-of-tensors sym-maker)
(labels ((reducer (x) (reduce #'union x :from-end t)))
(thread-last list-of-tensors
(mapcar sym-maker)
(reducer))))
(defun make-symmetries-in-effective-node-list (list-of-tensors sym-maker)
(make-symmetries-in-node-list (find-effective-nodes-list list-of-tensors)
sym-maker))
(assert-equal (find-effective-nodes-list
'((V (p q) (r s)) (T2 (a b) (c d))))
'(((p q) (r s)) ((a b) (c d))))
(assert-equal (find-effective-nodes-list '((V (p q) (r s))
(T2 (a b) (c d)) (T2 (a2 b2) (c2 d2))
(R2 (g h) (h2 g2))))
'(((P Q) (R S))
((A B) (C D) (A2 B2) (C2 D2))
((G H) (H2 G2))))
(assert-equal (find-effective-nodes-list '((V (p r) (q s))
(T1 (a i)) (T1 (aa ii))
(R1 (g e))))
'(((p r) (q s))
((a i) (aa ii))
((g e))))
(let* ((tensors '((V (h1 p1) (h2 p2))
(T2 (p3 h3) (p4 h4))
(T1 (p5 h5))))
(symmetries (make-symmetries-in-effective-node-list
tensors #'make-node-symmetry)))
(assert-equal symmetries
'(((H1 . H2) (P1 . P2))
((P3 . P4) (H3 . H4)))))
9.2. TODO Antisymmetry
For fermionic diagrams where the tensors are represented by real antisymmetric objects, there is a further symmetry to be considered.
(defun unzip (ls)
(loop for i below (apply #'min (mapcar #'length ls))
collect (mapcar (lambda (l) (nth i l)) ls)))
(defun make-antisymmetry-symmetry (nodes)
(assert (every (lambda (n) (eq (length n) (length (car nodes)))) nodes)
nil
"Antisymmetry is not expected to work for tensors with~%~4t~a~%~a"
nodes "an unequal number of legs per node")
(let* ((legs-list (unzip nodes))
(nlegs (length (car legs-list)))
(tpairs (triangle-pairs nlegs))
(single-symmetries
(mapcar (lambda (legs)
(mapcar (lambda (pair) (cons (nth (car pair) legs)
(nth (cadr pair) legs)))
tpairs))
legs-list)))
;;
;; TODO: think about including also the products or not
;;
;; (append (mapcar #'list (apply #'append single-symmetries))
;; (eval `(cartesian-product ,@single-symmetries)))
(mapcar #'list (apply #'append single-symmetries))))
(defun make-parity-symmetry--1 (nodes)
(let* ((legs-list (unzip nodes))
(iota (iota (length (car legs-list))))
(legs-transpositions (mapcar #'all-transpositions legs-list)))
(assert (eq (length legs-list) 2))
(mapcar (lambda (x) (reduce #'append x))
(eval `(cartesian-product ,@legs-transpositions)))))
(defun make-nodes-difference (nodes-a nodes-b)
(let (result)
(loop for node-a in nodes-a
for node-b in nodes-b
do (loop for a in node-a
for b in node-b
unless (eq a b)
unless (or (assoc b result)
(assoc a result))
do (push (cons a b) result)))
(reverse result)))
#+nil
(assert-equal (make-nodes-difference '((a b) (e f)) '((a c) (e h)))
'((b . c) (f . h)))
(progn
;; todo: improve this, it is too expensive
(defun make-parity-symmetry (nodes)
(let* ((node-symmetries (make-node-symmetry nodes))
;; apply node symmetries to the nodes
(new-nodes (cons nodes
(mapcar (lambda (sym)
(apply-symmetry-to-nodes sym nodes))
node-symmetries)))
;; for every node let us take all the parity symmetries
(parity-symmetries (mapcar #'make-parity-symmetry--1 new-nodes))
(nodes-with-parity
(mapcar (lambda (syms -nnodes)
(mapcar (lambda (sym)
(apply-symmetry-to-nodes sym -nnodes))
syms))
parity-symmetries new-nodes))
(node-differences (mapcar (lambda (-nodes)
(make-nodes-difference nodes -nodes))
(apply #'concatenate
(cons 'list nodes-with-parity)))))
(remove-if #'null node-differences)))
(make-parity-symmetry '((a i) (b j) (c k)))
(make-parity-symmetry '((a i) (b j))))
(apply 'concatenate '(list (a b c) (d e f)))
;(apply-symmetry-to-nodes '((a . b)) '(T2 a))
(apply-symmetry-to-nodes nil '(T2 a))
(make-parity-symmetry--1 '((a i) (b j)))
(make-parity-symmetry--1 '((a i)))
(make-parity-symmetry--1 '((a i) (b j) (c k)))
;; utility function
(assert-equal (unzip '((a b) (c d)))
'((a c) (b d)))
(assert-equal (unzip '((a b) (c d) (e f)))
'((a c e) (b d f)))
(assert-equal (make-antisymmetry-symmetry '((a i) (b j)))
'(((A . B)) ((I . J))))
(assert-equal (make-antisymmetry-symmetry '((a i) (b j) (c k)))
'(((A . B)) ((A . C)) ((B . C))
((I . J)) ((I . K)) ((J . K))))
(arponen::assert-equal
(arponen::make-symmetries-in-node-list '(((p2 h2) (h3 p3)) ((p1 h1) (p4 h4)))
#'arponen::make-antisymmetry-symmetry)
'(((H2 . P3)) ((P2 . H3)) ((P1 . P4)) ((H1 . H4))))
9.3. Filtering diagrams through contractions
Now the question in everyones minds is, wether or not we can restrict ourselves only to the contractions to apply the symmetries.
The following is a text depiction of a diagram where we have numbered the nodes from 1 to 5.
hh
V
pp
1 2
x--------------------x
--- ---
/ \ / \
/ \ / \
| | | |
. . ^ v ^ v
. . | | | |
v ^ \ / \ /
. . \ / \ /
. --- ---
o==============o ===o===
3 4 5
T T
2 1
In this case, the contractions will be
((h1 h4) (p1 p4) (h2 h5) (p2 p5))
The equivalent diagram linking through bubbles nodes \( (4, 2) \) and \( (1, 5) \) would be writted as
((h1 h5) (p1 p5) (h2 h4) (p2 p4))
If we are to apply the node symmetry of \( V \) to this contraction set we will get the original contraction as depicted in the diagram and thus it is enough to apply the symmetries to the contractions.
Indeed, the contractions are the differntiating element that distinguish diagrams, it is therefore understandable that through them we can also identify equal diagrams.
9.4. Filtering contractions through symmetries
Given a set of diagrams, we should decide which ones are equivalent through a set of symmetries and which ones arent.
The obvious way of checking the diagrams is through looping
through a set of diagrams and a set of symmetries
and check wether or not they are the same in terms of contractions
and in the sense of sets through such a function like
find-duplicate-set.
(defun find-duplicate-set (element lst)
(find element lst :test-not (lambda (-x -y)
(set-difference -x -y :test #'equal))))
(defun pair-set-difference (pa pb)
(check-type pa cons)
(check-type pb cons)
(flet ((c-to-list (c) (list (car c) (cdr c))))
(set-difference (c-to-list pa) (c-to-list pb))))
(defun find-duplicate-pair-set (element lst)
(find element lst :test-not
(lambda (x y)
(set-difference x y
:test-not #'pair-set-difference))))
(assert-equal (find-duplicate-set '#1=((a . b) (c . d))
'(((c . e) (a . b))
((c . d) (a . b))
#1#))
'((c . d) (a . b)))
;; it should also accept transposed variations of the
;; elements
(assert-equal (find-duplicate-pair-set '((a . b) (d . c))
'(((c . e) (a . b))
((c . d) (a . b))))
'((c . d) (a . b)))
(assert-equal (find-duplicate-pair-set '((a . b) (d . c))
'(((c . e) (a . b))
((c . d) (b . a))))
'((c . d) (b . a)))
However, this begs the question of given a set of symmetries as discussed so far, wether it is necessary to compute the minimal group containing them in order to discover equivalent diagrams.
The symmetry group of a diagram is exactly the product group of the individual symmetry groups of every piece of the diagram. Which means that in general we should have to compute the product group of the symmetry components. However, since in general we will look for repeated diagrams in a set of contractions that is already been created by computing all combinations of contractions, simply computing the direct sum of the symmetry sets will be enough for these cases.
In conclusion, the suitable function for filtering a set of contractions through a set of symmetries (which might be a group or not) is:
(defun filter-contractions-by-symmetries (symmetries contractions)
(let ((-contractions (copy-tree contractions)))
(do (result seen-contractions)
((null -contractions) result)
(let ((c (pop -contractions)))
(block :sym-searching
;; go through all symmetries
(loop for sym in (cons nil symmetries)
do (let ((new-c (apply-symmetry-to-nodes sym c)))
(when (find-duplicate-pair-set new-c seen-contractions)
(push new-c seen-contractions)
(logger "~&~a is the same as ~a by virtue of ~a"
c new-c sym)
(return-from :sym-searching))))
;; if I got here, then c is a new contraction
;; never seen before
(push c result)
(push c seen-contractions))))))
10. Contractions
Contraction rules should be something that tells us which contractions are not zero. For instance having
(v (j b)) (t (a i))
here we can see that
a bcan contract:(P 1 0)(i.e. first position and zeroth position)i jcan contract:(H 0 1)(i.e. zeroth position and first position)
A contraction is given by the format
((contraction ((a b))) (v (j b) (t (a i))))
and we can stich this contraction together to create a tensor
This is done by contraction-to-temp-tensor.
((contraction ((a b))) (v (j b) (t (a i)))) =>> (tv (j i)) which would match (_ (H H))
10.1. EXPLAIN Mergin nodes
In this section we work on the fact that when a contraction is made between legs, then these legs disappear from the resulting tensor object having in general two legs less, i.e., one node less.
TODO:: Think about why is not possible to contract '(a c) '(a b) '(c d)…
(defun stich-together (contraction node-a node-b)
;; contraction-assoc: ((c0 . x) (c1 . x))
(let ((contraction-assoc (mapcar (lambda (x) (cons x 'x)) contraction)))
(labels ((kill-matching (i) (sublis contraction-assoc i)))
(let* ((killed-a (kill-matching node-a))
(pos-a (position 'x killed-a))
(killed-b (kill-matching node-b))
(pos-b (position 'x killed-b)))
(when (or (equal killed-a node-a)
(equal killed-b node-b))
(error "The contraction ~a does not link nodes ~a and ~a"
contraction node-a node-b))
(if (eq pos-a pos-b) ;; NUCLEAR-TODO
(error "You are trying to contract ~a and ~a at the same position ~a"
node-a node-b pos-a)
(progn
(setf (nth pos-a node-a) (car (delete 'x killed-b)))
node-a))))))
(assert-equal (stich-together '(a d)
'(a b) '(c d))
'(c b))
(assert-equal (stich-together '(b c)
'(a b) '(c d))
'(a d))
(assert-errmsg (stich-together '(a c) '(a d) '(e f))
simple-error
"The contraction (A C) does not link nodes (A D) and (E F)")
(assert-errmsg (stich-together '(e c) '(a d) '(e f))
simple-error
"The contraction (E C) does not link nodes (A D) and (E F)")
(defun find-and-replace-matching-nodes (contraction tensor-nodes-list
&key killed-pair)
"tensor-nodes-list is a list of list of nodes"
(let* ((result (copy-tree tensor-nodes-list))
(all-nodes-flat (reduce #'append result)))
(loop for node in all-nodes-flat
do (case (length (intersection node contraction))
(0 (continue))
;; self-contraction
(2 (return (subst killed-pair node result :test #'equal)))
;; usual contraction
;; x--<>---
;; we should find exactly ONE OTHER PLACE where this
;; contraction is linked by the contraction
;; otherwise it is an error
(1 (let ((matching-nodes
(remove-if
(lambda (x) (or (equal x node)
(not (intersection x contraction))))
all-nodes-flat)))
(logger "~¤t: ~s matching: ~s through: ~s"
node matching-nodes contraction)
(case (length matching-nodes)
(0 (error "Unbound contractiong ~a with ~a"
node contraction))
(1 (let ((stiched (stich-together contraction
node
(car matching-nodes))))
(return (subst killed-pair
(car matching-nodes)
(subst stiched node result
:test #'equal)
:test #'equal))))
(t
(error "Contraction arity(~a) error ~a contracts with ~a"
(length matching-nodes) node matching-nodes)))
))))))
(macrolet ((assert-eq (index result)
`(assert (equal (find-and-replace-matching-nodes ,index
original
:killed-pair
'(x x))
,result))))
(let ((original '(((a b) (c d))
((e f) (g h))
((i j) (k l) (h1 h2)))))
;; 0-1 contraction
(assert-eq '(e h) '(((a b) (c d))
((g f) (x x))
((i j) (k l) (h1 h2))))
;; self contraction
(assert-eq '(k l) '(((a b) (c d))
((e f) (g h))
((i j) (x x) (h1 h2))))
;; 1-0 contraction
(assert-eq '(b k) '(((a l) (c d))
((e f) (g h))
((i j) (X X) (h1 h2))))
;; contraction with tripes
(assert-eq '(a h2) '(((h1 b) (c d))
((e f) (g h))
((i j) (k l) (x x))))
;; contraction within the tensor
(assert-eq '(a d) '(((c b) (X X))
((e f) (g h))
((i j) (k l) (h1 h2))))
;; todo: test error messages
))
This functions is a handy function to get from a contraction object
(defun get-contracted-nodes (contraction-tensor &key killed-pair)
;; todo replace with contraction-p
(assert (eq (caar contraction-tensor) 'contraction))
(let ((contracted-nodes (copy-list (mapcar #'cdr (cdr contraction-tensor))))
(contractions (cadar contraction-tensor)))
(loop for contraction in contractions
do
(setq contracted-nodes
(find-and-replace-matching-nodes contraction
contracted-nodes
:killed-pair killed-pair)))
contracted-nodes))
(assert-equal (get-contracted-nodes
'((contraction ((e d) (k j)))
(v (a b) (c d))
(h (e f) (g h))
(l (i j) (k l))) :killed-pair '(x x))
'(((A B) (C F))
((X X) (G H))
((I L) (X X))))
10.2. Effective temporary tensor
Given a contraction, we will want to know what kind of tensor it will result when the contraction gets applied.
(defun get-contracted-temp-tensor (contraction-tensor &key (name 'contracted))
(let* ((killed-pair '(x x))
(x-nodes (get-contracted-nodes contraction-tensor
:killed-pair killed-pair))
(flat-nodes (reduce (lambda (x y) (concatenate 'list x y))
x-nodes))
(cleaned-nodes (remove-if (lambda (x) (equal x killed-pair))
flat-nodes)))
`(,name ,@cleaned-nodes)))
(assert-equal (get-contracted-temp-tensor
'((contraction ((e d) (k j)))
(v (a b) (c d))
(h (e f) (g h))
(l (i j) (k l))))
'(contracted (A B) (C F) (G H) (I L)))
(assert-equal (get-contracted-temp-tensor
'((contraction ((b a) (j k)))
(V (J I) (A B))
(T (C K))
(R (G L))) :name '|v*t*r|)
'(|v*t*r| (C I) (G L)))
(assert-equal (get-contracted-temp-tensor
'((contraction nil)
(F (a i))) :name '|Fai|)
'(|Fai| (A I)))
10.3. Contraction discovery
10.3.1. Compatible contractions
This routing finds the possible contractions between two nodes. One could think that one should create all combinations of legs that belong to the node and then check according to the contraction rules. In fact, one just has to loop over the contraction rules and match every time against the two nodes since the position of the legs are encoded in the description of the contraction rules.
(defun compatible-contractions (node-a node-b &key
orbital-spaces
contraction-rules)
(declare (cons node-a) (cons node-b))
(assert (and (eq (length node-a) 2) (eq (length node-a) (length node-b))))
(remove-if
#'null
(mapcar (lambda (rule)
(destructuring-bind ((space-a space-b) pos-a pos-b) rule
(let ((a (nth pos-a node-a))
(b (nth pos-b node-b)))
(when (and (eq (find-space-name-by-leg a orbital-spaces)
space-a)
(eq (find-space-name-by-leg b orbital-spaces)
space-b))
(list a b)))))
contraction-rules)))
;; test
(let ((spaces '((H I J K L)
(P A B C D)
(G G))))
(let ((rules '(((H H) 0 1)
((P P) 1 0)))
(values '(((j i) (i a) . nil)
((j i) (i k) . ((j k)))
((a b) (c k) . ((b c)))
((i a) (g l) . ((i l)))
((i j) (k l) . ((i l)))
((i a) (b j) . ((i j) (a b)))
((a i) (j b) . nil)
((a b) (c d) . ((b c))))))
(loop for (a b . result) in values
do (assert (equal (compatible-contractions a b
:orbital-spaces spaces
:contraction-rules rules)
result))))
(let ((spaces '((H I J K L)
(P A B C D)
(G G)))
;; test with some absurd contraction rules
(rules '(((H H) 0 1)
((H P) 1 1)
((P H) 0 1)
((P G) 0 0)
((P P) 1 0)))
(values '(((j i) (i a) . ((i a)))
((j i) (i k) . ((j k)))
((a b) (c k) . ((a k) (b c)))
((a i) (g l) . ((a l) (a g)))
((i j) (k l) . ((i l)))
((i a) (b j) . ((i j) (a b))))))
(loop for (a b . result) in values
do (assert (equal (compatible-contractions a b
:orbital-spaces spaces
:contraction-rules rules)
result)))))
10.3.2. Checking for connectedness
To calculate if a diagram is connected, it is not enough to check if the contractions touch all diagrams, but we have to check that we can go to any diagram through a contraction path.
Therefore, we can simply
(defun is-connected-contraction (pair-combination node-pairs &key group-lengths)
(let* ((psums (mapcar (lambda (ls) (apply #'+ ls))
(maplist #'identity (reverse group-lengths))))
;; an interval represents a diagram
(intervals (mapcar #'cons psums (append (cdr psums) '(0))))
(diagrams-names (mapcar (lambda (i) (cons i (gensym "DIAGRAM-")))
intervals))
(node-indices (mapcar (lambda (pair-index) (nth pair-index node-pairs))
pair-combination)))
;; TODO: optimize this...
(labels ((diagram-of (i)
(cdr (assoc (find-if (lambda (interval)
(and (> (car interval) i)
(>= i (cdr interval))))
intervals)
diagrams-names))))
(block :main-routine
(loop
for node-permutation in (all-permutations node-indices)
do (let (path)
(block :current-permutation
(tagbody
(loop for node in node-permutation
do (let ((diagrams (mapcar #'diagram-of node)))
(if (equal (intersection diagrams path)
diagrams)
(return-from :current-permutation)
(progn
(setq path
(append
path
(set-difference diagrams path)))
(when (>= (length path)
(length group-lengths))
(return-from :main-routine t))))))
))
))))
))
(macrolet ((! (&rest pts)
`(mapcar (lambda (p)
(position p node-pairs :test #'equal)) ',pts)))
(let ((node-pairs
'((0 1) (0 2) (0 3) (0 4) (0 5) (0 6) (0 7) (0 8) ;; | 1st -> all
(1 4) (1 5) (1 6) (1 7) (1 8) ;; | 2nd diagram -> 3
(2 4) (2 5) (2 6) (2 7) (2 8) ;; |
(3 4) (3 5) (3 6) (3 7) (3 8)))) ;; |
;; this contraction only goes from the first diagram to the second
(assert! (is-connected-contraction (! (0 1) (0 2) (0 3))
node-pairs :group-lengths '(1 3 5)))
;; this contraction only goes from the 2nd diagram to the 3rc
(assert! (is-connected-contraction (! (1 4) (1 6) (3 4) (3 7) (2 6))
node-pairs :group-lengths '(1 3 5)))
;; this is quick, it just goes to from 1 to 2 and to 3 directly
(assert (is-connected-contraction (! (0 1) (2 5))
node-pairs :group-lengths '(1 3 5)))
;; this is less quick, it goes from 1 to 2 twice and then goes to 3
(assert (is-connected-contraction (! (0 1) (0 2) (2 5))
node-pairs :group-lengths '(1 3 5)))))
10.3.3. Finding contractions by number of legs
In this routine magic happens. So we have a target tensor with Nt operators and some product of tensors with Ni operators each. The number of contractions should be Nc, so filters for the number of contractions are
Nc = (Σi Ni) - Nt
If we need Nc contractions, we can get up to Nc pairs of indices, where every index has a single contraction. Therefore we need all ORDERED subsets of length up to Nc
Here we apply the norm simply Find contractions in a product. Some filters used are the number of contractions
2 * N-c = Sum (i) legs(product) - legs(target)
Some contractiosn might be combinatorially very expensive to go through but a quick check can solve the issue. Every contraction reduces the number of legs by two. If we count the number of legs in the target and we loop over contractions we can see if at all there can be contractions appearing, this can serve as a quick check for some terms that might be very expensive and where it is clear that the combinatorial search will have a negative result.
#+nil
(defun is-a-contraction-possible-by-number-of-legs
(target tensor-list &key
orbital-spaces
contraction-rules)
(let* ((N-c (/ (- (length (flatten-list (mapcar #'cdr tensor-list)))
(length (flatten-list (cdr target))))
2))
(all-nodes (copy-tree (reduce #'append (mapcar #'cdr tensor-list))))
(group-lengths (mapcar (lambda (tsr) (length (cdr tsr))) tensor-list))
;; '((1 1) (1 2) (2 2)) if length all-nodes = 2
(node-pairs (get-node-pairs (length all-nodes)
:group-lengths
(unless *allow-self-contractions*
group-lengths)))
(node-pair-combinations
(eval `(ordered-subsets-with-repetition ,N-c
,(length node-pairs))))
results)
))
(defun find-contractions-in-product-by-number-of-legs
(target tensor-list &key
orbital-spaces
contraction-rules)
(let* ((N-c (/ (- (length (flatten-list (mapcar #'cdr tensor-list)))
(length (flatten-list (cdr target))))
2))
(all-nodes (reduce #'append (mapcar #'cdr tensor-list)))
(group-lengths (mapcar (lambda (tsr) (length (cdr tsr))) tensor-list))
;; '((1 1) (1 2) (2 2)) if length all-nodes = 2
(node-pairs (get-node-pairs (length all-nodes)
:group-lengths
(unless *allow-self-contractions*
group-lengths)))
(node-pair-combinations
(eval `(ordered-subsets-with-repetition ,N-c
,(length node-pairs))))
results)
(logger "~&============")
(logger "~&N-contractions: ~s" N-c)
(logger "~&all nodes: ~s" all-nodes)
(logger "~&all node-pairs: ~s" node-pairs)
(logger "~&all combinations (of pairs) : ~s" node-pair-combinations)
(setq results
(flet
((indexing (indices lst) (mapcar (lambda (i) (nth i lst))
indices)))
(loop
for node-pair-combination in node-pair-combinations
nconcing
(block :pairs-discovery
(tagbody
(let* ((pairs (indexing node-pair-combination node-pairs))
(nodes (mapcar (lambda (x)
(indexing x all-nodes)) pairs))
(II 0)
top-contractions)
(logger "~&combination: ~s pairs: ~s [~s]"
node-pair-combination
pairs nodes)
(incf II)
(when *only-connected-diagrams*
(unless (is-connected-contraction node-pair-combination
node-pairs
:group-lengths
group-lengths)
(return-from :pairs-discovery)))
(loop for pair in pairs
collect
(let* ((vertices (indexing pair all-nodes))
(conts (compatible-contractions
(car vertices)
(cadr vertices)
:orbital-spaces orbital-spaces
:contraction-rules contraction-rules)))
(cond
((null conts) (return-from :pairs-discovery))
((equal conts
(intersection top-contractions conts
:test #'equal))
(logger "~&~30t⇐Exiting since ~a fully in ~a"
conts top-contractions)
(return-from :pairs-discovery))
(t
(logger "~&~8tvertices: ~s" vertices)
(logger "~&~24t appending contractions ~s" conts)
(push conts top-contractions)))))
;; START FILTERING
(return-from :pairs-discovery
(let (--result)
(mapc (lambda (real-contraction)
;; photons say: repeated letters must go!
(let ((letters (flatten-list real-contraction)))
(unless (symbols-repeated-p letters)
(pushnew real-contraction
--result
:test-not
(lambda (x y) (set-difference
x y
:test #'equal))))))
(eval `(cartesian-product
,@top-contractions)))
--result))
))))))
(let ((cleaned-results (remove-if #'null results))
symmetries)
(format t "~&~5tCompute symmetries")
(when *filter-node-symmetry*
(format t "~&~10tnode symms")
(setq symmetries
(append symmetries (make-symmetries-in-effective-node-list
tensor-list #'make-node-symmetry))))
(when *filter-parity-symmetry*
(format t "~&~10tparity symms")
(setq symmetries
(append symmetries (make-symmetries-in-node-list
(mapcar #'cdr tensor-list)
#'make-parity-symmetry))))
(format t "~&~5tResults BEFORE cleaning ~a" (length cleaned-results))
(setq cleaned-results
(filter-contractions-by-symmetries symmetries
cleaned-results))
(format t "~&~5tResults AFTER Cleaning ~a" (length cleaned-results))
cleaned-results)))
- Case study: Vijab with T1 and T2 coupling to singles excitations
Let us illustrage this function with some tests
(let ((orbital-spaces '((H i j k l m n o h1 h2 h3 h4 h5) (P a b c d e f g p1 p2 p3 p4 p5))) (contraction-rules '(((H H) 0 1) ((P P) 1 0)))) (labels ((with-rules (target tensor) (find-contractions-in-product-by-number-of-legs target tensor :orbital-spaces orbital-spaces :contraction-rules contraction-rules)) (with-rules-c (target tensor) (let ((*only-connected-diagrams* t) (*allow-self-contractions* nil)) (with-rules target tensor)))) (let ((*filter-node-symmetry* t)) (with-rules-c '(_ (P H) (P H)) '((V (h1 p1) (h2 p2)) (T2 (p3 h3) (p4 h4)) (T1 (p5 h5))))) )) #+(or) '((1 (P2 P5) (H2 H4) (H1 H3) (P1 P3)) (2 (H2 H5) (P2 P4) (H1 H3) (P1 P3)) (3 (H2 H5) (P2 P5) (H1 H3) (P1 P3)) (4 (P2 P5) (H2 H3) (H1 H4) (P1 P3)) (5 (H2 H5) (P2 P3) (P1 P4) (H1 H3)) (6 (P2 P5) (H2 H4) (P1 P4) (H1 H3)) (7 (H2 H5) (P2 P4) (H1 H4) (P1 P3)) (8 (H2 H5) (P2 P5) (P1 P4) (H1 H3)) (9 (H2 H5) (P2 P5) (H1 H4) (P1 P3)) (10 (P2 P4) (H2 H3) (H1 H5) (P1 P3)) (11 (H2 H4) (P2 P3) (P1 P5) (H1 H3)) (12 (P2 P5) (H2 H3) (H1 H5) (P1 P3)) (13 (H2 H5) (P2 P3) (P1 P5) (H1 H3)) (14 (H2 H4) (P2 P4) (P1 P5) (H1 H3)) (15 (H2 H4) (P2 P4) (H1 H5) (P1 P3)) (16 (P2 P5) (H2 H4) (H1 H5) (P1 P3)) (17 (H2 H5) (P2 P4) (P1 P5) (H1 H3)) (18 (P2 P5) (H2 H3) (H1 H4) (P1 P4)) (19 (H2 H5) (P2 P3) (H1 H4) (P1 P4)) (20 (H2 H5) (P2 P5) (H1 H4) (P1 P4)) (21 (H2 H3) (P2 P3) (P1 P5) (H1 H4)) (22 (H2 H3) (P2 P3) (H1 H5) (P1 P4)) (23 (P2 P4) (H2 H3) (P1 P5) (H1 H4)) (24 (H2 H4) (P2 P3) (H1 H5) (P1 P4)) (25 (P2 P5) (H2 H3) (H1 H5) (P1 P4)) (26 (H2 H5) (P2 P3) (P1 P5) (H1 H4)) (27 (P2 P5) (H2 H4) (H1 H5) (P1 P4)) (28 (H2 H5) (P2 P4) (P1 P5) (H1 H4)) (29 (H2 H3) (P2 P3) (H1 H5) (P1 P5)) (30 (P2 P4) (H2 H3) (H1 H5) (P1 P5)) (31 (H2 H4) (P2 P3) (H1 H5) (P1 P5)) (32 (H2 H4) (P2 P4) (H1 H5) (P1 P5))) (defun count-duplicates (lst) (mapcar (lambda (x) (count x lst :test-not (lambda (-x -y) (set-difference -x -y :test #'equal)))) lst)) (let* ((tensors '((V (h1 p1) (h2 p2)) (T2 (p3 h3) (p4 h4)) (T1 (p5 h5)))) (symmetries (make-symmetries-in-effective-node-list tensors #'make-node-symmetry)) (contractions '(((P2 P5) (H2 H4) (H1 H3) (P1 P3)) ((H2 H5) (P2 P4) (H1 H3) (P1 P3)) ((H2 H5) (P2 P5) (H1 H3) (P1 P3)) ((P2 P5) (H2 H3) (H1 H4) (P1 P3)) ((H2 H5) (P2 P3) (P1 P4) (H1 H3)) ((P2 P5) (H2 H4) (P1 P4) (H1 H3)) ((H2 H5) (P2 P4) (H1 H4) (P1 P3)) ((H2 H5) (P2 P5) (P1 P4) (H1 H3)) ((H2 H5) (P2 P5) (H1 H4) (P1 P3)) ((P2 P4) (H2 H3) (H1 H5) (P1 P3)) ((H2 H4) (P2 P3) (P1 P5) (H1 H3)) ((P2 P5) (H2 H3) (H1 H5) (P1 P3)) ((H2 H5) (P2 P3) (P1 P5) (H1 H3)) ((H2 H4) (P2 P4) (P1 P5) (H1 H3)) ((H2 H4) (P2 P4) (H1 H5) (P1 P3)) ((P2 P5) (H2 H4) (H1 H5) (P1 P3)) ((H2 H5) (P2 P4) (P1 P5) (H1 H3)) ((P2 P5) (H2 H3) (H1 H4) (P1 P4)) ((H2 H5) (P2 P3) (H1 H4) (P1 P4)) ((H2 H5) (P2 P5) (H1 H4) (P1 P4)) ((H2 H3) (P2 P3) (P1 P5) (H1 H4)) ((H2 H3) (P2 P3) (H1 H5) (P1 P4)) ((P2 P4) (H2 H3) (P1 P5) (H1 H4)) ((H2 H4) (P2 P3) (H1 H5) (P1 P4)) ((P2 P5) (H2 H3) (H1 H5) (P1 P4)) ((H2 H5) (P2 P3) (P1 P5) (H1 H4)) ((P2 P5) (H2 H4) (H1 H5) (P1 P4)) ((H2 H5) (P2 P4) (P1 P5) (H1 H4)) ((H2 H3) (P2 P3) (H1 H5) (P1 P5)) ((P2 P4) (H2 H3) (H1 H5) (P1 P5)) ((H2 H4) (P2 P3) (H1 H5) (P1 P5)) ((H2 H4) (P2 P4) (H1 H5) (P1 P5))))) ;; there are no duplicates (assert (every (lambda (x) (eq x 1)) (count-duplicates contractions))) (let ((sym-conts (filter-contractions-by-symmetries symmetries contractions))) (assert-equal sym-conts '(#| 32 |# ((H2 H4) (P2 P4) (H1 H5) (P1 P5)) #| 31 |# ((H2 H4) (P2 P3) (H1 H5) (P1 P5)) #| 28 |# ((H2 H5) (P2 P4) (P1 P5) (H1 H4)) #| 24 |# ((H2 H4) (P2 P3) (H1 H5) (P1 P4)) #| 23 |# ((P2 P4) (H2 H3) (P1 P5) (H1 H4)) #| 17 |# ((H2 H5) (P2 P4) (P1 P5) (H1 H3)) #| 16 |# ((P2 P5) (H2 H4) (H1 H5) (P1 P3)) #| 15 |# ((H2 H4) (P2 P4) (H1 H5) (P1 P3)) #| 14 |# ((H2 H4) (P2 P4) (P1 P5) (H1 H3)) #| 12 |# ((P2 P5) (H2 H3) (H1 H5) (P1 P3)) #| 8 |# ((H2 H5) (P2 P5) (P1 P4) (H1 H3)) #| 5 |# ((H2 H5) (P2 P3) (P1 P4) (H1 H3)) #| 4 |# ((P2 P5) (H2 H3) (H1 H4) (P1 P3)) #| 3 |# ((H2 H5) (P2 P5) (H1 H3) (P1 P3)) #| 2 |# ((H2 H5) (P2 P4) (H1 H3) (P1 P3)) #| 1 |# ((P2 P5) (H2 H4) (H1 H3) (P1 P3))))))
10.3.4. Finding contractions by target properties
(defun find-contractions-in-product-by-target
(target tensor-list &key
orbital-spaces
contraction-rules)
(let ((result (find-contractions-in-product-by-number-of-legs
target tensor-list :orbital-spaces orbital-spaces
:contraction-rules contraction-rules)))
(logger "~&CONTRACTIONS TO CHECK: ~a" result)
(remove-if (lambda (x) (eq x :no-match))
(loop for contraction in (cons nil result)
collect
(let* ((contraction-tensor `((contraction ,contraction)
,@(copy-list tensor-list)))
(contracted-tensor (get-contracted-temp-tensor
contraction-tensor)))
(logger "~&temp-tensor... ~a" contracted-tensor)
(if (match-target-with-tensor target
contracted-tensor
:orbital-spaces orbital-spaces)
contraction
:no-match))))))
(let ((*filter-node-symmetry* nil)
(orbital-spaces '((H I J K L h1 h2 h3)
(P A B C D p1 p2 p3)
(G g)))
(contraction-rules '(((H H) 0 1)
((P P) 1 0)))
(|_ H P H| '(_ (G H) (P H)))
(|P H P H| '(_ (P H) (P H)))
(|Vhhpp * Tpphh * Tpphh| '((V (i a) (j b))
(T (c k) (d l))
(T (p1 h1) (p2 h2))))
(|Vhphp * Thp * Rh| '((V (J I) (A B))
(T (C K))
(R (G L)))))
(macrolet ((assert-with-env (fun-applied value)
`(assert
(equal
,(concatenate 'list fun-applied '(:orbital-spaces
orbital-spaces
:contraction-rules
contraction-rules))
,value))))
;; with self-contractions
(let ((*allow-self-contractions* t))
(assert-with-env
(find-contractions-in-product-by-target |_ H P H| |Vhphp * Thp * Rh|)
'(((B A) (J I))
((B C) (J I))
((B A) (J K))
((B C) (J K))
((B A) (J L))
((B C) (J L))))
(assert-with-env
(find-contractions-in-product-by-target '(_ (P H))
'((f (a b)) (t (c i))))
'(((B A)) ((B C))))
(assert-with-env
(find-contractions-in-product-by-target '(_ (G H))
'((f (a b)) (t (c i))))
'())
(assert-with-env
(find-contractions-in-product-by-target '(_ (H P))
'((f (a b)) (t (c i))))
'()))))
(defun contract-expressions-by-target
(target expression &key orbital-spaces contraction-rules)
(let ((products (expr-to-lists expression))
sums)
(setq sums
(loop
for product in products
appending
(progn (print product)
(let ((contractions
(find-contractions-in-product-by-target target product
:orbital-spaces
orbital-spaces
:contraction-rules
contraction-rules)))
(mapcar (lambda (x) `((contraction ,x) ,@product))
contractions)))))
`(+ ,@sums)))
(let ((*allow-self-contractions* t)
(*filter-node-symmetry* nil))
(assert-equal
(contract-expressions-by-target '(_ (P H))
'(* (+ (f (a b)) (f (i j)))
(t (c k)))
:orbital-spaces
'((H i j k)
(P a b c))
:contraction-rules
'(((H H) 0 1)
((P P) 1 0)))
'(+ ((CONTRACTION ((B A))) (F (A B)) (T (C K)))
((CONTRACTION ((B C))) (F (A B)) (T (C K)))
((CONTRACTION ((I J))) (F (I J)) (T (C K)))
((CONTRACTION ((I K))) (F (I J)) (T (C K))))))
11. Help routines
TOOD: Explain that all indices must be different and so on
(defun space-subseq (&key orbital-spaces from-index)
(mapcar (lambda (space)
(handler-case `(,(car space)
,@(subseq (cdr space) from-index))
(condition ()
(error (concatenate
'string
"Dear user: "
"When partitioning tensors, all spaces "
"should have a long enough length to cut "
"through the leg names using from-index. "
"~&In this case "
"the space ~s needs at least more "
"than ~s elements "
"BUT it currently has ~s ")
space from-index (length (cdr space))))))
orbital-spaces))
(assert-equal (space-subseq :orbital-spaces '((H 1 2 3 4) (P a b c) (G g g2))
:from-index 2)
'((H 3 4) (P c) (G)))
TODO:: Explain how one to do the naming of tensors so that everything works well
(defun name-legs-by-space-name (tensor-description &key orbital-spaces (from-index 0))
(let ((orbital-spaces-copy (copy-tree
(space-subseq :orbital-spaces orbital-spaces
:from-index from-index))))
`(,(car tensor-description)
,@(loop for index-description in (cdr tensor-description)
collect
(loop for space-name in index-description
collect
(let ((space (find-space-by-name space-name orbital-spaces-copy)))
(if (cdr space)
(pop (cdr space))
(error "Not enough leg names given for space ~a~%"
space))))))
))
(let ((vals '((0 . (t (h1 p1) (p2 h2)))
(1 . (t (h2 p2) (p3 h3)))
(2 . (t (h3 p3) (p4 h4))))))
(loop for (from-index . result) in vals
do (assert (equal
(name-legs-by-space-name
'(t (H P) (P H))
:orbital-spaces '((H h1 h2 h3 h4) (P p1 p2 p3 p4))
:from-index from-index)
result))))
11.1. Partitions
TODO: Explain the concept of partitioning and the format
(defun partition-tensor (tensor &key orbital-spaces partition (from-index 0))
(let ((name (car tensor))
(indices (cdr tensor))
(orbital-spaces-copy (copy-tree
(space-subseq :orbital-spaces orbital-spaces
:from-index from-index)))
new-indices-unexpanded)
(setq
new-indices-unexpanded
(mapcar
(lambda (index)
(mapcar
(lambda (leg)
(let* ((space (find-space-by-leg leg orbital-spaces))
(space-name (car space))
(partition (find space-name partition :key #'car)))
(if partition
;; we found a partition
(mapcar (lambda (-space-name)
(let* ((space (find-space-by-name
-space-name
orbital-spaces-copy)))
(if (cdr space) ;; available leg names
(pop (cdr space))
(error "Not enough leg names given for space ~a~%"
space))))
;; elements of the partition (e.g H P)
(cdr partition))
(list leg))))
index))
indices))
(let ((new-indices (eval `(cartesian-product
,@(mapcar (lambda (index-set)
(eval `(cartesian-product ,@index-set)))
new-indices-unexpanded)))))
`(+ ,@(mapcar (lambda (ids) `(,name ,@ids))
new-indices)))))
(let ((orbital-spaces '((PQ p q r s)
(H i j k l)
(P a b c d)))
(partition '((PQ H P))))
(partition-tensor '(f (p q))
:orbital-spaces orbital-spaces
:partition partition)
(partition-tensor '(V (p q) (r s))
:orbital-spaces orbital-spaces
:partition partition))
12. API
In this section we will present most functions that will be useful using the library from a user perspective in order to contract expressions.
From the discussion up to know, some points should be noted:
- the names of the tensors is meaningful and they should be equal
only in reference to symmetries.
In general, one should name the many-body components
of a tensor differently, i.e. for instance
(T2 (a i) (b j))and(T1 (a i)). - All the legs in an expression should be named differently.
12.1. Particle hole picture
In order to design a good user interface, we should make it as clear and as modular as possible for the user, and reading these lines one should be convinced that it is not too much effort to design your own interface if this one does not suit you.
Generally we will want to define tensors generally in the real vacuum description and partition the tensor in holes and particles.
For our purposes let us denote fix the index space:
(PQ pq1 pq2 pq3 ...) ;; real vacuum indices (P p1 p2 p3 ...) ;; particle indices (H h1 h2 h3 ...) ;; hole indices
Then we will want to turn
(Vpqrs (PQ1 PQ2) (PQ3 PQ4))
into all 16 combinations
(Vhhpp (h1 p1) (h2 p2)) (Vhphp (h1 h2) (p1 p2)) ;; ...
Actually we will want less combinations since by virtue of node symmetry we should consider
(Vhphp (h1 h2) (p1 p2)) ;; and (Vhphp (p1 p2) (h1 h2))
equivalent.
(defpackage :arponen/hole-particle-picture
(:use :cl :arponen)
(:nicknames :hp))
(in-package :arponen/hole-particle-picture)
(defvar *default-orbital-spaces*
'((H) ;; holes
(P) ;; particles
(G) ;; general (or rather ghosts)
(PH)) ;; particle-holes (real vacuum)
"Orbital space for the default particle-hole picture")
(defvar *orbital-spaces* (copy-tree *default-orbital-spaces*))
(defvar *default-orbital-spaces-counter*
'((H . 0)
(P . 0)
(G . 0)
(PH . 0))
"Current index for the orbital spaces")
(defvar *orbital-spaces-counter* (copy-tree *default-orbital-spaces-counter*))
(defvar *default-space-partition*
'((PH H P)))
(defvar *space-partition* (copy-tree *default-space-partition*))
(defvar *contraction-rules* '(((H H) 0 1)
((P P) 1 0))
"The conctractions that are not zero.")
(defun reset-spaces ()
(setq *orbital-spaces-counter* (copy-tree *default-orbital-spaces-counter*))
(setq *orbital-spaces* (copy-tree *default-orbital-spaces*))
(values *orbital-spaces* *orbital-spaces-counter*))
;; simple useful functions for knowing which kind of index we have
(defun hole-p (idx) (char= (char (symbol-name idx) 0) #\H))
(defun particle-p (idx) (char= (char (symbol-name idx) 0) #\P))
(defun genindex (space-name)
(let ((counter (find space-name *orbital-spaces-counter* :key #'car))
(space (position space-name *orbital-spaces* :key #'car)))
(unless (and counter space) (error "~&The name ~s is not one of ~s"
space-name
(mapcar #'car *orbital-spaces*)))
(let ((new-index
(intern (format nil "~a~a" space-name (incf (cdr counter))))))
(setf (nth space *orbital-spaces*)
(append (nth space *orbital-spaces*) (list new-index)))
new-index)))
(defun name-legs-by-space-name-1 (tensor-description)
(arponen::traverse-legs #'genindex tensor-description))
(defun do-partition-node-description (node &key partition)
(eval `(arponen::cartesian-product
,@(mapcar (lambda (leg)
(let ((p (find leg partition :key #'car)))
(if p (cdr p) (list leg))))
node))))
(defun partition-tensor-description (tensor-description &key partition)
(destructuring-bind (name . nodes) tensor-description
(let* ((p-node-lists (mapcar (lambda (n)
(do-partition-node-description n
:partition partition)) nodes))
(new-node-lists (eval `(arponen::cartesian-product ,@p-node-lists))))
(mapcar (lambda (nodes) `(,name ,@nodes)) new-node-lists))))
(defun remove-1-in-product-list (prod-list)
(mapcar (lambda (product)
(remove-if (lambda (el) (or (eq el 1) (equal el '(1))))
product))
prod-list))
(defun filter-tensors-by-symmetries (symmetries-list tensor-list)
(let (result)
(mapc (lambda (sym tsr)
(let ((new-tsrs (arponen::apply-symmetries-to-nodes sym tsr)))
(unless (intersection (cons tsr new-tsrs) result :test #'equal)
(push tsr result))))
symmetries-list tensor-list)
(reverse result)))
(defun filter-tensors-by-symmetries-and-description
(symmetries tensor-list &key orbital-spaces)
(mapcar #'cadr
(remove-duplicates
(mapcar #'list symmetries tensor-list)
:test
(lambda (x y)
(let* ((all-x (cons (cadr x)
(arponen::apply-symmetries-to-nodes (car x) (cadr x))))
(all-y (cons (cadr y)
(arponen::apply-symmetries-to-nodes (car y) (cadr y))))
(x-descr (mapcar (lambda (-x)
(arponen::tensor-to-description -x
:orbital-spaces
orbital-spaces))
all-x))
(y-descr (mapcar (lambda (-y)
(arponen::tensor-to-description -y
:orbital-spaces
orbital-spaces))
all-y)))
(intersection x-descr y-descr :test #'equal))))))
(defun partition-symmetrize-and-filter (tensor-description &key unrestricted)
(let* ((tensors (mapcar #'name-legs-by-space-name-1
(partition-tensor-description tensor-description
:partition
*space-partition*)))
(arponen::*filter-parity-symmetry* unrestricted)
symmetries)
(print tensors)
(when arponen::*filter-node-symmetry*
(setq symmetries
(mapcar (lambda (x) (arponen::make-node-symmetry (cdr x)))
tensors)))
(when arponen::*filter-parity-symmetry*
(setq symmetries
(append symmetries
(mapcar (lambda (x)
(arponen::make-parity-symmetry (cdr x)))
(remove-duplicates
(reduce #'append
(arponen::apply-symmetries-to-nodes symmetries
tensors))
:test #'equal)))))
(filter-tensors-by-symmetries-and-description symmetries
tensors
:orbital-spaces
*orbital-spaces*)))
(defmacro ! (name &rest nodes)
`'(,name ,@nodes))
(defmacro !! (name &rest nodes)
(let ((tensor-description `(,name ,@nodes)))
`'(+ ,@(partition-symmetrize-and-filter tensor-description))))
(defmacro .* (&rest args)
`(list '* ,@args))
(defmacro .+ (&rest args)
`(arponen::tensor-sum ,@args))
;; TODO: node-symmetry ein und auschalten
(defun contract (target expression &key (node-symmetry t) (only-connected nil)
(unrestricted nil))
(let* ((expanded (remove-1-in-product-list (arponen::expr-to-lists expression)))
(n (length expanded))
(arponen::*only-connected-diagrams* only-connected)
(arponen::*allow-self-contractions* nil)
(arponen::*filter-parity-symmetry* unrestricted)
(i 0))
(remove-if #'null
(mapcar (lambda (tensor-product)
(format t "~&[~a/~a] ~a" (incf i) n tensor-product)
(let ((begin (get-internal-run-time))
(contractions
(arponen::find-contractions-in-product-by-target
target
tensor-product
:orbital-spaces
*orbital-spaces*
:contraction-rules
*contraction-rules*)))
(format t "~2t in (~,1f seconds)"
(/ (- (get-internal-run-time) begin)
internal-time-units-per-second))
(when contractions
(list `(contractions ,contractions) tensor-product))))
expanded))))
(defun save-contractions (file-name contractions &key format)
(with-open-file (s file-name :if-exists :supersede :direction :output)
(case format
(t (format s "~s" contractions)))))
(in-package :hp)
(import 'arponen::assert-equal)
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
(progn
(reset-spaces)
(print *default-orbital-spaces*)
(dotimes (i 10)
(when (zerop (mod i 2)) (genindex 'P))
(genindex 'H))
(assert (eq (genindex 'H) 'H11))
(assert (eq (genindex 'H) 'H12))
(assert (eq (genindex 'P) 'P6))
(assert (eq (genindex 'G) 'G1))
(assert (eq (genindex 'ph) 'ph1))
(reset-spaces)
(assert (eq (genindex 'H) 'H1))
(reset-spaces))
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
(assert-equal (do-partition-node-description '(H P)
:partition '((PH H P)))
'((H P)))
(assert-equal (do-partition-node-description '(H P)
:partition '((P |a| A)))
'((H |a|) (H A)))
(assert-equal (do-partition-node-description '(PH P)
:partition '((PH H P)))
'((H P) (P P)))
(assert-equal (do-partition-node-description '(PH PH)
:partition '((PH H P)))
'((H H) (H P) (P H) (P P)))
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
(reset-spaces)
(assert-equal (name-legs-by-space-name-1 '(t2 (P H) (P H)))
'(t2 (p1 h1) (p2 h2)))
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
(assert-equal (partition-tensor-description '(T2 (HP HP))
:partition '((HP P H)))
'((T2 (P P)) (T2 (P H)) (T2 (H P)) (T2 (H H))))
(assert-equal (partition-tensor-description '(T2 (HP H))
:partition '((HP P H)))
'((T2 (P H)) (T2 (H H))))
(assert-equal (partition-tensor-description '(T2 (HP HP) (HP HP))
:partition '((HP H P)))
'((T2 (H H) (H H))
(T2 (H H) (H P))
(T2 (H H) (P H))
(T2 (H H) (P P))
(T2 (H P) (H H))
(T2 (H P) (H P))
(T2 (H P) (P H))
(T2 (H P) (P P))
(T2 (P H) (H H))
(T2 (P H) (H P))
(T2 (P H) (P H))
(T2 (P H) (P P))
(T2 (P P) (H H))
(T2 (P P) (H P))
(T2 (P P) (P H))
(T2 (P P) (P P))))
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
(filter-tensors-by-symmetries '((((P . H) (H . P)))
(((H . P) (P . H))))
'((V (P H) (H P))
(V (H P) (P H))))
(assert-equal (arponen::apply-symmetries-to-nodes '(((H1 . H3) (H2 . H4))
((H1 . H2))
((H3 . H4)))
'(V (H1 H2) (H3 H4)))
'((V (H3 H4) (H1 H2)) (V (H2 H1) (H3 H4)) (V (H1 H2) (H4 H3))))
(arponen::make-node-symmetry '((P H) (H P)))
(arponen::make-node-symmetry '((P H) (P P) (A B)))
(arponen::apply-symmetry-to-nodes '((P . P) (H . P))
'(V (P H) (P P)))
(arponen::apply-symmetry-to-nodes '((P . H) (H . P))
'(V (P H) (H P)))
'((V (H1 H2) (H3 H4)) ;; Vijkl
(V (H13 P5) (H14 H15)) ;; Vijak
(V (H16 P6) (H17 P7)) ;; Vijab
(V (P13 H21) (H22 H23)) ;; Vaijk
(V (P14 H24) (H25 P15)) ;; Vaijb
(V (P16 H26) (P17 H27)) ;; Vabij
(V (P21 P22) (H29 H30)) ;; Vaibj
(V (P23 P24) (H31 P25)) ;; Vaibc
(V (P26 P27) (P28 H32)) ;; Vabci
(V (P29 P30) (P31 P32))) ;; Vabcd
(progn (reset-spaces)
(partition-symmetrize-and-filter '(V (PH PH) (PH PH)) :unrestricted t))
(assert-equal (! V (a b) (c d))
'(V (a b) (c d)))
(progn
(reset-spaces)
(let* ((arponen::*print-log* nil)
(t2 (!! t2 (P H) (P H)))
(vaibj (!! V (P P) (H H)))
(vaijb (!! V (P H) (H P)))
(V (.+ vaijb vaibj))
(H (.+ v))
;; t1 amplitudes
(t1 (!! T1 (P H)))
(fai (!! f (P H)))
;; the whole expression to contract
;; i.e. Hamiltonian times T amplitudes
(expression (.* H t1))
(expression-2 (.* (.+ fai)
(.+ 1 t1)))
;; targets for the diagrams search space
(%singles (! _ (P H)))
(%doubles (! _ (P H) (P H)))
(contractions (contract %singles expression
:unrestricted t
:only-connected t)))
(format nil "Processing singles~%")
(assert-equal (arponen::expr-to-lists expression)
'(((V (P3 H3) (H4 P4)) (T1 (P2 H2)))
((V (P5 P6) (H5 H6)) (T1 (P2 H2)))))
;; it should be possible to just find an uncontracted diagram
(assert-equal (contract %singles expression-2
:unrestricted t
:only-connected t)
'(((CONTRACTIONS (NIL)) ((F (P1 H1))))))
(assert-equal (arponen/hole-particle-picture::remove-1-in-product-list
(arponen::expr-to-lists expression-2))
'(((F (P1 H1))) ((F (P1 H1)) (T1 (P2 H2)))))
(assert-equal (contract %doubles (.* vaibj t2)
:unrestricted t
:only-connected t)
'(((CONTRACTIONS (((H5 H7) (P6 P7))))
((V (P5 P6) (H5 H6)) (T2 (P7 H7) (P8 H8))))))
(let ((c (contract %doubles (.* H t2)
:unrestricted t
:only-connected t)))
(assert-equal (length c) 2))
))
13. Examples
13.1. Ccsd equations
(load "arponen.lisp")
(in-package :hp)
(let* ((arponen::*print-log* nil)
;; define the hamiltonian through
;; a particle-hole partition PH
;;
;; one-body matrix
(f (!! f (PH PH)))
;; two-body matrix
(v (!! V (PH PH) (PH PH )))
;; the sum
(h (.+ v f))
;; t1 amplitudes
(t1 (!! T1 (P H)))
(t1- (!! T1 (P H)))
;; t2 amplitudes
(t2 (!! T2 (P H) (P H)))
(t2- (!! T2 (P H) (P H)))
;; the whole expression to contract
;; i.e. Hamiltonian times T amplitudes
(expression
(.* h
(.+ 1 t1 t2 (.* t1 t1-) (.* t1 t2) (.* t2 t2-))))
;; targets for the diagrams search space
(singles (! _ (P H)))
(doubles (! _ (P H) (P H))))
(format nil "Processing singles~%")
(save-contractions "singles.lisp"
(contract singles expression
:unrestricted t
:only-connected t))
(format nil "Processing doubles~%")
(save-contractions "doubles.lisp"
(contract doubles expression
:unrestricted t
:only-connected t)))
13.1.1. Singles
(defvar *ccsd-singles*
'(((CONTRACTIONS (((H18 H39) (H17 H40) (P9 P39)) ((H18 H40) (H17 H39) (P9 P39))))
((V (H17 P9) (H18 H19)) (T2 (P39 H39) (P40 H40))))
((CONTRACTIONS (((H18 H37) (H17 H38) (P9 P37)) ((H18 H38) (H17 H37) (P9 P37))))
((V (H17 P9) (H18 H19)) (T1 (P37 H37)) (T1 (P38 H38))))
((CONTRACTIONS
(((H21 H37) (P11 P37) (H20 H40) (P10 P40)) ((H21 H40) (P11 P37) (P10 P40) (H20 H39))
((P11 P40) (H21 H37) (H20 H40) (P10 P39)) ((H21 H37) (P11 P37) (H20 H40) (P10 P39))
((H21 H40) (P11 P37) (H20 H39) (P10 P39)) ((P11 P40) (H21 H37) (H20 H39) (P10 P39))
((H21 H40) (P11 P37) (P10 P40) (H20 H37)) ((H21 H40) (P11 P40) (H20 H39) (P10 P37))
((H21 H40) (P11 P40) (P10 P39) (H20 H37)) ((H21 H40) (P11 P39) (H20 H39) (P10 P37))
((P11 P40) (H21 H39) (P10 P39) (H20 H37)) ((H21 H40) (P11 P37) (P10 P39) (H20 H37))
((P11 P40) (H21 H37) (H20 H39) (P10 P37)) ((P11 P39) (H21 H37) (H20 H39) (P10 P37))
((P11 P40) (H21 H39) (H20 H37) (P10 P37)) ((H21 H39) (P11 P39) (H20 H37) (P10 P37))))
((V (H20 P10) (H21 P11)) (T1 (P37 H37)) (T2 (P39 H39) (P40 H40))))
((CONTRACTIONS (((H29 H37) (P19 P37)))) ((V (P18 H28) (H29 P19)) (T1 (P37 H37))))
((CONTRACTIONS (((H33 H37) (P26 P37)))) ((V (P25 P26) (H33 H34)) (T1 (P37 H37))))
((CONTRACTIONS (((H35 H40) (P29 P40) (P28 P39)) ((P29 P40) (H35 H39) (P28 P39))))
((V (P27 P28) (H35 P29)) (T2 (P39 H39) (P40 H40))))
((CONTRACTIONS (((H35 H38) (P29 P38) (P28 P37)) ((P29 P38) (H35 H37) (P28 P37))))
((V (P27 P28) (H35 P29)) (T1 (P37 H37)) (T1 (P38 H38))))
((CONTRACTIONS (((H1 H37)))) ((F (H1 H2)) (T1 (P37 H37))))
((CONTRACTIONS (((P1 P40) (H3 H39)) ((H3 H39) (P1 P39)))) ((F (H3 P1)) (T2 (P39 H39) (P40 H40))))
((CONTRACTIONS (((P1 P38) (H3 H37)))) ((F (H3 P1)) (T1 (P37 H37)) (T1 (P38 H38))))
((CONTRACTIONS (((P4 P37)))) ((F (P3 P4)) (T1 (P37 H37))))))
(defun latex-sum-contractions (contractions-tuples)
(format nil "\\sum_{~{~a~^,~}}" (mapcar #'cadr contractions-tuples)))
(defun latex-contractions (contractions)
(let* ((conts (cadar contractions))
(product `(* ,@(cadr contractions)))
(substitutions (mapcar (lambda (-conts)
(mapcar (lambda (c) (cons (car c) (cadr c)))
-conts))
conts))
(substituted-products (mapcar (lambda (-subs) (sublis -subs product))
substitutions))
(text-contractions (mapcar #'latex-sum-contractions conts))
(text-products (mapcar #'arponen::latex substituted-products)))
(format nil "~{~a~}" (mapcar (lambda (c p) (format nil "~a ~a~%" c p))
text-contractions text-products))))
(defun latex-all-contractions (conts)
(format nil "~{~a~}" (mapcar #'latex-contractions conts)))
(print (latex-all-contractions *ccsd-singles*))
\[ \sum_{H_{39},H_{40},P_{39}} V^{H_{40}H_{39}}_{P_{39}H_{19}} T2^{P_{39}P_{40}}_{H_{39}H_{40}} \]
\[ \sum_{H_{40},H_{39},P_{39}} V^{H_{39}H_{40}}_{P_{39}H_{19}} T2^{P_{39}P_{40}}_{H_{39}H_{40}} \]
\[ \sum_{H_{37},H_{38},P_{37}} V^{H_{38}H_{37}}_{P_{37}H_{19}} T1^{P_{37}}_{H_{37}} T1^{P_{38}}_{H_{38}} \]
\[ \sum_{H_{38},H_{37},P_{37}} V^{H_{37}H_{38}}_{P_{37}H_{19}} T1^{P_{37}}_{H_{37}} T1^{P_{38}}_{H_{38}} \]
\[ \sum_{H_{37},P_{37},H_{40},P_{40}} V^{H_{40}H_{37}}_{P_{40}P_{37}} T1^{P_{37}}_{H_{37}} T2^{P_{39}P_{40}}_{H_{39}H_{40}} \]
\[ \sum_{H_{40},P_{37},P_{40},H_{39}} V^{H_{39}H_{40}}_{P_{40}P_{37}} T1^{P_{37}}_{H_{37}} T2^{P_{39}P_{40}}_{H_{39}H_{40}} \]
\[ \sum_{P_{40},H_{37},H_{40},P_{39}} V^{H_{40}H_{37}}_{P_{39}P_{40}} T1^{P_{37}}_{H_{37}} T2^{P_{39}P_{40}}_{H_{39}H_{40}} \]
\[ \sum_{H_{37},P_{37},H_{40},P_{39}} V^{H_{40}H_{37}}_{P_{39}P_{37}} T1^{P_{37}}_{H_{37}} T2^{P_{39}P_{40}}_{H_{39}H_{40}} \]
\[ \sum_{H_{40},P_{37},H_{39},P_{39}} V^{H_{39}H_{40}}_{P_{39}P_{37}} T1^{P_{37}}_{H_{37}} T2^{P_{39}P_{40}}_{H_{39}H_{40}} \]
\[ \sum_{P_{40},H_{37},H_{39},P_{39}} V^{H_{39}H_{37}}_{P_{39}P_{40}} T1^{P_{37}}_{H_{37}} T2^{P_{39}P_{40}}_{H_{39}H_{40}} \]
\[ \sum_{H_{40},P_{37},P_{40},H_{37}} V^{H_{37}H_{40}}_{P_{40}P_{37}} T1^{P_{37}}_{H_{37}} T2^{P_{39}P_{40}}_{H_{39}H_{40}} \]
\[ \sum_{H_{40},P_{40},H_{39},P_{37}} V^{H_{39}H_{40}}_{P_{37}P_{40}} T1^{P_{37}}_{H_{37}} T2^{P_{39}P_{40}}_{H_{39}H_{40}} \]
\[ \sum_{H_{40},P_{40},P_{39},H_{37}} V^{H_{37}H_{40}}_{P_{39}P_{40}} T1^{P_{37}}_{H_{37}} T2^{P_{39}P_{40}}_{H_{39}H_{40}} \]
\[ \sum_{H_{40},P_{39},H_{39},P_{37}} V^{H_{39}H_{40}}_{P_{37}P_{39}} T1^{P_{37}}_{H_{37}} T2^{P_{39}P_{40}}_{H_{39}H_{40}} \]
\[ \sum_{P_{40},H_{39},P_{39},H_{37}} V^{H_{37}H_{39}}_{P_{39}P_{40}} T1^{P_{37}}_{H_{37}} T2^{P_{39}P_{40}}_{H_{39}H_{40}} \]
\[ \sum_{H_{40},P_{37},P_{39},H_{37}} V^{H_{37}H_{40}}_{P_{39}P_{37}} T1^{P_{37}}_{H_{37}} T2^{P_{39}P_{40}}_{H_{39}H_{40}} \]
\[ \sum_{P_{40},H_{37},H_{39},P_{37}} V^{H_{39}H_{37}}_{P_{37}P_{40}} T1^{P_{37}}_{H_{37}} T2^{P_{39}P_{40}}_{H_{39}H_{40}} \]
\[ \sum_{P_{39},H_{37},H_{39},P_{37}} V^{H_{39}H_{37}}_{P_{37}P_{39}} T1^{P_{37}}_{H_{37}} T2^{P_{39}P_{40}}_{H_{39}H_{40}} \]
\[ \sum_{P_{40},H_{39},H_{37},P_{37}} V^{H_{37}H_{39}}_{P_{37}P_{40}} T1^{P_{37}}_{H_{37}} T2^{P_{39}P_{40}}_{H_{39}H_{40}} \]
\[ \sum_{H_{39},P_{39},H_{37},P_{37}} V^{H_{37}H_{39}}_{P_{37}P_{39}} T1^{P_{37}}_{H_{37}} T2^{P_{39}P_{40}}_{H_{39}H_{40}} \]
\[ \sum_{H_{37},P_{37}} V^{P_{18}H_{37}}_{H_{28}P_{37}} T1^{P_{37}}_{H_{37}} \]
\[ \sum_{H_{37},P_{37}} V^{P_{25}H_{37}}_{P_{37}H_{34}} T1^{P_{37}}_{H_{37}} \]
\[ \sum_{H_{40},P_{40},P_{39}} V^{P_{27}H_{40}}_{P_{39}P_{40}} T2^{P_{39}P_{40}}_{H_{39}H_{40}} \]
\[ \sum_{P_{40},H_{39},P_{39}} V^{P_{27}H_{39}}_{P_{39}P_{40}} T2^{P_{39}P_{40}}_{H_{39}H_{40}} \]
\[ \sum_{H_{38},P_{38},P_{37}} V^{P_{27}H_{38}}_{P_{37}P_{38}} T1^{P_{37}}_{H_{37}} T1^{P_{38}}_{H_{38}} \]
\[ \sum_{P_{38},H_{37},P_{37}} V^{P_{27}H_{37}}_{P_{37}P_{38}} T1^{P_{37}}_{H_{37}} T1^{P_{38}}_{H_{38}} \]
\[ \sum_{H_{37}} F^{H_{37}}_{H_{2}} T1^{P_{37}}_{H_{37}} \]
\[ \sum_{P_{40},H_{39}} F^{H_{39}}_{P_{40}} T2^{P_{39}P_{40}}_{H_{39}H_{40}} \]
\[ \sum_{H_{39},P_{39}} F^{H_{39}}_{P_{39}} T2^{P_{39}P_{40}}_{H_{39}H_{40}} \]
\[ \sum_{P_{38},H_{37}} F^{H_{37}}_{P_{38}} T1^{P_{37}}_{H_{37}} T1^{P_{38}}_{H_{38}} \]
\[ \sum_{P_{37}} F^{P_{3}}_{P_{37}} T1^{P_{37}}_{H_{37}} \]
13.2. IP Equation of motion CCSD
(load "arponen.lisp")
(in-package :arponen)
(defun make-space (name prefix n)
`(,name ,@(mapcar (lambda (i)
(intern (format nil "~a~a" prefix i)))
(loop for i from 1 to n collect i))))
(defun remove-1 (prod-list)
(mapcar (lambda (product)
(remove-if (lambda (el) (or (eq el 1) (equal el '(1))))
product))
prod-list))
(defun contract-expression (target expr &key orbital-spaces contraction-rules)
(let* ((expanded (remove-1 (expr-to-lists expr)))
(n (length expanded))
(i 0))
(remove-if
#'null
(mapcar
(lambda (tensor-product)
(format t "~&[~a/~a] ~a" (incf i) n tensor-product)
(let ((begin (get-internal-run-time))
(contractions
(find-contractions-in-product-by-target target
tensor-product
:orbital-spaces
orbital-spaces
:contraction-rules
contraction-rules)))
(format t "~2t in (~,1f seconds)"
(/ (- (get-internal-run-time) begin)
internal-time-units-per-second))
(when contractions
(list `(contractions ,contractions) tensor-product))))
expanded))))
(defun tensor-sum (&rest args)
`(+ ,@(reduce (lambda (tsr rest) (ccase (car tsr)
('+ (append (cdr tsr) rest))
(t (cons tsr rest))))
args
:from-end t
:initial-value nil)))
(let* ((orbital-spaces
(list (append (make-space 'H 'h 20) '(i j k l m n))
(append (make-space 'P 'p 20) '(a b c d e f))
(make-space 'G 'g 20)
(append (make-space 'pq 'pq- 20) '(p q r s))))
(*only-connected-diagrams* t)
(partition '((PQ H P)))
(contraction-rules
'(((H H) 0 1)
((P P) 1 0)))
;; tensors
vpqrs f t1 t2 r1 r2
(r1-like '(_ (G H)))
(r2-like '(_ (G H) (P H)))
ccsd-expressions)
(setq *print-log* nil)
(setq vpqrs (partition-tensor '(V (p q) (r s))
:orbital-spaces orbital-spaces
:partition partition))
(setq f (partition-tensor '(f (p q))
:orbital-spaces orbital-spaces
:partition partition
:from-index 4))
(setq t1 (name-legs-by-space-name '(T1 (P H))
:orbital-spaces orbital-spaces
:from-index 6))
(setq -t1 (name-legs-by-space-name '(T1 (P H))
:orbital-spaces orbital-spaces
:from-index 7))
(setq t2 (name-legs-by-space-name '(T2 (P H) (P H))
:orbital-spaces orbital-spaces
:from-index 8))
(setq -t2 (name-legs-by-space-name '(T2 (P H) (P H))
:orbital-spaces orbital-spaces
:from-index 10))
(setq r1 (name-legs-by-space-name '(R1 (G H))
:orbital-spaces orbital-spaces
:from-index 12))
(setq r2 (name-legs-by-space-name '(R2 (G H) (P H))
:orbital-spaces orbital-spaces
:from-index 13))
(setq ccsd-expressions
`(* ,(tensor-sum f vpqrs)
(+ 1 ,t1 ,t2 (* ,t1 ,-t1) (* ,t1 ,t2) (* ,t2 ,-t2))
(+ ,r1 ,r2)))
(format t "~&======~&=> ~a" ccsd-expressions)
(format t "~&~& DOING R1")
(with-open-file (s "r1.lisp" :direction :output :if-exists :supersede)
(time (format s "~s"
(contract-expression r1-like
ccsd-expressions
:orbital-spaces orbital-spaces
:contraction-rules contraction-rules))))
(format t "~&~& DOING r2")
(with-open-file (s "r2.lisp" :direction :output :if-exists :supersede)
(time (format s "~s"
(contract-expression r2-like
ccsd-expressions
:orbital-spaces orbital-spaces
:contraction-rules contraction-rules))))
(format t "~&~&Done"))
14. Integration tests
14.1. Hirata
The tensor contraction engine of Hirata is a python based tensor contraction system that is able to produce many kinds of equations in an unrestricted formalism. It has been thoroughly tested and veted by many codes and researchers and it is a great candidate for testing against.
For this we wrote a parser to convert the hirata equations into arponen expressions, you can read the details here.
14.1.1. CCD
The CCD equations are represented in arponen using the following input
(in-package :hp)
(hp::reset-spaces)
(let* ((arponen::*print-log* nil)
(f (!! f (PH PH)))
(v (!! V (PH PH) (PH PH)))
(h (.+ v f))
;; \(t^{ab}_{ij}\) amplitudes
(t2 (!! T2 (P H) (P H)))
(t2- (!! T2_2 (P H) (P H)))
(expression
(.* h
(.+ t2 (.* t2 t2-))))
(doubles (! _ (P H) (P H))))
(format nil "Processing doubles~%")
(save-contractions "doubles.lisp"
(contract doubles expression
:unrestricted t
:only-connected t)))
14.1.2. MP2
The MP2 energy equation should be like
(in-package :hp)
(hp::reset-spaces)
(let* ((arponen::*print-log* nil)
(v (!! V (H P) (H P)))
;; \(t^{ab}_{ij}\) amplitudes
(t2 (!! T2 (P H) (P H)))
(expression (.* v t2))
(target (! _)))
(format nil "Processing doubles~%")
expression
(save-contractions "out.lisp"
(contract target expression
:unrestricted t
:only-connected t))
(contract target expression
:unrestricted t
:only-connected t))
15. Output formats
15.1. TeX
(defun latex-tensor (tensor)
(format nil "~a^{~a}_{~a}"
(car tensor)
(format nil "~{~a~}" (mapcar #'car (cdr tensor)))
(format nil "~{~a~}" (mapcar #'cadr (cdr tensor)))))
(defun latex (tensor-expression &optional (stream nil))
(case (car tensor-expression)
(+ (format stream "~&( ~{~a~^~%+ ~}~%)" (mapcar #'latex
(cdr tensor-expression))))
(* (format nil "~{~a ~}" (mapcar #'latex (cdr tensor-expression))))
(t (latex-tensor tensor-expression))))
TODO actually to the tests
(let ((orbital-spaces '((PQ p q r s)
(H i j k l)
(P a b c d)))
(partition '((PQ H P))))
(latex (partition-tensor '(f (p q))
:orbital-spaces orbital-spaces
:partition partition))
(latex (partition-tensor '(V (p q) (r s))
:orbital-spaces orbital-spaces
:partition partition)))
15.2. TODO Dot diagrams
15.2.1. herodot
(asdf:defsystem "herodot"
:description "Another s-expression language for dot"
:version "0.0.1"
:author "Alejandro Gallo <aamsgallo@gmail.com>"
:licence "GPLv3"
:depends-on (#:uiop)
:serial t
:components ((:file "herodot/herodot")
(:file "herodot/goldstone")))
- Main routines
(defpackage :herodot (:use :cl) (:export #:render-sdot #:render-sdot! #:save)) (in-package :herodot) (declaim (optimize (debug 3) (safety 3))) (defun render-sdot (graph &optional stream) (eval `(render-sdot! ,graph ,stream))) (defparameter *default-indentation* " ") (defun indent (text &key (indentation *default-indentation*)) (declare (string text) (string indentation)) (with-output-to-string (s) #1=(princ indentation s) (mapc (lambda (c) (princ c s) (when (char= c #\newline) #1#)) (coerce text 'list)))) (defun indent-list (textlist &key (indentation *default-indentation*)) (indent (format nil "~{~a~^~%~}" textlist) :indentation indentation)) (defmacro render-sdot! (graph &optional stream) `(labels ((:-- (from to &rest opts) (format ,stream "~a -- ~a ~a" from to (:-[options] opts))) (:-> (from to &rest opts) (apply #':-- (append (list from to) (cons '(dir "forward") opts)))) (:<- (from to &rest opts) (apply #':-- (append (list from to) (cons '(dir "back") opts)))) (:-render-atom (p) (if (symbolp p) (substitute #\_ #\- (string-downcase (symbol-name p))) p)) (:-[options] (opts) (format ,stream "[~{~a~^, ~}]" (:-options opts))) (:-options (opts) (mapcar (lambda (o) (etypecase o (atom (format ,stream "~a" o)) (list (:= (car o) (cadr o))))) opts)) (:scope (name &rest els) (format ,stream "~&{ /* begin: ~a */~%~a~%} /* end: ~a */" name (indent-list els) name)) (:= (key var) (format ,stream "~a = ~s" (:-render-atom key) (:-render-atom var))) (:el (name &rest opts) (format ,stream "~(~a~) ~a" name (:-[options] opts))) (:node (&rest opts) (apply #':el (cons "node" opts))) (:edge (&rest opts) (apply #':el (cons "edge" opts))) (:graph-env (&rest opts) (apply #':el (cons "graph" opts))) (:cluster (name &rest els) (format ,stream "subgraph cluster_~a ~a" name (apply #':scope (cons "" els)))) (:list (&rest els) (format nil "~&~{~a~%~}" (remove-if-not (lambda (x) (and (atom x) (not (functionp x)) (not (null x)))) els))) (:-named-scope (scope-name name els) (format ,stream "~a ~a {~%~a~%}" scope-name name (indent (apply #':list els)))) (:graph (name &rest els) (:-named-scope "graph" name els)) (:digraph (name &rest els) (:-named-scope "digraph" name els))) ,graph)) (defun save (file-basename graph &key (format :svg)) (let ((dot-string (render-sdot nil graph)) (out-file (format nil "~a.~(~a~)" file-basename format)) (dot-file (format nil "~a.dot" file-basename)) (dot-cmd (format nil "dot -T~(~a~)" format))) ;; save dot file (with-open-file (f dot-file :if-exists :supersede :direction :output) (princ dot-string f)) ;; create export (with-input-from-string (i dot-string) (with-open-file (f out-file :if-exists :supersede :direction :output) (princ (with-output-to-string (o) (uiop:run-program dot-cmd :output o :input i)) f))) out-file)) - Goldstone diagrams
(in-package :herodot) (defun hole (from to &rest attrs) (append `(:<- ,from ,to '(:color :black)) attrs)) (defun particle (from to) `(:-> ,from ,to '(:color :red))) (defun make-top (names &rest opts) `(:scope "top level" (:= :rank :sink) (:node '(shape point) '(width .0)) ,@(mapcar (lambda (x) `',x) opts) ,@names)) (defun tensor (order name label &rest opts) (let ((-label (case order (1 label) (t (format nil "~{~a~^|~}" (loop for i below order collect (format nil "<~a>~a" i (if (= i (floor order 2)) label "")))))))) `(:el ,name '(label ,-label) '(shape "record") '(heigth 0.01) ,@(mapcar (lambda (x) `',x) opts)))) (defun style () `(:list (:graph-env '(:fontname "CMU Serif") '(:rankdir "BT") '(:splines "line") '(:nodesep "equally")) (:node '(:fontname "CMU Serif")))) (defun plus () `(:el (gensym) '(:shape :none) '(label "+"))) (defun t-amplitude (n) "Create a stand-alone t-amplitude" (let* ((*tops* (loop for i below (* 2 n) collect `',(gensym "HEAVEN"))) (*t* (tensor n (format nil "t~a" n) "")) (*contractions* (let ((name (cadr *t*))) (loop for j below n with c = -1 appending (list (hole #1=(format nil "~a:~a" name j) #2=(nth (incf c) *tops*)) (particle #1# #2#)))))) `(:list ,(make-top *tops*) ,*t* ,@*contractions*)))- Examples
(format t "~a" (render-sdot `(:graph "Diagram" ,(style) (:scope "top" ,(make-top '(:top1 :top2 :top3 :top4 :top5 :top6))) (:scope "t3 diagram" ,(tensor 3 :t3 "T")) ,(hole "t3:0" :top1) ,(particle "t3:0" :top2) ,(hole "t3:1" :top3) ,(particle "t3:1" :top4) ,(hole "t3:2" :top5) ,(particle "t3:2" :top6))))
- Examples
- Render arponen contractions arponen
#| (save "v@c-pp-hp-t1-t2" `(graph "diagram" ,(style) ,(make-top '(:top1 :top2 :top3 :top4)) (scope "V" (node '(color blue)) ,(tensor 2 "v" "V")) ,(tensor 1 "t1" "T") ,(tensor 2 "t2" "T") (scope "t1 times v" ,(particle "t1:0" "v:0") ,(particle "v:0" :top1) ,(hole "t1:0" :top2)) (scope "t2 times V" ,(particle "t2:0" "v:1") ,(hole "t2:0" "v:1") ,(particle "t2:1" :top3) ,(hole "t2:1" :top4)))) |# (defpackage arponen/dot (:use :cl :herodot :arponen)) (in-package arponen/dot) (defun top (i) (intern (format nil "TOP~a" i) "KEYWORD")) (defun make-top (contraction) (let ((n (- (length (arponen::flatten-list (mapcar #'cdr (cadr contraction)))) (length (arponen::flatten-list (cdar contraction)))))) (loop for i from 1 to n collect (top i)))) (make-top '((CONTRACTIONS (((P116 P84) (P114 P83)))) ((V (P113 P114) (P115 P116)) (T2 (P83 H83) (P84 H84))))) (defun render-contraction-pair (cnt tensors) (let (c-tensors) ;; find tensors participating in the contraction (loop :for tsr :in tensors :if (intersection cnt (arponen::flatten-list (cdr tsr))) :do (push tsr c-tensors)) (labels ((coords (leg) (let* ((tsr-idx (position leg c-tensors :test (lambda (i r) (member i (arponen::flatten-list r))))) (tsr (nth tsr-idx c-tensors)) (inode (position leg (cdr tsr) :test (lambda (i r) (member i (arponen::flatten-list r)))))) (list :itensor tsr-idx :inode inode :label (let ((name (symbol-name (car tsr)))) (format nil "~(~a~):~a" name inode)) :hole-p (hp::hole-p leg))))) (let (left right hole-or-particle (max-itensor -1)) (loop :for leg :in cnt :do (destructuring-bind (&key itensor inode label hole-p) (coords leg) (setq hole-or-particle (if hole-p #'herodot::hole #'herodot::particle)) (if (> itensor max-itensor) (setq right left left label max-itensor itensor) (setq right label)))) (funcall hole-or-particle right left))))) (render-contraction-pair '(P116 P84) '((V (P113 P114) (P115 P116)) (T2 (P83 H83) (P84 H84)))) (defun render-heaven (contraction heaven) (let ((contracted-indices (arponen::flatten-list (cdar contraction)))) (flet ((is-contracted (idx) (member idx contracted-indices))) (let (coords) (loop :for tsr :in (cadr contraction) :for c :from 0 :collect (let ((ileg 0)) (arponen::traverse-legs (lambda (leg) (unless (is-contracted leg) (push (list :itensor c :ileg ileg :name (format nil "~(~a~):~a" (car tsr) (position leg (cdr tsr) :test (lambda (i x) (member i x)))) :top (pop heaven) :leg leg :hole-p (hp::hole-p leg)) coords)) (incf ileg)) tsr))) (loop :for coord :in (reverse coords) :collect (destructuring-bind (&key itensor top name ileg inode leg hole-p) coord (if hole-p (herodot::hole name top) (herodot::particle name top)))))))) (render-heaven #1='((CONTRACTIONS (((P116 P84) (P114 P83)))) ((V (P113 P114) (P115 P116)) (T2 (P83 H83) (P84 H84)))) (make-top #1#)) (defun render (contraction) "Render only one diagram, this means, do not pass a set of diagrams in the format of many contractions like ((contractions ((...) (...))) (tensors...))" (assert (eq (length (cdar contraction)) 1)) (herodot:render-sdot (let ((top (make-top contraction))) `(:graph "diagram" ,(herodot::style) ,(herodot::make-top top) ,@(mapcar (lambda (tsr) (let ((lbl (symbol-name (car tsr)))) (herodot::tensor (length (cdr tsr)) lbl lbl))) (cadr contraction)) (:scope "Contractions" ,@(let ((cts (cadar contraction)) (tsrs (cadr contraction))) (loop :for ct :in (car cts) :collect (render-contraction-pair ct tsrs)))) (:scope "Heaven" ,@(render-heaven contraction top)) )))) #+(or) (progn #1=(render '((CONTRACTIONS (((P116 P84) (P114 P83)))) ((V (P113 P114) (P115 P116)) (T2 (P83 H83) (P84 H84))))) (format t "~a" #1#)) (defun render-all (contractions) "Return a list of strings of dot graphics." (destructuring-bind (conts tensors) contractions (mapcar #+nil (lambda (cnt) (list :cnt cnt #'identity)) (lambda (cont) (render `((contractions (,cont)) ,tensors))) (cadar contractions)))) #+(or) (render-all '((CONTRACTIONS (((P91 P82) (H101 H83) (H100 H82) (P90 P83)) ((H101 H82) (P91 P82) (H100 H84) (P90 P83)) ((P91 P82) (H101 H81) (P90 P84) (H100 H83)) ((P91 P82) (H101 H83) (H100 H84) (P90 P83)) ((H101 H81) (P91 P83) (P90 P84) (H100 H83)) ((H101 H81) (P91 P81) (H100 H83) (P90 P83)) ((P91 P81) (H101 H84) (H100 H83) (P90 P83)))) ((V (H100 P90) (H101 P91)) (T2 (P83 H83) (P84 H84)) (T2 (P81 H81) (P82 H82)))))(progn (format t "~a" (render '((CONTRACTIONS (((P116 P84) (P114 P83)))) ((V (P113 P114) (P115 P116)) (T2 (P83 H83) (P84 H84)))))))(loop :for d :in (render-all '((CONTRACTIONS (((P91 P82) (H101 H83) (H100 H82) (P90 P83)) ((H101 H82) (P91 P82) (H100 H84) (P90 P83)) ((P91 P82) (H101 H81) (P90 P84) (H100 H83)) ((P91 P82) (H101 H83) (H100 H84) (P90 P83)) ((H101 H81) (P91 P83) (P90 P84) (H100 H83)) ((H101 H81) (P91 P81) (H100 H83) (P90 P83)) ((P91 P81) (H101 H84) (H100 H83) (P90 P83)))) ((V (H100 P90) (H101 P91)) (T2 (P83 H83) (P84 H84)) (T2_2 (P81 H81) (P82 H82))))) :for c :from 0 :do (with-open-file (s (format nil "ccd-diagram-~a.dot" c) :direction :output :if-exists :supersede) (format s "~a" d)))
15.2.2. old
This section defines some utilities to export the diagrams to dot format so that one can have some rough estimates for the diagrammatic form of the terms.
(in-package :arponen)
(defun tensor-name-to-sdot-name (tensor)
(format nil "~(~a~)" (car tensor)))
(declaim (optimize (debug 3)))
(defun contraction-to-sdot (contraction)
(destructuring-bind ((contraction cts) . tensors) contraction
(declare (ignore contraction))
(flet ((tensor-of-leg (leg)
(find leg tensors :key #'cdr
:test (lambda (l nodes)
(member l (flatten-list nodes))))))
(loop for ct in cts
collect
(let* ((tsr-a (tensor-of-leg (car ct)))
(name-a (tensor-name-to-sdot-name tsr-a))
(tsr-b (tensor-of-leg (cadr ct)))
(name-b (tensor-name-to-sdot-name tsr-b)))
`(edge ))))))
(contraction-to-sdot '((contraction ((a b)))
(V (a e)) (t2 (b e) (i j))))
(defun render-sdot (stream graph)
(eval `(render-sdot-macro ,stream ,graph)))
(defmacro render-sdot-macro (stream graph)
`(labels ((-- (from to &rest opts)
(format ,stream "~a -- ~a ~a" from to (-[options] opts)))
(-> (from to &rest opts)
(apply #'-- (append (list from to) (cons '(dir "forward") opts))))
(<- (from to &rest opts)
(apply #'-- (append (list from to) (cons '(dir "back") opts))))
(-[options] (opts) (format ,stream "[~{~a~^,~}]" (-options opts)))
(-options (opts) (mapcar (lambda (o)
(etypecase o
(atom (format ,stream "~a" o))
(list (:= (car o) (cadr o)))))
opts))
(scope (&rest els) (format ,stream "{~{~&~t~a~%~}}" els))
(:= (key var)
(format ,stream "~a = ~a" (if (symbolp key)
(string-downcase (symbol-name key))
key) var))
(-element (name &rest opts)
(format ,stream "~(~a~) ~a" name (-[options] opts)))
(node (&rest opts) (apply #'-element (cons "node" opts)))
(edge (&rest opts) (apply #'-element (cons "edge" opts)))
(:graph (&rest opts) (apply #'-element (cons "graph" opts)))
(cluster (name &rest els) (format ,stream
"cluster cluster_~a ~a"
name
(apply #'scope els)))
(-named-scope (scope-name name els)
(format ,stream "~a ~a {~%~{~2t~a~%~}}" scope-name name els))
(graph (name &rest els) (-named-scope "graph" name els))
(digraph (name &rest els) (-named-scope "digraph" name els)))
,graph))
(render-sdot-macro nil
(graph "This"
(:= 'rankdir 'BT)
(:graph '(rankdir 654))
(scope (-> 'i 'k)
(-> 'e 'k)
(<- 'e 'k))
(node '(dir backward))
(edge '(dir backward))
(cluster 'v (-> 'e 'k))
(-element 'node 'simple)
(-> 'a 'b)
(<- 'a 'b)
(-> 'c 'd)))
(let ((a '(-> 'a 'b)))
(render-sdot nil a))
(render-sdot nil
'(graph nil
(scope (-> 'i 'k)
(-> 'a 'b))
(-> 'a 'b)
(<- 'a 'b)
(-> 'c 'd)))
(defpackage :arponen/dot
(:use :cl))
(in-package :arponen/dot)
(defun symb (&rest args)
(intern (format nil "~{~a~}" (cl-user::flatten-list args))))
(defun subgraph (name &rest contents)
(format nil "~&subgraph cluster_~a {~%~{~t~a~^~%~}~%}" name contents))
(defun declare-legs (tensor)
(format nil "~{~a~^, ~} [shape = point]"
(mapcar (lambda (x) (symb (car tensor) x))
(cdr tensor))))
(defun contraction-to-graphviz (contraction)
;;(assert (cl-user::contraction? contraction))
(destructuring-bind ((contraction-symbol conts . nil) prods . nil) contraction
(loop
for tensor in prods
collect
(let ((tname (car tensor)))
(case tname
((v t2) (subgraph tname
(declare-legs tensor)))
(t1 (format nil "~&~tt1 [shape = box, height = 0.02, label=\"\""))))))
)
(contraction-to-graphviz
'((contraction ((h1 h5) (p1 p5) (h2 h3)))
((V (h1 h2 ) (p1 p2))
(t1 (p3 h3))
(t2 (p4 h4) (p5 h5)))))
An example output should be
graph {
rankdir = TB
{
node [width = .3]
rank = same
heaven1, heaven2, heaven3, heaven4 [shape = none, label="", height=.01]
}
subgraph cluster_v {
rank = same
vleft, vright [shape = "point", label=""];
{
vleft -- vright [style = dashed]
}
}
subgraph cluster_t2 {
rank = same
style = filled
t2left, t2right [shape = "point", label=""];
}
t1 [shape = "box", height=.02, label=""]
//
// contractions
//
vleft -- t2left [dir = forward]
vleft -- t2left [dir = back]
vright -- t1 [dir=back]
heaven3 -- t2right [dir = forward, label="i"]
heaven2 -- t2right [dir = back, label="a"]
heaven4 -- t1 [dir = forward, label="j"]
heaven1 -- vright [dir = back, label="b"]
}
15.3. TODO SVG
#+quicklisp (ql:quickload 'cl-svg)
#+asdf (asdf:load-system 'cl-svg)
(in-package :cl-svg)
(defvar test-contraction nil)
(setq test-contraction
'((contraction ((h1 h3) (h2 h4) (p1 p2) (p2 p5)))
(V (h1 p1) (h2 p2))
(T1 (p3 h3))
(T2 (p4 h4) (p5 h3))))
(defun svg-diagram (contraction &key (node-length 100) (diagram-height 100)
(diagram-box-height 20)
(out-file nil))
(labels ((box-pos (i) (cons (+ (* i node-length) (* (mod i 2) 10))
(* i diagram-height)))
(node-position (node) (let* ((-i (position node contraction
:key #'cdr :test #'find))
(i (- -i 1))
(tsr (nth -i contraction ))
(-j (position node tsr :test #'equal))
(j (- -j 1)))
(cons (+ (car (box-pos i))
(* j node-length))
(+ (cdr (box-pos i))
(/ diagram-box-height 2)))
))
(nodes-of-contraction (leg-a leg-b)
(let* ((tsr-a (find leg-a (cdr contraction)
:key #'cdr
:test (lambda (x where) (find x where :test #'member))))
(tsr-b (find leg-b (cdr contraction)
:key #'cdr
:test (lambda (x where) (find x where :test #'member))))
(a (find leg-a (cdr tsr-a) :test #'member))
(b (find leg-b (cdr tsr-b) :test #'member)))
(assert (and a b))
(list a b))))
(let* ((n-nodes (reduce (lambda (x n)
(+ n (length (cdr x))))
(cdr contraction)
:initial-value 0
:from-end t))
(n-diagrams (length (cdr contraction)))
(width (* n-nodes node-length))
(height (* n-diagrams diagram-height))
(scene (make-svg-toplevel 'svg-1.1-toplevel
:height height :width width))
(iotas (loop for i below n-nodes collect i)))
;; draw blocks
;; DRAW BODIES
(mapc (lambda (i tsr)
(let* ((nodes (cdr tsr))
(n (length nodes))
(box-width (if (> n 1)
(* node-length (- n 1))
(/ node-length 2)))
(box-pos (box-pos i)))
;;; DRAW BOX
(draw scene
(:rect :x (car box-pos)
:y (cdr box-pos)
:width box-width
:height diagram-box-height)
:fill "#000000")
;; TEXT
(text scene (:x (- (+ (car box-pos)
(/ box-width 2))
(/ diagram-box-height 2))
:y (+ (cdr box-pos)
diagram-box-height))
(tspan (:fill "red"
:font-weight "bold"
:font-size diagram-box-height)
(symbol-name (car tsr))))
;;; NODE POINTS
(mapc (lambda (node)
(let ((node-pos (node-position node)))
(draw scene (:circle :cx (car node-pos)
:cy (cdr node-pos)
:r (/ diagram-box-height 5))
:fill "#ff0000")))
nodes)))
iotas (cdr contraction))
;; DRAW CONTRACTIONS
(mapc (lambda (c)
(destructuring-bind (node-a node-b) (nodes-of-contraction (car c) (cadr c))
(let ((pos-a (node-position node-a))
(pos-b (node-position node-b)))
(draw scene (:line :x1 (car pos-a)
:y1 (cdr pos-a)
:x2 (car pos-b)
:y2 (cdr pos-b))
:stroke "green"))))
(cadar contraction))
(if out-file
(with-open-file (s out-file :direction :output :if-exists :supersede)
(stream-out s scene))
(with-output-to-string (s nil)
(stream-out s scene)
s)))))
(svg-diagram '((contraction ((h1 i) (h2 i2) (p1 a2)))
(V (h1 h2) (p1 p2))
(t2 (a i) (b j))
(t1 (a2 i2)))
:out-file "test.svg")
(svg-diagram test-contraction
:out-file "test.svg")